opti518 lecture 7 primary aberrations...6 standard -20.4942 -10.0135 16.69285 0 sto standard...

Prof. Jose SasianOPTI 518

Introduction to aberrations

Lecture 7

Fourth-order terms


Fourth-order terms

2400311220

2222131

2040,

HHWHHHWHHW

HWHWWHW

Spherical aberrationComaAstigmatism (cylindrical aberration!)Field curvatureDistortionPiston


Primary monochromatic aberrationsFourth-order terms


Key concept

22040 131 222

2

220 311 400

(6)

,W H W W H W H

W H H W H H H W H H

O

•Total wavefront deformation is the superposition (addition) of all aberration orders.

•The fourth-order group of aberrations is also know as the primary aberrations.

•For slow systems, ~F/8 or less, and for small fields of view, a few degrees, the primary aberrationsprovide a substantially accurate approximation to the wavefront deformation.


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


rs

srh

rs

shs

srh

rh

rhsh

s

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


24

2

422

118

118

112

12

][][][

rssy

rsry

rsyruy

PQOQPB

2

''

4

'2

4

'

22

''

118

118

112

12

][][]'[

rssy

rsry

rsyruy

PQOQPB

22

''

'4

''

2

4

''

2

'

11118

11118

111112

][]'[

rssn

rssny

rsn

rsn

ry

rsn

rsny

ruy

PBnPBnW




syu /

'' / syu

/ /A ni ny s ny rnu n

0 A

nuyAW 2

040 81


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:


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


Petzval field curvature

0 A

By inserting the Eqs. into the quadratic term of Eq. 8,:

22

''

'4

''

2

4

''

2

'

11118

11118

111112

][]'[

rssn

rssny

rsn

rsn

ry

rsn

rsny

ruy

PBnPBnW

22'

'

2 2 2' '

'2 ' 2

2 2'

2 2

6220

1 1 1 112 2

1 112 4 4

112 4

1 112 4

P

y uW y n nr S r S r

u Ж Жy nu nur n ri n ri

u Жy u ur Ar

u Жyr r n

W O

2

2201 14PЖWr n


Aberration function at CC

HHWWHW P2202

040,

SurfaceExit pupil

Chief ray

CC


Ideal image point

Reference sphereOld exit

pupil

New exitpupil

,newW H

' ', , , ,' ' '

old old old oldold old old new

old old old

y y H y yW H W H W H H W Hy y y

Stop shifting

,oldW H


Expansion about the new chief ray height

HHWWHW P2202

040,

HAAH

yy

E

Eshift


Graphical view

Old Pupil

New pupil

Ey

Old chief rayNew chief ray


Quadratic term

HHAAH

AA

HAAH

AA

shiftshift

2

2

2220

2

220

220220

2 HHWAAHHHW

AA

HHWHHW

PP

PP


Quartic term

243

22

2

22

42

44

2

2

HHAAHHH

AAHH

AA

HAAH

AA

HHAAH

AA

HHAAH

AA

shiftshift


All terms

2400311220

2222131

2040,

HHWHHHWHHW

HWHWWHW

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


Entrance and exit pupils


Exit pupil becomes entrance pupil for next surface

Exit pupil for surface J

Entrance pupil for surface J+1

OPD’s add

2040,A A A AW H W

A B B

2040,B B B BW H W

A B B


Exit pupil becomes entrance pupil for next surface

Pupil

OPD’s add

,A AW H

A B B

,B BW H

BB

B

Pupil

A B


Extrinsic terms

Entrance pupil for surface J+1

2 2020 040 040

020

2 2040 040

, , ,

, ,

2

...

....

A B B A A B B

A B B B B

A A A A A A B B B

A B B B B B B

A B B B B B B B B B

B B B B B

W H W H W H

W H W H

W W W

W

W W

A B B


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 third-order error in 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.


RephraseThere are no second order terms in the

aberration function 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


First-order chief and marginal rays

Imageplane

's

'u'uy

y

Stopaperture

Object and image space quantities


The slope alpha αNormal line

ry

α

Spherical surface

u u’

s s’

i

/ /A ni ny s ny rnu n

From the axis/normal to theray in a counter-clock wise rotationindicates a positive slope


Example : Cooke triplet lens


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


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


Summary

• Fourth-order terms• Derivation of the coefficients in terms of

systems parameters• Application of stop shifting• Rely on second-order structure• Example for a Cooke triplet

opti518 lecture 7 primary aberrations...6 standard -20.4942 -10.0135 16.69285 0 sto standard...

Documents