Bridging the worlds of OFDM (on which I spent the last year) and XPolM (on which I spent the three years before that), I thought it might be interesting to see what an OFDM channel does to an XPolM probe. This is an extension of all the stuff that was written in this post, in particular we shall look at the dispersion-related autocovariance function (ACovF), as the integral of this ACovF determines the magnitude of XPolM effects.

For reference, we start with Fig. 1 of the original post:

Initially, the ACovF drops quickly as $z_1$ separates from $z_2$ due to the walk-off of the interfering channel in the reference frame of the probe (within a walk-off length $L_{WO}$ of roughly 16 km for a 10G signal 50 GHz away from the probe in standard SMF). As $z_1$ and $z_2$ become larger, chromatic dispersion leads to pulse broadening, and thus the ACovF “smears out” a bit more – the distortion remains correlated over larger distances $|z_1 - z_2|$ but the peak value of the ACovF also becomes smaller.

Figure 2 shows the ACovF for an OFDM signal with a near-rectangular spectrum of 10 GHz width with the same power as the OOK signal and also offset 50 GHz from the probe.

At $z_1 = 0$ it looks very similar to the OOK signal (see Fig. 3). However, contrary to the OOK signal, the ACovF does not smear out around the diagonal for large $z$. The reason for this is the shape of the OFDM signal. With its Gaussian amplitude distribution and flat spectrum it is already a very noise-like signal. The amplitude statistics do not change with accumulated chromatic dispersion (in the OOK signal they become more and more Gaussian). While each particular sample may undergo some variations due to dispersion, the signal statistics and thus the ACovF remain unaffected. Interesting.

#### Supplemental

Following a recent discussion with a colleague, I thought I’d plot the full ACovFs for polarization-multiplexed signals (the graphs for the case $z_2 = 0$ were already shown in the other post), because there seemed to be somewhat of a peculiar difference between the case where both tributaries are aligned in time and where they are interleaved – the latter was shown by Xie to cause much less XPolM. So here they are, the ACovFs for 10 Gbaud PolDM-RZ-QPSK signals at a frequency spacing of 50 GHz from the probe.

The oscillation which was visible for the time-interleaved tributaries in Fig. 4 of this post is also present when accounting for dispersion-induced pulse shape variations, however, only initially when accumulated fiber dispersion is low. With significant accumulated dispersion (and consequent loss of RZ pulse shape) the oscillation disappears and the ACovF looks much like the one for time-aligned tributaries. Time-interleaving thus works especially well in dispersion maps with nearly full inline dispersion compensation. The maximum amplitude along the diagonal remains smaller than in the time-aligned case and the secondary extremum is a minimum instead of a maximum, indicating that interleaving indeed reduces XPolM, even when pulses no longer remain RZ-shaped.

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