## Abstract

Coherent crosstalk mechanisms induced by polarization mode dispersion in polarization multiplexed fiber transmission are examined by systems experiments, analysis and supporting modeling. Primary mechanisms include destructive interference, edge effects, and beat effects, leading to pulse distortion and to bit-error-rate penalties. Rules of thumb for tolerable crosstalk levels are developed.

©2000 Optical Society of America

## 1. Introduction

The promise of doubling fiber information capacity has led to the proposal and study of polarization multiplexing for NRZ terabit/sec transmission [1,2], solitons [3,4], and coherent lightwave systems [5,6]. Impairments observed in polarization multiplexed systems include coherent crosstalk due to misaligned polarizers or polarization beam splitters. Recently we have observed evidence of coherent crosstalk induced by polarization mode dispersion (PMD) [7]. Here we present a detailed study of the mechanisms underlying this crosstalk based on experimental evidence, analysis, and supporting modeling. Based on this new understanding, we aim to develop rules of thumb for estimating tolerable crosstalk levels.

An excellent example of such a rule of thumb is the 20/20 rule, in use since the 1996 Terabit Transmission Experiment [1] or possibly earlier. The 20/20 rule [8] has been a convenient measure for the experimenter and systems designer for estimating the system penalty due to coherent crosstalk from one multiplexed polarization to the other on the basis of intensity crosstalk measurements. The rule is simply stated: Consider two multiplexed channels *A* and *B* with approximately equal intensity levels *I*_{A}
and *I*_{B}
and examine the output at port *A*: measure the intensity crosstalk *I*_{BA}
from *B* to *A* while *I*_{A}
is switched off. If *I*_{BA}
is *20 dB* down then *I*_{A}
/*I*_{BA}
=100 and the upper rail of the eye pattern is reduced by *20%* for the worst-case coherent crosstalk. As a consequence there is a *1 dB* eye closure penalty due to crosstalk. This is the maximum tolerable limit.

The 20/20 rule applies to crosstalk caused by a misalignment of the polarization controllers preceding the polarization beam splitters at the receiver end or by imperfect extinction in the splitter. Note that this misalignment causes *frequency independent* coupling within the spectral band of interest. In contrast to this, the crosstalk produced by PMD is strongly dependent on frequency and, thus, its mechanisms need to be examined in greater detail. We will discuss these mechanisms on the basis of the results of systems experiments, accompanying analysis and supporting systems modeling.

## 2. Background and Assumptions of the 20/20 Rule

A simplified sketch of a polarization multiplexed system is shown in Fig. 1. The two multiplexed channels with complex amplitudes *A* and *B* enter at the polarization beam splitter (PBS) at the transmitter end, pass through polarization controllers and the fiber, and are separated by the PBS at the receiver end. The amplitude *A*_{out}
at the output of channel *A* is:

where we assume that the PBS misalignment causes a small coupling coefficient *κ*(|*κ*|≪1) so that one can neglect depletion of *A*. We allow for a phase difference Δφ between *A* and *B* as they enter the receiver PBS by setting

With this notation, the intensities in the two channels are:

The intensity coupled from the *B* channel to the *A* channel in the absence of *A* (*A*=*O*) is:

assuming real valued κ. This is the intensity crosstalk one monitors and enters into the 20/20 rule. Assuming complete spectral overlap of *A* and *B*, the intensity detected at the output of *A* is:

$$={I}_{A}+2\sqrt{{I}_{A}{I}_{BA}}\mathrm{sin}\Delta \phi ,$$

where we must neglect terms in *κ*
^{2} because we have also neglected depletion of *A*. For Δφ=±π/2, destructive or constructive interference occurs as indicated in Fig. 2. For φ=0 or π, the out-of-phase cases occur where there are no *AB* terms in eq. (5) implying zero coherent crosstalk. In the case of destructive interference for κ=1/10 we have *I*_{BA}
=BB*/100=*I*_{A}
/100 and a reduction of the upper rail of the detected signal by 20 percent, as mentioned. This coupling corresponds to a PBS misalignment of 11.5 degrees in Stokes space (e.g. a 5.75° rotation in the lab, i.e. in Jones space).

## 3. The Interchannel Phase Shift

The phase difference Δφ between *A* and *B* as expressed in eq. (2), deserves special note as an important parameter controlling coherent crosstalk. It can have a variety of causes. When one laser is used as a common source for *A* and *B* as in [1], its light is split into two optical paths for independent modulation before entering the transmitter PBS. A difference of those two paths can be one reason for the phase difference, while a difference in modulator chirp is another. Polarization controllers both at the transmitter and receiver end inserted to match the channel polarizations to the PBS can also cause phase differences between the orthogonal channels. Another cause of Δφ are optical nonlinearities in the fiber [9,10]. Finally, we need to consider the dynamic change in Δφ when two different lasers are used as sources for *A* and *B*. When there is a frequency difference *ω*_{B}
between them, Δφ will oscillate in time due to beating. Assuming perfect coherence we have to modify eq. (2) to:

to reflect the difference in carrier frequency, ω
_{B}
. With this modification, the detected intensity at the output of *A* becomes:

The difference in carrier frequencies causes a periodic fluctuation of the phase difference corresponding to the beat frequency between the channels, as expected. The coherent cross talk will cycle through destructive and constructive interference, and all the intermediate interference states as a function of time and, therefore, the amount of coherent crosstalk can vary from bit to bit.

## 4. Cross-talk Due to PMD

For the above derivation one assumes complete spectral overlap and frequency independent coupling κ. Therefore, there was no need for a careful distinction between the time-domain complex amplitudes *A*(*t*) and *B*(*t*) and their Fourier transforms, the spectral amplitudes *Ã*(*ω*) and *B̃*(*ω*). From now on we use the tilde to make this distinction.

Recall that there are several polarization-based multiplexing schemes: one where adjacent wavelength channels have orthogonal polarizations (i.e. polarization interleaving), one where adjacent bits in a single channel have orthogonal polarizations, and one using two polarizations at each wavelength channel (PDM). Clearly, for polarization interleaving we have a violation of the spectral overlap assumption. We have to distinguish between the *Ã*(*ω*) and *B̃*(*ω*) spectra having different carrier frequencies (or wavelengths), and eq. (2) no longer holds.

The second violation of the 20/20 rule assumptions is the fact that PMD causes frequency dependent coupling. Consider the simple case of a launch at 45° to the principal state of a (full-wave) birefringent plate. The coupling coefficient is [7]:

where Δ*τ* is the differential group delay (DGD) and *ω* is the angular frequency deviation from the carrier (Note that for PDM this launch assumes careful adjustment of PC2, the polarization controller immediately before the PBS, so that the *B* carrier is suppressed and does not couple to the *A* channel). As we continue to assume small coupling, we replace the sine-function by its argument and rewrite eq. (1) for the spectral amplitudes in the form

which restricts the analysis to first order PMD effects. At this stage it is tempting to insert the coupled power spectral density (Δτω/2)^{2}
*B̃**B̃** into the 20/20 rule, however this is a pitfall since the 20/20 rule is a relation between time-domain quantities! Recall the relation between the time derivative *Ȧ* and its Fourier transform (FT),

With this relation, we obtain as the Fourier transform of eq. (9) the expression for the complex amplitudes

The nature of the interference *Ḃ* term in eq. (11) raises at least five additional points for the simpler case of PDM:

1) Edge effects and overshoot — The time derivative of the interfering complex amplitude *B* implies that peak crosstalk occurs at the edges of the *B* pulses and not during the whole pulse period as before.

2) Sign-Reversal — The interference term *Ḃ* has the opposite sign at the leading and trailing edges of the pulse turning constructive interference at one edge into destructive interference at the other, etc.

3) Optical (and electronic) filters — can reduce overshoot effects significantly. Their parameters should, therefore, influence estimates of cross-talk penalties. The strong effect of filters is, of course, expected for the case of polarization interleaving. All amplitudes entered into our equations are those *after* the optical filter (such as *FT* (${B}_{\mathit{\text{in}}}^{\sim}$
·*F* for the filtered *B* channel, where *F*(*ω*) is the filter response and ${B}_{\mathit{\text{in}}}^{\sim}$
the input spectrum).

4) 90°-Rotation — Note that the imaginary unit *j* has disappeared from eq. (11). For the special case of real κ and the simultaneous presence of coupling due to PMD and PC-misalignment this means that the complex amplitudes of the corresponding interference terms are rotated 90° relative to each other.

5) Relative Timing Delay — So far it was tacitly assumed that the bit intervals of the *A* and *B* signals are synchronized. This should, indeed, lead to worst case interference for coupling due to a PBS misalignment. PMD coupling, however, should lead to worst-case crosstalk when the edge of a *B* pulse occurs near the center of the *A* pulse, e.g. for *B*(*t*)=*A*(*t*-*T*/2) in the case of an isolated “one” where *T* is the bit interval.

For the case of polarization interleaving or PDM with separate laser sources for *A* and *B*, a sixth important point arises: there is a

6) Second Coupling Term due to the Shifted Carrier — For simplicity assume that the *B* amplitude can be described as in eq. (6). Then, the *Ḃ* interference term in eq. (11) will split into two parts

one causing interference over the whole bit interval and the other at the pulse edges. Their two complex amplitudes are rotated 90° relative to each other as in Point 4.

## 5. Edge Effects in PDM Systems

As a simple illustration of edge effects in a non-interleaved PDM system consider *A* and *B* channels with the same peak amplitude *A*
_{0} and real amplitudes *A*(*t*) and *B*(*t*) . For the simple case of a sinusoidal edge we have an amplitude

in the leading edge of the pulse extending from *t*=0 to *t*=π·Δ*T*/2. The corresponding intensity is *I*_{B}
(*t*)=*B*
^{2} with a maximum slope of

where Δ*T* is the effective risetime. The maximum time derivative of the amplitude is

From eq. (11) we obtain the detected intensity

for the worst case scenarios of constructive and destructive interference. Maximum coherent crosstalk occurs when an edge of the *B* signal coincides with a peak of the *A* signal. Then eq. (16) becomes

This maximum crosstalk occurs at the time of the maximum slope *Ḃ*
_{max}
during the short duration of the *Ḃ* spike. Note that the sign of *Ḃ* reverses at the trailing edge of the pulse. As a consequence we get destructive interference at the trailing edge for a phase difference Δφ that leads to constructive interference at the leading edge.

Eq. (17) suggests a *simple rule of thumb*: a 20 percent reduction of the upper rail of the eye diagram occurs when the DGD Δτ is 20 percent of the (filtered!) risetime Δ*T*. As an illustration, the modeled edge effect for two different 40 Gb/s bit sequences *A* and *B* is shown in Fig. 3 for DGD values of 5.1 and 1.1 ps (see below for a more detailed description of the model).

## 6. Crosstalk Due to PMD Plus PBS Misalignment

In the laboratory one expects that the two idealized effects discussed above will occur in combination: there will be coherent crosstalk due to the combined effects of PMD and PBS misalignment. From the Jones Matrix for this situation we find that the simplified expression for the complex amplitude at the output takes the form

again assuming a 45° worst-case launch into the PM-fiber. Note that the coupling constant κ_{0} is, in general, a complex number whose phase depends on the nature of the misalignment. This phase enters in the complex vector addition of the PBS and PMD terms.

For in-phase addition the PBS misalignment can add significant coherent crosstalk: when the power coupling due to κ_{0} is 30 dB down, the misalignment contributes ≈6 percent to the lowering of the upper rail in the eye diagram, increasing a 20 percent lowering due to PMD to an overall 26 percent. Note that 30-dB coupling is already at the low end of the currently available 25 to 30 dB range for the extinction ratio of a PBS.

## 7. Beat Effects for Polarization Interleaving

In the interleaving case there are two significant interfering *B* channels of the form given in eq. (6) with carriers spaced ±*ω*_{B}
from the *A* carrier. This leads to interference terms of the type shown in eq. (12). The corresponding detected signal intensity *I*_{dA}
is somewhat more complex than in the PDM case. Edge effects proportional to Δ*τ*/Δ*T* are still present, but reduced in magnitude as the filter, centered on the *A* carrier, causes a considerable reduction of the coupled *B* amplitude. In addition, there is a coherent crosstalk term proportional to Δτ*ω*_{B}
which extends through the whole duration of the *B* pulse and is about three times larger for our parameters (*ω*_{B}
=2π·50 GHz, Δ*T*=12 ps). Finally, there is a modulation of the combined edge and Δτ*ω*_{B}
terms by the beating of the *A* and *B* carriers with a period of *τ*_{B}
=2π/*ω*_{B}
(20 ps in our case). This beat effect is illustrated in Fig. 4 showing the modeled output intensity for 40 Gb/s sequences with two *B* channels at ±50 GHz from *A*, zero relative timing delays and zero relative phases, and DGD values of 1.1 and 5.1 ps. We note that a narrow RF filter following detection could have a strong effect on this beating crosstalk. This was not included in our modeling at this stage.

## 8. Experimental and Modeling Results for Worst-Case Interference

The above analysis made several simplifying assumptions in order to highlight several key mechanisms affecting coherent crosstalk in polarization multiplexed systems. In order to provide a better and more practical perspective, we summarize here the principal results of experiments reported earlier [7] as well as supporting modeling results. In these studies, the polarization multiplexed *A* and *B* signals were transmitted through various lengths of polarization maintaining (PM) fiber with a DGD of 1.8 ps/m. To gain understanding of the worst-case scenario, the signals were launched at 45° to the axes of the PM fiber. The polarization controller after the PM fiber (PC2) was carefully adjusted to align the output signal with the PBS for maximum rejection of unwanted channels. The signals were modulated at 40 Gb/s data rates with considerable chirp in the modulators. Even for PDM, different lasers were used for the *A* and *B* channels. While their frequencies were set to be nominally identical, one expects differences up to about 2 GHz. These differences as well as the chirp should assure occurrence of worst-case Δφ conditions. Measurements made included coupled power spectral density, eye diagrams, and 10^{-9} BER power penalties due to coherent crosstalk for both the PDM and polarization interleaving cases. In order to compare transmission with equally high spectral efficiency (i.e. 0.8 bits/s/Hz), the PDM experiments were done with 100 GHz channel spacing while polarization interleaving was done with 50 GHz channel spacing.

The model calculations assumed parameters matching the experiment, i.e., 40 Gb/s pseudo random bit sequences (PRBS) with 6 ps (20 to 80 percent) risetime, an optical filter of 50 GHz width (FWHM) with a 21 GHz (20 to 80 percent) flank and no significant RF filtering. After the 50 GHz filter, the risetime of the modeled PRBS agreed closely with the 12 ps risetime of the experimental bit sequence. The model pseudo-independent PRBS’s for the *A* and *B* channels contained 2^{7}-1 bits each. For PDM, pseudo-independence of *B* was generated by time reversal of the *A* sequence. For interleaving, the lower-frequency *B* sequence was also generated by time reversal of *A*. To generate the higher-frequency *B* sequence, the lower-frequency *B* sequence was time-shifted 64 bit periods. The model includes depletion of *A* and is exact for a 45° launch into the PM fiber. The Jones matrix for the PM fiber leads to an output spectral amplitude of

with the simplifying assumption that the PM-fiber acts as a full-wave plate at the carrier frequency ω=0.

Fig. 5 shows the spectral power density coupled from the *B* to the *A* channel for the (non-interleaved) PDM case for six DGD values as measured after the optical filter. The spectral resolution of these measurements was 0.1 nm. We measured the level of the peak coupled spectral densities against the peak of the spectral spike of the *A* carrier. The results are listed in Table 1.

Fig. 6 shows modeling results for a supergaussian pulse with 25 ps width (FWHM) and 12 ps risetime for the case of polarization interleaving with two interfering *B* channels spaced ±50 GHz from the *A* channel. The calculated peaks of the coupled power densities are shown in Table 2 together with corresponding measured values. Note that the results for polarization interleaving are particularly sensitive to filter characteristics. The model includes a super-gaussian filter matching the experimental filter FWHM and flank. Fig. 7 shows an example of a modeled eye diagram obtained for the interleaving case with a DGD of 5.1 ps, a relative timing delay of 0 ps, and a differential phase of 0 between the *A* and *B* channels.

Experimental and modeling results for five DGD values from 1.1 ps to 5.1 ps for the two cases of PDM and polarization interleaving are summarized in Tables 1 and 2. Interleaving results are for two interfering independently modulated *B* channels with carriers shifted ±50 GHz relative to the *A* carrier. The tables show the peak coupled power spectral densities coupled from *B* to *A* for *A*=0. Modeled eye diagrams for both the PDM and polarization interleaving cases were examined for a range of relative timing delays and differential phases between the *A* and *B* channels. The worst case eye openings are listed in Tables 1 and 2.

The eye diagram in Fig. 7 shows that there is not only a lowering of the upper rail due to coherent PMD crosstalk, but also a considerable raising of the lower rail. We attribute this to spilling of the edges of *A* pulses into zeros due to strong filtering and coincident constructive interference from *Ḃ*. The corresponding experimental eye closures are difficult to measure precisely but were generally consistent with the model data. The experimental 10^{-9} BER power penalties are also shown in Tables 1 and 2.

We should stress that the modeled eye diagrams apply to pin-receivers while the experiments were done with optical preamplifiers resulting in additional eye closing. In the experiment, there is another addition to eye closing as a consequence of imperfect PBS alignment with an intensity crosstalk expected to be in the 25 to 30 dB range. We emphasize that the polarization interleaving results were very sensitive to filtering and showed significant improvements for larger carrier spacings.

With these additions, there appears to be good agreement between the measured and modeled systems impairments. In the PDM case of Table 1 a DGD value of 1.1 ps results in -33 dB peak coupled spectral intensity measured in the optical spectrum. This value corresponds to the measured 1 dB power penalty associated with a total eye closing of 20%. The -33 dB level is, of course, in total disagreement with the -20 dB level prediction of a 1 dB penalty obtained with an incorrectly used 20/20 rule. On the other hand, it appears that the rule of thumb based on eq. (17) and the filtered pulse risetime gives a good order of magnitude estimate, particularly when allowance is made for some raising of the lower rail and a reasonable PBS misalignment.

## 9. Discussion

Demultiplexing of polarization multiplexed channels in practical WDM systems will face several complexities in addition to the principal crosstalk mechanisms discussed above. In interleaving, for example, the output PBS is usually preceded by an arrayed waveguide router, splitting each waveguide channel to a different output port with its separate PBS (or polarizer) [2]. In PDM systems, each wavelength carries both multiplexed (*A* and *B*) channels. Here one can use a shared PBS for both channels, or choose to split the signal path and use a separate PBS for each of the output ports. A separate PBS per channel (say for *A*) allows adjustment for minimum crosstalk (from *B*), and is preferred to adjustment for maximum throughput of *A*. In our experiments (reported here and in [7]), we used a separate PBS for channel *A*. The polarization controller before the PBS (PC2) was adjusted by minimizing the crosstalk from *B* when the primary *A* channel was switched off.

In the presence of pure first-order PMD and when the interfering (*B*) spectra are symmetric relative to the *A* carrier (as in our simulations), the adjustments for minimum crosstalk and for maximum throughput are the same. An example of this is PDM when the frequencies of the *A* and *B* carriers are the same and orthogonality between the A and B *carriers* is maintained even in the presence of PMD. Spectral asymmetries arise when the two carrier frequencies differ. Other sources of asymmetries are unequal carrier spacings for interleaving, PBS misalignments, filters that are not centered on the primary carrier, and higher order PMD. In these practical systems, minimizing crosstalk of the unwanted (*B*) channels will result in a small reduction in the amplitude of the selected (*A*) channel.

## 10. Conclusions

We find by experiment, analysis, and modeling that there are several coherent crosstalk mechanisms leading to impairments in polarization multiplexed systems. In addition to the known effects due to PBS misalignment there are two mechanisms caused by PMD in the fiber. For PDM a key mechanism is due to an edge effect, while a beat effect plays a strong role in polarization interleaving. The worst-case 1-dB penalty limit of a single channel due to PMD is commonly estimated to be a mean DGD equaling 10 percent of the bit interval T. For 40 Gb/s this corresponds to a mean DGD of 2.5 ps or a peak DGD of about 8 ps. We find that polarization multiplexed systems are about five times more sensitive to PMD than a single channel. The 1 dB penalty limit for a 40 Gb/s NRZ system was found to be in the 1 to 2 ps range, with obvious consequences for fiber selection or PMD compensation in polarization multiplexed systems.

## Acknowledgements

The authors would like to thank Luc Boivin, Andy Chraplyvy, and Bill Shieh for valuable comments and discussion.

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