Prof. Jose Sasian

Higher-order aberrations

Lens Design OPTI 517

Prof. Jose Sasian

Higher order Aberrations

• Aberration function• Six-order terms (fifth-order transverse)• Wavefront shapes and field dependence• Coefficients• Pupil matching• References.

Prof. Jose Sasian

References

• 1) J. Sasian, “Introduction to Aberrations in optical imaging systems.”

• 2) O. N. Stavroudis, “Modular Optical Design”

• 3) Buchdahl, “Optical Aberration Coefficients”

• 4) M. Herzberger, “Modern Geometrical Optics”

Prof. Jose Sasian

“Back of the envelope”

Prof. Jose Sasian

Coordinate system and reference sphere

HyI

⋅'

ρ

⋅'Ey

Image plane

Exit pupil plane

Prof. Jose Sasian

Aberration function

Prof. Jose Sasian

Prof. Jose Sasian

Aberration orders

Roland Shack’s aberration diagram

Prof. Jose Sasian

• W240 Oblique spherical aberration• W331 Cubic coma• W422 Quartic astigmatism• W420 Six order field curvature• W511 Six order distortion• W600 Six order piston

• W060 Six order spherical aberration• W151

• W242

• W333

Terminology

Prof. Jose Sasian

Some earlier terminology• Oblique spherical aberration• Elliptical coma• Line coma• Secondary spherical aberration• Secondary coma• Lateral coma• Lateral image curvature/astigmatism• Trefoil

Prof. Jose Sasian

Wavefront deformation shapes

Prof. Jose Sasian

Spherical aberration: W060

Prof. Jose Sasian

W151 & W333

Prof. Jose Sasian

W242

Prof. Jose Sasian

Higher-order aberration coefficients

• Harder to derive/calculate than fourth-order

• Intrinsic coefficients• Extrinsic coefficients• Depend highly on coordinate system

Prof. Jose Sasian

Intrinsic spherical aberration

∆−=

nuyAW 2

040 81

2

060 040 040 0402

1 1 ' 822 2 'I

y u u y yW W A u W Wr n n r Ж y

− = − + + + ⋅

2

060 040 040 0402

1 1 ' 82 '2 2 'I

y u u y yW W A u W Wr n n r Ж y

+ = − + + − ⋅

Aperture vector at entrance pupil

Aperture vector at exit pupil

Prof. Jose Sasian

Pupil aberrations

( ) ( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )

000 200 111 020

2 2

040 131 222

2220 311 400

,W H W W W H W H H

W H H W H H H W H

W H H W H W

ρ ρ ρ ρ

ρ ρ

ρ ρ ρ ρ ρ ρ ρ

= + ⋅ + ⋅ + ⋅

+ ⋅ + ⋅ ⋅ + ⋅

+ ⋅ ⋅ + ⋅ ⋅ + ⋅

Prof. Jose Sasian

Distortion at entrance pupilrepresents a cross-section deformation

( )( ) ( ) ( ){ }( ) ( ) ( )

040 131

222 220 311

1 ,

4 21

2 2

HW HЖ

W H H H W H H H H

Ж W H W H W

ρ ρ

ρ ρ

ρ ρ ρ ρ ρ ρ ρ

∆ = − ∇

⋅ ⋅ + ⋅ + ⋅ ⋅ + = − ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅

ρ ρ+ ∆

Prof. Jose Sasian

Prof. Jose Sasian

Prof. Jose Sasian

Concept of pupil matching

• Not traditionally discussed.• Pupil matching concept is important.• Optical system connect: exit pupil of one

connects with the entrance pupil of the next.

• Any pupil mismatch produces an effect.• In general we have pupil mismatch

Prof. Jose Sasian

Example: f/#

y∆

3111 Wu

y =∆

/ #2ff

d y=

− ∆

P P

(fourth-order contribution)

Prof. Jose Sasian

Exit pupil becomes entrance pupil for next surface

Exit pupil for surface J

Entrance pupil for surface J+1

Exit pupil for surface J+1

∆+=∆+=

++++=

yyyyy

ycycycyY

1

...77

55

33

Prof. Jose Sasian

Extrinsic aberrations

( ) ( ) ( )1, , ,E A H BW H W H W HЖ ρρ ρ ρ= − ∇ ⋅∇

( ) ( ) ( )4 6, , ,B B BW H W H W Hρ ρ ρ= +

( ) ( ) ( )4 6, , ,A A AW H W H W Hρ ρ ρ= +

( ) ( ) ( ) ( )( ) ( )

4 4 4 4

4 4

, , , ,

, ,

A B A B A A

A B A

W H W H W H W H

W H W H

ρ ρ ρ ρ ρ ρ

ρ ρ ρ

+ ∆ = + ∆ − +

= ∇ ⋅∆ +

Prof. Jose Sasian

Prof. Jose Sasian

Buchdahl-Rimmerfifth-order aberrations

( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )

( )( ) ( ) ( )

3 2 2 3

5 4 2 3 25 1 2 1 2 3

2 2 3 4 51 2 5 5 5

cos 2 cos 2 3 cos

cos cos 2 cos cos

cos 5 cos

y B F H C H EH

B F F H M M M H

N N H C H E H

ε φ ρ φ ρ π φ ρ

φ ρ φ ρ φ φ ρ

φ ρ π φ ρ

= + + + + +

+ + + + + +

+ + + + +

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )

3 2 2

5 4 2 3 25 2 2 3

2 3 43 5 5

sin sin 2 sin

sin sin 2 cos sin

sin 2 sin

x B F H C H

B F H M M H

N H C H

ε φ ρ φ ρ π φ ρ

φ ρ φ ρ φ φ ρ

φ ρ π φ ρ

= + + +

+ + + +

+ + +

12 fifth-order terms

Prof. Jose Sasian

Aberration correction concepts• Destroy an aberration (early days)• Aberration correction (compensation): Add the

opposite amount to have a net zero residual• Aberration balancing: Add a different aberration

and minimize or trade-off performance; fourth vs. higher order.

• Minimize an aberration.• Do not generate an aberration• Main mechanism for aberration correction is

compensation and balancing

Prof. Jose Sasian

Aberration balancing: 4th order vs. higher order

Prof. Jose Sasian

Summary

• Higher-order aberrations• Pupil aberrations• Aberration correction and balancing