Abstract

A two-point, joint-probability-density function for the intensity of a laser-generated speckle field after propagation through turbulence is developed by using the method of compound distributions. The parameters of the joint-density function are derived in terms of the path length, strength of turbulence, wave number, beam size, and degree of focus. Closed-form expressions for some of the joint moments of the intensity are developed and compared with experimental data. Good agreement is obtained.

© 1990 Optical Society of America

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  1. V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity of a laser generated speckle pattern in the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1981), p. 12.
  2. V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the received intensity of a target generated speckle pattern in the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213–1218 (1982).
    [CrossRef]
  3. L. C. Andrews, R. L. Philips, “I-K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]
  4. J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  5. J. H. Churnside, S. F. Clifford, “Log-normal Rician probability density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  6. V. S. Rao Gudimetla, J. F. Holmes, “Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1986), paper FS7, pp. 19–25.
  7. J. W. Strohbehn, ed., Laser Beam Propagation through the Atmosphere (Springer-Verlag, New York, 1978).
    [CrossRef]
  8. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  9. M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.
  10. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
    [CrossRef]
  11. I. S. Gradshteyn, I. M. Rhyzhik, Tables of Integrals, Series and Products, (Academic, New York, 1965).
  12. F. J. 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]
  13. M. A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, D.C., 1970).

1987 (2)

1985 (1)

1982 (1)

1980 (1)

1975 (1)

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

Abramowitz, M. A.

M. A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, D.C., 1970).

Andrews, L. C.

Churnside, J. H.

Clifford, S. F.

Fante, R. L.

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

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Rhyzhik, Tables of Integrals, Series and Products, (Academic, New York, 1965).

Hill, R. J.

Holmes, F. J.

Holmes, J. F.

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the received intensity of a target generated speckle pattern in the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213–1218 (1982).
[CrossRef]

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity of a laser generated speckle pattern in the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1981), p. 12.

V. S. Rao Gudimetla, J. F. Holmes, “Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1986), paper FS7, pp. 19–25.

Kerr, J. R.

Lee, M. H.

Nakagami, M.

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.

Philips, R. L.

Rao Gudimetla, V. S.

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the received intensity of a target generated speckle pattern in the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213–1218 (1982).
[CrossRef]

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity of a laser generated speckle pattern in the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1981), p. 12.

V. S. Rao Gudimetla, J. F. Holmes, “Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1986), paper FS7, pp. 19–25.

Rhyzhik, I. M.

I. S. Gradshteyn, I. M. Rhyzhik, Tables of Integrals, Series and Products, (Academic, New York, 1965).

Stegun, I.

M. A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, D.C., 1970).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Proc. IEEE (1)

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

Other (7)

M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
[CrossRef]

I. S. Gradshteyn, I. M. Rhyzhik, Tables of Integrals, Series and Products, (Academic, New York, 1965).

V. S. Rao Gudimetla, J. F. Holmes, “Two-point joint density function of the intensity for a laser speckle pattern after propagation through the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1986), paper FS7, pp. 19–25.

J. W. Strohbehn, ed., Laser Beam Propagation through the Atmosphere (Springer-Verlag, New York, 1978).
[CrossRef]

M. A. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, D.C., 1970).

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity of a laser generated speckle pattern in the turbulent atmosphere,” in Digest of the Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1981), p. 12.

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Tables (1)

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Table 1 Comparison of Theoretical and Experimental Joint Moments of Intensitya

Equations (59)

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P a s ( I ) = 0 P s ( I λ = x ) P a ( x ) d x ,
P a s ( I 1 , I 2 ) = 0 0 P s ( I 1 , I 2 λ 1 = x 1 , λ 2 = x 2 ) P a ( x 1 , x 2 ) d x 1 d x 2 ,
P s ( I 1 , I 2 λ 1 , λ 2 ) = exp [ I 1 / λ 1 ( 1 ρ s ) ] λ 1 exp [ I 2 / λ 2 ( 1 ρ s ) ] λ 2 ( 1 ρ s ) × I 0 [ 2 1 ρ s ( ρ s I 1 I 2 λ 1 λ 2 ) 1 / 2 ] .
P a ( I 1 , I 2 ) = ( I 1 I 2 ) ( M 1 ) / 2 Γ ( M ) exp [ M I 1 / I 1 ( 1 ρ a ) ] I 1 ( M + 1 ) / 2 × exp [ M I 2 / I 2 ( 1 ρ a ) ] I 2 ( M + 1 ) / 2 M ( M + 1 ) ( 1 ρ a ) ρ a ( M 1 ) / 2 × I M 1 [ 2 M 1 ρ a ( ρ a I 1 I 2 I 1 I 2 ) 1 / 2 ] .
P a s ( I 1 , I 2 ) = ( I 1 I 2 ) ( M 1 ) / 2 M M + 1 Γ ( M ) ( 1 ρ a ) ρ a ( M 1 ) / 2 I 1 I 2 ( 1 ρ s ) × 0 0 exp [ M I 1 / x 1 ( 1 ρ a ) x 1 / I 1 × ( 1 ρ s ) ] x 1 ( M + 1 ) / 2 exp [ M I 2 / x 2 ( 1 ρ a ) x 2 / I 2 ( 1 ρ s ) ] x 2 ( M + 1 ) / 2 { I M 1 [ 2 M 1 ρ a × ( ρ a I 1 I 2 x 1 x 2 ) 1 / 2 ] } { I 0 [ 2 1 ρ s ( ρ s x 1 x 2 I 1 I 2 ) 1 / 2 ] } × d x 1 d x 2 .
I M 1 [ 2 M 1 ρ a ( ρ a I 1 I 2 x 1 x 2 ) 1 / 2 ] = N M M 1 + 2 N ( ρ a I 1 I 2 ) N + ( M 1 ) / 2 Γ ( N + 1 ) Γ ( N + M ) ( 1 ρ a ) M 1 + 2 N ( x 1 x 2 ) N ( M 1 ) / 2
I 0 [ 2 1 ρ s ( ρ s x 1 x 2 I 1 I 2 ) 1 / 2 ] = S ( ρ s x 1 x 2 ) S Γ 2 ( S + 1 ) Γ ( 1 ρ s ) 2 S ( I 1 I 2 ) S ,
P a s ( I 1 , I 2 ) = ( I 1 I 2 ) M 1 M M + 1 Γ ( M ) ( 1 ρ a ) ρ a ( M 1 ) / 2 I 1 I 2 ( 1 ρ s ) N S M M 1 + 2 N ρ a N + ( M 1 ) / 2 ( I 1 I 2 ) N ρ s S Γ ( N + 1 ) Γ ( N + M ) ( 1 ρ a ) M 1 + 2 N Γ 2 ( S + 1 ) Γ ( 1 ρ s ) 2 S ( I 1 I 2 ) S × d x 1 exp [ M I 1 x 1 ( 1 ρ a ) x 1 I 1 ( 1 ρ s ) ] x 1 M N + S d x 2 exp [ M I 2 x 2 ( 1 ρ a ) x 2 I 2 ( 1 ρ s ) ] x 2 M N + S .
Int 1 = 2 [ M I 1 I 1 ( 1 ρ s ) ( 1 ρ a ) ] ( S M N + 1 ) / 2 × K S M N + 1 { 2 [ M I 1 ( 1 ρ a ) ( 1 ρ s ) I 1 ] 1 / 2 }
Int 2 = 2 [ M I 2 I 2 ( 1 ρ s ) ( 1 ρ a ) ] ( S M N + 1 ) / 2 × K S M N + 1 { 2 [ M I 2 ( 1 ρ a ) ( 1 ρ s ) I 2 ] 1 / 2 } .
P a s ( I 1 , I 2 ) = N S 4 ( I 1 I 2 ) ( M + S + N 1 ) / 2 M M + S + N + 1 ρ s S ρ a N Γ ( M ) Γ ( N + 1 ) Γ ( N + M ) Γ 2 ( S + 1 ) ( 1 ρ a ) s + N + 1 ( 1 ρ s ) S + M + N ( I 1 I 2 ) ( S + M + N + 1 ) / 2 × K S M N + 1 { 2 [ M I 1 I 1 ( 1 ρ a ) ( 1 ρ s ) ] 1 / 2 } K S M N + 1 { 2 [ M I 2 I 2 ( 1 ρ a ) ( 1 ρ s ) ] 1 / 2 } .
P a s ( I 1 , I 2 ) = 4 ( I 1 I 2 ) ( M 1 ) / 2 M M + 1 Γ ( M ) ρ a ( M 1 ) / 2 ( 1 ρ a ) ( 1 ρ s ) M ( I 1 I 2 ) ( M + 1 ) / 2 × N S ( I 1 I 2 ) ( S + N ) / 2 M S + N ρ s S ρ s S ρ a N Γ ( N + 1 ) Γ ( N + M ) Γ 2 ( S + 1 ) ( 1 ρ a ) S + N ( 1 ρ s ) S + N ( I 1 I 2 ) ( S + N ) / 2 × K S M N + 1 { 2 [ M I 1 I 1 ( 1 ρ a ) ( 1 ρ s ) ] 1 / 2 } K S M N + 1 { 2 [ M I 2 I 2 ( 1 ρ a ) ( 1 ρ s ) ] 1 / 2 } .
d I 1 I 1 p + ( S + N + M 1 ) / 2 K s + 1 M N { 2 [ M I 1 I 1 ( 1 ρ s ) ( 1 ρ a ) ] 1 / 2 }
2 d x x 2 p + S + M + N K s + 1 M N { 2 x [ M I 1 ( 1 ρ a ) ( 1 ρ a ) ] 1 / 2 } = 1 2 [ M I 1 ( 1 ρ s ) ( 1 ρ a ) ] p ( S + N + M + 1 ) / 2 × Γ ( S + 1 + p ) Γ ( M + N + p ) .
I 1 p I 2 q I 1 p I 2 q = 1 Γ ( M ) N S ρ s S ρ a N Γ ( N + 1 ) Γ ( N + M ) Γ 2 ( S + 1 ) × M p q ( 1 ρ s ) p + q + 1 ( 1 ρ a ) p + q + M × Γ ( S + 1 + p ) Γ ( S + 1 + q ) × Γ ( M + N + p ) Γ ( M + N + q ) .
I 1 I 2 I 1 I 2 = ( 1 + ρ a ) ( 1 + ρ a M ) .
I 1 2 I 2 I 1 2 I 2 = 2 ( 1 + 2 ρ s ) ( 1 + 1 M ) ( 1 + 2 ρ a M ) .
I 1 3 I 2 I 1 2 I 2 = 6 ( 1 + 3 ρ s ) ( 1 + 1 M ) ( 1 + 2 ρ a M ) ( 1 + 3 ρ a M ) .
I 1 2 I 2 2 I 1 2 I 2 2 = 4 ( 1 + 4 ρ s + ρ s 2 ) ( 1 + 1 M ) × [ 1 + 1 M + 4 ρ a M ( 1 + 1 M ) + 2 ρ a 2 M 2 ] .
I 1 3 I 2 2 I 1 3 I 2 2 = 12 ( 1 + 6 ρ s + 3 ρ s 2 ) ( 1 + 1 M ) ( 1 + 2 M ) × [ 1 + 1 M + 6 ρ a M ( 1 + 1 M ) + 6 ρ a 2 M 2 ] .
σ I 2 = I 2 I 2 I 2 = 1 + 2 M .
M = 2 σ I 2 1 ,
C I ( p , v τ ) = I 1 I 2 I 1 I 2 I 1 I 2 = ( 1 + ρ s ) ( 1 + ρ a M ) 1 = ρ a M + ρ s + ρ s ρ a M ,
ρ s = C I C n 2 = 0 = exp { p 2 2 [ 1 α 0 2 + ( k α 0 L ) 2 ( 1 L F ) 2 ] } ,
ρ a = M ( C I ρ s ) 1 + ρ s ,
σ I 2 = ( k L ) 2 0 0 r d r ρ d ρ { 2 exp [ 4 C χ ( p , ρ , 0 ) ] 1 } J 0 ( k L ρ r ) f 2 ( r ) ,
C I = C I 1 + C I 2 ,
C I 1 = 1 2 π ( k L ) 2 0 2 π d θ 0 r d r 0 ρ d ρ × { exp [ C χ ( p , ρ , v τ ) ] 1 } J 0 ( k L ρ r ) f 2 ( r ) ,
C I 2 = 1 2 π ( k L ) 2 exp [ 2 ( p ρ 0 ) 5 / 3 ] × 0 2 d θ 0 r d r 0 ρ d ρ f 1 ( p , ρ , v τ ) J 0 ( k L ρ r ) f 2 ( r ) ,
f 1 p , ρ , v τ ) = exp { i k L ρ ρ 2 ( p ρ 0 ) 5 / 3 + 8 3 1 ρ 0 5 / 3 × [ 0 1 d t | p t + ( 1 t ) ρ v τ | 5 / 3 + 0 1 d t | p t ( 1 t ) ρ v τ | 5 / 3 ] + 2 C χ ( p , ρ , v τ ) + 2 C χ ( p ρ , v τ ) } ,
f 2 ( r ) = exp ( r 2 { 1 2 α 0 2 + 2 [ k α 0 2 L ( 1 L F ) ] 2 } 2 ( r ρ 0 ) 5 / 3 ) .
C χ ( p , ρ , v τ ) = 132 π 2 k 2 0 1 C n 2 ( t ) d t 0 d u u 8 / 3 × sin 2 [ u 2 t ( 1 t ) L 2 k ] J 0 [ u | p t + ρ ( 1 t ) v τ | )
ρ 0 = ( 0.545625 C n 2 L k 2 ) 3.5 .
σ I 2 = m b m { 2 exp [ 4 C χ ( 0 , L P m k A , 0 ) ] 1 } ,
C I = 1 2 π m b m d θ { exp [ 4 C χ ( p , L P m k A , 0 ) ] 1 + exp [ 2 ( P ρ 0 ) 5 / 3 ] f 1 ( p , L P m k A , 0 ) } ,
b m = 2 A 2 J 1 2 ( P m ) 0 A x f 2 ( x ) J 0 ( P m A X ) d x .
C χ ( p , ρ , v τ ) = 0.132 π 2 k 2 L C n 2 0 1 d t H ( a , b ) ,
H ( a , b ) = 3 b 5 / 3 Γ ( 1 / 3 ) 20 ( π ) 1 / 2 Γ ( 11 / 6 ) cos ( π 6 ) + 3 a 5 / 3 5 ( 2 ) 1 / 6 Γ ( 1 6 ) × { exp [ i ( π / 12 ) ] 2 i F 1 ( 5 6 ; 1 ; i b 2 8 a 2 ) exp [ i ( π / 12 ) ] 2 i F 1 ( 5 6 ; 1 ; i b 2 8 a 2 ) } ,
a 2 = t ( 1 t ) L 2 k
b = | p t + ρ ( 1 + t ) v τ | .
I = I s + I b , I 2 = I s 2 + I b 2 + 2 I s I b , I 3 = I s 3 + 3 I s 2 I b + 3 I s I b 2 + I b 3 ,
I 2 I 2 I 1 I 2
I 1 2 I 2 I 1 2 I 2
I 1 3 I 2 I 1 3 I 2
I 1 2 I 2 2 I 1 2 I 2 2
I 1 3 I 2 2 I 1 3 I 2 2
C χ ( p , ρ , v τ ) = 0.132 π 2 k 2 L 0 1 d t C n 2 ( t ) 0 d u u 8 / 3 × sin 2 [ u 2 t ( 1 t ) L 2 k ] J 0 [ u | p t + ρ ( 1 t ) v τ | ] .
C χ ( p , ρ , v τ ) = 0.132 π 2 k 2 L C n 2 0 1 d t H ( a , b ) ,
H ( a , b ) = 0 d u u 8 / 3 sin 2 ( a 2 u 2 ) J 0 ( b u ) ,
H ( a , b ) = 6 a 2 5 0 d u u 2 / 3 sin ( 2 a 2 u 2 ) J 0 ( b u ) 3 5 b 0 d u u 5 / 3 × sin 2 ( 2 a 2 u 2 ) J 1 ( b u ) .
J 0 ( b u ) = 2 π 0 1 d x 1 ( 1 x 2 ) 1 / 2 cos ( bux ) ,
J 1 ( b u ) = 2 b u π 0 1 d x ( 1 x 2 ) 1 / 2 cos ( bux ) ,
H ( a , b ) = 12 a 2 5 π 0 1 d x 1 ( 1 x 2 ) 1 / 2 0 d u u 2 / 3 sin ( 2 a 2 u 2 ) × cos ( bux ) 3 b 2 5 π 0 1 d x ( 1 x 2 ) 1 / 2 0 d u u 2 / 3 × [ 1 cos ( 2 a 2 u 2 ) ] cos ( bux ) .
0 d u u 2 / 3 cos ( 2 a 2 u 2 ) cos ( bux ) = lim 0 0 d u u 2 / 3 exp ( u 2 ) [ exp ( 2 i a 2 u 2 ) + exp ( 2 i a 2 u 2 ) ] 2 × cos ( bux ) = Γ ( 1 / 6 ) 4 [ ( 2 i a 2 ) 1 / 6 F l ( 1 6 ; 1 2 ; i b 2 x 2 8 a 2 ) + ( 2 i a 2 ) 1 / 6 F 1 ( 1 6 ; 1 2 ; i b 2 x 2 8 a 2 ) ] ,
H ( a , b ) = 0 1 d x ( 1 x 2 ) 1 / 2 3 a 2 5 i π Γ ( 1 6 ) [ ( 2 i a 2 ) 1 / 6 × F 1 ( 1 6 ; 1 2 ; i b 2 x 2 8 a 2 ) ( 2 i a 2 ) 1 / 6 F 1 ( 1 6 ; 1 2 ; i b 2 x 2 8 a 2 ) ] + 0 1 d x ( 1 x 2 ) 1 / 2 3 b 2 20 π Γ ( 1 6 ) [ ( 2 i a 2 ) 1 / 6 × F 1 ( 1 6 ; 1 2 ; i b 2 x 2 8 a 2 ) + ( 2 i a 2 ) 1 / 6 × F 1 ( 1 6 ; 1 2 ; i b 2 x 2 8 a 2 ) ] 0 1 d x ( 1 x 2 ) 1 / 2 3 b 2 5 π Γ ( 1 / 3 ) ( b x ) 1 / 3 × cos ( π 6 ) .
H ( a , b ) = 3 b 5 / 3 20 ( π ) 1 / 2 Γ 2 ( 1 / 3 ) Γ ( 11 / 6 ) cos ( π 6 ) + 3 a 5 / 3 5 ( 2 ) 1 / 6 × Γ ( 1 6 ) [ exp ( i π / 12 ) 2 i F 1 ( 1 6 ; 1 ; i b 2 8 a 2 ) exp ( i π / 12 ) 2 i F 1 ( 1 6 ; 1 ; i b 2 8 a 2 ) + b 2 8 a 2 exp ( i π / 12 ) 2 × F 1 ( 1 6 ; 2 ; i b 2 8 a 2 ) + b 2 8 a 2 exp ( i π / 12 ) 2 × F 1 ( 1 6 ; 2 ; i b 2 8 a 2 ) ] .
β F 1 ( α ; β ; z ) z F 1 ( α ; β + 1 ; z ) = β F 1 ( α 1 ; β ; z )
H ( a , b ) = 3 b 5 / 3 20 ( π ) 1 / 2 Γ 2 ( 1 / 3 ) Γ ( 11 / 6 ) cos ( π 6 ) + 3 a 5 / 3 5 ( 2 ) 1 / 6 Γ ( 1 6 ) × { [ exp ( i π / 12 ) ] 2 i F 1 ( 5 6 ; 1 ; i b 2 8 a 2 ) exp [ i ( π / 12 ) ] 2 i F 1 ( 5 6 ; 1 ; i b 2 8 a 2 ) } .
C χ ( p , v τ ) = 0.132 π 2 k 2 L C n 2 0 1 d t × H { [ 5 ( 1 t ) L 2 k ] 1 / 2 | t p + ( 1 t ) ρ v τ | } ,

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