Abstract

The propagation of spherical waves in a turbulent medium is considered. In particular, the case of nonstationary statistics is examined in general and then applied to the specific case of vertical propagation in the atmosphere. The analysis uses the Rytov approximation and the perturbation technique of J. B. Keller. The model of Cn2(h) variation is exponential and similar to that utilized by Tatarski.

The results are compared to other known results for plane and spherical waves in both homogeneous and locally homogeneous random media. In addition, the optimum aperture results of D. L. Fried are examined for this nonstationary case. The marked dependence on the height of the observer and parameters describing the turbulence distribution are noted.

© 1969 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).
  2. R. A. Silverman, IEEE Trans. Inform. Theory IT-3, 182 (1957).
    [Crossref]
  3. R. A. Silverman, Proc. Cambridge Phil. Soc. 54, 530 (1958).
    [Crossref]
  4. R. A. Silverman, Communs. Pure and Appl. Math. 12, 373 (1959).
    [Crossref]
  5. See Ref. 7, for a review of the validity of the Rytov procedure, particularly for the optical case.
  6. J. B. Keller, Proc. Sym. in Appl. Math., Am. Math. Soc. 13, 227 (1962).
    [Crossref]
  7. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968); D. L. Fried, Proc. IEEE 55, 57 (1967).
    [Crossref]
  8. F. P. Carlson, dissertation, University of Washington (1967).
  9. See Appendix A for a different approach to the terms in the Rytov approximation.
  10. A. D. Wheelon, J. Res. Natl. Bur. Std. (U. S.) Radio Prop. 63D, 1959.
  11. A. N. Kolmogorov, Compt. Rend. Doklady Akad. Nauk U.S.S.R. 30, 301 (1941), contained in S. K. Friedlander and L. Topper, Turbulence (Interscience Publishers, Inc., John Wiley & Sons, Inc., New York, 1961).
  12. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962).
  13. ∫0∞[1-J0(x)]x¯Pdx=π[2P(Γ(P+12))2×sin-1(π(P-1)2)]-1.
  14. D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966).
    [Crossref]
  15. W. E. Gordon, Proc. IRE 43, 23 (1955).
    [Crossref]
  16. I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
    [Crossref]
  17. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).
  18. W. Magnus and F. Oberhettinger, Formulas to Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949).
  19. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]

1968 (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968); D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

1967 (2)

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
[Crossref]

1966 (1)

1965 (1)

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[Crossref]

1962 (1)

J. B. Keller, Proc. Sym. in Appl. Math., Am. Math. Soc. 13, 227 (1962).
[Crossref]

1959 (2)

R. A. Silverman, Communs. Pure and Appl. Math. 12, 373 (1959).
[Crossref]

A. D. Wheelon, J. Res. Natl. Bur. Std. (U. S.) Radio Prop. 63D, 1959.

1958 (1)

R. A. Silverman, Proc. Cambridge Phil. Soc. 54, 530 (1958).
[Crossref]

1957 (1)

R. A. Silverman, IEEE Trans. Inform. Theory IT-3, 182 (1957).
[Crossref]

1955 (1)

W. E. Gordon, Proc. IRE 43, 23 (1955).
[Crossref]

1941 (1)

A. N. Kolmogorov, Compt. Rend. Doklady Akad. Nauk U.S.S.R. 30, 301 (1941), contained in S. K. Friedlander and L. Topper, Turbulence (Interscience Publishers, Inc., John Wiley & Sons, Inc., New York, 1961).

Carlson, F. P.

F. P. Carlson, dissertation, University of Washington (1967).

Chabot, A.

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[Crossref]

Fried, D. L.

Goldstein, I.

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[Crossref]

Gordon, W. E.

W. E. Gordon, Proc. IRE 43, 23 (1955).
[Crossref]

Keller, J. B.

J. B. Keller, Proc. Sym. in Appl. Math., Am. Math. Soc. 13, 227 (1962).
[Crossref]

Kolmogorov, A. N.

A. N. Kolmogorov, Compt. Rend. Doklady Akad. Nauk U.S.S.R. 30, 301 (1941), contained in S. K. Friedlander and L. Topper, Turbulence (Interscience Publishers, Inc., John Wiley & Sons, Inc., New York, 1961).

Magnus, W.

W. Magnus and F. Oberhettinger, Formulas to Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949).

Miles, P. A.

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[Crossref]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Formulas to Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949).

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Silverman, R. A.

R. A. Silverman, Communs. Pure and Appl. Math. 12, 373 (1959).
[Crossref]

R. A. Silverman, Proc. Cambridge Phil. Soc. 54, 530 (1958).
[Crossref]

R. A. Silverman, IEEE Trans. Inform. Theory IT-3, 182 (1957).
[Crossref]

Strohbehn, J. W.

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968); D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).

Wheelon, A. D.

A. D. Wheelon, J. Res. Natl. Bur. Std. (U. S.) Radio Prop. 63D, 1959.

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962).

Communs. Pure and Appl. Math. (1)

R. A. Silverman, Communs. Pure and Appl. Math. 12, 373 (1959).
[Crossref]

Compt. Rend. Doklady Akad. Nauk U.S.S.R. (1)

A. N. Kolmogorov, Compt. Rend. Doklady Akad. Nauk U.S.S.R. 30, 301 (1941), contained in S. K. Friedlander and L. Topper, Turbulence (Interscience Publishers, Inc., John Wiley & Sons, Inc., New York, 1961).

IEEE Trans. Inform. Theory (1)

R. A. Silverman, IEEE Trans. Inform. Theory IT-3, 182 (1957).
[Crossref]

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Std. (U. S.) Radio Prop. (1)

A. D. Wheelon, J. Res. Natl. Bur. Std. (U. S.) Radio Prop. 63D, 1959.

Proc. Cambridge Phil. Soc. (1)

R. A. Silverman, Proc. Cambridge Phil. Soc. 54, 530 (1958).
[Crossref]

Proc. IEEE (2)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968); D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[Crossref]

Proc. IRE (1)

W. E. Gordon, Proc. IRE 43, 23 (1955).
[Crossref]

Proc. Sym. in Appl. Math., Am. Math. Soc. (1)

J. B. Keller, Proc. Sym. in Appl. Math., Am. Math. Soc. 13, 227 (1962).
[Crossref]

Quart. Appl. Math. (1)

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Other (7)

W. Magnus and F. Oberhettinger, Formulas to Theorems for the Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949).

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962).

∫0∞[1-J0(x)]x¯Pdx=π[2P(Γ(P+12))2×sin-1(π(P-1)2)]-1.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).

F. P. Carlson, dissertation, University of Washington (1967).

See Appendix A for a different approach to the terms in the Rytov approximation.

See Ref. 7, for a review of the validity of the Rytov procedure, particularly for the optical case.

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Figures (3)

Fig. 1
Fig. 1

Pi factor relating locally homogeneous spherical waves to plane waves.

Fig. 2
Fig. 2

Camma factor relating locally homogeneous waves to homogeneous plane waves.

Fig. 3
Fig. 3

Plot of the change of optimum aperture size against turbulent-regime size, h0, or observer distance, L.

Equations (94)

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X 2 = 4 π k 2 0 K d K 0 L C n 2 ( r ) Φ n ( 0 ) ( K L η ) × sin 2 ( L ( L - η ) K 2 2 k η ) ( L η ) 2 d η ,
Φ n ( K ¯ , r ¯ ) = C n 2 ( r ¯ ) Φ n ( 0 ) ( K ¯ ) ,
Φ n ( 0 ) ( K ¯ ) = { 0.033 K - 11 / 3 K < K m = 5.48 l 0 0 K > K m .
X 2 = 0.132 π 2 k 2 0 K m K - 8 / 3 d K 0 L C n 2 ( r ¯ ) ( η L ) 5 / 3 × sin 2 [ L ( L - η ) K 2 2 k η ] d η .
X 2 = 7.37 l 0 - 7 / 3 0 L C n ( r ¯ ) ( L η ) 1 / 3 ( L - η ) 2 d η .
X 2 = 0.56 k 7 / 6 0 L C n 2 ( r ¯ ) [ L - η ] 5 / 6 ( η L ) 5 / 6 d η .
D μ ( x , y , z , x , y , z ) = [ μ ( x , y , z ) - μ ( x , y , z ) ] 2 ,
μ ( x , y , z ) = μ ( x , 0 , 0 ) + - [ 1 - exp { i ( K 2 y + K 3 z ) } ] × d ν ( K 2 , K 3 , x ) .
ψ 1 ( x , y , z ) = ψ 1 ( x , 0 , 0 ) + - [ 1 - exp { i ( K 2 y + K 3 z ) } ] × d Λ ( K 2 , K 3 , x ) ,
d Λ ( K 2 , K 3 , x ) = d K 2 d K 3 k 2 x 8 π 3 × - exp { [ i ( K 2 y + K 3 z ) ] } d y d z × v d v 0 g ( x , y , z x 0 , y 0 , z 0 ) [ μ ( x , 0 , 0 ) + - [ 1 - exp { i K 2 y + i K 3 z } ] × d ν ( K 2 , K 3 , x 0 ) ] ,
g ( x , y , z x 0 , y 0 , z 0 ) = exp [ - i k 2 x x 0 ( y y 0 + z z 0 ) - x 0 2 ( y 2 + z 2 ) - x 2 ( y 0 2 + z 0 2 ) 2 x x 0 ( x - x 0 ) ] x 0 ( x - x 0 ) .
d Λ ( K 2 , K 3 , L ) = - i k 0 L d x 0 exp [ i L ( L - x 0 ) 2 x 0 k K 2 ] × d ν ( K 2 L x 0 , K 3 L x 0 , x 0 ) .
F s ( K 2 , K 3 , 0 ) = 2 k 2 π 0 L C n 2 ( r ¯ ) ( L η ) 2 × cos [ L ( L - η ) K 2 2 k η ] Φ n ( 0 ) ( K L η ) d η
F A ( K 2 , K 3 , 0 ) = 2 k 2 π 0 L C n 2 ( r ¯ ) ( L η ) 2 × sin [ L ( L - η ) K 2 2 k η ] Φ n ( 0 ) ( K L η ) d η ,
D ( ρ ) = 4 π 0 [ 1 - J 0 ( K ρ ) ] F ( K ) K d K ,
D s ( ρ ) = 8 k 2 π 2 0 [ 1 - J 0 ( K ρ ) ] K d K 0 L C n 2 ( r ¯ ) ( L η ) 2 × cos 2 ( L ( L - η ) K 2 2 k η ) Φ n ( 0 ) ( K L η ) d η ,
D ( ρ ) = 8 k 2 π 2 0 [ 1 - J 0 ( K ρ ) ] K d K × 0 L C n 2 ( r ¯ ) ( L η ) 2 Φ n ( 0 ) ( K L η ) d η .
D ( ρ ) = 0.264 π 2 k 2 0 L C n 2 ( r ¯ ) d η × 0 K m [ 1 - J ( K ρ η L ) ] K - 8 / 3 d K ,
K = K L / η
K m = ( 5.48 / η ) ( L / l 0 ) .
[ 1 - J 0 ( K ρ η L ) ] 1 4 K 2 ρ 2 ( η L ) 2 ;
D ( ρ ) = 3.44 l 0 - 1 3 k 2 ρ 2 0 L C n 2 ( r ¯ ) ( η L ) 5 / 3 d η             ρ l 0 .
D ( ρ ) = 2.91 k 2 ρ 5 / 3 0 L C n 2 ( r ¯ ) ( η L ) 5 / 3 d η ,             ρ l 0 .
D s ( ρ ) D ( ρ ) .
Φ n ( 0 ) ( K L η ) = Γ ( 5 / 2 ) π π ¯ l 0 11 / 3 ( 1 + K 2 l 0 2 ) 5 2 .
C n 2 ( h ) = C n 0 2 exp { - h / h 0 } .
D ( ρ ) = 3.44 l 0 - 1 3 k 2 ρ 2 L C n 0 2 0 1 exp { - L z / h 0 } × ( 1 - z ) 5 / 3 d z ,             ρ l 0
D ( ρ ) = 2.91 k 2 ρ 5 / 3 L C n 0 2 0 1 exp { - L z / h 0 } × ( 1 - z ) 5 / 3 d z ,             ρ l 0
X 2 = 7.37 l 0 - 7 / 3 L 3 C n 0 2 0 1 exp { - L z / h 0 } × z 2 ( 1 - z ) - 1 3 d z ,             L l 0 2 λ
X 2 = 0.56 k 7 / 6 L 11 / 6 C n 0 2 0 1 exp { - L z / h 0 } × z 5 / 6 ( 1 - z ) 5 / 6 d z ,             L l 0 2 λ
D ( ρ ) = 3.44 k 2 l 0 - 1 3 ρ 2 L C n 0 2 exp { - L / h 0 } ( 3 8 ) F 1 1 ( 8 3 ; 11 3 ; L h 0 ) ,             ρ l 0
D ( ρ ) = 2.91 k 2 ρ 5 / 3 L C n 0 2 exp { - L / h 0 } ( 3 8 ) F 1 1 ( 8 3 ; 11 3 ; L h 0 ) ,             ρ l 0
X 2 = 7.37 l 0 - 7 / 3 L C n 0 2 exp { - L / h 0 } ( 0.675 ) F 1 1 ( 2 3 ; 11 3 ; L h 0 ) ,             L l 0 2 λ
X 2 = 0.56 k 7 / 6 L 11 / 6 C n 0 2 exp { - L / h 0 } ( 0.221 ) F 1 1 ( 11 6 ; 11 3 ; L h 0 ) ,             L l 0 2 λ .
X 2 = 7.37 l 0 - 7 / 3 L 3 C n 0 2 ( 1 3 ) exp { - L / h 0 } F 1 1 ( 1 ; 4 ; L h 0 ) ,             L l 0 2 λ
X 2 = 0.56 k 7 / 6 L 11 / 6 C n 0 2 ( 6 11 ) exp { - L / h 0 } F 1 1 ( 1 ; 17 6 ; L h 0 ) ,             L l 0 2 λ
D ( ρ ) = 2.91 k 2 ρ 5 / 3 L C n 0 2 exp { - L / h 0 } F 1 1 ( 1 , 2 , L h 0 )             ρ l 0
D ( ρ ) = 3.44 k 2 ρ 2 l 0 - 1 3 C n 0 2 L exp { - L / h 0 } F 1 1 ( 1 , 2 , L h 0 )             ρ l 0 .
D s p ( p ) = D p l ( ρ ) ( 3 8 ) F 1 1 ( 8 3 ; 11 3 ; L h 0 ) F 1 1 ( 1 ; 2 ; L h 0 ) = D p l ( ρ ) π 1 ,             ρ l 0 ρ l 0
X s p 2 = X p l 2 ( 2.03 ) F 1 1 ( 2 3 ; 11 3 ; L h 0 ) F 1 1 ( 1 , 4 , L h 0 ) X p l 2 π 2 ,             L l 0 2 λ
X s p 2 = X p l 2 ( 0.405 ) F 1 1 ( 11 6 ; 11 3 ; L h 0 ) F 1 1 ( 1 ; 17 6 ; L h 0 ) = X p l 2 π 3 ,             L l 0 2 λ .
D s p ( ρ ) = D p l ( ρ ) ( 3 8 ) F 1 1 ( 8 3 ; 11 3 ; L h 0 ) exp { - L / h 0 } = D p l ( ρ ) γ 1
X s p 2 = X p l 2 ( 0.675 ) F 1 1 ( 2 3 ; 11 3 ; L h 0 ) exp { - L / h 0 } = X p l 2 γ 2 ,             L l 0 2 λ
X s p 2 = X p l 2 ( 0.221 ) F 1 1 ( 11 6 ; 11 3 ; L h 0 ) exp { - L / h 0 } = X p l 2 γ 3 ,             L l 0 2 λ .
S N = 1 2 ( η A 0 A s G ) 2 R e G 2 η ( π / 4 ) D 2 A 0 2 R d ¯ r d ¯ r W ( r ¯ + 1 2 r ¯ ) × W ( r ¯ - 1 2 r ¯ ) exp [ l ( x ¯ ) + l ( x ¯ ) ± j ( ϕ s ( x ¯ ) - ϕ s ( x ¯ ) ) ] ,
S N = 2 η A ¯ s 2 e π D 2 d r ¯ d r ¯ W ( r ¯ + 1 2 r ¯ ) W ( r ¯ - 1 2 r ¯ ) × exp [ - 1 2 D ( x ¯ , x ¯ ) ] .
r 0 = ( 6.88 / A ) 3 / 5 ,
D ( ρ , z 0 ) = A ρ 5 / 3 .
A A = 2.91 k 2 L C n 0 2 exp { - L / h 0 } F 1 1 ( 3 8 , 11 3 , L h 0 ) 2.91 k 2 L C n 0 2 = 3 8 exp { - L / h 0 } F 1 1 ( 8 3 , 11 3 , L h 0 ) .
r 0 = 1.79 exp { + 3 L / 5 h 0 } [ F 1 1 ( 8 3 , 11 3 , L h 0 ) ] - 3 / 5 r 0 ,
A A = 2.91 k 2 L C n 0 2 exp { - L / h 0 } F 1 1 ( 1 , 2 , L h 0 ) 2.91 k 2 L C n 0 2 ,
r 0 = exp { + 3 L / 5 h 0 } [ F 1 1 ( 1 , 2 , L h 0 ) ] - 3 / 5 r 0 .
2 u ( r ¯ , ω ) + k 2 n 2 ( r ¯ , ω ) u ( r ¯ , ω ) = 0 ,
u ( r ¯ ) = e ψ ( r ¯ ) ,
2 ( ψ ) + ( ψ ) 2 + k 2 n 2 = 0 ,
n ( r ¯ ) = 1 + μ ( r ¯ )
ψ ( r ¯ ) = ψ 0 ( r ¯ ) + ψ 1 ( r ¯ ) + 2 ψ 2 ( r ¯ ) + σ ( 3 ) .
ψ 1 ( r ¯ ) = k 2 2 π V μ ( r ¯ 0 ) u 0 ( r ¯ 0 ) u 0 ( r ¯ ) exp [ i k r ¯ - r ¯ 0 ] r ¯ - r ¯ 0 d r ¯ 0
ψ 2 ( r ) = 1 4 π V { k 2 μ 2 + ψ 1 · ψ 1 } u 0 ( r ¯ 0 ) u 0 ( r ¯ ) × exp [ i k r ¯ - r ¯ 0 ] r ¯ - r ¯ 0 d r ¯ 0
ψ n ( r ¯ ) = 1 4 π i = 1 n - 1 { ψ i · ψ n - i } u 0 ( r ¯ 0 ) u 0 ( r ¯ ) × exp [ i k r ¯ - r ¯ 0 ] r ¯ - r ¯ 0 d r ¯ 0 .
I 1 = 0 1 exp { - L z / h 0 } ( 1 - z ) 5 / 3 d z
I 2 = 0 1 exp { - L z / h 0 } z 2 ( 1 - z ) - 1 3 d z
I 3 = 0 1 exp { - L z / h 0 } z 5 / 6 ( 1 - z ) 5 / 6 d z .
z = ( h 0 / L ) x ,
I 1 = ( h 0 / L ) 8 / 3 0 L / h 0 exp { - x } ( L / h 0 - x ) 5 / 3 d x
I 2 = ( h 0 / L ) 8 / 3 0 L / h 0 exp { - x } ( x ) 2 ( L / h 0 - x ) - 1 3 d x
I 3 = ( h 0 / L ) 8 / 3 0 L / h 0 exp { - x } ( x ) 5 / 6 ( 1 - x ) 5 / 6 d x .
F 1 1 ( a ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( c - a ) z 1 - c 0 z e t t a - 1 ( z - t ) c - a - 1 d t .
y = L / h 0 - x
I 1 = ( L / h 0 ) - 8 / 3 exp { - L / h 0 } 0 L / h 0 exp { y } ( y ) 5 / 3 d y
I 2 = ( L / h 0 ) - 8 / 3 exp { - L / h 0 } 0 L / h 0 exp { y } ( y ) - 1 3 × ( L / h 0 - y ) 2 d y
I 3 = ( L / h 0 ) - 8 / 3 exp { - L / h 0 } 0 L / h 0 exp { y } ( y ) 5 / 6 × ( L / h 0 - y ) 5 / 6 d y .
I 1 : a = 8 / 3             c = 11 / 3
I 2 : a = 2 / 3             c = 11 / 3
I 3 : a = 11 / 6             c = 11 / 3.
I 1 = exp { - L / h 0 } Γ ( 8 / 3 ) Γ ( 1 ) Γ ( 11 / 3 ) × F 1 1 ( 8 / 3 ; 11 / 3 ; L / h 0 )
I 2 = exp { - L / h 0 } Γ ( 2 / 3 ) Γ ( 3 ) Γ ( 11 / 3 ) F 1 1 ( 2 / 3 ; 11 / 3 ; L / h 0 )
I 3 = exp { - L / h 0 } Γ ( 11 / 6 ) Γ ( 11 / 6 ) Γ ( 11 / 3 ) × F 1 1 ( 11 / 6 ; 11 / 3 ; L / h 0 ) ;
K ( z ) = F 1 1 ( a ; c ; z ) = 1 + a c z 1 ! + a ( a + 1 ) c ( c + 1 ) z 2 2 ! + .
I 1 = 0.375 exp { - L h 0 } ( 1 + 8 11 ( L h 0 ) + 3 11 ( L h 0 ) 2 + 10 143 ( L h 0 ) 3 + )
I 2 = 0.675 exp { - L h 0 } ( 1 + 2 11 ( L h 0 ) + 1 44 ( L h 0 ) 2 + 1 429 ( L h 0 ) 3 + )
I 3 = 0.221 exp { - L h 0 } ( 1 + 1 2 ( L h 0 ) + 1 6 ( L h 0 ) 2 + 1 24 ( L h 0 ) 3 + ) .
exp [ l ( x ¯ ) + l ( x ¯ ) ± j ( ϕ s ( x ¯ ) - ϕ s ( x ¯ ) ) ]
α = l ( x ¯ ) + l ( x ¯ )
β = ϕ s ( x ¯ ) - ϕ s ( x ¯ )
exp [ α ± j β ] = 1 + ( α ± j β ) + [ ( α ± j β ) 2 / 2 ! ] + = 1 + α ± j β + [ α 2 / 2 ] ± α β - [ β 2 / 2 ] + .
exp [ α ± j β ] = 1 + α + α 2 2 - β 2 2 + .
exp [ 2 l ( x ¯ ) ] = 1.
exp [ 2 l ( x ¯ ) ] = 1 + 2 l ( x ¯ ) + 2 l 2 ( x ) + .
l ( x ¯ ) = - l 2 ( x ¯ ) .
exp [ α ± j β ] = 1 + l ( x ¯ ) + l ( x ¯ ) + l 2 ( x ¯ ) 2 + l ( x ¯ ) l ( x ¯ ) + l 2 ( x ¯ ) 2 - β 2 2 + .
exp [ α ± j β ] = 1 - 1 2 [ l 2 ( x ¯ ) - 2 l ( x ¯ ) l ( x ¯ ) + l 2 ( x ¯ ) ] - 1 2 [ ϕ s ( x ¯ ) - ϕ s ( x ¯ ) ] 2 + = 1 - 1 2 [ l ( x ¯ ) - l ( x ¯ ) ] 2 - 1 2 [ ϕ s ( x ¯ ) - ϕ s ( x ¯ ) ] 2 + = 1 - 1 2 D ( x ¯ , x ¯ ) + .
exp [ l ( x ¯ ) + l ( x ¯ ) ± j ( ϕ s ( x ¯ ) - ϕ s ( x ¯ ) ) ] exp [ - 1 2 D ( x ¯ , x ¯ ) ] .
0[1-J0(x)]x¯Pdx=π[2P(Γ(P+12))2×sin-1(π(P-1)2)]-1.