Abstract

We extend the concept of the degree of paraxiality, introduced recently for monochromatic fields, to the domain of stochastic fields. As an example we analytically evaluate the degree of paraxiality for a broad class of model stochastic fields, the Gaussian Schell-model fields, without and with truncation and twist phase. The dependence of the degree of paraxiality on the size and the state of coherence of the source as well as on the truncation parameter and the magnitude of twist phase is analyzed by a number of numerical examples.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  32. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  42. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
    [CrossRef]
  43. B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26, 862–869 (2009).
    [CrossRef]
  44. E. Marchand and E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
    [CrossRef]
  45. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
  46. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  47. 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]
  48. D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
    [CrossRef]

2009 (5)

2008 (6)

2007 (2)

2006 (3)

2005 (2)

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[CrossRef]

2002 (3)

2001 (2)

2000 (2)

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[CrossRef]

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

1999 (3)

1998 (3)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

1994 (2)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

1993 (3)

1992 (1)

1988 (2)

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]

W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

1986 (1)

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1980 (1)

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (1)

P. D. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

1978 (1)

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

1972 (1)

1954 (1)

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Ansari, N. A.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Breazeale, M. A.

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]

Cai, Y.

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17, 2450–2464 (2009).
[CrossRef]

F. Wang, Y. Cai, and Olga Korotkova, “Experimental observation of focal shifts in focused partially coherent beams,” Opt. Commun. 282, 3408–3412 (2009).
[CrossRef]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25, 2001–2010 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937–1944 (2007).
[CrossRef]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15, 15480–15492 (2007).
[CrossRef] [PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31, 685–687 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian Schell-model beams,” Opt. Lett. 27, 1672–1674 (2002).
[CrossRef]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
[CrossRef]

Chen, B.

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

Davidson, F. M.

Ding, D.

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Friberg, A. T.

Gawhary, O.

Goodman, W.

W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Gu, J.

Guattari, G.

P. D. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Hu, L.

Kaivola, M.

Korotkova, O.

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17, 2450–2464 (2009).
[CrossRef]

Korotkova, Olga

F. Wang, Y. Cai, and Olga Korotkova, “Experimental observation of focal shifts in focused partially coherent beams,” Opt. Commun. 282, 3408–3412 (2009).
[CrossRef]

Lin, Q.

Lindfors, K.

Liu, X.

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262–267 (2008).
[CrossRef]

D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Marchand, E.

Mei, Z.

Movilla, J. M.

Mukunda, N.

Nemoto, S.

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

Nugent, K. A.

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Paganin, D.

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Palma, C.

P. D. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Peschel, U.

Pu, J.

B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26, 862–869 (2009).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[CrossRef]

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262–267 (2008).
[CrossRef]

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

Ricklin, J. C.

Saghafi, S.

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Santis, P. D.

P. D. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Serna, J.

Setala, T.

Severini, S.

Sheppard, C. J. R.

Simon, R.

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Sundar, K.

Tervonen, E.

Turunen, J.

Wang, F.

Wang, X.

Wen, J. J.

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]

Wolf, E.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

E. Marchand and E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
[CrossRef]

Zhang, Z.

Zhou, G.

Zhu, S.

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Zubairy, M. S.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[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. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (15)

K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[CrossRef]

C. J. R. Sheppard, “High aperture beams,” J. Opt. Soc. Am. A 18, 1579–1587 (2001).
[CrossRef]

D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
[CrossRef]

B. Chen, Z. Zhang, and J. Pu, “Tight focusing of partially coherent and circularly polarized vortex beams,” J. Opt. Soc. Am. A 26, 862–869 (2009).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25, 2001–2010 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937–1944 (2007).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[CrossRef]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[CrossRef]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[CrossRef]

Opt. Commun. (7)

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

F. Wang, Y. Cai, and Olga Korotkova, “Experimental observation of focal shifts in focused partially coherent beams,” Opt. Commun. 282, 3408–3412 (2009).
[CrossRef]

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262–267 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lett. (9)

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Phys. Rev. E (1)

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

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Figures (4)

Fig. 1
Fig. 1

Dependence of the degree of paraxiality of a GSM beam on the normalized transverse beam width σ I 0 λ for different values of the coherence width σ g 0 .

Fig. 2
Fig. 2

Dependence of the degree of paraxiality of a truncated GSM beam on the normalized aperture radius a λ for different values of the coherence width σ g 0 with σ I 0 = 2.5 λ .

Fig. 3
Fig. 3

Dependence of the degree of paraxiality of a twisted GSM beam on the normalized transverse beam width σ I 0 λ for different absolute values of the twist factor μ 0 with λ = 632.8 nm and σ g 0 = λ .

Fig. 4
Fig. 4

Dependence of the divergence angle of a twisted GSM beam on the normalized transverse beam width σ I 0 λ for different absolute values of the twist factor μ 0 with λ = 632.8 nm and σ g 0 = λ .

Equations (33)

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P = p 2 + q 2 < 1 λ 2 | A 0 ( p , q ) | 2 1 λ 2 ( p 2 + q 2 ) d p d q | A 0 ( p , q ) | 2 d p d q ,
W 0 ( x 1 , y 1 , x 2 , y 2 ) = exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ I 0 2 ] exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 0 2 ] ,
A 0 ( p 1 , q 1 , p 2 , q 2 ) = W 0 ( x 1 , y 1 , x 2 , y 2 ) exp [ 2 π i ( p 1 x 1 + q 1 y 1 ) + 2 π i ( p 2 x 2 + q 2 y 2 ) ] d x 1 d y 1 d x 2 d y 2 .
A 0 ( p 1 , q 1 , p 2 , q 2 ) = 16 π 2 σ I 0 4 σ g 0 2 σ g 0 2 + 4 σ I 0 2 exp ( 4 π 2 σ I 0 2 ( 4 σ I 0 2 + 2 σ g 0 2 ) ( p 1 2 + q 1 2 + p 2 2 + q 2 2 ) 2 ( σ g 0 2 + 4 σ I 0 2 ) ) exp ( 16 π 2 σ I 0 4 ( p 1 p 2 + q 1 q 2 ) σ g 0 2 + 4 σ I 0 2 ) .
| A 0 ( p , q ) | 2 = A 0 ( p , q , p , q ) = 16 π 2 σ I 0 4 σ g 0 2 σ g 0 2 + 4 σ I 0 2 exp ( 8 π 2 σ I 0 2 σ g 0 2 ( p 2 + q 2 ) σ g 0 2 + 4 σ I 0 2 ) .
P = α 2 + β 2 < 1 exp ( 2 k 2 σ I 0 2 σ g 0 2 ( α 2 + β 2 ) σ g 0 2 + 4 σ I 0 2 ) 1 ( α 2 + β 2 ) d α d β exp ( 2 k 2 σ I 0 2 σ g 0 2 ( α 2 + β 2 ) σ g 0 2 + 4 σ I 0 2 ) d α d β ,
P = 0 1 0 2 π exp ( 2 k 2 σ I 0 2 σ g 0 2 r 2 σ g 0 2 + 4 σ I 0 2 ) 1 r 2 r d r d θ 0 0 2 π exp ( 2 k 2 σ I 0 2 σ g 0 2 r 2 σ g 0 2 + 4 σ I 0 2 ) r d r d θ .
P = 1 π exp ( γ ) 2 γ 1 2 Erf i [ γ ] .
0 1 exp [ a x ] 1 x d x = 1 a π exp ( a ) 2 a 3 2 Erf i [ a ] .
W 0 ( x 1 , y 1 , x 2 , y 2 ) = exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ I 0 2 ] exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 0 2 ] H ( x 1 , y 1 ) H * ( x 2 , y 2 ) .
H ( x , y ) = m = 1 N A m exp [ B m ( x 2 + y 2 ) a 2 ] ,
A ( p 1 , q 1 , p 2 , q 2 ) = m = 1 N n = 1 N π 2 A m A n * C m C n exp ( π 2 ( p 1 2 + q 1 2 ) C m ) exp [ π 2 p 2 2 C n π 2 p 1 2 4 C m 2 C n σ g 0 4 + π 2 p 1 p 2 C m C n σ g 0 2 ] × exp [ π 2 q 2 2 C n π 2 q 1 2 4 C m 2 C n σ g 0 4 + π 2 q 1 q 2 C m C n σ g 0 2 ] ,
| A 0 ( p , q ) | 2 = m = 1 N n = 1 N π 2 A m A n * C m C n exp [ λ 2 α m n ( p 2 + q 2 ) ] ,
P = k 2 m = 1 N n = 1 N A m A n * C m C n ( 1 2 α m n π exp ( α m n ) Erf i [ α m n ] 4 α m n 3 2 ) 4 σ I 0 2 ( 1 exp ( a 2 2 σ I 0 2 ) ) .
W 0 ( x 1 , y 1 , x 2 , y 2 ) = exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ I 0 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 0 2 ] exp [ i k μ 0 ( x 1 y 2 x 2 y 1 ) ] ,
W 0 ( r ̃ ) = exp ( i k 2 r ̃ T M 0 1 r ̃ ) ,
M 0 1 = ( ( i 2 k σ I 0 2 i k σ g 0 2 ) I i k σ g 0 2 I + μ 0 J i k σ g 0 2 I + μ 0 J T ( i 2 k σ I 0 2 i k σ g 0 2 ) I ) ,
J = ( 0 1 1 0 ) .
A 0 ( p ̃ ) = A 0 ( p 1 , p 2 ) = W 0 ( r ̃ ) exp ( i 2 π r ̃ T I ̃ p ̃ ) d r ̃ ,
I ̃ = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .
A 0 ( p ̃ ) = λ 2 [ det ( M 0 1 ) ] 1 2 exp ( i π λ p ̃ T I ̃ T M 0 I ̃ p ̃ ) ,
A 0 ( p 1 , p 2 , q 1 , q 2 ) = 16 π 2 σ g 0 2 σ I 0 4 4 σ I 0 2 + σ g 0 2 ( 4 k 2 σ I 0 4 μ 2 + 1 ) exp [ 16 i π 2 k σ I 0 4 σ g 0 2 μ 0 ( p 2 q 1 p 1 q 2 ) 4 σ I 0 2 + σ g 0 2 ( 4 k 2 σ I 0 4 μ 0 2 + 1 ) ] exp [ 8 π 2 σ I 0 4 ( ( p 1 p 2 ) 2 + ( q 1 q 2 ) 2 ) + 4 π 2 σ I 0 2 σ g 0 2 ( p 1 2 + p 2 2 + q 1 2 + q 2 2 ) 4 σ I 0 2 + σ g 0 2 ( 4 k 2 σ I 0 4 μ 2 + 1 ) ] .
| A 0 ( p , q ) | 2 = 16 π 2 σ g 0 2 σ I 0 4 4 σ I 0 2 + σ g 0 2 ( 4 k 2 σ I 0 4 μ 2 + 1 ) exp [ 8 π 2 σ I 0 2 σ g 0 2 ( p 2 + q 2 ) 4 σ I 0 2 + σ g 0 2 ( 4 k 2 σ I 0 4 μ 2 + 1 ) ] .
P = 1 π exp ( γ 1 ) 2 γ 1 1 2 Erf i [ γ 1 ] ,
W 0 ( x 1 , y 1 , x 2 , y 2 ) = exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ I 0 2 ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ g 0 2 ] exp [ i k μ 0 ( x 1 y 2 x 2 y 1 ) ] H ( x 1 , y 1 ) H * ( x 2 , y 2 ) .
W 0 ( r ̃ ) = m = 1 M n = 1 M A m A n * exp [ i k 2 r ̃ T ( B m n + M 0 1 ) r ̃ ] ,
B m n = 2 i k a 2 ( B m I 0 I 0 I B n * I ) .
A 0 ( p ̃ ) = m = 1 M n = 1 M A m A n * λ 2 [ det ( B m n + M 0 1 ) ] 1 2 exp [ i π λ p ̃ T I ̃ T ( B m n + M 0 1 ) 1 I ̃ p ̃ ] .
A 0 ( p 1 , q 1 , p 2 , q 2 ) = m = 1 N n = 1 N π 2 A m A n * D m D n exp [ ( π 2 k 2 μ 0 2 4 D m 2 D n π 2 4 D m 2 D n σ g 0 4 π 2 D m ) ( p 1 2 + q 1 2 ) ] exp [ π 2 ( p 2 2 + q 2 2 ) D n ] exp [ π 2 ( p 1 p 2 + q 1 q 2 ) D m D n σ g 0 2 ] exp [ i π 2 k μ 0 ( p 2 q 1 p 1 q 2 ) D m D n ] ,
D n = 1 ( 4 σ I 0 2 ) + 1 ( 2 σ g 0 2 ) + B n * a 2 1 ( 4 D m σ g 0 4 ) + k 2 μ 0 2 ( 4 D m ) ,
D m = 1 ( 4 σ I 0 2 ) + 1 ( 2 σ g 0 2 ) + B m a 2 .
| A 0 ( p , q ) | 2 = m = 1 N n = 1 N π 2 A m A n * D m D n exp [ λ 2 β m n ( p 2 + q 2 ) ] ,
P = k 2 m = 1 N n = 1 N A m A n * D m D n ( 1 2 β m n π exp ( β m n ) Erf i [ β m n ] 4 β m n 3 2 ) 4 σ I 0 2 ( 1 exp ( a 2 2 σ I 0 2 ) ) .

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