Prof. Jose SasianOPTI 517

Lens Design OPTI 517

Seidel aberration coefficients

Prof. Jose SasianOPTI 517

Fourth-order terms

( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )2400311220

2222131

2040,

HHWHHHWHHW

HWHWWHW

⋅+⋅⋅+⋅⋅+

⋅+⋅⋅+⋅=

ρρρ

ρρρρρρρ

Spherical aberrationComaAstigmatism (cylindrical aberration!)Field curvatureDistortionPiston

Prof. Jose SasianOPTI 517

Coordinate system

Prof. Jose SasianOPTI 517

Spherical aberration

We have a spherical surface of radius of curvature r, a ray intersecting the surface at point P, intersecting the reference sphere at B’, intersecting the

wavefront in object space at B and in image space at A’, and passing inimage space by the point Q’’ in the optical axis. The reference spherein object space is centered at Q and in image space is centered at Q’

][]'['][']'[' ' PBnPBnPAnPBnW −=−=

3

42

82 rh

rhZ +=

+= y

ruyh2

1

Question: how do we draw the first ordermarginal ray in image space?

Prof. Jose SasianOPTI 517

( )

−+

−+=

+

+−

+=

++−=+−=

rs

srh

rs

shs

sr

hr

hr

hshs

hZsZshZsPQ

14

11

4822

1

2][

22

4

2

22

2

2

4

3

422

2

222222

24

2

42 118

118

112

][][][

−+

−−

−−=

−=

rssh

rsrh

rsh

PQOQPB

+= y

ruyh2

1

Prof. Jose SasianOPTI 517

24

2

422

118

118

112

12

][][][

−+

−−

−

+−=

−=

rssy

rsry

rsy

ruy

PQOQPB

2

''

4

'2

4

'

22

''

118

118

112

12

][][]'[

−+

−−

−

+−=

−=

rssy

rsry

rsy

ruy

PQOQPB

−−

−+

−−

−−

−−

−

+−=

=−=

22

''

'4

''

2

4

''

2

'

11118

11118

111112

][]'[

rssn

rssny

rsn

rsn

ry

rsn

rsny

ruy

PBnPBnW

Spherical aberration

Prof. Jose SasianOPTI 517

Spherical aberration

syu /−='' / syu −=

( )/ /A ni n y s y r= = − −

{ } 0=∆ A

∆−=

nuyAW 2040 8

1

Prof. Jose SasianOPTI 517

Surface of radius r

StopS0y

S’

s s’

Objectplane

Image plane

Petzval field curvature PW220

We locate the aperture stop at the center of curvature of the spherical surface. With being the object height, then the inverse of the distancealong the chief ray from the off-axis object point to the surface is:

Prof. Jose SasianOPTI 517

Petzval field curvature

( ) ( )( )

( )( ) ( )

( )

2 2200

2

2200

2

2 20 0

2

2 20

2 2

1 1 1

1

1 111 11 22

1 1 1 11 11 12 2

1 1 112 2

S yr r s y r r sr s

yy sr r s r s sr s

y ys r s s s rs

s r

yu u Жs irs s y n ri

= =− − + − + − + − +

−

≅ =

− −− + − + −− = − + = − + = − −

= − + = − −

' 2

' ' 2 '2 '

1 1 12

u ЖS s y n ri

≅ − −−

Prof. Jose SasianOPTI 517

Petzval field curvature

{ } 0=∆ A

By inserting the 1/s in the quadratic term of W

−−

−+

−−

−−

−−

−

+−=

=−=

22

''

'4

''

2

4

''

2

'

11118

11118

111112

][]'[

rssn

rssny

rsn

rsn

ry

rsn

rsny

ruy

PBnPBnW

( )

( )

2'

'

2 2' '

'2 ' 2

2'

2

6220

1 1 1 12

1 14 4

14

1 14

P

yW n nS r S r

Ж Жn u nun ri n ri

Ж u uAr

Жr n

W O

= − − − − = = − − − =

= − − = = − ∆

= +

2

2201 14PЖWr n

= − ∆

Prof. Jose SasianOPTI 517

Aberration function at CC

( ) ( ) ( )( )ρρρρρ ⋅⋅+⋅= HHWWHW P2202040,

SurfaceExit pupil

Chief ray

CC

Prof. Jose SasianOPTI 517

Stop shifting

Ey

Ey

Old stop at CCNew stop

Marginal ray height at old pupil

New chiefray

Hyyy EEshiftE

+= ρρ

HAAH

yy

E

Eshift

+=+= ρρρ

New chief ray height at old pupil

Prof. Jose SasianOPTI 517

Expansion about chief ray height

AA

yy

E

E =

Ey

Hyyy EEshiftE

+= ρρ

HAAH

yy

E

Eshift

+=+= ρρρ

Ey

Prof. Jose SasianOPTI 517

Expansion about the new chief ray height

( ) ( ) ( )( )ρρρρρ ⋅⋅+⋅= HHWWHW P2202040,H

AAH

yy

E

Eshift

+=+= ρρρ

Prof. Jose SasianOPTI 517

Graphical view

Old Pupil

New pupil

Ey

Prof. Jose SasianOPTI 517

Quadratic term

HHAAH

AA

HAAH

AA

shiftshift

⋅

+⋅+⋅=

=

+⋅

+=⋅

2

2 ρρρ

ρρρρ

( )( ) ( )( )( )( ) ( )2220

2

220

220220

2 HHWAAHHHW

AA

HHWHHW

PP

PP

⋅

+⋅⋅+

+⋅⋅→⋅⋅

ρ

ρρρρ

Prof. Jose SasianOPTI 517

Quartic term

( )

( ) ( )( ) ( )

( )( ) ( )( ) ( )243

22

2

22

42

44

2

2

HHAAHHH

AAHH

AA

HAAH

AA

HHAAH

AA

HHAAH

AA

shiftshift

⋅

+⋅⋅

+⋅⋅+

⋅+⋅⋅+⋅=

=

⋅

+⋅+⋅

×

⋅

+⋅+⋅=⋅

ρρρ

ρρρρρρ

ρρρ

ρρρρρ

Prof. Jose SasianOPTI 517

All terms ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )2400311220

2222131

2040,

HHWHHHWHHW

HWHWWHW

⋅+⋅⋅+⋅⋅+

⋅+⋅⋅+⋅=

ρρρ

ρρρρρρρ

P

P

P

WAAW

AAW

WAAW

AAW

WWAAW

WAAW

WAAW

WW

220

2

040

4

400

220040

3

311

220040

2

220

040

2

222

040131

040040

24

2

4

4

+

=

+

=

+

=

=

=

=

Piston

Prof. Jose SasianOPTI 517

For as system of two surfacesExit pupil becomes entrance pupil for next surface.

Exit pupil for surface J

Entrance pupil for surface J+1

Prof. Jose SasianOPTI 517

Fourth-order contributionsFor a given system ray we add the OPD contributed by each surface.The problem is that because of pupilaberrations we do not know the pupil coordinates of the ray at previous exit pupils.However, the error in knowing the ray pupilcoordinate leads to six-order aberrations.

To fourth-order we do not haveother fourth-order terms to account for.

We are assuming we do not have second-order aberrations. Otherwise these will generateother fourth-order terms.

Prof. Jose SasianOPTI 517

Order of Error

• We know the ray heights to first-order• There is an error on the ray heights y and

y-bar of third order• If the third order error is accounted for it

leads to sixth-order terms

( ) ( ) ( )

3

4 4 6

4 4 6040 040 040

y y yy y y

W y W y W y

α

β

= +

→ +

→ +

Prof. Jose SasianOPTI 517

ConclusionAssume no second order terms in the

aberration functions of each surface

Then for a system of surfaces the fourth-order coefficients are the sum of the

coefficients contributed by each surface

There are no fourth-order extrinsic termsfrom previous aberration in the system

Prof. Jose SasianOPTI 517

Prof. Jose SasianOPTI 517

Prof. Jose SasianOPTI 517

Example : Cooke triplet lens

Prof. Jose SasianOPTI 517

Wave coefficients

1.0000 5.8831 16.4912 11.5569 37.9424 61.2785 -10.4705 -14.67532.0000 4.6978 -50.6336 136.4338 -0.1227 -366.9639 -6.5847 35.48543.0000 -22.3709 117.1708 -153.4247 -36.2070 295.7159 18.4586 -48.33984.0000 -9.6490 -65.3484 -110.6438 -40.4402 -324.2767 14.8117 50.15645.0000 1.6894 24.1504 86.3110 10.3704 382.5921 -4.7481 -33.93876.0000 22.0849 -42.6064 20.5492 43.9016 -52.2587 -12.3359 11.8993Totals 2.3352 -0.7760 -9.2175 15.4444 -3.9128 -0.8690 0.5872

W040 W131 W222 W220 W311 W020 W111

In waves at 0.000587 mm

Prof. Jose SasianOPTI 517

PrescriptionF/4, f=50 mm; FOV +/- 20 degrees

Stop at surface 3SURFACE DATA SUMMARY:Surf Type Radius Thickness Glass Diameter Conic CommentOBJ STANDARD Infinity Infinity 0 0

1 STANDARD 23.713 4.831 LAK9 20.26679 0 2 STANDARD 7331.288 5.86 18.17704 0 3 STANDARD -24.456 0.975 SF5 9.598584 0 4 STANDARD 21.896 4.822 9.909458 0 5 STANDARD 86.759 3.127 LAK9 16.07715 0 6 STANDARD -20.4942 -10.0135 16.69285 0

STO STANDARD Infinity 51.25 12.90717 0 IMA STANDARD Infinity 36.46158 0

Prof. Jose SasianOPTI 517

Summary

• Spherical aberration• Stop shifting• Off-axis aberrations• Entrance/exit pupil concatenation• Seidel sums• Actual coefficients computation• Information acquired

Lens Design OPTI 517��Seidel aberration coefficients��Fourth-order termsCoordinate systemSpherical aberrationSlide Number 5Spherical aberrationSpherical aberration Petzval field curvaturePetzval field curvaturePetzval field curvatureAberration function at CCStop shiftingExpansion about chief ray heightExpansion about the new �chief ray heightGraphical viewQuadratic termQuartic termAll terms For as system of two surfaces�Exit pupil becomes entrance pupil for next surface.�Fourth-order contributionsOrder of ErrorConclusionSlide Number 23Slide Number 24Example : Cooke triplet lensWave coefficientsPrescription�F/4, f=50 mm; FOV +/- 20 degrees�Stop at surface 3Summary