Abstract

The applicability and domains of validity for the Born and the Rytov methods in scattering theory are established, with mathematical rigor, by comparing successive terms of the Born and the Rytov series calculated for wave propagation in a homogeneous dielectric half-space and a homogeneous dielectric slab for which the permittivities are assumed to have a low contrast over that of free space. While the (first-order) Rytov approximation is superior to the (first-order) Born approximation when it is applied to estimate the scattered field in the homogeneous dielectric half-space, both approximations are inapplicable to estimate the scattered fields when the dielectric slab is thick.

© 1992 Optical Society of America

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  45. H. Ermert, M. Dohlus, “Microwave-diffraction-tomography of cylindrical objects using 3-dimensional wave-fields,” NTZ Archiv. 8, 111–117 (1986).
  46. M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
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  48. Z. Lu, “JKM perturbation theory, relaxation perturbation theory, and their applications to inverse scattering, I. Theory and reconstruction algorithms,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 722–730 (1986).
  49. R. L. Nowack, A. Keiiti, “Iterative inversion for velocity using waveform data,” Geophys. J. R. Astron. Soc. 87, 701–730 (1986).
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  50. M. Soumekh, M. Kaveh, “A theoretical study of model approximation errors in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 10–20 (1986).
    [Crossref]
  51. M. Soumekh, “An improvement to the Rytov approximation in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 394–401 (1986).
    [Crossref]
  52. B. J. Bates, S. M. Bates, “Stochastic simulation and first-order multiple scatter solutions for acoustic propagation through oceanic internal waves,”J. Acoust. Soc. Am. 82, 2042–2050 (1987).
    [Crossref]
  53. Y. Baykal, “Correlation and structure functions for multimode-laser-beam incidence in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 817–819 (1987).
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  54. A. J. Devaney, “Inverse scattering theory foundations of tomography with diffracting wavefields,” in Pattern Recognition and Acoustical Imaging, L. A. Ferrari, ed., Proc. Soc. Photo-Opt. Instrum. Eng.768, 2–6 (1987).
    [Crossref]
  55. L.-J. Gelius, J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng.808, 209–217 (1987).
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  57. W. B. Beydoun, A. Tarantola, “First Born and Rytov approximations: modeling and inversion conditions in a canonical example,”J. Acoust. Soc. Am. 83, 1045–1055 (1988).
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  58. T. F. Duda, S. M. Flatte, D. B. Creamer, “Modelling meter-scale acoustic intensity fluctuations from oceanic fine structure and microstructure,”J. Geophys. Res. 93, 5130–5142 (1988).
    [Crossref]
  59. N. Gorenflo, “Inversion formulae for first-order approximations in fixed-energy scattering by compactly supported potentials,” Inverse Problems 4, 1025–1035 (1988).
    [Crossref]
  60. O. Stoham, E. Goldner, A. Weitz, N. Ben-Yosef, “Aperture averaging effects on the two-color correlation of scintillations,” Appl. Opt. 27, 2157–2160 (1988).
    [Crossref]
  61. S. D. Rajan, G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysica 54, 864–871 (1989).
    [Crossref]
  62. H. T. Yura, S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564–575 (1989).
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  63. K. T. Ladas, G. A. Tsihrintzis, “Contour reconstruction in diffraction tomography,” Int. J. Imaging Syst. Technol. 2, 127–133 (1990).
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  64. F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
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  67. A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Processing 39, 672–682 (1991).
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1991 (2)

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Processing 39, 672–682 (1991).
[Crossref]

A. Mandelis, “Theory of photothermal wave diffraction tomography via spatial Laplace spectral decomposition,”J. Phys. A. Gen. Phys. 24, 2485–2505 (1991).
[Crossref]

1990 (4)

K. T. Ladas, G. A. Tsihrintzis, “Contour reconstruction in diffraction tomography,” Int. J. Imaging Syst. Technol. 2, 127–133 (1990).
[Crossref]

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[Crossref]

C.-B. Peng, Y. Chen, “A computer simulation study on the diffraction seismic tomography,” Acta Geophys. Sin. 33, 530–539 (1990) (in Chinese).

K. M. Yoo, R. R. Alfano, “Time-resolved coherent and incoherent components of forward light scattering in random media,” Opt. Lett. 15, 320–322 (1990).
[Crossref] [PubMed]

1989 (2)

H. T. Yura, S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564–575 (1989).
[Crossref]

S. D. Rajan, G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysica 54, 864–871 (1989).
[Crossref]

1988 (4)

W. B. Beydoun, A. Tarantola, “First Born and Rytov approximations: modeling and inversion conditions in a canonical example,”J. Acoust. Soc. Am. 83, 1045–1055 (1988).
[Crossref]

T. F. Duda, S. M. Flatte, D. B. Creamer, “Modelling meter-scale acoustic intensity fluctuations from oceanic fine structure and microstructure,”J. Geophys. Res. 93, 5130–5142 (1988).
[Crossref]

N. Gorenflo, “Inversion formulae for first-order approximations in fixed-energy scattering by compactly supported potentials,” Inverse Problems 4, 1025–1035 (1988).
[Crossref]

O. Stoham, E. Goldner, A. Weitz, N. Ben-Yosef, “Aperture averaging effects on the two-color correlation of scintillations,” Appl. Opt. 27, 2157–2160 (1988).
[Crossref]

1987 (3)

Y. Baykal, “Correlation and structure functions for multimode-laser-beam incidence in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 817–819 (1987).
[Crossref]

R.-S. Wu, M. N. Toksoz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysica 52, 11–25 (1987).
[Crossref]

B. J. Bates, S. M. Bates, “Stochastic simulation and first-order multiple scatter solutions for acoustic propagation through oceanic internal waves,”J. Acoust. Soc. Am. 82, 2042–2050 (1987).
[Crossref]

1986 (7)

H. Ermert, M. Dohlus, “Microwave-diffraction-tomography of cylindrical objects using 3-dimensional wave-fields,” NTZ Archiv. 8, 111–117 (1986).

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[Crossref]

J. F. Greenleaf, “A graphical description of scattering,” Ultrasound Med. Biol. 12, 603–609 (1986).
[Crossref] [PubMed]

Z. Lu, “JKM perturbation theory, relaxation perturbation theory, and their applications to inverse scattering, I. Theory and reconstruction algorithms,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 722–730 (1986).

R. L. Nowack, A. Keiiti, “Iterative inversion for velocity using waveform data,” Geophys. J. R. Astron. Soc. 87, 701–730 (1986).
[Crossref]

M. Soumekh, M. Kaveh, “A theoretical study of model approximation errors in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 10–20 (1986).
[Crossref]

M. Soumekh, “An improvement to the Rytov approximation in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 394–401 (1986).
[Crossref]

1985 (5)

1983 (1)

1978 (1)

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,”J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[Crossref]

1974 (1)

L. R. Brownlee, “Reply to comments on Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 55, 1339 (1974).
[Crossref]

1973 (3)

L. R. Brownlee, “Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 53, 156–161 (1973).
[Crossref]

D. A. de Wolf, “Comments on Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 54, 1109–1110 (1973).
[Crossref]

D. A. de Wolf, “Strong irradiance fluctuations in turbulent air: plane waves,”J. Opt. Soc. Am. 63, 171–179 (1973).
[Crossref]

1971 (1)

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tamarskii, “Status of the theory of propagation of waves in a radomly inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[Crossref]

1970 (4)

J. A. Neubert, “Asymptotic solution of the stochastic Helmholtz equation for turbulent water,”J. Acoust. Soc. Am. 48, 1203–1211 (1970).
[Crossref]

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE, 58, 140–141 (1970).
[Crossref]

K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE, 58, 1168–1169 (1970).
[Crossref]

K. Mano, “Symmetry associated with the Born and Rytov methods,” Proc. IEEE 58, 1405–1406 (1970).
[Crossref]

1969 (2)

1968 (3)

1967 (6)

1966 (2)

1965 (1)

1964 (1)

1960 (2)

V. V. Pisareva, “Limits of applicability of the method of, ‘smooth’ perturbations in the problem of radiation propagation through a medium containing inhomogeneities,” Sov. Phys. Acoust. 6, 81–86 (1960).

T. A. Shirokova, “Second approximation in the method of smooth perturbations,” Sov. Phys. Acoust. 5, 498–503 (1960).

1958 (1)

H. Scheffler, “Zur Berücksichtigung der Mehrfachstreuung in der Theorie der Szintillation optischer und radiofrequenter Stralhung,” Astron. Nachr. 285, 21–23 (1958) (in German).

Alfano, R. R.

Barabanenkov, Y. N.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tamarskii, “Status of the theory of propagation of waves in a radomly inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[Crossref]

Bates, B. J.

B. J. Bates, S. M. Bates, “Stochastic simulation and first-order multiple scatter solutions for acoustic propagation through oceanic internal waves,”J. Acoust. Soc. Am. 82, 2042–2050 (1987).
[Crossref]

Bates, R. H. T.

G. R. Dunlop, W. M. Boerner, R. H. T. Bates, “On an extended Rytov approximation and its comparison with the Born approximation,” in Digest of APS International Symposium (University of Massachusetts at Amherst, Amherst, Mass., 1976), pp. 587–591.

Bates, S. M.

B. J. Bates, S. M. Bates, “Stochastic simulation and first-order multiple scatter solutions for acoustic propagation through oceanic internal waves,”J. Acoust. Soc. Am. 82, 2042–2050 (1987).
[Crossref]

Baykal, Y.

Ben-Yosef, N.

Beydoun, W. B.

W. B. Beydoun, A. Tarantola, “First Born and Rytov approximations: modeling and inversion conditions in a canonical example,”J. Acoust. Soc. Am. 83, 1045–1055 (1988).
[Crossref]

Boerner, W. M.

G. R. Dunlop, W. M. Boerner, R. H. T. Bates, “On an extended Rytov approximation and its comparison with the Born approximation,” in Digest of APS International Symposium (University of Massachusetts at Amherst, Amherst, Mass., 1976), pp. 587–591.

Brown, W. P.

Brownlee, L. R.

L. R. Brownlee, “Reply to comments on Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 55, 1339 (1974).
[Crossref]

L. R. Brownlee, “Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 53, 156–161 (1973).
[Crossref]

Carter, W. H.

Chen, Y.

C.-B. Peng, Y. Chen, “A computer simulation study on the diffraction seismic tomography,” Acta Geophys. Sin. 33, 530–539 (1990) (in Chinese).

Chernov, L. A.

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

Chu, A.

J. F. Greenleaf, A. Chu, “Multifrequency diffraction tomography,” in Acoustical Imaging. Proceedings of the Thirteenth International Symposium, M. Kaveh, R. K. Mueller, J. F. Greenleaf, eds. (Plenum, New York, 1984), pp. 43–56.
[Crossref]

Clifford, S. F.

Coulman, C. E.

Creamer, D. B.

T. F. Duda, S. M. Flatte, D. B. Creamer, “Modelling meter-scale acoustic intensity fluctuations from oceanic fine structure and microstructure,”J. Geophys. Res. 93, 5130–5142 (1988).
[Crossref]

de Wolf, D. A.

Devaney, A. J.

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Processing 39, 672–682 (1991).
[Crossref]

A. J. Devaney, “Inverse scattering theory foundations of tomography with diffracting wavefields,” in Pattern Recognition and Acoustical Imaging, L. A. Ferrari, ed., Proc. Soc. Photo-Opt. Instrum. Eng.768, 2–6 (1987).
[Crossref]

Dohlus, M.

H. Ermert, M. Dohlus, “Microwave-diffraction-tomography of cylindrical objects using 3-dimensional wave-fields,” NTZ Archiv. 8, 111–117 (1986).

Duda, T. F.

T. F. Duda, S. M. Flatte, D. B. Creamer, “Modelling meter-scale acoustic intensity fluctuations from oceanic fine structure and microstructure,”J. Geophys. Res. 93, 5130–5142 (1988).
[Crossref]

Dunlop, G. R.

G. R. Dunlop, W. M. Boerner, R. H. T. Bates, “On an extended Rytov approximation and its comparison with the Born approximation,” in Digest of APS International Symposium (University of Massachusetts at Amherst, Amherst, Mass., 1976), pp. 587–591.

Ermert, H.

H. Ermert, M. Dohlus, “Microwave-diffraction-tomography of cylindrical objects using 3-dimensional wave-fields,” NTZ Archiv. 8, 111–117 (1986).

Feinberg, Y. L.

Y. L. Feinberg, Propagation of Radiowaves along the Earth’s Surface (Academy of SciencesUSSR, Moscow, 1961).

Feshbach, H.

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

Fiddy, M. A.

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[Crossref]

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[Crossref]

L. Zapalowski, S. Leeman, M. A. Fiddy, “Image reconstruction fidelity using the Born and Rytov approximations,” in Acoustical Imaging. Proceedings of the 14th International Symposium, A. J. Berkhout, J. Ridder, L. F. Van der Wal, eds. (Plenum, New York, 1985), pp. 295–304.
[Crossref]

F. C. Lin, M. A. Fiddy, “The Born–Rytov controversy: II. Applications in the direct- and the inverse-scattering problems,” J. Opt. Soc. Am. A (to be published).

Flatte, S. M.

T. F. Duda, S. M. Flatte, D. B. Creamer, “Modelling meter-scale acoustic intensity fluctuations from oceanic fine structure and microstructure,”J. Geophys. Res. 93, 5130–5142 (1988).
[Crossref]

Fried, D. L.

Frisk, G. V.

S. D. Rajan, G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysica 54, 864–871 (1989).
[Crossref]

Gelius, L.-J.

L.-J. Gelius, J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng.808, 209–217 (1987).
[Crossref]

Goldner, E.

Gorenflo, N.

N. Gorenflo, “Inversion formulae for first-order approximations in fixed-energy scattering by compactly supported potentials,” Inverse Problems 4, 1025–1035 (1988).
[Crossref]

Greenleaf, J. F.

J. F. Greenleaf, “A graphical description of scattering,” Ultrasound Med. Biol. 12, 603–609 (1986).
[Crossref] [PubMed]

J. F. Greenleaf, A. Chu, “Multifrequency diffraction tomography,” in Acoustical Imaging. Proceedings of the Thirteenth International Symposium, M. Kaveh, R. K. Mueller, J. F. Greenleaf, eds. (Plenum, New York, 1984), pp. 43–56.
[Crossref]

Gretzula, A.

Hadden, W. J.

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,”J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[Crossref]

Hanson, S. G.

Heidbreder, G. R.

Hill, R. J.

Hufnagel, R. E.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

Kaveh, M.

M. Soumekh, M. Kaveh, “A theoretical study of model approximation errors in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 10–20 (1986).
[Crossref]

M. Kaveh, M. Soumekh, “Algorithms and error analysis for diffraction tomography using the Born and Rytov approximations,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, H. Brand, L. A. Cram, D. T. Gjessing, A. K. Jordan, W. Keydel, G. Schwierz, M. Vogel, eds. (Reidel, Dordrecht, The Netherlands, 1983), pp. 1137–1146.

Keiiti, A.

R. L. Nowack, A. Keiiti, “Iterative inversion for velocity using waveform data,” Geophys. J. R. Astron. Soc. 87, 701–730 (1986).
[Crossref]

Keller, J. B.

Kravtsov, Y. A.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tamarskii, “Status of the theory of propagation of waves in a radomly inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[Crossref]

Ladas, K. T.

K. T. Ladas, G. A. Tsihrintzis, “Contour reconstruction in diffraction tomography,” Int. J. Imaging Syst. Technol. 2, 127–133 (1990).
[Crossref]

Leeman, S.

L. Zapalowski, S. Leeman, M. A. Fiddy, “Image reconstruction fidelity using the Born and Rytov approximations,” in Acoustical Imaging. Proceedings of the 14th International Symposium, A. J. Berkhout, J. Ridder, L. F. Van der Wal, eds. (Plenum, New York, 1985), pp. 295–304.
[Crossref]

Lin, F. C.

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[Crossref]

F. C. Lin, M. A. Fiddy, “The Born–Rytov controversy: II. Applications in the direct- and the inverse-scattering problems,” J. Opt. Soc. Am. A (to be published).

Lu, Z.

Z. Lu, “JKM perturbation theory, relaxation perturbation theory, and their applications to inverse scattering, I. Theory and reconstruction algorithms,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 722–730 (1986).

Mandelis, A.

A. Mandelis, “Theory of photothermal wave diffraction tomography via spatial Laplace spectral decomposition,”J. Phys. A. Gen. Phys. 24, 2485–2505 (1991).
[Crossref]

Mano, K.

K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE, 58, 1168–1169 (1970).
[Crossref]

K. Mano, “Symmetry associated with the Born and Rytov methods,” Proc. IEEE 58, 1405–1406 (1970).
[Crossref]

Mintzer, D.

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,”J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[Crossref]

Morse, P. M.

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

Neubert, J. A.

J. A. Neubert, “Asymptotic solution of the stochastic Helmholtz equation for turbulent water,”J. Acoust. Soc. Am. 48, 1203–1211 (1970).
[Crossref]

Nowack, R. L.

R. L. Nowack, A. Keiiti, “Iterative inversion for velocity using waveform data,” Geophys. J. R. Astron. Soc. 87, 701–730 (1986).
[Crossref]

Oristaglio, M. L.

Peng, C.-B.

C.-B. Peng, Y. Chen, “A computer simulation study on the diffraction seismic tomography,” Acta Geophys. Sin. 33, 530–539 (1990) (in Chinese).

Pisareva, V. V.

V. V. Pisareva, “Limits of applicability of the method of, ‘smooth’ perturbations in the problem of radiation propagation through a medium containing inhomogeneities,” Sov. Phys. Acoust. 6, 81–86 (1960).

Plonus, M. A.

Rajan, S. D.

S. D. Rajan, G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysica 54, 864–871 (1989).
[Crossref]

Rytov, S. M.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tamarskii, “Status of the theory of propagation of waves in a radomly inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[Crossref]

Sancer, M. I.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE, 58, 140–141 (1970).
[Crossref]

Scheffler, H.

H. Scheffler, “Zur Berücksichtigung der Mehrfachstreuung in der Theorie der Szintillation optischer und radiofrequenter Stralhung,” Astron. Nachr. 285, 21–23 (1958) (in German).

Schmeltzer, R. A.

R. A. Schmeltzer, “Means, variances, and covariances for laser beam propagation through a random medium,”Q. Appl. Math. 24, 339–354 (1967).

Shirokova, T. A.

T. A. Shirokova, “Second approximation in the method of smooth perturbations,” Sov. Phys. Acoust. 5, 498–503 (1960).

Soumekh, M.

M. Soumekh, M. Kaveh, “A theoretical study of model approximation errors in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 10–20 (1986).
[Crossref]

M. Soumekh, “An improvement to the Rytov approximation in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 394–401 (1986).
[Crossref]

M. Kaveh, M. Soumekh, “Algorithms and error analysis for diffraction tomography using the Born and Rytov approximations,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, H. Brand, L. A. Cram, D. T. Gjessing, A. K. Jordan, W. Keydel, G. Schwierz, M. Vogel, eds. (Reidel, Dordrecht, The Netherlands, 1983), pp. 1137–1146.

Stamnes, J.

L.-J. Gelius, J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng.808, 209–217 (1987).
[Crossref]

Stanley, N. R.

Stoham, O.

Strohbehn, J. W.

J. W. Strohbehn, “Comments on Rytov’s method,”J. Opt. Soc. Am. 58, 139–140 (1968).
[Crossref]

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[Crossref]

Sung, C. C.

Tamarskii, V. I.

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tamarskii, “Status of the theory of propagation of waves in a radomly inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[Crossref]

Tarantola, A.

W. B. Beydoun, A. Tarantola, “First Born and Rytov approximations: modeling and inversion conditions in a canonical example,”J. Acoust. Soc. Am. 83, 1045–1055 (1988).
[Crossref]

Tatarskii, V. I.

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

Taylor, L. S.

Toksoz, M. N.

R.-S. Wu, M. N. Toksoz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysica 52, 11–25 (1987).
[Crossref]

Tsihrintzis, G. A.

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Processing 39, 672–682 (1991).
[Crossref]

K. T. Ladas, G. A. Tsihrintzis, “Contour reconstruction in diffraction tomography,” Int. J. Imaging Syst. Technol. 2, 127–133 (1990).
[Crossref]

Varvatsis, A. D.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE, 58, 140–141 (1970).
[Crossref]

Weitz, A.

Wenzel, A. R.

A. R. Wenzel, “Comment on ‘second-order Rytov approximation’,” J. Opt. Soc. Am. A 2, 774–775 (1985).
[Crossref]

A. R. Wenzel, “Localization of the mean and mean squared intensities, and intensity fluctuations, of waves propagating in a one-dimensional random medium,” Wave Motion 7, 589–600 (1985).
[Crossref]

Wu, R.-S.

R.-S. Wu, M. N. Toksoz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysica 52, 11–25 (1987).
[Crossref]

Yoo, K. M.

Yura, H. T.

Zapalowski, L.

L. Zapalowski, S. Leeman, M. A. Fiddy, “Image reconstruction fidelity using the Born and Rytov approximations,” in Acoustical Imaging. Proceedings of the 14th International Symposium, A. J. Berkhout, J. Ridder, L. F. Van der Wal, eds. (Plenum, New York, 1985), pp. 295–304.
[Crossref]

Acta Geophys. Sin. (1)

C.-B. Peng, Y. Chen, “A computer simulation study on the diffraction seismic tomography,” Acta Geophys. Sin. 33, 530–539 (1990) (in Chinese).

Appl. Opt. (1)

Astron. Nachr. (1)

H. Scheffler, “Zur Berücksichtigung der Mehrfachstreuung in der Theorie der Szintillation optischer und radiofrequenter Stralhung,” Astron. Nachr. 285, 21–23 (1958) (in German).

Geophys. J. R. Astron. Soc. (1)

R. L. Nowack, A. Keiiti, “Iterative inversion for velocity using waveform data,” Geophys. J. R. Astron. Soc. 87, 701–730 (1986).
[Crossref]

Geophysica (2)

R.-S. Wu, M. N. Toksoz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysica 52, 11–25 (1987).
[Crossref]

S. D. Rajan, G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysica 54, 864–871 (1989).
[Crossref]

IEEE Trans. Signal Processing (1)

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Processing 39, 672–682 (1991).
[Crossref]

IEEE Trans. Ultrasonic Ferroelectrics Frequency Control (3)

Z. Lu, “JKM perturbation theory, relaxation perturbation theory, and their applications to inverse scattering, I. Theory and reconstruction algorithms,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 722–730 (1986).

M. Soumekh, M. Kaveh, “A theoretical study of model approximation errors in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 10–20 (1986).
[Crossref]

M. Soumekh, “An improvement to the Rytov approximation in diffraction tomography,”IEEE Trans. Ultrasonic Ferroelectrics Frequency Control UFFC-33, 394–401 (1986).
[Crossref]

Int. J. Imaging Syst. Technol. (2)

K. T. Ladas, G. A. Tsihrintzis, “Contour reconstruction in diffraction tomography,” Int. J. Imaging Syst. Technol. 2, 127–133 (1990).
[Crossref]

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[Crossref]

Inverse Problems (1)

N. Gorenflo, “Inversion formulae for first-order approximations in fixed-energy scattering by compactly supported potentials,” Inverse Problems 4, 1025–1035 (1988).
[Crossref]

J. Acoust. Soc. Am. (7)

B. J. Bates, S. M. Bates, “Stochastic simulation and first-order multiple scatter solutions for acoustic propagation through oceanic internal waves,”J. Acoust. Soc. Am. 82, 2042–2050 (1987).
[Crossref]

W. B. Beydoun, A. Tarantola, “First Born and Rytov approximations: modeling and inversion conditions in a canonical example,”J. Acoust. Soc. Am. 83, 1045–1055 (1988).
[Crossref]

J. A. Neubert, “Asymptotic solution of the stochastic Helmholtz equation for turbulent water,”J. Acoust. Soc. Am. 48, 1203–1211 (1970).
[Crossref]

L. R. Brownlee, “Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 53, 156–161 (1973).
[Crossref]

D. A. de Wolf, “Comments on Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 54, 1109–1110 (1973).
[Crossref]

L. R. Brownlee, “Reply to comments on Rytov’s method and large fluctuations,”J. Acoust. Soc. Am. 55, 1339 (1974).
[Crossref]

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,”J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[Crossref]

J. Geophys. Res. (1)

T. F. Duda, S. M. Flatte, D. B. Creamer, “Modelling meter-scale acoustic intensity fluctuations from oceanic fine structure and microstructure,”J. Geophys. Res. 93, 5130–5142 (1988).
[Crossref]

J. Opt. Soc. Am. (14)

H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second-order Rytov approximation,”J. Opt. Soc. Am. 73, 500–502 (1983).
[Crossref]

D. A. de Wolf, “Strong irradiance fluctuations in turbulent air: plane waves,”J. Opt. Soc. Am. 63, 171–179 (1973).
[Crossref]

J. B. Keller, “Accuracy and validity of the Born and Rytov approximations,”J. Opt. Soc. Am. 59, 1003–1004 (1969).

H. T. Yura, “Optical propagation through a turbulent medium,”J. Opt. Soc. Am. 59, 111–112 (1969).
[Crossref]

D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,”J. Opt. Soc. Am. 58, 961–969 (1968).
[Crossref]

J. W. Strohbehn, “Comments on Rytov’s method,”J. Opt. Soc. Am. 58, 139–140 (1968).
[Crossref]

L. S. Taylor, “Decay of mutual coherence in turbulent media,”J. Opt. Soc. Am. 57, 304–308 (1967).
[Crossref]

R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,”J. Opt. Soc. Am. 54, 52–61 (1964).
[Crossref]

D. A. de Wolf, “Wave propagation through quasi-optical irregularities,”J. Opt. Soc. Am. 55, 812–817 (1965).
[Crossref]

W. P. Brown, “Validity of the Rytov approximation in optical propagation calculations,”J. Opt. Soc. Am. 56, 1045–1052 (1966).
[Crossref]

C. E. Coulman, “Dependence of image quality on horizontal range in a turbulent atmosphere,”J. Opt. Soc. Am. 56, 1232–1238 (1966).
[Crossref]

W. P. Brown, “Validity of the Rytov approximation,”J. Opt. Soc. Am. 57, 1539–1543 (1967).
[Crossref]

D. L. Fried, “Test of the Rytov approximation,”J. Opt. Soc. Am. 57, 268–269 (1967).
[Crossref]

G. R. Heidbreder, “Multiple scattering and the method of Rytov,”J. Opt. Soc. Am. 57, 1477–1479 (1967).
[Crossref]

J. Opt. Soc. Am. A (6)

J. Phys. A. Gen. Phys. (1)

A. Mandelis, “Theory of photothermal wave diffraction tomography via spatial Laplace spectral decomposition,”J. Phys. A. Gen. Phys. 24, 2485–2505 (1991).
[Crossref]

J. Phys. D (1)

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[Crossref]

NTZ Archiv. (1)

H. Ermert, M. Dohlus, “Microwave-diffraction-tomography of cylindrical objects using 3-dimensional wave-fields,” NTZ Archiv. 8, 111–117 (1986).

Opt. Lett. (1)

Proc. IEEE (4)

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[Crossref]

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE, 58, 140–141 (1970).
[Crossref]

K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE, 58, 1168–1169 (1970).
[Crossref]

K. Mano, “Symmetry associated with the Born and Rytov methods,” Proc. IEEE 58, 1405–1406 (1970).
[Crossref]

Q. Appl. Math. (1)

R. A. Schmeltzer, “Means, variances, and covariances for laser beam propagation through a random medium,”Q. Appl. Math. 24, 339–354 (1967).

Radio Sci. (1)

L. S. Taylor, “On Rytov’s method,” Radio Sci. 2, 437–441 (1967).

Sov. Phys. Acoust. (2)

V. V. Pisareva, “Limits of applicability of the method of, ‘smooth’ perturbations in the problem of radiation propagation through a medium containing inhomogeneities,” Sov. Phys. Acoust. 6, 81–86 (1960).

T. A. Shirokova, “Second approximation in the method of smooth perturbations,” Sov. Phys. Acoust. 5, 498–503 (1960).

Sov. Phys. Usp. (1)

Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, V. I. Tamarskii, “Status of the theory of propagation of waves in a radomly inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[Crossref]

Ultrasound Med. Biol. (1)

J. F. Greenleaf, “A graphical description of scattering,” Ultrasound Med. Biol. 12, 603–609 (1986).
[Crossref] [PubMed]

Wave Motion (1)

A. R. Wenzel, “Localization of the mean and mean squared intensities, and intensity fluctuations, of waves propagating in a one-dimensional random medium,” Wave Motion 7, 589–600 (1985).
[Crossref]

Other (14)

L. Zapalowski, S. Leeman, M. A. Fiddy, “Image reconstruction fidelity using the Born and Rytov approximations,” in Acoustical Imaging. Proceedings of the 14th International Symposium, A. J. Berkhout, J. Ridder, L. F. Van der Wal, eds. (Plenum, New York, 1985), pp. 295–304.
[Crossref]

J. F. Greenleaf, A. Chu, “Multifrequency diffraction tomography,” in Acoustical Imaging. Proceedings of the Thirteenth International Symposium, M. Kaveh, R. K. Mueller, J. F. Greenleaf, eds. (Plenum, New York, 1984), pp. 43–56.
[Crossref]

A. J. Devaney, “Inverse scattering theory foundations of tomography with diffracting wavefields,” in Pattern Recognition and Acoustical Imaging, L. A. Ferrari, ed., Proc. Soc. Photo-Opt. Instrum. Eng.768, 2–6 (1987).
[Crossref]

L.-J. Gelius, J. Stamnes, “Computer experiments in acoustical diffraction tomography,” in Inverse Problems in Optics, E. R. Pike, ed., Proc. Soc. Photo-Opt. Instrum. Eng.808, 209–217 (1987).
[Crossref]

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

F. C. Lin, M. A. Fiddy, “The Born–Rytov controversy: II. Applications in the direct- and the inverse-scattering problems,” J. Opt. Soc. Am. A (to be published).

H. T. Yura, Electromagnetic Field and Intensity Fluctuations in a Weakly Inhomogeneous Medium, RM-5697-PR (RAND Corporation, Los Angeles, Calif., 1968).

H. T. Yura, The Second-Order Rytov Approximation, RM-5787-PR (RAND Corporation, Los Angeles, Calif., 1969).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

M. Kaveh, M. Soumekh, “Algorithms and error analysis for diffraction tomography using the Born and Rytov approximations,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, H. Brand, L. A. Cram, D. T. Gjessing, A. K. Jordan, W. Keydel, G. Schwierz, M. Vogel, eds. (Reidel, Dordrecht, The Netherlands, 1983), pp. 1137–1146.

G. R. Dunlop, W. M. Boerner, R. H. T. Bates, “On an extended Rytov approximation and its comparison with the Born approximation,” in Digest of APS International Symposium (University of Massachusetts at Amherst, Amherst, Mass., 1976), pp. 587–591.

Y. L. Feinberg, Propagation of Radiowaves along the Earth’s Surface (Academy of SciencesUSSR, Moscow, 1961).

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

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

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Figures (2)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Scattering geometry for the case of the 1-D dielectric slab.

Tables (2)

Tables Icon

Table 1 Domains of Validity of the Born and the Rytov Approximations in the Dielectric Half-Space Case for Finite k0 and V (V ≪ 1)a

Tables Icon

Table 2 Domains of Validity of the Born and the Rytov Approximations in the Dielectric-Slab Case for Finite k0 and V (V ≪ 1)a

Equations (72)

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

Ψ 0 ( 0 ) = Ψ 1 ( 0 ) ,
1 μ 0 d Ψ 0 ( z ) d z | z = 0 = 1 μ 1 d Ψ 1 ( z ) d z | z = 0 .
Ψ 0 ( z ) = exp ( i k 0 z ) + R ( k 1 ) exp ( - i k 0 z ) ,             z < 0 ,
Ψ 1 ( z ) = T ( k 1 ) exp ( i k 1 z ) ,             z 0 ,
R ( k 1 ) = k 0 - k 1 k 0 + k 1 ,
T ( k 1 ) = 2 k 0 k 0 + k 1 .
Ψ 0 ( z ) V 1 exp ( i k 0 z ) + ( - V 4 + V 2 8 - 5 V 3 64 + ) × exp ( - i k 0 x ) ,             z < 0 ,
Ψ 1 ( z ) V 1 ( 1 - V 4 + V 2 8 - 5 V 3 16 + ) exp ( i k 0 z ) × exp [ i ( V 2 - V 2 8 + 5 V 3 16 ) k 0 z ] , V 1 ( 1 - V 4 + i V 2 k 0 z + V 2 8 - i V 2 4 k 0 z - V 2 8 k 0 2 z 2 - 5 V 3 16 + i 5 V 3 32 k 0 z + 3 V 3 32 k 0 2 z 2 - i V 3 48 k 0 3 z 3 + ) × exp ( i k 0 z ) ,             z 0.
d 2 Ψ 0 ( z ) d z 2 + k 0 2 Ψ 0 ( z ) = 0 ,             z < 0 ,
d 2 Ψ 1 ( z ) d z 2 + k 1 2 Ψ 1 ( z ) = 0 ,             z 0.
d 2 Ψ 1 ( z ) d z 2 + k 0 2 Ψ 1 ( z ) = - k 0 2 V Ψ 1 ( z ) ,             z 0 ,
d 2 G ( z , z ) d z 2 + k 0 2 G ( z , z ) = δ ( z - z ) ,
Ψ 0 ( z ) = Ψ 0 ( 0 ) ( z ) - k 0 2 0 d z G ( z , z ) V Ψ 1 ( z ) ,             z < 0 ,
Ψ 1 ( z ) = Ψ 1 ( 0 ) ( z ) - k 0 2 0 d z G ( z , z ) V Ψ 1 ( z ) ,             z 0 ,
G ( z , z ) = - i 2 k 0 exp ( i k 0 z - z ) ,
Ψ 0 ( z ) Ψ 0 ( 0 ) ( z ) - k 0 2 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) + k 0 4 0 d z G ( z , z ) V 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) - k 0 6 0 d z G ( z , z ) V 0 d z G ( z , z ) V × 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) + Ψ 0 ( 0 ) ( z ) + Ψ 0 s 1 B A ( z ) + Ψ 0 s 2 B A ( z ) + Ψ 0 s 3 B A ( z ) + ,             z < 0 ,
Ψ 1 ( z ) Ψ 1 ( 0 ) ( z ) - k 0 2 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) + k 0 4 0 d z G ( z , z ) V 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) - k 0 6 0 d z G ( z , z ) V 0 d z G ( z , z ) V × 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) + Ψ 1 ( 0 ) ( z ) + Ψ 1 s 1 B A ( z ) + Ψ 1 s 2 B A ( z ) + Ψ 1 s 3 B A ( z ) + ,             z 0
Ψ 0 s 1 B A ( z ) = - V 4 exp ( - i k 0 z ) ,             z < 0 ,
Ψ 0 s 2 B A ( z ) = V 8 exp ( - i k 0 z ) ,             z < 0 ,
Ψ 0 s 3 B A ( z ) = - 5 V 3 64 exp ( - i k 0 z ) ,             z < 0 ,
Ψ 1 s 1 B A ( z ) = ( - V 4 + i V 2 k 0 z ) exp ( i k 0 z ) ,             z 0 ,
Ψ 1 s 2 B A ( z ) = ( V 2 8 - i V 2 4 k 0 z - V 2 8 k 0 2 z 2 ) exp ( i k 0 z ) ,             z 0 ,
Ψ 1 s 3 B A ( z ) = ( - 5 V 3 64 + i 5 V 3 32 k 0 z + 3 V 3 32 k 0 2 z 2 - i V 3 48 k 0 3 z 3 ) × exp ( i k 0 z ) ,             z 0.
Ψ 0 ( z ) Ψ 0 ( 0 ) ( z ) exp [ i φ 0 ( z ) ] ,             z < 0 ,
Ψ 1 ( z ) Ψ 1 ( 0 ) ( z ) exp [ i φ 1 ( z ) ] ,             z 0 ,
φ 0 ( z ) = - i ln [ Ψ 0 ( z ) Ψ 0 ( 0 ) ( z ) ] = - i ln [ 1 + R ( k 1 ) exp ( - i 2 k 0 z ) ] ,             z < 0 ,
φ 1 ( z ) = - i ln [ Ψ 1 ( z ) Ψ 1 ( 0 ) ( z ) ] = - i ln [ T ( k 1 ) ] + ( k 1 - k 0 ) z ,             z 0 ,
φ 0 ( z ) V 1 i ( V 4 - V 2 8 + 5 V 3 64 ) exp ( - i 2 k 0 z ) + i ( V 3 32 - V 3 32 ) × exp ( - i 4 k 0 z ) + i V 3 192 exp ( - i 6 k 0 z ) + ,             z < 0 ,
φ 1 ( z ) V 1 i ( V 4 - 3 V 2 32 + 5 V 3 96 ) + ( V 2 - V 2 8 + V 3 16 ) k 0 z + ,             z 0.
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 ) ,             z < 0 ,
d 2 d z 2 [ Ψ 1 ( 0 ) ( z ) φ 1 ( z ) ] + k 0 2 Ψ 1 ( 0 ) ( z ) φ 1 ( z ) = i k 0 2 V Ψ 1 ( 0 ) ( z ) - i [ d φ 1 ( z ) d z ] 2 Ψ 1 ( 0 ) ( z ) ,             z 0.
φ 0 ( z ) = - i Ψ 0 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) + i Ψ 0 ( 0 ) ( z ) 0 d z G ( z , z ) { k 0 2 V - [ d φ 1 ( z ) d z ] 2 } Ψ 1 ( 0 ) ( z ) ,             z < 0 ,
φ 1 ( z ) = - i Ψ 1 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) + i Ψ 1 ( 0 ) ( z ) 0 d z G ( z , z ) { k 0 2 V - [ d φ 1 ( z ) d z ] 2 } Ψ 1 ( 0 ) ( z ) ,             z 0.
d φ 0 ( z ) d z = - 2 k 0 R ( k 1 ) exp ( - i 2 k 0 z ) 1 + R ( k 1 ) exp ( - i 2 k 0 z ) ,             z < 0 ,
d φ 1 ( z ) d z = k 1 - k 0 ,             z 0.
φ 0 ( z ) i k 0 2 Ψ 0 ( 0 ) ( z ) 0 d z ( z , z ) V Ψ 1 ( 0 ) ( z ) - i Ψ 0 ( 0 ) ( z ) × - 0 d z G ( z , z ) [ d φ 0 1 R A ( z ) d z + d φ 0 2 R A ( z ) d z + d φ 0 3 R A ( z ) d z + ] 2 Ψ 0 ( 0 ) ( z ) - i Ψ 0 ( 0 ) ( z ) 0 d z G ( z , z ) × [ d φ 1 1 R A ( z ) d z + d φ 1 2 R A ( z ) d z + d φ 1 3 R A ( z ) d z + ] 2 Ψ 1 ( 0 ) ( z ) = φ 0 1 R A ( z ) + φ 0 2 R A ( z ) + φ 0 3 R A ( z ) + ,             z < 0 ,
φ 1 ( z ) i k 0 2 Ψ 1 ( 0 ) ( z ) 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) - i Ψ 1 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 1 R A ( z ) d z + d φ 0 2 R A d z + d φ 0 3 R A ( z ) d z + ] 2 Ψ 0 ( 0 ) ( z ) - i Ψ 1 ( 0 ) ( z ) 0 d z G ( z , z ) × [ d φ 1 1 R A ( z ) d z + d φ 1 2 R A ( z ) d z + d φ 1 3 R A ( z ) d z + ] 2 Ψ 1 ( 0 ) ( z ) = φ 1 1 R A ( z ) + φ 1 2 R A ( z ) + φ 1 3 R A ( z ) + ,             z 0 ,
φ 0 1 R A ( z ) = i k 0 2 Ψ 0 ( 0 ) ( z ) 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) = i V 4 exp ( - i 2 k 0 z ) ,             z < 0 ,
φ 1 1 R A ( z ) = i k 0 2 Ψ 1 ( 0 ) ( z ) 0 d z G ( z , z ) V Ψ 1 ( 0 ) ( z ) = i V 4 + V 2 k 0 z ,             z 0 ,
φ 0 2 R A ( z ) = - i Ψ 0 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 1 R A ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) - i Ψ 0 ( 0 ) ( z ) 0 d z G ( z , z ) [ d φ 1 1 R A ( z ) d z ] 2 Ψ 1 ( 0 ) ( z ) = - i V 2 8 exp ( - i 2 k 0 z ) + i V 2 32 exp ( - i 4 k 0 z ) ,             z < 0 ,
φ 1 2 R A ( z ) = - i Ψ 1 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 1 R A ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) - i Ψ 1 ( 0 ) ( z ) 0 d z G ( z , z ) [ d φ 1 1 R A ( z ) d z ] 2 Ψ 1 ( 0 ) ( z ) = - i 3 V 2 32 - V 2 8 k 0 z ,             z 0 ,
φ 0 3 R A ( z ) = - i 2 Ψ 0 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 1 R A ( z ) d z ] × [ d φ 0 2 R A ( z ) d z ] Ψ 0 ( 0 ) ( z ) - i 2 Ψ 0 ( 0 ) ( z ) 0 d z G ( z , z ) × [ d φ 1 1 R A ( z ) d z ] [ d φ 1 2 R A ( z ) d z ] Ψ 1 ( 0 ) ( z ) = i 5 V 3 64 exp ( - i 2 k 0 z ) - i V 3 32 exp ( - i 4 k 0 z ) + i V 3 192 exp ( - i 6 k 0 z ) ,             z < 0 ,
φ 1 3 R A ( z ) = - i 2 Ψ 1 ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 1 R A ( z ) d z ] × [ d φ 0 2 R A ( z ) d z ] Ψ 0 ( 0 ) ( z ) - i 2 Ψ 1 ( 0 ) ( z ) 0 d z G ( z , z ) × [ d φ 1 1 R A ( z ) d z ] [ d φ 1 2 R A d z ] Ψ 1 ( 0 ) ( z ) = i 5 V 3 96 + V 3 16 k 0 z ,             z 0.
Ψ m s 1 B A ( z ) = i Ψ m ( 0 ) ( z ) φ m 1 R A ( z ) ,
Ψ m s 2 B A ( z ) = Ψ m ( 0 ) ( z ) { [ i φ m 1 R A ( z ) ] 2 2 ! + i φ m 2 R A ( z ) } ,
Ψ m s 3 B A ( z ) = Ψ m ( 0 ) ( z ) { [ i φ m 1 R A ( z ) ] 3 3 ! - φ m 1 R A ( z ) φ m 2 R A ( z ) + i φ m 3 R A ( z ) } ,
Ψ 0 s 1 R A ( z ) = Ψ 0 ( 0 ) ( z ) { exp [ i φ 0 1 R A ( z ) ] - 1 } V 1 - V 4 exp ( - i k 0 z ) + O [ V 2 exp ( - i 3 k 0 z ) ] = Ψ 0 s 1 B A ( z ) + O [ V 2 exp ( - i 3 k 0 z ) ] ,             z < 0 ,
Ψ 0 s ( 1 + 2 ) R A ( z ) = Ψ 0 ( 0 ) ( z ) ( exp { i [ φ 0 1 R A ( z ) + φ 0 2 R A ( z ) ] } - 1 ) V 1 ( - V 4 + V 2 8 ) exp ( - i k 0 z ) + O [ V 3 exp ( - i 3 k 0 z ) ] = Ψ 0 s 1 B A ( z ) + Ψ 0 s 2 B A ( z ) + O [ V 3 exp ( - i 3 k 0 z ) ] ,             z < 0 ,
Ψ 0 s ( 1 + 2 + 3 ) R A ( z ) = Ψ 0 ( 0 ) ( z ) × ( exp { i [ φ 0 1 R A ( z ) + φ 0 2 R A ( z ) + φ 0 3 R A ( z ) ] } - 1 ) V 1 ( - V 4 + V 2 8 - 5 V 3 64 ) exp ( - i k 0 z ) + O [ V 4 exp ( - i 3 k 0 z ) ] = Ψ 0 s 1 B A ( z ) + Ψ 0 s 2 B A ( z ) + Ψ 0 s 3 B A ( z ) + O [ V 4 exp ( - i 3 k 0 z ) ] ,             z < 0 ,
Ψ 1 s 1 R A ( z ) = Ψ 1 ( 0 ) ( z ) { exp [ i φ 1 1 R A ( z ) ] - 1 } V 1 ( - V 4 + i V 2 k 0 z ) exp ( i k 0 z ) + O ( V 2 ) + O ( V 2 k 0 2 z 2 ) = Ψ 1 s 1 B A ( z ) + O ( V 2 ) + O ( V 2 k 0 2 z 2 ) ,             z 0 ,
Ψ 1 s ( 1 + 2 ) R A ( z ) = Ψ 1 ( 0 ) ( z ) ( exp { i [ φ 1 1 R A ( z ) + φ 1 2 R A ( z ) ] } - ) V 1 ( - V 4 + i V 2 k 0 z + V 2 8 - i V 2 4 k 0 z - V 2 8 k 0 2 z 2 ) × exp ( i k 0 z ) + O ( V 3 ) + O ( V 3 k 0 2 z 2 ) = Ψ 1 s 1 B A ( z ) + Ψ 1 s 2 B A ( z ) + O ( V 3 ) + O ( V 3 k 0 2 z 2 ) ,             z 0 ,
Ψ 1 s ( 1 + 2 + 3 ) R A ( z ) = Ψ 1 ( 0 ) ( z ) × ( exp { i [ φ 1 1 R A ( z ) + φ 1 2 R A ( z ) + φ 1 3 R A ( z ) ] } - 1 ) V 1 ( - V 4 + i V 2 k 0 z + V 2 8 k 0 z - V 2 8 k 0 2 z 2 - 5 V 3 64 + i 5 V 3 32 k 0 z + 3 V 3 32 k 0 2 z 2 - i V 3 48 k 0 3 z 3 ) × exp ( i k 0 z ) + O ( V 4 ) + O ( V 4 k 0 2 z 2 ) = Ψ 1 s 1 B A ( z ) + Ψ 1 s 2 B A ( z ) + Ψ 1 s 3 B A ( z ) + O ( V 4 ) + O ( V 4 k 0 2 z 2 ) ,             z 0.
lim n Ψ m s ( 1 + 2 + + n ) R A ( z ) = Ψ m ( 0 ) ( z ) { exp [ i j = 1 φ m j R A ( z ) ] - 1 } = j = 1 Ψ m s j B A ( z ) .
Ψ 0 ( 0 ) = Ψ 1 ( 0 ) ,
Ψ 1 ( d ) = Ψ 2 ( d ) ,
d Ψ 0 ( z ) d z | z = 0 = d Ψ 1 ( z ) d z | z = 0 ,
d Ψ 1 ( z ) d z | z = d = d Ψ 2 ( z ) d z | z = d ,
Ψ 0 ( z ) = exp ( i k 0 z ) + R ( k 1 ) [ 1 - exp ( i 2 k 1 d ) ] D ( k 1 ) exp ( - i k 0 z ) ,             z < 0 ,
Ψ 1 ( z ) = T ( k 1 ) D ( k 1 ) exp ( i k 1 z ) - R ( k 1 ) T ( k 1 ) exp ( i 2 k 1 d ) D ( k 1 ) × exp ( - i k 1 z ) ,             0 z d ,
Ψ 2 ( z ) = T ( k 1 ) [ 1 - R ( k 1 ) ] exp [ i ( k 1 - k 0 ) d ] D ( k 1 ) exp ( i k 0 z ) ,             z > d ,
Ψ 0 ( z ) V 1 V k 0 d 1 exp ( i k 0 z ) + { [ - V 4 + V 2 8 - 5 V 3 64 - V 3 64 exp ( i 2 k 0 d ) ] [ 1 - exp ( i 2 k 0 d ) ] + i ( V 2 4 - 3 V 3 16 ) k 0 d × exp ( i 2 k 0 d ) - V 3 8 k 0 2 d 2 exp ( i 2 k 0 d ) + } exp ( - i k 0 z ) ,             z < 0 ,
Ψ 1 ( z ) V 1 V k 0 d 1 [ 1 - V 4 + V 2 8 - 5 V 3 64 + ( V 2 16 - 5 V 3 64 ) × exp ( i 2 k 0 d ) + i V 3 16 k 0 ( d + z 2 ) exp ( i 2 k 0 d ) + i ( V 2 - V 2 4 + 5 V 3 32 ) k 0 z + ( - V 2 8 + 3 V 3 32 ) k 0 2 z 2 - i V 3 48 k 0 3 z 3 ] exp ( i k 0 z ) + [ V 4 - 3 V 2 16 + 9 V 3 64 + V 3 64 × exp ( i 2 k 0 d ) + i ( V 2 4 - V 3 4 ) k 0 ( d - z 2 ) - V 3 8 k 0 2 ( d - z 2 ) 2 ] exp ( i 2 k 0 d ) exp ( - i k 0 z ) + ,             0 z d ,
Ψ 2 ( z ) V 1 V k 0 d 1 { 1 + ( - V 2 16 + V 3 16 ) [ 1 - exp ( i 2 k 0 d ) ] + i ( V 2 - V 2 8 + V 3 32 ) k 0 d + i 3 V 3 32 k 0 d exp ( i 2 k 0 d ) + ( - V 2 8 + V 3 16 ) k 0 2 d 2 - i V 3 48 k 0 3 d 3 + } exp ( i k 0 z ) ,             z > d .
Ψ m ( z ) = Ψ m ( 0 ) ( z ) - k 0 2 0 d d z G ( z , z ) V Ψ 1 ( z )             ( m = 0 , 1 , 2 ) ,
Ψ m ( z ) Ψ m ( 0 ) ( z ) exp [ i φ m ( z ) ] ,
φ 0 ( z ) = - i ln { 1 + R ( k 1 ) [ 1 - exp ( i 2 k 1 d ) ] D ( k 1 ) exp ( - i 2 k 0 z ) } ,             z < 0 ,
φ 1 ( z ) = - i ln [ T ( k 1 ) D ( k 1 ) - R ( k 1 ) T ( k 1 ) exp ( i 2 k 1 d ) D ( k 1 ) × exp ( - i 2 k 1 z ) ] + ( k 1 - k 0 ) z ,             0 z d ,
φ 2 ( z ) = - i ln { T ( k 1 ) [ 1 - R ( k 1 ) ] D ( k 1 ) } + ( k 1 - k 0 ) d ,             z > d .
φ 0 ( z ) V 1 V k 0 d 1 { i [ V 4 - V 2 8 + 5 V 3 64 + V 3 64 exp ( i 2 k 0 d ) ] × [ 1 - exp ( i 2 k 0 d ) ] + ( V 2 4 - 3 V 3 16 ) k 0 d exp ( i 2 k 0 d ) + i V 3 8 k 0 2 d 2 exp ( i 2 k 0 d ) } exp ( - i 2 k 0 z ) + { i ( V 2 32 - V 3 32 ) [ 1 - exp ( i 2 k 0 d ) ] + V 3 16 k 0 d exp ( i 2 k 0 d ) } × [ 1 - exp ( i 2 k 0 d ) ] exp ( - i 4 k 0 z ) + i V 3 192 × [ 1 - exp ( i 2 k 0 d ) ] 3 exp ( - i 6 k 0 z ) + ,             z < 0 ,
φ 1 ( z ) V 1 V k 0 d 1 i ( V 4 - 3 V 2 32 + 5 V 3 96 ) + ( - i V 2 16 + i V 3 16 + V 3 16 k 0 d ) exp ( i 2 k 0 d ) + ( V 2 - V 2 8 + V 3 16 ) k 0 z + [ i ( - V 4 + V 2 8 - 5 V 3 64 ) + ( V 2 4 - V 3 16 ) k 0 ( d - z ) + i V 3 8 k 0 2 ( d - z ) 2 ] exp ( i 2 k 0 d ) exp ( - i 2 k 0 z ) + [ i ( V 2 32 - V 3 32 ) - V 3 16 k 0 ( d - z ) ] exp ( i 4 k 0 d ) × exp ( - i 4 k 0 z ) - i V 3 192 exp ( i 6 k 0 d ) exp ( - i 6 k 0 z ) + O ( V 4 ) + O ( V 4 k 0 2 d 2 ) + O [ V 4 k 0 3 ( d - z ) 3 ] + ,             0 z d ,
φ 2 ( z ) V 1 V k 0 d 1 ( V 2 - V 2 8 + V 3 16 - 5 V 4 128 ) k 0 d + i ( V 2 16 - V 3 16 + 29 V 4 512 ) + [ i ( - V 2 16 + V 3 16 - 7 V 4 128 ) + ( V 3 16 + 5 V 4 64 ) k 0 d + i V 4 32 k 0 2 d 2 ] exp ( i 2 k 0 d ) - i V 4 512 exp ( i 4 k 0 d ) + ,             z > d .
φ m ( z ) = - i Ψ m ( 0 ) ( z ) - 0 d z G ( z , z ) [ d φ 0 ( z ) d z ] 2 Ψ 0 ( 0 ) ( z ) + i Ψ m ( 0 ) ( z ) 0 d d z G ( z , z ) { k 0 2 V - [ d φ 1 ( z ) d z ] 2 } Ψ 1 ( 0 ) ( z ) - i Ψ m ( 0 ) ( z ) d d z G ( z , z ) [ d φ 2 ( z ) d z ] 2 Ψ 2 ( 0 ) ( z ) ,             ( m = 0 , 1 , 2 ) .

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