Modulational Instability in Optical Fibers with Randomly-Kicked Normal

Modulational instability in optical fibers with randomly-kicked normal dispersion G Dujardin, A Armaroli, Simona Nodari, A Mussot, A Kudlinski, S Trillo, M Conforti, Stephan de Bièvre

To cite this version:

G Dujardin, A Armaroli, Simona Nodari, A Mussot, A Kudlinski, et al.. Modulational instability in optical fibers with randomly-kicked normal dispersion. Physical Review A, American Physical Society, 2021, 103 (5), pp.053521. 10.1103/PhysRevA.103.053521. hal-03157350v2

HAL Id: hal-03157350 https://hal.archives-ouvertes.fr/hal-03157350v2 Submitted on 10 Jun 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW A 103, 053521 (2021)

Modulational instability in optical ﬁbers with randomly kicked normal dispersion

G. Dujardin ,1 A. Armaroli ,2 S. Rota Nodari ,3 A. Mussot,2 A. Kudlinski,2 S. Trillo,4 M. Conforti ,2,* and S. De Bièvre 1,† 1Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlevé, Inria, F-59000 Lille, France 2Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France 3Institut de Mathématiques de Bourgogne (IMB), CNRS, UMR 5584, Université Bourgogne Franche Comté, F-21000 Dijon, France 4Department of Engineering, University of Ferrara, I-44122 Ferrara, Italy

(Received 3 March 2021; accepted 7 May 2021; published 24 May 2021)

We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength. This perturbation leads to the appearance of low-frequency MI side lobes that grow with the strength of the perturbations, whereas they are faded by randomness in their position. If the random perturbations exhibit a finite average value, they can be compared with periodically perturbed fibers, where parametric resonance tongues (also called Arnold tongues) appear. In that case, increased randomness in the strengths of the variations tends to affect the Arnold tongues less than increased randomness in their positions.

DOI: 10.1103/PhysRevA.103.053521

I. INTRODUCTION constitutes an idealization that does not always provide a relevant modeling of the randomness that may occur in physical The combined effect of nonlinearity and group-velocity GVD fibers [30]. An attempt to consider a GVD perturbed by dispersion (GVD) may lead to the destabilization of the sta- a Gaussian noise with a finite correlation length was reported tionary states (plane or continuous waves) of a given physical in Ref. [31], but the analysis was not conclusive, and the system. This phenomenon, known under the name of mod- problem was solved only with numerical simulations [32]. ulational instability (MI), consists in the exponential growth The question then arises for which type of random GVD of small harmonic perturbations of a continuous wave [1]. MI processes does MI occur and how do the characteristics of has been pioneered in the 60s in the context of fluid mechanics such instabilities depend on the statistical properties of the [2,3], electromagnetic waves [4] as well as in plasmas [5], and process? In such generality, the question seems, however, out it has been observed in nonlinear fiber optics in the 80s [6]. In of reach and it is consequently of interest to study the problem uniform fibers, MI arises for anomalous (negative) GVD, but in a class of random fibers that is both experimentally accessi- it may also appear for normal GVD if polarization [7], higher- ble and theoretically tractable. Our focus here will therefore order modes [8], or higher-order dispersion are considered be on homogeneous fibers with a normal GVD perturbed [9]. A different kind of MI related to a parametric resonance by a set of random “kicks.” More specifically, the fibers we mechanism emerges when the dispersion or the nonlinearity consider, described in more detail in Sec. II, have a GVD of the fiber are periodically modulated [10–13]. Many studies given by were published to address generalizations to high-order dispersion [14,15], birefringence [16], fiber cavities [17–23], and − β = β + β λ δ z Zn . the nonlinear stage of MI [24]. 2(z) 2,ref 2 n (1) Zref The effect of a random variation of GVD on MI has been n∈Z studied extensively [25–29] for the particular case where the In our previous work [33], we investigated such fibers with GVD is perturbed by a Gaussian white noise, which is ex- a periodic modulation of the fiber dispersion induced by a plicitly solvable. Under these conditions a deformation of the Dirac comb, in which the Z = nZ are periodically placed conventional MI gain profile due to the random perturbation n ref along the fiber and the strengths λ of the δ functions are was found when the unperturbed fiber has an anomalous n all equal. Such GVD was shown to be well approximated in dispersion. In the case of normal dispersion, the generation experiments by a periodic series of short Gaussian-like pulses. of MI sidebands as the result of the random perturbation In this work, the Z are chosen to be random points along the was reported as well. White noise, however, which implies n fiber, and the λ are independent and identically distributed arbitrarily high variations of GVD on arbitrarily small scales, n centered random variables and Zref is a reference length. We limit our analysis to perturbations of fibers with a normal GVD because in that case the unperturbed fibers show no MI, *[email protected] and consequently any MI observed in the randomly perturbed †[email protected] fibers is entirely due to the randomness.

The perturbations of the GVD considered in Eq. (1) result the (in)stability of its fixed point at the originq ˆ = 0 = pˆ as in a nonautonomous nonlinear Schrödinger equation (NLSE) a function of the frequency ω and of the properties of β2(z). [see Eq. (2)] determining the evolution of the wave profile as Since, for general β2(z), its explicit solution cannot be com- a function of the longitudinal coordinate z along the fiber in puted analytically, this is not straightforward. We concentrate which the perturbation can be interpreted as a succession of on random fiber profiles, as detailed below. kicks taking place at points z = Zn. These systems therefore Note that if γ = 0, Eq. (3) is reminiscent of a har- β = 2 (z) ω2 bear an analogy to the paradigmatic problem of the kicked monic oscillator with random frequency k 2 (also rotor in classical and quantum mechanics [34–36], which is called multiplicative noise) for which a vast literature exists why we refer to them as randomly GVD kicked fibers. [30,39–43]. The focus of the present work is to study a “gen- We show that MI occurs in such fibers through a mech- eralized” random oscillator (including the term 2γ P) which anism that is familiar from Anderson localization theory accounts for the nonlinear effects, where the random fre- [37,38]. The MI gain can indeed be computed in terms of quency is modeled by the nonstationary stochastic processes a random product of transfer matrices in the same way as described by Eq. (1). The study of stationary colored noise, the localization length of the stationary eigenfunctions of the where classic perturbative techniques apply [30], will be the random Schrödinger operator in the Anderson model. We are subject of a future work [44]. then able to analyze how the properties of this gain depend on The modulational instability of the fiber is expressed in the features of the random process β2(z) and on the frequency terms of the sample MI gain G(ω), defined as follows: of the harmonic perturbation of the continuous wave. 1 The rest of the paper is organized as follows. In Sec. II G(ω) = lim ln xˆ(z,ω), (4) →+∞ we precisely describe the randomly kicked fibers under study, z z then we derive a general expression for the mean MI gain in where · designates the Euclidean norm. Here sample Sec. III. In Sec. IV, we develop a perturbative estimation of stands for a single realization of the random perturba- the mean MI gain in the case where the random perturbations tion. When G(ω) > 0, this indicates xˆ(z,ω) exp[G(ω)z], of the GVD vanish on average (λn = 0) and compare it with meaning that the stationary solution is unstable for perturba- numerical results. We establish in this manner the existence tions with frequency ω. One is interested in establishing for of MI at low frequencies of which we characterize the proper- which ω, if any, this occurs, and how large G(ω) is in that ties. We finally compare the MI gain with the gain computed case. from the solution of the nonlinear Schrödinger equation and Dispersion-kicked fibers are characterized by the expres- observe a good correspondence. In Sec. VI we consider MI sion of β2(z)giveninEq.(1). Here δ is a sharply peaked δ = in a randomly kicked homogeneous fiber of normal GVD in positive function satisfying R (z)dz 1; in our theoretical which the random perturbation does not vanish on average analysis below, we take δ to be a Dirac δ function; λn are in- (λn = 0). The situation is very different since, depending on dependent, identically distributed real random variables with λ > the nature of the point process Zn determining the positions of mean 0, and Zref 0 is a characteristic length associated the kicks along the fiber, there may or may not be a remnant of with the random sequence of points Zn. We write Arnold tongues, a signature of MI in periodic fibers. In Sec. V λ = λ + εδλ , (5) we show how GVD kicked fibers can approximate fibers with n n a white noise GVD. Conclusions are drawn in Sec. VII. with δλn = 0 and ε>0 a dimensionless parameter. We think of this as a fiber with irregularities in its diameter of random area, giving rise to effective dispersion kicks |λ |β Z , II. MODULATIONAL INSTABILITY IN RANDOMLY n 2 ref placed at the points Z along the fiber. These fibers can be KICKED FIBERS n physically fabricated by means of the state-of-the-art fiber- We consider the NLSE drawing techniques. Indeed, some examples of uniformly 1 2 2 dispersion-kicked fibers have been reported in Ref. [33] with a i∂zu − β2(z)∂ u + γ |u| u = 0, (2) 2 t period Zref = 10 m and a kick width w = 0.14 m and relative large kick strength max |β (z)|/β , ≈ 35. where γ>0 is the fiber nonlinear coefficient, and β2(z) its 2 2 ref GVD. We are interested in the√ modulational (in)stability of the To compute the MI gain in such fibers, we proceed as fol- δ stationary solution u (z) = P exp(iPγ z)ofEq.(2). We con- lows: We first note that, due to the presence of the functions, 0 ± ω = ±,ω , ±,ω sider a perturbation of u (z)intheformu(z, t ) = [v(z, t ) + the left and right limitsx ˆn ( ) (qˆ(Zn ) pˆ(Zn )) around 0 = ,ω 1]u (z), where the perturbation v(z, t ) satisfies |v|1. Writ- z Zn of the solutionx ˆ(z )atZn are different and are 0 + ω = − ω ing v = q + ip, with q and p real functions, inserting this related byx ˆn ( ) Knxˆn ( ), where the random matrix Kn is defined as expression into Eq. (2) and retaining only the linear terms, we obtain a linear system for q and p. Writing x(z, t ) = ω2 ω2 cos β2Zref λn − sin β2Zref λn √1 −iωt = 2 2 . (q(z, t ), p(z, t )) andx ˆ(z,ω) = x(z, t )e dt, one finds Kn ω2 ω2 (6) 2π β λ β λ sin 2Zref n 2 cos 2Zref n 2 β − 2 (z) ω2 0 On the other hand, for Zn < z < Zn+1,(3) is autonomous and ∂ xˆ(z,ω) = β 2 xˆ(z,ω). (3) z 2 (z) ω2 + 2γ P 0 straightforwardly solved; the solutionx ˆ(z,ω) is smooth in this 2 − = + range. One findsx ˆn+1 Lnxˆn , where now Note that this is, for each ω, a nonautonomous linear −μ Hamiltonian dynamical system in a two-dimensional phase = cos (k Zn ) sin (k Zn ) , Ln −1 (7) plane with canonical coordinates (q ˆ, pˆ). We wish to study μ sin (kZn ) cos (kZn )

053521-2 MODULATIONAL INSTABILITY IN OPTICAL FIBERS … PHYSICAL REVIEW A 103, 053521 (2021) with When all λn = 0 the above ﬁbers are homogeneous. We β β β 2,ref ω2 concentrate here on the defocusing regime, in which no mod- 2 2,ref 2 2,ref 2 2 k = ω ω + 2γ P ,μ= . (8) ulational instability occurs when λn = 0. 2 2 k Some examples of realizations of the random-walk and of

Note that k is real for all ω when β2,ref > 0 (normal dispersion the Poisson fibers are reported in Fig. 1, where the kicks are or defocusing NLSE), and that it is imaginary for small ω modeled by sharp Gaussian functions. Here and in all numer- β = β = γ = = = when β , < 0 (anomalous dispersion or focusing NLSE). ical examples we take 2 2,ref P Zref 1. 2 ref λ To sum up, we can now describe the evolution of this system In Secs. III and IV we consider random kick strengths n = − = − ω ∈ that vanish on average. Our focus is therefore on the question: between z Zn and z Zn+1 as follows. For all R and n ∈ N, what kind of random inhomogeneities of the GVD can produce MI in an otherwise modulationally stable homogeneous − = − = −. xˆn+1 LnKnxˆn nxˆn (9) fiber? − = In Sec. VI we then briefly discuss the case λ>0: in that Considering an initial conditionx ˆ0 −,ω , −,ω ∈ 2 −= situation, MI occurs in the form of Arnold tongues in the (qˆ(Z0 ) pˆ(Z0 )) R with xˆ0 1, one then finds limiting case when εZ = 0, resulting in a periodic GVD. We 1 1 − investigate the stability of these Arnold tongues under the G(ω) = lim ln − ··· xˆ . (10) →+∞ n n 1 1 0 / λ Zref n n random perturbations of the Zn and or n which occur when ε = 0 and/or ε = 0. Note that Z 2γ P 0 μ 1,μ 1 − + O(ω−4), lim μ = 1. 2 ω→+∞ β2,refω III. THE MEAN MODULATIONAL INSTABILITY GAIN OF A RANDOMLY KICKED FIBER Hence it follows that for large ω, both Kn and Ln are rotation matrices and consequently their product n is also a rotation Since the random-walk process Zn in Eq. (12) has indepen- matrix. Consequently, for large ω, the MI gain tends to zero. dent and identically distributed increments, the Furstenberg We finally describe the models we will consider for the theorem [45] (see Ref. [38] for a textbook treatment) asserts random positions Zn of the δ functions, and for their strengths that the limit in Eq. (10) exists for almost every realization λ λn.FortheZn,weset of the fiber; that is, for almost every choice of the n and the j ; it is in addition strictly positive and independent of the = ∀ ∈ , = + , n Z0 0 n N Zn+1 Zn Zref jn (11) realization. As a result, in such random fibers, there is always ω where Zref > 0 and jn is a sequence of independent and identi- MI at all values of . However, it is notoriously difficult to ob- cally distributed positive random variables with values in R+ tain analytical expressions for G(ω) as function of the model and with probability density ρ( j ). We assume that parameters, and hence to assess the strength of the sample MI n ω +∞ gain G( ). We will therefore follow Ref. [28] and introduce a suitable mean MI gain based on moments of Eq. (9) which we jn = jρ( j)dj = 1, (12) 0 will refer to as G2(ω). To determine the gain, we need to determine the second- so that Zn = (Zn+1 − Zn ) =Zref > 0. Hence Zn+1 = order moments of the components of the vectorx ˆ− in Eq. (9). Zn + Zn is a random walk with drift. We will principally n For that purpose, let consider two cases. First, introducing a new parameter εZ > 0, we consider an bn = + ε δ , n ≡ LnKn = . jn (1 Z jn ) (13) cn dn where now δjn is a sequence of independent, identically, and It is then straightforward to check that, for all n ∈ N, uniformly distributed random variables in (−1, 1) and εZ ∈ , ⎛ ⎞ ⎛ ⎞ [0 1]. In this random walk model the mean position of Zn is − 2 − 2 nZ and their variance qˆ(Zn+1) qˆ(Zn ) ref ⎝ − − ⎠ ⎝ − − ⎠ X + := qˆ(Z )ˆp(Z ) = M qˆ(Z )ˆp(Z ) , (15) − 2 = ε2 2 δ 2 , n 1 n+1 n+1 n n n (Zn nZref ) n Z Zref ( j) − 2 − 2 pˆ(Zn+1) pˆ(Zn ) grows with n. The increments are independent and identically distributed and their variance is given by where 2 2 2 2 ⎛ ⎞ (Zn − Zref ) =Z ε (δ j) . 2 2 ref Z an 2anbn bn = ⎝ + ⎠. We refer to this as the simple random-walk model. Mn ancn andn bncn bndn (16) c2 2c d d2 Second, we consider the Poisson model where the Zn = n n n n Zref jn are independent and identically distributed with expo- | − − | 1 { − 2 + − 2} nential density, Since qˆ(Zn )ˆp(Zn ) 2 [ˆq(Zn )] [ˆp(Zn )] ,wehave xˆ−2 X 3 xˆ−2, and it readily follows that ρ( j) = exp (− j). (14) n n 1 2 n

The Zn are now the arrival times of a Poisson process with 1 1 −1 G(ω) = lim ln X , →+∞ n 1 parameter Zref . We refer to this as the Poisson ﬁber. 2Zref n n

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FIG. 1. Realization of a simple random-walk fiber (left panel) and a Poisson fiber (central panel) with zero-mean kick strength λ¯ = 0, and of a simple random-walk fiber with λ¯ = 1 (right panel). The kicks are modeled with Gaussian pulses δ(z) = 1/(2πw2 )1/2 exp(−z2/ 2w2 )of width w = Zref /20. The position of the kicks is indicated by the vertical dotted lines; λn are uniform random variables in [−1, 1]. where now (x, y, z)1 =|x|+|y|+|z|. From Jensen’s in- is independent of n. Hence equality [46], it follows that 1 1 G2(ω) = lim ln Xn1 1 1 2Z n→+∞ n G(ω) = lim ln X ref →+∞ n 1 2Zref n n 1 1 n 1 = lim ln M X01 = ln |x|, (19) 1 1 2Z n→+∞ n 2Z lim ln X = G (ω). (17) ref ref →+∞ n 1 2 2Zref n n where x is the eigenvalue of M with largest modulus. We refer to G2(ω) as the mean MI gain. It is worth pointing The matrix M depends on the parameters of the model, in out that this corresponds to one half the growth rate of the particular on ω, the laws of λn, jn, ε and εZ . It follows that average power of the perturbations. G2(ω) can be computed from the spectrum of M, which is Equation (17) shows that the mean MI gain G2(ω) is larger easily determined numerically. The explicit analytical compu- than the sample MI gain G(ω) itself. However, it has the tation of closed formulas for its spectrum remains however distinctive advantage that it can be readily computed. For that complicated. As we show below, a perturbative treatment | − − | | − − | purpose, note that, since qˆ(Zn )ˆp(Zn ) qˆ(Zn )ˆp(Zn ) and yields an analytic expression provided that the perturbation | − − | 1 {| − |2 +| − |2} is not too large. qˆ(Zn )ˆp(Zn ) 2 qˆ(Zn ) pˆ(Zn ) we have Dropping the index n on λn, Zn, we introduce X X 3 X . n 1 n 1 2 n 1 ω2 θ = β2Zref λ , (20) Since the Mn are identically distributed and mutually indepen- n 2 X = M X − = M X dent, one has moreover n n n 1 0, where which corresponds to the phase acquired by the perturbation ⎛ ⎞ 2 2 vˆ (ω) at each kick, and write an 2anbn bn = ⎝ + ⎠, 2 2 M ancn andn bncn bndn (18) M = M+ cos θ + M0 sin (2θ ) + M− sin θ, (21) c2 2c d d2 n n n n where ⎛ ⎞ cos2(kZ) −μsin(2kZ) μ2sin2(kZ) ⎜ sin(2kZ) − μ sin(2kZ) ⎟ M+ = ⎝ 2μ cos(2k Z) 2 ⎠, sin2 (kZ) sin(2kZ) 2 μ2 μ cos (k Z) ⎛ ⎞ μ2sin2(kZ) μsin(2kZ) cos2(kZ) ⎜ − μ sin(2kZ) − sin(2kZ) ⎟ M− = ⎝ 2 cos(2k Z) 2μ ⎠, 2 − sin(2kZ) sin2 (kZ) cos (k Z) μ μ2 and ⎛ μ μ ⎞ − sin(2k Z) μ2 sin2(kZ) − cos2(kZ) sin(2k Z) ⎜ 2 2 ⎟ = cos(2kZ) − μ + 1 sin(2kZ) − cos(2kZ) . M0 ⎝ 2 μ 2 2 ⎠ sin(2kZ) 2 − sin2 (kZ) − sin(2kZ) 2μ cos (k Z) μ2 2μ

Note that all information on the randomness in the strengths the spacings between the kicks is encoded in the matrices of the kicks is contained in θ, whereas the randomness in M0, M±. From the independence of the random variables λn

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FIG. 2. Comparison of numerically computed MI gains in randomly kicked homogeneous fibers with λ = 0; εZ and ε are indicated in each panel. The random kicks are located at Zn as in Eq. (13) (random walk). The δλn and δjn are taken to be uniform in [−1, 1]. Shown are the numerically simulated sample MI gain G(ω) (dashed blue line), the mean MI gain G2(ω) computed directly from the largest eigenvalue of M (dotted red line), the perturbative approximation of the mean MI gain (dash-dotted black line), and the perturbations’ growth rates (black stars) calculated from numerical solutions of the NLSE. and Zn it follows that The MI gains of such random fibers with Zn as in Eqs. (13) and (14) are illustrated in Fig. 2 (for the random-walk model) and = 2 θ + θ + 2 θ. M M+cos M0sin 2 M−sin (22) in Fig. 3 (for the Poisson model) for various parameter values ε ε The expressions of the coefficients of the matrix M are and Z as indicated. It is worth reminding the reader that reported in Appendix B; the (in)stability depends on its the sample MI gain is independent of the sample if calculated spectrum. at infinite z. In the numerics we calculated an approximation = For a homogeneous randomly kicked fiber with average of the sample MI gain from Eq. (10)forn 250 kicks and normal GVD and vanishing mean kicking strength (λ = 0) we averaged over 500 realizations (dashed blue curves in the fig- resort to a perturbative analysis, presented below in Sec. IV. ures). To check that the sample MI gain correctly predicts the growth rate of the perturbations we computed the latter from a numerical solution of the NLSE using the same procedure IV. ZERO AVERAGE KICK AMPLITUDE (black stars in the figures). One observes a good agreement. We consider in this section randomly kicked fibers as in One notices that the random perturbation produces modu- Eq. (1) with λ as in Eq. (5) and λ = 0 and with Z a random lational instabilities that we now further analyze, for a given n n λ ω process as in Eq. (11), so that the random fiber can be seen as distribution of the Zn and the n, and at fixed . This can a perturbation of a homogeneous defocusing fiber (see Fig. 1). be done perturbatively in ε. Indeed, since λ = 0, we have We recall that the latter is known to be modulationally stable. λ = εδλ, which is small. We can therefore compute the largest

FIG. 3. Same as Fig. 2,butwithZn as in Eq. (14) (Poisson model). The parameter ε is indicated in the panels.

053521-5 G. DUJARDIN et al. PHYSICAL REVIEW A 103, 053521 (2021) eigenvalue of M perturbatively in ε; its logarithm will yield where ω the mean MI gain G2( ). The details of the computation are (1 − μ2 )2 given in Appendix A. x(1) =− , 4μ2 To set up the perturbation problem, we proceed as follows: 2 For ε = 0, M = M+ so that the first ingredient we need are 1 2S(2S − 1) + S x(2) = (1 − μ4)2 2 , (31) the eigenvalues and eigenvectors of M+. It is easy to check μ4 2 + 2 16 4S S2 that M+ has the eigenvalue x0 = 1. The corresponding right and left eigenvectors are with 2 1 (0) 2 T (0) 2 T S = kZ = ( − kZ ), S = kZ . ϕ = (μ 01) and ψ = (10μ ) , (23) sin ( ) 2 1 cos(2 ) 2 sin(2 ) (32) with ψ (0)T ϕ(0) = 2μ2. We always suppose ω = 0 so that μ = 0. The two remain- The corresponding mean MI gain is then, using Eq. (19), ing eigenvalues of M+ are easily determined to be 1 1 (1) 2 (2) +∞ G2(ω) = ln xη ≈ ln(1 + ηx + η x ). (33) 2Zref 2Zref x± = exp (±i2kZ) = exp (±i2kZref j)ρ( j)dj. (24) 0 To lowest order in εω2, we therefore find that, approximately, | | Since the exp(i2kZref j) lie on the unit circle, clearly x± for small ω, 1. To apply nondegenerate perturbation theory, we need that = − μ2 2 β2Z2 γ P x± 1. To investigate this condition in the two models that 1 (1 ) 1 2 ref 2 2 G2(ω) ≈ |η| ≈ (εω) δλ . 2 2 ω we investigate here for the Zn, let us first consider the random- εω 1 2 4μ 1 4 β2,ref δ walk model (13) and assume that the distribution of j is given (34) by a density σ so that The mean MI gain G2(ω), computed by evaluating the ±i2kZ ±2ikZ ε s x± = exp (±i2kZ) = e ref e ref Z σ (s)ds. (25) spectral radius of M numerically, as well as its approximation from Eq. (33), are shown in Fig. 2 for the simple random-walk We can then compute x± perturbatively in εZ as follows: Not- model and in Fig. 3 for the Poisson model. Note that the term ε = = ± 2 ing that, when Z 0, x± exp( i2kZref ) and remembering in η in (33) is crucial to obtain a good approximation of G2 that δj =0, one sees for values of ω in the range considered. There is, in both cases, 2 2 3 a MI side lobe. The perturbative approximation works well x± = exp (±i2kZref ) 1 − 2(kZref εZ ) δ j +O ε , (26) Z for the Poisson model, for all ω in the range considered (see ε so that, with increasing Z , the eigenvalues x± move radially dashed black curves in Fig. 3). It does not, however, capture inward, towards the origin. Hence |x±| < 1 and, a fortiori, the vanishing of the MI that occurs at a specific value of ω for ± = = ε = x x0 1for Z 0. For a Poisson fiber, where the Zn εZ = 0 or small in the random-walk model (see Fig. 2). There form a Poisson process, the eigenvalues x± can be computed is indeed a marked difference between the shape of these side explicitly: lobes, depending on which of the two random processes are 1 chosen for the Zn, which we now explain. x± = , (27) 1 ∓ i2kZ For that purpose, first consider the leftmost column of ref Fig. 2. There, ε = 0, which means the Z = nZ are dis- | | < ω = Z n ref so that x± 1 for all 0. tributed periodically along the fiber. The GVD is nevertheless We can therefore use, in the above cases, a nondegenerate not periodic, since the kick strengths λ are random. One ε n perturbation expansion to compute x0 as a function of .We notices on the figure that, in that case, both the sample MI gain ε will establish that x0 is real and a growing function of ,giving and the mean MI gain show a characteristic zero at a precise rise to a strictly positive mean MI gain. value of ω. This phenomenon can be explained as follows: Recalling that λ = εδλ [see Eq. (5), with λ = 0], we con- If ω = ω is such that kZref = π for some ∈ Z, then the sider the case where the probability distribution ν(δλ)ofδλ matrices Ln in Eq. (7) are all equal to the identity matrix, satisfies ν(−δλ) = ν(δλ) so that not only λ = ε sν(s)ds = and one immediately sees from Eq. (10) that G(ω) = 0, for 0 but also sin(2θ ) = 0, since sin θ is an odd function of δλ. all ε. Indeed the propagation through the constant dispersion Hence, we can write segments of the fiber does not change the perturbation and the M = M+ + Mη, (28) kicks act as random rotations, which only change the phase √ of the perturbationv ˆ, giving as a results a vanishing MI gain. η = θ − = π ν εβ ω2 − ν where cos(2 ) 1 2 [ˆ( 2Zref ) ˆ (0)] and Using Eq. (8), one sees that this corresponds to the specific 1 = + − − . ν M 2 (M M ) Here ˆ designates the Fourier trans- values of ω given by ω = ω, where form of ν. ⎛ ⎞ Finally, for small εω2, π 2 2 2 ⎝ 2 ⎠ ω = (γ P) + − γ P . (35) η −1 εβ ω2 2δλ2. β , Z 2 ( 2Zref ) (29) 2 ref ref

Therefore, the eigenvalue xη of M in Eq. (28) that emanates One furthermore readily checks that the eigenvalues of M from x = 1 can be expanded for small η as 0 in this case are given by 1, cos θ, sin θ, which are less than (1) 2 (2) xη ≈ 1 + ηx + η x , (30) or equal to one in absolute value. Therefore the mean MI

053521-6 MODULATIONAL INSTABILITY IN OPTICAL FIBERS … PHYSICAL REVIEW A 103, 053521 (2021) gain G2(ω) also vanishes for all ε at ω = ω.Again,thisis V. APPROXIMATING WHITE-NOISE GROUP-VELOCITY apparent from Fig. 2 at ω = ω1. DISPERSION WITH RANDOMLY KICKED Interestingly enough, the frequencies determined from GROUP-VELOCITY DISPERSION Eq. (35) fulfill the parametric resonance condition kZ = π ref Previous work on MI in random fibers has concentrated on [12,24,33]. These frequencies correspond to the location of GVD perturbed by white noise [26–28]. As pointed out above, the tips of Arnold tongues for any periodic fiber with pe- white noise is not necessarily physically pertinent since it re- riod Z and whose average GVD over one period equals ref quires arbitrarily large variations of the GVD over arbitrarily β , . Quite surprisingly, for kicks with random amplitudes 2 ref short distances. In this section we show that the kicked fibers and zero-mean, the situation is reversed and the system be- considered here can, in an appropriate parameter regime de- comes stable under perturbations precisely at these same termined below, and for sufficiently low frequencies, produce frequencies. a similar MI gain as white-noise GVD. Figure 4 illustrates our It is finally clear from Fig. 2 that the perturbative treat- findings. ment of the mean MI gain does not function well when For that purpose, we compute the two-point function ω approaches ω1 or ω2. This is as expected since, when [β2(z) − β2,ref[[β2(z ) − β2,ref] of a kicked GVD as in (1), ω = ω, the three eigenvalues of M+ coincide: x± = 1 = x . 0 with λ as in (5) and λ = 0. The nondegenerate perturbation theory used above does not n Using that the λn are independent and denoting by ρn(z) then apply. When ω = ω, this degeneracy is lifted and the the probability distribution function of Zn, one finds perturbation theory yields increasingly good results as εZ increases, even at ω = ω. As can be seen in the second [β (z) − β , ][β (z ) − β , ] 2 2 ref 2 2 ref column of Fig. 2, both the sample and mean MI gains are still diminished in the neighborhood of ω1 and ω2 when εZ is 2 2 = (β2Zrefε) δλ ρn(z) δ(z − z ). small, and this is well captured by the perturbative analysis n above. This phenomenon can be seen as a remnant of the underlying periodic structure of the random points Z that is Note that the fiber is therefore delta correlated in z,butit n ρ only partially destroyed when εZ is nonzero, but small. It com- is not stationary, since n n(z) is not a constant function. pletely disappears when ε approaches its maximal possible Nevertheless, one finds Z value, which is 1, as can be seen in the third column of the −1 lim ρn(z) = Z , figure. z→+∞ ref n Since for the homogeneous fiber [β2(z) = β2,ref, εZ = 0 = ε] there is no MI at all, it is clear that all MI is created by the so that it does becomes stationary for large z.Toshow randomness. Note, however, that, whereas increased fluctua- this last statement, one can proceed as follows: Introducing λ the random variable N(Z) = {n | Zn Z}, which counts the tions in the kicking strengths n increases the MI, increased Z number of Zn below Z, one sees that N(Z) since the fluctuations in the Zn tends to decrease it. Zref In Fig. 3 the MI for the Poisson model is illustrated. One mean distance between two successive values of Zn isZref .On = Z = δ − sees that, as for the simple random-walk model, there is a MI the other hand, N(Z) 0 n(z)dz, where n(z) m (z . side lobe starting at low frequencies, but the frequencies ω Zm ) So, since now no longer play a special role. The gain of this side lobe dN dN is comparable in width and height for the same value ε = 0.5 n(z) = (z), n(z) = (z), dz dz of the strength of the kicks, as in the random-walk fiber with ε = . = ρ Z 0 9. Note that, in the latter, this means successive Zn can it follows from n(z) m m(z) that be close, as in the Poisson fiber. The perturbative treatment of the previous section reproduces the mean MI gain quite dN 1 lim ρn(z) = lim n(z) = lim (z) = . accurately. z→+∞ z→+∞ z→+∞ dz Z n ref

FIG. 4. Comparison between the mean MI gain of the white-noise model (black dots) and the mean MI gain of the random-walk (left√ and center panels) and Poisson models (right panel). MI gain for white-noise model is calculated from Eq. (29)fromRef.[25] with A = 1(A ≡ P 2 2 in our notation) and σ = 0.1. Parameters ε and εZ as indicated and Zref as in Eq. (37) with δλ = 1/3.

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In conclusion, for z large, one has driven systems. The tongues appear as a sequence of expand- 2 ing unstable regions in the parameter space of the forcing [β (z) − β , ][β (z ) − β , ] ≈ 2σ δ(z − z ), 2 2 ref 2 2 ref frequency and amplitude. The first such Arnold tongue can where be observed in the top-left panel of Fig. 5. In the other panels ε = 2 2 2 of the first column, 0 so that the kick strengths are now 2σ = (β2ε) Zrefδλ . (36) random, while the Zn remain periodic. The corresponding One recognizes here the two-point function of Gaussian fibers can therefore be viewed as random perturbations of a white noise. In a kicked fiber, the fluctuations take place on periodic fiber. It appears from these data that the fluctuating a length scale comparable to Zref and have a strength propor- kick strengths only weakly affect the position and strength of tional to ε. This suggests that, if Zref is small and ε large, with the Arnold tongues. In the other columns of Fig. 5,onthe a scaling given by other hand, MI gains are shown for fibers for which there is randomness in both kick positions Zn and kick strengths Z = 2σ 2 β2ε2δλ2 , (37) ref 2 λn. It appears from these data that randomizing the positions then the kicked fiber will be statistically close to a white-noise of the kicks has a much stronger effect than randomizing fiber and the resulting MI will therefore be similar in both their strengths. As the randomness increases, one observes a fibers. This is indeed illustrated in Fig. 4. The mean MI gain marked decrease in the MI gain, with a widening of the Arnold of a white-noise fiber is plotted there (black circles) using tongue, while its position is less affected. Note that the MI is λ = an explicit formula for this gain obtained in Ref. [25]. It is considerably larger in these fibers with 1 than with those λ = compared with the mean MI gain of randomly kicked fibers, having 0 as can be seen by comparing the vertical scale of Fig. 5 with that of Figs. 2 and 3. In the first case, we are with parameters ε and εZ as indicated and with Zref as in (37), computed from the largest eigenvalue of M. As suggested by dealing with random perturbations of a periodic fiber, which the above argument, for sufficiently large ε, the MI gain of displays MI due to parametric resonance. In the second, the the kicked fibers converges to the one of the white-noise fiber. MI is, on the contrary, generated by the randomness. The agreement is best for small ω, a reflection of the fact that In Fig. 6 the numerically computed sample and mean MI ε the low-frequency perturbations are less sensitive to the rapid gains are displayed for a Poisson fiber with values of as ε = variations of the white noise. indicated. For 0 one observes the identical vanishing of ω ≈ . We make two further comments: First, the value of σ 2 = these gains at 2 5. This is actually a more general phe- 0.1 is chosen in the numerics because it is the right order of nomenon, not related to the distribution of the kicks, that is magnitude for the fibers used in the experiments described in easily understood as follows: Let us consider a defocusing fiber with, for all n, λ = λ>0 and with arbitrary values Ref. [33] in which both ε and Zref are of order 1. Note that n this means that the mean spacing between the kicks in these for Zn. They could in particular be periodic, quasiperiodic, or ω fibers is of the same order of magnitude as the nonlinear length random. If we now choose n as = γ −1 ZNL ( P) . Second, for higher but intermediate values of ω2 ε n , the MI lobe of the randomly kicked fibers can be higher and nπ = λβ2Zref , (38) wider than the one of the white-noise model. Third, it would 2 be a challenge to make fibers with a considerably larger value then we immediately see from (6) that Kn =±I . In other ε 2 of since they correspond to a small value of Zref . This means words, for these values of ω, the kicks have no effect on the that one would need to be able to put the sharp peaks and dips linearized solution of the equation of motion. One therefore in the fiber diameter very closely together. has n =±Ln and so the sample MI gain is equal to the sample MI gain of the unperturbed fiber. Since the latter is de- VI. NONZERO AVERAGE KICK AMPLITUDE focusing,√ the sample MI gain vanishes. When n = 1, one finds ω = 2π ≈ 2.5. That the mean MI gain must also vanish In this section we briefly discuss the effect on MI of ran- follows from observing that, if the above condition is satisfied, domly placed kicks with random strengths λn that are not then θ = nπ in Eq. (20). Hence, Eq. (21) implies M = M+ vanishing on average, so that λ = 0. We separately consider and consequently M = M+. But we saw that the eigenvalues the two cases where the Zn form a simple random walk or a of M+ are 1, x±, with |x±| 1. Hence the mean MI gain also Poisson process and that lead to different phenomena. vanishes. We first consider the simple random walk, as in Eq. (13), an Let us stress that this phenomenon is not limited to the example of which is reported in Fig. 1, rightmost panel. In the Poisson fiber, for which it can be observed in Fig. 6. In fact, first column of Fig. 5 the numerically computed sample and for the simple random-walk fiber it is visible in the top panels mean MI gains are displayed for such a fiber for which ε = 0, Z of Fig. 5. The phenomenon also appears in periodically kicked so that Z = nZ , i.e., the δ kicks are placed periodically. The n ref fibers where λ are chosen as in Eq. (5) with average λ = 1. Note that, n if in addition ε = 0, then this fiber is actually periodic: this z − nZref β2(z) = β2,ref + β2λ δ , is the situation in the top-left panel of the figure. It is well Zref known that periodic fibers display MI through a parametric n resonance mechanism, whose gain in shaped in the form of so that the spatial average of the GVD is so-called Arnold tongues [12,16,47–50], which have been experimentally shown in periodically kicked fibers [13,33]. 1 Zref β2,av = β2(z)dz = β2,ref + β2λ. Arnold tongues are a well-known feature of parametrically Zref 0

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FIG. 5. Sample and mean MI gains for a simple random-walk process and λ = 1andδλn uniform in [−1, 1]. Color code is the same as in Fig. 2.

It is then well known that the Arnold tongues for such a ﬁber Condition (38) is complementary to condition (35). The ω occur at values n deﬁned by former arises when the spacing between the kicks is constant, whereas the second appears when the amplitude of the kicks ω 2 n 2 2 is constant. The two conditions show that, whenever either β2,avZref = (γ PZref ) + (nπ ) − γ PZref . 2 the position or the amplitude of the kicks are determinis- ω < tic, particular values of the frequency exist where the gain Comparing this to Eq. (38), one observes that, for all n, n ω ω ω λ vanishes. n. Note that both n and n depend on .

FIG. 6. Sample and mean MI gains for a Poisson model with λ = 1. Color code is the same as in Fig. 2.

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VII. DISCUSSION AND CONCLUSIONS where ϕ(0), ψ (0) are defined in Eq. (23). Expanding the eigen- η We reported on the modulation instability phenomenon in value equation in powers of leads in the usual manner to the optical fibers where the GVD is a random process. Most stud- first correction to the eigenvalue: (0)T (0) ies of such random fibers have concentrated on the case when 1 ψ (M+ − M−)ϕ a homogeneous GVD is perturbed by stationary white noise. x(1) = 2 ψ (0)T ϕ(0) In that case various methods exist to compute the MI gain. White noise is, however, very particular and not always ade- 1 (1 − μ2 )2 = [2μ2 − (1 + μ4 )] =− . quate to model physically relevant scenarios. In this paper we 4μ2 4μ2 investigated the behavior of the MI gain in random fibers for which the GVD is of a very different nature. More specifically, Note that this correction does not at all depend on the distri- λ we considered a class of experimentally realizable [33] ran- bution of Zn nor of n. dom fibers in which a constant normal GVD is perturbed by a To obtain a satisfactory expression for the mean MI gain ω (2) sequence of δ kicks with random positions and amplitudes. G2( ), we need to obtain the second-order correction x . ϕ(1) The main result of our analysis is fourfold. First, we show For that purpose, we need the first correction to the that, in this situation, low frequency MI lobes always arise as eigenvector corresponding to xη, which is given by a result of such random perturbations, independently of the (1) 1 T (1) T (1) ϕ = ψ ϕ ϕ+ + ψ ϕ ϕ− , statistical distribution of the position and amplitudes of the T [( + ) ( − ) ] ψ+ ϕ+ kicks. We trace the occurrence of MI in random fibers to a mathematical mechanism familiar from the study of Ander- where son localization and of the occurrence of positive Lyapunov ψT ϕ(0) T (1) ± ( M) (0)T (1) exponents in chaotic dynamical systems, namely, the growth ψ± ϕ = ,ψ ϕ = 0. (A2) 1 − x± of the product of random two-by-two matrices, as described T by the Furstenberg theorem [45]. Second, we show that the Here ϕ± are the eigenvectors of M+ and of M+ corresponding specific shape of these side lobes depends on these statistical to the eigenvalues x±, given by properties and we provide expressions to determine them. In 2 T particular, we find that, if either the positions or the amplitudes ϕ± = (−μ ±iμ 1) , of the kicks are deterministic, then at special frequency values 2 T ψ± = (1 ±2iμ −μ ) , the MI gain is identically zero. At these same points, the MI (A3) T 2 remains suppressed when the random fluctuations of the con- with ψ± ϕ± =−4μ . Finally, the second-order correction to trol parameter remain small enough. Third, we show that the the eigenvalue is randomly kicked fibers considered behave, in a suitable pa- ψ (0)T MRMϕ(0) rameter regime that we identify, as a fiber with a white-noise x(2) = , (A4) GVD. Finally, we have observed that, for comparable param- ψ (0)T ϕ(0) eter regimes, the MI produced through parametric resonance with in fibers with a periodic GVD is considerable larger than the ϕ ψT ϕ ψT ϕ ψT MI gain obtained from random perturbations of the fibers. = 1 + + + − − = 2 + + . R T T Re ψ+ ϕ+ 1 − x+ 1 − x− ψ+ ϕ+ 1 − x+ ACKNOWLEDGMENTS Hence (0)T T (0) The work was supported in part the by the French govern- 2 ψ Mϕ+ψ+ Mϕ x(2) = Re ment through the Programme Investissement d’Avenir with (0)T (0) T ψ ϕ ψ+ ϕ+ 1 − x+ the Labex CEMPI (Grant ANR-11-LABX-0007-01) and the (0)T T (0) I-SITE ULNE (Grant ANR-16-IDEX-0004 ULNE, projects 1 ψ M−ϕ+ψ+ M−ϕ = Re . (A5) (0)T (0) T VERIFICO, EXAT, FUNHK) managed by the Agence Na- 2ψ ϕ ψ+ ϕ+ 1 − x+ tionale de la Recherche. The work was also supported by ψ (0)T ϕ = the Nord-Pas-de-Calais Regional Council and the European Here we used in the last line that M+ + 0 since ψ (0)T ϕ = ψT ϕ(0) = Regional Development Fund through the Contrat de Projets + 0 and similarly + M+ 0. One finds État-Région (CPER) and IRCICA. ψT ϕ = − μ4 ,ψT ϕ(0) = − μ4 . 0 M− + (1 ) + M− (1 )x+ Hence APPENDIX A: PERTURBATIVE ANALYSIS (2) 1 4 2 1 The eigenvalue xη of M in Eq. (28) that emanates from 1 x = (1 − μ ) 1 + Re μ4 − can be expanded as 16 x+ 1 2 (1) 2 (2) 3 1 2S(2S − 1) + S xη = 1 + ηx + η x + O(η ), (A1) = (1 − μ4)2 2 , (A6) 16μ4 4S2 + S2 T 2 and the corresponding eigenvectors of M and M , where (0) (1) 2 (2) 3 ϕη = ϕ + ηϕ + η ϕ + O(η ), = 2 = 1 − , = . S sin (k Z) 2 (1 cos(2k Z)) S2 sin(2k Z) ψ = ψ (0) + ηψ(1) + η2ψ (2) + η3 , η O( ) (A7)

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ε APPENDIX B: COMPUTATION OF M 2 1 1 sin(2kZref Z ) sin (kZ) = − cos(2kZref ) , 2 2 2kZref εZ We describe here how to calculate the coefﬁcients of the 3 × 3matrixM in Eq. (22). They depend on the following sin(2kε Z ) averages of trigonometric functions: For the three terms in- kZ = kZ Z ref , cos(2 ) cos(2 ref ) ε volving the kicks we have 2k Z Zref and 1 1 ω2 2 θ = 2 β λ + ε ε cos cos 2Zref ( x) dx = sin(2k Z Zref ). 2 −1 2 sin(2k Z) sin(2kZref ) 2kεZ Zref λβ ω2 εβ ω2 = 1 + 1 cos( 2Zref )sin( 2Zref ), For the four terms involving Z, we have for the Poisson εβ ω2 2 2 2Zref model

2 1 1 1 2 2 cos (kZ) = + , 1 1 cos(λβ2Zref ω )sin(εβ2Zref ω ) 2 2 1 + 4k2Z2 sin2 θ = − , ref 2 2 2 εβ2Zref ω 1 1 1 and sin2(kZ) = − , 2 2 1 + 4k2Z2 sin(λβ Z ω2 )sin(εβ Z ω2 ) ref sin 2θ = 2 ref 2 ref . εβ ω2 2Zref 1 cos(2kZ) = , + 2 2 For the four terms involving Z, we have for the random- 1 4k Zref walk model and 1 1 sin(2kZref εZ ) 2 = + , 2kZref cos (k Z) cos(2kZref ) sin(2kZ) = . 2 2 2kZref εZ + 2 2 1 4k Zref

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