For reference, we start with Fig. 1 of the original post:
Fig. 1: Two-dimensional autocovariance function (ACovF) of the Stokes vectors of a 10G OOK signal with 50 GHz frequency separation from the probe channel as determined using linear fiber simulations.$^2$ Green pixels correspond to positive, red pixels to negative values; higher opacity corresponds to higher magnitude. Integrating / summing over all values yields the nonlinear variance, in this case of the probe channel polarization states. The function is dominated by channel walk-off when moving away from the diagonal; the (slight) effect of pulse distortion due to dispersion can be seen along the diagonal.
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.
Fig. 2: Two-dimensional autocovariance function (ACovF) of the Stokes vectors of a 10 GHz (field modulated) OFDM signal with 50 GHz frequency separation from the probe channel as determined using linear fiber simulations.$^2$ Green pixels correspond to positive, red pixels to negative values; higher opacity corresponds to higher magnitude.
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.
Fig. 3: autocovariance function (ACovF) resulting from walk-off between probe and interferer for SPol-NRZ-OOK (red) and OFDM (blue) channels. The asymptotic ACovF for SPol-NRZ-OOK with rectangular pulses is shown dashed for reference, all channels have equal mean power. $L_{WO}$ is the walk-off length, meaning the transmission distance after which the immediately neighboring WDM channel has walked off by one symbol of duration $T_S$.
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.
Fig. 4: Two-dimensional autocovariance function (ACovF) of the Stokes vectors of a 10 Gbaud PolDM-RZ-QPSK signal with 50 GHz frequency separation from the probe channel and aligned polarization tributaries.
Fig. 5: Two-dimensional autocovariance function (ACovF) of the Stokes vectors of a 10 Gbaud PolDM-RZ-QPSK signal with 50 GHz frequency separation from the probe channel and interleaved polarization tributaries.
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.