Abstract

The statistical properties of three-dimensional normal and fractal speckle fields produced by two or three scattered waves crossed orthogonally are studied theoretically. The probability density function and the autocorrelation function of intensity are derived for speckle fields superposed with and without interference. It is shown that the spatial anisotropy of intensity distributions exists even when three scattered waves interfere with one another. This spatial anisotropy affects the power-law distribution of intensity correlation for fractal speckles and leads to intensity patterns that are not self-similar in two or three dimensions. A potential application of the superposed speckle field is proposed.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. 76, 2668-2670 (2000).
    [CrossRef]
  4. T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
    [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]
  14. W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
    [CrossRef]
  15. T. Okamoto and A. Fukuyama, “Light amplification from cantor and asymmetric multilayer resonators,” Opt. Express 13, 8122-8127 (2005).
    [CrossRef] [PubMed]
  16. J. Uozumi, “Fractality of the optical fields scattered by power-law-illuminated diffusers,” Proc. SPIE 4607, 257-267 (2001).
    [CrossRef]
  17. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359-367 (2008).
    [CrossRef]

2008

2005

2001

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

J. Uozumi, “Fractality of the optical fields scattered by power-law-illuminated diffusers,” Proc. SPIE 4607, 257-267 (2001).
[CrossRef]

2000

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. 76, 2668-2670 (2000).
[CrossRef]

1999

E. Kolenovic, W. Osten, and W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333-344 (1999).
[CrossRef]

W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
[CrossRef]

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

1998

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350-358 (1998).
[CrossRef]

1997

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

1993

1990

1989

Armstrong, R. L.

W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
[CrossRef]

Asakura, T.

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350-358 (1998).
[CrossRef]

T. Okamoto and T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 34, E.Wolf, ed. (Elsevier, 1995), pp. 183-248.
[CrossRef]

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

Bernabeu, E.

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Costard, E.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Fukuyama, A.

Gauthier-Lafaye, O.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007), Chap. 3.

Hanson, R. S.

Hanson, S. G.

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Ibrahim, M.

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350-358 (1998).
[CrossRef]

Iwamoto, S.

Jakobsen, M. L.

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

Juptner, W.

E. Kolenovic, W. Osten, and W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333-344 (1999).
[CrossRef]

Kawata, S.

S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. 76, 2668-2670 (2000).
[CrossRef]

Kim, W.

W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
[CrossRef]

Kirchner, M.

Kolenovic, E.

E. Kolenovic, W. Osten, and W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333-344 (1999).
[CrossRef]

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

Kowalczyk, M.

Leushacke, L.

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

Okamoto, T.

T. Okamoto and A. Fukuyama, “Light amplification from cantor and asymmetric multilayer resonators,” Opt. Express 13, 8122-8127 (2005).
[CrossRef] [PubMed]

T. Okamoto and T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 34, E.Wolf, ed. (Elsevier, 1995), pp. 183-248.
[CrossRef]

Osten, W.

E. Kolenovic, W. Osten, and W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333-344 (1999).
[CrossRef]

Rose, B.

Safonov, V. P.

W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
[CrossRef]

Shalaev, V. M.

W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
[CrossRef]

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Shoji, S.

S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. 76, 2668-2670 (2000).
[CrossRef]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Uozumi, J.

J. Uozumi, “Fractality of the optical fields scattered by power-law-illuminated diffusers,” Proc. SPIE 4607, 257-267 (2001).
[CrossRef]

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350-358 (1998).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359-367 (2008).
[CrossRef]

Yoshimura, T.

Yura, H. T.

Appl. Phys. Lett.

S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. 76, 2668-2670 (2000).
[CrossRef]

T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

J. Appl. Phys.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Phys.

D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359-367 (2008).
[CrossRef]

Nature

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Opt. Commun.

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350-358 (1998).
[CrossRef]

E. Kolenovic, W. Osten, and W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333-344 (1999).
[CrossRef]

Opt. Express

Phys. Rev. Lett.

W. Kim, V. P. Safonov, V. M. Shalaev, and R. L. Armstrong, “Fractals in microcavities: giant coupled, multiplicative enhancement of optical responses,” Phys. Rev. Lett. 82, 4811-4814 (1999).
[CrossRef]

Proc. SPIE

J. Uozumi, “Fractality of the optical fields scattered by power-law-illuminated diffusers,” Proc. SPIE 4607, 257-267 (2001).
[CrossRef]

Other

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007), Chap. 3.

T. Okamoto and T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 34, E.Wolf, ed. (Elsevier, 1995), pp. 183-248.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the theoretical model to analyze superposed speckle fields.

Fig. 2
Fig. 2

Probability density functions for (a) two and (b) three speckle waves. The solid curves denote the case of incoherent addition, while the dashed curves represent (a) coherent addition and (b) partially coherent addition.

Fig. 3
Fig. 3

Intensity correlation functions for a single speckle wave as a function of (a) Δ x or Δ y with Δ z = 0 and (b) Δ z with Δ x = Δ y = 0 . The values of parameters are λ = 600 nm , w = 2.5 mm , L = 100 mm , and z 2 = 0 .

Fig. 4
Fig. 4

Intensity distributions in the x y plane with the area of 120 × 120 μ m 2 for (a),(b) two speckle waves propagating in the x and z directions and (c),(d) three speckle waves propagating in the x, y, and z directions. Here (a) and (c) are for coherent addition, while (b) and (d) are for incoherent addition. The parameter values are λ = 600 nm , w = 2.5 mm , and L = L = L = 100 mm . The horizontal axis is the x axis.

Fig. 5
Fig. 5

Intensity correlation functions for (a),(b) coherent addition and (c),(d) incoherent addition of two speckle waves as a function of (a),(c) Δ x with x 2 = y 1 = y 2 = z 1 = z 2 = 0 and (b),(d) x 1 x 2 with x 1 = y 1 and x 2 = y 2 = z 1 = z 2 = 0 . The parameter values are the same as those in Fig. 4.

Fig. 6
Fig. 6

Intensity correlation functions for (a),(b) partially coherent addition and (c),(d) incoherent addition of three speckle waves as a function of (a),(c) Δ x with x 2 = y 1 = y 2 = z 1 = z 2 = 0 and (b),(d) x 1 x 2 with x 1 = y 1 and x 2 = y 2 = z 1 = z 2 = 0 . The parameter values are the same as those in Fig. 4.

Fig. 7
Fig. 7

Intensity correlation functions for a single fractal speckle wave as a function of (a) Δ x or Δ y with Δ z = 0 and (b) Δ z with Δ x = Δ y = 0 for D = 1.2 (solid line), 1.5 (dashed line), and 1.8 (dotted line). The parameter values are λ = 600 nm , L = 100 mm , and z 2 = 0 .

Fig. 8
Fig. 8

Intensity distributions of superposed fractal speckle waves propagating in the x and z directions for (a),(b) D = 1.2 ; (c),(d) 1.5; and (e),(f) 1.8. The area of 120 × 120 μ m 2 in the x y plane is shown. Here (a), (c), and (e) are for coherent addition, while (b), (d), and (f) are for incoherent addition. The parameter values are λ = 600 nm and L = L = 100 mm . The horizontal axis is the x axis.

Fig. 9
Fig. 9

Intensity correlation functions for (a) coherent addition and (b) incoherent addition of two fractal speckle waves as a function of Δ x with x 2 = y 1 = y 2 = z 1 = z 2 = 0 . The parameter values are the same as those in Fig. 8.

Fig. 10
Fig. 10

Intensity distributions of superposed fractal speckle waves propagating in the x, y, and z directions for (a),(b) D = 1.2 ; (c),(d) 1.5; and (e),(f) 1.8. The area of 120 × 120 μ m 2 in the x y plane is shown. Here (a), (c), and (e) are for coherent addition, while (b), (d), and (f) are for incoherent addition. The parameter values are λ = 600 nm and L = L = L = 100 mm . The horizontal axis is the x axis.

Fig. 11
Fig. 11

Intensity correlation functions for (a) partially coherent addition and (b) incoherent addition of three fractal speckle waves as a function of Δ x with x 2 = y 1 = y 2 = z 1 = z 2 = 0 . The parameter values are the same as those in Fig. 10.

Equations (83)

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A 1 ( x ) = j = 1 N 1 N a p j q exp ( i ϕ p j q ) ,
A 2 ( x ) = j = 1 N 1 N a p j q exp ( i ϕ p j q ) ,
A ( x ) = A 1 ( x ) + A 2 ( x ) .
p I ( I ) = 1 I e I I ,
I ( x 1 ) I ( x 2 ) = I ( x 1 ) I ( x 2 ) [ 1 + γ A ( x 1 , x 2 ) 2 ] ,
I ( x 1 ) = A ( x 1 ) A * ( x 1 ) ,
I ( x 2 ) = A ( x 2 ) A * ( x 2 ) ,
γ A ( x 1 , x 2 ) = A ( x 1 ) A * ( x 2 ) [ I ( x 1 ) I ( x 2 ) ] 1 2 .
A ( x 1 ) A * ( x 2 ) = A 1 ( x 1 ) A 1 * ( x 2 ) + A 2 ( x 1 ) A 2 * ( x 2 ) + A 1 ( x 1 ) A 2 * ( x 2 ) + A 2 ( x 1 ) A 1 * ( x 2 ) .
A 1 ( x ) = 1 i λ a 1 ( ξ ) exp ( i 2 π R λ ) R d ξ d η ,
A 2 ( x ) = 1 i λ a 2 ( η ) exp ( i 2 π R λ ) R d η d ζ ,
R = x ξ = [ ( x ξ ) 2 + ( y η ) 2 + ( z + L ) 2 ] 1 2 ,
R = x η = [ ( x + L ) 2 + ( y η ) 2 + ( z ζ ) 2 ] 1 2 .
R = z + L + ( x ξ ) 2 + ( y η ) 2 2 ( z + L ) ,
R = x + L + ( y η ) 2 + ( z ζ ) 2 2 ( x + L ) .
A 1 ( x ) = 1 i λ L exp [ i 2 π λ ( z + L ) ] a 1 ( ξ ) exp { i π [ ( x ξ ) 2 + ( y η ) 2 ] λ ( z + L ) } d ξ d η ,
A 2 ( x ) = 1 i λ L exp [ i 2 π λ ( x + L ) ] a 2 ( η ) exp { i π [ ( y η ) 2 + ( z ζ ) 2 ] λ ( x + L ) } d η d ζ ,
a 1 ( ξ 1 ) a 1 * ( ξ 2 ) = κ P ( ξ 1 ) P * ( ξ 2 ) δ ( ξ 1 ξ 2 ) ,
a 2 ( η 1 ) a 2 * ( η 2 ) = κ P ( η 1 ) P * ( η 2 ) δ ( η 1 η 2 ) ,
a 1 ( ξ 1 ) a 2 * ( η 2 ) = 0 ,
a 2 ( η 1 ) a 1 * ( ξ 2 ) = 0 ,
A 1 ( x 1 ) A 1 * ( x 2 ) = c exp ( i 2 π λ Δ z ) [ i π λ ( x 1 2 + y 1 2 z 1 + L x 2 2 + y 2 2 z 2 + L ) ] × P ( ξ ) 2 exp [ i π λ Δ z ( z 1 + L ) ( z 2 + L ) ( ξ 2 + η 2 ) ] exp [ i 2 π λ ( Δ x ξ + Δ y η ) ] d ξ d η ,
A 2 ( x 1 ) A 2 * ( x 2 ) = c exp ( i 2 π λ Δ x ) [ i π λ ( y 1 2 + z 1 2 x 1 + L y 2 2 + z 2 2 x 2 + L ) ] × P ( η ) 2 exp [ i π λ Δ x ( x 1 + L ) ( x 2 + L ) ( η 2 + ζ 2 ) ] × exp [ i 2 π λ ( Δ y η + Δ z ζ ) ] d η d ζ ,
A 1 ( x 1 ) A 2 * ( x 2 ) = 0 ,
A 2 ( x 1 ) A 1 * ( x 2 ) = 0 ,
c = κ λ 2 L 2 , c = κ λ 2 L 2 ,
Δ x = x 1 x 2 , Δ z = z 1 z 2 ,
Δ x = x 1 z 1 + L x 2 z 2 + L , Δ y = y 1 z 1 + L y 2 z 2 + L ,
Δ z = z 1 x 1 + L z 2 x 2 + L ,
Δ y = y 1 x 1 + L y 2 x 2 + L .
I ( x 1 ) = I ( x 2 ) I 1 + I 2 ,
I 1 = c P ( ξ ) 2 d ξ d η ,
I 2 = c P ( η ) 2 d η d ζ .
γ A ( x 1 , x 2 ) = A 1 ( x 1 ) A 1 * ( x 2 ) + A 2 ( x 1 ) A 2 * ( x 2 ) I 1 + I 2 .
p I ( I ) = N N I N 1 ( N 1 ) ! I N exp ( N I I ) .
p I ( I ) = 4 I I 2 exp ( 2 I I ) .
I ( x 1 ) I ( x 2 ) = [ I x ( x 1 ) + I y ( x 1 ) ] [ I x ( x 2 ) + I y ( x 2 ) ] = I x ( x 1 ) I x ( x 2 ) + I y ( x 1 ) I y ( x 2 ) + I x ( x 1 ) I y ( x 2 ) + I y ( x 1 ) I x ( x 2 ) ,
I x ( x 1 ) I y ( x 2 ) = I x ( x 1 ) I y ( x 2 ) = I 1 ( x 1 ) I 2 ( x 2 ) ,
I y ( x 1 ) I x ( x 2 ) = I y ( x 1 ) I x ( x 2 ) = I 2 ( x 1 ) I 1 ( x 2 ) ,
I ( x 1 ) I ( x 2 ) = I 1 ( x 1 ) I 1 ( x 2 ) + A 1 ( x 1 ) A 1 * ( x 2 ) 2 + I 2 ( x 1 ) I 2 ( x 2 ) + A 2 ( x 1 ) A 2 * ( x 2 ) 2 + I 1 ( x 1 ) I 2 ( x 2 ) + I 2 ( x 1 ) I 1 ( x 2 ) .
I ( x 1 ) I ( x 2 ) = I 1 2 [ 1 + γ A 1 ( x 1 , x 2 ) 2 ] + I 2 2 [ 1 + γ A 2 ( x 1 , x 2 ) 2 ] + 2 I 1 I 2 ,
γ A 1 ( x 1 , x 2 ) = A 1 ( x 1 ) A 1 * ( x 2 ) [ I 1 ( x 1 ) I 1 ( x 2 ) ] 1 2 = A 1 ( x 1 ) A 1 * ( x 2 ) I 1 ,
γ A 2 ( x 1 , x 2 ) = A 2 ( x 1 ) A 2 * ( x 2 ) [ I 2 ( x 1 ) I 2 ( x 2 ) ] 1 2 = A 2 ( x 1 ) A 2 * ( x 2 ) I 2 .
I ( x 1 ) I ( x 2 ) = I 2 4 [ 4 + γ A 1 ( x 1 , x 2 ) 2 + γ A 2 ( x 1 , x 2 ) 2 ] .
A 3 ( x ) = j = 1 N 1 N a p j q exp ( i ϕ p j q ) ,
A x ( x ) = 1 2 A 1 ( x ) + 1 2 A 3 ( x ) ,
A y ( x ) = 1 2 A 1 ( x ) + 1 2 A 2 ( x ) ,
A z ( x ) = 1 2 A 2 ( x ) + 1 2 A 3 ( x ) .
[ J ] = [ A ] [ A ] = [ A x A y A z ] [ A x * A y * A z * ] = [ I x I x I y γ x y I x I z γ x z I x I y γ x y * I y I y I z γ y z I x I z γ x z * I y I z γ y z * I z ] .
p I ( I ) = I λ 1 ( λ 1 λ 3 ) exp ( I λ 1 ) λ 3 ( λ 1 λ 3 ) 2 exp ( I λ 1 ) + λ 3 ( λ 1 λ 3 ) 2 exp ( I λ 3 ) = ( 12 I I 2 + 8 3 I ) exp ( 6 I I ) + 8 3 I exp ( 3 I 2 I ) .
I ( x 1 ) I ( x 2 ) = [ I x ( x 1 ) + I y ( x 1 ) + I z ( x 1 ) ] [ I x ( x 2 ) + I y ( x 2 ) + I z ( x 2 ) ] = I x ( x 1 ) I x ( x 2 ) + I y ( x 1 ) I y ( x 2 ) + I z ( x 1 ) I z ( x 2 ) + I x ( x 1 ) I y ( x 2 ) + I x ( x 1 ) I z ( x 2 ) + I y ( x 1 ) I x ( x 2 ) + I y ( x 1 ) I z ( x 2 ) + I z ( x 1 ) I x ( x 2 ) + I z ( x 1 ) I y ( x 2 ) ,
I x ( x 1 ) I y ( x 2 ) = I x ( x 1 ) I y ( x 2 ) + A x ( x 1 ) A y * ( x 2 ) 2 .
I ( x 1 ) I ( x 2 ) = I x 2 [ 1 + γ A x ( x 1 , x 2 ) 2 ] + I y 2 [ 1 + γ A y ( x 1 , x 2 ) 2 ] + I z 2 [ 1 + γ A z ( x 1 , x 2 ) 2 ] + 1 2 I 1 2 [ 1 + γ A 1 ( x 1 , x 2 ) 2 ] + 1 2 I 2 2 [ 1 + γ A 2 ( x 1 , x 2 ) 2 ] + 1 2 I 3 2 [ 1 + γ A 3 ( x 1 , x 2 ) 2 ] + 3 2 I 1 I 2 + 3 2 I 2 I 3 + 3 2 I 3 I 1 ,
I x = I 1 + I 3 2 , I y = I 1 + I 2 2 , I z = I 2 + I 3 2 ,
γ A x ( x 1 , x 3 ) = A 1 ( x 1 ) A 1 * ( x 2 ) + A 3 ( x 1 ) A 3 * ( x 2 ) I 1 + I 3 ,
γ A y ( x 1 , x 2 ) = A 1 ( x 1 ) A 1 * ( x 2 ) + A 2 ( x 1 ) A 2 * ( x 2 ) I 1 + I 2 ,
γ A z ( x 1 , x 2 ) = A 2 ( x 1 ) A 2 * ( x 2 ) + A 3 ( x 1 ) A 3 * ( x 2 ) I 2 + I 3 .
γ A 3 ( x 1 , x 2 ) = A 3 ( x 1 ) A 3 * ( x 2 ) [ I 3 ( x 1 ) I 3 ( x 2 ) ] 1 2 = A 3 ( x 1 ) A 3 * ( x 2 ) I 3 ,
A 3 ( x 1 ) A 3 * ( x 2 ) = c exp ( i 2 π λ Δ y ) [ i π λ ( z 1 2 + x 1 2 y 1 + L z 2 2 + x 2 2 y 2 + L ) ] × P ( ζ ) 2 exp [ i π λ Δ y ( y 1 + L ) ( y 2 + L ) ( ζ 2 + ξ 2 ) ] × exp [ i 2 π λ ( Δ z ζ + Δ x ξ ) ] d ζ d ξ ,
I 3 = c P ( ζ ) 2 d ζ d ξ ,
c = κ λ 2 L 2 ,
Δ y = y 1 y 2 ,
Δ x = x 1 y 1 + L x 2 y 2 + L , Δ z = z 1 y 1 + L z 2 y 2 + L .
I ( x 1 ) I ( x 2 ) = I 2 9 [ 9 + γ A x ( x 1 , x 2 ) 2 + γ A y ( x 1 , x 2 ) 2 + γ A z ( x 1 , x 2 ) 2 + 1 2 γ A 1 ( x 1 , x 2 ) 2 + 1 2 γ A 2 ( x 1 , x 2 ) 2 + 1 2 γ A 3 ( x 1 , x 2 ) 2 ] .
p I ( I ) = 27 I 2 2 I 3 exp ( 3 I I ) .
I ( x 1 ) I ( x 2 ) = I 1 ( x 1 ) I 1 ( x 2 ) + A 1 ( x 1 ) A 1 * ( x 2 ) 2 + I 2 ( x 1 ) I 2 ( x 2 ) + A 2 ( x 1 ) A 2 * ( x 2 ) 2 + I 3 ( x 1 ) I 3 ( x 2 ) + A 3 ( x 1 ) A 3 * ( x 2 ) 2 + I 1 ( x 1 ) I 2 ( x 2 ) + I 2 ( x 1 ) I 1 ( x 2 ) + I 2 ( x 1 ) I 3 ( x 2 ) + I 3 ( x 1 ) I 2 ( x 2 ) + I 3 ( x 1 ) I 1 ( x 2 ) + I 1 ( x 1 ) I 3 ( x 2 ) .
I ( x 1 ) I ( x 2 ) = I 1 2 [ 1 + γ A 1 ( x 1 , x 2 ) 2 ] + I 2 2 [ 1 + γ A 2 ( x 1 , x 2 ) 2 ] + I 3 2 [ 1 + γ A 3 ( x 1 , x 2 ) 2 ] + 2 I 1 I 2 + 2 I 2 I 3 + 2 I 3 I 1 .
I ( x 1 ) I ( x 2 ) = I 2 9 [ 9 + γ A 1 ( x 1 , x 2 ) 2 + γ A 2 ( x 1 , x 2 ) 2 + γ A 3 ( x 1 , x 2 ) 2 ] .
P ( ξ ) 2 = exp [ 2 ξ 2 w 2 ] ,
P ( η ) 2 = exp [ 2 η 2 w 2 ] ,
P ( ζ ) 2 = exp [ 2 ζ 2 w 2 ] ,
P ( ξ ) 2 = ξ D , P ( η ) 2 = η D , P ( ζ ) 2 = ζ D ,
c I ( x 1 , x 2 ) = γ A 1 ( x 1 , x 2 ) 2 ,
c I ( x 1 , x 2 ) = γ A ( x 1 , x 2 ) 2 ,
c I ( x 1 , x 2 ) = 1 2 [ γ A 1 ( x 1 , x 2 ) 2 + γ A 2 ( x 1 , x 2 ) 2 ] ,
c I ( x 1 , x 2 ) = 2 9 [ γ A x ( x 1 , x 2 ) 2 + γ A y ( x 1 , x 2 ) 2 + γ A z ( x 1 , x 2 ) 2 + 1 2 γ A 1 ( x 1 , x 2 ) 2 + 1 2 γ A 2 ( x 1 , x 2 ) 2 + 1 2 γ A 3 ( x 1 , x 2 ) 2 ] ,
c I ( x 1 , x 2 ) = 1 3 [ γ A 1 ( x 1 , x 2 ) 2 + γ A 2 ( x 1 , x 2 ) 2 + γ A 3 ( x 1 , x 2 ) 2 ] .
c I ( x 1 , x 2 ) = exp [ π 2 w 2 λ 2 ( z 0 + L ) 2 ( Δ x 2 + Δ y 2 ) ] ,
c I ( x 1 , x 2 ) = 1 + ( w 2 2 ) i π Δ z λ ( z 2 + L ) 2 + λ ( z 2 + L ) Δ z 2 .
c I ( r ) r 2 ( D 2 )
C ( r ) r ( d D s ) ,
c I ( x 1 , x 2 ) ( z 2 + L ) ( z 2 + Δ z + L ) Δ z ( i Δ z λ ( z 2 + L ) 2 + λ ( z 2 + L ) Δ z ) D 2 2 .
c I ( x 1 , x 2 ) Δ z D 2

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