Freeform, φ-Polynomial Optical Surfaces: Optical Design ...

Freeform, φ-Polynomial Optical Surfaces:

Optical Design, Fabrication and Assembly

by

Kyle Fuerschbach

Submitted in Partial Fulfillment of the

Requirements for the Degree

Doctor of Philosophy

Supervised by Professor Jannick Rolland

The Institute of Optics

Arts, Science and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of Rochester

Rochester, New York

2014

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Biographical Sketch

Kyle Fuerschbach is originally from Albuquerque, NM. After graduating high school, he

attended the University of Arizona in Tucson, AZ and graduated summa cum laude with

a Bachelor of Science degree in Optical Sciences and Engineering. He began doctoral

studies at The Institute of Optics at the University of Rochester in 2008. During his

tenure he was awarded the Robert L. and Mary L. Sproull University Fellowship in 2008,

the Frank J. Horton Research Fellowship from 2008-2014, and the Michael Kidger

Memorial Scholarship in Optical Design in 2011. He has also served as an elected

representative for the University of Rochester’s student chapter of SPIE. He pursued his

research in optical design and fabrication of optical systems with freeform optics under

the direction of Professor Jannick Rolland and co-direction of Dr. Kevin Thompson.

The following peer reviewed publications and patents were a result of work conducted

during doctoral study:

K. Fuerschbach, K. P. Thompson, and J. P. Rolland, "Assembly of an off-axis optical system employing three φ-polynomial, Zernike mirrors," Optics Letters (Accepted to appear April 2014).

K. Fuerschbach, K. P. Thompson, and J. P. Rolland, "Interferometric measurement of a concave, phi-polynomial, Zernike mirror," Optics Letters 39, 18-21 (2014).

J. P. Rolland and K. Fuerschbach, "Nonsymmetric optical system and design method for nonsymmetric optical system," US8616712 B2 (2013).

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, "Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces," Opt. Express 20, 20139-20155 (2012).

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, "A new family of optical systems employing phi-polynomial surfaces," Opt. Express 19, 21919-21928 (2011).

S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, "Airy beams: a geometric optics perspective," J. Opt. Soc. Am. A 27, 2574-2582 (2010).

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Acknowledgments

I would like to thank Professor Jannick Rolland and Dr. Kevin Thompson for their

support and guidance during the Ph.D. process. The original idea to work on a three

mirror freeform design came about when we were deciding what I could present at the

2010 International Optical Design Conference. At the time, I didn’t know it would

eventually become part of my thesis, but through their direction and my hard work, we

were able to explore many avenues in freeform optical surfaces that were all prompted by

the first “pamplemousse” design.

I would like to thank John Miller at the university machine shop and Gregg Davis and

Alan Hedges at II-VI Infrared for providing me with fabrication support. These men

helped translate my crazy ideas into tangible, working pieces of hardware that have been

critical to the success of my research.

I would like to thank all my labmates and officemates: Dr. Cristina Canavesi, Robert

Gray, Jinxin Huang, Jianing Yao, Eric Schiesser, Jacob Reimers, and Aaron Bauer.

Specifically, I would like to thank Aaron Bauer for answering all my questions

throughout the years. He was always willing to help me through a problem or read

something I had written. Thanks also to all the students who helped me with my research

in the lab: Eddie Lavilla, Jean Inard-Charvin, Johan Thivollet, and Isaac Trumper.

Without them, I’d still be in the lab working to get my experiments finished.

Thanks to Elizabeth for her support during my academic career. She made many

personal and professional sacrifices along the way and they have not gone unnoticed.

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Also, thanks to my parents, Phil and Marcie, for helping me get to this point. Without

their guidance, I may have never studied optics.

Lastly, I would like to thank my support, the Frank J. Horton Research Fellowship, the

II-VI Foundation, and the National Science Foundation (EECS-1002179) as well as Zygo

for their partnership in optical testing, Synopsys Inc. for the student license of CODE V,

and Photon Engineering for the student license of FRED.

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Abstract

Freeform optical surfaces are creating exciting new opportunities in optics for design,

fabrication, metrology, and assembly. While the term freeform is currently being applied

over a broad range of surface shapes, in our research on imaging with freeform optical

surfaces, a freeform is a surface whose sag varies not only with the radial component but

also with the azimuthal component, φ, also known as a φ-polynomial optical surface.

Interestingly, these surfaces are readily fabricated with techniques like single point

diamond turning; however, challenges remain in their optimization during optical design

and characterization after fabrication.

In this dissertation, we propose a more effective optical design approach based in

nodal aberration theory that considers the aberrations induced by a φ-polynomial optical

surface up to sixth order. Specifically, when a φ-polynomial overlay is placed on a

surface away from the aperture stop, there is both a field constant and field dependent

contribution to the net aberration field. These findings are validated through the design,

implementation, and wavefront measurement of an aberration generating Schmidt

telescope that employs a custom fabricated φ-polynomial plate. The measured wavefront

behavior is in good agreement with the theoretical predictions of nodal aberration theory

throughout the field of view.

The design methods are also applied to a specific example: a wide field, fast focal

ratio, long wave infrared, unobscured reflective imager. The system employs three, tilted

φ-polynomial surfaces to provide diffraction limited performance throughout the field of

view. The surfaces were fabricated with diamond turning and a novel metrology

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approach based on an inteferometric null is proposed for characterizing the figure error of

the fabricated surfaces. A mechanical design is also presented for the housing structure

that simplifies the system assembly. The as-built optical system maintains diffraction

limited performance throughout the field of view.

The work conducted in this dissertation provides a foundation for the efficient design

of optical systems employing freeform surfaces and demonstrates that a system based on

freeform surfaces is realizable in the long wave infrared and may be extended to shorter

wavelength regimes.

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Contributors and Funding Sources

This work was supervised by a dissertation committee consisting of Professors Jannick

Rolland (advisor) and Miguel Alonso of The Institute of Optics, Dr. Kevin Thompson of

Synopsys, and Professor Victor Genberg of The Department of Mechanical Engineering.

The original matlab code to plot the Full Field Displays in Chapter 3 and Chapter 4 was

developed by Dr. Christina Dunn. The fabrication of the components and experiments in

Chapter 4 were assisted by Isaac Trumper (undergraduate research assistant) and in part

by Edward Lavilla (summer research assistant). The experiments in Chapter 6 were

assisted in part by Johan Thivollet (graduate research assistant). The mirror surfaces and

optical housing in Chapter 7 were manufactured by II-VI Infrared. All other work

conducted for the dissertation was completed by the student independently. Graduate

study was supported by the Frank J. Horton Research Fellowship from the Laboratory for

Laser Energetics, the II-VI foundation, and the National Science Foundation

(EECS-1002179).

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Table of Contents

Biographical Sketch ............................................................................................................ ii

Acknowledgments.............................................................................................................. iii

Abstract ........................................................................................................................... v

Contributors and Funding Sources.................................................................................... vii

Table of Contents ............................................................................................................. viii

List of Figures .................................................................................................................. xiii

List of Tables ................................................................................................................. xxix

List of Acronyms ........................................................................................................... xxxi

Chapter 1. Introduction ....................................................................................................... 1

1.1 Off-Axis Reflective Systems .................................................................................. 1

1.1.1 Offset Aperture and/or Biased Field ................................................................. 2

1.1.2 Tilted Optical Surfaces ..................................................................................... 4

1.2 Freeform Optical Surfaces ...................................................................................... 6

1.3 Motivation ............................................................................................................... 9

1.4 Dissertation Outline .............................................................................................. 14

Chapter 2. Aberration Fields for Tilted and Decentered Optical Systems with

Rotationally Symmetric Components ............................................................. 16

2.1 Aberration Field Centers ....................................................................................... 16

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2.2 Wave Aberration Expansion in a Perturbed Optical System ................................ 17

2.3 Full Field Aberration Display ............................................................................... 24

Chapter 3. Aberration Fields in Optical Systems with φ-Polynomial Optical Surfaces ... 29

3.1 Formulating Nodal Aberration Theory for Freeform, ϕ-Polynomial Surfaces away

from the Aperture Stop ......................................................................................... 30

3.2 The Aberration Fields of ϕ-Polynomial Surface Overlays ................................... 36

3.2.1 Zernike Astigmatism ....................................................................................... 37

3.2.2 Zernike Coma .................................................................................................. 41

3.2.3 Zernike Trefoil (Elliptical Coma) ................................................................... 46

3.2.4 Zernike Oblique Spherical Aberration ............................................................ 49

3.2.5 Zernike Fifth Order Aperture Coma ............................................................... 53

3.3 APPLICATION: The Astigmatic Aberration Field Induced by Three Point

Mount-Induced Trefoil Surface Deformation on a Mirror of a Reflective

Telescope .............................................................................................................. 58

3.3.1 Astigmatic Reflective Telescope Configuration ( 222 0W ≠ ) in the Presence of a

Three Point Mount-Induced Surface Deformation on the Secondary Mirror . 60

3.3.2 Anastigmatic Reflective Telescope Configuration ( 222 0W = ) in the Presence of

a Three Point Mount-Induced Surface Deformation on the Secondary Mirror

......................................................................................................................... 64

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3.3.3 Validation of the Nodal Properties of a Reflective Telescope with Three Point

Mount-Induced Figure Error on the Secondary Mirror .................................. 64

3.4 Extending Nodal Aberration Theory to Include Decentered Freeform

ϕ-Polynomial Surfaces away from the Aperture Stop .......................................... 69

Chapter 4. Experimental Validation of Nodal Aberration Theory for φ-Polynomial

Optical Surfaces .............................................................................................. 72

4.1 Design of an Aberration Generating Schmidt Telescope ..................................... 72

4.2 Fabrication of the Aspheric Corrector/Nonsymmetric Plate ................................ 79

4.3 Experimental Setup of the Aberration Generating Schmidt Telescope ................ 82

4.4 Experimental Results ............................................................................................ 85

4.4.1 The Generated Field Conjugate, Field Linear Astigmatic Field ..................... 85

4.4.2 Rotation of the Aberration Generating Plate .................................................. 89

4.4.3 Lateral Displacement of the Aberration Generating Plate .............................. 90

Chapter 5. Design of a Freeform Unobscured Reflective Imager Employing

φ-Polynomial Optical Surfaces ....................................................................... 93

5.1 The New Method of Optical Design ..................................................................... 93

5.2 The Starting Form ................................................................................................. 95

5.3 The Unobscured Form .......................................................................................... 97

5.3.1 Creating Field Constant Aberration Correction .............................................. 98

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5.3.2 Creating Field Dependent Aberration Correction ......................................... 100

5.4 The Final Form ................................................................................................... 104

5.5 Mirror Surface Figures ........................................................................................ 107

Chapter 6. Interferometric Null Configurations for Measuring φ-Polynomial Optical

Surfaces ......................................................................................................... 109

6.1 Concave Surface Metrology ............................................................................... 109

6.1.1 First Order Design......................................................................................... 111

6.1.2 Optimization of the Interferometric Null System ......................................... 117

6.1.3 Experimental Setup of Interferometric Null System .................................... 119

6.1.4 Experimental Results .................................................................................... 124

6.2 Convex Surface Metrology ................................................................................. 127

Chapter 7. Assembly of an Optical System with φ-Polynomial Optical Surfaces ......... 132

7.1 Mechanical Design.............................................................................................. 132

7.1.1 Sensitivity Analysis ...................................................................................... 134

7.1.2 Stray Light Analysis ..................................................................................... 144

7.2 As-built Optical System ...................................................................................... 150

7.2.1 As-built Optical Performance ....................................................................... 152

Conclusion and Future Work .......................................................................................... 158

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Appendix A. Vector Multiplication and Its Vector Properties and Identities ............. 163

List of References ........................................................................................................... 166

xiii

List of Figures

Figure 1-1. Demonstration of how an on-axis optical system is made unobscured by

offsetting the aperture, biasing the field, or a combination of both. ....................... 2

Figure 1-2. Single point diamond turning surface roughness evolution through time. Each

color represents a lateral measurement of a part from a specific time period.

(Adapted from Schaefer [45]) ............................................................................... 10

Figure 1-3. Optical design space defined by the light collection (F/number), area

collection (FOV), and packaging for various surface representations. ................. 13

Figure 2-1. Coordinate system for aberration theory of a perturbed optical system where

both the pupil and field coordinate are represented as vectors. ............................ 19

Figure 2-2. Representation of the new effective field vector. ........................................... 20

Figure 2-3. Node locations for third order astigmatism in a perturbed optical system.

There are two points in the field where the aberration can be zero. ..................... 23

Figure 2-4. Full field display (FFD) showing (a) third order field quadratic in a centered

system and (b) in perturbed optical system that yields binodal astigmatism. ....... 28

Figure 3-1. (a) When the aspheric corrector plate of a Schmidt telescope is displaced

longitudinally from the aperture stop, the beam for any off-axis field point will

displace along the corrector plate. The displacement depends on the paraxial

quantities for the marginal ray height, y , chief ray height, y , chief ray angle, u ,

and the distance between the stop and plate, t . (b) Alternatively, the beam

displacement on the corrector plate can be thought of as a field dependent

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decenter of the aspheric corrector, h∆

, that modifies the mapping of the

normalized pupil coordinate from ρ to 'ρ . ........................................................... 33

Figure 3-2. Generation of coma and astigmatism as the aspheric corrector plate in a

Schmidt telescope is moved longitudinally (along the optical axis) from the

physical aperture stop located at the center of curvature of the spherical primary

mirror for various positions (a-d). For each field point in the FFD, the plot symbol

conveys the magnitude and orientation of the aberration. (e) Plots of the

magnitude of coma and astigmatism generated as the aspheric plate is moved

longitudinally for two field points, (0°, 2°) (blue square) and (0°, 4°) (red

triangle). ................................................................................................................ 35

Figure 3-3. Fringe Zernike polynomial set up to 5th order (6th order in wavefront). The set

includes Z1 (piston), Z2/3 (tilt), Z4 (defocus), Z5/6 (astigmatism), Z7/8 (coma), Z9

(spherical aberration), Z10/11 (elliptical coma or trefoil), Z12/13 (oblique spherical

aberration or secondary astigmatism), Z14/15 (fifth order aperture coma or

secondary coma), and Z16 (fifth order spherical aberration or secondary spherical

aberration). The φ-polynomials to be explored include Z5/6, Z7/8, Z10/11, Z12/13, and

Z14/15. ..................................................................................................................... 37

Figure 3-4. Surface map describing the freeform Zernike overlay for astigmatism on an

optical surface over the full aperture. The error is quantified by its magnitude

5/6FFz and its orientation 5/6FFξ that is measured clockwise with respect to the

y − axis. P and V denote where the surface error is a peak rather than a valley. .. 39

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Figure 3-5. The characteristic field dependence of field constant astigmatism that is

generated by a Zernike astigmatism overlay on an optical surface in an optical

system. This induced aberration is independent of stop position. ........................ 41

Figure 3-6. The characteristic field dependence of (a) field constant coma, (b) field

asymmetric, field linear astigmatism, and (c) field linear, field curvature that is

generated by a Zernike coma overlay on an optical surface away from the stop

surface. .................................................................................................................. 46

Figure 3-7. The characteristic field dependence of (a) field constant elliptical coma, (b)

field conjugate, field linear astigmatism, which is generated by a Zernike elliptical

coma overlay on an optical surface away from the stop surface. ......................... 49

Figure 3-8. The characteristic field dependence of (a) field constant oblique spherical

aberration, (b) field asymmetric, field linear trefoil, (c) field conjugate, field linear

coma, (d) field constant, field quadratic astigmatism, and (e) field quadratic, field

curvature that is generated by a Zernike oblique spherical aberration overlay on an

optical surface away from the stop surface. .......................................................... 53

Figure 3-9. The characteristic field dependence of (a) field constant, fifth order aperture

coma, (b) field linear medial oblique spherical aberration, (c) field asymmetric,

field linear oblique spherical aberration, (d) field quadratic trefoil, (e) field

quadratic coma, (f) field asymmetric, field cubed astigmatism, and (g) field cubic,

field curvature that is generated by a Zernike fifth order aperture coma overlay on

an optical surface away from the stop surface. ..................................................... 58

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Figure 3-10. (a) The nodal behavior for an optical system with conventional third order

field quadratic astigmatism and Zernike trefoil at a surface away from the stop,

e.g., a two mirror telescope with a three point mount-induced error on the

secondary mirror, is displayed in a reduced field coordinate,Π

, where the node

located by ( )2222 MNTERR x

has an orientation angle of 10/11MNTERRξ and a magnitude that

is proportional to 333,MNTERR SMC

. The two related nodes on the circle are then

advanced by 120º and 240º for this special case. (b) When the nodal solutions are

re-mapped to the conventional field coordinate, H

, the node located by

( )2222 MNTERR x

has an orientation angle of 10/11MNTERRξ and a magnitude that is

proportional to 3333,MNTERR SMC

. ................................................................................. 63

Figure 3-11. A measurement or simulation of the mount-induced error on the secondary

mirror yields the magnitude and orientation of 333,MNTERR SMC

. .................................. 63

Figure 3-12. (a) Layout for a F/8, 300 mm Ritchey-Chrétien telescope and (b) a Full Field

Display (FFD) of the RMS WFE of the optical system at 0.633 µm over a ±0.2°

FOV. Each circle represents the magnitude of the RMS WFE at a particular

location in the FOV. .............................................................................................. 66

Figure 3-13. Displays of the magnitude and orientation of Fringe Zernike astigmatism

(Z5/6) and Fringe Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for

(a) a Ritchey-Chrétien telescope in its nominal state and (b) the telescope when

0.5λ of three point mount-induced error oriented at 0° has been added to the

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secondary mirror. It is important to recognize that these displays of data are FFDs

that are based on a Zernike polynomial fit to real ray trace OPD data evaluated on

a grid of points in the FOV. For each field point, the plot symbol conveys the

magnitude and orientation of the Zernike coefficients pairs, Z5/6 on the left and

Z10/11 on the right. .................................................................................................. 66

Figure 3-14. (a) Layout for a JWST-like telescope geometry and (b) a Full Field Display

(FFD) of the RMS WFE of the optical system at 1.00 µm over a ±0.2° FOV. The

system utilizes a field bias (outlined in red) to create an accessible focal plane. . 68


(Z5/6) and Fringe Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for

(a) a JWST-like telescope in its nominal state and (b) the telescope when 0.5λ of

three point mount-induced error oriented at 0° has been added to the secondary

mirror. ................................................................................................................... 68


(Z5/6) and Fringe Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for a

JWST-like telescope with 0.5λ of three point mount-induced error oriented at 0°

on the off-axis tertiary mirror. .............................................................................. 71

Figure 4-1: Testing configuration for the Schmidt telescope to demonstrate the field

dependent aberration behavior of a freeform optical surface. A freeform, Zernike

plate can purposely be placed at or away from the stop surface to induce field

dependent aberrations. The aberration field behavior of the telescope is measured

xviii

interferometrically by acquiring the double pass wavefront over a two-

dimensional FOV with a scanning mirror. ............................................................ 73

Figure 4-2. Layout of the nominal Schmidt telescope configuration. The aspheric and

Zernike trefoil plate are both fabricated in NBK7 substrates and the primary

mirror is a commercially available 152.4 mm, F/1 concave, spherical mirror. .... 75

Figure 4-3. Simulated interferogram at a wavelength 632.8 nm of the 3 µm trefoil

deformation added on one surface of the 100 mm, NBK7 plate to be added into

the optical path of the nominal Schmidt telescope. .............................................. 76

Figure 4-4. (a) The predicted astigmatism (Z5/6) and (b) elliptical coma (Z10/11) FFDs over

a square, 5 degree full FOV for the Schmidt telescope system with the Zernike

trefoil plate oriented at 0° and located 120mm away from the stop surface. The

Zernike trefoil plate generates both field constant elliptical coma and field

conjugate, field linear astigmatism. ...................................................................... 77

Figure 4-5. The predicted magnitude of the (a) astigmatism (Z5/6) and (b) elliptical coma

(Z10/11) as a function of the Zernike trefoil plate position relative to the stop

surface for the ( )1, 0x yH H= = field point of Schmidt telescope configuration. ...... 77

Figure 4-6. First order layout demonstrating how the retro-reflector must be designed to

ensure that the pupil of the Schmidt telescope is conjugate to the pupil of the

concave mirror that sends the wavefront back towards the interferometer. ......... 79

Figure 4-7. (a) Measured surface departure of the aspheric corrector plate for the Schmidt

telescope and (b) residual error when the nominal optical design surface is

xix

subtracted from the measured surface. The error is about 0.56λ PV or 0.066λ

RMS at the testing wavelength of 632.8 nm. ........................................................ 81

Figure 4-8. (a) Measured surface departure of the Zernike trefoil plate and (b) residual

error when the nominal optical design surface is subtracted from the measured

surface. The error is about 0.30λ PV or 0.05λ RMS at the testing wavelength of

632.8 nm. .............................................................................................................. 82

Figure 4-9. Experimental setup of the Schmidt telescope system. The scanning mirror and

retro-reflector are motorized so that the FOV can be scanned over a two-

dimensional grid of points. The trefoil plate is also motorized so that effect of

plate position on magnitude of generated aberration field can be studied. ........... 85

Figure 4-10. (a) Measured interferograms after baseline subtraction for a 3x3 grid of field

points spanning a square, 5° degree diagonal FOV for the Schmidt telescope

system with the Zernike trefoil plate oriented at 0° and displaced roughly 100 mm

longitudinally away from the stop surface and (b) the 3x3 grid of wavefronts with

the field constant elliptical coma removed, revealing the generated field

conjugate, field linear astigmatism induced by the trefoil plate. .......................... 86

Figure 4-11. The measured Zernike astigmatism (Z5/6) FFD after baseline subtraction,

left, and theoretical Zernike astigmatism (Z5/6) FFD predicted by NAT, right, over

a 9x9 grid spanning a square, 5° full FOV for the Schmidt telescope system with

the Zernike trefoil plate oriented at 0° and located (a) 10.81 mm, (b) 53.31 mm,

and (c) 95.81 mm away from the stop surface. ..................................................... 88

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Figure 4-12. Plot of the mean magnitude of the Zernike trefoil and astigmatism after

baseline subtraction for two field points, ( )1, 0x yH H= = represented by the blue

circle and ( )1, 0x yH H= − = represented by the red star, for five measured plate

positions. The error bars on the data points represent plus or minus one standard

deviation from the mean value over the ten measurements acquired at each plate

position. In black, the magnitude of the Zernike trefoil and astigmatism based on

the theoretical predictions of NAT is plotted as a function of plate position. ...... 89

Figure 4-13. The (a) measured Zernike astigmatism (Z5/6) FFD after baseline subtraction

and (b) theoretical Zernike astigmatism Z5/6 FFD predicted by NAT over a 9x9

grid spanning a square, 5 degree full FOV for the Schmidt telescope system with

the Zernike trefoil plate oriented at 45° and located roughly 100 mm away from

the stop surface. .................................................................................................... 90

Figure 4-14. The (a) measured Zernike astigmatism (Z5/6) FFD after baseline subtraction

and (b) theoretical Zernike astigmatism (Z5/6) FFD predicted by NAT over a 9x9

grid spanning a square, 5 degree full FOV for the Schmidt telescope system with

the Zernike trefoil plate oriented at 0°, located roughly 100 mm away from the

stop surface, and displaced laterally 1 mm in the x-direction and -1 mm in the y-

direction. ............................................................................................................... 92

Figure 5-1. (a) Layout of U.S. Patent 5,309,276 consisting of three off-axis sections of

rotationally symmetric mirrors and a fourth fold mirror (mirror 3). The optical

system had, at the time of its design, the unique property of providing the largest

xxi

planar, circular input aperture in the smallest overall spherical volume for a

gimbaled application. (b) The new optical design based on tilted φ-polynomial

surfaces to be coupled to an uncooled microbolometer. ....................................... 95

Figure 5-2. (a) Layout for a fully obscured solution for a F/1.9, 10° full FOV LWIR

imager. The system utilizes three conic mirror surfaces. (b) A FFD of the RMS

WFE of the optical system. Each circle represents the magnitude of the RMS

wavefront at a particular location in the FOV. The system exhibits a RMS WFE

of < λ/250 over 10° full FOV. ............................................................................... 96

Figure 5-3. The lens layout, Zernike coma (Z7/8) and astigmatism (Z5/6) FFDs for a ±40˚

FOV for the (a) on-axis optical system, (b) halfway tilted, 50% obscured system,

and (c) fully tilted, 100% unobscured system. The region in red shows the field of

interest, a 10˚ diagonal FOV. ................................................................................ 98

Figure 5-4. The lower order spherical (Z9), coma (Z7/8), astigmatism (Z5/6), higher order,

elliptical coma (Z10/11), oblique spherical aberration (Z12/13), and fifth order

aperture coma (Z14/15) Zernike aberration contributions and RMS WFE FFDs over

a ±5 degree FOV for the fully unobscured, on-axis solution. It can be seen that the

system is dominated by field constant coma and astigmatism which are the largest

contributors to the RMS WFE of ~12λ. ................................................................ 99




xxii

a ±5 degree FOV for the optimized system where Zernike astigmatism and coma

were used as variables on the secondary (stop) surface. When the system is

optimized, the field constant contribution to astigmatism and coma are greatly

reduced improving the RMS WFE from ~12λ to ~0.75λ. .................................. 100



aperture coma (Z14/15) Zernike aberration and RMS WFE FFDs over a ±5 degree

FOV for the optimized system where Zernike coma is added as an additional

variable to the primary surface. The RMS WFE has been reduced from ~0.75λ to

~0.125λ. .............................................................................................................. 102




a ±5 degree FOV for the optimized system where Zernike coma is added as an

additional variable to the tertiary surface. The RMS WFE has been reduced from

~0.75λ to ~0.180λ. .............................................................................................. 103




a ±5 degree FOV for the optimized system where the mirror conic constants are

added as additional variables in addition to Zernike elliptical coma, oblique

xxiii

spherical aberration, fifth order aperture coma on the secondary surface. The

RMS WFE has been reduced from ~0.180λ to ~0.065λ. .................................... 105




a ±5 degree FOV for the optimized system where Zernike astigmatism, elliptical

coma, and oblique spherical aberration are added as additional variables to the

tertiary surface. The RMS WFE has been reduced from ~0.065λ to ~0.012λ. ... 106

Figure 5-10. (a) Layout of LWIR imaging system optimized with φ-polynomial surfaces

and (b) the RMS WFE of the final, optimized system, which is < λ/100 (0.01λ)

over a 10˚ diagonal full FOV. ............................................................................. 107

Figure 5-11. (a) Sag of the primary mirror surface various Zernike components removed

from the base sag, (b) sag of the secondary mirror surface various Zernike

components removed from the base sag, and (c) sag of the tertiary mirror surface

mirror surface various Zernike components removed from the base sag. When the

piston, power, and astigmatism are removed from the base sags of the three

mirrors, the asymmetry induced from the coma being added into the surface is

observed. ............................................................................................................. 108

Figure 6-1. (a) Sag of the secondary mirror surface with the piston, power, and tilt

Zernike components removed revealing the astigmatic contribution of the surface,

xxiv

(b) sag with the astigmatic component additionally removed, and (c) sag with the

spherical component additionally removed. ....................................................... 110

Figure 6-2. First order layout of the Offner null to compensate spherical aberration. The

rays in red show the illumination path for the testing wavefront whereas the rays

in blue show the imaging path for the pupils of the Offner null. ........................ 111

Figure 6-3. First order layout of the comatic and higher order null. A collimating lens is

uses to couple the wavefront to an actuated, deformable membrane mirror. The

rays in red show the illumination path for the testing wavefront whereas the rays

in blue show the imaging path for the pupils of the comatic null. ...................... 116

Figure 6-4. Layout of the optimized interferometric null for the concave, secondary

mirror to be coupled to a conventional Fizeau interferometer with a transmission

flat. The interferometric null is composed of three nulling subsystems: an Offner

null to null spherical aberration, a tilted geometry to null astigmatism, and a retro-

reflecting DM to null coma and any higher order aberration terms. .................. 119

Figure 6-5. Simulation of the double pass wavefront exiting the concave interferometric

null (a) before and (b) after the deformable null has been applied at a testing

wavelength of 632.8 nm. ..................................................................................... 119

Figure 6-6. (a) Layout of the setup to create the comatic and higher order null on the DM

surface. The setup uses a Shack-Hartmann wavefront sensor to run a closed loop

optimization to set the shape of the DM. The DM is also interrogated with a

Fizeau interferometer. (b) The setup realized in the laboratory. ......................... 121

xxv

Figure 6-7. (a) DM comatic null surface measured by the interferometer and (b) the

residual after the theoretical shape has been subtracted. The residual has a PV

error of 2 µm PV. ................................................................................................ 122

Figure 6-8. Custom designed kinematic indexing mount for counter rotating the test

mirror during alignment of the interferometric null. The plates are machined in

304 stainless steel and employ three hardened 440C stainless steel 7/16” spheres.

............................................................................................................................. 123

Figure 6-9. The interferometric null configuration realized in the laboratory. A rotation

stage with a rail affixed is used to create the tilted geometry. The secondary

mirror is measured using a Zygo Fizeau-type interferometer. ............................ 124

Figure 6-10. (a) Initial surface error map of the test mirror with power and (b) with the

power removed. The PV error of the surface residual before and after the power is

removed is 3.821 µm and 2.025 µm, respectively. (c) Final surface error map of

the test mirror after the software null has been subtracted (c) before and (d) after

the power has been removed. In this case, the PV error is 3.230 µm before and

1.140 µm after the power has been removed. ..................................................... 126

Figure 6-11. (a) Sag of the primary mirror surface with the piston, power, and tilt Zernike

components removed, (b) sag with the astigmatic component additionally

removed, and (c) sag with the spherical component additionally removed. With

the piston, power, tilt, astigmatism, and spherical components removed, the

asymmetry induced from the coma being added into the surface can be seen. .. 127

xxvi

Figure 6-12. Layout of the optimized interferometric null for the convex, Primary mirror

to be coupled to a conventional Fizeau interferometer with a transmission flat.

The interferometric null is composed of three nulling subsystems: an afocal

Offner null to null spherical aberration, a tilted geometry to null astigmatism and

coma, and a retro-reflecting DM to null any higher order aberration terms. ...... 131

Figure 6-13 Simulation of the double pass wavefront exiting the convex interferometric

null (a) before and (b) after the deformable null has been applied at a testing

wavelength of 632.8 nm. ..................................................................................... 131

Figure 7-1. (a) Layout of the housing structure of the three mirror freeform optical system

and (b) exploded view of the tertiary mirror subassembly consisting of the optical

mirror surface, adaptor plate, and steel dowel pins for alignment. ..................... 133

Figure 7-2. The tertiary mirror subassembly and values that determine its alignment,

namely, the pin hole position tolerances and their relative spacings. ................. 139

Figure 7-3. Cumulative probability as a function of as-built RMS WFE for the three

mirror optical system over nine field points assuming only passive alignment. 140

Figure 7-4. The astigmatism (Z5/6) and coma (Z7/8) Zernike aberration FFDs over an

8°x6° full FOV for the (a) nominal system and with 0.1° α tilt of the (b) primary,

(c) secondary, and (d) tertiary mirror surfaces. ................................................... 143

Figure 7-5. Cumulative probability as a function of as-built RMS WFE for the three

mirror optical system over nine field points assuming active alignment where

secondary mirror tilt and focal plane tilt are used as compensators. .................. 144

xxvii

Figure 7-6. The computed elevation log(PST) for the baseline optical housing with the

walls of the housing material assumed to be machined aluminum, resulting in a

near specular surface with 80% reflectance. ....................................................... 146

Figure 7-7. The computed elevation log(PST) for the optical system with blackened walls

in blue and the computed elevation log(PST) for the baseline optical housing in

gray. An improvement is observed when the walls of the housing are blackened

versus left machined aluminum. ......................................................................... 147

Figure 7-8. Cutaway of the optical system (a) without a baffle and (b) with a baffle and

its solid angle to the environment from the focal plane shown in red for each case.

With the baffle added to the housing, the solid angle to the environment goes to

zero. ..................................................................................................................... 148

Figure 7-9. The computed elevation log(PST) for the optical system with blackened walls

as well as baffling near the image plane in red and the computed elevation

log(PST) for the optical housing with blackened walls in light blue. A two order

of magnitude improvement is observed in the regions of large stray light when

baffling is added near the image plane. .............................................................. 149

Figure 7-10. The computed elevation log(PST) for the optical system with blackened

walls, baffling near the image plane, and baffling at the primary mirror in green

and the computed elevation log(PST) for the optical housing with blackened walls

and baffling near the image plane in light red. A two order of magnitude

xxviii

improvement is observed for large positive elevation angles where scattering is

the dominant contributor to stray light. ............................................................... 150

Figure 7-11. As-built subassemblies for the (a) primary, (b) secondary, and (c) tertiary

mirrors of the three mirror system that are to be mated to the optical housing.

Each subassembly mates to one face of the optical housing and rests on three

raised, diamond turned pads. .............................................................................. 151

Figure 7-12. Assembled three mirror optical system. The system consists of a housing

structure and three mirror subassemblies that are mated to the faces of the

housing. ............................................................................................................... 151

Figure 7-13. Experimental setup for measuring the full field performance of the as-built

optical system...................................................................................................... 153

Figure 7-14. Measured wavefronts for a 3x3 grid of field points spanning an

8 mm x 6 mm FOV for the directly assembled three mirror optical system. The

RMS WFE in microns displayed within the wavefront for each field. ............... 155


8 mm x 6 mm FOV for the directly assembled three mirror optical system with

the secondary mirror tilted roughly 1 arc minute with a 23 µm shim. The RMS

WFE in microns is displayed within the wavefront for each field. .................... 157

Figure 7-16. Sample LWIR image from the optical system ........................................... 157

Figure A-1. Concept of vector multiplication. ................................................................ 164

xxix

List of Tables

Table 2-1. Names of the aberration terms from the wavefront expansion up to fifth order.

............................................................................................................................... 18

Table 2-2. Summary of the first sixteen Fringe Zernike polynomials and their relation to

the standard Zernike set. ....................................................................................... 26

Table 2-3. Field dependence of the Zernike coefficients in terms of the wave aberration

coefficients. (Adapted from Gray et al. [35]) ....................................................... 27

Table 3-1. Field aberration terms that are generated from the longitudinal shift of an

aspheric plate from the stop surface in a Schmidt telescope. ............................... 33

Table 3-2. Image degrading aberration terms that are generated by a Zernike coma

overlay and how the terms link to existing concepts of NAT ............................... 45

Table 3-3. Image degrading aberration terms that are generated by a Zernike elliptical

coma overlay and how the terms link to existing concepts of NAT ..................... 47

Table 3-4. Image degrading aberration terms that are generated by a Zernike oblique

spherical aberration overlay and how the terms link to existing concepts of NAT

............................................................................................................................... 52

Table 3-5. Image degrading aberration terms that are generated by a Zernike fifth order

aperture coma overlay and how the terms link to existing concepts of NAT ....... 56

Table 4-1. Design specifications for the nominal aberration generating Schmidt telescope.

............................................................................................................................... 75

xxx

Table 7-1. Summary of the initial sensitivity analysis of the three mirror optical system.

For each tolerance, the change in RMS WFE from nominal is computed and the

RSS is compiled to provide the as-built RMS WFE. The RMS WFE is terms of

waves at the central operating wavelength of 10 µm. ......................................... 136

Table 7-2. Summary of the quantities used to derive the tolerances for the Monte Carlo

sensitivity analysis. The pin hole tolerances are used to derive the mirror x/y

decenter and mirror clocking angle. .................................................................... 139

xxxi

List of Acronyms

CGH Computer Generated Hologram

COTS Commercial Off The Shelf

DM Deformable Mirror

DOF(s) Degree(s) of Freedom

FFD Full Field Display

FOV(s) Field(s) of View

JWST James Webb Space Telescope

LWIR Long Wave InfraRed

MRF MagnetoRheological Finishing

NAT Nodal Aberration Theory

OAR Optical Axis Ray

OPD Optical Path Difference

PST Point Source Transmittance

PV Peak to Valley

RMS Root Mean Square

RSS Root Sum Square

TMA Three Mirror Anastigmat

WALRUS Wide Angle Large Reflective System

WFE Wavefront Error

1

Chapter 1. Introduction

In the introductory part of this dissertation, a brief history of off-axis reflective systems

and freeform optical surfaces is presented. Our motivation for this research is then

presented and the dissertation is outlined.

1.1 Off-Axis Reflective Systems

Reflective telescopes are commonly used for astronomical and earth based surveying

because they provide large apertures for light collection; however, most classical

telescope forms, i.e. Newtonian, Cassegrain, Gregorian, and Ritchey-Chrétien, have an

obscured aperture that will affect the overall image quality from diffraction of the

obscuration and its spider supports. The obscuration may also cause stray light in infrared

applications because the warm mechanical structure from the obscuration exists in the

beam path. Handling the obscuring aperture and creating an accessible image plane

becomes even more difficult when trying to design a system to correct the three primary

aberrations, i.e. spherical aberration, coma, and astigmatism, where three mirror surfaces

are required [1-3].

One way to avoid an obscured configuration is to operate off-axis creating an

unobscured form. Historically, there are two principal ways to operate off-axis. The first

is to take a nominally rotationally symmetric reflective form and either offset the

aperture, bias the field, or a combination of both [4]. In this configuration each optical

surface is a section of a larger parent surface where each parent surface lies on a common

optical axis. The other way to operate off-axis is to tilt the optical surfaces themselves to

create an unobscured form [5, 6]. In this fashion, each optical surface is not arranged

along a common optical axis. In some unobscured configurations that tilt the optical

2

surfaces, a nonsymmetric surface is employed to restore the optical performance after the

surfaces have been tilted [6].

1.1.1 Offset Aperture and/or Biased Field

A reflective telescope that is made unobscured by operating off-axis in aperture, in field,

or both, only uses part of a larger, rotationally symmetric optical system. As an example

of this concept, Figure 1-1 shows an F/5, inverse telephoto made unobscured by these

techniques. For the biased field system, the aberration performance does not change

because the incoming beam has only been tilted with respect to the optical axis. When the

aperture is offset, the stop surface has been decentered with respect to the optical axis;

therefore, the aberration performance of the optical system will change. Leveraging a

combination of both field bias and aperture offset is often required to find an optimal

unobscured solution with minimal impact on the aberration performance.

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On-Axis

Biased Field Offset Aperture

Biased Field and Offset Aperture

Figure 1-1. Demonstration of how an on-axis optical system is made unobscured by

offsetting the aperture, biasing the field, or a combination of both.

3

One of the earliest three mirror forms that used aperture offset and field bias was

introduced by Cook [7, 8] who took a three mirror anastigmat (TMA) form by Korsch [9]

off-axis in aperture and field. Each mirror is an off-axis conic section with the primary

and secondary elements forming a Cassegrainian pair that creates an intermediate image.

The image is then re-imaged with a tertiary element of approximately unit magnification.

The design has the degrees of freedom (DOFs) to provide an aplanatic, anastigmatic, and

flat image plane. This type of system is useful for infrared applications because it has an

accessible exit pupil, but is best suited for a narrow, strip field of view (FOV).

Another set of unobscured forms with large FOVs were created in the late 1970s by

using combinations of systems that are designed with symmetry principles in mind.

These principles are: stop at the center of curvature of a sphere (i.e. Schmidt telescope),

concentricity about the object/image or pupil (i.e. Schwarzschild objective), and confocal

(Mersenne) parabolas [10]. From these principles, several wide field obscured forms can

be derived from which an unobscured form is obtained. The basic form found by Baker

[11] combines confocal parabolas with a Schmidt telescope to form an aplanatic,

anastigmatic telescope objective. In this configuration, the secondary element, which is

located at the center of curvature of the tertiary, is aspherized to provide spherical

aberration correction. Two other variations of the form presented by Baker are the

reflective triplet [12], which is on-axis in aperture but off-axis in field, and the wide angle

large reflective system (WALRUS) [13], which is an inverse Baker design. These designs

yield larger FOVs than the TMA and maintain similar performance but sacrifice the

accessible exit pupil.

4

A still larger FOV is achieved in a three mirror unobscured form by using a principle

proposed by Brueggemann in which the stop surface is placed at one of the mathematical

foci of a conic mirror to remove astigmatism [14]. Following this principle Egdall formed

a three mirror objective called the three mirror long using a hyperboloidal primary and an

ellipsoidal tertiary [15]. A nearly flat secondary is placed at the common conic focus and

serves as the stop of the optical system to correct astigmatism. By aspherizing this

element, spherical aberration can also be corrected creating an aplanatic, anastigmatic

objective. The shortcomings of this configuration are its length, usually around four focal

lengths [16], and its mirrors, which will have a larger diameter than the entrance pupil of

the system when using a large FOV.

1.1.2 Tilted Optical Surfaces

Another approach to create an unobscured system is to tilt the optical surfaces directly.

For this method there are several approaches on how to create an unobscured optical

system from tilted components. One approach takes a well corrected system that is

nominally rotationally symmetric and adjusts the tilts of the optical surfaces to create an

unobscured form [6]. The performance of the optical system after applying the tilt is

restored by adjusting the system parameters or by adding additional DOFs to the optical

surfaces by changing their surface shape.

One of the first examples of a system designed with this method was introduced by

Kutter where he took a two mirror telescope and tilted the mirrors in a configuration that

removes the obscuration while keeping the astigmatism and coma at a minimum [17].

Leonard used similar principles to obtain a three mirror telescope he called the Yolo that

used two conic elements and one anamorphic conic (different curvatures in orthogonal

5

directions) [18]. The system is slow at roughly F/12 but provides good performance over

a 2° diameter field. Buchroeder [5] developed a modified Seidel aberration theory to

understand the behavior of tilted component systems. In his theory the net aberration

fields are still the superposition of the individual surface aberration field contributions;

however, each contribution will have its own center defined by its decentration or tilt.

Shack [19] then developed an expression for the wave aberration expansion that used the

concepts of Buchroeder. In the new expansion of the wave aberration, the aberration

types can have multiple points in the field where they may go to zero and these zeros are

called nodes. The theory of Shack, often called vector or nodal aberration theory (NAT),

was developed through fifth order by Thompson [20-25] and was applied to the

tolerancing of optical systems. Rogers applied NAT as a design technique for three

mirror telescope objectives [26-28]. In his method, two tilted optical components are

combined to yield a system with linear coma and constant astigmatism. Next, a third

optical element is added with some cylindrical power to eliminate axial astigmatism.

Lastly, the elements are aspherized to correct the residual coma and spherical aberration.

With this method systems of similar performance to the Yolo are obtained in a different

packaging geometry. The Yolo and the systems proposed by Rogers are slow (greater

than F/10), have small FOVs, and do not utilize freeform surfaces to improve

performance; rather, they are special configurations where the net aberration fields are

arranged to be near zero.

In another approach, the optical system is designed from the outset in an unobscured

form. Systems designed in this manner require a method to set up the initial system

parameters but give the designer the freedom to control the geometry, i.e. volume, while

6

selecting an initial design. One method to generate systems composed of three spherical

mirrors has been proposed by Howard [29]. In this method, the imaging properties about

a central ray are Taylor expanded. The coefficients of this expansion represent the first

order imaging properties and are used to constrain the system parameters like the

distances between elements, curvatures of the mirrors, and their tilts. Only solutions that

yield no first order blur are considered and these solutions are found using a systematic

search or a global optimization technique. Such a method allows the designer to explore a

larger design space more rapidly but does not guarantee a practical solution with useful

performance. Another three mirror, tilted component system has been proposed by

Nakano [30] in which the geometry is derived to maximize the compactness as well as

the input aperture. Setting the optical path configuration fixes the mirror positions and

then Cartesian surfaces are used to correct spherical aberration and minimize

astigmatism. Coma is minimized by adding higher than second order deformation to the

surfaces. The system achieves a compact geometry operating over a 4°x4° square FOV at

F/2.2.

1.2 Freeform Optical Surfaces

In the systems described above, the symmetry of the optical system is broken out of

necessity, either to avoid an obscuration or to meet the size and/or weight constraints of

the optical system. However, in general, unless special configurations are exploited, the

performance of the optical system degrades when the system symmetry is broken. As a

result, the surfaces of the optical system can be freeform to help recover from the

performance degradation. We define freeform surfaces as nonsymmetric surfaces that

include coma and potentially higher orders to their surface departure and go beyond

7

anamorphic. One of the first examples of an optical system that utilized a freeform

optical surface is the Polaroid SX-70 [31]. The commercial product was designed to be

collapsible and the need for flatness of the overall package prompted Baker, the lead

optical designer, to use mirrors rather than a penta-prism for the viewfinder. The

constraints on the system geometry forced the use of two freeform lenses that are

described by up to an eighth order power series in both the x and y directions of the

optical surface.

Around the same time, Tatian [32-34] began studying nonsymmetric surfaces for the

design of unobscured reflective systems. The surface representation dubbed the “unusual

optical surface” is described by a section of an aspheric surface with bilateral symmetry

in both the x and y directions where within the local origin of the section may exist up to

a tenth order power series in both the x and y directions. With this surface description,

Tatian was able to achieve roughly a 3X improvement in the root mean square (RMS)

wavefront error (WFE) of a three mirror WALRUS design with unusual surfaces versus

the same design with only aspheric surfaces.

Shafer also applied a nonsymmetric optical surface to the design of unobscured

systems. In his approach, he proposed a two-axis aspheric surface that is the summation

of two aspheres that are shifted relative to one another and may be anamorphically

stretched. In the region that these two aspheres overlap, lower order aberration

contributions like coma and astigmatism are generated. With this approach, special

optical configurations like a two-axis asphere at a pupil location can be exploited to yield

an unobscured two mirror optical system that is corrected for all third order aberrations.

Shafer mentions that these surfaces could be described and optimized with a

8

two-dimensional polynomial set over the entire surface, but the computational power

required to do so at the time was prohibitive. More recently, now that computational

power is no longer nearly as restrictive, two-dimensional polynomial sets to describe an

optical surface have started to appear. As mentioned in Section 1.1.2, Nakano [30] used

an orthogonal polynomial set called the Zernike polynomial set (described in detail in

Chapter 2) to describe an optical surface. The Zernike set is expressed in polar

coordinates and is desirable as it directly relates to the wavefront aberrations proposed by

Hopkins [35]. A related two-dimensional orthogonal polynomial set has been proposed

by Forbes [36] to describe freeform surfaces. Forbes’ set is also based on Jacobi

polynomials but arranged and normalized so that the slope of the optical surface can be

minimized. Also, rigid body terms like defocus and tilt have been eliminated from the

description. Since both the set proposed by Forbes and the Zernike polynomial set are

orthogonal, they can be used interchangeably to describe one another.

The surface representations described above consider the global surface shape so that

the variables describing the surface affect the entire surface. A more localized optical

representation based on a bicubic spline has been proposed for nonsymmetric optical

systems by Vogl et al. [37] and implemented further by Stacy [38] for the design of an

unobscured optical system. For a spline surface, the optical surface is sampled by a grid

of points. At each point, the surface deformation at that point becomes a variable that can

be optimized. The values between these mesh points are interpolated by a cubic

polynomial. A benefit of the spline surface is that the deformations at each point are only

partially correlated to surrounding points. Stacy applied the spline surface to a mirror

near the focal plane of a four mirror telescope to improve the field performance of the

9

system. The final surface shape exhibited strong oscillations that do cause image

degradation. Spline surfaces are computationally intensive because many variables are

required to describe them. Another approach at local shape control was proposed by

Cakmakci et al. [39] where the optical surface is written as a sum of basis functions, in

this case, a two-dimensional Gaussian. In this approach, the surface is sampled by a grid

of points where at each point, the Gaussian shape can be varied. This surface description

was applied to a single mirror head-worn display. In a local approach the key is to ensure

that the performance metric of the optical system is appropriately sampled throughout the

FOV [40, 41]. For this reason, a global or hybrid surface representation may be more

effective for a sparsely sampled field that is often the case during optimization in optical

design.

1.3 Motivation

The concept of a freeform optical surface is not new and was recognized early on as a

promising tool for the design of the nonsymmetric optical systems; however, unless the

surfaces can be manufactured, they are little more than an academic exercise. For

example, in 1972, Gelles when studying unobscured two mirror systems wrote that

“progress in surface generation will undoubtedly permit the use of exotic types of

surfaces in the future” [42]. Until recently, the fabrication capabilities did not exist to

manufacture these types of optical surfaces in a cost effective manner. One of these

recent advances has been in diamond turning technology where servos have been

integrated into the axes geometry in either a fast tool servo or slow slide servo

configuration [43, 44]. This integration allows for surfaces that are nonsymmetric to be

routinely manufactured. Moreover, the residual surface roughness after diamond turning

10

has been reduced so that post-polishing is no longer required, further reducing the cost of

manufacturing [45]. To demonstrate this progress, Figure 1-2 shows the improvement of

optical surface finish with single point diamond turning as a function of time. As a result

of this progress, freeform optical surfaces may be specified for application in the long

wave infrared (LWIR) with the technology continuing to push towards shorter

wavelength regimes as the residual surface roughness continues to get smaller.

Actual Measured Data

~1980 ~1986 ~1992 ~1998 ~2004

Figure 1-2. Single point diamond turning surface roughness evolution through time. Each

color represents a lateral measurement of a part from a specific time period. (Adapted

from Schaefer [45])

The other component to the fabrication of freeform surfaces is the form error that

results from the manufacturing process and how the final surface figure is quantified. The

metrology component of the manufacturing chain is the current limitation and cost driver

for freeform optics manufacturing as there are very few metrology techniques available.

One method available is profilometry where a probe, either contact or non-contact, is

scanned along the optical surface and the vertical displacement is recorded [46, 47]. This

method can be very accurate but it acquires the overall measurement on a point by point

basis. Therefore, the measurement process is time intensive, which relates directly to cost

11

on the manufacturing floor. Another method is based on the use of a computer generated

hologram (CGH) that acts as a nulling component in an interferometric arrangement [48].

The quality of the measurement obtained with the CGH depends strongly on the

fabrication of the CGH and the arrangement in which it is placed in the

interferometer [49]. Moreover, each CGH is unique to one specific surface and can be

cost prohibitive for multiple surfaces [50]. Another potential method is to arrange optical

elements (i.e. lenses or mirrors) in a null configuration. These methods exist for

measuring off-axis sections of conics and aspherics [51] but have not been developed for

freeform surfaces.

In addition to fabrication, one of the challenges with freeform optical surfaces is the

excess of variables introduced during optimization. If a global surface representation like

the Zernike polynomial set is used, the optical designer has access to an impractical

number of variables per surface during optimization. In a more localized approach, the

number of coefficients grows rapidly as the sampling is increased on the optical surface.

In 1978, Shafer recognized this point and to motivate his two-axis asphere approach over

a set of polynomials, he wrote, “…a Zernike set of aspheric coefficients would be able to

describe these surfaces and could be used to design systems. That, however, would be a

very cumbersome way to proceed, and would probably have a poor convergence rate

during optimization” [52]. Even with modern day computational power, where the time

per optimization cycle is minimal, a more efficient approach for choosing which surfaces

would benefit from a freeform surface and which variables to optimize on the surface is

desirable.

12

With the challenges described above for the design and fabrication of a freeform

surface, there has to be some direct benefit that cannot be achieved without a freeform

surface to justify their use in an optical system. To describe this benefit, consider the

specifications of an optical system. Any optical system will be required to meet some sort

of image quality metric with a certain light collection capability like F/number and with a

certain area coverage like FOV. Another more esoteric constraint may be the packaging

of the optical system. For example, the weight or size of the optical system might be

constrained for certain applications. These three items, F/number, FOV, and packaging,

define the design space for optical design. The extent of the design space that may be

covered by a particular surface representation is demonstrated in Figure 1-3. The most

restrictive optical design shape is the sphere. If the package is to be made smaller with

the same performance, thus widening the optical design space, conics or aspheric surfaces

are usually employed. Examples here are the use of conics in astronomical applications

[14, 53] and the use of aspheres for mobile phone optics [54]. If non-inline geometries

are considered like a tilted or decentered optical system, the aberration correction

capability is limited with conic or aspheric surfaces. Innovative packaging geometries are

the strength of freeform surfaces as they provide the necessary DOFs to operate in this

space, thus, increasing the optical design space.

13

F/#FOV

Packaging

Freeform

Spheres

Conics/Aspheres

Figure 1-3. Optical design space defined by the light collection (F/number), area collection (FOV), and packaging for various surface representations.

In this dissertation, our research is focused on exploring these innovative package

geometries that are enabled by freeform surfaces. We propose a method based in NAT

for describing the aberration field behavior of a freeform surface, specifically,

φ-polynomial (Zernike based) surfaces. With an analytical theory, the selection of

variables during optimization becomes structured and is no longer purely a brute-force

approach. In addition, we explore the state of the art in freeform manufacturing through

the development of a specific optical system. This system allows for each step in the

manufacturing chain of freeform optical surfaces to be studied and identify what links are

missing. In the case of metrology for freeform surfaces, we propose a new technique; in

particular, a new null based interferometric method for measuring freeform surfaces. An

end goal of the research is to demonstrate that a high performing optical system can be

designed, fabricated, and assembled with freeform optical surfaces. The principles

described in this work extend to a wide variety of applications.

14

1.4 Dissertation Outline

The dissertation is organized as follows:

Chapter 2 discusses NAT in the context of a perturbed optical system with rotationally

symmetric components. The misalignment induced aberration fields are reviewed through

fifth order with the concept of the aberration field center. Also, the concept of the full

field display, a visualization tool for studying the aberration behavior of a nonsymmetric

optical system, is described.

Chapter 3 presents a method for integrating freeform optical surfaces, specifically

φ-polynomial (Zernike) optical surfaces, into NAT. Using this method, the aberration

fields generated by a Zernike overlay away from the stop surface are derived up to sixth

order and linked to preexisting concepts of NAT. This theory is then applied to a specific

example, three-point mount induced error for both two and three mirror telescopes.

Chapter 4 experimentally validates the extension of NAT to freeform optical surfaces

by measuring the aberration behavior of a specially designed Schmidt telescope. The

Schmidt telescope is composed of two corrector plates, one to remove third order

spherical aberration, and the other to induce an aberration field known as field linear,

field conjugate astigmatism. The generated aberration field is studied under several

conditions including both axial and lateral displacement and rotation of the aberration

generating plate.

Chapter 5 presents the design of an unobscured three mirror imager that utilizes three,

tilted φ-polynomial optical surfaces. The design shows how the concepts derived in NAT

for freeform surfaces can be used to effectively choose variables for optimization. These

15

strategies target either field constant or field dependent aberration correction and utilize

the full field display as an analysis technique.

Chapter 6 demonstrates a new interferometric nulling technique for the measurement

of φ-polynomial optical surfaces. In this method, several adaptable subsystems are

combined that each null an aberration type present in the departure of the mirror surface.

This method is used to design configurations for measuring both convex and concave

optical surfaces. An experimental measurement of an as-fabricated concave,

φ-polynomial optical surface is also demonstrated.

Chapter 7 demonstrates the design and assembly of an optical housing for the optical

system described in Chapter 5. The mechanical housing and its sensitivity to

manufacturing error is studied as well as its susceptibility to stray light. Finally, the

as-built system is presented along with its as-built optical performance.

16

Chapter 2. Aberration Fields for Tilted and Decentered Optical Systems with Rotationally Symmetric Components

The wavefront expansion and surface contributions to the individual aberrations that

describe the imaging properties of an optical system have historically assumed the optical

system is rotationally symmetric [55]. In this case, the third order aberrations are the sum

of the individual surface contributions. For the unobscured reflective systems that were

described in Chapter 1, the symmetry has been broken by either offsetting the aperture or

tilting the optical components. As a result, a new foundation needs to be established that

can handle the imaging behavior of nonsymmetric optical systems.

2.1 Aberration Field Centers

The extension of aberration theory to nonsymmetric optical systems was approached by

Buchroeder [5] in which he proposed that the aberration fields of any optical system are

composed of the contributions of rotationally symmetric surfaces that may be aspheric

where each surface contributes rotationally symmetric aberration fields; however, the

center of the aberration fields will be offset and defined by the surface’s decentration or

tilt. As a result, the net aberration fields are still the summation of the shifted, individual

surface contributions. The shift of the aberration fields is relative to the center of the

Gaussian image plane that is located by the optical axis ray (OAR). The OAR

corresponds to the ray that connects the center of the object, to be chosen arbitrarily, with

the center of the aperture stop in the system [20, 22]. The intersection of the OAR with

the image plane defines the field center for each individual surface’s offset. Buchroeder

introduced a vector, sphjσ , to quantify the shift of the aberration field contributions for a

spherical surface j. Specifically, sphjσ represents a vector that lies in the plane of the image

and points to the intersection of a line that connects the center of curvature and the center

17

of a local entrance or exit pupil (image of the aperture stop in the local space) of surface j

with the image plane. For an aspheric cap on surface j, there is an additional asphjσ

parameter that is defined by the intersection of a line that connects the vertex of the

asphere (relative to the OAR) and the local pupil of surface j with the image plane.

2.2 Wave Aberration Expansion in a Perturbed Optical System

In a centered, rotationally symmetric system, the common way to express the aberrations

of the system is through the use of the scalar wave expansion of Hopkins [56], which is

represented in the form

( ) ( )cos ,k l mklm j

j p n mW W H ρ ϕ

∞ ∞ ∞

= ∑∑∑∑ (2.1)

where

2 , 2 ,k p m l n m= + = + (2.2)

W is the total wave aberration and is the sum of all the individual surface contributions, H

is the normalized field coordinate, ρ is the normalized pupil coordinate, and φ is the

azimuthal coordinate in the pupil. The scalar expansion assumes rotational symmetry so

only terms containing powers of 2H , 2ρ , and cos( )Hρ ϕ are valid. If Eq. (2.1) is expanded

through sixth order (fifth order in transverse ray aberration), W takes the form

( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

( )

2 4 3 2 220 11 040 131 220

2 2 2 3 6 1 5222 311 060 151

2 4 2 4 2 3 3240 242 331

3 3 3 4 2 4 2 2333 420 422

5511

cos cos

cos cos cos

cos cos

cos cos

cos ,

S

S

S

W W W H W W H W H

W H W H W W H

W H W H W H

W H W H W H

W H

ρ ρ φ ρ ρ φ ρ

ρ φ ρ φ ρ ρ φ

ρ ρ φ ρ φ

ρ φ ρ ρ φ

ρ φ

= ∆ + ∆ + + +

+ + + +

+ + +

+ + +

+

(2.3)

where

.jklm klm

jW W= ∑ (2.4)

The total wave aberration in Eq. (2.4) is a summation over all the intrinsic surface

contributions that are derived with paraxial quantities. For the fifth order aberrations, the

18

surface contributions consist of both intrinsic and induced contributions. The induced

contributions at a surface depend on a sum of the third order image and pupil aberrations

at the previous surface, though, they are still calculated from paraxial quantities [57, 58].

A set of naming conventions commonly used in optical design to refer to the terms of the

wavefront expansion in Eq. (2.3) is presented in Table 2-1. Note the name for each term

refers to the transverse ray aberration at the image plane so the pupil order is one order

lower than the wavefront order.

Table 2-1. Names of the aberration terms from the wavefront expansion up to fifth order.

kH lρ ( )cosm φ Coeff. Transverse Ray Aberration Name 2 0 20W∆ Defocus 1 1 11W∆ Tilt

4th Order Wave Aberration Type 0 4 0 040W 3rd order spherical aberration 1 3 1 131W 3rd order coma 2 2 0 220SW 3rd order sagittal focal surface 2 2 2 222W 3rd order astigmatism 3 1 1 311W 3rd order distortion

6th Order Wave Aberration Type 0 6 0 060W 5th order spherical aberration 1 5 1 151W 5th order field linear coma 2 4 0 240SW 5th order sagittal focal surface for

oblique spherical aberration 2 4 2 242W 5th order oblique spherical aberration 3 3 1 331W 5th order field cubed coma 3 3 3 333W 5th order elliptical coma 4 2 0 420SW 5th order sagittal focal surface 4 2 2 442W 5th order astigmatism 5 1 1 511W 5th order distortion

Now, to describe the aberrations of a nonsymmetric optical system, Shack [19]

proposed allowing both H and ρ to have independent orientation angles, θ and φ , so

19

that the vector H

represents the normalized position in the two-dimensional field and ρ

represents the normalized position in the two-dimensional pupil as shown in Figure 2-1.

ρx

ρy z

Pupil

Imageφ

θ

ρ

HHx

Hy

Figure 2-1. Coordinate system for aberration theory of a perturbed optical system where both the pupil and field coordinate are represented as vectors.

Updating the scalar expansion of Hopkins to the vector form proposed by Shack, the

representation in Eq. (2.1) takes the following form

( ) ( ) ( ) ( ) .p mn

klm jj p n m

W W H H Hρ ρ ρ∞ ∞ ∞

= ∑∑∑∑

(2.5)

With Buchroeder’s insight that each surface has its own aberration field defined by jσ ,

the aberration offset is included in the expansion, so Eq. (2.5) is modified to the final

form

( ) ( ) ( ) ( ) ( ) ,p mn

klm j j jjj p n m

W W H H Hσ σ ρ ρ σ ρ∞ ∞ ∞

= − − − ∑∑∑∑

(2.6)

where a new effective field component has been defined for surface j that accounts for

that surface’s aberration offset. This new effective field vector is written as

,Aj jH H σ= −

(2.7)

and is illustrated at the image plane in Figure 2-2. Depending on the surface of interest,

there may be an effective field vector for the spherical part of the surface and the aspheric

cap resulting in two contributions at the surface.

20

H

j AjH

xH

yH

Figure 2-2. Representation of the new effective field vector.

If Eq. (2.6), which now has the effective field vector in the expansion, is expanded

through fifth order following Thompson [59], W takes the form

2

20 11 040

131 220

22

222 311

3 2

060 151

1

2

j

j j

j j

j j

j

j M j j

j j

j j j j

j j

j

j j

W W W H W

W H W H H

W H W H H H

W W H

W

2

240

22

242

331

233

333 420

2

422

1

2

1

4

1

2

j

j

j

j j

j

M j j

j

j

j

M j j j

j

j M j j

j j

j j j

j

H H

W H

W H H H

W H W H H

W H H H

2

2

511 ,j j j j

j

W H H H

(2.8)

where through the use of several vector identities described in Appendix A

220 220 222

240 240 242

331 331 333

420 420 422

1,

2

1,

2

3,

4

1,

2

M S

M S

M

M S

W W W

W W W

W W W

W W W

(2.9)

21

have been defined. The identities used to arrive at the terms in Eq. (2.9) and the final

wavefront expansion in Eq. (2.8) have made use of an operation known as vector

multiplication. This operation is essential to represent the aberration fields of

nonsymmetric optical systems and can be thought of as an extension to the mathematics

of complex numbers. The properties of vector multiplication are described in more detail

in Thompson [20, 59] and are summarized in Appendix A. Also, it is important to

re-emphasize that the aberration surface contributions observed in Eq. (2.9) are not

affected by a perturbation because they are functions of paraxial quantities.

Though Eq. (2.8) is complete, it does not provide any insight into the behavior of the

aberrations in a nonsymmetric optical system. This behavior is revealed by performing a

summation over the surfaces to get the total aberration effect. To simplify the notation,

Thompson [59] proposed using substitutions for the summations as follows

( )

( )

( )

( )

( )

2 2

3 3

2

2 2

2,

j

j

j

j

j

j

j

j

klm klm jj

klm klm j jj

klm klm jj

klm klm j j jj

klm klm jj

klm klm j jj

klm klm j j jj

klm klm j j jj

A W

B W

B W

C W

C W

D W

D W

E W

σ

σ σ

σ

σ σ σ

σ

σ σ

σ σ σ

σ σ σ

=

=

=

=

=

=

=

=

∑

∑

∑

∑

∑

∑

∑

∑

(2.10)

creating a set of image plane perturbation vectors that are only created when the

symmetry of the optical system is broken. With the perturbation vectors defined, the

wavefront expansion in Eq. (2.8) is expanded further and simplified following

22

Thompson [59] to reveal how the field behavior of rotationally symmetric aberration

fields are modified when the symmetry of the optical system is broken, which is given as,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

220 11 040 131 131

220 220 220

2 2 2222 222 222

2 *311 311 311 311 311 311

060

2

1 22

2 2

j

M M M

W W W H W W H A

W H H H A B

W H HA B

W H H H H A H B H H H A B H C

W

ρ ρ ρ ρ ρ ρ ρ ρ

ρ ρ

ρ

ρ

= ∆ + ∆ + + − + − +

+ − +

+ − + − + −

+

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

3 2151 151

2240 240 240

2 2 2242 242 242

331 331 331 331

2 *331 331

2

1 22

2 2

14

M M M

M M M M

M M

W H A

W H H H A B

W H HA B

W H H H H A H B H H H A

B H C

ρ ρ ρ ρ ρ

ρ ρ

ρ ρ ρ

ρ ρ ρ

+ − + − +

+ − +

− + − + + −

+

( )( ) ( )( ) ( )( ) ( )

( )

( ) ( ) ( ) ( )

3 2 2 3 3333 333 333 333

420 420 420

2 2420 420 420

2 2 2422 422 422 422

3 *422

3 3

4 4

2 4

2 3 212 3

M M M

M M M

W H H A HB C

W H H H H H H H A B H H

H B H C D

W H H H H H HA H H B H A H

C H B

ρ

ρ ρ

− + −

− + + + − +

− + −+

− +

( )( ) ( )( ) ( )( ) ( ) ( )( )( ) ( )

2

2 2422 422 422

511 511 511

2 2511 511 511 511

2 * 2 * 3 2* 2 *511 511 511 511 511 511

3

4 6

2 4 3

2 4 2

H HC D

W H H H H H H H H A H B H H H

H B H H C H D H H H H H A

H H B H H H C H C C H D H E

ρ − +

− +

+ + − + −

+ − − − + −

.ρ

(2.11)

In Eq. (2.11), for each rotationally symmetric aberration type, several additional

components of the aberration are induced when the symmetry is broken. The perturbation

induced aberration components have a field dependence of lower order than their parent

rotationally symmetric aberration and the induced aberration components change how the

aberration behaves throughout the field. More specifically, the zero location of the field

dependent aberrations will change [23-25]. For a centered optical system, the field

dependent aberrations are always zero on-axis and increase from that zero point

23

depending on their inherent field dependence. When the optical system is perturbed, the

zero location is altered by the perturbation vectors and may result in the aberration going

to zero at more than one field location. The order of the field dependence of the

rotationally symmetric aberration determines how many zeros (or nodes) may exist when

the system is perturbed. If the field component of the various aberration types is solved

for its zeros, the nodes can be analytically predicted.

As an example, consider the case of third order astigmatism where the wavefront

aberration in a nonsymmetric optical system is

2 2 2222 222 222

1 2 .2

W W H HA B ρ = − +

(2.12)

To compute the nodes of this aberration, the field component is set to zero, as follows

2 2222 222 2222 0.W H HA B− + =

(2.13)

With the concept of vector multiplication, the quadratic formula is used as if the vectors

were scalar quantities and the nodes locations are solved for as

2 2

222 222 222 222

222

,A i W B A

HW

± −=

(2.14)

where i± defines a rotation of the vector by ±90°. A graphical interpretation of these

node locations for third order astigmatism is shown in Figure 2-3.

xH

yH

2 2222 222 222

222

W B Ai

W−

+

222

222

AW

2 2222 222 222

222

W B Ai

W−

−

Figure 2-3. Node locations for third order astigmatism in a perturbed optical system. There are two points in the field where the aberration can be zero.

24

In addition to the general case, there are several special cases to be addressed for this

aberration as well. The first case is when the system is corrected for third order

astigmatism, that is, 222 0W = . In this case, the wavefront aberration becomes

2 2222 222

1 2 .2

W HA B ρ = − +

(2.15)

If both 222A

and 2222B

are non-zero, the aberration takes the form of linear astigmatism (i.e.

astigmatism that depends linearly with field) and a node will exist at

2222

222

.2BHA

=

(2.16)

Lastly, if 222 0W = and 222 0A =

then the wavefront aberration becomes

2 2222

1 ,2

W B ρ=

(2.17)

where the astigmatism is constant throughout the field in both magnitude and orientation

and governed by 2222B

. Similar methodology is applied for the other aberrations to reveal

their characteristic behavior throughout the field and to analytically determine their node

locations for different tilt and decenter perturbations as detailed in [23-25].

2.3 Full Field Aberration Display

In a centered optical system with rotationally symmetric components, the aberrations

need only be assessed in one field direction (historically, the +y-field). With the

aberrations known in this field direction, they are also known in every other field

direction since the aberrations will also be rotationally symmetric. In most optical design

software packages, a common way to assess the aberration performance is through a

transverse ray aberration plot. In this plot, the transverse ray aberration is computed as a

function of pupil position for various field heights along one field direction. This plot is

useful for rotationally symmetric systems, but if the optical system symmetry is broken,

25

the transverse ray aberration plot is no longer useful because the aberrations are no longer

known for every field direction. A more useful plot would be one that computes the

aberrations over a two-dimensional field. One such plot developed by Thompson [60] is

the full field display (FFD). This plot computes the aberrations over a two-dimensional

grid of field points and then displays those aberrations using symbols to represent the

magnitude and orientation of the aberration at a particular field point. The method for

computing the aberrations is based on an orthogonal polynomial fit to the wavefront at

the exit pupil. The orthogonal polynomial set used is known as the Fringe Zernike

polynomial set [61], a modified form of the standard Zernike polynomial set [61, 62].

These sets have several benefits over other polynomials sets including the fact that they

are orthogonal and complete over a unit radius circular pupil, they represent balanced

aberrations, and they can be equated to the aberrations of the Hopkins’ wavefront

expansion. More specifically, the standard Zernike polynomial set is given by

( ) ( )( )( )

cos, ,

sinm m

n n

m for mZ R

m for m

φρ φ ρ

φ±

+= −

(2.18)

where m is a positive integer (or zero) and ( )mnR ρ is the radial component given by

( ) ( )( )( ) 22

0

1 !.

! ! !2 2

n mm n sn

s

n sR

n m n ms s sρ ρ

−−

=

− −=

+ − − −

∑ (2.19)

The Fringe Zernike set was developed by John Loomis [63] and is based on the

standard Zernike polynomial set but has a specific ordering that is more aligned with that

of aberration theory. The first 16 Fringe Zernike terms, their relationship to the standard

Zernike set, and their naming convention are summarized in Table 2-2.

26

Table 2-2. Summary of the first sixteen Fringe Zernike polynomials and their relation to the standard Zernike set.

Fringe ( ),jZ ρ φ

Standard ( ),m

nZ ρ φ± Zernike Polynomial Name

Z1 Z00 1 Piston

Z2 Z11 ( )cosρ φ

Tilt Z3 Z1-1 ( )sinρ φ

Z4 Z20 22 1ρ − Defocus

Z5 Z22 ( )2 cos 2ρ φ

Pri. Astigmatism Z6 Z2-2 ( )2 sin 2ρ φ

Z7 Z31 ( ) ( )33 2 cosρ ρ φ−

Pri. Coma Z8 Z3-1 ( ) ( )33 2 sinρ ρ φ−

Z9 Z40 4 26 6 1ρ ρ− + Pri. Spherical

Z10 Z33 ( )3 cos 3ρ φ Trefoil

(Elliptical Coma) Z11 Z3-3 ( )3 sin 3ρ φ

Z12 Z42 ( ) ( )4 24 3 cos 2ρ ρ φ− Sec. Astigmatism

(Oblique Spherical) Z13 Z4-2 ( ) ( )4 24 3 sin 2ρ ρ φ−

Z14 Z51 ( ) ( )5 310 12 3 cosρ ρ ρ φ− +

Sec. Coma Z15 Z5-1 ( ) ( )5 310 12 3 sinρ ρ ρ φ− +

Z16 Z60 6 4 220 30 12 1ρ ρ ρ− + − Sec. Spherical

The relationship of the Fringe Zernike set to the Hopkins wavefront expansion was

presented by Gray et al. [35] and the resulting relationships are displayed in Table 2-3. In

Table 2-3 it is seen that the Zernike terms do not directly relate to the wave aberration

coefficients. Some low order Zernike coefficients, like Z5/6 and Z7/8, are composed of

both third and fifth order wave aberration types. Therefore, when evaluating a FFD, it is

important to understand that the display does not isolate a single aberration type but will

display the dominant aberration characteristics. The higher order Zernike terms appear to

be directly related to a wave aberration type, though, it is only because the wavefront

27

expansion is up to fifth order. If seventh order components are considered, additional

factors will exist for these terms as well.

Table 2-3. Field dependence of the Zernike coefficients in terms of the wave aberration coefficients. (Adapted from Gray et al. [35])

Fringe ( ),jZ ρ φ

Wavefront Expansion Coefficient Function Through 5th order

Z1 2 2 420 040 060 220 240 420

1 1 1 1 1 12 3 4 2 3 2M M MW W W W H W H W H∆ + + + + +

Z2/3 ( )( )

2 2 411 131 151 311 331 511

cos2 1 23 2 3 sinMW W W W H W H W H H

θ

θ

∆ + + + + +

Z4 2 2 420 040 060 220 240 420

1 1 9 1 1 12 2 20 2 2 2M M MW W W W H W H W H∆ + + + + +

Z5/6 ( )( )

2 2222 242 422

cos 21 3 12 8 2 sin 2

W W W H Hθ

θ

+ +

Z7/8 ( )( )

2131 151 331

cos1 2 13 5 3 sinMW W W H H

θ

θ

+ +

Z9 2040 060 240

1 1 16 4 6 MW W W H+ +

Z10/11 ( )( )

3333

cos 314 sin 3

W Hθ

θ

Z12/13 ( )( )

2242

cos 218 sin 2

W Hθ

θ

Z14/15 ( )( )151

cos110 sin

W Hθ

θ

Z16 060120

W

As an example of the FFD, Figure 2-4 (a-b) shows the Zernike pair for astigmatism

(Z5/6) for the case when an optical system is centered so only field quadratic astigmatism

is present, as shown in Figure 2-4 (a), and when the optical system is perturbed to create

a binodal response, as shown in Figure 2-4 (b). With the FFD, the two nodes are readily

visible. Note that the Zernike pair, Z5/6, is plotted together so both the magnitude of the

28

entire aberration and its orientation, in this case, at the image plane, can be visualized

with the FFD.

(a) (b)

Figure 2-4. Full field display (FFD) showing (a) third order field quadratic in a centered system and (b) in perturbed optical system that yields binodal astigmatism.

29

Chapter 3. Aberration Fields in Optical Systems with φ-Polynomial Optical Surfaces

Nodal aberration theory (NAT) describes the aberration fields of optical systems when

the constraint of rotational symmetry is not imposed. Historically the theory, discovered

by Shack [19] and developed by Thompson [20], has been limited to optical imaging

systems made of rotationally symmetric components, or offset aperture portions thereof,

that are tilted and/or decentered. Recently, the special case of an astigmatic optical

surface located at the aperture stop (or pupil) was introduced into NAT by

Schmid et al. [64] and analyzed for the case of a primary mirror in a two mirror

telescope. At the stop surface, the beam footprint is the same for all field points, so all

field angles receive the same contribution from the astigmatic surface. The net astigmatic

field dependence, as predicted by NAT, and as validated by real ray tracing, takes on

characteristic nodal features that allow the presence and magnitude/orientation of

astigmatic figure error to be readily distinguished from the presence and

magnitude/orientation of any misalignment of the secondary mirror.

In this chapter, a path based in NAT is presented for developing an analytic theory for

the aberration fields of nonsymmetric optical systems with freeform surfaces. With this

extension to NAT, the zeros (or nodes) of the aberration contributions, which are

distributed throughout the FOV, can be anticipated analytically and targeted directly for

the correction or control of the aberrations in an optical system with freeform surfaces.

We consider an optical surface defined by a conic plus a ϕ-polynomial overlay, where the

sag of the overlay depends on the radial component, ρ, as well as the azimuthal

component, φ, within the aperture of the surface. Significantly, the freeform overlay can

be placed anywhere within the optical imaging system. Under these more general

30

conditions, it will be shown that the aberration contributions of the freeform surface

contribute both field constant and field dependent terms to the net aberration field of the

optical system. These aberration terms are derived for a specific ϕ-polynomial set, the

Zernike polynomial set up to sixth order. For each term in this subset, the aberration

behavior throughout the field is examined. Unexpectedly, we find that the impact of

integrating ϕ-polynomial freeform surfaces into NAT does not introduce new forms of

field dependence; rather, the freeform parameters link directly with the terms presented

for the generally multinodal field dependence of the sixth order wavefront aberrations

derived for tilted and decentered rotationally symmetric surfaces as reviewed in

Chapter 2. As an example of the types of analyses that can now be carried out with NAT,

the impact of three point mount-induced error (trefoil) on the field dependence of

astigmatism is presented here.

3.1 Formulating Nodal Aberration Theory for Freeform, ϕ-Polynomial Surfaces away from the Aperture Stop

To analytically characterize the impact of a ϕ-polynomial optical surface away from the

stop on the net aberration field, first consider a classical Schmidt telescope configuration.

The telescope is composed of a rotationally symmetric third order (fourth order in

wavefront) aspheric corrector plate in coincidence with a mechanical aperture that is the

stop of the optical system, located at the center of curvature of a spherical mirror. In such

a configuration, the net aberration contribution of the aspheric corrector plate, ,Corrector StopW ,

is described by the overall third order spherical aberration it induces, given by

( ) ( )2, 040

ASPHCorrector StopW W ρ ρ=

(3.1)

31

where ( )040

ASPHW denotes the spherical aberration wave aberration contribution from the

aspheric corrector plate and ρ is a normalized two-dimensional pupil vector that denotes

a location in the pupil of the Schmidt telescope.

Nominally, the Schmidt telescope is corrected for third order spherical aberration by

the corrector plate and for third order coma and astigmatism by locating the stop at the

center of curvature of the spherical mirror, leaving only field curvature as the limiting

third order aberration. The case where an aspheric corrector plate located at the stop or

pupil of an optical system is decentered from the optical axis was previously treated in

the context of NAT by Thompson [65] and was more recently revisited by

Wang et al. [66]. If the aspheric plate is instead shifted axially (i.e. longitudinally along

the optical axis) relative to a physical aperture stop, as shown in Figure 3-1 (a), the beam

for any off-axis field point will begin to displace across the aspheric plate. The amount of

relative beam displacement, h∆

, is given by

,y uth H Hy y

∆ ≡ =

(3.2)

where y is the paraxial marginal ray height on the aspheric plate, y is the paraxial chief

ray height on the aspheric plate, u is the paraxial chief ray angle, t is the distance between

the aspheric corrector plate and the mechanical aperture that is the optical system stop,

and H

is the normalized two-dimensional field vector that locates the field point of

interest in the image plane (i.e. 0≤| H

|≤1).

Conceptually, the beam displacement on the corrector plate when it is shifted away

from the stop can be thought of as a field dependent decenter of the aspheric corrector

when it is located at the aperture stop as shown in Figure 3-1 (b) where the mapping of

32

the normalized pupil coordinate is modified from ρ to 'ρ . Therefore, the net aberration

contribution of the aspheric corrector described by Eq. (3.1) must be modified to account

for this effect. By replacing ρ with ' hρ + ∆

and expanding the pupil dependence leads to

a modified aberration contribution, ,Corrector Not StopW , that is given by

( ) ( ) ( )

( )( ) ( )( ) ( )( )

( ) ( )( ) ( )

2

, 040

2

040 22 2

' '

4 4,

2 4

ASPHCorrector Not Stop

ASPH

W W h h

h h hW

h h h h h h

ρ ρ

ρ ρ ρ ρ ρ ρ ρ

ρ ρ

= + ∆ + ∆ + ∆ + ∆ ∆ = + ∆ + ∆ ∆ ∆ + ∆ ∆

(3.3)

where it is recognized that the measurement is done in the shifted pupil coordinate and

the primes have been dropped from the final expression of Eq. (3.3). As can be seen from

Eq. (3.3), the original spherical aberration contribution from the aspheric plate generates

lower order field dependent aberration components as the plate is shifted away from the

stop. Note that the operation of vector multiplication, introduced in [20], is being used in

this expansion. The aberration terms that are generated by this expansion are the

conventional third order field aberration terms summarized in Table 3-1, which could be

anticipated since the field aberrations are the product of spherical aberration in the

presence of a stop shift from the center of curvature.

33

16:56:15

Flat-field Schmidt Scale: 0.80 ORA 12-Jun-12

31.25 MM

yHy

t

uH

Stop Corrector Plate

SphericalMirror

h '

(a) (b)

Figure 3-1. (a) When the aspheric corrector plate of a Schmidt telescope is displaced

longitudinally from the aperture stop, the beam for any off-axis field point will displace

along the corrector plate. The displacement depends on the paraxial quantities for the

marginal ray height, y , chief ray height,

y , chief ray angle, u , and the distance between

the stop and plate, t . (b) Alternatively, the beam displacement on the corrector plate can

be thought of as a field dependent decenter of the aspheric corrector, h , that modifies

the mapping of the normalized pupil coordinate from to ' .

Table 3-1. Field aberration terms that are generated from the longitudinal shift of an

aspheric plate from the stop surface in a Schmidt telescope.

Terms in Eq. (3.3) 3

rd Order Vector Aberration

using Eq. (3.1) and (3.3)

3rd

Order Naming

Convention

2

040

ASPHW

2

040

ASPHW

Spherical

Aberration

0404ASPH

W h 0404ASPH y

W Hy

Coma

0404ASPH

W h h 2

0404ASPH y

W H Hy

Field Curvature

2 2

0402ASPH

W h 2

2 2

0402ASPH y

W Hy

Astigmatism

0404ASPH

W h h h 3

0404ASPH y

W H H Hy

Distortion

2

040

ASPHW h h

42

040

ASPH yW H H

y

Piston

Figure 3-2 (a-d) demonstrates the generation of astigmatism and coma for an example

F/1.4 Schmidt telescope analyzed using a FFD over a ±4° FOV. The aberration

34

components of the displays are calculated based on real ray optical data using either a

generalized Coddington close skew ray trace for astigmatism [67] or a Fringe Zernike

polynomial fit to the wavefront optical path difference (OPD) data in the exit pupil for

coma and any higher order aberration terms. In Figure 3-2 (e), the magnitude of the

generated coma and astigmatism is evaluated at two specific field points for several

longitudinal positions of the fourth order aspheric corrector plate. From Figure 3-2 it can

be seen that as the plate moves longitudinally away from the aperture stop along the

optical axis, third order field linear coma is generated linearly with the distance from the

aperture stop. In addition, third order field quadratic astigmatism is generated

quadratically with distance from the aperture stop, matching the predictions described in

Table 3-1. These observed dependencies parallel observations made by Burch [68] when

he introduced his “see-saw diagram” concept and by Rakich [69] when he used the

“see-saw diagram” to simplify the third order analysis of optical systems.

What has been recognized for the first time in the context of NAT is that this method

for generating the aberration terms displayed in Eq. (3.3) is not restricted to rotationally

symmetric corrector plates and it can be applied, with interpretation, to the general class

of ϕ-polynomial surfaces. This approach is a pathway for melding freeform optical

surfaces into NAT. More significantly, the outcome is that freeform surfaces in the

ϕ-polynomial family fit directly into the existing discoveries for the characteristic

aberration fields of a perturbed (i.e. tilted or decentered) optical system through sixth

order that are developed in [20, 23-25].

35

08:36:17

Flat-field Schmidt

ORA 30-May-12

ASTIGMATIC LINE IMAGE

vs

FIELD ANGLE IN OBJECT SPACE

Minimum = 0

Maximum = 0.55844

Average = 0.19928

Std Dev = 0.12784

1.992mm

-4 -2 0 2 4

X Field Angle in Object Space - degrees

-4

-2

0

2

4

Y Field Angle in Object Space - degrees

08:36:21

Flat-field Schmidt

ORA 30-May-12

FRINGE ZERNIKE PAIR Z7 AND Z8

vs


Minimum = 0.10098e-10

Maximum = 58.486

Average = 33.692

Std Dev = 12.373

150waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


08:36:17

Flat-field Schmidt ORA 30-May-12

25.00 MM

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

08:36:04


25.00 MM

08:36:04

Flat-field Schmidt

ORA 30-May-12


vs


Minimum = 0

Maximum = 0.24281

Average = 0.08755

Std Dev = 0.055768

1.992mm

-4 -2 0 2 4


-4

-2

0

2

4


08:36:08

Flat-field Schmidt

ORA 30-May-12


vs


Minimum = 0.89056e-12

Maximum = 40.116

Average = 22.744

Std Dev = 8.4663

150waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-408:35:51

Flat-field Schmidt

ORA 30-May-12


vs


Minimum = 0

Maximum = 0.060055

Average = 0.021751

Std Dev = 0.013804

1.992mm

-4 -2 0 2 4


-4

-2

0

2

4


08:35:55

Flat-field Schmidt

ORA 30-May-12


vs


Minimum = 0.92121e-13

Maximum = 19.979

Average = 11.303

Std Dev = 4.2147

150waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


16:17:34


25.00 MM

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

08:35:42

Flat-field Schmidt

ORA 30-May-12


vs


Minimum = 0.3321e-11

Maximum = 0.40548

Average = 0.2324

Std Dev = 0.08561

150waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


08:35:39

Flat-field Schmidt

ORA 30-May-12


vs


Minimum = 0

Maximum = 0.28862e-4

Average = 0.10046e-4

Std Dev = 0.65357e-5

1.992mm

-4 -2 0 2 4


-4

-2

0

2

4


16:17:22


25.00 MM

Coma Astigmatism

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

Y O

bj.

Fie

ld (

de

g.)

4

2

0

-2

-4

X Obj. Field (deg.)-4 -2 0 2 4








(a) 150λ (0.587µm)

09:17:02

Flat-field Schmidt

ORA 12-Jan-12


vs


Minimum = 0.55999

Maximum = 0.57967

Average = 0.56928

Std Dev = 0.0057599

4waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


150λ (0.587µm)

09:17:02

Flat-field Schmidt

ORA 12-Jan-12


vs


Minimum = 0.55999

Maximum = 0.57967

Average = 0.56928

Std Dev = 0.0057599

4waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


(b)

150λ (0.587µm)

09:17:02

Flat-field Schmidt

ORA 12-Jan-12


vs


Minimum = 0.55999

Maximum = 0.57967

Average = 0.56928

Std Dev = 0.0057599

4waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


(c)

(d) 150λ (0.587µm)

09:17:02

Flat-field Schmidt

ORA 12-Jan-12


vs


Minimum = 0.55999

Maximum = 0.57967

Average = 0.56928

Std Dev = 0.0057599

4waves ( 587.6 nm)

-4 -2 0 2 4


-4

-2

0

2

4


(e)

0 60 120 1800

10

20

30

40

50

Plate Postion from Stop (mm)

Astigm

atism

(w

aves)

0 60 120 1800

10

20

30

40

50

Plate Postion from Stop (mm)

Com

a (

waves)

Figure 3-2. Generation of coma and astigmatism as the aspheric corrector plate in a

Schmidt telescope is moved longitudinally (along the optical axis) from the physical

aperture stop located at the center of curvature of the spherical primary mirror for various

positions (a-d). For each field point in the FFD, the plot symbol conveys the magnitude

and orientation of the aberration. (e) Plots of the magnitude of coma and astigmatism

generated as the aspheric plate is moved longitudinally for two field points, (0°, 2°) (blue

square) and (0°, 4°) (red triangle).

36

3.2 The Aberration Fields of ϕ-Polynomial Surface Overlays

The set of ϕ-polynomial overlays to be placed on an optical surface is the Fringe Zernike

polynomial set that is presented in Figure 3-3 up to sixth order. This set differs from other

Zernike polynomial sets in its arrangement of terms where they are ordered by wavefront

expansion order, with the third order aberration components appearing before the fifth

order components. Out of the sixteen terms displayed in Figure 3-3, twelve are

nonsymmetric, ϕ-polynomial types and of the twelve nonsymmetric terms, ten will blur

the image if they are placed on a surface of an optical system. Moreover, these ten terms

form five pairs to be explored, namely, they are Zernike astigmatism (Z5/6), Zernike coma

(Z7/8), Zernike trefoil or elliptical coma (Z10/11), Zernike secondary astigmatism or

oblique spherical aberration (Z12/13), and Zernike secondary coma or fifth order aperture

coma (Z14/15).

In Section 3.1, when describing the aspheric corrector plate of the Schmidt telescope,

it was found that the aberration contribution from the corrector is field constant when the

plate is located at the aperture stop and develops a field dependent contribution as the

surface is shifted longitudinally away from the aperture stop. For the aspheric corrector

plate of the Schmidt telescope, the field constant aberration that results is third order

spherical aberration. By analogy, if a plate placed at the stop is deformed by one of the

Zernike terms described above, it will also introduce a field constant aberration. By

utilizing the vector pupil dependence of the Zernike overlay terms, the induced field

constant aberration is predicted by NAT and it can be added to the total aberration field.

37

Z1

Z3 Z2

Z5Z4

Z8 Z10

Z6

Z12

Z11 Z7

Z9Z13

Z16

Z14Z15

Figure 3-3. Fringe Zernike polynomial set up to 5th order (6th order in wavefront). The set includes Z1 (piston), Z2/3 (tilt), Z4 (defocus), Z5/6 (astigmatism), Z7/8 (coma), Z9 (spherical aberration), Z10/11 (elliptical coma or trefoil), Z12/13 (oblique spherical aberration or secondary astigmatism), Z14/15 (fifth order aperture coma or secondary coma), and Z16 (fifth order spherical aberration or secondary spherical aberration). The φ-polynomials to be explored include Z5/6, Z7/8, Z10/11, Z12/13, and Z14/15.

3.2.1 Zernike Astigmatism

In order of increasing radial dependence, the first freeform overlay term to consider is

astigmatism. In optical metrology terminology, Zernike astigmatism (Fringe polynomial

terms 5Z and 6Z ) is given by

( )( )

255

26 6

cos 2,

sin 2

zZZ z

ρ φ

ρ φ

=

(3.4)

where 5z and 6z are the coefficient values for the astigmatism term, ρ is the normalized

radial coordinate, and φ represents the azimuthal angle on the surface. In optical testing,

the Fringe Zernike set is described in a right-handed coordinate system with φ measured

38

counter-clockwise from the x − axis. The magnitude, 5/6FFz , and orientation, 5/6

TestFFξ , of the

freeform Zernike astigmatism overlay is then calculated from the coefficients by

2 25/6 5 6FF

z z z= + (3.5)

1 65/6

5

1 tan ,2

TestFF

zz

ξ − =

(3.6)

where the superscript Test denotes the optical testing coordinate system.

Zernike astigmatism can be introduced in the vector multiplication environment of

NAT with the following observation, which is the basis for NAT,

( )( )

( )( )

2 2sin sin 2

, ,cos cos 2

if thenφ φ

ρ ρ ρ ρφ φ

= =

(3.7)

where consistent with commercial optical design raytrace programs, a right-handed

coordinate system is employed with φ measured clockwise from the y − axis. To

implement a coordinate system for the overlay term that is consistent with its generated

aberration field within the context of the real ray based environment of NAT, the

orientation in Eq. (3.6) must be modified. A new orientation, 5/6FFξ , is defined and is

displayed in Figure 3-4 and given by

1 65/6

5

1 tan .2 2FF

zz

πξ − = −

(3.8)

39

x pupil coordinate0.0 0.5 1.0-0.5-1.0

0.0

-0.5

-1.0

0.5

1.0

y pup

il coo

rdin

ate

-

-

-

-

-

0

0

1

1

2

2

+1.0λ (P)

-1.0λ (V)

V

PV

P

5/6FFξ

Figure 3-4. Surface map describing the freeform Zernike overlay for astigmatism on an optical surface over the full aperture. The error is quantified by its magnitude 5/6FF

z and

its orientation 5/6FFξ that is measured clockwise with respect to the y − axis. P and V denote where the surface error is a peak rather than a valley.

From the vector pupil dependence in Eq. (3.7), it is deduced that the astigmatism

overlay will induce field constant astigmatism that is predicted by NAT when the optical

surface is placed at the aperture stop. Based on this observation, it is added to the total

aberration field as

( )2 2222, 222

1 ,2Stop FFW B ρ=

(3.9)

where 2222FF B

is a two-dimensional vector that describes the magnitude and orientation of

the astigmatic overlay, which is related to the overall Zernike astigmatism by

( ) ( )2222 5/6 5/62 1 exp 2 ,FF FFFF

B n z i ξ≡ − −

(3.10)

where n is the index of refraction of the substrate medium.

If a surface with a Zernike astigmatism overlay is now placed away from the stop, the

beam footprint for an off-axis field angle will begin to displace across the surface

resulting in the emergence of a number of field dependent terms. Replacing ρ with

' hρ + ∆

in Eq. (3.9), expanding the pupil dependence, and simplifying leads to a specific

40

set of additive terms for the wavefront expansion when a surface is located away from the

stop,

( )2

2222, 222

2 2 2 2 2222 222 222

1 '21 2 ,2

Not Stop FF

FF FF FF

W B h

B B h B h

ρ

ρ ρ

= + ∆

= + ∆ + ∆

(3.11)

where, as in Section 3.1, the primes on the pupil coordinate have been dropped from the

final expression.

To map the impact of these additive terms on the overall field dependent wave

aberration expansion of an optical system, the pupil dependence needs to be converted

into existing aberration types. To this end, an additional vector operation, introduced

in [20], is used,

* ,A BC AB C=

(3.12)

where *B

is a conjugate vector with the standard properties of a conjugate variable in the

mathematics of complex numbers

( )* ˆ ˆexp .x yB B i B x B yβ= − = − +

(3.13)

By applying the vector identity of Eq. (3.12), Eq. (3.11) takes the form

2 2 2 * 2 2222, 222 222 222

1 2 .2Not Stop FF FF FFW B B h B hρ ρ = + ∆ + ∆

(3.14)

In Eq. (3.14) two additional field dependent aberration terms are generated in addition to

the anticipated field constant astigmatism term. The second and third terms, however, are

a tilt and piston that do not affect the image quality but affect the mapping and phase.

Here we are focusing on the image quality; therefore, these terms will not be directly

addressed for this or any subsequent Zernike overlay terms. In this case, the only image

degrading aberration is field constant astigmatism that is independent of where the

Zernike astigmatism overlay is located with respect to the stop. A magnitude and

41

orientation plot for the field constant astigmatism contribution is illustrated in Figure 3-5.

Since the aberration has no dependence on the field vector, the magnitude and orientation

are the same everywhere throughout the field.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

Figure 3-5. The characteristic field dependence of field constant astigmatism that is generated by a Zernike astigmatism overlay on an optical surface in an optical system. This induced aberration is independent of stop position.

The field constant astigmatism contribution can be added to the existing concepts of

NAT by re-defining the field constant astigmatic term of NAT, 2222B

, as

2 2 2222 222 222,

1,

N

ALIGN FF jj

B B B=

= +∑

(3.15)

where the summation and index j has been introduced to generalize the result to include a

multi-element optical system where the Zernike astigmatism overlay is on the jth optical

surface and 2222ALIGN B

defines the existing astigmatic component that may result from a

misalignment. From the definition of 2222B

in Eq. (3.15), the conventional strategies of NAT

can be applied to solve for the nodal properties of the astigmatic aberration field when a

Zernike astigmatism overlay is placed on an optical surface of a multi-element optical

system.

3.2.2 Zernike Coma

The next freeform overlay term in order of pupil dependence is Zernike coma (Fringe

polynomial terms 7Z and 8Z ) that, in optical metrology terminology, is written as

42

( ) ( )( ) ( )

377

38 8

3 cos 2 cos,

3 sin 2 sin

zZZ z

ρ φ ρ φ

ρ φ ρ φ

− = − (3.16)

where 7z and 8z are the coefficient values for the coma term. Within this term there is

cubic aperture ( 3ρ ) coma term and a linear aperture ( ρ ) tilt term. The tilt term is

inherently built into Zernike coma to minimize the RMS WFE of the aberration

polynomial over the aperture, a property of the Zernike polynomial set. In order to

generate coma that can be introduced in the vector multiplication environment of NAT,

an adjusted Zernike coma is used that combines both Zernike coma and Zernike tilt and is

written as

( )( )

377 7 2

38 38 8

3 cos2.

2 3 sin

AdjAdj

Adj Adj

zZ Z ZZ ZZ z

ρ φ

ρ φ

+ = = + (3.17)

Similar to Zernike astigmatism, the magnitude, 7/8Adj

FFz , and orientation, 7/8

AdjFFξ , of the

freeform, adjusted Zernike coma overlay term is then calculated from the coefficients by

( ) ( )2 2

7/8 7 8Adj Adj Adj

FFz z z= + (3.18)

1 87/8

7

tan ,2

AdjAdj

FF Adj

zz

πξ − = −

(3.19)

where the orientation in Eq. (3.19) creates an orientation consistent within the real ray

based environment of NAT. The overlay term in Eq. (3.17) can be linked to the vector

multiplication environment of NAT, with the following observation,

( )( )

( )( )( )

3sin sin

, ,cos cos

if thenφ φ

ρ ρ ρ ρ ρ ρφ φ

= =

(3.20)

where a right-handed coordinate system is employed with φ measured clockwise from

the y − axis. From the vector pupil dependence in Eq. (3.20), it is deduced that the

43

overlay will induce field constant coma when located at the stop surface and is added to

the total aberration field as

( )( )131, 131 ,Stop FFW A ρ ρ ρ=

(3.21)

where 131FF A


the Zernike coma overlay, which is related to the overall Zernike coma by

( ) ( )131 7/8 7/83 1 exp .Adj AdjFF FFFF

A n z i ξ≡ − −

(3.22)

Now replacing ρ with ' hρ + ∆

in Eq. (3.21), expanding the pupil dependence, and

simplifying leads to a specific set of additive terms for the wavefront expansion when a

surface with a Zernike coma overlay is located away from the stop,

( ) ( ) ( )( )( ) ( )( )

( )( ) ( )( )

131, 131

2131 131 131

* 2131 131 131

' ' '

2,

2

Not Stop FF

FF FF FF

FF FF

W A h h h

A A h A h

h h A A h A h h h

ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ

= + ∆ + ∆ + ∆ + ∆ + ∆ = + ∆ ∆ + ∆ + ∆ ∆ ∆

(3.23)


final expression. As can be seen from Eq. (3.23), five additional field dependent

aberration terms are generated in addition to the anticipated field constant coma term.

The first, field constant term is added into NAT by re-defining the field constant coma

term in NAT, 131A

, as

131 131 131,1

,N

ALIGN FF jj

A A A=

= −∑

(3.24)

where 131ALIGN A

is any comatic contribution from misalignment. The second term is

recognized to be an astigmatic term based on the 2ρ aperture dependence. When Eq. (3.2)

is used to replace h∆

in the astigmatic term of Eq. (3.23), it becomes

2 2131 131, .j

FF FF jj

yA h A H

yρ ρ

∆ =

(3.25)

44

Equation (3.25) is a form of field asymmetric, field linear astigmatism that was first seen

in the derivation for the nodal structure of third order (fourth order in wavefront)

astigmatism by Thompson [20, 21]. This contribution is added to the field linear

astigmatism contribution of NAT, 222A

, as

222 222 131,1

,N

jALIGN FF j

j j

yA A A

y=

= −

∑

(3.26)

where 222ALIGN A

defines the existing astigmatic component that may result from a

misalignment. The third term is recognized to be a field curvature term based on the

( )ρ ρ

aperture dependence and when h∆

is replaced in it, it takes the form

( )( ) ( )( )131 1312 2 .jFF FF

j

yA h A H

yρ ρ ρ ρ

∆ =

(3.27)

Equation (3.27) is now recognized as a form of field curvature, seen in the derivation for

the nodal structure of third order field curvature by Thompson [20], that yields a tilted

focal surface relative to the Gaussian image plane and is added to the field linear, field

curvature contribution of NAT, 220MA

, as

220 220 131,1

.M M

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

(3.28)

where 220MALIGN A

defines the existing field linear, field curvature component that may result

from a misalignment. The process of linking the aberration terms generated by a Zernike

coma overlay to existing concepts of NAT is summarized in Table 3-2 where the

aberration terms from Eq. (3.23) are displayed in column one with h∆

replaced by its

form using Eq. (3.2), column two displays the NAT analog term that has the same field

and pupil behavior as the generated terms in column one, and column three displays how

45

the NAT analog term is re-defined to include both the misalignment and freeform overlay

components.

Table 3-2. Image degrading aberration terms that are generated by a Zernike coma overlay and how the terms link to existing concepts of NAT

Aberration Terms for a Zernike Coma Overlay NAT Analog Addition of overlay term into

NAT

( )( )131,FF jA ρ ρ ρ

( )( )131A ρ ρ ρ−

131 131 131,1

N

ALIGN FF jj

A A A=

= −∑

2131,

jFF j

j

yA H

yρ

( )2222

1 22

A H ρ−

222 222 131,1

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

( )( )131,2 jFF j

j

yA H

yρ ρ

( )( )2202M

A H ρ ρ−

220 220 131,1

M M

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

The magnitude and orientation plots of the aberration terms generated by a Zernike

coma overlay, summarized in Table 3-2, are depicted throughout the field in

Figure 3-6 (a-c). In Figure 3-6 (a), the field constant comatic contribution from a Zernike

coma overlay is displayed. The magnitude and orientation are the same everywhere

throughout the field and are governed by the vector describing the overlay term, 131FF A

. In

Figure 3-6 (b), the astigmatic contribution from a Zernike coma overlay away from the

stop is displayed. As can be seen from the line images, the aberration is asymmetric with

field while increasing linearly from a single node. Lastly, in Figure 3-6 (c), the field

curvature contribution is displayed. This form of field curvature increases linearly with

field in the direction of the vector describing the overlay term, 131FF A

.

46

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

(a) (b) (c)

Figure 3-6. The characteristic field dependence of (a) field constant coma, (b) field

asymmetric, field linear astigmatism, and (c) field linear, field curvature that is generated

by a Zernike coma overlay on an optical surface away from the stop surface.

3.2.3 Zernike Trefoil (Elliptical Coma)

The next freeform overlay term that has the same pupil dependence as coma but a higher

order azimuthal dependence is Zernike trefoil (Fringe polynomial terms 10Z and

11Z ) that,

in optical metrology terminology, is written as

3

1010

311 11

cos 3,

sin 3

zZ

Z z

(3.29)

where 10z and

11z are the coefficient values for the trefoil term. The magnitude, 10/11FFz ,

and orientation, 10/11FF , of the Zernike trefoil is then calculated from the coefficients by

2 2

10/11 10 11FFz z z (3.30)

1 11

10/11

10

1tan .

2 3FF

z

z

(3.31)

The overlay term in Eq. (3.29) can be linked to the vector multiplication environment

of NAT, with the following observation

3 3

sin sin 3, ,

cos cos 3if then

(3.32)

where a right-handed coordinate system is employed with measured clockwise from

the y axis. From the vector pupil dependence in Eq. (3.32), it is deduced that the trefoil

47

deformation will induce field constant, elliptical coma when located at the stop surface

and is added to the total aberration field as

( )3 3333, 333

1 ,4Stop FFW C ρ=

(3.33)

where 3333FF C


field constant elliptical coma, which is related to the overall Zernike trefoil by

( ) ( )3333 10/11 10/114 1 exp 3 .FF FFFF

C n z i ξ≡ − −

(3.34)




surface with a Zernike trefoil overlay is located away from the stop,

( )3

3333, 333

3 3 3 * 2333 333

3 *2 3 3333 333

1 '4

31 ,4 3

Not Stop FF

FF FF

FF FF

W C h

C C h

C h C h

ρ

ρ ρ

ρ

= + ∆ + ∆

= + ∆ + ∆

(3.35)


final expression. In Eq. (3.35), three additional field dependent aberration terms are

generated in addition to the anticipated field constant elliptical coma (trefoil) term.

Following the method outlined for the Zernike coma overlay, Table 3-3 displays the

image degrading aberration terms generated by the Zernike trefoil overlay with h∆

replaced in each term and shows how each term links to existing concepts of NAT.

Table 3-3. Image degrading aberration terms that are generated by a Zernike elliptical coma overlay and how the terms link to existing concepts of NAT

Aberration Terms for a Zernike Trefoil Overlay NAT Analog Addition of overlay term into

NAT

3 3333,

14 FF jC ρ

3 3333

14

C ρ−

3 3 3333 333 333,

1

N

ALIGN FF jj

C C C=

= −∑

3 * 2333,

34

jFF j

j

yC H

yρ

( )3 * 2422

12

C H ρ−

3 3 3422 422 333,

1

32

Nj

ALIGN FF jj j

yC C C

y=

= −

∑

48

In Table 3-3, it can be seen that the field constant elliptical coma term pairs with

3333ALIGN C

which is a fifth order (sixth order in wavefront) misalignment induced aberration

component. Normally, since 3333ALIGN C

is a cubic vector, this contribution is small and

dominated by lower order misalignment contributions. However, with the use of freeform

overlays, particularly any overlay of equal or higher order than Zernike trefoil, the fifth

order aberration space and their misalignment induced aberration components like 3333C

can be roughly equal to or greater than the third order misalignment induced aberration

components of NAT.

The second term from Eq. (3.35) is seen to be an astigmatic term based on the

2ρ aperture dependence and it is a form of field linear astigmatism that was first seen in

the derivation for the nodal structure of field quartic fifth order astigmatism by

Thompson [25], and reported in Table 3-3 (second row, second column). This linear

astigmatism term has not previously been isolated as an observable field dependence and

it represents the first time any aberration with conjugate field dependence has been linked

to an observable quantity [70]. In Chapter 4, this aberration and its characteristic field

behavior are experimentally validated through the design and implementation of an

aberration generating telescope.


trefoil overlay, summarized in Table 3-3, are depicted throughout the field in

Figure 3-7 (a-b). In Figure 3-7 (a), the field constant elliptical coma contribution from a

Zernike trefoil overlay is displayed. The magnitude and orientation are the same

everywhere throughout the field and are governed by the vector describing the overlay

term, 3333FF C

. In Figure 3-7 (b), the astigmatic contribution from a Zernike trefoil overlay

49

away from the stop is displayed. The aberration is of the same order as field asymmetric,

field linear astigmatism but it depends on the conjugate vector so it takes on a different

orientation throughout the field. This form of astigmatic field dependence was reported in

the literature by Stacy [71], but its analytical origin has remained unexplained until now.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

(a) (b)

Figure 3-7. The characteristic field dependence of (a) field constant elliptical coma, (b)

field conjugate, field linear astigmatism, which is generated by a Zernike elliptical coma

overlay on an optical surface away from the stop surface.

3.2.4 Zernike Oblique Spherical Aberration

Moving to the next pupil order, the next freeform overlay term is Zernike oblique

spherical (Fringe polynomial terms 12Z and

13Z ) that, in optical metrology terminology, is

written as

4 2

1212

4 213 13

4 cos 2 3 cos 2,

4 sin 2 3 sin 2

zZ

Z z

(3.36)

where 12z and

13z are the coefficient values for the oblique spherical term. Within this

term there is a quartic aperture ( 4 ) oblique spherical aberration term and a quadratic

aperture ( 2 ) astigmatism term. Similar to the case of Zernike coma, there is an included

astigmatic term to minimize the RMS WFE of the oblique spherical aberration term. In

order to generate oblique spherical aberration that can be introduced in the vector

multiplication environment of NAT, an adjusted Zernike oblique spherical aberration is

50

used that combines both Zernike oblique spherical aberration and Zernike astigmatism

and is written as

( )( )

41212 512

412 613 13

4 cos 23.

3 4 sin 2

AdjAdj

Adj Adj

zZ ZZZ ZZ z

ρ φ

ρ φ

+ = = + (3.37)

The magnitude, 12/13Adj


AdjFFξ , of the freeform, adjusted Zernike oblique

spherical aberration overlay term is then calculated from the coefficients by

( ) ( )2 2

12/13 12 13Adj Adj Adj

FFz z z= + (3.38)

1 1312/13

12

tan .2

AdjAdj

FF Adj

zz

πξ − = −

(3.39)

The overlay term in Eq. (3.37) can be linked to the vector multiplication environment of

NAT, with the following observation

( )( )

( )( )( )

2 4sin sin 2

, ,cos cos 2

if thenφ φ


= =

(3.40)


the y − axis. From the vector pupil dependence in Eq.(3.40), it is deduced that the oblique

spherical overlay will induce field constant, oblique spherical aberration when located at

the stop surface and is added to the total aberration field as

( )( )2 2242, 242

1 ,2Stop FFW B ρ ρ ρ=

(3.41)

where 2242FF B


field constant oblique spherical aberration, which relates to adjusted Zernike oblique

spherical aberration by

( ) ( )2242 12/13 12/138 1 exp 2 .Adj Adj

FF FFFFB n z i ξ≡ − −

(3.42)

51


in Eq. (3.41) , expanding the pupil dependence, and


surface with a Zernike oblique spherical aberration overlay is located away from the stop,

( ) ( ) ( )( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )

22

242, 242

2 2 2 * 2 3242 242 242

2 2 2 2242 242

2 2 2 *242 242

1 ' ' '2

3

3 312 2 2

Not Stop FF

FF FF FF

FF FF

FF FF

W B h h h

B B h B h

h h B B h

B h h h h B h

ρ ρ ρ

ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ

ρ

= + ∆ + ∆ + ∆

+ ∆ + ∆

+ ∆ ∆ + ∆=

+ ∆ ∆ + ∆ ∆ ∆

( )( )( )2 2

242

,

FFh h B h

ρ

+ ∆ ∆ ∆

(3.43)


final expression. As can be seen from Eq. (3.43), seven additional field dependent

aberration terms are generated in addition to the anticipated field constant oblique

spherical aberration term. Table 3-4 displays the image degrading aberration terms

generated by the Zernike oblique spherical aberration overlay with h∆

replaced in each

term and shows how each term links to existing concepts of NAT. In order of decreasing

pupil dependence, the first field dependent term is identified as an elliptical coma

aberration based on the 3ρ dependence, where, the elliptical coma is linear throughout the

field. The second term is identified as a comatic aberration based on the

( )ρ ρ ρ

dependence. The aberration field is linear with conjugate field dependence and

belongs with the misalignment induced aberrations of field cubed coma. The third term is

a fifth order astigmatic aberration based on the 2ρ dependence where the aberration is

quadratic with field from the ( )H H

component; however, since this quantity is a scalar,

the orientation only depends on the vector 2242,FF jB

and, as a result, the orientation is

constant throughout the field. The final term is a fifth order field curvature aberration

52

based on the ( )ρ ρ

dependence that yields a saddle shaped focal surface relative to the

Gaussian image plane.

Table 3-4. Image degrading aberration terms that are generated by a Zernike oblique spherical aberration overlay and how the terms link to existing concepts of NAT

Aberration Terms for a Zernike Oblique Spherical

Aberration Overlay NAT Analog Addition of overlay term into

NAT

( )( )2 2242,

12 FF jB ρ ρ ρ

( )( )2 2242

12

B ρ ρ ρ

2 2 2242 242 242,

1

N

ALIGN FF jj

B B B=

= +∑

2 3242,

12

jFF j

j

yB H

yρ

( )2 3333

1 34

B H ρ

2 2 2333 333 242,

1

23

Nj

ALIGN FF jj j

yB B B

y=

= +

∑

( )( )2 *242,

32

jFF j

j

yB H

yρ ρ ρ

( )( )2 *331M

B H ρ ρ ρ

2 2 2331 331 242,

1

32M M

Nj

ALIGN FF jj j

yB B B

y=

= +

∑

( )( )2

2 2242,

32

jFF j

j

yH H B

yρ

( )( )2 2422

1 32

H H B ρ

2

2 2 2422 422 242,

1

Nj

ALIGN FF jj j

yB B B

y=

= +

∑

( )( )2

2 2242,

32

jFF j

j

yB H

yρ ρ

( )( )2 24202

MB H ρ ρ

2

2 2 2420 420 242

1

34M M

Nj

ALIGN FFj j

yB B B

y=

= +

∑


oblique spherical aberration overlay, summarized in Table 3-4, are depicted throughout

the field in Figure 3-8 (a-e). In Figure 3-8 (a), the field constant oblique spherical

aberration contribution from a Zernike oblique spherical aberration overlay is displayed.

The magnitude and orientation are the same everywhere throughout the field and are

governed by the vector describing the overlay term, 2242FF B

. In Figure 3-8 (b), the elliptical

coma contribution from a Zernike oblique spherical aberration overlay away from the

stop is displayed. The field behavior of this elliptical coma term is of the same form as

field asymmetric, field linear astigmatism. Figure 3-8 (c) displays the field cubed comatic

contribution where the conjugate field dependence of the aberration yields a unique

53

orientation when compared to conventional third order field linear coma. The fifth order

astigmatic contribution, Figure 3-8 (d), exhibits a field constant orientation while the

magnitude of the aberration varies quadratically with the field vector. Lastly,

Figure 3-8 (e), displays a fifth order field curvature contribution that equates to a saddle

shaped focal plane as the aberration curves up in one direction and down in the other.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

(a) (b) (c)

(d) (e)

Figure 3-8. The characteristic field dependence of (a) field constant oblique spherical

aberration, (b) field asymmetric, field linear trefoil, (c) field conjugate, field linear coma,

(d) field constant, field quadratic astigmatism, and (e) field quadratic, field curvature that

is generated by a Zernike oblique spherical aberration overlay on an optical surface away

from the stop surface.

3.2.5 Zernike Fifth Order Aperture Coma

The next pupil order and last freeform overlay term is Zernike fifth order aperture coma

(Fringe polynomial terms 14Z and

15Z ) that, in optical metrology terminology, is written as

5 3

1414

5 315 15

10 cos 12 cos 3 cos,

10 sin 12 sin 3 sin

zZ

Z z

(3.44)

54

where 14z and 15z are the coefficient values for the fifth order coma term. Within this term

there is a quintic aperture ( 5ρ ) coma term, a cubic aperture ( 3ρ ) coma term, and a linear

aperture ( ρ ) tilt term to minimize the RMS WFE of the fifth order aperture coma term.

To generate a fifth order aperture coma that can be introduced in the vector multiplication

environment of NAT, an adjusted Zernike fifth order coma is used that combines Zernike

fifth order aperture coma, Zernike coma, and Zernike tilt and is written as

( )( )

51414 7 214

515 8 315 15

10 cos4 5.

4 5 10 sin

AdjAdj

Adj Adj

zZ Z ZZZ Z ZZ z

ρ φ

ρ φ

+ + = = + + (3.45)

The magnitude, 14/15Adj


AdjFFξ , of the freeform, adjusted Zernike fifth

order aperture coma overlay term is then calculated from the coefficients by

( ) ( )2 2

14/15 14 15Adj Adj Adj

FFz z z= + (3.46)

1 1514/15

14

tan .2

AdjAdj

FF Adj

zz

πξ − = −

(3.47)

The overlay term in Eq. (3.37) can be linked to the vector multiplication environment of

NAT, with the following observation

( )( )

( )( )( )

2 5sin sin

, ,cos cos

if thenφ φ


= =

(3.48)


the y − axis. From the vector pupil dependence in Eq.(3.40), it is deduced that the fifth

order aperture coma overlay will induce field constant, fifth order aperture coma when

located at the stop surface and is added to the total aberration field as

( )( )2151, 151 ,Stop FFW A ρ ρ ρ=

(3.49)

where 151FF A

is a two-dimensional vector describing the magnitude and orientation of field

constant, fifth order aperture coma, which relates to adjusted Zernike fifth order coma by

55

( ) ( )151 14/15 14/1510 1 exp .Adj AdjFF FFFF

A n z i ξ≡ − −

(3.50)




surface with a Zernike fifth order aperture coma overlay is located away from the stop,

( ) ( ) ( )( )( ) ( )( ) ( )

( ) ( ) ( )

( ) ( )

2

151, 151

2 2 2 3151 151 151

151 151

2 2151 151

15

' ' '

3

6 3

2 2

2

Not Stop FF

FF FF FF

FF FF

FF FF

FF

W A h h h

A A h A h

A h h h h A

A h h h h A h

A

ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ ρ

ρ

= + ∆ + ∆ + ∆

+ ∆ + ∆

+ ∆ ∆ + ∆ ∆ + ∆ ∆ + ∆ ∆ ∆

=+

( )( ) ( )( )( )

( ) ( )( )( ) ( )

21 151

2

151 151

2

151

,6

4

FF

FF FF

FF

h h h A h

h h A A h h h h

h h A h

ρ ρ ρ ρ ρ

ρ

∆ + ∆ ∆ ∆

+ ∆ ∆ + ∆ ∆ ∆ ∆ + ∆ ∆ ∆

(3.51)


final expression. As can be seen from Eq. (3.43), eleven additional field dependent

aberration terms are generated in addition to the anticipated field constant fifth order

aperture coma term. Table 3-5 displays the image degrading aberration terms generated

by the Zernike fifth order aperture coma overlay with h∆

replaced in each term and shows

how each term links to existing concepts of NAT. In order of decreasing pupil

dependence, the first field dependent term is identified as medial oblique spherical

aberration and it equates to a tilted medial surface for oblique spherical aberration

relative to the Gaussian image plane. The second term is identified as oblique spherical

aberration based on the ( ) 2ρ ρ ρ

dependence where the aberration is linear throughout the

field. The third term is an elliptical coma aberration based on the 3ρ dependence where

the aberration is quadratic with field. The fourth and fifth terms are both identified as a

form of coma based on the ( )ρ ρ ρ

dependence and are found as a misalignment induced

56

aberration of fifth order, field cubed coma. Likewise, the six and seventh terms are a form

astigmatism based on the 2ρ dependence and are found as a misalignment induced

aberration of fifth order, astigmatism. Lastly, the eighth term is a fifth order field

curvature term that manipulates the focal surface relative to the Gaussian image plane.

Table 3-5. Image degrading aberration terms that are generated by a Zernike fifth order aperture coma overlay and how the terms link to existing concepts of NAT

Aberration Terms for a Zernike Fifth Order

Aperture Coma Overlay NAT Analog Addition of overlay term into

NAT

( )( )2151,FF jA ρ ρ ρ

( )( )2151A ρ ρ ρ−

151 151 151,1

N

ALIGN FF jj

A A A=

= −∑

( )( )2151,3 j

jj

yH A

yρ ρ

( )( )22402

MH A ρ ρ−

240 240 151,1

32M M

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

( )( )2151,2 j

FF jj

yA H

yρ ρ ρ

( )( )2242

1 22

HA ρ ρ ρ −

242 242 151,1

2N

jALIGN FF j

j j

yA A A

y=

= −

∑

2

2 3151,

jFF j

j

yA H

yρ

( )2 3333

1 34

A H ρ−

2

333 333 151,1

43

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

( )( )( )2

151,6 jFF j

j

yA H H

yρ ρ ρ

( )( )( )3312M

H A H ρ ρ ρ−

2

331 331 151,1

3M M

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

( )( )( )2

151,3 jFF j

j

yH H A

yρ ρ ρ

( )( )( )331MH H A ρ ρ ρ−

( )( )3

2 2151,2 j

FF jj

yA H H

yρ

( )( )2 2422

1 22

H A H ρ −

3

422 422 151,1

2N

jALIGN FF j

j j

yA A A

y=

= −

∑

( )( )3

2151,2 j

FF jj

yH H A H

yρ

( )( )2422

1 22

H H HA ρ −

( )( )( )3

151,6 jFF j

j

yH H A H

yρ ρ

( )( )( )4204M

H H A H ρ ρ−

3

420 420 151,1

32M M

Nj

ALIGN FF jj j

yA A A

y=

= −

∑

57


fifth order aperture coma overlay, summarized in Table 3-5, are depicted throughout the

field in Figure 3-9 (a-g). In Figure 3-9 (a), the field constant fifth order aperture coma

contribution from a Zernike fifth order aperture coma overlay is displayed. The

magnitude and orientation are the same everywhere throughout the field and are governed

by the vector describing the overlay term, 151FF A

. In Figure 3-9 (b), the medial oblique

spherical aberration contribution from a Zernike fifth order aperture coma overlay away

from the stop is displayed. The field behavior of this term resembles that of the field

curvature term generated by a Zernike coma overlay. Figure 3-9 (c) displays the oblique

spherical aberration contribution where the field behavior resembles that of field

asymmetric, field linear astigmatism. The elliptical coma contribution, Figure 3-9 (d), is

field quadratic and depending on the vector describing the overlay term, 151FF A

, the

aberration orientation may appear rotationally symmetric as is depicted in Figure 3-9 (d).

In Figure 3-9 (e), the two field quadratic coma contributions are displayed together

resulting in a net field asymmetric aberration. Similarly, Figure 3-9 (f) displays together

the two field cubed, fifth order astigmatism contributions, resulting in a net aberration

that is field asymmetric. Lastly, Figure 3-9 (g), displays a fifth order field curvature

contribution that equates to a cubic shaped focal plane.

58

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

X Field

Y F

ield

(a) (b) (c)

(d) (e) (f)

(g)

Figure 3-9. The characteristic field dependence of (a) field constant, fifth order aperture

coma, (b) field linear medial oblique spherical aberration, (c) field asymmetric, field

linear oblique spherical aberration, (d) field quadratic trefoil, (e) field quadratic coma, (f)

field asymmetric, field cubed astigmatism, and (g) field cubic, field curvature that is

generated by a Zernike fifth order aperture coma overlay on an optical surface away from

the stop surface.

3.3 APPLICATION: The Astigmatic Aberration Field Induced by Three Point

Mount-Induced Trefoil Surface Deformation on a Mirror of a Reflective Telescope

With a theoretical framework in place for understating the aberration behavior of a

freeform overlay, deformations that exist in an as-built configuration of a telescope can

59

be studied. The deformation of particular interest here is the self weight deflection of an

optic located away from the aperture stop being held at three points, a kinematically

stable condition. An error of this nature is usually measured interferometrically by

measuring the optic in its on-axis, null configuration while in its in-use mounting

configuration; or, the error can be simulated by the use of finite element methods [72]. In

either the measured or simulated case, the deformation is quantified based on the values

of its Fringe Zernike coefficients. The predominant surface error that arises with this

mount configuration is trefoil, in optical testing terminology, Fringe polynomial terms

10Z and 11Z . In Section 3.2, the field aberration influence of a trefoil surface overlay was

described. The next step is to apply these results to various reflective telescope forms to

observe the impact of the mount-induced trefoil deformation from the nominal telescope

configuration.

Depending on the telescope optical configuration, the third order aberrations, i.e.

spherical aberration, coma, and astigmatism, may or may not be corrected. For the case of

a two mirror telescope, the system is corrected for third order spherical aberration and

may be corrected for third order coma depending on the conic distribution of the mirrors.

Whether or not coma is corrected, third order astigmatism remains uncorrected. If a third

mirror is added, the telescope system may also be corrected for third order astigmatism.

In either the two or three mirror case, when the secondary mirror is deformed by a three

point mount, it will generate a field dependent astigmatic contribution, assuming the

secondary mirror is not the stop surface. Under these conditions, the astigmatic response

of the telescope is of interest because it reveals information into the as-built state of the

60

telescope. In the case described above, the astigmatic response, ASTW , of the telescope

takes the nodal form

2 3 * 2222 333,

1 3 ,2 4

SMAST MNTERR SM

SM

yW W H C H

yρ

= +

(3.52)

where the subscript SM signifies that the mount-induced trefoil deformation is on the

secondary mirror surface and depending on whether the telescope is anastigmatic,

222W may or may not be equal to zero.

To emphasize, Eq. (3.52) presents the magnitude and orientation of the astigmatic

Fringe Zernike coefficients (Z5/6) that would be measured if an interferogram was

collected at the field point H

in the FOV of the perturbed telescope. The perturbation, in

this case, is a three point kinematic mount deformation on the secondary mirror,

characterized by 3333,MNTERR SMC

, and is directly related to the measured values of the Fringe

Zernike trefoil (Z10/11) following Eq. (3.34).

To exploit the strength of NAT for developing insight into the relationships between

alignment, fabrication, uncorrected aberration fields, and now mount-induced errors, the

next step is to understand the nodal response of the astigmatism to these deviations from

a nominal design depending on whether the system is corrected for third order

astigmatism.

3.3.1 Astigmatic Reflective Telescope Configuration ( 222 0W ≠ ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror

In order to determine the possible nodal geometry for the case where residual third order

astigmatism exists, the term inside the brackets of Eq. (3.52) is set equal to zero, as

represented in Eq. (3.53),

61

2 3 *222 333,

1 3 0.2 4

SMMNTERR SM

SM

yW H C H

y

+ =

(3.53)

The first step in solving the vector formulation represented in Eq. (3.53) is to establish

a path for arranging 2H

and *H

in a form that can be solved, ideally using previously

developed techniques. This step is accomplished by multiplying both sides of Eq. (3.53)

by unity in the form of

* *

2 2ˆ ˆ0 1 ,HH H Hi j

H H+ = =

(3.54)

where a vector multiplication relation presented in [20] has been applied. Since Eq. (3.54)

is a unit, vector formulation, it does not affect the magnitude or orientation of either

vector in Eq. (3.53). Multiplying the identity in Eq. (3.54) through Eq. (3.53) yields

3 3 * *222 333,2

1 1 3 0.2 4

SMMNTERR SM

SM

yW H C H H H

yH

+ =

(3.55)

Again, making use of the identity in Eq. (3.54), Eq. (3.55) takes the form

3

3 *222 333,2

1 3 0.2 4

SMMNTERR SM

SM

yHW C HyH

+ =

(3.56)

It can now be seen based on the powers of H

that there is a quadranodal astigmatic

response in the FOV to a mount-induced trefoil deformation on the secondary mirror with

the term in the brackets of Eq. (3.56) exhibiting equilateral trinodal behavior with a

fourth zero located on-axis at 0H =

. In order to find the nodal response, the term inside

the bracket of Eq. (3.56) is rearranged, and set to zero, taking the form

3

3333,2

222

3 0.2

SMMNTERR SM

SM

yH CW yH

+ =

(3.57)

The first term of Eq. (3.57) is substituted with a new reduced field vector 3Π

written in

complex notation as

62

3 3 33 1

33 3 32 2 ,

ii i

H eH H e H eH H

θ

θ θ Π ≡ = = =

(3.58)

where the new vector represented in Eq. (3.58) has the same orientation, θ, as H

but with

a magnitude equal to the cube root of H

. In this new form, Eq. (3.57) takes the form

3 3333,

222

3 0.2

SMMNTERR SM

SM

yC

W y

Π + =

(3.59)

Following the method proposed by Thompson and detailed in [23] for solving the

nodes of a cubic vector equation, that has been applied to the case of elliptical coma and

fifth order astigmatism [23, 25] in tilted and decentered systems, the node locations for a

trinodal form are governed by two vectors, x

and x

, which, in this case are equal, and

given by

( )1

3 13 3

222 222 333,222

3 .2

SMMNTERR MNTERR MNTERR SM

SM

yx x C

W y

= = −

(3.60)

In terms of these cubic equation solution vectors 222MNTERR x

and 222MNTERR x

, which are best

kept independent for later generalizations, the three node locations referenced to the

intersection of the OAR with the image plane are, for this case, equidistant from the

on-axis node with 0ASTW = at

( ) ( ) ( ) ( ) ( )222 222 222 222 2222 , 3 , 3 .MNTERR MNTERR MNTERR MNTERR MNTERRx x i x x i x− + − −

(3.61)

The four field points at which astigmatism is found to be zero are illustrated in

Figure 3-10 (a) where the solutions are plotted in the Π

reduced field coordinate. In

Figure 3-10 (b), the four nodal solutions have been re-mapped into the conventional

H

field coordinate. The solution vectors follow a notation introduced in [23, 25] for

characterizing the cubic nodal behavior of elliptical coma and fifth order astigmatism. In

this case, the vectors are proportional to 333,MNTERR SMC

, which is directly computed from a

63

measurement or simulation of the mount-induced trefoil deformation on the secondary

mirror, as visualized in Figure 3-11.

x

y

xH

yH

(a) (b)

1

33

333,MNTERR SMC

3

333,MNTERR SMC

10/11MNTERR

10/11MNTERR

2222

MNTERRx

222MNTERRx

2223

MNTERRi x

2223

MNTERRi x

Figure 3-10. (a) The nodal behavior for an optical system with conventional third order

field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a

two mirror telescope with a three point mount-induced error on the secondary mirror, is

displayed in a reduced field coordinate, , where the node located by 2222 MNTERR x has

an orientation angle of 10/11MNTERR and a magnitude that is proportional to 333,MNTERR SMC .

The two related nodes on the circle are then advanced by 120º and 240º for this special

case. (b) When the nodal solutions are re-mapped to the conventional field

coordinate, H , the node located by 2222 MNTERR x has an orientation angle of 10/11MNTERR

and a magnitude that is proportional to 3

333,MNTERR SMC .

Waves

0.0000

1.0000

0.5000

WAVEFRONT ABERRATION

Cassegrain Ritchey-Chretien

Field = ( 0.000, 0.000) DegreesWavelength = 632.8 nmDefocusing = 0.000000 mm

λ (0.633µm)

10/113

MNTERR

10/11MNTERR

,

3

333MNTERR SMC

,333MNTERR SMC

1.0

0.5

0.0

x

y

2222x

222x

2223i x

2223i x

x

y

(a) (b)

10/11MNTERR

Figure 3-11. A measurement or simulation of the mount-induced error on the secondary

mirror yields the magnitude and orientation of 333,MNTERR SMC .

64

3.3.2 Anastigmatic Reflective Telescope Configuration ( 222 0W = ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror

For the case where the telescope configuration is corrected for third order astigmatism,

the first term inside the brackets of Eq. (3.52) is set to zero yielding

3 * 2333,

3 .4

SMAST MNTERR SM

SM

yW C H

yρ

=

(3.62)

In Eq. (3.62) it can be seen that the only astigmatic contribution is now from the

mount-induced perturbation on the secondary mirror. In this case, the nodal solution is

trivial where if the term inside the brackets of Eq. (3.62) is set to zero, the only solution is

located on-axis at 0H =

.

For both the astigmatic and anastigmatic cases presented above, the astigmatism takes

on a unique distribution throughout the FOV when there is a mount-induced error on the

secondary mirror. These unique distributions are significant because by measuring only

the Fringe Zernike pair (Z5/6) and reconstructing the nodal geometry from these

measurements, it can be determined whether the as-built telescope is dominated by mount

error versus other errors like alignment or residual figure error.

3.3.3 Validation of the Nodal Properties of a Reflective Telescope with Three Point Mount-Induced Figure Error on the Secondary Mirror

3.3.3.1 Astigmatic Reflective Telescope Configuration ( 222 0W ≠ ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror

As a validation of the predicted nodal behavior summarized in Figure 3-11 (a) for the

case of a two mirror telescope with a mount-induced perturbation on the secondary

mirror, an F/8, 300 mm Ritchey-Chrétien telescope, displayed in Figure 3-12 (a), has

been simulated in commercially available lens design software, in this case, CODE V®.

The aberration performance throughout the FOV in terms of a total measure of image

65

quality, the RMS WFE is displayed in Figure 3-12 (b). The RMS WFE increases as a

function of FOV because of the uncorrected field quadratic astigmatism.

When it comes to assembling and aligning an optical system of this type, it is

becoming increasingly common to measure the system interferometrically and use

information that is available about significant characteristic aberrations through a

polynomial fit to the wavefront OPD. Figure 3-13 (a-b) displays separately the Fringe

Zernike astigmatism (Z5/6) and Fringe Zernike trefoil (Z10/11) that would be measured at

selected, discrete points in the FOV. As can be seen from Figure 3-13 (a), the system

suffers from third order astigmatism. The higher order aberrations, like elliptical coma,

are near zero, which is expected for a system with a modest F/number and FOV. When a

0.5λ, 0° orientation, trefoil mount error is added to the secondary mirror, the aberration

displays are modified as shown in Figure 3-13 (b). The astigmatic contribution has

developed a quadranodal behavior and there is now a field constant contribution to the

elliptical coma. The astigmatic behavior matches the general case shown in

Figure 3-11 (a) where the orientation angle, , has been set to zero. A quantitative

evaluation of the zeros in the display for astigmatism from Figure 3-13 (b) confirms the

predictions made by NAT described in Section 3.3.1. The displays are based on real ray

data and the zero locations for the astigmatic contribution are independent of NAT so

they are an excellent validation of the theoretical developments presented in Section 3.3.1

and 3.3.2.

10/11MNTERR

66

16:08:28

telescope_102209

KPT 05-Jun-12

RMS WAVEFRONT ERROR

vs

REAL RAY IMAGE HEIGHT

Minimum = 0.015705

Maximum = 0.20333

Average = 0.082848

Std Dev = 0.043672

1waves ( 632.8 nm)

-10 -5 0 5 10

X Real Ray Image Height - mm

-10

-5

0

5

10

Y Real Ray Image Height - mm

14:24:20

telescope_102209

KPT 13-May-12


vs



Maximum = 0.44379

Average = 0.16275

Std Dev = 0.10257

1waves ( 632.8 nm)

-10 -5 0 5 10


-10

-5

0

5

10


1λ (0.633µm)

(a)

(b)

Y Fi

eld

An

gle

(deg

)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

16:02:10

telescope_102209 Scale: 0.33 KPT 05-Jun-12

75.00 MM

16:02:10

telescope_102209 Scale: 0.33 KPT 05-Jun-12

75.00 MM

(b)

75.00 mm

Figure 3-12. (a) Layout for a F/8, 300 mm Ritchey-Chrétien telescope and (b) a Full Field

Display (FFD) of the RMS WFE of the optical system at 0.633 µm over a ±0.2° FOV.

Each circle represents the magnitude of the RMS WFE at a particular location in the

FOV.

14:24:20

telescope_102209

KPT 13-May-12


vs



Maximum = 0.44379

Average = 0.16275

Std Dev = 0.10257

1waves ( 632.8 nm)

-10 -5 0 5 10


-10

-5

0

5

10


14:16:32

telescope_102209

KPT 13-May-12


vs


Minimum = 0.3246e-7

Maximum = 0.59586

Average = 0.18471

Std Dev = 0.12847

1waves ( 632.8 nm)

-10 -5 0 5 10


-10

-5

0

5

10


14:03:40

telescope_102209

KPT 13-May-12


vs


Minimum = 0.28221e-15

Maximum = 0.00014343

Average = 0.36672e-4

Std Dev = 0.31462e-4

1waves ( 632.8 nm)

-10 -5 0 5 10


-10

-5

0

5

10


14:11:34

telescope_102209

KPT 13-May-12


vs


Minimum = 0.98331

Maximum = 0.98381

Average = 0.98345

Std Dev = 0.98902e-4

1waves ( 632.8 nm)

-10 -5 0 5 10


-10

-5

0

5

10


Y Fi

eld

An

gle

(deg

)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Z10/11Z5/6

(a)

(b)

Y Fi

eld

An

gle

(deg

)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Y Fi

eld

An

gle

(deg

)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Y Fi

eld

An

gle

(deg

)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

14:24:20

telescope_102209

KPT 13-May-12


vs



Maximum = 0.44379

Average = 0.16275

Std Dev = 0.10257

1waves ( 632.8 nm)

-10 -5 0 5 10


-10

-5

0

5

10


1λ (0.633µm)


(Z5/6) and Fringe Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a

Ritchey-Chrétien telescope in its nominal state and (b) the telescope when 0.5λ of three

point mount-induced error oriented at 0° has been added to the secondary mirror. It is

important to recognize that these displays of data are FFDs that are based on a Zernike

polynomial fit to real ray trace OPD data evaluated on a grid of points in the FOV. For

each field point, the plot symbol conveys the magnitude and orientation of the Zernike

coefficients pairs, Z5/6 on the left and Z10/11 on the right.

67

3.3.3.2 Anastigmatic Reflective Telescope Configuration ( 222 0W = ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror

In the case of an anastigmatic telescope with a mount-induced perturbation on the

secondary mirror, the nodal behavior is simplified as discussed in Section 3.3.2 where the

node is on-axis at 0H =

. As a validation for this prediction, a relevant TMA geometry

based on the James Webb Space Telescope (JWST) [73] has been simulated and analyzed

for a trefoil perturbation on the secondary mirror. The optical system operates at F/20

with a 6.6 m entrance pupil diameter and is shown in Figure 3-14 (a). In order to yield an

accessible focal plane, the FOV is biased so that an off-axis portion of the tertiary mirror

is utilized. The RMS WFE of the system is displayed in Figure 3-14 (b) over a

±0.2° FOV and the portion of the field that is utilized for the biased system is bounded by

the red rectangle. In the center of the on-axis FOV, the RMS WFE is well behaved

because the third order aberrations are well corrected. The performance does increase at

the edge of the FOV due to higher order aberration contributions.

Following a similar approach to that outlined in Section 3.3.3.1, the individual

aberration contributions that make up the total RMS WFE can be evaluated over the

FOV. Figure 3-15 displays separately the Fringe Zernike astigmatism (Z5/6) and Fringe

Zernike trefoil (Z10/11) that would be measured at selected, discrete points in the FOV for

the JWST-like system. As can be seen from Figure 3-15 (a), the system is anastigmatic

and the elliptical coma is near zero throughout the FOV. If a 0.5λ, 0° orientation, trefoil

error is added to the secondary mirror, the aberration displays are modified as shown in

Figure 3-15 (b). The astigmatic contribution has developed field linear, field conjugate

astigmatism with a single node centered on-axis. The node lies outside the usable FOV

for the field biased telescope. As with the previous case, there is also a field constant

68

contribution to the elliptical coma. Both contributions match the theoretical developments

presented in Section 3.3.1 and 3.3.2.

-0.2000 -0.1000 0.0000 0.1000 0.2

000

000

000

000

000

1250.00 MM

(a) (b)

1250.00 mm

1

0.25λ (1.000µm)

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Figure 3-14. (a) Layout for a JWST-like telescope geometry and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at 1.00 µm over a ±0.2° FOV. The system utilizes a field bias (outlined in red) to create an accessible focal plane.

-0.2000 -0.1000 0.0000 0.1000 0.2000

000

000

000

000

000

-0.2000 -0.1000 0.0000 0.1000 0.2000

000

000

000

000

000

-0.2000 -0.1000 0.0000 0.1000 0.2

000

000

000

000

000

-0.2000 -0.1000 0.0000 0.1000 0.2000


000

000

000

000

000

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Z10/11Z5/6

(a)

(b)

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

1

1λ (1.000µm)

Figure 3-15. Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z5/6) and Fringe Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a JWST-like telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror.

69

3.4 Extending Nodal Aberration Theory to Include Decentered Freeform ϕ-Polynomial Surfaces away from the Aperture Stop

In the case of the JWST-like geometry in Figure 3-14 (a), the tertiary mirror is an off-axis

section of a larger rotationally symmetric surface. If a trefoil deformation is to be applied

to the tertiary mirror, the error must be centered with respect to the off-axis portion of the

surface, not the larger parent surface. Therefore, an additional parameter must be defined

that accounts for a shift of the nonsymmetric deformation from the reference axis that is

defined to be the OAR [22]. Following the method used in [22] for the decenter of an

aspheric cap of an optical surface, the nonsymmetric deformation is treated as a

zero-power thin plate. When the nonsymmetric deformation is shifted, there is a freeform

sigma vector ( )FF jσ that is expressed as

( )( )*

,FF j

FF jj

v

y

δσ =

(3.63)

where ( )*FF j

vδ is the distance between the OAR and the freeform departure vertex. For the

case of a freeform, φ-polynomial surface, the freeform vertex corresponds to the origin of

the unit circle that bounds the polynomial set. To compute the overall aberration field

from the shifted freeform deformation, a new effective aberration field height ( )FF jH

is

defined, following the notation of [20], as

( ) ( ) .FF FF jjH H σ= −

(3.64)

The astigmatic response of a telescope with a mount-induced trefoil deformation can

now be modified to account for the new effective aberration field height. Updating Eq.

(3.52) with the effective field height ( )FF jH

and generalizing the perturbation to be on the

jth optical surface, ASTW takes the form

70

( )2 3 * 2222 333,

1 3 .2 4

jAST MNTERR j FF j

j

yW W H C H

yρ

= +

(3.65)

The nodal solution for the astigmatic response represented in Eq. (3.65) is best found

numerically and may be quadranodal but degenerates to special cases where only three or

two nodes exist. For the anastigmatic case where the third order astigmatism is zero,

Eq. (3.65) simplifies to

( )3 * 2333,

3 ,4

jAST MNTERR j FF j

j

yW C H

yρ

=

(3.66)

where there is a single node located at ( )FF jH σ=

.

As a validation of these predictions, the JWST-like system evaluated in

Section 3.3.3.2 is reevaluated where the 0.5λ, 0° orientation, trefoil error is now added to

the off-axis section of the tertiary mirror. In this case, the aberration displays are

modified as shown in Figure 3-16. The astigmatic contribution has developed field linear,

field conjugate astigmatism with a single node now centered off-axis. The node has

moved off-axis because the trefoil deformation is no longer located along the OAR and

now lies in the center of the field biased FOV. It is also interesting to note that for this

configuration, the induced astigmatic contribution is larger than the induced field

constant contribution to the elliptical coma. At the tertiary mirror, the beam footprints for

each field are widely spread about the optical surface; as a result, the field dependent

contribution has a larger net effect than the field constant contribution.

71

-0.2000 -0.1000 0.0000 0.1000 0.2000


0.2000

0.1000

0.0000

0.1000

0.2000

-0.2000 -0.1000 0.0000 0.1000 0.2000


-0.2000

-0.1000

0.0000

0.1000

0.2000

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

Y Fi

eld

Angl

e (d

eg)

0.2

0.1

0.0

-0.1

-0.2

X Field Angle (deg)-0.2 -0.1 0.0 0.1 0.2

1

1λ (1.000µm)

Z10/11Z5/6

Figure 3-16. Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z5/6) and Fringe Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for a JWST-like telescope with 0.5λ of three point mount-induced error oriented at 0° on the off-axis tertiary mirror.

72

Chapter 4. Experimental Validation of Nodal Aberration Theory for φ-Polynomial Optical Surfaces

Chapter 3 presented a theoretical foundation for the general, unrestricted aberration

theory for optical systems that employ φ-polynomial surfaces. In this chapter, this

theoretical foundation is verified experimentally by the design and implementation of an

aberration generating telescope. Within the basic telescope framework, a surface with a

φ-polynomial departure is placed in the optical path. When the surface is displaced

axially from the aperture stop of the optical system, aberrations of lower radial order than

the nonsymmetric departure of the surface are generated. The particular nonsymmetric

departure to be studied in this chapter is elliptical coma or Zernike trefoil. As will be

verified, when the trefoil surface is displaced from the stop surface, field conjugate, field

linear astigmatism is generated throughout the FOV. It will be shown that the aberration

is centered in the image plane about a point that depends on the lateral offset of the

Zernike trefoil vertex from the OAR of the telescope. Moreover, it will be verified that

the magnitude of the astigmatic aberration is generated linearly with relative axial

distance from the stop surface as predicted theoretically in Chapter 3.

4.1 Design of an Aberration Generating Schmidt Telescope

In its simplest form, a Schmidt telescope is composed of a spherical mirror, stop, and

aspheric corrector plate. The stop lies at the center of curvature of the spherical mirror so

that the system is corrected for third order coma and astigmatism. By placing a sixth

order aspheric corrector plate at the stop surface, the system is also corrected for third and

fifth order spherical aberration. In this configuration, the telescope is corrected for all

third order aberrations except field curvature and, as such, makes an excellent baseline

optical system for introducing controlled amounts of individual, isolated aberration types

73

to study their field behavior. In this particular case, depicted in Figure 4-1, the aberrations

are generated by inserting an additional plate with Fringe Zernike trefoil (or elliptical

coma) polished directly into the surface, yielding a Zernike, freeform surface. When the

plate is shifted away from the stop surface, aberrations of lower radial order than trefoil,

which is cubic, are generated. The aberration fields induced by the plate are observed by

evaluating the wavefront with a 100 mm aperture, Zygo Fizeau-type He-Ne laser

interferometer. The wavefront is reflected back to the interferometer by the use of a

re-imaging retro-reflecting component placed near the image plane. In order to evaluate

the wavefront across a two-dimensional FOV, a scanning mirror is introduced into the

path between the output of the interferometer and the entrance aperture of the Schmidt

telescope. When the FOV is scanned, the retro-reflector must follow the beam to track the

image displacement, including field curvature, created by the off-axis field angle.

Interferometer Field of viewgenerator

Spherical Primary

Zernike Plate

Corrector Plate

Retro-reflector

Figure 4-1: Testing configuration for the Schmidt telescope to demonstrate the field

dependent aberration behavior of a freeform optical surface. A freeform, Zernike plate

can purposely be placed at or away from the stop surface to induce field dependent

aberrations. The aberration field behavior of the telescope is measured interferometrically

by acquiring the double pass wavefront over a two-dimensional FOV with a scanning

mirror.

74

The specifications for the nominal Schmidt system are summarized in Table 4-1 and

are based on the constraints of pre-existing optical/mechanical components. The focal

length is constrained by the selection of the primary mirror. In this case, an existing

152.4 mm diameter, 152.4 mm focal length spherical mirror is selected to maintain a

small package for the telescope. The pupil size is constrained by the 100 mm aperture of

the interferometer. Because the FOV scanning mirror does not lie in a pupil plane, the

beam will displace on the focal plane of the interferometer. As a result, a 70 mm entrance

pupil diameter is selected so that the entire FOV can be measured by the interferometer

without vignetting. The scanning of the retro-reflector limits the achievable FOV. The

stages that move the retro-reflector are limited to ±6.5 mm of motion so the maximum

measureable diagonal full FOV is roughly 5°. Lastly, the departure of the Fringe Zernike

trefoil plate is constrained to 3 µm so a discernible amount of aberration is induced by the

plate but ensures that the surface is still manufacturable with available resources.

The nominal Schmidt design without the Fringe Zernike trefoil deformation on the

plate is shown in Figure 4-2. The aspheric corrector plate is concave and has about 37 µm

of departure from planar or 9.5 µm of departure from the best fit sphere of -19670 mm.

The overall system is diffraction limited throughout the FOV (relative to a curved focal

plane) with a maximum RMS WFE of 0.013λ at 632.8 nm.

75

Table 4-1. Design specifications for the nominal aberration generating Schmidt telescope.

Parameter Target Value

Type

Schmidt telescope

Spherical, primary mirror

4th

and 6th

order NBK7 corrector

NBK7, 3 µm Fringe Zernike trefoil

plate at 50 mm normalization radius

Diagonal Full FOV (deg.) 5

F/# 2.2

Focal Length (mm) 152.4

Entrance pupil diameter (mm) 70

Wavelength (nm) 632.8 (He-Ne)

RMS Wavefront Error on a

curved image plane, R=-151 mm

(waves @ λ = 632.8 nm)

[nominal design]

On axis < 0.07

0.7 FOV < 0.07

1.0 FOV < 0.07

Image Quality [nominal design] Zero 3

rd and 5

th order spherical, 3

rd order

coma, 3rd

order astigmatism

16:57:01

F/2, Schmidt Scale: 0.60 KHF 02-Jan-14

41.67 MM

76.2 mm Dia.8 mm thick , NBK7 window

100 mm Dia.10 mm thick , NBK7 window

152.4 mm Dia.304.8 mm ROC Spherical Mirror

Figure 4-2. Layout of the nominal Schmidt telescope configuration. The aspheric and

Zernike trefoil plate are both fabricated in NBK7 substrates and the primary mirror is a

commercially available 152.4 mm, F/1 concave, spherical mirror.

76

When the 3 µm trefoil deformation, as shown in Figure 4-3, is added to the plate, the

aberration behavior of the optical system is altered. In Figure 4-4 (a-b), the FFDs for

astigmatism (Z5/6) and elliptical coma (Z10/11) are simulated across a square, 5° diagonal

FOV for the trefoil plate oriented at 0° and located 120 mm away from the stop surface.

With this visualization, a line symbol is used at each field point to represent the

magnitude and orientation of the aberration. In the presence of the trefoil plate, the

wavefront of the optical system now exhibits field conjugate, field linear astigmatism,

shown in Figure 4-4 (a), and field constant elliptical coma, shown in Figure 4-4 (b), as

predicted by Eq. (3.35). Moreover, if the magnitude of the Z5/6 and Z10/11 contributions to

the wavefront for the ( )1, 0x yH H= = field point is tracked as a function of the plate

position as shown in Figure 4-5, it can be seen that the magnitude of the astigmatism

increases linearly with plate position and the elliptical coma term remains roughly

constant as a function of plate position, as expected from the equations in Table 3-3. Any

discrepancy in the trend of the aberration magnitudes as a function of the plate position is

attributable to the residual higher order aberrations present in the nominal Schmidt

telescope design.

Waves @ 632.8 nm 1.0

0.5

0.0

Figure 4-3. Simulated interferogram at a wavelength 632.8 nm of the 3 µm trefoil deformation added on one surface of the 100 mm, NBK7 plate to be added into the optical path of the nominal Schmidt telescope.

77

09:59:19

F/2, Schmidt

KHF 10-May-13


vs


Minimum = 0.84881

Maximum = 0.84937

Average = 0.84905

Std Dev = 0.00014319

1waves ( 632.8 nm)

-2 -1 0 1 2


-2

-1

0

1

2


09:59:19

F/2, Schmidt

KHF 10-May-13


vs


Minimum = 0.84881

Maximum = 0.84937

Average = 0.84905

Std Dev = 0.00014319

1waves ( 632.8 nm)

-2 -1 0 1 2


-2

-1

0

1

2


09:59:43

F/2, Schmidt

KHF 10-May-13


vs



Maximum = 0.41885

Average = 0.24015

Std Dev = 0.08969

1waves ( 632.8 nm)

-2 -1 0 1 2


-2

-1

0

1

2


09:59:43

F/2, Schmidt

KHF 10-May-13


vs



Maximum = 0.41885

Average = 0.24015

Std Dev = 0.08969

1waves ( 632.8 nm)

-2 -1 0 1 2


-2

-1

0

1

2


09:59:19

F/2, Schmidt

KHF 10-May-13


vs


Minimum = 0.84881

Maximum = 0.84937

Average = 0.84905

Std Dev = 0.00014319

1waves ( 632.8 nm)

-2 -1 0 1 2


-2

-1

0

1

2


Z5/6 Z10/11

(a) (b)

09:59:43

F/2, Schmidt

KHF 10-May-13


vs



Maximum = 0.41885

Average = 0.24015

Std Dev = 0.08969

1waves ( 632.8 nm)

-2 -1 0 1 2


-2

-1

0

1

2


Figure 4-4. (a) The predicted astigmatism (Z5/6) and (b) elliptical coma (Z10/11) FFDs over

a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil

plate oriented at 0° and located 120mm away from the stop surface. The Zernike trefoil

plate generates both field constant elliptical coma and field conjugate, field linear

astigmatism.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 40 80 120 160 200

Wav

es

(@ 6

32

.8 n

m)

Trefoil Plate Position Relative to Stop (mm)

Z5/6 Z10/11

Figure 4-5. The predicted magnitude of the (a) astigmatism (Z5/6) and (b) elliptical coma

(Z10/11) as a function of the Zernike trefoil plate position relative to the stop surface for

the 1, 0x yH H field point of Schmidt telescope configuration.

78

In order to couple the Schmidt telescope to the interferometer, the beam must be

retro-reflected at or near the image plane. When the beam is retro-reflected, it is

important that the beam traverses the same path heading back to the interferometer. In

this case, the beam must strike normal to the retro-reflector so that the light reflects back

on itself and to minimize re-trace errors, the pupil of the telescope must be conjugate to

the retro-reflector. Several retro-reflector configurations can be designed to meet these

criteria, however, some of them require the fabrication of custom optical components. In

order to employ commercial, off-the-shelf (COTS) components, a retro-reflector that

utilizes a concave mirror and plano-convex field lens is selected. A first order layout of

this retro-reflector is shown in Figure 4-6 where the illumination light (red rays) is

retro-reflected by the concave mirror and the field lens ensures the concave mirror and

aperture stop of the Schmidt telescope are conjugate to one another by imaging the image

of the aperture stop as seen through the primary mirror onto the concave mirror

(blue rays). More specifically, knowing that the aperture stop is 310.93 mm away from

the primary mirror with a radius of curvature 304.8 mm, the image of the aperture stop is

found to be 298.91 mm in front of the primary mirror or roughly collocated with the

aperture stop. Next, based on readily available COTS components, the concave, retro

mirror is selected to be a 12.7 mm diameter, 19 mm radius of curvature mirror. With the

concave mirror defined, the field lens focal length is calculated to be 21.83 mm. Based on

readily available COTS components, the most similar lens that can be found is an NBK7,

plano-convex, 9 mm diameter, 18 mm focal length lens.

79

ConcaveMirror

Field Lens

Rmirror

Image of Stop as seen through Primary

From Primary Mirror

zstop

Figure 4-6. First order layout demonstrating how the retro-reflector must be designed to ensure that the pupil of the Schmidt telescope is conjugate to the pupil of the concave mirror that sends the wavefront back towards the interferometer.

4.2 Fabrication of the Aspheric Corrector/Nonsymmetric Plate

With a completed optical design in place, the next step is to procure and fabricate the

optical components. The spherical mirror is an existing COTS component; however, the

aspheric and trefoil plates are nonstandard optical components that require custom

fabrication. Moreover, conventional full aperture lapping techniques cannot be employed

to fabricate these components without special tooling; these components require the use

of a sub-aperture polishing process. One accessible sub-aperture polishing process at the

University of Rochester in the Robert E. Hopkins Center is magnetorheological

finishing (MRF). In this process a magnetic, abrasive impregnated fluid is pumped over a

polishing wheel. At the apex of the wheel there is a magnetic field. When the fluid

encounters the magnetic field, the fluid hardens. An optical surface is set into this

hardened region, creating a small polishing zone. The removal of the material is

determined by the dwell of the optic in the fluid, and through computer control, a

multitude of shape profiles can be polished into the surface. The MRF machine located at

the University of Rochester is the QED Q22-XE that is capable of polishing both

80

rotationally symmetric and nonsymmetric surfaces up to 100 mm in diameter and is well

suited for polishing the aspheric and trefoil plates.

The NBK7 substrates for fabrication are COTS pre-polished flats to λ/4 or better. The

aspheric plate substrate is 76.2 mm in diameter and 8 mm thick whereas the trefoil plate

substrate is 100 mm in diameter and 10 mm thick. For each plate only one surface is to be

polished by the MRF machine. In order to create the dwell maps for polishing, the

removal within the polishing zone must be known. To determine this removal, four spots

are polished into a sacrificial 50 mm NBK7 substrate and characterized by a Zygo laser

interferometer with a transmission reference flat attached. Once the removal of the

polishing zone has been characterized, the machine computes from the initial surface

shape a dwell map to create the desired final surface shape. Because a large amount of

material needs to be removed for both plates and the removal rate of MRF is generally

small, the polishing is split into multiple runs. During each run only part of the overall

departure is polished into the surface. With this method, the polishing runtimes are

shorter, providing better stability of the polishing parameters during fabrication. After

each polishing run, the plates are measured with an interferometer in reflection or

transmission if the departure of the surface from planar is too great. The measurement

after each polishing run serves as the input for the subsequent polishing cycle.

The final surface shape of the third and fifth order aspheric correcting plate and its

residual from the theoretical design are shown in Figure 4-7 (a-b) over a 70 mm clear

aperture. As depicted in Figure 4-7 (a), roughly 26.6 µm peak-to-valley (PV) of departure

has been polished into the substrate material over the course of fourteen polishing runs

that were each removing 2 µm PV. The residual error after polishing, Figure 4-7 (b), is

81

about 0.56λ PV or 0.066λ RMS at the testing wavelength of 632.8 nm. There is some

coma present in the surface but most of the residual is higher order and present at the

edges where the slopes are largest. A small center feature as well as a spoking pattern (a

mid-spatial frequency error) is observed. Both features are a residual from the sub-

aperture MRF polishing process that is difficult to correct once polished into the part.

However, since the measured errors are higher order and not low order astigmatism, it

will not impact or prevent any features to be observed during the measurement of the

assembled Schmidt optical system.

(a) (b)

Figure 4-7. (a) Measured surface departure of the aspheric corrector plate for the Schmidt telescope and (b) residual error when the nominal optical design surface is subtracted from the measured surface. The error is about 0.56λ PV or 0.066λ RMS at the testing wavelength of 632.8 nm.

For the trefoil plate, the final shape and its residual from theoretical are shown in

Figure 4-8 (a-b) over a 98 mm clear aperture. The surface departure of the final surface,

Figure 4-8 (a), is 5.75 µm PV and has been polished in over the course of seven runs that

each removed about 1.2 µm of material. The residual for this surface, Figure 4-8 (b), is

0.30λ PV or 0.05λ RMS. Similar to the case with the asphere, the residual is primarily

higher order with a center artifact and a mid-spatial frequency spoking pattern. Some

82

fringe features are observed in the residual and are most likely caused by interference

from reflections of the back surface. Similar to the case of the asphere, since the residual

features are primarily higher order, they will not impact the low order astigmatism that is

to be measured by the Schmidt telescope.

(a) (b)

Figure 4-8. (a) Measured surface departure of the Zernike trefoil plate and (b) residual error when the nominal optical design surface is subtracted from the measured surface. The error is about 0.30λ PV or 0.05λ RMS at the testing wavelength of 632.8 nm.

4.3 Experimental Setup of the Aberration Generating Schmidt Telescope

The assembled Schmidt telescope is displayed in Figure 4-9. The test wavefront from the

interferometer is reflected off the motorized, FOV generating fold mirror nominally at

90° where it enters the telescope. The telescope is composed of the aperture stop,

aspheric plate, trefoil plate, primary mirror, and retro-reflector. The mount that holds the

aspheric plate also serves as the aperture stop of the optical system. The trefoil plate is

motorized so that the position of the plate relative to the stop can be varied. Moreover,

the trefoil plate and aspheric plate are assembled so that they can be moved as close to

each other as possible. The illumination light is focused by the primary mirror and

reflected back through the system by the motorized retro-reflector. In this configuration, a

83

wide variety of FOVs can be directed into the telescope and the retro-reflector is

re-positioned in x, y, and z to send the wavefront back towards the interferometer without

defocus or tilt. The entire optical system is computer controlled by a custom written

LabVIEW program so that the FOV can be scanned over a grid of points, acquiring the

double pass wavefront at each point, for multiple plate positions. The software uses a

lookup table for the actuator positions of the FOV mirror and retro-reflector. These

lookup tables are created by using the relationship between image displacement and FOV

since the focal length of the primary mirror is known. The lookup table for the focus

position of the retro-reflector is determined from its x and y position and the radius of

curvature of the image plane for the nominal Schmidt telescope.

During initial alignment of the telescope, the Zernike trefoil plate is replaced with a

λ/10, flat NBK7 plate of the same center thickness as the trefoil plate. This plate provides

the correct optical path length between optical elements but does not impart any

additional aberrations into the telescope. With this plate in place, the aberration free,

on-axis field angle of the telescope is found. This angle lies parallel to the axis that

connects the center of curvature of primary mirror and the aperture stop. To find this

point, the aperture stop and aspheric plate combination are longitudinally displaced from

the center of curvature of the primary mirror. When the aperture stop and aspheric plate

are displaced from the center of curvature, third order coma and astigmatism are

generated. For a small displacement of the aperture stop and aspheric plate, the generated

comatic contribution is largest. By observing the generated third order coma and inherent

field curvature in the double pass wavefront with the interferometer as a function of input

field angle, the field angle at which both aberrations go to zero is found. This field angle

84

corresponds to the aberration free, on-axis field angle. Once the on-axis field is found, the

wavefront of an off-axis field angle is observed with the interferometer. The aspheric

plate is now translated longitudinally until the coma in the wavefront is zero at the

interrogated field point as well as throughout the entire FOV. This step ensures that the

aperture stop and aspheric plate are now again at the center of curvature of the primary

mirror.

With the telescope aligned and the on-axis field angle determined, the Zernike plate is

placed into the optical path, replacing the surrogate NBK7 plate. The Zernike plate is

aligned so that the generated field conjugate, linear astigmatism for a plate position away

from the stop is roughly zero on-axis. In addition, the orientation of the Zernike plate is

determined by evaluating the Z10 contribution to Zernike trefoil and adjusting the

orientation of the plate until this term is zero. This equates to an orientation of 0° for the

Zernike plate. With the plate aligned, the experiment proceeds. The LabVIEW program

cycles through five longitudinal Zernike plate positions. The first plate position is

roughly 10 mm away from the stop surface and the other plate positions are equally

spaced at roughly 20 mm increments. At each plate position, the program acquires ten

wavefront measurements with a one second difference between measurements over a 9x9

grid throughout the FOV. Averaging multiple measurements over time helps reduce the

effects of vibration and air turbulence on the overall measurement since the

interferometric cavity is long and there are many mounted optical components that lie in

the optical path.

In addition to acquiring a set of interferograms over a 9x9 grid of field points with the

Zernike plate in the Schmidt telescope, a baseline measurement is also acquired with the

85

flat plate in the telescope over the same field grid. This baseline measurement is

subtracted from the Zernike plate measurement set as it accounts for any residual

misalignment induced aberrations present in the telescope configuration as well as the

surface figure error of the folding mirror that generates the FOV for the telescope. This

baseline subtraction ensures that the analyzed wavefront only reflects the aberrations

induced by the trefoil plate.

Zygo Interferometer

Primary Mirror

TrefoilPlate

AsphericPlate

FOV Mirror

Retro-Reflector

Figure 4-9. Experimental setup of the Schmidt telescope system. The scanning mirror and retro-reflector are motorized so that the FOV can be scanned over a two-dimensional grid of points. The trefoil plate is also motorized so that effect of plate position on magnitude of generated aberration field can be studied.

4.4 Experimental Results

4.4.1 The Generated Field Conjugate, Field Linear Astigmatic Field

As an example of the measurement process, a 3x3 grid of wavefronts spanning a square,

5° diagonal FOV is shown in Figure 4-10 (a). The data shown is the difference at each

field point between the baseline measurement with the flat plate and the actual

measurement with the Zernike plate oriented at 0° and displaced approximately 100 mm

86

longitudinally away from the stop surface. The obscuration present in the wavefront is

from the retro-reflector mounted on a half inch optical post. In evaluating the structure of

the wavefront, there is a field constant elliptical coma contribution to the wavefront. In

Figure 4-10 (b) the field constant elliptical coma is subtracted from the wavefront

revealing the generated astigmatic contribution. The astigmatic contribution is recognized

as field conjugate, field linear astigmatism by evaluating the orientation and magnitude of

the wavefront throughout the FOV.

(a) (b)

Figure 4-10. (a) Measured interferograms after baseline subtraction for a 3x3 grid of field points spanning a square, 5° degree diagonal FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at 0° and displaced roughly 100 mm longitudinally away from the stop surface and (b) the 3x3 grid of wavefronts with the field constant elliptical coma removed, revealing the generated field conjugate, field linear astigmatism induced by the trefoil plate.

Another way to visualize the generated astigmatic field is to plot the astigmatism, Z5/6,

FFD throughout the FOV. In Figure 4-11 (a-c), left, the measured astigmatism with the

baseline subtracted are plotted over a 9x9 grid of field points spanning a square, 5°

diagonal FOV at three different plate positions: 10.81 mm, 53.31 mm, and 95.81 mm. As

a point of comparison, the theoretical aberration fields predicted by NAT are displayed as

87

well in Figure 4-11 (a-c), right. In evaluating Figure 4-11 (a-c), there is good agreement

in both magnitude and orientation between the experimental and theoretical predictions

of NAT for all three plate positions. To create a more quantitative comparison,

Figure 4-12 compares the magnitude of the measured versus theoretical Zernike trefoil

and Zernike astigmatism for two field points, ( )1, 0x yH H= = and ( )1, 0x yH H= − = , as a

function of plate position. As demonstrated earlier in Figure 4-5, the magnitude of the

Zernike trefoil remains constant as a function of plate position and the magnitude of the

generated Zernike astigmatism increases linearly with plate position. For the five

measured plate positions, the measured and theoretical data agree within the uncertainty

in the measurement for both field points analyzed. Any error in the measurement and

deviation from theoretical is attributed to air turbulence and vibration in the measurement

that has the greatest effect for small plate offsets where the magnitude of the generated

astigmatism is small.

88

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj Field (deg)

Y O

bj F

ield

(deg)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg)

Y O

bj. F

ield

(deg)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj Field (deg)

Y O

bj F

ield

(deg)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg)

Y O

bj. F

ield

(deg)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj Field (deg)

Y O

bj F

ield

(deg)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg)

Y O

bj. F

ield

(deg)

1λ (λ=0.632.8nm)

Experimental Theoretical

(a)

(b)

(c)

Measured

Figure 4-11. The measured Zernike astigmatism (Z5/6) FFD after baseline subtraction,

left, and theoretical Zernike astigmatism (Z5/6) FFD predicted by NAT, right, over a 9x9

grid spanning a square, 5° full FOV for the Schmidt telescope system with the Zernike

trefoil plate oriented at 0° and located (a) 10.81 mm, (b) 53.31 mm, and (c) 95.81 mm

away from the stop surface.

89

0 10 20 30 40 50 60 70 80 90 100 110

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Plate Position (mm)

Mag

nitu

de (w

aves

at 6

32.8

nm

)

H=(-1,0)H=( 1,0)Theoretical

Zernike Trefoil

Zernike Astigmatism

Figure 4-12. Plot of the mean magnitude of the Zernike trefoil and astigmatism after baseline subtraction for two field points, ( )1, 0x yH H= = represented by the blue circle

and ( )1, 0x yH H= − = represented by the red star, for five measured plate positions. The error bars on the data points represent plus or minus one standard deviation from the mean value over the ten measurements acquired at each plate position. In black, the magnitude of the Zernike trefoil and astigmatism based on the theoretical predictions of NAT is plotted as a function of plate position.

4.4.2 Rotation of the Aberration Generating Plate

In Section 4.4.1, the Zernike plate in the Schmidt telescope configuration was aligned

on-axis and oriented at 0°. If the Zernike plate is now rotated, the orientation of the

astigmatic line images throughout the FOV will also rotate. To demonstrate this effect,

the Zernike plate in the Schmidt telescope is rotated 45° and the experiment is repeated to

acquire the wavefront throughout the FOV. Figure 4-13 (a) displays the measured results

for the rotated Zernike plate at 45° when the plate is roughly 100 mm away from the stop

surface and Figure 4-13 (b) displays the theoretical predictions from NAT. Similar to the

results shown earlier for a plate on-axis and oriented at 0°, there is very good agreement

between the measured astigmatic field and the theoretical simulations. In both cases, the

zero of the field conjugate, field linear astigmatism stays nearly on-axis and the line

90

images rotate with the orientation of the Zernike plate. If the astigmatic field for the 45°

oriented Zernike plate shown in Figure 4-13 is compared to the 0° oriented Zernike plate,

shown in Figure 4-11 (c), it can be seen that the field structure is the same except that the

entire aberration field is rotated by 45°.

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg.)

Y O

bj. F

ield

(deg

.)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg.)

Y O

bj. F

ield

(deg

.)

1λ (λ=0.632.8nm)(a) (b)

Figure 4-13. The (a) measured Zernike astigmatism (Z5/6) FFD after baseline subtraction and (b) theoretical Zernike astigmatism Z5/6 FFD predicted by NAT over a 9x9 grid spanning a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at 45° and located roughly 100 mm away from the stop surface.

4.4.3 Lateral Displacement of the Aberration Generating Plate

The Zernike trefoil plate has been initially aligned so that the vertex of the Zernike

deformation is coincident with the on-axis field point. If the Zernike plate is now shifted

laterally, the generated astigmatic aberration field will shift. The shift of the aberration

field is predicted by NAT as outlined in Chapter 3 with the introduction of a freeform

sigma vector that modifies the astigmatic contribution.

For the Schmidt telescope configuration, the Zernike plate is displaced +1 mm in the

x-direction and -1 mm in the y-direction. Based on the telescope configuration with the

Zernike plate approximately 100 mm away from the stop surface, the freeform sigma

vector is computed as

91

* 1 0.32011 ,

1 0.32013.124FF

FF

mmvmmy mm

δσ

+ + = = = − −

(4.1)

where the freeform sigma vector defines a new effective field height FFH

defined as,

.FF FFH H σ= −

(4.2)

Since the only astigmatic aberration generated is field conjugate, field linear astigmatism,

NAT predicts a single node at FFH σ=

. To verify this prediction, the experiment to

acquire the wavefront throughout the FOV proceeds as described above with, in this case,

a laterally shifted Zernike plate. Figure 4-14 (a) displays the measured results for the

shifted Zernike plate and Figure 4-14 (b) displays the theoretical predictions from NAT.

Similar to the results shown earlier for a plate on-axis, there is very good agreement

between the measured astigmatic field and the theoretical simulations predicted by NAT.

In both cases, the zero of the field conjugate, field linear astigmatism has moved off-axis

with the zero approximately at

0.57deg.

0.57H

+ = −

(4.3)

92

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg.)

Y O

bj. F

ield

(deg

.)

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X Obj. Field (deg.)

Y O

bj. F

ield

(deg

.)

1λ (λ=0.632.8nm)(a) (b)

Figure 4-14. The (a) measured Zernike astigmatism (Z5/6) FFD after baseline subtraction and (b) theoretical Zernike astigmatism (Z5/6) FFD predicted by NAT over a 9x9 grid spanning a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at 0°, located roughly 100 mm away from the stop surface, and displaced laterally 1 mm in the x-direction and -1 mm in the y-direction.

93

Chapter 5. Design of a Freeform Unobscured Reflective Imager Employing φ-Polynomial Optical Surfaces

Historically, optical designers had a reputation for designing optical systems that exceed

the industry capabilities for fabrication and/or assembly. In general, these systems were

intrinsically rotationally symmetric using spheres, aspheres, or off-axis segments of a

rotationally symmetric surface (other than the occasional use of cylindrical or toric

surfaces for special case anamorphic systems). Recently, the optical fabrication industry

changed this paradigm by implementing a capability to fabricate diamond turned, optical

quality surfaces in the LWIR that are not rotationally symmetric. In particular, it is now

possible to fabricate an optical surface that is defined as a conic plus the lower order

terms of a Zernike polynomial (less than Fringe term 16).

In this chapter, an optical system design that is composed of tilted Zernike polynomial

mirrors is optimized to create a compact, LWIR optical system that will couple to a

320x256 pixel, 25 µm pixel pitch, uncooled microbolometer detector. The optimization

strategies that are employed during the optical design of this nonsymmetric system use

the principles of NAT applied to Zernike polynomial optical surfaces, as described in

Chapter 3. With this new understanding of freeform optical surfaces and their intrinsic

aberration fields, it is now possible to apply a NAT based optical analysis approach to

optimization.

5.1 The New Method of Optical Design

In the 1960s, the first optical designs that involved three or more mirrors in an

unobscured configuration started to be declassified and began to appear in limited

distribution government reports [74]. Motivated by the advance in LWIR detectors and

the accompanying need for stray light control, a number of systems were designed as

94

concept designs for missile defense. While many of these systems appear to lack

rotational symmetry, detailed analysis reveals that any successful design with a

significant FOV was, in fact, based on a rotationally symmetric design with an offset

aperture, a biased field, or both. Analysis shows that this fact could be anticipated, as

many systems that depart from rotational symmetry immediately display on-axis coma,

where the axis for a nonsymmetric system is defined by the OAR [22]. While there are

special configurations that eliminate axial coma, there are very few practical forms that

do not reduce to a rotational symmetric form.

In 1994, an optical system designed by Rodgers was patented that had the property of

providing the largest planar, circular input aperture in the smallest overall spherical

volume [75]. A design attempting to meet similar constraints can be found in 2005 by

Nakano [30]. The particular form embodied in the patent of [75] is shown in

Figure 5-1 (a). This optical design is a 9:1 afocal relay that operates over a 3˚ full FOV

using four mirrors. It provides a real, accessible exit pupil that is often a requirement in

earlier infrared systems requiring cooled detectors. In use, it is coupled with a fast

F/number refractive component in a dewar near the detector. It is based on using off-axis

sections of rotationally symmetric conic mirrors that are folded into the spherical volume

by using one fold mirror (mirror 3).

95

(a) (b)

14:37:35

AFOCAL 9:1 PUPIL DEMAG, 3x3 deg. FULL SCALE ORA 17-Sep-11

25.00 MM

14:25:51

Zernike Polynomial FULL SCALE KHF 17-Sep-11

25.00 MM

14:25:51


25.00 MM

14:25:51


25.00 MM

Figure 5-1. (a) Layout of U.S. Patent 5,309,276 consisting of three off-axis sections of

rotationally symmetric mirrors and a fourth fold mirror (mirror 3). The optical system

had, at the time of its design, the unique property of providing the largest planar, circular

input aperture in the smallest overall spherical volume for a gimbaled application. (b) The

new optical design based on tilted φ-polynomial surfaces to be coupled to an uncooled

microbolometer.

As is often the case, many applications would exploit a larger FOV if it were available

with usable performance. In addition, if an optical form could be developed at a fast

enough F/number, it becomes feasible to transition to an uncooled detector thereby

abandoning the need for the reimaging configuration, the external exit pupil, and the

refractive component in the dewar. Using the new paradigm of tilted freeform,

φ-polynomial optical surfaces, a three mirror, F/1.9 form with a 10˚ diagonal full FOV

has been developed using the methods of NAT for the optimization. The nominal optical

design is shown in Figure 5-1 (b) and has an overall RMS WFE of less than λ/100 at a

wavelength of 10 µm over a 10° full FOV where the overall RMS WFE is computed as

the average plus one standard deviation RMS WFE for all field points. The remainder of

this chapter will detail how this solution was developed using the tools and concepts of

NAT applied to tilted φ-polynomial surfaces.

5.2 The Starting Form

The first step in the new design process is to design a well corrected rotationally

symmetric optical form without regard for the fact that no light can pass through the

96

system because of the obscuration by the mirrors. This step corrects the spherical

aberration, coma, and astigmatism and creates a basic configuration with conic mirrors to

minimize the use of the Zernike terms, which can challenge the testing program.

Figure 5-2 (a-b) shows the result of this step for a system with aggressive goals for the

F/number and FOV. The primary and tertiary mirrors are oblate ellipsoids whereas the

secondary mirror is hyperbolic and is also the stop surface. The system is well corrected

throughout the FOV where the overall average RMS WFE over the 10° full FOV, as

displayed in Figure 5-2 (b), is less than λ/250 (0.004λ). The next step is to make this

fictitious starting point design unobscured. Typically, the solution to creating an

unobscured design from an obscured form is to go off-axis in aperture and/or bias the

input field [4]. It is difficult to do so with this geometry because the primary mirror is

smaller than the secondary and tertiary mirrors. With the knowledge that there is a path to

removing axial coma by using the new design DOF, machining coma directly onto the

surface, the new strategy is to simply tilt the surfaces until the light clears the mirrors.

(a) (b)

16:10:05

Zernike Polynomial Scale: 1.70 KHF 03-Jun-11

14.71 MM

kPri=4.95

kSec=-4.65 kTer=0.3

16:10:05

Zernike Polynomial Scale: 1.70 KHF 03-Jun-11

14.71 MM

11:56:12

Zernike Polynomial

KHF 09-Jun-11

RMS WAVEFRONT ERROR

vs


Minimum = 0.0028776

Maximum = 0.0038846

Average = 0.0031063

Std Dev = 0.00022442

0.05waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


11:56:12

Zernike Polynomial

KHF 09-Jun-11

RMS WAVEFRONT ERROR

vs


Minimum = 0.0028776

Maximum = 0.0038846

Average = 0.0031063

Std Dev = 0.00022442

0.05waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


11:56:12

Zernike Polynomial

KHF 09-Jun-11

RMS WAVEFRONT ERROR

vs


Minimum = 0.0028776

Maximum = 0.0038846

Average = 0.0031063

Std Dev = 0.00022442

0.05waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


Figure 5-2. (a) Layout for a fully obscured solution for a F/1.9, 10° full FOV LWIR

imager. The system utilizes three conic mirror surfaces. (b) A FFD of the RMS WFE of

the optical system. Each circle represents the magnitude of the RMS wavefront at a

particular location in the FOV. The system exhibits a RMS WFE of < λ/250 over 10° full

FOV.

97

5.3 The Unobscured Form

Tilting the on-axis solution breaks the rotational symmetry of the system and changes

where the aberration field zeros (nodes) are located for each aberration type. The shift of

the aberration fields drastically degrades the overall performance of the system. A

strategy for tracking the evolution of the nodal structure as the unobscured design form

unfolds is to oversize the FOV to many times the intended FOV. As an example of this

strategy, Figure 5-3 (a-c) shows the design form at 0%, 50% and 100% unobscured

accompanied by an evaluation of Zernike coma (Z7/8) and Zernike astigmatism (Z5/6)

across a ±40˚ field (note there is a 12X scale change between Figure 5-3 (a) and

Figure 5-3 (b-c) so the nodal behavior can be seen for each tilt position). As can be seen

from Figure 5-3 (a), the on-axis solution is well corrected for astigmatism and coma

within the 10˚ diagonal full FOV (sub-region in red) and the nodes (blue star and green

dot) are centered on the optical axis (zero field). As the system is tilted halfway to an

unobscured solution, shown in Figure 5-3 (b), the node for coma has moved immediately

beyond the field being evaluated resulting in what is a field constant coma. For this

intermediate tilt, one of the two astigmatic nodes remains within the extended analysis

field, moving linearly with tilt. When the system is tilted to a fully unobscured solution,

shown in Figure 5-3 (c), field constant coma is increased while the astigmatic node also

moves out of the 8X oversized analysis field leaving the appearance of a field constant

astigmatism. The first significant observation regarding formulating a strategy for

correction is that in the unobscured configuration the nodes have moved so far out in the

field that the astigmatism and coma contributions within the region of interest, a 10˚ full

FOV, are nearly constant.

98

(a)

(b)

(c)

14:37:35

Zernike Polynomial KHF 08-Jun-11

25.40 MM

14:37:34

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 2.8641

Average = 0.87527

Std Dev = 0.68384

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:37:35

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 0.14889

Average = 0.076407

Std Dev = 0.049123

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


ZAstigZComa

14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


2.5λ (10µm)

14:05:06


25.40 MM

14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.004978

Maximum = 16.328

Average = 9.2986

Std Dev = 3.0961

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.7002

Maximum = 7.1567

Average = 4.4879

Std Dev = 0.59061

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

14:06:27


25.40 MM

14:06:26

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.1297

Maximum = 85.517

Average = 29.836

Std Dev = 9.5017

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40Y Field Angle in Object Space - degrees

14:06:27

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 7.0185

Maximum = 53.123

Average = 11.697

Std Dev = 5.3486

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

40

20

0

-40

-20

-40 -20 0 4020

(a)

(b)

(c)

14:37:35


25.40 MM

14:37:34

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 2.8641

Average = 0.87527

Std Dev = 0.68384

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:37:35

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 0.14889

Average = 0.076407

Std Dev = 0.049123

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


ZAstigZComa

14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


2.5λ (10µm)

14:05:06


25.40 MM

14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.004978

Maximum = 16.328

Average = 9.2986

Std Dev = 3.0961

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.7002

Maximum = 7.1567

Average = 4.4879

Std Dev = 0.59061

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

14:06:27


25.40 MM

14:06:26

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.1297

Maximum = 85.517

Average = 29.836

Std Dev = 9.5017

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:06:27

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 7.0185

Maximum = 53.123

Average = 11.697

Std Dev = 5.3486

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20


30λ (10µm)

40

20

0

-40

-20

-40 -20 0 4020

40

20

0

-40

-20

-40 -20 0 4020

40

20

0

-40

-20

-40 -20 0 4020

40

20

0

-40

-20

-40 -20 0 4020

40

20

0

-40

-20

-40 -20 0 4020

(a)

(b)

(c)

14:37:35


25.40 MM

14:37:34

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 2.8641

Average = 0.87527

Std Dev = 0.68384

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:37:35

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 0.14889

Average = 0.076407

Std Dev = 0.049123

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


ZAstigZComa

14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


2.5λ (10µm)

14:05:06


25.40 MM

14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.004978

Maximum = 16.328

Average = 9.2986

Std Dev = 3.0961

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.7002

Maximum = 7.1567

Average = 4.4879

Std Dev = 0.59061

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

14:06:27


25.40 MM

14:06:26

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.1297

Maximum = 85.517

Average = 29.836

Std Dev = 9.5017

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:06:27

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 7.0185

Maximum = 53.123

Average = 11.697

Std Dev = 5.3486

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

(a)

(b)

(c)

14:37:35


25.40 MM

14:37:34

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 2.8641

Average = 0.87527

Std Dev = 0.68384

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:37:35

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 0.14889

Average = 0.076407

Std Dev = 0.049123

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


ZAstigZComa

14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


2.5λ (10µm)

14:05:06


25.40 MM

14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.004978

Maximum = 16.328

Average = 9.2986

Std Dev = 3.0961

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.7002

Maximum = 7.1567

Average = 4.4879

Std Dev = 0.59061

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

14:06:27


25.40 MM

14:06:26

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.1297

Maximum = 85.517

Average = 29.836

Std Dev = 9.5017

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:06:27

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 7.0185

Maximum = 53.123

Average = 11.697

Std Dev = 5.3486

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

(a)

(b)

(c)

14:37:35


25.40 MM

14:37:34

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 2.8641

Average = 0.87527

Std Dev = 0.68384

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:37:35

Zernike Polynomial

KHF 08-Jun-11


vs



Maximum = 0.14889

Average = 0.076407

Std Dev = 0.049123

2.5waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


ZAstigZComa

14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


2.5λ (10µm)

14:05:06


25.40 MM

14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.004978

Maximum = 16.328

Average = 9.2986

Std Dev = 3.0961

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:05:06

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.7002

Maximum = 7.1567

Average = 4.4879

Std Dev = 0.59061

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

14:06:27


25.40 MM

14:06:26

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 3.1297

Maximum = 85.517

Average = 29.836

Std Dev = 9.5017

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:06:27

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 7.0185

Maximum = 53.123

Average = 11.697

Std Dev = 5.3486

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


14:04:21

Zernike Polynomial

KHF 08-Jun-11


vs


Minimum = 0.22109

Maximum = 8.4653

Average = 3.9119

Std Dev = 1.4341

30waves (10000.0 nm)

-40 -20 0 20 40


-40

-20

0

20

40


30λ (10µm)

YFi

eld

(d

eg.)

X Field (deg.)

YFi

eld

(d

eg.)

X Field (deg.)

YFi

eld

(d

eg.)

X Field (deg.)

YFi

eld

(d

eg.)

X Field (deg.)

YFi

eld

(d

eg.)

X Field (deg.)

YFi

eld

(d

eg.)

X Field (deg.)

2.5λ (10µm)

30λ (10µm)

30λ (10µm)

Figure 5-3. The lens layout, Zernike coma (Z7/8) and astigmatism (Z5/6) FFDs for a ±40˚

FOV for the (a) on-axis optical system, (b) halfway tilted, 50% obscured system, and (c)

fully tilted, 100% unobscured system. The region in red shows the field of interest, a 10˚

diagonal FOV.

5.3.1 Creating Field Constant Aberration Correction

With a baseline unobscured system established, the next step is to use the new DOFs,

efficiently and effectively, to create a usable performance over the 10˚ diagonal full FOV

and at an F/number that allows the use of an uncooled microbolometer (less than F/2).

Now that the nodal evolution has been established, it is more effective to return to an

analysis only over the target FOV. Figure 5-4 shows that when the field performance is

99

evaluated over a smaller field, ±5˚, the field constant behavior is clearly observed for

both coma and astigmatism as well as for the higher order aberration contributions, such

as elliptical coma (Z10/11) and oblique spherical aberration (Z12/13). It is worth noting that

Figure 5-4 shows that the spherical aberration (Z9) is nearly unchanged even for this

highly tilted system. An evaluation of the RMS WFE is also added (far lower right) to

determine when adequate correction is achieved. For this starting point, the RMS WFE is

~12λ and is predominately due to the astigmatism and coma contributions both of which

are, significantly, field constant.

13:52:59

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 31.614

Maximum = 34.206

Average = 32.948

Std Dev = 0.74182

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4


13:52:59

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 10.975

Maximum = 12.996

Average = 11.927

Std Dev = 0.62792

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:53:00

Zernike Polynomial

KHF 23-Sep-13

FRINGE ZERNIKE COEFFICIENT Z9

vs


Minimum = -0.2004

Maximum = -0.17296

Average = -0.18392

Std Dev = 0.0080848

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:53:00

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.92293

Maximum = 1.3468

Average = 1.123

Std Dev = 0.13169

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:53:00

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.26485

Maximum = 0.53057

Average = 0.3919

Std Dev = 0.080087

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:53:00

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.17107

Maximum = 0.22424

Average = 0.19594

Std Dev = 0.016572

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:53:01

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 11.294

Maximum = 12.351

Average = 11.905

Std Dev = 0.29646

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6Y

Fie

ld (

deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

ZEllip. Coma

ZObl .Spher. RMS WFE

ZAstigZComaZSpher

Z5th Coma

14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


10.0λ (10 µm)

13:53:01

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 11.294

Maximum = 12.351

Average = 11.905

Std Dev = 0.29646

10waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



elliptical coma (Z10/11), oblique spherical aberration (Z12/13), and fifth order aperture coma

(Z14/15) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for

the fully unobscured, on-axis solution. It can be seen that the system is dominated by

field constant coma and astigmatism which are the largest contributors to the RMS WFE

of ~12λ.

It is possible to correct the field constant aberrations shown in Figure 5-4 by using the

fact that the stop location for this optical system is the secondary mirror. In Chapter 3, it

was shown that when a Zernike polynomial overlay is placed at the stop location, a field

constant aberration is induced. In this design case, Zernike coma and astigmatism are

100

added as variables to the secondary conic surface, so they will introduce, when

optimized, the opposite amount of field constant coma and astigmatism present from

tilting the optical system to create an unobscured form. The effect of optimizing the

optical system with these variables is shown in Figure 5-5 where the field constant coma

and astigmatism have been removed. The RMS WFE has gone from ~12λ for the tilted

system without φ-polynomials to ~0.75λ for the tilted system with Zernike coma and

astigmatism on the secondary surface (note that there is a 10X scale change from

Figure 5-4 to Figure 5-5 to show the residual terms in further detail).

13:57:21

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.054353

Maximum = 5.0395

Average = 2.7771

Std Dev = 1.0497

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:57:21

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0052108

Maximum = 0.47304

Average = 0.25737

Std Dev = 0.11196

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:57:21

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.031014

Maximum = 0.10608

Average = 0.067278

Std Dev = 0.023332

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:57:22

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.027263

Maximum = 1.0754

Average = 0.57087

Std Dev = 0.26785

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:57:22

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.052021

Maximum = 0.18089

Average = 0.11437

Std Dev = 0.038553

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:57:22

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.063068

Maximum = 0.095524

Average = 0.078182

Std Dev = 0.010147

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:57:22

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.13731

Maximum = 1.2474

Average = 0.71749

Std Dev = 0.25169

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

Y F

ield

(deg.)

6

4

2

0

2-

4-

6-

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

ZEllip. Coma


ZAstigZComaZSpher

Z5th Coma

14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


1.00λ (10 µm)

13:57:22

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.13731

Maximum = 1.2474

Average = 0.71749

Std Dev = 0.25169

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6





the optimized system where Zernike astigmatism and coma were used as variables on the

secondary (stop) surface. When the system is optimized, the field constant contribution to

astigmatism and coma are greatly reduced improving the RMS WFE from ~12λ to

~0.75λ.

5.3.2 Creating Field Dependent Aberration Correction

By studying the residual behavior of the optical system after optimization of Zernike

coma and astigmatism on the secondary surface, it can be seen from the displays, shown

101

in Figure 5-5, that the dominant aberration contribution is Zernike astigmatism and it is

the largest contributor to the RMS WFE of ~0.75λ. Moreover, the astigmatism has taken

the form of field linear, field asymmetric astigmatism. In Chapter 3, it was discovered

that a Zernike coma overlay displaced axially away from the stop surface will introduce

field linear, field asymmetric astigmatism as well as field linear, field curvature. Using

this result, Zernike coma is placed on an optical surface away from the stop location, that

is, the primary or tertiary surface, and optimized to reduce (and in some cases eliminate)

the residual field linear, field asymmetric astigmatism. However, in order to effectively

use this added variable, the tilt angle of the focal plane must also be varied to compensate

the induced field linear, field curvature component also introduced by the Zernike coma

overlay.

Since both the primary and tertiary surfaces lie away from the stop surface, a Zernike

coma overlay can be placed on either surface. If the equation for the generated field

linear, field asymmetric astigmatism from a Zernike coma overlay away from the stop is

investigated, Eq. (3.25), it can be seen that the magnitude of the aberration depends

linearly on the ratio of the chief to marginal ray on the optical surface. For the current

configuration, the ratio at the primary surface is roughly 0.170 whereas the ratio is 0.086

on the tertiary surface. Based on this first order analysis, it appears that the primary

surface will be a much more effective variable for removing the residual field linear, field

asymmetric astigmatism since less comatic departure will be required to create the

equivalent induced astigmatic aberration.

The effectiveness of the Zernike coma overlay on the primary mirror surface is

demonstrated in Figure 5-6 where the relevant aberration contributions after optimization

102

are shown. As can be seen from Figure 5-6, which is on the same scale as Figure 5-5, the

astigmatism contribution has been reduced and the RMS WFE has been improved by

another factor of 6X going from 0.750λ to roughly 0.125λ. The astigmatism contribution

has not been eliminated completely by the use of the Zernike coma overlay because while

the Zernike coma overlay is used to correct the residual third order, field linear, field

asymmetric astigmatism, it also induces higher order aberration components on

subsequent surfaces that will impact the overall RMS WFE. The optimized solution is a

balance of these two effects.

13:58:22

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0073

Maximum = 0.4213

Average = 0.21883

Std Dev = 0.091257

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4


13:58:22

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.033723

Maximum = 0.56428

Average = 0.30152

Std Dev = 0.12926

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:58:22

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.072859

Maximum = 0.086048

Average = 0.082145

Std Dev = 0.0038394

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:58:23

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0020488

Maximum = 0.80295

Average = 0.4397

Std Dev = 0.17715

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:58:23

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0048849

Maximum = 0.053164

Average = 0.028583

Std Dev = 0.014843

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:58:23

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.036374

Maximum = 0.046397

Average = 0.042713

Std Dev = 0.0031971

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:58:24

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.04427

Maximum = 0.21659

Average = 0.12552

Std Dev = 0.041275

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6Y

Fie

ld (

deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

ZEllip. Coma


ZAstigZComaZSpher

Z5th Coma

14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


1.00λ (10 µm)

13:58:24

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.04427

Maximum = 0.21659

Average = 0.12552

Std Dev = 0.041275

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6




(Z14/15) Zernike aberration and RMS WFE FFDs over a ±5 degree FOV for the optimized

system where Zernike coma is added as an additional variable to the primary surface. The

RMS WFE has been reduced from ~0.75λ to ~0.125λ.

If the Zernike coma overlay is instead added onto the tertiary mirror surface and

optimized, there is a similar improvement to the RMS WFE where the average

performance is around 0.180λ but the relevant aberration contributions after optimization,

shown in Figure 5-7, are much different. In this configuration the residual astigmatism is

103

no longer field linear, field asymmetric and now resembles field conjugate, field linear

astigmatism, a fifth order aberration. Moreover, the residual coma and elliptical coma

contributions are smaller than the case where the Zernike coma overlay is applied to the

primary surface; however, the Zernike spherical and oblique spherical aberration

contributions are larger. The difference in these higher order aberration components is a

result of induced aberrations that stem from the arrangement of Zernike overlay terms on

the mirror surfaces. While the overlay terms help correct residual aberrations in the

design, they may, depending on their location in the optical system, induce higher order

aberrations since the beam shape and ray angles on subsequent surfaces will change. 13:59:50

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0057161

Maximum = 0.68543

Average = 0.28603

Std Dev = 0.1636

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:59:51

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.013632

Maximum = 0.36969

Average = 0.14397

Std Dev = 0.10007

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:59:51

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.21641

Maximum = -0.14896

Average = -0.18044

Std Dev = 0.020724

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:59:51

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.44899

Maximum = 0.5808

Average = 0.52974

Std Dev = 0.036596

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:59:51

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.10484

Maximum = 0.14582

Average = 0.12655

Std Dev = 0.012196

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:59:51

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.010167

Maximum = 0.023382

Average = 0.016188

Std Dev = 0.0040848

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


13:59:52

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.14727

Maximum = 0.21404

Average = 0.18221

Std Dev = 0.01856

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

ZEllip. Coma


ZAstigZComaZSpher

Z5th Coma

14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


1.00λ (10 µm)

13:59:52

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.14727

Maximum = 0.21404

Average = 0.18221

Std Dev = 0.01856

1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4





the optimized system where Zernike coma is added as an additional variable to the

tertiary surface. The RMS WFE has been reduced from ~0.75λ to ~0.180λ.

104

5.4 The Final Form

With the successful creation of a nearly compliant unobscured form, the remaining

optimization proceeds with additional use of Zernike coefficients for either field constant

or field dependent correction. Continuing from the configuration with Zernike

astigmatism and coma on the secondary mirror and Zernike coma on the tertiary surface,

Figure 5-7 shows that the optical system is now limited by field constant aberrations,

namely, field constant oblique spherical aberration that shows up in the Z9 and Z12/13

FFDs, field constant elliptical coma that shows up in the Z10/11 FFD, and field constant

fifth order aperture coma that shows up in the Z14/15 FFD. These field constant aberrations

are reduced by adding the conic constants of the surfaces as additional variables as well

as adding variables for elliptical coma (Z11), oblique spherical aberration (Z12), and fifth

order aperture coma (Z15) at the secondary mirror. When the system is optimized with

these additional variables, the field constant aberrations are decreased as shown in

Figure 5-8 and the RMS WFE has improved from 0.180λ to 0.065λ (note that there is a

4X scale change from Figure 5-7 to Figure 5-8 to show the residual terms in further

detail).

105

14:28:48

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.012678

Maximum = 0.50061

Average = 0.27048

Std Dev = 0.11969

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:28:48

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0070809

Maximum = 0.19644

Average = 0.088709

Std Dev = 0.04005

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:28:48

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.025261

Maximum = 0.034217

Average = 0.007247

Std Dev = 0.018505

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:28:48

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0029427

Maximum = 0.075872

Average = 0.029481

Std Dev = 0.016632

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:28:48

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.00016525

Maximum = 0.042151

Average = 0.025016

Std Dev = 0.0093532

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:28:48

Zernike Polynomial

KHF 23-Sep-13


vs



Maximum = 0.0088598

Average = 0.0043835

Std Dev = 0.002246

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:28:49

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.027109

Maximum = 0.11106

Average = 0.064292

Std Dev = 0.019996

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6Y

Fie

ld (

deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

YF

ield

(deg.)

ZEllip. Coma


ZAstigZComaZSpher

Z5th Coma

14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


0.25λ (10 µm)

14:28:49

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.027109

Maximum = 0.11106

Average = 0.064292

Std Dev = 0.019996

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6





the optimized system where the mirror conic constants are added as additional variables

in addition to Zernike elliptical coma, oblique spherical aberration, fifth order aperture

coma on the secondary surface. The RMS WFE has been reduced from ~0.180λ to

~0.065λ.

After optimization to remove the higher order field constant aberrations, the dominant

residual aberrations are now fifth order, field conjugate, field linear astigmatism and a

fifth order comatic contribution that resembles field conjugate, field linear coma. In

Chapter 3, it was discovered that a Zernike trefoil overlay displaced axially away from

the stop surface will introduce field conjugate, field linear astigmatism so it is added as

an additional variable on the tertiary surface. To compensate the comatic contribution, a

Zernike oblique spherical overlay is added at the tertiary surface as it primarily induces

field conjugate, field linear coma when placed away from the stop. Because a Zernike

oblique spherical overlay has a Zernike astigmatism component built into its term, it also

helps to add Zernike astigmatism to the tertiary surface as an additional independent

variable so the oblique spherical aberration term can be independently controlled relative

106

to the astigmatism. Figure 5-9 shows the resulting aberration contributions after

optimization with these three additional variables. There has been a drastic improvement

in the RMS WFE going from 0.065λ to 0.012λ.

14:37:40

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0030928

Maximum = 0.10011

Average = 0.032326

Std Dev = 0.021346

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0012659

Maximum = 0.046904

Average = 0.01482

Std Dev = 0.0091452

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.00011767

Maximum = 0.035384

Average = 0.011416

Std Dev = 0.0077174

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0012925

Maximum = 0.028797

Average = 0.016026

Std Dev = 0.0062967

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = 0.0001119

Maximum = 0.0060283

Average = 0.0032851

Std Dev = 0.0013844

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:37:42

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.0066824

Maximum = 0.021031

Average = 0.012715

Std Dev = 0.0041642

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


ZEllip. Coma


ZAstigZComaZSpher

Z5th Coma

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

6 4 2 0 642- - -X Field (deg.)

6-

4-

2-

0

2

4

6

Y F

ield

(deg.)

14:37:41

Zernike Polynomial

KHF 23-Sep-13


vs


Minimum = -0.015258

Maximum = 0.012736

Average = 0.00036325

Std Dev = 0.0086811

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


0.25λ (10 µm)

14:37:42

Zernike Polynomial

KHF 23-Sep-13

RMS WAVEFRONT ERROR

vs


Minimum = 0.0066824

Maximum = 0.021031

Average = 0.012715

Std Dev = 0.0041642

0.25waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6





the optimized system where Zernike astigmatism, elliptical coma, and oblique spherical

aberration are added as additional variables to the tertiary surface. The RMS WFE has

been reduced from ~0.065λ to ~0.012λ.

Ultimately, further optimization leads to the system shown in Figure 5-10 (a) where

the overall average RMS WFE over the 10° full FOV, as displayed in Figure 5-10 (b), is

less than λ/100 (0.01λ), well within the diffraction limit (0.07λ). In this final

optimization, the Zernike distribution on the three mirror surfaces is manipulated as it

helps to alter the induced aberration components. Since the induced aberration behavior

is not currently predicted by NAT, the optimizer is useful for distributing the Zernike

contributions about the mirror surfaces; however, the method presented here is useful for

determining which variables will be effective for reducing the intrinsic aberration

components. As a point of comparison for the final system performance, if the field and

107

F/number of the unobscured, conic only solution presented in Figure 5-3 (c) are reduced

to produce a diffraction limited system, the field must be reduced to a 3° diagonal full

FOV and the system speed must be reduced to F/22. Thus with the φ-polynomial surface,

there is a substantial advance in usable FOV and light collection capability in this design

space.

(a) (b)

14:25:51


25.00 MM

14:25:51


25.00 MM

16:54:26

Zernike Polynomial

KHF 26-Sep-11

RMS WAVEFRONT ERROR

vs


Minimum = 0.0049516

Maximum = 0.015382

Average = 0.0085467

Std Dev = 0.0017325

0.1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


16:54:26

Zernike Polynomial

KHF 26-Sep-11

RMS WAVEFRONT ERROR

vs


Minimum = 0.0049516

Maximum = 0.015382

Average = 0.0085467

Std Dev = 0.0017325

0.1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


16:54:26

Zernike Polynomial

KHF 26-Sep-11

RMS WAVEFRONT ERROR

vs


Minimum = 0.0049516

Maximum = 0.015382

Average = 0.0085467

Std Dev = 0.0017325

0.1waves (10000.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


Figure 5-10. (a) Layout of LWIR imaging system optimized with φ-polynomial surfaces

and (b) the RMS WFE of the final, optimized system, which is < λ/100 (0.01λ) over a 10˚

diagonal full FOV.

5.5 Mirror Surface Figures

The sags of three mirrors for the final design are displayed in Figure 5-11 (a-c) where

they are evaluated with different Zernike components removed from the base sag. In

Figure 5-11 (a), the sags are evaluated with the piston, tilt, and power Zernike

contributions removed so that the dominant astigmatic contribution present in the

surfaces can be seen. When the astigmatism is also removed from the surface sags,

Figure 5-11 (b), the remaining sag components are observed. An asymmetry is now seen

in the sags that results from the comatic departure present in the surfaces. If the spherical

aberration is now removed, Figure 5-11 (c), the comatic departure on the surfaces is more

readily visible. The primary mirror surface has the smallest amount of Zernike departure,

which is on the order of 50 µm and is primarily composed of higher order coma. The

108

secondary mirror surface has roughly 120 µm of Zernike departure where most of the

departure is composed of astigmatism. Since the secondary mirror is the stop surface, it

makes sense that the surface is primarily astigmatic. The tertiary has the most Zernike

departure out of the three mirror surfaces, which is on the order of 700 µm. Similar to the

secondary mirror, the tertiary surface is largely composed of astigmatism with the next

most dominant contribution being coma.

(a)

(b)

(c)

Sag of Secondary minus Piston/Power/Tilt in µm

Sag of Secondary minus Piston/Power/Tilt/Astig. in µm

Sag of Secondary minus Piston/Power/Tilt/Astig./Spher. in µm

Sag of Tertiary minus Piston/Power/Tilt in µm

Sag of Tertiary minus Piston/Power/Tilt/Astig. in µm

Sag of Tertiary minus Piston/Power/Tilt/Astig./Spher. in µm

Sag of Primary minus Piston/Power/Tilt in µm

Sag of Primary minus Piston/Power/Tilt/Astig. in µm

Sag of Primary minus Piston/Power/Tilt/Astig./Spher. in µm

Figure 5-11. (a) Sag of the primary mirror surface various Zernike components removed from the base sag, (b) sag of the secondary mirror surface various Zernike components removed from the base sag, and (c) sag of the tertiary mirror surface mirror surface various Zernike components removed from the base sag. When the piston, power, and astigmatism are removed from the base sags of the three mirrors, the asymmetry induced from the coma being added into the surface is observed.

109

Chapter 6. Interferometric Null Configurations for Measuring φ-Polynomial Optical Surfaces

As seen in the Chapter 5, a φ-polynomial surface will usually have some amount of

spherical aberration, astigmatism, coma, and some higher order aberration terms placed

into the surface departure. As a result, a conventional interferometer that is designed for

measuring spherical surfaces has insufficient dynamic range to measure the as-fabricated

surface because the departure between the spherical reference wavefront and the test

wavefront reflected off the surface of the mirror is too great. However, if the test

wavefront is manipulated to null or partially null each aberration type present in the

mirror, the departure between the test and measurement wavefronts can be minimized

and brought within the dynamic range of the interferometer. This chapter presents a

method for measuring concave and convex φ-polynomial surfaces by utilizing a series of

adaptive subsystems that each null a particular aberration type present in the departure of

the freeform surface.

6.1 Concave Surface Metrology

As a demonstration of a realizable null configuration for a concave optical surface, the

secondary mirror of the optical system designed in Chapter 5 will be used as an example.

This mirror was diamond turned by II-VI Infrared, as were the other two mirror surfaces,

in a copper substrate with a gold protective coating. For reference, the sag of the

secondary mirror surface of the three mirror design is shown in Figure 6-1 (a-c) where it

is evaluated with different Zernike components removed from the base sag. In

Figure 6-1 (a), the sag is evaluated with the piston, power, and tilt Zernike contributions

removed so that the dominant astigmatic contribution present in the surface can be seen.

When the astigmatism is removed from the surface sag, shown in Figure 6-1 (b), the

110

remaining sag components are observed. An asymmetry is seen in the sag that results

from the comatic departure present in the surface. If the spherical aberration, the next

most dominant contribution, is removed from the surface sag, shown in Figure 6-1 (c),

the comatic departure of the surface is more readily visible.

(a) (b) (c)

Sag of Secondary minus Piston/Power/Tilt in µm

Sag of Secondary minus Piston/Power/Tilt/Astig. in µm

Sag of Secondary minus Piston/Power/Tilt/Astig./Spher. in µm

Figure 6-1. (a) Sag of the secondary mirror surface with the piston, power, and tilt Zernike components removed revealing the astigmatic contribution of the surface, (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed.

The goal of the interferometric null is to systematically subtract the spherical

aberration, astigmatism, and coma present in the secondary mirror surface. The first step

in designing the null configuration is to select either a planar or spherical reference

wavefront out of the interferometer. For this particular design, a planar wavefront,

translating to a flat reference surface at the output of the interferometer, is chosen

because the alignment of the null to the interferometer will be less critical since it can lie

anywhere within the aperture of the interferometer/transmission flat. From the output of

the interferometer, the aberration terms can be nulled in multiple configurations. For this

design, the spherical aberration component is first nulled by the use of a refractive Offner

null [76]. Next, the astigmatic component is removed by tilting the test surface. Lastly,

the residual comatic and higher order terms are nulled by adding their opposite departure

on an adaptive mirror that also acts as retro-reflector to send the light back towards the

111

measurement interferometer [77]. Together these three components form a configuration

that enables the optical surface to be measured with a conventional interferometer. In the

section below, the first order design of the null components is described and finally the

entire optimized, null system is presented.

6.1.1 First Order Design

6.1.1.1 Spherical Null

One of the common methods for creating a spherical null is to implement a refractive

Offner null lens consisting of two refractive elements as shown in Figure 6-2. The first

element focuses the planar wavefront from the interferometer in such a manner that the

exiting wavefront has the same amount of spherical aberration present in the test mirror

so that the beam incident the mirror is normal to the surface. The second element is a

field lens placed at or near the focus of the first null lens and it images the pupil of the

null lens to the pupil of the test mirror as shown by the blue dashed ray in Figure 6-2.

Conjugating the pupil between the null lens and the test surface ensures that higher order

aberrations are not generated as the beam propagates through the null system.

Test MirrorField LensNull Lens

fnull zmirror

ynull

ymirror

Figure 6-2. First order layout of the Offner null to compensate spherical aberration. The rays in red show the illumination path for the testing wavefront whereas the rays in blue show the imaging path for the pupils of the Offner null.

The focal lengths of the two lenses depend on the first order parameters of the testing

configuration. For this mirror the diameter of the pupil located at the null lens, nully , is

selected to be 45 mm so that the lens is not overly difficult to fabricate. In addition, the

112

region of interest on the mirror, mirrory , is 70 mm and to keep the overall length of the null

system small, the distance between the focus of the null lens and the test mirror, mirrorz , is

chosen to be at or near the radius of curvature of the mirror under test, which for this

system has been chosen to be 367.5 mm. Based on these parameters, the focal length of

the null lens, nullf , is computed using the magnification as,

,mirrornull null

mirror

zf y

y

=

(6.1)

and yields for the parameters described above, a null lens focal length of 234.6 mm. From

the null lens focal length, the focal length of the field lens is computed from the thin lens

equation as,

1

1 1 ,fieldmirror null

fz f

−

= −

(6.2)

and yields a field lens focal length of 143.2 mm.

Now that the first order parameters of the Offner null have been computed, the next

step is to select the curvatures for the null lens that will yield the correct amount of

spherical aberration to create a null for the mirror under test. Since the null lens and

mirror are conjugate to one another, the Fringe Zernike spherical contribution on the

mirror surface, 9Mirrorz , is related to the required null lens spherical wave aberration, 040

NullW ,

by taking the opposite transmitted wavefront aberration of the mirror calculated as,

040 96 ( 1 1).Null MirrorW z= − − − (6.3)

where the refractive index of the mirror is assumed to be -1. For the 2.5 µm of Fringe

Zernike spherical present in the test mirror, 30 µm of spherical wave aberration must be

created by the null lens. From the required wave aberration of the null lens, the equation

113

for spherical aberration of a thin lens at the stop surface [55] is rearranged to compute the

required shape factor, Nullβ , for the null lens and is given by,

( ) ( )1/2

22 232 1

040 42 1

2 1 132,

2 1 2 2

Null NullNull Null

Null Null NullNull

n n nfR R n nWn n n nR R y

β − − + = = + − + + − + +−

(6.4)

where 1NullR and 2

NullR are the front and back radius of curvature of the null lens and n is the

refractive index of the lens. For ease of fabrication, NBK7 glass is chosen, which has a

refractive index of 1.515 at interferometer testing wavelength of 632.8nm. Substituting

all known parameters in Eq. (6.4) leads to a shape factor of roughly 2.81, which indicates

a strong meniscus for the null lens. Finally using the relation of the curvature of the null

lens to the focal length and shape factor, the front and back radius of curvature is

computed as,

( )( )1/2

2 1,

1Null f n

Rβ

−=

± (6.5)

where after substitution, the two radii come to 62.29 mm and 128.57 mm, respectively.

With all the first order parameters of the Offner null calculated, the next step is to

calculate the required tilt angle of the test mirror to null astigmatism.

6.1.1.2 Astigmatic Null

When a spherical mirror is operated off-axis at or near the center of curvature, the

dominant aberration is third order astigmatism. The astigmatism is minimized by adding

a toroidal shape to the mirror. The principal radii of curvature, xR and yR , that determine

the ideal toroidal mirror to minimize astigmatism are found by the Coddington equations

[78] that are expressed for a mirror as,

( )

1 1 2' costT T R i+ = (6.6)

and

114

( )2cos1 1 ,' x

iS S R

+ = (6.7)

where T and S are the distances from the tangential (T) and sagittal (S) astigmatic focal

surfaces of the object to the mirror, T’ and S’ are the distances from the mirror to the

astigmatic focal surfaces of the image, and i is the angle of the mirror with respect to the

optical axis in the YZ plane. If the object and image are to be anastigmatic, T must be

equivalent to S as well as T’ must also be equal to S’. In this special case, Eq. (6.6)

and (6.7) reduce to,

( )2cos ,x

y

Ri

R= (6.8)

where x yR R< for there to be a valid mirror angle, i . For the case where y xR R< , the

mirror angle, i , must be re-defined in the XZ plane, so that Eq. (6.8) becomes,

( )2cos .y

x

Ri

R= (6.9)

For the mirror of interest, a toroidal shape has been intentionally polished into the

surface by prescribing some combination of primary and secondary Zernike astigmatism.

Following Eq. (6.8), in order to null the toroidal shape, the mirror must be tilted at an

angle where the angle depends on the principal radii of curvature. The principal radii of

curvature are derived from the Zernike terms present in the mirror surface by following a

method proposed by Schwiegerling et al. [79]. In this method, the sag of the optical

surface with the Fringe Zernike overlay is approximated as parabolic, so that the sag

along the x-direction is written as,

2 2 2 22 2

4 8 5 122 6 3 ,2 2x

x B N N N N

sag z z z zR R R R R Rρ ρ ρ ρ ρ ρ

= = + − + −

(6.10)

and the sag in the y-direction is written as,

javascript:searchAuthor('Schwiegerling,%20J')

115

2 2 2 22 2

4 8 5 122 6 3 ,2 2y

y B N N N N

sag z z z zR R R R R Rρ ρ ρ ρ ρ ρ

= = + − − +

(6.11)

where BR is the base radius of curvature of the mirror, ρ is the radial coordinate, NR is the

normalizing radius of Fringe Zernike overlay, and 4,8,5,12z are the Fringe Zernike

coefficients for power, spherical aberration, astigmatism, and oblique spherical

aberration. By simplifying and manipulating Eq. (6.10) and (6.11), xR and yR are

computed as,

2

/ 2

4 8 5 12

.2 2 6

2

Nx y

N

B

RR

R z z z zR

=

+ − ±

(6.12)

After substituting the prescription parameters of the mirror in Eq. (6.12), xR and yR are

calculated to be -394.2 mm and -371.4 mm leading to a tilt angle from Eq. (6.8) of

13.91°. More specifically, if the mirror under test is tilted at 13.91° then the astigmatic

contribution from the mirror will be nulled. However, an obstacle to overcome with a

tilted geometry is that the reflected wavefront from the mirror will no longer be reflected

back on itself and requires the use of an additional element to return the test wavefront

back to the interferometer. In this case, a deformable mirror (DM) that is nominally flat

but can be deformed into a wide variety of shapes is employed. It provides

retro-reflection without inversion and also nulls the residual coma and any higher order

aberrations present in the test wavefront.

6.1.1.3 Comatic and Higher Order Null

In order to use the quasi-flat DM, the wavefront reflected off the test mirror must be

collimated with the use of a collimating lens. In addition, the DM with a clear aperture of

15 mm must also be conjugate to the test mirror. This configuration is diagramed in

Figure 6-3 where the illumination path from the interferometer is shown in red and the

116

imaging path between the test mirror and DM is shown in blue. As can be seen from

Figure 6-3, the collimating lens is performing two first order imaging functions.

Test Mirror Collimating Lens

fDMz'mirror

yDM

ymirror

Deformable Mirror

z‘DM

Figure 6-3. First order layout of the comatic and higher order null. A collimating lens is uses to couple the wavefront to an actuated, deformable membrane mirror. The rays in red show the illumination path for the testing wavefront whereas the rays in blue show the imaging path for the pupils of the comatic null.

With the constraints laid out above, it is possible to derive the first order parameters

for this section of the interferometric null. The focal length of the collimating lens is

found using the magnification between the test mirror and DM, calculated as,

1

1 2' ,DM DMDM mirror

mirror mirror B mirror

y yf zy z R y

− −

= = +

(6.13)

where 'mirrorz is the image distance of the test mirror that is related to the object distance of

the mirror mirrorz by the thin lens imaging equation. Based on the values for the distances

and sizes of the optics, the focal length of the collimating lens is found to be roughly 81.7

mm. The other parameter that needs to be calculated is the distance between the

collimating lens and the DM, 'DMz , ensuring both the test mirror and DM are conjugate to

one another. Using the thin lens equation, this distance is calculated as,

11

1 1 1 1' ,'DM

DM DM DM mirror DM

zf z f z f

−−

= + = + +

(6.14)

and yields for the parameters above a distance of 99.2 mm. The comatic and higher order

departure that needs to be applied to the DM is negative two times the departure present

on the test mirror since the test wavefront is reflected off the mirror twice. Also, because

of the imaging condition between the test mirror and DM, the Zernike contribution on the

117

DM must be rotated by 180°. With all paraxial parameters for the inteferometric null now

established, the null configuration can now be optimized using commercially available

lens design software to provide a final thick lens solution.

6.1.2 Optimization of the Interferometric Null System

The paraxial solution described in the section above creates a starting point for further

optimization. The end goal for optimization is to produce a double pass, thick lens

solution that provides a null or quasi-null wavefront exiting the interferometric system.

Using CODE V, user defined constraints are written for nulling the Fringe Zernike

spherical aberration, astigmatism, coma, and any higher order aberration terms while

maintaining conjugates between nulling components. As for the parameters allowed to

vary during optimization, the radii of the null lens are roughly set by the first and third

order constraints but are allowed to vary to account for variations from the lens thickness;

however, the focal length is kept fixed to its first order value. For the field lens, the lens

has been chosen to be NBK7. Since the size of the beam footprint at the field lens is

small, it introduces little spherical aberration, so for ease of fabrication, positioning, and

alignment, its shape is chosen to be bi-convex. The focal length is allowed to vary from

its paraxial value because the field lens parameters can be used to minimize higher order

spherical aberration. The collimating lens near the DM is chosen to be a high index

material, SF6, so it introduces less spherical aberration. For ease of fabrication and to

introduce as little spherical aberration as possible, the shape is chosen to be plano-convex

as it is near the shape factor for minimum spherical aberration. The focal length is kept

roughly the same as its first order value to ensure that the size of the beam on the DM

does not exceed 15 mm. The mirror tilt about the y-axis is allowed to vary from its

118

paraxial value and the Fringe Zernike tilt (Z3), coma (Z8), trefoil (Z11), oblique spherical

aberration (Z12), and higher order coma (Z14) are allowed to vary at the DM surface.

The final, optimized system is shown in the XZ plane in Figure 6-4. As can be seen

from the figure, an aspect ratio of at least 7:1 is selected for each lens to aid in

manufacturability. The overall package of the interferometric null is roughly

600 mm x 225 mm. The theoretical wavefront exiting the interferometric null is shown in

Figure 6-5 (a) before the DM is active and in Figure 6-5 (b) after the comatic and higher

order null has been applied. In Figure 6-5 (a), the astigmatism and spherical aberration

have been nulled from the wavefront but there is still a departure of 38λ PV at the testing

wavelength of 632.8 nm in the double pass wavefront. After the DM has been applied,

the residual present is on the order of 4λ PV or 0.46λ RMS. At the operating wavelength

of around 10 µm, the residual in the double pass null wavefront corresponds to 0.25λ PV

and 0.03λ RMS. The residual is non-zero because of the tilt angle required to null the

astigmatic part of the surface. With a tilted geometry, the pupils cannot be perfectly

conjugate to one another since a tilted object must be imaged to a tilted image per the

Scheimpflug principle [80]. Moreover, the beam incident on the test mirror is slightly

elliptical and will alter the Zernike composition of the wavefront. The residual in the

exiting wavefront can be compensated either in hardware or software by using the DM to

subtract the residual or simulating a software null in CODE V to subtract from the

measured data.

119

75.00 MM

Output of Interferometer

Offner Null

DeformableMirror

Mirror UnderTest

Collimating Lens

Figure 6-4. Layout of the optimized interferometric null for the concave, secondary mirror to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an Offner null to null spherical aberration, a tilted geometry to null astigmatism, and a retro-reflecting DM to null coma and any higher order aberration terms.

0.0λ

0.5λ

1.0λ

0.0λ

0.5λ

1.0λ

(a) (b)0

1

0

0

1

0

Figure 6-5. Simulation of the double pass wavefront exiting the concave interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 632.8 nm.

6.1.3 Experimental Setup of Interferometric Null System

The first step in assembling the interferometric null system is to create the comatic and

higher order null on the DM surface. The DM selected for this project is mirao™ 52-e, a

fifty two actuator reflective membrane mirror, from Imagine Eyes. On the underside of

the membrane surface, small magnets are affixed at each actuator site. The actuation of

the surface is achieved through variation of the voltage in a small coil that creates a

magnetic field at the actuator site. The magnetic field influences the magnet either

120

pushing or pulling the membrane surface depending on the applied voltage. This type of

DM is capable of achieving large deformations of the surface and is well suited for

creating the comatic null for the interferometric null system.

The system for setting this shape is shown in Figure 6-6 (a) where a 3:1 afocal

telescope relays collimated light from a 632.8 nm Zygo laser interferometer through a

cube beamsplitter and onto the DM. The wavefront then reflects off the surface of the

mirror and half the light is directed back to the interferometer and the other half is

directed through a 4:1 afocal telescope that images the DM surface onto a 4.8 x 3.6 mm

Shack-Hartmann wavefront sensor. Using the wavefront sensor to interrogate the DM

surface, it is operated in a closed loop configuration where the influence functions of the

actuators on the DM are known a priori and they are iteratively adjusted in software to

converge to a desired shape. The optimized Fringe Zernike coefficients of the comatic

null from the lens design are the target for the closed loop optimization. The laser

interferometer is used as an additional aid to measure the shape of the comatic null as the

DM is adjusted to its final form. The assembled optical system for the DM calibration is

shown in Figure 6-6 (b). Any aberrations induced from the afocal telescopes can be

subtracted from the measured wavefront by first replacing the DM with a flat reference

mirror of high quality and using this measurement as a baseline.

121

Deformable Mirror

Wavefront Sensor

3:1 Afocal Relay

4:1 Afocal Relay

DeformableMirror

4F Afocal Relay (3:1)

4F AfocalRelay (4:1)

Shack-Hartmann Wavefront Sensor

Zygo DynaFizInterferometer

(a)(b)

Figure 6-6. (a) Layout of the setup to create the comatic and higher order null on the DM surface. The setup uses a Shack-Hartmann wavefront sensor to run a closed loop optimization to set the shape of the DM. The DM is also interrogated with a Fizeau interferometer. (b) The setup realized in the laboratory.

The shape of the comatic null measured by the interferometer is shown in

Figure 6-7 (a). The dynamic range of the DM is capable of creating this large departure

null with a surface PV of roughly 12 µm. However, when the theoretical shape is

subtracted from the actual comatic null, there is a large residual as displayed in

Figure 6-7 (b). The residual is on the order of 2 µm PV and is mostly composed of higher

order deformations that result from the local deformation at or near the actuator sites. The

voltages of the actuators are near their maximum for this surface shape so some residual

is to be expected. Since the deformable surface has been measured, it can be applied as a

hitmap in CODE V and the residual wavefront can be simulated to create a software null

in CODE V to subtract from the measured data.

122

(a) (b)

+5.565µm

-6.310µm

-6.0

+0.0

-4.0

-2.0

+2.0

+4.0

+1.352µm

-0.588µm

+1.2

+0.3

-0.3

+0.0

+0.6

+0.9

Figure 6-7. (a) DM comatic null surface measured by the interferometer and (b) the residual after the theoretical shape has been subtracted. The residual has a PV error of 2 µm PV.

With the correct shape set on the DM, the rest of the interferometric null is assembled.

The three lens components, the Offner null lens, field lens, and collimating lens, have

been fabricated by Optimax Systems. Each lens is coated with a V-coating that ensures

back reflections to the interferometer are minimized. The optics are mounted with

commercially available optomechanical components that provide four DOF movement

(x/y decenter and tip/tilt). Moreover, each optical component sits on a kinematic base that

provides stable six DOF positioning. With kinematic couplings, each element can be

removed for the alignment of subsequent components and after alignment, the element

can be replaced repeatably.

The tilted geometry for the test mirror, collimating lens, and DM is created by using a

precision rotation stage with a rail attached to the tabletop of the stage. Since the mirror is

to be tilted at α, the optical axis or rail must be rotated by 2α. The mirror, whose vertex

lies at the axis of rotation of the rotation stage, also rotates by 2α and must be

counter-rotated by α. The counter-rotation of the test mirror is made possible by the use

of a custom kinematic base known as a Kelvin clamp. The two part base, shown in

123

Figure 6-8, has a bottom plate with three conical cups milled about a radius separated by

120°. In these cups sit three spheres. On the top plate, sets of 120° spaced vee grooves are

milled into the plate. Each sphere of the bottom plate sits in one vee groove, constraining

two DOFs. In total, all six DOFs are uniquely determined. Each set of grooves defines

one index (rotation) of the top plate. For the test mirror measurement there are three vee

sets milled into the plate: 0, α, and 2α. Using the specialized base, the mirror under test

can first be aligned perpendicular to the optical axis in the zero index position. When the

stage is rotated by 2α, the base can be indexed to the α position, bisecting the rotation

angle of the stage. Since the stage is kinematic, the positioning will be repeatable for

multiple iterations of positioning.

Figure 6-8. Custom designed kinematic indexing mount for counter rotating the test mirror during alignment of the interferometric null. The plates are machined in 304 stainless steel and employ three hardened 440C stainless steel 7/16” spheres.

The assembled and aligned interferometric null configuration is displayed in

Figure 6-9. The interferometric null is coupled to a phase shifting Zygo DynaFiz

interferometer with a reference transmission flat. The optical axis of the null (shown in

red) is defined by a line that interests the vertex of the test mirror and is normal to the

transmission flat of the interferometer. Because the vertex of the mirror is not readily

accessible, a precision external target that couples to the mechanical alignment features

124

of the test mirror is used to locate the vertex. Using this target and the transmission flat,

an alignment telescope is aligned to these datums and it defines the optical axis for

subsequent alignment of the other optical components. The alignment telescope sits

behind the test mirror in the interferometric null. During alignment the test mirror is

removed to provide an unobstructed view of the other components. In this fashion the

Offner null is easily aligned. In order to align the collimating lens and DM, the rotation

stage and rail are first aligned to the optical axis defined by the alignment telescope. In

this case, the Offner null components are removed. Once the components have been

aligned, the rotation stage is set to its angle and the Offner null components are replaced.

Offner Null

DeformableMirror

SecondaryMirror

Zygo DynaFizInterferometer

Figure 6-9. The interferometric null configuration realized in the laboratory. A rotation stage with a rail affixed is used to create the tilted geometry. The secondary mirror is measured using a Zygo Fizeau-type interferometer.

6.1.4 Experimental Results

With the interferometric null aligned, the Zygo interferometer is used to acquire an

interferogram of the optical surface. One important consideration for the raw

interferogram acquired by the interferometer is the scaling factor, or the relationship

between the fringe pattern on the wavefront and surface error on the mirror. A typical

125

Fizeau measurement is double pass, resulting in one fringe being equivalent to 2λ surface

error. For this test configuration, the test wavefront reflects off the tilted mirror twice, so,

in this case, one fringe on the wavefront is estimated as

( )cos4λ α (6.15)

on the surface, where α is the angle of incidence on the mirror with respect to the optical

axis. The cosine term is included to account for the projection of surface height from the

tilted plane back to a normal condition. Moreover, the raw interferogram acquired by the

interferometer is rotated 180° from the actual surface of the mirror since the light passes

through an intermediate focus in the Offner null. Taking these items into consideration,

the initial surface error of the test mirror surface is shown in Figure 6-10 (a-b) where the

surface error maps are presented in microns. In Figure 6-10 (a), the surface error is

presented with the residual power present in the surface. The PV error is 3.821 µm and

0.819 µm RMS. When the dominant power is subtracted from the measurement,

Figure 6-10 (b), the PV error goes to 2.025 µm and 0.235 µm RMS. With the power

subtracted, the less dominant features of the residual can be discerned and these errors

resemble the residual of the comatic null from its theoretical state presented in

Figure 6-10 (b).

In order to observe the errors of the mirror surface and not the errors of the comatic

null, a software null is created in CODE V. The software null simulates the wavefront at

the exit pupil and it includes the effects of the residual aberrations present in the testing

setup and incorporates a hitmap of the comatic null. The surface error maps after

subtracting the software null from the measured data are depicted in Figure 6-10 (c-d).

When the power is present in the surface error, Figure 6-10 (c), the PV error is now

126

3.230 µm and 0.798 µm RMS. After the power is subtracted, Figure 6-10 (d), the PV

error is reduced to 1.140 µm and 0.156 µm RMS. At a wavelength of 10 µm, the center

operating wavelength of the optical system, the PV error is 0.114λ and 0.016λ RMS.

Therefore, for an LWIR application, the surface is almost a tenth wave. In evaluating the

features in the surface error, it can be seen that, while small, the error is mostly

astigmatism that may be a residual from the mounting process during fabrication.

(c) (d)

(a) (b)

-0.559µm

+0.580µm

+0.0

-0.4

-0.2

+0.2

+0.4

+2.573µm

-1.249µm

-1.0

+0.0

+1.0

+2.0

-1.301µm

+0.5

-0.5

+0.0

+1.0

+1.5

-1.0

+1.929µm

-0.821µm

+0.3

-0.3

+0.0

+0.6

+0.9

-0.6

+1.203µm

Figure 6-10. (a) Initial surface error map of the test mirror with power and (b) with the power removed. The PV error of the surface residual before and after the power is removed is 3.821 µm and 2.025 µm, respectively. (c) Final surface error map of the test mirror after the software null has been subtracted (c) before and (d) after the power has been removed. In this case, the PV error is 3.230 µm before and 1.140 µm after the power has been removed.

127

6.2 Convex Surface Metrology

As a demonstration of a realizable null configuration for a convex optical surface, the

primary mirror of the optical system designed in Chapter 5 will be used as an example.

For reference the sag of the primary mirror surface of the three mirror design is shown in

Figure 6-11 (a-c) where it is evaluated with different Zernike components removed from

the base sag. In Figure 6-11 (a), the sag is evaluated with the piston, power, and tilt

Zernike contributions removed. For this particular surface, the dominant aberration

component is coma. This fact can more readily be seen by additionally subtracting the

astigmatism from the surface sag, as shown in Figure 6-11 (b), and also the spherical

aberration, as shown in Figure 6-11 (c). When these two aberration components are

additionally removed, there is little change in the surface sag residual and based on the

characteristic asymmetric behavior, the residual is recognized as coma.

(a) (b) (c)

Sag of Primary minus Piston/Power/Tilt in µm

Sag of Primary minus Piston/Power/Tilt/Astig. in µm

Sag of Primary minus Piston/Power/Tilt/Astig./Spher. in µm

Figure 6-11. (a) Sag of the primary mirror surface with the piston, power, and tilt Zernike components removed, (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed. With the piston, power, tilt, astigmatism, and spherical components removed, the asymmetry induced from the coma being added into the surface can be seen.

Similar to the concave null configuration, the aberration components will be nulled

with a series of subsystems starting with a planar wavefront, translating to a flat reference

surface at the output of the interferometer. From the output of the interferometer, the

aberration terms can be nulled in multiple configurations. For this design, the spherical

128

aberration component is first nulled by the use of an afocal, refractive Offner null,

consisting of two refractive elements, one of which is a null lens that introduces the

opposite amount of spherical aberration present in the mirror under test and the other is a

field lens that collimates the beam and conjugates the null lens to the mirror under test.

Next, the astigmatic component is removed by operating the mirror off-axis, or tilting the

mirror. Unlike for the case of a concave test mirror where the wavefront converges to a

point, the beam exiting the tilted, convex mirror will diverge so an additional auxiliary

optic is needed to focus the wavefront. In this case, a large concave mirror is used similar

to a Hindle sphere [81]. This additional mirror adds another DOF so that its tilt is used to

remove the residual comatic contribution present in the wavefront. Finally, similar to the

concave null configuration, the residual higher order terms are nulled by adding their

opposite departure on a DM that also acts as a reimaging retro-reflector to send the light

back towards the measurement interferometer. In order to couple the wavefront to the

quasi-flat DM, the wavefront is collimated with the use of an additional lens. Together

these three components form a configuration that allows the optical surface to be

measured with a conventional interferometer.

The sizing of the various optical components constrains the layout of the

interferometric null. The region of interest on the test mirror is 45 mm in diameter while

the output from the Zygo interferometer is 101.6 mm. To keep the components of the

refractive null readily commercially available, a 1:1 afocal Offner null is selected so only

45 mm of the 101.6 mm aperture is used. After the wavefront passes through the null lens

and reflects off the test mirror, its beam size will grow rapidly as the wavefront is

diverging. As a result, the distance between the test mirror and auxiliary mirror should be

129

minimized to keep the beam footprint on the auxiliary mirror small. The auxiliary mirror

clear aperture and its beam footprint are constrained to 150 mm diameter maximum,

which is the largest mirror size that is readily commercially available at relatively fast

focal ratios. The focal ratio must be fast to keep the length of the null small. An F/1,

150 mm COTS sphere is used to meet these constraints. The distance between the

auxiliary sphere and the mirror under test also impacts how the wavefront traverses

through the entire null configuration. The beam exiting the afocal Offner null must pass

by the auxiliary sphere and the sphere should not obscure the incoming beam. Similarly,

after the beam has reflected off the auxiliary sphere, it passes by the mirror under test and

the mirror should not obscure the beam either. The distance between the test and

auxiliary mirror is set so that when the two mirrors are tilted to null both astigmatism and

coma, they do not obscure any part of the beam. Lastly, the DM has a 15 mm clear

aperture so the collimating lens after the auxiliary sphere must be arranged to meet this

constraint. With all these constraints in mind, a solution is optimized in CODE V with

user defined constraints to null the Fringe Zernike spherical aberration, astigmatism,

coma, and any higher order aberration terms while ensuring that the clear aperture

limitations are met, the beam is not obscured, and the imaging conjugates between

components are maintained.

The final, optimized system is shown in the YZ plane in Figure 6-12. The overall

package of the interferometric null is roughly 700 mm x 350 mm. All the lens

components are plano-convex making them readily commercially available. The

theoretical interferogram exiting the interferometric null is shown in Figure 6-13 (a)

before the DM is active and in Figure 6-13 (b) after the higher order null has been

130

applied. In Figure 6-13 (a), the spherical aberration, astigmatism, and coma have been

nulled from the wavefront but there is still about 45λ PV of departure present in the

double pass wavefront at the testing wavelength of 632.8 nm where most of the departure

resembles that of Zernike trefoil. The Zernike trefoil that is present in the wavefront is

not all from the amount present in the mirror surface. The fast auxiliary sphere is tilted at

a fairly large angle so the beam footprint has become elliptical. The elliptical beam on the

auxiliary sphere results in the generation of elliptical coma also known as Zernike trefoil.

This residual that is a result of the testing configuration is subtracted at the DM null.

After the higher order null has been applied, the residual is on the order of 4λ PV or 1.0λ

RMS. At the operating wavelength of around 10 µm, the residual in the double pass null

wavefront corresponds to 0.25λ PV and 0.10λ RMS. The residual is non-zero for several

reasons. The ideal shape factor for the null lens in the spherical null is near plano-convex

but not perfectly plano-convex. To aid in commercial availability, the lens was forced to

be plano-convex at the cost of some residual spherical aberration. Moreover, the beam

incident on the test mirror is slightly elliptical and will alter the Zernike composition of

the wavefront. The residual in the exiting wavefront can be compensated either in

hardware or software by using the DM to subtract the residual or by simulating a

software null in the lens design software to subtract from the measured data.

131

Afocal Offner Null

DeformableMirror

Mirror UnderTest

Auxiliary Sphere

Output of Interferometer

Figure 6-12. Layout of the optimized interferometric null for the convex, Primary mirror to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an afocal Offner null to null spherical aberration, a tilted geometry to null astigmatism and coma, and a retro-reflecting DM to null any higher order aberration terms.

0.0λ

0.5λ

1.0λ

a e

0

1

0

0.0λ

0.5λ

1.0λ

(a) (b)

Figure 6-13 Simulation of the double pass wavefront exiting the convex interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 632.8 nm.

132

Chapter 7. Assembly of an Optical System with φ-Polynomial Optical Surfaces

The optical system described in Chapter 5 opens a new space for optical design where a

freeform overlay may be utilized on an optical surface to enable a non-inline, tilted

geometry of the overall optical system. New fabrication and assembly challenges arise

when building an optical system of this type because conventional methods of fabrication

must be abandoned to enable these new optical design forms. Chapter 6 showed how

interferometric metrology can be configured to measure this new class of optical

surfaces. When it comes to assembling an optical system of this type, the mounting and

fiducialization of the optical surfaces becomes critical. In particular, the optical surfaces

must be oriented in a particular manner with respect to the optical housing and

constrained in all six DOFs because of their nonsymmetric shape. In this chapter, the

mechanical design of the optical design in Chapter 5 is presented and the sensitivity of

the design to assembly alignment residuals is evaluated. In addition to a sensitivity

analysis, the mounting structure is evaluated for stray light and the problems are

mitigated through baffles and surface preparation. Lastly, the assembled optical system

and its optical performance are presented.

7.1 Mechanical Design

The housing structure of the three mirror system is displayed in Figure 7-1 (a) and was

developed in collaboration with II-VI Infrared. It is constructed from an aluminum block

with the faces of the block machined to the required tilt angle for each mirror. The

mirrors are designed to be back surface mounted so an adaptor plate, as shown in

Figure 7-1 (b), is used to couple the mirror to its corresponding face. Steel dowel pins are

used to position the mirror correctly within the mechanical housing. These dowel pins

133

provide a good mechanical datum to the optical surface because during the fabrication

process they register the optical surface to a tooling plate with a reference flat that is

trued to the diamond turning machine. In total there are two mechanical connections for

each mirror subassembly. The first connection is between the optical surface and the

adaptor plate and the second connection is between the adaptor plate and the housing

face. At each connection three diamond turned raised pads are used as the mounting

interface to provide a quasi-kinematic condition when the two surfaces are mated

together. In total, the pin connections constrain the x decenter, y decenter, and clocking

angle of the optical surface with the pads and screws providing preload thus constraining

any in-plane movement. Paths are bored through the housing and are sized to ensure the

light passes through the housing without vignetting.

Mirror

Dowel Pins

Adaptor Plate

(a) (b)

Raised Pads

Figure 7-1. (a) Layout of the housing structure of the three mirror freeform optical system and (b) exploded view of the tertiary mirror subassembly consisting of the optical mirror surface, adaptor plate, and steel dowel pins for alignment.

134

7.1.1 Sensitivity Analysis

As is the case with any piece of hardware, there is some tolerance on how well the

mirrors can be positioned in the housing relative to their nominal value. The key is to

ensure that within the manufacturing tolerances, the as-built optical system remains

diffraction limited. In addition to hardware tolerances, the assembly method of the optical

system may impact the manufacturing tolerances. If the optical system is to be passively

aligned, that is, no adjustments are made with the exception of focus, the manufacturing

tolerances will have to be tighter. If the system is to be actively aligned, that is, a

compensator is used to restore the optical performance during assembly, the assessment

of the performance during alignment is important and the mechanical complexity of the

housing will have to increase because a DOF must now be made adjustable. In this

section, both approaches are explored.

7.1.1.1 Passive Alignment

In a passive alignment approach, the three mirrors must be constrained in their x/y

translation, tip/tilt, and clocking angle. There are also two vertex spacings that must be

held between the three mirrors. The spacing between the tertiary mirror and focal plane is

used as a focus adjustment after assembly. The focus compensation is performed by

shimming the detector in 12.5 µm steps and determining through an optical assessment

the shim that provides the best performance. The detector must also be held in tip and tilt

relative to the housing and there is a separate tip and tilt tolerance for the focal plane

relative to the mounting fixtures on the detector. In total, there are 21 positioning

tolerances to consider for this assembly.

To check the sensitivity of the optical housing to manufacturing errors, each tolerance

is perturbed by its expected error value and the change in the RMS WFE after focus

135

compensation is recorded for several field points. Once each tolerance and its resulting

change in performance is computed, the total change in performance for each field is

computed as the root sum square (RSS) of all the tolerances. The results of this analysis

are displayed in Table 7-1 where the change in RMS WFE is displayed for each tolerance

at two field points: on-axis (0°, 0°) and the most sensitive field (4°, 3°). For this analysis

the x decenter, y decenter, and despace of the mirrors is assumed to be ±50 µm, the tip

and tilt of the mirrors and detector (α and β tilt) is assumed to be ±1 arc min or ±0.017°,

and the clocking angle (γ rotation) is assumed to be larger at ±0.1°. From the mechanical

drawing of the detector, the focal plane tip and tilt tolerances are calculated to be roughly

±0.56°. With all the tolerances considered, the as-built RMS WFE is found to be roughly

0.056λ for the (0°, 0°) field and 0.060λ for the (4°, 3°) field, both of which are within the

diffraction limit of 0.07λ. Looking at the tolerances on a term by term basis, the primary

contributors to the overall loss in performance are the tilt tolerances on the secondary

mirror, tertiary mirror, and focal plane. Therefore, the initial tolerances selected are

sufficient for meeting the performance specification; however, they do not consider how

the optical components will actually be mated together. Since the components are to be

assembled with pin connections, an alternative analysis would be to model the sensitivity

of the connections directly. The position of the dowel pin holes affects the x decenter,

y decenter, and clocking angle of the mirror surfaces.

136

Table 7-1. Summary of the initial sensitivity analysis of the three mirror optical system. For each tolerance, the change in RMS WFE from nominal is computed and the RSS is compiled to provide the as-built RMS WFE. The RMS WFE is terms of waves at the central operating wavelength of 10 µm.

Tolerance Δ RMS WFE

(waves) Field: (0°,0°)

Δ RMS WFE (waves)

Field: (4°,3°) Pri. Mirror x decenter ±50 µm 0.002 0.002 y decenter ±50 µm 0.006 0.005 α tilt ±0.017° 0.003 0.001 β tilt ±0.017° 0.001 0.001 γ rotation ±0.1° 0.000 0.000 Pri.-Sec. despace ±50 µm 0.001 0.001 Sec. Mirror x decenter ±50 µm 0.003 0.004 y decenter ±50 µm 0.010 0.006 α tilt ±0.017° 0.023 0.018 β tilt ±0.017° 0.014 0.015 γ rotation ±0.1° 0.003 0.004 Sec.-Ter. despace ±50 µm 0.000 0.000 Ter. Mirror x decenter ±50 µm 0.000 0.000 y decenter ±50 µm 0.003 0.003 α tilt ±0.017° 0.013 0.008 β tilt ±0.017° 0.006 0.007 γ rotation ±0.1° 0.031 0.031 Detector α tilt ±0.017° 0.000 0.000 β tilt ±0.017° 0.000 0.000 Focal Plane α tilt ±0.56° 0.000 0.015 β tilt ±0.56° 0.000 0.023

RSS 0.045 0.050 Nominal 0.011 0.010

Predicted As-Built 0.056 0.060

For each set of dowel pins, the diametrical true position of the two dowel pin holes

must be considered with respect to a reference datum. Specifically, the diametrical true

position defines a region in which the dowel pin hole must lie. Since it is a diametrical

137

tolerance zone, as the x decenter of the dowel pin hole increases, the y decenter must

decrease accordingly. Furthermore, if the top dowel pin hole is not collinear with the

bottom dowel pin hole, the mirror will be rotated.

For the analysis, an initial diametrical true position tolerance, φ , is selected for the

position of the top and bottom dowel pin holes. From this tolerance a random x decenter

of the hole is selected from a position within the tolerance zone, computed as

( )/ 2 ,2 2

T Bdecx RANDφ φ

= − + (7.1)

where /T Bdecx is the x decenter of either the top or bottom hole and RAND is a normally

distributed random number between 0 and 1. From the x decenter, the maximum possible

y decenter, /maxT B

decy , is calculated as

( )2

2/ /max .

2T B T B

dec decy xφ = −

(7.2)

From Eq. (7.2) a random y decenter of either the top or bottom hole is computed as

( )/ / /max max2 .T B T B T B

dec dec decy y RAND y= − + (7.3)

With the x and y decenter computed for both the top and bottom dowel pin holes, the total

x and y decenter of the set is computed as

,2

T Bdec dec

decx x

x+

= (7.4)

and

.2

T Bdec dec

decy y

y+

= (7.5)

Lastly, the clocking angle error of the set, γ , is computed as

1tan ,T Bdec dec

pin

x xd

γ − −

=

(7.6)

138

where pind is the spacing between the two pins. From Eq. (7.6) it can be seen that if the

pin spacing is increased, the clocking angle error of the set will decrease for the same

diametrical true position tolerance.

For each mirror there are two pin connections that yield four sets of dowel pin holes.

Therefore, the total mirror x decenter, y decenter, and clocking angle error is the

summation of the decenter and clocking angle tolerances of the four sets. These totals are

computed as

1 2 3 4 ,mirror pin pin pin pindec dec dec dec decx x x x x= + + + (7.7)

1 2 3 4 ,mirror pin pin pin pindec dec dec dec decy y y y y= + + + (7.8)

and

1 2 3 4 .mirror pin pin pin pindec dec dec decγ γ γ γ γ= + + + (7.9)

As an example, the quantities required for deriving an overall random x decenter,

y decenter, and clocking angle error are shown in Figure 7-2 for the tertiary mirror and

are tabulated for all the mirror surfaces in Table 7-2. The diametrical tolerance zone is

different depending on the mating interface. For example, the housing has the loosest

tolerance on the position of the dowel pin holes because it is difficult to machine a hole

into a tilted plane with a high level of accuracy. Also, while the tolerance on the position

of the dowel pin holes are the same for each mirror, the pin spacings, pind , are different

because the primary mirror is a different physical size than secondary and tertiary. As a

result, the clocking angle of primary mirror will be larger than the secondary and tertiary

mirrors because the pin spacing is the smallest for this mirror.

139

dplate/housing = 100 mm

dmirror/plate = 60 mm

A

B

φplate/housing=25 µm A B

φmirror/plate=25 µm A B

φmirror=12.5 µm A B

Figure 7-2. The tertiary mirror subassembly and values that determine its alignment, namely, the dowel pin hole position tolerances and their relative spacings.

Table 7-2. Summary of the quantities used to derive the tolerances for the Monte Carlo sensitivity analysis. The dowel pin hole position tolerances are used to derive the mirror x/y decenter and mirror clocking angle.

Item Value Dowel Pin Hole Position Tolerance (All Mirrors)

Mirror 12.50 µm Adaptor Plate to Mirror 25.00 µm Adaptor Plate to Housing 25.00 µm Housing to Adaptor Plate 62.50 µm

Dowel Pin Hole Spacing (Pri.) Mirror to Adaptor Plate 35 mm Adaptor Plate to Housing 60 mm

Dowel Pin Hole Spacing (Sec. and Ter.) Mirror to Adaptor Plate 60 mm Adaptor Plate to Housing 100 mm

From these tolerance ranges and the other tolerances mentioned in Table 7-1, a Monte

Carlo sensitivity analysis is performed. In this analysis, the DOFs of the system are

perturbed and the optical performance is computed. From multiple trials the probability

of the as-built optical system meeting a given performance metric may be determined.

Figure 7-3 displays the results of a simulation with 500 trials where the cumulative

probability of the as-built optical system meeting a given RMS WFE at the operating

wavelength of 10 µm is displayed for nine field points across the FOV. As to be expected

from the prior sensitivity analysis, the edge of field is most heavily impacted and the

140

change in performance is most likely dominated by the large focal plane tilt tolerance. If

a cumulative 95% is selected as the passing metric, all the field points lie below an

as-built RMS WFE of 0.06λ. These results are in good agreement with the prior analysis

where the as-built performance was around 0.06λ. From these two analyses, it can be

concluded that the optical system can be machined and assembled with standard machine

shop tolerances and the final system will remain diffraction limited, less than 0.07λ,

throughout the FOV.

0%

20%

40%

60%

80%

100%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Cum

ulat

ive P

erce

ntag

e

As-Built RMS Wavefront Error (waves at 10µm)

(+0°,+0°)

(-4°,+0°)

(+4°,+0°)

(-3°,+0°)

(+3°,+0°)

(+4°,+3°)

(+4°,-3°)

(-4°,+3°)

(-4°,-3°)

Figure 7-3. Cumulative probability as a function of as-built RMS WFE for the three mirror optical system over nine field points assuming only passive alignment.

7.1.1.2 Active Alignment

The two analyses performed in section 7.1.1.1 assumed that the optical system is to be

passively aligned where the as-built optical performance is determined by the as-built

manufacturing tolerances and a shim for focus compensation. If now some additional

DOFs are allowed to vary during the assembly process, it may be possible to improve the

as-built optical performance. Active alignment becomes more likely as the optical design

form is considered for shorter wavebands.

141

The previously shown sensitivity analyses use the overall RMS WFE as the

performance metric to assess the as-built optical performance; however, it does not

provide any information on which aberration contributions are limiting the performance

and how they vary throughout the FOV. If the DOFs are perturbed a known amount and

now the Zernike aberration contributions are monitored throughout the FOV using the

FFD, additional insight can be gathered on alignment strategies as well as which DOFs

are going to be best for compensators. From the results of the sensitivity analysis in

Table 7-1, it is seen that for the optical system, in general, there is a greater loss in

performance when the mirror components are tilted versus decentered, so an effective

compensator will be the mirror tilt. Figure 7-4 (a-d) displays the FFDs for Zernike

astigmatism (Z5/6) and Zernike coma (Z7/8) for the nominal optical system, shown in

Figure 7-4 (a), and for the case where the mirrors are individually tilted +0.1° in the YZ

plane, shown in Figure 7-4 (b-d). When each mirror is tilted, the primary aberration

component induced is field constant astigmatism. Some field constant coma is induced as

well; though, its magnitude is about ten times less than that of the astigmatism.

From this result, several conclusions can be drawn. First, since field constant

astigmatism is primarily induced when the telescope is misaligned, during active

alignment only one field point needs to be monitored to get a good representation of how

the other field points are behaving. Second, for the same tilt of the three mirrors, the

secondary is most sensitive to the perturbation. Consequently, this mirror may make the

most effective compensator because a small perturbation of the mirror will have a large

net effect on the overall system performance, thus requiring less mechanical movement

of the compensator, assuming the compensator has enough mechanical resolution. The

142

tilt of the focal plane is also a key compensator. It reduces the focus variation across the

FOV that will not be compensated by tilting the secondary mirror which primarily

compensates astigmatism. Having selected the two most effective compensators, the

Monte Carlo simulation is re-analyzed with the addition of the secondary mirror tilt and

focal plane tilt as compensators. Figure 7-5 displays the results of a simulation with 500

trials where the cumulative probability of the as-built optical system meeting a given

RMS WFE at the operating wavelength of 10 µm is displayed for nine field points across

the FOV. In comparing these results to those of Figure 7-3, it can be seen that when these

compensators are allowed to vary, the variance of the RMS WFE has decreased and now

all the field points lie below an as-built RMS WFE of 0.025λ if the cumulative 95% point

is used as the passing metric. In this case, the as-built optical performance is near

nominal so there is little degradation in performance when assembly tolerances are

considered with the secondary mirror tilt and focal plane tilt being used as compensators.

In this case, the tolerances on the components could be relaxed if any of the tolerances

were challenging to meet during fabrication or driving the cost of the components.

The sensitivity analyses shown above have not considered the irregularity of the

optical surfaces. In Chapter 6 where the one of the optical surfaces was measured

interferometrically, the predominant residual error in the surface was found to be

astigmatism. As a result, the system performance is going to be degraded by a field

constant astigmatic aberration as explained in Chapter 3. In this case, active alignment

can be used to restore the optical functionality of the optical system. More specifically,

the secondary mirror tilt is used to introduce the opposite amount of field constant

astigmatism that results from the three fabricated optical surfaces. Ultimately, this

143

property makes the entire optical system robust to both misalignment and fabrication

induced errors.

16:04:40

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.15967

Maximum = 0.21519

Average = 0.19827

Std Dev = 0.013871

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


16:04:41

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.020385

Maximum = 0.033847

Average = 0.028041

Std Dev = 0.0033958

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.

)

15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.

)

16:03:10

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.30753

Maximum = 0.39565

Average = 0.33898

Std Dev = 0.024489

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


16:04:02

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.01667

Maximum = 0.022854

Average = 0.019084

Std Dev = 0.001515

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.

)

15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.

)

15:58:49

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.036542

Maximum = 0.07623

Average = 0.060924

Std Dev = 0.01086

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:59:05

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00016152

Maximum = 0.0089318

Average = 0.003117

Std Dev = 0.0021271

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.)

15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.)

15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


14:44:01

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00016152

Maximum = 0.0089318

Average = 0.003117

Std Dev = 0.0021271

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.

)

15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6



4

0

2

-2

-4

Y O

bj.

Fie

ld (d

eg.

)

(c)

(d)

(b)

(a)

15:15:47

Zernike Polynomial

KHF 01-Nov-13


vs


Minimum = 0.00097077

Maximum = 0.026129

Average = 0.011091

Std Dev = 0.0056026

0.25waves (10600.0 nm)

-6 -4 -2 0 2 4 6


-6

-4

-2

0

2

4

6


0.25λ (10µm)

ZAstig ZComa

Figure 7-4. The astigmatism (Z5/6) and coma (Z7/8) Zernike aberration FFDs over an

8°x6° full FOV for the (a) nominal system and with 0.1° α tilt of the (b) primary, (c)

secondary, and (d) tertiary mirror surfaces.

144

0%

20%

40%

60%

80%

100%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Cum

ulat

ive P

erce

ntag

e

As-Built RMS Wavefront Error (waves at 10µm)

(+0°,+0°)

(-4°,+0°)

(+4°,+0°)

(-3°,+0°)

(+3°,+0°)

(+4°,+3°)

(+4°,-3°)

(-4°,+3°)

(-4°,-3°)

Figure 7-5. Cumulative probability as a function of as-built RMS WFE for the three mirror optical system over nine field points assuming active alignment where secondary mirror tilt and focal plane tilt are used as compensators.

7.1.2 Stray Light Analysis

In addition to considering the manufacturing tolerances of the mechanical design that

relate to the alignment of the optical system, another consideration for the design is its

susceptibility to stray light. Therefore, an additional component to the mechanical design

is to limit the light outside the FOV from reaching the detector through the use of baffling

and surface preparation. Similar to the case of the sensitivity analysis, a figure of merit is

established that measures how well the mechanical structure is rejecting unwanted

radiation. For this analysis, the figure of merit is the point source

transmittance (PST) [82]. The PST computes the ratio of the average detector irradiance

to the incident source irradiance as a function of input angle of the source. In an ideal

system, the PST would be one everywhere within the FOV and zero elsewhere. However,

in reality, some radiation from a source outside the intended FOV will reach the detector

either through direct paths to the focal plane or through multiple bounces or scattering off

the mechanical and optical surfaces. For this optical system, since the system is

145

nonsymmetric, the PST will also be nonsymmetric. Furthermore, the optical system is

tilted only in the XZ plane so the primary contributor to the PST is the elevation angle of

the source. Using FRED, a non-sequential raytrace program from Photon Engineering,

the elevation PST of the optical system and base mechanical design presented in

Figure 7-1 is computed. In the software, rays from a 501x501, 120 mm diameter, 10 µm

source at the input aperture are traced in 1° increments over a 180° elevation. At each

input angle, the PST is computed. Figure 7-6 shows the log(PST) as a function of input

elevation angle for the base optical system where the optical surfaces are assumed to be

perfect reflectors and the walls of the optical housing are assumed to be near specular

with 80% reflectivity, which is a good representation for the specular component of

machined aluminum in the LWIR [83]. In Figure 7-6, it can be seen that the stray light

rejection of the base optical system is poor. There is a large region of stray light from

roughly 20° to 45° that results from a direct path to the focal plane from the input

aperture. A similar region of stray light is observed from -45° to -20° as the light in this

region reaches the focal plane by reflecting off one of the input faces of housing thus

creating a mirror image of the region between 20° and 45. These problem regions should

be mitigated to improve the overall signal-to-noise ratio of the optical system, if possible.

146

-10

-8

-6

-4

-2

0

-90 -60 -30 0 30 60 90

Log1

0(PS

T)

Input Angle (deg.)

Figure 7-6. The computed elevation log(PST) for the baseline optical housing with the walls of the housing material assumed to be machined aluminum, resulting in a near specular surface with 80% reflectance.

As a first step to mitigate the stray light, the walls of the optical housing are made less

specular by blackening the walls with a suitable paint. In this fashion, the walls of the

housing now become scattering surfaces. In FRED, the surface preparation is modeled as

a flat black paint with a reflectance coefficient of 0.1. Moreover, importance sampling is

added that preferentially traces rays that are scattered towards the focal plane, primary

mirror, and tertiary mirror, which are the most direct ray paths to the focal plane. When

this surface preparation is added to the walls of the optical housing and the PST is

re-computed, the elevation PST is improved as shown in Figure 7-7 where the new PST

is shown in blue and the previous PST with near specular walls is shown in gray. With

the walls of the housing less specular, the amount of radiation reaching the focal plane is

lessened. The large region of stray light that resulted from a reflection off the input face

is no longer present because the surface now scatters the incoming light. The main

147

contributor to the stray light is now the direct path to the focal plane from the input

aperture of the optical system.

-10

-8

-6

-4

-2

0

-90 -60 -30 0 30 60 90

Log1

0(PS

T)

Input Angle (deg.)

Figure 7-7. The computed elevation log(PST) for the optical system with blackened walls in blue and the computed elevation log(PST) for the baseline optical housing in gray. An improvement is observed when the walls of the housing are blackened versus left machined aluminum.

To lessen the direct path to the focal plane from the input aperture of the optical

housing, a baffle is added near the image plane that blocks most of the input radiation

from the source. The effect of the baffle near the image plane is observed in Figure 7-8

(a-b), where Figure 7-8 (a) shows a cutaway of the optical system without the baffle and

rays are drawn from the focal plane to the limiting mechanical structure to demonstrate

the solid angle of the outside environment that can be seen by the focal plane. When the

baffle, which is a hemispherical aluminum mask, is added as shown in Figure 7-8 (b), the

solid angle of the environment seen by the focal plane is zero with only a direct view to

the input aperture of the housing. An additional baffle is added on the other side of the

image plane to block a region of stray light observed in Figure 7-7 around -20° that is

148

caused by radiation from the source reflecting off the tertiary and primary and reaching

the focal plane.

Image Plane Image Plane

(a) (b)

Solid Angle

Figure 7-8. Cutaway of the optical system (a) without a baffle and (b) with a baffle and its solid angle to the environment from the focal plane shown in red for each case. With the baffle added to the housing, the solid angle to the environment goes to zero.

The effect of adding these baffles to the optical housing is observed by re-computing

the PST as shown in Figure 7-9 where the PST with the baffles is shown in red and the

PST without the baffles is shown in light blue. In comparing the two PST plots, it can be

seen that the magnitude of the stray light around -20° and between 25° and 40° has

decreased by about two orders of magnitude. These regions of stray light have not

completely vanished because some scattered radiation still reaches the focal plane

through these ray paths. As a final step to decrease some of the light that reaches the focal

plane through scattering, an additional baffle is added at the primary mirror. This baffle is

a cylindrical sleeve that fits around the clear aperture of the primary mirror as observed in

Figure 7-8 (b). Its effect is observed by re-computing the PST, as shown in Figure 7-10,

where the PST with the primary baffle is shown in green and the PST without the baffle

is shown in light red. There is a slight decrease to the stray light around -20° and a two

149

order of magnitude decrease in the scattered stray light for elevation angles greater than

40° when compared to the previous PST plot. Overall, through the use of several baffles

and a black surface preparation on the walls of the optical housing, the optical surface is

much better suited for the rejection of stray light that will improve the signal-to-noise

ratio of the optical system when operated.

-10

-8

-6

-4

-2

0

-90 -60 -30 0 30 60 90

Log1

0(PS

T)

Input Angle (deg.)

Figure 7-9. The computed elevation log(PST) for the optical system with blackened walls as well as baffling near the image plane in red and the computed elevation log(PST) for the optical housing with blackened walls in light blue. A two order of magnitude improvement is observed in the regions of large stray light when baffling is added near the image plane.

150

-10

-8

-6

-4

-2

0

-90 -60 -30 0 30 60 90

Log1

0(PS

T)

Input Angle (deg.)

Figure 7-10. The computed elevation log(PST) for the optical system with blackened walls, baffling near the image plane, and baffling at the primary mirror in green and the computed elevation log(PST) for the optical housing with blackened walls and baffling near the image plane in light red. A two order of magnitude improvement is observed for large positive elevation angles where scattering is the dominant contributor to stray light.

7.2 As-built Optical System

Working with II-VI Infrared, the optical housing has been manufactured offsite and

directly assembled at the University of Rochester. In Figure 7-11 (a-c) the subassemblies

of the three mirrors are shown for the primary, secondary, and tertiary. Within each

subassembly, the three diamond turned raised pads and dowel pins can be seen that mate

the subassembly to the optical housing. The secondary mirror subassembly, shown in

Figure 7-11 (b), differs from the other two subassemblies as it includes the aperture stop

of the optical system. The elliptical knife edge rests above the secondary mirror and

ensures the correct ray bundle enters the optical system.

151

(a) (b) (c)

Figure 7-11. As-built subassemblies for the (a) primary, (b) secondary, and (c) tertiary mirrors of the three mirror system that are to be mated to the optical housing. Each subassembly mates to one face of the optical housing and rests on three raised, diamond turned pads.

The as-built optical system with the three subassemblies mated to the housing is

shown in Figure 7-12. With the use of the slip fit steel dowel pins, the subassemblies

readily mate to the faces of the optical housing. To minimize mounting distortion of the

mirror components, the screws are tightened just enough to ensure that the subassemblies

are secure to the housing as well as the mirrors secure to the adaptor plates. 1/4-20 tapped

holes are machined into both sides of the optical housing so that the housing can be

secured to other mechanical components. The layout of the hole pattern is 1” on center

and is designed to be perpendicular to the input face of the optical system.

Figure 7-12. Assembled three mirror optical system. The system consists of a housing structure and three mirror subassemblies that are mated to the faces of the housing.

152

7.2.1 As-built Optical Performance

The as-built full field performance of the assembled optical system is measured

interferometrically with a Zygo 632.8 nm wavelength DynaFiz laser interferometer in a

double pass configuration as shown in Figure 7-13. The interferometer is affixed with an

F/1.50 transmission sphere, providing a spherical wavefront output that overfills the

F/1.90 optical system. With this interferometer configuration, the optical system is

oriented backwards, that is, the output face of the optical system faces the interferometer.

The point source focus of the interferometer is located at the image plane of the optical

system by adjusting the position of the optical system which is mounted on a z-axis

translation stage. When the point source is at the correct image plane location, the beam

exiting the optical system is collimated. A 150 mm diameter, λ/20 high quality flat mirror

is inserted at the input of the optical system so the wavefront is retro-reflected back

towards the interferometer. The mirror must be oversized relative to the 30 mm entrance

pupil diameter of the optical system because as the point source is scanned along the

image plane surface, the angle of the exiting beam will change and at the retro-reflector,

the beam will displace along the mirror surface. The retro-reflector mirror mount has

variable tip and tilt and the optical system is mounted on both an x-axis and y-axis

translation stage so various field points on the focal plane surface can be measured with

the interferometer.

153

Figure 7-13. Experimental setup for measuring the full field performance of the as-built optical system.

Before any measurements are made with the optical system in the interferometric

configuration, the system must be aligned to the interferometer. Specifically, the output

face of the optical system must be aligned normal to the output of the interferometer.

Without this alignment step, the measurements acquired throughout the FOV will not be

relative to the correct image plane and the field curvature present in the measured

wavefronts will be incorrect. To perform this alignment, the transmission sphere is first

replaced with a transmission flat and the flat is aligned to the interferometer. Using the

reflection from the output face of the optical housing as a guide, the tip and tilt of the

optical system is adjusted until it is nulled relative to the interferometer. With the

alignment complete, the transmission sphere is replaced and aligned relative to the

interferometer.

The next step is to find the image plane location that corresponds to the on-axis field

point of the telescope. This point is found by placing a reference optical flat on the input

154

face of optical housing, which is designed to be perpendicular to the on-axis field angle.

The wavefront reflected back towards the interferometer from the reference flat is used to

adjust the x, y, and z position of the optical housing until the wavefront is nulled in both

tilt and defocus. With the on-axis field point found, the rest of the FOV is measured. In

total, a 3x3 grid of field points is measured over the 8 mm x 6 mm image plane. At each

field point, the retro-reflecting mirror must be re-positioned in tip and tilt to null the tilt

present in the resulting interferogram. No adjustments are made to the focus of the optical

system during the measurement process to ensure that the field curvature of the optical

system is appropriately measured relative to the nominally designed image plane.

The measured 3x3 grid of wavefronts for the directly assembled system are shown

below in Figure 7-14 where the RMS WFE at each field point is displayed inside the

wavefront in microns. As can be seen in the structure of the wavefronts, the as-built

system does suffer from field constant astigmatism oriented at 0°; however, the

magnitude of the aberration is small and the RMS WFE throughout the FOV is less than

0.06λ at 10 µm, below the diffraction limit of 0.07λ. Based on the analysis performed in

Section 7.1.1.2, the residual field constant astigmatism could be a result of either

misalignment, figure error, or both. The as-built measured data of the optical housing

from II-VI Inc. can be analyzed to determine if the residual aberration in the optical

system is the result of a misalignment. At II-VI using a coordinate measuring machine,

the angular errors of the optical housing faces were measured. For the three mirror faces,

the magnitude of angle error was found to be 2.5 arcsec for the primary mirror, 13 arcsec

for the secondary mirror, and 12 arcsec for the tertiary mirror. Based on these

magnitudes, which are very small, it is concluded that the observed field constant

155

astigmatism is not the result of a manufacturing error. On the other hand, the surface

characterization of the mirror surface in Chapter 6 did reveal astigmatism as the

predominant surface error and would suggest that figure error of the mirror surfaces is the

main contributing factor to the residual field constant astigmatism observed in the overall

system. If the optical system is to be pushed to a shorter wavelength regime, further

alignment will be required that compensates for the residual field constant astigmatism.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

-1.5

-1

-0.5

0

0.5

1

1.5

0.364 µm

0.448 µm

0.420 µm

0.544 µm

0.395 µm

0.486 µm 0.449 µm

0.469 µm

0.559 µm

Wavefront Error (µm)

0 4-4

0

3

-3

X Image Plane Height (mm)

Y Im

age

Pla

ne

He

igh

t (m

m)


8 mm x 6 mm FOV for the directly assembled three mirror optical system. The RMS

WFE in microns displayed within the wavefront for each field.

The field constant astigmatism present in the directly assembled optical system that

results from figure error of the as-fabricated surfaces can be removed by tilting the

secondary mirror as discussed in Section 7.1.1.2. The required tilt of the secondary mirror

is dictated by the amount of field constant astigmatism present in the optical system.

Using the on-axis field point as the reference point, the Zernike astigmatism (Z5/6) is

measured for the as-built system and compared to its nominal value. The measured Z5

156

and Z6 astigmatism for the as-built system is -0.805 µm and -0.315 µm compared to

0.247 µm and 0.000 µm for the nominal system. This difference is simulated in a

commercial lens design software package, in this case CODE V, by adding -1.052 µm of

Z5 astigmatism and -0.315 µm of Z6 astigmatism to the entrance pupil. Next, in CODE V,

the secondary mirror tilt is re-optimized to remove the residual field constant

astigmatism. Only the tilt in the YZ plane is allowed to vary because the XZ plane tilt is

difficult to implement in the actual as-built system where shims will have to be used.

From the simulation the optimum tilt is found to be -0.0175° and it improves the overall

performance so that it is near nominal with a maximum RMS WFE of 0.024λ at a

wavelength of 10 µm. However, there is a tradeoff for this improvement as the boresight

of the optical system does change and the image shifts down 70 µm in the y-direction.

For a camera with 25 µm pixels, the secondary tilt results in a boresight error of three

pixels. Based on the optical housing geometry, the shim required to be placed underneath

the raised pad where the secondary mirror subassembly mounts to the optical housing is

roughly 30 µm. As an example of this implementation, a 23 µm shim at the secondary

mirror has been implemented. Figure 7-15 shows the resulting measured wavefronts for

the 3x3 grid of field points with the corresponding RMS WFE at each field point

displayed inside the wavefront in microns. The maximum RMS WFE has improved by a

factor of two to 0.03λ at a wavelength of 10 µm. Further improvement is still possible if

the shim size is increased to 30 µm and the mirror is shimmed out of plane to remove the

residual field constant astigmatism oriented at 45°. However, even at this stage in the

alignment, the optical system is well within the diffraction limit of 0.07λ at a wavelength

of 10 µm and would perform well if operated at 1 µm. As an example to demonstrate the

157

image quality of the assembled optical system, Figure 7-16 shows a sample image of the

optical system affixed with the 8 mm x 6 mm, 25 µm pixel pitch uncooled

microbolometer detector.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

0.157 µm

0.213 µm

0.192 µm

0.297 µm

0.201 µm

0.298 µm 0.242 µm

0.208 µm

0.295 µm

Wavefront Error (µm)

0 4-4

0

3

-3

X Image Plane Height (mm)

Y Im

age

Pla

ne

He

igh

t (m

m)


8 mm x 6 mm FOV for the directly assembled three mirror optical system with the

secondary mirror tilted roughly 1 arc minute with a 23 µm shim. The RMS WFE in

microns is displayed within the wavefront for each field.

Figure 7-16. Sample LWIR image from the optical system

158

Conclusion and Future Work

In this work nodal aberration theory (NAT) has been extended to describe the emerging

class of optical systems that are nonsymmetric and employ nonsymmetric or freeform

optical surfaces. We find that the aberration fields of freeform surfaces in the

ϕ-polynomial family fit directly into the existing discoveries for the characteristic

aberration fields of tilted and/or decentered optical systems. These theoretical findings

have been verified with an experimental setup that employs a custom fabricated freeform

plate and demonstrates that the predictions of NAT for freeform surfaces are valid and

relate to observable quantities. Also, it has been shown how strategies based in this new

branch of NAT can be applied to the design of a freeform optical system. This optical

design has been fabricated and assembled to demonstrate that freeform optical systems

are realizable with current fabrication technologies for use in the infrared. The research

presented in this dissertation is comprised of three major contributions:

1) A method for integrating freeform optical surfaces, particularly those related to

ϕ-polynomial surfaces, including Zernike polynomial surfaces, with NAT has been

developed. When a freeform optical surface is placed at a surface away from the aperture

stop, there is the anticipated field constant contribution as well as a field dependent

contribution to the net aberration field. This method has been applied to describe the

aberration behavior of a Fringe Zernike polynomial overlay up to sixth order. This

behavior has been studied in detail for both the case of a two mirror and three mirror

telescope with a three point mount-induced trefoil deformation on the secondary or

tertiary mirror. The deformation induces a previously unobserved new type of astigmatic

field dependence, field conjugate, field linear astigmatism, which in the presence of

159

conventional third order field quadratic astigmatism yields quadranodal behavior. With

this new development in NAT, mount-induced error, misalignment induced error, and

astigmatic figure error can all be analytically described. This result is directly relevant to

the alignment and control of large astronomical telescopes.

2) An aberration generating Schmidt telescope has been designed and assembled to

validate the predicted aberration field behavior of freeform optical surfaces with NAT.

Specifically, a Zernike trefoil plate has been custom fabricated and implemented into the

Schmidt telescope to show that when the trefoil plate is moved away from the aperture

stop of the telescope, field conjugate, field linear astigmatism is generated. This

experiment not only validates the use of NAT for freeform surfaces but also verifies

many of the mathematical constructs in the theory like vector multiplication and the

conjugate vector. This experiment is also the first time that an aberration field associated

with NAT has been specifically isolated. The experiment demonstrates that the concepts

in NAT can be directly visualized through a measurement of the wavefront of an optical

system throughout the FOV. With these measurements and the resulting aberration field

signature, the state of the optical system can be assessed.

3) A φ-polynomial type optical surface, specifically based on Zernike polynomials, has

been successfully implemented in the design, fabrication, and assembly of an unobscured,

LWIR reflective three mirror imager based on tilted components. The optical design

utilizes a strategy based in NAT applied to freeform surfaces to create an efficient path to

a solution with minimum added complexity and testable surfaces. In this approach,

nonsymmetric terms are placed on an optical surface depending on the limiting

aberrations and their characteristics throughout the FOV that were described in

160

Chapter 3. The final design reaches a diffraction limited solution in the LWIR and

operates at a fast focal ratio of F/1.9 over a 10° diagonal full FOV allowing the system to

couple to an uncooled microbolometer detector.

The optical surfaces of this design have been diamond turned and a novel metrology

approach has been developed to measure the as-fabricated surface figure of the optical

surfaces. The measurement utilizes an interferometric null configuration that is a

combination of subsystems each addressing a specific aberration type present in the

departure of the mirror surface, namely, spherical aberration, astigmatism, and coma. The

metrology setup is capable of adapting and measuring a wide variety of surface shapes as

the astigmatism can be varied by changing the tilt angle of the mirror surface being

measured and the comatic departure can be varied by changing the shape on an adaptive

mirror. This approach has successfully been implemented for the measurement of the

as-fabricated secondary mirror in the three mirror imager where it has been found that the

PV surface error is around a tenth wave in the LWIR and the dominant error in the mirror

surface is astigmatism.

Finally, an optical housing has been designed and built that allows for the optical

system to be snap together assembled with diffraction limited performance in the LWIR.

The as-built optical system achieves a measured RMS WFE of less than 0.06λ in the

LWIR over the entire FOV. If the system is actively aligned, even better as-built

performance can be achieved with a final RMS WFE of less than 0.03λ over the entire

FOV. The as-built optical system demonstrates that a freeform optical system can be

designed, fabricated, and built to meet its design requirements. Furthermore, it

161

demonstrates an all freeform optical design that has been carried through the entire

optical manufacturing process.

Looking towards future research paths, there are several areas to be explored in more

detail. First, the link between NAT and freeform, φ-polynomial overlays has only been

carried out through sixth order (the first sixteen Zernike polynomials). For the optical

system described in Chapter 5, these terms on the mirror surface were sufficient to

provide a diffraction limited optical system; however, as freeform surfaces are pushed to

shorter wavelengths, more correction may be needed that may require polynomial

overlays greater than sixth order. To describe the aberration characteristics of these

overlays would first require NAT to be fully derived through seventh order. Some

polynomial terms and their aberration characteristics may be derived through inference

by studying prior derivations of lower order terms but for the complete description of the

polynomial set, NAT should be fully extended to encompass seventh order aberration

components.

Another path to be explored in NAT is the full description of an optical system that

suffers from misalignment, mount induced error, and figure error. These aberration

components are each understood in parts but the complete nodal description that

encompasses all of these errors has not been studied in detail. If these errors are further

investigated, the as-built parameters of the optical system can be better understood during

the design and tolerancing phase that will save time and money later on in the project

when the system is being integrated. Also, the state of an as-built system may be reverse

engineered with this complete description and a path based in NAT can be developed to

perform correction of the optical system and restore its optical functionality.

162

Beyond NAT, an interesting aspect of this work has been the fabrication and

metrology of freeform surfaces. A flexible approach has been presented that utilizes an

active or deformable membrane mirror surface. While the DM is able to create a wide

variety of shapes, it suffers large surface errors resulting from local deformations at the

actuator sites. Based on the success of fabricating freeform components with sub-aperture

polishing processes like MRF (see Chapter 4), one area to be explored is to fabricate

using MRF the comatic null shape that has historically been placed on the active mirror.

With this implementation, the need for a software null to remove the residual error

present from the active mirror in the final measurement may be eliminated.

Lastly, in this dissertation, only one example of freeform optical surfaces for

unobscured reflective design has been presented. An interesting exercise would be to

explore the design space further where the tools developed in this dissertation are utilized

to guide the addition of freeform overlay terms on the optical surfaces. Also, it would be

useful to find solutions for shorter wavelength regimes where the focal ratio does not

need to be as fast. Based on the work presented in this dissertation, the fabrication

capabilities exist to assemble an optical system for the short wave infrared with near

diffraction limited performance.

163

Appendix A. Vector Multiplication and Its Vector Properties and Identities

Nodal aberration theory makes use of a vector operation known as vector multiplication.

The resultant is a coplanar vector and the operation requires the use of an absolute

coordinate system. In this appendix, vector multiplication is introduced and some of its

properties and identities are summarized following the work of Thompson [20, 59].

Vector Multiplication

When describing the concepts of vector multiplication, the analogy of multiplying two

complex numbers is often used. Consider two vectors A

and B

that are expressed as

ˆ ˆ,ix yA ae a i a jα= = +

(A.1)

and

ˆ ˆ,ix yB be b i b jβ= = +

(A.2)

where

sin , cos ,

sin , cos ,x y

x y

a a a ab b b b

α α

β β

= =

= = (A.3)

and the coordinate system in Figure A-1 is used. Multiplying these vectors gives

( ) ( )( ) ˆ ˆ.iy x x y y y x xAB abe a b a b i a b a b jα β+= = + + −

(A.4)

The resultant AB

is a vector with a magnitude equal to the product of the magnitudes of

vectors A

and B

and an orientation that is the sum of the orientations of vectors A

and B

.

The operation is further illustrated in Figure A-1.

164

A

B

AB

α

βi

j

Figure A-1. Concept of vector multiplication.

Conjugate Vector

Another operation that is introduced in NAT to maintain pupil dependence is the

conjugate vector. The operation is simply a reflection of the vector about the y-axis or a

sign change in the exponent. More specifically, the conjugate of the vector A

in

Eq. (A.1) is written as

* ˆ ˆ.ix yA ae a i a jα−= = − +

(A.5)

Vector Identities

With vector multiplication and the conjugate vector described, several useful vector

identities can be written that appear often when deriving the aberration terms that appear

in NAT. They are as follows:

( )( ) ( )( ) 22 ,A B A C A A B C A BC= +

(A.6)

*

,A BC AB C=

(A.7)

( )( ) ( )( ) ( )( )2 2 2 22 ,A B AB C A A B C B B A C= +

(A.8)

( )( ) ( )( )2 2 2 3 22 .A B A C A A AB C A BC= +

(A.9)

As an example of how to use these identities, take the third order field curvature and

astigmatism component from the perturbed wavefront expansion written as

165

( ) ( ) ( ) ( ).2

220 222 .j jS j j j

j jW W H H W Hσ σ ρ ρ σ ρ = − − + − ∑ ∑

(A.10)

Now, using Eq. (A.6) where ( )A H σ= −

and B C ρ= =

, the third order astigmatism component

is re-written so that the wavefront takes the following form

( ) ( ) ( ) ( ) ( ) ( )

( )

220 222

2 2222

12

1 .2

j j

j

S j j j jj j

jj

W W H H W H H

W H

σ σ ρ ρ σ σ ρ ρ

σ ρ

= − − + − −

+ −

∑ ∑

∑

(A.11)

The first two terms in Eq. (A.11) relate to the medial focal surface that is defined as

220 220 2221 ,2M SW W W= + (A.12)

so that the wavefront expansion is re-written as

( ) ( ) ( ) ( )2 2220 222

1 .2j jM j j j

j jW W H H W Hσ σ ρ ρ σ ρ = − − + − ∑ ∑

(A.13)

166

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