Abstract

The study of the statistical effects of atmospheric turbulence on coherent radiation scattered from a diffuse target is extended, both theoretically and experimentally, to include the effects of target glint. Theoretically, expressions are developed for the first- and second-order moments of irradiance for targets containing multiple glints. The expression for the normalized variance of irradiance for a diffuse target with a single glint illuminated by a coherent source over a horizontal atmospheric path is evaluated numerically for comparison with experimental results. Experimental measurements of both the variance and covariance of irradiance for a coherent source illuminating a diffuse target containing a single glint are presented.

© 1980 Optical Society of America

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References

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  1. J. F. Holmes, Myung H. Lee, and 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]
  2. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [Crossref] [PubMed]
  3. 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]
  4. Myung 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]
  5. J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–185 (1978).
    [Crossref]
  6. J. L. Bufton, R. S. Iyer, and L. S. Taylor, “Scintillation statistics caused by atmospheric turbulence and speckle in satellite laser ranging,” Appl. Opt. 16, 2408–2413 (1977).
    [Crossref] [PubMed]
  7. H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
    [Crossref]
  8. R. F. Lutomirski and R. E. Warren, “Atmospheric distortions in a retroreflected laser signal,” Appl. Opt. 14, 840–846 (1975).
    [Crossref] [PubMed]
  9. Myung H. Lee, J. F. Holmes, and J. R. Kerr, “Generalized spherical wave mutual coherence function,” J. Opt. Soc. Am. 67, 1279–1281 (1977).
    [Crossref]
  10. J. R. Kerr, R. A. Elliott, M. E. Fossey, J. F. Holmes, Myung H. Lee, and P. A. Pincus, “Propagation of multi-wavelength laser radiation through atmospheric turbulence,” Rome Air Development Center report RADC-TR-77-18, Air Force Systems Command, Griffiss Air Force Base, New York 13441 (unpublished).
  11. J. R. Dunphy and J. R. Kerr, “Turbulence effects on target illumination by laser sources: phenomenological analysis and experimental results,” Appl. Opt. 16, 1345–1358 (1977).
    [Crossref] [PubMed]
  12. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975), pp. 9–75.
    [Crossref]
  13. C. M. McIntyre, J. R. Kerr, Myung H. Lee, and J. H. Churnside, “Enhanced variance of irradiance from target glint,” Appl. Opt. 18, 3211–3212 (1979).
    [Crossref] [PubMed]
  14. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [Crossref]

1980 (1)

1979 (1)

1978 (1)

1977 (3)

1976 (1)

1975 (2)

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[Crossref]

R. F. Lutomirski and R. E. Warren, “Atmospheric distortions in a retroreflected laser signal,” Appl. Opt. 14, 840–846 (1975).
[Crossref] [PubMed]

1972 (1)

1971 (1)

1970 (1)

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

Bufton, J. L.

Churnside, J. H.

Dunphy, J. R.

Elliott, R. A.

J. R. Kerr, R. A. Elliott, M. E. Fossey, J. F. Holmes, Myung H. Lee, and P. A. Pincus, “Propagation of multi-wavelength laser radiation through atmospheric turbulence,” Rome Air Development Center report RADC-TR-77-18, Air Force Systems Command, Griffiss Air Force Base, New York 13441 (unpublished).

Fossey, M. E.

J. R. Kerr, R. A. Elliott, M. E. Fossey, J. F. Holmes, Myung H. Lee, and P. A. Pincus, “Propagation of multi-wavelength laser radiation through atmospheric turbulence,” Rome Air Development Center report RADC-TR-77-18, Air Force Systems Command, Griffiss Air Force Base, New York 13441 (unpublished).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975), pp. 9–75.
[Crossref]

Holmes, J. F.

Iyer, R. S.

Kerr, J. R.

Lawrence, R. S.

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

Leader, J. C.

Lee, Myung H.

Lutomirski, R. F.

McIntyre, C. M.

Pedersen, H. M.

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[Crossref]

Pincus, P. A.

J. R. Kerr, R. A. Elliott, M. E. Fossey, J. F. Holmes, Myung H. Lee, and P. A. Pincus, “Propagation of multi-wavelength laser radiation through atmospheric turbulence,” Rome Air Development Center report RADC-TR-77-18, Air Force Systems Command, Griffiss Air Force Base, New York 13441 (unpublished).

Strohbehn, J. W.

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

Taylor, L. S.

Warren, R. E.

Yura, H. T.

Appl. Opt. (6)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[Crossref]

Proc. IEEE (1)

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

Other (2)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975), pp. 9–75.
[Crossref]

J. R. Kerr, R. A. Elliott, M. E. Fossey, J. F. Holmes, Myung H. Lee, and P. A. Pincus, “Propagation of multi-wavelength laser radiation through atmospheric turbulence,” Rome Air Development Center report RADC-TR-77-18, Air Force Systems Command, Griffiss Air Force Base, New York 13441 (unpublished).

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Figures (12)

FIG. 1
FIG. 1

Illuminator, target, and receiver configuration.

FIG. 2
FIG. 2

Normalized variance of irradiance versus log amplitude variance for a diffuse target illuminated by a coherent source over a 500-m path. Solid line is the theoretical curve and ⊕ represents experimental data.

FIG. 3
FIG. 3

Normalized variance of irradiance versus log amplitude variance for a diffuse target with a single glint illuminated by a coherent source over a 500-m path. Each curve corresponds to a different size glint with the 1/e radius of the model Gaussian glint given. The glint is located 1 cm off the optical axis.

FIG. 4
FIG. 4

Normalized variance of irradiance versus log amplitude variance for a diffuse target with a single glint illuminated by a coherent source over a 500-m path. Each curve corresponds to a different size glint with the 1/e radius of the model Gaussian glint given. The glint is located 0.5 cm off the optical axis.

FIG. 5
FIG. 5

Normalized variance of irradiance versus log amplitude variance for a diffuse target with a single glint illuminated by a coherent source over a 500-m path. Each curve corresponds to a different size glint with the 1/e radius of the model Gaussian glint given. The glint is located on the optical axis.

FIG. 6
FIG. 6

Normalized variance of irradiance versus log amplitude variance for a diffuse target with a single glint illuminated by a coherent source over a 500-m path. The curves are for a model Gaussian glint with a 1/e radius of 0.19 mm located on axis, 0.5 cm off axis, and 1 cm off axis.

FIG. 7
FIG. 7

Normalized variance of irradiance verus log amplitude variance for a diffuse target with a single glint illuminated by a coherent source over a 500-m path. The curves are for a model Gaussian glint with a 1/e radius of 0.19 mm located on axis, 0.5 cm off axis, and 1 cm off axis. Experimental points are indicated on the figure.

FIG. 8
FIG. 8

Normalized variance of irradiance versus log amplitude variance for a diffuse target with a single glint illuminated by a coherent source over a 500-m path. The curves are for a model Gaussian glint with a 1/e radius of 0.144 mm located on axis, 0.5 cm off axis, and 1 cm off axis. Experimental points are indicated on the figure.

FIG. 9
FIG. 9

Normalized covariance of irradiance versus separation (ρ) as a function of glint strength for a fixed turbulence level: solid line—no glint (diffuse target only); solid circles—spherical glint; open squares—0.48-mm-diameter flat glint; open triangles—1.00-mm-diameter flat glint. Path length = 500 m, σ χ 2 = 0.85.

FIG. 10
FIG. 10

Normalized covariance of Irradiance versus separation (ρ) for a 0.45-mm-diameter flat glint: solid circles σ χ 2 = 0.0014; open squares σ χ 2 = 0.015; open triangles σ χ 2 = 0.16. Path length = 500 m.

FIG. 11
FIG. 11

Normalized covariance of irradiance versus separation (ρ) for a 2.38-mm-diameter flat glint and fixed turbulence level: solid circles—path length = 1500 m, D/ρ0 = 1.0; open squares—path length = 500 m, D/ρ0 = 0.19.

FIG. 12
FIG. 12

Normalized covarlance of Irradiance versus separation (ρ): solid circles—2.38-mm-diameter flat glint, D/ρ0 = 0.35, σ χ 2 = 0.08; open squares—1.00-mm-diameter flat glint, D/ρ0 = 0.41, σ χ 2 = 0.10.

Tables (1)

Tables Icon

TABLE I Normalized variance of irradiance ( σ I N 2 ) for a diffuse target with a single glint.

Equations (72)

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U 0 ( r ) = U 0 exp ( r 2 2 α 0 2 i k 2 F r 2 ) ,
U i ( ρ ) = k 2 π i L exp ( i k L ) × U 0 ( r ) exp ( i k | ρ r | 2 2 L + ψ 1 ( ρ , r ) ) d r ,
U s ( ρ ) = U d ( ρ ) + U g ( ρ ) ,
U d ( ρ ) = R 1 / 2 U i ( ρ ) ,
U g ( ρ ) = m = 1 M a m r m ( ρ ) U i * ( ρ m ) ,
r m ( ρ ) = exp ( | ρ ρ m | 2 ( Δ ρ m ) 2 ) ,
U r ( p ) = k 2 π i L exp ( i k L ) U s ( ρ ) × exp ( i k | p ρ | 2 2 L + ψ 2 ( p , ρ ) ) d ρ ,
Γ ( p 1 , p 2 ) = U r ( p 1 ) U r * ( p 2 ) = ( k 2 π L ) 2 d ρ 1 d ρ 2 U s ( ρ 1 ) U s * ( ρ 2 ) × exp ( i k | p 1 ρ 1 | 2 2 L i k | p 2 ρ 2 | 2 2 L ) × exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 2 , ρ 2 ) ] ,
U r ( ρ 1 ) U r * ( ρ 2 ) = [ U d ( ρ 1 ) + U g ( ρ 1 ) ] [ U d ( ρ 2 ) + U g ( ρ 2 ) ] * = U d ( ρ 1 ) U d * ( ρ 2 ) + U d ( ρ 1 ) U g * ( ρ 2 ) + U g ( ρ 1 ) U d * ( ρ 2 ) + U g ( ρ 1 ) U g * ( ρ 2 ) .
U d ( ρ 1 ) U d * ( ρ 2 ) = ( 4 π / k 2 ) I d ( ρ 1 ) δ ( ρ 1 ρ 2 ) ,
U d ( ρ 1 ) U g * ( ρ 2 ) = U g ( ρ 1 ) U d * ( ρ 2 ) = 0 .
U s ( ρ 1 ) U s * ( ρ 2 ) = 4 π R k 2 I i ( ρ 1 ) δ ( ρ 1 ρ 2 ) + m 1 = 1 M m 2 = 1 M a m 1 a m 2 * U i * ( ρ m 1 ) U i ( ρ m 2 ) × exp ( | ρ 1 ρ m 1 | 2 ( Δ ρ m 1 ) 2 | ρ 2 ρ m 2 | 2 ( Δ ρ m 2 ) 2 ) ,
Γ ( p 1 , p 2 ) = Γ d ( p 1 , p 2 ) + Γ g ( p 1 , p 2 ) ,
Γ d ( p 1 , p 2 ) = R π L 2 d ρ 1 d ρ 2 I i ( ρ 1 ) δ ( ρ 1 ρ 2 ) × exp ( i k | p 1 ρ 1 | 2 2 L i k | p 2 ρ 2 | 2 2 L ) , × exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 2 , ρ 2 ) ]
Γ g ( p 1 , p 2 ) = ( k 2 π L ) 2 m 1 = 1 M m 2 = 1 M × d ρ 1 d ρ 2 U i * ( ρ m 1 ) U i ( ρ m 2 ) × exp ( 1 ( Δ ρ ω ) 2 ( | ρ 1 ρ m 1 | 2 + | ρ 2 ρ m 2 | 2 ) ) × exp ( i k 2 L ( | p 1 ρ 1 | 2 | p 2 ρ 2 | 2 ) ) × exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 2 , ρ 2 ) ] .
I r ( p ) | M = 1 = R | U 0 | 2 α 0 2 L 2 + ( Δ ρ ω ) 4 ( k 2 L ) 2 I i ( ρ g ) ,
C I ( p 1 , p 2 ) = I r ( p 1 ) I r ( p 2 ) I r ( p 1 ) I r ( p 2 ) .
B I ( p 1 , p 2 ) = I r ( p 1 ) I r ( p 2 ) = U r ( p 1 ) U r * ( p 1 ) U r ( p 2 ) U r * ( p 2 )
B I ( p 1 , p 2 ) = ( k 2 π L ) 4 d ρ 1 d ρ 2 d ρ 3 d ρ 4 × U s ( ρ 1 ) U s * ( ρ 2 ) U s ( ρ 3 ) U s * ( ρ 4 ) × exp ( i k 2 L ( | p 1 ρ 1 | 2 | p 1 ρ 2 | 2 + | p 2 ρ 3 | 2 | p 2 ρ 4 | 2 ) ) × exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 1 , ρ 2 ) + ψ 2 ( p 2 , ρ 3 ) + ψ 2 * ( p 2 , ρ 4 ) ] .
U s ( ρ 1 ) U s * ( ρ 2 ) U s ( ρ 3 ) U s * ( ρ 4 ) = U d ( ρ 1 ) U d * ( ρ 2 ) U d ( ρ 3 ) U d * ( ρ 4 ) + U d ( ρ 1 ) U d * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) + U g ( ρ 1 ) U g * ( ρ 2 ) U d ( ρ 3 ) U d * ( ρ 4 ) + U g ( ρ 1 ) U d * ( ρ 2 ) U d ( ρ 3 ) U g * ( ρ 4 ) + U d ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) U d * ( ρ 4 ) + U g ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) n = 1 6 A n ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) ,
B I ( p 1 , p 2 ) = n = 1 6 B I n ( p 1 , p 2 ) ,
B I n ( p 1 , p 2 ) = ( k 2 π L ) 4 d ρ 1 d ρ 2 d ρ 3 d ρ 4 A n ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) × exp ( i k 2 L ( | p 1 ρ 1 | 2 | p 1 ρ 2 | 2 + | p 2 ρ 3 | 2 | p 2 ρ 4 | 2 ) ) × exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 1 , ρ 2 ) + ψ 2 ( p 2 , ρ 3 ) + ψ 2 * ( p 2 , ρ 4 ) ] .
H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 1 , ρ 2 ) + ψ 2 ( p 2 , ρ 3 ) + ψ 2 * ( p 2 , ρ 4 ) ] = exp { ( 1 / 2 ) [ D ψ ( ρ 1 ρ 2 ) D ψ ( p 1 p 2 , ρ 1 ρ 3 ) + D ψ ( p 1 p 2 , ρ 1 ρ 4 ) + D ψ ( p 1 p 2 , ρ 2 ρ 3 ) D ψ ( p 1 p 2 , ρ 2 ρ 4 ) + D ψ ( ρ 3 ρ 4 ) ] + 2 C l ( p 1 p 2 , ρ 1 ρ 3 ) + 2 C l ( p 1 p 2 , ρ 2 ρ 4 ) } ,
σ I 2 ( p ) = σ I d 2 ( p ) + σ I g 2 ( p ) + 2 C I d I g ( p , p ) + 2 C U d g ( p , p ) ,
σ I d 2 ( p ) = C I d ( p 1 , p 2 ) | p 1 = p 2 = p
σ I g 2 ( p ) = C I g ( p 1 , p 2 ) | p 1 = p 2 = p .
σ I N 2 ( p ) = σ I 2 ( p ) / I r ( p ) 2 ,
σ I N 2
| p j ρ j | 2 = ( p j ρ j ) ( p j ρ j )
exp [ ψ 2 ( p 1 , ρ 2 ) + ψ 2 * ( p 2 , ρ 2 ) ] = exp [ ( | p 1 p 2 | ρ 0 ) 5 / 3 ] ,
Γ d ( p 1 , p 2 ) = R π L 2 exp [ i k ( p 1 2 p 2 2 ) 2 L ] d ρ 2 | U i ( ρ 2 ) | 2 × exp ( i k L ( p 1 p 2 ) ρ 2 ) exp [ ( | p 1 p 2 | ρ 0 ) 5 / 3 ] ,
| U i ( ρ 2 ) | 2 = ( k 2 π L ) 2 d r 1 d r 2 U 0 ( r 1 ) U 0 * ( r 2 ) × exp ( i k | ρ 2 r 1 | 2 2 L i k | ρ 2 r 2 | 2 2 L ) × exp [ ψ 1 ( ρ 2 , r 1 ) + ψ 1 * ( ρ 2 , r 2 ) ] .
| U i ( ρ 2 ) | 2 = ( k 2 π L ) 2 | U 0 | 2 d r 1 d r 2 exp ( r 1 2 + r 2 2 2 α 0 2 ) × exp [ i k 2 L ( 1 L F ) ( r 1 2 r 2 2 ) ] × exp [ i k L ρ 2 ( r 1 r 2 ) ] × exp [ ( | r 1 r 2 | 5 / 3 ρ 0 ) ] ,
Γ d ( p 1 , p 2 ) = R | U 0 | 2 α 0 2 L 2 exp ( i k 2 L ( p 1 2 p 2 2 ) | p 1 p 2 | 2 4 α 0 2 2 | p 1 p 2 ρ 0 | 5 / 3 ) .
exp [ ψ 2 ( p 1 , ρ 1 ) + ψ 2 * ( p 2 , ρ 2 ) ] = exp [ ( 1 / 2 ) D ( p 1 p 2 , ρ 1 ρ 2 ) ] ,
Γ g ( p 1 , p 2 ) = ( k 2 π L ) 2 exp [ i k 2 L ( p 1 2 p 2 2 ) ] × m 1 = 1 M m 2 = 1 M U i * ( ρ m 1 ) U i ( ρ m 2 ) d ρ 1 d ρ 2 × exp ( 1 ( Δ ρ ω ) 2 ( | ρ 1 ρ m 1 | 2 + | ρ 2 ρ m 2 | 2 ) + ( i k / 2 L ) [ ( ρ 1 2 2 p 1 ρ 1 ) ( ρ 2 2 2 p 2 ρ 2 ) ] ( 1 / 2 ) D ( p 1 p 2 , ρ 1 ρ 2 ) ) .
ρ = ρ 1 ρ 2
2 R = ρ 1 + ρ 2 ,
Γ g ( p 1 , p 2 ) = π ( Δ ρ ω ) 2 2 ( k 2 π L ) 2 m 1 = 1 M m 2 = 1 M U i * ( ρ m 1 ) U i ( ρ m 2 ) × d ρ exp [ ( Δ ρ ω ) 2 8 ( k L ) 2 | ρ p | 2 + i k 2 L ( ρ m 1 + ρ m 2 ) ( ρ p ) 1 2 ( Δ ρ ω ) 2 | ρ ρ m 1 + ρ m 2 | 2 i k 2 L ( p 1 + p 2 ) ( ρ p ) 1 2 D ( p , ρ ) ] ,
1 π ( Δ ρ ω ) 2 exp ( | ρ ρ m | 2 ( Δ ρ ω ) 2 ) δ ( ρ ρ m )
Γ g ( p 1 , p 2 ) ( Δ ρ ω ) 4 ( k 2 L ) 2 m 1 = 1 M m 2 = 1 M U i * ( ρ m 1 ) U i ( ρ m 2 ) × exp [ i k 2 L ( | ρ m 1 p 1 | 2 | ρ m 2 p 2 | 2 ) ( Δ ρ ω ) 2 8 ( k L ) 2 | ρ m 1 ρ m 2 p | 2 ( 1 / 2 ) D ( p , ρ m 1 ρ m 2 ) ] .
I r ( p ) = Γ ( p , p ) = R | U 0 | 2 α 0 2 L 2 + ( Δ ρ ω ) 2 4 ( k L ) 2 × m 1 = 1 M m 2 = 1 M U i * ( ρ m 1 ) U i ( ρ m 2 ) × ρ d ρ J 0 ( ρ | k L ( p ρ m + ) + i ρ m ( Δ ρ ω ) 2 | ) × exp [ ( Δ ρ ω ) 2 8 ( k L ) 2 ρ 2 ρ 2 + ρ m 2 2 ( Δ ρ ω ) 2 ( ρ ρ 0 ) 5 / 3 ] ,
ρ m = ρ m 1 ρ m 2
2 ρ m + = ρ m 1 + ρ m 2
I r ( p ) = R | U 0 | 2 α 0 2 L 2 + ( Δ ρ ω ) 4 ( k 2 L ) 2 m 1 = 1 M m 2 = 1 M U i * ( ρ m 1 ) U i ( ρ m 2 ) × exp [ i k L ρ m ( ρ m + p ) ( Δ ρ ω ) 2 8 ( k L ) 2 ρ m 2 ( ρ m ρ 0 ) 5 / 3 ] .
I r ( p ) | M = 1 = R | U 0 | 2 α 0 2 L 2 + ( Δ ρ ω ) 4 ( k 2 L ) 2 I i ( ρ g ) ,
I i ( ρ g ) = U i * ( ρ g ) U i ( ρ g ) .
I r ( p ) | M = 2 = R | U 0 | 2 α 0 2 L 2 + ( Δ ρ ω ) 4 ( k 2 L ) 4 × { I i ( ρ g 1 ) + I i ( ρ g 2 ) + U i * ( ρ g 1 ) U i ( ρ g 2 ) × exp [ i k 2 L ( ρ g 1 ρ g 2 ) ( ρ g 1 + ρ g 2 2 p ) ( Δ ρ ω ) 2 8 ( k L ) 2 | ρ g 1 ρ g 2 | 2 ( | ρ g 1 ρ g 2 | ρ 0 ) 5 / 3 ] + U i * ( ρ g 1 ) U i ( ρ g 2 ) exp [ i k 2 L ( ρ g 2 ρ g 1 ) ( ρ g 2 + ρ g 1 2 p ) ( Δ ρ ω ) 2 8 ( k L ) 2 ( | ρ g 2 ρ g 1 | 2 ) ( | ρ g 2 ρ g 1 | ρ 0 ) 5 / 3 ] } .
A 1 ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = U d ( ρ 1 ) U d * ( ρ 2 ) U d ( ρ 3 ) U d * ( ρ 4 ) = U d ( ρ 1 ) U d * ( ρ 2 ) U d ( ρ 3 ) U d * ( ρ 4 ) + U d ( ρ 1 ) U d * ( ρ 4 ) U d * ( ρ 2 ) U d ( ρ 3 ) = ( 4 π k 2 ) 2 [ I d ( ρ 1 ) I d ( ρ 3 ) δ ( ρ 1 ρ 2 ) δ ( ρ 3 ρ 4 ) + I d ( ρ 1 ) I d ( ρ 3 ) δ ( ρ 1 ρ 4 ) δ ( ρ 3 ρ 2 ) ]
B I 1 ( p 1 , p 2 ) = 1 π 2 L 4 d ρ 2 d ρ 4 I d ( ρ 2 ) I d ( ρ 4 ) × { exp [ 4 C l ( p 1 p 2 , ρ 2 ρ 4 ) ] + exp [ ( i k / L ) ( p 1 p 2 ) ( ρ 2 ρ 4 ) ] × H ( p 1 , p 2 ; ρ 4 , ρ 2 , ρ 2 , ρ 4 ) } .
A 2 ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = U d ( ρ 1 ) U d * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) = U d ( ρ 1 ) U d * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) = ( 4 π / k 2 ) I d ( ρ 1 ) U g ( ρ 3 ) U g * ( ρ 4 ) δ ( ρ 1 ρ 2 )
B I 2 ( p 1 , p 2 ) = k 2 4 π 3 L 4 × d ρ 2 d ρ 3 d ρ 4 I d ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) × exp [ ( i k / 2 L ) ( ρ 3 + ρ 4 2 p 2 ) ( ρ 3 ρ 4 ) ] × H ( p 1 , p 2 ; ρ 2 , ρ 2 , ρ 3 , ρ 4 ) .
A 3 ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = U g ( ρ 1 ) U g * ( ρ 2 ) U d ( ρ 3 ) U d * ( ρ 4 ) = U g ( ρ 1 ) U g * ( ρ 2 ) U d ( ρ 3 ) U d * ( ρ 4 ) = ( 4 π / k 2 ) U g ( ρ 1 ) U g * ( ρ 2 ) I d ( ρ 3 ) δ ( ρ 3 ρ 4 )
B I 3 ( p 1 , p 2 ) = k 2 4 π 3 L 4 × d ρ 1 d ρ 2 d ρ 4 I d ( ρ 4 ) U g ( ρ 1 ) U g * ( ρ 2 ) × exp [ ( i k / 2 L ) ( ρ 1 + ρ 2 2 p 1 ) ( ρ 1 ρ 2 ) ] × H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 4 , ρ 4 ) .
A 4 ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = U g ( ρ 1 ) U d * ( ρ 2 ) U d ( ρ 3 ) U g * ( ρ 4 ) = U g ( ρ 1 ) U g * ( ρ 4 ) U d * ( ρ 2 ) U d ( ρ 3 ) = ( 4 π / k 2 ) U g ( ρ 1 ) U g * ( ρ 4 ) I d ( ρ 3 ) δ ( ρ 3 ρ 2 )
B I 4 ( p 1 , p 2 ) = k 2 4 π 3 L 4 × d ρ 1 d ρ 2 d ρ 4 I d ( ρ 4 ) U g ( ρ 1 ) U g * ( ρ 2 ) × exp { ( i k / 2 L ) [ ( ρ 1 + ρ 2 2 p 1 ) ( ρ 1 ρ 2 ) + ( ρ 2 + ρ 4 2 p 2 ) ( ρ 2 ρ 4 ) ] } H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 2 , ρ 4 ) .
A 5 ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = U d ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) U d * ( ρ 4 ) = U d ( ρ 1 ) U d * ( ρ 4 ) U g * ( ρ 2 ) U g ( ρ 3 ) = ( 4 π / k 2 ) I d ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) δ ( ρ 1 ρ 4 )
B I 5 ( p 1 , p 2 ) = k 2 4 π 3 L 4 × d ρ 2 d ρ 3 d ρ 4 I d ( ρ 2 ) U g * ( ρ 2 ) U g ( ρ 3 ) × exp { ( i k / 2 L ) [ ( ρ 4 + ρ 2 2 p 1 ) ( ρ 4 ρ 2 ) + ( ρ 3 + ρ 4 2 p 2 ) ( ρ 3 ρ 4 ) ] } H ( p 1 , p 2 ; ρ 4 , ρ 2 , ρ 3 , ρ 4 ) .
A 6 ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) = U g ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 )
B I 6 ( p 1 , p 2 ) = ( k 2 π L ) 4 d ρ 1 d ρ 2 d ρ 3 d ρ 4 × U g ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) × exp { ( i k / 2 L ) [ | p 1 ρ 1 | 2 | p 1 ρ 2 | 2 + | p 2 ρ 3 | 2 | p 2 ρ 4 | 2 } H ( p 1 , p 2 ; ρ 1 , ρ 2 , ρ 3 , ρ 4 ) .
I r ( p ) = I d ( p ) + I g ( p ) = I d ( p ) + I g ( p ) ,
I r ( p 1 ) I r ( p 2 ) = I d ( p 1 ) I d ( p 2 ) + I d ( p 1 ) I g ( p 2 ) + I g ( p 1 ) I d ( p 2 ) + I g ( p 1 ) I g ( p 2 ) .
C I d ( p 1 , p 2 ) = I d ( p 1 ) I d ( p 2 ) I d ( p 1 ) I d ( p 2 ) = B I 1 ( p 1 , p 2 ) I d ( p 1 ) I d ( p 2 ) ,
C I g ( p 1 , p 2 ) = I g ( p 1 ) I g ( p 2 ) I g ( p 1 ) I g ( p 2 ) = B I 6 ( p 1 , p 2 ) I g ( p 1 ) I g ( p 2 ) ,
C I d I g ( p 1 , p 2 ) = I d ( p 1 ) I g ( p 2 ) I d ( p 1 ) I g ( p 2 ) = B I 2 ( p 1 , p 2 ) I d ( p 1 ) I g ( p 2 ) ,
C I d I g ( p 2 , p 1 ) = I d ( p 2 ) I g ( p 1 ) I d ( p 2 ) I g ( p 1 ) = B I 3 ( p 1 , p 2 ) I d ( p 2 ) I g ( p 1 ) ,
C U d g ( p 1 , p 2 ) = U d ( p 1 ) U g * ( p 1 ) U d * ( p 2 ) U g ( p 2 ) = B I 5 ( p 1 , p 2 ) ,
C U d g * ( p 1 , p 2 ) = U d * ( p 1 ) U g ( p 1 ) U d ( p 2 ) U g * ( p 2 ) = B I 4 ( p 1 , p 2 ) ,
U g ( ρ 1 ) U g * ( ρ 2 ) U g ( ρ 3 ) U g * ( ρ 4 ) = m 1 = 1 M m 2 = 1 M m 3 = 1 M m 4 = 1 M U i * ( ρ m 1 ) U i ( ρ m 2 ) U i * ( ρ m 3 ) U i ( ρ m 4 ) × exp ( 1 ( Δ ρ ω ) 2 [ ( ρ 1 ρ m 1 ) 2 + ( ρ 2 ρ m 2 ) 2 + ( ρ 3 ρ m 3 ) 2 + ( ρ 4 ρ m 4 ) 2 ] ) ,
U i * ( ρ m 1 ) U i ( ρ m 2 ) U i * ( ρ m 3 ) U i ( ρ m 4 ) = ( k 2 π L ) × d r 1 d r 2 d r 3 d r 4 U * ( r 1 ) U ( r 2 ) U * ( r 3 ) U ( r 4 ) × exp ( i k 2 L [ ( ρ m 1 r 1 ) 2 ( ρ m 2 r 2 ) 2 + ( ρ m 3 r 3 ) 2 ( ρ m 4 r 4 ) 2 ) × H ( ρ m 1 , ρ m 2 , ρ m 3 , ρ m 4 ; r 1 , r 2 , r 3 , r 4 ) .
C I ( p 1 , p 2 ) = C I d ( p 1 , p 2 ) + C I g ( p 1 , p 2 ) + C I d I g ( p 1 , p 2 ) + C I d I g ( p 2 , p 1 ) + C U d g ( p 1 , p 2 ) + C U d g * ( p 1 , p 2 ) .
σ I 2 ( p ) = σ I d 2 ( p ) + σ I g 2 ( p ) + 2 C I d I g ( p , p ) + 2 C U d g ( p , p ) ,