Polarization mode dispersion (PMD) [1] describes the group velocity dependence on the state of polarization, and it has been widely recognized as one of the major factors limiting high bit-rate communication systems. Polarization dependent loss (PDL) is a varying insertion loss arising from the dependence of a component’s transmission coefficient on the state of the polarization. Most passive optical components have some level of anisotropy with a transmission coefficient that is sensitive to the state of polarization. In a complex system with optical fibers, couplers, isolators, filters, multiplexers/demultiplexers, combiner/splitters, variable optical attenuators (VOAs), Erbium doped optical amplifiers (EDFA) and optical add/drop multiplexing switches (OADMS), the key question is: what is the combined effect of PMD and PDL? Gisin and Huttner were the first to consider such a question in their seminal work [2]. They showed that the combination of PMD and PDL will lead to a complex principal state vector. The autocorrelation function for such a complex principal state vector has been reported by several groups [3, 4, 5]. However, those calculations either assume a zero correlation between local PMD and PDL polarization directional [3] or strong correlation [4, 5]. The object of this work is to generalize the principal state vector autocorrelation considering arbitrary local PMD and PDL directional correlations. Experimental measurements are compared with our analytic results.

We start our investigation by using the standard waveplate model. Because we are going to take the continuum limit at the end we consider the following simple arrangement: each PMD element β⃗_j is followed by a PDL element α⃗_j leading to the following transmission Jones matrix, T_j (ω)=exp[-iωβ⃗_j ·σ⃗/2]exp[α⃗_j ·σ⃗/2]. The output electric field E⃗_out (ω) is connected with the input electric field E⃗_in (ω) by:

where A_N =exp[-α ₀-${\sum }_{j=1}^N$ α_j ], and α ₀ represents the polarization independent attenuation; φ_CD (ω) stands for the total chromatic dispersion (CD) of the system; β⃗_j =β_j$\widehat{\beta }$_j represents j-th PMD waveplate having differential group delay (DGD) β_j and the fast axis polarization is expressed by the unit vector $\widehat{\beta }$_j in the Stokes space; α⃗_j =α_j$\widehat{\alpha }$_j stands for the j-th PDL waveplate with value expressed in decibel by 20|α_j |log₁₀ e and the maximum transmission polarization is denoted by the unit vector $\widehat{\alpha }$_j in the Stokes space; and σ⃗ are the standard Pauli matrices. The complex principal state of polarization (PSP) vector W⃗_N (ω) is accordingly defined by the following [2]:

which leads to the following recursion relationship:

Writing the complex PSP vector explicitly in its real and imaginary part W⃗_N (ω)=Ω⃗_N(ω)+iΛ⃗_N(ω), we would like to find its autocorrelation functions <Ω⃗_N(ω)·Ω⃗_N(ω′)>, <Ω⃗_N(ω)·Λ⃗_N(ω′)> and <Λ⃗_N(ω)·Λ⃗_N(ω′)>. Where 〈…〉 means the average over statistical fluctuations of the PMD and PDL element. However, before we go any further we would like to mention the main object of this work, namely the local correlation <$\widehat{\alpha }$_j·$\widehat{\beta }$_j> between the PMD unit vector $\widehat{\beta }$_j and the PDL unit vector $\widehat{\alpha }$_j, could be neither zero [3] nor ±1 [4, 5]. In what follows we will assume a general local correlation between the PMD and PDL polarization directions. Now, it is convenient to recall some mathematical properties for averaging over a random unit vector n̂ (e.g., vectors $\widehat{\alpha }$_j and $\widehat{\beta }$_j) for given constant vectors A⃗ and B⃗:

Furthermore, considering the local correlation between $\widehat{\alpha }$_j and $\widehat{\beta }$_j, we also have:

where we have used the generic expression <$\widehat{\alpha }$_j·$\widehat{\beta }$_j> to represent the local PMD and PDL directional correlation.

To go to the continuum limit, we adopt the strategy first introduced by Karlsson and Brentel [6], namely we are going to assume in the waveplate model that each section has the same DGD, and PDL, i.e., β_j =β and α_j =α, except treating their directions $\widehat{\alpha }$_j and $\widehat{\beta }$_j as random unit vectors. The continuum limit is taken in such a way that N→∞, α→0, β→0 while keeping Nα ²=〈η ²〉, Nβ ²=〈Δτ ²〉 constants. Using the above strategy, we can easily get the following:

with q=[1+2cosβ(ω-ω′)]/3, and remembering 〈W⃗ ₀(ω)·W⃗ ₀(ω′)〉=0. Taking the continuum limit as described above, we have:

where Δω=ω-ω′. Similarly we can find another complex correlation function:

with

Likewise, taking the continuum limit, we would find:

with the abbreviation g=[4<η ²>-<Δτ ²>(Δω)²]/3 and $h=\frac{4<\hat{\alpha }·\hat{\beta }>\sqrt{〈\Delta {\tau }^2〉〈{\eta }^2〉}\left(\Delta \omega \right)}{3}$ . Substituting the complex PSP vector by their real and imaginary parts in the above two complex autocorrelations Eq. (7) and Eq. (10), we could get the following:

These are the main results of this work. They agree with the published result [3] when one takes <$\widehat{\alpha }$·$\widehat{\beta }$>=0, and it also agrees with the result [5] when one takes <$\widehat{\alpha }$·$\widehat{\beta }$>=1. It is interesting to note the fact that PMD and PDL polarization directional correlation is nonzero and can be linked with the finite values of the cross autocorrelation between the real and imaginary parts of the PSP vector.

We now discuss a few interesting aspect of our results. First, because of the PMD and PDL interaction the parameter 〈Δτ ²〉 is not equal to the measured average square DGD, rather it corresponds to the pristine average square DGD (PASDGD). Neither is 〈η ²〉 proportional to the measured average square PDL in dB unit due to the mutual interaction of PMD and PDL. Again, we call 〈η ²〉 the pristine average squared PDL (PASPDL). However, PASDGD 〈Δτ ²〉 and PASPDL 〈η ²〉 can be calculated by the measurement of the autocorrelation at Δω=0:

Lastly the parameter, 〈$\widehat{\alpha }$·$\widehat{\beta }$〉, describing the local PMD and PDL directional correlation, can be linked to the derivative of the cross autocorrelation at Δω=0:

It is worthwhile to note that complex PSP vector can be directly measured via the equation of motion method [7], thus all the parameters introduced in this paper can be determined. Lastly the parameters PASDGD 〈Δτ ²〉, PASPDL 〈η ²〉 and <$\widehat{\alpha }$·$\widehat{\beta }$> are only system parameters needed to exactly evaluate the eye diagram for given pulse sequence in a highly mode coupled fiber optic link. We will publish this result in a future work.

In order to check if the local PMD and PDL polarization directional correlation <$\widehat{\alpha }$·$\widehat{\beta }$> is relevant we have performed an experiment on a 52km field fiber with both buried and aerial segments. Two measurements (10 days apart) were performed by launching three states of polarization and scanning (3GHz steps) across optical frequencies from 186.2 to 196.6THz (1526 to 1611nm). The fiber was observed to have an average DGD of 1 ps. Figure 1 shows the experimental autocorrelation functions and their corresponding analytical fits for two different instants: one was taken during the daytime ((a), (b), and (c)), and the other during the nighttime ((d), (e), and (f)). Figure 1(a) plots the autocorrelation between the real parts of the PSP vector, Fig. 1(b) plots the autocorrelation between the imaginary parts of the PSP vector, and Fig. 1(c) plots the autocorrelation between the real and the imaginary parts of the PSP vector. While Figs. 1(d), (e), and (f) show the same autocorrelations for another instant. Note the choice of the fitting parameter <$\widehat{\alpha }$·$\widehat{\beta }$>=1 and <$\widehat{\alpha }$·$\widehat{\beta }$>=0.15 shown in the inset of Fig. 1(c); clearly <$\widehat{\alpha }$·$\widehat{\beta }$>=1 does not fit. Thus, one has to consider finite PMD and PDL directional correlation in field fiber. Overall the agreement between the experiment and the analytic result is satisfactory. However, notice in Fig. 1(c) that the experimental value is not zero at Δω=0, this might be related to several effects like the experimental accuracy as well as the fact that the whole link might be not treated as a continuum highly coupled limit. The fitting parameters chosen for 〈Δτ ²〉=1.31ps² and 〈η ²〉=0.073 (equivalent PDL 2.2dB) are obtained from solving Eq. (12), while the value <$\widehat{\alpha }$·$\widehat{\beta }$>=0.15 was chosen based on eying the overall fit. The positive value of the fitting parameter <$\widehat{\alpha }$·$\widehat{\beta }$>=0.15 demonstrate that the fast PMD vector lies more closely in the direction of maximum transmission polarization. Now it is interesting to note the same comparison of the fiber at different instants (here is the nighttime data) as plotted in the Figs. 1(d), 1(e), and 1(f). However, it is found that <$\widehat{\alpha }$·$\widehat{\beta }$>=-0.32 seems to fit the experiment the best. This means that the fast PMD vector lies more closely in the direction of minimum transmission polarization. The other fitting parameter 〈Δτ ²〉=1.26ps² and 〈η ²〉=0.011 (equivalent PDL 0.84 dB) were obtained similarly by solving Eq. (12). This time one notices more obvious oscillatory behaviors in the Fig. 1(e) for the imaginary part of the PSP autocorrelation despite their overall small magnitude. Again, this could be linked to the finite number of optical components in the fiber span, signaling the deviation from our analytic result which is only valid in the continuum limit. It is also interesting to notice the significant variations in the fitting parameter PASPDL 〈η ²〉 between the two instants of the experimental measurement, while the parameter PASDGD 〈Δτ ²〉 seems to be relatively constant between those two instants. This indicates the importance of taking PDL fluctuations into consideration in real field fiber links.

 

Fig. 1. The autocorrelation functions for a buried field fiber of 52km (thick lines) are compared with the analytical fitting curves (thin lines). (a), (b) and (c) are correspond to one instant, and (d), (e) and (f) to another instant. The overall agreement demonstrates the necessity of finite PMD and PDL directional correlation. Note: ω ₁=Δω+ω ₂ here.

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Because of the relatively small PDL values observed in the field fiber link we have constructed a PMD and PDL emulator made of 14 sections of polarization maintaining fiber spliced together (PDL is contributed by the splicing). Figure 2 plots the three experimental autocorrelation functions and their analytical fits. Clearly the overall agreement between theory and experiment are much better than those shown in Fig. 1. Note the fitting parameter PASPDL 〈η ²〉=2 (equivalent PDL 11.3 dB) is relatively large as well the fitting parameter PASDGD 〈Δτ ²〉=46.4 ps² is bigger than those in the Fig. 1. From this result one can conclude that the emulator constructed seems to be closer to the continuum limit that is required by the analytical result presented here.

 

Fig. 2. The autocorrelation functions (lines with dots) of an emulator are compared with analytical fitting curves(solid lines). The overall agreement is very good.

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Fig. 3. The analytical autocorrelation functions are plotted for large PASPDL with finite PMD and PDL polarization directional correlation. Notice the oscillatory behavior.

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Figure 3 illustrates the last interesting consequence of the finite PMD and PDL directional correlation (i.e., <$\widehat{\alpha }$·$\widehat{\beta }$>≠0). Analytically it is clear that all three autocorrelation functions show oscillatory behavior as a function of Δω=ω ₁-ω ₂ for large PASPDL 〈η ²〉. This is illustrated by the case 〈Δτ ²〉=64.5 ps², 〈η ²〉=10 and <$\widehat{\alpha }$·$\widehat{\beta }$>=1.

In conclusion, we have derived analytical autocorrelation functions between complex PSP vectors considering the finite local PMD and PDL directional correlation. Analytical results are compared with a field fiber span and an emulator. Both experimental results illustrate the importance of considering the effect of PMD and PDL directional correlation.

We wish to acknowledge the financial support from Canadian funding agency: NSERC and the Canadian networks of excellence: CIPI, AAPN.