The Relation Between RSOP and PSP Rotation Rates and an Effective Algorithm for Monitoring PSP Rotation

Abstract

We begin by theoretically analyzing the relationship between the rotation rates of the rotation of state of polarization (RSOP) and the principal state of polarization (PSP) in a fiber link where both polarization mode dispersion (PMD) and time-varying RSOP are present. The theoretical analysis is validated through numerical simulations. Our findings reveal that the rotation rates of both the input and output PSPs significantly differ from the channel’s RSOP rate in most scenarios. Moreover, under varying RSOP distribution scenarios within the channel, the relationships among the rotation rates of input PSP, output PSP and RSOP also differ, and therefore rotation rates of input or output PSPs can reflect the changes of RSOP, indicating that monitoring PSP rotation rate can enable a better understanding of RSOP. Furthermore, we propose a DSP-based algorithm for monitoring PSPs and their rotation rates. By jointly applying a sliding-window median filter and a modulus judgment procedure, the algorithm yields more accurate PSP trajectories and rotation rate estimates than the existing approaches in literature, while relying solely on the existing DSP module without requiring any additional hardware. The recovered PSP orientation and rotation rate information can then be fed into the CMA equalizer, enhancing its compensation performance and thereby improving the overall stability and performance of the coherent optical system.

1. Introduction

In polarization division multiplexing (PDM) coherent optical communication systems, polarization effects are among the most significant impairments causing signal degradation in the channel [1]. Common polarization effects include the PMD and the RSOP. PMD introduces differential group delay (DGD) between the two orthogonal principal states of polarization (PSPs). As a result, an optical pulse will be broadened or even split after transmission, leading to misinterpretation of the signal code stream by the optical receiver and, consequently, higher bit-error rates (BER) [2]. RSOP causes random change in the state of polarization (SOP) and coupling between the two polarization tributaries, resulting in crosstalk between the signals encoded on the two polarization states [3,4]. Therefore, polarization impairments equalization to eliminate the damage caused by the polarization effects on the signal is essential in high-speed PDM optical communication systems.

Since compensating PMD involves addressing the time delay between the two orthogonal PSPs, visualizing the dynamic trajectory of PSPs is crucial for understanding the channel’s polarization effects. PMD compensation in the optical domain can be divided into DGD compensation using time-delay lines and polarization demultiplexing using polarization controllers (PCs), and PSP monitoring provides valuable feedback for PMD compensation in the optical domain. External environment factors, such as vibrations caused by the passing trains [5,6,7] or fiber Faraday effect from lightning strikes [4,8,9,10,11], can dramatically increase the RSOP rate in the fiber. This rapid RSOP vibration may cause polarization demultiplexing algorithm to fail, resulting in communication interruptions. To prevent such disruptions, estimating the RSOP rate in the channel is necessary to enable link rerouting before communication is compromised [12,13]. Although both RSOP and PMD originate from random birefringence in optical fibers, existing studies lack a clear quantitative analysis of the relationship between RSOP and PSP rotation rate. Furthermore, many approaches focus solely on output SOP changes while neglecting PSP variations, which are critical for PMD compensation. In some cases, the two concepts are even confused.

The existing literature primarily employs four methods for PSP or RSOP rate monitoring: the polarimeter method [14,15], the 3D-Stokes method [16,17,18], the power analysis method [19,20], and the DSP-based channel estimation method [21,22,23,24,25,26,27,28,29,30]. Polarimeter and 3D-Stokes methods offer accurate trajectories of output SOP and its rate. However, for high-order modulation format signals, a sufficient number of constellation points is required to obtain an accurate output SOP in order to calculate the normal vector in the plane where the constellation points are located. When the RSOP rate is high, the reduced number of available constellation points within a certain spatial range degrades the accuracy of the calculated output SOP. As a result, these methods are only suitable for scenarios with relatively low RSOP rates. The power analysis method uses the received optical signal to calculate the relative intensity error (RIE) for direct RSOP rate estimation but cannot provide detailed information on SOP or PSP. DSP-based monitoring methods have been widely adopted because FIR filters and other DSP algorithms are already extensively used in coherent optical communication systems [31]. Leveraging the existing DSP module allows multiple polarization impairments to be monitored simultaneously without introducing any additional hardware. However, current DSP-based approaches often directly use equalizer parameters without further processing, which can introduce significant errors from non-polarization-related factors.

In this paper, we theoretically analyze the relationship between RSOP rotation rate and PSP rotation rate in optical fibers and propose a PSP rotation rate estimation algorithm based on the multi-input multi-output (MIMO) digital signal processing (DSP) algorithm with trajectory smoothing and jump-elimination processing. When considering the fiber channel as a whole, RSOP effect can be mainly concentrated near the input or output, influencing the temporal rotation rates of the corresponding PSPs. Numerical simulations reveal that, in a statistical sense, PSP rotation rate approximates the RSOP rotation rate, with the input PSP rate more closely matching the RSOP of the entire link when RSOP effects dominate near the input, and the output PSP rate aligning more closely with the RSOP of the entire link when the RSOP effect dominate near the output. Monitoring input and output PSPs and their rotation rates, therefore, enables better understanding of PMD evolution, RSOP rate estimation, and we propose a PSP rotation rate monitoring algorithm based on MIMO tap coefficients. By using sliding-window median filtering and a modulus judgment method, our algorithm mitigates the problem of jitter and jumping of the monitoring PSPs caused by noise and other factors, accurately recovering time-varying PSP trajectories and rotation rates.

The rest of this paper is organized as follows. In Section 2, we theoretically analyze the relationship between RSOP and PSP rotation rates and validate our findings through Monte Carlo simulations. In Section 3, we present our PSP rotation rate monitoring algorithm with window-sliding median filtering and modulus judgment methods, and verify its effectiveness through numerical simulations. In Section 4, we summarize our main results.

2. Analysis of the Relation Between the Rotation Rates of RSOP and PSP

RSOP and PMD both originate from the birefringence of optical fibers, establishing an intrinsic relationship between them. For the convenience of analysis, the frequency-dependent components of fiber birefringence are abstracted as PMD, while the frequency-independent components are attributed to RSOP. Under the first-order approximation, PMD introduces a differential group delay (DGD) between two orthogonal principal states of polarization (PSPs) and is related to frequency [2,32]. The PSPs vary over time due to time-varying changes in the fiber’s birefringence, manifesting as a rotation rate in the Stokes space [3]. PMD is a characteristic of the fiber channel and is independent of the transmitted signal. In contrast, RSOP, as its name suggests, describes the rotation of the SOP exhibited in the Stokes space. Given a specific input SOP at the transmitter, the birefringence of the fiber causes a different SOP to be received at the receiver. Importantly, the received SOP, or output SOP, depends on the input SOP transmitted by the transmitter. If the input SOP changes, even under identical channel conditions, the resulting output SOP and the RSOP rate will also differ. This highlights that the RSOP rotation rate is strongly influenced by the SOP of the input light (input SOP). Monitoring the RSOP rate requires measurement of the output SOP and calculation of its rotation rate in the Stokes space [18,19,20,29,30]. Based on the analysis above, we theoretically derive the RSOP rotation rate and the PSP rotation rate in optical fibers, and compare their differences.

2.1. Theoretical Analysis

According to the principal state of polarization (PSP) theory proposed by Poole and Wagner [32], the first-order PMD in fiber links is modeled as

$${\mathbf{U}}_{\mathrm{PMD}}={\mathbf{R}}_{\mathrm{out}}{\mathbf{\Lambda }}_{\mathrm{DGD}}{\mathbf{R}}_{\mathrm{in}}^{-1},$$

(1)

where the columns of matrices ${\mathbf{R}}_{\mathrm{in}}$ and ${\mathbf{R}}_{\mathrm{out}}$ represent the input and output PSPs, respectively. The birefringence matrix ${\mathbf{\Lambda }}_{\mathrm{DGD}}$, which introduces a time-delay $\Delta \tau$ between the two PSPs, is expressed as

$${\mathbf{\Lambda }}_{\mathrm{DGD}}=\left(\begin{array}{cc}{e}^{j\Delta \omega \Delta \tau /2}& 0\\ 0& e^{-j\Delta \omega \Delta \tau /2}\end{array}\right).$$

(2)

In the presence of RSOP in fiber links, the channel matrix is represented as

$$\mathbf{H}\left(t,\omega \right)={\mathbf{J}}_2\left(t\right)\mathbf{U}\left(\omega \right){\mathbf{J}}_1\left(t\right),$$

(3)

where $\mathbf{J}$ represents the time-varying RSOP matrix. Given the input SOP $\left|E_{\mathrm{in}}\right.〉$, the output SOP $\left|E_{\mathrm{out}}\right.〉$ is described as

$$\left|E_{\mathrm{out}}\right.〉=\mathbf{H}\left(t,\omega \right)\left|E_{\mathrm{in}}\right.〉.$$

(4)

The output SOP can be further transformed into Stokes space using the Pauli matrices

$${\mathbf{S}}_{\mathrm{out}}=\left(\begin{array}{c}\langle E_{\mathrm{out}}\left|{\mathbf{\sigma }}_1\right|E_{\mathrm{out}}\rangle \\ \langle E_{\mathrm{out}}\left|{\mathbf{\sigma }}_2\right|E_{\mathrm{out}}\rangle \\ \langle E_{\mathrm{out}}\left|{\mathbf{\sigma }}_3\right|E_{\mathrm{out}}\rangle \end{array}\right)=\left(\begin{array}{c}\langle E_{\mathrm{in}}\left|{\mathbf{H}}^{†}{\mathbf{\sigma }}_1\mathbf{H}\right|E_{\mathrm{in}}\rangle \\ \langle E_{\mathrm{in}}\left|{\mathbf{H}}^{†}{\mathbf{\sigma }}_2\mathbf{H}\right|E_{\mathrm{in}}\rangle \\ \langle E_{\mathrm{in}}\left|{\mathbf{H}}^{†}{\mathbf{\sigma }}_3\mathbf{H}\right|E_{\mathrm{in}}\rangle \end{array}\right),$$

(5)

where ${\mathbf{\sigma }}_i$ are Pauli matrices, as defined in Ref. [2].

Assuming that there are only PMD and RSOP impairments in the fiber channel, the channel matrix $\mathbf{H}\left(t\right)$ is unitary. Therefore, the output SOP ${\mathbf{S}}_{\mathrm{out}}$ can be treated as a unit vector. By analogy with spherical motion, the angular velocity of ${\mathbf{S}}_{\mathrm{out}}$ is defined as the magnitude of the time derivative ${\displaystyle \frac{d{\mathbf{S}}_{\mathrm{out}}}{dt}}$. As shown in Equation (5), only the channel matrix $\mathbf{H}\left(t\right)$ is time-dependent in the expression of ${\mathbf{S}}_{\mathrm{out}}$. Therefore, applying the chain rule, the rotation rate of the output SOP can be calculated as

$$∥{\displaystyle \frac{d{\mathbf{S}}_{\mathrm{out}}}{dt}}∥=∥\left(\begin{array}{c}{\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d\mathbf{H}}{dt}}\right)}^{†}{\mathbf{\sigma }}_1\mathbf{H}|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\mathbf{H}}^{†}{\mathbf{\sigma }}_1\frac{d\mathbf{H}}{dt}|E_{\mathrm{in}}\rangle }\\ {\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d\mathbf{H}}{dt}}\right)}^{†}{\mathbf{\sigma }}_2\mathbf{H}|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\mathbf{H}}^{†}{\mathbf{\sigma }}_2\frac{d\mathbf{H}}{dt}|E_{\mathrm{in}}\rangle }\\ {\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d\mathbf{H}}{dt}}\right)}^{†}{\mathbf{\sigma }}_3\mathbf{H}|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\mathbf{H}}^{†}{\mathbf{\sigma }}_3\frac{d\mathbf{H}}{dt}|E_{\mathrm{in}}\rangle }\end{array}\right)∥.$$

(6)

It is evident from Equation (6) that the rotation rate of the output SOP is directly associated with the input SOP. Generally speaking, the signals emitted from the transmitter in fiber-optic communication systems are loaded on the x- and y-linearly polarized light, and therefore it is a reasonable choice to assume that $\left|E_{\mathrm{in}}\right.〉=\left(1,0\right)$ when calculating the rotation rate of the output SOP. However, it should be reminded that the above formula for calculating the rotation rate of the output SOP is applicable to arbitrary input SOP.

An example is provided to illustrate the calculation of the RSOP rate. In Ref. [3], we proposed a three-parameter RSOP model given by

$$\mathbf{J}=\left(\begin{array}{cc}cos\kappa e^{j\xi }& -sin\kappa e^{j\eta }\\ sin\kappa e^{-j\eta }& cos\kappa e^{-j\xi }\end{array}\right),$$

(7)

where $\kappa$ is the azimuth rotation angle, and $\xi$ and $\eta$ are the phase rotation angles. To generate a time-varying RSOP matrix, these three independent parameters are defined as functions of time t,

$$\begin{array}{cc}\hfill & \kappa ={\omega }_{\kappa }t+{\kappa }_0, \xi ={\omega }_{\xi }t+{\xi }_0, \eta ={\omega }_{\eta }t+{\eta }_0\hfill \end{array},$$

(8)

where ${\omega }_{\kappa }$, ${\omega }_{\xi }$, and ${\omega }_{\eta }$ represent the rotation rates of the respective parameters, while ${\kappa }_0$, ${\xi }_0$, and ${\eta }_0$ are the initial angles, which follow a random distribution within the range $\left[0,2\pi \right]$. When the input SOP is the x-linearly polarized light $|E_{\mathrm{in}}\rangle ={\left(\begin{array}{c}1,0\end{array}\right)}^{\mathrm{T}}$, substituting $|E_{\mathrm{in}}\rangle$ into Equation (6) and taking its magnitude yields the rotation rate of the output SOP as

$$∥{\displaystyle \frac{d{\mathbf{S}}_{\mathrm{out}}}{dt}}∥=\sqrt{4{\omega }_{\kappa }^2+{sin}^2\left[2\left({\omega }_{\kappa }t+{\kappa }_0\right)\right]{\left({\omega }_{\xi }+{\omega }_{\eta }\right)}^2}.$$

(9)

Equation (9) indicates that the rotation rate exhibited by the RSOP hinge depends not only on the respective rotation rates of the three parameters, but also on the initial state of the azimuthal rotation angle ${\kappa }_0$ and the evolution time t. It is worth noting that an RSOP matrix initially being a non-identity matrix can be treated as a time-varying RSOP matrix initially being an identity matrix by first transforming the input SOP into a different SOP. Consequently, the rotation rate of the output SOP for an arbitrary input SOP passing through an RSOP matrix with an initial identity state can be converted into the rotation rate of the output SOP for an x-linearly polarized input SOP passing through an RSOP matrix with a specific initial state. This approach allows the rotation rate to be efficiently calculated using Equation (9), thereby simplifying the computation.

Next, we discuss the calculation of the rotation rate of the output PSPs. By substituting Equation (1) into Equation (3), we obtain

$$\begin{array}{cc}\hfill \mathbf{H}(t,\omega )& =\left({\mathbf{J}}_2{\mathbf{R}}_{\mathrm{out}}\right)\mathbf{\Lambda }{\left({\mathbf{J}}_2{\mathbf{R}}_{\mathrm{in}}\right)}^{-1}\left({\mathbf{J}}_2{\mathbf{J}}_1\right)\hfill \\ \hfill & ={\mathbf{R}}_{\mathrm{out}}^{\prime }\mathbf{\Lambda }{\left({\mathbf{R}}_{\mathrm{in}}^{\prime }\right)}^{-1}{\mathbf{J}}^{\prime }\hfill \end{array},$$

(10)

where the RSOP matrices ${\mathbf{J}}_1$ and ${\mathbf{J}}_2$ are combined into a new RSOP matrix ${\mathbf{J}}^{\prime }$, and the combination of ${\mathbf{J}}_2$ and ${\mathbf{R}}_{\mathrm{out}}$ produces the new output PSP matrix ${\mathbf{R}}_{\mathrm{out}}^{\prime }$, while ${\mathbf{J}}_2$ and ${\mathbf{R}}_{\mathrm{in}}$ produces the new input PSP matrix ${\mathbf{R}}_{\mathrm{in}}^{\prime }$. It can be observed that the output PSPs also vary over time due to the time-varying RSOP, and this variation contains part of the RSOP information within the channel. Therefore, the rotation rate of PSP approximately reflects the RSOP rotation rate in the channel. The two columns of the unitary matrix ${\mathbf{R}}_{\mathrm{out}}^{\prime }$ represent the two rotating output PSPs. The two columns of ${\mathbf{R}}_{\mathrm{out}}^{\prime }$ are orthogonal to each other, meaning that the two output PSPs share the same rotation rate. By selecting the first column as the output PSP for the rotation rate calculation, we can observe that extracting the first column of ${\mathbf{R}}_{\mathrm{out}}^{\prime }$ is equivalent to the output SOP that the x-polarized light processed by ${\mathbf{R}}_{\mathrm{out}}^{\prime }$. In other words, the rotation rate of the output PSP is equivalent to that of the output SOP when the RSOP is present only at the output side. Similar to the calculation of the output SOP rotation rate in Equation (6), the rotation rate of the output PSP, denoted as ${\mathbf{P}}_{\mathrm{out}}$, can be calculated in the Stokes space as

$$∥{\displaystyle \frac{d{\mathbf{P}}_{\mathrm{out}}}{dt}}∥=∥\left(\begin{array}{c}{\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d{\mathbf{R}}_{\mathrm{out}}^{\prime }}{dt}}\right)}^{†}{\mathbf{\sigma }}_1{\mathbf{R}}_{\mathrm{out}}^{\prime }|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\left({\mathbf{R}}_{\mathrm{out}}^{\prime }\right)}^{†}{\mathbf{\sigma }}_1\frac{d{\mathbf{R}}_{\mathrm{out}}^{\prime }}{dt}|E_{\mathrm{in}}\rangle }\\ {\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d{\mathbf{R}}_{\mathrm{out}}^{\prime }}{dt}}\right)}^{†}{\mathbf{\sigma }}_2{\mathbf{R}}_{\mathrm{out}}^{\prime }|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\left({\mathbf{R}}_{\mathrm{out}}^{\prime }\right)}^{†}{\mathbf{\sigma }}_2\frac{d{\mathbf{R}}_{\mathrm{out}}^{\prime }}{dt}|E_{\mathrm{in}}\rangle }\\ {\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d{\mathbf{R}}_{\mathrm{out}}^{\prime }}{dt}}\right)}^{†}{\mathbf{\sigma }}_3{\mathbf{R}}_{\mathrm{out}}^{\prime }|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\left({\mathbf{R}}_{\mathrm{out}}^{\prime }\right)}^{†}{\mathbf{\sigma }}_3\frac{d{\mathbf{R}}_{\mathrm{out}}^{\prime }}{dt}|E_{\mathrm{in}}\rangle }\end{array}\right)∥.$$

(11)

Here, the equivalent input polarization state $|E_{\mathrm{in}}\rangle$ should be constrained to x- or y-linear polarization to extract one of the columns of the output PSP matrix ${\mathbf{R}}_{\mathrm{out}}^{\prime }$. Comparing Equations (6) and (11), it can be observed that the rotation rate of the output SOP is determined by the transmission matrix of the entire channel, whereas the rotation rate of the output PSP is only influenced by the RSOP matrix at the output side. However, it is important to note that the RSOP matrix at the output side is influenced by the channel’s transmission matrix. As a result, the rotation rate of the output PSP can serve as an indirect indicator of the overall transmission performance of the channel.

Similarly to Equation (10), an alternative transformation form of the channel matrix can be derived,

$$\begin{array}{cc}\hfill \mathbf{H}(t,\omega )& =\left({\mathbf{J}}_2{\mathbf{J}}_1\right){\left({\mathbf{R}}_{\mathrm{out}}^{-1}{\mathbf{J}}_1\right)}^{-1}\mathbf{\Lambda }\left({\mathbf{R}}_{\mathrm{in}}{\mathbf{J}}_1\right)\hfill \\ \hfill & ={\mathbf{J}}^{\prime }{\left({\mathbf{R}}_{\mathrm{out}}^{″}\right)}^{-1}\mathbf{\Lambda }{\mathbf{R}}_{\mathrm{in}}^{″}\hfill \end{array},$$

(12)

where the combination of the RSOP matrix ${\mathbf{J}}_2$ and the inversion of the output PSP matrix ${\mathbf{R}}_{\mathrm{out}}^{-1}$, or the input PSP matrix ${\mathbf{R}}_{\mathrm{in}}$, produces the new output PSP matrix ${\mathbf{R}}_{\mathrm{out}}^{″}$ or the new input PSP matrix ${\mathbf{R}}_{\mathrm{in}}^{″}$, respectively. Consequently, the exhibited rotation rate of the input PSP across the entire fiber link reflects the RSOP rate concentrated at the input side. Similar to the calculation of the output PSP rotation rate, the rotation rate of the input PSP can be regarded as the rotation rate of the output SOP after the x- or y-linearly polarized light passes through the RSOP element ${\mathbf{J}}_1$ at the input side. Referring to the calculation of the output PSP rotation rate in Equation (11), the rotation rate of the input PSP can be expressed as

$$∥{\displaystyle \frac{d{\mathbf{P}}_{\mathrm{in}}}{dt}}∥=∥\left(\begin{array}{c}{\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d{\mathbf{R}}_{\mathrm{in}}^{″}}{dt}}\right)}^{†}{\mathbf{\sigma }}_1{\mathbf{R}}_{\mathrm{in}}^{″}|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\left({\mathbf{R}}_{\mathrm{in}}^{″}\right)}^{†}{\mathbf{\sigma }}_1\frac{d{\mathbf{R}}_{\mathrm{in}}^{″}}{dt}|E_{\mathrm{in}}\rangle }\\ {\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d{\mathbf{R}}_{\mathrm{in}}^{″}}{dt}}\right)}^{†}{\mathbf{\sigma }}_2{\mathbf{R}}_{\mathrm{in}}^{″}|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\left({\mathbf{R}}_{\mathrm{in}}^{″}\right)}^{†}{\mathbf{\sigma }}_2\frac{d{\mathbf{R}}_{\mathrm{in}}^{″}}{dt}|E_{\mathrm{in}}\rangle }\\ {\displaystyle \langle E_{\mathrm{in}}|{\left({\displaystyle \frac{d{\mathbf{R}}_{\mathrm{in}}^{″}}{dt}}\right)}^{†}{\mathbf{\sigma }}_3{\mathbf{R}}_{\mathrm{in}}^{″}|E_{\mathrm{in}}\rangle +\langle E_{\mathrm{in}}|{\left({\mathbf{R}}_{\mathrm{in}}^{″}\right)}^{†}{\mathbf{\sigma }}_3\frac{d{\mathbf{R}}_{\mathrm{in}}^{″}}{dt}|E_{\mathrm{in}}\rangle }\end{array}\right)∥,$$

(13)

where again, the equivalent input polarization state $|E_{\mathrm{in}}\rangle$ should also be constrained to x- or y-linear polarization to extract one of the columns from the matrix of the input PSPs ${\mathbf{R}}_{\mathrm{in}}^{″}$.

2.2. Verification by Numerical Simulations

To validate the analysis of the relationship between the rotation rates of PSP and SOP, a simulation platform was constructed, as illustrated in Figure 1. The fiber channel consists of N spans, with each span including a DGD element, two time-varying RSOP connection units, and an amplified spontaneous emission (ASE) noise source. The three-parameter model is employed to construct the RSOP connection units, as described in Equation (7), while the temporal evolution rule is defined by Equation (8).

To calculate the true RSOP rate of the channel, an x-linearly polarized continuous wave (CW) signal is transmitted through the fiber link, and the output SOP is measured. The rotation rate of the output SOP (i.e., the RSOP rate of the channel) can then be determined in the Stokes space. The PMD of the fiber link can be calculated using the eigenmatrix $\mathbf{Q}=j{\mathbf{H}}_{\omega }{\mathbf{H}}^{†}$, where $\mathbf{H}$ is the transmission matrix of the channel and ${\mathbf{H}}_{\omega }$ is the derivative of $\mathbf{H}$ with respect to $\omega$. The eigenvalues of $\mathbf{Q}$ correspond to the DGD, while the eigenvectors represent the PSPs. The rotation rates of SOPs and PSPs are calculated for every sampling moment, and the mean value of all rotation rates over a period of time is taken as the reported rate. A comparison is then made between the reported rates of the output SOP and PSP. It is worth noting that in practical optical communication systems, it is neither necessary nor feasible to report the rotation rates for each symbol. Instead, the average rate within a specific time interval is typically reported. In our simulation platform, the time interval for calculating the output SOP or PSP is set to 30 ps, and the total number of calculations in a single stochastic simulation is 1000. Although the simulation duration is relatively short compared to the reporting interval in real systems, the number of monitoring samples is statistically sufficient.

To statistically observe the relationship between the rotation rates of PSP and SOP, 5000 stochastic simulations were conducted under four different channel scenarios. For each random simulation, the initial phase of the RSOP were randomly assigned according to Equation (8). The comparisons of the distributions of the rotation rates for each scenario are presented in Figure 2. The four scenarios are as follows: (a) A one-span model (i.e., $N=1$ in Figure 1), where the rotation rate of the RSOP hinge at the input side is set to 0, and the rotation rate at the output side is set as 13 Mrad/s; (b) A 12-span model (i.e., $N=12$ in Figure 1), where the RSOP hinge rotation rates are reduced from 13 Mrad/s at the input side to 0 at the output side; (c) A 12-span model, where the RSOP hinge rotation rates increase from 0 at the input side to 13 Mrad/s at the output side; (d) A 12-span model, where the RSOP hinge rotation rates increase from 0 at the input side to 13 Mrad/s at the middle position and then decrease to 0 at the output side. In all four scenarios, the mean DGD is set to 3 ps. These scenarios comprehensively verify the theoretical analysis presented in Section 2.1. Among them, scenario (c) most closely resembles actual fiber links. In real communication systems, both the transmitting and the receiving sides are typically located in stable environments, such as server rooms, where RSOP rates are relatively low. In contrast, the middle position of the link, which is farther from the server rooms, is more exposed to external environmental influences and therefore experiences higher RSOP rates.

From Figure 2, it is evident that the statistical distributions of the output SOP and PSP rotation rates vary across different scenarios. Specifically, in the one-span model of scenario (a), where a single RSOP connection unit located at the receiver side, the rotation rate distributions of the output PSP and SOP are identical. Figure 2 further illustrates that in scenario (b), where the RSOP rotation rate dominates near the output side, the statistical distributions of the rotation rates for the output PSP and SOP are more closely aligned. In contrast, in scenario (c), where the RSOP rotation rate dominates near the input side, the statistical distributions of the output SOP and output PSP rotation rates show greater deviations, while the rotation rate distributions of the input PSP and output SOP are more closely aligned. Finally, in scenario (d), where the RSOP rotation rate at the middle position dominates, the statistical distributions of the input and the output PSP rotation rates are closely aligned, while both deviate significantly from the statistical distribution of the output SOP rotation rates. Additionally, as shown in Figure 2, it is evident that the relationship between the mean value of the rotation rates of the output SOP, input PSP and output PSP follows similar patterns across four different scenarios. In scenario (a), the average rotation rate of the output SOP is almost identical to that of the output PSP. In scenario (b), the average rotation rate of the input PSP is closer to that of the output SOP. In scenario (c), the average rotation rate of the output PSP is closer to that of the output SOP. In scenario (d), the average rotation rates of the input and output PSPs are closer to each other, both showing significant differences compared to the output SOP’s average rotation rate. Figure 3 further illustrates the variations in the average rotation rates across the frequency domain. It can be observed that the average rotation rate is largely independent of frequency. At all frequency points, the relationship between the average rotation rates of the input PSP, output PSP and output SOP aligns with the conclusions described above.

3. Algorithm for PSP Rotation Rate Monitoring

As mentioned in Introduction, monitoring the rotation rate of PSPs provides critical feedback for PMD compensation and facilitates better understanding of RSOP variations in fiber links, and in Section 2, we derived the quantitative relationship between the rotation rates of RSOP and PSP. As shown in Figure 1, since PSPs are intrinsic channel properties and independent of the transmitted signal, calculating their rotation rates require knowledge of the channel transmission matrix. In this section, we employ a DSP-based PSP monitoring algorithm to estimate a preliminary trajectory of PSPs, and we propose algorithms for trajectory smoothing and jump elimination to refine the PSP trajectory, enabling the derivation of a more accurate PSP rotation rate.

3.1. Monitoring Algorithm

Consider a channel model incorporating chromatic dispersion (CD), PMD and RSOP, represented by $D\left(\omega \right)$, $\mathbf{U}\left(\omega \right)$ and $\mathbf{J}\left(t\right)$, respectively. The transfer function $\mathbf{H}(\omega ,t)$ of the fiber can be expressed as

$$\mathbf{H}\left(\omega ,t\right)=D\left(\omega \right){\mathbf{J}}_2\left(t\right)\mathbf{U}\left(\omega \right){\mathbf{J}}_1\left(t\right),$$

(14)

where the scalar function $D\left(\omega \right)=e^{-j{\omega }^2{\beta }_{2}L/2}$ represents the chromatic dispersion. Once the constant modulus algorithm (CMA) converges and polarization equalization is achieved, the monitoring matrix $\mathbf{M}$ can be constructed in the frequency domain by applying the discrete Fourier transforms (DFT) to the tap coefficients of the CMA. $\mathbf{M}$ is regarded as an approximate representation of the channel transfer matrix $\mathbf{H}$, retaining the information of all linear impairments.

The Jones Matrix Eigenvalue Analysis (JMEA) method is employed to monitor the PMD in the channel [33,34,35]. For the output PSPs $\left|{\widehat{\epsilon }}_{\mathrm{out}\pm }\right.〉$, the characteristic equation can be expressed in the Jones space as

$$j{\mathbf{M}}_{\omega }{\mathbf{M}}^{†}|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle =\pm {\displaystyle \frac{\Delta \tau \left(\omega \right)}{2}}|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle ,$$

(15)

where, when the frequency interval is sufficiently small, the derivative of the channel transmission matrix with respect to angular frequency can be approximated using the forward difference. Thus, Equation (15) can be reformulated as

$$j{\displaystyle \frac{\mathbf{M}(\omega +\Delta \omega )-\mathbf{M}\left(\omega \right)}{\Delta \omega }}{\mathbf{M}}^{†}\left(\omega \right)|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle =\pm {\displaystyle \frac{\Delta \tau \left(\omega \right)}{2}}|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle .$$

(16)

Since the channel transmission matrix $\mathbf{M}$ is unitary, satisfying ${\mathbf{M}}^{†}={\mathbf{M}}^{-1}$, the characteristic equation can be rearranged into the following expression,

$$\mathbf{M}(\omega +\Delta \omega ){\mathbf{M}}^{-1}\left(\omega \right)|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle =\left[1\mp j{\displaystyle \frac{\Delta \tau \left(\omega \right)}{2}}\Delta \omega \right]|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle .$$

(17)

To apply the JMEA method, the matrix in the characteristic equation is constructed, and the expression of the channel transmission matrix in Equation (15) is substituted to obtain

$$\mathbf{M}(\omega +\Delta \omega ){\mathbf{M}}^{-1}\left(\omega \right)=D(\omega +\Delta \omega )D^{*}\left(\omega \right){\mathbf{J}}_2\mathbf{U}(\omega +\Delta \omega ){\mathbf{U}}^{†}\left(\omega \right){\mathbf{J}}_2^{-1}.$$

(18)

By substituting Equations (1) and (2) into Equation (18), the expression simplifies to

$$\mathbf{M}(\omega +\Delta \omega ){\mathbf{M}}^{-1}\left(\omega \right)=c\left({\mathbf{J}}_2{\mathbf{R}}_{\mathrm{out}}\right)\mathbf{\Lambda }(\Delta \omega ){\left({\mathbf{J}}_2{\mathbf{R}}_{\mathrm{out}}\right)}^{-1},$$

(19)

where the scalar term $D(\omega +\Delta \omega )D^{*}\left(\omega \right)$, related to CD, can be expressed as c. Both ${\mathbf{R}}_{\mathrm{out}}$ and ${\mathbf{J}}_2$ take the form of unitary matrices, and their combination represents a new PSP that changes with time at the rotation rate of ${\mathbf{J}}_2$. The two eigenvectors $|{\widehat{\epsilon }}_{\mathrm{out}\pm }\rangle$ in the Jones space correspond the output PSPs.

Analogous to the derivation above, the characteristic equation for calculating the input PSPs can be expressed as

$${\mathbf{M}}^{-1}\left(\omega \right)\mathbf{M}(\omega +\Delta \omega )|{\widehat{\epsilon }}_{\mathrm{in}\pm }\rangle =\left[1\mp j{\displaystyle \frac{\Delta \tau \left(\omega \right)}{2}}\Delta \omega \right]|{\widehat{\epsilon }}_{\mathrm{in}\pm }\rangle .$$

(20)

The corresponding monitoring matrix can then be expressed as

$${\mathbf{M}}^{-1}\left(\omega \right)\mathbf{M}(\omega +\Delta \omega )=c\left({\mathbf{J}}_1^{-1}{\mathbf{R}}_{\mathrm{in}}\right)\mathbf{\Lambda }(\Delta \omega ){\left({\mathbf{J}}_1^{-1}{\mathbf{R}}_{\mathrm{in}}\right)}^{-1}.$$

(21)

Here, the combination of ${\mathbf{R}}_{\mathrm{in}}$ and ${\mathbf{J}}_1^{-1}$ forms the matrix of the new PSP, which is related to the input PSP and rotates with time at the rate of ${\mathbf{J}}_1$. The two eigenvectors of the monitoring matrix $|{\widehat{\epsilon }}_{\mathrm{in}\pm }\rangle$ correspond to the input PSPs of the channel.

To calculate the temporal rotation rate of the output PSP, we first compute the output PSPs at different instances. Tap coefficients are regularly extracted to obtain the Stokes vectors of the output PSPs $\mathbf{p}\left(t\right)$ on the Poincaré sphere. The rotation rate of PSP can then be calculated in the 3-dimensional space as

$$\mathrm{rotation}\mathrm{rate}={\displaystyle \frac{arccos\left[{\displaystyle \frac{\mathbf{p}\left(t\right)·\mathbf{p}(t+\Delta t)}{\left|\mathbf{p}\right(t\left)\right|\left|\mathbf{p}\right(t+\Delta t\left)\right|}}\right]}{\Delta t}}.$$

(22)

It is important to note that the time interval for calculating the PSP rotation rate should be sufficiently small. This is because the output PSP may follow a curved trajectory, and the spatial angle between the two monitoring PSPs at adjacent moments may be small. If the time interval is too large, it fails to accurately capture the true trajectory of the PSP, resulting in significantly underestimated rotation rates.

In actual communication systems, the trajectory of the monitored PSP on the Poincaré sphere can exhibit significant vibrations due to noise, causing the calculated rotation rate to be much greater than the actual rotation rate. The commonly used mean-value filtering method, often employed to remove the influence of noise, is unsuitable for smoothing PSP trajectories. This is because the modulus of the mean PSP – calculated using the mean values of the three Stokes parameters for all PSPs being processed – may not necessarily equal 1. For instance, in an extreme scenario, the mean PSP of two PSPs located at opposite positions on the Poincaré sphere would lie at the center of the sphere, where its modulus is clearly not 1.

To address this issue, the sliding-window median filtering method is applied to smooth the monitored PSP trajectory. The detailed procedure is outlined in Algorithm 1: using the first Stokes parameter ($s_1$) as the reference parameter (other parameters such as $s_2$ or $s_3$ are also applicable), the median value of $s_1$ within a sliding window of length L at time k is selected, and the entire PSP vector corresponding to this $s_1$ is used as the filtering result. The sliding step length $\nu$ must be carefully chosen, as it directly impacts the filtering effect. If the step length is too large, details of the PSP trajectories may be lost; if too small, the jitter might remain.

3.2. Numerical Simulations

To validate the effectiveness of the proposed PSP rotation rate monitoring method, we constructed a channel emulation platform incorporating PMD and time-varying RSOP, along with a coherent optical communication system operating on it, as depicted in Figure 4. The channel model is consistent with Figure 1, featuring a total of 12 spans. At the transmitter, symbols are generated at a rate of 140 GBaud. The mean DGD of the fiber link is set to $0.8T_0$, where $T_0$ denotes the symbol period, and the RSOP rotation rate of each hinge is set to 0.26 Mrad/s. At the receiver, polarization demultiplexing is performed using CMA, followed by the application of the proposed PSP monitoring method.

Algorithm 1 PSP Monitoring with Median Filtering
Input: Initial index k, step length $\nu$, sliding window length L Output: Filtered PSP trajectories $\mathbf{P}$, rotation rates $\mathit{\zeta }$
	1: Perform CMA and obtain tap coefficient matrix $\mathbf{H}$   2: $\mathbf{M}\left(\omega \right)={\left\{\mathrm{DFT}\left[\mathbf{H}\right]\right\}}^{-1}$   3: Calculate $\mathbf{M}\left(\omega +\Delta \omega \right){\mathbf{M}}^{-1}\left(\omega \right)$           // Equations (18) and (19)   4: Extract eigenvalues ${\rho }_i$ and eigenvectors ${\mathbf{p}}_i$ of $\mathbf{M}\left({\omega }_0+\Delta \omega \right){\mathbf{M}}^{-1}\left({\omega }_0\right)$   5: Stokes vector $\mathbf{p}=\left[s_1,s_2,s_3\right]$ ← Jones vector $\mathbf{p}$   6: count = 0                         // Tracking the filtering process   7: ${\widehat{s}}_1=\mathrm{median}\left(s_{1,k},\dots ,s_{1,k+L-1}\right)$   8: for $l=0$ to $L-1$ do   9:     if $s_{1,l}=={\widehat{s}}_1$ then 10:           ${\mathbf{p}}_{\mathrm{out}}\left(k\right)=\left[s_{1,l},s_{2,l},s_{3,l}\right]$ 11:     end if 12: end for 13: $\mathbf{P}\left(:,\mathrm{count}\right)={\mathbf{p}}_{\mathrm{out}}\left(k\right)$             // Put the filtered PSPs together 14: $k\leftarrow k+\nu$ 15: $\mathbf{\zeta }\left(\mathrm{count}\right)\leftarrow \mathrm{Compute}\mathrm{the}\mathrm{rotation}\mathrm{rate}\mathrm{by}\mathrm{Equation}\left(22\right)$ 16: $\mathrm{count}\leftarrow \mathrm{count}+1$

The trajectories of the true PSP, the monitoring PSP before filtering, and after filtering are shown in Figure 5a–c, respectively. Due to the influence of noise and the limited accuracy of the CMA, the PSP trajectories directly calculated from the tap coefficients of the CMA exhibit severe jitter, as illustrated in Figure 5b, resulting in significantly overestimated rotation rates. After applying median filtering, the PSP trajectory becomes smoother and closely resembles the true PSP trajectory, as shown in Figure 5c. This demonstrates that the calculated rotation rate after median filtering more accurately reflects the actual PSP rotation rate.

Another critical factor affecting the monitoring accuracy is the ambiguity of the monitoring PSPs, i.e., the position of the two calculated orthogonal PSPs may interchange, as shown in Figure 6. During the interchange process, the angle between the monitored PSPs at two adjacent moments in the Stokes space approaches $\pi$, leading to abnormally large rotation rate values given the small monitoring time interval. One intuitive solution is to filter out these spikes in the time-domain spectrum of the rotation rate. However, this approach risks erroneously ignoring genuinely large rotation rates, which may occur during rapid polarization state changes, leading to a failure in detecting sudden events. Figure 6 shows that during the interchange process, the monitored PSP is located inside the Poincaré sphere, with its modulus deviating from 1 (indicating that the degree of polarization is not 1). Based on this observation, it is possible to identify the interchange process by checking the modulus of the monitored PSP and determine whether to retain the calculated rotation rate at that moment. The processing procedure is detailed in Algorithm 2. Specifically, if the modulus of the median-filtered PSP falls below a predefined threshold (denoted as the target PSP), the two rotation rates associated with this PSP are discarded. These include the rotation rates between the target PSP and the preceding and subsequent PSPs. The average value of the remaining rates within the processing time range is then reported as the final rotation rate.

Algorithm 2 PSP Rotation Rates Selection by Modulus Judgment Method
Input: median-filtered monitored output PSPs $\mathbf{P}$ and rotation rates $\mathit{\zeta }$ Output: processed output PSP rotation rates $\widehat{\mathit{\zeta }}$, mean processed rotation rate ${\widehat{\zeta }}_m$
	1: Set the modulus threshold m.   2: $N_p$ ← length$\left(\mathbf{P}\right)$   3: for $i=0$ to $N_p-1$ do   4:     if norm$\left[\mathbf{P}\left(:,1\right)\right]<m$ then   5:           $\widehat{\mathit{\zeta }}\leftarrow$ delete $\mathit{\zeta }\left(i\right)$ and $\mathit{\zeta }\left(i+1\right)$   6:     end if   7: end for   8: ${\widehat{\zeta }}_m\leftarrow \mathrm{mean}\left(\widehat{\mathit{\zeta }}\right)$                   // the reporting rotation rate of PSP

To validate the effectiveness of the proposed PSP monitored trajectory smoothing and PSP interchanging elimination algorithm, we performed 500 stochastic numerical simulations with the simulation system shown in Figure 4. We calculated the error between the estimated PSP rotation rate values, both before and after trajectory processing, and the actual PSP rotation rates in the channel. The simulation results, expressed as percentages, are shown in Figure 7. The results indicate that the average estimation error of the PSP rotation rates before trajectory processing reached 236.6%, which is several times larger than the true rotation rates. After applying the trajectory processing algorithm, the average error was reduced to approximately 12.4%. This demonstrates that the proposed trajectory processing algorithm effectively mitigates the influence of noise and other factors, resulting in significantly more accurate rotation rate estimations.

Another key metric for evaluating an algorithm is its computational complexity, which directly affects the required hardware resources and the achievable real-time performance. Computational complexity is typically characterized by three basic hardware units: multipliers, adders, and look-up tables (LUTs), among which multipliers are the most resource-intensive and expensive to implement. Therefore, in this work, we focus on the number of multipliers as the primary indicator for assessing implementation complexity. The proposed PSP rotation-speed monitoring algorithm is designed based on the CMA structure. As a result, its complexity scales linearly with the number of equalizer taps N, and the total number of required multipliers per symbol can be expressed as $42N+13$. For comparison, another widely used polarization equalization method in coherent optical systems, the Extended Kalman Filter (EKF), exhibits a different complexity behavior: its multiplier count depends on the dimensionality of the tracked state vector n, with a per-symbol complexity of $[n^3+6n^2+(4L_w+13)n+(4{log}_2{L}_w+18)L_w]/\Delta s$, where $L_w$ is the length of the sliding window, and $\Delta s$ represents the sliding step length. For example, in a joint PMD and RSOP compensation scenario, the EKF requires tracking six state variables. Considering a sliding window with $L_w=16$ and $\Delta s=2$, the per-symbol multiplier count of the EKF is 751.5. In contrast, for a CMA equalizer with $N=15$, the proposed scheme achieves both polarization compensation and PSP rotation rate monitoring with a total multiplier count of 643, which is significantly lower than that required by the EKF under the same conditions.

4. Conclusions

In this work, we have theoretically analyzed the relationship between the input and output PSP rotation rates, which are intrinsic channel attributes, and the output SOP rotation rate, which characterizes the optical signals in fiber-optic communication systems. Our theoretical analysis reveals that both the input and output PSP rotation rates can partially reflect the output SOP rotation rate of the fiber channel. Specifically, the input PSP rotation rate is indicative of the RSOP dominating near the transmitter, while the output PSP rotation rate reflects the RSOP dominating near the receiver. This theoretical analysis has been validated statistically through multiple Monte Carlo simulations across four types of channel environments.

Since the DGD represents the time delay between the two orthogonal PSPs, monitoring the rotation rate of PSPs is critical for compensating the first-order PMD. Moreover, due to the relationship between the rotation rates of PSP and RSOP, monitoring of the PSP rotation rates can provide a better understanding of RSOP changes in the fiber link. To address this, we proposed a DSP-based PSP rotation rate monitoring algorithm employing median filtering and modulus judgment. Numerical simulations demonstrate that our algorithm effectively mitigates jitter and trajectory jumps caused by noise and algorithmic inaccuracies, reliably recovering the time-varying PSP trajectory, and yields relatively accurate PSP rotation rate estimates. Compared to existing methods, our algorithm offers the following key advantages: (1) it directly calculates the PSP rotation rate using the tap coefficient matrices of the polarization demultiplexing algorithm, eliminating the need for additional equipment, and (2) it provides accurate information on PSP orientation and rotation rates to the CMA, which directly enhances the compensation performance. More effective compensation of polarization impairments such as PMD increases the usable signal bandwidth that can be reliably transmitted through the fiber channel.