Abstract

The Wigner distribution function (WDF) matrix is introduced, and its analytical propagation expressions in free space and the corresponding far-field forms are derived for the first time. It is shown that the propagation expressions for the WDF matrix have general applicable advantages and reduce to those for the WDF of partially coherent scalar paraxial beams within the paraxial regime. The application of the WDF matrix to the partially coherent vectorial nonparaxial beams is illustrated with vectorial nonparaxial Gaussian Schell-model beams.

© 2005 Optical Society of America

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  1. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  2. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [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, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
    [CrossRef]
  5. M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215-1224 (1981).
    [CrossRef]
  6. M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 82, 173-181 (1989).
  7. D. Mendlovic and Z. Zalevsky, "Definition, properties and applications of the generalized temporal-spatial Wigner distribution function," Optik (Stuttgart) 107, 49-56 (1997).
  8. G. Ding and B. Lü, "Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems," J. Mod. Opt. 47, 1483-1499 (2000).
  9. G. Ding and B. Lü, "Propagation of twisted Gaussian Schell-model beams through a misaligned first-order optical system," J. Mod. Opt. 48, 1617-1621 (2001).
    [CrossRef]
  10. P. Avi, D. Wang, A. W. Lohmann, and A. A. Friesem, "Wigner formulation of optical processing with light of arbitrary coherence," Appl. Opt. 40, 249-256 (2001).
    [CrossRef]
  11. G. Ding and B. Lü, "Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system," Opt. Quantum Electron. 35, 91-100 (2003).
    [CrossRef]
  12. K. B. Wolf, M. A. Alonso, and G. W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A 16, 2476-2479 (1999).
    [CrossRef]
  13. C. J. R. Sheppard and K. G. Larkin, "Wigner function for highly convergent three-dimensional wave fields," Opt. Lett. 26, 968-970 (2001).
    [CrossRef]
  14. K. Duan and B. Lü, "Partially coherent vectorial nonparaxial beams," J. Opt. Soc. Am. A 21, 1924-1932 (2004).
    [CrossRef]
  15. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
    [CrossRef]
  17. F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 241-243 (1998).
    [CrossRef]
  18. R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, 1970).
  19. R. Borghi, A. Ciattoni, and M. Santarsiero, "Exact axial electronmagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions," J. Opt. Soc. Am. A 19, 1207-1211 (2002).
    [CrossRef]
  20. Y. Zhang and B. Lü, "Propagation of the Wigner distribution function for partially coherent nonparaxial beams," Opt. Lett. 29, 2710-2712 (2004).
    [CrossRef] [PubMed]

2004 (2)

2003 (2)

G. Ding and B. Lü, "Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system," Opt. Quantum Electron. 35, 91-100 (2003).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

2002 (1)

2001 (3)

2000 (1)

G. Ding and B. Lü, "Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems," J. Mod. Opt. 47, 1483-1499 (2000).

1999 (1)

1998 (1)

1997 (1)

D. Mendlovic and Z. Zalevsky, "Definition, properties and applications of the generalized temporal-spatial Wigner distribution function," Optik (Stuttgart) 107, 49-56 (1997).

1989 (1)

M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 82, 173-181 (1989).

1981 (1)

M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215-1224 (1981).
[CrossRef]

1979 (1)

1978 (1)

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

1968 (1)

1932 (1)

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

Alonso, M. A.

Avi, P.

Bastiaans, M. J.

M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 82, 173-181 (1989).

M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215-1224 (1981).
[CrossRef]

M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
[CrossRef]

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

Bellman, R.

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, 1970).

Borghi, R.

Ciattoni, A.

Ding, G.

G. Ding and B. Lü, "Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system," Opt. Quantum Electron. 35, 91-100 (2003).
[CrossRef]

G. Ding and B. Lü, "Propagation of twisted Gaussian Schell-model beams through a misaligned first-order optical system," J. Mod. Opt. 48, 1617-1621 (2001).
[CrossRef]

G. Ding and B. Lü, "Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems," J. Mod. Opt. 47, 1483-1499 (2000).

Duan, K.

Forbes, G. W.

Friesem, A. A.

Gori, F.

Larkin, K. G.

Lohmann, A. W.

Lü, B.

K. Duan and B. Lü, "Partially coherent vectorial nonparaxial beams," J. Opt. Soc. Am. A 21, 1924-1932 (2004).
[CrossRef]

Y. Zhang and B. Lü, "Propagation of the Wigner distribution function for partially coherent nonparaxial beams," Opt. Lett. 29, 2710-2712 (2004).
[CrossRef] [PubMed]

G. Ding and B. Lü, "Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system," Opt. Quantum Electron. 35, 91-100 (2003).
[CrossRef]

G. Ding and B. Lü, "Propagation of twisted Gaussian Schell-model beams through a misaligned first-order optical system," J. Mod. Opt. 48, 1617-1621 (2001).
[CrossRef]

G. Ding and B. Lü, "Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems," J. Mod. Opt. 47, 1483-1499 (2000).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

Mendlovic, D.

D. Mendlovic and Z. Zalevsky, "Definition, properties and applications of the generalized temporal-spatial Wigner distribution function," Optik (Stuttgart) 107, 49-56 (1997).

Santarsiero, M.

Sheppard, C. J.

Walther, A.

Wang, D.

Wigner, E.

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

Wolf, E.

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

Wolf, K. B.

Zalevsky, Z.

D. Mendlovic and Z. Zalevsky, "Definition, properties and applications of the generalized temporal-spatial Wigner distribution function," Optik (Stuttgart) 107, 49-56 (1997).

Zhang, Y.

Appl. Opt. (1)

J. Mod. Opt. (2)

G. Ding and B. Lü, "Propagation characteristics of the ten-parameter family of partially coherent general anisotropic Gaussian Schell-model (AGSM) beams passing through first-order optical systems," J. Mod. Opt. 47, 1483-1499 (2000).

G. Ding and B. Lü, "Propagation of twisted Gaussian Schell-model beams through a misaligned first-order optical system," J. Mod. Opt. 48, 1617-1621 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (1)

M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215-1224 (1981).
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (3)

Opt. Quantum Electron. (1)

G. Ding and B. Lü, "Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system," Opt. Quantum Electron. 35, 91-100 (2003).
[CrossRef]

Optik (Stuttgart) (2)

M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 82, 173-181 (1989).

D. Mendlovic and Z. Zalevsky, "Definition, properties and applications of the generalized temporal-spatial Wigner distribution function," Optik (Stuttgart) 107, 49-56 (1997).

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. (1)

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

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, 1970).

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Figures (1)

Fig. 1
Fig. 1

Schematic of the position vectors ρ j 0 ( j = 1 , 2), and the field E j ( ρ j 0 , 0 ) at the plane z = 0 . E x ( ρ j 0 , 0 ) and E y ( ρ j 0 , 0 ) are the components of E j ( ρ j 0 , 0 ) .

Equations (79)

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W ̂ 0 ( ρ 10 , ρ 20 , 0 ) = [ W x x 0 ( ρ 10 , ρ 20 , 0 ) W x y 0 ( ρ 10 , ρ 20 , 0 ) 0 W y x 0 ( ρ 10 , ρ 20 , 0 ) W y y 0 ( ρ 10 , ρ 20 , 0 ) 0 0 0 0 ] ,
W ̂ ( ρ 1 , ρ 2 , z ) = [ W x x ( ρ 1 , ρ 2 , z ) W x y ( ρ 1 , ρ 2 , z ) W x z ( ρ 1 , ρ 2 , z ) W y x ( ρ 1 , ρ 2 , z ) W y y ( ρ 1 , ρ 2 , z ) W y z ( ρ 1 , ρ 2 , z ) W z x ( ρ 1 , ρ 2 , z ) W z y ( ρ 1 , ρ 2 , z ) W z z ( ρ 1 , ρ 2 , z ) ] ,
F ̂ ( ρ 0 , z = 0 ; q 0 ) = [ F x x ( ρ 0 , 0 ; q 0 ) F x y ( ρ 0 , 0 ; q 0 ) 0 F y x ( ρ 0 , 0 ; q 0 ) F y y ( ρ 0 , 0 ; q 0 ) 0 0 0 0 ] ,
F ̂ ( ρ , z ; q ) = [ F x x ( ρ , z ; q ) F x y ( ρ , z ; q ) F x z ( ρ , z ; q ) F y x ( ρ , z ; q ) F y y ( ρ , z ; q ) F y z ( ρ , z ; q ) F z y ( ρ , z ; q ) F z y ( ρ , z ; q ) F z z ( ρ , z ; q ) ] ,
F α β ( ρ 0 , 0 ; q 0 ) = W α β 0 ( ρ 0 + ρ 0 2 , ρ 0 ρ 0 2 , 0 ) exp ( i q 0 ρ 0 ) d ρ 0 ,
F μ ν ( ρ , z ; q ) = W μ ν ( ρ + ρ 2 , ρ ρ 2 , z ) exp ( i q ρ ) d ρ .
W μ ν ( ρ 1 , ρ 2 , z ) = 1 4 π 2 F μ ν ( ρ 1 + ρ 2 2 , z ; q ) exp { [ i ( ρ 1 ρ 2 ) q ] } d q .
J μ ν ( ρ , z ) = 1 4 π 2 F μ ν ( ρ , z ; q ) d q ,
I ( ρ , z ) = μ = ν Tr [ J μ ν ( ρ , z ) ] = 1 4 π 2 Tr [ F ̂ ( ρ , z ; q ) ] d q ,
W μ ν * ( ρ 1 , ρ 2 , z ) = W ν μ ( ρ 2 , ρ 1 , z ) ,
F μ ν * ( ρ , z ; q ) = F ν μ ( ρ , z ; q ) .
F α β ( ρ , z ; q ) = K α β ( ρ , q , ρ 0 , q 0 ) F α β ( ρ 0 , 0 ; q 0 ) d ρ 0 d q 0 ,
F α z ( ρ , z ; q ) = [ K α , z x ( ρ , q , ρ 0 , q 0 ) F α x ( ρ 0 , 0 ; q 0 ) + K α , z y × ( ρ , q , ρ 0 , q 0 ) F α y ( ρ 0 , 0 ; q 0 ) ] d ρ 0 d q 0 ,
F z z ( ρ , z ; q ) = [ K z x , z x ( ρ , q , ρ 0 , q 0 ) F x x ( ρ 0 , 0 ; q 0 ) + K z x , z y ( ρ , q , ρ 0 , q 0 ) F x y ( ρ 0 , 0 ; q 0 ) + K z y , z x ( ρ , q , ρ 0 , q 0 ) F y x ( ρ 0 , 0 ; q 0 ) + K z y , z y ( ρ , q , ρ 0 , q 0 ) F y y ( ρ 0 , 0 ; q 0 ) ] d ρ 0 d q 0 ,
K α β ( ρ , q , ρ 0 , q 0 ) = 1 4 π 2 h α ( ρ + ρ 2 , ρ 0 + ρ 0 2 ) h β * ( ρ ρ 2 , ρ 0 ρ 0 2 ) exp [ i ( q ρ q 0 ρ 0 ) ] d ρ d ρ 0 ,
K α , z β ( ρ , q , ρ 0 , q 0 ) = 1 4 π 2 h α ( ρ + ρ 2 , ρ 0 + ρ 0 2 ) h z β * ( ρ ρ 2 , ρ 0 ρ 0 2 ) exp [ i ( q ρ q 0 ρ 0 ) ] d ρ d ρ 0 ,
K z α , z β ( ρ , q , ρ 0 , q 0 ) = 1 4 π 2 h z α ( ρ + ρ 2 , ρ 0 + ρ 0 2 ) h z β * ( ρ ρ 2 , ρ 0 ρ 0 2 ) exp [ i ( q ρ q 0 ρ 0 ) ] d ρ d ρ 0 .
K ϕ ψ * ( ρ , q , ρ 0 , q 0 ) = K ψ ϕ ( ρ , q , ρ 0 , q 0 ) .
h α ( ρ , ρ 0 ) = 1 2 π δ α ( x 0 ξ ) δ α ( y 0 η ) G z d ξ d η ,
h α z ( ρ , ρ 0 ) = 1 2 π δ α ( x 0 ξ ) δ α ( y 0 η ) G α d ξ d η ,
G = exp ( i k R ) R
G z z i k R 2 exp ( i k R ) ,
G α ( α α 0 ) i k R 2 exp ( i k R ) .
R r + ξ 2 + η 2 2 x ξ 2 y η 2 r .
h α ( ρ , ρ 0 ) i z λ r 2 exp ( i k r ) exp ( i k x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) ,
h z α ( ρ , ρ 0 ) i ( α α 0 ) λ r 2 exp ( i k r ) exp ( i k x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) ,
h α ( ρ + ρ 2 , ρ 0 + ρ 0 2 ) i z λ r 2 exp { i k [ ( x + x 2 ) 2 + ( y + y 2 ) 2 + z 2 ] 1 2 } exp { i k 2 r [ ( x 0 + x 0 2 ) 2 + ( y 0 + y 0 2 ) 2 2 ( x + x 2 ) ( x 0 + x 0 2 ) 2 ( y + y 2 ) ( y 0 + y 0 2 ) ] } ,
h α ( ρ ρ 2 , ρ 0 ρ 0 2 ) i z λ r 2 exp { i k [ ( x x 2 ) 2 + ( y y 2 ) 2 + z 2 ] 1 2 } exp { i k 2 r [ ( x 0 x 0 2 ) 2 + ( y 0 y 0 2 ) 2 2 ( x x 2 ) ( x 0 x 0 2 ) 2 ( y y 2 ) ( y 0 y 0 2 ) ] } .
[ ( x + x 2 ) 2 + ( y + y 2 ) 2 + z 2 ] 1 2 r + 1 2 r ( x x + y y + x 2 + y 2 4 ) ,
[ ( x x 2 ) 2 + ( y y 2 ) 2 + z 2 ] 1 2 r + 1 2 r ( x x y y + x 2 + y 2 4 ) .
h α ( ρ + ρ 2 , ρ 0 + ρ 0 2 ) h α * ( ρ ρ 2 , ρ 0 ρ 0 2 ) z 2 λ 2 r 4 exp ( i k x x + y y r ) exp [ i k x x 0 + x 0 ( x x 0 ) + y y 0 + y 0 ( y y 0 ) r ] = z 2 λ 2 r 4 exp [ i k ( x 0 x ) ( x 0 x ) + ( y 0 y ) ( y 0 y ) r ] .
exp ( i ξ x ) d x = 2 π δ ( ξ ) ,
K α β ( ρ , q , ρ 0 , q 0 ) = z 2 4 π 2 λ 2 r 4 exp [ i k ( x 0 x ) ( x 0 x ) + ( y 0 y ) ( y 0 y ) r ] exp [ i ( q ρ q 0 ρ 0 ) ] d ρ d ρ 0 = z 2 4 π 2 λ 2 r 4 exp [ i k ( x 0 x ) x r i u x ] d x exp [ i k ( y 0 y ) y r i v y ] d y exp [ i k ( x 0 x ) x 0 r + i u x 0 ] d x 0 exp [ i k ( y 0 y ) y 0 r + i v y 0 ] d y 0 = k 2 z 2 r 4 δ ( a + q ) δ ( a + q 0 ) ,
K α , z x ( ρ , q , ρ 0 , q 0 ) = k 2 z 2 r 4 [ 2 ( x x 0 ) δ ( a + q ) δ ( a + q 0 ) i δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) i δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) ] ,
K α , z y ( ρ , q , ρ 0 , q 0 ) = k 2 z 2 r 4 [ 2 ( y y 0 ) δ ( a + q ) δ ( a + q 0 ) i δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) i δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) ] ,
K z x , z x ( ρ , q , ρ 0 , q 0 ) = k 2 1 4 r 4 [ 4 ( x x 0 ) 2 δ ( a + q ) δ ( a + q 0 ) 2 δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a x + u ) δ ( a y + v ) + δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) + δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) ] ,
K z y , z y ( ρ , q , ρ 0 , q 0 ) = k 2 1 4 r 4 [ 4 ( y y 0 ) 2 δ ( a + q ) δ ( a + q 0 ) 2 δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a x + u ) δ ( a y + v ) + δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) + δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) ] ,
K z x , z y ( ρ , q , ρ 0 , q 0 ) = k 2 r 4 ( x x 0 ) ( y y 0 ) δ ( a + q ) δ ( a + q 0 ) i k 2 2 r 4 ( x x 0 ) [ δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) + δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) ] + i k 2 2 r 4 ( y y 0 ) [ δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) + δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) ] + k 2 4 r 4 [ δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) + δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) + δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a x + u ) δ ( a y + v ) + δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a x + u ) δ ( a y + v ) ] ,
F α β ( ρ , z ; q ) = z 2 r 2 F α β ( p , 0 ; q ) ,
F x z ( ρ , z ; q ) = z r k [ u F x x ( p , 0 ; q ) + v F x y ( p , 0 ; q ) ] i z 2 r k [ F x x ( 1 , 0 , 0 , 0 ) ( p , 0 ; q ) + F x y ( 0 , 1 , 0 , 0 ) ( p , 0 ; q ) ] ,
F y z ( ρ , z ; q ) = z r k [ u F y x ( p , 0 ; q ) + v F y y ( p , 0 ; q ) ] i z 2 r k [ F y x ( 1 , 0 , 0 , 0 ) ( p , 0 ; q ) + F y y ( 0 , 1 , 0 , 0 ) ( p , 0 ; q ) ] ,
F z z ( ρ , z ; q ) = 1 k 2 [ u 2 F x x ( p , 0 ; q ) + v 2 F y y ( p , 0 ; q ) ] + 1 r 2 [ F x x ( 0 , 0 , 2 , 0 ) ( p , 0 ; q ) + F y y ( 0 , 0 , 0 , 2 ) ( p , 0 ; q ) ] + 1 4 k 2 [ F x x ( 2 , 0 , 0 , 0 ) ( p , 0 ; q ) + F y y ( 0 , 2 , 0 , 0 ) ( p , 0 ; q ) ] + 2 u v k 2 F x y ( p , 0 ; q ) + 1 2 k 2 F x y ( 1 , 1 , 0 , 0 ) ( p , 0 ; q ) ,
F α β ( s , t , m , n ) ( ρ , z ; q ) = s x s t y t m u m n v n F α β ( ρ , z ; q ) .
h α F ( ρ , ρ 0 ) i z λ r 2 exp ( i k r ) exp ( i k x x 0 + y y 0 r ) ,
h α z F ( ρ , ρ 0 ) i ( α α 0 ) λ r 2 exp ( i k r ) exp ( i k x x 0 + y y 0 r ) .
F α β f ( ρ , z ; q ) = z 2 r 2 F α β ( p , 0 ; b ) ,
F x z f ( ρ , z ; q ) = z r k [ u F x x ( p , 0 ; b ) + v F x y ( p , 0 ; b ) ] + i z 2 r 2 × [ F x x ( 0 , 0 , 1 , 0 ) ( p , 0 ; b ) + F x y ( 0 , 0 , 0 , 1 ) ( p , 0 ; b ) ] i z 2 r k [ F x x ( 1 , 0 , 0 , 0 ) ( p , 0 ; b ) + F x y ( 0 , 1 , 0 , 0 ) ( p , 0 ; b ) ] ,
F y z f ( ρ , z ; q ) = z r k [ u F y x ( p , 0 ; b ) + v F y y ( p , 0 ; b ) ] + i z 2 r 2 × [ F y x ( 0 , 0 , 1 , 0 ) ( p , 0 ; b ) + F y y ( 0 , 0 , 0 , 1 ) ( p , 0 ; b ) ] i z 2 r k [ F y x ( 1 , 0 , 0 , 0 ) ( p , 0 ; b ) + F y y ( 0 , 1 , 0 , 0 ) ( p , 0 ; b ) ] ,
F z z f ( ρ , z ; q ) = 1 k 2 [ u 2 F x x ( p , 0 ; b ) + v 2 F y x ( p , 0 ; b ) + 2 u v F x y ( p , 0 ; q ) ] + 1 2 r k [ F x x ( 1 , 0 , 1 , 0 ) ( p , 0 ; b ) + F y y ( 0 , 1 , 0 , 1 ) ( p , 0 ; b ) ] + 1 4 r 2 [ F x x ( 0 , 0 , 2 , 0 ) ( p , 0 ; b ) + F y y ( 0 , 0 , 0 , 2 ) ( p , 0 ; b ) ] + 1 4 k 2 [ F x x ( 2 , 0 , 0 , 0 ) ( p , 0 ; b ) + F y y ( 0 , 2 , 0 , 0 ) ( p , 0 ; b ) ] + 1 2 k r [ F x y ( 1 , 0 , 0 , 1 ) ( p , 0 ; b ) + F x y ( 0 , 1 , 1 , 0 ) ( p , 0 ; b ) ] + 1 2 k 2 F x y ( 1 , 1 , 0 , 0 ) ( p , 0 ; b ) + 1 2 r 2 F x y ( 0 , 0 , 1 , 1 ) ( p , 0 ; b ) ,
F α β ( ρ , z ; q ) = F α β ( ρ z k q , 0 ; q ) ,
F α β f ( ρ , z ; q ) = F α β ( ρ z k q , 0 ; k z ρ ) .
W α β 0 ( ρ 10 , ρ 20 , z = 0 ) = { exp ( ρ 10 ρ 20 2 2 σ 0 2 ) exp ( ρ 10 2 + ρ 20 2 2 w 0 2 ) α = β = x 0 otherwise } .
F α β ( ρ 0 , z = 0 ; q 0 ) = { 4 π Q exp ( Q q 0 2 ) exp ( k 2 f 2 ρ 0 2 ) , α = β = x 0 otherwise } ,
F x x ( ρ , z ; q ) = 4 π Q z 2 r 2 exp ( Q q 2 ) exp [ k 2 f 2 p 2 ] ,
F x z ( ρ , z ; q ) = 4 π Q z r k [ 2 i f 2 k 2 p x u ] exp ( Q q 2 ) exp [ k 2 f 2 p 2 ] ,
F z z ( ρ , z ; q ) = 2 π Q { 2 u 2 k 2 + 4 π Q 1 r 2 ( 2 Q u 2 1 ) + f 2 [ 2 f 2 k 2 p x 2 1 ] } exp ( Q q 2 ) exp k 2 f 2 p 2 .
W x x ( ρ 1 , ρ 2 , z ) = k 2 z 2 4 s 1 s 2 + k 4 f σ 4 exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 exp ( k 2 4 s 1 x 1 2 + y 1 2 r 1 2 ) exp { k 2 s 1 4 s 1 s 2 + k 4 f σ 4 [ ( x 2 r 2 k 2 f σ 2 2 s 1 x 1 r 1 ) 2 + ( y 2 r 2 k 2 f σ 2 2 s 1 y 1 r 1 ) 2 ] } ,
W z z ( ρ 1 , ρ 2 , z ) = 2 k 4 f σ 2 s 1 ( 4 s 1 s 2 + k 4 f σ 4 ) 3 { ( 2 s 2 + k 4 f σ 4 2 s 1 ) + k 2 ( x 2 r 2 k 2 f σ 2 2 s 1 x 1 r 1 ) 2 + i k ( 2 s 2 + k 4 f σ 4 2 s 1 ) ( x 2 r 2 k 2 f σ 2 2 s 1 x 1 r 1 ) [ x 2 + ( 1 + f 2 f σ 2 ) x 1 ] ( 1 + f 2 f σ 2 ) ( 2 s 2 + k 4 f σ 4 2 s 1 ) 2 x 1 x 2 } exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 exp ( k 2 4 s 1 x 1 2 + y 1 2 r 1 2 ) exp { k 2 s 1 4 s 1 s 2 + k 4 f σ 4 [ ( x 2 r 2 k 2 f σ 2 2 s 1 x 1 r 1 ) 2 + ( y 2 r 2 k 2 f σ 2 2 s 1 y 1 r 1 ) 2 ] } ,
W x z ( ρ 1 , ρ 2 , z ) = 8 π 2 z s 1 λ 2 ( 4 s 1 s 2 + k 4 f σ 4 ) 2 exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 exp ( k 2 4 s 1 x 1 2 + y 1 2 r 1 2 ) [ i k ( k 2 f σ 2 2 s 1 x 1 r 1 x 2 r 2 ) + 4 s 1 s 2 + k 4 f σ 2 2 s 1 x 2 ] exp { k 2 s 1 4 s 1 s 2 + k 4 f σ 2 × [ ( k 2 f σ 2 2 s 1 x 1 r 1 x 2 r 2 ) 2 + ( k 2 f σ 2 2 s 1 y 1 r 1 y 2 r 2 ) 2 ] } ,
s i = i k 2 r 1 + ( 1 ) i 1 k 2 2 ( f 2 + f σ 2 ) ( i = 1 , 2 ) .
I ( ρ , z ) = { 1 A z 2 r 2 + 1 A 3 [ A f σ 2 + x 2 k 2 f σ 2 ( f σ 2 f 2 ) B k 2 x 2 1 + k 2 ( f 2 + f σ 2 ) 2 r 2 ] } exp [ k 2 f 2 A ( x 2 + y 2 ) ] ,
ϵ ( r ) = k 2 f 2 ( f 2 + 2 f σ 2 ) r 2 ,
A = 1 + ϵ ( r ) ,
B = A f σ 2 ( f 2 + 2 f σ 2 ) A 2 ( f 2 + f σ 2 ) 2 + r 2 k 2 f σ 6 ( f 2 + f σ 2 ) .
F x x f ( ρ , z ; q ) = 4 π Q z 2 r 2 exp ( Q b 2 ) exp [ k 2 f 2 p 2 ] ,
F x z f ( ρ , z ; q ) = 4 π Q z r ( u k i Q b x r + i f 2 k p x ) exp ( Q b 2 ) exp [ k 2 f 2 p 2 ] ,
F z z f ( ρ , z ; q ) = 2 π Q { 2 u 2 k 2 + 4 Q k f 2 b x r p x + Q r 2 ( 2 Q b x 2 1 ) + f 2 [ 2 f 2 k 2 p x 2 1 ] } exp ( Q b 2 ) exp k 2 f 2 p 2 ,
W x x f ( ρ 1 , ρ 2 , z ) = k 2 z 2 k 4 f σ 4 4 s 2 exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 exp ( k 2 4 s x 1 2 + y 1 2 r 1 2 ) exp { k 2 s k 4 f σ 4 4 s 2 [ ( x 2 r 2 k 2 f σ 2 2 s x 1 r 1 ) 2 + ( y 2 r 2 k 2 f σ 2 2 s y 1 r 1 ) 2 ] } ,
W z z f ( ρ 1 , ρ 2 , z ) = 2 k 4 f σ 2 s ( k 4 f σ 4 4 s 2 ) 3 { k 4 f σ 4 4 s 2 2 s + k 2 ( x 2 r 2 k 2 f σ 2 2 s x 1 r 1 ) 2 + ( k 4 f σ 4 4 s 2 2 s ) 2 x 1 x 2 k 2 f σ 2 ( i k r 1 2 s ) i k k 4 f σ 4 4 s 2 2 s ( k 2 f σ 2 2 s x 1 r 1 x 2 r 2 ) [ x 2 x 1 k 2 f σ 2 ( i k r 1 2 s ) ] } exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 exp ( k 2 4 s x 1 2 + y 1 2 r 1 2 ) exp { k 2 s k 4 f σ 4 4 s 2 [ ( x 2 r 2 k 2 f σ 2 2 s x 1 r 1 ) 2 + ( y 2 r 2 k 2 f σ 2 2 s y 1 r 1 ) 2 ] } ,
W x z f ( ρ 1 , ρ 2 , z ) = 8 π 2 z s λ 2 ( k 4 f σ 4 4 s 2 ) 2 exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 exp ( k 2 4 s x 1 2 + y 1 2 r 1 2 ) [ i k ( k 2 f σ 2 2 s x 1 r 1 x 2 r 2 ) + k 4 f σ 4 4 s 2 2 s x 2 ] exp { k 2 s k 4 f σ 4 4 s 2 [ ( k 2 f σ 2 2 s x 1 r 1 x 2 r 2 ) 2 + ( k 2 f σ 2 2 s y 1 r 1 y 2 r 2 ) 2 ] } ,
I f ( ρ , z ) = 1 k 4 f 4 ( f 2 + 2 f σ 2 ) 3 [ ( f 2 + 2 f σ 2 ) f σ 2 + f 2 x 2 r 2 + k 2 f 2 ( f 2 + 2 f σ 2 ) 2 ( x 2 + z 2 ) ] 1 r 4 × exp ( 1 f 2 + 2 f σ 2 x 2 + y 2 r 2 ) ,
s = k 2 2 ( f 2 + f σ 2 ) .
h α ( ρ + ρ 2 , ρ 0 + ρ 0 2 ) h z x * ( ρ ρ 2 , ρ 0 ρ 0 2 ) z λ 2 r 4 ( x x 0 + x 0 x 2 ) × exp [ i k ( x 0 x ) ( x 0 x ) + ( y 0 y ) ( y 0 y ) r ] .
K α , z x ( ρ , q , ρ 0 , q 0 ) = z 4 π 2 λ 2 r 4 ( x x 0 + x 0 x 2 ) exp [ i k ( x 0 x ) ( x 0 x ) + ( y 0 y ) ( y 0 y ) r ] × exp [ i ( q ρ q 0 ρ 0 ) ] d ρ d ρ 0 = z 4 π 2 λ 2 r 4 { ( x x 0 ) exp [ i k ( x 0 x ) ( x 0 x ) + ( y 0 y ) ( y 0 y ) r ] exp [ i ( q ρ q 0 ρ 0 ) ] d ρ d ρ 0 + 1 2 exp [ i k ( x 0 x ) x + ( y 0 y ) y r ] exp ( i q ρ ) d ρ × x 0 exp [ i k ( x 0 x ) x 0 + ( y 0 y ) y 0 r ] exp ( i q 0 ρ 0 ) d ρ 0 1 2 x exp [ i k ( x 0 x ) x + ( y 0 y ) y r ] exp ( i q ρ ) d ρ × exp [ i k ( x 0 x ) x 0 + ( y 0 y ) y 0 r ] exp ( i q 0 ρ 0 ) d ρ 0 } .
x exp ( i ξ x ) d x = i 2 π δ ( ξ )
f ( x ) δ ( x ξ ) d x = f ( ξ ) ,
F 1 ( ρ , z ; q ) = [ K x , z x ( ρ , q , ρ 0 , q 0 ) F x x ( ρ 0 , 0 ; q 0 ) ] d ρ 0 d q 0 = z k 2 2 r 4 [ 2 ( x x 0 ) δ ( a + q ) δ ( a + q 0 ) i δ ( a x + u ) δ ( a y + v ) δ ( a + q 0 ) i δ ( a x + u 0 ) δ ( a y + v 0 ) δ ( a + q ) ] F x x ( ρ 0 , 0 ; q 0 ) d ρ 0 d q 0 = z k 2 2 r 4 { 2 u r 3 k 3 F x x ( p , 0 ; q ) i [ r 2 k 2 F x x ( 0 , 0 , 1 , 0 ) ( p , 0 ; q ) r 3 k 3 F x x ( 1 , 0 , 0 , 0 ) ( p , 0 ; q ) ] + i r 2 k 2 F x x ( 0 , 0 , 1 , 0 ) ( p , 0 ; q ) } = z 2 r k [ 2 u F x x ( p , 0 ; q ) + i F x x ( 1 , 0 , 0 , 0 ) ( p , 0 ; q ) ] .
F 2 ( ρ , z ; q ) = [ K x , z y ( ρ , q , ρ 0 , q 0 ) F x y ( ρ 0 , 0 ; q 0 ) ] d ρ 0 d q 0 = z 2 r k [ 2 v F x y ( p , 0 ; q ) + i F x y ( 0 , 1 , 0 , 0 ) ( p , 0 ; q ) ] .
F x z ( ρ , z ; q ) = F 1 ( ρ , z ; q ) + F 2 ( ρ , z ; q ) = z r k [ u F x x ( p , 0 ; q ) + v F x y ( p , 0 ; q ) ] i z 2 r k [ F x x ( 1 , 0 , 0 , 0 ) ( p , 0 ; q ) + F x y ( 0 , 1 , 0 , 0 ) ( p , 0 , q ) ] .

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