Abstract

New representations are defined for describing electromagnetic wave fields in free space exactly in terms of rays for any wavelength, level of coherence or polarization, and numerical aperture, as long as there are no evanescent components. These representations correspond to tensors assigned to each ray such that the electric and magnetic energy densities, the Poynting vector, and the polarization properties of the field correspond to simple integrals involving these tensors for the rays that go through the specified point. For partially coherent fields, the ray-based approach provided by the new representations can reduce dramatically the computation times for the physical properties mentioned earlier.

© 2004 Optical Society of America

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References

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  1. M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  2. L. Cohen, Time-Frequency Analysis (Prentice Hall, Englewood Cliffs, N.J., 1995).
  3. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).
  4. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  5. A. T. Friberg, Vol. Ed., Selected Papers on Coherence and Radiometry, Milestone Series Vol. MS69 (SPIE Optical Engineering Press, Bellingham, Wash., 1993).
  6. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).
  7. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  8. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1980).
    [CrossRef]
  9. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.
  10. Yu. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, Vol. XXXVI, E. Wolf, ed. (North Holland, New York, 1996), pp. 179–244.
  11. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  12. K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. B 4, 1233–1236 (1987).
    [CrossRef]
  13. A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. A 9, 1386–1393 (1992).
    [CrossRef]
  14. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  15. 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]
  16. 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]
  17. M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial coherence,” J. Opt. Soc. Am. A 18, 2502–2511 (2001).
    [CrossRef]
  18. C. J. R. Sheppard, K. G. Larkin, “Wigner function for highly convergent three-dimensional wave fields,” Opt. Lett. 26, 968–970 (2001).
    [CrossRef]
  19. M. A. Alonso, “Exact representation of free electromagnetic wave fields in terms of rays,” Opt. Express 11, 3128–3135 (2003).
    [CrossRef] [PubMed]
  20. Notice that His used here as the magnetic field, as opposed to B, which was used in Ref. 19. This change of convention causes some changes in the constant factors in front of several expressions common to both papers.
  21. L. E. Vicent, M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002).
    [CrossRef]
  22. J. Ellis, A. Dogariu, S. Pomonarenko, E. Wolf, “Polarization of statistically stationary electromagnetic fields,” manuscript available from the authors (jdellis@mail.ucf.edu).
  23. T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]

2003

2002

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

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

2001

1999

1992

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. A 9, 1386–1393 (1992).
[CrossRef]

1987

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. B 4, 1233–1236 (1987).
[CrossRef]

1984

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

1981

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

1980

1968

1964

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

1932

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

Agarwal, G. S.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. A 9, 1386–1393 (1992).
[CrossRef]

Alonso, M. A.

Apresyan, L. A.

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

Bastiaans, M. J.

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice Hall, Englewood Cliffs, N.J., 1995).

Dogariu, A.

J. Ellis, A. Dogariu, S. Pomonarenko, E. Wolf, “Polarization of statistically stationary electromagnetic fields,” manuscript available from the authors (jdellis@mail.ucf.edu).

Dolin, L. S.

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

Ellis, J.

J. Ellis, A. Dogariu, S. Pomonarenko, E. Wolf, “Polarization of statistically stationary electromagnetic fields,” manuscript available from the authors (jdellis@mail.ucf.edu).

Foley, J. T.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. A 9, 1386–1393 (1992).
[CrossRef]

Forbes, G. W.

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. A 9, 1386–1393 (1992).
[CrossRef]

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

A. T. Friberg, Vol. Ed., Selected Papers on Coherence and Radiometry, Milestone Series Vol. MS69 (SPIE Optical Engineering Press, Bellingham, Wash., 1993).

Hillery, M.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Kim, K.

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. B 4, 1233–1236 (1987).
[CrossRef]

Kravtsov, Yu. A.

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

Larkin, K. G.

Mandel, L.

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

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Pomonarenko, S.

J. Ellis, A. Dogariu, S. Pomonarenko, E. Wolf, “Polarization of statistically stationary electromagnetic fields,” manuscript available from the authors (jdellis@mail.ucf.edu).

Schleich, W. P.

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Sheppard, C. J. R.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Vicent, L. E.

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

Walther, A.

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

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

Wolf, E.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. A 9, 1386–1393 (1992).
[CrossRef]

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. B 4, 1233–1236 (1987).
[CrossRef]

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

J. Ellis, A. Dogariu, S. Pomonarenko, E. Wolf, “Polarization of statistically stationary electromagnetic fields,” manuscript available from the authors (jdellis@mail.ucf.edu).

Wolf, K. B.

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. B 4, 1233–1236 (1987).
[CrossRef]

Opt. Acta

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

Opt. Commun.

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

Opt. Express

Opt. Lett.

Phys. Rep.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Phys. Rev.

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

Phys. Rev. E

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Other

A. T. Friberg, Vol. Ed., Selected Papers on Coherence and Radiometry, Milestone Series Vol. MS69 (SPIE Optical Engineering Press, Bellingham, Wash., 1993).

L. Cohen, Time-Frequency Analysis (Prentice Hall, Englewood Cliffs, N.J., 1995).

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).

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

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

J. Ellis, A. Dogariu, S. Pomonarenko, E. Wolf, “Polarization of statistically stationary electromagnetic fields,” manuscript available from the authors (jdellis@mail.ucf.edu).

Notice that His used here as the magnetic field, as opposed to B, which was used in Ref. 19. This change of convention causes some changes in the constant factors in front of several expressions common to both papers.

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Figures (2)

Fig. 1
Fig. 1

The vector w is constrained to the plane perpendicular to the unit vector u. As θ increases, w rotates around u.

Fig. 2
Fig. 2

Change of variables described in Eq. (3.1). Here the unit vector u bisects the unit vectors u1 and u2, α is the angle between these two vectors, and w is a unit vector perpendicular to u and coplanar with u, u1, and u2.

Equations (72)

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

U(r)=0,
V(r)=0,
×U(r)=iωμ0V(r),
×V(r)=-iω0U(r).
EE(r)=08π |E(r, t)|2=08π |U(r)|2,
EM(r)=μ08π |H(r, t)|2=μ08π |V(r)|2.
S(r)Re[E*(r, t)×B(r, t)]4π=Re[U*(r)×V(r)]4π=U*(r)×V(r)-V*(r)×U(r)8π.
U(r)=10p=±14πϕ(u, p)wu, p π4×exp(iku  r)dΩ,
V(r)=1μ0p=±14πϕ(u, p)u×wu, p π4×exp(ikur)dΩ.
WE(r1, r2)=U*(r1)U(r2)=10p1,p2=±14πΞ(u1, p1, u2, p2)×wu1, p1π4wu2, p2π4×exp[ik(u2r2-u1r1)]dΩ1dΩ2,
GE(u1, u2)  p1,p2=±1Ξ(u1, p1, u2, p2)×wu1, p1π4wu2, p2π4.
WE(r1, r2)=104πGE(u1, u2)×exp[ik(u2r2-u1r1)]dΩ1dΩ2.
WM(r1, r2)=1μ04πGM(u1, u2)×exp[ik(u2r2-u1r1)]dΩ1dΩ2,
GM(u1, u2)  p1,p2=±1Ξ(u1, p1, u2, p2)×[u1×w(u1, p1π4)][u2×w(u2, p2π4)]=-u1×GE(u1, u2)×u2.
EE(r)=08πTr[WE(r, r)]=18π4πTr[GE(u1, u2)]×exp[ik(u2-u1)r]dΩ1dΩ2,
EM(r)=μ08πTr[WM(r, r)]=18π4πTr[GM(u1, u2)]×exp[ik(u2-u1)r]dΩ1dΩ2=18π4π{u1u2Tr[GE(u1, u2)]-u2GE(u1, u2)u1}×exp[ik(u2-u1)r]dΩ1dΩ2,
S(r)  U*(r)×V(r)-V*(r)×U(r)8π=c8π4πp1,p2=±1Ξ(u1, p1, u2, p2)×exp[ik(u2-u1)r][w1×(u2×w2)-(u1×w1)×w2]dΩ1dΩ2=c8π4πp1,p2=±1Ξ(u1, p1, u2, p2)×exp[ik(u2-u1)r]×[w1w2(u1+u2)-(w1w2u1+u2w1w2)]dΩ1dΩ2
=c8π4π{Tr[GE(u1, u2)](u1+u2)-[GE(u1, u2)u1+u2GE(u1, u2)]}×exp[ik(u2-u1)r]dΩ1dΩ2,
S(r)=c8π4π{Tr[GE(u1, u2)](u1+u2)-[GE(u1, u2)(u1+u2)+(u1+u2)GE(u1, u2)]}×exp[ik(u2-u1)r]dΩ1dΩ2=c8π4π{Tr[GE(u1, u2)](u1+u2)-[GE(u1, u2)+GET(u1, u2)](u1+u2)}×exp[ik(u2-u1)r]dΩ1dΩ2=c8π4π{Tr[GE(u1, u2)](u1+u2)-[GE(u1, u2)+GE*(u1, u2)](u1+u2)}×exp[ik(u2-u1)r]dΩ1dΩ2=c8π4π{Tr[GE(u1, u2)](u1+u2)-2 Re[GE(u1, u2)](u1+u2)}×exp[ik(u2-u1)r]dΩ1dΩ2,
u21=ucos α/2±w(u, θ)sin α/2,
WE(r, r)=8π04πBE(r, u)dΩ,
BE(r, u)=18π0π2πGEucosα2-w(u, θ)sinα2, ucosα2+w(u, θ)sinα2exp2ikrw(u, θ)sinα2sin αdθdα.
(u)BE(r, u)=0.
BE(r, u)=18π0π2πGEucosα2-w(u, θ+π)sinα2, ucosα2+w(u, θ+π)sinα2×exp2ikrw(u, θ+π)sinα2sin αdθdα=18π0π2πGEucosα2+w(u, θ)sinα2, ucosα2-w(u, θ)sinα2×exp-2ikrw(u, θ)sinα2sin αdθdα=BE(r, u),
EE(r)=Tr4πBE(r, u)dΩ=4πBE(r, u)dΩ,
EM(r)=4πBM(r, u)dΩ,
BM(r, u)=Tr[BM(r, u)]=18π0π2πTrGMucosα2-w(u, θ)×sinα2, ucosα2+w(u, θ)sinα2×exp2ikrw(u, θ)sinα2sin αdθdα.
BM(r, u)=18π0π2π{(uc-ws)(uc+ws)×Tr[GE(uc-ws, uc+ws)]-(uc+ws)GE(uc-ws, uc+ws)(uc-ws)}exp[2ikrws]sin αdθdα=18π0π2π{(c2-s2)×Tr[GE(uc-ws, uc+ws)]-(2uc)GE(uc-ws, uc+ws)(2uc)}exp[2ikrws]sin αdθdα=18π0π2π{(1-2s2)×Tr[GE(uc-ws, uc+ws)]-4(1-s2)uGE(uc-ws, uc+ws)u}×exp[2ikrws]sin αdθdα,
2BE(r, u)=18π0π2π(-4k2s2)×GE(uc-ws, uc+ws)×exp(2ikrws)sin αdθdα.
BM(r, u)=1+12k2 2Tr[BE(r, u)]-41+14k2 2uBE(r, u)u.
S(r)=c4πuBR(r, u)dΩ+4πBV(r, u)dΩ.
BR(r, u)=2 Tr[BP(r, u)],
BV(r, u)=-4c Re[BP(r, u)]u,
BP(r, u)=18π0π2πGEucosα2-w(u, θ)×sinα2, ucosα2+w(u, θ)sinα2×exp2ikrw(u, θ)sinα2sin α cosα2dθdα.
cosα2=1-sin2α21/2=j=0(-1)j(2j-3)!!2jj!sin2jα2,
BP(r, u)=-j=0(2j-3)!!23jj!k2j 2jBE(r, u)=BE(r, u)-18k2 2BE(r, u)+ .
(u)B(r, u)=0.
M(L, u)=B(u×L+τu, u)=B(u×L, u).
M(r×u, u)=B[r-(ru)u, u]=B(r, u).
ME(L, u)  08π0π2πGEucosα2-w(u, θ)sinα2, ucosα2+w(u, θ)sinα2×exp2ik(u×L)w(u, θ)sinα2sin αdθdα=08π0π2πGEucosα2-w(u, θ)sinα2, ucosα2+w(u, θ)sinα2×exp-2ikLwu, θ+π2×sinα2sin αdθdα,
ME(L, u)d2L=π2k2 GE(u, u),
4πME(r×u, u)dΩ=08π WE(r, r).
ME(L, u)  Tr[ME(L, u)],
MM(L, u)  1+12k2 L2Tr[ME(L, u)]-41+14k2 L2uME(L, u)u,
MR(L, u)  -j=0(2j-3)!!23j-1j!k2j L2jTr[ME(L, u)],
MV(L, u)  cj=0(2j-3)!!23j-3j!k2j L2jRe[ME(L, u)]u,
GE(u1, u2)Fu1+u2|u1+u2|g(|u2-u1|)×expikΦu1+u2|u1+u2|, u2-u1,
ME(L, u)=18π F(u)0π2πg2 sinα2×exp2ik(u×L)w(u, θ)sinα2+ikΦu, 2w(u, θ)sinα2sin αdθdα=18π F(u)022πg(β)×exp{ik(u×L)w(u, θ)β+ikΦ[u, w(u, θ)β]}βdθdβ,
ME(L, u)18π F(u)02πg(β)exp{ik[u×L+IIΦ(u, 0)][βw(u, θ)]}βdθdβ=14 F(u)g̑[k|u×L+IIΦ(u, 0)-uIIΦ(u, 0)u|],
g̑(|q|)=12πg(|β|)exp(iqβ)d2β=0g(β)J0(|q|β)dβ.
BE(r, u)=ME(r×u, u)14 F(u)g̑[k|u×(r×u)+IIΦ(u, 0)-uIIΦ(u, 0)u|]=14 F(u)g̑[k|r-(ru)u|],
Ξ(u1, p1, u2, p2)=S0δ(1-u1u2)π δp1,p2,
GE(u1, u2)  S0δ(1-u1u2)π×p=±1wu1, p π4wu1, p π4,
ME(L, u)=S08π0π2πδ(2 sin2 α/2)π×p=±1wu, p π4wu, p π4sin αdθdα=S08πp=±1wu, p π4wu, p π4,
BE=ME=BM=MM=BR/2=MR/2=S0/4π,
BV=MV=0.
WE(r¯1, r¯2)=10fhGE(u1, u2)exp[ik(u2r¯2-u1r¯1)]dΩ1dΩ2,
GE(u1, u2)=0u1zu2zk2π4WE(r¯1, r¯2)×exp[-ik(u2r¯2-u1r¯1)]d2r¯1d2r¯2.
GE(u1, u2)=GE(u1, u2)-u2z-1GE(u1, u2)u2-u1z-1u1GE(u1, u2)u1z-1u2z-1u1GE(u1, u2)u2,
WE(r¯1, r¯2)=P0Sr¯1+r¯22μ(r¯2-r¯1),
GE(u1, u2)=0k22π2u1zu2zP0,-u1zP0u2-u2zu1P0,u1P0u2×μ˜k u1+u22S˜[k(u2-u1)],
f˜(p)  12πf(r¯)exp(-ir¯p)d2r¯.
BE(r, u)=08πk22π2022πμ˜(kuγ)S˜(kwβ)exp(ikrwβ)×uz2γ2-wz2β24P0,-uzγ-wzβ2P0uγ+wβ2-uzγ+wzβ2uγ-wβ2P0,uγ-wβ2P0uγ+wβ2βdθdβ,
μ˜(kuγ)=μ˜(ku)-18uμ˜(ku)kβ2+O(k2β4)=μ˜(ku)-18uμ˜(ku)k(|w|2+wz2)β2+O(k2β4),
uz2γ2-wz2β24P0,-uzγ-wzβ2P0uγ+wβ2-uzγ+wzβ2uγ-wβ2P0,uγ-wβ2P0uγ+wβ2
=uz2P0,-uzP0u-uzuP0,uP0u+0,P0wz2u-uz2wuz2w-wz2uP0,iuIm(P0)wβ+O(β2),
BE(r, u)08πk22π2×uz2P0,-uzP0u-uzuP0,uP0uμ˜(ku)×022πS˜(kwβ)exp(ikrwβ)βdθdβ.
BE(r, u)08πk22π2uzP0,-P0u-uP0,uP0uuzμ˜(ku)×Q(u)S˜(kq)exp[ik(r-zu/uz)q]d2q.
BE(r, u)BE(0)(r, u) uz0k2(4π)2P0,-P0uuz-uP0uz,uP0uuz2×μ˜(ku)Sr-zuuz.
BE(r, u)BM(r, u)BP(r, u)2uz0k2(4π)2Tr(P0)+uP0uuz2μ˜(ku)×Sr-zuuz,
BE(1a)(r, u)  -uz0k(4π)2P0,-P0uuz-uP0uz,uP0uuz2uμ˜(ku)2+uuz2S,
BE(1b)(r, u)  uz0k(4π)20,i2P0uuuz+uzIS-i2 Suuuz+uzIP0,uIm(P0)Sμ˜(ku),

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