Abstract

We present a theoretical and experimental study of the scattering of light by double passage through a system that consists of a strong diffuser, a piece of birefringent crystal, and a plane mirror. We show that this arrangement can produce not only enhanced backscattering and satellite peaks but also satellite dips in the angular distribution of the mean intensity. The experiments are in agreement with theoretical results based on scalar diffraction theory in the paraxial approximation.

© 2001 Optical Society of America

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References

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  1. Yu. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
    [CrossRef]
  2. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  3. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  4. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  5. E. Jakeman, “Enhanced backscattering through a deep ran-dom phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
    [CrossRef]
  6. A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough dielectric diffuser,” Opt. Commun. 72, 279–285 (1989).
    [CrossRef]
  7. V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded media with disorder,” Phys. Lett. A 193, 467–470 (1994).
    [CrossRef]
  8. V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
    [CrossRef]
  9. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
    [CrossRef]
  10. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
    [CrossRef]
  11. J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
    [CrossRef]
  12. A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
    [CrossRef]
  13. V. D. Freilikher, E. Kanziper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–205 (1997).
    [CrossRef]
  14. A. R. McGurn, A. A. Maradudin, R. F. Wallis, “Enhanced backscattering in a magnetic field,” Waves Random Media 1, 43–57 (1991).
    [CrossRef]
  15. E. R. Méndez, E. I. Chaikina, H. M. Escamilla, “Observation of satellite peaks and dips in the scattering of light in a double-pass geometry,” Opt. Lett. 24, 705–707 (1999).
    [CrossRef]
  16. Model 03 PPD 001, Melles Griot Inc., Irvine, California 92606.
  17. H. M. Escamilla, E. R. Méndez, D. F. Hotz, “Angular intensity correlations in the double passage of waves through a random phase screen,” Appl. Opt. 32, 2734–2743 (1993).
    [CrossRef] [PubMed]
  18. H. M. Escamilla, E. R. Méndez, A. Martı́nez, “Image formation in the double-passage scattering configuration,” J. Opt. Soc. Am. A 13, 1439–1447 (1996).
    [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 84.

1999 (1)

1997 (2)

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

V. D. Freilikher, E. Kanziper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–205 (1997).
[CrossRef]

1996 (1)

1995 (2)

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

1994 (3)

V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded media with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

1993 (1)

1991 (1)

A. R. McGurn, A. A. Maradudin, R. F. Wallis, “Enhanced backscattering in a magnetic field,” Waves Random Media 1, 43–57 (1991).
[CrossRef]

1989 (1)

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough dielectric diffuser,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

1988 (1)

1987 (1)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

1985 (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

1984 (1)

1982 (1)

Yu. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Celli, V.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Chaikina, E. I.

Escamilla, H. M.

Freilikher, V. D.

V. D. Freilikher, E. Kanziper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–205 (1997).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded media with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
[CrossRef]

Freilikher, V. R.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 84.

Hotz, D. F.

Ishimaru, A.

Jakeman, E.

Kanziper, E.

V. D. Freilikher, E. Kanziper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–205 (1997).
[CrossRef]

Kravtsov, Yu. A.

Yu. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Kuga, Y.

Lu, Jun Q.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Madrazo, A.

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Maradudin, A. A.

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

V. D. Freilikher, E. Kanziper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–205 (1997).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
[CrossRef]

A. R. McGurn, A. A. Maradudin, R. F. Wallis, “Enhanced backscattering in a magnetic field,” Waves Random Media 1, 43–57 (1991).
[CrossRef]

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough dielectric diffuser,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Marti´nez, A.

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, R. F. Wallis, “Enhanced backscattering in a magnetic field,” Waves Random Media 1, 43–57 (1991).
[CrossRef]

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough dielectric diffuser,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Méndez, E. R.

O’Donnell, K. A.

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Pustilnik, M.

V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded media with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Saichev, A. I.

Yu. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Wallis, R. F.

A. R. McGurn, A. A. Maradudin, R. F. Wallis, “Enhanced backscattering in a magnetic field,” Waves Random Media 1, 43–57 (1991).
[CrossRef]

Yurkevich, I.

V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded media with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Satellite peak in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate,” J. Mod. Opt. 43, 435–452 (1995).
[CrossRef]

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

Opt. Commun. (4)

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

A. Madrazo, A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

A. R. McGurn, A. A. Maradudin, “An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough dielectric diffuser,” Opt. Commun. 72, 279–285 (1989).
[CrossRef]

V. D. Freilikher, M. Pustilnik, I. Yurkevich, A. A. Maradudin, “Wave scattering from a thin film with volume disorder: reflection and transmission,” Opt. Commun. 110, 263–269 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Wave scattering from a bounded media with disorder,” Phys. Lett. A 193, 467–470 (1994).
[CrossRef]

Phys. Rep. (1)

V. D. Freilikher, E. Kanziper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–205 (1997).
[CrossRef]

Phys. Rev. B (3)

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. R. Freilikher, “Transmission of electromagnetic waves through thin metal films with randomly rough surface,” Phys. Rev. B 51, 17100–17115 (1995).
[CrossRef]

J. A. Sánchez-Gil, A. A. Maradudin, Jun Q. Lu, V. D. Freilikher, M. Pustilnik, I. Yurkevich, “Electromagnetic waves from a bounded media with a random surface,” Phys. Rev. B 50, 15353–15368 (1994).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Sov. Phys. Usp. (1)

Yu. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Waves Random Media (1)

A. R. McGurn, A. A. Maradudin, R. F. Wallis, “Enhanced backscattering in a magnetic field,” Waves Random Media 1, 43–57 (1991).
[CrossRef]

Other (2)

Model 03 PPD 001, Melles Griot Inc., Irvine, California 92606.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 84.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the scattering geometry. The angles θ0 and θs are taken as positive in the direction of the arrows.

Fig. 2
Fig. 2

Illustrative diagram showing the propagation of light with two orthogonal polarizations through the beam displacing prism.

Fig. 3
Fig. 3

Illustrative diagram showing the unfolded optical system. RPS, random phase screen; BDP, beam displacing prism; LP, linear polarizer.

Fig. 4
Fig. 4

Calculated in-plane mean intensity as a function of the scattering angle for three cases of the polarization. The wavelength was λ=0.6328 µm, the aperture half-width W=2 mm, and the distance d=48 cm. We also set α=45°, Δ=2.7 mm, and m=0.

Fig. 5
Fig. 5

Calculated in-plane mean intensity as a function of the scattering angle for three cases of the polarization. The wavelength was λ=0.6328 µm, the aperture half-width W=2 mm, and the distance d=44 cm. We also set α=45°, Δ=2.7 mm, and m=0.

Fig. 6
Fig. 6

Calculated in-plane mean intensity as a function of the scattering angle and the distance d from the random phase screen to the mirror for three cases of the polarization. The wavelength was λ=0.6328 µm and the aperture half-width W=2 mm. We also set α=45°, Δ=2.7 mm, and m=0.

Fig. 7
Fig. 7

Schematic diagram of the experimental setup.

Fig. 8
Fig. 8

Mean scattered intensity detected by the CCD in the absence of a polarizer between the BDP and the mirror.

Fig. 9
Fig. 9

p-polarized component of the mean scattered intensity showing two satellite peaks.

Fig. 10
Fig. 10

p-polarized component of the mean scattered intensity showing a satellite peak and a satellite dip.

Equations (39)

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

E(r1, k)=Ep(r1)eˆp+Es(r1)eˆs=(A0peˆp+A0seˆs)exp(ik·r1),
Ep(r2)=A0p exp(i(ω/c)d1)iλd1 - exp(ik·r1)P(r1)D(r1)×expi ωc |r1-r2-Δr|22d1d2r1,
Es(r2)=A0s exp(i(ω/c)d1)iλd1 - exp(ik·r1)P(r1)D(r1)×expi ωc |r1-r2|22d1d2r1,
Ep(r2)=cos α[Ep(r2)cos α+Es(r2)sin α],
Es(r2)=sin α[Ep(r2)cos α+Es(r2)sin α].
Ep,s(r)=exp(i(ω/c)d2)iλd2 -Ep,s(r2)×expi ωc |r2-r|22d2d2r2,
- expi ωc |r1-r2|22d1expi ωc |r2-r|22d2d2r2
=iλ d1d2(d1+d2) expi ωc |r1-r|22(d1+d2),
Ep(r)=cos α[A0p cos αA1(k, r+Δr)+A0s sin αA1(k, r)],
Es(r)=sin α[A0p cos αA1(k, r+Δr)+A0s sin αA1(k, r)],
A1(k, r)=exp(i(ω/c)d)iλd - exp(ik·r1)P(r1)D(r1)×expi ωc |r1-r|22dd2r1,
Ep,s(r3)=exp(i(ω/c)d2)iλd2 -Ep,s(r)exp[iϕm(r)]×expi ωc |r-r3|22d2d2r,
ϕm(r)=2i ωc m·r,
m=-nxnz, -nymz,
Ep(r4)=exp(i(ω/c)d1)iλd1 -Ep(r3)×expi ωc |r3-r4+Δr|22d1d2r3,
Es(r4)=exp(i(ω/c)d1)iλd1 -Es(r3)×expi ωc |r3-r4|22d1d2r3,
Rp,s(k|q)=-Ep,s(r4)P(r4)D(r4)exp(-iq·r4)d2r4,
Rp(k|q)=cos α-[A0p cos αA1(k, r+Δr)+A0s sin αA1(k, r)]A2(q, r+Δr)×exp[iϕm(r)]d2r,
Rs(k|q)=sin α-[A0p cos αA1(k, r+Δr)+A0s sin αA1(k, r)]A2(q, r)×exp[iϕm(r)]d2r,
A2(q, r)=exp[i(ω/c)d]iλd - exp(-iq·r4)×P(r4)D(r4)expi ωc |r4-r|22dd2r4.
Is(k|q)=sin2 α--[A0p cos αA1(k, r+Δr)+A0s sin αA1(k, r)]A2(q, r)×[A0p* cos αA1*(k, r+Δr)+A0s* sin αA1*(k, r)]A2*(q, r)×exp{i[ϕm(r)-ϕm(r)]}d2rd2r,
T1(k|q)=sin2 α cos2 α|A0p|2--A1(k, r+Δr)×A2(q, r)A1*(k, r+Δr)A2*(q, r)×exp{i[ϕm(r)-ϕm(r)]}d2rd2r,
T1(k|q)=sin2 αcos2 α|A0p|2-- exp{i[ϕm(r)-ϕm(r)]}[A1(k, r+Δr)×A1*(k, r+Δr)A2(q, r)A2*(q, r)+A1(k, r+Δr)A2*(q, r)A2(q, r)×A1*(k, r+Δr)]d2rd2r.
A1(k, r+Δr)A2*(q, r)
=1(λd)2 --P(r1)P*(r4)D(r1)D*(r4)×exp[i(k·r1+q·r4)]×expi ωc |r1-r-Δr|2-|r4-r|22dd2r1d2r4,
D(r1)D*(r4)δ(r1-r4),
A1(k, r+Δr)A2*(q, r)
=1(λd)2 expi ωc |r+Δr|2-r22d-|P(r1)|2×expik+q-ωc r+Δr-rd·r1d2r1.
T1(k|q)=sin2 α cos2 α|A0p|21λd4-|P(r1)|2×expik+q-ωc Δrd·r1d2r12+-|P(r1)|2d2r12.
P(r1)=exp(-r12/W2),
Is(k|q)=B sin4 α|A0s|21+exp-W24|k+q|2+B sin2 α cos2 α|A0p|2×1+exp-W24 k+q-ωc Δrd2+2B ReA0pA0s* expi ωc |Δr|24d-iΔϕm×sin3 α cos αexp-W22ωc Δr2d2+exp-W24 k+q-ωc Δr2d2
B=πW24λd2.
Rs(k|q)=exp{-iΔϕm}sin α-[A0p cos αA1(k, r)+A0s sin αA1(k, r-Δr)]×A2(q, r-Δr)exp(iϕm(r))d2r.
Ip(k|q)=B cos4 α|A0p|21+exp-W24|k+q|2+B cos2 α sin2 α|A0s|2×1+exp-W24 k+q+ωc Δrd2+2B ReA0sA0p* expi ωc |Δr|24d+iΔϕm×cos3 α sin αexp-W22ωc Δr2d2+exp-W24 k+q+ωc Δr2d2.
Ip(k|q)=B|A0p|21+exp-W24|k+q|2.
Ip,s(k|q)=B|A0|24 1+exp-W24|k+q|2
+1+exp-W24|k+q±p|2
+2 cosωc |Δr|24d±Δϕmexp-W24 p24
+exp-W24 k+q±p22,

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