Stokes-space analysis of modal dispersion in fibers with multiple mode transmission

Cristian Antonelli; Antonio Mecozzi; Mark Shtaif; Peter J. Winzer

doi:10.1364/OE.20.011718

1. Introduction

With the fiber-optic communications industry facing an imminent capacity crunch [1, 2], the search for new, scalable technologies has become unavoidable. One of the most promising approaches for preventing the predicted crisis is the use of spatially multiplexed transmission [3] in a combination of multi-core (MCF) [4] and multi-mode (MMF) fibers [5, 6]. Indeed, a notable fraction of regular and post-deadline papers presented in the last two largest conferences on optical communications (the Optical Fiber Communications conference (OFC), Los Angeles 2011, and the European Conference on Optical Communications (ECOC), Geneva 2011) were devoted to spatially multiplexed optical transmission.

Conceptually, provided that all fiber modes are selectively addressed at transmitter and receiver, spatially multiplexed fiber-optic transmission is a direct extension of polarization multiplexed transmission in single mode fibers that is being used today. To describe a fiber with N spatial modes, the 2-dimensional polarization vector needs to be replaced by a 2N dimensional vector whose elements represent the excitation of the various modes (with the factor of 2 accounting for the fact that each mode supports two polarizations). Correspondingly, the 2 × 2 transfer matrix that describes linear transmission in single mode fibers needs to be replaced by a matrix whose dimension is 2N × 2N. This situation calls for the extension of the concepts that are at the heart of polarization studies. Thus, rather than referring to the phenomena of polarization mode dispersion (PMD) and polarization dependent loss (PDL), which describe the relevant consequence of polarization coupling, one now has to deal with the more general concepts of modal dispersion (MD) and mode dependent loss (MDL). Relevant work on this topic has already been reported, where the concept of principal states of polarizations was generalized to the multi-mode case [7] and where the statistics of the corresponding group delays was derived [8]. Important observations on the dynamics of MDL were also reported in [9]. Nonetheless, the understanding of the physical nature of these phenomena can greatly benefit from the generalization of the analytical tools, which were invaluable in polarization studies.

Polarization vectors, which have two complex-number components, reside in a so-called Jones space. This Jones space is isomorphic to a so-called Stokes space which consists of 3 dimensional real-valued column vectors [10]. Thus, every state of polarization can be uniquely represented either as a Jones vector, or as a vector in Stokes space. In addition, unitary operations in Jones space can be represented by a Stokes vector, whose direction is the axis of rotation and whose modulus equals the rotation angle. For this reason, the effect of birefringence (which is a unitary operation in Jones space) is customarily represented by a 3-dimensional vector β⃗ in Stokes space, known as the birefringence vector. Specifying the birefringence of a fiber therefore comes down to specifying the birefringence vector β⃗(z) that represents it at every point z along the fiber. Similarly, the effect of PMD is customarily represented by a vector τ⃗in Stokes space, known as the PMD vector, capturing birefringence-induced waveform distortion. The Stokes representations of the slowest and fastest states of polarization (known as the principal states of polarization) coincide with ±τ̂ (where τ̂ = τ⃗/τ is a unit vector), and the differential group delay (DGD) between the slowest and fastest polarization components is equal to τ = |τ⃗|. Even non-unitary phenomena such as PDL are typically represented in terms of vectors in Stokes space [11]. It is thus natural for the analysis of spatially multiplexed transmission to seek an extension of the above concepts to the multi-mode case.

In this paper, we introduce an extended D = 4N² – 1 dimensional Stokes space, that allows the representation of multi-mode 2N-dimensional state-vectors (which are the extension of the 2-dimensional Jones vectors in the case of polarization). While generalized Stokes representations have been considered previously in other contexts [12, 13], the properties of the generalized Stokes space and the relation to unitary dynamics (which is of particular importance in our case) have not been fully formulated previously, to the best of our knowledge. We show that while the extended Stokes-space lacks some of the properties of the usual Stokes-space describing polarization, many properties can be generalized in a way that sheds light on the physics of multi-mode propagation. In particular, focusing on the case where MDL is negligible, we show that the local birefringence vector β⃗ and the PMD vector τ⃗ can be extended to the D-dimensional space so as to capture the effects of local mode coupling and MD. As in the case of PMD, the MD vector τ⃗ determines the temporal spread of propagating pulses, although the relation between its magnitude and the differential mode delays is somewhat less direct in the multi-mode case, as we shall see. In order to demonstrate the use of the proposed formalism we consider the case of quasi-degenerate modes, namely transmission over multiple spatial modes with varying degrees of coupling, but with small deterministic difference between the various group delays, such that the delay spread observed at the output is always dominated by the random coupling caused by perturbations. In addition, our theory applies to the case of arbitrary deterministic group delay difference, as long as the coupling between the modes is strong. The statistics of mode dispersion in this scenario are identical to that of strongly coupled degenerate modes. This framework includes real-world scenarios in which groups of modes propagating at different group velocities are processed separately [14]. For example, this corresponds to the case in which the four degenerate LP₁₁ modes of a step-index fiber are processed separately from its two LP₀₁ modes. Larger groups of degenerate modes can be encountered in situations where multiple MMFs are combined in a MCF structure. Our analysis does not rely on a specific choice of the set of modes that are used to describe signal propagation in the optical fiber and it is valid as long as the quasi-degeneracy conditions are satisfied. For example, in the case of a single-core multi-mode fiber one could choose to work with either the exact (hybrid) fiber modes, or with the linearly polarized mode approximation, with no consequence to the validity of the results.

The source of MD within a group of degenerate modes is random fiber imperfection and environmental perturbation that remove the degeneracy and cause group-delay spread. While MD can in principle be compensated for using digital signal processing (DSP) [15], the requirements from the DSP hardware and algorithm, as well as the overall system performance strongly depend on the statistical properties of MD, which we extract in this paper. In what follows we will use the acronym MMF in order to refer collectively to all structures in which light can propagate in multiple spatial modes, whether by means of a single multi-mode waveguide, or through multiple cores.

2. Main results

Prior to delving into the analytical details of the proposed model, we devote the present section to a summary of the main results.

2.1. Generalized Stokes space and modal dispersion vector

An important outcome of this work is the introduction of a generalized D-dimensional Stokes space (with D = 4N² – 1) and the definition of a D-dimensional MD vector τ⃗, as derived in Section 3. Knowledge of τ⃗ allows the extraction of the principal modes of the system as well as their various group delays, although the relations between those quantities and τ⃗ are less transparent than they are in the case of PMD. As shown in Section 3.2, the quantity τ², which is the square modulus of τ⃗, is proportional to the sum of the squares of the 2N individual group delays t_i,

τ^{2} = 2 N \sum_{i = 1}^{2 N} t_{i}^{2} .

The individual group delays are defined with the mode averaged delay set to 0, ∑_j t_j = 0. The MD vector τ⃗ evolves as a Gaussian vector, such that its modulus is chi distributed (its square modulus obeys the chi-square distribution). If no spatial mode coupling exists (while still allowing for polarization coupling within each spatial mode), only 3N of the components of τ⃗ are different from zero, and the distribution of the modulus τ is chi distributed with 3N degrees of freedom. In the other extreme, i.e. in the presence of significant random coupling among all N spatial modes (as well as among the 2 polarizations within each spatial mode), the number of degrees of freedom is 4N² – 1 implying that the orientation of τ⃗ is homogenously distributed in the entire generalized Stokes space, and the chi distribution has 4N² – 1 degrees of freedom. Note that the chi distribution is a special case of the Nakagami distribution, which is famous in the context of wireless communications [16]. Interestingly, in the presence of partial coupling, the distribution of τ can be well approximated as a chi distribution with an effective number of degrees of freedom D_eff, ranging between 3N and 4N² – 1. This transition is illustrated in Fig. 1, where we plot the probability density function (PDF) of τ for different coupling regimes for the case N = 2. In (a) the spatial modes are uncoupled (D_eff = 3N) and in (d) there is full coupling (D_eff = 4N² – 1); plots (b) and (c) represent different degrees of partial coupling. The dots are the results of Monte Carlo simulations and the solid curves correspond to chi probability density functions with D_eff degrees of freedom. The details of the computation are discussed in Section 4.

Fig. 1 Probability density function of τ for N = 2, normalized to its root-mean square value. The coupling magnitude increases from no coupling in (a) to full coupling in (d). A similar behavior has been verified for a number of larger values of N.

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2.2. Maximum modal delay spread

A quantity of particular relevance in the analyzed scenario is the largest differential group delay, defined as the difference between the largest and the smallest group delays, T = max_k{t_k} – min_k{t_k}, as discussed in Section 3.2. This parameter defines the maximum spread of the propagating waveforms and is therefore related to the required complexity of the digital signal processing hardware (e.g. the number of taps of a finite impulse response equalizer). While the derivation of the distribution of T in the general case seems to be a very difficult task [8, 9], the chi distribution turns out to provide an excellent approximation, as illustrated in Fig. 2 for the case of full coupling. The parameters 〈T²〉 and K_N characterizing the distribution were obtained numerically and are plotted in the top panel of Fig. 2 as a function of the number of spatial modes N. The term 〈T²〉 is the mean square value of T, whereas K_N is what we refer to as the shape parameter of the chi-distribution used to fit the PDF of T [17]. The dependence of 〈T²〉 and K_N on N is very well approximated by the following heuristic expressions

K_{N} ≃ 3 + ⌈ 10.39 \times {(N - 1)}^{1.36} ⌉

〈 T^{2} 〉 ≃ \frac{{(N - 1)}^{2} + 24.7 \times (N - 1) + 16.14}{0.2532 \times {(N - 1)}^{2} + 7.401 \times (N - 1) + 16.14} 〈 τ^{2} 〉 / N^{2},

where ⌈x⌉ denotes the smallest integer greater than x. The approximate curves are not displayed in Fig. 2, as they are practically indistinguishable from the actual curves in the plotted scale. The curves in the bottom panels illustrate the accuracy of this approximation on a logarithmic scale for three values of N. The plots show that, as the number of spatial modes increases, the chi distribution becomes more and more symmetric around the root-mean square value of T, asymptotically approaching the Gaussian distribution. The analytic expression provided in [8] for the case of 2N = 3 (which unfortunately does not correspond to a practical multi-mode scenario) also agrees very closely with the chi approximation, the differences between the two curves being practically unnoticeable.

Fig. 2 Top panel. The solid line represents the shape parameter K_N of the chi PDF that provides the best approximation of the distribution of the largest DGD T as a function of N; the dashed line shows the mean square value of T normalized to 〈τ²〉/N². Bottom panels. Probability density function of T for (A) N = 3, (B) N = 50 and (C) N = 100. Dots are the results of Monte-Carlo simulations, solid lines are the plot of chi PDFs with parameters taken from the top panel.

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2.3. Autocorrelation functions

The spectral behavior of MD is conveniently quantified in terms of the autocorrelation functions of the MD-vector and of its modulus. The width of these functions is a measure of the bandwidth across which the distortion affecting different spectral components of the transmitted optical field are statistically correlated, and it is also defined as the bandwidth across which the first-order approximation [10] applies to the interpretation of MD effects. In the regime of strong coupling we were able to derive analytic expressions for the frequency autocorrelation function of the MD vector τ⃗,

R_{\vec{τ}} (ω) = 〈 \vec{τ} (Ω + ω) \cdot \vec{τ} (Ω) 〉 = \frac{D}{ω^{2}} [1 - exp (- \frac{ω^{2} 〈 τ^{2} 〉}{D})],

as well as that of its square modulus τ²,

R_{τ^{2}} (ω) = 〈 τ^{2} (Ω + ω) τ^{2} (Ω) 〉 = {〈 τ^{2} 〉}^{2} + \frac{4 〈 τ^{2} 〉}{ω^{2}} - \frac{4 D}{ω^{4}} [1 - exp (- \frac{ω^{2} 〈 τ^{2} 〉}{D})] .

These expressions generalize the expressions obtained in the analysis of PMD [18, 19] to which they reduce in the case of N = 1, that is D = 3. The ACF of τ⃗, Eq. (4), normalized to its peak value, is plotted versus ω(〈τ²〉/D)^1/2 by solid lines in the top panels of Fig. 3 for N = 2, N = 3 and N = 4. The stars are the results of Monte Carlo simulations that we performed to test the analytical result. The corresponding ACF of τ², Eq. (5), after subtraction of its asymptotic value for ω → ∞ and normalization to its peak value, is plotted by solid lines in the bottom panels of Fig. 3, where the stars are again the results of Monte Carlo simulations. We numerically verified that the autocorrelation function of τ, R_τ(ω) = 〈τ(Ω + ω)τ(Ω)〉, whose analytic expression is not available, is approximated very well by Eq. (5), provided that their asymptotic values are removed and that they are normalized to their peak values. More remarkably, we verified that Eq. (5), re-scaled and modified by replacing D with K_N, gives an excellent approximation of the autocorrelation function of the largest differential group delay. The ACF of T is plotted in the bottom panels of Fig. 3 by circles and the modified Eq. (5) by dot-dashed lines: although the modification of the re-scaled ACF of τ² is practically unnoticeable for small values of N, an excellent match between simulations and analytical expressions characterizes all cases. The bandwidth of the largest differential group delay B_T can thus be very well approximated as the 3dB width of the modified ACF (5)

B_{T} ≃ 3.2 \sqrt{K_{N} / 〈 τ^{2} 〉}

, which, for a moderate, yet practically important, number of spatial modes (∼ N < 5), is only negligibly different from

B_{T} ≃ 3.2 \sqrt{D / 〈 τ^{2} 〉}

.

Fig. 3 Top panels: normalized autocorrelation function of τ⃗ according to theory (solid curve) and simulations (stars). Bottom panels: the normalized autocorrelation function of τ² according to theory (solid curve) and simulations (stars) as well as the autocorrelation function of T (circles). Dot-dashed curves represent the re-scaled ACF of τ² Eq. (5), modified by replacing D with K_N as from Fig. 2.

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Analytical expressions corresponding to Eq. (4) and (5) in the regime of partial coupling are not available. However, we found through numerical simulations that the intriguing result illustrated in Fig. 1 (i.e., the good match to a chi distribution independent of D_eff) extends somewhat to the MD correlation properties. Indeed, the shape of the autocorrelation function of τ⃗ and τ² is not affected by the coupling magnitude, whereas its bandwidth increases as the coupling magnitude increases and scales, yet only approximately, with $\sqrt{D_{eff}}$ .

2.4. Non-degenerate modes

Figures 1(d), 2 and 3 were plotted for the case of N degenerate fully coupled modes with no MDL. This situation differs somewhat from reality, where there are groups of degenerate modes that are characterized by deterministically different wave-numbers and hence they couple to each other significantly less as a result of perturbations. Using the characteristic example of the coupling between the LP₀₁ and the LP₁₁ groups of modes in step-index fibers [20], we show that the results presented above remain practically unchanged. To illustrate this point, we show in Fig. 4(a) the PDF of the length of the D_LP₁₁ = 15 dimensional MD vector τ⃗_LP₁₁ that represents the four LP₁₁ modes and the PDF of the length of the D_LP₀₁ = 3 dimensional MD vector τ⃗_LP₀₁ representing the two polarization modes of LP₀₁. In Figs. 4(b) and 4(c) we show the corresponding MD vector ACFs. In order to plausibly emulate the situation in a real fiber [20], we adjusted the magnitude of the difference between the wave-numbers corresponding to the two groups of modes to the point where the overall output power in the LP₁₁ modes was about 10dB lower than the output power in the LP₀₁ mode, given that only the LP₀₁ mode was excited. As is evident from the figures, the numerically obtained distribution of τ_LP₁₁ and τ_LP₀₁ (symbols) is still very well fitted by chi distributions of D_LP₁₁ = 15 and D_LP₀₁ = 3 degrees of freedom respectively (solid lines). At the same time the numerical ACF (symbols) still matches the theoretical curves (solid lines) given by Eq. (4) very well. The individual MD vectors were extracted by considering the unitary part of the corresponding 4 × 4 and 2 × 2 blocks along the main diagonal of the simulated overall (6 × 6) transfer matrix.

Fig. 4 (a) Probability density function of the length of the MD vector τ⃗_LP₁₁ that represents the four LP₁₁ modes (subspace dimension D_LP₁₁ = 15) and of the length of the MD vector τ⃗_LP₀₁ that represents the two LP₀₁ modes (subspace dimension D_LP₁₁ = 3). The degree of coupling between the two groups of modes is discussed in the text. Figures (b) and (c) show the ACFs of the the two MD vectors. Symbols refer to numerical simulations, solid lines to the theory.

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3. Theory

Following [10], we use the notation |ψ(z,ω)〉 to represent the state of the field in what we refer to as the generalize Jones space, at angular frequency ω at a given point z along the optical fiber. One should interpret |ψ(z)〉 as a column vector with 2N complex components representing the excitation of the various modes in the fiber. The corresponding row vector, which is the hermitian conjugate of |ψ(z)〉, is denoted by 〈ψ(z)| = (|ψ(z)〉)^†. For convenience we will also assume in what follows that the state vector |ψ(z)〉 is normalized, such that 〈ψ(z)|ψ(z)〉 = 1. Propagation between the positions z₀ and z along the fiber is represented by a linear operator U(z,z₀) so that

| ψ (z) 〉 = U (z, z_{0}) | ψ (z_{0}) 〉,

where the dependence of U(z,z₀) and |ψ(z)〉 on ω was suppressed for brevity of notation. When considering the process of propagation in the frequency domain, U(z,z₀) can be conveniently represented by a 2N × 2N frequency dependent matrix, which is unitary at each frequency in the absence of MDL, such that U(z₀, z) = U⁻¹(z,z₀) = U^†(z, z₀). The space and frequency evolution of U(z,z₀) can be expressed most generally as

\frac{d U (z, z_{0})}{d z} = i B (z) U (z, z_{0})

\frac{d U (z, z_{0})}{d ω} = i T (z, z_{0}) U (z, z_{0}),

where B(z) and T(z,z₀) are frequency dependent Hermitian matrices. Their Hermiticity is required for the unitarity of U(z,z₀) to be preserved at all z and ω. The multi-mode propagation problem of Eqs. (7) and (8) can be conveniently approached by extending the Pauli-matrix formalism to the multi-mode case as we show in the section that follows.

3.1. Trace orthogonal matrices and the generalized Stokes space

A generic 2N × 2N Hermitian matrix H contains 4N² entries. Of these, 2N are necessarily real (on the main diagonal) and the remaining 2N(2N – 1) come in complex conjugate pairs. Thus 4N² real numbers are necessary to completely define a 2N ×2N Hermitian matrix. This is hence the number of basis elements in the linear space of 2N × 2N Hermitian matrices. A convenient basis for this space is the one consisting of the identity matrix 1 and 4N² – 1 matrices Λ_j having zero trace and satisfying the condition of trace-orthogonality

\frac{1}{2 N} Trace {Λ_{i} Λ_{j}} = δ_{i, j},

where δ_i,j is the Kronecker delta function. Equation (9) can be shown to define an inner product in the space of Hermitian matrices. An example of a trace-orthogonal basis in the case of N = 1 is the identity with the three Pauli matrices that are used for the study of PMD [10]. The choice of a trace-orthogonal basis is never unique, and many algorithms for constructing one are possible. The Gell-Mann method [21] is one known method of construction. Its modification, which is slightly more convenient for our purposes, is described in the appendix. As should be expected, the choice of basis is immaterial for any of the physically meaningful results, and the only guideline for its selection is the simplicity of calculations. Once a basis is chosen, the matrix H is given by a superposition of the basis elements. A compact representation is the following

H = \frac{1}{2 N} (η_{0} 1 + \vec{η} \cdot \vec{Λ}),

where η⃗ is a vector of 4N² – 1 real elements η_j, Λ⃗ is a vector whose elements are the matrices Λ_j, and the dot operator is implemented as

\vec{η} \cdot \vec{Λ} = \sum_{j = 1}^{4 N^{2} - 1} η_{j} Λ_{j} .

The components of η⃗ can be extracted by using the trace-orthogonality of the Λ_j matrices,

\begin{matrix} η_{i} = trace {Λ_{i} H} & and & η_{0} = trace {H} . \end{matrix}

Note that the term proportional to η₀ represents a mode independent quantity, whereas all mode related phenomena are captured by the traceless content of H, given by the term η⃗ · Λ⃗. The vectors η⃗ reside in a 4N² – 1 dimensional space which is a generalization of the Stokes space used in the analysis of polarizations (N = 1). In what follows we will refer to vectors in this space as to generalized Stokes vectors, to be distinguished from the vectors |ψ〉 to which we refer as state-vectors. Since a projection operator |ψ〉〈ψ| onto a normalized state-vector |ψ〉 is a Hermitian operator, it too can be expressed in the form

| ψ 〉 〈 ψ | = \frac{1}{2 N} (1 + \vec{ψ} \cdot \vec{Λ}) .

The generalized Stokes vector ψ⃗ is obtained from |ψ〉 via Eq. (12), such that its components are given by

ψ_{j} = trace {Λ_{j} | ψ 〉 〈 ψ |} = 〈 ψ | Λ_{j} | ψ 〉,

identical to the usual definition of Stokes vectors in the two-polarization case. Equations (13) and (14) can be used to point out a number of interesting properties of generalized Stokes vectors. Noting that |〈ψ|ϕ〉|² = trace{|ψ〉〈ψ|ϕ〉〈ϕ|} and substituting the right-hand-side of (13) instead of the individual projection operators, one finds that

{| 〈 ψ | ϕ 〉 |}^{2} = \frac{1}{2 N} (1 + \vec{ψ} \cdot \vec{ϕ}),

where we have used the fact that trace{1} = 2N, trace{Λ_j} = 0, as well as the identity

\frac{1}{2 N} trace {(\vec{A} \cdot \vec{Λ}) (\vec{B} \cdot \vec{Λ})} = \vec{A} \cdot \vec{B},

which can readily be proven using Eqs. (9) and (11). Using |ϕ〉 = |ψ〉 in Eq. (15), one finds that |ψ⃗|² = 2N − 1, implying that only in the case N = 1, the Stokes representation of a normalized state-vector is a unit vector. If the state vectors are orthogonal to each other, Eq. (15) indicates that the product of their generalized Stokes vectors is equal to −1. Consequently, the angle α between two generalized Stokes vectors that correspond to orthogonal states satisfies cos(α) = − (2N − 1)⁻¹, and the vectors are antipodal only in the case N = 1. The fact that 0 ≤ |〈ψ|ϕ〉|² ≤ 1 implies that −1 ≤ ψ⃗ · ϕ⃗ ≤ 2N − 1. The left equality occurs when the state-vectors are orthogonal to each other, whereas the right equality occurs when |ψ〉 = |ϕ〉. It is evident that in the case N > 1 not all generalized Stokes vectors with length

\sqrt{2 N - 1}

have a state vector that corresponds to them. For example, if ϕ⃗ = −ψ⃗ the product ψ⃗ · ϕ⃗ = −2N + 1 is smaller than −1 so that at least one of the two generalized Stokes vectors cannot represent a legitimate state. This observation can also be made directly from Eq. (13), noting that the projection operator |ψ〉〈ψ| has only two eigenvalues. A non-degenerate eigenvalue equal to 1 corresponds to the eigenstate |ψ〉 itself and another 2N − 1 fold degenerate eigenvalue is equal to 0. The eigenstates of the latter eigenvalue are the 2N − 1 state vectors that are orthogonal to |ψ〉. Thus, ψ⃗ is a legitimate representation of a state-vector only if the matrix ψ⃗ · Λ⃗ has a 2N −1 degenerate eigenvalue equal to −1 and exactly one non-degenerate eigenvalue, equal to 2N − 1. It is obvious that only a small fraction of Stokes vectors satisfies this requirement when N > 1.

As discussed in Section 3.2, a relevant question concerns the relation between a generalized Stokes vector S⃗ and the eigenvalues of the matrix S⃗ · Λ⃗. In the case N = 1 the two eigenvalues are ±|S⃗| and the corresponding eigenvectors are those whose Stokes representation is parallel, or anti-parallel to the direction of S⃗. In the case of general N, there is no such simple relationship. The total number of eigenvalues is 2N, and the orientation of S⃗ usually does not correspond to a legitimate state-vector. Nonetheless, the following relationship can be established. Given that |ψ_i〉 is an eigenvector of S⃗ · Λ⃗, thereby satisfying S⃗ · Λ⃗|ψ_i〉 =θ_i|ψ_i〉, where θ_i is the corresponding eigenvalue, multiplication by 〈ψ_i| from the left and using (14) yields

\vec{S} \cdot {\vec{ψ}}_{i} = θ_{i},

implying that the largest eigenvalue corresponds to the eigenvector whose generalized Stokes representation is best aligned with the direction of S⃗, whereas the smallest eigenvalue corresponds to the eigenvector whose generalized Stokes representation is least aligned with it. A more useful relationship follows from noting that

θ_{i}^{2}

is an eigenvalue of (S⃗ · Λ⃗)² and hence that

\sum θ_{i}^{2} = Trace {{(\vec{S} \cdot \vec{Λ})}^{2}}

. Using Eq. (16), this equality produces the relation

2 N {| \vec{S} |}^{2} = \sum θ_{i}^{2} .

As we shall see in what follows, this relation gives a physical meaning to the vectors that will be used to represent the matrices associated with propagation in the fiber.

3.2. Mode coupling and the mode-dispersion vector

Using Eq. (10), Eqs. (7) and (8) can be conveniently reexpressed as

\frac{d U (z, z_{0})}{d z} = \frac{i}{2 N} (β_{0} 1 + \vec{β} \cdot \vec{Λ}) U (z, z_{0})

\frac{d U (z, z_{0})}{d ω} = \frac{i}{2 N} (τ_{0} 1 + \vec{τ} \cdot \vec{Λ}) U (z, z_{0}),

where the dependence of β₀, τ₀, β⃗ and τ⃗ on z and ω was suppressed for the simplicity of notation. Note that β₀ and τ₀ introduce a phase and group delay that is common to all the propagation modes in the fiber. These terms are therefore immaterial to our study of MD and their values can be set to zero. The vectors β⃗ and τ⃗ are the two most important quantities for the study of modal dispersion in optical fibers. The first vector, β⃗ is a generalization of the birefringence vector, as it describes the local coupling between the various modes. If one chooses to use the basis presented in Appendix A, then the first N triplets of the components of β⃗ correspond to the birefringence vectors of the N individual propagation modes. Namely, (β₁, β₂, β₃) is the birefringence vector of the first mode, (β₄, β₅, β₆) is the birefringence vector of the second mode and so on. The remaining 4N² − 3N − 1 terms of β⃗ represent coupling between spatial modes, or the difference in the spatial modes’ propagation vectors. The vector τ⃗ is the direct generalization of the PMD vector [10] and it captures the essentials of MD.

Equations (19) and (20) can be used to establish the concatenation rule satisfied by the MD vector τ⃗. To this end let us consider an assembly of two concatenated fiber sections characterized by the two matrices U₁ and U₂. The matrix describing their concatenation is U = U₂U₁. One may express the combined MD vector τ⃗ of the two fibers as

\vec{τ} \cdot \vec{Λ} = - 2 N i \frac{d U}{d ω} U^{†} = - 2 N i \frac{d U_{2}}{d ω} U_{2}^{†} + U_{2} (- 2 N i \frac{d U_{1}}{d ω} U_{1}^{†}) U_{2}^{†} = {\vec{τ}}_{2} \cdot \vec{Λ} + U_{2} ({\vec{τ}}_{1} \cdot \vec{Λ}) U_{2}^{†},

where τ⃗₁ and τ⃗₂ are the MD vectors of the first and second fiber section, respectively. The concatenation rule (21) can be stated as follows: The joint MD vector equals the MD vector of the first section rotated by the coupling matrix of the second section and supplemented by the MD vector of the second section alone. This description is identical to that used in the case of PMD [10], although the word rotation is applied more loosely in the present case, indicating an operation that does not change the length of a generalized stokes vector. To see that this is indeed the case with the second term in (21), one may define

{\vec{τ}}_{1}^{(R)} \cdot \vec{Λ} = U_{2} ({\vec{τ}}_{1} \cdot \vec{Λ}) U_{2}^{†}

and take advantage of identity (16) with A⃗ = B⃗ to show that

{| {\vec{τ}}_{1}^{(R)} |}^{2} = \frac{1}{2 N} trace {{[U_{2} ({\vec{τ}}_{1} \cdot \vec{Λ}) U_{2}^{†}]}^{2}} = {| {\vec{τ}}_{1} |}^{2},

where we made use of the fact that the trace of a product is invariant under cyclic permutations. It is instructive to consider the case in which U₁ represents propagation from z₀ to z whereas U₂ represents an incremental evolution between the points z and z + dz. Using Eq. (19) one finds that to first order in dz, U₂ = 1 + (i/2N)β⃗(z,ω) · Λ⃗dz. Substituting this into (21) produces

\frac{\partial}{\partial z} (\vec{τ} \cdot \vec{Λ}) = {\vec{β}}_{ω} \cdot \vec{Λ} + \frac{i}{2 N} [(\vec{β} \cdot \vec{Λ}) (\vec{τ} \cdot \vec{Λ}) - (\vec{τ} \cdot \vec{Λ}) (\vec{β} \cdot \vec{Λ})],

with β⃗_ω = ∂β⃗/∂ω. This expression reduces to Eq. (6.15) of [10] in the case of N = 1. The solution to Eq. (23) is formally identical to Eq. (7.22) of [10],

\vec{τ} (L) = \int_{0}^{L} R (L, z) {\vec{β}}_{ω} d z

where R(L,z) is a D × D norm-preserving matrix accounting for the evolution of the infinitesimal contribution β⃗_ω(z)dz from z to the fiber end at z = L and related to U(L,z) by

U (L, z) ({\vec{β}}_{ω} \cdot \vec{Λ}) U^{†} (L, z) = (R (L, z) {\vec{β}}_{ω}) \cdot \vec{Λ} .

The physical meaning of τ⃗ is provided by Eqs. (17), (18). Indeed, if we use S⃗ = τ⃗/(2N), then the eigenvalues θ_i coincide with the group delays of the individual principal states t_i and Eq. (17) becomes t_i = τ⃗ · ψ⃗_i/(2N), thus showing that these are bounded between

- τ \sqrt{2 N - 1} / (2 N)

and

τ \sqrt{2 N - 1} / (2 N)

. On the other hand, Eq. (18) becomes

τ^{2} = 2 N \sum_{i = 1}^{2 N} t_{i}^{2},

which gives the magnitude of the MD vector as proportional to the sum of the square group delays. For N = 1, τ is equal to the differential group delay |t₂ − t₁|, as expected.

An alternative and much simpler formulation notationally follows by defining a cross-product operation in the generalized Stokes space. The Hermiticity and the zero-trace property of the matrices Λ_i imply that i (Λ_iΛ_j − Λ_jΛ_i) /(2N) is also a traceless Hermitian matrix. Hence it can be expanded as i(Λ_iΛ_j − Λ_jΛ_i) /(2N) = ∑_k f_i,j,kΛ_k, with f_i,j,k being a set of real-valued structure constants,

f_{i, j, k} = \frac{i}{{(2 N)}^{2}} trace {Λ_{k} (Λ_{i} Λ_{j} - Λ_{j} Λ_{i})} .

Using this representation we define a generalized cross-product as

\vec{A} \times \vec{β} = \sum_{i, j, k} f_{i, j, k} A_{i} B_{j} {\vec{e}}_{k}

where e⃗_i are the orthogonal unit-length basis vector of the D-dimensional generalized Stokes space [24]. Equation (28) can be used to re-express the second term on the right-hand-side of Eq. (23) as (β⃗ × τ⃗) · Λ⃗, so that Eq. (23) simplifies to

\frac{\partial \vec{τ}}{\partial z} = {\vec{β}}_{ω} + \vec{β} \times \vec{τ},

generalizing the familiar equation for the evolution of the PMD vector [10] to the multimode case. By differentiating S⃗ = 〈ψ(0)|U(z,0)^†Λ⃗U(z,0)|ψ(0)〉 with respect to z and ω, and by using Eqs. (19) and (20), the evolution of the generalized Stokes vector in space and frequency can be expressed as

\begin{matrix} \frac{\partial \vec{S}}{\partial z} = \vec{β} \times \vec{S}, & \frac{\partial \vec{S}}{\partial ω} = \vec{τ} \times \vec{S}, \end{matrix}

generalizing, once again, the corresponding equations obtained in the single-mode fiber case.

4. Statistical analysis

In order to account for the statistical properties of MD, we assume that the distribution of the mode coupling vector β⃗ can be adequately modeled as white Gaussian noise. Although this assumption is not strictly satisfied even in single-mode optical fibers [22], it was shown to produce a very accurate description of PMD when the overall fiber length is significantly larger than the correlation length of the local birefringence. In principle, the white noise model is viable whenever the correlation length of the mode coupling vector is very small relative to the overall system length. Since the source of perturbations (geometrical distortions and mechanical stress) are similar to those experienced by single-mode fibers, it is expected that the correlation length of β⃗(z) is of a similar order of magnitude as the correlation length of the polarization birefringence vector in single-mode fibers. Using this assumption, it is quite obvious that the MD vector τ⃗ is Gaussian and evolves as a Wiener process in z, which is the continuous limit of a random walk. This conclusion can be readily drawn from Eq. (24) which shows that τ⃗ is the sum of rotated and statistically independent increments of β⃗_ω. Exploiting similar arguments to those made in the analysis of PMD statistics [23], one can conclude that the vector τ⃗(ω,z) experiences a D dimensional isotropic random walk, hence it is Gaussian distributed and its modulus τ obeys a chi distribution. An important consequence of Eq. (24) is that the mean-square MD value 〈τ²〉 is equal to 〈τ²〉 = γ²z, where γ² is defined through the relation 〈β⃗_ω(z)·β⃗_ω(z′)〉 = γ²δ(z−z′). In the case of partial coupling, τ⃗ is still Gaussian if β⃗(z) is a white noise process, yet its distribution may not be isotropic, and the equation 〈τ²〉 = γ²z still rigorously applies. In this case, numerical modeling shows that τ approximately obeys a chi distribution with an effective number of degrees of freedom D_eff < D.

The derivation of the autocorrelation functions Eqs. (4) and (5) is performed most simply by using the relations introduced in Eq. (29). Just as in the procedure followed in the study of PMD [19], we switch to a rotating reference frame, where, after a first-order expansion in frequency, β⃗ is replaced by ωβ⃗_ω with ω denoting the offset from the central frequency of the propagating signal. The spatial dependence of β⃗_ω is modeled as white Gaussian noise. The customary procedure in this case is to represent the integral of β⃗_ω over the propagation distance as an isotropic D-dimensional Wiener process W⃗(z). The increments of W⃗(z) are characterized by 〈dW_i(z)dW_j(z)〉= D⁻¹γ²δ_i,jdz, where γ is a constant proportional to the mean-square magnitude of the generalized birefringence, such that 〈τ²(z)〉 = γ²z. Equation (29) can be re-expressed as a stochastic differential equation

d \vec{τ} = d \vec{W} + ω d \vec{W} \times \vec{τ},

which is to be interpreted in the Stratonovich sense. The corresponding Ito equation involves an additive correction term − (γ²ω²/D)τ⃗dz, which is obtained after some algebra. From this point on, considering that our cross-product possesses the property τ⃗ · (dW⃗ × τ⃗) = 0, the derivation of the autocorrelation functions is identical to the one presented in [19]. The only modification is that the term D generalizes the factor 3 appearing in the corresponding term in [19].

Finally, we note that the scaling with the number of degrees of freedom is such that the shape of the autocorrelation functions is uniquely determined by 〈τ²〉/D, a quantity that has the meaning of the mean square MD parameter per dimension in the extended Stokes space. Polarization mode dispersion in single-mode fibers is a special case corresponding to D = 3.

4.1. Numerical simulations

In the numerical simulations a fiber is modeled as an assembly of N_s wave-plates, such that the k-th element is characterized by a generalized real-valued D-dimensional birefringence vector b⃗_det +b⃗_k, where b⃗_det describes the deterministic birefringence and b⃗_k accounts for perturbation-induced birefringence. In the simulation of quasi-degenerate modes b⃗_det = 0, whereas in the study of non-degenerate modes reported in Fig. 4 b⃗_det is used to tune the degree of deterministic coupling between different groups of quasi-degenerate modes. Since b⃗_det describes wave-vector mismatch between different modes, as discussed in Appendix A, only the last N − 1 components of b⃗_det can be nonzero. Expanding b⃗_k to first order in the optical frequency, and omitting the frequency-independent zeroth-order term (whose effect on the statistics of modal dispersion is immaterial) we write b⃗_k = ωb⃗′_k, where ω denotes the offset from the central frequency of the transmitted signal. The transfer matrix of the k-th section is therefore given by U_k = exp [i(b⃗_det +ωb⃗′_k) · Λ⃗/(2N)]. The frequency independent part of b⃗_k was verified to have no effect on the statistics of MD and hence it was omitted in the simulations. A similar omission of the frequency independent part of the random birefringence is customary in the study of polarization effects and it is known to have no consequences on the statistics when characterizing linear propagation problems. The vectors b⃗′_k are selected independently of one another and with each having D statistically independent zero-mean Gaussian components.

We start from discussing the case of quasi-degenerate modes, when b⃗_det = 0. In this case, using Eq. (21), the average square-modulus of the MD vector can be shown to be 〈τ²〉 = N_s〈|b⃗′_k|²〉. To link this notation to the one used in prior sections we note that b⃗_k is equivalent to β⃗(z_k)Δz_k with z_k being the position of the k-th fiber increment and with Δz_k being its length.

In the case of full coupling, all components of the vector b⃗′_k are identically distributed and their variance is taken to be σ² = 〈τ²〉/(N_sD). While the modeling of the full coupling case is relatively straightforward, the case of partial coupling is significantly more involved. Most importantly, the regime of partial coupling cannot be captured unequivocally in terms of a single parameter. Recent experimental work [20] characterized mode coupling between modes i and j in terms of the fraction of power that is coupled into mode j at the end of the link when mode i is excited. While this is a practical and useful parameter for quantifying mode coupling, it may not provide a unique characterization of the MD statistics, as the same coupling can be generated in a variety of ways. While a detailed description of the partial coupling regime in multi-mode fibers requires further experimental investigation, we model this situation in two ways. If partial coupling is caused by the phase mismatch between non-degenerate modes, then the magnitude of b⃗_det controls the degree of coupling. The results reported in Fig. 4 refer to this situation. Otherwise, in the case of quasi-degenerate modes, partial coupling can only be caused by an asymmetry of the coupling strength itself. This situation is modeled by attaching different values to the variance of different components of b⃗′_k. In particular, using our choice of the matrices constructing Λ⃗ (see Appendix A), the first 3N components of b⃗′_k are responsible for polarization coupling within the same spatial mode, whereas the remaining components are responsible for coupling different spatial modes with one another. Thus the degree of inter-modal coupling is controlled by varying the variance $σ_{c}^{2}$ of the last D − 3N components of b⃗′_k, as compared to the variance of the first 3N components, which we denote by $σ_{p}^{2}$ . In this case the average square modulus of the MD vector is given by $〈 τ^{2} 〉 = [3 N σ_{p}^{2} + (D - 3 N) σ_{c}^{2}] N_{s}$ .

5. Conclusions

Inspired by the analogy between MD and PMD, we developed a formalism to characterize modal dispersion in multimode fibers. The proposed formalism is based on the definition of a set of D = 4N² − 1 matrices — which are the generalization of the Pauli spin matrices — as a basis for the expansion of the generalized Jones matrix U describing the propagation of N spatial modes. In the absence of mode-dependent loss, MD is characterized by a real D-dimensional MD vector τ⃗, whose components are the coefficients of the expansion of the traceless part of the matrix T = −i(dU/dω)U^†. The eigenstates of such a matrix are generalized principal states and its eigenvalues are the corresponding group delays; the square length of τ⃗ is the sum of the square group delays. In the regime of strong mode coupling the vector τ⃗ is Gaussian distributed and its modulus follows a chi distribution with D degrees of freedom. We found that to an excellent degree of approximation, the same distribution, yet with a smaller number of degrees of freedom, also characterizes the largest differential group delay, that is the difference between the delays of the fastest and the slowest principal states. Finally, we studied the frequency autocorrelation of the MD vector, of the square modulus of the MD vector and of the largest differential group delay. Extensive numerical simulations were used to check the analytical results, showing good agreement in all cases.

A. Algorithm to build the Λ matrices

For N = 1 the matrices Λ_i coincide with the Pauli matrices, namely

\begin{matrix} Λ_{1} = σ_{1} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}), & Λ_{2} = σ_{2} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), & Λ_{3} = σ_{3} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) . \end{matrix}

For N > 1 the algorithm which gives the 4N² − 1 matrices proceeds as follows.

From Λ₁ to Λ₃_N : Each 2 × 2 block in the main diagonal, from the leftmost to the rightmost, is sequentially set to either one of the three Pauli matrices, while the other elements are set to zero. This gives 3N matrices. The trace of such matrices squared is 2 and hence a normalization coefficient $\sqrt{N}$ is required.
From Λ₃_N₊₁ to Λ_4N²–N : Each element outside the 2 × 2 blocks in the main diagonal is sequentially set to either 1 or i and the symmetric element to 1 or −i respectively, while the other elements are set to zero. This gives 4N² – 4N matrices. The trace of such matrices squared is 2 and hence a normalization coefficient $\sqrt{N}$ is required.
From Λ_4N²–N+1 to Λ_4N²–1: We denote these matrices as Λ_4N²–N+n, with n = 1,..., N −1. The 2 × 2 blocks in the main diagonal from the first to the n-th are set to σ₀ and the (n + 1)-th to −nσ₀, while the other elements are set to zero. The trace of such matrix squared is 2(n² + n) and hence a normalization coefficient $\sqrt{N / (n^{2} + n)}$ is required.

It is apparent from the definition that the matrices Λ₁ . . . Λ₃_N do not couple different spatial modes. Correspondingly, the components β₁ . . . β₃_N of the generalized birefringence vector β⃗ account for polarization-mode coupling within the same spatial mode. The matrices Λ₃_N₊₁ to Λ_4N²–N couple individual polarization modes of different spatial modes, whose magnitude is defined by the corresponding components of β⃗. Finally, the matrices Λ_4N²–N to Λ_4N²–1 describe wave-vector mismatch between different spatial modes.

As an example, we list below the Λ_i matrices for the case N = 2.

\begin{array}{l} Λ_{1} = \sqrt{2} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{2} = \sqrt{2} (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{3} = \sqrt{2} (\begin{matrix} 0 & - i & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), \\ Λ_{4} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}), & Λ_{5} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}), & Λ_{6} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - i \\ 0 & 0 & i & 0 \end{matrix}), \\ Λ_{7} = \sqrt{2} (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{8} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}), & Λ_{9} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), \\ Λ_{10} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}), & Λ_{11} = \sqrt{2} (\begin{matrix} 0 & 0 & i & 0 \\ 0 & 0 & 0 & 0 \\ - i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{12} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - i & 0 & 0 & 0 \end{matrix}), \\ Λ_{13} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & - i & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{14} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & - i & 0 & 0 \end{matrix}), & Λ_{15} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}) . \end{array}

Acknowledgments

This work has been carried out within an agreement funded by Alcatel-Lucent in the framework of Green Touch (www.greentouch.org). MS also acknowledges financial support from Tera Santa Consortium and from the University of L’Aquila under project “Progetto speciale multiasse Reti per la conoscenza e l’orientamento tecnico-scientifico per lo sviluppo della competitività (Re.C.O.Te.S.S.C.) P.O.R. Regione Abruzzo F.S.E. 2007–2013 Piano 2007–2008.”

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Stokes-space analysis of modal dispersion in fibers with multiple mode transmission

Abstract

1. Introduction

2. Main results

2.1. Generalized Stokes space and modal dispersion vector

2.2. Maximum modal delay spread

2.3. Autocorrelation functions

2.4. Non-degenerate modes

3. Theory

3.1. Trace orthogonal matrices and the generalized Stokes space

3.2. Mode coupling and the mode-dispersion vector

4. Statistical analysis

4.1. Numerical simulations

5. Conclusions

A. Algorithm to build the Λ matrices

Acknowledgments

References and links

Cited By

Figures (4)

Equations (33)

Optics Express