Aberration Coefficients of Plane Symmetric Systems

P. J. Sands

doi:10.1364/JOSA.62.001211

Journal of the Optical Society of America
Vol. 62,
Issue 10,
pp. 1211-1220
(1972)
•https://doi.org/10.1364/JOSA.62.001211

Aberration Coefficients of Plane Symmetric Systems

P. J. Sands

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P. J. Sands, "Aberration Coefficients of Plane Symmetric Systems," J. Opt. Soc. Am. 62, 1211-1220 (1972)

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Abstract

Given an optical system with a plane of symmetry and an arbitrarily chosen ray contained in this plane, it is shown that it is possible to determine exact analytic formulas for a set of aberration coefficients characterizing the image-forming properties of bundles of rays in the neighborhood (but not necessarily centered on) the given ray. The foundations of the theory are laid, all formulas needed to compute the coefficients of orders one, two, and three are derived, and those simplifications which arise when a surface is a sphere are considered in detail.

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ν even			ν odd
Term	Coefficients		Term	Coefficients
Y	${\hat{q}}_{000}^{(1)}$	${\hat{q}}_{a}$	Z	${\hat{q}}_{110}^{(1)}$	${\hat{q}}_{c}$
V	${\hat{q}}_{100}^{(1)}$	${\hat{q}}_{b}$	W	${\hat{q}}_{111}^{(1)}$	${\hat{q}}_{d}$
Y²	${\hat{q}}_{000}^{(2)}$	ŝ_q₁	YZ	${\hat{q}}_{110}^{(2)}$	ŝ_q₃
YV	${\hat{q}}_{100}^{(2)}$	ŝ_q₂	YW	${\hat{q}}_{111}^{(2)}$	ŝ_q₄
V²	${\hat{q}}_{200}^{(2)}$	ŝ_q₅	VZ	${\hat{q}}_{210}^{(2)}$	ŝ_q₆
Z²	${\hat{q}}_{220} (2)$	ŝ_q₈	VW	${\hat{q}}_{211}^{(2)}$	ŝ_q₇
ZW	${\hat{q}}_{221}^{(2)}$	ŝ_q₉
W²	${\hat{q}}_{222}^{(2)}$	ŝ_q₁₀
Y³	${\hat{q}}_{000}^{(3)}$	${\hat{t}}_{q 1}$	Y²Z	${\hat{q}}_{110}^{(3)}$	${\hat{t}}_{q 3}$
Y²V	${\hat{q}}_{100}^{(3)}$	${\hat{t}}_{q 2}$	Y²W	${\hat{q}}_{111}^{(3)}$	${\hat{t}}_{q 4}$
YV²	${\hat{q}}_{200}^{(3)}$	${\hat{t}}_{q 5}$	YVZ	${\hat{q}}_{210}^{(3)}$	${\hat{t}}_{q 6}$
YZ²	${\hat{q}}_{220}^{(3)}$	${\hat{t}}_{q 8}$	YVW	${\hat{q}}_{211}^{(3)}$	${\hat{t}}_{q 7}$
YZW	${\hat{q}}_{221}^{(3)}$	${\hat{t}}_{q 9}$	V²Z	${\hat{q}}_{310}^{(3)}$	${\hat{t}}_{q 12}$
YW²	${\hat{q}}_{222}^{(3)}$	${\hat{t}}_{q 10}$	V²W	${\hat{q}}_{311}^{(3)}$	${\hat{t}}_{q 13}$
V³	${\hat{q}}_{300}^{(3)}$	${\hat{t}}_{q 11}$	Z³	${\hat{q}}_{330}^{(3)}$	${\hat{t}}_{q 17}$
VZ²	${\hat{q}}_{320}^{(3)}$	${\hat{t}}_{q 14}$	z²W	${\hat{q}}_{331}^{(3)}$	${\hat{t}}_{q 18}$
VZW	${\hat{q}}_{321}^{(3)}$	${\hat{t}}_{q 15}$	ZW²	${\hat{q}}_{322}^{(3)}$	${\hat{t}}_{q 19}$
VW²	${\hat{q}}_{322}^{(3)}$	${\hat{t}}_{q 16}$	W³	${\hat{q}}_{333}^{(3)}$	${\hat{t}}_{q 20}$

${\hat{x}}_{a} = - n_{0_{y}} q$	${\hat{x}}_{b} = 0$
$ŝ_{x 1} = \frac{1}{2} q^{3} c_{y}$	${\hat{S}}_{x 2} = {\hat{x}}_{a}^{2}$
ŝ_x₅ = 0	$ŝ_{x 8} = \frac{1}{2} q c_{z}$
ŝ_x₉ = 0	ŝ_x₁₀ = 0
${\hat{t}}_{x 1} = q^{2} (c_{y} {\hat{V}}_{0} ŝ_{x 1} + q^{2} θ_{30})$	${\hat{t}}_{x 2} = 3 {\hat{x}}_{a} ŝ_{x 1}$
${\hat{t}}_{x 5} = {\hat{x}}_{a} ŝ_{x 2}$	${\hat{t}}_{x 8} = q^{2} (c_{y} {\hat{V}}_{0} ŝ_{x 8} + θ_{32})$
${\hat{t}}_{x 9} = 2 {\hat{x}}_{a} ŝ_{x 8}$	${\hat{t}}_{x 10} = 0$
${\hat{t}}_{x 11} = 0$	${\hat{t}}_{x 14} = {\hat{x}}_{a} ŝ_{x 8}$
${\hat{t}}_{x 15} = 0$	${\hat{t}}_{x 16} = 0$

î_a = qc_y	î_c = c_z
î_b = 1	î_d = 1
$ŝ_{i 1} = 3 q^{2} θ_{30} + c_{y} {\hat{V}}_{0} ŝ_{x 1}$	ŝ_i₃ = 2qθ₃₂
$ŝ_{i 2} = c_{y} q {\hat{x}}_{a}$	$ŝ_{i 4} = c_{2} {\hat{x}}_{a}$
ŝ_i₅ = 0	ŝ_i₆ = 0
$ŝ_{i 8} = θ_{32} + c_{y} {\hat{V}}_{0} ŝ_{x 8}$	ŝ_i₇ = 0
ŝ_i₉ = 0
ŝ_i₁₀ = 0
${\hat{t}}_{i 1} = c_{y} {\hat{V}}_{0} {\hat{t}}_{x 1} + 6 θ_{30} {\hat{V}}_{0} q ŝ_{x 1} + 4 θ_{40} q^{3}$	${\hat{t}}_{i 3} = 2 {\hat{V}}_{0} θ_{32} ŝ_{x 1} + 2 θ_{42} q^{2}$
${\hat{t}}_{i 2} = c_{y} {\hat{V}}_{0} {\hat{t}}_{x 2} + c_{y} ŝ_{x 1} + 6 θ_{30} q^{2} {\hat{x}}_{a}$	${\hat{t}}_{i 4} = c_{z} ŝ_{x 1} + 2 θ_{32} q {\hat{x}}_{a}$
${\hat{t}}_{i 5} = c_{y} q ŝ_{x 2}$	${\hat{t}}_{i 6} = 2 θ_{32} q {\hat{x}}_{a}$
${\hat{t}}_{i 8} = c_{y} {\hat{V}}_{0} {\hat{t}}_{x 8} + 6 θ_{30} {\hat{V}}_{0} q ŝ_{x 8} + 2 θ_{42} q$	${\hat{t}}_{i 7} = c_{z} ŝ_{x 2}$
${\hat{t}}_{i 9} = c_{y} {\hat{V}}_{0} {\hat{t}}_{x 9} + 2 θ_{32} {\hat{x}}_{a}$	${\hat{t}}_{i 12} = 0$
${\hat{t}}_{i 10} = 0$	${\hat{t}}_{i 13} = 0$
${\hat{t}}_{i 11} = 0$	${\hat{t}}_{i 17} = 2 {\hat{V}}_{0} θ_{32} ŝ_{x 8} + 4 θ_{44}$
${\hat{t}}_{i 14} = c_{y} q ŝ_{x 8}$	${\hat{t}}_{i 18} = c_{z} ŝ_{x 8}$
${\hat{t}}_{i 15} = 0$	${\hat{t}}_{i 19} = 0$
${\hat{t}}_{i 16} = 0$	${\hat{t}}_{i 20} = 0$

ρ₁ = q′/2kα₀α₀′ ρ₂ = kα₀q′/2kα₀′ $ρ_{3} = - 2 ρ_{2} {\bar{R}}_{0}$
${\hat{σ}}_{a} = q ρ_{2} {\hat{V}}_{0} c_{y}$	${\hat{τ}}_{a} = q ρ_{3} Î_{0 y} c_{y}$
${\hat{σ}}_{b} = - 2 ρ_{1} α_{0} β_{0} + ρ_{2} ({\hat{V}}_{0} + Î_{0 y})$	${\hat{τ}}_{b} = {\bar{R}}_{0} (ρ_{2} n_{0 y} - {\hat{σ}}_{b})$
$ŝ_{σ 1} = ρ_{2} {\hat{V}}_{0} ŝ_{i 1}$	$ŝ_{τ 1} = ρ_{3} (Î_{0 y} ŝ_{i 1} + \frac{1}{2} q^{2} {c_{y}}^{2})$
$ŝ_{σ 2} = ρ_{2} ({\hat{V}}_{0} ŝ_{i 2} + q c_{y})$	$ŝ_{τ 2} = ρ_{3} (Î_{0 y} ŝ_{i 2} + q c_{y})$
ŝ_σ₅ = ρ₂−ρ₁α₀²	$ŝ_{τ 5} = - {\bar{R}}_{0} ŝ_{σ 5}$
$ŝ_{σ 8} = ρ_{2} {\hat{V}}_{0} ŝ_{i 8}$	$ŝ_{τ 8} = ρ_{3} (Î_{0 y} ŝ_{i 8} + \frac{1}{2} {c_{z}}^{2})$
ŝ_σ₉ = ρ₂c_z	ŝ_τ₉ = ρ₃c_z
ŝ_σ₁₀ = ŝ_σ6	ŝ_τ₁₀ = ŝ_τ5
${\hat{t}}_{σ 1} = ρ_{2} {\hat{V}}_{0} {\hat{t}}_{i 1}$	${\hat{t}}_{τ 1} = ρ_{3} (Î_{0 y} {\hat{t}}_{i 1} + q c_{y} ŝ_{i 1})$
${\hat{t}}_{σ 2} = ρ_{2} ({\hat{V}}_{0} {\hat{t}}_{i 2} + ŝ_{i 1})$	${\hat{t}}_{τ 2} = ρ_{3} (Î_{0 y} {\hat{t}}_{i 2} + q c_{y} ŝ_{i 2} + ŝ_{i 1})$
${\hat{t}}_{σ 5} = ρ_{2} q ŝ_{i 2}$	${\hat{t}}_{τ 5} = ρ_{3} (Î_{0 y} {\hat{t}}_{i 5} + ŝ_{i 2})$
${\hat{t}}_{σ 8} = ρ_{2} {\hat{V}}_{0} {\hat{t}}_{i 8}$	${\hat{t}}_{τ 8} = ρ_{3} (Î_{0 y} {\hat{t}}_{i 8} + q c_{y} ŝ_{i 8} + c_{z} ŝ_{i 3})$
${\hat{t}}_{σ 9} = ρ_{2} ({\hat{V}}_{0} {\hat{t}}_{i 9} + ŝ_{i 3})$	${\hat{t}}_{τ 9} = ρ_{3} (Î_{0 y} {\hat{t}}_{i 9} + c_{z} ŝ_{i 4} + ŝ_{i 3})$
${\hat{t}}_{σ 10} = ρ_{2} ŝ_{i 4}$	${\hat{t}}_{τ 10} = ρ_{3} ŝ_{i 4}$
${\hat{t}}_{σ 11} = 0$	${\hat{t}}_{τ 11} = 0$
${\hat{t}}_{σ 14} = ρ_{2} ({\hat{V}}_{0} {\hat{t}}_{i 14} + ŝ_{i 8})$	${\hat{t}}_{τ 14} = ρ_{3} (Î_{0 y} {\hat{t}}_{i 14} + ŝ_{i 8})$
${\hat{t}}_{σ 15} = 0$	${\hat{t}}_{τ 15} = 0$
${\hat{t}}_{σ 16} = 0$	${\hat{t}}_{τ 16} = 0$

${\hat{r}}_{a} = {\hat{ϕ}}_{a}$	${\hat{r}}_{b} = {\hat{ϕ}}_{b}$
$ŝ_{r 1} = ŝ_{ϕ 1} + {\hat{r}}_{a} {\hat{ψ}}_{a}$	ŝ_τ₈ = ŝ_ϕ₈
$ŝ_{r 2} = ŝ_{ϕ 2} + {\hat{r}}_{a} {\hat{ψ}}_{b} + {\hat{r}}_{b} {\hat{ψ}}_{a}$	ŝ_τ₉ = ŝ_ϕ₉
$ŝ_{r 5} = ŝ_{ϕ 5} + {\hat{r}}_{b} {\hat{ψ}}_{b}$	ŝ_τ₁₀ = ŝ₁₀
${\hat{t}}_{r 1} = {\hat{t}}_{ϕ 1} + {\hat{r}}_{a} (ŝ_{ψ 1} + {\hat{τ}}_{a} {\hat{r}}_{a}) + ŝ_{τ 1} {\hat{ψ}}_{a}$
${\hat{t}}_{r 2} = {\hat{t}}_{ϕ 3} + {\hat{r}}_{a} (ŝ_{ψ 2} + {\hat{τ}}_{a} {\hat{r}}_{b} + {\hat{τ}}_{b} {\hat{r}}_{a}) + {\hat{r}}_{b} (ŝ_{ψ 1} + {\hat{τ}}_{a} {\hat{r}}_{a}) + {\hat{ψ}}_{a} ŝ_{r 2} + {\hat{ψ}}_{a} ŝ_{r 1}$
${\hat{t}}_{r 5} = {\hat{t}}_{ϕ 5} + {\hat{r}}_{a} (ŝ_{ψ 5} + {\hat{τ}}_{a} {\hat{r}}_{b}) + {\hat{r}}_{b} (ŝ_{ψ 2} + {\hat{τ}}_{a} {\hat{r}}_{b} + {\hat{τ}}_{b} {\hat{r}}_{a}) + {\hat{ψ}}_{a} ŝ_{r 5} + {\hat{ψ}}_{b} ŝ_{r 2}$
${\hat{t}}_{r 8} = {\hat{t}}_{ϕ 8} + {\hat{r}}_{a} ŝ_{ψ 8} + {\hat{ψ}}_{a} ŝ_{r 8}$
${\hat{t}}_{r 9} = {\hat{t}}_{ϕ 9} + {\hat{r}}_{a} ŝ_{ψ 9} + {\hat{ψ}}_{a} ŝ_{r 9}$
${\hat{t}}_{r 10} = {\hat{t}}_{ϕ 10} + {\hat{r}}_{a} ŝ_{ψ 10} + {\hat{ψ}}_{a} ŝ_{r 10}$
${\hat{t}}_{r 11} = {\hat{t}}_{ϕ 11} + {\hat{r}}_{b} (ŝ_{ψ 5} + {\hat{τ}}_{b} {\hat{r}}_{b}) + {\hat{ψ}}_{b} ŝ_{r 5}$
${\hat{t}}_{r 14} = {\hat{t}}_{ϕ 14} + {\hat{r}}_{b} ŝ_{ϕ 8} + {\hat{ψ}}_{b} ŝ_{r 8}$
${\hat{t}}_{r 15} = {\hat{t}}_{ϕ 15} + {\hat{r}}_{b} ŝ_{ψ 9} + {\hat{ψ}}_{b} ŝ_{r 9}$
${\hat{t}}_{r 16} = {\hat{t}}_{ϕ 16} + {\hat{r}}_{b} ŝ_{ψ 10} + {\hat{ψ}}_{b} ŝ_{r 10}$

${\hat{p}}_{a} = {\hat{x}}_{a}$	${\hat{p}}_{b} = 0$
$ŝ_{p 1} = ŝ_{x 1} + {\hat{x}}_{a} {\hat{r}}_{a}$	$ŝ_{p 2} = ŝ_{x 2} + {\hat{x}}_{a} {\hat{r}}_{b}$
ŝ_p₅ = 0	ŝ_p₈ = ŝ_x₈
ŝ_p₉ = 0	ŝ_p₁₀ = 0
${\hat{t}}_{p 1} = {\hat{t}}_{x 1} + {\hat{r}}_{a} ŝ_{z 1} + {\hat{x}}_{a} ŝ_{r 1}$	${\hat{t}}_{p 2} = {\hat{t}}_{x 2} + {\hat{r}}_{a} ŝ_{x 2} + {\hat{r}}_{b} ŝ_{x 1} + {\hat{x}}_{a} ŝ_{r 2}$
${\hat{t}}_{p 5} = {\hat{t}}_{x 5} + {\hat{r}}_{a} ŝ_{x 2} + {\hat{x}}_{a} ŝ_{r 5}$	${\hat{t}}_{p 8} = {\hat{t}}_{x 8} + {\hat{r}}_{a} ŝ_{x 8} + {\hat{x}}_{a} ŝ_{r 8}$
${\hat{t}}_{p 9} = {\hat{t}}_{x 9} + {\hat{x}}_{a} ŝ_{r 9}$	${\hat{t}}_{p 10} = {\hat{x}}_{a} ŝ_{r 10}$
${\hat{t}}_{p 11} = 0$	${\hat{t}}_{p 14} = {\hat{t}}_{x 14} + {\hat{r}}_{b} ŝ_{x 8}$
${\hat{t}}_{p 15} = 0$	${\hat{t}}_{p 16} = 0$

$ŝ_{e 1} = R_{0} ŝ_{i 1} + Î_{0 y} ŝ_{r 1} + î_{a} {\hat{r}}_{a}$	$ŝ_{e 3} = R_{0} ŝ_{i 3} + î_{c} {\hat{r}}_{a}$
$ŝ_{e 2} = R_{0} ŝ_{i 2} + Î_{0 y} ŝ_{r 2} + î_{a} {\hat{r}}_{b} + î_{b} {\hat{r}}_{a}$	$ŝ_{e 4} = R_{0} ŝ_{i 4} + î_{d} {\hat{r}}_{a}$
$ŝ_{e 5} = R_{0} ŝ_{i 5} + Î_{0 y} ŝ_{r 5} + î_{b} {\hat{r}}_{b}$	$ŝ_{e 6} = R_{0} ŝ_{i 6} + î_{c} {\hat{r}}_{b}$
ŝ_eα = R₀ŝ_iα+Î₀_yŝ_rα (α=8,9,10)	$ŝ_{e 7} = R_{0} ŝ_{i 7} + î_{d} {\hat{r}}_{b}$
${\hat{t}}_{e 1} = R_{0} {\hat{t}}_{i 1} + Î_{0 y} {\hat{t}}_{r 1} + î_{a} ŝ_{r 1} + ŝ_{a} ŝ_{i 1}$
${\hat{t}}_{e 2} = R_{0} î_{i 2} + Î_{0 y} {\hat{t}}_{r 2} + î_{a} ŝ_{r 2} + î_{b} ŝ_{r 1} + {\hat{r}}_{a} ŝ_{i 2} + {\hat{r}}_{b} ŝ_{i 1}$
${\hat{t}}_{e 5} = R_{0} {\hat{t}}_{i 5} + Î_{0 y} {\hat{t}}_{r 5} + î_{a} ŝ_{r 5} + î_{a} ŝ_{r 2} + î_{b} ŝ_{r 2} + {\hat{r}}_{a} ŝ_{i 5} + {\hat{r}}_{b} ŝ_{i 2}$
$\begin{array}{l} {\hat{t}}_{e α} = R_{0} {\hat{t}}_{i α} + Î_{0 y} {\hat{t}}_{r α} + î_{a} ŝ_{r α} + {\hat{r}}_{a} ŝ_{i α} & (α = 8, 9, 10) \end{array}$
${\hat{t}}_{e 11} = R_{0} {\hat{t}}_{i 11} + Î_{0 y} {\hat{t}}_{r 11} + î_{b} ŝ_{r 5} + {\hat{r}}_{b} ŝ_{i 5}$
${\hat{t}}_{e 14} = R_{0} {\hat{t}}_{i 14} + Î_{0 y} {\hat{t}}_{r 14} + î_{b} ŝ_{r 8} + {\hat{r}}_{b} ŝ_{i 8}$
${\hat{t}}_{e 15} = R_{0} {\hat{t}}_{i 15} + Î_{0 y} {\hat{t}}_{r 15} + î_{b} ŝ_{r 9} + {\hat{r}}_{b} ŝ_{i 9}$
${\hat{t}}_{e 16} = R_{0} {\hat{t}}_{i 16} + Î_{0 y} {\hat{t}}_{r 16} + î_{b} ŝ_{r 10} + {\hat{r}}_{b} ŝ_{i 10}$
${\hat{t}}_{e 3} = R_{0} {\hat{t}}_{i 3} + î_{c} ŝ_{r 1} + {\hat{r}}_{a} ŝ_{i 3}$
${\hat{t}}_{e 4} = R_{0} {\hat{t}}_{i 4} + î_{d} ŝ_{r 1} + {\hat{r}}_{a} ŝ_{i 4}$
${\hat{t}}_{e 6} = R_{0} {\hat{t}}_{i 6} + î_{c} ŝ_{r 2} + {\hat{r}}_{a} ŝ_{i 6} + {\hat{r}}_{b} ŝ_{i 3}$
${\hat{t}}_{e 7} = R_{0} {\hat{t}}_{i 7} + î_{d} ŝ_{r 2} + {\hat{r}}_{a} ŝ_{i 7} + {\hat{r}}_{b} ŝ_{i 4}$
${\hat{t}}_{e 12} = R_{0} {\hat{t}}_{i 12} + î_{c} ŝ_{r 5} + {\hat{r}}_{b} ŝ_{i 6}$
${\hat{t}}_{e 13} = R_{0} {\hat{t}}_{i 13} + î_{d} ŝ_{r 5} + {\hat{r}}_{b} ŝ_{i 7}$
${\hat{t}}_{e 17} = R_{0} {\hat{t}}_{i 17} + î_{c} ŝ_{r 8}$
${\hat{t}}_{e 18} = R_{0} {\hat{t}}_{i 18} + î_{c} ŝ_{r 9} + î_{d} ŝ_{r 8}$
${\hat{t}}_{e 19} = R_{0} {\hat{t}}_{i 19} + î_{c} ŝ_{r 10} + î_{d} ŝ_{r 9}$
${\hat{t}}_{e 20} = R_{0} {\hat{t}}_{i 20} + î_{d} ŝ_{r 10}$

β	ν even, A = a or c

α	1	2		5
1	y_a²	y_aυ_a		υ_a²
2	2y_ay_b	y_aυ_b+y_bυ_a		2υ_aυ_b
5	y_b²	y_bυ_b		υ_b²
β

α	8	9		10
8	z_c²	z_cw_c		w_c²
9	2z_cz_d	z_cw_d+z_dw_c		2w_cw_d
10	z_d²	z_dw_d		w_d²
β	ν odd, A = a or c

α	3	4	6	7
3	y_az_c	y_aw_c	υ_az_c	υ_aw_c
4	y_az_d	y_aw_d	υ_az_d	υ_aw_d
6	y_bz_c	y_bw_c	υ_bz_c	υ_bw_c
7	y_bz_d	y_bw_d	υ_bz_d	υ_bw_d

β	ν even, A = a or c

α	1		2	5		11
1	y_a³		y_a²υ_a	y_aυ_a²		υ_a³
2	3y_a²y_b		y_a(y_aυ_b+2y_bυ_a)	υ_a(2y_aυ_b+ y_bυ_a)		3υ_a²υ_b
5	3y_ay_b²		y_b(2y_aυ_b+y_bυ_a)	υ_b(y_aυ_b+ 2y_bυ_a)		3υ_ab_b²
11	y_b³		y_b²υ_b	y_bυb²		υ_b³
β

α	8	9	10	14	15	16
8	y_az_c²	y_az_cw_c	y_aw_c²	υ_az_c²	υ_az_cw_c	υ_aw_c²
9	2y_az_cz_d	y_a(z_cw_d+z_dw_c)	2y_aw_cw_d	2υ_az_cz_d	υ_a(z_cw_d+z_dw_c)	2υ_aw_cw_d
10	y_az_d²	y_az_dw_d	y_aw_d²	υ_az_d²	υ_az_dw_d	υ_aw_d²
14	y_bz_c²	y_bz_cw_c	y_bw_c²	υ_bz_c²	υ_bz_cw_c	υ_bw_c²
15	2y_bz_cz_d	y_b(z_cw_d+z_dw_c)	2y_bw_cw_d	2υ_bz_cz_d	υb(z_cw_d+z_dw_c)	2υ_bw_cw_d
16	y_bz_d²	y_bz_dw_d	ybwd²	υ_bz_d²	υ_bz_dw_d	υ_bw_d²
β	ν odd, A = b or d

α	3	4	6	7	12	13
3	y_a²z_c	y_a²w_c	y_aυ_az_c	y_aυ_aw_c	υ_a²z_c	υ_a²w_c
4	y_a²z_d	y_a²w_d	y_aυ_az_d	y_aυ_aw_d	υ_a²z_d	υ_a²w_d
6	2y_ay_bz_c	2y_ay_bw_c	(y_aυ_b+y_bυ_a)z_c	(y_aυ_b+y_bυ_a)w_c	2υ_aυ_bz_c	2υ_aυ_bw_c
7	2y_ay_bz_d	2y_ay_bw_d	(y_aυ_b+y_bυ_a)z_d	(y_aυ_b+y_bυ_a)w_d	2υ_aυ_bz_d	2υ_aυ_bw_d
12	y_b²z_c	y_b²w_c	y_bυ_bz_c	y_bυ_bw_c	υ_b²z_c	υ_b²w_c
13	y_b²z_d	y_b²w_d	y_bυ_bz_d	y_bυ_bw_d	υ_b²z_d	υ_b²w_d
β

α	17		18	19	20
17	z_c³		w_cz_c²	z_cw_c²	w_c³
18	3z_c²zd		z_c(z_cw_d+2w_cz_d)	w_c(2z_cw_d+w_cz_d)	3w_c²w_d
19	3z_cz_d²		z_d(2z_cw_d+z_dw_c)	w_d(z_cw_d+2w_cz_d)	3w_cw_d²
20	z_d³		zd²w_d	z_dw_d²	w_d³

${\hat{x}}_{a} = - n_{0 y} q$	${\hat{x}}_{b} = 0$
$ŝ_{x 1} = \frac{1}{2} c q^{3}$	$ŝ_{x 2} = {\hat{x}}_{a}^{2}$
ŝ_x₅ = 0	$ŝ_{x 8} = \frac{1}{2} c q$
ŝ_x₉ = 0	ŝ_x₁₀ = 0
${\hat{t}}_{x 1} = c Î_{0 y} q^{2} ŝ_{x 1}$	${\hat{t}}_{x 2} = {\hat{x}}_{a} ŝ_{x 1}$
${\hat{t}}_{x 5} = {\hat{x}}_{a} ŝ_{x 2}$	${\hat{t}}_{x 8} = c Î_{0 y} q^{2} ŝ_{x 8}$
${\hat{t}}_{x 9} = 2 {\hat{x}}_{a} ŝ_{x 8}$	${\hat{t}}_{x 10} = 0$
${\hat{t}}_{x 11} = 0$	${\hat{t}}_{x 14} = {\hat{x}}_{a} ŝ_{x 8}$
${\hat{t}}_{x 15} = 0$	${\hat{t}}_{x 16} = 0$

ρ₁ = q′/2kα₀α₀′ ρ₂ = kα₀q′/2α₀′ $ρ_{3} = - 2 {\bar{R}}_{0} n_{0 x}$
${\hat{σ}}_{a} = c (ρ_{2} {\hat{V}}_{0} + ρ_{1} {\hat{x}}_{a})$	${\hat{τ}}_{a} = ρ_{3} {\hat{σ}}_{a} + 2 c {\bar{R}}_{0} ρ_{2} n_{0 y}$
${\hat{σ}}_{b} = ρ_{2} (Î_{0 y} + n_{0 x} {\hat{V}}_{0}) - 2 ρ_{1} α_{0} β_{0} n_{0 x}$	${\hat{τ}}_{b} = \frac{1}{2} ρ_{3} ({\hat{σ}}_{b} - ρ_{2} n_{0 y})$
ŝ_σ₁=cρ₁ŝ_z₁	$ŝ_{τ 1} = ρ_{3} ŝ_{σ 1} + c^{2} {\bar{R}}_{0} (ρ_{1} ŝ_{x 2} - ρ_{3})$
$ŝ_{σ 2} = c ρ_{2} + c ρ_{1} (2 α_{0} β_{0} {\hat{x}}_{a} + ŝ_{x 2})$	ŝ_τ₂=ρ₃ŝ_σ₂
ŝ_σ₅=n₀_x(ρ₂−α₀²ρ₁)	$ŝ_{τ 5} = \frac{1}{2} ρ_{3} ŝ_{σ 5}$
ŝ_σ₈=cρ₁ŝ_x₈	$ŝ_{τ 8} = ρ_{3} ŝ_{σ 8} - {\bar{R}}_{0} c^{2} ρ_{2}$
ŝ_σ₉ = cρ₂	ŝ_τ₉=ρ₃ŝ_σ₉
ŝ_σ₁₀=ŝ_σ₅	ŝ_τ₁₀=ŝ_r₅
${\hat{t}}_{σ 1} = c ρ_{1} {\hat{t}}_{x 1}$	${\hat{t}}_{τ 1} = ρ_{3} {\hat{t}}_{σ 1} + 2 c^{2} {\bar{R}}_{0} ρ_{1} ŝ_{x 1} {\hat{x}}_{a}$
${\hat{t}}_{σ 2} = c ρ_{1} ({\hat{t}}_{x 2} + 2 α_{0} β_{0} ŝ_{x 1})$	${\hat{t}}_{τ 2} = ρ_{3} {\hat{t}}_{σ 2} + 2 c^{2} {\bar{R}}_{0} ρ_{1} ŝ_{x 2} ({\hat{x}}_{a} + α_{0} β_{0})$
${\hat{t}}_{σ 5} = c ρ_{1} ({\hat{t}}_{x 5} + 2 α_{0} β_{0} ŝ_{x 2} + {α_{0}}^{2} {\hat{x}}_{a})$	${\hat{t}}_{τ 5} = ρ_{3} {\hat{t}}_{σ 5}$
${\hat{t}}_{σ 8} = c ρ_{1} {\hat{t}}_{x 8}$	${\hat{t}}_{τ 8} = ρ_{3} {\hat{t}}_{σ 8} + c {\bar{R}}_{0} {\hat{t}}_{σ 8}$
${\hat{t}}_{σ 9} = c ρ_{1} {\hat{t}}_{x 9}$	${\hat{t}}_{τ 9} = ρ_{3} {\hat{t}}_{σ 9}$
${\hat{t}}_{σ 10} = c ρ_{1} {α_{0}}^{2} {\hat{x}}_{a}$	${\hat{t}}_{τ 10} = ρ_{3} {\hat{t}}_{σ 10}$
${\hat{t}}_{σ 11} = 0$	${\hat{t}}_{τ 11} = 0$
${\hat{t}}_{σ 14} = c ρ_{1} ({\hat{t}}_{x 14} + 2 α_{0} β_{0} ŝ_{x 8})$	${\hat{t}}_{τ 14} = ρ_{3} {\hat{t}}_{σ 14}$
${\hat{t}}_{σ 15} = 0$	${\hat{t}}_{τ 15} = 0$
${\hat{t}}_{σ 16} = 0$	${\hat{t}}_{τ 16} = 0$

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Journal of the Optical Society of America