Abstract

Current commercial software for analysis and design of optical systems use finite difference (FD) approximation methodology to estimate the gradient matrix of a ray with respect to system variables. However, FD estimates are intrinsically inaccurate, subject to gross error when the denominator is excessively small relative to the numerator. We avoid these problems and determine these gradients by the application of Snell’s law. We give the background and basics for determining the first-order gradients of skew rays of optical systems, whereby the differential vector of any ray can be estimated by the product of the developed gradient matrix and differential changes of system variables. The most important application is for optical design by use of optimization methods where the merit function is defined as the spot size. FD used for such optimization is slow for large systems and subject to inaccuracy. The presented methodology is shown to be accurate and computationally faster than traditional FD. Two illustrative examples are provided to validate the proposed method.

© 2007 Optical Society of America

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  1. P. D. Lin and C. K. Sung, "Camera calibration based on Snell's law," J. Dyn. Syst., Meas., Control 128, 548-557 (2006).
    [CrossRef]
  2. B. D. Stone, "Determination of initial ray configurations for asymmetric systems," J. Opt. Soc. Am. A 14, 3415-3429 (1997).
    [CrossRef]
  3. D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976, abstract).
    [CrossRef]
  4. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
    [CrossRef] [PubMed]
  5. C. J. Progler and D. M. Byrne, "Merit functions for lithographic lens design," J. Vac. Sci. Technol. B 14, 3714-3718 (1996).
    [CrossRef]
  6. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).
  7. P. D. Lin and C. H. Lu, "Analysis and design of optical systems by use of sensitivity analysis of skew ray tracing," Appl. Opt. 43, 796-807 (2004).
    [CrossRef] [PubMed]

2006 (1)

P. D. Lin and C. K. Sung, "Camera calibration based on Snell's law," J. Dyn. Syst., Meas., Control 128, 548-557 (2006).
[CrossRef]

2004 (1)

1997 (1)

1996 (1)

C. J. Progler and D. M. Byrne, "Merit functions for lithographic lens design," J. Vac. Sci. Technol. B 14, 3714-3718 (1996).
[CrossRef]

1989 (1)

1976 (1)

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976, abstract).
[CrossRef]

Burkhard, D. G.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
[CrossRef] [PubMed]

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976, abstract).
[CrossRef]

Byrne, D. M.

C. J. Progler and D. M. Byrne, "Merit functions for lithographic lens design," J. Vac. Sci. Technol. B 14, 3714-3718 (1996).
[CrossRef]

Kassim, A. M.

Lin, P. D.

P. D. Lin and C. K. Sung, "Camera calibration based on Snell's law," J. Dyn. Syst., Meas., Control 128, 548-557 (2006).
[CrossRef]

P. D. Lin and C. H. Lu, "Analysis and design of optical systems by use of sensitivity analysis of skew ray tracing," Appl. Opt. 43, 796-807 (2004).
[CrossRef] [PubMed]

Lu, C. H.

Paul, R. P.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

Progler, C. J.

C. J. Progler and D. M. Byrne, "Merit functions for lithographic lens design," J. Vac. Sci. Technol. B 14, 3714-3718 (1996).
[CrossRef]

Shealy, D. L.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
[CrossRef] [PubMed]

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976, abstract).
[CrossRef]

Stone, B. D.

Sung, C. K.

P. D. Lin and C. K. Sung, "Camera calibration based on Snell's law," J. Dyn. Syst., Meas., Control 128, 548-557 (2006).
[CrossRef]

Appl. Opt. (2)

J. Dyn. Syst., Meas., Control (1)

P. D. Lin and C. K. Sung, "Camera calibration based on Snell's law," J. Dyn. Syst., Meas., Control 128, 548-557 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976, abstract).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Vac. Sci. Technol. B (1)

C. J. Progler and D. M. Byrne, "Merit functions for lithographic lens design," J. Vac. Sci. Technol. B 14, 3714-3718 (1996).
[CrossRef]

Other (1)

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

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Figures (5)

Fig. 1
Fig. 1

Ray tracing through a flat boundary surface.

Fig. 2
Fig. 2

Ray tracing through a spherical boundary surface.

Fig. 3
Fig. 3

Variables of a boundary surface.

Fig. 4
Fig. 4

Variables of an element.

Fig. 5
Fig. 5

General notation for ray tracing through a system with k elements and n boundary surfaces.

Tables (4)

Tables Icon

Table 1 Proposed Gradient Method Compared with the Traditional FD Method

Tables Icon

Table 2 Parameters and Variables of a Camera with k = 3 (Unit, millimeter)

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Table 3 Results from Optimization Method Based on Rms Spot Size

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Table 4 Parameters and Variables of a Camera with k = 6 (Unit, millimeter)

Equations (47)

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r i i = [ β i C α i 0 β i S α i 1 ] T
n i i = s i [ 0 1 0 0 ] T
A 0 b i = Trans ( t b i x , t b i y , t b i z ) Rot ( z , ω b i z ) Rot ( y , ω b i y ) Rot ( x , ω b i x ) = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ]
n i = [ n i x n i y n i z 0 ] T = A b i 0 n i i = ( A 0 b i ) 1 n i i = s i [ I i y J i y K i y 0 ] T ,
n i = [ n i x n i y n i z 0 ] T = A b i 0 n i i = ( A 0 b i ) 1 n i i = s i [ I i x C β i C α i + I i y C β i S α i + I i z S β i J i x C β i C α i + J i y C β i S α i + J i z S β i K i x C β i C α i + K i y C β i S α i + K i z S β i 0 ] T .
P b i = [ P i x P i y P i z 1 ] T = [ P i 1 x + l i 1 x λ i P i 1 y + l i 1 y λ i P i 1 z + l i 1 z λ i 1 ] T ,
λ i = ( I i y P i 1 x + J i y P i 1 y + K i y P i 1 z + t i y ) I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ,
λ i = D i ± D i 2 E i ,
D i = t i x ( I i x l i 1 x + J i x l i 1 y + K i x l i 1 z ) + t i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z )
+ t i z ( I i z l i 1 x + J i z l i 1 y + K i z l i 1 z ) + P i 1 x l i 1 x + P i 1 y l i 1 y + P i 1 z l i 1 z ,
E i = P i 1 x 2 + P i 1 y 2 + P i 1 z 2 + t i x 2 + t i y 2 + t i z 2 R i 2 + 2 t i x ( I i x P i 1 x + J i x P i 1 y + K i x P i 1 z )
+ 2 t i y ( I i y P i 1 x + J i y P i 1 y + K i y P i 1 z ) + 2 t i z ( I i z P i 1 x + J i z P i 1 y + K i z P i 1 z ) .
α i = arctan [ I i y ( P i 1 x + l i 1 x λ i ) + J i y ( P i 1 y + l i 1 y λ i ) + K i y ( P i 1 z + l i 1 z λ i ) + t i y I i x ( P i 1 x + l i 1 x λ i ) + J i x ( P i 1 y + l i 1 y λ i ) + K i x ( P i 1 z + l i 1 z λ i ) + t i x ] ,
β i = arcsin [ I i z ( P i 1 x + l i 1 x λ i ) + J i z ( P i 1 y + l i 1 y λ i ) + K i z ( P i 1 , z + l i 1 z λ i ) + t i z R i ] .
C θ i = l b i 1 T n i = s i ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) ,
C θ i = l b i 1 T n i = s i [ l i 1 x ( I i x C β i C α i + I i y C β i S α i + I i z S β i ) + l i 1 y ( J i x C β i C α i + J i y C β i S α i + J i z S β i ) + l i 1 z ( K i x C β i C α i + K i y C β i S α i + K i z S β i ) ] .
S θ ̱ i = ( ξ i 1 ξ i ) S θ i = N i S θ i ,
l b i = [ l i x l i y l i z 0 ] = [ n i x 1 N i 2 + ( N i C θ i ) 2 + N i ( l i 1 x + n i x C θ i ) n i y 1 N i 2 + ( N i C θ i ) 2 + N i ( l i 1 y + n i y C θ i ) n i z 1 N i 2 + ( N i C θ i ) 2 + N i ( l i 1 z + n i z C θ i ) 0 ] ,
[ P b n l b n ] T = [ P e k l e k ] T = [ P n 1 x + l n 1 x λ n P n 1 y + l n 1 y λ n P n 1 z + l n 1 z λ n l n x l n y l n z ] T .
X b i
= [ t b i x t b i y t b i z ω b i x ω b i y ω b i z ξ i 1 R i ] T
[ Δ P b i Δ l b i ] = [ P b i P b i 1 P b i l b i 1 l b i P b i 1 l b i l b i 1 ] [ Δ P b i 1 Δ l b i 1 ] + [ P b i X b i l b i X b i ] Δ X b i + [ P b i ξ i l b i ξ i ] Δ ξ i = M i [ Δ P b i 1 Δ l b i 1 ] + [ P b i X b i l b i X b i ] Δ X b i + [ P b i ξ i l b i ξ i ] Δ ξ i ,
Q i = [ I i x I i y I i z J i x J i y J i z K i x K i y K i z t i x t i y t i z ξ i 1 R i ] T .
A 0 e j = A 0 b 2 j 1 = Trans ( t e j x , t e j y , t e j z ) Rot ( z , ω e j z ) Rot ( y , ω e j y ) Rot ( x , ω e j x ) .
A e j b 2 j = Trans ( 0 , 0 , d j ) ,
A 0 b 2 j = A e j b 2 j A 0 e j = Trans ( t e j x , t e j y , t e j z ) Rot ( z , ω e j z ) Rot ( y , ω e j y ) Rot ( x , ω e j x ) Trans ( 0 , 0 , d j ) .
X e j = [ t e j x t e j y t e j z ω e j x ω e j y ω e j z ξ 2 j 2 R 2 j 1 ξ 2 j 1 q j R 2 j ] T .
[ Δ P b 2 j 1 Δ l b 2 j 1 ] = M 2 j 1 [ Δ P b 2 j 2 Δ l b 2 j 2 ] + [ P b 2 j 1 X b 2 j 1 l b 2 j 1 X b 2 j 1 ] Δ X b 2 j 1 + [ P b 2 j 1 ξ 2 j 1 l b 2 j 1 ξ 2 j 1 ] Δ ξ 2 j 1 ,
[ Δ P b 2 j Δ l b 2 j ] = M 2 j [ Δ P b 2 j 1 Δ l b 2 j 1 ] + [ P b 2 j X b 2 j l b 2 j X b 2 j ] Δ X b 2 j + [ P b 2 j ξ 2 j l b 2 j ξ 2 j ] Δ ξ 2 j .
[ Δ P e j Δ l e j ] = M 2 j M 2 j 1 [ Δ P e j 1 Δ l e j 1 ] + M 2 j [ P b 2 j 1 X b 2 j 1 l b 2 j 1 X b 2 j 1 ] Δ X b 2 j 1 + [ P b 2 j X b 2 j l b 2 j X b 2 j ] Δ X b 2 j + M 2 j [ P b 2 j 1 ξ 2 j 1 l b 2 j 1 ξ 2 j 1 ] Δ ξ 2 j 1 + [ P b 2 j ξ 2 j l b 2 j ξ 2 j ] Δ ξ 2 j .
[ Δ P e j Δ l e j ] = M 2 j M 2 j 1 [ Δ P e j 1 Δ l e j 1 ] + [ P e j X e j l e j X e j ] Δ X e j + [ P e j ξ 2 j l e j ξ 2 j ] Δ ξ 2 j .
[ Δ P e k Δ l e k ] = M 2 k 1 [ Δ P b 2 k 2 Δ l b 2 k 2 ] + [ P b 2 k 1 X b 2 k 1 l b 2 k 1 X b 2 k 1 ] Δ X b 2 k 1 = M 2 k 1 [ Δ P e k 1 Δ l e k 1 ] + ( [ P b 2 k 1 X b 2 k 1 l b 2 k 1 X b 2 k 1 ] [ I 8 × 8 0 8 × 3 ] ) Δ X e k = M 2 k 1 [ Δ P e k 1 Δ l e k 1 ] + [ P b 2 k 1 X e k l b 2 k 1 X e k ] Δ X e k .
[ Δ P e k Δ l e k ] = M 2 k 1 M 2 k 2 M 2 k 3 [ Δ P e k 2 Δ l e k 2 ] + M 2 k 1 [ P b 2 k 2 X e k 1 l b 2 k 2 X e k 1 ] Δ X e k 1 + M 2 k 1 [ P b 2 k 2 ξ 2 k 2 l b 2 k 2 ξ 2 k 2 ] Δ ξ 2 k 2 + [ P b 2 k 1 X e k l b 2 k 1 X e k ] Δ X e k .
[ Δ P e k Δ l e k ] = M 2 k 1 M 2 k 2 M 2 k 3 [ Δ P e k 2 Δ l e k 2 ] + M 2 k 1 [ P e k 1 X e k 1 l e k 1 X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k .
[ Δ P e k Δ l e k ] = [ P e k X e 0 l e k X e 0 ] Δ X e 0 + [ P e k X e 1 l e k X e 1 ] Δ X e 1 + + [ P e k X e k 1 l e k X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k = [ P e k X s y s t e m l e k X s y s t e m ] Δ X s y s t e m ,
[ Δ P e 0 Δ l e o ] = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 S β 0 C α 0 C β 0 S α 0 0 0 0 S β 0 S α 0 C β 0 S α 0 0 0 0 C β 0 0 ] [ Δ P 0 x Δ P 0 y Δ P 0 z Δ β 0 Δ α 0 ] = M 0 Δ X e 0 .
Φ = 1 q Φ P e 0 = 1 q 1 m [ ( P k x P k x , c h i e f ) 2 + ( P k z P k z , c h i e f ) 2 ] .
( P e 0 = [ 20 55 12 1 ] T , P e 0 = [ 22.5 55 13.5 1 ] T , P e 0 = [ 25 55 15 1 ] T ) .
Trans ( t b i x , t b i y , t b i z ) = [ 1 0 0 t b i x 0 1 0 t b i y 0 0 1 t b i z 0 0 0 1 ] ,
Rot ( x , ω b i x ) = [ 1 0 0 0 0 C ω b i x S ω b i x 0 0 S ω b i x C ω b i x 0 0 0 0 1 ] ,
Rot ( y , ω b i y ) = [ C ω b i y 0 S ω b i y 0 0 1 0 0 S ω b i y 0 C ω b i y 0 0 0 0 1 ] ,
Rot ( z , ω b i z ) = [ C ω b i z S ω b i z 0 0 S ω b i z C ω b i z 0 0 0 0 1 0 0 0 0 1 ] ,
M 2 j [ P b 2 j 1 X b 2 j 1 l b 2 j 1 X b 2 j 1 ] Δ X b 2 j 1 + [ P b 2 j X b 2 j l b 2 j X b 2 j ] Δ X b 2 j + M 2 j [ P b 2 j 1 ξ 2 j 1 l b 2 j 1 ξ 2 j 1 ] Δ ξ 2 j 1
= M 2 j [ P b 2 j 1 X b 2 j 1 l b 2 j 1 X b 2 j 1 ] [ I 8 × 8 0 8 × 3 ] Δ X e j + [ P b 2 j X e 2 j l b 2 j X e 2 j ] [ I 6 × 6 0 6 × 2 0 6 × 1 0 6 × 1 0 6 × 1 0 1 × 6 0 1 × 2 1 0 0 0 1 × 6 0 1 × 2 0 0 1 ] Δ x e j + M 2 j [ P b 2 j 1 ξ 2 j 1 l b 2 j 1 ξ 2 j 1 ] [ 0 1 × 8 1 0 1 × 2 ] Δ X e j = [ P b 2 j X e j l b 2 j X e j ] Δ X e j .
[ Δ P e k Δ l e k ] = M 2 k 1 M 2 k 2 M 2 k 3 M 2 k 4 M 2 k 5 [ Δ P e k 3 Δ l e k 3 ] + M 2 k 1 M 2 k 2 M 2 k 3 [ P e k 2 X e k 2 l e k 2 X e k 2 ] Δ X e k 2
+ M 2 k 1 M 2 k 2 M 2 k 3 [ P e k 2 ξ 2 k 4 l e k 2 ξ 2 k 4 ] Δ ξ 2 k 4 + M 2 k 1 [ P e k 1 X e k 1 l e k 1 X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k .
[ Δ P e k Δ l e k ] = M 2 k 1 M 2 k 2 M 2 k 3 M 2 k 4 M 2 k 5 [ Δ P e k 3 Δ l e k 3 ] + M 2 k 1 M 2 k 2 M 2 k 3 [ P e k 2 X e k 2 l e k 2 X e k 2 ] Δ X e k 2 + [ P e k X e k 1 l e k X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k = = M 2 k 1 M 2 k 2 M 2 j M 2 j 1 [ Δ P e j 1 Δ l e j 1 ] + M 2 k 1 M 2 k 2 M 2 j + 2 M 2 j + 1 [ P e j X e j l e j X e j ] Δ X e j + M 2 k 1 M 2 k 2 M 2 j M 2 j + 1 [ P e j ξ 2 j l e j ξ 2 j ] Δ ξ 2 j + M 2 k 1 M 2 k 2 M 2 j + 3 [ P e j + 1 X e j + 1 l e j + 1 X e j + 1 ] Δ X e j + 1 + [ P e k X e j + 2 l e k X e j + 2 ] Δ X e i + 2 + + [ P e k X e k 1 l e k X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k = = M 2 k 1 M 2 k 2 M 1 M 0 Δ X e 0 + M 2 k 1 M 2 k 2 M 4 M 3 [ P e 1 X e 1 l e 1 X e 1 ] Δ X e 1 + + [ P e k X e k 1 l e k X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k = [ P e k X e 0 l e k X e 0 ] Δ X e 0 + [ P e k X e 1 l e k X e 1 ] Δ X e 1 + + [ P e k X e k 1 l e k X e k 1 ] Δ X e k 1 + [ P e k X e k l e k X e k ] Δ X e k = [ P e k X s y s t e m l e k X s y s t e m ] Δ X s y s t e m .

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