Abstract

The heterodyne efficiency of a coherent free-space optical (FSO) communication model under the effects of atmospheric turbulence and misalignment is studied in this paper. To be more general, both the transmitted beam and local oscillator beam are assumed to be partially coherent based on the Gaussian Schell model (GSM). By using the derived analytical form of the cross-spectral function of a GSM beam propagating through atmospheric turbulence, a closed-form expression of heterodyne efficiency is derived, assuming that the propagation directions for the transmitted and local oscillator beams are slightly different. Then the impacts of atmospheric turbulence, configuration of the two beams (namely, beam radius and spatial coherence width), detector radius, and misalignment angle over heterodyne efficiency are examined. Numerical results suggest that the beam radius of the two overlapping beams can be optimized to achieve a maximum heterodyne efficiency according to the turbulence conditions and the detector radius. It is also found that atmospheric turbulence conditions will significantly degrade the efficiency of heterodyne detection, and compared to fully coherent beams, partially coherent beams are less sensitive to the changes in turbulence conditions and more robust against misalignment at the receiver.

© 2012 Optical Society of America

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References

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  1. X. M. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
    [CrossRef]
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
  3. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
    [CrossRef]
  4. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
    [CrossRef]
  5. A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16, 14151–14162 (2008).
    [CrossRef]
  6. K. Tanaka and N. Ohta, “Effects of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632(1987).
    [CrossRef]
  7. M. Salem and J. P. Rolland, “Heterodyne efficiency of a detection system for partially coherent beams,” J. Opt. Soc. Am. A 27, 1111–1119 (2010).
    [CrossRef]
  8. T. Takenaka, K. Tanaka, and O. Fukumitsu, “Signal-to-noise ratio in optical heterodyne detection for Gaussian fields,” Appl. Opt. 17, 3466–3471 (1978).
    [CrossRef]
  9. J. Salzman and A. Katzir, “Heterodyne detection SNR: calculations with matrix formalism,” Appl. Opt. 23, 1066–1074 (1984).
    [CrossRef]
  10. K. K. Das, K. M. Iftekharuddin, and A. Mohammad, “Improved heterodyne mixing efficiency and signal-to-noise ratio with an array of hexagonal detectors,” Appl. Opt. 36, 7023–7026 (1997).
    [CrossRef]
  11. K. Tanaka and N. Saga, “Maximum heterodyne efficiency of optical heterodyne detection in the presence of background radiation,” Appl. Opt. 23, 3901–3904(1984).
    [CrossRef]
  12. T. Tanaka, M. Taguchi, and K. Tanaka, “Heterodyne efficiency for a partially coherent optical signal,” Appl. Opt. 31, 5391–5394 (1992).
    [CrossRef]
  13. D. M. Chambers, “Modeling heterodyne efficiency for coherent laser radar in the presence of aberrations,” Opt. Express 1, 60–67 (1997).
    [CrossRef]
  14. M. S. Belenkii, “Effect of atmospheric turbulence on heterodyne lidar performance,” Appl. Opt. 32, 5368–5372 (1993).
    [CrossRef]
  15. R. G. Frehlich and J. M. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef]
  16. S. F. Clifford and S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981).
    [CrossRef]
  17. G. Guérit, P. Drobinski, P. H. Flamant, and B. Augère, “Analytical empirical expressions of the transverse coherence properties for monostatic and bistatic lidars in the presence of moderate atmospheric refractive-index turbulence,” Appl. Opt. 40, 4275–4285 (2001).
    [CrossRef]
  18. A. Belmonte, “Analyzing the efficiency of a practical heterodyne lidar in the turbulent atmosphere: telescope parameters,” Opt. Express 11, 2041–2046 (2003).
    [CrossRef]
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
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    [CrossRef]
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    [CrossRef]
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2008).
  24. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1962).

2010 (1)

2009 (1)

2008 (1)

2006 (1)

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
[CrossRef]

2003 (1)

2002 (2)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[CrossRef]

X. M. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

2001 (1)

1997 (2)

D. M. Chambers, “Modeling heterodyne efficiency for coherent laser radar in the presence of aberrations,” Opt. Express 1, 60–67 (1997).
[CrossRef]

K. K. Das, K. M. Iftekharuddin, and A. Mohammad, “Improved heterodyne mixing efficiency and signal-to-noise ratio with an array of hexagonal detectors,” Appl. Opt. 36, 7023–7026 (1997).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

1987 (1)

1984 (2)

1981 (1)

1978 (1)

1972 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Augère, B.

Belenkii, M. S.

Belmonte, A.

Chambers, D. M.

Clifford, S. F.

Das, K. K.

K. K. Das, K. M. Iftekharuddin, and A. Mohammad, “Improved heterodyne mixing efficiency and signal-to-noise ratio with an array of hexagonal detectors,” Appl. Opt. 36, 7023–7026 (1997).
[CrossRef]

Davidson, F. M.

Drobinski, P.

Flamant, P. H.

Frehlich, R. G.

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Fukumitsu, O.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2008).

Guérit, G.

Iftekharuddin, K. M.

K. K. Das, K. M. Iftekharuddin, and A. Mohammad, “Improved heterodyne mixing efficiency and signal-to-noise ratio with an array of hexagonal detectors,” Appl. Opt. 36, 7023–7026 (1997).
[CrossRef]

Kahn, J. M.

A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16, 14151–14162 (2008).
[CrossRef]

X. M. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Katzir, A.

Kavaya, J. M.

Kiasaleh, K.

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
[CrossRef]

Koyama, Y.

Kunimori, H.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mohammad, A.

K. K. Das, K. M. Iftekharuddin, and A. Mohammad, “Improved heterodyne mixing efficiency and signal-to-noise ratio with an array of hexagonal detectors,” Appl. Opt. 36, 7023–7026 (1997).
[CrossRef]

Ohta, N.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Ricklin, J. C.

Rolland, J. P.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2008).

Saga, N.

Salem, M.

Salzman, J.

Shoji, Y.

Taguchi, M.

Takayama, Y.

Takenaka, H.

Takenaka, T.

Tanaka, K.

Tanaka, T.

Toyoshima, M.

Wandzura, S.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1962).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yura, H. T.

Zhu, X. M.

X. M. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Appl. Opt. (11)

T. Takenaka, K. Tanaka, and O. Fukumitsu, “Signal-to-noise ratio in optical heterodyne detection for Gaussian fields,” Appl. Opt. 17, 3466–3471 (1978).
[CrossRef]

J. Salzman and A. Katzir, “Heterodyne detection SNR: calculations with matrix formalism,” Appl. Opt. 23, 1066–1074 (1984).
[CrossRef]

K. K. Das, K. M. Iftekharuddin, and A. Mohammad, “Improved heterodyne mixing efficiency and signal-to-noise ratio with an array of hexagonal detectors,” Appl. Opt. 36, 7023–7026 (1997).
[CrossRef]

K. Tanaka and N. Saga, “Maximum heterodyne efficiency of optical heterodyne detection in the presence of background radiation,” Appl. Opt. 23, 3901–3904(1984).
[CrossRef]

T. Tanaka, M. Taguchi, and K. Tanaka, “Heterodyne efficiency for a partially coherent optical signal,” Appl. Opt. 31, 5391–5394 (1992).
[CrossRef]

M. S. Belenkii, “Effect of atmospheric turbulence on heterodyne lidar performance,” Appl. Opt. 32, 5368–5372 (1993).
[CrossRef]

R. G. Frehlich and J. M. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef]

S. F. Clifford and S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981).
[CrossRef]

G. Guérit, P. Drobinski, P. H. Flamant, and B. Augère, “Analytical empirical expressions of the transverse coherence properties for monostatic and bistatic lidars in the presence of moderate atmospheric refractive-index turbulence,” Appl. Opt. 40, 4275–4285 (2001).
[CrossRef]

K. Tanaka and N. Ohta, “Effects of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632(1987).
[CrossRef]

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]

IEEE Trans. Commun. (2)

X. M. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54, 604–607 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (4)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2008).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1962).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Heterodyne detection model of a coherent FSO link through atmospheric turbulence. (b) Abstract model of mixing of signal beam and LO beam.

Fig. 2.
Fig. 2.

(a) Heterodyne efficiency against transmitted beam radius w 0 S ( w 0 L = 2 cm , R = 2 cm , C n 2 = 5 × 10 14 m 2 / 3 , L = 5 km , ϑ = 0 mrad ). (b) Heterodyne efficiency against LO beam radius w 0 L ( w 0 S = 2 cm , R = 2 cm , C n 2 = 5 × 10 14 m 2 / 3 , L = 5 km , ϑ = 0 mrad ). (c) Heterodyne efficiency against beam radius of transmitted beam and LO beam ( R = 2 cm , C n 2 = 5 × 10 14 m 2 / 3 , L = 5 km , ϑ = 0 mrad ).

Fig. 3.
Fig. 3.

Heterodyne efficiency against turbulence strength ( R / ρ 0 ) for (a) different spatial coherence widths ( w 0 S = 2 cm , w 0 L = 2 cm , R = 2 cm , C n 2 = 1 × 10 14 m 2 / 3 , L = 5 km , ϑ = 0 mrad ) and (b) different misalignment angles ϑ ( w 0 S = 2 cm , w 0 L = 2 cm , R = 2 cm , C n 2 = 1 × 10 14 m 2 / 3 , L = 5 km . σ S = , σ LO = ).

Fig. 4.
Fig. 4.

Heterodyne efficiency against detector radius R for different σ LO and σ S under (a)  ϑ = 0 mrad ; (b)  ϑ = 0.05 mrad ( w 0 S = 2 cm , w 0 L = 2 cm , C n 2 = 1 × 10 14 m 2 / 3 , L = 5 km ).

Fig. 5.
Fig. 5.

Heterodyne efficiency against misalignment angle ϑ for different σ LO and σ S ( w 0 S = 2 cm , w 0 L = 2 cm , R = 2 cm , C n 2 = 1 × 10 14 m 2 / 3 , L = 5 km ).

Equations (36)

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P total = 2 Re [ η ( r 1 ) η ( r 2 ) W LO ( r 1 , r 2 ) W S * ( r 1 , r 2 ) exp ( j ( k · r 1 k · r 2 ) ) ] d 2 r 1 d 2 r 2 ,
P max = 2 η ( r ) W LO ( r , r ) d r 2 η ( r ) W S ( r , r ) d r 2 .
= Re [ W LO ( r 1 , r 2 ) W S * ( r 1 , r 2 ) exp ( j ( k · r 1 k · r 2 ) ) ] d 2 r 1 d 2 r 2 W LO ( r , r ) d r 2 W S ( r , r ) d r 2 .
= θ 1 = 0 2 π θ 1 = 0 2 π r 1 = 0 R r 2 = 0 R Re [ W L 0 ( r 1 , r 2 ) W S * ( r 1 , r 2 ) e r 1 2 + r 2 2 W 2 exp ( j ( k r 1 ϑ cos θ 1 k r 2 ϑ cos θ 2 ) ) ] d 2 r 1 d 2 r 2 W L 0 ( r , r ) e r 2 / W 2 d r 2 W S ( r , r ) e r 2 / W 2 d r 2 ,
W LO ( r 1 , r 2 ) = exp [ 1 w 0 L 2 ( r 1 2 + r 2 2 ) j k 2 R LO ( r 1 2 r 2 2 ) ( r 1 r 2 ) 2 2 σ LO 2 ] ,
W S ( r 1 , r 2 , L ) = 1 ( λ L ) 2 d r 1 2 d r 2 2 W S ( r 1 , r 2 , 0 ) exp [ ψ ( ρ 1 , r 1 ) + ψ * ( ρ 2 , r 2 ) ] × exp { j k 2 L [ ( r 1 ρ 1 ) 2 + ( r 2 ρ 2 ) 2 ] } ,
r s = 1 2 ( r 1 + r 2 ) , r d = r 1 r 2 , ρ s = 1 2 ( ρ 1 + ρ 2 ) , ρ d = ρ 1 ρ 2 .
W S ( r s , r d , 0 ) = exp { 1 2 w 0 S 2 ( r d 2 + 4 r s 2 ) j k 2 R S ( 2 r d · r s ) r d 2 2 σ S 2 } ,
exp { j k 2 L [ ( r 1 ρ 1 ) 2 + ( r 2 ρ 2 ) 2 ] } = exp { j k L [ ( r s ρ s ) · ( r d ρ d ) ] } .
exp [ φ ( ρ 1 , r 1 ) + φ * ( ρ 2 , r 2 ) ] exp [ 1 ρ 0 2 ( r d 2 + r d · ρ d + ρ d 2 ) ] ,
W S ( ρ s , ρ d , L ) = 1 ( λ L ) 2 d 2 r d d 2 r s exp ( 2 r s 2 w 0 S 2 ) × exp [ j k r s · r d R 0 + j k r s · ( r d ρ d ) L ] × exp [ r d 2 2 w 0 S 2 r d 2 2 σ S 2 r d 2 + r d · ρ d + ρ d 2 ρ 0 2 j k ρ s · ( r d ρ d ) L ] .
W S ( r 1 , r 2 , L ) = w 0 S 2 w ς 2 ( L ) exp { ( r 1 r 2 ) 2 ( 1 ρ 0 2 + 1 2 w 0 S 2 z 2 ϕ 2 2 w ς 2 ( L ) ) j ( r 1 2 r 2 2 ) ( ϕ w ς 2 ( L ) 1 w 0 S 2 z ) } × exp ( ( r 1 + r 2 ) 2 2 w ς 2 ( L ) ) ,
M = w 0 S 2 w ς 2 ( L ) 0 0 0 2 π 0 2 π Re [ exp { A r 1 2 B r 2 2 C r 1 · r 2 } × exp ( j ( k r 1 ϑ cos θ 1 k r 2 ϑ cos θ 2 ) ) ] r 1 r 2 d r 1 d r 2 d θ 1 d θ 2
α = 1 w 0 L 2 + 1 ρ 0 2 + 1 2 w 0 S 2 z 2 + 1 2 σ LO 2 + 1 ϕ 2 2 w ς 2 ( L ) + 2 R 2 , β = k 2 R LO ϕ w ς 2 ( L ) + 1 w 0 S 2 z , γ = 2 ρ 0 2 + 1 w 0 S 2 z 2 ϕ 2 w ς 2 ( L ) + 1 σ LO 2 1 w ς 2 ( L ) ,
M = π 2 w 0 S 2 w ς 2 ( L ) ( e D 2 / 2 τ e ( D 2 + γ 2 D 2 / τ 2 ) / 4 υ τ υ e γ D 2 / 2 τ υ + e D 2 / 2 τ * e ( D 2 + γ 2 D 2 / τ * 2 ) / 4 υ * τ * υ * e γ D 2 / 2 τ * υ * ) ,
χ LO = W L 0 ( r , r ) e r 2 / W 2 d r 2 , χ S = W S ( r , r ) e r 2 / W 2 d r 2 ,
W LO ( r , r ) = exp ( 2 r 2 w 0 L 2 ) ; W S ( r , r ) = w 0 S 2 w ς 2 ( L ) exp ( 2 r 2 w ς 2 ( L ) ) .
χ LO = π w 0 L 2 R 2 2 ( w 0 L 2 + R 2 ) ; χ S = w 0 S 2 w ς 2 ( L ) · π w ς 2 ( L ) R 2 2 ( w ς 2 ( L ) + R 2 ) .
= M N = 4 ( w 0 L 2 + R 2 ) ( w ς 2 ( L ) + R 2 ) w 0 L 2 w ς 2 ( L ) R 4 × ( e D 2 / 2 τ e ( D 2 + γ 2 D 2 / τ 2 ) / 4 υ τ υ e γ D 2 / 2 τ υ + e D 2 / 2 τ * e ( D 2 + γ 2 D 2 / τ * 2 ) / 4 υ * τ * υ * e γ D 2 / 2 τ * υ * ) .
M = w 0 S 2 w ς 2 ( L ) 0 0 0 2 π 0 2 π e α r 1 2 α r 2 2 + γ r 1 r 2 cos ( θ 1 θ 2 ) × Re [ e j ( β r 1 2 β r 2 2 k ϑ ( r 1 cos θ 1 r 2 cos θ 2 ) ) ] r 1 r 2 d r 1 d r 2 d θ 1 d θ 2 ,
M = w 0 S 2 2 w ς 2 ( L ) ( M 1 + M 2 ) ,
M 1 = 0 0 0 2 π 0 2 π e A r 1 2 B r 2 2 + C r 1 r 2 cos ( θ 1 θ 2 ) exp ( j k ϑ ( r 1 cos θ 1 r 2 cos θ 2 ) ) r 1 r 2 d r 1 d r 2 d θ 1 d θ 2 ,
M 2 = 0 0 0 2 π 0 2 π e A * r 1 2 B * r 2 2 + C r 1 r 2 cos ( θ 1 θ 2 ) exp ( j k ϑ ( r 1 cos θ 1 r 2 cos θ 2 ) ) r 1 r 2 d r 1 d r 2 d θ 1 d θ 2 .
M 1 = 0 0 0 2 π e A r 1 2 B r 2 2 e j D r 2 cos θ 2 × 0 2 π exp ( cos θ 1 ( C r 1 r 2 cos θ 2 + j D r 1 ) + sin θ 1 ( C r 1 r 2 sin θ 2 ) ) d θ 1 r 1 r 2 d r 1 d r 2 d θ 2 ,
0 2 π e α cos ϕ + β sin ϕ d ϕ = 2 π I 0 ( α 2 + β 2 ) , I 0 ( x 2 + y 2 2 x y cos ( ϕ φ ) ) = ( 1 ) m I m ( x ) I m ( y ) cos m ( ϕ φ ) ,
M 1 = 2 π ( 1 ) m 0 0 e A r 1 2 B r 2 2 I m ( C r 1 r 2 ) I m ( j D r 1 ) r 1 r 2 d r 1 d r 2 × 0 2 π exp ( j D r 2 ) cos m ( ϕ 2 π ) d ϕ 2 .
0 2 π e α cos ϕ + β sin ϕ cos m ( ϕ φ ) d ϕ = 2 π I m ( α 2 + β 2 ) cos m ( φ tan 1 ( β / α ) ) ,
M 1 = 4 π 2 ( 1 ) m cos m φ 0 e A r 1 2 J m ( D r 1 ) r 1 d r 1 0 e B r 2 2 I m ( C r 1 r 2 ) J m ( D r 2 ) r 2 d r 2 .
0 2 π x e α 2 x 2 I n ( β x ) J n ( γ x ) d x = 1 2 α 2 exp ( β 2 γ 2 4 α 2 ) J n ( β γ 2 α 2 ) ;
M 1 = 4 π 2 exp ( D 2 / 2 τ ) τ ( 1 ) m cos m φ 0 r 1 e υ r 1 2 J m ( D r 1 ) J m ( γ D τ r 1 ) d r 1 ,
0 2 π x e α 2 x 2 J n ( β x ) J n ( γ x ) d x = 1 2 α 2 exp ( β 2 + γ 2 4 α 2 ) I n ( β γ 2 α 2 ) ,
M 1 = 4 π 2 e D 2 / 2 τ e ( D 2 + γ 2 D 2 / τ 2 ) / 4 υ 2 τ υ ( 1 ) m cos m φ I m ( γ D 2 2 τ υ ) .
( 1 ) n cos n φ I n ( x ) = exp ( x cos φ ) ,
M 1 = 4 π 2 e D 2 / 2 τ e ( D 2 + γ 2 D 2 / τ 2 ) / 4 υ 2 τ υ e γ D 2 / 2 τ υ .
M 2 = 4 π 2 e D 2 / 2 τ * e ( D 2 + γ 2 D 2 / τ * 2 ) / 4 υ * 2 τ * υ * e γ D 2 / 2 τ * υ * .
M = π 2 w 0 S 2 w ς 2 ( L ) ( e D 2 / 2 τ e ( D 2 + γ 2 D 2 / τ 2 ) / 4 υ τ υ e γ D 2 / 2 τ υ + e D 2 / 2 τ * e ( D 2 + γ 2 D 2 / τ * 2 ) / 4 υ * τ * υ * e γ D 2 / 2 τ * υ * ) .

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