Higher-order aberrations - University of Arizona · Higher order Aberrations • Aberration function • Six-order terms (fifth-order transverse) • Wavefront shapes and field dependence
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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
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