Abstract

A new analytical evaluation of polarization-induced error probability for optical systems consisting of both PMD and PDL is presented. Using a simplified model containing a lumped PMD-PDL fiber, an amplifier with ASE noise, an idealized optical filter, and an electrical filter with integrate-and-dump response, a closed-form of the probability density of the filtered current is obtained. This allows us to evaluate the BER affected by the PMD and PDL. Based on this, two polarization related effects, i.e., the PMD and PDL directional coupling and the polarization-induced intersymbol interference (ISI), are studied. We show that the PMD and PDL directional coupling can be strongest when the PMD vector perpendicularly correlates with both the PDL vector and the input signal polarization in the 3D Stokes space. Besides, its impact on bit-error-rate (BER) strongly depends on the PDL value. We also find that, for an optical system with realistic parameters, the impact of polarization-induced ISI on the BER is mostly caused by the two closest neighbors of the desired bit. Related with these two polarization effects is the PMD value fluctuation. Large PMD value variation can play an overwhelming role in impairing the optical performance.

© 2007 Optical Society of America

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References

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  1. B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
    [CrossRef]
  2. N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
    [CrossRef]
  3. A. Mecozzi and M. Shtaif, "Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization," J. Lightwave Technol. 22, 1856-1871 (2004).
    [CrossRef]
  4. I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, "A receiver model for optical fiber communication systems with arbitrarily polarized noise," J. Lightwave Technol. 23, 1478-1490 (2005).
    [CrossRef]
  5. D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
    [CrossRef]
  6. P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
    [CrossRef]
  7. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
    [CrossRef]
  8. R. Holzohner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightwave Technol. 20, 389-400 (2002).
    [CrossRef]
  9. J. Wang and J. M. Kahn, "Accurate bit-error-ratio computation in nonlinear CRZ-OOK and CRZ-DPSK systems," IEEE Photon. Technol. Lett. 14, 2165- 2167 (2004).
    [CrossRef]
  10. J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
    [CrossRef]
  11. M. Shtaif and O. Rosenberg, "Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems," J. Lightwave Technol. 23, 923-930 (2005).
    [CrossRef]
  12. J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
    [CrossRef]
  13. C. Xie and L. Moller, "The accuracy assessment of different polarization mode dispersion models," Opt. Fiber Technol. 12, 101-109 (2006).
    [CrossRef]
  14. E. Forestieri and G. Prati, "Exact analytical evaluation of the second-order PMD impact on the outage probability for a compensated system," J. Lightwave Technol. 22, 988-996 (2004).
    [CrossRef]
  15. P. Lu, L. Chen, and X. Bao, "Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fiber," J. Lightwave Technol. 19, 856-859 (2001).
    [CrossRef]
  16. H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
    [CrossRef]
  17. L. Chen S. Hadjifaradji, D. S. Waddy and X. Bao, "Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation," Opt. Express 11, 3141-3146 (2003).
    [CrossRef] [PubMed]
  18. L. Chen and X. Bao, "Polarization-dependent loss-induced pulse narrowing in birefringent optical fiber with finite differential group delay," J. Lightwave Technol. 18, 665-667 (2000)
    [CrossRef]

2006 (1)

C. Xie and L. Moller, "The accuracy assessment of different polarization mode dispersion models," Opt. Fiber Technol. 12, 101-109 (2006).
[CrossRef]

2005 (3)

2004 (4)

2003 (2)

2002 (2)

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

R. Holzohner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightwave Technol. 20, 389-400 (2002).
[CrossRef]

2001 (1)

2000 (3)

1991 (1)

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

1990 (1)

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

Azizoglu, M.

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Bao, X.

Cartaxo, A. V. T.

J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
[CrossRef]

Carter, G. M.

Chen, L.

Forestieri, E.

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Gisin, N.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Gordon, J. P.

Grigoryan, V. S.

Holzohner, R.

Humblet, P. A.

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Jiao, H.

Kath, W. L.

Kim, N. Y.

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

Kogelnik, H.

Lee, D.

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

Lima, A. O.

Lima, I. T.

Marcuse, D.

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

Moller, L.

C. Xie and L. Moller, "The accuracy assessment of different polarization mode dispersion models," Opt. Fiber Technol. 12, 101-109 (2006).
[CrossRef]

Nelson, L. E.

P. Lu,

Park, J.

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

Park, N.

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

Prati, G.

Rebola, J. L.

J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
[CrossRef]

Rosenberg, O.

Shtaif, M.

Sun, Y.

Xie, C.

C. Xie and L. Moller, "The accuracy assessment of different polarization mode dispersion models," Opt. Fiber Technol. 12, 101-109 (2006).
[CrossRef]

Yoon, H.

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

Zweck, J.

IEE Proc. Optoelectron. (1)

J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

N. Y. Kim, D. Lee, H. Yoon, J. Park, and N. Park, "Limitation of PMD compensation due to polarizationdependent loss in high-speed optical transmission links," IEEE Photon. Technol. Lett. 14, 104-106 (2002).
[CrossRef]

J. Wang and J. M. Kahn, "Accurate bit-error-ratio computation in nonlinear CRZ-OOK and CRZ-DPSK systems," IEEE Photon. Technol. Lett. 14, 2165- 2167 (2004).
[CrossRef]

J. Lightwave Technol. (12)

J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
[CrossRef]

E. Forestieri and G. Prati, "Exact analytical evaluation of the second-order PMD impact on the outage probability for a compensated system," J. Lightwave Technol. 22, 988-996 (2004).
[CrossRef]

A. Mecozzi and M. Shtaif, "Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization," J. Lightwave Technol. 22, 1856-1871 (2004).
[CrossRef]

M. Shtaif and O. Rosenberg, "Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems," J. Lightwave Technol. 23, 923-930 (2005).
[CrossRef]

I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, "A receiver model for optical fiber communication systems with arbitrarily polarized noise," J. Lightwave Technol. 23, 1478-1490 (2005).
[CrossRef]

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

P. Lu, L. Chen, and X. Bao, "Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fiber," J. Lightwave Technol. 19, 856-859 (2001).
[CrossRef]

L. Chen and X. Bao, "Polarization-dependent loss-induced pulse narrowing in birefringent optical fiber with finite differential group delay," J. Lightwave Technol. 18, 665-667 (2000)
[CrossRef]

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
[CrossRef]

R. Holzohner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightwave Technol. 20, 389-400 (2002).
[CrossRef]

H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. (1)

C. Xie and L. Moller, "The accuracy assessment of different polarization mode dispersion models," Opt. Fiber Technol. 12, 101-109 (2006).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the lightwave system used for BER evaluation.

Fig. 2.
Fig. 2.

Probability density function W B 0 (y) for B 0 = 0 (solid) and B 0 = 1 (dashed) plotted for illustration. For optimum threshold, the shaded area represents twice of the BER defined by (20).

Fig. 3.
Fig. 3.

BER as a function of φ = φ ατ -φ sτ c τ with θ ατ = θ sτ = 90° and M = 3. Other parameters used to obatain BERs in (a) are τ/T = 0.3, α = 0.5dB, Ĩ 1 = 75. Parameters in (b) are same as those in (a), except that α = 1.0dB for the dashed , τ/T = 0.5 for the dotted, Ĩ 1 = I 1/I 0 = 90 for the dash-dotted, respectively.

Fig. 4.
Fig. 4.

BER as the function of PDL with Ĩ 1 = 85 and M = 3. Thick and thin solid curves are obtained with fixed DGD of τ = 0.25T and τ = 0.35T respectively. The dashed curve, which is 106 ~ 107 times of the thick solid, is obtained by performing Maxwellian average for Δτ = 0.25T. Inset: BERs at Δτ = 0.25T with truncation ratio t ≡ τ max /Δτ = 3.3 ~ ∞(dashed), t = 2.5 (dotted), and t = 1.5 (dash-dotted).

Fig. 5.
Fig. 5.

The pdf of BER with Ĩ 1 = 85, Δτ = 0.25T and M = 3. Thick solid (thin dotted) curve is obtained from (22) with Np = 106 and α = 0.25dB (α = 0.75dB), respectively.

Fig. 6.
Fig. 6.

Conditional BERs as functions of PDL, with (a) Δτ/T = 0.35, desired bit B 0 = 0, (b) Δτ/T = 0.35, B 0 = 1, (c) Δτ/T = 0.7, B 0 = 0. Other parameters are same as those given in the caption of Fig. 4, except Ĩ 1 = 100 in (c).

Equations (43)

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E in ( t ) = E s ( t ) e s = j = B j E 0 ( t jT ) exp ( c ( t jT ) ) e s ,
e ( t ) = ν = ν 1 ν 1 + M c ν e i ω ν t e ν 0
E in ( t ) = E in ( ω ) e iωt 2 π e s
E out ( t ) F 1 T ( ω ) F E in ( t ) ,
σ × σ = 2 i σ ,
T ( ω ) = T PDL T PMD ( ω ) = exp ( α 2 ) exp ( α σ 2 ) exp ( τ σ 2 ) .
y′ = I ¯ = K 0 T E out ( t ) + e ( t ) E out ( t ) + e ( t ) dt T .
W ˜ B 0 ( y ) = G B 0 ( ξ ) e iξy ( 2 π ) ,
G B 0 ( ξ ) = < e iξy′ > ,
( σ A ) ( σ B ) = A B + i σ ( A × B ) ,
e s σ A e s = e s A ,
e s e s = [ 1 + σ e s ] 2 .
τ 0 T PMD ( ω ) τ 0 = e τ 2 , τ T PMD ( ω ) τ = e τ 2 , τ 0 T PMD ( ω ) τ = 0 ,
α 0 T PDL α 0 = 1 , α T PDL α = e α , α 0 T PDL α = 0 ,
E out = a α 0 + a α
e = ν = ν 1 ν 1 + M [ c αν α 0 + c ⊥ν α ] e i ω ν t .
a = α 0 E out ( t ) = α 0 τ 0 τ 0 e s E s ( t + τ 2 ) + α 0 τ τ e s E s ( t τ 2 )
a = α E out ( t ) = e α [ α τ 0 τ 0 e s E s ( t + τ 2 ) + α τ τ e s E s ( t τ 2 ) ] .
G B 0 ( ξ ) = e iξK a 2 + a 2 ¯ [ 1 2 i σ 2 ] 2 M e 2 σ 2 K 2 ξ 2 ν = ν 1 ν 1 + M [ a αν ¯ 2 + a ⊥ν ¯ 2 ] 1 2 i σ 2 .
G B 0 ( ξ ) = e i ξ I 1 λ 1 [ 1 i I 0 ξ M ] 2 M e I 0 I 1 ξ 2 λ 2 M 1 i I 0 ξ M ,
λ 1 ( τ , α , e s ) = [ b ¯ τ cosh α + Δ b τ sinh α cos θ ατ + ( Δ b τ cosh α + b ¯ τ sinh α cos θ ατ ) cos θ
+ b ± τ sinh α sin θ ατ sin θ cos ( φ ατ φ ω c τ ) ] e α
λ 2 ( τ , α , e s ) = [ β ¯ τ cosh α + Δ β τ sinh α cos θ ατ + ( Δ β τ cosh α + β ¯ τ sinh α cos θ ατ ) cos θ
+ β ± τ sinh α sin θ ατ sin θ cos ( φ ατ φ ω c τ ) ] e α .
W ˜ B 0 ( y ) = M I 0 e M I 0 [ y + I 1 ( 2 λ 2 λ 1 ) ] n = 0 ( M I 1 I 0 λ 2 ) n { M I 0 [ λ + I 1 ( λ 2 λ 1 ) ] } 2 M + n 1 n ! ( 2 M + n 1 ) !
= M exp [ M ( X 1 X 2 ) 2 ] [ X 1 X 2 ] 2 M 1 I ˜ 2 M 1 ( 2 M X 1 X 2 ) I 0 ,
W B 0 ( y ) = 0 p τ 0 p α × 0 π p ατ 2 sin θ ατ d θ ατ 0 π p 2 sin θ 0 2 π p φ 2 π W ˜ B 0 ( y , α , τ θ ατ , θ , φ ) ,
BER = [ P 0 ( I d ) + P 1 ( I d ) ] 2 ,
P 0 ( I d ) = I d W 0 ( y ) dy , P 1 ( I d ) = 0 I d W 1 ( y ) dy ,
pdf ( BER ) = lim Δ BER 0 Δ N i N p 1 Δ BER ,
BER pa = j = 1 N p BER j N p = 0 1 ( BER ) pdf ( BER ) d ( BER ) ,
W B 0 ( y ) = 0 τ max p τ′ dτ′ 0 π sin θ ατ 2 ατ 0 π sin θ 2 0 2 π 1 2 π W ˜ B 0 ( y ; τ , θ τα , θ τs , φ ) .
a 2 + a 2 ¯ = 0 T dt T ( a 2 + a 2 ) .
0 T E s ( t + τ 2 ) 2 dt T = 0 T { τ 2 } B E 0 2 ( t + { τ 2 } ) dt T + T { τ 2 } T B j τ + 1 E 0 2 ( t + { τ 2 } T ) dt T
= E 1 2 [ B ( T { τ 2 } ) + B + 1 { τ 2 } ] T E 1 2 b + τ
0 T E s ( t τ 2 ) 2 dt T = 0 { τ 2 } B 1 E 0 2 ( t { τ 2 } + T ) dt T + { τ 2 } T B j τ E 0 2 ( t { τ 2 } ) dt T
= E 1 2 [ B ( T { τ 2 } ) + B 1 { τ 2 } ] T E 1 2 b τ .
b ± τ = { ( B B 1 + B B + 1 ) T { τ 2 } T + B + 1 B 1 2 { τ 2 } T T , ( { τ 2 } > T 2 ) ; ( B B 1 + B B + 1 ) { τ 2 } T + B B T 2 { τ 2 } T , ( { τ 2 } < T 2 ) .
a 2 + a 2 = E 1 2 { [ b + τ A ˜ ( τ 0 , α 0 ) + b τ A ˜ ( τ 0 , α 0 ) + b ± τ 2 B ˜ + [ b + τ A ˜ ( τ 0 , α 0 ) + b τ A ˜ ( τ 0 , α 0 ) b ± τ 2 B ˜ ] e 2 α } E 1 2 λ 1 ,
a αν ¯ = E 1 [ α 0 τ 0 τ 0 e s b + τν e i τ 4 ( ω c + ω ν ) + α 0 τ τ e s b τν e i τ 4 ( ω c + ω ν ) ]
a ⊥ν ¯ = E 1 e α [ α τ 0 τ 0 e s b + τν e i τ 4 ( ω c + ω ν ) + α τ τ e s b τν e i τ 4 ( ω c + ω ν ) ] ,
ν = ν 1 ν 1 + M a αν ¯ 2 + a ⊥ν ¯ 2 = E 1 2 { β + τ A ˜ ( τ 0 , α 0 ) + β τ A ˜ ( τ 0 , α 0 ) + β ± τ 2 B ˜ + [ β + τ A ˜ ( τ 0 , α 0 ) + β τ A ˜ ( τ 0 , α 0 ) β ± τ 2 B ˜ ] e 2 α } E 1 2 λ 2 ,
β ± τ = ν = ν c ν c + m b + τν b τν cos τ ( ω ν ω c ) 2 .

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