Abstract

Some of the advantages and limitations of Moliere’s high-energy solution of the Schröddinger scattering equation are discussed in connection with the study of wave propagation in a turbulent medium. Certain inconsistencies in the conventional Born and Rytov methods are shown to be closely related to an unwarranted extension of the Moliere solution to higher order in the stationary-phase approximation. The error associated with the extended solution is shown to increase as the penetration distance decreases; its estimated magnitude may be restricted to be relatively small by adoption of a lower range limit for the validity of the Born-Rytov approximation. This restriction, when combined with the standard upper limit associated with the first-order refractive-index approximation, defines a range interval, bounded from above and below, outside of which the Born-Rytov approximation is inconsistent. For optical propagation in the atmosphere, the lower limit exceeds the upper limit, and the Born-Rytov approximation is nowhere consistent.

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  1. G. Moliere, Z. Naturforsch. 2, 133 (1947).
  2. I. I. Gol'dman and A. B. Midgal, JETP 1, 304 (1955).
  3. L. I. Schiff, Phys. Rev. 103, 443 (1956).
  4. R. J. Glauber, in Lectures in Theoretical Physics, edited by W. E. Britten and L. G. Dunham (Wiley–Interscience, New York, 1959).
  5. The WKB approximation is ψ~ψW=exp[i∫(k2-U2ds], with the integration along the classical trajectory.
  6. Schiff3 has extended the Moliere approximation for use in large-angle scattering.
  7. The WKB approximation is ψW=exp(iknds), with the integration along the ray path.
  8. It will be assumed that the mean value of n1 satisfies 〈n1〉 = 0, and that 〈n12〉 is constant.
  9. The use of the Fresnel or stationary-phase approximation to evaluate the Rytov integral was introduced by A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz., No. 2, 155 (1959).
  10. L. A. Chernov, Wave Propagation in a Random Medium McGraw-Hill, New York, 1960), especially p. 66 and Appendix I.
  11. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), especially p. 127.
  12. Although several aspects of the Rytov approximation have undergone considerable investigation in recent years, the use the Fresnel stationary-phase approximation for its evaluation continues to be widely accepted. See, for example, V. I. Tatarski, Propagation of Waves in a Turbulent Atmosphere (Nauka, Moscow, 1967), especially pp. 283–7 (in Russian).
  13. Yu. N. Barabanekov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, Usp. Fiz. Nauk 102, 3 (1970), especially p. 9.
  14. More precisely, the Fresnel paraboloid of revolution (the region of stationary phase). See Ref. 3 and the discussion by C. Eckhart, Rev. Mod. Phys. 20, 399 (1948).
  15. The separation of the scattered field into refractive and diffractive parts is discussed by H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 103. The distinction between coherent and incoherent scattering in random media is discussed by M. Lax, Rev. Mod. Phys. 23, 287 (1951), especially p. 290.
  16. G. Modesitt, On the Upper and Lower Limits to the Rytov Approximation (The Rand Corporation, Santa Monica, Calif., 1969), RM-6090.
  17. This condition is neither necessary nor sufficient for the validity of the Rytov equation; it has been derived many times with various approaches, some of which are discussed in Ref. 16.
  18. A closely related inconsistency is that which arises from the assertion (see, for example, Ref. 12, p. 127) that, for z small compared to ka2, the Rytov equation may be further approximated and render valid the equations of linear (in n1) geometric optics. This view, when combined with the widely adopted upper limit given above (γz <ka2) implies, inconsistently, the existence when γ > 1 of an interval γ-1 <z/ka2 < 1 in which the Rytov equation is invalid and the less general geometric equation is valid.

Barabanekov, Yu. N.

Yu. N. Barabanekov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, Usp. Fiz. Nauk 102, 3 (1970), especially p. 9.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium McGraw-Hill, New York, 1960), especially p. 66 and Appendix I.

Eckhart, C.

More precisely, the Fresnel paraboloid of revolution (the region of stationary phase). See Ref. 3 and the discussion by C. Eckhart, Rev. Mod. Phys. 20, 399 (1948).

Glauber, R. J.

R. J. Glauber, in Lectures in Theoretical Physics, edited by W. E. Britten and L. G. Dunham (Wiley–Interscience, New York, 1959).

Gol’dman, I. I.

I. I. Gol'dman and A. B. Midgal, JETP 1, 304 (1955).

Kravtsov, Yu. A.

Yu. N. Barabanekov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, Usp. Fiz. Nauk 102, 3 (1970), especially p. 9.

Midgal, A. B.

I. I. Gol'dman and A. B. Midgal, JETP 1, 304 (1955).

Modesitt, G.

G. Modesitt, On the Upper and Lower Limits to the Rytov Approximation (The Rand Corporation, Santa Monica, Calif., 1969), RM-6090.

Moliere, G.

G. Moliere, Z. Naturforsch. 2, 133 (1947).

Obukhov, A. M.

The use of the Fresnel or stationary-phase approximation to evaluate the Rytov integral was introduced by A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz., No. 2, 155 (1959).

Rytov, S. M.

Yu. N. Barabanekov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, Usp. Fiz. Nauk 102, 3 (1970), especially p. 9.

Schiff, L. I.

L. I. Schiff, Phys. Rev. 103, 443 (1956).

Tatarski, V. I.

Yu. N. Barabanekov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, Usp. Fiz. Nauk 102, 3 (1970), especially p. 9.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), especially p. 127.

Although several aspects of the Rytov approximation have undergone considerable investigation in recent years, the use the Fresnel stationary-phase approximation for its evaluation continues to be widely accepted. See, for example, V. I. Tatarski, Propagation of Waves in a Turbulent Atmosphere (Nauka, Moscow, 1967), especially pp. 283–7 (in Russian).

van de Hulst, H. C.

The separation of the scattered field into refractive and diffractive parts is discussed by H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 103. The distinction between coherent and incoherent scattering in random media is discussed by M. Lax, Rev. Mod. Phys. 23, 287 (1951), especially p. 290.

Other (18)

G. Moliere, Z. Naturforsch. 2, 133 (1947).

I. I. Gol'dman and A. B. Midgal, JETP 1, 304 (1955).

L. I. Schiff, Phys. Rev. 103, 443 (1956).

R. J. Glauber, in Lectures in Theoretical Physics, edited by W. E. Britten and L. G. Dunham (Wiley–Interscience, New York, 1959).

The WKB approximation is ψ~ψW=exp[i∫(k2-U2ds], with the integration along the classical trajectory.

Schiff3 has extended the Moliere approximation for use in large-angle scattering.

The WKB approximation is ψW=exp(iknds), with the integration along the ray path.

It will be assumed that the mean value of n1 satisfies 〈n1〉 = 0, and that 〈n12〉 is constant.

The use of the Fresnel or stationary-phase approximation to evaluate the Rytov integral was introduced by A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz., No. 2, 155 (1959).

L. A. Chernov, Wave Propagation in a Random Medium McGraw-Hill, New York, 1960), especially p. 66 and Appendix I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), especially p. 127.

Although several aspects of the Rytov approximation have undergone considerable investigation in recent years, the use the Fresnel stationary-phase approximation for its evaluation continues to be widely accepted. See, for example, V. I. Tatarski, Propagation of Waves in a Turbulent Atmosphere (Nauka, Moscow, 1967), especially pp. 283–7 (in Russian).

Yu. N. Barabanekov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarski, Usp. Fiz. Nauk 102, 3 (1970), especially p. 9.

More precisely, the Fresnel paraboloid of revolution (the region of stationary phase). See Ref. 3 and the discussion by C. Eckhart, Rev. Mod. Phys. 20, 399 (1948).

The separation of the scattered field into refractive and diffractive parts is discussed by H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 103. The distinction between coherent and incoherent scattering in random media is discussed by M. Lax, Rev. Mod. Phys. 23, 287 (1951), especially p. 290.

G. Modesitt, On the Upper and Lower Limits to the Rytov Approximation (The Rand Corporation, Santa Monica, Calif., 1969), RM-6090.

This condition is neither necessary nor sufficient for the validity of the Rytov equation; it has been derived many times with various approaches, some of which are discussed in Ref. 16.

A closely related inconsistency is that which arises from the assertion (see, for example, Ref. 12, p. 127) that, for z small compared to ka2, the Rytov equation may be further approximated and render valid the equations of linear (in n1) geometric optics. This view, when combined with the widely adopted upper limit given above (γz <ka2) implies, inconsistently, the existence when γ > 1 of an interval γ-1 <z/ka2 < 1 in which the Rytov equation is invalid and the less general geometric equation is valid.

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