Abstract

The statistical properties of a wave emitted by an incoherent source with a sharp edge embedded in a turbulent medium are examined by use of the path-integral formulation for weak- and strong-scintillation conditions. The asymptotic analysis of the problem is discussed in detail, and the energy conservation principle for instantaneous field realization is derived from the path-integral formulation. It is shown that the wave from the model infinite incoherent source with a sharp edge reveals no amplitude fluctuations but only phase fluctuations for an arbitrary turbulence level.

© 1996 Optical Society of America

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References

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  1. V. I. Tatarskii, Theory of Fluctuation Phenomena in the Propagation of Waves in a Turbulent Atmosphere (Akademizdat, Moscow, 1959) (in Russian); Wave Propagation in a Random Medium (McGraw-Hill, New York, 1961).
  2. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  3. V. U. Zavorotnyi, “Image intensity fluctuations of an incoherent source observed through a turbulent medium,” Radiophys. Quantum Electron. 28, 972–977 (1985).
    [CrossRef]
  4. V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys JETP 46, 252–260 (1977).
  5. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  6. M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of the Fifth Symposium on Laser Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, Russia, 1979), pp. 74–78 (in Russian).
  7. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
    [CrossRef]
  8. M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
    [CrossRef]
  9. V. I. Tatarskii, “Light propagation in a medium with random index refraction inhomogeneities in the Markov process approximation,” Sov. Phys. JETP 29, 1133–1138 (1969).
  10. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).
  11. V. I. Klyatskin, Stochastic Equations and Waves in Random Inhomogeneous Media (Nauka, Moscow, 1980) (in Russian); Ondes et Equations Stochastique dans les Milieux Aléatoirement non Homogenes (Editions de Physique, Besançon Cedex, France, 1985).
  12. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–266.
    [CrossRef]
  13. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  14. V. I. Klyatskin, V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828–833 (1970).
    [CrossRef]
  15. V. I. Klyatskin, V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–342 (1970).
  16. V. U. Zavorotny, “Origin of intensity fluctuations in the image of an incoherent source observed through a turbulent medium,” Opt. Spectrosc. (USSR) 65(4), 575–576 (1988).
  17. 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]
  18. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
    [CrossRef]
  19. V. I. Tatarskii, V. U. Zavorotny, “Strong fluctuations in light propagation in a randomly inhomogeneous media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 18, pp. 205–265.
    [CrossRef]
  20. I. G. Yakushkin, “Intensity fluctuations during small-angle scattering of wave field (review),” Radiophys. Quantum Electron. 28, 365–389 (1985).
    [CrossRef]
  21. R. Dashen, G. Y. Hang, “Asymptotic scheme for waves in random media,” Opt. Lett. 17, 91–93 (1992).
    [CrossRef] [PubMed]
  22. M. M. Dubovikov, “On a class of non-Gaussian path integrals,” Theor. Math. Phys. 58, 215–220 (1984).
    [CrossRef]
  23. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 504.

1994 (1)

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

1992 (1)

1991 (1)

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
[CrossRef]

1988 (1)

V. U. Zavorotny, “Origin of intensity fluctuations in the image of an incoherent source observed through a turbulent medium,” Opt. Spectrosc. (USSR) 65(4), 575–576 (1988).

1986 (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

1985 (2)

V. U. Zavorotnyi, “Image intensity fluctuations of an incoherent source observed through a turbulent medium,” Radiophys. Quantum Electron. 28, 972–977 (1985).
[CrossRef]

I. G. Yakushkin, “Intensity fluctuations during small-angle scattering of wave field (review),” Radiophys. Quantum Electron. 28, 365–389 (1985).
[CrossRef]

1984 (1)

M. M. Dubovikov, “On a class of non-Gaussian path integrals,” Theor. Math. Phys. 58, 215–220 (1984).
[CrossRef]

1983 (1)

1979 (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

1977 (1)

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys JETP 46, 252–260 (1977).

1974 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
[CrossRef]

1970 (2)

V. I. Klyatskin, V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828–833 (1970).
[CrossRef]

V. I. Klyatskin, V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–342 (1970).

1969 (1)

V. I. Tatarskii, “Light propagation in a medium with random index refraction inhomogeneities in the Markov process approximation,” Sov. Phys. JETP 29, 1133–1138 (1969).

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 504.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
[CrossRef]

Charnotskii, M. I.

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
[CrossRef]

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of the Fifth Symposium on Laser Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, Russia, 1979), pp. 74–78 (in Russian).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–266.
[CrossRef]

Clifford, S. F.

Codona, J. L.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

Dashen, R.

R. Dashen, G. Y. Hang, “Asymptotic scheme for waves in random media,” Opt. Lett. 17, 91–93 (1992).
[CrossRef] [PubMed]

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

Dubovikov, M. M.

M. M. Dubovikov, “On a class of non-Gaussian path integrals,” Theor. Math. Phys. 58, 215–220 (1984).
[CrossRef]

Flatté, S. M.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

Frehlich, R. G.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

Gochelashvili, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
[CrossRef]

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–266.
[CrossRef]

Hang, G. Y.

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

Hill, R. J.

Klyatskin, V. I.

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys JETP 46, 252–260 (1977).

V. I. Klyatskin, V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828–833 (1970).
[CrossRef]

V. I. Klyatskin, V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–342 (1970).

V. I. Klyatskin, Stochastic Equations and Waves in Random Inhomogeneous Media (Nauka, Moscow, 1980) (in Russian); Ondes et Equations Stochastique dans les Milieux Aléatoirement non Homogenes (Editions de Physique, Besançon Cedex, France, 1985).

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
[CrossRef]

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 504.

Sung, C. C.

Tatarskii, V. I.

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys JETP 46, 252–260 (1977).

V. I. Klyatskin, V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–342 (1970).

V. I. Klyatskin, V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828–833 (1970).
[CrossRef]

V. I. Tatarskii, “Light propagation in a medium with random index refraction inhomogeneities in the Markov process approximation,” Sov. Phys. JETP 29, 1133–1138 (1969).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

V. I. Tatarskii, Theory of Fluctuation Phenomena in the Propagation of Waves in a Turbulent Atmosphere (Akademizdat, Moscow, 1959) (in Russian); Wave Propagation in a Random Medium (McGraw-Hill, New York, 1961).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–266.
[CrossRef]

V. I. Tatarskii, V. U. Zavorotny, “Strong fluctuations in light propagation in a randomly inhomogeneous media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 18, pp. 205–265.
[CrossRef]

Yakushkin, I. G.

I. G. Yakushkin, “Intensity fluctuations during small-angle scattering of wave field (review),” Radiophys. Quantum Electron. 28, 365–389 (1985).
[CrossRef]

Yura, H. T.

Zavorotny, V. U.

V. U. Zavorotny, “Origin of intensity fluctuations in the image of an incoherent source observed through a turbulent medium,” Opt. Spectrosc. (USSR) 65(4), 575–576 (1988).

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys JETP 46, 252–260 (1977).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–266.
[CrossRef]

V. I. Tatarskii, V. U. Zavorotny, “Strong fluctuations in light propagation in a randomly inhomogeneous media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 18, pp. 205–265.
[CrossRef]

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Image intensity fluctuations of an incoherent source observed through a turbulent medium,” Radiophys. Quantum Electron. 28, 972–977 (1985).
[CrossRef]

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

V. U. Zavorotny, “Origin of intensity fluctuations in the image of an incoherent source observed through a turbulent medium,” Opt. Spectrosc. (USSR) 65(4), 575–576 (1988).

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1974).
[CrossRef]

Radio Sci. (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 29–48 (1986).
[CrossRef]

Radiophys. Quantum Electron. (3)

V. U. Zavorotnyi, “Image intensity fluctuations of an incoherent source observed through a turbulent medium,” Radiophys. Quantum Electron. 28, 972–977 (1985).
[CrossRef]

V. I. Klyatskin, V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828–833 (1970).
[CrossRef]

I. G. Yakushkin, “Intensity fluctuations during small-angle scattering of wave field (review),” Radiophys. Quantum Electron. 28, 365–389 (1985).
[CrossRef]

Sov. Phys JETP (1)

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys JETP 46, 252–260 (1977).

Sov. Phys. JETP (2)

V. I. Klyatskin, V. I. Tatarskii, “The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities,” Sov. Phys. JETP 31, 335–342 (1970).

V. I. Tatarskii, “Light propagation in a medium with random index refraction inhomogeneities in the Markov process approximation,” Sov. Phys. JETP 29, 1133–1138 (1969).

Theor. Math. Phys. (1)

M. M. Dubovikov, “On a class of non-Gaussian path integrals,” Theor. Math. Phys. 58, 215–220 (1984).
[CrossRef]

Waves Random Media (2)

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991).
[CrossRef]

Other (8)

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of the Fifth Symposium on Laser Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, Russia, 1979), pp. 74–78 (in Russian).

V. I. Tatarskii, Theory of Fluctuation Phenomena in the Propagation of Waves in a Turbulent Atmosphere (Akademizdat, Moscow, 1959) (in Russian); Wave Propagation in a Random Medium (McGraw-Hill, New York, 1961).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

V. I. Klyatskin, Stochastic Equations and Waves in Random Inhomogeneous Media (Nauka, Moscow, 1980) (in Russian); Ondes et Equations Stochastique dans les Milieux Aléatoirement non Homogenes (Editions de Physique, Besançon Cedex, France, 1985).

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–266.
[CrossRef]

V. I. Tatarskii, V. U. Zavorotny, “Strong fluctuations in light propagation in a randomly inhomogeneous media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 18, pp. 205–265.
[CrossRef]

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 504.

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Figures (3)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Average-intensity profile for a Gaussian aperture. 1, free space; 2, D(2a) = 4; 3, D(2a) = 10; 4, D(2a) = 20; 5, D(2a) = 40.

Fig. 3
Fig. 3

Average-intensity profile for a circular aperture. 1, free space; 2, D(2a) = 4; 3, D(2a) = 10; 4, D(2a) = 20; 5, D(2a) = 40.

Equations (70)

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U ( x , r ) = d 2 r U ( x , r ) G ( x , r ; x , r ) .
I IM ( R 1 ) = d 2 R 0 O ( R 0 ) d 2 R d 2 r G ( L , R 0 ; 0 , R + r 2 ) × G * ( L , R 0 ; 0 , R - r 2 ) exp [ - i k F ( R · r ) ] × A ( R + r 2 ) A ( R - r 2 ) G 0 ( 0 , R + r 2 ; - 1 , R 1 ) × G 0 * ( 0 , R - r 2 ; - 1 , R 1 ) ,
G 0 ( x , r ; x , r ) = - i k 2 π x - x exp ( i k 2 x - x r - r 2 ) .
G ( L , r 0 ; 0 , r ) = G 0 ( L , r 0 ; 0 , r ) g ( L , r 0 , r ) ,
I IM ( R 1 ) = k 2 4 π 2 L 2 k 2 4 π 2 l 2 d 2 R 0 O ( R 0 ) × d 2 R d 2 r A ( R + r 2 ) A ( R - r 2 ) × exp [ - i k L ( R 0 · r ) - i k l ( R 1 · r ) ] × g ( L , R 0 , R + r 2 ) g * ( L , R 0 , R - r 2 ) .
g ( L , r 0 , r ) = 2 i π L k D 2 v ( ξ ) exp [ i k 2 0 L v 2 ( ξ ) d ξ ] × δ [ 0 L v ( ξ ) d ξ ] exp { i k 2 0 L [ x , r 0 x L + r ( 1 - x L ) + x L v ( ξ ) d ξ ] d x } ,
r ( x ) = x L v ( ξ ) d ξ ,
D 2 v ( ξ ) exp [ i k 2 0 L v 2 ( ξ ) d ξ ] = 1.
g ( L , r 0 , r ) g * ( L , r 0 , r ) = exp { - π k 2 4 0 L d x H [ x , ( r 0 - r 0 ) x L + ( r - r ) ( 1 - x L ) ] } .
H ( x , r ) = 2 d 2 p Φ ( x , p ) [ 1 - cos ( p · r ) ] ,
Φ ( x , p ) = 0.033 C 2 ( x ) p - 11 / 3 ,
H ( x , r ) = 0.46 C 2 ( x ) r 5 / 3 .
I I M 2 ( R 1 ) = ( k 2 4 π 2 l 2 ) 2 k 2 4 π 2 L 2 d 2 R 0 d 2 r 0 O ( R 0 + r 0 2 ) × O ( R 0 + r 0 2 ) d 2 R d 2 r 1 d 2 r 2 d 2 p A ( R , r 1 , r 2 , p ) × exp [ - i k L ( R 0 · p + r 0 · r 2 ) - i k l ( R 1 · p ) ] × D 2 v 1 ( ξ ) D 2 v 2 ( ξ ) exp [ i k 2 0 L v 1 ( ξ ) v 2 ( ξ ) d ξ ] × δ [ 0 L v 1 ( ξ ) d ξ ] δ [ 0 L v 2 ( ξ ) d ξ ] exp { - Ψ [ r 0 x L + r 1 ( 1 - x L ) + x L v 1 ( ξ ) d ξ , r 2 ( 1 - x L ) + x L v 2 ( ξ ) d ξ , p ( 1 - x L ) ] } ,
A ( R , r 1 , r 2 , p ) = A ( R + r 1 2 + r 2 2 + p 4 ) A ( R + r 1 2 - r 2 2 - p 4 ) × A ( R - r 1 2 + r 2 2 - p 4 ) A ( R - r 1 2 - r 2 2 + p 4 )
Ψ [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] = π k 2 4 0 L d x { H [ x , S 1 ( x ) + s ( x ) 2 ] + H [ x , S 1 ( x ) - s ( x ) 2 ] + H [ x , S 2 ( x ) + s ( x ) 2 ] + H [ x , S 2 ( x ) - s ( x ) 2 ] - H [ x , S 1 ( x ) + S 2 ( x ) ] - H [ x , S 1 ( x ) - S 2 ( x ) ] } .
P ( x ) = d 2 r I ( x , r ) = d 2 r 0 I ( 0 , r 0 ) = P ( 0 ) ,
I IM ( R 1 ) = d 2 R 0 O ( R 0 ) d 2 R d 2 r G ( L , R 0 ; 0 , R + r 2 ) × G * ( L , R 0 ; 0 , R - r 2 ) U B ( R + r 2 , R 1 ) × U B * ( R - r 2 , R 1 ) ,
U B ( R ; R 1 ) = - i k 2 π l A ( R ) exp [ - i k 2 L ( R - R 1 ) 2 ] ,
G ( x , r ; x , r ) = G ( x , r ; x , r ) ,
I IM ( R 1 ) = d 2 R 0 O ( R 0 ) I B ( L , R 0 ; R 1 ) .
exp { - Ψ [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] } 1 - Ψ + Ψ 2 / 2 - .
I IM 2 ( R 1 ) = W 0 + W 1 + W 2 + .
D ( r ) = π k 2 2 0 L d x H [ x , r ( 1 - x L ) ] .
ρ 0 = [ 0.364 k 2 0 L d x C 2 ( x ) ( 1 - x L ) 5 / 3 ] - 3 / 5 .
exp { - Ψ [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] } = exp ( - π k 2 4 0 L d x { H [ x , S 1 ( x ) + s ( x ) 2 ] + H [ x , S 1 ( x ) - s ( x ) 2 ] } ) { 1 - Q [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] + 1 2 ! Q 2 [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] - } ,
Q [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] = π k 2 4 0 L d x { H [ x , S 2 ( x ) + s ( x ) 2 ] + H [ x , S 2 ( x ) - s ( x ) 2 ] - H [ x , S 1 ( x ) + S 2 ( x ) ] - H [ x , S 1 ( x ) + S 2 ( x ) ] } .
exp { - Ψ [ S 1 ( x ) , S 2 ( x ) , s ( x ) ] } = exp ( - π k 2 4 0 L d x { H [ x , S 2 ( x ) + s ( x ) 2 ] + H [ x , S 2 ( x ) - s ( x ) 2 ] } ) { 1 - Q [ S 2 ( x ) , S 1 ( x ) , s ( x ) ] + 1 2 ! Q 2 [ S 2 ( x ) , S 1 ( x ) , s ( x ) ] - } ,
I IM 2 ( R 1 ) = A 0 + A 1 + + M 0 + M 1 + ,
C 2 k 7 / 6 L 11 / 6 1.
M 0 = I IM ( R 1 ) 2 .
O ( R 0 ) = ( y 0 , z 0 ) = O 0 [ 1 + ϑ sign ( y 0 ) ] .
ϑ = O MAX - O MIN O MAX + O MIN
O ( R 0 ) = O ( y 0 , z 0 ) = O 0 exp ( y 0 2 + z 0 2 b 2 ) [ 1 + ϑ sign ( y 0 ) ] ,
Γ AP ( R , r ) = U ( 0 , R + r 2 ) U * ( 0 , R - r 2 ) = k 2 4 π 2 L 2 d 2 R 0 O ( R 0 ) × exp [ i k L ( R - R 0 ) · r - D ( r ) 2 ] .
Γ SP ( L , R , r ) = exp ( i k L R · r )
γ ( L , r ) = k 2 4 π 2 L 2 d 2 R 0 O ( R 0 ) exp [ - i k L R 0 · r - D ( r ) 2 ]
γ ( L , r ) = γ ( L , y , z ) = O 0 δ ( y ) δ ( z ) + O 0 ϑ 1 i π y δ ( z ) exp [ - D ( y , 0 ) / 2 ) ] .
I IM ( R 1 ) = k 2 4 π 2 L 2 k 2 4 π 2 l 2 d 2 R 0 O ( R 0 ) d 2 r K A ( r ) × exp [ - i k L ( R 0 · r ) - i k l ( R 1 · r ) ] × exp ( - D ( r ) 2 ) ,
K A ( r ) = d 2 R A ( R + r 2 ) A ( R - r 2 )
O ^ ( q ) = d 2 R 0 O ( R 0 ) exp ( i q · R 0 ) , I ^ IM ( q ) = d 2 R 1 I IM ( R 1 ) exp ( i q · R 1 ) ,
I ^ IM ( q ) = k 2 4 π 2 L 2 O ^ ( - q l L ) K A ( q l k ) × exp [ - 1 2 D ( q l k ) ] .
I IM ( y 1 , z 1 ) = O 0 k 2 4 π 2 l 2 Σ - O 0 ϑ k 2 2 π 3 l 2 0 d y K A ( y , 0 ) × exp [ - D ( y , 0 ) 2 ] 1 y sin ( k l y 1 y ) .
lim y 1 + I IM ( y 1 ) = O 0 k 2 4 π 2 l 2 Σ ( 1 - ϑ ) = I - ,
lim y 1 - I IM ( y 1 ) = O 0 k 2 4 π 2 l 2 Σ ( 1 + ϑ ) = I + .
J ( u ) = I IM ( u l ) I IM ( 0 ) = 1 - ϑ 2 π 0 d t exp ( - t 2 ) × exp [ - ( 2 a t ρ 0 ) 5 / 3 ] 1 t sin ( u u d t ) ,
J ( u ) = 1 - ϑ 4 π 2 0 1 d t [ arccos ( t ) - t 1 - t 2 ] × exp [ - ( 2 a t ρ 0 ) 5 / 3 ] 1 t sin ( u u d t ) .
I AP 2 ( R ) = ( k 2 4 π 2 L 2 ) d 2 R 0 d 2 r 0 O ( R 0 + r 0 2 ) O ( R 0 - r 0 2 ) × D 2 v 1 ( ξ ) D 2 v 2 ( ξ ) exp [ i k 2 0 L v 1 ( ξ ) · v 2 ( ξ ) d ξ ] × δ [ 0 L v 1 ( ξ ) d ξ ] δ [ 0 L v 2 ( ξ ) d ξ ] × exp { - Ψ [ r 0 x L + x L v 1 ( ξ ) d ξ , x L v 2 ( ξ ) d ξ , 0 ] } .
M 0 = [ k 2 4 π 2 L 2 d 2 R 0 O ( R 0 ) ] 2 = L AP ( R ) 2 ,
M 1 = 0 L d x d 2 R 0 d 2 r 0 O ( R 0 + r 0 2 ) O ( R 0 - r 0 2 ) × d 2 p Φ ( x , p ) { 1 - cos [ p 2 x k ( 1 - x L ) ] } × exp ( i p · r 0 x L ) exp { - π k 2 2 0 L d ξ H [ ξ , r ^ ( x , ξ ) ] } .
r ^ ( x , ξ ) = - p k min ( x , ξ ) [ 1 - max ( x , ξ ) L ] .
σ AP 2 = I AP 2 ( R ) - I AP ( R ) 2 I AP ( R ) 2 ,
M 0 = ( O 0 2 k 2 b 2 / 4 π L 2 ) 2 ,
M 1 M 0 = 2 π k 2 0 L d x d μ d ν Φ ( x , p ) × sin 2 [ p 2 x 2 k ( 1 - x L ) ] exp ( - x 2 p 2 b 2 2 L 2 ) × [ 1 + ϑ 2 x 2 μ 2 b 2 π L 2 F 1 1 ( 1 2 , 3 2 , x 2 μ 2 b 2 2 L 2 ) ] × exp { - π k 2 2 0 L d ξ H [ ξ , r ^ ( x , ξ ) ] } ,
σ AP 2 = 0.033 π b - 7 / 3 L 0 L C 2 ( L t ) ( 1 - t ) 2 t - 1 / 3 d t × [ π Γ ( 7 / 6 ) 2 1 / 6 + c 0 ϑ 2 ] ,
c 2 = π - 1 d x d y x 2 ( x 2 + y 2 ) 1 / 6 × exp ( - y 2 2 ) F 1 1 2 ( 1 , 3 2 , - x 2 4 ) .
P ( L ) = d 2 R I B ( L , R ) = d 2 R d 2 R 0 d 2 r 0 Γ B ( 0 , R 0 , r 0 ) × exp [ i k L ( R 0 - R ) · r 0 ] D 2 V ( ξ ) D 2 v ( ξ ) × exp [ i k 2 0 L V ( ξ ) · v ( ξ ) d ξ ] δ [ 0 L V ( ξ ) d ξ ] × δ [ 0 L v ( ξ ) d ξ ] exp [ i k 0 L ( { x , ( R 0 + r 0 2 ) × ( 1 - x L ) + R x L + x L [ V ( ξ ) + v ( ξ ) 2 ] d ξ } - { x , ( R 0 - r 0 2 ) ( 1 - x L ) + R x L + x L [ V ( ξ ) - v ( ξ ) 2 ] d ξ } ) ] .
P ( L ) = P 0 + P 1 + P 2 + .
( x , r ) = d 2 p ^ ( x , p ) exp ( i pr )
P 1 = - k 0 L d x d 2 R d 2 R 0 d 2 r 0 d 2 p Γ B ( 0 , R 0 , r 0 ) × ^ ( x , p ) D 2 V ( ξ ) D 2 v ( ξ ) exp [ i k L ( R 0 - R ) · r 0 + i p · R x L + i p · R 0 ( 1 - x L ) ] × sin { p 2 · [ r 0 ( 1 - x L ) + x L v ( ξ ) d ξ ] } × exp [ i k 0 L V ( ξ ) · v ( ξ ) d ξ + i p · x L V ( ξ ) d ξ ] × δ [ 0 L V ( ξ ) d ξ ] δ [ 0 L v ( ξ ) d ξ ] .
d 2 R exp ( - i k L R · r 0 + i p · R x L ) = 4 π 2 L 2 k 2 δ ( r 0 - x k p ) .
1 4 π 2 d 2 n D 2 V ( ξ ) exp { i k 0 L V ( ξ ) · [ v ( ξ ) + p k θ ( ξ - x ) + n k ] d ξ } .
D 2 V ( ξ ) exp [ i k 0 L V ( ξ ) · W ( ξ ) d ξ ] = δ [ W ( ξ ) ] .
D 2 V ( ξ ) δ [ V ( ξ ) - W ( ξ ) ] F [ V ( ξ ) ] = F [ W ( ξ ) ] .
δ [ v ( ξ ) + p k θ ( ξ - x ) - p k ( 1 - x L ) ] .
P n = ( - k ) n 0 L d x 1 0 x 1 d x 2 0 x n - 1 d x n d 2 p 1 d 2 p n × ^ ( x 1 , p 1 ) ^ ( x n , p n ) d 2 R d 2 R 0 d 2 r 0 Γ B ( 0 , R 0 , r 0 ) × ^ ( x , p ) D 2 V ( ξ ) D 2 v ( ξ ) exp { i k L ( R 0 - R ) · r 0 + i R · [ p 1 x 1 L + + p n x n L ] + i R 0 · [ p 1 ( 1 - x 1 L ) + + p n ( 1 - x n L ) ] } × sin { p 1 2 · [ r 0 ( 1 - x 1 L ) + x 1 L v ( ξ ) d ξ ] } × sin { p n 2 · [ r 0 ( 1 - x n L ) + x n L v ( ξ ) d ξ ] } × exp [ i k 0 L V ( ξ ) · v ( ξ ) d ξ + i p · x L V ( ξ ) d ξ ] × δ [ 0 L V ( ξ ) d ξ ] δ [ 0 L v ( ξ ) d ξ ] .
4 π 2 L 2 k 2 δ ( r 0 - L k [ p 1 x 1 L + + p n x n L ] ) .
δ { v ( ξ ) + [ p 1 k θ ( ξ - x 1 ) + + p n k θ ( ξ - x n ) ] - 1 k [ p 1 ( 1 - x 1 L ) + + p n ( 1 - x n L ) ] } .
sin ( p i 2 k · { p 1 [ max ( x 1 , x i ) - x i ] + + p n [ max ( x n , x i ) - x i ] } ) .
P 2 ( L ) = d 2 R d 2 R I B ( L , R ) I B ( L , R ) = P 0 2 .
M 0 = P 0 2 ; M n = 0             for n > 0.

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