Abstract

When monochromatic light is scattered from an optically rough surface a complicated three-dimensional (3D) field is generated. These fields are often described by reference to the 3D volume (extent) of their speckles, leading to the definition of lateral (x,y) and longitudinal speckle sizes (z). For reasons of mathematical simplicity the longitudinal speckle size is often derived by examining the decorrelation of the speckle field for a single point lying on axis, i.e., x=y=0, and this size is generally assumed to be representative for other speckles that lie further off-axis. Some recent theoretical results, however, indicate that in fact longitudinal speckle size gets smaller as the observation position moves to off-axis spatial locations. In this paper (Part I), we review the physical argument leading to this conclusion and support this analysis with a series of robust numerical simulations. We discuss, in some detail, computational issues that arise when simulating the propagation of speckle fields numerically, showing that the spectral method is not a suitable propagation algorithm when the autocorrelation of the scattering surface is assumed to be delta correlated. In Part II [J. Opt. Soc. Am. A 28, 1904 (2011)] of this paper, experimental results are provided that exhibit the predicted variation of longitudinal speckle size as a function of position in x and y. The results are not only of theoretical interest but have practical implications, and in Part II a method for locating the optical system axis is proposed and experimentally demonstrated.

© 2011 Optical Society of America

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    [CrossRef] [PubMed]
  11. D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394–3396 (2007).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011 (3)

2010 (1)

2009 (3)

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316 (2009).
[CrossRef]

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. 79, 053831 (2009).
[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. 26, 1855–1864 (2009).
[CrossRef]

2008 (2)

S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems,” J. Opt. Soc. Am. 25, 2338–2346(2008).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

2007 (1)

2006 (4)

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32–34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef] [PubMed]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

2002 (1)

1999 (2)

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. 16, 1402–1412 (1999).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

1993 (1)

T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. 10, 324–328 (1993).
[CrossRef]

1992 (1)

1990 (1)

L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. 7, 827–832 (1990).
[CrossRef]

1987 (1)

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

1986 (2)

T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. 3, 1032–1054 (1986).
[CrossRef]

D. W. Li and F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. 3, 1023–1031 (1986).
[CrossRef]

1982 (1)

H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982).
[CrossRef]

1981 (2)

J. Ohtsubo, “The second-order statistics of speckle patterns,” J. Opt. 12, 129–142 (1981).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1979 (1)

G. W. Goodman, “Role of coherence concepts in the study of speckle,” Proc. SPIE 194, 86–94 (1979).

1977 (1)

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Chang, N.

N. Chang, N. George, and W. Chi, “Wavelength decorrelation of speckle in propagation through a thick diffuser,” J. Opt. Soc. Am. 28, 245–254 (2011).
[CrossRef]

Chi, W.

N. Chang, N. George, and W. Chi, “Wavelength decorrelation of speckle in propagation through a thick diffuser,” J. Opt. Soc. Am. 28, 245–254 (2011).
[CrossRef]

Chiang, F. P.

Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt. 31, 6287–6291 (1992).
[CrossRef] [PubMed]

D. W. Li and F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. 3, 1023–1031 (1986).
[CrossRef]

Duncan, D.

Ezawa, T.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Ferri, F.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

Gamble, W. L.

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Gao, Z.

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316 (2009).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Gatti, A.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

George, N.

N. Chang, N. George, and W. Chi, “Wavelength decorrelation of speckle in propagation through a thick diffuser,” J. Opt. Soc. Am. 28, 245–254 (2011).
[CrossRef]

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Goodman, G. W.

G. W. Goodman, “Role of coherence concepts in the study of speckle,” Proc. SPIE 194, 86–94 (1979).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2007).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Gopinathan, U.

Halford, C. E.

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Hansen, R. S.

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. 16, 1402–1412 (1999).
[CrossRef]

Hanson, S. G.

S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems,” J. Opt. Soc. Am. 25, 2338–2346(2008).
[CrossRef]

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. 16, 1402–1412 (1999).
[CrossRef]

Hennelly, B. M.

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef] [PubMed]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32–34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Iwamoto, S.

T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. 10, 324–328 (1993).
[CrossRef]

Jakobsen, M. L.

S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems,” J. Opt. Soc. Am. 25, 2338–2346(2008).
[CrossRef]

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, 1st ed. (Springer, 2004).

Kelly, D. P.

D. Li, D. P. Kelly, and J. T. Sheridan “Three-dimensional static speckle fields: part II. Experimental investigation,” J. Opt. Soc. Am. A 28, 1904–1908 (2011).
[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. 26, 1855–1864 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394–3396 (2007).
[CrossRef] [PubMed]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32–34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2011), paper DTuC5.

Kirchner, M.

L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. 7, 827–832 (1990).
[CrossRef]

Kirkpatrick, S.

Kogelnik, H.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).

Leushacke, L.

L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. 7, 827–832 (1990).
[CrossRef]

Li, D.

Li, D. W.

D. W. Li and F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. 3, 1023–1031 (1986).
[CrossRef]

Li, Q. B.

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).

Liu, Y.

Magatti, D.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Meinecke, T.

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2011), paper DTuC5.

Miyamoto, Y.

Naik, D. N.

O’Neill, F. T.

Ohtsubo, J.

J. Ohtsubo, “The second-order statistics of speckle patterns,” J. Opt. 12, 129–142 (1981).
[CrossRef]

Patten, R. F.

Pedersen, H. M.

H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Rose, B.

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. 16, 1402–1412 (1999).
[CrossRef]

Sabitov, N.

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2011), paper DTuC5.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, 1st ed. (Springer, 2004).

Sheridan, J. T.

Singh, R. K.

Sinzinger, S.

D. P. Kelly, N. Sabitov, T. Meinecke, and S. Sinzinger, “Some considerations when numerically calculating diffraction patterns,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2011), paper DTuC5.

Stoffregen, B.

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Takeda, M.

D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19, 1408–1421(2011).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18, 13782–13787(2010).
[CrossRef] [PubMed]

S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems,” J. Opt. Soc. Am. 25, 2338–2346(2008).
[CrossRef]

Wang, W.

S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems,” J. Opt. Soc. Am. 25, 2338–2346(2008).
[CrossRef]

Ward, J. E.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. 26, 1855–1864 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394–3396 (2007).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. 23, 2861–2870 (2006).
[CrossRef]

Weigelt, G. P.

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Yamaguchi, I.

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Yoshimura, T.

T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. 10, 324–328 (1993).
[CrossRef]

T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. 3, 1032–1054 (1986).
[CrossRef]

Yura, H. T.

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. 16, 1402–1412 (1999).
[CrossRef]

Zhao, X.

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316 (2009).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IEEE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

J. Opt. (1)

J. Ohtsubo, “The second-order statistics of speckle patterns,” J. Opt. 12, 129–142 (1981).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Free-space propagation geometry for static speckle for mation. The illuminating spot on the diffuse object is circular with diameter 2 w .

Fig. 2
Fig. 2

Illustration in the x - o - z plane of the orientation property of the static speckle grains. In the figure, an on-axis speckle and two off-axis speckles have been drawn (three ellipses). The off-axis speckle sizes in the directions that pass through the system origin (denoted by dashed lines) have on average the same projection length onto the optical axis (z).

Fig. 3
Fig. 3

Lateral speckle correlation coefficients between an off-axis field at position Q 1 = ( x 1 , 0 , z 0 ) in the z = z 0 plane and fields in a longitudinally displaced plane z = z 0 + ε (mm), as a function of γ. (a) Plane wave illumination. (b) Gaussian beam illumination.

Fig. 4
Fig. 4

Longitudinal speckle correlation coefficients between the field at Q = ( x , 0 , z 0 ) in the z = z 0 plane and a longi tudinally displaced field in plane z = z 0 + ε , as a function of ε. x is in units of millimeters. (a) Plane wave illumination. (b) Gaussian beam illumination.

Fig. 5
Fig. 5

Longitudinal speckle decorrelation as a function of radial offset r , between fields from two longitudinally displaced planes z = z 0 and z = z 0 + ε .

Fig. 6
Fig. 6

Simulation of longitudinal on-axis decorrelation trend using the SM with different choices of sample interval Δ ξ . (Gaussian beam illumination case).

Equations (14)

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I ( r ) I ( r ) = A ( r ) A * ( r ) A ( r ) A * ( r ) , I ( r ) I ( r ) = I ( r ) I ( r ) + | A ( r ) A * ( r ) | 2 ,
I ( r ) I ( r ) I ( r ) I ( r ) = 1 + | μ 12 ( r , r ) | 2 ,
μ 12 ( r , r ) = A ( r ) A * ( r ) [ A ( r ) A * ( r ) A ( r ) A * ( r ) ] 1 / 2 .
A ( r ) = 1 i λ A 0 ( p ) z 0 | r p | exp ( i k | r p | ) | r p | d ξ d η ,
A ( r ) A * ( r ) = ( z λ ) 2 A 0 ( p 1 ) A 0 * ( p 2 ) exp [ i k ( | r p 1 | | r p 2 | ) ] | r p 1 | 2 | r p 2 | 2 d ξ 1 d η 1 d ξ 2 d η 2 .
A 0 ( p 1 ) A 0 * ( p 2 ) = C 0 E ( p 1 ) E * ( p 2 ) δ ( p 1 p 2 ) ,
A ( r ) A * ( r ) = C 0 ( z λ ) 2 | E ( p ) | 2 exp [ i k ( | r p | | r p | ) ] | r p | 2 | r p | 2 d ξ d η .
| μ 12 ( r , r ) | 2 = | | E ( p ) | 2 exp [ i k ( | r p | | r p | ) ] d ξ d η | E ( p ) | 2 d ξ d η | 2 .
| μ 12 ( x , y , z 0 ; γ , δ , ε ) | 2 = | 2 n = 0 i n ( 2 n + 1 ) j n ( u / 4 ) J 2 n + 1 ( υ ) / υ | 2 ,
| E ( p ) | = ( w 0 w ) 2 exp ( | p | w 2 2 ) ,
| μ 12 ( x , y , z 0 ; γ , δ , ε ) | 2 = 1 1 + ( ε / l z ) 2 exp { ( 1 r s ) 2 [ ( ε z 0 x γ ) 2 + ( ε z 0 y δ ) 2 ] } ,
γ = ε ( x z 0 ) .
c ( g , l ) = m , n [ f ( m , n ) f ¯ g , l ] [ t ( m g , n l ) t ¯ ] { m , n [ f ( m , n ) f ¯ g , l ] 2 m , n [ t ( m g , n l ) t ¯ ] 2 } 1 / 2 ,
Δ x = λ z 0 N Δ ξ .

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