Abstract

The free-space propagation of paraxial, partially coherent stationary fields can be described in a simple and intuitive way through the use of the Wigner function. In this context, this function plays the role of a generalized radiance that is constant along straight lines or rays. The effect of diffraction by transverse planar opaque obstacles or apertures is considered here for this representation, and a simple analytic approximate formula is given for the case when the incident field is quasi-homogeneous, at least in the neighborhood of the obstacle’s edges. In this result, diffraction is accounted for by including rays emanating from the obstacle’s edges.

© 2009 Optical Society of America

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  1. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
    [CrossRef]
  2. Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1998).
  3. M. A. Alonso, “Rays and waves,” in Phase Space Optics: Fundamentals and Applications, B.Hennelly, J.Ojeda-Castañeda, and M.Testorf, eds. (McGraw-Hill, to be published in 2009), Chap. 8.
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995), pp. 287-307.
  5. A.T.Friberg, vol. ed., Selected Papers on Coherence and Radiometry, Milestone Series Vol. MS69 (SPIE Press, 1993).
  6. Yu. A. Kravtsov and L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics Vol. XXXVI, E.Wolf, ed. (North Holland, 1996) pp. 179-244.
    [CrossRef]
  7. L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer. Statistical and Wave Aspects (Gordon & Breach, 1996).
  8. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), p. 13.
  9. L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
    [CrossRef]
  10. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  11. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).
  12. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710-1716 (1980).
    [CrossRef]
  13. M. J. Bastiaans, “Wigner function,” in Phase Space Optics: Fundamentals and Applications, B.Hennelly, J.Ojeda-Castañeda, and M.Testorf, eds. (McGraw-Hill, to be published in 2009), Chap. 1.
  14. A. Torre, Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture (Elsevier, 2005).
    [PubMed]
  15. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261-277 (1981).
    [CrossRef]
  16. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007) pp. 90-95.
  17. G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419-1421 (1972).
  18. M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial coherence,” J. Opt. Soc. Am. A 18, 2502-2511 (2001), and references therein.
    [CrossRef]
  19. M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space,” J. Opt. Soc. Am. A 21, 2233-2243 (2004), and references therein.
    [CrossRef]
  20. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  21. R.C.Hansen, vol. ed., Geometric Theory of Diffraction (IEEE Press, New York, 1981).
  22. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986).
  23. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
    [CrossRef]
  24. R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
    [CrossRef]
  25. R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, E76-E87 (2008).
    [CrossRef] [PubMed]
  26. This follows from the Hermiticity of the cross-spectral density, i.e., W(r⃗1;r⃗2)=W*(r⃗2;r⃗1)4. The fact that B is real is then shown by considering the complex conjugate of both sides of Eq. and changing the variable of integration according to x″=−x′.
  27. E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024-2026 (1994).
    [CrossRef] [PubMed]

2008 (1)

2007 (1)

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

2004 (1)

2002 (1)

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

2001 (1)

1994 (1)

1981 (1)

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261-277 (1981).
[CrossRef]

1980 (1)

1974 (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

1972 (1)

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419-1421 (1972).

1964 (1)

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).

1962 (1)

1932 (1)

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

Alonso, M. A.

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

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

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

M. A. Alonso, “Rays and waves,” in Phase Space Optics: Fundamentals and Applications, B.Hennelly, J.Ojeda-Castañeda, and M.Testorf, eds. (McGraw-Hill, to be published in 2009), Chap. 8.

Apresyan, L. A.

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer. Statistical and Wave Aspects (Gordon & Breach, 1996).

Yu. A. Kravtsov and L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics Vol. XXXVI, E.Wolf, ed. (North Holland, 1996) pp. 179-244.
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710-1716 (1980).
[CrossRef]

M. J. Bastiaans, “Wigner function,” in Phase Space Optics: Fundamentals and Applications, B.Hennelly, J.Ojeda-Castañeda, and M.Testorf, eds. (McGraw-Hill, to be published in 2009), Chap. 1.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), p. 13.

Carrasquilla, J.

R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, E76-E87 (2008).
[CrossRef] [PubMed]

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

Castañeda, R.

R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, E76-E87 (2008).
[CrossRef] [PubMed]

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

Dolin, L. S.

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).

Friberg, A. T.

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261-277 (1981).
[CrossRef]

Herrera, J.

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986).

Keller, J. B.

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Kravtsov, Yu. A.

Yu. A. Kravtsov and L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics Vol. XXXVI, E.Wolf, ed. (North Holland, 1996) pp. 179-244.
[CrossRef]

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1998).

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer. Statistical and Wave Aspects (Gordon & Breach, 1996).

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995), pp. 287-307.

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1998).

Ovchinnikov, G. I.

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419-1421 (1972).

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Tatarskii, V. I.

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419-1421 (1972).

Torre, A.

A. Torre, Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture (Elsevier, 2005).
[PubMed]

Vicent, L. E.

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

Wigner, E. P.

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

Wolf, E.

E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024-2026 (1994).
[CrossRef] [PubMed]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007) pp. 90-95.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995), pp. 287-307.

Appl. Opt. (1)

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419-1421 (1972).

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559-563 (1964).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261-277 (1981).
[CrossRef]

Opt. Commun. (2)

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

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

Proc. IEEE (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Other (14)

This follows from the Hermiticity of the cross-spectral density, i.e., W(r⃗1;r⃗2)=W*(r⃗2;r⃗1)4. The fact that B is real is then shown by considering the complex conjugate of both sides of Eq. and changing the variable of integration according to x″=−x′.

M. J. Bastiaans, “Wigner function,” in Phase Space Optics: Fundamentals and Applications, B.Hennelly, J.Ojeda-Castañeda, and M.Testorf, eds. (McGraw-Hill, to be published in 2009), Chap. 1.

A. Torre, Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture (Elsevier, 2005).
[PubMed]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007) pp. 90-95.

R.C.Hansen, vol. ed., Geometric Theory of Diffraction (IEEE Press, New York, 1981).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986).

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1998).

M. A. Alonso, “Rays and waves,” in Phase Space Optics: Fundamentals and Applications, B.Hennelly, J.Ojeda-Castañeda, and M.Testorf, eds. (McGraw-Hill, to be published in 2009), Chap. 8.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995), pp. 287-307.

A.T.Friberg, vol. ed., Selected Papers on Coherence and Radiometry, Milestone Series Vol. MS69 (SPIE Press, 1993).

Yu. A. Kravtsov and L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics Vol. XXXVI, E.Wolf, ed. (North Holland, 1996) pp. 179-244.
[CrossRef]

L. A. Apresyan and Yu. A. Kravtsov, Radiative Transfer. Statistical and Wave Aspects (Gordon & Breach, 1996).

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), p. 13.

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Figures (7)

Fig. 1
Fig. 1

At the plane of the screen (indicated by the vertical black line), each incident ray that hits the screen’s plane at x > x 0 spreads into a fan of rays with alternating positive and negative weights (indicated by shades of gray that are lighter and darker, respectively, than the background). The directional spread of this fan is inversely proportional to the distance from the edge. Rays that hit this plane at x < x 0 are fully blocked. While the figure shows only incident horizontal rays, this spreading/blocking is experienced by rays incident at any angle.

Fig. 2
Fig. 2

Wigner phase space near x = x 0 for a Gaussian-correlated field incident on a semi-infinite opaque screen at the plane z = z 0 . Shown are (a) B t , (b) B r , and (c) B d . In all cases, brighter shades of gray represent higher values of B, with the shade of gray at the left section ( x < x 0 ) of each plot corresponding to zero.

Fig. 3
Fig. 3

(a) The weight distribution of the rays scattered by the edge of the aperture is related to that of the rays incident at the edge approximately through Eq. (28). (b) Weight distributions for the rays incident at the edge (dashed curve) and for those scattered from it (solid curve), for a Gaussian-correlated incident field.

Fig. 4
Fig. 4

Plots of z ¯ I d I 0 (dashed curves) for z ¯ = ( z z 0 ) k Δ x c 2 equal to the indicated values, compared with the corresponding plot for the analytic approximation in Eq. (32) (thick gray curve). These results depend only on the dimensionless coordinates x ¯ = ( x x 0 ) Δ x c and z ¯ . The factor of z ¯ and the scaling of the transverse coordinate as x ¯ z ¯ are included in order to make the curve for the approximate analytic estimate the same at all distances.

Fig. 5
Fig. 5

(a) Opaque screen at the plane z = z 0 with circular aperture of radius R centered at x = 0 . (b) The product of the aperture functions being Fourier transformed in Eq. (34) equals unity for x inside the white area and zero elsewhere. (c) Magnitude of the Fourier transform of the function in (b), where black corresponds to zero. (d) Diagram showing that the scattering of rays is preferentially in the direction normal to the aperture’s edge at its closest point.

Fig. 6
Fig. 6

Contour plots of (a) I ¯ r I 0 and (b) I ¯ d I 0 , where the vertical axis corresponds to x R and the horizontal one to Δ p c ( z z 0 ) R . Notice that the distribution in (b) diverges at x = R , z = z 0 .

Fig. 7
Fig. 7

(a) On-axis plots of I ¯ r and I ¯ d . On-axis comparisons of I t I r (dashed curve) and I d (gray curve) for w = Δ x c R equal to (b) 0.2, (c) 0.1, and (d) 0.05.

Equations (56)

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W ( r 1 ; r 2 ) = U * ( r 1 ) U ( r 2 ) ,
i k W z 1 , 2 + 1 2 k 2 ( 2 x 1 , 2 2 + 2 y 1 , 2 2 ) W + W = 0 .
B ( x , p ; z ) = ( k 2 π ) 2 W ( x x 2 , z ; x + x 2 , z ) exp ( i k x p ) d 2 x ,
I ( r ) = W ( r , r ) = B ( x , p ; z ) d 2 p ,
B ( x , p ; z ) = B [ x ( z z 0 ) p , p ; z 0 ] .
B ( x , p ; z ) = k 2 π W ( x x 2 , z ; x + x 2 , z ) exp ( i k x p ) d x = k 2 π F ̂ x p W ( x x 2 , z ; x + x 2 , z ) ,
f ̃ ( p ) = F ̂ x p f ( x ) = k 2 π f ( x ) exp ( i k x p ) d x .
W ( x 1 , z 0 ; x 2 , z 0 ) I ( x 1 + x 2 2 , z 0 ) μ ( x 2 x 1 , z 0 ) ,
B ( x , p ; z ) k 2 π I ( x , z ) μ ̃ ( p , z ) ,
W t ( x 1 , z 0 ; x 2 , z 0 ) = W i ( x 1 , z 0 ; x 2 , z 0 ) A ( x 1 ) A ( x 2 ) ,
B t ( x , p ; z 0 ) = B i ( x , p ; z 0 ) B A ( x , p p ) d p = B i ( x , p ; z 0 ) p B A ( x , p ) ,
B A ( x , p ) = k 2 π A ( x x 2 ) A ( x + x 2 ) exp ( i k x p ) d x = k 2 π F ̂ x p [ A ( x x 2 ) A ( x + x 2 ) ] .
B A ( x , p ) = k 2 π Θ ( x x 0 ) F ̂ x p rect [ x 4 ( x x 0 ) ] = Θ ( x x 0 ) sin [ 2 k p ( x x 0 ) ] π p ,
B t ( x , p ; z 0 ) = Θ ( x x 0 ) F ̂ x p { rect [ x 4 ( x x 0 ) ] F ̂ p x 1 B i ( x , p ; z 0 ) } ,
B t ( x , p ; z 0 ) = B r ( x , p ; z 0 ) + B d ( x , p ; z 0 ) ,
B r ( x , p ; z 0 ) = Θ ( x x 0 ) B i ( x , p ; z 0 ) ,
B d ( x , p ; z 0 ) = Θ ( x x 0 ) F ̂ x p ( { rect [ x 4 ( x x 0 ) ] 1 } F ̂ p x 1 B i ( x , p ; z 0 ) ) .
I t ( x , z ) = B t ( x , p ; z ) d p = B t [ x p ( z z 0 ) , p ; z 0 ] d p = I r ( x , z ) + I d ( x , z ) ,
I r ( x , z ) = B r [ x p ( z z 0 ) , p ; z 0 ] d p = ( x x 0 ) ( z z 0 ) B i [ x p ( z z 0 ) , p ; z 0 ] d p ,
I d ( x , z ) = B d [ x p ( z z 0 ) , p ; z 0 ] d p .
B d ( x , p ; z 0 ) Θ ( x x 0 ) F ̂ x p ( { rect [ x 4 ( x x 0 ) ] 1 } F ̂ p x 1 B i ( x 0 , p ; z 0 ) ) .
I d ( x , z ) k 2 π ( x x 0 ) ( z z 0 ) { rect [ x 4 ( x p z + p z 0 x 0 ) ] 1 } exp ( i k x p ) d p F ̂ p x 1 B i ( x 0 , p ; z 0 ) d x = F ̂ x ( x x 0 ) ( z z 0 ) { 2 k x sin [ k x x 4 ( z z 0 ) ] exp [ i k x x 4 ( z z 0 ) ] × F ̂ p x 1 B i ( x 0 , p ; z 0 ) } ,
F ̂ p x 1 B i ( x 0 , p ; z 0 ) = k 2 π W ( x 0 x 2 , z 0 ; x 0 + x 2 , z 0 ) ,
2 k x sin [ k x x 4 ( z z 0 ) ] exp [ i k x x 4 ( z z 0 ) ] x 2 ( z z 0 ) ,
x z z 0 k .
I d ( x , z ) 1 2 ( z z 0 ) F ̂ x ( x x 0 ) ( z z 0 ) [ x F ̂ p x 1 B i ( x 0 , p ; z 0 ) ] = 1 2 ( z z 0 ) p H ̂ p B i ( x 0 , p ; z 0 ) p = ( x x 0 ) ( z z 0 ) ,
z z 0 k Δ c 2 ,
H ̂ p f ̃ ( p ) = i F ̂ x p [ sgn ( x ) F ̂ p x 1 f ̃ ( p ) ] ,
B d ( x , p ; z 0 ) δ ( x x 0 ) x 0 B d ( x , p ; z 0 ) d x ,
B d ( x , p ; z 0 ) δ ( x x 0 ) D ̂ p B i ( x 0 , p ; z 0 ) ,
D ̂ p B i = 1 2 H ̂ p p B i = 1 2 p H ̂ p B i = 1 2 F ̂ x p ( x F ̂ p x 1 B i ) .
B i ( x , p ; z 0 ) = I 0 2 π Δ p c exp ( p 2 2 Δ p c 2 ) .
B d ( x , p ; z 0 ) δ ( x x 0 ) I 0 2 π k Δ p c 2 [ π 2 p Δ p c Erfi ( p 2 Δ p c ) exp ( p 2 2 Δ p c 2 ) 1 ] ,
I d ( x , z ) I 0 2 π k ( z z 0 ) Δ p c 2 [ π 2 x x 0 ( z z 0 ) Δ p c Erfi ( x x 0 2 ( z z 0 ) Δ p c ) exp ( ( x x 0 ) 2 2 ( z z 0 ) 2 Δ p c 2 ) 1 ] .
B t ( x , p ; z 0 ) = B i ( x , p ; z 0 ) p B A ( x , p ) ,
B A ( x , p ) = ( k 2 π ) 2 A ( x x 2 ) A ( x + x 2 ) exp ( i k x p ) d x d y = k 2 π F ̂ x p [ A ( x x 2 ) A ( x + x 2 ) ] ,
B t ( x , p ; z 0 ) = F ̂ x p [ A ( x x 2 ) A ( x + x 2 ) F ̂ p x 1 B i ( x , p ; z 0 ) ] .
B t ( x , p ; z 0 ) = B r ( x , p ; z 0 ) + B d ( x , p ; z 0 ) ,
B r ( x , p ; z 0 ) = A ( x ) B i ( x , p ; z 0 ) ,
B d ( x , p ; z 0 ) = F ̂ x p { [ A ( x x 2 ) A ( x + x 2 ) A ( x ) ] F ̂ p x 1 B i ( x , p ; z 0 ) } .
I t ( x , z ) = B t ( x , p ; z ) d 2 p = I r ( x , z ) + I d ( x , z ) ,
I r ( x , z ) = B r ( x , p ; z ) d 2 p = P ( x , z ) B i [ x p ( z z 0 ) , p ; z 0 ] d 2 p ,
I d ( x , z ) = B d [ x p ( z z 0 ) , p ; z 0 ] d 2 p .
B d ( x , p ; z 0 ) δ ( x R ) R 0 1 B d ( r x 0 , p , z 0 ) r d r ,
B d ( x , p ; z 0 ) δ ( x R ) R F ̂ x p 0 1 F ̂ p x 1 B i ( r x 0 , p ; z 0 ) [ A ( r x 0 x 2 ) A ( r x 0 + x 2 ) A ( r x 0 ) ] r d r δ ( x R ) R F ̂ x p { F ̂ p x 1 B i ( x 0 , p ; z 0 ) 0 1 [ A ( r x 0 x 2 ) A ( r x 0 + x 2 ) 1 ] r d r } ,
0 1 [ A ( r x 0 x 2 ) A ( r x 0 + x 2 ) 1 ] r d r = r 0 2 ( x 0 , x ) 1 2 ,
r 0 ( x 0 , x ) = 1 + ( x 0 x ) 2 R 2 x 2 4 R 4 x 0 x 2 R 2 .
0 1 [ A ( r x 0 x 2 ) A ( r x 0 + x 2 ) 1 ] r d r = x 0 x 2 R 2 + 2 ( x 0 x ) 2 R 2 x 2 8 R 4 + O ( x 3 ) .
B d ( x , p ; z 0 ) δ ( x R ) [ D ̂ p + 1 8 k 2 R 2 ( 2 p 2 2 p 2 ) + ] B i ( x , p ; z 0 ) ,
B d ( x , p ; z 0 ) δ ( x R ) D ̂ p B i ( x , p ; z 0 ) .
B i ( x , p ; z 0 ) = I 0 2 π Δ p c 2 exp ( p 2 2 Δ p c 2 ) .
I t ( x , z ) I ¯ r ( x R , Δ p c z z 0 R ) + w I ¯ d ( x R , Δ p c z z 0 R ) ,
I ¯ r = I 0 { 1 exp [ R 2 2 Δ p c 2 ( z z 0 ) 2 ] } ,
I ¯ d = I 0 R 3 Δ p c 3 ( z z 0 ) 3 { 1 2 Erfi [ R 2 Δ p c ( z z 0 ) ] exp [ R 2 2 Δ p c 2 ( z z 0 ) 2 ] Δ p c ( z z 0 ) 2 π R } .
W t ( x 1 , z 1 ; x 2 , z 2 ) = exp [ i k ( z 2 z 1 ) ] B t ( x 1 + x 2 2 , p ; z 1 + z 2 2 ) exp { i k [ ( x 2 x 1 ) p ( z 2 z 1 ) p 2 2 ] } d 2 p = exp [ i k ( z 2 z 1 ) ] B t [ x 1 + x 2 2 ( z 1 + z 2 2 z 0 ) p , p ; z 0 ] exp { i k [ ( x 2 x 1 ) p ( z 2 z 1 ) p 2 2 ] } d 2 p .
x 2 x 1 ( z 1 + z 2 ) 2 z 0 k Δ x c , z 2 z 1 [ ( z 1 + z 2 ) 2 z 0 ] 2 k Δ x c 2 .

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