Abstract

We derive a general analytic expression for the coupling efficiency when a partially coherent, partially polarized beam is coupled into a multimode optical fiber. We adopt the Gaussian–Schell model for incident electromagnetic beams and use our general result to discuss the effects of the partial coherence and partial polarization on the coupling efficiency of an optical beam focused onto a step-index, single-mode fiber with a lens. Our results should be useful for any application requiring coupling of partially coherent beams into optical fibers.

© 2009 Optical Society of America

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  1. R. Wagner and W. Tomlinson, Appl. Opt. 21, 2671 (1982).
    [CrossRef] [PubMed]
  2. D. N. Christodoulides, L. A. Reith, and M. A. Saifi, J. Lightwave Technol. 5, 1623 (1987).
    [CrossRef]
  3. K. Chen and D. Kerps, J. Lightwave Technol. 5, 1600 (1887).
    [CrossRef]
  4. P. Winzer and W. Leeb, Opt. Lett. 23, 986 (1998).
    [CrossRef]
  5. S. Withington and G. Yassin, J. Opt. Soc. Am. A 18, 3061 (2001).
    [CrossRef]
  6. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, Opt. Lett. 33, 1389 (2008).
    [CrossRef] [PubMed]
  7. E. Wolf, Opt. Lett. 28, 1078 (2003).
    [CrossRef] [PubMed]
  8. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  9. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991), Chap. 2.
  10. D. Marcuse, J. Opt. Soc. Am. A 68, 103 (1978).
    [CrossRef]
  11. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
    [CrossRef]
  12. G. P. Agarwal, Fiber-Optic Communication Systems, 3rd ed. (John Wiley and Sons, 2002).
    [CrossRef]
  13. O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
    [CrossRef]

2008 (1)

2007 (1)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

2004 (1)

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

2003 (1)

2002 (1)

G. P. Agarwal, Fiber-Optic Communication Systems, 3rd ed. (John Wiley and Sons, 2002).
[CrossRef]

2001 (2)

S. Withington and G. Yassin, J. Opt. Soc. Am. A 18, 3061 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

1998 (1)

1991 (1)

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991), Chap. 2.

1987 (1)

D. N. Christodoulides, L. A. Reith, and M. A. Saifi, J. Lightwave Technol. 5, 1623 (1987).
[CrossRef]

1982 (1)

1978 (1)

D. Marcuse, J. Opt. Soc. Am. A 68, 103 (1978).
[CrossRef]

1887 (1)

K. Chen and D. Kerps, J. Lightwave Technol. 5, 1600 (1887).
[CrossRef]

Agarwal, G. P.

G. P. Agarwal, Fiber-Optic Communication Systems, 3rd ed. (John Wiley and Sons, 2002).
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Cai, Y.

Chen, K.

K. Chen and D. Kerps, J. Lightwave Technol. 5, 1600 (1887).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides, L. A. Reith, and M. A. Saifi, J. Lightwave Technol. 5, 1623 (1987).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Kerps, D.

K. Chen and D. Kerps, J. Lightwave Technol. 5, 1600 (1887).
[CrossRef]

Korotkova, O.

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Leeb, W.

Lu, X.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991), Chap. 2.

D. Marcuse, J. Opt. Soc. Am. A 68, 103 (1978).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Reith, L. A.

D. N. Christodoulides, L. A. Reith, and M. A. Saifi, J. Lightwave Technol. 5, 1623 (1987).
[CrossRef]

Saifi, M. A.

D. N. Christodoulides, L. A. Reith, and M. A. Saifi, J. Lightwave Technol. 5, 1623 (1987).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Tomlinson, W.

Wagner, R.

Wang, F.

Wang, Y.

Winzer, P.

Withington, S.

Wolf, E.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef] [PubMed]

Yassin, G.

Zhao, C.

Appl. Opt. (1)

J. Lightwave Technol. (2)

D. N. Christodoulides, L. A. Reith, and M. A. Saifi, J. Lightwave Technol. 5, 1623 (1987).
[CrossRef]

K. Chen and D. Kerps, J. Lightwave Technol. 5, 1600 (1887).
[CrossRef]

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

Opt. Commun. (1)

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Opt. Lett. (3)

Other (4)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991), Chap. 2.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

G. P. Agarwal, Fiber-Optic Communication Systems, 3rd ed. (John Wiley and Sons, 2002).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Illustrating notation related to the coupling of stochastic electromagnetic beam into an optical fiber.

Fig. 2
Fig. 2

Variation of the coupling efficiency with the NA. The parameters of the single-mode fiber are given in the text. The incident beam is assumed to be symmetric and has the following parameters: σ x = σ y = 0.7 mm , A x = A y = 1 , B x y = B y x = 1 , and δ x x = δ y y = δ x y and taking a different value for each curve.

Fig. 3
Fig. 3

Effect of degree of polarization on the coupling efficiency of a beam with δ x x = δ y y = δ x y = . Other parameters are the same as in Fig. 2.

Equations (15)

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E ( ρ , ω ) = m [ C m x F m x ( ρ , ω ) x ̂ + C m y F m y ( ρ , ω ) y ̂ ] ,
E μ ( ρ , ω ) = m C m μ F m μ ( ρ , ω ) , ( μ = x , y ) .
W μ ν ( ρ 1 , ρ 2 , ω ) = E μ * ( ρ 1 , ω ) E ν ( ρ 2 , ω ) = m n C m μ * C n ν F m μ * ( ρ 1 , ω ) F n ν ( ρ 2 , ω ) ,
F m μ * ( ρ , ω ) F n ν ( ρ , ω ) d 2 ρ = δ n m δ μ ν .
C m μ * C n ν = W μ ν ( ρ 1 , ρ 2 , ω ) F m μ ( ρ 1 , ω ) F n ν * ( ρ 2 , ω ) d 2 ρ 1 d 2 ρ 2 .
P c m = Tr [ W ( ρ 1 , ρ 2 , ω ) . F m ( ρ 1 , ρ 2 , ω ) ] d 2 ρ 1 d 2 ρ 2 ,
F m ( ρ 1 , ρ 2 , ω ) = ( F m x * ( ρ 1 , ω ) F m x ( ρ 2 , ω ) F m x * ( ρ 1 , ω ) F m y ( ρ 2 , ω ) F m y * ( ρ 1 , ω ) F m x ( ρ 2 , ω ) F m y * ( ρ 1 , ω ) F m y ( ρ 2 , ω ) ) .
η c m = Tr [ W ( ρ 1 , ρ 2 , ω ) . F m ( ρ 1 , ρ 2 , ω ) ] d 2 ρ 1 d 2 ρ 2 2 Tr [ W ( ρ , ρ , ω ) ] d 2 ρ .
W i j ( ρ 1 , ρ 2 ; ω ) = S i ( ρ 1 ; ω ) S j ( ρ 2 ; ω ) μ i j ( ρ 2 ρ 1 ; ω ) ,
S j ( ρ ; ω ) = A j 2 exp [ ρ 2 2 σ j 2 ] , ( j = x , y ) ,
μ i j ( ρ 2 ρ 1 ; ω ) = B i j exp [ ( ρ 2 ρ 1 ) 2 2 δ i j 2 ] , ( i = x , y ; j = x , y ) .
F j A = 2 π w j A 1 exp ( ρ 2 w j A 2 ) ,
η C = i = x , y π w i 2 j = x , y B i j A i A j [ C i j C j i ( 1 4 δ i j 4 ) ] 1 ( λ f W ) 2 i = x , y A i 2 σ i 2 ( W 2 + 2 σ i 2 ) 1 ,
C i j = 1 4 σ i 2 + 1 2 δ i j 2 + 1 w i A 2 + 1 W 2 .
DOP = ( A x 2 A y 2 ) 2 + 4 A x 2 A y 2 | B x y | 2 A x 2 + A y 2 .

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