Polarization Drift Channel Model for Coherent Fibre-Optic Systems

TL;DR

Coherent fibre-optic systems suffer time-varying SOP that is not well captured by static models. The authors propose a stochastic polarization drift model based on a sequence of random Jones rotations $J(\\pmb{\\alpha})$ with $\\pmb{\\alpha} = \\theta \\mathbf{a}$, producing a three-dimensional isotropic random walk on the Poincaré sphere and expressible in Jones, Stokes, and 4D real formalisms. They prove isotropy, derive the 3D SOP-distribution for small step variance, and obtain autocorrelation functions for both Jones/4D and Stokes representations, including a phase-noise ACF; they validate the model against long-term measurements. They demonstrate that the model enables realistic simulations and analysis of polarization effects in polarization-multiplexed transmission, informing DSP design and performance quantification for future systems.

Abstract

A theoretical framework is introduced to model the dynamical changes of the state of polarization during transmission in coherent fibre-optic systems. The model generalizes the one-dimensional phase noise random walk to higher dimensions, accounting for random polarization drifts, emulating a random walk on the Poincaré sphere, which has been successfully verified using experimental data. The model is described in the Jones, Stokes and real four-dimensional formalisms, and the mapping between them is derived. Such a model will be increasingly important in simulating and optimizing future systems, where polarization-multiplexed transmission and sophisticated digital signal processing will be natural parts. The proposed polarization drift model is the first of its kind as prior work either models polarization drift as a deterministic process or focuses on polarization-mode dispersion in systems where the state of polarization does not affect the receiver performance. We expect the model to be useful in a wide-range of photonics applications where stochastic polarization fluctuation is an issue.

Figures (4)

• Figure 1: Phase noise pdf evolution. The pdf of $e^{i\phi}$ for $\phi=\pi/4$ and $\sigma_\nu^2=0.0025$ is shown. In ($\mathbf{a}$), $k=1$ corresponds to a single innovation and illustrates the second and third properties, i.e., the pdf is symmetric around the current state (the vertical line) and the peak of the pdf is at the current state. In ($\mathbf{b}$), $k=5$ and the pdf spreads over the circle. In ($\mathbf{c}$), $k=8000$ and the pdf approaches the uniform pdf, which supports the last property.
• Figure 2: Random walk. The evolution of a random SOP drift obtained by equation (\ref{['eq:st_sys_mod']}), without additive noise, for a fixed input $\mathbf{s}_{\mathbf{u}_{}}=(0,0,1)^\mathrm{T}$ and $\sigma_p^2=6\cdot 10^{-4}$ is shown. The trajectories for ($\mathbf{a}$) $k=1,\ldots, 300$, ($\mathbf{b}$) $k=1,\ldots, 1500$ and ($\mathbf{c}$) $k=1,\ldots, 3000$ are plotted.
• Figure 3: The histograms of $\mathbf{M}_{k} \mathbf{s}_{\mathbf{u}_{}}$ for different steps $k$ and a fixed $\mathbf{s}_{\mathbf{u}_{}}=[1,1,1]^\mathrm{T}/\sqrt{3}$ obtained from the model (top row) and from measurements (bottom row) are shown. The highest density is represented by dark red and the lowest by dark blue, the outer part of the density. The parameters of the simulated drift in equation (\ref{['eq:rand_alp']}) are $T=2.2$ h (set by the measurements) and $\Delta p=60$ nHz (obtained by fitting the dash-dotted ACF line in Fig. \ref{['fig:autocorr']}). In ($\mathbf{a}$) and ($\mathbf{d}$), $k=2$ innovation steps are plotted, whereas $k=8$ in ($\mathbf{b}$), ($\mathbf{e}$) and $k=16$ in ($\mathbf{c}$), ($\mathbf{f}$). Gaussian-like isotropic distributions can be noted in all cases, simulations and measurements, leading to a good (visual) agreement. The spread over the sphere increases with $k$ and the pdf will become uniform if we let $k$ grow large enough. Unfortunately, our measured data do not cover a long enough time period such that uniformity is achieved.
• Figure 4: ACF comparison. The normalized ACF of the phase noise and SOP drift is plotted versus normalized time. Solid/dashed lines refer to the analytic expressions, whereas the triangles are extracted from measurements. We observe excellent agreement between the experiment and theory, except in the tails of the experimental ACF. This inconsistency can be caused by the lack of accuracy in that region.

Theorems & Definitions (4)

• Theorem 1
• Lemma 1
• proof
• proof