Abstract

We develop a mathematical method for determining the optical path length (OPL) gradient matrix relative to all the system variables such that the effects of variable changes can be evaluated in a single pass. The approach developed avoids the requirement for multiple ray-tracing operations and is, therefore, more computationally efficient. By contrast, the effects of variable changes on the OPL of an optical system are generally evaluated by utilizing a ray-tracing approach to determine the OPL before and after the variable change and then applying a finite-difference (FD) approximation method to estimate the OPL gradient with respect to each individual variable. Utilizing a Petzval lens system for verification purposes, it is shown that the approach developed reduces the computational time by around 90% compared to that of the FD method.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983-990 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
  5. T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. 49, 402-404 (1959).
    [CrossRef]
  6. P. D. Lin and C. Y. Tsai, “First-order gradients of skew rays of axis- symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776-784 (2007).
    [CrossRef]
  7. M. Laikin, Lens Design (Marcel Dekker, 1995).
  8. P. D. Lin and C. H. Lu, “Analysis and design of optical system by use of sensitivity analysis of skew ray tracing,” Appl. Opt. 43, 796-807 (2004).
    [CrossRef] [PubMed]
  9. P. D. Lin and C. K. Sung, “Camera calibration based on Snell's law,” J. Dyn. Syst. Meas. Control 128, 548-557 (2006).
    [CrossRef]
  10. P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621-628 (2008).
    [CrossRef]

2008 (1)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621-628 (2008).
[CrossRef]

2007 (1)

2006 (2)

M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane-parallel uniaxial plate,” J. Opt. Soc. Am. A 23, 926-932 (2006).
[CrossRef]

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)

1999 (1)

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983-990 (1999).
[CrossRef]

1995 (1)

M. Laikin, Lens Design (Marcel Dekker, 1995).

1985 (1)

1968 (1)

1959 (1)

Avendaño-Alejo, M.

Fortunato, G.

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Journet, B.

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Laikin, M.

M. Laikin, Lens Design (Marcel Dekker, 1995).

Lin, P. D.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621-628 (2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First-order gradients of skew rays of axis- symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776-784 (2007).
[CrossRef]

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 system by use of sensitivity analysis of skew ray tracing,” Appl. Opt. 43, 796-807 (2004).
[CrossRef] [PubMed]

Lu, C. H.

Meiron, J.

Purnet, S.

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Rosete-Aguilar, M.

Sharma, A.

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]

Suzuki, T.

Tsai, C. Y.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621-628 (2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First-order gradients of skew rays of axis- symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776-784 (2007).
[CrossRef]

Uwoki, I.

Appl. Opt. (3)

Appl. Phys. B (1)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621-628 (2008).
[CrossRef]

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)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983-990 (1999).
[CrossRef]

Other (1)

M. Laikin, Lens Design (Marcel Dekker, 1995).

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Figures (5)

Fig. 1
Fig. 1

Petzval lens system with k = 6 elements and n = 11 boundary surfaces [7].

Fig. 2
Fig. 2

Skew ray tracing at spherical boundary surface.

Fig. 3
Fig. 3

Skew ray tracing at flat boundary surface.

Fig. 4
Fig. 4

The jth element with thickness q e j .

Fig. 5
Fig. 5

The OPL between two points is the product of their geometric length and the refractive index.

Tables (3)

Tables Icon

Table 1 Variables of Petzval Lens System [7] a

Tables Icon

Table 2 Comparison of Results Obtained for OPL system / X e 1 Using the FD Method and the Proposed Method

Tables Icon

Table 3 Results Obtained from the FD Method for OPL system / ω e 1 x as a Function of Δ ω e 1 x a

Equations (45)

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A 0 i = Trans ( t i x , t i y , t i z ) Rot ( z , ω i z ) Rot ( y , ω i y ) Rot ( x , ω 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 ] ,
X i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i R i ] T
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 ) .
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 air ξ e j R 2 j 1 R 2 j q e j ] T .
X 2 j 1 = [ 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 air ξ e j R 2 j 1 ] T .
[ Δ P 0 Δ 0 ] = [ 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 C 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 ] .
Δ OPL i = [ OPL i P i 1 OPL i i 1 ] [ Δ P i 1 Δ i 1 ] = ξ i 1 Δ λ i = ξ i 1 [ λ i P i 1 λ i i 1 ] [ Δ P i 1 Δ i 1 ] .
( f , g ) ( u , v ) = [ f u f v g u g v ] .
Δ OPL i = λ i Δ ξ i 1 + ξ i 1 λ i X i Δ X i ,
Δ OPL i = ξ i 1 [ λ i P i 1 λ i i 1 ] [ Δ P i 1 Δ i 1 ] + λ i Δ ξ i 1 + ξ i 1 λ i X i Δ X i = ξ i 1 [ λ i P i 1 λ i i 1 ] [ Δ P i 1 Δ i 1 ] + λ i [ 0 0 0 0 0 0 1 0 0 ] Δ X i + ξ i 1 λ i X i Δ X i = [ OPL i P i 1 OPL i i 1 ] [ Δ P i 1 Δ i 1 ] + OPL i X i Δ X i .
Δ OPL e j = Δ OPL 2 j 1 + Δ OPL 2 j .
Δ OPL 2 j 1 = ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] [ Δ P 2 j 2 Δ 2 j 2 ] + λ 2 j 1 Δ ξ 2 j 2 + ξ 2 j 2 λ 2 j 1 X 2 j 1 Δ X 2 j 1 ,
Δ OPL 2 j = ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] [ Δ P 2 j 1 Δ 2 j 1 ] + λ 2 j Δ ξ 2 j 1 + ξ 2 j 1 λ 2 j X 2 j Δ X 2 j .
[ Δ P 2 j 1 Δ 2 j 1 ] = M 2 j 1 [ Δ P 2 j 2 Δ 2 j 2 ] + [ P 2 j 1 / X 2 j 1 2 j 1 / X 2 j 1 ] Δ X 2 j 1 ,
Δ X 2 j 1 = [ X 2 j 1 / X e j ] Δ X e j ,
Δ X 2 j = [ X 2 j / X e j ] Δ X e j .
X 2 j 1 X e j = [ I 6 × 6 0 6 × 3 0 6 × 2 0 3 × 6 I 3 × 3 0 3 × 2 ] .
X 2 j X e j = [ 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 c 2 , 9 c 2 , 10 c 2 , 11 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ] .
Δ ξ 2 j 2 = Δ ξ air = [ 0 1 × 6 1 0 1 × 4 ] Δ X e j ,
Δ ξ 2 j 1 = Δ ξ e j = [ 0 1 × 7 1 0 1 × 3 ] Δ X e j .
Δ OPL 2 j 1 = ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] [ Δ P 2 j 2 Δ 2 j 2 ] + λ 2 j 1 [ 0 1 × 6 1 0 1 × 4 ] Δ X e j + ξ 2 j 2 λ 2 j 1 X 2 j 1 [ I 6 × 6 0 6 × 3 0 6 × 2 0 3 × 6 I 3 × 3 0 3 × 2 ] Δ X e j = ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] [ Δ P 2 j 2 Δ 2 j 2 ] + OPL 2 j 1 X e j Δ X e j .
Δ OPL 2 j = ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] M 2 j 1 [ Δ P 2 j 2 Δ 2 j 2 ] + ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] [ P 2 j 1 / X 2 j 1 2 j 1 / X 2 j 1 ] [ I 6 × 6 0 6 × 3 0 6 × 2 0 3 × 6 I 3 × 3 0 3 × 2 ] Δ X e j + λ 2 j [ 0 1 × 7 1 0 1 × 3 ] Δ X e j + ξ 2 j 1 λ 2 j X 2 j X 2 j X e j Δ X e j = ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] M 2 j 1 [ Δ P 2 j 2 Δ 2 j 2 ] + OPL 2 j X e j Δ X e j .
Δ OPL e j = Δ OPL 2 j 1 + Δ OPL 2 j = { ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] + ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] M 2 j 1 } [ Δ P 2 j 2 Δ 2 j 2 ] + ( OPL 2 j 1 X e j + OPL 2 j X e j ) Δ X e j ,
Δ OPL system = j = 1 j = k 1 Δ OPL e j + Δ OPL 2 k 1 = j = 1 j = k 1 ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] [ Δ P 2 j 2 Δ 2 j 2 ] + j = 1 j = k 1 ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] M 2 j 1 [ Δ P 2 j 2 Δ 2 j 2 ] + j = 1 j = k 1 ( OPL 2 j 1 X e j + OPL 2 j X e j ) Δ X e j + ξ 2 k 2 [ λ 2 k 1 P 2 k 2 λ 2 k 1 2 k 2 ] [ Δ P 2 k 2 Δ 2 k 2 ] + OPL 2 k 1 X e k Δ X e k = j = 1 j = k ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] [ Δ P 2 j 2 Δ 2 j 2 ] + j = 1 j = k 1 ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] M 2 j 1 [ Δ P 2 j 2 Δ 2 j 2 ] + j = 1 j = k OPL 2 j 1 X e j X e j X system Δ X system + j = 1 j = k 1 OPL 2 j X e j X e j X system Δ X system .
[ Δ P 2 j 2 Δ 2 j 2 ] = M 2 j 2 M 2 j 3 [ Δ P 2 j 4 Δ 2 j 4 ] + [ P 2 j 2 / X 2 j 2 2 j 2 / X 2 j 2 ] Δ X 2 j 2 .
Δ OPL system = j = 1 j = 6 ξ 2 j 2 [ λ 2 j 1 P 2 j 2 λ 2 j 1 2 j 2 ] [ Δ P 2 j 2 Δ 2 j 2 ] + j = 1 j = 5 ξ 2 j 1 [ λ 2 j P 2 j 1 λ 2 j 2 j 1 ] M 2 j 1 [ Δ P 2 j 2 Δ 2 j 2 ] + j = 1 j = 6 O P L 2 j 1 X e j X e j X system Δ X system + j = 1 j = 5 O P L 2 j X e j X e j X system Δ X system .
n i = [ n i x n i y n i z 0 ] T = A i 0 n i i = ( A 0 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 ,
n i = [ n i x n i y n i z 0 ] T = A i 0 n i i = ( A 0 i ) 1 n i i = s i [ I i y J i y K i y 0 ] T ,
P i = [ P i x P i y P i z 1 ] T = [ P i 1 x + i 1 x λ i P i 1 y + i 1 y λ i P i 1 z + i 1 z λ i 1 ] T ,
λ i = D i ± [ D i 2 E i ] 1 / 2 ,
D i = t i x ( I i x i 1 x + J i x i 1 y + K i x i 1 z ) + t i y ( I i y i 1 x + J i y i 1 y + K i y i 1 z ) + t i z ( I i z i 1 x + J i z i 1 y + K i z i 1 z ) + P i 1 x i 1 x + P i 1 y i 1 y + P i 1 z 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 ) .
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 i 1 x + J i y i 1 y + K i y i 1 z ) = B i / G i .
α i = arctan ( I i y ( P i 1 x + i 1 x λ i ) + J i y ( P i 1 y + i 1 y λ i ) + K i y ( P i 1 z + i 1 z λ i ) + t i y I i x ( P i 1 x + i 1 x λ i ) + J i x ( P i 1 y + i 1 y λ i ) + K i x ( P i 1 z + i 1 z λ i ) + t i x ) ,
β i = arcsin ( I i z ( P i 1 x + i 1 x λ i ) + J i z ( P i 1 y + i 1 y λ i ) + K i z ( P i 1 , z + i 1 z λ i ) + t i z R i ) .
C θ i = i 1 T n i = s i [ 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 ) + 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 ) + 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 ) ] ,
C θ i = i 1 T n i = s i ( I i y i 1 x + J i y i 1 y + K i y i 1 z ) .
S θ _ i = ( ξ i 1 / ξ i ) S θ i = N i S θ i ,
i = [ i x i y i z 0 ] = [ n i x [ 1 N i 2 + ( N i C θ i ) 2 ] 1 / 2 + N i ( i 1 x + n i x C θ i ) n i y [ 1 N i 2 + ( N i C θ i ) 2 ] 1 / 2 + N i ( i 1 y + n i y C θ i ) n i z [ 1 N i 2 + ( N i C θ i ) 2 ] 1 / 2 + N i ( i 1 z + n i z C θ i ) 0 ] ,
Rot ( x , ω i x ) = [ 1 0 0 0 0 C ω i x S ω i x 0 0 S ω i x C ω i x 0 0 0 0 1 ] .
Rot ( y , ω i y ) = [ C ω i y 0 S ω i y 0 0 1 0 0 S ω i y 0 C ω i y 0 0 0 0 1 ] .
Rot ( z , ω i z ) = [ C ω i z S ω i z 0 0 S ω i z C ω i z 0 0 0 0 1 0 0 0 0 1 ] .
Trans ( t i x , t i y , t i z ) = [ 1 0 0 t i x 0 1 0 t i y 0 0 1 t i z 0 0 0 1 ] .
A e j 2 j = Trans ( 0 , d e j , 0 ) ,
A 0 2 j = A e j 2 j A 0 e j = Trans ( t e j x , t e j y + d e j , t e j z ) Rot ( z , ω e j z ) Rot ( y , ω e j y ) Rot ( x , ω e j x ) .
X 2 j = [ t e j x t e j y + d e j t e j z ω e j x ω e j y ω e j z ξ e j ξ air R 2 j ] T .

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