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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 11718–11733
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Stokes-space analysis of modal dispersion in fibers with multiple mode transmission

Cristian Antonelli, Antonio Mecozzi, Mark Shtaif, and Peter J. Winzer  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 11718-11733 (2012)
http://dx.doi.org/10.1364/OE.20.011718


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Abstract

Modal dispersion (MD) in a multimode fiber may be considered as a generalized form of polarization mode dispersion (PMD) in single mode fibers. Using this analogy, we extend the formalism developed for PMD to characterize MD in fibers with multiple spatial modes. We introduce a MD vector defined in a D-dimensional extended Stokes space whose square length is the sum of the square group delays of the generalized principal states. For strong mode coupling, the MD vector undertakes a D-dimensional isotropic random walk, so that the distribution of its length is a chi distribution with D degrees of freedom. We also characterize the largest differential group delay, that is the difference between the delays of the fastest and the slowest principal states, and show that it too is very well approximated by a chi distribution, although in general with a smaller number of degrees of freedom. Finally, we study the spectral properties of MD in terms of the frequency autocorrelation functions of the MD vector, of the square modulus of the MD vector, and of the largest differential group delay. The analytical results are supported by extensive numerical simulations.

© 2012 OSA

1. Introduction

With the fiber-optic communications industry facing an imminent capacity crunch [1

1. A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).

, 2

2. R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010). [CrossRef]

], 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

3. Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011). [CrossRef] [PubMed]

] in a combination of multi-core (MCF) [4

4. S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

] and multi-mode (MMF) fibers [5

5. S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011). [CrossRef] [PubMed]

, 6

6. E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

]. 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.

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

10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000). [CrossRef] [PubMed]

]. 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

11. N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995). [CrossRef]

]. 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

12. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001). [CrossRef]

, 13

13. M. Nazarathy and E. Simony, “Generalized Stokes parameters shift keying approach to multichip differential phase encoded optical modulation formats,” Opt. Lett. 31, 435–437 (2006). [CrossRef] [PubMed]

], 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

14. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol. 30, 618–623 (2012). [CrossRef]

]. 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

15. S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

], 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 ti,
τ2=2Ni=12Nti2.
(1)

The individual group delays are defined with the mode averaged delay set to 0, ∑j tj = 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

16. D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002). [CrossRef]

]. 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 Deff, 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 (Deff = 3N) and in (d) there is full coupling (Deff = 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 Deff 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.

2.2. Maximum modal delay spread

Fig. 2 Top panel. The solid line represents the shape parameter KN 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.

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

10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000). [CrossRef] [PubMed]

] 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,
Rτ(ω)=τ(Ω+ω)τ(Ω)=Dω2[1exp(ω2τ2D)],
(4)
as well as that of its square modulus τ2,
Rτ2(ω)=τ2(Ω+ω)τ2(Ω)=τ22+4τ2ω24Dω4[1exp(ω2τ2D)].
(5)
These expressions generalize the expressions obtained in the analysis of PMD [18

18. M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999). [CrossRef]

, 19

19. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000). [CrossRef]

] 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, Rτ(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 KN, 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 BT can thus be very well approximated as the 3dB width of the modified ACF (5) BT3.2KN/τ2, which, for a moderate, yet practically important, number of spatial modes (∼ N < 5), is only negligibly different from BT3.2D/τ2.

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 KN as from Fig. 2.

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 Deff) 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 Deff.

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

20. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). [CrossRef]

], 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 DLP11 = 15 dimensional MD vector τ⃗LP11 that represents the four LP11 modes and the PDF of the length of the DLP01 = 3 dimensional MD vector τ⃗LP01 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

20. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). [CrossRef]

], 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 τLP11 and τLP01 (symbols) is still very well fitted by chi distributions of DLP11 = 15 and DLP01 = 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 τ⃗LP11 that represents the four LP11 modes (subspace dimension DLP11 = 15) and of the length of the MD vector τ⃗LP01 that represents the two LP01 modes (subspace dimension DLP11 = 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.

3. Theory

Following [10

10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000). [CrossRef] [PubMed]

], 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,z0)|ψ(z0),
(6)
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
dU(z,z0)dz=iB(z)U(z,z0)
(7)
dU(z,z0)dω=iT(z,z0)U(z,z0),
(8)
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

As discussed in Section 3.2, a relevant question concerns the relation between a generalized Stokes vector S⃗ and the eigenvalues of the matrix S· 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 |ψi〉 is an eigenvector of S⃗ · Λ⃗, thereby satisfying S⃗ · Λ⃗|ψi〉 =θi|ψi〉, where θi is the corresponding eigenvalue, multiplication by 〈ψi| from the left and using (14) yields
Sψi=θi,
(17)
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 θi2 is an eigenvalue of (S⃗ · Λ⃗)2 and hence that θi2=Trace{(SΛ)2}. Using Eq. (16), this equality produces the relation
2N|S|2=θi2.
(18)
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

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 Λi imply that iiΛj − ΛjΛi) /(2N) is also a traceless Hermitian matrix. Hence it can be expanded as iiΛj − ΛjΛi) /(2N) = ∑k fi,j,kΛk, with fi,j,k being a set of real-valued structure constants,
fi,j,k=i(2N)2trace{Λk(ΛiΛjΛjΛi)}.
(27)
Using this representation we define a generalized cross-product as
A×β=i,j,kfi,j,kAiBjek
(28)
where e⃗i are the orthogonal unit-length basis vector of the D-dimensional generalized Stokes space [24

24. The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

]. Equation (28) can be used to re-express the second term on the right-hand-side of Eq. (23) as (β⃗ × τ⃗) · Λ, so that Eq. (23) simplifies to
τz=βω+β×τ,
(29)
generalizing the familiar equation for the evolution of the PMD vector [10

10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000). [CrossRef] [PubMed]

] 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
Sz=β×S,Sω=τ×S,
(30)
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

22. A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001). [CrossRef]

], 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 β⃗ω. Exploiting similar arguments to those made in the analysis of PMD statistics [23

23. F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990). [CrossRef]

], 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 〈β⃗ω(zβ⃗ω(z′)〉 = γ2δ(zz′). 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 Deff < 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

19. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000). [CrossRef]

], we switch to a rotating reference frame, where, after a first-order expansion in frequency, β⃗ is replaced by ωβ⃗ω with ω denoting the offset from the central frequency of the propagating signal. The spatial dependence of β⃗ω is modeled as white Gaussian noise. The customary procedure in this case is to represent the integral of β⃗ω over the propagation distance as an isotropic D-dimensional Wiener process W⃗(z). The increments of W⃗(z) are characterized by 〈dWi(z)dWj(z)〉= D−1γ2δi,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
dτ=dW+ωdW×τ,
(31)
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

19. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000). [CrossRef]

]. The only modification is that the term D generalizes the factor 3 appearing in the corresponding term in [19

19. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000). [CrossRef]

].

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

5. Conclusions

A. Algorithm to build the Λ matrices

For N = 1 the matrices Λi coincide with the Pauli matrices, namely
Λ1=σ1=(1001),Λ2=σ2=(0110),Λ3=σ3=(0ii0).
(32)
For N > 1 the algorithm which gives the 4N2 − 1 matrices proceeds as follows.

  • From Λ1 to Λ3N : 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 N is required.
  • From Λ3N+1 to Λ4N2N : 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 N is required.
  • From Λ4N2N+1 to Λ4N2–1: We denote these matrices as Λ4N2N+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 −0, while the other elements are set to zero. The trace of such matrix squared is 2(n2 + n) and hence a normalization coefficient N/(n2+n) is required.

It is apparent from the definition that the matrices Λ1 . . . Λ3N do not couple different spatial modes. Correspondingly, the components β1 . . . β3N of the generalized birefringence vector β⃗ account for polarization-mode coupling within the same spatial mode. The matrices Λ3N+1 to Λ4N2N couple individual polarization modes of different spatial modes, whose magnitude is defined by the corresponding components of β⃗. Finally, the matrices Λ4N2N to Λ4N2–1 describe wave-vector mismatch between different spatial modes.

As an example, we list below the Λi matrices for the case N = 2.
Λ1=2(1000010000000000),Λ2=2(0100100000000000),Λ3=2(0i00i00000000000),Λ4=2(0000000000100001),Λ5=2(0000000000010010),Λ6=2(00000000000i00i0),Λ7=2(0010000010000000),Λ8=2(0001000000001000),Λ9=2(0000001001000000),Λ10=2(0000000100000100),Λ11=2(00i00000i0000000),Λ12=2(000i00000000i000),Λ13=2(000000i00i000000),Λ14=2(0000000i00000i00),Λ15=(1000010000100001).

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.”

References and links

1.

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).

2.

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010). [CrossRef]

3.

Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011). [CrossRef] [PubMed]

4.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

5.

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011). [CrossRef] [PubMed]

6.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

7.

S. Fan and J. M. Kahn, “Principal modes in multi-mode waveguides”, Opt. Lett. 30, 135–137 (2005). [CrossRef] [PubMed]

8.

K-P. Ho and J. M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011). [CrossRef]

9.

K-P. Ho and J. M Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011). [CrossRef] [PubMed]

10.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000). [CrossRef] [PubMed]

11.

N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995). [CrossRef]

12.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001). [CrossRef]

13.

M. Nazarathy and E. Simony, “Generalized Stokes parameters shift keying approach to multichip differential phase encoded optical modulation formats,” Opt. Lett. 31, 435–437 (2006). [CrossRef] [PubMed]

14.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol. 30, 618–623 (2012). [CrossRef]

15.

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

16.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002). [CrossRef]

17.

In the literature the shape parameter KN is typically referred to as the number of degrees of freedom of the chi distribution. We refrain from this terminology so as to avoid confusion with the number of degrees of freedom Deff representing the dimension of the manyfold spanned by the MD vector in the generalized Stokes space.

18.

M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999). [CrossRef]

19.

M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000). [CrossRef]

20.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). [CrossRef]

21.

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962). [CrossRef]

22.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001). [CrossRef]

23.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990). [CrossRef]

24.

The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.4230) Fiber optics and optical communications : Multiplexing
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 2, 2012
Revised Manuscript: March 28, 2012
Manuscript Accepted: March 29, 2012
Published: May 9, 2012

Citation
Cristian Antonelli, Antonio Mecozzi, Mark Shtaif, and Peter J. Winzer, "Stokes-space analysis of modal dispersion in fibers with multiple mode transmission," Opt. Express 20, 11718-11733 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11718


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References

  1. A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).
  2. R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J.14, 3–9 (2010). [CrossRef]
  3. Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express19, 16680–16696 (2011). [CrossRef] [PubMed]
  4. S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).
  5. S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express19, 16697–16707 (2011). [CrossRef] [PubMed]
  6. E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).
  7. S. Fan and J. M. Kahn, “Principal modes in multi-mode waveguides”, Opt. Lett.30, 135–137 (2005). [CrossRef] [PubMed]
  8. K-P. Ho and J. M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol.29, 3119–3128 (2011). [CrossRef]
  9. K-P. Ho and J. M Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express19, 16612–16635 (2011). [CrossRef] [PubMed]
  10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA97, 4541–4550 (2000). [CrossRef] [PubMed]
  11. N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun.114, 399–405 (1995). [CrossRef]
  12. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A64, 052312 (2001). [CrossRef]
  13. M. Nazarathy and E. Simony, “Generalized Stokes parameters shift keying approach to multichip differential phase encoded optical modulation formats,” Opt. Lett.31, 435–437 (2006). [CrossRef] [PubMed]
  14. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.30, 618–623 (2012). [CrossRef]
  15. S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).
  16. D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun.20, 1247–1257 (2002). [CrossRef]
  17. In the literature the shape parameter KN is typically referred to as the number of degrees of freedom of the chi distribution. We refrain from this terminology so as to avoid confusion with the number of degrees of freedom Deff representing the dimension of the manyfold spanned by the MD vector in the generalized Stokes space.
  18. M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett.24, 939–941 (1999). [CrossRef]
  19. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett.25, 707–709 (2000). [CrossRef]
  20. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol.30, 521–531 (2012). [CrossRef]
  21. M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev.125, 1067–1084 (1962). [CrossRef]
  22. A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett.26, 962–964 (2001). [CrossRef]
  23. F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol.8, 1162–1166 (1990). [CrossRef]
  24. The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

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