Abstract

A theoretical expression for the covariance of the received intensity of a partially coherent laser speckle pattern after propagation through the turbulent atmosphere is developed. It is shown that the atmospheric perturbation on a partially coherent speckle pattern can be decomposed into a coherent term and an incoherent term. The dependence of contributions of these components on the level of turbulence, vacuum speckle-contrast ratio, and detector spacing is studied in detail and the results are compared with the available experimental data.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. H. Lee, J. F. Holmes, J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]
  2. B. A. Sotskii, A. M. Goncharenko, “On the relationship between the radiation coherence and the number of modes of a laser,” Opt. Spectrosc. (USSR) 19, 435–437 (1965).
  3. J. F. Holmes, V. S. Rao Gudimetla, “Variance of intensity for a discrete spectrum polychromatic speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 71, 1176–1179 (1981).
    [CrossRef]
  4. J. W. Goodman, “Role of coherence concepts in the study of speckle,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 86–94 (1979).
  5. J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
    [CrossRef]
  6. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
    [CrossRef]
  7. V. S. Rao Gudimetla, J. F. Holmes, R. A. Elliott, “Two-point joint density function of the intensity of a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. A 7, 1008–1014 (1990).
    [CrossRef]
  8. P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
    [CrossRef]
  9. J. F. Holmes, M. H. Lee, J. R. Kerr, “Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
    [CrossRef]
  10. M. H. Lee, J. F. Holmes, J. R. Kerr, “Generalized spherical wave mutual coherence function,” J. Opt. Soc. Am. 67, 1279–1281 (1977).
    [CrossRef]
  11. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  12. M. E. Fossey, “Statistical measurements of speckle propagation through the turbulent atmosphere,” M.S. thesis (Department of Applied Physics and Electronic Science, Oregon Graduate Center, Beaverton, Ore., 1976).
  13. J. F. Holmes, “Speckle propagation through the turbulent atmosphere,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 20–27 (1980).
  14. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation through the Turbulent Atmosphere, J. W. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
    [CrossRef]
  15. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vols. 1 and 2 (Academic, New York, 1978).
  16. A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
    [CrossRef]
  17. R. L. Fante, “Multiple frequency mutual coherence function for a beam in a random medium,” IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).
    [CrossRef]
  18. V. S. Rao Gudimetla, “Statistics of polychromatic speckle propagation through the turbulent atmosphere,” Ph.D. dissertation (Department of Applied Physics and Electrical Engineering, Oregon Graduate Center, Beaverton, Ore., 1982).
  19. J. D. Jackson, Classical ElectrodynamicsWiley, New York, 1975).
  20. P. A. Pincus, J. R. Kerr, “Spectral covariance of scintillations,” Appl. Opt. 15, 2305 (1976).
    [CrossRef] [PubMed]
  21. D. L. Fried, “Spectral and angular covariance of scintillation for propagation in a randomly homogeneous medium,” Appl. Opt. 10, 721–731 (1971).
    [CrossRef] [PubMed]
  22. J. F. Holmes, J. R. Kerr, R. A. Elliott, M. H. Lee, P. A. Pincus, M. E. Fossey, Experimental Feasibility Pulsed Laser Remote Cross Wind Measurement Systems—Feasibility and Design (Part V), Rep. No. ARSCD-CR-79-007 (U.S. Army Armament Research and Development Command, Dover, New Jersey, September1978).
  23. S. F. Clifford, H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
    [CrossRef]

1990 (1)

1981 (1)

1980 (1)

1979 (1)

1978 (2)

P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
[CrossRef]

R. L. Fante, “Multiple frequency mutual coherence function for a beam in a random medium,” IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).
[CrossRef]

1977 (1)

1976 (2)

1974 (1)

1972 (2)

H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
[CrossRef] [PubMed]

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
[CrossRef]

1971 (1)

1967 (1)

1965 (1)

B. A. Sotskii, A. M. Goncharenko, “On the relationship between the radiation coherence and the number of modes of a laser,” Opt. Spectrosc. (USSR) 19, 435–437 (1965).

Clifford, S. F.

S. F. Clifford, H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
[CrossRef]

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation through the Turbulent Atmosphere, J. W. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

Elliott, R. A.

V. S. Rao Gudimetla, J. F. Holmes, R. A. Elliott, “Two-point joint density function of the intensity of a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. A 7, 1008–1014 (1990).
[CrossRef]

J. F. Holmes, J. R. Kerr, R. A. Elliott, M. H. Lee, P. A. Pincus, M. E. Fossey, Experimental Feasibility Pulsed Laser Remote Cross Wind Measurement Systems—Feasibility and Design (Part V), Rep. No. ARSCD-CR-79-007 (U.S. Army Armament Research and Development Command, Dover, New Jersey, September1978).

Fante, R. L.

R. L. Fante, “Multiple frequency mutual coherence function for a beam in a random medium,” IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).
[CrossRef]

Fossey, M. E.

P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
[CrossRef]

M. E. Fossey, “Statistical measurements of speckle propagation through the turbulent atmosphere,” M.S. thesis (Department of Applied Physics and Electronic Science, Oregon Graduate Center, Beaverton, Ore., 1976).

J. F. Holmes, J. R. Kerr, R. A. Elliott, M. H. Lee, P. A. Pincus, M. E. Fossey, Experimental Feasibility Pulsed Laser Remote Cross Wind Measurement Systems—Feasibility and Design (Part V), Rep. No. ARSCD-CR-79-007 (U.S. Army Armament Research and Development Command, Dover, New Jersey, September1978).

Fried, D. L.

Goncharenko, A. M.

B. A. Sotskii, A. M. Goncharenko, “On the relationship between the radiation coherence and the number of modes of a laser,” Opt. Spectrosc. (USSR) 19, 435–437 (1965).

Goodman, J. W.

J. W. Goodman, “Role of coherence concepts in the study of speckle,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 86–94 (1979).

Holmes, J. F.

V. S. Rao Gudimetla, J. F. Holmes, R. A. Elliott, “Two-point joint density function of the intensity of a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. A 7, 1008–1014 (1990).
[CrossRef]

J. F. Holmes, V. S. Rao Gudimetla, “Variance of intensity for a discrete spectrum polychromatic speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 71, 1176–1179 (1981).
[CrossRef]

J. F. Holmes, M. H. Lee, J. R. Kerr, “Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
[CrossRef]

P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
[CrossRef]

M. H. Lee, J. F. Holmes, J. R. Kerr, “Generalized spherical wave mutual coherence function,” J. Opt. Soc. Am. 67, 1279–1281 (1977).
[CrossRef]

M. H. Lee, J. F. Holmes, J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
[CrossRef]

J. F. Holmes, “Speckle propagation through the turbulent atmosphere,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 20–27 (1980).

J. F. Holmes, J. R. Kerr, R. A. Elliott, M. H. Lee, P. A. Pincus, M. E. Fossey, Experimental Feasibility Pulsed Laser Remote Cross Wind Measurement Systems—Feasibility and Design (Part V), Rep. No. ARSCD-CR-79-007 (U.S. Army Armament Research and Development Command, Dover, New Jersey, September1978).

Ishimaru, A.

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
[CrossRef]

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

Jackson, J. D.

J. D. Jackson, Classical ElectrodynamicsWiley, New York, 1975).

Kerr, J. R.

Leader, J. C.

Lee, M. H.

Pincus, P. A.

P. A. Pincus, M. E. Fossey, J. F. Holmes, J. R. Kerr, “Speckle propagation through turbulence: experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
[CrossRef]

P. A. Pincus, J. R. Kerr, “Spectral covariance of scintillations,” Appl. Opt. 15, 2305 (1976).
[CrossRef] [PubMed]

J. F. Holmes, J. R. Kerr, R. A. Elliott, M. H. Lee, P. A. Pincus, M. E. Fossey, Experimental Feasibility Pulsed Laser Remote Cross Wind Measurement Systems—Feasibility and Design (Part V), Rep. No. ARSCD-CR-79-007 (U.S. Army Armament Research and Development Command, Dover, New Jersey, September1978).

Rao Gudimetla, V. S.

Sotskii, B. A.

B. A. Sotskii, A. M. Goncharenko, “On the relationship between the radiation coherence and the number of modes of a laser,” Opt. Spectrosc. (USSR) 19, 435–437 (1965).

Yura, H. T.

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (2)

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
[CrossRef]

R. L. Fante, “Multiple frequency mutual coherence function for a beam in a random medium,” IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (1)

Opt. Spectrosc. (USSR) (1)

B. A. Sotskii, A. M. Goncharenko, “On the relationship between the radiation coherence and the number of modes of a laser,” Opt. Spectrosc. (USSR) 19, 435–437 (1965).

Other (8)

J. W. Goodman, “Role of coherence concepts in the study of speckle,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 86–94 (1979).

V. S. Rao Gudimetla, “Statistics of polychromatic speckle propagation through the turbulent atmosphere,” Ph.D. dissertation (Department of Applied Physics and Electrical Engineering, Oregon Graduate Center, Beaverton, Ore., 1982).

J. D. Jackson, Classical ElectrodynamicsWiley, New York, 1975).

M. E. Fossey, “Statistical measurements of speckle propagation through the turbulent atmosphere,” M.S. thesis (Department of Applied Physics and Electronic Science, Oregon Graduate Center, Beaverton, Ore., 1976).

J. F. Holmes, “Speckle propagation through the turbulent atmosphere,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 20–27 (1980).

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation through the Turbulent Atmosphere, J. W. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

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

J. F. Holmes, J. R. Kerr, R. A. Elliott, M. H. Lee, P. A. Pincus, M. E. Fossey, Experimental Feasibility Pulsed Laser Remote Cross Wind Measurement Systems—Feasibility and Design (Part V), Rep. No. ARSCD-CR-79-007 (U.S. Army Armament Research and Development Command, Dover, New Jersey, September1978).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Normalized covariance of the received intensity of a low-contrast speckle pattern generated by a multimode Nd:YAG laser versus the log-amplitude deviation. Dots indicate experimental data. The solid curve with open circles indicates the numerical values.

Fig. 2
Fig. 2

Normalized covariance of the received intensity of a low-contrast speckle pattern generated by a multimode argon laser versus the detector spacing. Dots indicate experimental data. The solid curve indicates the numerical values from theory.

Fig. 3
Fig. 3

Normalized covariance of the received intensity of a low-contrast speckle pattern generated by a multimode argon laser versus the detector spacing for different values of the VSCR at a low turbulence level (theory).

Fig. 4
Fig. 4

Normalized covariance of the received intensity of a low-contrast speckle pattern generated by a multimode argon laser versus the detector spacing for different values of the VSCR in the unsaturated region of turbulence (theory).

Fig. 5
Fig. 5

Normalized covariance of the received intensity of a low-contrast speckle pattern generated by a multimode argon laser versus the detector spacing for different values of the VSCR for a turbulence level just at saturation (theory).

Fig. 6
Fig. 6

Normalized covariance of the received intensity of a low-contrast speckle pattern generated by a multimode Nd:YAG laser versus the log-amplitude standard deviation for three different values of the VSCR used to study sensitivity.

Equations (45)

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

V ( x , t ) = j = 1 N exp [ i ( k j x ω j t ) ] .
Γ ( x 1 , t 1 ; x 2 , t 2 ) = V ( x 1 , t 1 ) V * ( x 2 , t 2 ) = j = 1 N m = 1 N exp [ i ( k j k m ) x i ( ω j ω m ) t ] ,
γ 12 ( x , τ ) = Γ 12 ( x , τ ) Γ 12 ( 0 , 0 ) = ( 1 N ) j = 1 N exp [ i ( k j x ω j τ ) ] .
k s = k 0 + Δ k s ,
Δ k s k 0 = θ s = λ 0 s ( 4 α 0 n ) ,
| γ 12 ( x , τ ) | 2 = sin 2 ( π N x / 4 α 0 n ) N 2 sin 2 ( π x / 4 α 0 n ) .
ω q ω s = π c / [ n L ( q s ) ] ,
| γ 12 ( x , τ ) | 2 = sin 2 ( π N c τ / 2 L n ) N 2 sin 2 ( π c τ / 2 n L ) .
| γ 12 ( x , τ ) | 2 = ( 1 N 2 ) { N + 2 n > s = 1 N cos [ ( k n k s ) x ( w n w s ) τ ] } .
| γ 12 ( x , τ ) | 2 = 1 N .
B I ( p 2 ¯ : p 1 ¯ ) = I ( p 2 ¯ ) I ( p 1 ¯ ) ,
U 0 ( r ) = j = 1 N U 0 ( r , k j ) = j = 1 N U 0 j exp ( r 2 2 α 0 2 i k j r 2 2 F ) ,
U ( ρ ) = j = 1 N [ ( k j U 0 j ) / ( i 2 π L ) ] exp [ i k j ( L + ρ 2 / 2 L ) ] × exp [ ( r 2 / 2 α 0 2 i r 2 k j / 2 L ) ( 1 L / F ) i k j ( r ¯ · ρ ¯ ) / L + ψ 1 ( ρ , r , k j ) ] d r ¯ ,
U ( p ) = j = 1 N [ ( k j / i 2 π L ) ] exp [ i k j ( L + p 2 / 2 L ) ] d ρ ¯ U ( ρ , k j ) × exp [ i ( k j / 2 L ) ( ρ 2 2 p ρ ) + ψ 2 ( p , ρ , k j ) ] .
ψ = χ + i ϕ ,
B I ( p 1 , p 2 ) = U ( p 1 ¯ , 0 ) U * ( p 1 ¯ , 0 ) U ( p 2 ¯ U * ( p 2 ¯ ) = i N j N k i 2 k j 2 ( 2 π L ) 4 d ρ 1 ¯ d ρ 2 ¯ d ρ 3 ¯ d ρ 4 ¯ × U ( ρ 1 ¯ , k i ) U * ( ρ 2 ¯ , k i ) U ( ρ 3 ¯ , k j ) U * ( ρ 4 ¯ , k j ) × exp { i k i / 2 L ) [ ρ 1 2 ρ 2 2 2 p 1 ¯ ( ρ ¯ 1 ρ ¯ 2 ) ] + ( i k j / 2 L ) [ ρ 3 2 ρ 4 2 2 p 2 ¯ ( ρ ¯ 3 ρ ¯ 4 ) ] } × H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 3 , ρ 4 ; k i , k j ) ,
H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 3 , ρ 4 ; k i , k j ) = exp [ ψ ( p 1 , ρ 1 , k i ) + ψ* ( p 1 , ρ 2 , k i ) + ψ ( p 2 , ρ 3 , k j ) + ψ* ( p 2 , ρ 4 , k j ) ] = exp { 1 / 2 [ D ψ ( 0 , ρ 2 ρ 1 , k i , k i ) D ψ ( p 2 ρ 1 , ρ 3 ρ 1 , k i , k j ) + D ψ ( p 2 p 1 , ρ 4 ρ 1 , k i , k j ) + D ψ ( p 2 p 1 , ρ 3 ρ 2 , k i , k j ) D ψ ( p 2 p 1 , ρ 4 ρ 2 , k i , k j ) + D ψ ( 0 , ρ 4 ρ 3 , k j , k j ) ] + 2 C χ ( p 2 p 1 , ρ 3 ρ 1 , k i , k j ) + 2 C χ ( p 2 p 1 , ρ 4 ρ 2 , k i , k j ) } .
U ( ρ 1 ¯ , k i ) U * ( ρ 2 ¯ , k i ) U ( ρ 3 ¯ , k j ) U * ( ρ 4 ¯ , k j ) = ( 4 π k i k j ) 2 I ( ρ 2 ¯ , k i ) I ( ρ 3 ¯ , k j ) δ ( ρ 1 ¯ ρ 2 ¯ ) δ ( ρ 3 ¯ ρ 4 ¯ ) + ( 4 π k i k j ) 2 I ( ρ 4 ¯ , k i ) I ( ρ 2 ¯ , k j ) δ ( ρ 1 ¯ ρ 4 ¯ ) δ ( ρ 3 ¯ ρ 2 ¯ ) δ i j ,
B I ( p , τ ) = C I 1 ( p ) + C I 2 ( p ) ,
C I 1 ( p ) = i N j N 1 π 2 L 4 d ρ 2 ¯ d ρ 4 ¯ I ( ρ 2 ¯ , k i ) I ( ρ 4 ¯ , k j ) × exp [ 4 C χ ( p 1 p 2 , ρ 2 ρ 4 , k i , k j ) ] ,
C I 2 = i N 1 π 2 L 4 d ρ 2 ¯ d ρ 4 ¯ I ( ρ 4 ¯ , k i ) I ( ρ 2 ¯ , k i ) × exp [ ( i k / L ) p ρ ] H ( p 1 , p 2 ; ρ 2 , ρ 2 , ρ 4 , ρ 4 ; k i , k i )
C I 1 ( p ) = i N j N ( α 0 4 2 π L 4 ) U 0 i 2 U 0 j 2 d ρ ¯ d r ¯ J 0 ( ρ r ) × exp { ( r 2 L 2 / 4 α 0 2 ) ( 1 / k i 2 + 1 / k j 2 ) [ ( r L / ( k i ρ 0 i ) ] 5 / 3 [ ( r L / ( k j ρ 0 j ) ] 5 / 3 r 2 ( α 0 2 / 2 ) × ( 1 L F ) 2 } exp [ 4 C χ ( p , ρ , k i , k j ) ] ,
C I 2 ( p ) = i N ( α 0 4 4 π 2 L 4 ) U 0 i 4 d ρ ¯ d r ¯ × exp [ ( r 2 L 2 / 2 α 0 2 k 2 ) D ψ ( 0 , L r k i ) r 2 ( α 0 2 / 2 ) ( 1 L F ) 2 ] exp [ i k i L ρ ¯ ( p + r ¯ ) ] H 2 ( p ¯ , ρ ¯ ) ,
H 2 ( ρ ¯ , p ¯ ) = exp [ D ψ ( 0 , ρ ¯ , k i ) D ψ ( p ¯ , 0 k i ) + ( 1 / 2 ) D ψ ( p ¯ , ρ ¯ , k i ) + ( 1 / 2 ) D ψ ( p ¯ , ρ ¯ , k i ) + 2 C χ ( p ¯ , ρ ¯ , k i ) + 2 C χ ( p ¯ , ρ ¯ , k i ) ] .
I = j = 1 N I j ,
I j = U 0 j 2 ( α 0 2 L 2 ) .
C I ( p ) = B I ( p ) I 2 = C I 1 ( p ) + C I 2 ( p ) I 2 .
C I 1 ( P ) = i N j N I i I j 2 π ρ d ρ r d r d θ ρ J 0 × ( ρ r ) f 1 ( r , k i , k j ) exp [ 4 C χ ( p , ρ , k i k j ) ] ,
f 1 ( r , k i , k j ) = exp { ( r 2 L 2 / 4 α 0 4 ) ( 1 / k i 2 + 1 / k j 2 ) [ ( r L ) / ( k i ρ 0 i ) ] 5 / 3 [ ( r L ) / ( k j ρ 0 j ) ] 5 / 3 r 2 ( α 0 2 / 2 ) ( 1 L F ) 2 } ,
C I 2 ( P ) = i N I i 2 2 π ρ d ρ r d r d θ ρ f 2 ( r , k i ) J 0 ( ρ r ) × exp [ i k i L ρ P cos ( θ P θ ρ ) ] H 2 ( P ¯ , ρ ¯ ) ,
f 2 ( r , k i ) = exp { r 2 ( L 2 / 2 α 0 2 ) k i 2 2 [ ( r L ) / ( k i ρ 0 i ) ] 5 / 3 r 2 ( α 0 2 / 2 ) ( 1 L F ) 2 } ,
H 2 ( ρ ¯ , p ¯ ) = exp [ D ψ ( 0 , ρ ¯ , k i ) D ψ ( p ¯ , 0 , k i ) + ( 1 / 2 ) D ψ ( p ¯ , ρ ¯ , k i ) + ( 1 / 2 ) D ψ ( p ¯ , ρ ¯ , k i ) + 2 C χ ( p ¯ , ρ ¯ , k ι ˙ ) + 2 C χ ( p ¯ , ρ ¯ , k i ) ] ,
f 1 ( r , k i , k j ) = Σ m b m ( k i , k j ) J 0 ( P m r A 1 ( k i , k j ) ) ,
b m ( k i , k j ) = [ 2 A 1 2 ( k i , k j ) J 1 2 ( p m ) ] 0 A 1 r d r f 1 ( r , k i , k j ) J 0 [ P m r A 1 ( k i , k j ) ]
C I 1 ( p ) = i N j N I i I j 2 π d θ ρ ( Σ m b m ( k i , k j ) exp × { 4 C χ [ p , ρ = P m A 1 ( k i , k j ) k i , k j ] } ) .
f 2 ( r , k i ) = m C m ( k i ) J 0 [ P m r A 2 ( k i ) ] ,
C m ( k i ) = [ 2 A 2 2 ( k i ) J 1 2 ( P m ) ] 0 A 2 r d r f 2 ( r , k i ) J 0 [ P m r A 2 ( k i ) ] ,
C I 2 ( p ) = i N I i 2 2 π d θ ρ [ Σ m C m ( k i ) ] × exp [ i k i L P m A 2 ( k i ) cos ( θ P θ ρ ) ] H 2 [ p , ρ = P m A 2 ( k i ) ] .
C I N ( p ) = C I 1 ( p ) + C I 2 ( p ) I 2 1 .
C I N = i j I i I j ( i I i ) 2 A I N T 1 + i I i 2 ( i I i ) 2 A I N T 2 1 = A I N T 1 + i I i 2 ( i I i ) 2 A I N T 2 1 ,
A I N T 1 = 1 2 π 0 2 π d θ ρ [ m b m ] exp [ 4 C χ ( p , ρ = P m A , k ) ] ,
A I N T 2 = 1 2 π 0 2 π d θ ρ [ m C m ] exp [ i k L P m A cos ( θ p θ ρ ) ] × H 2 ( p , ρ = P m A , k ) .
C I N = σ I N 2 = ( VSCR ) 2 = i I i 2 ( i I i ) 2 ,
C I N ( p ) = A I N T 1 + ( VSCR ) 2 A I N T 2 1 ,
C I σ ( p ) = C I ( p ) I 2 I 2 σ I 2 = C I ( p ) σ I 2 ,

Metrics