Abstract

The effect of an apertured optical system on Wigner distribution can be expressed as a superposition integral of the input Wigner distribution function and the double Wigner distribution function of the apertured optical system. By introducing a hard aperture function into a finite sum of complex Gaussian functions, the double Wigner distribution functions of a first-order optical system with a hard aperture outside and inside it are derived. As an example of application, the analytical expressions of the Wigner distribution for a Gaussian beam passing through a spatial filtering optical system with an internal hard aperture are obtained. The analytical results are also compared with the numerical integral results, and they show that the analytical results are proper and ascendant.

© 2008 Optical Society of America

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References

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  1. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  2. A. Walther, "Propagation of the generalized radiance through lenses," J. Opt. Soc. Am. 68, 1606-1610 (1978).
    [CrossRef]
  3. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
    [CrossRef]
  4. M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
    [CrossRef]
  5. E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6-17 (1978).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. M. J. Bastiaans and P. G. J. van de Mortel, "Wigner distribution function of a circular aperture," J. Opt. Soc. Am. A 13, 1698-1703 (1996).
    [CrossRef]
  9. D. Zhao, H. Mao, and D. Sun, "Representation of the Wigner distribution function for light beams passing through apertured optical systems," Proc. SPIE 5642, 100-107 (2005).
    [CrossRef]
  10. D. Sun and D. Zhao, "Wigner distribution function of Hermite-cosine-Gaussian beams through an apertured optical system," J. Opt. Soc. Am. A 22, 1683-1690 (2005).
    [CrossRef]
  11. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
    [CrossRef]
  12. M. H. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, 2000).
  13. J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  14. A. Pacut, W. J. Kolodziej, and A. Said, "Discrete domain Wigner distributions--a comparison and an implementation," in Proceedings of the IEEE International Symposium on Circuits and Systems (IEEE, 1989), pp. 1264-1267.
    [CrossRef]

2005 (2)

D. Zhao, H. Mao, and D. Sun, "Representation of the Wigner distribution function for light beams passing through apertured optical systems," Proc. SPIE 5642, 100-107 (2005).
[CrossRef]

D. Sun and D. Zhao, "Wigner distribution function of Hermite-cosine-Gaussian beams through an apertured optical system," J. Opt. Soc. Am. A 22, 1683-1690 (2005).
[CrossRef]

1996 (1)

1988 (1)

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1986 (1)

1979 (1)

1978 (4)

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6-17 (1978).
[CrossRef]

A. Walther, "Propagation of the generalized radiance through lenses," J. Opt. Soc. Am. 68, 1606-1610 (1978).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

1970 (1)

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

Phys. Rev. (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Proc. SPIE (1)

D. Zhao, H. Mao, and D. Sun, "Representation of the Wigner distribution function for light beams passing through apertured optical systems," Proc. SPIE 5642, 100-107 (2005).
[CrossRef]

Other (2)

M. H. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, 2000).

A. Pacut, W. J. Kolodziej, and A. Said, "Discrete domain Wigner distributions--a comparison and an implementation," in Proceedings of the IEEE International Symposium on Circuits and Systems (IEEE, 1989), pp. 1264-1267.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical system with a hard aperture placed externally.

Fig. 2
Fig. 2

Optical system with a hard aperture placed internally.

Fig. 3
Fig. 3

Gaussian beam passing through a spatial filtering optical system.

Fig. 4
Fig. 4

(a) Normalized Wigner distribution function of Gaussian beam at input plane, (b) projection of (a).

Fig. 5
Fig. 5

Normalized Wigner distribution functions at the output plane of a fundamental order Gaussian beam passing through the spatial filtering optical system calculated by approximate analytical Eq. (19): (a) a = 0.1 , (c) a = 0.2 , and (e) a = 0.4 ; (b), (d), and (f) are the projections of (a), (c), and (e), respectively.

Fig. 6
Fig. 6

Same as Fig. 5 but calculated by using the numerical integral formula of Eqs. (1) and (6) directly.

Fig. 7
Fig. 7

Absolute errors of the analytical results compared with the numerically integrated ones: (a) a = 0.1 , (b) a = 0.2 , and (c) a = 0.4 .

Fig. 8
Fig. 8

(a) Intensity and (b) power frequency distribution function at the output plane from Wigner distribution functions by integrating over x and u for different size filtering apertures.

Tables (2)

Tables Icon

Table 1 Coefficients A n and B n with N = 10

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Table 2 CPU Time Used for Simulating the Wigner Distribution of a Gaussian Beam

Equations (24)

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W ( x , u ) = E ( x + x 2 ) E * ( x x 2 ) exp ( i 2 π u x ) d x ,
E 2 ( x 2 ) = h ( x 2 , x 1 ) E ( x 1 ) d x 1 .
W 2 ( x 2 , u 2 ) = K h ( x 2 , u 2 ; x 1 , x 1 ) W 1 ( x 1 , u 1 ) d x 1 d u 1 ,
K h ( x 2 , u 2 ; x 1 , u 1 ) = h ( x 2 + x 2 2 , x 1 + x 1 2 ) × h * ( x 2 x 2 2 , x 1 x 1 2 ) × exp [ i 2 π ( u 2 x 2 u 1 x 1 ) ] × d x 2 d x 1 .
K h 3 ( x 2 , u 2 ; x 1 , u 1 ) = K h 2 ( x 2 , u 2 ; x 2 , u 2 ) × K h 1 ( x 2 , u 2 ; x 1 , u 1 ) d x 2 d u 2 .
E 2 ( x 2 ) = i λ B E 1 ( x 1 ) × exp [ i π λ B ( A x 1 2 2 x 1 x 2 + D x 2 2 ) ] d x 1 ,
h ( x 2 , x 1 ) = i λ B   exp [ i π λ B ( A x 1 2 2 x 1 x 2 + D x 2 2 ) ] .
K h ( x 2 , u 2 ; x 1 , u 1 ) = δ ( x 1 D x 2 + B u 2 ) δ ( u 1 + C x 2 A u 2 ) .
A p ( x ) = { 1 , | x | a 0 , others ,
h p ( x 2 , x 1 ) = A p ( x 1 ) i λ B   exp [ i π λ B ( A x 1 2 2 x 1 x 2 + D x 2 2 ) ] .
A p ( x 1 ) = n = 1 N A n   exp ( B n a 2 x 1 2 ) ,
exp ( p 2 x 2 ± q x ) d x = π p   exp ( q 2 4 p 2 ) ,
K h p ( x 2 , u 2 ; x 1 , u 1 ) = n = 1 N m = 1 N 2 a α π β   exp ( β P 1 2 a 2 ) × exp [ 4 π 2 a 2 β ( u 1 + P 2 ) 2 ] δ ( x 1 + P 1 ) ,
α = A n A m * , β = B n + B m * , β = B n B m * ,
P 1 = D x 2 + λ B u 2 , P 2 = ( C λ + i β D 2 π a 2 ) x 2 ( A + i β λ B 2 π a 2 ) u 2 .
K h ( x 2 , u 2 ; x 1 , u 1 ) = K h 2 ( x 2 , u 2 ; x 2 , u 2 ) × K h 1 ( x 2 , u 2 ; x 1 , u 1 ) d x 2 d u 2 = n = 1 N m = 1 N 2 a α π β   exp ( β P 21 2 a 2 ) × exp [ 4 π 2 a 2 β ( C 1 λ x 1 + D 1 u 1 + P 22 ) 2 ] × δ ( A 1 x 1 + λ B 1 u 1 + P 21 ) ,
P 21 = D 2 x 2 + λ B 2 u 2 ,
P 22 = ( C 2 λ + i β D 2 2 π a 2 ) x 2 ( A 2 + i β λ B 2 2 π a 2 ) u 2 .
[ A B C D ] = [ A 1 B 1 C 1 D 1 ] ¦ [ A 2 B 2 C 2 D 2 ] = [ 0 f 1 1 f 1 1 d 1 f 1 ] ¦ [ 1 d 2 f 2 f 2 1 f 2 0 ] ,
K h ( x 2 , u 2 ; x 1 , u 1 ) = n = 1 N m = 1 N 2 a α π β   exp ( β P 21 2 a 2 ) × exp [ 4 π 2 a 2 β ( 1 λ f 1 x 1 + ( 1 d 1 f 1 ) u 1 + P 22 ) 2 ] δ ( λ f 1 u 1 + P 21 ) ,
P 21 = λ f 2 u 2 , P 22 = 1 λ f 2 x 2 ( 1 d 2 f 2 + i β λ f 2 2 π a 2 ) u 2 .
E 1 ( x 1 ) = exp ( x 1 2 ω 0 2 ) .
W 1 ( x 1 , u 1 ) = 2 π ω 0   exp ( 2 x 1 2 ω 0 2 ) exp ( 2 π 2 u 1 2 ω 0 2 ) .
W 2 ( x 2 , u 2 ) = K h ( x 2 , u 2 ; x 1 , u 1 ) W 1 ( x 1 , u 1 ) d x 1 d u 1 = n = 1 N m = 1 N 2 π 3 / 2 a α ω 0 2 Q   exp [ ( β a 2 + 2 π 2 ω 0 2 λ 2 B 1 2 ) P 21 2 ] exp [ 4 π 2 a 2 Q ( ( 1 d 1 f 1 ) P 21 + λ f 1 P 22 ) 2 ] ,

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