## Abstract

A technique is demonstrated for polarization demultiplexing of arbitrary complex-modulated signals. The technique is based entirely on the observation of samples in Stokes space, does not involve demodulation and is modulation format independent. The data in Stokes space is used to find the best fit plane and the normal to it which contains the origin. This normal identifies the two orthogonal polarization states of transmission and the desired polarization alignment transformation matrix. The technique is verified experimentally and is compared with the constant modulus algorithm.

©2010 Optical Society of America

## 1. Introduction

Complex modulation formats that utilize multiple discrete levels of amplitude and phase are well known from the transmission of electrical signals. Thus, their introduction to optical communications allows reuse of many existing algorithms related to the recovery of clock, carrier and carrier phase; and to the compensation of distortions related to chromatic and polarization mode dispersion [1–3]. An interesting new feature of optical transmission is polarization multiplexing, i.e., the use of two orthogonal polarization states for simultaneous transmission of two optical signals at the same wavelength. Since the polarization states of the optical waves that propagate through the optical fiber are not preserved, they typically arrive at the receiver misaligned. The nature of this misalignment is defined with respect to the polarizing beam splitter that is typically a part of the receiver [4–6]. Thus, the splitter defines the system of coordinates and local orthogonal linear polarization states. The received optical wave is split into two orthogonal optical waves. Each of these contains a linear combination of the transmitted optical waves. Therefore, in order to successfully demodulate these received waves, they have to be transformed back to the original transmitted optical waves.

In this paper a new method of polarization transformation that leads to demultiplexing is proposed. The method does not require knowledge of the modulation format. A unique transformation matrix is found from examination of objects created by the multiplexed optical waves in Stokes space prior to any signal processing. Thus, the method separates the task of polarization alignment from that of actual demodulation. Demodulation can be performed after polarization alignment on the individual polarization channels recovered from the polarization-multiplexed optical signal. The analysis in Stokes space utilizes all the sampled data points, is independent of the carrier frequency, and does not require knowledge of the clock. The method is based on a least squares algorithm, and in this sense it is optimal. It can operate on frames of data or it can be recursive. It will be shown that the convergence of the proposed method is substantially quicker than that of the constant modulus algorithm (CMA). Thus, the technique is especially suitable for applications that examine short frames of data.

This paper is organized as follows. First, a brief review of the constant modulus and decision directed algorithms is provided in Section 2. Both of these methods are later used in the analysis of the experimental data in Section 4. Section 3 begins with a review of the unitary transformation in connection with propagation of light in an ideal single mode fiber. Then complex modulation signals are described in terms of Jones and Stokes vectors. The key equations relating Jones vectors to Stokes space are given to allow visualization of the polarization multiplexed data in Stokes space. Examination of ideal polarization-multiplexed quadrature phase-shift keying (QPSK) signals reveals the existence of a lens-like object in Stokes space. This defines a plane and a normal pointing uniquely to two polarization states of transmission. The equations that describe boundaries of the disk in Stokes space are then derived for a hypothetical modulation format that is contained in a unit circle in the complex plane. The conditions required for preserving the lens-like object during the propagation of the polarization-multiplexed signal over the optical fiber are briefly discussed. In Section 4, the developed polarization alignment procedure is applied to experimental 40Gb/s polarization-multiplexed QPSK data. The complex signals recovered through the polarization alignment procedure are demodulated to show proper recovery of the constituent orthogonal components of the polarization-multiplexed optical signal. In Section 5 the convergence of the new algorithm is discussed and compared to the commonly used CMA technique as applied to polarization alignment. The paper ends with conclusions in Section 6.

## 2. Constant modulus and decision directed algorithms

Transformation of the polarization state can be accomplished in hardware or in software. The hardware solution implements polarization state transformation by means of a polarization controller [7,8]. This effectively aligns the transmission polarization states with the receiver’s polarizing beam splitter. Alternatively, software or firmware solutions can be used to perform a linear transformation of the received optical signals [4–6,9]. The most popular method of polarization demultiplexing is based on a generalization of the constant modulus algorithm [6,10] that accounts for two polarization-multiplexed optical waves. Since each of the received linearly-polarized optical waves contains a linear combination of the transmitted optical waves, the resulting apparent modulation becomes more complex. For example, a simple modulation scheme like QPSK contains only symbols equidistant from the center of the complex plane (the same modulus) but a misaligned receiver observes a linear combination of the transmitted optical waves as a multi-level signal. Thus, by enforcing the condition that the modulus is constant, the transformation can be found that restores the original single amplitude (constant modulus) nature of the modulation and leads to successful demultiplexing. This key idea remains the same even as the modulation format is generalized beyond QPSK. Then, demultiplexing involves analysis of the received symbols and transforming them in such a way that the resulting symbols map with minimal errors onto the expected modulation symbols. This, for example, takes place in a decision-directed algorithm that compares the received symbols with the desired template of the modulation format [6]. Mathematically, both constant modulus and decision-directed algorithms search for the solution using a well known complex gradient formula [1, 11]:

where*i*denotes the iteration number, ${W}_{i}$ is the complex butterfly filter in matrix form,

*μ*is the step size, ${\epsilon}_{i}$ is a measure of deviation from constant modulus or the nearest symbol, ${X}_{i}$ is the vector containing sampled values from the received polarization channels and superscript

*H*denotes a complex conjugate transpose. Use of the preceding equation will be illustrated for a single complex signal and for polarization demultiplexing in Section 4.

The CMA and the closely related decision-directed method, as used for polarization demultiplexing, show some undesirable properties. Both methods are recursive and therefore require examination of the individual received symbols in order to determine the next step of the iteration. In consequence, polarization demultiplexing becomes an integral part of the demodulation process. Furthermore, the demodulation process is intimately related to the modulation format. Thus, the details of the demodulation algorithm need to be modified if the modulation format changes. It is also well known that convergence of the algorithm may be relatively slow. Consequently, if only short frames of data are examined, finding the proper polarization alignment may be challenging. In addition, the algorithms may lead to recovery of the same polarization channel twice. This problem is known in the literature as a singularity of the CMA [12]. Singularity takes place when rows of the estimated butterfly filter ${W}_{i}$ are repeated. In summary, there is a need for an alternative method that does not suffer from the above limitations.

## 3. Complex modulated signals in Stokes space

The system of coordinates used in this paper is defined by the receiver. For simplicity and without loss of generality, assume initially that the transmitter and the receiver are in the same system of coordinates and are aligned. This simplifying assumption is not strictly necessary and will be removed in Section 3.2.

#### 3.1 Misalignment between transmitter and receiver

Typically, the optical wave from a transmitter laser is split into two waves that are independently modulated and then recombined in orthogonal polarization states at the polarization beam combiner [5]. As the optical waves propagate over the optical fiber, their polarization states evolve due to birefringence but remain nearly orthogonal. This is a simple consequence of optical fiber being relatively well approximated by a unitary matrix [13]. The optical effects that may break this simple and desirable property will be examined in more detail in Section 3.3. For now assume that the optical fiber can be represented by a unitary matrix, *M*, that is independent of the optical frequency:

*a*and

*b*are complex and the determinant of the matrix

*M*is equal to 1. The optical waves launched into an optical fiber in orthogonal linear polarization states ${(1,\text{\hspace{0.17em}}0)}^{T}$ and ${(0,\text{\hspace{0.17em}}1)}^{T}$ remain orthogonal while propagating through the optical fiber and emerge at the receiver in the orthogonal polarization states ${J}_{1}={\left(a,-{b}^{*}\right)}^{T}$ and ${J}_{2}={\left(b,\text{\hspace{0.17em}}{a}^{*}\right)}^{T}$. The orthogonality of the received optical waves represented by the Jones vectors ${J}_{1}$ and ${J}_{2}$ is verified by noting that the product ${J}_{1}^{H}{J}_{2}$is equal to zero. The process of aligning the polarization states of the emerging optical waves with the receiver is equivalent to finding an inverse matrix ${M}^{-1}$. The inverse matrix ${M}^{-1}$ is used to transform the measured polarized optical waves, represented by the Jones vectors ${J}_{1}$ and ${J}_{2}$, back to linear horizontal and linear vertical polarization states ${(1,\text{\hspace{0.17em}}0)}^{T}$ and ${(0,\text{\hspace{0.17em}}1)}^{T}$. Since the matrix

*M*is unitary, its inverse is equal to its complex conjugate transpose:It is immediately evident from the preceding equation that the rows of the matrix ${M}^{-1}$ are represented by ${J}_{1}^{H}$and ${J}_{2}^{H}$. Thus, identifying polarization states of transmission at the receiver provides all the required information to construct the inverse matrix ${M}^{-1}$. To learn how to recognize the polarization states of transmission it is convenient to examine in detail known polarization-multiplexed signals as described hereafter.

#### 3.2 Polarization demultiplexing

Let us denote the received horizontal and vertical optical waves that emerge from the receiver’s polarizing beam splitter by ${e}_{x}$ and ${e}_{y}$, respectively. For simplicity detailed analysis of the coherent receiver that allows the recovery of the two linear orthogonal components of the optical field is omitted, since this topic is well documented in the literature [4–6]. The Jones vector that represents the received optical wave can be written in the following form:

*ω*is within the bandwidth of the receiver. The carrier frequency is not exactly zero and it may fluctuate over time. The Jones vector,

*E*, of Eq. (4) is transformed into the Stokes vector,

*S*, by the following set of equations [14]:

*E*. The first component of the Stokes vector, ${s}_{0}$, represents the total power; the remaining three components ${\left({s}_{1},{s}_{2},{s}_{3}\right)}^{T}$ represent 0° linear, 45° linear, and circularly polarized light, respectively. The vector ${\left({s}_{1},{s}_{2},{s}_{3}\right)}^{T}$ allows for visualization of the polarization state in three dimensional space, often on the Poincare sphere. It is important to note that the Stokes vector,

*S*, of Eq. (5) is not normalized. Thus, in three dimensional plots in Stokes space, the distance from the origin represents the power of the optical signal. It is also important to note that the carrier frequency

*ω*is no longer present in Eq. (5). Thus, analysis in Stokes space is independent of the residual carrier and its fluctuations.

It is expedient to initially assume that the receiver and the transmitter are aligned, so that the optical waves at the receiver, ${e}_{x}$ and ${e}_{y}$, are the same as the transmitted optical waves. Furthermore, for simplicity, assume that the carrier frequency *ω* is mixed down to zero and that the amplitudes of the optical waves are equal (${a}_{x}={a}_{y}=1$). Assume also that the phases ${\varphi}_{x}$ and ${\varphi}_{y}$take the four discrete values of QPSK modulation: $\pi /4+n\pi /2$, where *n* takes integer values from 0 to 3. The QPSK modulated optical waves ${e}_{x}$ and ${e}_{y}$ are illustrated in the complex plane in the inset of Fig. 1
. Ideal linear transitions are included between the symbols. Figure 1 also illustrates the multiplexed optical waves in Stokes space obtained from Eq. (5). To produce the image one hundred random symbols were used to capture all the possible transitions between the symbols. The image of Fig. 1 reveals several interesting properties. The four QPSK symbols from the complex plane are also present in Stokes space. The transitions that were linear in the complex plane take more complicated trajectories in Stokes space and clearly outline a lens-like object. The lens-like object clearly defines a plane whose normal contains polarization states of transmission, i.e., linear horizontal and linear vertical polarization states labeled as H and V. Consequently, it is the normal that identifies polarization states of transmission.

To understand why the formation of this lens-like object in three dimensional Stokes space is not dependent on the modulation format, consider a modulation technique that is confined to a unit circle in a complex plane. This hypothetical modulation represents a general modulation format that covers all possible modulation scenarios. This hypothetical format does not restrict a modulation technique in any way besides stipulating that the signal is bounded and can be normalized to fit into a unit circle in the complex plane. Since all realizable signals are always bounded and never take infinite values, the assumption that they fit into the unit circle is always correct. Furthermore, since all values from within the unit circle are allowed, all formats are possible. Thus, showing that the unit circle of the hypothetical modulation format is mapped into the lens-like structure in Stokes space constitutes a complete mathematical proof that the presented method works for all modulation formats. To derive an equation that describes a boundary in the Stokes space one can assume that ${e}_{x}=1$ while ${e}_{y}$ takes any value from within the unit circle in the complex plane. This choice will become clear in the course of analysis. The Jones vector that represents this optical wave is:

The key observation used in deriving Eq. (7) was that the lens-like object surface is created by polarization multiplexing of an optical wave of maximum power (${e}_{x}=1$ or ${e}_{y}=1$) with an arbitrary complex signal (optical wave) from within the unit circle in the orthogonal polarization state. If both optical waves take complex values that are inside the unit circle ($\left|{e}_{x}\right|<1$ and $\left|{e}_{y}\right|<1$) the optical power is reduced and the corresponding point in Stokes space is inside the lens-like object of Fig. 2. Thus, the surfaces of the lens-like object define boundaries that enclose a volume containing all points of an arbitrary constellation of polarization-multiplexed signals. The existence of the lens-like object in Stokes space uniquely identifies a symmetry plane, its normal, and the polarization states of transmission. The symmetry plane can be found from a least squares fit of the data including symbols and transitions. The normal is chosen to contain the origin. Thus, the polarization states identified by the normal are orthogonal and always positioned on the opposite sides of the sphere. Identification of the transmission polarization states does not require demodulation of the data. Details of the polarization alignment procedure are presented in Section 4. The procedure works for all modulation formats that are not confined to a line or a point in a complex plane. Notable exceptions confined to a line are binary phase shift-keying (BPSK) and duobinary modulation. An example of a degenerate modulation that is confined to a single point is the data stream that contains only one symbol in each polarization state.

Since, in general, the transmitter and the receiver are not aligned and polarization state can freely evolve within the single mode fiber, the lens-like object may freely rotate in Stokes space. This rotation simply indicates that the arriving polarization multiplexed signal is not aligned with the receiver. Finding the normal identifies the polarization states of transmission. Aligning the identified polarization states of transmission with horizontal and vertical polarization states, as defined by the receiver, leads to polarization demultiplexing.

The confinement of the data to a lens-like object may be, at first glance, somewhat surprising. It would seem that all possible output polarization states should cover the entire surface of the sphere. However, due to the independence of powers in the two polarization channels and the unnormalized form of the Stokes vector of Eq. (5), the total power is not constant over the disk surface. For example, in the case of a hypothetical modulation format contained within the unit circle in the complex plane, when power in one of the polarization channels is very low, the polarization state of the combined optical waves is nearly equal to that of the orthogonal polarization channel, while power drops to about $1/2$of the maximum value. If however, powers of both polarization channels are at maximum, the distance from the origin is equal to 1, while the polarization states of the combined optical waves are in a plane that is normal to the axis defined by the transmission polarization states. This forms the edge of the disk. In Fig. 2, linear horizontal and linear vertical polarization states of transmission define the axis H-V. If power in one of the polarization channels drops to zero, the polarization state of the combined optical waves becomes horizontal or vertical, while the distance from the origin becomes $1/2$. The edge of the disk that touches the surface of the sphere corresponds to maximum power in both polarization channels and defines the plane that contains the $\pm {45}^{\circ}$ linear polarization states and the circular polarization states. The normalized diameter of the disk is equal to 2 and the normalized thickness is equal to 1.

#### 3.3 Brief consideration of optical fiber impairments

A detailed analysis of optical link impairments is beyond the scope of this paper. However, this section briefly considers the effects of polarization mode dispersion (PMD), chromatic dispersion (CD), and polarization dependent loss (PDL). Before considering different impairments let us first consider an ideal fiber described by the unitary matrix of Eq. (2). It is well known that birefringence described by Eq. (2) defines an axis in Stokes space [15]. The axis is defined by the eigenstates of polarization (eigenvectors of the matrix), i.e., polarization states that are maintained within the birefringent medium. All other polarization states evolve over the length of the birefringent medium along arcs whose angular measures correspond to the value of the birefringence and whose center is the birefringence axis. Clearly in this type of birefringent medium the shape of the disk is preserved as all its points are rotated around the same axis by the same angle.

The above simplified scenario corresponds to a situation in which the matrix of Eq. (2) remains approximately constant over the spectral width of the modulated signal. This process begins to break down when the matrix of Eq. (2) changes over this spectral width. In that case, the elements of the disk may rotate differently depending on their frequency content. This leads to the dispersion of the disk in the three dimensional space. Nevertheless, in modeling and in our experimental work, the polarization alignment procedure of this paper has been successfully used for birefringent elements whose differential group delay is smaller than half of the clock cycle.

Another optical effect that may cause deformations of the disk is polarization dependent loss (PDL). This is intuitively obvious since in three dimensional Stokes space, the objects are attracted by the polarization state that has the lowest loss. Therefore, PDL effects shift the disk in the three dimensional space and alter its original shape. However, the alignment procedure presented here still provides a valid solution that contains PDL related distortions. Moreover, from the shift of the disk it is possible to determine the direction and strength of the PDL and to arrive at appropriate compensation.

Finally, it is important to mention chromatic dispersion. Since chromatic dispersion affects both polarization states in the same way, its effects are primarily visible in the complex plane of the constellation plots. However, as discussed in the preceding section, the method presented in this paper is suitable for any modulation format that is not confined to a line or a point. The distortions of the constellation are mapped into Stokes space and lead to dispersion of the otherwise well delineated symbols. However, this does not affect the boundaries of the disk, and consequently, this alignment procedure is immune to chromatic dispersion.

## 4. Experimental verification

By examining the objects plotted in Stokes space it was shown in Section 3 that it is possible to estimate a proper mathematical transformation that allows for separation of the complex variables describing two multiplexed optical waves. In this section this procedure is applied to experimental data. Figure 3 shows experimental data that is QPSK modulated and polarization multiplexed; the transmission rate is about 40Gb/s (10GBaud). The figure contains about ${2}^{15}\approx 32000$ points sampled at a rate of 40GSa/s.

As expected, the lens-like object that defines a plane and its normal *J*
_{1}-*J*
_{2} are clearly visible. The inset shows the same set of data after a transformation whose details are provided below. A least squares plane fit to the experimental data identifies the normal shown in the figure and the two orthogonal polarization states of transmission: $\pm {\left(\begin{array}{ccc}\text{-0 .23,}& \text{0 .34,}& \text{0 .91}\end{array}\right)}^{T}$. The resulting orthogonal Stokes vectors transformed to orthogonal Jones vectors are:

*x*and

*y*resulting in angles ranging from $-\pi $ to

*π*. This yields the inverse transformation matrix:

The demodulated data in Fig. 4 are obtained by estimating clock frequency and phase from the power spectrum of ${e}_{h}{e}_{h}^{*}$ and ${e}_{v}{e}_{v}^{*}$, using the Viterbi-Viterbi algorithm to estimate the carrier [2], and using the decision-directed algorithm [6], described by Eq. (1), to remove carrier phase fluctuations. If the clock frequency and phase are known, the complex signals ${e}_{h}$ and ${e}_{v}$ are interpolated at the desired time points, i.e.: ${x}_{i}={e}_{h,v}\left({t}_{i}\right)$. Then, the decision-directed algorithm seeks a scalar complex multiplier ${w}_{i}$ that removes carrier phase and adjusts the scaling to position the demodulated symbol at the expected location. The demodulated symbols ${y}_{i}$ are found from the product ${w}_{i}\text{\hspace{0.05em}}{x}_{i}$. The scalar error term $\epsilon i$ in the recursion formula of Eq. (1) is defined as a distance from the nearest symbol *s*:

As illustrated in Fig. 4 this demodulation procedure yields correct constellation results. Thus, the task of transforming the measured optical waves into transmitted optical waves is accomplished without recursive examination of the individual symbols. Demodulation is then independently applied to the constituent orthogonal components of the polarization demultiplexed optical signal.

## 5. Speed of convergence

In order to understand the nature and speed of convergence, a comparison is now presented between the demodulation procedure of this paper and the generalized CMA technique of Eq. (1) that aligns polarization state while recursively examining individual symbols. It is important to mention that for polarization demultiplexing the dimensionality of Eq. (1) changes; its variables are no longer scalars, as in the preceding section, but vectors. As in the previous scalar case, one can assume that the clock frequency and phase are properly recovered by known techniques, e.g., by power spectrum analysis. The simple procedure of this section is possible when the clocks of the polarization multiplexed signals are in phase. Then, the complex variables ${e}_{x}$ and ${e}_{y}$ can be interpolated at the desired points of time to define the column vector ${x}_{i}={\left({e}_{x}\left({t}_{i}\right),\text{\hspace{0.17em}}{e}_{y}\left({t}_{i}\right)\right)}^{T}$. The matrix ${W}_{i}$is a 2 × 2 complex matrix that is expected to be similar in its properties to the transformation matrix ${M}^{-1}$. The complex variables of vector ${x}_{i}$ are transformed by a matrix ${W}_{i}$ to yield a column vector ${y}_{i}={W}_{i}{x}_{i}={\left({y}_{h},\text{\hspace{0.17em}}{y}_{v}\right)}^{T}$. The error vector ${\epsilon}_{i}$ of Eq. (1) is a column vector that depends on the distance of the demodulated symbols from the unit circle (constant modulus) [1,6]:

The above procedure is used to solve for the transformation matrix ${W}_{i}$ for different values of step-size *μ.* To illustrate the speed of convergence, the absolute values of the first row elements of the estimated 2 × 2 complex matrix are plotted as the algorithm iterates over each symbol. The results are shown in Fig. 5
. As expected, the speed of convergence depends on the step size *μ*. If the step size is too large ($\mu ={10}^{-1}$), convergence to an approximately constant value is not obtained. As the step size is reduced the stability of the estimate is improved; however, the convergence is slower. For $\mu ={10}^{-2}$approximately constant values of ${W}_{i}$ are reached after about 400 symbols.

This result is now compared with the convergence of polarization alignment in Stokes space. The normal that identifies the polarization states of transmission is calculated for frame sizes that increase in length and contain increasingly larger number of symbols. The absolute values of the first row elements of the matrix ${M}^{-1}$ are plotted to analyze the speed of convergence. The results are shown in Fig. 6 . Approximately constant values are obtained after about 50 symbols. Thus, the alignment in Stokes space in not only simple and independent of the modulation format but also converges substantially quicker than the CMA.

Quick convergence is especially important in analysis of short frames of data. Use of short frames is common for a test instrument that does not receive the data continuously (burst-mode operation) but only samples short segments of transmitted data to monitor its quality. Since the frames are acquired at relatively low rates, their polarization misalignment may randomly vary and use of the prior estimates may not be useful. The polarization alignment algorithm must be robust enough to tolerate random polarization states of individual short frames.

## 6. Conclusions

A polarization demultiplexing technique has been demonstrated that relies on the geometrical alignment of lens-like objects in three dimensional Stokes space. The technique does not require data demodulation and is independent of modulation format, residual carrier frequency, and clock frequency. The technique is free of the type of singularity that is well known for the CMA. The concepts of polarization alignment were explained for the QPSK modulation format and for a hypothetical modulation format that contains all states within the unit circle of the complex plane. In fact, this hypothetical modulation format corresponds to an arbitrary modulation technique. The parametric equations describing the lens-like object containing all modulation formats in Stokes space were derived. The effects of optical fiber impairments on the evolution and distortions of the disk were briefly discussed. The new polarization alignment concept was verified experimentally for 40Gb/s polarization-multiplexed QPSK data. Successful demodulation of the demultiplexed polarization channels was also demonstrated. Finally, the speed of convergence of the new method was compared to the commonly used CMA technique. The comparison indicates about an order of magnitude improvement in the speed of convergence over the CMA technique.

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