Abstract

The Born and Rytov methods are applied to a study of the propagation and the scattering of waves in a one-dimensional (1-D) half-space medium with a permittivity either depending on the wave intensity or having a stochastic property. Both Born and Rytov series in the 1-D nonlinear half-space case are compared with their counterparts in a 1-D homogeneous dielectric half-space case. The multiple scattering in a 1-D random half-space, consisting of a host material and randomly distributed scatterers, is treated with a random medium model. We have derived the autocorrelation function for the case in which 1-D scatterers are randomly distributed in a host medium. We show that the autocorrelation function has an exponential-decay shape and depends on the fractional length, the average physical length, and the scattering strength of these scatterers. Under the bilocal approximation the effective permittivity for this 1-D random half-space medium is shown to have an oscillatory dependence on range. When it is applied to estimate the ensemble-averaged scattered fields in the 1-D random half-space case, the Rytov method is not as useful as the Born method because the Rytov method suffers a more serious divergence problem than the Born method.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  2. F. C. Lin, M. A. Fiddy, “The Born–Rytov controversy: I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A 9, 1102–1110 (1992).
    [Crossref]
  3. P. W. Smith, P. J. Maloney, A. Ashkin, “Use of a liquid suspension of dielectric spheres as an artificial Kerr medium,” Opt. Lett. 7, 347–349 (1982).
    [Crossref] [PubMed]
  4. S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1152–1228.
  5. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960).
  6. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  7. R. C. Bourret, “Stochastically perturbed fields, with applications to wave propagation in random media,” Nuovo Cimento 26, 1–31 (1962).
    [Crossref]
  8. R. C. Bourret, “Propagation of randomly perturbed fields,” Can. J. Phys. 40, 782–790 (1962).
    [Crossref]
  9. J. Kampé de Fériet, “Statistical mechanics of continuous media,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 165–198.
    [Crossref]
  10. J. B. Keller, “Wave propagation in random media,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 227–246.
    [Crossref]
  11. R. H. Kraichnan, “The closure problem of turbulence theory,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 199–225.
    [Crossref]
  12. K. Furutsu, “On the statistical theory of electromagnetic waves in a fluctuation medium (I),”J. Res. Natl. Bur. Stand. (USA) 67, 303–323 (1963).
  13. W. C. Hoffman, “Wave propagation in a general random continuous medium,” in Proceedings of the 16th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1964), pp. 117–144.
    [Crossref]
  14. F. C. Karl, J. B. Keller, “Elastic, electromagnetic, and other waves in a random medium,”J. Math. Phys. 5, 537–547 (1964).
    [Crossref]
  15. J. B. Keller, “Stochastic equations and wave propagation in random media,” in Proceedings of the 16th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1964), pp. 145–170.
    [Crossref]
  16. V. I. Tatarskii, “Propagation of electromagnetic waves in a medium with strong dielectric constant fluctuations,” Sov. Phys. JETP 19, 946–953 (1964).
  17. W. P. Brown, “Coherent field in a random medium—effective refractive index,” in Proceedings of the Symposium on Modern Optics (Polytechnic, Brooklyn, 1967), pp. 717–742.
  18. C. H. Liu, “Wave propagation in a random medium with parabolic background,” Radio Sci. 2, 961–977 (1967).
  19. C. H. Liu, “Effective dielectric tensor and propagation constant of plane waves in a random anisotropic medium,”J. Math. Phys. 8, 2236–2242 (1967).
    [Crossref]
  20. U. Frisch, “Wave propagation in a random medium,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), Vol. 1, pp. 75–198.
  21. J. B. Keller, “A survey of the theory of wave propagation in continuous random media,” in Proceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic, Brooklyn, 1968), pp. 131–142.
  22. S. Rosenbaum, “On energy-conserving formulations in a randomly fluctuating medium,” in Proceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic, Brooklyn, 1968), pp. 163–185.
  23. Y. A. Ryzhov, V. V. Tamoikin, “Radiation and propagation of electromagnetic waves in randomly inhomogeneous media,” Radiophys. Quantum Electron. 13, 273–300 (1970).
    [Crossref]
  24. V. I. Tatarskii, The Effects of the Thrbulent Atmosphere on Wave Propagation, TT-68-50464 (National Technical Information Service, Springfield, Va., 1971).
  25. D. Dence, J. E. Spence, “Wave propagation in random anisotropic media,” in Probabilistic Methods in Applied and Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1973), Vol. 3, pp. 121–181.
  26. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).
  27. A. Ashkin, J. M. Dziedzic, P. W. Smith, “Continuous-wave self-focusing and self-trapping of light in artificial Kerr media,” Opt. Lett. 7, 276–278 (1982).
    [Crossref] [PubMed]
  28. A. D. Varvatsis, M. I. Sancer, “On the renormalization method in random wave propagation,” Radio Sci. 6, 87–97 (1971).
    [Crossref]
  29. M. L. Oristaglio, “Accuracy of the Born and Rytov approximations for reflection and refraction at a plane interface,” J. Opt. Soc. Am. A 2, 1987–1993 (1985).
    [Crossref]
  30. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).
  31. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  32. F. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, 4806–502 (1949).
    [Crossref]
  33. F. Dyson, “The Smatrix in quantum electrodynamics,” Phys. Rev. 75, 1736–1755 (1949).
    [Crossref]
  34. R. P. Feynman, “The theory of positrons,” Phys. Rev. 76, 749–759 (1949).
    [Crossref]
  35. R. P. Feynman, “Space-time approach to quantum electrodynamics,” Phys. Rev. 76, 769–789 (1949).
    [Crossref]
  36. F. C. Lin, “Theoretical models for microwave remote sensing of snow-covered sea ice,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).
  37. P. Debye, H. R. Anderson, H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
    [Crossref]
  38. F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
    [Crossref]
  39. E. Kreyszig, Advanced Engineering Mathematics, 4th ed. (Wiley, New York, 1979).
  40. A. Stogryn, “The bilocal approximation for the effective dielectric constant of an isotropic random medium,”IEEE Trans. Antennas Propag. AP-32, 517–520 (1984).
    [Crossref]
  41. A. D. Yaghjian, “Maxwellian and cavity electromagnetic fields within continuous sources,” Am. J. Phys. 53, 859–863 (1985).
    [Crossref]
  42. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  43. A. Ishimaru, L. Tsang, “Backscattering enhancement of random discrete scatterers of moderate sizes,” J. Opt. Soc. Am. A 5, 228–236 (1988).
    [Crossref]
  44. A. K. Jordan, S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6, 1206–1212 (1989).
    [Crossref]
  45. L. S. Tamil, A. K. Jordan, “Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,” Proc. IEEE 79, 1519–1528 (1991).
    [Crossref]

1992 (1)

1991 (1)

L. S. Tamil, A. K. Jordan, “Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,” Proc. IEEE 79, 1519–1528 (1991).
[Crossref]

1989 (1)

1988 (2)

A. Ishimaru, L. Tsang, “Backscattering enhancement of random discrete scatterers of moderate sizes,” J. Opt. Soc. Am. A 5, 228–236 (1988).
[Crossref]

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

1985 (2)

A. D. Yaghjian, “Maxwellian and cavity electromagnetic fields within continuous sources,” Am. J. Phys. 53, 859–863 (1985).
[Crossref]

M. L. Oristaglio, “Accuracy of the Born and Rytov approximations for reflection and refraction at a plane interface,” J. Opt. Soc. Am. A 2, 1987–1993 (1985).
[Crossref]

1984 (1)

A. Stogryn, “The bilocal approximation for the effective dielectric constant of an isotropic random medium,”IEEE Trans. Antennas Propag. AP-32, 517–520 (1984).
[Crossref]

1982 (2)

1971 (1)

A. D. Varvatsis, M. I. Sancer, “On the renormalization method in random wave propagation,” Radio Sci. 6, 87–97 (1971).
[Crossref]

1970 (1)

Y. A. Ryzhov, V. V. Tamoikin, “Radiation and propagation of electromagnetic waves in randomly inhomogeneous media,” Radiophys. Quantum Electron. 13, 273–300 (1970).
[Crossref]

1967 (2)

C. H. Liu, “Wave propagation in a random medium with parabolic background,” Radio Sci. 2, 961–977 (1967).

C. H. Liu, “Effective dielectric tensor and propagation constant of plane waves in a random anisotropic medium,”J. Math. Phys. 8, 2236–2242 (1967).
[Crossref]

1964 (2)

F. C. Karl, J. B. Keller, “Elastic, electromagnetic, and other waves in a random medium,”J. Math. Phys. 5, 537–547 (1964).
[Crossref]

V. I. Tatarskii, “Propagation of electromagnetic waves in a medium with strong dielectric constant fluctuations,” Sov. Phys. JETP 19, 946–953 (1964).

1963 (1)

K. Furutsu, “On the statistical theory of electromagnetic waves in a fluctuation medium (I),”J. Res. Natl. Bur. Stand. (USA) 67, 303–323 (1963).

1962 (2)

R. C. Bourret, “Stochastically perturbed fields, with applications to wave propagation in random media,” Nuovo Cimento 26, 1–31 (1962).
[Crossref]

R. C. Bourret, “Propagation of randomly perturbed fields,” Can. J. Phys. 40, 782–790 (1962).
[Crossref]

1957 (1)

P. Debye, H. R. Anderson, H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
[Crossref]

1949 (4)

F. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, 4806–502 (1949).
[Crossref]

F. Dyson, “The Smatrix in quantum electrodynamics,” Phys. Rev. 75, 1736–1755 (1949).
[Crossref]

R. P. Feynman, “The theory of positrons,” Phys. Rev. 76, 749–759 (1949).
[Crossref]

R. P. Feynman, “Space-time approach to quantum electrodynamics,” Phys. Rev. 76, 769–789 (1949).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1152–1228.

Anderson, H. R.

P. Debye, H. R. Anderson, H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
[Crossref]

Arcone, S. A.

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

Ashkin, A.

Bourret, R. C.

R. C. Bourret, “Stochastically perturbed fields, with applications to wave propagation in random media,” Nuovo Cimento 26, 1–31 (1962).
[Crossref]

R. C. Bourret, “Propagation of randomly perturbed fields,” Can. J. Phys. 40, 782–790 (1962).
[Crossref]

Brown, W. P.

W. P. Brown, “Coherent field in a random medium—effective refractive index,” in Proceedings of the Symposium on Modern Optics (Polytechnic, Brooklyn, 1967), pp. 717–742.

Brumberger, H.

P. Debye, H. R. Anderson, H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
[Crossref]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Debye, P.

P. Debye, H. R. Anderson, H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
[Crossref]

Dence, D.

D. Dence, J. E. Spence, “Wave propagation in random anisotropic media,” in Probabilistic Methods in Applied and Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1973), Vol. 3, pp. 121–181.

Dyson, F.

F. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, 4806–502 (1949).
[Crossref]

F. Dyson, “The Smatrix in quantum electrodynamics,” Phys. Rev. 75, 1736–1755 (1949).
[Crossref]

Dziedzic, J. M.

Feshback, H.

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Feynman, R. P.

R. P. Feynman, “The theory of positrons,” Phys. Rev. 76, 749–759 (1949).
[Crossref]

R. P. Feynman, “Space-time approach to quantum electrodynamics,” Phys. Rev. 76, 769–789 (1949).
[Crossref]

Fiddy, M. A.

Frisch, U.

U. Frisch, “Wave propagation in a random medium,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), Vol. 1, pp. 75–198.

Furutsu, K.

K. Furutsu, “On the statistical theory of electromagnetic waves in a fluctuation medium (I),”J. Res. Natl. Bur. Stand. (USA) 67, 303–323 (1963).

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Gow, A. J.

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

Hoffman, W. C.

W. C. Hoffman, “Wave propagation in a general random continuous medium,” in Proceedings of the 16th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1964), pp. 117–144.
[Crossref]

Ishimaru, A.

Jordan, A. K.

L. S. Tamil, A. K. Jordan, “Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,” Proc. IEEE 79, 1519–1528 (1991).
[Crossref]

A. K. Jordan, S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6, 1206–1212 (1989).
[Crossref]

Kampé de Fériet, J.

J. Kampé de Fériet, “Statistical mechanics of continuous media,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 165–198.
[Crossref]

Karl, F. C.

F. C. Karl, J. B. Keller, “Elastic, electromagnetic, and other waves in a random medium,”J. Math. Phys. 5, 537–547 (1964).
[Crossref]

Keller, J. B.

F. C. Karl, J. B. Keller, “Elastic, electromagnetic, and other waves in a random medium,”J. Math. Phys. 5, 537–547 (1964).
[Crossref]

J. B. Keller, “Stochastic equations and wave propagation in random media,” in Proceedings of the 16th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1964), pp. 145–170.
[Crossref]

J. B. Keller, “A survey of the theory of wave propagation in continuous random media,” in Proceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic, Brooklyn, 1968), pp. 131–142.

J. B. Keller, “Wave propagation in random media,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 227–246.
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1152–1228.

Kong, J. A.

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

Kraichnan, R. H.

R. H. Kraichnan, “The closure problem of turbulence theory,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 199–225.
[Crossref]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics, 4th ed. (Wiley, New York, 1979).

Lakshmanasamy, S.

Lin, F. C.

F. C. Lin, M. A. Fiddy, “The Born–Rytov controversy: I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A 9, 1102–1110 (1992).
[Crossref]

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

F. C. Lin, “Theoretical models for microwave remote sensing of snow-covered sea ice,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

Liu, C. H.

C. H. Liu, “Wave propagation in a random medium with parabolic background,” Radio Sci. 2, 961–977 (1967).

C. H. Liu, “Effective dielectric tensor and propagation constant of plane waves in a random anisotropic medium,”J. Math. Phys. 8, 2236–2242 (1967).
[Crossref]

Maloney, P. J.

Morse, P. M.

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Oristaglio, M. L.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Rosenbaum, S.

S. Rosenbaum, “On energy-conserving formulations in a randomly fluctuating medium,” in Proceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic, Brooklyn, 1968), pp. 163–185.

Ryzhov, Y. A.

Y. A. Ryzhov, V. V. Tamoikin, “Radiation and propagation of electromagnetic waves in randomly inhomogeneous media,” Radiophys. Quantum Electron. 13, 273–300 (1970).
[Crossref]

Sancer, M. I.

A. D. Varvatsis, M. I. Sancer, “On the renormalization method in random wave propagation,” Radio Sci. 6, 87–97 (1971).
[Crossref]

Shin, R. T.

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

Smith, P. W.

Spence, J. E.

D. Dence, J. E. Spence, “Wave propagation in random anisotropic media,” in Probabilistic Methods in Applied and Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1973), Vol. 3, pp. 121–181.

Stogryn, A.

A. Stogryn, “The bilocal approximation for the effective dielectric constant of an isotropic random medium,”IEEE Trans. Antennas Propag. AP-32, 517–520 (1984).
[Crossref]

Sukhorukov, A. P.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1152–1228.

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).

Tamil, L. S.

L. S. Tamil, A. K. Jordan, “Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,” Proc. IEEE 79, 1519–1528 (1991).
[Crossref]

Tamoikin, V. V.

Y. A. Ryzhov, V. V. Tamoikin, “Radiation and propagation of electromagnetic waves in randomly inhomogeneous media,” Radiophys. Quantum Electron. 13, 273–300 (1970).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, “Propagation of electromagnetic waves in a medium with strong dielectric constant fluctuations,” Sov. Phys. JETP 19, 946–953 (1964).

V. I. Tatarskii, The Effects of the Thrbulent Atmosphere on Wave Propagation, TT-68-50464 (National Technical Information Service, Springfield, Va., 1971).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Tsang, L.

Varvatsis, A. D.

A. D. Varvatsis, M. I. Sancer, “On the renormalization method in random wave propagation,” Radio Sci. 6, 87–97 (1971).
[Crossref]

Yaghjian, A. D.

A. D. Yaghjian, “Maxwellian and cavity electromagnetic fields within continuous sources,” Am. J. Phys. 53, 859–863 (1985).
[Crossref]

Am. J. Phys. (1)

A. D. Yaghjian, “Maxwellian and cavity electromagnetic fields within continuous sources,” Am. J. Phys. 53, 859–863 (1985).
[Crossref]

Can. J. Phys. (1)

R. C. Bourret, “Propagation of randomly perturbed fields,” Can. J. Phys. 40, 782–790 (1962).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. Stogryn, “The bilocal approximation for the effective dielectric constant of an isotropic random medium,”IEEE Trans. Antennas Propag. AP-32, 517–520 (1984).
[Crossref]

J. Appl. Phys. (1)

P. Debye, H. R. Anderson, H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its application,” J. Appl. Phys. 28, 679–683 (1957).
[Crossref]

J. Geophys. Res. (1)

F. C. Lin, J. A. Kong, R. T. Shin, A. J. Gow, S. A. Arcone, “Correlation function study for sea ice,”J. Geophys. Res. 93, 14055–14063 (1988).
[Crossref]

J. Math. Phys. (2)

F. C. Karl, J. B. Keller, “Elastic, electromagnetic, and other waves in a random medium,”J. Math. Phys. 5, 537–547 (1964).
[Crossref]

C. H. Liu, “Effective dielectric tensor and propagation constant of plane waves in a random anisotropic medium,”J. Math. Phys. 8, 2236–2242 (1967).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Res. Natl. Bur. Stand. (USA) (1)

K. Furutsu, “On the statistical theory of electromagnetic waves in a fluctuation medium (I),”J. Res. Natl. Bur. Stand. (USA) 67, 303–323 (1963).

Nuovo Cimento (1)

R. C. Bourret, “Stochastically perturbed fields, with applications to wave propagation in random media,” Nuovo Cimento 26, 1–31 (1962).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (4)

F. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, 4806–502 (1949).
[Crossref]

F. Dyson, “The Smatrix in quantum electrodynamics,” Phys. Rev. 75, 1736–1755 (1949).
[Crossref]

R. P. Feynman, “The theory of positrons,” Phys. Rev. 76, 749–759 (1949).
[Crossref]

R. P. Feynman, “Space-time approach to quantum electrodynamics,” Phys. Rev. 76, 769–789 (1949).
[Crossref]

Proc. IEEE (1)

L. S. Tamil, A. K. Jordan, “Spectral inverse scattering theory for inhomogeneous dielectric waveguides and devices,” Proc. IEEE 79, 1519–1528 (1991).
[Crossref]

Radio Sci. (2)

C. H. Liu, “Wave propagation in a random medium with parabolic background,” Radio Sci. 2, 961–977 (1967).

A. D. Varvatsis, M. I. Sancer, “On the renormalization method in random wave propagation,” Radio Sci. 6, 87–97 (1971).
[Crossref]

Radiophys. Quantum Electron. (1)

Y. A. Ryzhov, V. V. Tamoikin, “Radiation and propagation of electromagnetic waves in randomly inhomogeneous media,” Radiophys. Quantum Electron. 13, 273–300 (1970).
[Crossref]

Sov. Phys. JETP (1)

V. I. Tatarskii, “Propagation of electromagnetic waves in a medium with strong dielectric constant fluctuations,” Sov. Phys. JETP 19, 946–953 (1964).

Other (21)

W. P. Brown, “Coherent field in a random medium—effective refractive index,” in Proceedings of the Symposium on Modern Optics (Polytechnic, Brooklyn, 1967), pp. 717–742.

J. B. Keller, “Stochastic equations and wave propagation in random media,” in Proceedings of the 16th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1964), pp. 145–170.
[Crossref]

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

W. C. Hoffman, “Wave propagation in a general random continuous medium,” in Proceedings of the 16th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1964), pp. 117–144.
[Crossref]

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1152–1228.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

J. Kampé de Fériet, “Statistical mechanics of continuous media,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 165–198.
[Crossref]

J. B. Keller, “Wave propagation in random media,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 227–246.
[Crossref]

R. H. Kraichnan, “The closure problem of turbulence theory,” in Proceedings of the 13th Symposium on Applied Mathematics (American Mathematical Society, Providence, R.I., 1962), pp. 199–225.
[Crossref]

V. I. Tatarskii, The Effects of the Thrbulent Atmosphere on Wave Propagation, TT-68-50464 (National Technical Information Service, Springfield, Va., 1971).

D. Dence, J. E. Spence, “Wave propagation in random anisotropic media,” in Probabilistic Methods in Applied and Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1973), Vol. 3, pp. 121–181.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).

U. Frisch, “Wave propagation in a random medium,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), Vol. 1, pp. 75–198.

J. B. Keller, “A survey of the theory of wave propagation in continuous random media,” in Proceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic, Brooklyn, 1968), pp. 131–142.

S. Rosenbaum, “On energy-conserving formulations in a randomly fluctuating medium,” in Proceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic, Brooklyn, 1968), pp. 163–185.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

F. C. Lin, “Theoretical models for microwave remote sensing of snow-covered sea ice,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

E. Kreyszig, Advanced Engineering Mathematics, 4th ed. (Wiley, New York, 1979).

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

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.


Figures (6)

Fig. 1
Fig. 1

Scattering geometry for the 1-D dielectric half-space case.

Fig. 2
Fig. 2

(a) Symbolic definitions of the mean fields and all related variables, (b) Feynman diagram for 〈Ψ0(z)〉, (c) Feynman diagram for 〈Ψ1(z)〉.

Fig. 3
Fig. 3

(a) Symbolic definitions of the mass operator, (b) Dyson’s equation for 〈Ψ0(z)〉, (c) Dyson’s equation for 〈Ψ1(z)〉.

Fig. 4
Fig. 4

Scattering geometry and permittivity profile of the 1-D random medium having a total length L.

Fig. 5
Fig. 5

(a) Symbolic definitions of ηp(z) and Γpq(z, z′) (p, q = 0,1), (b) Feynman diagram for η0(z), (c) Feynman diagram for η1(z).

Fig. 6
Fig. 6

Scattering geometry for the 1-D homogeneous dielectricslab case.

Equations (104)

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

d 2 G p q ( z , z ) d z 2 + k p 2 G p q ( z , z ) = δ p q δ ( z - z )             ( p , q = 0 , 1 ) ,
G p p ( z , z ) z = 0 = G q p ( z , z ) z = 0 ,
d G p p ( z , z ) d z | z = 0 = d G q p ( z , z ) d z | z = 0
G 00 ( z , z ) = ( 1 / i 2 k 0 ) { exp ( i k 0 z - z ) + R ( k 1 ) exp [ - i k 0 ( z + z ) ] } ,
G 10 ( z , z ) = ( 1 / i 2 k 0 ) T ( k 1 ) exp ( i k 1 z ) exp ( - i k 0 z ) ,
G 11 ( z , z ) = ( 1 / i 2 k 1 ) { exp ( i k 1 z - z ) - R ( k 1 ) exp [ i k 1 ( z + z ) ] } ,
G 01 ( z , z ) = ( 1 / i 2 k 0 ) T ( k 1 ) exp ( - i k 0 z ) exp ( i k 1 z ) ,
R ( k 1 ) = ( k 0 - k 1 ) / ( k 0 + k 1 ) ,
T ( k 1 ) = 2 k 0 / ( k 0 + k 1 ) ,
D ( k 1 ) = 1 - [ R ( k 1 ) ] 2 exp ( i 2 k 1 d ) .
G p q ( z , z ) = G q p ( z , z )             ( p , q = 0 , 1 ) .
Δ n ( z ) = ( n 2 / 2 n 1 ) Ψ 1 ( z ) 2 ,
d 2 Ψ 0 ( z ) d z 2 + k 0 2 Ψ 0 ( z ) = 0 ,
d 2 Ψ 1 ( z ) d z 2 + k 1 2 Ψ 1 ( z ) = - k 1 2 { [ n 0 ( z ) + n 2 2 n 1 Ψ 1 ( z ) 2 ] 2 - 1 } × Ψ 1 ( z ) ,
Ψ 0 ( z ) = exp ( i k 0 z ) + R d exp ( - i k 0 z ) ,
Ψ 1 ( z ) = T d exp ( i k d z ) ,
Ψ 0 ( 0 ) = Ψ 1 ( 0 ) ,
d Ψ 0 ( z ) d z | z = 0 = d Ψ 1 ( z ) d z | z = 0 ,
1 + R d = T d ,
1 - R d = ( k 1 / k 0 ) [ 1 + ( n 2 / 2 n 1 ) T d 2 ] T d .
n 2 T d 3 + [ 4 / T ( k 1 ) ] T d - 4 = 0 ,
( T d ) 1 = { 2 n 2 + [ 64 27 n 2 3 T 3 ( k 1 ) + 4 n 2 2 ] 1 / 2 } 1 / 3 - 4 3 n 2 T ( k 1 ) { 2 n 2 + [ 64 27 n 2 3 T 3 ( k 1 ) + 4 n 2 2 ] 1 / 2 } 1 / 3 ,
( T d ) ± = - 1 2 [ { 2 n 2 + [ 64 27 n 2 3 T 3 ( k 1 ) + 4 n 2 2 ] 1 / 2 } 1 / 3 - 4 3 n 2 T ( k 1 ) { 2 n 2 + [ 64 27 n 2 3 T 3 ( k 1 ) + 4 n 2 2 ] 1 / 2 } 1 / 3 ] ± i 3 2 [ { 2 n 2 + [ 64 27 n 2 3 T 3 ( k 1 ) + 4 n 2 2 ] 1 / 2 } 1 / 3 + 4 3 n 2 T ( k 1 ) { 2 n 2 + [ 64 27 n 2 3 T 3 ( k 1 ) + 4 n 2 2 ] 1 / 2 } 1 / 3 ] ,
( T d ) 1 n 2 1 T ( k 1 ) - n 2 4 T 4 ( k 1 ) + 3 n 2 2 16 T 7 ( k 1 ) - 3 n 2 3 16 T 10 ( k 1 ) + ,
( T d ) ± n 2 1 - 1 2 [ T ( k 1 ) - n 2 4 T 4 ( k 1 ) + 3 n 2 2 16 T 7 ( k 1 ) - 3 n 2 3 16 T 10 ( k 1 ) + ] ± i 3 2 { 4 [ 3 T ( k 1 ) n 2 ] 1 / 2 + [ 3 T 5 ( k 1 ) n 2 ] 1 / 2 8 - 35 [ 3 T 11 ( k 1 ) n 2 3 ] 1 / 2 512 + } ,
Ψ 0 ( z ) n 2 1 exp ( i k 0 z ) + [ R ( k 1 ) - n 2 4 T 4 ( k 1 ) + 3 n 2 2 16 T 7 ( k 1 ) - ] exp ( - i k 0 z ) ,
Ψ 1 ( z ) n 2 1 [ T ( k 1 ) - n 2 4 T 4 ( k 1 ) + i n 2 2 T 3 ( k 1 ) k 0 z + 3 n 2 2 16 T 7 ( k 1 ) - i 3 n 2 2 8 T 6 ( k 1 ) k 0 z - n 2 2 8 T 5 ( k 1 ) k 0 2 z 2 - ] exp ( i k 1 z ) .
Ψ p ( z ) = Ψ p ( 0 ) ( z ) - k 1 2 0 d z G p 1 ( z , z ) × [ n 2 n 1 Ψ 1 ( z ) 2 + n 2 2 4 n 1 2 Ψ 1 ( z ) 4 ] Ψ 1 ( z ) .
Ψ 0 ( 0 ) ( z ) = exp ( i k 0 z ) + R ( k 1 ) exp ( - i k 0 z ) ,
Ψ 1 ( 0 ) ( z ) = T ( k 1 ) exp ( i k 1 z ) ,
Ψ p ( z ) Ψ p ( 0 ) ( z ) + Ψ p s 1 BA ( z ) + Ψ p s 2 BA ( z ) + Ψ p s 3 BA ( z ) + = Ψ p ( 0 ) ( z ) - k 1 2 0 d z G p 1 ( z , z ) × [ n 2 n 1 Ψ 1 ( 0 ) ( z ) + Ψ 1 s 1 BA ( z ) + Ψ 1 s 2 BA ( z ) + 2 + n 2 2 4 n 1 2 Ψ 1 ( 0 ) ( z ) + Ψ 1 s 1 BA ( z ) + Ψ 1 s 2 BA ( z ) + 4 ] × [ Ψ 1 ( 0 ) ( z ) + Ψ 1 s 1 BA ( z ) + Ψ 1 s 2 BA ( z ) + ] .
Ψ 0 s 1 BA ( z ) = - n 2 4 T 4 ( k 1 ) exp ( - i k 0 z ) ,
Ψ 0 s 2 BA ( z ) = 3 n 2 2 16 T 7 ( k 1 ) exp ( - i k 0 z ) ;
Ψ 1 s 1 BA ( z ) = [ - n 2 4 T 4 ( k 1 ) + i n 2 2 T 3 ( k 1 ) k 0 z ] exp ( i k 1 z ) ,
Ψ 1 s 2 BA ( z ) = [ 3 n 2 2 16 T 7 ( k 1 ) - i 3 n 2 2 8 T 6 ( k 1 ) k 0 z - n 2 2 8 T 5 ( k 1 ) k 0 2 z 2 ] exp ( i k 1 z ) ,
d 2 d z 2 [ Ψ 0 ( 0 ) ( z ) φ 0 ( z ) ] + k 0 2 Ψ 0 ( 0 ) ( z ) φ 0 ( z ) = - i [ d φ 0 ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) ,
d 2 d z 2 [ Ψ 1 ( 0 ) ( z ) φ 1 ( z ) ] + k 1 2 Ψ 1 ( 0 ) ( z ) φ 1 ( z ) = - i [ d φ 1 ( z ) d z ] 2 Ψ 1 ( 0 ) ( z ) + i k 1 2 ( n 2 n 1 Ψ 1 ( 0 ) ( z ) 2 × exp { - 2 Im [ φ 1 ( z ) ] } + n 2 2 4 n 1 2 Ψ 1 ( 0 ) ( z ) 4 × exp { - 4 Im [ φ 1 ( z ) ] } ) Ψ 1 ( 0 ) ( z ) ,
φ p ( z ) = - i Ψ p ( 0 ) ( z ) - 0 d z G p 0 ( z , z ) [ d φ 0 ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) - i Ψ p ( 0 ) ( z ) 0 d z G p 1 ( z , z ) [ d φ 1 ( z ) d z ] 2 Ψ 1 ( 0 ) ( z ) + i k 1 2 Ψ p ( 0 ) ( z ) 0 d z G p 1 ( z , z ) ( n 2 n 1 Ψ 1 ( 0 ) ( z ) 2 × exp { - 2 Im [ φ 1 ( z ) ] } + n 2 2 4 n 1 2 Ψ 1 ( 0 ) ( z ) 4 × exp { - 4 Im [ φ 1 ( z ) ] } ) Ψ 1 ( 0 ) ( z ) .
φ 0 1 RA ( z ) = i n 2 4 Ψ 0 ( 0 ) ( z ) T 4 ( k 1 ) exp ( - i k 0 z ) ,
φ 0 2 RA ( z ) = - i n 2 2 16 Ψ 0 ( 0 ) ( z ) [ 3 - 1 2 Ψ 0 ( 0 ) ( z ) T ( k 1 ) exp ( - i k 0 z ) ] × T 7 ( k 1 ) exp ( - i k 0 z ) ,
φ 1 1 RA ( z ) = i n 2 4 T 3 ( k 1 ) + n 2 2 T 2 ( k 1 ) k 0 z ,
φ 1 2 RA ( z ) = - i 5 n 2 2 32 T 6 ( k 1 ) - n 2 2 4 T 5 ( k 1 ) k 0 z .
Ψ p ( z ) = Ψ p ( 0 ) ( z ) - k 1 2 0 d z G p 1 ( z , z ) V ( z ) Ψ 1 ( z )             ( p = 0 , 1 ) ,
Ψ p ( z ) = Ψ p ( 0 ) ( z ) - k 1 2 0 d z 1 G p 1 ( z , z 1 ) V ( z 1 ) Ψ 1 ( 0 ) ( z 1 ) + k 1 4 0 d z 1 G p 1 ( z , z 1 ) V ( z 1 ) × 0 d z 2 G 11 ( z 1 , z 2 ) V ( z 2 ) Ψ 1 ( 0 ) ( z 2 ) - k 1 6 0 d z 1 G p 1 ( z , z 1 ) V ( z 1 ) × 0 d z 2 G 11 ( z 1 , z 2 ) V ( z 2 ) × 0 d z 3 G 11 ( z 2 , z 3 ) V ( z 3 ) Ψ 1 ( 0 ) ( z 3 ) +             ( p = 0 , 1 ) .
Ψ p ( z ) = Ψ p ( 0 ) ( z ) + k 1 4 0 d z 1 G p 1 ( z , z 1 ) 0 d z 2 G 11 ( z 1 , z 2 ) × V ( z 1 ) V ( z 2 ) Ψ 1 ( 0 ) ( z 2 ) + k 1 8 0 d z 1 G p 1 ( z , z 1 ) × 0 d z 2 G 11 ( z 1 , z 2 ) V ( z 1 ) V ( z 2 ) 0 d z 3 G 11 ( z 2 , z 3 ) × 0 d z 4 G 11 ( z 3 , z 4 ) V ( z 3 ) V ( z 4 ) Ψ 1 ( 0 ) ( z 4 ) + k 1 8 × 0 d z 1 G p 1 ( z , z 1 ) 0 d z 2 G 11 ( z 1 , z 2 ) × 0 d z 3 G 11 ( z 2 , z 3 ) V ( z 1 ) V ( z 3 ) 0 d z 4 G 11 ( z 3 , z 4 ) × V ( z 2 ) V ( z 4 ) Ψ 1 ( 0 ) ( z 4 ) + k 1 8 0 d z 1 G p 1 ( z , z 1 ) × 0 d z 2 G 11 ( z 1 , z 2 ) 0 d z 3 G 11 ( z 2 , z 3 ) V ( z 2 ) V ( z 3 ) × 0 d z 4 G 11 ( z 3 , z 4 ) V ( z 1 ) V ( z 4 ) Ψ 1 ( 0 ) ( z 4 ) + = Ψ p ( 0 ) ( z ) + Ψ p ( 2 , 1 ) ( z ) + Ψ p ( 4 , 1 ) ( z ) + Ψ p ( 4 , 2 ) ( z ) + Ψ p ( 4 , 3 ) ( z ) + ,
P 11 ( ζ ) + P 12 ( ζ ) = 1 ,
P 21 ( ζ ) + P 22 ( ζ ) = 1 ,
f 1 P 12 ( ζ ) = f 2 P 21 ( ζ ) ,
P p q ( ζ ) = f q Γ ( ζ ) ,
P p p ( ζ ) = 1 - f q Γ ( ζ ) ,
V ( z 1 ) V ( z 2 ) = p = 1 2 q = 1 2 f p P p q ( ζ ) V p V q ,
V ( z 1 ) V ( z 2 ) = f 1 f 2 ( V 1 - V 2 ) 2 [ 1 - Γ ( ζ ) ] .
Δ P 22 ( ζ ) = - P 22 ( ζ ) ( N Δ ζ / f 2 L ) + P 21 ( ζ ) ( N Δ ζ / f 1 L ) .
d d z [ 1 - Γ ( ζ ) ] = - N f 1 f 2 L [ 1 - Γ ( ζ ) ] .
Γ ( ζ ) = 1 - exp ( - ζ / l ) ,
l = ( 1 - N l s / L ) l s .
V ( z 1 ) V ( z 2 ) = V 2 exp ( - z 1 - z 2 / l ) ,
Ψ p ( z ) BIA = Ψ p ( 0 ) ( z ) + k 1 4 0 d z 1 G p 1 ( z , z 1 ) × 0 d z 2 G 11 ( z 1 , z 2 ) V ( z 1 ) V ( z 2 ) Ψ p ( z 2 ) BIA ,
d 2 Ψ 1 ( z ) BIA d z 2 + k 1 2 Ψ 1 ( z ) BIA - k 1 4 0 d z 1 G 11 ( z , z 1 ) × V ( z ) V ( z 1 ) Ψ 1 ( z 1 ) BIA = 0.
d 2 Ψ 1 ( z ) BIA d z 2 + k 1 2 [ 1 - k 1 2 0 d z 1 G 11 ( z , z 1 ) V ( z ) V ( z ) V ( z 1 ) ] × Ψ 1 ( z ) BIA = 0.
k eff ( z ) = k 1 [ 1 - k 1 2 0 d z 1 G 11 ( z , z 1 ) V ( z ) V ( z 1 ) ] 1 / 2 = k 1 ( 1 + i V 2 k 1 l 2 ( 1 + k 1 2 l 2 ) { 2 ( 1 + i k 1 l ) - [ 1 + i k 1 l - ( 1 - i k 1 l ) R ( k 1 ) ] × exp ( i k 1 z - z l ) - 2 R ( k 1 ) exp ( i 2 k 1 z ) } ) 1 / 2 .
k eff = k 1 [ 1 + i V 2 k 1 l ( 1 + i k 1 l ) 1 + k 1 2 l 2 ] 1 / 2 .
Ψ 0 ( 2 , 1 ) ( z ) = k 1 4 0 d z 1 G 01 ( z , z 1 ) 0 d z 2 G 11 ( z 1 , z 2 ) × V ( z 1 ) V ( z 2 ) Ψ 1 ( 0 ) ( z 2 ) = i V 2 k 1 2 l ( 1 + i 2 k 1 l ) 8 k 0 ( 1 + 4 k 1 2 l 2 ) T 2 ( k 1 ) [ R ( k 1 ) - 2 ] × exp ( - i k 0 z ) ,
Ψ 1 ( 2 , 1 ) ( z ) = k 1 4 0 d z 1 G 11 ( z , z 1 ) 0 d z 2 G 11 ( z 1 , z 2 ) × V ( z 1 ) V ( z 2 ) Ψ 1 ( 0 ) ( z 2 ) = i V 2 k 1 l 4 ( 1 + 4 k 1 2 l 2 ) T ( k 1 ) [ A 1 + A 2 k 1 z + A 3 × exp ( - z l ) + A 4 exp ( i 2 k 1 z ) ] exp ( i k 1 z ) ;
Ψ 0 ( 4 , 1 ) ( z ) = k 1 8 0 d z 1 G 01 ( z , z 1 ) 0 d z 2 G 11 ( z 1 , z 2 ) × V ( z 1 ) V ( z 2 ) 0 d z 3 G 11 ( z 2 , z 3 ) × 0 d z 4 G 11 ( z 3 , z 4 ) V ( z 3 ) V ( z 4 ) Ψ 1 ( 0 ) ( z 4 ) = - V 4 k 1 4 l T 2 ( k 1 ) 16 k 0 ( 1 + 4 k 1 2 l 2 ) ( A 1 B 1 + A 2 B 2 + A 3 B 3 + A 4 B 4 ) exp ( - i k 0 z ) ,
Ψ 1 ( 4 , 1 ) ( z ) = k 1 8 0 d z 1 G 11 ( z , z 1 ) 0 d z 2 G 11 ( z 1 , z 2 ) × V ( z 1 ) V ( z 2 ) 0 d z 3 G 11 ( z 2 , z 3 ) × 0 d z 4 G 11 ( z 3 , z 4 ) V ( z 3 ) V ( z 4 ) Ψ 1 ( 0 ) ( z 4 ) = - V 4 k 1 3 l T ( k 1 ) 16 ( 1 + 4 k 1 2 l 2 ) [ C 1 + C 2 k 1 z + C 3 k 1 2 z 2 + C 4 exp ( - z l ) + C 5 k 1 z exp ( - z l ) + C 6 exp ( i 2 k 1 z ) + C 7 k 1 z exp ( i 2 k 1 z ) + C 8 exp ( i 2 k 1 z - z l ) + C 9 exp ( i 4 k 1 z ) ] × exp ( i k 1 z ) ,
( n 2 / n 1 ) Ψ 1 ( 0 ) ( z ) 2 exp { - 2 Im [ φ 1 ( z ) ] } + ( n 2 2 / 4 n 1 2 ) Ψ 1 ( 0 ) ( z ) 4 × exp { - 4 Im [ φ 1 ( z ) ] }
η p ( z ) [ d φ p ( z ) d z ] 2             ( p = 0 , 1 ) ,
Γ p q ( z , z ) = - i d d z [ G p q ( z , z ) Ψ p ( 0 ) ]             ( p , q = 0 , 1 ) ,
η p ( z ) η p ( 2 ) ( z ) + η p ( 3 ) ( z ) + η p ( 4 ) ( z ) + = k 1 4 [ 0 d z 1 Γ p 1 ( z , z 1 ) V ( z 1 ) Ψ 1 ( 0 ) ( z 1 ) ] 2 + { - 0 d z 1 Γ p 0 ( z , z 1 ) [ η 0 ( 2 ) ( z 1 ) + η 0 ( 3 ) ( z 1 ) + η 0 ( 4 ) ( z 1 ) + ] Ψ 0 ( 0 ) ( z 1 ) } 2 + { 0 d z 1 Γ p 1 ( z , z 1 ) [ η 1 ( 2 ) ( z 1 ) + η 1 ( 3 ) ( z 1 ) + η 1 ( 4 ) ( z 1 ) + ] Ψ 1 ( 0 ) ( z 1 ) } 2 + 2 - 0 d z 1 Γ p 0 ( z , z 1 ) [ η 0 ( 2 ) ( z 1 ) + η 0 ( 3 ) ( z 1 ) + η 0 ( 4 ) ( z 1 ) + ] Ψ 0 ( 0 ) ( z 1 ) × 0 d z 1 Γ p 1 ( z , z 1 ) [ η 1 ( 2 ) ( z 1 ) + η 1 ( 3 ) ( z 1 ) + η 1 ( 4 ) ( z 1 ) + ] Ψ 1 ( 0 ) ( z 1 ) - 2 k 1 2 0 d z 1 Γ p 1 ( z , z 1 ) V ( z 1 ) Ψ 1 ( 0 ) ( z 1 ) - 0 d z 2 Γ p 0 ( z , z 2 ) [ η 0 ( 2 ) ( z 2 ) + η 0 ( 3 ) ( z 2 ) + η 0 ( 4 ) ( z 2 ) + ] Ψ 0 ( 0 ) ( z 2 ) - 2 k 1 2 0 d z 1 Γ p 1 ( z , z 1 ) V ( z 1 ) Ψ 1 ( 0 ) ( z 1 ) 0 d z 2 Γ p 1 ( z , z 2 ) [ η 1 ( 2 ) ( z 2 ) + η 1 ( 3 ) ( z 2 ) + η 1 ( 4 ) ( z 2 ) + ] Ψ 1 ( 0 ) ( z 2 ) ,
φ p ( z ) φ p ( 1 ) ( z ) + φ p ( 2 ) ( z ) + φ p ( 3 ) ( z ) + = i k 1 2 Ψ p ( 0 ) ( z ) 0 d z G p 1 ( z , z ) V ( z ) Ψ 1 ( 0 ) ( z ) - i Ψ p ( 0 ) ( z ) - 0 d z G p 0 ( z , z ) [ η 0 ( 2 ) ( z ) + η 0 ( 3 ) ( z ) + η 0 ( 4 ) ( z ) + ] Ψ 0 ( 0 ) ( z ) - i Ψ p ( 0 ) ( z ) 0 d z G p 1 ( z , z ) [ η 1 ( 2 ) ( z ) + η 1 ( 3 ) ( z ) + η 1 ( 4 ) ( z ) + ] Ψ 1 ( 0 ) ( z ) ,
N ( m ) = 2 N ( m - 1 ) + { 4 i = 2 ( m - 1 ) / 2 N ( m - i ) N ( i ) , m odd 4 i = 2 m / 2 - 1 N ( m - i ) N ( i ) + N ( m 2 ) [ 2 N ( m 2 ) + 1 ] , m even ,
Ψ 0 ( z ) = exp ( i k 0 z ) + k 0 - k 1 ( 1 + V ) 1 / 2 k 0 + k 1 ( 1 + V ) 1 / 2 exp ( - i k 0 z ) ,
Ψ 1 ( z ) = 2 k 0 k 0 + k 1 ( 1 + V ) 1 / 2 exp [ i k 1 z ( 1 + V ) 1 / 2 ] .
Ψ 0 ( z ) V 1 exp ( i k 0 z ) + [ R ( k 1 ) - V k 1 4 k 0 T 2 ( k 1 ) + V 2 k 1 ( k 0 + 3 k 1 ) 32 k 0 2 T 3 ( k 1 ) + ] exp ( - i k 0 z ) ,
Ψ 1 ( z ) V 1 [ T ( k 1 ) - V k 1 4 k 0 T 2 ( k 1 ) + i V 2 T ( k 1 ) k 1 z + V 2 k 1 ( k 0 + 3 k 1 ) 32 k 0 2 T 3 ( k 1 ) - i V 2 ( k 0 + 3 k 1 ) 16 k 0 × T 2 ( k 1 ) k 1 z - V 2 8 T ( k 1 ) k 1 2 z 2 + ] exp ( i k 1 z ) .
φ 0 ( z ) V 1 i V k 1 4 k 0 Ψ 0 ( 0 ) ( z ) T 2 ( k 1 ) exp ( - i k 0 z ) - i V 2 k 1 ( k 0 + 3 k 1 ) 32 k 0 2 Ψ 0 ( 0 ) ( z ) T 3 ( k 1 ) exp ( - i k 0 z ) + i V 2 k 1 2 32 k 0 2 [ Ψ 0 ( 0 ) ( z ) ] 2 T 4 ( k 1 ) exp ( - i 2 k 0 z ) ,
φ 1 ( z ) V 1 i V k 1 4 k 0 T ( k 1 ) + V 2 k 1 z - i V 2 k 1 ( k 0 + 2 k 1 ) 32 k 0 2 T 2 ( k 1 ) - V 2 8 k 1 z +
A 1 = ( 1 + i 2 k 1 l ) [ - ½ R 2 ( k 1 ) + ( 2 - i k 1 l ) R ( k 1 ) - 1 - 2 k 1 2 l 2 ] ,
A 2 = i 2 ( 1 + i 2 k 1 ( 1 - i k 1 l ) ,
A 3 = 2 k 1 2 l 2 [ 1 + i 2 k 1 l - R ( k 1 ) ] ,
A 4 = - ½ R 2 ( k 1 ) ,
B 1 = i l ( 1 + i 2 k 1 l ) 2 k 1 ( 1 + 4 k 1 2 l 2 ) [ 2 - R ( k 1 ) ] ,
B 2 = - l ( 1 + i 2 k 1 l ) 2 8 k 1 ( 1 + 4 k 1 2 l 2 ) 2 × [ 4 ( 1 - i 3 k 1 l ) - ( 1 - i 4 k 1 l ) R ( k 1 ) ] ,
B 3 = l 2 2 [ 2 ( 1 + i 2 k 1 l ) 1 + 4 k 1 2 l 2 + 1 + i k 1 l 1 + k 1 2 l 2 ] × [ 1 + i 2 k 1 l 1 + 4 k 1 2 l 2 - 1 + i 4 k 1 l 1 + 16 k 1 2 l 2 R ( k 1 ) ] ,
B 4 = i l 12 k 1 ( 1 + i 2 k 1 l 1 + 4 k 1 2 l 2 + 1 + i 4 k 1 l 1 + 16 k 1 2 l 2 ) [ 3 - 2 R ( k 1 ) ] ,
C 1 = A 1 i l ( 1 + i 2 k 1 l ) k 1 ( 1 + 4 k 1 2 l 2 ) × [ 1 + 2 k 1 2 l 2 - ( 2 - i k 1 l ) R ( k 1 ) + R 2 ( k 1 ) ] + A 2 l ( 1 + i 2 k 1 l ) 2 2 k 1 ( 1 + 4 k 1 2 l 2 ) 2 [ - 1 + i 3 k 1 l - i 4 k 1 3 l 3 - 8 k 1 4 l 4 + ( 2 - i 7 k 1 l - 2 k 1 2 l 2 ) R ( k 1 ) - 1 - i 4 k 1 l 4 R 2 ( k 1 ) ] + A 3 l 2 ( 1 + i k 1 l ) 2 ( 1 + k 1 2 l 2 ) × [ 3 - i 2 k 1 l - 2 ( 1 + i 2 k 1 l ) 2 ( 3 - i 6 k 1 l - 2 k 1 2 l 2 ) ( 1 + 4 k 1 2 l 2 ) 2 × R ( k 1 ) + ( 1 + i 2 k 1 l ) ( 1 + i 4 k 1 l ) ( 3 - i 4 k 1 l ) ( 1 + 4 k 1 2 l 2 ) ( 1 + 16 k 1 2 l 2 ) × R 2 ( k 1 ) ] + A 4 i l ( 1 + i 4 k 1 l ) k 1 ( 1 + 16 k 1 2 l 2 ) × [ 1 - i 2 k 1 l - 2 - i 3 k 1 l + 10 k 1 2 l 2 - i 8 k 1 3 l 3 2 ( 1 + 4 k 1 2 l 2 ) R ( k 1 ) + ( 1 + i 2 k 1 l ) ( 1 - i 3 k 1 l ) 3 ( 1 + 4 k 1 2 l 2 ) R 2 ( k 1 ) ] ,
C 2 = l ( 1 - i k 1 l ) ( 1 + i 2 k 1 l ) k 1 ( 1 + 4 k 1 2 l 2 ) × [ 2 A 1 + A 2 i ( 1 + i 2 k 1 l ) ( 1 - i 2 k 1 l + 4 k 1 2 l 2 ) 1 + 4 k 1 2 l 2 ] ,
C 3 = A 2 l ( 1 - i k 1 l ) ( 1 + i 2 k 1 l ) k 1 ( 1 + 4 k 1 2 l 2 ) ,
C 4 = i 2 k 1 l 3 1 + 4 k 1 2 l 2 { A 1 [ - 1 - i 2 k 1 l + R ( k 1 ) ] + A 2 [ k 1 l ( 1 + i 2 k 1 l ) - k 1 l ( 1 - i 2 k 1 l ) 1 + 4 k 1 2 l 2 R ( k 1 ) ] + A 3 [ ( 1 + i k 1 l ) ( 1 + i 2 k 1 l ) 2 ( 5 - i 10 k 1 l - 4 k 1 2 l 2 ) 2 ( 1 + k 1 2 l 2 ) ( 1 + 4 k 1 2 l 2 ) + ( 1 - i 2 k 1 l ) R ( k 1 ) ] + A 4 [ - 1 + ( 1 + i 2 k 1 l ) ( 1 - i 4 k 1 l ) 1 + 16 k 1 2 l 2 R ( k 1 ) ] } ,
C 5 = A 3 i 2 l 2 ( 1 + i 2 k 1 l ) 1 + 4 k 1 2 l 2 ,
C 6 = i l 2 k 1 ( 1 + 4 k 1 2 l 2 ) [ A 1 R ( k 1 ) + A 2 i ( 3 + 28 k 1 2 l 2 ) 4 ( 1 + 4 k 1 2 l 2 ) R ( k 1 ) - A 4 ( 1 - i k 1 l ) ( 1 - i 2 k 1 l ) ( 1 - i 4 k 1 l ) 1 + 16 k 1 2 l 2 ] ,
C 7 = A 2 i l 2 k 1 ( 1 + 4 k 1 2 l 2 ) R ( k 1 ) ,
C 8 = A 3 - l 2 ( 1 + i k 1 l ) ( 1 + i 2 k 1 l ) ( 1 + i 4 k 1 l ) ( 1 + k I 2 l 2 ) ( 1 + 4 k 1 2 l 2 ) ( 1 + 16 k 1 2 l 2 ) R ( k 1 ) ,
C 9 = A 4 i l 6 k 1 ( 1 + 16 k 1 2 l 2 ) R ( k 1 ) .
G 00 ( z , z ) = 1 i 2 k 0 { exp ( i k 0 z - z + R ( k 1 ) [ 1 - exp ( i 2 k 1 d ) ] D ( k 1 ) × exp [ - i k 0 ( z + z ) ] } ,
G 10 ( z , z ) = 1 i 2 k 0 T ( k 1 ) D ( k 1 ) [ exp ( i k 1 z ) - R ( k 1 ) exp ( i 2 k 1 d ) × exp ( - i k 1 z ) ] exp ( - i k 0 z ) ,
G 20 ( z , z ) = 1 i 2 k 0 T ( k 1 ) [ 1 - R ( k 1 ) ] exp [ i ( k 1 - k 0 ) d ] D ( k 1 ) × exp [ i k 0 ( z - z ) ] ,
G 11 ( z , z ) = 1 i 2 k 1 { exp ( i k 1 z - z ) - R ( k 1 ) D ( k 1 ) × exp [ i k 1 ( z + z ) - R ( k 1 ) exp ( i 2 k 1 d ) D ( k 1 ) × exp [ - i k 1 ( z + z ) ] + 2 R 2 ( k 1 ) exp ( i 2 k 1 d ) D ( k 1 ) × cos [ k 1 ( z - z ) ] } ,
G 21 ( z , z ) = 1 i 2 k 0 T ( k 1 ) exp [ i ( k 1 - k 0 ) d ] D ( k 1 ) exp ( i k 0 z ) × [ - R ( k 1 ) exp ( i k 1 z ) + exp ( - i k 1 z ) ] ,
G 01 ( z , z ) = 1 i 2 k 0 T ( k 1 ) D ( k 1 ) exp ( - i k 0 z ) × [ exp ( i k 1 z ) - R ( k 1 ) exp ( i 2 k 1 d ) 0 ( - i k 1 z ) ] ,
G 22 ( z , z ) = 1 i 2 k 0 { exp ( i k 0 z - z + R ( k 1 ) [ 1 - exp ( i 2 k 1 d ) ] exp ( - i 2 k 0 d ) D ( k 1 ) × exp [ i k 0 ( z + z ) ] } ,
G 12 ( z , z ) = 1 i 2 k 0 T ( k 1 ) exp [ i ( k 1 - k 0 ) d ] D ( k 1 ) × [ - R ( k 1 ) exp ( i k 1 z ) + exp ( - i k 1 z ) ] exp ( i k 0 z ) ,
G 02 ( z , z ) = 1 i 2 k 0 T ( k 1 ) [ 1 - R ( k 1 ) ] exp [ i ( k 1 - k 0 ) d ] D ( k 1 ) × exp [ - i k 0 ( z - z ) ] ,

Metrics