## Abstract

We show that light propagation in a group of degenerate modes of a multi-mode optical fiber in the presence of random mode coupling is described by a multi-component Manakov equation, thereby making multi-mode fibers the first reported physical system that admits true multi-component soliton solutions. The nonlinearity coefficient appearing in the equation is expressed rigorously in terms of the multi-mode fiber parameters.

© 2012 Optical Society of America

## 1. Introduction

Several years after the publication of the famous Zakharov and Shabat paper [1] which showed that the scalar nonlinear Schrödinger equation (NLSE) is integrable, Manakov [2] discovered that a special form of a two-component vector NLSE in which the nonlinearity is isotropic shares the same property. However, no practical physical system was described by this equation until Wai, Menyuk and Chen [3] found that Manakov’s equation accurately describes the propagation of a polarized optical field in single-mode birefringent optical fibers in the presence of random mode coupling. This implied that such fibers support the existence of vector solitons, as predicted by Manakov almost two decades earlier.

Recently, with fiber-communications exhausting the capacity of single-mode fibers, multimode fibers are being considered for transmission, with the purpose of increasing the throughput through spatial multiplexing. This new paradigm calls for the extension of the existing theoretical framework to the multi-mode fiber case. Random coupling of degenerate or quasi degenerate modes is a distinctive feature of these fibers [4, 5]. We have addressed the linear problem of mode coupling and modal dispersion in multi-mode fibers in a recent paper [6]. In this paper we focus on the nonlinear propagation regime, where we show that the evolution of the electric field in a group of degenerate modes of a multi-mode optical fiber is described by a generalized multi-components Manakov equation [7]. The central feature of the generalized Manakov equation is that it is integrable [7] and hence supports the propagation of solitons. Those are particle-like waveforms that remain unchanged in the process of propagation and preserve their identity when colliding with each other. It is the latter property that distinguishes between solitons and generic solitary waves, whose shape is also unaltered by propagation, but becomes corrupted when two, or more, such waveforms collide. Interestingly, solitary waves in certain multi-mode optical systems were shown to be possible using specific combinations of physical parameters [8, 9]. Yet, unlike the case which we present here, these systems are modeled by non-integrable equations and hence they do not support true soliton propagation.

We consider propagation over a group of degenerate modes in a multi-mode optical fiber. The degeneracy of the modes implies that fiber imperfections in the form of mechanical stress or manufacturing distortions produce strong random coupling between them on a length-scale typically much shorter than the length-scale of the nonlinear evolution [4]. On the other hand, the distinctive difference in wave vector between non degenerate groups of modes makes coupling between them significantly smaller and hence we neglect it in our analysis. The four degenerate LP_{11} modes [4] of a step-index fiber are a relevant example for a group of degenerate modes of the kind that we consider here. Larger groups of degenerate modes can be encountered in situations where multiple multi-mode fibers are combined in a multi-core fiber structure.

Starting from the coupled NLSE and assuming random mode coupling, we generalize the standard Manakov equation [3] to the multi-mode case. We verify the accuracy of the generalized Manakov equation and demonstrate the existence of multidimansional vector solitons, by solving the complete set of coupled NLSE numerically.

## 2. Analysis

The electric field in a group of *N* degenerate spatial modes is represented by a 2*N*-dimensional complex valued vector *E⃗*(*z*,*t*), which is constructed by stacking the Jones vectors of the *N* individual spatial modes one on top of the other. The term spatial mode is used here and throughout the paper to refer to the set of two polarization modes sharing, in the weakly guiding fiber approximation [10], the same lateral field profile. The components of *E⃗*(*z*,*t*) then satisfy the following set of coupled NLSE [11, 12]

**B**

^{(i)}(

*z*),

*i*= 0, 1, 2 are 2

*N*× 2

*N*Hermitian matrices of components ${\beta}_{jm}^{(i)}(z)$ [6] and where we use

*ê*with

_{j}*j*= 1, . . . , 2

*N*to denote the set of complex orthogonal unit vectors used to represent the electric field.

The fourth term on the right-hand side of Eq. (1) represents the coupling induced by the Kerr nonlinearity of the fiber. The coefficient *γ* = *ω*_{0}*n*_{2}/*cA*_{eff} is identical to the usual nonlinearity coefficient appearing in the scalar NLSE [11] of a single mode fiber, where *n*_{2} is the Kerr coefficient of glass, *c* is the speed of light in vacuum, and *A*_{eff} is the effective area of the fundamental mode at central frequency *ω*_{0}. The dimensionless constants *C _{jhkm}* depend on the details of the spatial mode profiles and are obtained as follows [12]. Defining

*F⃗*(

_{n}*x*,

*y*) describes the lateral mode profiles, and

*n*(

_{f}*x*,

*y*) is the refractive index, we obtain ${C}_{jhkm}={A}_{0}\left[2{D}_{jhkm}^{(1)}+{D}_{jhkm}^{(2)}\right]/3$, with ${A}_{0}^{-1}=\left[2{D}_{1111}^{(1)}+{D}_{1111}^{(2)}\right]/3={D}_{1111}^{(1)}$, so that

*C*

_{1111}=

*C*

_{2222}= 1, where the indices 1 and 2 are used to denote the two polarizations of the fundamental mode. Note that ${A}_{0}\simeq {n}_{\text{eff}}^{2}{A}_{\text{eff}}$, where

*n*

_{eff}is the effective refractive index of the fundamental mode at the central frequency

*ω*

_{0}.

The first three terms on the right-hand side of Eq. (1) account for linear propagation and coupling [6]. While in the ideal case the matrices **B**^{(i)}(*z*) (*i* = 0, 1, 2) are proportional to the identity, unavoidable position dependent perturbations result in the presence of off-diagonal terms, producing linear coupling between the various modes. The strongest coupling results from the term proportional to **B**^{(0)}, and in most cases of practical interest, its characteristic length-scale is shorter by orders of magnitude than the length-scale that characterizes the nonlinear evolution [11]. Under these conditions the orientation of the electric field vector, defined by *E⃗*/|*E⃗*| must be uniform, implying that the probability density of the complex field *E⃗* may depend only on its modulus |*E⃗*|. Field propagation is then approximated by averaging the nonlinear terms with respect to this distribution. In the case of *N* = 1, this condition is rigorously equivalent to the fields Stokes vector’s orientation being uniformly distributed on the surface of the Poincaré sphere. As can be anticipated based on symmetry and dimensionality arguments, the averaged nonlinear term of (1) must reduce to the form
${\sum}_{jhkm}{C}_{jhkm}{E}_{h}^{*}{E}_{k}{E}_{m}{\widehat{e}}_{j}=\kappa {\left|\overrightarrow{E}\right|}^{2}\overrightarrow{E}$, where *κ* is a dimensionless parameter that depends only on the nonlinear coupling coefficients *C _{jhkm}*, whose expression can be derived by performing scalar multiplication of both sides by

*E⃗*and averaging with respect to the field’s orientation. This yields

*κ*= ∑

*, where*

_{jhkm}C_{jhkm}Q_{jhkm}*ℰ*denotes statistical averaging and ${Q}_{jhkm}=\mathcal{E}\left[{E}_{h}^{*}{E}_{k}{E}_{m}{E}_{j}^{*}\right]/{\left|\overrightarrow{E}\right|}^{4}$. Since

*E⃗*can be considered as a constant modulus vector whose orientation is uniformly distributed, the statistical average can be performed introducing an auxiliary random vector

*X⃗*with 2

*N*complex, statistically independent components. The real and imaginary parts of each component

*X*of

_{i}*X⃗*are statistically independent standard Gaussian variables having zero-mean and unit variance. It can be argued that ${Q}_{jhkm}=\mathcal{E}\left[{X}_{h}^{*}{X}_{k}{X}_{m}{X}_{j}^{*}|{\left|\overrightarrow{X}\right|}^{2}\right]/{\left|\overrightarrow{X}\right|}^{4}$. Multiplying both sides by |

*⃗X*|

^{4}and performing another average (with respect to the square modulus |

*X⃗*|

^{2}), we find that ${Q}_{jhkm}=\mathcal{E}\left[{X}_{h}^{*}{X}_{k}{X}_{m}{X}_{j}^{*}\right]/\mathcal{E}\left[{\left|\overrightarrow{X}\right|}^{4}\right]$. Since all components of

*X⃗*are statistically independent standard complex Gaussian variables, the numerator is 4 (

*δ*+

_{hk}δ_{jm}*δ*), where

_{hm}δ_{jk}*δ*is Kronecker’s delta function. The denominator is the second moment of a chi-square random variable having 4

_{ij}*N*degrees of freedom and hence its value is 4

*N*(4

*N*+ 2). This yields

*N*component generalization of the well-known Manakov-PMD equation describing the propagation in a single-mode optical fiber with random polarization coupling [13]. Indeed, when

*N*= 1, we have

*C*

_{1221}=

*C*

_{2112}= 2/3 and

*C*

_{1111}=

*C*

_{2222}= 1 [11], yielding

*κ*= 8/9, as expected [13]. The non-diagonal terms in the matrices

**B**

^{(1)}and

**B**

^{(2)}are due to the frequency dependence of the mode coupling, which is typically only a small correction to the coupling contained in the frequency independent part

**B**

^{(0)}. By neglecting these terms and transforming Eq. (4) to a reference frame evolving with

**B**

^{(0)}, the final form of the generalized Manakov equation is obtained

*β*′ and

*β*″ are the inverse group velocity and the dispersion coefficient, respectively. The form of Eq. (5) is the multi-component version of the familiar Manakov equation [2].

## 3. Results

The generalized Manakov equation (5) is integrable by the inverse scattering transform [7], and hence it admits the propagation of solitons. Unlike generic solitary waveforms that were shown to exist in multi-mode fibers with certain parameter combinations [9], true solitons can only exist in the strong coupling regime, where propagation is accurately described by Eq. (5). Thus, in order to test the accuracy of Eq. (5) in describing multi-mode propagation, we verify that the fundamental soliton waveform, which is a rigorous analytical solutions of Eq. (5), indeed forms a solitary solution when the coupled NLSE (1) are solved numerically in the strong mode-coupling regime. In addition, we check that when two such waveforms are launched into different fiber modes and at different central optical frequencies, they collide elastically without leaving a trace in the form of a dispersive wave. As noted earlier, this phenomenon is a distinctive feature of solitons [1] distinguishing them from generic solitary waves. We find that Manakov solitons can be observed when the length-scale characterizing the correlation of mode coupling is smaller than the soliton length *L _{s}* =

*τ*

^{2}/|

*β*″| (related to the soliton period

*z*by

_{s}*z*=

_{s}*πL*/2).

_{s}In the simulations we consider propagation in the two degenerate LP_{11} modes of a step-index optical fiber,
${\text{LP}}_{11}^{(a)}$ and
${\text{LP}}_{11}^{(b)}$ [14]. We used a core radius of 7.5*μ*m, a core refractive index of 1.4621, a refractive index step of 9.7 × 10^{−3} and a dispersion coefficient *β*″ = −25ps^{2}/km. The nonlinear coefficient in our computations was *n*_{2} = 2.6 × 10^{−20}m^{2}W^{−1}, corresponding to *γ* ≃ 0.835 W^{−1}km^{−1}, *κ* ≃ 0.76, and the effective area was of 126*μ*m^{2} for the fundamental mode.

In Fig. 1(a) we consider the case in which we launch the waveform describing the fundamental soliton of the Manakov equation (5),
${P}_{0}^{1/2}\text{sech}(t/\tau )$ with *P*_{0} = |*β*″|/(*γκτ*^{2}), into each of the two modes
${\text{LP}}_{11}^{(a)}$ and
${\text{LP}}_{11}^{(b)}$. The two launched soliton waveforms are separated by *T* = 30*τ* in time and by Δ*ω* = 0.4/*τ* in frequency. As random coupling of polarizations within each spatial mode is always very fast [3], the launch polarization states of the two waveforms is immaterial. We assume a soliton half-width *τ _{h}* = 52.9ps corresponding to

*τ*= 30ps. The soliton length is

*L*≃ 36km and the peak power

_{s}*P*

_{0}= 44mW. The correlation length of the mode coupling was taken to be

*L*= 100m (

_{c}*L*/

_{c}*L*≃ 2.8 × 10

_{s}^{−3}). It is evident that the launched waveforms are indeed solitary solutions in this regime and the collision between them is elastic, as expected from true soliton behavior. The collision dynamics shown in Fig. 1(a) is indistinguishable from that obtained integrating the Manakov equation with the same initial conditions.

In Fig. 1(b) we consider the regime in which the spatial modes are perfectly isolated so that no coupling between them occurs, although full coupling occurs between polarizations in each mode. In this case we launch the same sech-waveform, but with *P*_{0} = |*β*″|/(8*γC*_{3333}*γτ*^{2}/9), where *C*_{3333} ≃ 1.07 corresponds to self-phase modulation in the LP_{11} modes (see definitions around Eq. (2), where 3 is the index of mode
${\text{LP}}_{11}^{(a)}$). Since the launched waveform is solitary for the individual modes
${\text{LP}}_{11}^{(a)}$ and
${\text{LP}}_{11}^{(b)}$, the pulses propagate unperturbed until their collision. Yet, upon collision they stick to each other, forming a new (and non-solitary) pulse, while some energy is lost in the form of dispersive radiation. This behavior is in clear contrast with the elastic collision observed in Fig. 1(a), which corresponds to the strong coupling regime, in which the generalized Manakov equation holds.

In Figs. 2(a) and 2(b) we consider cases where the correlation length of the random mode coupling is *L _{c}* = 10km (

*L*/

_{c}*L*≃ 2.8 × 10

_{s}^{−1}) and

*L*= 100km (

_{c}*L*/

_{c}*L*≃ 2.8), respectively. In the case of

_{s}*L*= 10km shown in Fig. 2(a), the pulse evolution is similar to that observed in the strong coupling case of Fig. 1(a), although some perturbations to the propagating waveforms can be observed due to the insufficient averaging of the random mode coupling in this regime. These perturbations are further exacerbated when the correlation length of the mode coupling is extended to 100km, as shown in Fig. 2(b). In both Figs. 2(a) and 2(b), the launched waveforms were Manakov-equation solitons, identical to those used in Fig. 1(a).

_{c}The fact that light propagating through degenerate modes in a multi-mode fiber satisfies the generalized Manakov equation can be exploited for the prediction of various properties of signal propagation. For example, the isotropy of the Manakov equation implies that collisions between two polarized pulses (i.e. pulses whose orientation *E⃗*(*t*)/|*E⃗*(*t*)| is time independent) can always be considered as occurring in a two dimensional (complex) subspace of the 2*N* dimensional space spanned by the entire set of modes [15]. Consequently, the description of two-pulse interactions becomes identical to the description of two-pulse interactions in single-mode fibers, allowing use of the conventional Stokes-space representation [16]. Pulse collisions can be modeled as a precession of the Stokes vector of each of the pulses about the Stokes vector of the other pulse, identically to the single-mode case described in [17, 18]. When the colliding pulses are orthogonal (implying that their Stokes vectors are antiparallel) their orientations in the 2*N*-dimensional space of the electric field remain unaltered. Conversely, when the input pulses are not orthogonal, their Stokes vectors are not aligned and precession in Stokes space implies that the pulses’ orientations are modified by the collision, and the distribution of the overall optical power among the various fiber modes changes. This effect, which could not be easily predicted using Eq. (1), may be of importance to space and wavelength multiplexed systems when optical nonlinearities are non negligible.

## 4. Conclusions

To conclude, we showed that propagation in a degenerate group of randomly coupled spatial modes of a multi-mode optical fiber is the first reported physical scenario described by the generalization [7] of Manakov’s equation [2], and hence admits vector soliton solutions. The nonlinear parameter of the equation was expressed in terms of the standard parameters of a generic optical fiber. This equation constitutes a starting point for analytical and numerical studies of nonlinear effects in multi-mode transmission. Its importance is emphasized by the massive recent interest in spatially multiplexed transmission using multi-mode and multi-core optical fibers.

## Acknowledgment

This work has been carried out within an agreement funded by Alcatel-Lucent in the framework of Green Touch. MS also acknowledges financial support from Tera Santa Consortium and from the University of L’Aquila under project Re.C.O.Te.S.S.C. – P.O.R. Regione Abruzzo F.S.E. 2007–2013 Piano 2007–2008.”

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