Abstract

The distribution of the ray density of the spot diagram formed in the image plane is called the geometric point spread function (PSF). It plays an important role in the image formation theory, since it describes the impulse response of an optical system to a source point. However, the literature contains very few techniques for deriving the PSF of optical systems. Accordingly, this study presents a method based on an irradiance model for computing the geometric PSF of an optical system by considering the energy conservation along a single light ray. It is shown that the proposed method obtains a reliable and accurate estimate of the PSF and enables the computation of the centroid and root-mean-square radius of the focus spot on the image plane. In addition, compared to existing ray-counting methods presented in the literature, in which the quality of the PSF solution depends on the number of rays traced and the grid size used to mesh the image plane, the proposed irradiance-based method requires just one tracing operation. Overall, the results presented in this study confirm that the proposed method provides an ideal solution for calculating the merit function and modulation transfer function of an optical system.

© 2010 Optical Society of America

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References

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  1. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.
  2. V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics (SPIE Press, 1998).
  3. 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]
  4. C. C. Hsueh and P. D. Lin, “Computationally efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893-902 (2009).
    [CrossRef] [PubMed]
  5. D. L. Shealy and D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 76 (1976).
    [CrossRef]
  6. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601-606(1989).
    [CrossRef] [PubMed]
  7. J. W. Foreman, “Computation of RMS spot radii by ray tracing,” Appl. Opt. 13, 2585-2588 (1974).
    [CrossRef] [PubMed]
  8. T. B. Andersen, “Evaluating RMS spot radii by ray tracing,” Appl. Opt. 21, 1241-1248 (1982).
    [CrossRef] [PubMed]
  9. Berlyn Brixner, “Lens design merit functions: rms image spot size and rms optical path difference,” Appl. Opt. 17, 715-716 (1978).
    [CrossRef] [PubMed]
  10. R. P. Paul, Robot Manipulators--Mathematics, Programming and Control, (MIT Press, 1982).
  11. M. Laikin, Lens Design (Marcel Dekker1995), pp. 71-72.

2009 (1)

2007 (1)

1989 (1)

1982 (1)

1978 (1)

1976 (1)

1974 (1)

Andersen, T. B.

Brixner, Berlyn

Burkhard, D. G.

Foreman, J. W.

Hsueh, C. C.

Kassim, A. M.

Laikin, M.

M. Laikin, Lens Design (Marcel Dekker1995), pp. 71-72.

Lin, P. D.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics (SPIE Press, 1998).

Paul, R. P.

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

Shealy, D. L.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.

Tsai, C. Y.

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Figures (10)

Fig. 1
Fig. 1

Skew ray tracing at a flat boundary surface.

Fig. 2
Fig. 2

Skew ray tracing at a spherical boundary surface.

Fig. 3
Fig. 3

Petzval lens system with n = 11 boundary surfaces [11].

Fig. 4
Fig. 4

Representation of unit directional vector 0 0 originating from source point P 0 0 .

Fig. 5
Fig. 5

Source point radiating uniformly with constant intensity.

Fig. 6
Fig. 6

Distribution of PSF on the image plane as computed by the proposed method.

Fig. 7
Fig. 7

Distribution of PSF on the image plane computed by the ray-counting method (192,539 rays; grid size, 1 / 300 mm × 1 / 300 mm ) and the proposed method.

Fig. 8
Fig. 8

Distribution of PSF on the image plane as computed by ray-counting method (2409 rays; grid size, 1 / 300 mm × 1 / 300 mm ).

Fig. 9
Fig. 9

Optical system containing a double-convex lens.

Fig. 10
Fig. 10

Distribution of PSF on the image plane of the system shown in Fig. 9 by the ray-counting method (7,119,482 rays; grid size, 1 / 100 mm × 1 / 100 mm ) and the proposed method.

Tables (2)

Tables Icon

Table 1 Variables of Petzval Lens System [11] in Millimeters

Tables Icon

Table 2 PSF Values of Points in Regions 0 and 1

Equations (47)

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r i i = [ x i y i z i 1 ] T = [ β i C α i 0 β i S α i 1 ] T ( 0 β i , 0 α i < 2 π ) ,
r i i = [ x i y i z i 1 ] T = [ R i C β i C α i R i C β i S α i R i S β i 1 ] T ( π / 2 β i π / 2 , 0 α i < 2 π ) ,
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 ] ,
n i 0 = [ n i x 0 n i y 0 n i z 0 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 ,
n i 0 = [ n i x 0 n i y 0 n i z 0 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 .
P i 0 = [ P i x 0 P i y 0 P i z 0 1 ] T = [ P i 1 x 0 + i 1 x 0 λ i P i 1 y 0 + i 1 y 0 λ i P i 1 z 0 + i 1 z 0 λ i 1 ] T ,
λ i = ( I i y P i 1 x 0 + J i y P i 1 y 0 + K i y P i 1 z 0 + t i y ) / ( I i y i 1 x 0 + J i y i 1 y 0 + K i y i 1 z 0 ) ,
λ i = D i ± D i 2 E i ,
D i = t i x ( I i x i 1 x 0 + J i x i 1 y 0 + K i x i 1 z 0 ) + t i y ( I i y i 1 x 0 + J i y i 1 y 0 + K i y i 1 z 0 ) + t i z ( I i z i 1 x 0 + J i z i 1 y 0 + K i z i 1 z 0 ) + P i 1 x 0 i 1 x 0 + P i 1 y 0 i 1 y 0 + P i 1 z 0 i 1 z 0 ,
E i = P i 1 x 2 0 + P i 1 y 2 0 + P i 1 z 2 0 + 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 0 + J i x P i 1 y 0 + K i x P i 1 z 0 ) + 2 t i y ( I i y P i 1 x 0 + J i y P i 1 y 0 + K i y P i 1 z 0 ) + 2 t i z ( I i z P i 1 x 0 + J i z P i 1 y 0 + K i z P i 1 z 0 ) .
α i = a tan 2 ( x , y ) ( I i z ( P i 1 x 0 + i 1 x 0 λ i ) J i z ( P i 1 y 0 + i 1 y 0 λ i ) K i z ( P i 1 z 0 + i 1 z 0 λ i ) t i z , I i x ( P i 1 x 0 + i 1 x 0 λ i ) + J i x ( P i 1 y 0 + i 1 y 0 λ i ) + K i x ( P i 1 z 0 + i 1 z 0 λ i ) + t i x ) ,
β i = { [ I i x ( P i 1 x 0 + i 1 x 0 λ i ) + J i x ( P i 1 y 0 + i 1 y 0 λ i ) + K i x ( P i 1 z 0 + i 1 z 0 λ i ) + t i x ] 2 + [ I i z ( P i 1 x 0 + i 1 x 0 λ i ) + J i z ( P i 1 y 0 + i 1 y 0 λ i ) + K i z ( P i 1 z 0 + i 1 z 0 λ i ) + t i z ] 2 } 1 2 .
α i = a tan 2 ( x , y ) ( I i y ( P i 1 x 0 + i 1 x 0 λ i ) + J i y ( P i 1 y 0 + i 1 y 0 λ i ) + K i y ( P i 1 z 0 + i 1 z 0 λ i ) + t i y , I i x ( P i 1 x 0 + i 1 x 0 λ i ) + J i x ( P i 1 y 0 + i 1 y 0 λ i ) + K i x ( P i 1 z 0 + i 1 z 0 λ i ) + t i x ) ,
β i = a tan 2 ( x , y ) ( I i z ( P i 1 x 0 + i 1 x 0 λ i ) + J i z ( P i 1 y 0 + i 1 y 0 λ i ) + K i z ( P i 1 z 0 + i 1 z 0 λ i ) + t i z , [ I i x ( P i 1 x 0 + i 1 x 0 λ i ) + J i x ( P i 1 y 0 + i 1 y 0 λ i ) + K i x ( P i 1 z 0 + i 1 z 0 λ i ) + t i x ] 2 + [ I i y ( P i 1 x 0 + i 1 x 0 λ i ) + J i y ( P i 1 y 0 + i 1 y 0 λ i ) + K i y ( P i 1 z 0 + i 1 z 0 λ i ) + t i y ] 2 ¯ ) .
β i = arcsin ( I i z ( P i 1 x 0 + i 1 x 0 λ i ) + J i z ( P i 1 y 0 + i 1 y 0 λ i ) + K i z ( P i 1 z 0 + i 1 z 0 λ i ) + t i z R i ) .
C θ i = i 1 T 0 n i 0 = s i ( I i y i 1 x 0 + J i y i 1 y 0 + K i y i 1 z 0 ) ,
C θ i = i 1 T 0 n i 0 = s i [ i 1 x 0 ( 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 0 ( 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 0 ( 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 ,
i 0 = [ i x 0 i y 0 i z 0 0 ] = [ i 1 x 0 + 2 n i x 0 C θ i i 1 y 0 + 2 n i y 0 C θ i i 1 z 0 + 2 n i z 0 C θ i 0 ] ,
i 0 = [ i x 0 i y 0 i z 0 0 ] = [ n i x 0 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 x 0 + n i x 0 C θ i ) n i y 0 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 y 0 + n i y 0 C θ i ) n i z 0 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 z 0 + n i z 0 C θ i ) 0 ] ,
d x i d z i = | x i / α 0 x i / β 0 z i / α 0 z i / β 0 | d α 0 d β 0 = | ( x i , z i ) ( α 0 , β 0 ) | d α 0 d β 0 ,
d α i d β i = | α i / α 0 α i / β 0 β i / α 0 β i / β 0 | d α 0 d β 0 = | ( α i , β i ) ( α 0 , β 0 ) | d α 0 d β 0 .
[ Δ P i 0 Δ i 0 ] = [ P i 0 / X 0 i 0 / X 0 ] Δ X 0 = [ ( P i x 0 , P i y 0 , P i z 0 ) ( P 0 x 0 , P 0 y 0 , P 0 z 0 ) 3 × 3 ( P i x 0 , P i y 0 , P i z 0 ) ( β 0 , α 0 ) 3 × 2 ( i x 0 , i y 0 , i z 0 ) ( P 0 x 0 , P 0 y 0 , P 0 z 0 ) 3 × 3 ( i x 0 , i y 0 , i z 0 ) ( β 0 , α 0 ) 3 × 2 ] [ Δ P 0 x 0 Δ P 0 y 0 Δ P 0 z 0 Δ β 0 Δ α 0 ] ,
[ d P i x i d P i z i ] = [ d x i d z i ] = [ P i x i / α 0 P i x i / β 0 P i z i / α 0 P i z i / β 0 ] [ d α 0 d β 0 ] = [ x i / α 0 x i / β 0 z i / α 0 z i / β 0 ] [ d α 0 d β 0 ] ,
x i α 0 = P i x i α 0 = I i x P i x 0 α 0 + J i x P i y 0 α 0 + K i x P i z 0 α 0 , x i β 0 = P i x i β 0 = I i x P i x 0 β 0 + J i x P i y 0 β 0 + K i x P i z 0 β 0 , z i α 0 = P i z i α 0 = I i z P i x 0 α 0 + J i z P i y 0 α 0 + K i z P i z 0 α 0 , z i β 0 = P i z i β 0 = I i z P i x 0 β 0 + J i z P i y 0 β 0 + K i z P i z 0 β 0 .
[ d x i d z i ] = [ x i / α i x i / β i z i / α i z i / β i ] [ α i / α 0 α i / β 0 β i / α 0 β i / β 0 ] [ d α 0 d β 0 ] = [ R i C β i S α i R i S β i C α i 0 R i C β i ] [ α i / α 0 α i / β 0 β i / α 0 β i / β 0 ] [ d α 0 d β 0 ] = [ x i / α 0 x i / β 0 z i / α 0 z i / β 0 ] [ d α 0 d β 0 ] .
( α i , β i ) ( α 0 , β 0 ) = [ R i C β i S α i R i S β i C α i 0 R i C β i ] 1 [ x i / α 0 x i / β 0 z i / α 0 z i / β 0 ] = 1 R i C β i 2 S α i [ C β i S β i C α i 0 C β i S α i ] [ x i / α 0 x i / β 0 z i / α 0 z i / β 0 ] .
[ P i 0 i 0 ] T = [ P i x 0 P i y 0 P i z 0 i x 0 i y 0 i z 0 ] T = [ P 0 x 0 P 0 y 0 P 0 z 0 C β 0 C α 0 C β 0 S α 0 S β 0 ] T
d F 0 = I 0 d Ω 0 = I 0 C β 0 d α 0 d β 0 .
F 0 = I 0 C β 0 d α 0 d β 0 = I 0 Ω 0 ,
I 0 = 1 / Ω 0 .
d π i = d x i d z i = | ( x i , z i ) ( α 0 , β 0 ) | d α 0 d β 0 ,
d π i = R i 2 C β i d α i d β i = R i 2 C β i | ( α i , β i ) ( α 0 , β 0 ) | d α 0 d β 0 ,
d F 0 = I 0 C β 0 d α 0 d β 0 = d F i = B i ( x i , z i ) d π i = B i ( x i , z i ) d x i d z i ,
d F 0 = I 0 C β 0 d α 0 d β 0 = d F i = B i ( α i , β i ) d π i = B i ( α i , β i ) R i 2 C β i d α i d β i .
B i ( x i , z i ) d π i = 1 ,
B i ( α i , β i ) d π i = 1.
PSF = B i ( x i , z i ) = C β 0 Ω 0 | x i α 0 z i β 0 x i β 0 z i α 0 | = C β 0 Ω 0 | ( x i , z i ) ( α 0 , β 0 ) | .
PSF = B i ( α i , β i ) = C β 0 Ω 0 R i 2 C β i | α i α 0 β i β 0 α i β 0 β i α 0 | = C β 0 Ω 0 R i 2 C β i | ( α i , β i ) ( α 0 , β 0 ) | .
[ x i c y i c z i c 1 ] = [ x i B i d π i / B i d π i y i B i d π i / B i d π i z i B i d π i / B i d π i 1 ] = [ x i B i d π i y i B i d π i z i B i d π i 1 ] .
[ x i c y i c z i c 1 ] = 1 Ω 0 [ x i C β 0 d α 0 d β 0 0 z i C β 0 d α 0 d β 0 1 ] .
[ x i c y i c z i c 1 ] = R i 2 Ω 0 [ C β i C α i C β 0 d α 0 d β 0 C β i S α i C β 0 d α 0 d β 0 S β i C β 0 d α 0 d β 0 1 ] .
rms i 2 = [ ( x i x i c ) 2 + ( z i z i c ) 2 ] B i ( x i , z i ) d π i = ( x i 2 + z i 2 ) B i ( x i , z i ) d π i ( x i c 2 + z i c 2 ) = 1 Ω 0 ( x i 2 + z i 2 ) C β 0 d α 0 d β 0 ( x i c 2 + z i c 2 ) .
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 ] .

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