## Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss

Optics Express, Vol. 16, Issue 18, pp. 13918-13932 (2008)

http://dx.doi.org/10.1364/OE.16.013918

Acrobat PDF (464 KB)

### Abstract

The ultimate limits introduced by polarization dependent loss (PDL) in coherent polarization multiplexed systems using advanced signal processing are studied. An analytical framework for effectively assessing the penalties is established and applied to systems with and without dynamically optimized launch polarization control. In systems with no launch polarization control, the PDL induced penalty is described by a simple formula and it is independent of the choice of constellation, or modulation format. The gain from optimizing launch polarizations is studied numerically and the mechanisms limiting it are described.

© 2008 Optical Society of America

## 1. Introduction

1. J. Renaudier, G. Charlet, M. Salsi, O.B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver,” J. Lightwave Technol. **26**, 36–42 (2008). [CrossRef]

4. H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express **16**, 873–879 (2008) [CrossRef] [PubMed]

5. A. Meccozzi 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]

6. M. Shtaif and A. Mecozzi, “Polarization-dependent loss and its effect on the signal-to-noise ratio in fiber-optic systems,” IEEE Photon. Technol. Lett. **16**, 671–673 (2004). [CrossRef]

7. I.T. Lima, A.O Lima, Yu Sun, Hua 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 (2004). [CrossRef]

3. C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O’Sullivan, “WDM Performance and PMD Tolerance of a Coherent 40-Gbit/s Dual-Polarization QPSK Transceiver,” J. Lightwave Technol. **26**, 168–175 (2008). [CrossRef]

4. H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express **16**, 873–879 (2008) [CrossRef] [PubMed]

## 2. Theory

*N*

*amplified spans and exploits an optimally designed receiver. It is assumed that the distortion of the Gaussian noise statistics as a result of the optical nonlinearity of the fiber is small and can be neglected. The system is assumed to contain*

_{s}*N*

*amplified spans and the matrix describing transmission from the output of the*

_{s}*j*-th amplifier to the receiver is denoted by

**T**

*. Thus*

_{j}**T**

_{0}is the matrix describing transmission over the entire link from transmitter to receiver. The signal launched into the fiber on the transmitter side is denoted by a Jones vector [

*s*

_{1}(

*t*),

*s*

_{2}(

*t*)]

*where the superscript*

^{t}*t*denotes “transposed” and where

*f*(

*t*) being a unit energy pulse representing the transmitted symbol. The terms

*a*

^{(k)}

_{1}and

*a*

^{(k)}

_{2}are complex numbers that represent the digital information carried by the

*k*-th symbol in each one of the two polarizations. It is assumed that the two polarizations are modulated independently while using identical modulation formats. For example, in the case of Mary PSK (M-PSK) transmission

*l*

_{1}and

*l*

_{2}are two independent integers between 0 and

*M*-1. The terms Δ

*φ*and Δ

*t*denote possible differences in optical phase and temporal alignment of the two signals, respectively. Typically, both those parameters are slowly varying functions of time. The optical signal reaching the receiver can be expressed as

*n*

^{(j)}

_{1}and

*n*

^{(j)}

_{2}being two independent circular Gaussian processes whose power density is given by

*P*

*.*

_{n}*t*/

*T*and assuming optimal signaling, a given symbol in each data stream overlaps with two symbols in the other data-stream, such that the number of dimensions that must be taken into account in the analysis of optimal detection needs to be doubled. This is in fact a form of inter-symbol interference generated by loss of orthogonality that is induced by PDL. Notice that with the effects of PMD compensated for, misalignment between the data streams is generated mainly at the transmitter and may be caused by phase misalignment between the clocks driving the two signals. A situation that can in principle be fixed in practical implementations. Thus, for the tractability of the analysis, we consider only the case of full temporal overlap between the data streams in the calculations that follow. The relevance of the calculations in providing an estimate for the performance degradation in the presence of PDL should not be noticeably affected by this limitation.

*h*(

*t*) the receiver filter’s impulse response, one may now replace the time dependent quantities in the column vectors in Eq. (2) by their filtered samples

*r*

_{1,2}=∫

_{T}*r*

_{1,2}(

*t*)

*h*(-

*t*)d

*t*and

*n*

^{(j)}

_{1,2}=∫

_{T}*n*

^{(j)}

_{1,2}(

*t*)

*h*(-

*t*)d

*t*are independent identically distributed circular Gaussian random variables with variance equal to

*P*

*/2*

_{n}*N*

*. The term*

_{s}*η*is the overlap integral

*η*=∫

_{T}*f*(

*t*)

*h*(-

*t*)d

*t*and it assumes its maximum value of 1 for the case of matched filtering, i.e. when

*h*(

*t*)=

*f*(-

*t*). The superscript

*k*denoting the symbol number above

*a*

_{1}and

*a*

_{2}in Eq. (1) has been omitted for clarity. In a shortened notation (3) can be conveniently rewritten as

*r*

_{1}and

*r*

_{2}are the decision variables, based on which, a decision regarding the identity of the received symbols is to be made. The statistics of the accumulated Gaussian noise in Eq. (3) is completely described in terms of its coherency matrix, which can be readily obtained from Eq. (2) and it is given by

**T**

_{j}**T**

^{†}

*are hermitian, the coherency matrix can be conveniently expressed in the form*

_{j}*g*⃗

*can be shown to satisfy |*

_{j}*g*⃗

*|≤*

_{j}*g*

^{(0)}

*with equality obtained only when the matrix*

_{j}**T**

*represents a perfect polarizer. The term σ⃗ is the Pauli-matrix vector σ⃗=[σ*

_{j}_{1},σ

_{2},σ

_{3}], with σi (i=1,2,3) being the pauli-matrices, commuted as in [9] for consistency with the conventional polarization terminology. The rightmost expression in Eq. (5) follows the definitions

**I**represents the 2 by 2 identity matrix. Next, we denote by

**U**the matrix that diagonalizes the coherency matrix Λ

*. Clearly, since Λ*

_{n}*is Hermitian,*

_{n}**U**is a unitary matrix satisfying

**U**

^{-1}=

**U**

^{†}. The diagonalized coherency matrix can be expressed as

_{1}is the first Pauli matrix in the notation of [9

9. J. P. Gordon and H. Kogelnik, “PMD fundamentals,” Proc. Natl. Acad. Sci. **97**, 4541–4550 (2000). [CrossRef] [PubMed]

*in the physical spin literature). It is now convenient to normalize the decision vector [*

_{z}*r*

_{1},

*r*

_{2}]

*as follows*

^{t}*n*̱̃=[

*n*̃

_{1},

*n*̃

_{2}]

*are statistically independent circular-complex Gaussian variables with zero mean and unit variance. The term*

^{t}*Q*(

*d*

*/2) where*

_{min}*u*̱

_{1}and

*u*̱

_{2}, the difference between which is Δ

*u*̱=[Δ

*a*

_{1},exp(

*i*Δ

*φ*)Δ

*a*

_{2}]

*. The corresponding square distance between those points in the re-normalized constellation space is*

^{t}**U**

^{†}(

**I**-Γ′σ

_{1})

**U**=

**I**-Γ⃗′·σ⃗ and also that

**T**

^{†}

_{0}

**T**

_{0}can be expressed as

**T**

^{†}

_{0}

**T**

_{0}=

*g*

_{0}(

**I**+Γ⃗

_{0}·σ⃗) with Γ⃗

_{0}being the PDL vector of the entire link and with

*g*

_{0}denoting the polarization averaged loss. With these relations, the square distance can be re-expressed as

*=Δ*

_{in}*u*̱

^{†}σ⃗Δ

*u*̱ and Δ⃗

*=Δ*

_{out}*u*̱

^{†}

**T**

^{†}

_{0}σ⃗

**T**

_{0}Δ

*u*are the Stokes vectors that correspond to the difference between the two constellation points at the input and output of the link, respectively. Namely, these are the Stokes representations of the difference between the Jones vectors corresponding to the two constellation points at the input, and output the link, respectively. One must therefore avoid misinterpreting these quantities as the difference between the Stokes vectors of the two constellation points. The term Δ

*denotes the modulus of Δ⃗*

_{in}*(which is equal to Δ*

_{in}*=|Δ*

_{in}*a*

_{1}|

^{2}+|Δ

*a*

_{2}|

^{2}), whereas

*represents its orientation. Using the relation Δ*

_{in}*=Δ*

_{out}*u*̱

^{†}

**T**

^{†}

_{0}

**T**

_{0}Δ

*u*̱=Δ

_{in}*g*

_{0}(1+Γ⃗

_{0}·

*) and denoting by*

_{in}*α*⃗

*and*

_{j}*α*

^{(0)}

*denote the local PDL vector [11*

_{j}11. B. Huttner, C. Geiser, and N. Gisin, Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. **6**, 317–329 (2000). [CrossRef]

*β*⃗

*is the birefringence vector. Since we do not consider dispersive effects, all quantities are taken to be independent of optical frequency. The previously defined transmission matrices*

_{j}**T**

*are then given by*

_{i}*M*

*are ordered with the smallest index appearing on the rightmost side and where*

_{j}*i*is between 0 and

*N*

*. The noise of each amplifier is modelled as if it is added to the signal at the amplifier output and so the noise of the*

_{s}*j*th amplifier sees the PDL of only the spans and amplifiers that follow (see Fig. 1). Although in practice, in multi-stage amplifiers the noise is generated adiabatically along the amplifying medium, this simplified view (which is critical from the analytical standpoint) is quite reliable in systems consisting of a large number of spans. It is assumed that the first amplifier is positioned at the end of the first span and that the last amplifier is immediately in front of the receiver so that the noise emitted by it does not experience PDL. Therefore, the last matrix

*M*

*for*

_{j}*j*=

*N*

*is the identity matrix. The vectors*

_{s}*α*⃗

*are picked from an isotropic Gaussian distribution with [8*

_{j}8. A. Mecozzi and M. Shtaif, “The statistics of polarization dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. **14**, 313–315 (2002). [CrossRef]

10. A. Galtarossa and L. Palmieri, “Spatially Resolved PMD Measurements,” J. Lightwave Technol. **22**, 1103–1105 (2004). [CrossRef]

## 2.1. The relation to performance degradation

*u*̱|

^{2}=|Δ

*a*

_{1}|

^{2}+|Δ

*a*

_{2}|

^{2}which is smallest when either Δ

*a*

_{1}, or Δ

*a*

_{2}is set to 0. Since the two polarization signals are assumed to be identically modulated, the smallest square distance between two constellation points that differ in both polarization components is larger by exactly a factor of 2 than the minimum square distance over all points. Therefore, for a pair of points that differ in both polarizations to become minimally distant in the re-normalized constellation space,

*η*

*must reach values lower than 0.5, implying a fairly large PDL. Unless the input states of polarization are optimized dynamically, with feedback from the receiver, a scenario that we consider separately in the next section, such high values of PDL cannot be encountered with any meaningful probability in normally operating systems. If it were not so, then penalties in excess of 3dB would be observed with approximately the same probability. For this reason, the minimally distant pairs of constellation points in the re-normalized space would always be minimally distant in the original space, corresponding to either*

_{r}*=(1,0,0) when Δ*

_{in}*a*

_{2}=0, or

*=(-1,0,0) when Δ*

_{in}*a*

_{1}=0. The smaller of the two values obtained from Eq. (10) with

*=(±1,0,0) and the corresponding*

_{in}*is the SNR penalty, defined as the ratio between the PDL impaired SNR and the SNR in the absence of PDL. In what follows we use the symbol ηSNR to denote this quantity. Note that we find it convenient to use linear units, in which case*

_{out}*η*

*receives values in the range of 0 to 1, with 1 corresponding to no penalty. It is important to note that in this regime the PDL induced SNR penalty is independent of the choice of constellation. In other words, the SNR will be degraded by the same amount regardless of which modulation format is applied to the individual polarizations, provided only that coherent detection with joint signal processing is performed.*

_{SNR}*g*

_{0}/

*g*′ and expanded the rest of the expression to second order in Γ⃗

_{0}and Γ⃗′. In this expansion we have used the relation

*⋍*

_{out}*+Γ⃗*

_{in}^{⊥}

_{0}, which is correct to first order in Γ⃗

_{0}, as can be shown based on the dynamic Eq. of the unit Stokes vectors in fibers with PDL [5

5. A. Meccozzi 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]

_{0}denotes the component of Γ⃗

_{0}that is orthogonal to

*. Additionally, for small PDL and with a the above specified equalization strategy,*

_{in}*η*

*can be described in terms a quadratic form of Gaussian variables. On the other hand, truncating the expression after the first term on the right hand-side leads to a Gaussian approximation. While the distribution of quadratic forms can be addressed rigorously, the resulting expressions are very cumbersome and unnecessary for the approximation level that we need here. Instead, we use the Gaussian approximation with a small correction to the mean value based on the two second order terms. As we show by comparison to numerical simulations, the resulting Gaussian distribution very accurately predicts the behavior of the studied systems in the relevant range of PDL values. Thus, after some algebra we find that the mean of*

_{r}*η*

*is*

_{r}^{‖}

_{0}and Γ⃗

^{‖}are the components of Γ⃗

_{0}and Γ⃗′ in the direction of Δ

*and with 〈|*

_{in}*α*⃗

*|*

_{j}^{2}〉 specified by Eq. (11) and it is equal approximately to the mean PDL in decibels divided by 64

*N*

*. The SNR reduction ratio is*

_{s}*η*

*=*

_{SNR}*m*

*-|Γ⃗‖*

_{r}_{0}-Γ⃗′′|,

*η*

*is shown, whereas Fig. 2(b) shows the cumulated distribution of the SNR penalty, namely the probability that*

_{SNR}*η*

*is smaller than a given value. The numerical results were obtained by simulating Eq. (10) directly, without any assumptions, or approximations and for all relevant pairs of points in an 8-PSK constellation. Identical results were obtained for the case of Q-PSK, demonstrating the independence on the exact constellation. For the clarity of the Fig., the displayed numerical data corresponds only to case of 8-PSK. Monte-Carlo simulations were performed with 10*

_{SNR}^{6}fiber realizations and for the case of

*N*

*=10. The results correspond to average PDL values of 1,2 and 3 dB and in all cases the excellent agreement with the analytical results of (16) is evident. This agreement provides final evidence to the fact that the choice of modulation format is immaterial. Also, in this situation our early assumption that there is full temporal overlap between the symbols transmitted in the two polarization does not introduce any inaccuracy to the obtained results. Only in the case of 〈*

_{s}*ρ*〉=3 dB can a small discrepancy between the approximate theory and simulation be observed in the tails of the distribution, and even then it is of insignificant magnitude. The system margin that needs to be allocated for PDL is easily obtained from the above theory. For a specified outage probability the margin can be defined as the SNR degradation whose probability of occurrence is equal to the outage probability. Thus, based on Eq. (16) we express the system margin in decibels as

## 2.2. Dynamically optimized launch polarization

_{0}is always well below unity for relevant PDL values, the first-order term is by far the most significant contributor to performance degradation. This is particularly true since the second-order term in (12) is independent of the launch polarization. This reasoning suggests that

*should be in the plane that is orthogonal to the vector Γ⃗*

_{in}_{0}-Γ⃗′, but it does not specify its direction in that plane. As we show in what follows, the exact choice of direction is not very significant. That is because in this regime, with the first order term eliminated, performance degradation is dictated not so much by the high-order terms in Eq. (12), but rather by the increased probability of events in which non-minimally distant constellation points in the original space, such that differ in both polarization components at the input, become minimally distant following the effect of PDL.

*a*

_{1}|=|Δ

*a*

_{2}| are equal to the minimum distance. For these pairs of points, the input unit Stokes vector

^{1}is of the form

^{′}

*=(0, sin(*

_{in}*θ*),cos(

*θ*) where

*θ*can assume different values, depending on the modulation format that is used. Note that the orientation of

*′*

*in this case is in a plane that is orthogonal to the direction of*

_{in}*that corresponds to constellation points differing in only one polarization. Thus, launching the input signals in polarization states that are approximately orthogonal to Γ⃗*

_{in}_{0}-Γ⃗′ for the purpose of minimizing the first-order penalty term, approximately maximizes the probability that pairs of points differing in both polarizations become minimally distant. To illustrate this point further, we consider two possible choices for

*. The first choice is to have*

_{in}*point in the direction specified by ±Γ⃗*

_{in}_{0}×Γ⃗′ so that it is orthogonal to both vectors. The second choice is to place

*in the same plane in which Γ⃗*

_{in}_{0}and Γ⃗′ reside, but such that it is orthogonal to the difference between them. In both cases the SNR penalty can be expressed as

*and over all allowed values of*

_{in}*θ*in

*. The factor of 2 in the second term in Eq. (18) is due to the fact that for these events the PDL-free square distance is twice larger than the minimum distance between points differing in only one polarization.*

_{in}^{1}Of course, technically we use a reference frame in which the launch states coincide with the

*x*and

*y*axes. Variation of launch polarization in the actual physical space therefore translates formally into the corresponding opposite rotation of the physical vectors Γ⃗

_{0}, Γ⃗′,

*α*⃗

*etc.*

_{j}*η*

*in these two cases, corresponding to mean PDL of 4 dB and to 8-PSK modulation. These curves were obtained from Monte-Carlo simulations of Eq. (10) similarly to the numerical calculations described in the context of Fig. 2 and performed for all the pairs of constellation points participating in the scheme. The dashed curves represent the simulation of Eq. (10) while ignoring pairs of constellation points differing in both polarizations. It illustrates the effect of the high-order terms in Eq. (12). The solid curves correspond to a simulation that takes all relevant pairs of constellation points into account. Notice how the solid curves are characterized by two distinct behaviors. While, for small penalties the high order terms dominate (as can be seen from the comparison between the dashed and solid curves), the more relevant range of high penalties is dominated by events in which the minimal distance corresponds to pairs that differ in both polarization. In this range the two solid curves overlap almost perfectly, suggesting that as long as*

_{SNR}*is orthogonal to Γ⃗*

_{in}_{0}-Γ⃗′, the further optimization of its orientation has little significance. In what follows, the choice of

*orthogonal to both Γ⃗*

_{in}_{0}and Γ⃗′ is assumed. Figure 3(b) shows the cumulated distributions of

*η*

*for PDL values of 2, 3 and 4 dB, indicating fairly large penalties.*

_{SNR}^{-5}. The case in which the launch polarization is optimized is represented by the top two curves. These curves correspond to 8-PSK and to

*Q*-

*PSK*modulation. The curve corresponding to the case without optimizing launch polarizations is also shown in the figure as we discussed earlier. Note that the improvement in PDL tolerance that is achieved by optimal polarization control is only of the order of ~2dB.

*θ*in Eq. (18) increases and thus the minimum distance can attain a smaller value. For example, in the case of M-PSK, the allowed values of

*θ*are

*θ*=2

*π*

*j*/

*M*with

*j*=0, ‖,

*M*-1, so that the vector

*may point in M equally spaced angles in the plane*

_{in}*S*̂

_{1}=0. Thus, the larger

*M*the more options there are for

*to coincide with Γ⃗*

_{in}_{0}-Γ⃗′. This notion has a very important and interesting implication and that is that for a given mean PDL, the outage probability may scale at most linearly with

*M*. In practice linear scaling is quite insignificant for outage probabilities in the relevant range. Therefore even in cases where optimal state launch is performed, the importance of the exact constellation is minimal in terms PDL tolerance. This is evident from Fig. 4(a), where the difference between the Q-PSK and 8-PSK curves is shown to be very small, and it is also illustrated in Fig. 4(b), where the cumulated probability curves corresponding to the cases of Q-PSQ and 8-PSK are plotted for 〈

*ρ*〉=4 dB. Consistently with the above argument, the probability of a given penalty in the Q-PSK case is found to be smaller by slightly less than a factor of 2 in the wings of the distribution.

## 3. Conclusions

## Acknowledgment

## References and links

1. | J. Renaudier, G. Charlet, M. Salsi, O.B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver,” J. Lightwave Technol. |

2. | L. E. Nelson, S. L. Woodward, M.D. Feuer, X. Zhou, P.D. Magill, S. Foo, D, Hanson, D. McGhan, H. Sun, M. Moyer, and M. O’Sullivan, “ |

3. | C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O’Sullivan, “WDM Performance and PMD Tolerance of a Coherent 40-Gbit/s Dual-Polarization QPSK Transceiver,” J. Lightwave Technol. |

4. | H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express |

5. | A. Meccozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” J. Lightwave Technol. |

6. | M. Shtaif and A. Mecozzi, “Polarization-dependent loss and its effect on the signal-to-noise ratio in fiber-optic systems,” IEEE Photon. Technol. Lett. |

7. | I.T. Lima, A.O Lima, Yu Sun, Hua 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. |

8. | A. Mecozzi and M. Shtaif, “The statistics of polarization dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. |

9. | J. P. Gordon and H. Kogelnik, “PMD fundamentals,” Proc. Natl. Acad. Sci. |

10. | A. Galtarossa and L. Palmieri, “Spatially Resolved PMD Measurements,” J. Lightwave Technol. |

11. | B. Huttner, C. Geiser, and N. Gisin, Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. |

12. | This is what happens when each channel passes through different optical routes before being multiplexed into the transmission fiber. |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.1660) Fiber optics and optical communications : Coherent communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: May 12, 2008

Revised Manuscript: June 25, 2008

Manuscript Accepted: July 22, 2008

Published: August 25, 2008

**Citation**

Mark Shtaif, "Performance degradation in coherent polarization multiplexed systems as a
result of polarization dependent loss," Opt. Express **16**, 13918-13932 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13918

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### References

- J. Renaudier, G. Charlet, M. Salsi, O.B. Pardo, H. Mardoyan, P. Tran, S. Bigo, "Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver," J. Lightwave Technol. 26, 36-42 (2008). [CrossRef]
- L. E. Nelson, S. L. Woodward, M.D. Feuer, X. Zhou, P.D. Magill, S. Foo, D , Hanson, D. McGhan, H. Sun, M. Moyer, M. O�??Sullivan, "Performance of a 46Gbps dual polarization QPSK transceiver in a high-PMD fiber transmission experiment," Optical Fiber Communications conference, Paper PDP9, OFC San Diego (2008).
- C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O�??Sullivan, "WDM Performance and PMD Tolerance of a Coherent 40-Gbit/s Dual-Polarization QPSK Transceiver," J. Lightwave Technol. 26, 168-175 (2008). [CrossRef]
- H. Sun, K.-T. Wu, and K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Express 16, 873-879 (2008) [CrossRef] [PubMed]
- A. Meccozzi and M. Shtaif, "Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization," J. Lightwave Technol. 221856-1871 (2004). [CrossRef]
- M. Shtaif and A. Mecozzi, "Polarization-dependent loss and its effect on the signal-to-noise ratio in fiber-optic systems," IEEE Photon. Technol. Lett. 16, 671-673 (2004). [CrossRef]
- I.T. Lima, A.O Lima,Yu Sun, Hua Jiao,J. Zweck, C.R. Menyuk, G.M. Carter,"A receiver model for optical fiber communication systems with arbitrarily polarized noise," J. Lightwave Technol. 23, 1478-1490 (2004). [CrossRef]
- A. Mecozzi and M. Shtaif, "The statistics of polarization dependent loss in optical communication systems," IEEE Photon. Technol. Lett. 14, 313-315 (2002). [CrossRef]
- J. P. Gordon and H. Kogelnik, "PMD fundamentals," Proc. Natl. Acad. Sci. 97, 4541-4550 (2000). [CrossRef] [PubMed]
- A. Galtarossa and L. Palmieri, "Spatially Resolved PMD Measurements," J. Lightwave Technol. 22, 1103-1105 (2004). [CrossRef]
- Q1. B. Huttner, C. Geiser, and N. Gisin, Polarization-induced distortions in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000). [CrossRef]
- This is what happens when each channel passes through different optical routes before being multiplexed into the transmission fiber.

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