Abstract

We give an analytical expression for the probability density function of the differential group delay for a concatenation of Maxwellian fiber sections and an arbitrary number of lumped elements with constant and isotropically oriented birefringence. When the contribution of the average squared of the constant birefringence elements is a significant fraction of the total, we show that the outage probability can be significantly overestimated if the probability density function of the differential group delay is approximated by a Maxwellian distribution.

© 2004 Optical Society of America

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References

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  1. F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).
  2. P. K. A. Wai and C. R. Menyuk, J. Lightwave Technol. 14, 148 (1996).
    [CrossRef]
  3. M. Karlsson, J. Lightwave Technol. 19, 324 (2001).
    [CrossRef]
  4. J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
    [CrossRef]
  5. C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” submitted to IEEE Photon. Technol. Lett.

2001 (1)

2000 (1)

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

1996 (1)

P. K. A. Wai and C. R. Menyuk, J. Lightwave Technol. 14, 148 (1996).
[CrossRef]

1990 (1)

F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).

Antonelli, C.

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” submitted to IEEE Photon. Technol. Lett.

Curti, F.

F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).

Daino, B.

F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).

De Marchis, G.

F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).

Gordon, J. P.

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

Karlsson, M.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

Matera, F.

F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).

Mecozzi, A.

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” submitted to IEEE Photon. Technol. Lett.

Menyuk, C. R.

P. K. A. Wai and C. R. Menyuk, J. Lightwave Technol. 14, 148 (1996).
[CrossRef]

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, J. Lightwave Technol. 14, 148 (1996).
[CrossRef]

J. Lightwave Technol. (3)

F. Curti, B. Daino, G. De Marchis, and F. Matera, J. Lightwave Technol. 9, 1439 (1990).

P. K. A. Wai and C. R. Menyuk, J. Lightwave Technol. 14, 148 (1996).
[CrossRef]

M. Karlsson, J. Lightwave Technol. 19, 324 (2001).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[CrossRef]

Other (1)

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” submitted to IEEE Photon. Technol. Lett.

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Figures (2)

Fig. 1
Fig. 1

PDF of the DGD for a fiber made of a Maxwellian part and five lumped elements with uneven birefringence. The mean-square DGD is equally distributed between the Maxwellian part and the lumped elements, Δτ02=n=15Δτn2 with Δτn=nΔτ1. The solid curve is the plot of the analytical expression, and the filled circles are the results of a Monte Carlo simulation with 106 realizations. The dotted curve is the Maxwellian with the same total mean-square DGD.

Fig. 2
Fig. 2

Relative error Eout versus the ratio between the average DGDs of the Maxwellian part, σ02=Δτ02, and the part coming from the constant birefringence elements, σN2=n=1NΔτn2. The solid and dashed curves are for N=10 and N=5 elements with equal birefringence, respectively. The dotted–dashed and dotted curves are for N=10 and N=5 elements with birefringence Δτn=nΔτ1, respectively. Inset, portions of the same curves for σ02/σN23 in a semilogarithmic scale.

Equations (12)

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τkω=Δτk+expωΔτk×Rkτk-1ω,
pτ=τuτ2M-1M-2!Δτ1Δτ2ΔτM×k=12M-1ξktk-τM-2utk-τ,
tk=±Δτ1±Δτ2±±ΔτM,
pτΔτ0=τuτ2NN-1!Δτ0Δτ1Δτ2ΔτN×k=12Ns=01-1ξk+stk,s-τN-1utk,s-τ,
tk,s=tk+-1sΔτ0,
pΔτ0=24π3Δτ022σ03exp-3Δτ022σ02,
pτ=54τσ0N-2uτπ2NN-1!Δτ1Δτ2ΔτN×k=12N-1ξkINtk-τσ0,
INx=-xtx+tN-1exp-32t2dt.
IN+2x=N+133xNIN+1x+INx,
pτ=54τσ0N-2uτπ2NN-1!ΔτN×n=0N-1nNnINN-2nΔτ-τσ0.
Fτ=54σ0N-1π2NN+1!Δτ1Δτ2ΔτNk=12N-1ξk×N+1τIN+1tk-τσ0+σ0IN+2tk-τσ0.
τ=36σ0N+1π2N-1N+2!Δτ1Δτ2ΔτN×k=12N-1ξkIN+3tkσ0.

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