Journal of Lightwave Technology
Abstract: Starting from a model of random nonlinear polarization rotations for the effect of cross-polarization modulation in DWDM systems, we derive the mean distribution of the time-dependent polarization states at an arbitrary location within an optical link. We show that this distribution is fully parameterized by the degree of polarization of the particular wavelength channel, and we derive expressions to approximate this parameter for general optical links consisting of multiple optically amplified and dispersion-compensated spans, as well as related power thresholds. From the analytical expressions we derive a method to significantly reduce the detrimental effects.
Reference:
M. Winter, C.-A. Bunge, D. Setti, K. Petermann, “A Statistical Treatment of Cross-Polarization Modulation in DWDM Systems,” Journal of Lightwave Technology, vol. 27, no. 17, pp. 3739–3751, Sep 2009.
Download the paper
Note: This is not the original paper available on IEEEXplore. The JLT rasterizes all formulas and graphs in their PDFs, which looks less than nice. This is my original submission with scalable math and figures, augmented by the JLT publication data. The contents is exactly the same, but the layout may vary somewhat.
August 2nd, 2010
In a recent discussion with Curtis Menyuk, Alexei Pilipetskii, and Chongjin Xie I was made aware of a slightly inexact statement in the JLT published last year in which I was a co-author. It concerned the separation of XPM and XPolM in the nonlinear phase evolution equation. Since we were writing about XPolM at the time, we didn’t care too much about XPM and so this slipped our attention. The term describing XPolM actually also contains the relative polarization-dependent XPM and the term describing XPM is only the average over all polarizations. Here, I’d like to make this separation more exact.
We want to separate the Kerr nonlinearity evolution equation$^1$
$\newcommand{\vect}[1]{\mathbf{#1}}$ $\newcommand{\bra}[1]{\langle #1|}$ $\newcommand{\ket}[1]{|#1\rangle}$ $\newcommand{\braket}[2]{\langle #1|#2\rangle}$
$$\partial_z \ket{u} - i \bar \gamma \braket{u}{u}\ket{u} = 0\tag{A1}$$
with a DWDM signal
$$\ket{u} = \sum_{\nu=1}^N \ket{u_\nu} \exp \bigl[i \Delta \omega_\nu t\bigr]\tag{11}$$
into terms for SPM, XPM, and XPolM (ignoring FWM) that appear within the probe channel $\rho$ at $\Delta\omega_\rho$. The terms remaining from the triple sum which results when inserting (11) into (A1) and setting $\Delta\omega_\rho = 0$ are (cf. (12) and (13) in [1])
$$\partial_z \ket{u_\rho} - i \bar\gamma \Bigl[ \braket{u_\rho}{u_\rho}\ket{u_\rho} + \braket{u_\nu}{u_\nu}\ket{u_\rho} + \braket{u_\nu}{u_\rho}\ket{u_\nu} \Bigr] = 0\tag{A2}$$
These terms were summarized in [1] as
$$ \partial_\zeta\ket{u_\rho} - i \bar\gamma \Bigl[ \underbrace{\braket{u_\rho}{u_\rho}\vphantom{\Bigg|}}_\mathrm{SPM} + \sum_{\nu \ne \rho} \Bigl( \underbrace{\frac{3}{2}\, \braket{u_\nu}{u_\nu}\vphantom{\Bigg|}}_\mathrm{XPM} + \underbrace{\frac{1}{2}\, \vect U_\nu\cdot \vec\sigma\vphantom{\Bigg|}}_\mathrm{XPolM} \Bigr)\Bigr] \ket{u_\rho} = 0\tag{14}$$
However, it is important to notice that the term labeled ‘XPM’ is only the average XPM over all relative polarization states of probe $\rho$ and interferer $\nu$ and the term labeled ‘XPolM’ also contains the portion of XPM which depends on the relative SOPs of $\rho$ and $\nu$. Physically, XPM from co-polarized interferers must be larger than from orthogonal interferers because of the coherent mixing term that appears as a result of the inner product in (A1) or the third term in the square brackets of (A2). To examine this mathematically, we write
$$\ket{u_\nu} = u_\nu \ket{e_1} \quad \text{and} \quad \ket{u_\rho} = u_\rho \braket{e_1}{e_\rho} \ket{e_1} + u_\rho \braket{e_2}{e_\rho} \ket{e_2}\tag{A3}$$
with
$$\braket{e_n}{e_m} = \delta_{nm} \quad n,m \in \lbrace 1,2\rbrace$$
so that $\ket{e_1}$ and $\ket{e_2}$ form an orthonormal basis in Jones space which is determined by the SOP of $\ket{u_\nu}$. What we have done in (A3) is to separate the probe field into a part that is co-polarized with $\ket{u_\nu}$ and a part that is orthogonal to it by using the projection operators $\ket{e_1}\bra{e_1}$ and $\ket{e_2}\bra{e_2}$ on $\ket{u_\rho}$. We now expand the terms of (A2) that contain the cross-channel nonlinearities by using (A3) and have
$$\begin{aligned}
&\braket{u_\nu}{u_\nu}\ket{u_\rho} + \braket{u_\nu}{u_\rho}\ket{u_\nu}\vphantom{\frac{3}{2}}\\
&\quad = u_\nu^* \, u_\nu \, u_\rho \braket{e_1}{e_\rho} \braket{e_1}{e_1}\ket{e_1} + u_\nu^* \, u_\nu \, u_\rho \braket{e_2}{e_\rho} \braket{e_1}{e_1}\ket{e_2}\\
&\quad\quad +\, u_\nu^* \, u_\nu \, u_\rho \braket{e_1}{e_\rho} \braket{e_1}{e_1}\ket{e_1} + u_\nu^* \, u_\nu \, u_\rho \braket{e_2}{e_\rho} \braket{e_1}{e_2}\ket{e_1}\vphantom{\frac{3}{2}}\\
&\quad = 2 u_\nu^* \, u_\nu \, u_\rho \braket{e_1}{e_\rho} \ket{e_1} + u_\nu^* \, u_\nu \, u_\rho \braket{e_2}{e_\rho} \ket{e_2}\\
&\quad = \frac{3}{2} u_\nu^* \, u_\nu \, \ket{u_\rho} + \frac{1}{2} u_\nu^* \, u_\nu \, u_\rho \Bigl[ \braket{e_1}{e_\rho} \ket{e_1} - \braket{e_2}{e_\rho }\ket{e_2} \Bigr]\tag{A4}
\end{aligned}$$
Hence, the co-polarized part of the probe channel experiences twice as much phase shift as the orthogonal part (cf. the second equality above). In the last line of (A4) we have separated the phase shifts into a mean and a differential part – we could also have obtained that relation by doing the $\sigma$-expansion of (14) and using (A3). The differential phase shift on one hand gives rise to XPolM (and can be regarded as a nonlinear birefringence) and on the other hand causes the XPM to become dependent on the relative SOP between probe and interferer. To clearly separate both effects, we expand the differential using the product rule and obtain
$$\partial_z \ket{u_\rho} = \partial_z u_\rho \cdot \ket{e_\rho} + u_\rho \cdot \partial_z \ket{e_\rho}\tag{A5}$$
where $\ket{e_\rho}$ denotes the SOP of the probe. Thus by separating (A4) into a part aligned with $\ket{e_\rho}$ (which leads to no SOP change and is thus pure XPM) and a part that is orthogonal to it (leading to a change of the SOP only) we can achieve this separation. For the former we use the projection operator for the SOP of the probe, $\ket{e_\rho}\bra{e_\rho}$, and for the latter its orthogonal equivalent $\ket{e’_\rho}\bra{e’_\rho}$, with $\ket{e_\rho}$ and $\ket{e’_\rho}$ forming an orthonormal basis. We have
$$\begin{aligned}
\partial_z u_\rho \cdot \ket{e_\rho} &= i \bar \gamma u_\rho^* u_\rho \ket{u_\rho} + i \bar \gamma \, \frac{3}{2} u_\nu^* \, u_\nu \, \ket{u_\rho}\\
&\quad +\, i \bar \gamma \, \frac{1}{2} u_\nu^* \, u_\nu \Bigl[ \braket{e_1}{e_\rho} \braket{e_\rho}{e_1} - \braket{e_2}{e_\rho } \braket{e_\rho}{e_2} \Bigr] \ket{u_\rho}
\end{aligned}\tag{A6}$$
which is the same as
$$\begin{aligned}
\partial_z u_\rho &= i \bar \gamma u_\rho^* u_\rho \, u_\rho + i \bar \gamma \, \frac{3}{2} u_\nu^* \, u_\nu \, u_\rho\\
&\quad +\, i \bar \gamma \, \frac{1}{2} u_\nu^* \, u_\nu \Bigl[ \bigl|\braket{e_1}{e_\rho}\bigr|^2 - \bigl|\braket{e_2}{e_\rho }\bigr|^2 \Bigr] u_\rho
\end{aligned}\tag{A7}$$
which Karlsson and Sunnerud in [2, eq. (10)] have written as
$$\partial_z u_\rho = i \bar \gamma u_\rho^* u_\rho \, u_\rho + i \bar \gamma \, \frac{3 u_\nu^* \, u_\nu + \hat{\vect{S}}_\rho \cdot \vect{S}_\nu}{2} u_\rho$$
which has been adapted to current notation and where $\vect{S}$ is a Stokes vector and $\hat{\vect{S}}$ is an SOP (Stokes vector normalized to unit length). One can show if one so desires that these two are equivalent. Karlsson and Sunnerud’s expression requires the knowledge of the associated Stokes vectors whereas (A7) is given completely in Jones coordinates. The corresponding expression for the SOP change is
$$\partial_z \ket{e_\rho} = i \bar \gamma \, \frac{1}{2} u_\nu^* \, u_\nu \Bigl[ \braket{e_1}{e_\rho} \braket{e’_\rho}{e_1} - \braket{e_2}{e_\rho} \braket{e’_\rho}{e_2} \Bigr] \ket{e’_\rho}\tag{A8}$$
This is zero whenever $\ket{e_\rho} = \ket{e_1}$ or $\ket{e_\rho} = \ket{e_2}$, in agreement with the Stokes space description (15) in the original paper [1].
Again, I’d like to thank Chongjin, Curtis and Alexei for pointing this out to me.
1 I will omit the attenuation term that appears in the paper consequently throughout to make the equations shorter. Also, any equations whose number does not begin with A correspond to the equation in [1] with the same number.
[1] M. Winter, C.-A. Bunge, D. Setti, K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” Journal of Lightwave Technology, vol. 27, no. 17, pp. 3739–3751, Sep 2009.
[2] M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” Journal of Lightwave Technology, vol. 24, no. 11, pp. 4127–4137, Nov 2006.
