1.5.8 Orders of Polynomials

1.5.8 Seidel AberrationsAberration W(0,ρy) W(ρx,0) εy(0,ρy) εx(ρx,0) h

Tilt linear zero constant zero linear

Defocus parabolic parabolic linear linear constant

Spherical quartic quartic cubic cubic constant

Coma cubic parabolic parabolic zero linear

Astigmatism parabolic zero linear zero parabolic

Field Curvature

parabolic parabolic linear linear parabolic

Distortion linear zero constant zero cubic

Doublet Wavefront Error

Doublet Transverse Ray Error

1.5.9.1 Non‐Rotationally Symmetric Systems

Edmundoptics.com Edmundoptics.com

www.davidmullenasc.com

Essilor

1.5.9.1 Non‐Rotationally Symmetric Systems

Linhof.de

Desmoineseyecare.com

1.5.9.3 Zernike Polynomials

1D Curve Fitting

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

Low‐order Polynomial Fit

y = 9.9146x + 2.3839R2 = 0.9383

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

In this case, the error is the vertical distance between the line andthe data point. The sum of the squares of the error is minimized.

High‐order Polynomial Fit

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5-400

-200

0

200

400

600

800

1000

1200

1400

1600

1800

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

y = a0 + a1x + a2x2 + … a16x16

Fit to W(x) = xj aj

0 π

1 -22 03 -14 05 -0.66666 07 -0.58 09 -0.410 0

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

W(x)1 term3 terms11 terms

Extension to Two Dimensions

In many cases, wavefronts take on a complex shape defined over a circularregion and we wish to fit this surface to a series of simpler components.

Wavefront Fitting

=

-0.003 x

+ 0.002 x

+ 0.001 x

Zernike Polynomials

Azimuthal Frequency, θ

Rad

ialP

olyn

omia

l, ρ

Z00

Z11Z1

1−

Z20

Z31− Z3

1

Z40 Z4

2

Z22

Z42−

Z33− Z3

3

Z44Z4

4−

Z22−

Zernike Polynomials ‐ Single Index ANSI

STANDARD

Azimuthal Frequency, θ

Rad

ialP

olyn

omia

l, ρ

Z0

Z1

Z4 Z5Z3

Z9Z8Z7Z6

Z10 Z11 Z12 Z13 Z14

Z2

Starts at 0Left-to-RightTop-to-Bottom

Other Single Index Schemes

Z1

Z3

Z4 Z6Z5

Z10Z8Z7 Z9

Z15Z13Z11 Z12 Z14

Z2

NON-STANDARDStarts at 1cosines are even termssines are odd terms

Noll, RJ. Zernike polynomials and atmospheric turbulence. J Opt Soc Am 66; 207-211 (1976).

Also Mahajan and Zemax “Standard Zernike Coefficients”

Other Single Index Schemes

Zemax “Zernike Fringe Coefficients”

Also, Air Force or University of Arizona

Z1

Z3

Z4 Z5 Z6

Z10Z7 Z8 Z11

Z18Z13Z9 Z12 Z17

Z2

NON-STANDARDStarts at 1increases along diagonalcosine terms first35 terms plus two extraspherical aberration terms.No Normalization!!!

Zeroth Order Zernike Polynomials

Z00

This term is called Piston and is usually ignored.The surface is constant over the entire circle, sono error or variance exists.

First Order Zernike Polynomials

Z11Z1

1−

These terms represent a tilt in the wavefront.

max11

max11

1111

1111

1111

rxa

rya

cosasina),(Za),(Za

+=

θρ+θρ=θρ+θρ

−

−

−−

Combining these terms results in ageneral equation for a plane, thusby changing the coefficients, a planeat any orientation can be created.This rotation of the pattern is truefor the sine/cosine pairs of Zernikes

Second Order Zernike Polynomials

Z20 Z2

2Z22−

Z(2,0) is a paraboloid, so it represents defocus. Z(2,2) and Z(2,-2) aresaddle shaped surfaces rotated 45 degrees to one another. Much liketilt, appropriate weighting of these two terms can create a saddle shaperotated through any angle.

Third Order Zernike Polynomials

Z31− Z3

1Z33− Z3

3

The inner two terms are coma and the outer two terms are trefoil. These terms represent asymmetric.

Fourth Order Zernike Polynomials

Z40 Z4

2Z42− Z4

4Z44−

SphericalAberration

4th orderAstigmatism

4th orderAstigmatismQuadroil Quadroil

These terms represent more complex shapes of the wavefront.

1.5.10 Through‐Focus PSF and the Star Test

ParaxialFocus

Defocus

Spherical Aberration

Seidel Spherical Aberration

Zernike Spherical Aberration Z(4,0)

ParaxialFocus

MarginalFocus

AstigmatismSeidel Astigmatism

Zernike Astigmatism Z(2,2)

ParaxialFocus

TangentialFocus Sagittal

Focus

ComaSeidel Coma

Zernike Coma Z(3,-1)

ParaxialFocus

1.5.11 Barrel Distortion

www.recipester.org

Pincushion Distortion

Percent Distortion

Fischer, Optical System Design

Anamorphic Distortion

www.davidmullenasc.com

F‐θ Lens

Fischer, Optical System Design

Keystone Distortion

Keystone Correction

www.lonestardigital.com/perspective_correction.htm

Scheimpflug Imaging

http://users.gsinet.net/pjwhite/tilt-shift.htm