Abstract

The efficiency of coupling a plane wave into a single-mode fiber can be reduced by both the aperture obstruction of receivers and the turbulence-induced degradation of optical coherence. Using the Gaussian approximation to the mutual coherence function of the incident optical field, we derived an analytical solution for the fiber-coupling efficiency when a plane wave, propagating through atmospheric turbulence, is received by an annular-aperture receiver and coupled into a single-mode fiber. It is a function of the coupling geometry, the aperture-radius-to-coherence-radius ratio (ARCRR), and the aperture-obstruction parameter. It is found by the numerical optimization method that the optimal coupling geometry depends on both the ARCRR and the aperture-obstruction parameter. The results obtained are useful for analyzing and designing a fiber-coupling system influenced by atmospheric turbulence.

© 2011 Optical Society of America

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References

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  1. M. Lazzaroni and F. E. Zocchi, “Optical coupling from plane wave to step-index single-mode fiber,” Opt. Commun. 237, 37–43 (2004).
    [CrossRef]
  2. P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23, 986–988 (1998).
    [CrossRef]
  3. C. Ruilier, “A study of degraded light coupling into single-mode fibers,” Proc. SPIE 3350, 319–329 (1998).
    [CrossRef]
  4. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44, 4946–4952 (2005).
    [CrossRef] [PubMed]
  5. M. Toyoshima, “Maximum fiber coupling efficiency and optimum beam size in the presence of random angular jitter for free-space laser systems and their applications,” J. Opt. Soc. Am. A 23, 2246–2250 (2006).
    [CrossRef]
  6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), p. 190.
  7. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), p. 340.
  8. R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals (Cambridge University, 2001), pp. 83, 293.
  9. R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
    [CrossRef]
  10. A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Chapman & Hall, 2007), p. 1121.
  11. G. Fikioris, Mellin-Transform Method for Integral Evaluation (Morgan & Claypool, 2007), pp. 8, 17.

2006 (1)

2005 (1)

2004 (1)

M. Lazzaroni and F. E. Zocchi, “Optical coupling from plane wave to step-index single-mode fiber,” Opt. Commun. 237, 37–43 (2004).
[CrossRef]

1998 (2)

C. Ruilier, “A study of degraded light coupling into single-mode fibers,” Proc. SPIE 3350, 319–329 (1998).
[CrossRef]

P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23, 986–988 (1998).
[CrossRef]

1993 (1)

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), p. 190.

Davidson, F. M.

Dikmelik, Y.

Fikioris, G.

G. Fikioris, Mellin-Transform Method for Integral Evaluation (Morgan & Claypool, 2007), pp. 8, 17.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), p. 340.

Kaminski, D.

R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals (Cambridge University, 2001), pp. 83, 293.

Lazzaroni, M.

M. Lazzaroni and F. E. Zocchi, “Optical coupling from plane wave to step-index single-mode fiber,” Opt. Commun. 237, 37–43 (2004).
[CrossRef]

Leeb, W. R.

Manzhirov, A. V.

A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Chapman & Hall, 2007), p. 1121.

Paris, R. B.

R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals (Cambridge University, 2001), pp. 83, 293.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), p. 190.

Polyanin, A. D.

A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Chapman & Hall, 2007), p. 1121.

Ruilier, C.

C. Ruilier, “A study of degraded light coupling into single-mode fibers,” Proc. SPIE 3350, 319–329 (1998).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), p. 340.

Sasiela, R. J.

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

Shelton, J. D.

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

Toyoshima, M.

Winzer, P. J.

Zocchi, F. E.

M. Lazzaroni and F. E. Zocchi, “Optical coupling from plane wave to step-index single-mode fiber,” Opt. Commun. 237, 37–43 (2004).
[CrossRef]

Appl. Opt. (1)

J. Math. Phys. (1)

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

M. Lazzaroni and F. E. Zocchi, “Optical coupling from plane wave to step-index single-mode fiber,” Opt. Commun. 237, 37–43 (2004).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

C. Ruilier, “A study of degraded light coupling into single-mode fibers,” Proc. SPIE 3350, 319–329 (1998).
[CrossRef]

Other (5)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), p. 190.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), p. 340.

R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals (Cambridge University, 2001), pp. 83, 293.

A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Chapman & Hall, 2007), p. 1121.

G. Fikioris, Mellin-Transform Method for Integral Evaluation (Morgan & Claypool, 2007), pp. 8, 17.

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Figures (4)

Fig. 1
Fig. 1

Fiber-coupling efficiency in terms of coupling geometry with various values of ε γ combination.

Fig. 2
Fig. 2

Three-dimensional surface plot of a opt as a function of both ε and γ.

Fig. 3
Fig. 3

Optimal fiber-coupling efficiency η c - opt in terms of γ with various ε.

Fig. 4
Fig. 4

Optimal fiber-coupling efficiency η c - opt in terms of ε with various γ.

Equations (25)

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η c = P c P a = | A U i ( r ) U m * ( r ) d r | 2 A | U i ( r ) | 2 d r ,
P ( r ) = { 1 , ε D / 2 r D / 2 0 , otherwise ,
U m ( r ) = 2 π W m λ f exp [ ( π W m r λ f ) 2 ] ,
η c = P a 1 × A Γ 2 ( r 1 , r 2 ) P ( r 1 ) P * ( r 2 ) U m * ( r 1 ) U m ( r 2 ) d r 1 d r 2 ,
Γ 2 ( r 1 , r 2 ) = exp [ ( | r 1 r 2 | ρ 0 ) 5 / 3 ] ,
η c = 8 a 2 π ( 1 ε 2 ) ε 1 ε 1 exp [ a 2 ( x 1 2 + x 2 2 ) ] F ( D 2 4 ρ 0 2 ( x 1 2 + x 2 2 ) , 2 x 1 x 2 x 1 2 + x 2 2 ) x 1 x 2 d x 1 d x 2 ,
F ( v , u ) = 0 π exp { v 5 / 6 [ 1 u cos ( θ ) ] 5 / 6 } d θ .
η c = 8 a 2 ( 1 ε 2 ) ε 1 ε 1 exp [ ( γ 2 + a 2 ) ( x 1 2 + x 2 2 ) ] I 0 ( 2 γ 2 x 1 x 2 ) x 1 x 2 d x 1 d x 2 ,
η c = 8 a 2 ( 1 ε 2 ) 1 [ Q ( γ 2 + a 2 , 2 γ 2 , 1 , 1 ) ε 2 Q ( γ 2 + a 2 , 2 γ 2 ε , 1 , ε ) ε 2 Q ( γ 2 + a 2 , 2 γ 2 ε , ε , 1 ) + ε 4 Q ( γ 2 + a 2 , 2 γ 2 ε 2 , ε , ε ) ] ,
Q ( μ , β , ξ 1 , ξ 2 ) = 0 1 0 1 exp [ μ ( ξ 1 2 x 2 + ξ 2 2 y 2 ) ] I 0 ( β x y ) x y d x d y , ( μ 0 , β 0 , ξ 1 0 , ξ 2 0 ) .
Q ( μ , β , ξ 1 , ξ 2 ) = 1 ( 2 π i ) 2 C 1 C 2 H ( 1 s , 1 t ; μ , ξ 1 , ξ 2 ) G ( s , t ; β ) d s d t ,
H ( 1 s , 1 t ; μ , ξ 1 , ξ 2 ) = 1 4 μ 2 + s / 2 + t / 2 ξ 1 2 + s ξ 2 2 + t Γ ( 1 s / 2 ) Γ ( 1 t / 2 ) , ( Re { s } < 2 , Re { t } < 2 ) ,
G ( s , t ; β ) = 1 s t · F 3 2 ( t / 2 , s / 2 ; 1 , t / 2 + 1 , s / 2 + 1 ; β 2 / 4 ) , ( Re { s } > 0 , Re { t } > 0 ) ,
Q ( μ , β , ξ 1 , ξ 2 ) = 1 4 p = 0 ( β 2 / 4 ) p [ Γ ( 2 + p ) ] 2 × F 1 1 ( 1 + p ; 2 + p ; μ ξ 1 2 ) × F 1 1 ( 1 + p ; 2 + p ; μ ξ 2 2 ) ,
η c = 2 a 2 ( 1 ε 2 ) p = 0 { γ 2 p Γ ( 2 + p ) [ F 1 1 ( 1 + p ; 2 + p ; γ 2 a 2 ) F 1 1 ( 1 + p ; 2 + p ; γ 2 ε 2 a 2 ε 2 ) ε 2 p + 2 ] } 2 .
F q p [ ( a ) ; ( b ) ; x ] = Γ ( b 1 ) Γ ( b q ) Γ ( a 1 ) Γ ( a p ) 1 2 π i η i η + i d s ( x ) s Γ ( a 1 + s ) Γ ( a p + s ) Γ ( s ) Γ ( b 1 + s ) Γ ( b q + s ) ,
( a ) = a 1 , a 2 , , a p , ( b ) = b 1 , b 2 , , b q ,
Γ ( z + 1 ) = z Γ ( z ) ,
Q = 1 ( 2 π i ) 3 ξ 1 2 ξ 2 2 16 μ 2 C 1 C 2 C 3 Γ [ 1 s / 2 , 1 t / 2 , t / 2 + r , s / 2 + r , r 1 + r , t / 2 + 1 + r , s / 2 + 1 + r ] μ s / 2 μ t / 2 ξ 1 s ξ 2 t ( β 2 4 ) r d r d s d t ,
Γ [ a 1 , , a k b 1 , , b l ] = Π i = 1 k Γ ( a i ) Π j = 1 l Γ ( b j ) .
Q = 1 ( 2 π i ) 3 ξ 1 2 ξ 2 2 4 μ 2 C 1 C 2 C 3 Γ [ s + r , t + r , 1 s , 1 t , r 1 + r , t + 1 + r , s + 1 + r ] ( μ ξ 1 2 ) s ( μ ξ 2 2 ) t ( β 2 4 ) r d r d t d s .
Q = 1 4 n = 0 m = 0 p = 0 ( μ ) n + m n ! m ! p ! × ( β 2 / 4 ) p ξ 1 2 n ξ 2 2 m Γ ( 1 + p ) ( 1 + n + p ) ( 1 + m + p ) = 1 4 p = 0 ( β 2 / 4 ) p p ! Γ ( 1 + p ) Γ ( 1 + p ) Γ ( 2 + p ) n = 0 ( μ ξ 1 2 ) n ( 1 + p ) n n ! ( 2 + p ) n Γ ( 1 + p ) Γ ( 2 + p ) m = 0 ( μ ξ 2 2 ) m ( 1 + p ) m m ! ( 2 + p ) m ,
( a ) k = Γ ( a + k ) Γ ( a ) , k = 0 , 1 , 2 , .
F p q ( α 1 , , α p ; β 1 , , β q ; z ) = n = 0 ( α 1 ) n ( α 2 ) n ( α p ) n ( β 1 ) n ( β 2 ) n ( β q ) n z n n ! ,
Q = 1 4 p = 0 ( β 2 / 4 ) p [ Γ ( 2 + p ) ] 2 × F 1 1 ( 1 + p ; 2 + p ; μ ξ 1 2 ) × F 1 1 ( 1 + p ; 2 + p ; μ ξ 2 2 ) .

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