Grand Unified Theory

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:

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.


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