Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach

doi:10.1364/OE.15.002106

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Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach

Liang Chen, Zhongxi Zhang, and Xiaoyi Bao »View Author Affiliations

Optics Express, Vol. 15, Issue 5, pp. 2106-2119 (2007)
http://dx.doi.org/10.1364/OE.15.002106

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▸ Article Outline
– Abstract
1. Introduction
2. System modeling and BER calculation
3. Applications
4. Comments and Conclusions
– References and links

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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.

1. Introduction

It is well known that at high data rates (typically ≥ 10Gbits/s) polarization effects can severely impair optical communication system performance. Conventional polarization effects include polarization-mode dispersion (PMD), which causes the differential group delay (DGD) between the principal states of polarization (PSPs), and polarization-dependent loss (PDL), which causes polarization-dependent attenuation of the propagating signal. Apart from individual PMD and PDL effects, the combined PMD-PDL interaction can further degrade the system performance. Due to the interference between the fast and slow modes, the combined PMD-PDL interaction makes an optical system more complicated than PMD or PDL alone. It may result in anomalous dispersion and causes additional signal distortion [1

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

]. In Ref. [2

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

], numerical simulation for 10-Gbits/s transmission systems showed that even a small PDL (~ 0.7dB) could significantly limit the effectiveness of PMD compensation. While many bit-error-ratio (BER) evaluations have focused on various optical links with PMD alone, systems with both PMD and PDL have been attracting growing theoretical attention in recent years. Ref. [1

] pointed out the complexity of this problem and showed a number of examples demonstrating features of waveforms experiencing the combined effects of PMD and PDL. In Refs. [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]

], the PDL induced noise enhancement and its impact on signal-to-noise ratio (SNR) and Q factor were investigated. In this work we further consider the combined PMD-PDL impact on BERs in simplified optical systems.

The BERs can be predicted either by Monte Carlo simulation or by analytical evaluation. The first relies on the speed of the signal propagation equation solver. It is easier to be implemented but is computationally expensive when dealing with extremely rare events. To improve the simulation efficiency one needs detailed system configuration knowledge and advanced simulation techniques. As regards the second approach, while its computation can be faster than the first by a few orders of magnitude, the validity of its result strongly depends on how the system is modeled. Despite the fact that the theoretical modeling of system performance is lagging behind the rapid development of commercial systems, remarkable progress has been made in recent years [4

]–[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]

]. In Ref. [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]

], Marcuse reported his pioneering progress in analytical BER evaluation based on a simplified model with amplitude-shift-keying pulse modulation. By using the Karhunen-Loève series expansion (KLSE) method to describe the filtered amplified spontaneous emission (ASE) noise, Forestieri proposed a computationally efficient approach to evaluate BERs in optically preamplified systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering [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]

]. Authors of Refs. [8

R. Holzohner, V. S. Grigoryan, C. R. Menyuk, and 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]

] and [9

Jin 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

Jin 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]

] respectively studied the nonlinear noise effects and extended Forestieri’s work to various modulation formats. Ref. [12

] developed Forestieri’s approach by considering the interaction between two orthogonal noise components for the system with arbitrarily polarized noise and negligible PMD.

Here we extend Marcuse’s work[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]

] by taking into account the first order PMD and PDL effects and approximating them with a lumped PMD-PDL model to analytically evaluate the combined PMD-PDL impact on the system performance. In principle, an optical system with polarization effects can be modeled as a concatenation of many PMD-PDL sections with each section having the PMD and PDL that change randomly on a time scale of milliseconds to days. Motivated by the conclusion of Ref. [13

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

] that a wave-plate (an all-order) model for a pure PMD system with no PMD compensator can be well approximated by the lumped (first-order) PMD model, we use the lumped PMD-PDL model to approximate the polarization related effects. By taking the advantage that this simplified model is convenient for theoretical study, we quantitatively evaluate the impact of the PMD and PDL directional coupling on the BER and investigate its dependence on the related parameters, such as PDL and PMD vectors, the input signal polarization, and the optical signal-to-noise ratio (OSNR). Another polarization related effect, the polarization-induced intersymbol interference (ISI) is also quantitatively analyzed. Closely related with these two polarization effects is the DGD statistical distribution. Our calculations show its dominant role in impairing the system performance.

2. System modeling and BER calculation

Figure 1 shows the simplified model used for our BER evaluation. The optical signal E⃗_in (t) is launched into the lumped PMD-PDL fiber in the linear regime. At the receiver end it is amplified by a flat gain amplifier G which adds ASE noise. The amplifier output is optically filtered prior to photodetection. Then the detected current I(t) = K|E⃗_out (t)+e⃗(t)|2 is electrically filtered and sampled.

For binary ON-OFF keying (OOK) systems, we can express the nonreturn-to-zero (NRZ) signal with bit-time interval T and carrier frequency ω_c at the PMD input as (without causing any significant error, we take e^iω_cT = 1.)

∣ \vec{E_{in}} (t) 〉 = E_{s} (t) ∣ \vec{e_{s}} 〉 = \sum_{j = - \infty}^{\infty} B_{j} E_{0} (t - jT) \exp (- {iω}_{c} (t - jT)) ∣ \vec{e_{s}} 〉,

(1)

where the polarization direction is described by a complex unit vector in 2D Jones space |e⃗_s 〉=[x,y]T and E ₀(t) is the rectangular pulse function, i.e., during 0 <t < T, E ₀(t) = E ₁ and outside this time interval E ₀(t) is zero. Each coefficient B_j is either 0 or 1, depending on the input bit configuration. Note that in (1), we use |E⃗_in (t)〉 to denote the Jones representation of the input signal E⃗_in (t) in Fig. 1. Similar notations are used in (2) and (4) to represent Jones vectors of ASE noise e⃗(t) and output signal E⃗_out (t) in Fig. 1. To trace the roles of signal, noise, and the PMD-PDL interaction in the BER, we employ the following approximations proposed by Ref. [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]

]. Firstly, the optical filter shown in Fig. 1 is treated as an ideal filter with bandwidth of M/T (M = 1,2,3…) and a transfer function being unit magnitude inside the passband and zero magnitude outside of it. (Its phase effect on noise drops out of the analysis.) Based on this, the filtered ASE noise is simplified as

∣ \vec{e} (t) 〉 = \sum_{ν = ν 1}^{ν 1 + M} c_{ν} e^{- i ω_{ν} t} ∣ {\vec{e}}_{ν 0} 〉

(2)

with ω_ν = 2πν/T and |e⃗ _ν0〉 the unitary vectors representing the polarizations of noise components with angular frequencies ω_ν . Noise expansion coefficients c_ν are approximated as complex independent and identically distributed (i.i.d.) random variables (r.v.) with Gaussian pdfs of zero mean < c_ν >= 0 and variance σ [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]

]–[7

],[10

],[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]

]. Because both |e⃗ _ν0〉 and c_ν change randomly, each noise component with ω_ν in (2) has four degrees of freedom. Correlations between orthogonal noise polarizations [4

, 12

] have been neglected in this work. Secondly, the influence of the optical filter on the signal has been neglected so that |E⃗_out (t)〉 relates with the input

Fig. 1. Schematic of the lightwave system used for BER evaluation.

∣ {\vec{E}}_{in} (t) 〉 = \int E_{in} (ω) e^{- iωt} \frac{dω}{\sqrt{2 π}} ∣ {\vec{e}}_{s} 〉

(3)

by E_out (ω) ≈ T(ω)E_in (ω) or

∣ {\vec{E}}_{out} (t) 〉 \approx F^{- 1} T (ω) F ∣ {\vec{E}}_{in} (t) 〉,

(4)

where F denotes the Fourier-transform operation and F|E⃗_in (t)〉 = E_in (ω)|e⃗_s 〉. With the help of the Pauli spin matrices σ⃗, which obeys the commutation relation

\vec{σ} \times \vec{σ} = 2 i \vec{σ},

(5)

the lumped PMD-PDL transfer matrix T(ω) can be expressed as

T (ω) = T_{PDL} T_{PMD} (ω) = \exp (- \frac{α}{2}) \exp (\vec{α} ∙ \vec{\frac{σ}{2}}) \exp (- iω \vec{τ} ∙ \frac{\vec{σ}}{2}) .

(6)

Notice that σ⃗ or σ_i (i=1,2,3) are the standard physics notations for Pauli matrices, whereas σ, a commutable number, is the standard notation for the variance of a Gaussian random variable. Expressions of σ_i (i = 1,2, 3) given by Ref. [15

Ping 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]

] are typical forms of Pauli spin matrices. They are different from those given by Refs. [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]

, 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]

]. Such notation difference is not important provided that σ₂ of Ref. [15

Ping 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]

] (or σ₃ of Ref. [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]

, 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]

]) is used for the circular polarization component in the 3D Stokes space. This means that, for the given expression of T_PMD (ω) = exp(-iωt⃗∙σ⃗/2), the coordinate axes in 3D Stokes space are defined in different ways. This does not affect the commutation relation (5) as well as the analytical result given below. In general, the equivalent PDL vector α⃗ in (6) may be frequency dependent. This will influence the PDL induced waveform distortion effect in an optic link [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]

]. Considering that such frequency dependent waveform distortion is not so important in a system with realistic parameters [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]

], we approximate α⃗ in (6) as a frequency independent vector . Thirdly, the post-detect filter is modeled as the integrate-and-dump type with its output the random variable

y′ = \bar{I} = K \int_{0}^{T} 〈 {\vec{E}}_{out} (t) + \vec{e} (t) ∣ {\vec{E}}_{out} (t) + \vec{e} (t) 〉 \frac{dt}{T} .

(7)

Although many real communication systems with forward-error correction (FEC) can tolerate raw BERs of up to 10^-3, typical BERs for optical systems are in the range of 10^-9-10^-15 and below, within which the usual Gaussian fitting Q-factor approximation does not work well. In this case, the probability density function (pdf) of the filtered current, W̃_B₀ (y), becomes the key part of the BER evaluation. [As shown in (1), B ₀ is the value of the desired bit (j = 0).] Usually W̃ _{B
₀} (y) is obtained from its characteristic function, which relates with W̃ _{B
₀} (y) by

{\tilde{W}}_{B_{0}} (y) = \int_{- \infty}^{\infty} G_{B_{0}} (ξ) e^{- iξy} \frac{dξ}{(2 π)},

(8)

while the characteristic function G _{B
₀} (ξ) is by definition the statistical average of e ^iξy′

G_{B_{0}} (ξ) = < e^{iξy′} >,

(9)

where < … > means noise average and y′ is given by (7).

To get W̃ _{B
₀} (y) in this way, we designate a vector in 2D Jones complex space by the Dirac notation, e.g., |e⃗_s 〉 = [x,y]^T , while its mapping in 3D Stokes space by the standard vector notation, e.g., e⃗_s . According to the basic property (5), one can show that [3

, 15

Ping 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]

], for any two vectors A⃗, B⃗, and any unit vector, e.g.,e⃗_s ,

(\vec{σ} ∙ \vec{A}) (\vec{σ} ∙ \vec{B}) = \vec{A} ∙ \vec{B} + i \vec{σ} ∙ (\vec{A} \times \vec{B}),

〈 {\vec{e}}_{s} ∣ \vec{σ} ∙ \vec{A} ∣ {\vec{e}}_{s} 〉 = {\vec{e}}_{s} ∙ \vec{A},

∣ {\vec{e}}_{s} 〉 〈 {\vec{e}}_{s} ∣ = \frac{[1 + \vec{σ} ∙ {\vec{e}}_{s}]}{2} .

(10)

T_{PMD} (ω) = \exp (\frac{- iω \vec{τ} ∙ \vec{σ}}{2}) = \cos \frac{ωτ}{2} - i \sin \frac{ωτ}{2} ({\vec{τ}}_{0} ∙ \vec{σ})

, which leads to

〈 {\vec{τ}}_{0} ∣ T_{PMD} (ω) ∣ {\vec{τ}}_{0} 〉 = e^{- iω \frac{τ}{2}}, 〈 {\vec{τ}}_{⊥} ∣ T_{PMD} (ω) ∣ {\vec{τ}}_{⊥} 〉 = e^{iω \frac{τ}{2}}, 〈 {\vec{τ}}_{0} ∣ T_{PMD} (ω) ∣ {\vec{τ}}_{⊥} 〉 = 0,

(11)

whereτ⃗_⊥ is the unit vector satisfying 〈τ⃗_⊥|τ⃗₀〉 = 〈τ⃗₀|τ⃗_⊥〉 = 0. From (11), one can see that τ⃗₀ and τ⃗_⃗ are the normalized fast and slow PSPs in the PMD component. Similarly, by introducing α⃗₀≡α⃗/|α| and α⃗_⊥ satisfying 〈α⃗_⊥|α⃗₀〉 = 〈α⃗₀|α⃗_⊥〉 = 0, one can obtain

〈 {\vec{α}}_{0} ∣ T_{PDL} ∣ {\vec{α}}_{0} 〉 = 1, 〈 {\vec{α}}_{⊥} ∣ T_{PDL} ∣ {\vec{α}}_{⊥} 〉 = e^{- α}, 〈 {\vec{α}}_{0} ∣ T_{PDL} ∣ {\vec{α}}_{⊥} 〉 = 0,

(12)

which means that α⃗₀ and α⃗_⊥ are the two eigenvectors representing the minimum and maximum attenuation states in the PDL component. Basic relations from (10) to (12) are useful for the following derivation.

To get G _{B
₀} (ξ) defined by (9), we decompose the output signal |E⃗_out (t)〉 and the filtered noise |e⃗ (t)〉 in (2) as

∣ {\vec{E}}_{out} 〉 = a ∣ {\vec{α}}_{0} 〉 + a_{⊥} ∣ {\vec{α}}_{⊥} 〉

∣ \vec{e} 〉 = \sum_{ν = ν_{1}}^{ν_{1} + M} [c_{αν} ∣ {\vec{α}}_{0} 〉 + c_{⊥ν} ∣ {\vec{α}}_{⊥} 〉] e^{- i ω_{ν} t} .

(13)

With the help of (3), (4), (10)-(12), as well as relations |k⃗ ₀〉〈k⃗ ₀| +|k⃗ _⊥〉〈k⃗ _⊥| = 1 and k⃗ _⊥∙k⃗ ₀ = -1 (k⃗ = α⃗, τ⃗), one can express the signal expansion coefficients in (13) as

a = 〈 {\vec{α}}_{0} ∣ {\vec{E}}_{out} (t) 〉 = 〈 {\vec{α}}_{0} ∣ {\vec{τ}}_{0} 〉 〈 {\vec{τ}}_{0} ∣ {\vec{e}}_{s} 〉 E_{s} (t + \frac{τ}{2}) + 〈 {\vec{α}}_{0} ∣ {\vec{τ}}_{⊥} 〉 〈 {\vec{τ}}_{⊥} ∣ {\vec{e}}_{s} 〉 E_{s} (t - \frac{τ}{2})

a_{⊥} = 〈 {\vec{α}}_{⊥} ∣ {\vec{E}}_{out} (t) 〉 = e^{- α} [〈 {\vec{α}}_{⊥} ∣ {\vec{τ}}_{0} 〉 〈 {\vec{τ}}_{0} ∣ {\vec{e}}_{s} 〉 E_{s} (t + \frac{τ}{2}) + 〈 {\vec{α}}_{⊥} ∣ {\vec{τ}}_{⊥} 〉 〈 {\vec{τ}}_{⊥} ∣ {\vec{e}}_{s} 〉 E_{s} (t - \frac{τ}{2})] .

(14)

As discussed, the noise expansion coefficients c _αν and c _⊥ν appearing in (13) are approximated as complex i.i.d. Gaussian random variables consisting of real and imaginary parts c_jν = c^r _jν + icⁱ _jν (j = α,⊥). Based on Eqs. (7), (9), and (13), we have 〈E⃗_out | E⃗_out 〉 = |a|²+|a _⊥|², 〈e⃗|E⃗_out 〉+ c.c. = ∑^ν₁+M _ν=ν₁ 2[c ^r _αν a ^r _αν+c ⁱ _αν a ⁱ _αν+c ^r _⊥ν a ^r _⊥ν+c ⁱ _⊥ν a ⁱ _⊥ν] (a _αν≡ ae ^{iω
_ν
t} , a _⊥ν ≡ a _⊥ e ^{iω
_ν
t}), 〈e⃗|e⃗〉 = ∑^ν₁+M _ν₁ [c ^r2 _αν+c ⁱ² _αν +c ^r2 _⊥ν+c ⁱ² _⊥ν], and

G_{B_{0}} (ξ) = e^{iξK \bar{{∣ a ∣}^{2} + {∣ a_{⊥} ∣}^{2}}} {[1 - 2 i σ^{2} Kξ]}^{- 2 M} e^{\frac{- 2 σ^{2} K^{2} ξ^{2} \sum_{ν = ν_{1}}^{ν_{1} + M} [{∣ \bar{a_{αν}} ∣}^{2} + {∣ \bar{a_{⊥ν}} ∣}^{2}]}{1 - 2 i σ^{2} Kξ}} .

(15)

Here the formula

< e^{iK ξ (c^{2} + 2 ca)} = \int \frac{dc}{\sqrt{2 π σ^{2}}} e^{- \frac{c^{2}}{2 σ^{2}}} e^{iK ξ (c^{2} + 2 ca)} = {[1 - i 2 σ^{2} K ξ]}^{- \frac{1}{2}} e^{- \frac{2 σ^{2} K^{2} ξ^{2} a^{2}}{1 - i 2 σ^{2} K ξ}}

has been used and c.c. stands for complex conjugation. The time average term

\bar{{∣ a ∣}^{2} + {∣ a_{⊥} ∣}^{2}}

in (15) is due to the electrically filtered signal-signal beating, while another time average term

\sum_{ν = ν_{1}}^{ν_{1} + M} [{∣ \bar{a_{αν}} ∣}^{2} + {∣ \bar{a_{⊥ν}} ∣}^{2}]

is caused by the electrically filtered signal-noise beating. Further derivation is given in the Appendix.

By denoting I ₀ ≡ 2σ² KM, I ₁ ≡ KE ² ₁ and substituting (28) and (30) into (15), we obtain

G_{B_{0}} (ξ) = e^{i ξ I_{1} λ_{1}} {[1 - \frac{i I_{0 ξ}}{M}]}^{- 2 M} e^{- \frac{\frac{I_{0} I_{1} ξ^{2} λ_{2}}{M}}{\frac{1 - i I_{0} ξ}{M}}},

(16)

where, according to the discussion in the Appendix, λ_i (i = 1, 2) can be expressed as

λ_{1} (\vec{τ}, \vec{α}, {\vec{e}}_{s}) = [{\bar{b}}_{τ} \cosh α + Δ b_{τ} \sinh α \cos θ_{ατ} + (Δ b_{τ} \cosh α + {\bar{b}}_{τ} \sinh α \cos θ_{ατ}) \cos θ_{sτ}

+ b_{\pm τ} \sinh α \sin θ_{ατ} \sin θ_{sτ} \cos (φ_{ατ} - φ_{sτ} - ω_{c} τ)] e^{- α}

λ_{2} (\vec{τ}, \vec{α}, {\vec{e}}_{s}) = [{\bar{β}}_{τ} \cosh α + Δ β_{τ} \sinh α \cos θ_{ατ} + (Δ β_{τ} \cosh α + {\bar{β}}_{τ} \sinh α \cos θ_{ατ}) \cos θ_{sτ}

+ β_{\pm τ} \sinh α \sin θ_{ατ} \sin θ_{sτ} \cos (φ_{ατ} - φ_{sτ} - ω_{c} τ)] e^{- α} .

(17)

In (17), k̄ _τ ≡ (k _+τ +k _-τ)/2, Δk _τ ≡ (k _+τ-k _-τ)/2 (k = b,β), θ_ατ (θ_sτ) and φ_ατ (φ_sτ) are polar and azimuthal angles between α⃗ (e⃗ _s) and τ⃗ in 3D Stokes space. Substituting (16) into (8) and making use of the relation

\frac{1}{2 πi} \int \frac{e^{p z}}{z^{n}} dz = \frac{p^{n - 1}}{(n - 1)!},

, we get the analytic form of W̃ _{B
₀} (y)

{\tilde{W}}_{B_{0}} (y) = \frac{M}{I_{0}} e^{- \frac{M}{I_{0}} [y + I_{1} (2 λ_{2} - λ_{1})]} \sum_{n = 0}^{\infty} \frac{{(\frac{M I_{1}}{I_{0}} λ_{2})}^{n} {\frac{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}] {[\frac{X_{1}}{X_{2}}]}^{2 M - 1} {\tilde{I}}_{2 M - 1} \frac{(2 M X_{1} X_{2})}{I_{0}},

(18)

with

X_{1} \equiv \sqrt{\tilde{y} + (λ_{2} - λ_{1}) {\tilde{I}}_{1}}

and

X_{2} \equiv \sqrt{λ_{2} {\tilde{I}}_{1}}

. In (18), Ĩ_2M-1 is the (2M-1)th-order exponentially scaled modified Bessel function defined as

{\tilde{I}}_{ν} (x) = e^{- x} \sum_{n = 0}^{\infty} \frac{{(\frac{x}{2})}^{2 n + ν}}{n! (n + ν)!}

. Here ỹ ≡ y/I ₀ is the normalized detected current and parameter Ĩ ₁ ≡ I ₁/I ₀ is related with OSNR. We can show this by considering I ₁/I ₀ of a non-polarization system (τ = α = 0), where b̄ _τ =

\bar{β}

_τ = b _±τ = β_±τ = B ₀, Δb _τ = Δβ _τ = 0 and (18) with B ₀ = 0 (B ₀ = 1) yields Ī ₀ = ∫ W̃ ₀(y)ydy = 2I ₀ (Ī ₁ = ∫ W̃ ₁(y)ydy = 2I ₀+I ₁), or I ₀ = Ī ₀/2 (I ₁ = Ī ₁-Ī ₀), respectively.

Note that λ ₁ in (16) is caused by the signal itself, while λ ₂ is related with the interaction between signal and noise. Also, the last terms of λ ₁ and λ ₂ in (17), named f-s beat terms here, are caused by the interference between the fast mode and the slow mode. For the lumped PMD-PDL model, they are caused by the PMD-PDL interaction. Therefore, to explore the properties of such interaction, one can just pay attention to the f-s beat terms in (17). Expressions of the f-s beat terms in (17) show that in the case of θ _ατ = 0 or θ _αs = 0, there is no PMD-PDL interaction, even if both τ and α are non-zero. On the other hand, when θ _ατ = θ _αs = π/2, i.e., both PDL vector and signal polarization are perpendicular to the PMD vector in the 3D Stokes space, it is posible to get the strongest PMD-PDL interaction, depending on the value of φ = φ _ατ -φ _sτ-ω_cτ. In this sense the PMD-PDL interaction is a kind of directional coupling effect.

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).

Due to statistical properties of PMD and PDL, the pdf given by (18) needs to be further averaged as

W_{B_{0}} (y) = \int_{0}^{\infty} p_{τ} dτ \int_{0}^{\infty} p_{α} dα \times \int_{0}^{π} \frac{p_{ατ}}{2} \sin θ_{ατ} {d θ}_{ατ} \int_{0}^{π} \frac{p_{sτ}}{2} \sin θ_{sτ} {dθ}_{sτ} \int_{0}^{2 π} \frac{p_{φ}}{2 π} dφ {\tilde{W}}_{B_{0}} (y, α, τ θ_{ατ}, θ_{sτ}, φ),

(19)

where φ = φ _ατ -φ _sτ-ω_cτ and p _τ , p _α, p _ατ , p _sτ and p _φ are statistical weight factors of τ,α,θ _ατ ,θ _sτ and φ. Assuming bits ONE and ZERO are sent with equal probability, the BER can be easily obtained from (19) and

BER = \frac{[P_{0} (I_{d}) + P_{1} (I_{d})]}{2},

(20)

with

P_{0} (I_{d}) = \int_{I_{d}}^{\infty} W_{0} (y) dy, P_{1} (I_{d}) = \int_{0}^{I_{d}} W_{1} (y) dy,

(21)

where I_d is the threshold current. In this work, it is set to be optimal , i.e., to be the solution of the equation W ₀(I_d ) =W ₁(I_d ). We can see from Fig. 2 and (20)–(21) that, for the given W ₀(y) and W ₁(y), the BER is minimal when the threshold is optimal.

Considering that, in reality, a single BER measurement can be made in a time that is less than the timescale over which the DGD and PDL of the system drift, more useful information on BER variance can be obtained by considering the pdf of BER (or the histogram of BER), which needs to calculate BERs for different sample pairs, with each pair of samples (B ₀ = 0, 1) being specified by independent and random variables τ,θ _ατ ,θ _sτ and φ that obey the same statistical distributions as those in (19). Thus the pdf of BER can be obtained by

pdf (BER) = lim_{Δ BER \to 0} \frac{Δ N_{i}}{N_{p}} \frac{1}{Δ BER},

(22)

Fig. 3. BER as a function of φ = φ _ατ -φ _sτ-ω_cτ 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.

where ΔN_i is the number of sample pairs whose BERs are located between BER_i and BER_i + ΔBER. Provided that the total pair number N_p is large enough, the average BER obtained from this pdf,

{BER}_{pa} = \sum_{j = 1}^{N_{p}} \frac{{BER}_{j}}{N_{p}} = \int_{0}^{1} (BER) pdf (BER) d (BER),

(23)

should be equal to the average BER obtained from (19)–(21).

Parameters and statistical weights in (19) can be adjusted to study different systems. For the single mode fiber (SMF), unit vectors τ⃗₀ and α⃗₀ are usually assumed to be uniformly and independently distributed over their Poincaré spheres. However, both theoretical and experimental studies have shown evidence of directional correlation between them [17

L. Chen, S. Hadjifaradji, D. S. Waddy, and Xiaoyi 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]

]. Because they are almost perpendicularly correlated in the 3D Stokes space, directional weighting factors in (19) should not be treated as constants. To estimate the directional coupling effect in subsection 3.1, we set p _ατ = δ(θ _ατ -π/2) and p _sτ = δ(θ _ατ -π/2) for BER calculations. We are interested in this because it can show us the strongest PMD-PDL interaction. In other subsections below, the usual directional assumption for the SMF is applied. The DGD distribution, p _τ in (19), will be Maxwellian, except in the subsection 3.1, where we fix the DGD according to Ref. [2

] so that our results can be compared with theirs. Also, in this work, α is assumed to be equal to its variance, i.e., p _α = δ(α -Δα), for simplicity.

3. Applications

3.1. Directional coupling effect

As discussed, the PMD-PDL interaction can be observed by setting θ _ατ = θ _sτ = 90° and changing the φ = φ _ατ -φ _sτ-ω_cτ from 0° to 180°. This means, in 3D Stokes space, α⃗ and e⃗ _s are limited in the plane perpendicular to τ⃗. As indicated in Fig. 3(a), when φ = 90°, the BER is not affected by the PMD-PDL interaction, since f-s beat terms in (17) are zero. When φ approaches to zero, we get the minimum BER, because the DGD effect is partly cancelled by the combined PMD-PDL interaction [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]

]. On the other hand, at φ = 180°, we get the maximum BER. The PMD-PDL interaction induced BER variation, represented by BER_max /BER_min , is around one order of magnitude. To show how the BER is affected by different parameters that determine the PMD-PDL interaction, we plot relevant curves in Fig. 3(b), where results in Fig. 3(a) are replotted as the solid curve for comparison. Parameters of the dashed and solid curves are same, except that PDL value α has been increased from 0.5dB to 1.0dB. The dashed curve variation is around two order of magnitudes, since the f-s beat terms in (17) are proportional to sinh α. Also, the dotted (τ/T = 0.5) and solid (τ/T = 0.3) curves show that the effect of the combined PMD-PDL interaction is not very sensitive to the PMD value. This is because coefficients b _±τ and β _±τ in the f-s beat terms change linearly with τ. In fact, according to the physical meaning of b _±τ given by (27) [also β _±τ given by (31)], larger DGD means less overlap between fast and slow signal modes in the time window [0,T]. (In this work, apart from the last subsection, the BER is calculated by assuming that the value of the desired bit B ₀ is different from the value of its closest neighbors B _±1) This results in smaller b _±τ and β _±τ or less BER variation. Moreover, the dash-dotted curve shows that the PMD-PDL interaction induced BER is rather sensitive to the OSNR. This is because, as shown in (18), the f-s beat terms in (17) are always multiplied by Ĩ ₁ (OSNR). Therefore, larger Ĩ ₁ results in stronger PMD-PDL interaction induced effect.

In the above calculation, we choose M = 3. Our model thus obtained agrees well with the power margin condition of Ref. [2

], where BER without PMD and PDL is 8.6×10^-18 and power margin without polarization is 2.43dB at 10^-9 BER. (Ours is 2.43dB at 0.4×10^-9 BER.) Curves in Fig. 3 agree reasonably well with Monte Carlo results plotted in Fig. 5 of Ref. [2

], except for the difference in the initial phase of φ.

3.2. BERs with directional average

In this subsection, the BER calculation is based on conventional directional assumption for the SMF. Also the input polarization e⃗_s is treated as the third independent random variable. These two assumptions results in p _ατ = p _sτ = p _φ = 1 in (19). Therefore the average BER (20) is obtained based on standard statistical average (19) or

W_{B_{0}} (y) = \int_{0}^{τ_{max}} p_{τ′} dτ′ \int_{0}^{π} \frac{\sin θ_{ατ}}{2} {dθ}_{ατ} \int_{0}^{π} \frac{\sin θ_{sτ}}{2} {dθ}_{sτ} \int_{0}^{2 π} \frac{1}{2 π} {dφ \tilde{W}}_{B_{0}} (y; τ', θ_{τα}, θ_{τs}, φ) .

(24)

For a system with PMD compensation, τ_max in (24) should be finite and the corresponding p _τ′ depends on how the PMD is compensated in the system. On the other hand, for a system without any dynamical control of PMD, p _τ′ in (24) can be approximated as

p_{τ′} = \frac{N_{m}}{Δτ} {(\frac{τ′}{Δτ})}^{2} e^{- \frac{3}{2} {(\frac{τ′}{Δτ})}^{2}}

with τ_max →∞. Normalization constant

N_{m ∣ τ_{max \to \infty}} = 3 \sqrt{\frac{6}{π}}

. A system with fixed τ can be modeled by setting p′_τ = δ(τ′ -τ). Figure 4 shows some results with and without Maxwellian average. According to Ref. [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]

], parameter Ĩ ₁ in the caption of Fig. 4 is determined by letting the BER with τ = α = 0 equal to BER ₀ = 10^-12 and setting 3dB power margin. Thick (τ/T = 0.25) and thin (τ/T =0.35) solid curves in Fig. 4 are very close to the Monte Carlo results (without PMD compensation) plotted in Fig. 3 of Ref. [2

]. The discrepancy between thick solid (τ/T = 0.25) and dashed (Δτ/T = 0.25) curves shows remarkable impact due to the DGD variation, because the DGD variation affects W̃ _{B
₀} (y) (B ₀ = 0, 1) in (24) more than the Maxwellian weight factor. In the inset of Fig. 4, we plot BERs vs PDL with different values of τ_max, from which one can see that, with the suppressed DGD variance, the BER can be improved by orders of magnitude, depending on the DGD residual limit. Obviously, this BER improvement is more important than the effect of PMD and PDL directional coupling, which changes BER by two orders of magnitude or less.

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

The BER variance can be shown clearly by calculating the pdf of BER with (22). In Fig. 5, the pdfs of BER with α = 0.25dB (thick solid) and α = 0.75dB (thin dotted) are plotted (Δτ =0.25T). The corresponding pdf average BREs obtained from (23) are BER_pa = 2.9×10^-6 (α =0.25dB) and BER_pa =3.9×10^-6 (α =0.75dB), which agree well with the average BERs plotted in Fig. 4 (dashed curve). Note that for each given curve in Fig. 5, due to the very large BER variance, the BER with maximal pdf is orders of magnitude smaller than the BER_pa .

3.3. Polarization-induced ISI effect

Intersymbol interference (ISI) is a neighboring-bit effect. It consists of DGD, CD, and filter induced ISI effects. To evaluate the BER, one needs to know, for a given bit sequence, how many neighbors of the desired bit B ₀ will influence the BER. Here we only consider the DGD-induced ISI effect. For a system with a fixed DGD satisfying DGD< 2T, it is easy to see that only two nearest neighbors of the desired bit, B _±1, will affect the BER. However, for a realistic system, since its DGD is statistically distributed, the DGD-induced ISI effect needs to be studied quantitatively. In this work, we analyze this effect by calculating and comparing BERs associated with different bit configurations. Because the optimum threshold I_d in (21) is determined by both W ₀(y) and W ₁(y) shown in Fig. 2, we calculate the conditional BER for a given bit pattern by modifying (21), i.e., in the case of desired bit B ₀ = 1 (B ₀ = 0), the pdf W ₀(y) [W ₁(y)] in (21) is treated as the pdf averaging over all possible bit patterns {B_j } with B ₀ = 0 (B ₀ = 1), while the W ₁(y) [W ₀(y)] is the conditional pdf for the given bit configuration. As in the previous subsection, the pdf W _{B
₀} (y) for each bit pattern is obtained by averaging over random parameters according to (19) (with τ_max →∞). Curves in Figs. 6(a) and 6(b) show BERs corresponding to all possible bit configurations as functions of PDL with Δτ/T = 0.35, Ĩ ₁ = 85, (a) B ₀ = 0 and (b) B ₀ = 1. As expected, the lowest (highest) BER curve corresponds to the case when all nearby bits take the same (different) value as B ₀. Although there are a lot of bit configurations, only three different BER curves are observed in each figure. This is because the Maxwellian weight factor in (19) strongly suppresses the contribution from the non-nearest neighboring bits, which means that the Maxwellian impact on BER discussed in previous subsection mostly comes from the the two nearest neighbors B _±1. To observe contributions from B _±2, we increase the DGD and the OSNR to Δτ/T = 0.7 and Ĩ ₁ = 100 in Fig. 6(c), where the dotted BER (B ₂ = B _-2 = 1) is about 1.3 times of the solid BER (B ₂ = B _-2 = 0). Note that this is the case of the special bit pattern with B ₀ = B _±1 = 0. Further calculation shows that the BER obtained by averaging over all possible bit configurations with B ₀ = 0 and 0 < α < 0.75dB is 0.03 ~ 0.04 [not indicated in (c)], in which the contribution from B _±2 to such average BER is ~ 10^-5, whereas the contribution from B _±1 is around 0.01. Therefore, influences of B _±2 on the bit average BER can be neglected.

4. Comments and Conclusions

We use the generalized Marcuse model to analytically evaluate BERs in systems consisting of both PMD and PDL. Based on this model, the PMD-PDL directional coupling effect, the DGD Maxwellian impact on BER, and the DGD-induced ISI effect have been studied. The directional coupling effect can be strongest when the PMD vector perpendicularly correlates with both the PDL vector and the input polarization in the 3D Stokes space. It depends strongly on the value of PDL, then the OSNR, and finally the DGD. The impact of large DGD variation is much more important than the impact of the PMD-PDL directional coupling. For an optical system with realistic parameters, only two nearest neighbors of the desired bit show significant ISI influence on the average BER.

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).

Strictly speaking, the model used here requires large enough optical filter bandwith M [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]

] and it cannot accurately represent some features of real systems. We are interested in it because it makes it possible to trace the combined PMD-PDL effect and to analytically evaluate its impact on the system performance. In fact, with optical filter bandwidth being three times the bit rate (M = 3), our numerical calculations agree well with other simulation results [2

]. Moreover, our approach used to deal with the joint PMD and PDL interaction can be generalized to systems with chromatic dispersion, arbitrary filtering, and other modulation formats.

In the transoceanic undersea system where the PMD is negligibly small while the link PDL is very large, the PDL induced noise enhancement caused by correlations between orthogonal noise polarizations should be considered [3

, 4

, 12

]. Our evaluation of the combined PMD-PDL effect neglects such noise correlation effect. Rigorous evaluation of noise correlation impact on characteristic functions or moment-generating functions (MGFs) for real systems with nonnegligible PMD and PDL is a subject of separate investigation.

Appendices

Appendix: Derivation of electrically filtered signal-signal and signal-noise beatings

According to (7) and (9), the electrically filtered signal-signal beating in (15) is given by

\bar{{∣ a ∣}^{2} + {∣ a_{⊥} ∣}^{2}} = \int_{0}^{T} \frac{dt}{T} ({∣ a ∣}^{2} + {∣ a_{⊥} ∣}^{2}) .

(25)

Before substituting (14) into (25) we denote j _τ ≡int[τ/(2T)] and {τ/2} ≡ τ/2 - j _τ T for the case of rectangular pulse input E ₀(t) to calculate fast-fast and slow-slow mode beatings as

\int_{0}^{T} {∣ E_{s} (t + \frac{τ}{2}) ∣}^{2} \frac{dt}{T} = {\int_{0}^{T}}^{- {\frac{τ}{2}}} B_{jτ} E_{0}^{2} (t + {\frac{τ}{2}}) \frac{dt}{T} + \int_{T_{- {\frac{τ}{2}}}}^{T} B_{j_{τ} + 1} E_{0}^{2} (t + {\frac{τ}{2}} - T) \frac{dt}{T}

= \frac{E_{1}^{2} [B_{jτ} (T - {\frac{τ}{2}}) + B_{jτ + 1} {\frac{τ}{2}}]}{T} \equiv E_{1}^{2} b_{+ τ}

\int_{0}^{T} {∣ E_{s} (t - \frac{τ}{2}) ∣}^{2} \frac{dt}{T} = \int_{0}^{{\frac{τ}{2}}} B_{- jτ - 1} E_{0}^{2} (t - {\frac{τ}{2}} + T) \frac{dt}{T} + \int_{{\frac{τ}{2}}}^{T} B_{- j_{τ}} E_{0}^{2} (t - {\frac{τ}{2}}) \frac{dt}{T}

= \frac{E_{1}^{2} [B_{- jτ} (T - {\frac{τ}{2}}) + B_{- jτ - 1} {\frac{τ}{2}}]}{T} \equiv E_{1}^{2} b_{- τ} .

(26)

As indicated near (1), E ₁ is the amplitude of the input pulse. The physical meaning of (26) can be seen by considering a simple case of 0 < τ < 2T, which means j _τ = 0. Thus two bits (B ₀ and B ₁ in b _+τ or B ₀ and B _-1 in b _-τ) are located in the time window [0,T] and they contribute to b _+τ and b _-τ according to (26). Similarly, the fast-slow mode beating gives

\int_{0}^{T} E_{s}^{*} (t - \frac{τ}{2}) E_{s} (t + \frac{τ}{2}) \frac{dt}{T} \equiv E_{1}^{2} b_{\pm τ}

with

b_{\pm τ} = {\begin{matrix} (B_{jτ} B_{- jτ - 1} + B_{- jτ} B_{jτ + 1}) \frac{T - {\frac{τ}{2}}}{T} + B_{jτ + 1} B_{- jτ - 1} \frac{2 {\frac{τ}{2}} - T}{T}, ({\frac{τ}{2}} > \frac{T}{2}); \\ (B_{jτ} B_{- jτ - 1} + B_{- jτ} B_{jτ + 1}) \frac{{\frac{τ}{2}}}{T} + B_{jτ} B_{- jτ} \frac{T - 2 {\frac{τ}{2}}}{T}, ({\frac{τ}{2}} < \frac{T}{2}) . \end{matrix}

(27)

With the help of (10), (14) and (25)–(27), one can express the time average signal-signal beating in (15) as

\overset{}{{∣ a ∣}^{2} + {∣ a_{⊥} ∣}^{2}} = E_{1}^{2} {[b_{+ τ} \tilde{A} ({\vec{τ}}_{0}, {\vec{α}}_{0}) + b_{- τ} \tilde{A} ({- \vec{τ}}_{0}, {\vec{α}}_{0}) + \frac{b_{\pm τ}}{2} \tilde{B} + [b_{+ τ} \tilde{A} ({\vec{τ}}_{0}, {- \vec{α}}_{0}) + b_{- τ} \tilde{A} ({- \vec{τ}}_{0}, {- \vec{α}}_{0}) - \frac{b_{\pm τ}}{2} \tilde{B}] e^{- 2 α}} \equiv E_{1}^{2} λ_{1,}

(28)

where Ã(τ⃗₀,α⃗₀)≡(1+τ⃗₀∙α⃗₀)(1+τ⃗₀∙e⃗ _s)/4, B̃ = C̃ cos ω_c τ+ D̃ sinω_c τ, C̃ ≡e⃗ _s ∙ [τ⃗₀×(α⃗₀×τ⃗₀)], D̃≡τ⃗₀∙ (e⃗ _s×α⃗₀). Formula (28) shows that the filtered signal-signal beating is composed of the filtered fast-fast mode beating (b _+τ), the slow-slow mode beating (b _-τ) and the fast-slow mode beating (b _±τ) according to (28).

The time average of the noise-signal beating in (15) can be calculated in a similar way, except noticing that, in general, frequencies of the noise and signal are different. Notations near (15) and

\bar{a_{kν}} \equiv \int_{0}^{T} \frac{a_{kν} dt}{T} (k = α, ⊥)

yield

\bar{a_{αν}} = E_{1} [〈 {\overset{}{\vec{α}}}_{0} ∣ {\overset{}{\vec{τ}}}_{0} 〉 〈 {\overset{}{\vec{τ}}}_{0} ∣ {\overset{}{\vec{e}}}_{s} 〉 b_{+ τν} e^{- i \frac{τ}{4} (ω_{c} + ω_{ν})} + 〈 {\overset{}{\vec{α}}}_{0} ∣ {\overset{}{\vec{τ}}}_{⊥} 〉 〈 {\overset{}{\vec{τ}}}_{⊥} ∣ {\overset{}{\vec{e}}}_{s} 〉 b_{- τν} e^{i \frac{τ}{4} (ω_{c} + ω_{ν})}]

\bar{a_{⊥ν}} = E_{1} e^{- α} [〈 {\overset{}{\vec{α}}}_{⊥} ∣ {\overset{}{\vec{τ}}}_{0} 〉 〈 {\overset{}{\vec{τ}}}_{0} ∣ {\overset{}{\vec{e}}}_{s} 〉 b_{+ τν} e^{- i \frac{τ}{4} (ω_{c} + ω_{ν})} + 〈 {\overset{}{\vec{α}}}_{⊥} ∣ {\overset{}{\vec{τ}}}_{⊥} 〉 〈 {\overset{}{\vec{τ}}}_{⊥} ∣ {\overset{}{\vec{e}}}_{s} 〉 b_{- τν} e^{i \frac{τ}{4} (ω_{c} + ω_{ν})}],

(29)

which lead to

\sum_{ν = ν_{1}}^{ν_{1} + M} {∣ \bar{a_{αν}} ∣}^{2} + {∣ \bar{a_{⊥ν}} ∣}^{2} = E_{1}^{2} {β_{+ τ} \tilde{A} ({\vec{τ}}_{0}, {\vec{α}}_{0}) + β_{- τ} \tilde{A} (- {\vec{τ}}_{0}, {\vec{α}}_{0}) + \frac{β_{\pm τ}}{2} \tilde{B} + [β_{+ τ} \tilde{A} ({\vec{τ}}_{0}, {- \vec{α}}_{0}) + β_{- τ} \tilde{A} (- {\vec{τ}}_{0}, {- \vec{α}}_{0}) - \frac{β_{\pm τ}}{2} \tilde{B}] e^{- 2 α}} \equiv E_{1}^{2} λ_{2},

(30)

where

β_{+ τ} = \sum_{ν = ν_{1}}^{ν 1 + M} b_{+ τ ν}^{2}, β_{- τ} = \sum_{ν = ν_{1}}^{ν 1 + M} b_{- τ ν}^{2}

b_{+ τ ν} = 2 (B_{jτ + 1} - B_{jτ}) \frac{\sin [(ω_{ν} - ω_{c}) \frac{τ}{4}]}{(ω_{ν} - ω_{c}) T}

and

b_{- τ ν} = 2 (B_{- jτ - 1} - B_{- jτ}) \frac{\sin [(ω_{ν} - ω_{c}) \frac{τ}{4}]}{(ω_{ν} - ω_{c})}

. Factors

e^{\pm i \frac{τ}{4} (ω_{c} + ω_{ν})}

in (29) are caused by DGD induced signal phase shifts. If noise frequencies ν are centerted at ν_c [ν_c ≡ ω_c T/(2π)], the β _±τ in (30) can be expressed as

β_{\pm τ} = \sum_{ν = ν_{c}}^{ν_{c} + m} b_{+ τν} b_{- τν} \cos \frac{τ (ω_{ν} - ω_{c})}{2} .

(31)

In principle, by performing coordinate transformation, we can let the Poincaré sphere’s polar axis be parallel to τ⃗₀ so that B⃗ in (28) and (30) can be simplified as B⃗ =sinθ _ατ sinθ _sτ cos(φ _ατ -φ _sτ -ω_cτ). Thus λ ₁ in (28) and λ ₂ in (30) can be expressed by (17).

Acknowledgment

This work was supported by Canadian funding agencies: NSERC and the Centers of excellence program: AAPN. We would like to thank Dr. John Cameron for reading the manuscript.

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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, and 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.	Jin 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.	Jin 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.	Ping 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 Xiaoyi 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]

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(260.2030) Physical optics : Dispersion
(260.5430) Physical optics : Polarization

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: November 14, 2006
Revised Manuscript: January 23, 2007
Manuscript Accepted: February 20, 2007
Published: March 5, 2007

Citation
Liang Chen, Zhongxi Zhang, and Xiaoyi Bao, "Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach," Opt. Express 15, 2106-2119 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2106

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References

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]
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]
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]
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]
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]
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. 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]
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]
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]
C. Xie and L. Moller, "The accuracy assessment of different polarization mode dispersion models," Opt. Fiber Technol. 12, 101-109 (2006). [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]
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]
H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003). [CrossRef]
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]
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]

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