Theory of sixth-order wave aberrations

José Sasián

doi:10.1364/AO.49.000D69

Theory of sixth-order wave aberrations

José Sasián

Applied Optics
Vol. 49,
Issue 16,
pp. D69-D95
(2010)
•https://doi.org/10.1364/AO.49.000D69

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José Sasián, "Theory of sixth-order wave aberrations," Appl. Opt. 49, D69-D95 (2010)

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Related Topics
Table of Contents Category
- Optical Aberrations
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Aberrations (global) (080.1010)
- Geometric optical design (080.2740)
- Aberration expansions (080.1005)
- Image formation theory (110.2990)
- Propagation (350.5500)

About this Article
History
- Original Manuscript: November 6, 2009
- Manuscript Accepted: December 11, 2009
- Published: March 17, 2010

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Abstract

A sixth-order theory of wave aberrations for axially symmetric systems is developed. Specific formulas for the sixth-order extrinsic and intrinsic wave aberration coefficients are given, as well as relations between pupil and image aberrations. Equations are developed for the wavefront propagation to the sixth order of approximation. The concept of the irradiance function is developed, and the second-order irradiance coefficients are found via conservation of flux at the pupils of the optical system and in terms of pupil aberrations. From purely geometrical considerations a generalized irradiance transport equation that describes irradiance changes in an optical system is derived. Confirming the aberration coefficients with real ray-tracing data was found to be indispensable.

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Corrections

José Sasián, "Theory of sixth-order wave aberrations: errata," Appl. Opt. 49, 6502-6503 (2010)
http://proxy.osapublishing.org/ao/abstract.cfm?uri=ao-49-33-6502

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Aberration	Vector Form	Algebraic Form	j	m	n
Zero-order
Uniform piston	$W_{000}$	$W_{000}$	0	0	0
Second-order	Gauss
Quadratic piston	$W_{200} (\vec{H} \cdot \vec{H})$	$W_{200} H^{2}$	1	0	0
Magnification	$W_{111} (\vec{H} \cdot \vec{ρ})$	$W_{111} H ρ \cos (ϕ)$	0	1	0
Focus	$W_{020} (\vec{ρ} \cdot \vec{ρ})$	$W_{020} ρ^{2}$	0	0	1
Fourth order	Seidel
Spherical aberration	$W_{040} (\vec{ρ} \cdot \vec{ρ})^{2}$	$W_{040} ρ^{4}$	0	0	2
Coma	$W_{131} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})$	$W_{131} H ρ^{3} \cos (ϕ)$	0	1	1
Astigmatism	$W_{222} (\vec{H} \cdot \vec{ρ})^{2}$	$W_{222} H^{2} ρ^{2} \cos^{2} (ϕ)$	0	2	0
Field curvature	$W_{220} (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ})$	$W_{220} H^{2} ρ^{2}$	1	0	1
Distortion	$W_{311} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ})$	$W_{311} H^{3} ρ \cos (ϕ)$	1	1	0
Quartic piston	$W_{400} (\vec{H} \cdot \vec{H})^{2}$	$W_{400} H^{4}$	2	0	0
Sixth order	Schwarzschild
Oblique spherical aberration	$W_{240} (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ})^{2}$	$W_{240} H^{2} ρ^{4}$	1	0	2
Coma	$W_{331} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})$	$W_{331} H^{3} ρ^{3} \cos (ϕ)$	1	1	1
Astigmatism	$W_{422} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ})^{2}$	$W_{422} H^{4} ρ^{2} \cos^{2} (ϕ)$	1	2	0
Field curvature	$W_{420} (\vec{H} \cdot \vec{H})^{2} (\vec{ρ} \cdot \vec{ρ})$	$W_{420} H^{4} ρ^{2}$	2	0	1
Distortion	$W_{511} (\vec{H} \cdot \vec{H})^{2} (\vec{H} \cdot \vec{ρ})$	$W_{511} H^{5} ρ \cos (ϕ)$	2	1	0
Piston	$W_{600} (\vec{H} \cdot \vec{H})^{3}$	$W_{600} H^{6}$	3	0	0
Spherical aberration	$W_{060} (\vec{ρ} \cdot \vec{ρ})^{3}$	$W_{060} ρ^{6}$	0	0	3
Unnamed	$W_{151} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})^{2}$	$W_{151} H ρ^{5} \cos (ϕ)$	0	1	2
Unnamed	$W_{242} (\vec{H} \cdot \vec{ρ})^{2} (\vec{ρ} \cdot \vec{ρ})$	$W_{242} H^{2} ρ^{4} \cos^{2} (ϕ)$	0	2	1
Unnamed	$W_{333} (\vec{H} \cdot \vec{ρ})^{3}$	$W_{333} H^{3} ρ^{3} \cos^{3} (ϕ)$	0	3	0

Aberration	Seidel Sum
$W_{040} = \frac{1}{8} S_{I}$	$S_{I} = - \sum_{}^{} A^{2} y Δ (\frac{u}{n})$
$W_{131} = \frac{1}{2} S_{II}$	$S_{II} = - \sum_{}^{} A \bar{A} y Δ (\frac{u}{n})$
$W_{222} = \frac{1}{2} S_{III}$	$S_{III} = - \sum_{}^{} {\bar{A}}^{2} y Δ (\frac{u}{n})$
$W_{220} = \frac{1}{4} S_{IV}$	$S_{IV} = - \sum_{}^{} Ψ^{2} P - \sum_{}^{} {\bar{A}}^{2} y Δ (\frac{u}{n})$
$W_{311} = \frac{1}{2} S_{V}$	$S_{V} = - \sum_{}^{} \frac{\bar{A}}{A} [Ψ^{2} P + {\bar{A}}^{2} y Δ (\frac{u}{n})], S_{V} = - \sum_{}^{} \bar{A} [{\bar{A}}^{2} Δ (\frac{1}{n^{2}}) y - (Ψ + \bar{A} y) \bar{y} P]$
$W_{400} = \frac{1}{8} S_{VI}$	$S_{VI} = - \sum_{}^{} {\bar{A}}^{2} \bar{y} Δ (\frac{\bar{u}}{n})$

Quantity	Formula
Refraction-invariant marginal ray	$A = n i = n u + n y c$
Refraction- invariant chief ray	$\bar{A} = n \bar{i} = n \bar{u} + n \bar{y} c$
Lagrange invariant	$Ψ = n \bar{u} y - n u \bar{y} = \bar{A} y - A \bar{y}$
Surface curvature	$c = \frac{1}{r}$
Petzval sum term	$P = c \cdot Δ (\frac{1}{n})$

Name	Identity between Pupil and Image Aberration Coefficients
Pupil spherical aberration	${\bar{W}}_{040} = W_{400}$
Pupil coma	${\bar{W}}_{131} = W_{311} + \frac{1}{2} Ψ \cdot Δ {{\bar{u}}^{2}}$
Pupil astigmatism	${\bar{W}}_{222} = W_{222} + \frac{1}{2} Ψ \cdot Δ {u \bar{u}}$
Pupil sagittal field curvature	${\bar{W}}_{220} = W_{220} + \frac{1}{4} Ψ \cdot Δ {u \bar{u}}$
Pupil distortion	${\bar{W}}_{311} = W_{131} + \frac{1}{2} Ψ \cdot Δ {u^{2}}$
Pupil piston	${\bar{W}}_{400} = W_{040}$

With Aperture Vector $\vec{ρ}$ at Entrance Pupil	With Aperture Vector $\vec{ρ}$ at Exit Pupil
$W_{060 E}^{-} = \frac{1}{Ψ} {4 W_{040}^{B} {\bar{W}}_{311}^{A}}$	$W_{060 E}^{+} = - \frac{1}{Ψ} {4 W_{040}^{A} {\bar{W}}_{311}^{B}}$
$W_{151 E}^{-} = {\frac{1}{Ψ} \begin{matrix} 3 W_{131}^{B} {\bar{W}}_{311}^{A} + 8 W_{040}^{B} {\bar{W}}_{220}^{A} \begin{matrix} + 8 W_{040}^{B} {\bar{W}}_{222}^{A}} \end{matrix} \end{matrix}$	$W_{151 E}^{+} = - \frac{1}{Ψ} {\begin{matrix} 3 W_{131}^{A} {\bar{W}}_{311}^{B} + 8 W_{040}^{A} {\bar{W}}_{220}^{B} \begin{matrix} + 8 W_{040}^{A} {\bar{W}}_{222}^{B}} \end{matrix} \end{matrix}$
$W_{242 E}^{-} = \frac{1}{Ψ} {\begin{matrix} 2 W_{222}^{B} {\bar{W}}_{311}^{A} + 4 W_{131}^{B} {\bar{W}}_{220}^{A} \begin{matrix} + 6 W_{131}^{B} {\bar{W}}_{222}^{A} + 8 W_{040}^{B} {\bar{W}}_{131}^{A}} \end{matrix} \end{matrix}$	$W_{242 E}^{+} = - \frac{1}{Ψ} {\begin{matrix} 2 W_{222}^{A} {\bar{W}}_{311}^{B} + 4 W_{131}^{A} {\bar{W}}_{220}^{B} \begin{matrix} + 6 W_{131}^{A} {\bar{W}}_{222}^{B} + 8 W_{040}^{A} {\bar{W}}_{131}^{B}} \end{matrix} \end{matrix}$
$W_{333 E}^{-} = \frac{1}{Ψ} {4 W_{131}^{B} {\bar{W}}_{131}^{A} + 4 W_{222}^{B} {\bar{W}}_{222}^{A}}$	$W_{333 E}^{+} = - \frac{1}{Ψ} {4 W_{131}^{A} {\bar{W}}_{131}^{B} + 4 W_{222}^{A} {\bar{W}}_{222}^{B}}$
$W_{240 E}^{-} = \frac{1}{Ψ} {\begin{matrix} 2 W_{131}^{B} {\bar{W}}_{220}^{A} + 2 W_{220}^{B} {\bar{W}}_{311}^{A} \begin{matrix} + 4 W_{040}^{B} {\bar{W}}_{131}^{A}} \end{matrix} \end{matrix}$	$W_{240 E}^{+} = - \frac{1}{Ψ} {2 W_{131}^{A} {\bar{W}}_{220}^{B} + 2 W_{220}^{A} \begin{matrix} {\bar{W}}_{311}^{B} + 4 W_{040}^{A} \begin{matrix} {\bar{W}}_{131}^{B}} \end{matrix} \end{matrix}$
$W_{331 E}^{-} = \frac{1}{Ψ} {5 W_{131}^{B} {\bar{W}}_{131}^{A} + 4 W_{220}^{B} \begin{matrix} {\bar{W}}_{220}^{A} + 4 W_{220}^{B} {\bar{W}}_{222}^{A} + 4 W_{222}^{B} {\bar{W}}_{220}^{A} + W_{311}^{B} {\bar{W}}_{311}^{A} + 16 W_{040}^{B} \begin{matrix} {\bar{W}}_{040}^{A}} \end{matrix} \end{matrix}$	$W_{331 E}^{+} = - \frac{1}{Ψ} {5 W_{131}^{A} {\bar{W}}_{131}^{B} + 4 W_{220}^{A} \begin{matrix} {\bar{W}}_{220}^{B} + 4 W_{220}^{A} {\bar{W}}_{222}^{B} + 4 W_{222}^{A} \begin{matrix} {\bar{W}}_{220}^{B} + W_{311}^{A} {\bar{W}}_{311}^{B} + 16 W_{040}^{A} \begin{matrix} {\bar{W}}_{040}^{B}} \end{matrix} \end{matrix} \end{matrix}$
$W_{422 E}^{-} = \frac{1}{Ψ} {2 W_{311}^{B} {\bar{W}}_{222}^{A} + 4 W_{220}^{B} {\bar{W}}_{131}^{A} + 6 W_{222}^{B} {\bar{W}}_{131}^{A} + 8 W_{131}^{B} {\bar{W}}_{040}^{A}}$	$W_{422 E}^{+} = - \frac{1}{Ψ} {2 W_{311}^{A} {\bar{W}}_{222}^{B} + 4 W_{220}^{A} {\bar{W}}_{131}^{B} + 6 W_{222}^{A} {\bar{W}}_{131}^{B} + 8 W_{131}^{A} {\bar{W}}_{040}^{B}}$
$W_{420 E}^{-} = \frac{1}{Ψ} {2 W_{220}^{B} {\bar{W}}_{131}^{A} + 2 W_{311}^{B} {\bar{W}}_{220}^{A} + 4 W_{131}^{B} {\bar{W}}_{040}^{A}}$	$W_{420 E}^{+} = - \frac{1}{Ψ} {2 W_{220}^{A} {\bar{W}}_{131}^{B} + 2 W_{311}^{A} {\bar{W}}_{220}^{B} + 4 W_{131}^{A} {\bar{W}}_{040}^{B}}$
$W_{511 E}^{-} = \frac{1}{Ψ} {3 W_{311}^{B} {\bar{W}}_{131}^{A} + 8 W_{220}^{B} {\bar{W}}_{040}^{A} + 8 W_{222}^{B} {\bar{W}}_{040}^{A}}$	$W_{511 E}^{+} = - \frac{1}{Ψ} {3 W_{311}^{A} {\bar{W}}_{131}^{B} + 8 W_{220}^{A} {\bar{W}}_{040}^{B} + 8 W_{222}^{A} {\bar{W}}_{040}^{B}}$
$W_{600 E}^{-} = \frac{1}{Ψ} {4 W_{311}^{B} {\bar{W}}_{040}^{A}}$	$W_{600 E}^{+} = - \frac{1}{Ψ} {4 W_{311}^{A} {\bar{W}}_{040}^{B}}$
Added terms when the center of the reference sphere is located at the intersection of the chief ray with the Gaussian image plane
$Δ W_{331 E}^{-} = - W_{311}^{B} Δ^{A} {u^{2}} / 2$	$Δ W_{331 E}^{+} = + W_{311}^{A} Δ^{B} {u^{2}} / 2$
$Δ W_{422 E}^{-} = - W_{311}^{B} Δ^{A} {u \bar{u}}$	$Δ W_{422 E}^{+} = + W_{311}^{A} Δ^{B} {u \bar{u}}$
$Δ W_{420 E}^{-} = - W_{311}^{B} Δ^{A} {u \bar{u}} / 2$	$Δ W_{420 E}^{+} = + W_{311}^{A} Δ^{B} {u \bar{u}} / 2$
$Δ W_{511 E}^{-} = - 3 W_{311}^{B} Δ^{A} {{\bar{u}}^{2}} / 2$	$Δ W_{511 E}^{+} = + 3 W_{311}^{A} Δ^{B} {{\bar{u}}^{2}} / 2$

$4 W_{040} \frac{\bar{y}}{y} = {\bar{W}}_{311} - {\bar{W}}_{311}^{0}$	$4 {\bar{W}}_{040} \frac{y}{\bar{y}} = W_{311} - W_{311}^{0}$
$W_{131} \frac{\bar{y}}{y} = {\bar{W}}_{222} - {\bar{W}}_{311}^{0} \frac{\bar{y}}{y}$	${\bar{W}}_{131} \frac{y}{\bar{y}} = W_{222} - W_{311}^{0} \frac{y}{\bar{y}}$
$W_{222} \frac{\bar{y}}{y} = {\bar{W}}_{131} + W_{311}^{0}$	${\bar{W}}_{222} \frac{y}{\bar{y}} = W_{131} + {\bar{W}}_{311}^{0}$
$W_{220} \frac{\bar{y}}{y} = \frac{1}{2} {\bar{W}}_{131} - \frac{1}{2} {\bar{W}}_{311}^{0} (\frac{\bar{y}}{y})^{2}$	${\bar{W}}_{220} \frac{y}{\bar{y}} = \frac{1}{2} W_{131} - \frac{1}{2} W_{311}^{0} (\frac{y}{\bar{y}})^{2}$
$W_{311} \frac{\bar{y}}{y} = 4 {\bar{W}}_{040} + W_{311}^{0} \frac{\bar{y}}{y}$	${\bar{W}}_{311} \frac{y}{\bar{y}} = 4 W_{040} + {\bar{W}}_{311}^{0} \frac{y}{\bar{y}}$
$W_{311}^{0} = W_{311}^{y = 0} = \frac{1}{2} \frac{1}{R} Ψ \bar{A} \bar{y} Δ {\frac{1}{n}} = \frac{1}{2} Ψ \bar{α} ({\bar{u}}^{'} - \bar{u})$	${\bar{W}}_{311}^{0} = {\bar{W}}_{311}^{\bar{y} = 0} = - \frac{1}{2} \frac{1}{R} Ψ A y Δ {\frac{1}{n}} = - \frac{1}{2} Ψ α (u^{'} - u)$
$α = \frac{y}{r}$	$\bar{α} = \frac{\bar{y}}{r}$

$W_{151}$	$- 0.4193360087176732$	$- 0.4193360087176732$
$W_{242}$	1.3879576582604869	1.3879576582604876
$W_{333}$	$- 1.3737719852861099$	$- 1.3737719852861099$
$W_{240}$	0.1318037474153832	0.1318037474153833
$W_{331}$	$- 1.5376298938401587$	$- 1.5376298938401600$
$W_{422}$	0.4115466232465593	0.4115466232465597
$W_{420}$	2.3841052150066186	2.3841052150066186
$W_{511}$	$- 2.2205687849607498$	$- 2.2205687849607463$

Sixth-Order Aberration Coefficients
$W_{040}$	$W_{131}$	$W_{222}$	$W_{220}$	$W_{311}$
3.095E-013	4.842E-013	7.749E-014	$- 1.033 E - 013$	1.2695E-012
$W_{240}$	$W_{331}$	$W_{422}$	$W_{420}$	$W_{511}$
4.3519E-012	$- 1.3257 E - 013$	4.2807E-014	3.4504E-014	8.6500E-013
$W_{060}$	$W_{151}$	$W_{242}$	$W_{333}$
1.011E-013	7.878E-014	$- 1.176 E - 012$	$- 8.605 E - 014$
Fifth-Order Transverse Aberration Coefficients (mm)
B	F	C	Pi	E
$- 6.71 E - 15$	$- 2.522 E - 015$	$- 3.537 E - 016$	1.414E-015	$- 6.209 E - 015$
B5	F1	F2	M1	M2
$- 3.5428 E - 15$	$- 7.6113 E - 017$	$- 5.187 E - 017$	1.2030E-014	$- 9.6599 E - 014$
N1	N2	N3	C5	Pi5
$- 2.536 E - 016$	$- 4.7702 E - 016$	$- 4.952 E - 017$	$- 1.22 E - 016$	$- 1.0157 E - 016$
M3	E5
1.252E-014	$- 4.851 E - 015$

$W_{040}^{cap} = Δ {n} \cdot A_{4} \cdot y^{4}$	${\bar{W}}_{040}^{cap} = Δ {n} \cdot A_{4} \cdot {\bar{y}}^{4}$
$W_{131}^{cap} = 4 Δ {n} \cdot A_{4} \cdot y^{3} \cdot \bar{y}$	${\bar{W}}_{131}^{cap} = 4 Δ {n} \cdot A_{4} \cdot {\bar{y}}^{3} \cdot y$
$W_{222}^{cap} = 4 Δ {n} \cdot A_{4} \cdot y^{2} \cdot {\bar{y}}^{2}$	${\bar{W}}_{222}^{cap} = 4 Δ {n} \cdot A_{4} \cdot {\bar{y}}^{2} \cdot y^{2}$
$W_{220}^{cap} = 2 Δ {n} \cdot A_{4} \cdot y^{2} \cdot {\bar{y}}^{2}$	${\bar{W}}_{220}^{cap} = 2 Δ {n} \cdot A_{4} \cdot {\bar{y}}^{2} \cdot y^{2}$
$W_{311}^{cap} = 4 Δ {n} \cdot A_{4} \cdot y \cdot {\bar{y}}^{3}$	${\bar{W}}_{311}^{cap} = 4 Δ {n} \cdot A_{4} \cdot \bar{y} \cdot y^{3}$
$W_{400}^{cap} = Δ {n} \cdot A_{4} \cdot {\bar{y}}^{4}$	${\bar{W}}_{400}^{cap} = Δ {n} \cdot A_{4} \cdot y^{4}$

Surface Object	Radius	Thickness 10,000	Glass	$A_{4}$	$A_{6}$
1	255.635318	56.8473	BK7	$- 5.051563 e - 07$	$- 3.2061469 e - 11$
2	62.646002	23.6149		3.5999265e-007	9.4325832e-010
3	74.494599	15.7912	BK7	$- 1.0708598 e-6$	1.151724e-010
4	$- 58.717274$	89.4891
5	$- 42.87839$	5	BK7	4.4688245e-007	$- 3.256491 e - 010$
6	$- 203.330401$	4.2124
7		$- 65.250745$
Stop Image		65.249628