Abstract

We have obtained a sufficient condition under which the ladder approximation may be used to decouple the moment equations (e.g., the Bethe-Salpeter equation) that result when one analyzes the propagation of pptical waves in a randomly inhomogeneous medium. Simply stated, this means that the mean free path for multiple scattering by the inhomogeneities is required to be large in comparison with the size of the inhomogeneities.

© 1982 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Source, Springfield, Virginia, 1971); Secs. 60 and 61.
  2. Y. N. Barabanenkov et al., "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1852–1860 (1972).
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); see Chap. 14, where Twersky's theory is summarized.
  4. R. Fante, "Relationship between radiative transport theory and Maxwell's equations in random media," J. Opt. Soc. Am. 71, 460–468 (1981).
  5. L. Tsang and J. Kong, "Wave theory for microwave remote sensing of a half-space random medium with three dimensional variations," Radio Sci. 14, 359–369 (1979).
  6. V. Frisch, "Wave propagation in random media," in Probabilistic Methods in Applied Mathematics, A. Barucha-Reid, ed. (Academic, New York, 1968).
  7. K. Furutsu, "Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation," Radio Sci. 10, 29–44 (1975).
  8. It is demonstrated in Refs. 1 and 2 that the approximation of Eq. (1) by formula (2) is equivalent to the ladder approximation to the Bethe-Salpeter equation. Consequently, we have given this decomposition the name ladder approximation.
  9. For example, when evaluating 〈EE*〉 by taking moments of the integral form of the wave equation, we find that we need to evaluate terms of the form 〈∊(r′)∊(r′)E*(r)〉, where ∊(r′) is the relative permittivity fluctuation at r′. If ∊ is a Gaussian random variable, the Novikov-Furutsu theorem (see Ref. 1) gives [equation].where δ/δ∊ is a variational derivative. It is readily demonstrated that δE(r)/δ∊(r″)= -k2E(r″G(r, r″), where G(r, r″) is therandom Green's function and satisfies {∇2+ k2[1 + ∊(r)]}G(r,r′)= δ(r - r′). Therefore 〈∊(r′)E(r′)E*(r)〉=-k2∋∋∞∞∋d3r″〈∊(r′)×(r)G(r, r)E(r)E*(r)+G*(r,r)E(r′)E*(r″)〉so we see that terms of the form of Eq. (1) must be evaluated.
  10. V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).
  11. R. Lutomirski and H. Yura, "Propagation of a finite optical beam in an inhomogeneous medium," Appl. Opt. 10, 1652–1658 (1971).
  12. R. Fante, "Electromagnetic beam propagation in turbulent media: an update," Proc. IEEE 68, 1424–1443 (1980).
  13. R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).
  14. It is demonstrated in Sec. 37 of Ref. 1 that the representation in formula (4) is valid provided that the spatial scale size of the inhomogeneities is large in comparison with the signal wavelength. In particular, it is shown that formula (4) approximately includes multiple small-angle scatterings. Clearly, however, formula (4) cannot be used in the limit when the inhomogeneities are small in comparison with a wavelength, because both formulas (3) and (4) assume that individual backscatterings are negligible. This is equivalent to Twersky's assumption (see Ref. 3, Section 14-1) that there are no propagation paths that go through a scatterer more than once.
  15. It is not necessary to assume that ø has zero mean. When ø has a nonzero mean we can simply lump it in with G0〈…〉 in Eq. (7). That is, 〈G(r′, r″)〉 = [G0(r′, r″)exp(i-〈ø〉)]exp(-½〈ø2 (r′, r″)〉), where ø is now the fluctuation of the phase about the mean value 〈ø〉
  16. V. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), Chap. 6.
  17. The geometric-optics method gives valid results for the real and imaginary parts of the phase over propagation paths less than l02λ, where l0 is the scale size of the smallest inhomogeneities and λ is the wavelength. For propagation paths that are large in comparison with l02λ, the geometric-optics method gives incorrect results for the imaginary part of ø but gives solutions for the real part of ø that are in agreement with results obtained by both the method of smooth perturbations and the Markov approximation method. Consequently, geometric optics gives useful results for the real part of ø even for large propagation lengths (i.e., paths large compared with l02λ).
  18. It is shown is Sec. 39 [see formula (23)] of Ref. 1 that the ray paths can be approximated by straight lines if 〈n2〉≪1.
  19. These conclusions do not hold in the backscatter cone (i.e., in a cone of angle L/R « 1 centered about θ + γ = 0 in Fig. 1). However, this is unimportant because (1) it represents only a small small fraction of the total integral over ρ and ρ′, and (2) backscattering usually is insignificant in the limit when l is larger than a wavelength, as we have assumed.
  20. R. Fante, "Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence," J. Opt. Soc. Am. 64, 592–598 (1974); seeEq. (24).
  21. For the case when kl < 1 we expect that a sufficient condition for 〈Ei(r″)Ej*(r)Gkl(r′, r″)〉 〈(Ei (r″)Ej*(r)〉 〈Gkl(r′, r″)〉 is that 〈n2〉 ≪ 1, where Ei is the ith component of the vector E, Gij is the ij component of the Green's tensor, etc. (We did not need to consider this tensor case when kl »≫ 1 because cross polarization is then tinimportant.) Probably for kl sufficiently small in comparison with unity the ladder approximation is valid even for 〈n2〉 > 1, but we have been unable to prove this rigorously.

1981

1980

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

1979

L. Tsang and J. Kong, "Wave theory for microwave remote sensing of a half-space random medium with three dimensional variations," Radio Sci. 14, 359–369 (1979).

1975

K. Furutsu, "Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation," Radio Sci. 10, 29–44 (1975).

1972

Y. N. Barabanenkov et al., "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1852–1860 (1972).

1971

1967

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Barabanenkov, Y. N.

Y. N. Barabanenkov et al., "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1852–1860 (1972).

Fante, R.

R. Fante, "Relationship between radiative transport theory and Maxwell's equations in random media," J. Opt. Soc. Am. 71, 460–468 (1981).

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

R. Fante, "Electromagnetic beam propagation in turbulent media: an update," Proc. IEEE 68, 1424–1443 (1980).

R. Fante, "Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence," J. Opt. Soc. Am. 64, 592–598 (1974); seeEq. (24).

Feizulin, V.

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Frisch, V.

V. Frisch, "Wave propagation in random media," in Probabilistic Methods in Applied Mathematics, A. Barucha-Reid, ed. (Academic, New York, 1968).

Furutsu, K.

K. Furutsu, "Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation," Radio Sci. 10, 29–44 (1975).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); see Chap. 14, where Twersky's theory is summarized.

Kong, J.

L. Tsang and J. Kong, "Wave theory for microwave remote sensing of a half-space random medium with three dimensional variations," Radio Sci. 14, 359–369 (1979).

Kravtsov, Y.

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Lutomirski, R.

Tatarskii, V.

V. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), Chap. 6.

V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Source, Springfield, Virginia, 1971); Secs. 60 and 61.

Tsang, L.

L. Tsang and J. Kong, "Wave theory for microwave remote sensing of a half-space random medium with three dimensional variations," Radio Sci. 14, 359–369 (1979).

Yura, H.

Appl. Opt.

J. Opt. Soc. Am.

Radio Sci.

L. Tsang and J. Kong, "Wave theory for microwave remote sensing of a half-space random medium with three dimensional variations," Radio Sci. 14, 359–369 (1979).

K. Furutsu, "Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation," Radio Sci. 10, 29–44 (1975).

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

Radiophys. Quantum Electron.

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Y. N. Barabanenkov et al., "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1852–1860 (1972).

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); see Chap. 14, where Twersky's theory is summarized.

V. Frisch, "Wave propagation in random media," in Probabilistic Methods in Applied Mathematics, A. Barucha-Reid, ed. (Academic, New York, 1968).

R. Fante, "Electromagnetic beam propagation in turbulent media: an update," Proc. IEEE 68, 1424–1443 (1980).

It is demonstrated in Refs. 1 and 2 that the approximation of Eq. (1) by formula (2) is equivalent to the ladder approximation to the Bethe-Salpeter equation. Consequently, we have given this decomposition the name ladder approximation.

For example, when evaluating 〈EE*〉 by taking moments of the integral form of the wave equation, we find that we need to evaluate terms of the form 〈∊(r′)∊(r′)E*(r)〉, where ∊(r′) is the relative permittivity fluctuation at r′. If ∊ is a Gaussian random variable, the Novikov-Furutsu theorem (see Ref. 1) gives [equation].where δ/δ∊ is a variational derivative. It is readily demonstrated that δE(r)/δ∊(r″)= -k2E(r″G(r, r″), where G(r, r″) is therandom Green's function and satisfies {∇2+ k2[1 + ∊(r)]}G(r,r′)= δ(r - r′). Therefore 〈∊(r′)E(r′)E*(r)〉=-k2∋∋∞∞∋d3r″〈∊(r′)×(r)G(r, r)E(r)E*(r)+G*(r,r)E(r′)E*(r″)〉so we see that terms of the form of Eq. (1) must be evaluated.

It is demonstrated in Sec. 37 of Ref. 1 that the representation in formula (4) is valid provided that the spatial scale size of the inhomogeneities is large in comparison with the signal wavelength. In particular, it is shown that formula (4) approximately includes multiple small-angle scatterings. Clearly, however, formula (4) cannot be used in the limit when the inhomogeneities are small in comparison with a wavelength, because both formulas (3) and (4) assume that individual backscatterings are negligible. This is equivalent to Twersky's assumption (see Ref. 3, Section 14-1) that there are no propagation paths that go through a scatterer more than once.

It is not necessary to assume that ø has zero mean. When ø has a nonzero mean we can simply lump it in with G0〈…〉 in Eq. (7). That is, 〈G(r′, r″)〉 = [G0(r′, r″)exp(i-〈ø〉)]exp(-½〈ø2 (r′, r″)〉), where ø is now the fluctuation of the phase about the mean value 〈ø〉

V. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), Chap. 6.

The geometric-optics method gives valid results for the real and imaginary parts of the phase over propagation paths less than l02λ, where l0 is the scale size of the smallest inhomogeneities and λ is the wavelength. For propagation paths that are large in comparison with l02λ, the geometric-optics method gives incorrect results for the imaginary part of ø but gives solutions for the real part of ø that are in agreement with results obtained by both the method of smooth perturbations and the Markov approximation method. Consequently, geometric optics gives useful results for the real part of ø even for large propagation lengths (i.e., paths large compared with l02λ).

It is shown is Sec. 39 [see formula (23)] of Ref. 1 that the ray paths can be approximated by straight lines if 〈n2〉≪1.

These conclusions do not hold in the backscatter cone (i.e., in a cone of angle L/R « 1 centered about θ + γ = 0 in Fig. 1). However, this is unimportant because (1) it represents only a small small fraction of the total integral over ρ and ρ′, and (2) backscattering usually is insignificant in the limit when l is larger than a wavelength, as we have assumed.

R. Fante, "Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence," J. Opt. Soc. Am. 64, 592–598 (1974); seeEq. (24).

For the case when kl < 1 we expect that a sufficient condition for 〈Ei(r″)Ej*(r)Gkl(r′, r″)〉 〈(Ei (r″)Ej*(r)〉 〈Gkl(r′, r″)〉 is that 〈n2〉 ≪ 1, where Ei is the ith component of the vector E, Gij is the ij component of the Green's tensor, etc. (We did not need to consider this tensor case when kl »≫ 1 because cross polarization is then tinimportant.) Probably for kl sufficiently small in comparison with unity the ladder approximation is valid even for 〈n2〉 > 1, but we have been unable to prove this rigorously.

V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Source, Springfield, Virginia, 1971); Secs. 60 and 61.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.