Nerd Rage

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.

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last posts in XPolM:

• Grand Unified Theory
• XPolM Covariance Visualized
• catching up on XPolM