# Modulational instability in randomly dispersion-managed fiber links

Andrea Armaroli  Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes
et Molécules, F-59000 Lille, France  Guillaume Dujardin  Univ. Lille, Inria,
CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France  Alexandre
Kudlinski  Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et
Molécules, F-59000 Lille, France  Arnaud Mussot  Univ. Lille, CNRS, UMR
8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
Stephan De Bièvre  Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul
Painlevé, F-59000 Lille, France  Matteo Conforti  Univ. Lille, CNRS, UMR
8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France

###### Abstract

We study modulational instability in a dispersion-managed system where the
sign of the group-velocity dispersion is changed at uniformly distributed
random distances around a reference length. An analytical technique is
presented to estimate the instability gain from the linearized nonlinear
Schrödinger equation, which is also solved numerically. The comparison of
numerical and analytical results confirms the validity of our approach.
Modulational instability of purely stochastic origin appears. A competition
between instability bands of periodic and stochastic origin is also discussed.
We find an instability gain comparable to the conventional values found in a
homogeneous anomalous dispersion fiber.

Modulational instability (MI) is a pervasive phenomenon in the physics of
nonlinear dispersive waves. It manifests itself as the destabilization of a
uniform wavepacket by the exponential growth of small harmonic perturbations
around the carrier frequency of the wavepacket [1]. Its study originated in
hydrodynamics [2, 3], but analogous phenomena were also discovered in
electromagnetic waves [4] and optical fibers [5]. The main ingredients to
observe MI are focusing cubic nonlinearity (like the Kerr effect in silica)
and anomalous (negative) group-velocity dispersion (GVD).

Notwithstanding, MI can be found also in normal (positive) GVD, if higher-
order dispersion [6] or birefringence [7] are considered. Moreover, in single-
mode fibers, the periodic variation of GVD along the fiber length can also
give rise to MI in the normal GVD regime. This effect is similar to the
destabilization of a parametrically excited harmonic oscillator and is denoted
as parametric MI [8, 9, 10, 11, 12].

Optical fibers featuring random GVD variations were also extensively studied.
In the late 90s the exactly solvable white noise process was considered [9,
13, 14, 15]. More recently some of the present authors focused on different
processes such as localized GVD kicks [16] and coloured processes of low-pass
and band-pass type [17].

So far, both periodic and random fluctuations have been mostly assumed to
occur around an average GVD different from zero (and more often normal, to
avoid competition with conventional MI, which exhibits much higher MI gain).
The fluctuations can be large, though.

On the contrary, systems with zero average GVD have attracted a lot of
attention for the suppression of the dispersion-induced pulse broadening and
the optimization of nonlinear pulse transmission [18, 19]. This approach is
commonly denoted as dispersion-management (DM): segments of positive and
negative GVD alternate along the fiber. The study of MI in periodic DM fiber
links [20] shows a behavior different from the parametric MI: a threshold is
found for the segment lengths below which no MI appears. The amplitude of GVD
variations influences the MI spectral range, but has no effect on this
threshold.

Here we consider random fluctuations of the DM segment lengths. While pulse
propagation in a similar system was analyzed in Ref. [21], we focus here on
MI, by applying the technique developed in Ref. [14, 16]. After deriving some
analytical relations for uniformly distributed fluctuations around the
periodic arrangement, we compare them to numerical solutions. We find MI bands
of purely stochastic origin and characterize the transition from periodic to
stochastic DM.

Figure 1:  Schematic representation of the GVD profile in a typical fiber
realisation.

We consider the propagation of optical pulses ruled by the nonlinear
Schrödinger equation (NLSE)[18],

$i\partial_{z}U-\frac{1}{2}\beta_{2}(z)\partial_{tt}U+\gamma|U|^{2}U=0,$ (1)

where $U(t,z)$ is the complex envelope of the optical field, ($t$,$z$) are the
physical time and propagation distance in a frame moving at the group velocity
of the fiber mode, $\gamma$ the (constant) nonlinear coefficient, and
$\beta_{2}(z)=\pm\beta_{2}^{0}$ ($\beta_{2}^{0}>0$) is the GVD, which takes
only two values. As schematically illustrated in Fig. 1, the sign changes
occur at $z_{1},z_{2},\ldots,z_{N}$, where $z_{n}=z_{n-1}+L_{n}$,
$n=1,2,\ldots,N$ and $z_{0}=0$. The lengths $L_{n}$ are independent,
identically distributed random variables with uniform probability distribution
function in $[\bar{L}(1-\varepsilon),\bar{L}(1+\varepsilon)]$, where $\bar{L}$
is the average length of one fiber segment (half of the DM period) and
$\varepsilon$ the amplitude of the fluctuation.

Equation (1) has a continuous wave ($t$-independent) solution
$U_{0}(z)=\sqrt{P}\exp(i\gamma Pz)$. In order to study its stability, we
insert in (1) the Ansatz
${U}(z,t)=\left[\sqrt{P}+\check{x}_{1}(z,t)+i\check{x}_{2}(z,t)\right]\exp(i\gamma
Pz)$, where $\check{x}_{1,2}$ are assumed to be small, linearize and Fourier-
transform the resulting equation with respect to $t$ ($\omega$ is used as the
associated angular frequency detuning from the carrier $U_{0}$). We obtain

$\frac{\mathrm{d}x}{\mathrm{d}z}=\begin{bmatrix}0&-g(z)\\\
h(z)&0\end{bmatrix}x,$ (2)

with $x\equiv(x_{1},x_{2})^{\mathrm{T}}$ (functions of $\omega$ and $z$),
$g(z)=\beta_{2}(z)\frac{\omega^{2}}{2}$ and $h(z)=g(z)+2\gamma P$. (2) is a
system of stochastic differential equations (SDEs) for each value $\omega$.

Equation (2) is solved in the interval $(z_{0},z_{N})$ as a product of random
matrices, depending on the random variables $L_{n}$ as

$\displaystyle x(z_{N})$ $\displaystyle=T_{-}(L_{N})T_{+}(L_{N-1})\ldots
T_{-}(L_{2})T_{+}(L_{1})x(z_{0}),$ (3) $\displaystyle T_{\pm}(L_{n})$
$\displaystyle=\begin{bmatrix}\cos(k_{\pm}L_{n})&-\mu_{\pm}\sin(k_{\pm}L_{n})\\\
\mu_{\pm}^{-1}\sin(k_{\pm}L_{n})&\cos(k_{\pm}L_{n})\end{bmatrix},$ (4)

with
$k_{\pm}^{2}=\pm\frac{\beta_{2}^{0}\omega^{2}}{2}\left(\pm\frac{\beta_{2}^{0}\omega^{2}}{2}+2\gamma
P\right)$, $\mu_{\pm}=\pm\frac{\beta_{2}^{0}\omega^{2}}{2k_{\pm}}$. The sign
$\pm$ is chosen according to the sign of GVD in the corresponding segment. The
wave-number $k_{+}$ is always real and positive, whereas $k_{-}$ is purely
imaginary in the conventional MI band $0\leq\omega\leq\sqrt{\frac{4\gamma
P}{\beta_{2}^{0}}}$.

If the DM link is periodic, i.e., $\varepsilon=0$, we can apply Floquet theory
[20]. The unit cell of DM to be periodically replicated is represented by one
positive and one negative GVD trait of length $\bar{L}$. The MI gain is
defined as
$G_{1}(\omega)\equiv\frac{1}{2\bar{L}}\ln\max\\{|\tilde{\lambda}|,1\\}$, where
$\tilde{\lambda}$ is the eigenvalue of the monodromy matrix associated to (2)
of largest modulus. This corresponds to $T_{2\bar{L}}\equiv
T_{-}(\bar{L})T_{+}(\bar{L})$. In [20], it was observed that a critical value
$\bar{L}\approx 1.07$ exists, below which $G_{1}=0$ identically. The MI gain
is represented as a false-color map in Fig. 2 for $\omega\geq 0$ (for
$\omega\leq 0$ we obtain its mirror image). The MI gain exhibits several
lobes, in general.

For random $L_{n}$, in Ref. [14, 16] it was shown that we have to resort to
the Lyapunov exponent of the random linear map: the sample gain
$G_{\mathrm{S}}(\omega)\equiv\lim_{z_{N}\to\infty}\frac{1}{z_{N}}\ln\lVert
T_{+}(L_{N})T_{-}(L_{n-1})\ldots
T_{+}(L_{2})T_{-}(L_{1})x(0,\omega)\rVert^{2}$ converges for almost all
realizations of the fiber and is a deterministic quantity.

By taking the average of (3) and letting $N\rightarrow\infty$, we can estimate
the MI gain as
$G_{1}(\omega)\equiv\frac{1}{2\bar{L}}\ln\max\left\\{|\lambda|,1\right\\}$,
where $\lambda$ is the largest modulus eigenvalue of
$\overline{T}\equiv\langle T_{+}\rangle\langle T_{-}\rangle$ and the angle
brackets denote the expectation operation over the random lengths $L_{n}$. We
can split the averages because $L_{n}$ are all mutually independent.
Elementary integration gives

$\displaystyle\langle\cos(mk_{\pm}n)\rangle$
$\displaystyle=\cos(mk_{\pm}\bar{L})\frac{\sin
mk_{\pm}\varepsilon}{mk_{\pm}\varepsilon}$ (5)
$\displaystyle\langle\sin(mk_{\pm}n)\rangle$
$\displaystyle=\sin(mk_{\pm}\bar{L})\frac{\sin
mk_{\pm}\varepsilon}{mk_{\pm}\varepsilon},$

thus

$\langle T_{\pm}\rangle=\frac{\sin
k_{\pm}\varepsilon}{k_{\pm}\varepsilon}\begin{bmatrix}\cos(k_{\pm}\bar{L})&-\mu_{\pm}\sin(k_{\pm}\bar{L})\\\
\mu_{\pm}^{-1}\sin(k_{\pm}\bar{L})&\cos(k_{\pm}\bar{L})\end{bmatrix}$ (6)

and $\overline{T}$ can be easily obtained as

Figure 2:  MI gain for a periodic DM fiber as a function of detuning $\omega$
and period $L$. The red dashed horizontal line identifies $\bar{L}=1.07$ (MI
threshold), while the cyan dotted line denotes the reference value
$\bar{L}=1.15$, used in the inset and below in the random length fluctuation
examples.

$\overline{T}=\frac{\sin(k_{-}\varepsilon)}{k_{-}\varepsilon}\frac{\sin(k_{+}\varepsilon)}{k_{+}\varepsilon}T_{2\bar{L}}.$
(7)

As common in random dynamical systems [22], we may have $G_{1}=0$ for all
$\omega$, implying that $x_{1,2}$ decay on average. Particularly, it is
apparent that for $\bar{L}<1.07$, where both eigenvalues of $T_{2\bar{L}}$ are
on the unit circle, (7) implies that $G_{1}(\omega)=0$ identically, because
$\overline{T}$ differs from $T_{2\bar{L}}$ only by a factor no larger than
one.

Nevertheless, a different kind of instabilitity may occur, to understand which
the study of second moments is required. We let $X_{1}=x_{1}^{2}$,
$X_{2}=x_{2}^{2}$, and $X_{3}=x_{1}x_{2}$ and derive from (2)

$\frac{\mathrm{d}}{\mathrm{d}z}{X}=\begin{bmatrix}0&0&-2g(z)\\\ 0&0&2h(z)\\\
h(z)&-g(z)&0\end{bmatrix}{X},$ (8)

with $X\equiv(X_{1},X_{2},X_{3})^{\mathrm{T}}$. (8) can be again solved in
terms of transfer matrices $X(z_{n})=M_{\pm}(L_{n})X(z_{n-1})$, with

$M_{\pm}(L_{n})=\begin{bmatrix}\cos^{2}k_{\pm}L_{n}&\mu_{\pm}^{2}\sin^{2}k_{\pm}L_{n}&-\mu_{\pm}\sin
2k_{\pm}L_{n}\\\
\mu_{\pm}^{-2}\sin^{2}k_{\pm}L_{n}&\cos^{2}k_{\pm}L_{n}&\mu_{\pm}^{-1}\sin
2k_{\pm}L_{n}\\\ \frac{\mu_{\pm}^{-1}}{2}\sin
2k_{\pm}L_{n}&-\frac{\mu_{\pm}}{2}\sin 2k_{\pm}L_{n}&\cos
2k_{\pm}L_{n}\end{bmatrix}.$ (9)

The generic DM unit cell is associated to $M_{-}(L_{n})M_{+}(L_{n-1})$. After
Ref. [14, 16], we define
$G_{2}(\omega)\equiv\frac{1}{4\bar{L}}\ln\max\\{|\kappa|,1\\}$, where here
$\kappa$ is the largest modulus eigenvalue of $\overline{M}\equiv\langle
M_{+}\rangle\langle M_{-}\rangle$. In the periodic limit, the monodromy matrix
associated to (8) is $M_{2\bar{L}}\equiv M_{-}(\bar{L})M_{+}(\bar{L})$ and
$G_{2}=G_{1}$, for every $\omega$.

In the random case, we can easily find the expression of $\overline{M}$
analytically by using (5), but the expression is rather lengthy and it is not
reported here.

In contrast to $\overline{T}$, $\overline{M}$ is not, in general, trivially
proportional to $M_{2\bar{L}}$. This simple algebraic consideration implies
that we may have $G_{2}>0$ even when $G_{1}=0$. Therefore new MI sidebands of
purely stochastic origin exist. The eigenvalues $\kappa$ of $\overline{M}$ may
be found analytically, too. Their expression is very involved, though; we thus
rely on a numerical routine.

In order to assess the accuracy of our estimates, we also solve (2)
numerically, by taking a fixed number, $N=20$, of fiber segments. We take
$(x_{1}(0),x_{2}(0))^{T}=(1,0)$ [equivalently, $X(0)^{\mathrm{T}}=(1,0,0)$]
and multiply by the transfer matrix (4) $N$ times alternating the GVD sign
according to (3). We compute
$P_{\mathrm{out}}=x_{1}^{2}(z_{N})+x_{2}^{2}(z_{N})$ (obviously,
$P_{\mathrm{in}}=x_{1}^{2}(0)+x_{2}^{2}(0)=1$). We repeat this calculation
taking $N_{\mathrm{iter}}$ different fiber realizations. The mean gain is
defined as either [23, 16, 17]

$\displaystyle\overline{G}_{1}(\omega;N)$
$\displaystyle\equiv\frac{1}{N\bar{L}}\ln\left(\left|\langle
x_{1}(z_{N})\rangle\right|+\left|\langle x_{2}(z_{N})\rangle\right|\right)$
(10) $\displaystyle\overline{G}_{2}(\omega;N)$
$\displaystyle\equiv\frac{1}{2N\bar{L}}\ln\left\langle\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\right\rangle,$
(11)

which are compared to either $G_{1}$ or $G_{2}$, respectively.

Figure 3: MI gain curves for different values of $\bar{L}$ and $\varepsilon$,
for $\omega\geq 0$. (a) $\bar{L}=1$, $\varepsilon=0.2$; (b) $\bar{L}=1$,
$\varepsilon=0.5$; (c) $\bar{L}=1.15$, $\varepsilon=0.2$; (d) $\bar{L}=1.15$,
$\varepsilon=0.5$. Blue crosses (resp. purple circles) represent
$\overline{G}_{2}$ (resp. $\overline{G}_{1}$) from numerical data, the solid
yellow (resp. dash-dotted red) lines the theoretical estimates $G_{2}$ (resp.
$G_{1}$), the dashed blue (resp. dotted purple) lines the semi-analytical
estimates $\tilde{G}_{2}$ (resp. $\tilde{G}_{1}$). In (c)-(d), we include as
thin green dashed line the MI gain in the periodic case $\varepsilon=0$.

For definiteness, we take $\gamma=P=\beta_{2}^{0}=1$, which amounts to
introduce the normalized distance $z/z_{\mathrm{nl}}\rightarrow z$, time
$t/t_{0}\rightarrow t$, and field $U/\sqrt{P}\rightarrow U$ , where
$z_{\mathrm{nl}}=(\gamma P)^{-1}$ is the so-called nonlinear length and
$t_{0}=\sqrt{\beta_{2}^{0}z_{\mathrm{nl}}}$ is a characteristic time. The
conventional MI in anomalous GVD thus reaches the maximum value of
$G_{1,\mathrm{max}}=G_{2,\mathrm{max}}=1$ at $\omega=\sqrt{2}$; the MI
sidelobes are found in $0\leq|\omega|\leq 2$. We show in Fig. 3 four
illustrative examples of MI sidelobes. We notice that, in general,
$\overline{G}_{2}$ converges to $G_{2}$ for $N_{\mathrm{iter}}=1\times
10^{6}$, whereas in general a much larger $N_{\mathrm{iter}}=1\times
10^{7}-1\times 10^{8}$ is required to achieve a stable and reliable estimate
of $\overline{G}_{1}$.

We notice that $G_{1}=G_{2}=0$ is expected at $\omega=0$. $G_{1}$ exhibits a
single lobe (dash-dotted red lines), while $G_{2}$ grows monotonically to
reach a maximum value at around $\omega\approx 2$, then decays in an
oscillatory way (solid yellow lines). This is not the case in numerical
results (blue crosses and purple circles), which are affected by the limited
size of the numerical domain and present a finite $\overline{G}_{1,2}$ at
$\omega=0$. This can be quantified by deriving alternative semi-analytical
estimates of gain. We notice that $T_{-}(L_{N})\ldots
T_{+}(L_{1})\approx\overline{T}^{N/2}$ (resp. $M_{-}(L_{N})\ldots
M_{+}(L_{1})\approx\overline{M}^{N/2}$) and define

$\displaystyle\tilde{G}_{1}$
$\displaystyle\equiv\frac{1}{N\bar{L}}\ln\left[\left(\overline{T}^{\frac{N}{2}}\right)_{11}+\left(\overline{T}^{\frac{N}{2}}\right)_{21}\right],$
(12) $\displaystyle\tilde{G}_{2}$
$\displaystyle\equiv\frac{1}{2N\bar{L}}\ln\left[\left(\overline{M}^{\frac{N}{2}}\right)_{11}+\left(\overline{M}^{\frac{N}{2}}\right)_{21}\right],$
(13)

where matrix power are computed numerically and the subscripts refer to the
corresponding numerically computed matrix elements. We notice in Fig. 3 that
in every case this estimate performs very well not only for $\omega\approx 0$,
but in the whole domain.

In Fig. 3(a)-(b), $\bar{L}=1<1.07$ and $\varepsilon$ increases from $0.2$ to
$0.5$. The MI is of purely stochastic origin. The local maximum value achieved
by $\overline{G}_{2}$ at $\omega=\omega_{2,\mathrm{max}}>0$, say
$G_{2,\mathrm{max}}$, increases with $\varepsilon$ and becomes of the same
order of magnitude as the conventional MI (i.e., for constant anomalous GVD).
The width of the sidelobes increases significantly as well.

In Fig. 3(c)-(d), $\bar{L}=1.15>1.07$. There is thus a competition between the
periodic and the stochastic effects. We include also the periodic-DM MI
sidelobe for comparison (thin green dashed line).

For $\varepsilon=0.2$ [Fig. 3(c)] the random fluctuations yield a broadened MI
sidelobe. In contrast to the case of constant anomalous GVD perturbed by
white-noise, where the broadening is accompanied by a reduction of
$G_{2,\mathrm{max}}$ [9], here this value is slightly enhanced. We notice also
that $\overline{G}_{1}$ is always less than its periodic counterpart,
consistently with (7). Comparison of Fig. 3(a) and Fig. 3(c) shows that the
main sidelobe appearing in the former is located in the same region of the
periodic sidelobe. The random fluctuations facilitate the emergence of the MI
sidelobes in a range that coincides with the periodic DM.

For a larger fluctuation ($\varepsilon=0.5$), Fig. 3(d), the residual effect
of periodicity is completely erased and we obtain a single wide lobe similar
to the corresponding below-threshold example of Fig. 3(b), but with a larger
$G_{2,\mathrm{max}}$.

Figure 4: (a) Maximum gain values and (b) their corresponding $\omega$ as a
function of $\varepsilon$. The cross (resp. plus) markers correspond to the
maxima $G_{2,\mathrm{max}}$ for $\bar{L}=1$ (resp. $\bar{L}=1.15$). The dashed
(resp. solid) blue lines correspond to the maxima of $G_{2}$ for $\bar{L}=1$
(resp. $\bar{L}=1.15$). The green dash-dotted lines report for reference the
constant values found in the periodic limit, i.e., the maxima of $G_{1}=G_{2}$
for $\varepsilon=0$. Finally the red circles show the values of
$G_{1,\mathrm{max}}$ and $\omega_{1,\mathrm{max}}$ for $\bar{L}=1.15$, for
which red dotted lines illustrate the corresponding maxima of $G_{1}$. In
panel (b) the line stops at $\varepsilon\approx 0.54$ (cut-off for $G_{1}$).

In order to summarize our findings, we show in Fig. 4(a)-(b), respectively,
$G_{j,\mathrm{max}}$ and the corresponding $\omega$ value,
$\omega_{j,\mathrm{max}}$, with $j=1,2$, as a function of $\varepsilon$.
Obviously, $G_{1,\mathrm{max}}$ is the local maximum value of
$\overline{G}_{1}$ at $\omega=\omega_{1,\mathrm{max}}>0$. We compare them to
their corresponding theoretical estimates, i.e., the maxima of $G_{1}$ (resp.
$G_{2}$). Solid blue lines, crosses, red lines and circles correspond to
$\bar{L}=1.15$, while dashed lines and pluses correspond to $\bar{L}=1$. In
Fig. 4(a) we notice that $G_{2,\mathrm{max}}$ increases monotonically with
$\varepsilon$. As expected, for $\varepsilon\to 0$ the gain vanishes for
$\bar{L}=1$ and converges to the periodic $G_{1,\mathrm{max}}=0.31$ (for
$\varepsilon=0$, dash-dotted green line) for $\bar{L}=1.15$. For large
$\varepsilon$ they converge to two very similar values, which is around 30$\%$
smaller than the conventional MI value. The dotted red line in Fig. 4(a) shows
the maxima of $G_{1}$ obtained from (7). For $\varepsilon>0.54$, $G_{1}=0$ for
every $\omega$. For this value of the fluctuation amplitudes, randomness
completely overrule the effects of periodicity. This is corroborated by the
values ${G}_{1,\mathrm{max}}$ (red circles), which match very well with the
theoretical estimates (provided that a large enough $N_{\mathrm{iter}}$ is
chosen). In Fig. 4(b) we observe that for both $\bar{L}$,
$\omega_{2,\mathrm{max}}$ decreases with $\varepsilon$ and converges to a
value larger than the conventional MI value. It is always below its periodic
counterpart for $\bar{L}=1.15$, apart from numerical fluctuations. We also
notice that ${\omega}_{2,\mathrm{max}}<{\omega}_{1,\mathrm{max}}<2.18$, i.e.,
slightly below the $\varepsilon=0$ value. For $\bar{L}=1$,
$\omega_{2,\mathrm{max}}$ is above its periodic counterpart and crosses it for
$\varepsilon\approx 0.3$. The theoretical estimates work very well for every
considered value of $\varepsilon$.

To conclude, we investigated the effect of uniformly distributed random
fluctuations of the _length_ of DM fiber links. We considered the MI problem
and developed an analytical technique to estimate the instability gain. The MI
gain attains values comparable with the conventional ones in a homogeneous
anomalous GVD fiber and up to 50$\%$ larger those found for the periodic
arrangement. Comparison to the direct numerical solution in a Monte Carlo
fashion confirm the soundness of the method.

This may be of of interest for tailoring and control of MI sidebands for
telecommunications and parametric sources.

## Acknowledgments

The present research was supported by IRCICA (USR 3380 CNRS), Agence Nationale
de la Recherche (Programme Investissements d’Avenir, I-SITE VERIFICO, Labex
CEMPI); Ministry of Higher Education and Research; Hauts de France Council;
European Regional Development Fund (Photonics for Society P4S, Wavetech), CNRS
(IRP FELANI).

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