Abstract

We present a theory of propagation of a partially coherent and partially polarized electromagnetic beam through a multilayered stratified medium. The analysis shows that spatial coherence and polarization properties of the beam change, in general, on propagation through such a medium. We illustrate the results by an example.

© 2013 Optical Society of America

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References

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  1. F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).
  2. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).
  3. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).
  4. F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
    [CrossRef]
  5. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).
  6. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  7. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  9. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  10. Another definition of spatial degree of coherence is proposed in [25]. There have been some discussions relating to this definition [26,27]. However, for the problem addressed in this paper, it does not matter which of the definitions is used.
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  12. The formula is obtained as immediate generalization of the corresponding scalar version presented in Ref. [11], Section 5.6. In Eq. (5), we have neglected the contributions from evanescent waves. It is a reasonable assumption for this problem.
  13. In the case when the normal is not along the direction of stratification, one additional rotation matrix needs to be introduced in the analysis.
  14. M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
    [CrossRef]
  15. It is to be noted that the matrix U↔(0) is, in general, not unitary. For a discussion on this see Ref. [14].
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).
  17. Formulas for s(M) and q(M) are obtained by following the same technique used in Ref. [14], Appendix D.
  18. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
    [CrossRef]
  19. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
    [CrossRef]
  20. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
    [CrossRef]
  21. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  22. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
    [CrossRef]
  23. M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
    [CrossRef]
  24. The formula used [see, for example, [28], Section 3.323, Eq. (2)]: ∫−∞∞dte−β2t2e±qt=(π/β)eq2/4β2,Re{β2}>0.
  25. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef]
  26. E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1714 (2004).
    [CrossRef]
  27. T. Setälä, J. Tervo, and A. T. Friberg, “Reply to comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713 (2004).
    [CrossRef]
  28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

2012 (1)

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

2008 (1)

2006 (1)

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

2005 (2)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

2004 (4)

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

1977 (2)

1963 (1)

F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
[CrossRef]

1950 (2)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).

1948 (1)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).

Abelès, F.

F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
[CrossRef]

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).

Alonso, M.

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Friberg, A. T.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).

Hong, C.-S.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).

Korotkova, O.

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

Lahiri, M.

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

Mandel, L.

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

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Ramírez-Sánchez, V.

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Setälä, T.

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Tervo, J.

Wolf, E.

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1714 (2004).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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

Yariv, A.

Yeh, P.

Ann. Phys. (3)

F. Abelès, “Sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. 3, 504–520 (1948).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640 (1950).

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces (second part),” Ann. Phys. 5, 707–782 (1950).

J. Mod. Opt. (1)

M. Alonso, O. Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van Cittert–Zernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006).
[CrossRef]

J. Opt. A (2)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86, 043815 (2012).
[CrossRef]

Prog. Opt. (1)

F. Abelès, “Methods for determining optical parameters of thin films,” Prog. Opt. 2, 249–288 (1963).
[CrossRef]

Other (12)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).

It is to be noted that the matrix U↔(0) is, in general, not unitary. For a discussion on this see Ref. [14].

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 2004).

Formulas for s(M) and q(M) are obtained by following the same technique used in Ref. [14], Appendix D.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Another definition of spatial degree of coherence is proposed in [25]. There have been some discussions relating to this definition [26,27]. However, for the problem addressed in this paper, it does not matter which of the definitions is used.

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

The formula is obtained as immediate generalization of the corresponding scalar version presented in Ref. [11], Section 5.6. In Eq. (5), we have neglected the contributions from evanescent waves. It is a reasonable assumption for this problem.

In the case when the normal is not along the direction of stratification, one additional rotation matrix needs to be introduced in the analysis.

The formula used [see, for example, [28], Section 3.323, Eq. (2)]: ∫−∞∞dte−β2t2e±qt=(π/β)eq2/4β2,Re{β2}>0.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

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Figures (9)

Fig. 1.
Fig. 1.

Illustrating notation relating to the (xv,xh,xp) coordinate system. The thick arrows indicate the beam propagation direction.

Fig. 2.
Fig. 2.

Illustrating the geometry of beam propagation in a stratified medium. The principal transmitted beam, which has no contribution from reflected fields, is shown in the layer m.

Fig. 3.
Fig. 3.

Illustrating the coordinate systems (xv(m),xh(m),xp(m)).

Fig. 4.
Fig. 4.

Illustrating the plane of incidence B (shaded) of a typical plane wave component of the incident beam. The plane A (dashed line) is formed by the axis of the incident beam and the z axis.

Fig. 5.
Fig. 5.

Illustrating the (xv(0),xh(0),xp(0)) and the (xv(1),xh(1),xp(1)) coordinate systems.

Fig. 6.
Fig. 6.

Modulus of the spectral degree of coherence (|η|) of the principal transmitted beam, at a pair of radially opposite points separated by 8×104m in the layer m=3, plotted against the angle θ(0) (solid line). The dotted line represents |η| at equivalent points on the source plane. The dashed line represents |η| at equivalent points in a beam cross section as distance D from the source plane in absence of the stratified medium. The chosen parameters are given in the text.

Fig. 7.
Fig. 7.

Modulus of the spatial degree of coherence (|η|) of the principal transmitted beam (solid line), at a pair of radially opposite points, plotted against R, for θ(0)=30°. |η| at the source plane (dotted line) and at a beam cross section, optical path length D away from the source plane (dashed line) in absence of the stratified medium, are also plotted.

Fig. 8.
Fig. 8.

Degree of polarization (P) of the principal transmitted beam (solid line), at a point 5×103m away from center of the beam cross section, plotted against the angle θ(0). P is also plotted at the source plane (dotted line), and at a beam cross section, optical path length D away from the source plane (dashed line) in absence of the stratified medium.

Fig. 9.
Fig. 9.

Degree of polarization (P) of the principal transmitted beam (solid line), at a point on the xh(3) axis, plotted against the distance R between the point and the center of the beam cross section, for θ(0)=30°. P is also plotted at the source plane (dotted line), and at a beam cross section, optical path length D away from the source plane (dashed line) in absence of the stratified medium.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

Wlm(r,r;ω)El*(r;ω)Em(r;ω),
η(r,r;ω)TrW(r,r;ω)TrW(r,r;ω)TrW(r,r;ω),
Wlm(r,r;ω)El*(r;ω)Em(r;ω).
P(r;ω)14DetW(r,r;ω)[TrW(r,r;ω)]2,
El(r;ω)=s2+q2k2dsdqAl(s,q;ω)×exp[i(sxv+qxh+wxp)],
wk[1(s2+q2)/(2k2)].
E(r;ω)(Ev(r;ω)Eh(r;ω))=(Ev(r;ω)Eh(r;ω))T
W(r,r;ω)=E*(r;ω)ET(r;ω),
W(r,r;ω)=dsdqdsdqWA(s,q,s,q;ω)×exp{i(sxv+qxh+wxp+sxv+qxhwxp)},
WA(s,q,s,q;ω)=A*(s,q;ω)AT(s,q;ω)
W(r,r)=d2fd2fWA(f,f)exp[i(f·ρ+f·ρ)]×exp[i(wxpwxp)],
W(r,r)|xp=0,xp=0=d2fd2fWA(f,f)exp[i(f·ρ+f·ρ)],
W(r,r)exp[ik(xpxp)]WA(f,f)exp[i(f·ρ+f·ρ)]×exp[i2k(|f|2xp|f|2xp)]d2fd2f.
E(m)(r)=ds(m)dq(m)A(m)(s(m),q(m))×exp[i(s(m)xv(m)+q(m)xh(m)+w(m)xp(m))],
(xv(m)xh(m)xp(m))=(1000cos[θ(m1)θ(m)]sin[θ(m1)θ(m)]0sin[θ(m1)θ(m)]cos[θ(m1)θ(m)])(xv(m1)xh(m1)xp(m1)d(m1)cos[θ(m1)]).
n(0)sinθ˜(0)=n(m)sinθ˜(m),m=1,2,3,.
(Av(0)(s(0),q(0);ω)Ah(0)(s(0),q(0);ω))=U(0)(s(0),q(0);ω)(Av(0)(s(0),q(0);ω)Ah(0)(s(0),q(0);ω)),
U(0)=(cosαsinαcosθ(0)sinαcosθ(0)cosαcosθ(0)cosθ˜(0)+sinθ(0)sinθ˜(0)).
(Av(1)(s(1),q(1))Ah(1)(s(1),q(1)))=exp[ik(0)d(0)/cosθ˜(0)]T0,1(Av(0)(s(0),q(0))Ah(0)(s(0),q(0))),
T0,1=(Tv0,1(θ˜(0))00Th0,1(θ˜(0))).
Tv0,1=2n(0)cosθ˜(0)n(0)cosθ˜(0)+μ(0)μ(1)[n(1)]2[n(0)]2sin2θ˜(0),
Th0,1=2n(0)n(1)cosθ˜(0)μ(0)μ(1)[n(1)]2cosθ˜(0)+n(0)[n(1)]2[n(0)]2sin2θ˜(0).
(Av(M)(s(M),q(M))Ah(M)(s(M),q(M)))=exp[ik0m=0M1n(m)d(m)cosθ˜(m)]{m=0M1Tm,m+1}(Av(0)(s(0),q(0))Ah(0)(s(0),q(0))),
s(M)=s(0),q(M)q(0)cosθ(0)cosθ(M),
(Av(M)Ah(M))={U(M)}(Av(M)Ah(M)),
(Av(M)Ah(M))=exp[ik0m=0M1n(m)d(m)cosθ˜(m)]P0,M(Av(0)Ah(0)),
P0,M={U(M)}{m=0M1Tm,m+1(θ˜(m))}U(0).
(Av(M)Ah(M))exp[ik0m=0M1n(m)d(m)cosθ(m)]P0,M(Av(0)Ah(0))
P0,Mm=0M1Tm,m+1(θ(m)).
P0,M(m=0M1Tvm,m+1(θ(m))00m=0M1Thm,m+1(θ(m))),
Wjl(0)(ρ0,ρ0;ω)=AjAlBjlexp[(ρ02+ρ02)/(4σ2)]×exp[(ρ0ρ0)2/(2δjl2)],
Wjl(3)(r,r;ω)=exp[ik(3)(xp(3)xp(3))]{m=02Tjm,m+1Tlm,m+1}×AjAlBjl16(ajl2bjl2)(γjl(3)γjl(3)βjl2)(γ˜jl(3)γ˜jl(3)β˜jl2)×exp{γjl(3)(xv(3))2+γjl(3)(xv(3))22βjlxv(3)xv(3)4(γjl(3)γjl(3)βjl2)}×exp{γ˜jl(3)(xh(3))2+γ˜jl(3)(xh(3))22β˜jlxh(3)xh(3)4(γ˜jl(3)γ˜jl(3)β˜jl2)},
D=m=03n(m)d(m)/cosθ(m).
W(M)(r,r)exp[ik(M)(xp(M)xp(M))]×d2f(M)d2f(M)WA(M)(f(M),f(M))exp[i(f(M)·ρ(M)+f(M)·ρ(M))]exp[i2k(M)(|f(M)|2xp(M)|f(M)|2xp(M))],
WA(M)(s(M),q(M),s(M),q(M))=(P0,M)*WA(0)(s(0),q(0),s(0),q(0))(P0,M)T.
Wjl(M)(r,r)exp[ik(M)(xp(M)xp(M))]×{m=0M1(Tjm,m+1)*Tlm,m+1}×ds(M)dq(M)ds(M)dq(M){WA(0)(s(0),q(0))}jlexp{i[s(M)xv(M)+q(M)xh(M)+s(M)xv(M)+q(M)xh(M)]}exp{i2k(M)[({s(M)}2+{q(M)}2)xp(M)({s(M)}2+{q(M)}2)xp(M)]},
{WA(0)(s(0),q(0))}jl=AjAlBjl16π2(ajl2bjl2)exp{[ξjl({s(0)}2+{q(0)}2)+ξjl({s(0)}2+{q(0)}2)+2βjl(s(0)s(0)+q(0)q(0))]},
ajl=12δjl2+14σ2,bjl=12δjl2,
ξjl=ajl4(ajl2bjl2),βjl=bjl4(ajl2bjl2).
{WA(0)}jl=AjAlBjl16π2(ajl2bjl2)×exp{[ξjl({s(M)}2+{s(M)}2)+2βjls(M)s(M)]}×exp{[ξ˜jl({q(M)}2+{q(M)}2)+2β˜jlq(M)q(M)]},
ξ˜jl=cos2θ(M)cos2θ(0)ξjl,β˜jl=cos2θ(M)cos2θ(0)βjl.
Wjl(M)(r,r)exp[ik(M)(xp(M)xp(M))]×(m=0M1(Tjm,m+1)*Tlm,m+1)AjAlBjl16π2(ajl2bjl2){ds(M)ds(M)exp[i(s(M)xv(M)+s(M)xv(M))]exp[{ξjl({s(M)}2+{s(M)}2)+2βjls(M)s(M)}]exp[i2k(M)({s(M)}2xp(M){s(M)}2xp(M))]}{dq(M)dq(M)exp[i(q(M)xh(M)+q(M)xh(M))]exp[{ξ˜jl({q(M)}2+{q(M)}2)+2β˜jlq(M)q(M)}]exp[i2k(M)({q(M)}2xp(M){q(M)}2xp(M))]}.
Wjl(M)(r,r;ω)=exp[ik(M)(xp(M)xp(M))]{m=0M1(Tjm,m+1)*Tlm,m+1}AjAlBjl16(ajl2bjl2)(γjl(M)γjl(M)βjl2)(γ˜jl(M)γ˜jl(M)β˜jl2)×exp{γjl(M)(xv(M))2+γjl(M)(xv(M))24(γjl(M)γjl(M)βjl2)+2βjlxv(M)xv(M)4(γjl(M)γjl(M)βjl2)}×exp{γ˜jl(M)(xh(M))2+γ˜jl(M)(xh(M))24(γ˜jl(M)γ˜jl(M)β˜jl2)+2β˜jlxh(M)xh(M)4(γ˜jl(M)γ˜jl(M)β˜jl2)},
γjl(M)=ξjlixp(M)2k(M),γjl(M)=ξjl+ixp(M)2k(M),
γ˜jl(M)=ξ˜jlixp(M)2k(M),γ˜jl(M)=ξ˜jl+ixp(M)2k(M).

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