We study random coupling induced crosstalk between groups of degenerate modes in spatially multiplexed optical transmission. Our analysis shows that the average crosstalk is primarily determined by the wavenumber mismatch, by the correlation length of the random perturbations, and by the coherence length of the degenerate modes, whereas the effect of a deterministic group velocity difference is negligible. The standard deviation of the crosstalk is shown to be comparable to its average value, implying that crosstalk measurements are inherently noisy.
© 2013 Optical Society of America
Spatially multiplexed transmission is considered to be a promising solution for increasing the information throughput of fiber-optic systems , and significant efforts are being invested into exploring the properties of fiber structures supporting several propagation modes. A typical situation occurring in multimode structures is that groups of modes are characterized by similar propagation constants, implying strong coupling between them . We refer to such modes as degenerate. On the other hand, modes that belong to different groups characterized by notably different propagation constants are weakly coupled, as has been observed in a series of experiments [3, 4]. This reality is nicely exemplified in the case of single-core fibers operating in the weakly guiding regime, where the linearly polarized (LP) mode approximation is valid . A multi-core fiber forms another example as, depending on the overall core number and geometry, multiple degeneracies between the resultant super-modes may exist .
Our goal in this paper is to model the coupling between non-degenerate groups of modes and to understand its dependence on various fiber parameters. We obtain an expression for the crosstalk and find that the degree of coupling is mainly determined by the mismatch between the wavenumbers of the two groups, whereas the effect of the mismatch on group velocities is negligible. In addition, we show that the standard deviation of the crosstalk is comparable to its average value, a property that has to be taken in proper consideration when characterizing crosstalk in multimode fibers.
As elaborated in , linear propagation in a fiber supporting N orthogonal spatial modes is described by the equation8]. The vector has 4N2 − 1 real-valued components and it generalizes the familiar birefringence vector used in the modeling of polarization effects in single mode fibers . The tilde above the vector sign serves to distinguish between generalized birefringence vectors  and electric field vectors such as E⃗. The term Λ⃗(2N) is the generalized Pauli matrix vector  and its elements are 2N × 2N traceless Hermitian matrices Λi, which reduce to the standard Pauli matrices  in the case of the single mode fiber (N = 1). These matrices constitute a basis for the space of 2N × 2N traceless Hermitian matrices. The scalar product is to be interpreted as , and (when is assigned the appropriate statistics) accounts for the effect of mode coupling resulting from random perturbations in the fiber structure. For ease of representation, it is convenient to decompose the electric field vector E⃗ into vectors E⃗j of smaller dimension 2gj, whose components are the field envelopes of the degenerate members of the j-th group. In this representation, the first 2g1 terms of the main diagonal of the matrix B are equal to β1 (the wavenumber of the modes in the first group), the following 2g2 terms are equal to β2 (the wavenumber of the second group) and so on.
We now proceed by dividing the matrix into rectangular blocks denoted by blj whose dimension is 2gj × 2gl, which can be demonstrated to have the property . With this notation, and substituting for convenience e⃗l(z) = exp(−iβlz)E⃗l(z), Eq. (1) can be rewritten as a set of coupled equationsEq. (2) is given by
2.1. Coupling between groups of modes
We now study the signal leaking from one group of degenerate modes to another. To that end, we evaluate E⃗l(z) when the input signal is injected only into group q, with q ≠ l. Since we expect the coupling between different groups to be small due to the large wavenumber difference, we extract E⃗l(z) via a first-order perturbation analysis, which yieldsEq. (4) is rather intuitive. The electric field generated in group l at position z′ is proportional to the product of the random coupling matrix blq(z′) by the field e⃗q(z′) = Uq(z′, 0)e⃗q(0). The matrix Ul(z, z′) accounts for the propagation of the field in group l from z′ to z and the term exp(iβqlz′) accounts for the mode-independent phase that it accumulates. In what follows we will assume that the input vector is normalized, so that |e⃗q(0)| = 1.
It is of interest to note that the inverse Fourier transform of Eq. (4) is the impulse response of the system describing coupling between the q-th and the l-th groups of modes. When the delay spread in the fiber is dominated by the deterministic walk-off between the groups, the impulse response measured after a distance z will be limited to a time interval whose duration is |zβql,1|, where βql,1 is the frequency derivative of βql at the carrier frequency, namely at ω = 0. This picture is in agreement, in terms of the general shape of the impulse response, with the experimental results reported in  and it can be deduced from Eq. (4) by approximating βql ≃ βql,0 + βql,1ω and by neglecting the frequency dependence of the perturbation vector , which is responsible for random modal dispersion .
In order to quantify the extent of coupling between different groups of modes we consider the total energy received in the l-th group when a signal is transmitted in the q-th group. This quantity is given by
2.2. Statistics of mode coupling
We first calculate the average crosstalk 〈ulq〉. This involves calculating first 〈|e⃗l(z, ω)|2〉, which in turn involves calculating , as a consequence of the double integral appearing when evaluating |e⃗l(z, ω)|2 from Eq. (4). Since the statistics of mode coupling should be stationary with respect to frequency within the telecom bandwidth, we may set the offset from the carrier frequency to zero, namely ω = 0, in when performing the average. Since the matrices bll and bqq (that determine Ul and Uq) are statistically independent of each other  and of the coupling blocks blm, the averaging of the inner term can be performed independently of the outer terms. By isotropy the desired average should be proportional to the identity matrix Il, and we may write 〈Ul(z, 0)〉 = exp(−z/Ll)Il, where we assumed exponential decorrelation of the electric field vector. The physical meaning of the correlation length Ll is seen when expressing the longitudinal autocorrelation function of the field vector as . We thus refer to Ll as the coherence length of the electric field .
We are then left with the calculation of , where the inner product can be averaged separately as it is independent of Uq. To accomplish this task, we refer the reader to the detailed construction of the Λ⃗(2N) matrix vector, which is presented in the appendix of . Moreover, since the matrices blq represent off-diagonal blocks of the matrix , the real and imaginary parts of each of the elements of blq are proportional to different components of the vector , with the proportionality coefficient equal to . Hence, since we model the generalized birefringence vector as consisting of statistically independent components, there is statistical independence between the elements of blq and between the real and imaginary parts of each element. Denoting by f(z) the autocorrelation function of the components of , namely 〈bj(z′)bk(z″)〉 = f(z′ − z″)δjk, we obtain . This leaves us with the calculation of , which follows the same lines as in the calculation performed for Ul, with the result:13]. The average coupled energy can thus be written as Eq. (7) can be approximated by z/K, which shows that the degree of coupling can be reduced by increasing the wavenumber mismatch. Expanding Eq. (7) to fourth order in , and approximating βlq ≃ βlq,0 + ωβlq,1, yields 3], and for the much greater value βlq,1 = 4.35 ns/km , which gives a very small number since βlq,0 is by orders of magnitude larger than 1 m−1. This means that with a large enough wavenumber mismatch the group velocity difference is expected to have a negligible effect on system performance .
We now estimate the standard deviation of the crosstalk σulq. Inspection of Eq. (4) suggests that, if the propagation distance is much longer than the correlation length of blq, then the output field is the sum of many independent, random contributions, and hence by the central-limit theorem e⃗l(z) becomes a Gaussian vector with 2gl independent complex components. The crosstalk ulq is proportional to the frequency integral of |e⃗l(z)|2 and its statistical properties are affected by the frequency-dependence of the coupling. In the special case where fiber characterization is performed with a continuous-wave (CW) signal, ulq can be approximated by a chi-squared distributed variable with 4gl degrees of freedom and standard deviation given by
In order to validate the above theory, one would like to compare its predictions with the simulation of a realistic fiber structure, supporting multiple groups of modes. Such a procedure would however be prohibitively inefficient because of the multiple length scales involved in the processes of propagation and coupling. On the one hand, the beat-length associated with the wavenumber difference between any two groups of modes can easily be of the order of a millimeter or less, while on the other hand, the correlation lengths characterizing perturbations in the fiber are likely to be on the multiple meters scale (as can be expected based on polarization studies in single-mode fibers). A meaningful statistical study requires simulation of multiple correlation lengths, a prohibitively time-consuming procedure when a step-size of a fraction of a millimeter needs to be used. In order to bypass this difficulty we present in Fig. 1 the results of a computation performed where the correlation length of the perturbations Lc is much shorter than in reality and equal to only fifteen beat lengths ( ). In addition, the coherence lengths of the fields in the two groups of modes were assumed to be Ll = Lq = Lc/10. We assumed degeneracy factors of gq = 1 and gl = 2, for the two groups of modes and the perturbation vector was assigned Gaussian statistically independent components. Since we do not seek to resolve the details of the formation of strong coupling within each of the two groups of modes, we take the components of that are responsible for such coupling as white noise processes in the longitudinal dimension z. It can be shown that their power spectral density (which is constant) is equal to . The other components of , which are responsible for coupling between the two groups of modes, were produced as Ornstein-Uhlenbeck processes , with the above specified correlation length Lc and with a variance n0 which was chosen such that a coupling of ∼ 5 × 10−2 is predicted by Eq. (8) at L = 10Lc.
In Fig. 1(a) we plot the average crosstalk 〈ulq〉 as a function of propagation distance, assuming transmission of a CW signal. The thick solid lines (black) represent the full expression Eq. (7) and the thin lines (red) represent the results of Monte Carlo simulations with 50,000 fiber realizations. The dashed lines represent the simplified expression Eq. (8). The excellent agreement between the analytical solutions and the simulations is evident. The small deviation observed when the crosstalk level rises towards 5% represents the saturation of the first order analysis that we used. We note that the accuracy of the analytical results depends only on the overall level of coupling and not on the exact parameter combinations. To demonstrate this, the dotted green curve in Fig. 1(a), which overlaps with the red curve, was calculated with different parameters; and Ll = Lq = Lc/5. The shaded area surrounding the curves marks one standard deviation as given by Eq. (9). Figure 1(b) shows the standard deviation σulq as a function of propagation distance. Once again, the thick black and the thin red lines represent the analytical expression Eq. (9) and the numerical results, respectively. In Fig. 1(c) we plot the crosstalk probability density function for the displayed values of the propagation distance. Symbols represent Monte Carlo simulations, solid lines are the plot of a chi-squared distribution with 4g2 = 8 degrees of freedom and mean value given by Eq. (7).
We studied the crosstalk between groups of degenerate modes induced by random perturbations in multimode fibers. We showed that the crosstalk is determined almost exclusively by the wavenumber difference between the groups, whereas the effect of the group-velocity difference is negligible. In addition, the standard deviation of the crosstalk was found to be comparable in magnitude with the average crosstalk. This result is of major significance for experimental fiber characterization as it questions the reliability of isolated crosstalk measurements.
This work was funded by Alcatel-Lucent in the framework of Green Touch. Mark Shtaif also acknowledges financial support from Israel Science Foundation (grant 737/12). Cristian Antonelli and Antonio Mecozzi acknowledge financial support from the Italian Ministry of University and Research through ROAD-NGN project (PRIN 2010–2011). Discussions with Herwig Kogelnik are gratefully acknowledged.
References and links
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