Abstract

A nonparaxial generalization of the ambiguity function that retains several properties of its paraxial counterpart is presented, in both two and three dimensions. This generalization is used to extend into the nonparaxial regime a scheme for the recovery of the coherence properties of scalar partially coherent fields in two-dimensional space.

© 2011 Optical Society of America

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References

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  1. P. Woodward, “The transmitted radar signal,” in Probability and Information Theory with Application to Radar(Pergamon, 1953), pp. 115–125.
  2. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  3. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
    [CrossRef]
  4. J.-P. Guigay, “Ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
    [CrossRef]
  5. K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “Ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  6. K. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
    [CrossRef] [PubMed]
  7. K. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
    [CrossRef]
  8. M. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  9. D. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183(1995).
    [CrossRef] [PubMed]
  10. A. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  11. C. Tran, A. Peele, A. Roberts, K. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A 22, 1691–1700(2005).
    [CrossRef]
  12. C. Tran, A. Mancuso, B. Dhal, and K. Nugent, “Phase-space reconstruction of focused x-ray fields,” J. Opt. Soc. Am. A 23, 1779–1786 (2006).
    [CrossRef]
  13. A. Camara, T. Alieva, J. Rodrigo, and M. L. Calvo, “Phase space tomography reconstruction of the Wigner distribution for optical beams separable in Cartesian coordinates,” J. Opt. Soc. Am. A 26, 1301–1306 (2009).
    [CrossRef]
  14. K. Vogel and H. Risken, “Determination of quasi-probability distributions in terms of probability distributions for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
    [CrossRef] [PubMed]
  15. J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949(1997).
    [CrossRef]
  16. J. Tu and S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
    [CrossRef]
  17. M. A. Alonso, “Radiometry and wide-angle wave fields: I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).
    [CrossRef]
  18. M. A. Alonso, “Radiometry and wide-angle wave fields: II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
    [CrossRef]
  19. M. A. Alonso, “Radiometry and wide-angle wave fields: III. Partial coherence,” J. Opt. Soc. Am. A 18, 2502–2511 (2001).
    [CrossRef]
  20. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487(1999).
    [CrossRef]
  21. M. A. Alonso, “Exact description of free electromagnetic wave fields in terms of rays,” Opt. Express 11, 3128–3135 (2003).
    [CrossRef] [PubMed]
  22. M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized, electromagnetic wave fields in free space,” J. Opt. Soc. Am. 21, 2233–2243 (2004).
    [CrossRef]
  23. R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik 35, 237–246 (1972).
  24. J. Fienup, “Phase-retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  25. J. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  26. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 276 –287.
  27. C. J. R. Sheppard and K. G. Larkin, “Wigner function and ambiguity function for nonparaxial wavefields,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.Caulfield, ed. (SPIE Press, 2002), pp. 37–56.
  28. K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
    [CrossRef]

2009

2007

2006

2005

2004

M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized, electromagnetic wave fields in free space,” J. Opt. Soc. Am. 21, 2233–2243 (2004).
[CrossRef]

2003

2002

C. J. R. Sheppard and K. G. Larkin, “Wigner function and ambiguity function for nonparaxial wavefields,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.Caulfield, ed. (SPIE Press, 2002), pp. 37–56.

2001

1999

1998

1997

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949(1997).
[CrossRef]

1995

1994

M. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993

1992

K. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

1989

K. Vogel and H. Risken, “Determination of quasi-probability distributions in terms of probability distributions for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

1983

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “Ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1982

1978

J.-P. Guigay, “Ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

1974

1972

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik 35, 237–246 (1972).

1955

H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
[CrossRef]

1953

P. Woodward, “The transmitted radar signal,” in Probability and Information Theory with Application to Radar(Pergamon, 1953), pp. 115–125.

Alieva, T.

Alonso, M. A.

Beck, M.

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “Ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Calvo, M. L.

Camara, A.

Clarke, L.

Dhal, B.

Fienup, J.

Forbes, G. W.

Gerchberg, R.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik 35, 237–246 (1972).

Guigay, J.-P.

J.-P. Guigay, “Ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

Hopkins, H.

H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Wigner function and ambiguity function for nonparaxial wavefields,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.Caulfield, ed. (SPIE Press, 2002), pp. 37–56.

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

Lohmann, A.

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “Ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Mancuso, A.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 276 –287.

Mayer, A.

McAlister, D.

McNulty, I.

Nugent, K.

Ojeda-Castaneda, J.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “Ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Papoulis, A.

Paterson, D.

Peele, A.

Raymer, M.

M. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Raymer, M. G.

Risken, H.

K. Vogel and H. Risken, “Determination of quasi-probability distributions in terms of probability distributions for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Roberts, A.

Rodrigo, J.

Saxton, W.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik 35, 237–246 (1972).

Sheppard, C. J. R.

C. J. R. Sheppard and K. G. Larkin, “Wigner function and ambiguity function for nonparaxial wavefields,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.Caulfield, ed. (SPIE Press, 2002), pp. 37–56.

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

Tamura, S.

J. Tu and S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949(1997).
[CrossRef]

Tran, C.

Tu, J.

J. Tu and S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949(1997).
[CrossRef]

Vogel, K.

K. Vogel and H. Risken, “Determination of quasi-probability distributions in terms of probability distributions for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 276 –287.

Wolf, K. B.

Woodward, P.

P. Woodward, “The transmitted radar signal,” in Probability and Information Theory with Application to Radar(Pergamon, 1953), pp. 115–125.

Appl. Opt.

J. Opt. Soc. Am.

M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized, electromagnetic wave fields in free space,” J. Opt. Soc. Am. 21, 2233–2243 (2004).
[CrossRef]

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

J. Opt. Soc. Am. A

K. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
[CrossRef]

A. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

C. Tran, A. Peele, A. Roberts, K. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A 22, 1691–1700(2005).
[CrossRef]

C. Tran, A. Mancuso, B. Dhal, and K. Nugent, “Phase-space reconstruction of focused x-ray fields,” J. Opt. Soc. Am. A 23, 1779–1786 (2006).
[CrossRef]

A. Camara, T. Alieva, J. Rodrigo, and M. L. Calvo, “Phase space tomography reconstruction of the Wigner distribution for optical beams separable in Cartesian coordinates,” J. Opt. Soc. Am. A 26, 1301–1306 (2009).
[CrossRef]

J. Tu and S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields: I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).
[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields: II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields: III. Partial coherence,” J. Opt. Soc. Am. A 18, 2502–2511 (2001).
[CrossRef]

K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487(1999).
[CrossRef]

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

Opt. Commun.

J.-P. Guigay, “Ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “Ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik 35, 237–246 (1972).

Phys. Rev. A

K. Vogel and H. Risken, “Determination of quasi-probability distributions in terms of probability distributions for the quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Phys. Rev. E

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949(1997).
[CrossRef]

Phys. Rev. Lett.

M. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

K. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

Proc. R. Soc. London A

H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
[CrossRef]

Other

P. Woodward, “The transmitted radar signal,” in Probability and Information Theory with Application to Radar(Pergamon, 1953), pp. 115–125.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 276 –287.

C. J. R. Sheppard and K. G. Larkin, “Wigner function and ambiguity function for nonparaxial wavefields,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.Caulfield, ed. (SPIE Press, 2002), pp. 37–56.

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Figures (4)

Fig. 1
Fig. 1

Illustration of u 1 , 2 ( = u 1 v 2 v ) in the integrand of Eq. (27). The forward-propagating assumption implies a zero value of A * ( u 1 ) A ( u 2 ) unless the z components of both u 1 and u 2 are positive.

Fig. 2
Fig. 2

Basic principle of the phase-space tomography method of Tu and Tamura. From the knowledge of the intensity at (a) a line of fixed z = z i , the ambiguity function (paraxial or nonparaxial) can be calculated at (b) a line of slope z i through the origin. Therefore, from the knowledge of the intensity at a sufficiently large number of lines, the complete ambiguity function can be estimated through interpolation.

Fig. 3
Fig. 3

Illustration of (a) spectral density, (b) reconstructed nonparaxial ambiguity function and (c) its normalized error in x and τ , (d) the reconstructed angular spectrum correlation and (e) its normalized error, (f) the recovered cross-spectral density and (g) its normalized error, for the angular spectrum correlation in Eq. (34) with σ θ = 1.0 and ϵ = 0.25 .

Fig. 4
Fig. 4

Illustration of (a) spectral density, (b) reconstructed ambiguity function, (c) reconstructed cross-spectral density and (d) its normalized error, for the angular spectrum correlation in Eq. (34) with σ θ = 1.0 and ϵ = 0.25 .

Equations (36)

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W ( r 1 , r 2 ; ω ) = U * ( r 1 ; ω ) U ( r 2 ; ω ) ,
S ( r ; ω ) = W ( r , r ; ω ) = | U ( r ; ω ) | 2 .
A ( x , z ; p ) = ( k 2 π ) 2 W ( x x 2 , z ; x + x 2 , z ) exp ( i k x · p ) d 2 x ,
A ( x , z ; p ) = A ( x z p , 0 ; p ) ;
S ( x , z ) = A ( 0 , z ; p ) exp ( i k p · x ) d 2 p ;
A ( x , z ; p ) = ( k 2 π ) 2 B ( x , z ; p ) exp [ i k ( x · p x · p ) ] d 2 x d 2 p ,
B ( x , z ; p ) = ( k 2 π ) 2 W ( x x 2 , z ; x + x 2 , z ) exp ( i k x · p ) d 2 x .
U ( r ) = k 2 π π / 2 π / 2 A ( θ ) exp [ i k u ( θ ) · r ] d θ ,
W ( x 1 , z ; x 2 , z ) = k 2 π A * ( θ 1 ) A ( θ 2 ) exp { i k [ x 2 sin θ 2 x 1 sin θ 1 + z ( cos θ 2 cos θ 1 ) ] } d θ 1 d θ 2 ,
S ( x , z ) = k 2 π A * ( θ 1 ) A ( θ 2 ) exp { i k [ x ( sin θ 2 sin θ 1 ) + z ( cos θ 2 cos θ 1 ) ] } d θ 1 d θ 2 .
A np ( x , z ; τ ) = A np ( x z τ , 0 ; τ ) ;
S ( x , z ) = A np ( 0 , z ; τ ) exp ( i k τ x ) d τ ;
A np ( 0 , z ; τ ) = k 2 π S ( x , z ) exp ( i k x τ ) d x .
S ( x , z ) = k 2 π π / 2 π / 2 π + 2 | θ | π 2 | θ | A * ( θ α 2 ) A ( θ + α 2 ) × exp [ 2 i k ( x cos θ z sin θ ) sin α 2 ] d α d θ .
A np ( 0 , z ; τ ) = H ( 2 | τ | ) k π θ ¯ ( τ ) θ ¯ ( τ ) A * [ θ α ¯ ( τ , θ ) 2 ] A [ θ + α ¯ ( τ , θ ) 2 ] × exp ( i k τ z tan θ ) 4 cos 2 θ τ 2 d θ ,
α ¯ ( τ , θ ) = 2 arcsin ( τ 2 cos θ ) , θ ¯ ( τ ) = arccos | τ | 2 .
A np ( x , 0 ; τ ) = H ( 2 | τ | ) k π θ ¯ ( τ ) θ ¯ ( τ ) A * [ θ α ¯ ( τ , θ ) 2 ] A [ θ + α ¯ ( τ , θ ) 2 ] × exp ( i k x tan θ ) 4 cos 2 θ τ 2 d θ .
A np ( x , z ; τ ) = H ( 2 | τ | ) k π θ ¯ ( τ ) θ ¯ ( τ ) A * [ θ α ¯ ( τ , θ ) 2 ] A [ θ + α ¯ ( τ , θ ) 2 ] exp [ i k ( x z τ ) tan θ ] 4 cos 2 θ τ 2 d θ .
B np ( x , z ; τ ) = 1 1 + τ 2 k 2 π π π A * [ θ ^ ( τ ) α 2 ] A [ θ ^ ( τ ) + α 2 ] × exp ( 2 i k x z τ 1 + τ 2 sin α 2 ) d α ,
B np ( x , z ; τ ) = B np ( x z τ , 0 ; τ ) ;
S ( x , z ) = B np ( x , z ; τ ) d τ .
A np ( x , z ; τ ) = k 2 π B np ( x , z ; τ ) exp [ i k ( x τ x τ ) ] d x d τ .
U ( r ) = k 2 π u z 0 A ( u ) exp ( i k u · r ) d Ω u ,
A np ( x , z ; τ ) = A np ( x z τ , 0 ; τ ) ;
S ( x , z ) = A np ( 0 , z ; τ ) exp ( i k τ · x ) d τ ;
A np ( x , z ; τ ) = ( k 2 π ) 2 B np ( x , z ; τ ) exp [ i k ( x · τ x · τ ) ] d 2 x d 2 τ ,
B np ( x , z ; τ ) = 1 ( 1 + | τ | 2 ) 3 2 ( k π ) 2 D A * ( u 1 v 2 v ) A ( u 1 v 2 + v ) exp ( 2 i k r · v ) d 2 v ,
A np ( x , z ; τ ) = ( k 2 π ) 2 exp ( i k x · τ ) ( 1 + | τ | 2 ) 3 2 × D A * ( u 1 v 2 v ) A ( u 1 v 2 + v ) × δ ( v τ 2 ) exp ( 2 i k z v z ) d 2 v d 2 τ ,
A np ( x , z ; τ ) = H ( 2 | τ | ) ( k 2 π ) 2 A * ( u ¯ 1 v ¯ 2 v ¯ ) × A ( u ¯ 1 v ¯ 2 + v ¯ ) H ( u ¯ z 1 v ¯ 2 | v ¯ z | ) 1 + | τ | 2 × exp [ i k ( x z τ ) · τ ] d 2 τ ,
A ( z p , 0 ; p ) = k 2 π S ( x , z ) exp ( i k p x ) d x .
W ( x x 2 , 0 ; x + x 2 , 0 ) = A ( x , 0 ; p ) exp ( i k x p ) d p .
A np ( z τ , 0 ; τ ) = k 2 π S ( x , z ) exp ( i k τ x ) d x .
A * [ θ α ¯ ( τ , θ ) 2 ] A [ θ + α ¯ ( τ , θ ) 2 ] = 4 cos 2 θ τ 2 2 cos 2 θ × A np ( x , 0 ; τ ) exp ( i k x tan θ ) d x .
A * ( θ 1 ) A ( θ 2 ) = I 0 σ s σ θ cos θ 1 cos θ 2 exp [ ( sin θ 2 sin θ 1 ) 2 2 ϵ 2 sin 2 θ 1 + sin 2 θ 2 2 σ θ 2 ] ,
W ( x 1 , 0 ; x 2 , 0 ) = I 0 exp ( x 1 2 + x 2 2 2 σ s 2 ) exp [ ( x 2 x 1 ) 2 2 δ 2 ] ,
σ θ 2 = δ 2 + 2 σ s 2 k 2 σ s 2 δ 2 , ϵ 2 = δ 2 + 2 σ s 2 k 2 σ s 4 .

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