Analytical expressions of the imaging and aberration coefficients of a general form surface

Liu Yang; Jin Wei Qi; Zhu Bin

doi:10.1364/JOSAA.34.002077

Journal of the Optical Society of America A
Vol. 34,
Issue 12,
pp. 2077-2095
(2017)
•https://doi.org/10.1364/JOSAA.34.002077

Analytical expressions of the imaging and aberration coefficients of a general form surface

Liu Yang, Jin Wei Qi, and Zhu Bin

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Liu Yang, Jin Wei Qi, and Zhu Bin, "Analytical expressions of the imaging and aberration coefficients of a general form surface," J. Opt. Soc. Am. A 34, 2077-2095 (2017)

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Abstract

A theoretical development is presented in this paper for describing and understanding the imaging and aberrations of a general form surface. The development is based on the Taylor expansion of an arbitrary ray trace from the object reference plane to the image reference plane, which is called the base mapping of the general form surface in this paper. The base mapping is expressed as two Taylor series of the object and pupil coordinates and the imaging and aberration coefficients in the third-order scope are derived and presented as analytical expressions relevant to the optic parameters, invoking no approximations. The situation with tilted object and observing plane is also considered, and the mapping from a tilted object to a tilted observing plane is derived via simple mathematical manipulations based on the base mapping.

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Equations (89)

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Quantity	Definition
$n$	Refractive index in the object space
$\bar{n}$	Refractive index in the image space
$q$	Sine of the base ray’s angle of incidence at the general form surface, positive (negative) if clockwise (counterclockwise) rotation required to make the base ray reach the surface normal in the object space
$m$	Cosine of the base ray’s angle of incidence at the general form surface
$\bar{q}$	Sine of the base ray’s angle of refraction (reflection) at the general form surface, positive (negative) if clockwise (counterclockwise) rotation required to make the base ray reach the surface normal in the image space
$\bar{m}$	Cosine of the base ray’s angle of refraction (reflection) at the general form surface
$(X, Y, Z)$	Coordinate of the intersection point of an arbitrary ray with the general form surface in $O_{2}$
$(P_{0}, Q_{0}, M_{0})$	Direction cosines of the arbitrary ray before refraction (reflection) in $O_{2}$
$(P, Q, M)$	Direction cosines of the surface normal at the intersection point of the arbitrary ray with the general form surface in $O_{2}$
$(P_{1}, Q_{1}, M_{1})$	Direction cosines of the arbitrary ray after refraction (reflection) in $O_{2}$

	$A_{i j h k}$	$B_{i j h k}$
0001	0	$\frac{L}{L m - m S}$
0010	$\frac{L}{L - S}$	0
0100	0	$B_{0001} \| (L ⇄ S)$
1000	$A_{0010} \| (L ⇄ S)$	0
0002	0	$\frac{L q (L C_{02} - m)}{m^{3} {(L - S)}^{2}}$
0011	$- \frac{L q}{m {(L - S)}^{2}}$	$- \frac{L^{2} q C_{11}}{m^{2} {(L - S)}^{2}}$
0020	0	$\frac{L^{2} q C_{20}}{m {(L - S)}^{2}}$
0101	0	$\frac{q (m (L + S) - 2 LS C_{02})}{m^{3} {(L - S)}^{2}}$
0110	$\frac{q S}{m {(L - S)}^{2}}$	$\frac{L q S C_{11}}{m^{2} {(L - S)}^{2}}$
1001	$A_{0110} \| (L ⇄ S)$	$B_{0110} \| (L ⇄ S)$
1010	0	$- \frac{2 L q S C_{20}}{m {(L - S)}^{2}}$
0200	0	$B_{0002} \| (L ⇄ S)$
1100	$A_{0011} \| (L ⇄ S)$	$B_{0011} \| (L ⇄ S)$
2000	0	$B_{0020} \| (L ⇄ S)$
0003	0	$\frac{L (m^{2} q^{2} - 2 L^{2} C_{02}^{2} q^{2} + L m ((- 2 q^{2} - 1) C_{02} + L q C_{03}))}{m^{5} {(L - S)}^{3}}$
0012	$\frac{L (m q^{2} - L C_{02})}{m^{3} {(L - S)}^{3}}$	$\frac{L^{2} (3 L q^{2} C_{02} C_{11} - m (2 C_{11} q^{2} - L C_{12} q + C_{11}))}{m^{4} {(L - S)}^{3}}$
0021	$\frac{L^{2} C_{11}}{m^{2} {(L - S)}^{3}}$	$\frac{L^{2} (L q (q (C_{11}^{2} + 2 C_{02} C_{20}) + m C_{21}) - m (2 q^{2} + 1) C_{20})}{m^{3} {(L - S)}^{3}}$
0030	$- \frac{L^{2} C_{20}}{m {(L - S)}^{3}}$	$- \frac{L^{3} q (q C_{11} C_{20} + m C_{30})}{m^{2} {(L - S)}^{3}}$
0102	0	$\frac{q^{2} S m^{3} + 2 L ((2 q^{2} + 1) S C_{02} - m q^{2}) m^{2}}{m^{6} {(L - S)}^{3}}$ $+ \frac{L^{2} (- 3 q S C_{03} m^{2} + 6 q^{2} S C_{02}^{2} m + (2 q^{4} - q^{2} - 1) C_{02})}{m^{6} {(L - S)}^{3}}$
0111	$\frac{2 LS C_{02} - m q^{2} (L + S)}{m^{3} {(L - S)}^{3}}$	$\frac{L (m (L + 3 S) C_{11} q^{2} - 6 LS C_{02} C_{11} q^{2} - 2 L m S C_{12} q + m (L + S) C_{11})}{m^{4} {(L - S)}^{3}}$
0120	$- \frac{LS C_{11}}{m^{2} {(L - S)}^{3}}$	$- \frac{L (2 m S C_{20} q^{2} + L (S (C_{11}^{2} + 2 C_{02} C_{20}) q^{2} - m S C_{21} q + m C_{20}))}{m^{3} {(L - S)}^{3}}$
1002	$\frac{L (L C_{02} - m q^{2})}{m^{3} {(L - S)}^{3}}$	$\frac{L (m (q^{2} + 1) S C_{11} + L q (m q C_{11} - 3 q S C_{02} C_{11} - m S C_{12}))}{m^{4} {(L - S)}^{3}}$
1011	$- \frac{L (L + S) C_{11}}{m^{2} {(L - S)}^{3}}$	$- \frac{2 L ((q^{4} - 1) S C_{20} + L m q (q (S C_{11}^{2} + m C_{20} + 2 S C_{02} C_{20}) - m S C_{21}))}{m^{4} {(L - S)}^{3}}$
1020	$\frac{L (L + 2 S) C_{20}}{m {(L - S)}^{3}}$	$\frac{3 L^{2} q S (q C_{11} C_{20} + m C_{30})}{m^{2} {(L - S)}^{3}}$
0201	0	$B_{0102} \| (L ⇄ S)$
0210	$A_{1002} \| (L ⇄ S)$	$B_{1002} \| (L ⇄ S)$
1101	$A_{0111} \| (L ⇄ S)$	$B_{0111} \| (L ⇄ S)$
1110	$A_{1011} \| (L ⇄ S)$	$B_{1011} \| (L ⇄ S)$
2001	$A_{0120} \| (L ⇄ S)$	$B_{0120} \| (L ⇄ S)$
2010	$A_{1020} \| (L ⇄ S)$	$B_{1020} \| (L ⇄ S)$
0300	0	$B_{0003} \| (L ⇄ S)$
2100	$A_{0021} \| (L ⇄ S)$	$B_{0021} \| (L ⇄ S)$
1200	$A_{0012} \| (L ⇄ S)$	$B_{0012} \| (L ⇄ S)$
0300	$A_{0003} \| (L ⇄ S)$	$B_{0003} \| (L ⇄ S)$

Quantity	Definition
$n$	Refractive index in the object space
$\bar{n}$	Refractive index in the image space
$q$	Sine of the base ray’s angle of incidence at the general form surface, positive (negative) if clockwise (counterclockwise) rotation required to make the base ray reach the surface normal in the object space
$m$	Cosine of the base ray’s angle of incidence at the general form surface
$\bar{q}$	Sine of the base ray’s angle of refraction (reflection) at the general form surface, positive (negative) if clockwise (counterclockwise) rotation required to make the base ray reach the surface normal in the image space
$\bar{m}$	Cosine of the base ray’s angle of refraction (reflection) at the general form surface
$(X, Y, Z)$	Coordinate of the intersection point of an arbitrary ray with the general form surface in $O_{2}$
$(P_{0}, Q_{0}, M_{0})$	Direction cosines of the arbitrary ray before refraction (reflection) in $O_{2}$
$(P, Q, M)$	Direction cosines of the surface normal at the intersection point of the arbitrary ray with the general form surface in $O_{2}$
$(P_{1}, Q_{1}, M_{1})$	Direction cosines of the arbitrary ray after refraction (reflection) in $O_{2}$

	$A_{i j h k}$	$B_{i j h k}$
0001	0	$\frac{L}{L m - m S}$
0010	$\frac{L}{L - S}$	0
0100	0	$B_{0001} \| (L ⇄ S)$
1000	$A_{0010} \| (L ⇄ S)$	0
0002	0	$\frac{L q (L C_{02} - m)}{m^{3} {(L - S)}^{2}}$
0011	$- \frac{L q}{m {(L - S)}^{2}}$	$- \frac{L^{2} q C_{11}}{m^{2} {(L - S)}^{2}}$
0020	0	$\frac{L^{2} q C_{20}}{m {(L - S)}^{2}}$
0101	0	$\frac{q (m (L + S) - 2 LS C_{02})}{m^{3} {(L - S)}^{2}}$
0110	$\frac{q S}{m {(L - S)}^{2}}$	$\frac{L q S C_{11}}{m^{2} {(L - S)}^{2}}$
1001	$A_{0110} \| (L ⇄ S)$	$B_{0110} \| (L ⇄ S)$
1010	0	$- \frac{2 L q S C_{20}}{m {(L - S)}^{2}}$
0200	0	$B_{0002} \| (L ⇄ S)$
1100	$A_{0011} \| (L ⇄ S)$	$B_{0011} \| (L ⇄ S)$
2000	0	$B_{0020} \| (L ⇄ S)$
0003	0	$\frac{L (m^{2} q^{2} - 2 L^{2} C_{02}^{2} q^{2} + L m ((- 2 q^{2} - 1) C_{02} + L q C_{03}))}{m^{5} {(L - S)}^{3}}$
0012	$\frac{L (m q^{2} - L C_{02})}{m^{3} {(L - S)}^{3}}$	$\frac{L^{2} (3 L q^{2} C_{02} C_{11} - m (2 C_{11} q^{2} - L C_{12} q + C_{11}))}{m^{4} {(L - S)}^{3}}$
0021	$\frac{L^{2} C_{11}}{m^{2} {(L - S)}^{3}}$	$\frac{L^{2} (L q (q (C_{11}^{2} + 2 C_{02} C_{20}) + m C_{21}) - m (2 q^{2} + 1) C_{20})}{m^{3} {(L - S)}^{3}}$
0030	$- \frac{L^{2} C_{20}}{m {(L - S)}^{3}}$	$- \frac{L^{3} q (q C_{11} C_{20} + m C_{30})}{m^{2} {(L - S)}^{3}}$
0102	0	$\frac{q^{2} S m^{3} + 2 L ((2 q^{2} + 1) S C_{02} - m q^{2}) m^{2}}{m^{6} {(L - S)}^{3}}$ $+ \frac{L^{2} (- 3 q S C_{03} m^{2} + 6 q^{2} S C_{02}^{2} m + (2 q^{4} - q^{2} - 1) C_{02})}{m^{6} {(L - S)}^{3}}$
0111	$\frac{2 LS C_{02} - m q^{2} (L + S)}{m^{3} {(L - S)}^{3}}$	$\frac{L (m (L + 3 S) C_{11} q^{2} - 6 LS C_{02} C_{11} q^{2} - 2 L m S C_{12} q + m (L + S) C_{11})}{m^{4} {(L - S)}^{3}}$
0120	$- \frac{LS C_{11}}{m^{2} {(L - S)}^{3}}$	$- \frac{L (2 m S C_{20} q^{2} + L (S (C_{11}^{2} + 2 C_{02} C_{20}) q^{2} - m S C_{21} q + m C_{20}))}{m^{3} {(L - S)}^{3}}$
1002	$\frac{L (L C_{02} - m q^{2})}{m^{3} {(L - S)}^{3}}$	$\frac{L (m (q^{2} + 1) S C_{11} + L q (m q C_{11} - 3 q S C_{02} C_{11} - m S C_{12}))}{m^{4} {(L - S)}^{3}}$
1011	$- \frac{L (L + S) C_{11}}{m^{2} {(L - S)}^{3}}$	$- \frac{2 L ((q^{4} - 1) S C_{20} + L m q (q (S C_{11}^{2} + m C_{20} + 2 S C_{02} C_{20}) - m S C_{21}))}{m^{4} {(L - S)}^{3}}$
1020	$\frac{L (L + 2 S) C_{20}}{m {(L - S)}^{3}}$	$\frac{3 L^{2} q S (q C_{11} C_{20} + m C_{30})}{m^{2} {(L - S)}^{3}}$
0201	0	$B_{0102} \| (L ⇄ S)$
0210	$A_{1002} \| (L ⇄ S)$	$B_{1002} \| (L ⇄ S)$
1101	$A_{0111} \| (L ⇄ S)$	$B_{0111} \| (L ⇄ S)$
1110	$A_{1011} \| (L ⇄ S)$	$B_{1011} \| (L ⇄ S)$
2001	$A_{0120} \| (L ⇄ S)$	$B_{0120} \| (L ⇄ S)$
2010	$A_{1020} \| (L ⇄ S)$	$B_{1020} \| (L ⇄ S)$
0300	0	$B_{0003} \| (L ⇄ S)$
2100	$A_{0021} \| (L ⇄ S)$	$B_{0021} \| (L ⇄ S)$
1200	$A_{0012} \| (L ⇄ S)$	$B_{0012} \| (L ⇄ S)$
0300	$A_{0003} \| (L ⇄ S)$	$B_{0003} \| (L ⇄ S)$

Abstract

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Equations (89)

Journal of the Optical Society of America A