General method for the determination of matrix coefficients for high-order optical system modeling

José B. Almeida

doi:10.1364/JOSAA.16.000596

Journal of the Optical Society of America A
Vol. 16,
Issue 3,
pp. 596-601
(1999)
•https://doi.org/10.1364/JOSAA.16.000596

General method for the determination of matrix coefficients for high-order optical system modeling

José B. Almeida

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José B. Almeida, "General method for the determination of matrix coefficients for high-order optical system modeling," J. Opt. Soc. Am. A 16, 596-601 (1999)

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Abstract

The nonlinear transformations incurred by the rays in an optical system can be suitably described by matrices to any desired order of approximation. In systems composed of uniform refractive-index elements each individual ray refraction or translation has an associated matrix, and a succession of transformations corresponds to the product of the respective matrices. A general method is described to find the matrix coefficients for translation and surface refraction, irrespective of the surface shape or the order of approximation. The choice of coordinates is unusual, as the orientation of the ray is characterized by the direction cosines rather than by the slopes; this is shown to greatly simplify and generalize coefficient calculation. Two examples are shown in order to demonstrate the power of the method: The first is the determination of seventh-order coefficients for spherical surfaces, and the second is the determination of third-order coefficients for a toroidal surface.

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Monomial	Refraction $(S^{'})$	Forward Offset $(X_{1})$	Backward Offset $(X_{2})$
X	$(ν - 1) / r$	1	1
S	ν	0	0
$X^{2} X^{*}$	$ν (ν - 1) / 2 r^{3}$	0	0
$X^{2} S^{*}$	$ν (ν - 1) / 2 r^{2}$	0	0
${XX}^{*} S$	$ν (ν - 1) / 2 r^{2}$	$1 / 2 r$	$- 1 / 2 r$
${XSS}^{*}$	$ν (ν - 1) / 2 r$	0	0
$X^{*} S^{2}$	0	0	0
$S^{2} S^{*}$	0	0	0
$X^{3} X^{* 2}$	$ν (ν^{3} - 1) / 8 r^{5}$	0	0
$X^{3} X^{} S^{}$	$ν^{2} (ν^{2} - 1) / 4 r^{4}$	0	0
$X^{3} S^{* 2}$	$ν^{2} (ν^{2} - 1) / 8 r^{3}$	0	0
$X^{2} X^{* 2} S$	$ν^{2} (ν^{2} - 1) / 4 r^{4}$	$1 / 8 r^{3}$	$- 1 / 8 r^{3}$
$X^{2} X^{} {SS}^{}$	$ν (2 ν^{3} - 3 ν + 1) / 4 r^{3}$	0	$1 / 4 r^{2}$
$X^{2} {SS}^{* 2}$	$ν^{2} (ν^{2} - 1) / 4 r^{2}$	0	0
${XX}^{* 2} S^{2}$	$ν^{2} (ν^{2} - 1) / 8 r^{3}$	0	$1 / 4 r^{2}$
${XX}^{} S^{2} S^{}$	$ν^{2} (ν^{2} - 1) / 4 r^{2}$	$1 / 4 r$	$- 1 / 4 r$
${XS}^{2} S^{* 2}$	$ν (ν^{3} - 1) / 8 r$	0	0
$X^{* 2} S^{3}$	0	0	0
$X^{} S^{3} S^{}$	0	0	0
$S^{3} S^{* 2}$	0	0	0
$X^{4} X^{* 3}$	$ν (ν^{5} - 1) / 16 r^{7}$	0	0
$X^{4} X^{* 2} S^{*}$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{6}$	0	0
$X^{4} X^{} S^{ 2}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{5}$	0	0
$X^{4} S^{* 3}$	$ν^{4} (ν^{2} - 1) / 16 r^{4}$	0	0
$X^{3} X^{* 3} S$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{6}$	$1 / 16 r^{5}$	$- 1 / 16 r^{5}$
$X^{3} X^{* 2} {SS}^{*}$	$ν (9 ν^{5} - 10 ν^{3} + 1) / 16 r^{5}$	0	$3 / 16 r^{4}$
$X^{3} X^{} {SS}^{ 2}$	$ν^{2} (9 ν^{4} - 11 ν^{2} + 2) / 16 r^{4}$	0	$- 1 / 8 r^{3}$
$X^{3} {SS}^{* 3}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{3}$	0	0
$X^{2} X^{* 3} S^{2}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{5}$	0	$3 / 16 r^{4}$
$X^{2} X^{* 2} S^{2} S^{*}$	$ν^{2} (9 ν^{4} - 11 ν^{2} + 2) / 16 r^{4}$	$1 / 16 r^{3}$	$- 7 / 16 r^{3}$
$X^{2} X^{} S^{2} S^{ 2}$	$ν (9 ν^{5} - 10 ν^{3} + 1) / 16 r^{3}$	0	$1 / 4 r^{2}$
$X^{2} S^{2} S^{* 3}$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{2}$	0	0
${XX}^{* 3} S^{3}$	$ν^{4} (ν^{2} - 1) / 16 r^{4}$	0	$- 1 / 8 r^{3}$
${XX}^{* 2} S^{3} S^{*}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{3}$	0	$1 / 4 r^{2}$
${XX}^{} S^{3} S^{ 2}$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{2}$	$3 / 16 r$	$- 3 / 16 r$
${XS}^{3} S^{* 3}$	$ν (ν^{5} - 1) / 16 r$	0	0
$X^{* 3} S^{4}$	0	0	0
$X^{* 2} S^{4} S^{*}$	0	0	0
$X^{} S^{4} S^{ 2}$	0	0	0
$S^{4} S^{* 3}$	0	0	0

Monomial	Refraction $(s^{'})$	Refraction $(t^{'})$	Forward Offset $(x^{'})$	Forward Offset $(y^{'})$
x	$\frac{ν - 1}{r_{1}}$	0	1	0
y	0	$\frac{ν - 1}{r_{1} + r_{2}}$	0	1
s	0	0	0	0
t	0	0	0	0
$x^{3}$	$\frac{ν (ν - 1)}{2 r_{1}^{3}}$	0	0	0
$x^{2} y$	0	$\frac{ν (ν - 1)}{2 r_{1}^{2} (r_{1} + r_{2})}$	0	0
$x^{2} s$	$\frac{ν (ν - 1)}{r_{1}^{2}}$	0	$\frac{1}{2 r_{1}}$	0
$x^{2} t$	0	0	$\frac{1}{2 r_{1}}$	0
${xy}^{2}$	$\frac{ν (ν - 1)}{2 r_{1} (r_{1} + r_{2})^{2}}$	0	0	0
xys	0	$\frac{ν (ν - 1)}{r_{1} (r_{1} + r_{2})}$	0	0
xyt	$\frac{ν (ν - 1)}{r_{1} (r_{1} + r_{2})}$	0	0	0
${xs}^{2}$	$\frac{ν (ν - 1)}{2 r_{1}}$	0	0	0
xst	0	0	0	0
${xt}^{2}$	$\frac{ν (ν - 1)}{2 r_{1}}$	0	0	0
$y^{3}$	0	$\frac{ν (ν - 1)}{2 (r_{1} + r_{2})^{3}}$	0	0
$y^{2} s$	0	0	0	$\frac{1}{2 (r_{1} + r_{2})}$
$y^{2} t$	0	$\frac{ν (ν - 1)}{(r_{1} + r_{2})^{2}}$	0	$\frac{1}{2 (r_{1} + r_{2})}$
${ys}^{2}$	0	$\frac{ν (ν - 1)}{2 (r_{1} + r_{2})}$	0	0
yst	0	0	0	0
${yt}^{2}$	0	$\frac{ν (ν - 1)}{2 (r_{1} + r_{2})}$	0	0
$s^{3}$	0	0	0	0
$s^{2} t$	0	0	0	0
${st}^{2}$	0	0	0	0
$t^{3}$	0	0	0	0

Monomial	Refraction $(S^{'})$	Forward Offset $(X_{1})$	Backward Offset $(X_{2})$
X	$(ν - 1) / r$	1	1
S	ν	0	0
$X^{2} X^{*}$	$ν (ν - 1) / 2 r^{3}$	0	0
$X^{2} S^{*}$	$ν (ν - 1) / 2 r^{2}$	0	0
${XX}^{*} S$	$ν (ν - 1) / 2 r^{2}$	$1 / 2 r$	$- 1 / 2 r$
${XSS}^{*}$	$ν (ν - 1) / 2 r$	0	0
$X^{*} S^{2}$	0	0	0
$S^{2} S^{*}$	0	0	0
$X^{3} X^{* 2}$	$ν (ν^{3} - 1) / 8 r^{5}$	0	0
$X^{3} X^{} S^{}$	$ν^{2} (ν^{2} - 1) / 4 r^{4}$	0	0
$X^{3} S^{* 2}$	$ν^{2} (ν^{2} - 1) / 8 r^{3}$	0	0
$X^{2} X^{* 2} S$	$ν^{2} (ν^{2} - 1) / 4 r^{4}$	$1 / 8 r^{3}$	$- 1 / 8 r^{3}$
$X^{2} X^{} {SS}^{}$	$ν (2 ν^{3} - 3 ν + 1) / 4 r^{3}$	0	$1 / 4 r^{2}$
$X^{2} {SS}^{* 2}$	$ν^{2} (ν^{2} - 1) / 4 r^{2}$	0	0
${XX}^{* 2} S^{2}$	$ν^{2} (ν^{2} - 1) / 8 r^{3}$	0	$1 / 4 r^{2}$
${XX}^{} S^{2} S^{}$	$ν^{2} (ν^{2} - 1) / 4 r^{2}$	$1 / 4 r$	$- 1 / 4 r$
${XS}^{2} S^{* 2}$	$ν (ν^{3} - 1) / 8 r$	0	0
$X^{* 2} S^{3}$	0	0	0
$X^{} S^{3} S^{}$	0	0	0
$S^{3} S^{* 2}$	0	0	0
$X^{4} X^{* 3}$	$ν (ν^{5} - 1) / 16 r^{7}$	0	0
$X^{4} X^{* 2} S^{*}$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{6}$	0	0
$X^{4} X^{} S^{ 2}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{5}$	0	0
$X^{4} S^{* 3}$	$ν^{4} (ν^{2} - 1) / 16 r^{4}$	0	0
$X^{3} X^{* 3} S$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{6}$	$1 / 16 r^{5}$	$- 1 / 16 r^{5}$
$X^{3} X^{* 2} {SS}^{*}$	$ν (9 ν^{5} - 10 ν^{3} + 1) / 16 r^{5}$	0	$3 / 16 r^{4}$
$X^{3} X^{} {SS}^{ 2}$	$ν^{2} (9 ν^{4} - 11 ν^{2} + 2) / 16 r^{4}$	0	$- 1 / 8 r^{3}$
$X^{3} {SS}^{* 3}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{3}$	0	0
$X^{2} X^{* 3} S^{2}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{5}$	0	$3 / 16 r^{4}$
$X^{2} X^{* 2} S^{2} S^{*}$	$ν^{2} (9 ν^{4} - 11 ν^{2} + 2) / 16 r^{4}$	$1 / 16 r^{3}$	$- 7 / 16 r^{3}$
$X^{2} X^{} S^{2} S^{ 2}$	$ν (9 ν^{5} - 10 ν^{3} + 1) / 16 r^{3}$	0	$1 / 4 r^{2}$
$X^{2} S^{2} S^{* 3}$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{2}$	0	0
${XX}^{* 3} S^{3}$	$ν^{4} (ν^{2} - 1) / 16 r^{4}$	0	$- 1 / 8 r^{3}$
${XX}^{* 2} S^{3} S^{*}$	$3 ν^{4} (ν^{2} - 1) / 16 r^{3}$	0	$1 / 4 r^{2}$
${XX}^{} S^{3} S^{ 2}$	$ν^{2} (3 ν^{4} - 2 ν^{2} - 1) / 16 r^{2}$	$3 / 16 r$	$- 3 / 16 r$
${XS}^{3} S^{* 3}$	$ν (ν^{5} - 1) / 16 r$	0	0
$X^{* 3} S^{4}$	0	0	0
$X^{* 2} S^{4} S^{*}$	0	0	0
$X^{} S^{4} S^{ 2}$	0	0	0
$S^{4} S^{* 3}$	0	0	0

Monomial	Refraction $(s^{'})$	Refraction $(t^{'})$	Forward Offset $(x^{'})$	Forward Offset $(y^{'})$
x	$\frac{ν - 1}{r_{1}}$	0	1	0
y	0	$\frac{ν - 1}{r_{1} + r_{2}}$	0	1
s	0	0	0	0
t	0	0	0	0
$x^{3}$	$\frac{ν (ν - 1)}{2 r_{1}^{3}}$	0	0	0
$x^{2} y$	0	$\frac{ν (ν - 1)}{2 r_{1}^{2} (r_{1} + r_{2})}$	0	0
$x^{2} s$	$\frac{ν (ν - 1)}{r_{1}^{2}}$	0	$\frac{1}{2 r_{1}}$	0
$x^{2} t$	0	0	$\frac{1}{2 r_{1}}$	0
${xy}^{2}$	$\frac{ν (ν - 1)}{2 r_{1} (r_{1} + r_{2})^{2}}$	0	0	0
xys	0	$\frac{ν (ν - 1)}{r_{1} (r_{1} + r_{2})}$	0	0
xyt	$\frac{ν (ν - 1)}{r_{1} (r_{1} + r_{2})}$	0	0	0
${xs}^{2}$	$\frac{ν (ν - 1)}{2 r_{1}}$	0	0	0
xst	0	0	0	0
${xt}^{2}$	$\frac{ν (ν - 1)}{2 r_{1}}$	0	0	0
$y^{3}$	0	$\frac{ν (ν - 1)}{2 (r_{1} + r_{2})^{3}}$	0	0
$y^{2} s$	0	0	0	$\frac{1}{2 (r_{1} + r_{2})}$
$y^{2} t$	0	$\frac{ν (ν - 1)}{(r_{1} + r_{2})^{2}}$	0	$\frac{1}{2 (r_{1} + r_{2})}$
${ys}^{2}$	0	$\frac{ν (ν - 1)}{2 (r_{1} + r_{2})}$	0	0
yst	0	0	0	0
${yt}^{2}$	0	$\frac{ν (ν - 1)}{2 (r_{1} + r_{2})}$	0	0
$s^{3}$	0	0	0	0
$s^{2} t$	0	0	0	0
${st}^{2}$	0	0	0	0
$t^{3}$	0	0	0	0

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Equations (20)

Journal of the Optical Society of America A