Abstract

A source having a deterministic beam-wave amplitude distribution with a spatially random phase variation is assumed. The source distibution simulates laser reflectance from (or transmission through) a rough surface with arbitrary height deviation and a correlation yielding a Gaussian intensity covariance. The intensity covariance resulting from the assumed source propagating through atmospheric turbulence is calculated using a formalism developed previously. The resultant eight-fold intergral is evaluated in closed form retaining all phase, log-amplitude, and cross phaselog-amplitude structure functions by employing the quadratic approximation for the complex phase. Limiting case conditions of (i) a field from a partially coherent source propagating in vacuo (speckle) and (ii) a coherent beam-wave propagating through turbulence are examined. Speckle contrast calculations replicate published data using less restrictive assumptions than formerly employed, while turbulent atmosphere beam-wave calculations appear more physically reasonable than results of Ishimaru. General-case calculations show that the normalized intensity variance (contrast or fluctuation parameter) increases less rapidly with increasing turbulence as the phase variance of the source increases. A saturation phenomenon is observed at high turbulence levels as the coherence decreases. The inability to sustain high spatial frequency speckle in turbulence is reflected in the calculated intensity covariance function.

© 1979 Optical Society of America

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References

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  1. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” IEEE Proc. 63, 1669–1692 (1975).
    [CrossRef]
  2. A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
    [CrossRef]
  3. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirchhoff method in problems of laser-beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172–174 (1977).
    [CrossRef] [PubMed]
  4. J. C. Leader, “Spatial coherence measurements of Rayleigh-scattered light,” J. Opt. Soc. Am. 65, 740–741 (1975).
    [CrossRef]
  5. J. C. Leader, “The generalized partial coherence of a radiation source and its far-field,” Optica Acta 25, 395–413 (1978).
    [CrossRef]
  6. J. C. Leader, “An analysis of the spatial coherence of laser light scattered from a surface with two scales of roughness,” J. Opt. Soc. Am. 66, 536–546 (1976).
    [CrossRef]
  7. J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–185 (1978).
    [CrossRef]
  8. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.
  9. J. C. Leader, “An analysis of the frequency spectrum of laser light scattered from moving rough objects,” J. Opt. Soc. Am. 67, 1091–1098 (1977).
    [CrossRef]
  10. A. Sommerfeld, “Optics”, Lectures on Theoretical Physics (Academic, New York, 1954) Vol. IV, pp 197–201.
  11. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp 311–319.
  12. Z. I. Feizulin and Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).
  13. R. L. Fante, “Two-source spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 66, 74 (1976).
    [CrossRef]
  14. A. V. Artem’ev and A. S. Gurvich, “Experimental study of coherence function spectra,” Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
    [CrossRef]
  15. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [CrossRef]
  16. J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
    [CrossRef] [PubMed]
  17. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).
  18. The unit vectors are evaluated using the formulas in Appendix C of Ref. 7.
  19. J. C. Dainty, Laser Speckle (Springer-Verlag, New York, 1975).
  20. H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Comm. 16, 68–72 (1976).
    [CrossRef]
  21. The phase deviation for transmission experiments is one half the value predicted by Eq. (21). The experiments reported in Ref. 16 were transmission experiments.
  22. A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
    [CrossRef]
  23. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.
  24. Ibid., pp 504–535.
  25. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” IEEE Proc. 58, 1523–1545 (1970).
    [CrossRef]
  26. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR.Otdelenie Okeanologii, Fiziki, Atmosfery;Geografii. (Preprint, Moscow, (1973).Aerospace Corp., translation No. LRG-73-T-28).
  27. Note added in review.
  28. P. A. Pincus, M. E. Fossey, J. F. Holmes, and J. R. Kerr, “Speckle propagation through turbulence: Experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
    [CrossRef]
  29. A phase deviation of σϕ = 5.0 and coherence length of ρu = 5.0 μm was assumed commensurate with the diffuse nature of the scattering target (Scotchlite).
  30. M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]

1978 (3)

1977 (3)

1976 (5)

1975 (3)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” IEEE Proc. 63, 1669–1692 (1975).
[CrossRef]

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

J. C. Leader, “Spatial coherence measurements of Rayleigh-scattered light,” J. Opt. Soc. Am. 65, 740–741 (1975).
[CrossRef]

1971 (1)

A. V. Artem’ev and A. S. Gurvich, “Experimental study of coherence function spectra,” Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
[CrossRef]

1970 (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” IEEE Proc. 58, 1523–1545 (1970).
[CrossRef]

1969 (1)

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

1967 (1)

Z. I. Feizulin and Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.

Artem’ev, A. V.

A. V. Artem’ev and A. S. Gurvich, “Experimental study of coherence function spectra,” Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
[CrossRef]

Asakura, T.

J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Comm. 16, 68–72 (1976).
[CrossRef]

Banakh, V. A.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.

Dainty, J. C.

J. C. Dainty, Laser Speckle (Springer-Verlag, New York, 1975).

Fante, R. L.

R. L. Fante, “Two-source spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 66, 74 (1976).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” IEEE Proc. 63, 1669–1692 (1975).
[CrossRef]

Feizulin, Z. I.

Z. I. Feizulin and Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Fossey, M. E.

Fujii, H.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Comm. 16, 68–72 (1976).
[CrossRef]

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR.Otdelenie Okeanologii, Fiziki, Atmosfery;Geografii. (Preprint, Moscow, (1973).Aerospace Corp., translation No. LRG-73-T-28).

Gurvich, A. S.

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

A. V. Artem’ev and A. S. Gurvich, “Experimental study of coherence function spectra,” Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
[CrossRef]

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR.Otdelenie Okeanologii, Fiziki, Atmosfery;Geografii. (Preprint, Moscow, (1973).Aerospace Corp., translation No. LRG-73-T-28).

Holmes, J. F.

Ishimaru, A.

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR.Otdelenie Okeanologii, Fiziki, Atmosfery;Geografii. (Preprint, Moscow, (1973).Aerospace Corp., translation No. LRG-73-T-28).

Kerr, J. R.

Kravtsov, Yu. A.

Z. I. Feizulin and Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” IEEE Proc. 58, 1523–1545 (1970).
[CrossRef]

Leader, J. C.

Lee, M. H.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp 311–319.

Mironov, V. L.

Ohtsubo, J.

Pakasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR.Otdelenie Okeanologii, Fiziki, Atmosfery;Geografii. (Preprint, Moscow, (1973).Aerospace Corp., translation No. LRG-73-T-28).

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.

Pedersen, H. M.

Pincus, P. A.

Shindo, Y.

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Comm. 16, 68–72 (1976).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Optics”, Lectures on Theoretical Physics (Academic, New York, 1954) Vol. IV, pp 197–201.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” IEEE Proc. 58, 1523–1545 (1970).
[CrossRef]

Tatarskii, V. I.

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

Appl. Opt. (1)

IEEE Proc. (2)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” IEEE Proc. 63, 1669–1692 (1975).
[CrossRef]

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” IEEE Proc. 58, 1523–1545 (1970).
[CrossRef]

J. Opt. Soc. Am. (8)

Opt. Comm. (1)

H. Fujii, T. Asakura, and Y. Shindo, “Measurements of surface roughness properties by means of laser speckle techniques,” Opt. Comm. 16, 68–72 (1976).
[CrossRef]

Opt. Lett. (1)

Optica Acta (1)

J. C. Leader, “The generalized partial coherence of a radiation source and its far-field,” Optica Acta 25, 395–413 (1978).
[CrossRef]

Radio Sci. (2)

A. S. Gurvich and V. I. Tatarskii, “Coherence and intensity fluctuations of light in the turbulent atmosphere,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Radiophys. and Quantum Electron. (USSR) (1)

Z. I. Feizulin and Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. and Quantum Electron. (USSR) 10, 68–73 (1967).

Radiophys. Quantum Electron. (USSR) (1)

A. V. Artem’ev and A. S. Gurvich, “Experimental study of coherence function spectra,” Radiophys. Quantum Electron. (USSR) 14, 580–583 (1971).
[CrossRef]

Other (12)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, New York, 1963).

The unit vectors are evaluated using the formulas in Appendix C of Ref. 7.

J. C. Dainty, Laser Speckle (Springer-Verlag, New York, 1975).

A. Sommerfeld, “Optics”, Lectures on Theoretical Physics (Academic, New York, 1954) Vol. IV, pp 197–201.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), pp 311–319.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ, 1964), pp 7–11, 27–35.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1968), pp 556–566.

Ibid., pp 504–535.

The phase deviation for transmission experiments is one half the value predicted by Eq. (21). The experiments reported in Ref. 16 were transmission experiments.

A phase deviation of σϕ = 5.0 and coherence length of ρu = 5.0 μm was assumed commensurate with the diffuse nature of the scattering target (Scotchlite).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pakasov, “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR.Otdelenie Okeanologii, Fiziki, Atmosfery;Geografii. (Preprint, Moscow, (1973).Aerospace Corp., translation No. LRG-73-T-28).

Note added in review.

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Figures (11)

FIG. 1
FIG. 1

Propagation geometry.

FIG. 2
FIG. 2

Comparison of theoretical speckle-contrast calculations (solid line) with measurements of Fujii et al.20 Measurements were performed with a helium-neon laser (λ = 0.6328 μm). Calculations assumed a spot radius of L = 15 μm (as indicated by ancillary experiments) and reported roughness statistics.

FIG. 3
FIG. 3

Comparison of theoretical speckle-contrast calculations with measurements of Ohtsubo and Asakura.16 Measurements were performed with a helium–neon laser (λ = 0.6328 μm) on a sample yielding a phase deviation of σϕ = 1.86 with a reported correlation length of ρu = 6.0

FIG. 4
FIG. 4

Comparison of theoretical speckle-contrast calculations with measurements of Ohtsubo and Asakura.16 Measurements were performed with a helium-neon laser (λ = 0.6328 μm) on a sample with a spot radius of 142 μm.

FIG. 5
FIG. 5

Comparison of extended theory fluctuation parameter predictions with results of Ishimaru22 [Eq. (44)] for axial locations on a beam with L = 0.5 cm, λ = 0.5 μm propagating in moderately strong turbulence C n 2 = 10 15 m 2 / 3. Rytov-theory plane wave and spherical wave results are also shown.

FIG. 6
FIG. 6

Comparison of extended theory fluctuation parameter predictions with results of Ishimaru22 [Eq. (44)] for axial locations on a beam with L = 2.5 cm, λ = 1.0 μm propagating in moderately strong turbulence C n 2 = 10 15 m 2 / 3. Rytov-theory plane wave and spherical wave results are also shown.

FIG. 7
FIG. 7

Comparison of extended theory fluctuation parameter predictions with results of Ishimaru22 [Eq. (44)] for radial displacements on a beam with λ = 0.5 μm, L = 0.5 cm propagating 1 km through turbulence with a strength C n 2 = 10 15 m 2 / 3.

FIG. 8
FIG. 8

Comparison of fluctuation parameter predictions of the extended theory and Ishimaru theory [Eq. (44)] with experimental data of Gracheva et al.26 A helium-neon laser (λ = 0.6328) with an effective collimated aperture of L = 15 cm was used in propagation experiments at ranges of 250 and 1750 m.

FIG. 9
FIG. 9

Calculated fluctuation parameter variation as a function of turbulence strength, C n 2, and source-phase deviation, σϕ, for a source radius of (a) L = 2.5 cm and (b) L = 5.0 cm. Calculations assume a fixed range (R+ = 3 km) wavelength (λ = 1.0 μm) and correlation length (ρu = 5 μm) for axial propagation (θ = 0°).

FIG. 10
FIG. 10

Normalized covariance functional forms [Eq. (40)] calculated for a range of phase deviations σϕ for a source radius of (a) L = 2.5 cm and (b) L = 5.0 cm. Calculations assume a fixed range (R+ = 3 km) wavelength (λ = 1.0 μm), turbulence strength ( C n 2 = 5.0 × 10 15 m 2 / 3 ) and correlation length (ρu = 5 μm) for axial propagation (θ = 0°).

FIG. 11
FIG. 11

Calculated values of the normalized covariance [Eq. (40)] for a fixed separation distance (δR+ = 5 cm) as a function of the turbulence strength, C n 2, and source phase deviation σϕ. Calculations for two source dimensions (a) L = 2.5 cm and (b) L = 5.0 cm are shown for a fixed range (R+ = 3 km) wavelength (λ = 1.0 μm) and correlation length (ρu = 5 μm) for axial propagation (θ = 0°).

Equations (94)

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U turb ( P ) cos θ i k 2 π R d 2 r U ( r ) e ik | r P | + ψ ( r , P ) ,
I turb ( P ) I turb ( P ) = U turb ( P ) U * turb ( P ) U turb ( P ) U * turb ( P ) = k 4 cos 2 θ cos 2 θ ( 2 π ) 4 ( R R ) 2 d 2 r 1 d 2 r 2 d 2 r 3 d 2 r 4 × Γ s ( r 1 , r 2 , r 3 , r 4 ) × exp [ ψ ( r 1 , P ) + ψ + ( r 2 , P ) + ψ ( r 3 , P ) + ψ * ( r 4 , P ) ] × exp { ik [ | r 1 P | | r 2 P | + | r 3 P | | r 4 P | ] } ,
Γ s ( r 1 , r 2 , r 3 , r 4 ) = U ( r 1 ) U * ( r 2 ) U ( r 3 ) U * ( r 4 )
exp { ik [ ] } exp { i k [ r 12 · r 12 + R r 12 · s + r 34 · r 34 + R r 34 · ŝ ] } , = exp { i k [ r 12 · r 12 + R + r 34 · r 34 + R ( r 12 + r 34 ) · ŝ + ( r 12 + r 34 ) · ŝ 2 ] }
r n m + = ( r m + r n ) / 2 , r n m = r m r n ,
ŝ + = ( ŝ + ŝ ) / 2 , ŝ = ŝ ŝ ,
ŝ = P / | P | and ŝ = P / | P |
H ( r 1 , r 2 , r 3 , r 4 , P , P ) = exp [ ψ ( r 1 , P ) + ψ * ( r 2 , P ) + ψ ( r 3 , P ) + ψ * ( r 4 , P ) ] = exp { 1 2 D 1 ( r 12 , 0 ) 1 2 D 1 ( r 14 , P ) 1 2 D 1 ( r 23 , P ) 1 2 D 1 ( r 34 , 0 ) + 1 2 D 1 ( r 24 , P ) + 1 2 D 1 ( r 13 , P ) + 2 B χ ( r 24 , P ) + 2 B χ ( r 13 , P ) + i D χ S ( r 24 , P ) i D χ S ( r 13 , P ) } ,
D 1 ( r mn , P ) = D χ ( r m n , P ) + D S ( r m n , P ) | ψ ( r m , P ) ψ ( r n , P ) | 2 ,
D χ S ( r m n , P ) | χ ( r m , P ) χ ( r n , P ) | | S ( r m , P ) S ( r n , P ) | ,
B χ ( r m n , P ) χ ( r m , P ) χ ( r n , P ) χ 2 = σ χ 2 1 2 D χ ( r n m , P ) ,
D χ ( r m n , P ) D S ( σ x y , P ) } = 1.455 k 2 R + C n 2 × 0 1 d t | t p ( P ) + ( 1 t ) p ( r m n ) | 5 / 3 0.875 k 13 / 6 ( R + ) 5 / 6 C n 2 × 0 1 d t [ t ( 1 t ) ] 1 / 6 | t p ( P ) + ( 1 t ) p ( r m n ) | 2 ,
D χ S ( r m n , P ) = 0.234 k 13 / 6 ( R + ) 5 / 6 C n 2 × 0 1 d t [ t ( 1 t ) ] 1 / 6 | t p ( P + ( 1 t ) p ( r m n ) | 2 ,
R + = ( R + R + ) / 2 , R = R R , P = P P ,
D 1 ( r m n , P ) 6 ρ 1 2 0 1 d t | t p ( P + ( 1 t ) p ( r m n | 2 ,
ρ 1 = ( 3 / 8 × 1.455 k 2 R + C n 2 ) 3 / 5 ,
D 1 ( r m n , P ) 2 ρ 1 2 [ ( x m 2 + x n 2 ) + ( y m 2 + y n 2 ) cos 2 θ + + x m ( R + β x n ) x n ( R + β + x m ) y m cos θ + ( R + δ + y n cos θ + ) + y n cos θ + ( R + δ y m cos θ + ) + ( R + ) 2 ( β 2 + δ 2 ) ]
β = ω sin θ + , δ = θ θ ,
θ + = ( θ + θ ) / 2 ,
U S ( r ) = U 0 e i ϕ ( r ) | r | 2 / L 2 ,
ϕ 2 ( r ) = σ ϕ 2 ,
σ ϕ = 2 k σ h cos θ ,
C I + ( r 1 , r 2 ) = U ( r 1 ) U ( r 2 ) U ( r 1 ) U ( r 2 )
C I ( r 1 , r 2 ) = U ( r 1 ) U * ( r 2 ) U ( r 1 ) U * ( r 2 ) ,
C I + ( r 1 , r 2 ) C I + ( | r 12 | = 0 ) e | r 12 | 2 / ρ u 2 C I ( r 1 , r 2 ) C I ( | r 12 | = 0 ) e | r 12 | 2 / ρ u 2 ,
Γ s ( r 1 , r 2 , r 3 , r 4 ) = U ( r 1 ) U * ( r 2 ) U ( r 3 ) U * ( r 4 ) + U ( r 1 ) U * ( r 4 ) U * ( r 2 ) U ( r 3 ) + U ( r 1 ) U ( r 3 ) × U * ( r 2 ) U * ( r 4 ) 2 U ( r 1 ) U * ( r 2 ) U ( r 3 ) U * ( r 4 ) .
U s ( r ) = U 0 e σ ϕ 2 / 2 | r | 2 / L 2
U s ( r m ) U s * ( r n ) = U 0 2 e ( | r m | 2 + | r n | 2 ) / L 2 e σ ϕ 2 [ 1 C ( r m n ) ] ,
U s ( r m ) U s ( r n ) = U 0 2 e ( | r m | 2 + | r n | 2 ) / L 2 e σ ϕ 2 [ 1 + C ( r m n ) ] ,
Γ s ( r 1 , r 2 , r 3 , r 4 ) = U 0 4 exp [ ( | r 1 | 2 + | r 2 | 2 + | r 3 | 2 + | r 4 | 2 ) / L 2 2 σ ϕ 2 ] × { ( χ + ) 2 exp [ ( | r 12 | 2 + | r 34 | 2 ) / ρ u 2 ] + ( χ + ) ( e | r 12 | | 2 / ρ u + e | r 34 | 2 / ρ u 2 + e | r 14 | 2 / ρ u 2 + e | r 23 | 2 / ρ u 2 ) + ( χ + ) 2 exp [ ( | r 14 | 2 + | r 23 | 2 ) / ρ u 2 ] + ( χ ) 2 exp [ ( | r 13 | 2 + | r 24 | 2 / ρ u 2 ] + ( χ ) ( e | r 13 | 2 / ρ u 2 + e | r 24 | 2 / ρ u 2 ) + 1 } ,
χ + = ( e σ ϕ 2 1 ) χ = ( e σ ϕ 2 1 ) ,
J = · · d x 1 d x 2 d x 3 d x 4 d y 1 d y 2 d y 3 d y 4 × exp ( m = 0 4 n = 0 4 ( a m n x m x n + b m n y m y n ) ) ,
x 0 = y 0 = 1 , a m n a n m , m n b m n b n m , m n a m m < 0 , b m m < 0 , a m n > 0 , m n b m n > 0 , m n a o o > 0 b o o > 0 .
J = π 4 exp [ 1 2 P 00 ( x ) + 1 2 P 00 ( y ) + ( P 04 ( x ) ) 2 / 4 ( 2 a 44 1 2 P 44 ( x ) ) + ( P 04 ( y ) ) 2 / 4 ( 2 b 44 P 44 ( y ) ) ] [ a 11 b 11 ( 2 a 22 1 2 M 22 ( x ) ) ( 2 b 22 1 2 M 22 ( y ) ) ( 2 a 33 1 2 N 33 ( x ) ) ( 2 b 33 1 2 N 33 ( y ) ) ( 2 a 44 1 2 P 44 ( x ) ) ( 2 b 44 1 2 P 44 ( y ) ) ] 1 / 2
M m n ( x , y ) = 2 ( α m n ( x , y ) + α 1 m ( x , y ) α 1 n ( x , y ) / α 11 ( x , y ) ) ,
N m n ( x , y ) = M m n x , y ) + M 2 m ( x , y ) M 2 n ( x , y ) 2 ( α 22 ( x , y ) α 12 ( x , y ) 2 / α 11 ( x , y ) ) ,
P m n ( x , y ) = N m n ( x , y ) + N 3 m ( x , y ) N 3 n ( x , y ) / 2 [ α 33 ( x , y ) α 13 ( x , y ) 2 / α 11 ( x , y ) ( M 23 ( x , y ) ) 2 4 ( α 22 ( x , y ) α 12 ( x , y ) 2 / α 11 ( x , y ) ) , ] ,
α m n x = a m n , α m n y = b m n .
I ( P ) I ( P ) = U 0 4 k 4 cos 2 θ cos 2 θ e 4 χ 2 ( 2 R + ) 4 × [ 1 ( R / 2 R + ) 2 ] 2 l = 1 10 J ( ( l ) α m n ( x , y ) ) .
( l ) α m n ( x , y ) = α m n ( x , y ) | propagation + ( l ) α m n x , y ) | source ,
B I ( P , P ) = I ( P ) I ( P ) I ( P ) I ( P ) ,
σ I 2 ( P ) = B I ( P , P ) / I ( P ) 2 ,
b I ( P , P ) = B I ( P , P ) [ B I ( P , P ) B I ( P , P ) ] 1 / 2 .
I ( P ) = cos 2 θ 2 ( k L 2 R ) 2 × { ( χ ) ( Q x Q x ) 1 / 2 exp [ ( 1 2 k sin θ ) 2 / Q x ] + e σ ϕ 2 ( S x S x ) 1 / 2 exp [ ( 1 2 k sin θ ) 2 / S x ] } ,
Q x = [ ρ u 2 + ρ 1 2 + 1 2 L 2 + ( ½ ) ( k L / 2 R ) 2 ] , Q x = [ ρ u 2 + ρ 1 2 cos 2 θ + ( ½ ) L 2 + 1 2 ( k L / 2 R ) 2 ] ,
S x = Q x ρ u 2 , S x = Q x ρ u 2 .
χ 2 = π 2 ( 0.033 C n 2 ) Γ ( 5 6 ) k 7 / 6 R 11 / 6 ( α R 1 + ( α R ) 2 ) 5 / 6 × [ 3 8 ¹ F 1 ( 5 6 , 1 ; 2 ρ 2 L 2 [ 1 + ( α R ) 2 ] ) g ( α R ) ] ,
g ( α R ) = Re [ 6 11 ( 1 + i α R i α R ) 5 / 6 · ² F 1 ( 5 6 , 1 ; 17 6 ; i α R ) ] ,
α = λ / π L 2 ,
σ I = [ e 4 χ 2 1 ] 1 / 2
σ P = ( 1.23 k 7 / 6 C n 2 R 11 / 6 ) 1 / 2 .
4 χ s 2 = 0.5 k 7 / 6 C n 2 R 11 / 6 .
a 33 = a 11 = [ 2 ρ 1 2 A ρ S 2 + i ( A ρ χ S 2 k / 2 R + ) ] ,
a 44 = a 22 = a 11 * ,
b 11 = b 33 = [ 2 ρ 1 2 A ρ S 2 + i ( A ρ χ S 2 sec 2 θ k / 2 R + ) ] cos 2 θ ,
b 44 = b 22 = b 11 * ,
a 10 = { 1 2 R + β [ ρ 1 2 + ( B 2 A ) ρ S 2 ] + i 1 2 [ ρ χ S 2 ( R + β ) ( B 2 A ) + k 2 ω sin θ + ] } ,
a 20 = a 11 * ,
a 30 = a 10 ,
a 40 = a 30 * ,
b 10 = { 1 2 R + δ cos θ + [ ρ 1 2 ( B 2 A ) ρ S 2 ] + i 1 2 [ ρ χ S 2 ( R + δ × cos θ + ) ( B 2 A ) k 2 ( δ cos θ + 2 sin θ + ) ] } ,
b 20 = b 10 * ,
b 30 = { 1 2 R + δ cos θ + [ ρ 1 2 + ( B 2 A ) ρ S 2 ] + i 1 2 [ ρ χ S 2 ( R + δ cos θ + ) ( B 2 A ) + k 2 ( δ cos θ + + 2 sin θ + ) ] } ,
b 40 = b 30 * ,
a 12 = a 14 = ρ 1 2 ,
a 13 = A ( ρ S 2 + i ρ χ S 2 ) ,
b 12 = b 14 = ρ 1 2 cos 2 θ + ,
b 13 = A cos 2 θ + ( ρ S 2 + i ρ χ S 2 ) ,
a 23 = a 34 = ρ 1 2 ,
b 23 = b 34 = ρ 1 2 cos 2 θ + ,
a 24 = a 13 * ,
b 24 = b 13 * ,
a 00 = 2 ( R + β ) 2 ( ρ 1 2 A ρ S 2 ) ,
b 00 = 2 ( R + δ ) 2 ( ρ 1 2 A ρ S 2 ) ,
ρ S = ( 0.875 k 13 / 6 R + 5 / 6 C n 2 ) 1 / 2 ,
ρ χ S = ( 0.234 k 13 / 6 R + 5 / 6 C n 2 ) 1 / 2 ,
A = 0 1 d t t 11 / 6 ( 1 t ) 1 / 6 = 0.485 18 ,
B = 2 0 1 d t t 5 / 6 ( 1 t ) 1 / 6 = 0 1 d t [ t ( 1 t ) ] 1 / 6 = 1.411 43 .
( l ) b m n = ( l ) a m , n for all l ,
( l ) a 12 = ρ u 2 for l = 1 , 4 ,
( l ) a 34 = ρ u 2 for l = 1 , 5 ,
( l ) a 14 = ρ u 2 for l = 2 , 6 ,
( l ) a 23 = ρ u 2 for l = 2 , 7 ,
( l ) a 13 = ρ u 2 for l = 3 , 8 ,
( l ) a 24 = ρ u 2 for l = 3 , 9 ,
( l ) a 00 = σ ϕ 2 + ln χ + for l = 1 , 2 ,
( 3 ) a 00 = σ ϕ 2 + ln χ ,
( l ) a 00 = σ ϕ 2 + ½ ln χ + for l = 4 7 ,
( l ) a 00 = σ ϕ 2 + ½ ln χ for l = 8 , 9 ,
( 10 ) a 00 = σ ϕ 2 ,
( l ) a 11 = { L 2 + ρ u 2 for l = 1 4 , 6 , 8 , L 2 for l = 5 , 7 , 9 , 10 ,
( l ) a 22 = { L 2 + ρ u 2 for l = 1 4 , 7 , 9 , L 2 for l = 5 , 6 , 8 , 10 ,
( l ) a 33 = { L 2 + ρ u 2 for l = 1 3 , 5 , 7 , 8 , L 2 for l = 4 , 6 , 9 , 10 ,
( l ) a 44 = { L 2 + ρ u 2 for l = 1 3 , 5 , 6 , 9 , L 2 for l = 4 , 7 , 8 , 10 .