Space-demultiplexing based on higher-order Poincar&#x000E9; spheres

Gil M. Fernandes; Nelson J. Muga; Armando N. Pinto

doi:10.1364/OE.25.003899

1. Introduction

Space-division multiplexing (SDM) has been proposed to increase capacity in fiber-optic communication systems. SDM allows to transmit several parallel data streams through a single fiber using orthogonal modes of a few-mode fiber (FMF) or multiple cores in a multicore fibers (MCF) [1]. For both approaches, the optical signals suffer from crosstalk like the two quasi-degenerated modes in standard single-mode fibers [2,3]. This crosstalk can be equalized in the coherent receiver using digital mode/core-demultiplexing techniques [1,2]. Nevertheless, the development of algorithms with faster adaptation and tolerable computational complexity is still needed.

In a broad sense, the concept of space diversity is a generalization of the widely used concept of polarization diversity [1]. In polarization-division multiplexing (PDM) systems both orthogonal polarization states are used to transmit two independent tributary signals at the same wavelength [3]. However, the propagation through the fiber rotates the polarization states. It is worth mentioning that these random rotation can be compensated in the optical domain, or in the digital domain. In the first case, the compensation is based on electro-optic techniques and can include, for instance, the employment of dynamic polarization controllers [4]. Whereas, in the second case, after a complete recover of the amplitude and phase of the signal through coherent detection, powerful equalization techniques are used in the digital domain [5,6]. Digital polarization demultiplexing (PolDemux) is usually based on multiple-input and multiple-output (MIMO) algorithms [3]. The constant modulus algorithm (CMA) is one of the most popular PolDemux algorithms due to its simplicity and robustness [3]. However, this algorithm is modulation format dependent, which may represent a major problem in some flexible networks where the baud rate and the modulation format of the signals are dynamically defined by software. Moreover, it also has a moderate convergence speed and a singularity problem [7]. An alternative PolDemux algorithm based on Stokes space was proposed to overcome the singularity problem improving at the same time the convergence speed [8]. This algorithm is also robust against the residual frequency offset [8,9]. In the PolDemux algorithm based on Stokes space the demultiplexing matrix is defined using the spatial orientation of the best fit plane to the received samples in the Stokes space [5–8,10]. Nevertheless, Stokes space based techniques assume a low value of polarization-mode dispersion (PMD) because the PMD can break the symmetry plane [11]. It was shown that when the symbol period becomes comparable to the differential group delay (DGD) the symmetry plane may be distorted and the performance of the technique decreases. [8]. The PMD can be mitigated by using special fibers [12] or by using pre-compensation PMD techniques. In SDM transmission systems, which is the focus of this paper, the modal dispersion (the counterpart of PMD in the single-mode case) can also be mitigated, namely by using new fibers with optimized index profiles [13], by link engineering [2,14,15] and even by pre-compensation [16]. In [17], it is proposed a sequentially algorithm in which the PMD is compensated neglecting the state of polarization and, afterwards, a PolDemux algorithm is employed to demultiplexing both polarizations. Using error signal estimation in the Stokes space, adaptive butterfly equalizers able to deal with PMD were also proposed in the time [18] and frequency [19] domains. The frequency domain equalizer was recently extended to mode-division multiplexing (MDM) systems by considering all the tributary signals in the update function of the proposed filter [20].

The 3-dimensional Stokes space was also used to represent multimodal signals carrying orbital angular momentum (OAM) using the Jones circular polarization basis [21]. Thereby, the Jones circular polarization basis is replaced by several sets of Jones circular basis of vector modes given rise to signal analyses in higher-order Poincaré spheres [21]. In [22], a hybrid spatio-polarization description for such modes is proposed considering two independent Poincaré spheres to represent simultaneously the polarization and spatial degree of freedom. Afterwards, the 3-dimensional Stokes space was extended to the multimodal case by considering a set of generalized Pauli matrices. The generalized Stokes space was initially used to investigate the spectral properties of the modal dispersion in terms of the frequency autocorrelation function [23]. In [24], it was proposed a method to determine the principal modes of a FMF by written the mode dispersion operator in the generalized Stokes space. Furthermore, a propagation model in the generalized Stokes space was also developed to investigate the evolution of the group delay (GD) in compensated links [15].

By properly handling the generators of the generalized Stokes space, this paper shows how a multimodal signal can be represented in higher-order Poincaré spheres. Based on the signal analyses in higher-order Poincaré spheres, we propose a time domain algorithm for space-demultiplexing (i.e., for few-mode and multicore approach) that is modulation format agnostic and free of training sequences. For MCF with significant crosstalk the proposed algorithm allows to fully reverse the crosstalk; however, for the few-mode approach a digital tool for mode-dispersion equalization is also required. We demonstrate that a given signal can be decomposed in sets of two tributary signals and each set of signals can be represented symmetrically to a given plane in a higher-order Poincaré sphere. Although the crosstalk changes the spatial orientation of this plane, its original position can be retrieved by reversing the crosstalk between any pair of tributary signals. By properly equalizing the crosstalk among all the pairs of tributary signals, we can calculate the inverse channel matrix and fully reverse the crosstalk.

This paper is organized in four sections. In section 2, we describe an arbitrary complex-modulated signal in the Jones and in the generalized Stokes space. The generalized Stokes space is divided in several higher-order Poincaré spheres and it is shown that any arbitrary pair of complex-modulated signals can be represented as a lens-like object in a higher-order Poincaré sphere. In section 3, we propose a space-demultiplexing algorithm based on the signal representation in higher-order Poincaré spheres, and present numerical results for different modulation formats. Finally, the main conclusions are presented in section 4.

2. Higher-order Poincaré spheres

Throughout this paper we make use of the Dirac notation to represent the signal in the Jones space, |ψ〉 [11]. Since all the tributary signals are coupled, |ψ〉 allows to describe a signal transmitted through FMF or MCF with significant crosstalk. In the following sections, and without loss of generality, it is assumed a SDM system based on FMF supporting n orthogonal modes in which a single mode comprises two orthogonal polarization states. In the Jones space, the electrical field vector of a given multimodal signal can be represented as a 2n dimensional complex ket vector [1]

| ψ 〉 = {(υ_{x}^{1}, υ_{y}^{1}, \dots υ_{x}^{n}, υ_{y}^{n})}^{T},

where

υ_{l}^{j}

, with j = 1, 2…n and l = x, y, represents the electrical field of a tributary signal multiplexed in the j mode and in the l polarization state, and T means transpose. This signal can be rewritten as

| ψ 〉 = {(υ_{1}, υ_{2}, \dots υ_{h}, \dots υ_{2 n})}^{T},

being υ_i the electric field of the i tributary signal, with i = 1…2n, and

υ (z, t) = a {(z, t)}^{e^{[- i (ω t + ϕ (z, t))]}},

where t is the temporal coordinate, z is the longitudinal coordinate, ω is the angular frequency of the carrier, a is the envelop amplitude and ϕ is the phase. Note that, the phase noise and the carrier frequency offsets can be considered in ϕ as a perturbation. It is more convenient write the electrical field vector in the form of Eq. (2), this choice will become clear in the course of this analysis.

The optical signal can be represented in the generalized Stokes space by using a particular set of generalized Pauli matrices (Λ_i, where i = 1, 2 … D, with D = 4n² − 1) chosen to facilitate the interpretation of the signal in this space [23]. These D matrices can be− grouped in three different subsets, and are generated as follows:

the first subset contains 3n matrices which can be sequentially written as the three Pauli matrices in the main diagonal, from the leftmost to the rightmost, and normalized by $\sqrt{n}$ , while the other elements are set to zero;
the second subset, with 4n² − 4n matrices, is written by considering the elements outside of the 2 × 2 blocks in the main diagonal and sequentially set to either 1 or $i = \sqrt{- 1}$ and the symmetric element to 1 or −i. The remaining elements are set to zero and the normalization coefficient $\sqrt{n}$ is again required;
the third subset contains n − 1 matrices that are generated by written 2 × 2 blocks in the main diagonal. The first n_l diagonal elements, with n_l = 1, 2 … n 1, are set to a 2 × 2 unitary matrix, σ₀, and the (n_l + 1) 2 ×2 diagonal elements are set to −n_lσ₀. The remaining elements are set to zero and, in this case, the normalization coefficient is $\sqrt{n / (n_{l}^{2} + n_{l})}$ .

In the generalized Stokes space, the signal |ψ〉 is written as a D-length real vector [23]

\vec{Ψ} = {(Ψ_{1}, Ψ_{2} \dots Ψ_{D})}^{T},

where the Stokes parameters, Ψi, can be calculated as,

Ψ_{i} = 〈 ψ | Λ_{i} | ψ 〉 .

The above subsets describe how the generalized Pauli matrices, i.e. the generators of the generalized Stokes space, can be obtained. In the next subsections, these subsets are called to properly explain the decomposition of the generalized Stokes space in several 3-dimensional subspaces, i.e. higher-order Poincaré spheres. Higher-order Poincaré spheres allow to represent all the possible combinations of pairs of tributary signals; including pairs of tributaries from the same mode, i.e. the intramode case, and the pairs of tributaries from distinct modes, i.e. intermode case.

2.1. Definition of higher-order Poincaré spheres

The generalized Pauli matrices, Λ_i, can be also used to define the structure of a Lie algebra of a special unitary group of dimensionality 2n, SU(2n) [24]. Therefore, the mode dispersion and the crosstalk operator can be expanded by the generators of a Lie algebra of SU(2n). Special unitary groups with dimensionality larger than 2 are semi-simple and, hence, can be decomposed in $g_{s} = (\frac{2 n}{2}) = n (2 n - 1)$ subgroups of SU(2) [24]. The structure of the Lie algebra of each subgroup SU(2) has an isomorphism in SO(3), i.e. a given pair of tributary signals in the Jones space, SU(2), can be represented in a higher-order Poincaré sphere, SO(3). Using a semi-simple representation of the Jones space, the generalized Stokes space can be decomposed in g_s subspaces. This g_s higher-order Poincaré spheres give a hybrid spatio-polarization description which allows represents simultaneously the polarization and spatial degree of freedom. In each subspace, the relative state between two arbitrary tributary signals can be represented by the respective Stokes vectors,

{\vec{Ψ}}^{(f, g)} = {(Ψ_{1}^{(f, g)}, Ψ_{2}^{(f, g)}, Ψ_{3}^{(f, g)})}^{T},

where

Ψ_{1}^{(f, g)}

,

Ψ_{2}^{(f, g)}

, and

Ψ_{3}^{(f, g)}

are the Stokes parameters. In Eq. (6), the superscript indices, g and f, indicate two arbitrary tributary signals, υ_f and υ_g, selected from the signal |ψ〉 considered in Eq. (2). Each Stokes parameter represented in Eq. (6) can be obtained from |ψ〉 as follows [21],

Ψ_{1}^{(f, g)} = | 〈 e_{f} | ψ 〉 |^{2} - | 〈 e_{g} | ψ 〉 |^{2},

Ψ_{2}^{(f, g)} = 2 Re ({〈 e_{f} | ψ 〉}^{*} 〈 e_{g} | ψ 〉),

Ψ_{3}^{(f, g)} = 2 Im ({〈 e_{f} | ψ 〉}^{*} 〈 e_{g} | ψ 〉),

being 〈e_h|, with h = f, g, the complex conjugated of |e_h〉. The vector |e_h〉 defines the basis of the higher-order Poincaré spheres and, therefore such spheres can be used to describe signals transmitted over MCF or FMF considering linear polarized (LP) modes, OAM modes or vector modes [25,26]. For more details, the reader can see in section 2.3 the calculation of the Stokes parameters for the particular case of two modes.

In order to realize the physical meaning of the Stokes parameters pointed out in Eq. (7), one considers that a given signal is made to pass through a mode filter system (to select only the signals υ_f and υ_g) followed by a polarizer. The parameter $Ψ_{1}^{(f, g)}$ describes the difference between the emerging powers obtained when the υ_f and υ_g signals are made to pass through a linear polarizer aligned at 0° and a linear polarizer aligned at 90°, respectively. It gives the preponderance of linear horizontal light over linear vertical light. The parameter $Ψ_{2}^{(f, g)}$ describes the difference between the emerging powers obtained when the signals υ_g and υ_f are made to pass through a linear polarizer aligned at an angle of 45^° and a linear polarizer aligned at an angle of −45^°, respectively; i.e., it gives the preponderance of 45^° linear polarized light over −45^° linear polarized light. The parameter $Ψ_{3}^{(f, g)}$ describes the difference between the emerging powers obtained when the signals υ_f and υ_g are made to pass through a right-circular polarizer (RCP) and a left-circular polarizer (LCP), i.e., it gives the preponderance of RCP light over LCP light.

2.1.1. Intramode case

As previously mentioned, the first subset of 3n matrices pointed out in (a) is obtained from the Pauli matrices and, therefore, we can write ${\vec{Λ}}^{(f, g)} = {(Λ_{1}^{(f, g)}, Λ_{2}^{(f, g)}, Λ_{3}^{(f, g)})}^{T}$ , with

Λ_{1}^{(f, g)} (k, l) = {\begin{array}{l} \sqrt{n} & if k = g, l = g \\ - \sqrt{n} & if k = f, l = f, \\ 0 & otherwise \end{array}

Λ_{2}^{(f, g)} (k, l) = {\begin{array}{l} \sqrt{n} & if k = f, l = g \\ \sqrt{n} & if k = g, l = j, \\ 0 & otherwise \end{array}

Λ_{3}^{(f, g)} (k, l) = {\begin{array}{l} - i \sqrt{n} & if k = f, l = g \\ i \sqrt{n} & if k = g, l = f, \\ 0 & otherwise \end{array}

where the subscript index g = f + 1, with f taking all the odd values in the range [1 : 2n − 1], i.e. in Eq. (8), the superscript indexes f and g denote the x and y polarization of a given mode. Like the Pauli spin vector in Stokes space, the vector

{\vec{Λ}}^{(f, g)}

can be used to calculate the Stokes parameters in a given higher-order Poincaré sphere.

Using Eq. (8) in Eq. (1), the relative state between the signals υ_f and υ_g (and its evolution) can be represented in a 3-dimensional higher-order Poincaré sphere with radius $\sqrt{n} (a_{g}^{2} + a_{f}^{2})$ , and Stokes parameters

Ψ_{1}^{(f, g)} = 〈 ψ | Λ_{2}^{(f, g)} | ψ 〉 = \sqrt{n} (a_{f}^{2} - a_{g}^{2}),

Ψ_{2}^{(f, g)} = 〈 ψ | Λ_{2}^{(f, g)} | ψ 〉 = 2 \sqrt{n} a_{f} a_{g} \cos δ_{f g},

Ψ_{3}^{(f, g)} = 〈 ψ | Λ_{3}^{(f, g)} | ψ 〉 = 2 \sqrt{n} a_{f} a_{g} \sin δ_{f g},

where δ_fg = ϕ_f − ϕ_g represents the phase difference between the signals υ_f and υ_g in the Jones space.

2.1.2. Intermode case

In order to fully describe the space-multiplexed signal in higher-order Poincaré spheres, the remaining matrices Λ_i must be handle to produce g_s − n vectors ${\vec{Λ}}^{(f, g)}$ . In these spheres, one consideres the remaining pairs of signals accounting for the polarization states from distinct modes. In this case, we identify two distinct sets; firstly, the subscript index f can take all the available values between 1 and 2n − 2 (f = 1, 2, 3, …2n − 2) ; meanwhile, for each f value, the subscript index g changes between f + 2 and 2n (g = f +2, …2n) . Secondly, the subscript index g = f + 1, with f taking all the even values in the range [1 : 2n − 2 ].

The matrices $Λ_{2}^{(f, g)}$ and $Λ_{3}^{(f, g)}$ are presented in the set of generalized Pauli matrices introduced in section 2, i.e. the second subset of 4n² − 4n matrices previously pointed out in (b). However, the matrix $Λ_{1}^{(f, g)}$ is not explicitly written in the generalized Pauli matrices. In that way, the third subset of n − 1 matrices pointed out in (c) must be properly decomposed in order to produce a set of linearly independent matrices in which each matrix acts only on two tributary signals, i.e. the matrices only have two non-null elements in the main diagonal. Since all these matrices are linearly independent, it is possible to extend this kind of decomposition to any dimension of the generalized Stokes space. Having properly decomposed these matrices, they can be written as

Λ_{1}^{(f, g)} (k, l) = {\begin{array}{l} \sqrt{\frac{n}{n_{l}^{2} + n_{l}}} k & if k = g, l = g \\ - \sqrt{\frac{n}{n_{l}^{2} + n_{l}}} k & if k = f, l = f, \\ 0 & otherwise \end{array}

Λ_{2}^{(f, g)} (k, l) = {\begin{array}{l} \sqrt{n} & if k = f, l = g \\ \sqrt{n} & if k = g, l = f, \\ 0 & otherwise \end{array}

Λ_{3}^{(f, g)} (k, l) = {\begin{array}{l} - i \sqrt{n} & if k = f, l = g \\ i \sqrt{n} & if k = g, l = f, \\ 0 & otherwise \end{array}

being the parameter κ a normalization constant introduced by the projection of the generalized Pauli matrix in the set of linearly independent matrices. Note that, we explain in detail this decomposition for the particular case of two modes in subsection 2.3.

Using Eq. (10) in Eq. (1), the Stokes parameters for signals υ_g and υ_f can be written as

Ψ_{1}^{(f, g)} = 〈 ψ | Λ_{1}^{(f, g)} | ψ 〉 = \sqrt{\frac{n}{n_{l}^{2} + n_{l}}} κ (a_{f}^{2} - a_{g}^{2}),

Ψ_{2}^{(f, g)} = 〈 ψ | Λ_{2}^{(f, g)} | ψ 〉 = 2 \sqrt{n} a_{f} a_{g} \cos δ_{f g},

Ψ_{3}^{(f, g)} = 〈 ψ | Λ_{3}^{(f, g)} | ψ 〉 = 2 \sqrt{n} a_{f} a_{g} \sin δ_{f g} .

In this case, the Poincaré sphere is replaced by an ellipsoid because the maximum value of the Stokes parameter

Ψ_{1}^{(f, g)}

is smaller than the other Stokes parameters

Ψ_{2}^{(f, g)}

and

Ψ_{3}^{(f, g)}

.

In Fig. 1, it is show a schematic representation of the higer-order Poincaré spheres here proposed. Using a semi-simple representation, the Jones space is represented in n spheres and g_s − n ellipsoids, or higer-order Poincaré spheres. In the n spheres, it are represented the intramode cases; whereas, in the g_s − n ellipsoids are represented the intermode cases. In each sphere/ellipsoid, it is possible represent any relative state between the two signal e_f and e_g. Note that, both signals considered are orthogonal even in the case of MCF it is assumed that the overlap integral between distinct cores is null. In order to illustrate the signal representation in higher order Poincaré spheres, in Fig. 1 are represented the diagonal basis written as,

| D 〉 = \frac{1}{\sqrt{2}} (| e_{f} 〉 + | e_{g} 〉),

| A 〉 = \frac{1}{\sqrt{2}} (| e_{f} 〉 - | e_{g} 〉),

and the circular basis written as

| R 〉 = \frac{1}{\sqrt{2}} (| e_{f} 〉 + i | e_{g} 〉),

| L 〉 = \frac{1}{\sqrt{2}} (| e_{f} 〉 - i | e_{g} 〉),

in both cases, inter and intramode. In the intermode case, the signal representation in a ellipsoid allows to connect the higher order Poincaré spheres to the generalized Stokes space representation by a properly summing the parameters

Ψ_{1}^{(f, g)}

. The norm of the Stokes vector in such ellipsoids is not constant or unitary, although the norm of the generalized Stokes vector remains constant.

Fig. 1 Schematic representation of the decomposition of the Jones space in g_s SO(3) subspaces, i.e., higher-order Poincaré spheres. The tributary signals from the same spatial channel are represented in a sphere (i.e., intramode case), whereas the tributary signals from distinct spatial channels are represented in a ellipsoid. Note that, the subindexes f and g are defined in subsection (2.1.1) and (2.1.2) for the intra and intermode cases, respectively.

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2.2. Complex-modulated signals in higher Poincaré spheres

In order to represent an arbitrary complex-modulated format, we assume a hypothetical modulation format that is confined to an unit circle in the complex plane. Although the signal needs be normalized, such condition does not restrict the modulation format in any way because all realizable signals are always bounded [8]. In that way, we assume an amplitude normalized mode-multiplexed signal,

| ϒ 〉 = \frac{1}{\sqrt{2 n}} {(1, \dots, 1, r e^{- i φ})}^{T},

where the parameters r, with 0 < r ≤ 1 and ϕ, with 0 ≤ φ < 2π, represent the amplitude and phase of the signal, respectively. In Eq. (14), the hypothetical modulated signal is considered in the last entry, while the remaining signals are assumed to have constant phase and amplitude. By replacing Eq. (14) in Eq. (11), the set of parametric equations can be obtained

Ψ_{1}^{(f, g)} = \frac{κ}{2} \sqrt{\frac{1}{n (n_{l}^{2} + n_{l})}} (1 - r^{2}),

Ψ_{2}^{(f, g)} = \frac{r}{\sqrt{n}} \cos φ,

Ψ_{3}^{(f, g)} = \frac{r}{\sqrt{2}} \sin φ .

In Fig. 2, it is shown the paraboloidal surface described by Eq. (15) in a given higher-order Poincaré sphere for the region Ψ₁ > 0. Note that, a given point in the higher-order Poincaré sphere represents the relative state between two signals. When a signal is assumed with constant amplitude and phase, it allows to calculate the boundary of such object in the higher-order Poincaré sphere, without loss of generality we use this assumption to visualize the lens-like objects. For the set ${\vec{Λ}}^{(g, 2 n)}$ , with g ≠ 2n − 1, the other paraboloidal surface in the region Ψ₁ < 0 can be obtained in a similar way replacing

| ϒ 〉 = \frac{1}{\sqrt{2 n}} {(1, \dots, r e^{- i φ}, \dots, 1)}^{T}

in Eq. (11). This leads to the equation of the second surface and hence to the symmetrical lens-like object inscribed in higher-order Poincaré spheres. Both surfaces are quite similar with the exception that Ψ₁ and Ψ₃ have opposite signs. The flips in Ψ₁ invert the paraboloidal surface resulting in the complete lens-like object and therefore clearly defining a symmetry plane. Likewise, using Eq. (14) in Eq. (9) a similar lens-like objects for the remaining sets

{\vec{Λ}}^{(f, g)}

can be written as,

Ψ_{1}^{(f, g)} = \frac{1}{2 \sqrt{n}} (1 - r^{2}),

Ψ_{2}^{(f, g)} = \frac{r}{\sqrt{n}} \cos φ,

Ψ_{3}^{(f, g)} = \frac{r}{\sqrt{n}} \sin φ .

Note that, such lens-like objects are independent of the considered tributary signals and respective modulation formats. Since the tributary signal have the same baud rate, the algorithm proposed is able to demultiplexes tributaries signals carrying distinct modulation formats because the available values for the parameters

Ψ_{1}^{(f, g)}

in Eq. (9) and Eq. (11) show a symmetric distribution relatively to

Ψ_{1}^{(f, g)} = 0

. In addition, the residual frequency offset and the phase noise only affect the Stokes parameters

Ψ_{2}^{(f, g)}

and

Ψ_{3}^{(f, g)}

. The parameter

Ψ_{1}^{(f, g)}

remains unaffected because they only depends on the amplitude of both signals considered in a given higher-order Poincaré sphere, like in the single mode case [8,9,18]. MIMO algorithms based on the signal representation based on higher-order Poincaré spheres are therefore robust against the frequency offset and phase noise.

Fig. 2 Boundaries of the signals in a higher-order Poincaré sphere produced by the set of matrices Λ⁽^g,²ⁿ⁾, with g denoting an arbitrary tributary signal comprised in (11). Insets (a) and (b) show the unit circle which contains an arbitrary modulation format.

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2.3. Representation of a two-mode multiplexed signal in higher Poincaré spheres

In order to illustrate the signal representation in higher-order Poincaré spheres, and without loss of generality, we consider two distinct modes, each one supporting two orthogonal polarizations. For this particular case, the eigenvectors can be written as

| e_{1} 〉 = {(1, 0, 0, 0)}^{T},

| e_{2} 〉 = {(0, 1, 0, 0)}^{T},

| e_{3} 〉 = {(0, 0, 1, 0)}^{T},

| e_{4} 〉 = {(0, 0, 0, 1)}^{T},

where |e₁〉 and |e₂〉 represent the x and y polarizations of the first mode, respectively. In the same manner, |e₃〉 and |e₄〉 represent the x and y polarizations of the second mode, respectively. In this case, the set of the generalized Pauli matrices is comprised by 15 matrices. The two first subset of matrices,

{\vec{Λ}}^{(1, 2)}

and

{\vec{Λ}}^{(3, 4)}

, can be arranged by the examination of the set of generalized Pauli matrices, as previously pointed out in (a). For the remaining subsets

{\vec{Λ}}^{(f, g)}

only the matrices

Λ_{2}^{(f, g)}

and

Λ_{3}^{(f, g)}

are presented in this set, see the second subset of matrices pointed out in (b). Therefore, the last subset of matrices pointed out in (c) must be decomposed in four matrices,

\begin{array}{l} (\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}) = \frac{1}{2} (\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}) + \\ \frac{1}{2} (\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array}) + \frac{1}{2} (\begin{array}{l} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}) + \frac{1}{2} (\begin{array}{l} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array}), \end{array}

where the factor 1/2 results from the projection of the matrix represented on the left-hand side of Eq. (19) into the considered set of linearly independent matrices. The matrices on the right-hand side only take into account the interaction between two given signals and therefore can be used as

Λ_{1}^{(f, g)}

. For instance, the subset of matrices

{\vec{Λ}}^{(1, 3)}

can be obtained by the examination of the remaining elements of the generalized Pauli matrices and taking into account Eq. (19),

Λ_{1}^{(1, 3)} = \frac{1}{2} (\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}),

Λ_{2}^{(1, 3)} = \sqrt{2} (\begin{array}{l} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}),

Λ_{3}^{(1, 3)} = \sqrt{2} (\begin{array}{l} 0 & 0 & - i & 0 \\ 0 & 0 & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}) .

The remaining three subsets

{\vec{Λ}}^{(f, g)}

can be arranged in the same way considering the following pairs of signals: (1, 4), (2, 3) and (2, 4). In the four signals we assume a pseudo-random binary sequence of 2¹⁸ bits mapped into the I and Q components of a quadrature phase-shift keying (QPSK) signal with an optical signal-to-noise ratio (SNR) of 17 dB. Using Eq. (9) and Eq. (11), the Stokes parameters are calculated and represented in higher-order Poincaré spheres, see Fig. 3. In each sphere, it is clear that the symbols are symmetrically distributed around the best fit plane which coincides with

Ψ_{1}^{(f, g)} = 0

.

Fig. 3 Four tributary QPSK signals, i.e. two modes with two polarization-multiplexed tributaries, are represented in the higher-order Poincaré spheres with the symmetric plane (i.e. the best fit plane) to the samples. The sub-captions indicate tributary signals, (f, g), represented in each subspace.

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3. Space-demultiplexing algorithm

3.1. Channel modeling

In order to emulate the signal propagation in a SDM link, we use the multi-section model [27]. Both approaches, FMF and MCF, can be properly described by this model considering a section with a length slightly longer than the correlation length and the dispersive properties for each spatial channel. The channel impulse response can be written as

| ψ_{o u t} 〉 = M_{tot} (ω) | ψ_{i n} 〉,

where |ψ_out〉 and |ψ_in〉 represent the output and the input signals, respectively. Henceforth, we use the multi-section model to describe the signal propagation in a FMF link. Thereby, the channel impulse response is given by

M_{tot} (ω) = M_{MD} (ω) e^{\frac{i}{2} ω^{2} {\bar{β}}_{2} L},

being L the total length of the link,

{\bar{β}}_{2}

the mode-averaged chromatic dispersion per unit length, and M_MD(ω) represents the mode-dependent effects; such as mode-differential loss (MDL), modal dispersion, and modal crosstalk [27]. This link is divided in n_s spans,

M_{MD} (ω) = \prod_{k = 1}^{n_{s}} M_{MD}^{k} (ω),

where

M_{MD}^{k}

represents the mode-dependent effects in the k span. Assuming that each span can be subdivided into n_step sections and that the MDL can be compensated by inline amplifiers, Eq. (23) can be rewritten as

M_{MD}^{k} (Ω) = diag (e^{\frac{g_{1}^{k}}{2}} \dots e^{\frac{g_{2 n}^{k}}{2}}) \prod_{l = 1}^{n_{step}} V_{k l} Θ (Ω) U_{k l}^{H},

with the superscript H denoting the Hermitian conjugate operator. The V_kl and U_kl matrices denote the frequency-independent random unitary matrices (which represent the crosstalk caused by random mode coupling) and

g_{i}^{k}

, with i = 1, 2 … 2n, represents the gain/loss for each tributary signal. Matrix Θ(Ω) accounts for the modal dispersion in a single section,

Θ (Ω) = diag (e^{i ω τ_{1}} \dots e^{i ω τ_{i}} \dots e^{i ω τ_{2 n}}),

where τ_i is the uncoupled modal group delay for the i signal.

3.2. Algorithm description

The idea behind the space-demultiplexing based on higher-order Poincaré spheres proposed in this paper lies in the sequential signal equalization between pairs of tributary signals. As previous pointed out in section 2.1, the crosstalk operator can be expanded in g_s operators in which each operator take into account the crosstalk between a given pair of tributary signals. Therefore, the channel matrix can be written as

M_{tot} = \prod_{g = 1}^{2 n - 1} \prod_{f = g + 1}^{2 n} ℳ^{(f, g)},

where

ℳ^{(f, g)}

represents the crosstalk between an arbitrary pair of tributary signals. Therefore, we need to properly apply g_s sequential space-demultiplexing steps to equalize the crosstalk among the g_s pairs of tributary signals. To assess the efficiency of the algorithm proposed, we only considered the crosstalk induced by the SDM link. In this way, we are assuming a previous compensation of the remaining linear impairments, e.g. chromatic dispersion, mode differential loss, and delay. This algorithm tends to be insensitivity to frequency offset and phase noise and, hence, they are not considered. The dispersive effects (e.g., chromatic dispersion and differential mode group delay) are not considered because it is assumed that they can be compensated before space-demultiplexing. We also assume that the tributary signals are synchronized before the space-demultiplexing. In MCF based transmission systems, the deviation from the ideal sampling instant are quite similar to the ones observed in standard communications systems based on SMF. Therefore, in such transmission systems the clock recovery and downsampling can be used after the space demultiplexing based on higher-order Poincaré spheres. Conversely, in MDM transmission systems the differential mode group delay can desynchronize the tributary signals. Nevertheless, these signals can be again synchronized by using a subsystem which allows to compensate the differential mode group delay; like in [17] the PMD is compensate before the polarization demultiplexing.

First of all, the Stokes parameters, ${\vec{Ψ}}^{(f, g)}$ , are calculated for the sampled signal. These samples are represented in the higher-order Poincaré spheres and the best fit plane (and their normal) is computed through a linear least square regression by minimizing the sum of squared residuals, i.e. the squared of the offsets among the points and the plane. The plane equation is written as

a Ψ_{1}^{(f, g)} + b Ψ_{2}^{(f, g)} + c Ψ_{3}^{(f, g)} = 0,

where

{\vec{n}}_{p} = {(a, b, c)}^{T}

represents its normal. Like in the single-mode case, this normal can be rotated in the 3-dimensional subspace to match (1, 0, 0)^T and, in that way, to reverse the crosstalk between the pair of signals considered. Such rotation is applied to the samples in the Jones space,

| φ_{o u t} 〉 = F^{(f, g)} | φ_{i n} 〉,

where |φ_out〉 and |φ_in〉 represent the signal after and before the demultiplexing stage, respectively. The unit matrix F⁽^f,^g⁾

F^{(f, g)} (k, l) = {\begin{array}{l} \cos (p) e^{i \frac{q}{2}} & if k = g, l = g \\ \sin (p) e^{- i \frac{q}{2}} & if k = g, l = f \\ - \sin (p) e^{i \frac{q}{2}} & if k = f, l = g \\ \cos (p) e^{- i \frac{q}{2}} & if k = f, l = f \\ 1 & if k = l and k, l \neq f, g \\ 0 & otherwise \end{array},

allows to reverse the crosstalk between the tributary signal υ_g and υ_f. The parameters p = arctan

(a, \sqrt{b^{2} + c^{2}})

and q = arctan (b, c) are the polar and azimuthal angles between the normal to the best fit plane and (1, 0, 0)^T. In Fig. 4(a), it is schematically represented this procedure, henceforth, designated by “space-demultiplexing step”. Afterwards, the output samples are launched in the next space-demultiplexing step and the process is repeated g_s times. At each step, we take into account the two possible spatial orientation of the normal, i.e.

{\vec{n}}_{p}

and

- {\vec{n}}_{p}

Note that, both normals allow the space-demultiplexing. However, an unsuitable direction of the normal leads to a signal commutation between the signals υ_g and υ_f. Such signal commutation can hinder the fully space-demultiplexing because two given tributary signals are demultiplexed two times, while the interaction with other tributary signal is not taked into account. Furthermore, distinct sections of fiber do not commute and therefore the space-demultiplexing performance depends on the sequence of the space-demultiplexing steps. The smaller residual the larger symmetry, using this heuristic a suitable sequence to apply the g_s space-demultiplexing steps can be found by minimizing the sum of the absolute value of the residuals. Figure 4(b) shows a schematic representation of all the steps comprising the algorithm. Initially, all possible sequences of space-demultiplexing steps are computed and their residuals are evaluated. For each sequence of space-demultiplexing steps, the parameter ζ is calculated considering the sum of the absolute value of the residuals. This parameter can be used to quantify the symmetry of the samples relatively to the best fit plane for a given sequence of space-demultiplexing steps. For a particular sequence of g_s steps, lower values of ζ means larger symmetries of the samples to best fit plane. Therefore, the sequence of space-demultiplexing steps with lower value of ζ is the chosen. Note that, to use the residuals minimization as a reliable criterion all the residuals must be equivalent and therefore the residuals provided by Eq. (10a) must be multiplied by

\sqrt{n_{l}^{2} + n_{l}} / κ

.

Fig. 4 a) Schematic of the proposed space-demultiplexing step algorithm. The samples are represented in a higher-order Poincaré sphere in order to estimate the normal to the best fit plane. Then, the matrix F⁽^f,g⁾ is calculated and applied to the signal. b) Schematic of the proposed space-demultiplexing algorithm. The received samples are sequentially launched in g_s space-demultiplexing steps. Then, the suitable sequence of these steps is chosen by the minimization the sum of the absolute value of the residuals, i.e. the parameter ζ.

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3.3. Mode-demultiplexing of a two-mode multiplexed signal

In this subsection, we investigate the performance of the proposed demultiplexing algorithm by numerically calculating of the imposed penalties on demultiplexing of different complex-modulation formats (PM-QPSK, PM-16QAM, and PM-64QAM). For the SDM transmission system, and without loss of generality, we are going to consider that all linear and nonlinear transmission impairments can be fully compensated. Therefore, the considered SDM signals will be only affected by optical noise and by the linear crosstalk.

The following method description considers the QPSK signal for a question of simplicity, but it can be also extended to the other higher-order modulation formats. Figure 5 shows the representation of the received QPSK signal in higher-order Poincaré spheres, i.e. the signal represented in Fig. 3 after propagated trough the SDM link discussed above. In order to demultiplex the four QPSK tributaries (two modes with two polarization-multiplexed signals) the proposed algorithm requires six steps. In each step, we must provide the signal and the vector ${\vec{Λ}}^{(f, g)}$ , with (f, g) = (1, 2), (1, 3),(1, 4), (2, 3), (2, 4) and (3, 4) Then, the Stokes parameters are calculated signal are represented in the(respective higher-order Poincaré sphere. By means of a linear regression, the best fit plane to those samples is computed and the residual provided by the linear regression is saved. The normal to the best fit plane is used to calculate the matrix F⁽^f,g⁾, according to Eq. (30), which is therefore applied to the signal in the Jones space in order to demultiplex both tributary signals υ_g and υ_f, as shown in Fig. 4(a). After that, the four tributary signals are launched in the next space-demultiplexing step where the above process is repeated. When all the sequences of space-demultiplexing steps are applied to the signal, we choose the sequence of space-demultiplexing steps with lower ζ, as shown in Fig. 4(b). By applying such sequence of space-demultiplexing steps to the QPSK signal under analysis, we obtained the results represented in Fig. 6. The output signal shown in Fig. 6 compares well with the input signal shown in Fig. 3, which confirms the successful demultiplication of the signal. In Fig. 7, it is shown the values of the components for the first line of matrix M as function of the number of sample considered in the calculations of M_tot. Results show that the algorithm converges to the expected results, dash line, after only 100 samples.

Fig. 5 Representation in the higher-order Poincaré spheres of the QPSK signal, shown in Fig. 3, after transmission through a FMF. In each figure, the deeper red color disks represent the higher order Poincaré sphere.

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Fig. 6 Representation in the higher-order Poincaré spheres of the four tributary QPSK signals after mode-demultiplexing. These signals are obtained from the signal represented in Fig. 5 and the output constellations are well compared with input constellations present in Fig. 3.

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Fig. 7 Convergence of the coefficients of the demultiplexing matrix as function of the number of samples for a QPSK signal. The dash line repressents the coefficients of the channel matrix, M_tot.

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The performance of this algorithm can be assessed in terms of the penalty induced in the SNR of the demultiplexed signal,

Δ = {SNR}_{i n} - {SNR}_{o u t},

with Δ denoting the penalty added by the algorithm, and SNR_out and SNR_in being the SNR for the demultiplexed signal and for the signal launched in the link, respectively. The SNR_out is calculated considering the error vector magnitude (EVM) of the tributary signal with higher EVM. Although the SNR penalty has been calculated by means of EVM, this analysis can be also performed by analyzing the bit error rate (BER) since EVM and BER are correlated [28]. In the calculations of the SNR, it is assumed a perfect phase and frequency recovery [28],

S N R = \frac{1}{E V M^{2}} .

In Fig. 8(a), it is shown the SNR penalty as function of the number of samples considered in the calculations of the inverse channel matrix for a QPSK, a 16-quadrature amplitude modulation (QAM) and a 64-QAM signal. For the QPSK signal, the algorithm needs ~ 500 samples to achieve a penalty as lower as Δ = 0.1 dB. In Figs. 8(b) and 8(c) is shown the constellation diagrams for the four tributary QPSK signals before and after demultiplexing, respectively. For the 16-QAM signal, the algorithm needs ∼ 12000 samples to achieve a penalty as lower as ∆ = 0.1 dB. The constellation diagram for the four tributary 16-QAM signal before and after demultiplexing are shown in the Fig. 8(d) and 8(e), respectively. Regarding the 64-QAM signal, the algorithm needs ∼ 20000 samples to achieve a penalty of Δ = 0.2 dB. Figure 8(d) and 8(e) show the constellation diagram for the four tributary 64-QAM signal before and after demultiplexing, respectively. These results show that higher-order modulation formats require more samples to accurately calculate the best fit plane and, therefore, the space-demultiplexing algorithm also requires a larger number of samples to find a suitable sequence of space-demultiplexing steps. These results are in good agreement with discussion reported in [5, 7], where the convergence analysis of Stokes space based PolDemux in single-mode fibers shown a slower convergence ratio for the higher order constellations.

Fig. 8 (a) SNR penalty induced by the space-demultiplexing algorithm as function of the number of samples considered in the calculations of the inverse channel matrix. Inset show in log scale the SNR penalty as function of the number of samples. The QPSK signals are assumed with an optical SNR of 17 dB. The 16-QAM and the 64-QAM signals are assumed with an optical SNR of 23 and 30 dB, respectively. Figure 8(b), (d) and (f) show the constellation diagram before demultiplexing for the QPSK, the 16-QAM and the 64-QAM, respectively. Figure 8(c), (e) and (g) show the constellation diagram after demultiplexing for the QPSK, the 16-QAM and the 64-QAM, respectively. Note that, the number of samples used to calculate the inverse channel matrix are pointed out in Fig. 8(a).

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Since both tributary signals remain polarized this algorithm is quite robust to additive Gaussian noise because the noise increase the dispersion of the points without break the symmetry proprieties of complex modulated signal in the higher-order Poincaré spheres. Nevertheless, the performance of the space-demultiplexing algorithm can be constrained by the optical SNR, namely in terms of convergence speed. Lower optical SNR in the received signal may require a large number of samples in the demultiplexing algorithm to calculate a suitable sequence of space-demultiplexing stages. In table 1, it is shown the SNR penalty as function of the SNR of the received signal for a QPSK signal. We assume 10000 samples in the calculation of the inverse channel matrix. Note that, even for lower SNR the algorithm proposed is able to successfully demultiplex the received signal.

Table 1. Penalty induced by the space-demultiplexing algorithm as function of the SNR for a QPSK signal. We assume 10000 samples in the calculation of the inverse channel matrix. The results are presented in decibels.

View Table

4. Conclusions

The signal representation in higher-order Poincaré spheres was obtained by properly handling the generators of the generalized Stokes space. Supported by this representation, we propose a modulation format agnostic space-demultiplexing algorithm that does not require training sequences. This algorithm can be used on transmission systems based on both MFC or FMF. We prove that an arbitrary complex-modulated signal can be described by a set of parametric equations, i.e. a lens-like object, in a higher-order Poincaré sphere. In order to demultiplex a given pair of tributaries, the symmetry plane of these lens-like objects, i.e. the best fit plane, can be realign with the initial plane. The inverse channel matrix is calculated through a progressive space-demultiplexing of all pairs of signals in which the sequence of space-demultiplexing steps is found by minimizing the sum of the absolute value of the residuals. Although the calculation of all the possible sequences can be parallelizable, further work is needed to decrease the computationally complexity of this algorithm. In a scenario where linear impairments are fully compensated, results show a negligible SNR penalty for the QPSK and 16-QAM signals, and SNR penalty as lower as 0.5 dB for the 64-QAM signal.

Funding

This work is funded by FCT/MEC through national funds and when applicable co-funded by FEDER - PT2020 partnership agreement under the project UID/EEA/50008/2013 (action SoftTransceiver and OPTICAL-5G), Ph.D. grant SFRH/BD/102631/2014 and the postdoctoral grant SFRH/BPD/77286/2011.

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Space-demultiplexing based on higher-order Poincaré spheres

Abstract

1. Introduction

2. Higher-order Poincaré spheres

2.1. Definition of higher-order Poincaré spheres

2.1.1. Intramode case

2.1.2. Intermode case

2.2. Complex-modulated signals in higher Poincaré spheres

2.3. Representation of a two-mode multiplexed signal in higher Poincaré spheres

3. Space-demultiplexing algorithm

3.1. Channel modeling

3.2. Algorithm description

3.3. Mode-demultiplexing of a two-mode multiplexed signal

4. Conclusions

Funding

References and links

Cited By

Figures (8)

Tables (1)

Equations (52)

Optics Express