Abstract

Two naturally linked pairs of nondimensional parameters are identified such that either pair, together with wavelength and path length, completely specifies the diffractive propagation environment for a lowest-order paraxial Gaussian beam. Both parameter pairs are intuitive, and within the context of locally homogeneous and isotropic turbulence they reflect the long-recognized importance of the Fresnel zone size in the behavior of Rytov propagation statistics. These parameter pairs, called, respectively, the transmitter and receiver parameters, also provide a change in perspective in the analysis of optical turbulence effects on Gaussian beams by unifying a number of behavioral traits previously observed or predicted, and they create an environment in which the determination of limiting interrelationships between beam forms is especially simple. The fundamental nature of the parameter pairs becomes apparent in the derived analytical expressions for the log-amplitude variance and the wave structure function. These expressions verify general optical turbulence-related characteristics predicted for Gaussian beams, provide additional insights into beam-wave behavior, and are convenient tools for beam-wave analysis.

© 1993 Optical Society of America

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References

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  1. L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1960).
  2. V I. Tatarski, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).
  3. R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).
  4. D. L. Fried, J. B. Seidman, “Laser beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
    [CrossRef]
  5. A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
    [CrossRef]
  6. J. R. Kerr, R. Eiss, “Transmitter-size and focus effects on scintillations,” J. Opt. Soc. Am. 62, 682–684 (1972).
    [CrossRef]
  7. J. R. Kerr, J. R. Dunphy, “Experimental effects of finite transmitter apertures on scintillation,” J. Opt. Soc. Am. 63, 1–8 (1973).
    [CrossRef]
  8. T. L. Ho, “Log-amplitude fluctuations of a laser beam in a turbulent atmosphere,” J. Opt. Soc. Am. 59, 385–390 (1969).
    [CrossRef]
  9. T. L. Ho, “Coherence degradation of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am. 60, 667–673 (1970).
    [CrossRef]
  10. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  11. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
    [CrossRef]
  12. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  13. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.
  14. A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
    [CrossRef]
  15. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
    [CrossRef]
  16. V. E. Zuev, Laser Beams in the Atmosphere, J. S. Wood, trans. (Consultants Bureau, New York, 1982).
    [CrossRef]
  17. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).
  18. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  19. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  20. R. T. Lutomirski, H. T. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. 61, 482–487 (1971).
    [CrossRef]
  21. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulence medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  22. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

1980 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

1975 (2)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

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

1973 (1)

1972 (2)

1971 (1)

1970 (2)

T. L. Ho, “Coherence degradation of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am. 60, 667–673 (1970).
[CrossRef]

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

1969 (2)

T. L. Ho, “Log-amplitude fluctuations of a laser beam in a turbulent atmosphere,” J. Opt. Soc. Am. 59, 385–390 (1969).
[CrossRef]

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

1967 (2)

R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

D. L. Fried, J. B. Seidman, “Laser beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
[CrossRef]

1966 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

Andrews, L. C.

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1960).

Dunphy, J. R.

Eiss, R.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

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

Fried, D. L.

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Ho, T. L.

Ishimaru, A.

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

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

A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
[CrossRef]

Kerr, J. R.

Lawrence, R. S.

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

Lutomirski, R. T.

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Schmeltzer, R. A.

R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Seidman, J. B.

Shishov, V I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

Strohbehn, J. W.

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

Tatarski, V I.

V I. Tatarski, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).

Yura, H. T.

Zuev, V. E.

V. E. Zuev, Laser Beams in the Atmosphere, J. S. Wood, trans. (Consultants Bureau, New York, 1982).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (7)

Proc. IEEE (4)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

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

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

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

Q. Appl. Math. (1)

R. A. Schmeltzer, “Means, variances and covariances for laser beam propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Radio Sci. (1)

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

Other (8)

L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1960).

V I. Tatarski, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).

V. E. Zuev, Laser Beams in the Atmosphere, J. S. Wood, trans. (Consultants Bureau, New York, 1982).
[CrossRef]

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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

A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
[CrossRef]

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (10)

Fig. 1
Fig. 1

Scaled log-amplitude variance σχ2(L,ρ)/Cn2k7/6L11/16 for a collimated beam (Ω0 = 1) as a function of the Fresnel ratio at the transmitter Ω = λL/πW02.

Fig. 2
Fig. 2

Scaled log-amplitude variance σχ2(L, ρ)/Cn2k7/6L11/6 for a divergent beam (Ω0 = 2) as a function of the Fresnel ratio at the transmitter Ω = λL/πW02.

Fig. 3
Fig. 3

Scaled longitudinal component of the log-amplitude variance σχ,l2(L)/Cn2k7/6L11/6 for divergent beams with various Ω0 values as a function of the Fresnel ratio at the transmitter Ω = λL/πW02.

Fig. 4
Fig. 4

Fresnel ratio at the receiver Λ = λL/πW2 as a function of the Fresnel ratio at the transmitter Ω = λL/πW02 for convergent beams with various values of |Ω0|.

Fig. 5
Fig. 5

Absolute value of the receiver curvature parameter |Θ| = 1 + L/R for convergent beams with various values of |Ω0| as a function of the Fresnel ratio at the transmitter Ω = λL/πW02.

Fig. 6
Fig. 6

Scaled log-amplitude variance σχ2(L,ρ)/Cn2k7/6L11/16 for a perfectly focused beam (Ω0 = 0) as a function of the Fresnel ratio at the transmitter Ω = λL/πW02 for various radial distances.

Fig. 7
Fig. 7

Scaled logitudinal component of the log-amplitude variance σχ,l2(L)/Cn2k7/6L11/16 for convergent beams with various values of Ω0 ≥ 0 as a function of the Fresnel ratio at the transmitter Ω = λL/πW02.

Fig. 8
Fig. 8

Same as Fig. 7 but for Ω0 ≤ 0.

Fig. 9
Fig. 9

Scaled WSF D(L, ρd)/Ds(L,ρd) of a collimated beam as a function of the Fresnel ratio at the transmitter Ω = λL/πW02. The curve marked a corresponds to the case ρ1 = −ρ2, and the curve marked b is that for ρ2 = 0. In both cases ρd = |ρ1ρ2|.

Fig. 10
Fig. 10

Scaled WSF D(L, ρd)/Ds(L,ρd) of a divergent beam with various Ω0 values as a function of the Fresnel ratio at the transmitter Ω = λL/πW02.

Tables (2)

Tables Icon

Table 1 Exact Expressions for Log-Amplitude Variance: σχ2(L,ρ)/Cn2k7/6L11/6

Tables Icon

Table 2 Approximate Expressions for Log-Amplitude Variance: σχ2(L,ρ)/Cn2k7/6L11/6

Equations (76)

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U ( 0 , ρ ) = exp ( 0.5 k α ρ 2 ) ,
α = λ / π W 0 2 + j ( 1 / R o ) .
U ( L , ρ ) = 1 1 + j α L × exp [ jkL 1 1 + j α L ( 1 W 0 2 + j k 2 R 0 ) ρ 2 ] = 1 1 + j α L exp [ jkL ( 1 W 2 + j k 2 R ) ρ 2 ] ,
I ( L , ρ ) = | U ( L , ρ ) | 2 = W 0 2 W 2 exp ( 2 ρ 2 W 2 ) ,
1 + j α L = Ω 0 + j Ω ,
Ω 0 = 1 L / R 0 , Ω = λ L / π W 0 2 .
1 / ( 1 + j α L ) = 1 / ( Ω 0 + j Ω ) = Θ j Λ ,
Θ = Ω 0 / ( Ω 0 2 + Ω 2 ) , Λ = Ω / ( Ω 0 2 + Ω 2 ) .
Θ = 1 + L / R , Λ = λ L / π W 2 ,
α x 2 ( L , ρ ) = 2 π 2 0 L 0 G ( ρ , κ , η ) ϕ n ( κ ) κ d κ d η ,
G ( ρ , κ , η ) = I 0 ( 2 γ i κ ρ ) | H ( κ , η ) | 2 + Re [ H 2 ( κ , η ) ] ,
γ = 1 + j α η 1 + j α L = γ r j γ i ,
| H ( κ , η ) | 2 = k 2 exp [ γ i κ 2 ( L η ) k ] ,
H 2 ( κ , η ) = k 2 exp [ j γ κ 2 ( L η ) k ] .
σ χ 2 ( L , ρ ) = 2 π 2 k 2 L × 0 1 0 { I 0 ( 2 Λ ρ ξ κ ) cos [ L κ 2 k ( 1 Θ ξ ) ξ ] } × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ ,
Θ = 1 Θ = L / R
σ χ 2 ( L , ρ ) = σ χ , l 2 ( L ) + σ χ , r 2 ( L , ρ ) ,
σ χ , l 2 ( L ) = 4 π 2 k 2 L 0 1 0 sin 2 [ L κ 2 2 k ( 1 Θ ξ ) ξ ] × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ ,
σ χ , r 2 ( L , ρ ) = 2 π 2 k 2 L 0 1 0 [ I 0 ( 2 Λ ρ ξ κ ) 1 ] × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ .
ϕ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
σ χ , r 2 ( L , ρ ) = ( 3 / 8 ) π 2 ( 0.033 Γ ( 5 / 6 ) C n 2 k 7 / 16 L 11 / 6 Λ 5 / 6 × [ F 1 1 ( 5 / 6 ; 1 ; 2 ρ 2 / W 2 ) 1 ] = 0.816 C n 2 k 7 / 6 L 11 / 6 Λ 5 / 6 × [ 1 F 1 1 ( 5 / 6 ; 1 ; 2 ρ 2 / W 2 ) ] ,
σ χ , r ( L , ρ ) { 1.36 C n 2 k 7 / 6 L 11 / 6 Λ 5 / 6 ( ρ / W ) 2 [ 1 + 0.083 ( ρ / W ) 2 ] , ρ W 0.034 C n 2 k 7 / 6 L 11 / 6 Λ 5 / 6 ( W / ρ ) 11 / 3 exp ( 2 ρ 2 / W 2 ) , ρ W
σ χ , l 2 ( L ) = ( 3 / 8 ) π 2 ( 0.033 ) Γ ( 5 / 6 ) C n 2 k 7 / 6 L 11 / 6 [ Λ 5 / 6 f ( Θ , Λ ) ] = 0.816 C n 2 k 7 / 6 L 11 / 6 [ f ( Θ , Λ ) Λ 5 / 6 ] ,
f ( Θ , Λ ) = Re [ ( 16 / 11 ) j 5 / 6 F 2 1 ( 5 / 6 , 11 / 6 ; 17 / 6 ; Θ + j Λ ) ]
σ χ 2 ( L , ρ ) = 0.816 C n 2 k 7 / 6 L 11 / 6 × [ f ( Θ , Λ ) Λ 5 / 6 F 1 1 ( 5 / 6 ; 1 ; 2 ρ 2 / W 2 ) ] ,
f ( Θ , Λ ) = 16 11 n = 0 ( 5 / 6 ) n ( 11 / 6 ) n ( 17 / 6 ) n n ! ( Θ 2 + Λ 2 ) n / 2 × cos [ n tan 1 ( Λ Θ ) + 5 π 12 ] , | Θ + j Λ | 1 ,
F 2 1 ( a , b , c ; x ) = Γ ( c ) Γ ( b a ) Γ ( b ) Γ ( c a ) x a × F 2 1 ( a , 1 + a c , 1 + a b ; 1 / x ) + Γ ( c ) Γ ( a b ) Γ ( a ) Γ ( c b ) x b × F 2 1 ( b , 1 + b c , 1 + b a ; 1 / x )
f ( Θ , Λ ) = 0.338 ( Θ 2 + Λ 2 ) 11 / 12 cos [ 11 6 tan 1 ( Λ Θ ) π 4 ] + ( Θ 2 + Λ 2 ) 5 / 12 n = 0 ( 5 / 6 ) n ( 8 / 3 ) n ( 5 / 3 ) n n ! ( Θ 2 + Λ 2 ) n / 2 × cos [ ( n 5 6 ) tan 1 ( Λ Θ ) + 5 π 12 ] , | Θ + j Λ | > 1 .
σ χ , l 2 ( L ) C n 2 k 7 / 6 L 11 / 6 = 1.187 n = 0 ( 5 / 6 ) n ( 11 / 6 ) n ( 17 / 6 ) n n ! ( Ω 2 1 + Ω 2 ) n / 2 × cos [ n tan 1 ( 1 Ω ) + 5 π 12 ] 0.816 ( Ω 1 + Ω 2 ) 5 / 6 .
σ χ , l 2 ( L ) C n 2 k 7 / 6 L 11 / 6 = 0.479 ( 1 + Ω 2 Ω 2 ) 11 / 12 sin ( 11 6 tan 1 Ω ) 0.816 ( Ω 2 1 + Ω 2 ) 5 / 6 + 1.187 Ω 11 / 6 × n = 1 ( 5 / 6 ) n ( 11 / 6 ) n ( 17 / 6 ) n n ! × ( 1 + Ω 2 ) n / 2 sin ( tan 1 Ω ) ,
σ χ , l 2 ( L ) C n 2 k 7 / 6 L 11 / 6 = 1.187 n = 0 ( 5 / 6 ) n ( 11 / 6 ) n ( 17 / 6 ) n n ! ( Θ 2 + Λ 2 ) n / 2 × cos [ n tan 1 ( Λ Θ ) + 5 π 12 ] 0.816 Λ 5 / 6 .
Λ max = 1 / 2 | Ω 0 | , Ω 0 0 .
| Θ | max = 1 / | Ω 0 | , Ω = 0 , Ω 0 0 .
σ χ , l 2 ( L ) C n 2 k 7 / 6 L 11 / 6 = 0.276 ( Θ 2 + Λ 2 ) 11 / 12 cos [ 11 6 tan 1 ( Λ Θ ) π 4 ] 0.816 Λ 5 / 6 + 0.816 ( Θ 2 + Λ 2 ) 5 / 12 × n = 0 ( 5 / 6 ) n ( 8 / 3 ) n ( 5 / 3 ) n n ! ( Θ 2 + Λ 2 ) n / 2 × cos [ ( n 5 6 ) tan 1 ( Λ Θ ) + 5 π 12 ] , | Θ + j Λ | > 1 .
σ χ , l 2 ( L ) C n 2 k 7 / 6 L 11 / 6 = 0.276 ( Ω 2 1 + Ω 2 ) 11 / 12 cos [ 11 6 tan 1 ( 1 Ω ) π 4 ] 0.816 Ω 5 / 6 + 0.816 ( 1 + Ω 2 Ω 2 ) 5 / 12 × n = 0 ( 5 / 6 ) n ( 8 / 3 ) n ( 5 / 3 ) n n ! ( Ω 2 1 + Ω 2 ) n / 2 × cos [ ( n 5 6 ) tan 1 ( 1 Ω ) + 5 π 12 ] .
σ 1 2 ( L ) > 0.15 Ω 5 / 6
σ χ 2 ( L , W ) = σ χ , l 2 ( L ) + σ χ , r 2 ( L , W ) 1 .
σ 1 2 ( L ) < ( 2 | Ω 0 | ) 5 / 6 ,
σ 1 2 ( L ) ( | Θ | 5 / 6 + 6.44 Λ 5 / 6 ) > 1 .
D ( L , ρ 1 , ρ 2 ) = 8 π 2 0 L 0 Re { [ 1 2 I 0 ( 2 γ i κ ρ 1 ) + 1 2 I 0 ( 2 γ i κ ρ 2 ) J 0 ( κ P ) ] H 2 ( κ , η ) } × ϕ n ( κ ) κ d κ d η ,
P = [ ( γ x 1 γ * x 2 ) 2 + ( γ y 1 γ * y 2 ) 2 ] 1 / 2
ρ 1 = ( x 1 2 + y 1 2 ) 1 / 2 , ρ 2 = ( x 2 2 + y 2 2 ) 1 / 2 .
D ( L , ρ 1 , ρ 2 ) = 2 [ σ χ , r 2 ( L , ρ 1 ) + σ χ , r 2 ( L , ρ 2 ) ] + D 12 ( L , ρ 1 , ρ 2 ) ,
D 12 ( L , ρ 1 , ρ 2 ) = 8 π 2 k 2 L 0 L 0 { 1 Re [ J 0 ( κ P ) ] } × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ ,
P 2 = ( 1 Θ ξ ) 2 [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ] Λ 2 ξ 2 [ ( x 1 + x 2 ) 2 + ( y 1 + y 2 ) 2 ] 2 j Λ ξ ( 1 Θ ξ ) [ ( x 1 2 x 2 2 ) + ( y 1 2 y 2 2 ) ] .
P = ( 1 Θ ξ ) ρ d , ρ d = | ρ 1 ρ 2 | ,
D ( L , ρ d ) = 8 π 2 k 2 L 0 1 0 { 1 J 0 [ ( 1 Θ ξ ) κ ρ d ] } ϕ n ( κ ) κ d κ d ξ .
D s ( L , ρ d ) = 1.093 C n 2 k 2 L ρ d 5 / 3 .
D p ( L , ρ d ) = 2.914 C n 2 k 2 L ρ d 5 / 3 .
D ( L , ρ d ) = 1.093 a C n 2 k 2 L ρ d 5 / 3 ,
a = { ( 1 Θ 8 / 3 ) / ( 1 Θ ) Θ 0 ( 1 + | Θ | 8 / 3 ) / ( 1 Θ ) Θ < 0 .
D ( L , ρ d ) 1.093 | Θ | 5 / 3 C n 2 k 2 L ρ d 5 / 3 , | Θ | 1 , Ω = 0 .
D ( L , ρ d ) = 4 σ χ , r 2 ( L , ρ d 2 ) + 8 π 2 k 2 L 0 1 0 { 1 J 0 [ ( 1 Θ ξ ) κ ρ d ] } × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ .
D ( L , ρ d ) = 1.093 a C n 2 k 2 L ρ d 5 / 3 3.263 C n 2 k 7 / 6 L 11 / 6 Λ 5 / 6 × F 1 1 ( 5 6 ; 1 ; ρ d 2 2 W 2 ) + 2.914 C n 2 k 2 L ρ d 5 / 3 × n = 1 N ( 5 / 6 ) n ( 5 / 6 ) n ( 1 / 2 ) n ( 3 / 2 ) n n ! ( 2 Λ W ρ d ) 2 n × F 2 1 ( 2 n 5 3 , 2 n + 1 ; 2 n + 2 ; Θ ) ,
D ( L , ρ d ) = 1.093 a C n 2 k 2 L ρ d 5 / 3 , 0 Ω < .
D ( L , ρ d ) D s ( L , ρ d ) = a = { 1 Θ 8 / 3 1 Θ Θ 0 1 + | Θ | 8 / 3 1 Θ Θ < 0 ,
P = [ 1 ( Θ + j Λ ) ξ ] ρ .
D ( L , ρ ) = 1.093 b C n 2 k 2 L ρ 5 / 3 ,
b = ( Θ 2 + Λ 2 ) 1 / 2 { cos [ tan 1 ( Λ Θ ) ] ( Θ 2 + Λ 2 ) 4 / 3 × cos [ 8 3 tan 1 ( Λ Θ ) + tan 1 ( Λ Θ ) ] } .
b = [ ( 1 + Ω 2 ) / Ω 2 ] 1 / 2 × [ sin ( tan 1 Ω ) + ( 1 + Ω 2 ) 4 / 3 sin ( 5 / 3 tan 1 Ω ) ] ,
0.816 ( Ω 1 + Ω 2 ) 5 / 6 F 1 1 ( 5 6 ; 1 ; 2 ρ 2 W 2 ) + 1.187 n = 0 ( 5 / 6 ) n ( 11 / 6 ) n ( 17 / 6 ) n n ! ( Ω 2 1 + Ω 2 ) n / 2 cos [ n tan ( 1 Ω ) + 5 π 12 ]
0.307 F 2 1 ( 5 6 , 11 6 ; 17 6 ; Θ )
0.816 Λ 5 / 6 F 1 1 ( 5 6 ; 1 ; 2 ρ 2 W 2 ) + 1.187 n = 0 ( 5 / 6 ) n ( 11 / 6 ) n ( 17 / 6 ) n n ! ( Θ 2 + Λ 2 ) n / 2 cos [ n tan ( Λ Θ ) + 5 π 12 ]
{ 0.307 Θ 5 / 6 F 2 1 ( 5 6 , 1 ; 17 6 ; 1 1 Θ ) , Ω 0 > 0 0.195 | Θ | 11 / 6 + 0.211 | Θ | 5 / 6 F 2 1 ( 5 6 , 8 3 ; 5 3 ; 1 | Θ | ) , Ω 0 < 0
0.816 Λ 5 / 6 F 1 1 ( 5 6 ; 1 ; 2 ρ 2 W 2 ) + 0.276 ( Θ 2 + Λ 2 ) 11 / 12 cos [ 11 6 tan 1 ( Λ Θ ) π 4 ] + 0.816 ( Θ 2 + Λ 2 ) 5 / 12 n = 0 ( 5 / 6 ) n ( 8 / 3 ) n ( 5 / 3 ) n n ! ( Θ 2 + Λ 2 ) n / 2 cos [ ( n 5 6 ) tan 1 ( Λ Θ ) + 5 π 12 ]
{ 0 , ρ = 0 , ρ > 0
0.816 Λ 5 / 6 F 1 1 ( 5 6 ; 1 ; 2 ρ 2 W 2 ) + 0.276 ( Ω 2 1 + Ω 2 ) 11 / 12 cos [ 11 6 tan 1 ( 1 Ω ) π 4 ] + 0.816 ( 1 + Ω 2 Ω 2 ) 5 / 12 n = 0 ( 5 / 6 n ( 8 / 3 ) n ( 5 / 3 ) n n ! ( Ω 2 1 + Ω 2 ) n / 2 cos [ ( n 5 6 ) tan 1 ( 1 Ω ) + 5 π 12 ]
1.36 ( Ω 1 + Ω 2 ) 5 / 6 ρ 2 W 2 + { 0.307 ( 1 + 1.98 Ω 2.66 Ω 5 / 6 ) , 0 Ω < 2 0.479 ( 1 + Ω 2 Ω 2 ) 11 / 12 sin ( 11 6 tan 1 Ω ) 0.816 Ω 5 / 6 , 2 Ω <
0.307 ( 1 55 102 Θ ) , | Θ | 1
1.36 Λ 5 / 6 ρ 2 W 2 + 0.307 0.640 ( Θ 2 + Λ 2 ) 1 / 2 cos [ tan 1 ( Λ Θ ) + 5 π 12 ] 0.033 ( Θ 2 + Λ 2 ) cos [ 2 tan 1 ( Λ Θ ) + 5 π 12 ] 0.816 Λ 5 / 6
D ( L , ρ d ) = 4 σ χ , r 2 ( L , ρ d 2 ) + 8 π 2 k 2 L × 0 1 0 { 1 J 0 [ ( 1 Θ ξ ) κ ρ d ] } × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ ,
D ( L , ρ d ) = 3.2638 C n 2 k 7 / 6 L 11 / 6 Λ 5 / 6 F 1 1 ( 5 6 ; 1 ; ρ d 2 2 W 2 ) 8 π 2 k 2 L 0 1 0 J 0 [ ( 1 Θ ξ ) κ ρ d ] × [ exp ( Λ L ξ 2 κ 2 k ) ] ϕ n ( κ ) κ d κ d ξ .
0 κ 8 / 3 J 0 [ ( 1 Θ ξ ) κ ρ d ] exp ( Λ L ξ 2 κ 2 k ) d κ = 1 2 Γ ( 5 6 ) ( Λ L ξ 2 k ) 5 / 6 F 1 1 [ 5 6 ; 1 ; ( 1 Θ ξ ) 2 k ρ d 2 4 Λ L ξ 2 ] .
F 1 1 [ 5 6 ; 1 ; ( 1 Θ ξ ) 2 k ρ d 2 4 Λ L ξ 2 ] 1 Γ ( 11 / 6 ) [ ( 1 Θ ξ ) 2 k ρ d 2 4 Λ L ξ 2 ] 5 / 6 n = 0 N ( 5 / 6 ) n ( 5 / 6 ) n n ! × [ 4 Λ L ξ 2 ( 1 Θ ξ ) 2 k ρ d 2 ] n ,
D ( L , ρ d ) = 3.263 C n 2 k 7 / 6 L 11 / 6 Λ 5 / 6 F 1 1 ( 5 6 ; 1 ; ρ d 2 2 W 2 ) + 2.914 C n 2 k 2 L ρ d 5 / 3 [ 3 8 ( 1 Θ 8 / 3 1 Θ ) n = 1 N ( 5 / 6 ) n ( 5 / 6 ) n ( 1 / 2 ) n ( 3 / 2 ) n n ! ( 2 Λ W ρ d ) 2 n × F 2 1 ( 2 n 5 3 , 2 n + 1 ; 2 n + 2 ; Θ ) ] .
D ( L , ρ d ) = 1.093 C n 2 k 2 L ρ d 5 / 3 ( 1 Θ 8 / 3 1 Θ ) , Θ 0 .

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