## Abstract

In optical fibers, stimulated Brillouin scattering are usually investigated in the regime of resonance. Whereas, in this paper, we discover for the first time that, without participation of Kerr effect, frequency detuning from resonance can give rise to rich dynamical behaviors for stimulated Brillouin scattering in optical fibers. Distinct from the dynamics presented in the conventional Brillouin lasers, this kind of phenomena does not need external feedback at all but also presents a variety of classifiable dynamical features for continuous-wave pumping, including steady state, periodic state and chaos. We analyze that the main mechanisms responsible for these dynamical behaviors include the transient response of acoustic wave, relaxation oscillation, frequency mixing effect induced by three-wave coherent coupling and Brillouin gain-induced group velocity change. Moreover, it should be pointed that it is the first time to discover in theory that there exists the frequency mixing effect induced by three-wave coherent coupling in the regime of non-resonance for the stimulated Brillouin scattering process, which as a consequence determines the periodic state.

© 2015 Optical Society of America

## 1. Introduction

In optical fibers, stimulated Brillouin scattering (SBS) process can be described classically as a nonlinear interaction between the pump and backscattered Stokes fields through a moving acoustic wave generated through the effect of electrostriction. The acoustic wave in turn generates a refractive-index grating which scatters the comoving pump light through Bragg diffraction. Thus, the backscattered Stokes light is downshifted in frequency because of the Doppler effect. Various fundamental and applied aspects of SBS in optical fibers have been studied in the past decades. Remarkable dynamics related to SBS include relaxation oscillation in fibers with and without feedback [1, 2], self-pulsing and deterministic chaos in continuous-wave pumped Brillouin fiber lasers [3–10], and passive *Q* switching and hybrid-gain assisted mode locking in rare earth ions doped fiber lasers [11–15]. The list of application areas where SBS is relevant is also abundant. For example, SBS has been extensively studied and exploited to realize Brillouin amplifiers for fiber-optic communication [16, 17], distributed optical fiber sensing [18, 19], pulse compression [20–22], signal-processing techniques, such as stored light [23] and slow and fast light [24–28], and various kinds of new light sources [3, 11, 12].

In cases of various kinds of Brillouin lasers, the Stokes signal always comes from the spontaneous Brillouin scattering through the thermal noise, and as a result, the central frequencies of the Stokes (*ω*_{s}) and pump (*ω*_{p}) waves satisfy automatically along the fiber the resonance condition, that is *ω*_{s} = *ω*_{p}-Ω_{B}, where Ω_{B} is the SBS resonant frequency shift. Whereas, in many other cases where SBS dominates, such as Brillouin amplification, Brillouin optical fiber sensing, pulse compression, SBS-based stored light, and slow and fast light, the Stokes wave is generally injected from outside and hence its carried frequency may deviate from the exact resonant frequency determined by the material properties of the fiber due to intentional arrangements or uncontrollable external factors. Especially in a Brillouin amplifier in the optical communication systems, due to inhomogeneous temperature and strain fluctuations distributed along the whole long fiber, SBS resonant frequency shift may present nontrivial change along the fiber. As a consequence, it may give rise to frequency detuning from resonance in the SBS process when the Stokes wave propagates along the fiber. Moreover, the situation of an off-resonant Stokes wave has been treated in an Yb-doped fiber amplifier, but the focus there is on the suppression of SBS instead of the dynamics [29, 30]. Therefore, it is necessary and meaningful to study the influence of frequency detuning from resonance on the dynamics of SBS in the optical fiber without external feedback. To our knowledge, detailed investigations have not been carried through up to now.

In this paper, we have studied the influence of frequency detuning from resonance on the dynamics of SBS in the optical fiber without external feedback for continuous-wave (CW) pumping. It is found numerically and analytically that the frequency detuning from resonance can give rise to rich dynamical behaviors of SBS, including steady state, periodic state and deterministic chaos. Furthermore, we have analyzed that the main mechanisms responsible for the dynamics mainly include the transient response of acoustic wave, relaxation oscillation, three-wave coherent coupling-based frequency mixing effect and SBS-induced group velocity change. Moreover, it is pointed out that the periodic state of the three-wave coupled system is defined by the frequency mixing effect.

## 2. Modeling of dynamical property

As shown in Fig. 1, we consider a Brillouin ber amplier where the Brillouin pump wave counterpropagates through the ber with respect to the Stokes wave. We assume that effects of self-phase modulation and cross-phase modulation are not considered. With the slowly varying envelope approximation, the SBS process can be described by the following set of three-wave coupled equations with the transient response of acoustic wave taken into account [31]

*A*(

_{p}*z*,

*t*),

*A*(

_{s}*z*,

*t*) and

*Q*(

*z*,

*t*) are the complex amplitudes of the forward pump wave ( +

*z*direction), backward Stokes wave (-

*z*direction), and acoustic wave, respectively;

*v*

_{g}is the group velocity for the pump and Stokes waves (it is assumed that the group velocity of the Stokes wave keeps the same with that of the pump wave due that the frequency shift between them is small enough);

*α*is the loss coefcient of the ber;

*κ*

_{1}and

*κ*

_{2}are the coupling coefficients of the SBS process; Γ

_{B}/2

*π*is the bandwidth of the Brillouin gain spectrum; ΔΩ = Ω

_{B}-Ω = Ω

_{B}-(

*ω*

_{p}-

*ω*

_{s}) is the frequency detuning from the Brillouin resonance;

*ω*

_{p}(

*ω*

_{s}) is the center angular frequency of the pump (Stokes) wave. Moreover, it should be noted that the

*z*derivative of

*Q*(

*z*,

*t*) in Eq. (1c) has been neglected in practice because of a much lower speed of an acoustic wave compared with that of an optical wave (

*v*

_{A}/

*v*

_{g}< 4 × 10

^{−5}). Furthermore, we have verified the aforementioned assumption numerically and found that the influence of ∂

*Q*/∂

*z*is indeed tiny and then can be neglected.

As mentioned above, in the previous works, most of attentions are paid on the Brillouin resonance case, that is *ω*_{s} = *ω*_{s0} = *ω*_{p}-Ω_{B}. However, when the frequency of the injected Stokes wave deviates from the resonant frequency *ω*_{s0}, the output Stokes wave will present rich dynamical behaviors in the CW-pumping regime, which is distinct from that in the resonance case. Based on the three-wave model presented above, we simulated numerically the dynamical behaviors in the CW-pumping regime under different parameter conditions when the frequency detuning from resonance is nonzero. The simulated results are shown in Fig. 2. The parameters used in the simulation are as follows: the fiber attention *α* = 0.2dB/km, the frequency detuning ΔΩ/2π = 30MHz, the fiber length *L* = 50m, the injected Stokes power *P _{sL}* = |

*A*(

_{s}*z*=

*L*)|

^{2}= 0.01W, the effective mode area

*A*= 46.6μm

_{eff}^{2}, the phonon lifetime T

_{B}= 1/Γ

_{B}= 5ns,

*g*= 4

_{B}*κ*

_{1}

*κ*

_{2}

*A*/Γ

_{eff}_{B}= 5.0 × 10

^{−11}m/W, the single-pass transition time T

_{r}=

*L*/

*v*

_{g},

*v*

_{g}= 2.1 × 10

^{8}m/s. It should be noted that all the spectra presented in the paper are the magnitude squared of the Fourier transforms of the output Stokes wave instead of the Stokes intensity.

First, as shown in Fig. 2(a), when the injected pump power *P _{p0}* = 0.25W, the output Stokes wave eventually tends to a rigorously CW state, and it is also found from the corresponding output Stokes spectrum that the output Stokes frequency keeps the same with that of the injected Stokes wave and there are no new frequency components emerged. Moreover, further calculation suggests that the aforementioned CW state strictly corresponds to the time-independent steady state of Eq. (1).

Then, increasing the injected pump power *P _{p0}* further until the critical power${P}_{1}^{cr}$of 0.512W, the output Stokes wave still tends to the CW state finally. However, when the injected pump power

*P*exceeds ${P}_{1}^{cr}$, as shown in Fig. 2(b) where

_{p0}*P*= 0.6W, the output Stokes wave eventually tends to a periodic state instead of the CW one. It is seen clearly from the output Stokes spectrum that there are some new frequency components occurred around the originally injected one. Further study indicates that these frequency components are distributed with the same frequency spacing of about 30.3MHz, which is approximately equal to the frequency detuning ΔΩ/2π of 30MHz. Moreover, it is also found that this frequency spacing changes with the injected pump power and keeps a little difference with the frequency detuning ΔΩ/2π.

_{p0}When the injected pump power *P _{p0}* increases further but is below the second critical power ${P}_{2}^{cr}$ of 2.899W, the output Stokes wave still tends to the periodic state eventually. However, once the injected pump power

*P*increases above ${P}_{2}^{cr}$, the output Stokes wave evolves to the periodic state no longer but presents chaotic behavior continually as shown in Fig. 2(c) where

_{p0}*P*= 3W. Moreover, in the corresponding output Stokes spectrum, there are so many new frequencies emerged that the whole spectrum is mixed together and contains very complicated fine structures, which is evidently different from that of the periodic state.

_{p0}## 3. Analysis of physical mechanisms

In Section 2, we have depicted the SBS dynamics in a Brillouin ber amplier with a nonzero frequency detuning from resonance, and concluded that the output Stokes wave presents different dynamical features for different parameter conditions. In this section, we will analyze the physical mechanisms responsible for the aforementioned phenomena.

#### 3.1 CW state

As presented in Section 2, when the injected pump power is below ${P}_{1}^{cr}$, the output Stokes wave eventually tends to the CW state which corresponds to the steady-state solution of Eq. (1). However, it is found that, during the evolution process to the CW state, the output Stokes wave presents a series of amplitude-damped wave packets with approximately equal time interval. Moreover, each single wave packet is made up of the interference between different frequency components. Figure 3 presents the temporal characteristics and corresponding optical spectrum of the output Stokes wave during the evolution process when the injected pump power is 0.4W.

As shown in Fig. 3(a), the time interval of the wave packets is approximately about 2.3T_{r}, which corresponds to the characteristic frequency of 1.87MHz emerged in the spectrum shown in Fig. 3(b). Moreover, the fine structure within the wave packet is induced by the interference between the new frequency component located around the resonant frequency *ω*_{s0} = *ω*_{s}-ΔΩ, as shown in Fig. 3(b), and the originally injected frequency component *ω*_{s}.

First, we elucidate the mechanism responsible for the characteristic time scale of 2.3T_{r}. In Fig. 3(a), we also give the temporal trace of the output pump intensity shown as the green dashed line. The output pump also presents intensity modulation with the same period of 2.3T_{r}. Therefore, it suggests that this characteristic time scale of 2.3T_{r} is resulted from the relaxation oscillation process during the Stokes amplification [2]. On the other hand, the relaxation-oscillation period is a bit larger than 2T_{r}, which can be attributed to the SBS-induced group velocity change at the Stokes resonant frequency component. As presented in [31], the Stokes resonant frequency component propagates in the fiber at a lower speed than that expected in the absence of SBS gain. As a result, the actual relaxation-oscillation period will exceed the 2T_{r} anticipated without the SBS gain. In order to study this phenomenon further, we calculated the relaxation-oscillation frequency at different pump powers for the CW state. The relative changes of relaxation-oscillation frequency are presented in Fig. 4. The other simulation parameters used are kept the same with those employed in Fig. 2.

As shown in Fig. 4, the actual relaxation-oscillation frequency is less than *f*_{r0} for all different pump powers, and decreases with the increase of the pump power. According to [31], the dependence of the SBS-induced relaxation-oscillation frequency change on frequency can be expressed as

*δ*= 2(

*ω*

_{s0}-

*ω*

_{s})/Γ

_{B}is a normalized detuning parameter. From Eq. (2), we can learn that the real relaxation-oscillation frequency at the resonant frequency is less than

*f*

_{r0}and the amplitude of change is proportional to the pump power. Due that Eq. (2) is obtained under the small signal approximation, the actual change of

*f*

_{r}in the calculation is far less than that predicted by Eq. (2), but the evolution tendency with the increase of pump power agrees between them. Moreover, we also calculate the relaxation-oscillation frequency at different fiber lengths for the CW state. The relative changes of relaxation-oscillation frequency are presented in Fig. 5. The pump power is set to be 0.5W and the other simulation parameters used are kept the same with those employed in Fig. 2. Likewise, the change trend agrees with that predicted by Eq. (2).

Then, we analyze how the new resonant frequency component of the Stokes wave emerges. At the initial moment, the forward pump and backward Stokes waves are injected into the fiber and counterpropagate with respect to each other. Consequently, the pump and Stokes encounter at a certain point within the fiber, and then begin to excite the acoustic wave through the process of electrostriction. Due to the transient response of the acoustic wave, the rapid growth of the acoustic wave will introduce new Fourier frequency components with a spectral width inversely proportional to the characteristic growth time of the acoustic wave [32]. Because the Stokes wave is amplified through the scattering of the pump wave with respect to the acoustic wave, as a result, the Stokes wave will also introduce new frequency components around the initially injected frequency component *ω*_{s}, thus providing seed for the emergence of the Stokes resonant frequency component. By solving Eq. (1) in the spectral domain with the undepleted pump approximation, the propagation of the Stokes wave along the fiber can be described analytically by

*z*=

*L*, and ${\tilde{g}}_{s}\left(\Delta \Omega ,\omega \right)$ is the effective gain spectrum which is given by

*ω*= -∆Ω which corresponds to the Stokes resonant frequency

*ω*

_{s0}. It is clearly seen that the resonant frequency component of the Stokes wave experiences the maximum Brillouin gain. Therefore, once the seed for the Stokes resonant frequency component is generated due to the transient response of acoustic wave, the seed and other frequency components will be amplified exponentially and deplete the pump wave largely resulting to occurrence of the relaxation oscillation process. When the depleted portion of the pump wave reaches to the Stokes injected port, the intense amplitude fluctuation of the pump wave will result in strong fluctuation of the excited acoustic wave, which will eventually introduce new Fourier frequency components for the next stage. The aforementioned process repeats itself until the steady state.

In order to verify the aforementioned deduction, we solve numerically the coupled equations Eq. (1) under the assumption that the acoustic wave has decayed to its steady-state value, and then observe whether the phenomena presented in Fig. 2 will occur. When the acoustic wave tends to its steady state, the three-wave coupled equations Eq. (1) are simplified as

Comparison between the curves given in Fig. 3(a) and 6(a) shows that wave packets with fine structures of interference are not observed but the relaxation oscillation still occurs when the transient response of acoustic wave is neglected. On the other hand, it is found from the corresponding spectrum shown in Fig. 6(b) that there is no Stokes resonant frequency component emerged without the transient response of acoustic wave considered. Therefore, based on the aforementioned results, it can be concluded that the emergence of Stokes resonant frequency component is resulted from the transient response of acoustic wave.

Moreover, the phenomenon that the wave packet in the evolution process experiences the amplitude damping is considered. Based on the analysis presented above, we have learned that the wave packet is resulted mainly from the interference between the Stokes resonant and originally injected frequency components. In addition, as shown in Fig. 2(a), we learn that there is only the originally injected frequency component included when the output Stokes wave eventually tends to the CW state. Therefore, the amplitude damping of the wave packets means that the proportion of the resonant frequency component within the whole output Stokes wave keeps decreasing with time relative to the originally injected one. As a result, we believe that the qualitative process of the amplitude damping of the wave packets can be elucidated by the following simplified model. First, the seed for the Stokes resonant frequency component, accompanied with the originally injected one, is amplified along the fiber and depletes the pump wave largely around the pump injected port. Then the depleted portion of the pump wave reaches to the Stokes injected port and results to new generation of the Stokes resonant frequency seed for the next stage. However, because the injected pump power is not large enough, the Stokes resonant frequency component is not amplified fully so that the extent of the fluctuation of the depleted portion of the pump is not strong enough. As a result, with the transient response of the acoustic wave, the newly generated seed for the Stokes resonant frequency component is weaker than the previous one. Consequently, when the Stokes wave reaches to the pump injected port, the amplitude of the second wave packet also decreases relative to the previous one. This process repeats itself until the Stokes resonant frequency component vanishes.

Actually, for the three-wave coupled equations (Eq. (1)), the time-independent steady-state solution, namely the CW state presented above, always exists under any parameter conditions, according to which the frequency-detuned Stokes wave experiences the Brillouin amplification without any change of frequency composition. However, when the injected pump power increases above the critical power ${P}_{1}^{cr}$, the output Stokes wave eventually presents self-oscillating in spite of the constant input. Therefore, it means that there exits transition points (for example, the critical pump power ${P}_{1}^{cr}$) which distinguish the stable and unstable regions for the CW state. We have made a linear stability analysis for the steady-state solution and found that there are four free system parameters determining the state of system output, namely the input pump and Stokes powers (*P _{p}*

_{0}and

*P*

_{s}

*), fiber length*

_{L}*L*, and frequency detuning ΔΩ/2π. In order to depict intuitively the transition points for the CW state, we plot in Fig. 7 the instability threshold pump power as a function of the input Stokes power for example. The region below the red line is characterized by a stable steady-state solution, and the region above the red line is characterized by the self-oscillating.

#### 3.2 Periodic state

As mentioned in Section 2, when the injected pump power increases above ${P}_{1}^{cr}$, the output Stokes wave evolves to a periodic state, which is featured by a series of separate frequency components as shown in Fig. 2(b). Moreover, the output Stokes spectrum includes not only the resonant frequency component but also other harmonic frequency components. And the frequency spacing between these components is not exactly equal to the frequency detuning ΔΩ/2*π*. Moreover, effect of relaxation oscillation is only observed during the evolution process to the periodic state but vanishes finally, which is similar to the case of the CW state.

First, we explain how the Stokes resonant frequency component is determined in the periodic state. Similar with the case of CW state, the Stokes resonant frequency component in the periodic state also originates from the SBS relaxation oscillation, and as a result, the magnitude of Stokes resonant frequency component must be an integral multiple of the relaxation-oscillation frequency. In order to verify this point intuitively, we present in Fig. 8 the optical spectrum corresponding to the evolution process to the periodic state presented in Fig. 2(b).

As shown in the Fig. 8, during the evolution process, the Stokes wave experiences the relaxation oscillation and consequently incorporates the relaxation-oscillation frequency in the spectrum. Furthermore, the Stokes resonant frequency component is found to originate from the relaxation oscillation process and hence has to be a certain integral multiple of the relaxation-oscillation frequency to make sure its magnitude the most close to the frequency detuning ΔΩ/2π. That is why the Stokes resonant frequency component locates at the position of −30.3MHz instead of −28.5MHz. Furthermore, in addition to the resonant frequency component, other harmonic frequency components also emerge in the evolution process, as presented in Fig. 8. The frequency spacing between these harmonic frequency components is equal to the magnitude of the resonant component and hence presents a little difference from the frequency detuning ΔΩ/2*π*. However, as long as the injected pump power is less than ${P}_{2}^{cr}$, the amplification of the original component for the Stokes wave still dominates in the SBS process. As a result, similar with the case of the CW state, the relaxation oscillation damps continually and eventually vanishes, as presented in Fig. 2(b). Whereas, what is different is that the Stokes resonant component and other harmonic ones are sustained due to the combined action of interference among these frequency components and transient response of the acoustic wave.

Next, we analyze how these harmonic frequency components are generated in the evolution process to the periodic state. The qualitative process can be elucidated by the following model. Due to the relaxation oscillation and transient response of acoustic wave, the Stokes resonant frequency component emerges and is amplified along the fiber. Thus, the Stokes wave totally includes two separate frequency components, namely the originally injected (*ω _{s}*) and resonant component (${\omega}_{s}^{-1}={\omega}_{s}-\Delta {\Omega}^{\prime}$, $\Delta {\Omega}^{\prime}$ is an integral multiple of the characteristic frequency of relaxation oscillation and hence different from ΔΩ). Excited by the Stokes (including ${\omega}_{s}$ and ${\omega}_{s}^{-1}$) and the pump with the central frequency of

*ω*, the acoustic wave totally includes the frequency components of Ω and ${\Omega}^{+1}={\omega}_{p}-{\omega}_{s}^{-1}=\Omega +\Delta {\Omega}^{\prime}$. Then the pump and Stokes begin to interact with each other through this acoustic wave. For the pump, the dissipation item of$i{\kappa}_{1}{A}_{s}Q$in the right side of Eq. (1a) suggests that the Stokes with the frequency components of

_{p}*ω*

_{s}and ${\omega}_{s}^{-1}$ interacts with the acoustic wave including Ω and${\Omega}^{\text{+}1}$, which consequently results in that the total frequency components of the pump are expanded as ${\omega}_{p}^{-1}={\omega}_{s}^{-1}+\Omega ={\omega}_{p}-\Delta {\Omega}^{\prime}$,

*ω*and ${\omega}_{p}^{+1}={\omega}_{s}+{\Omega}^{+1}={\omega}_{p}+\Delta {\Omega}^{\prime}$. For the Stokes, in the same way, it is learned from the gain item of $i{\kappa}_{1}{A}_{p}{Q}^{\ast}$ that the pump (including ${\omega}_{p}^{-1}$, ${\omega}_{p}$ and ${\omega}_{p}^{+1}$) interacts with the acoustic wave (including Ω and ${\Omega}^{\text{+}1}$), consequently resulting in that the total frequency components of the Stokes are expanded as ${\omega}_{s}^{-2}$, ${\omega}_{s}^{-1}$, ${\omega}_{s}$ and ${\omega}_{s}^{+1}$. Thus, the newly generated acoustic wave should include more frequency components as a result of the source item of $i{\kappa}_{2}{A}_{p}{A}_{s}^{\ast}$. The aforementioned process of frequency mixing will sustain continually. As a result, the output pump and Stokes waves will include a series of separate harmonic frequency components. In order to verify this deduction, Fig. 9 presents the respective spectrum for the output pump and Stokes wave in the periodic state. The simulation parameters are kept the same with those employed in Fig. 2(b).

_{p}Moreover, in order to study whether the conclusion that the Stokes resonant component must be a certain integral multiple of the relaxation-oscillation frequency to make sure its magnitude the most close to the frequency detuning ΔΩ/2π is correct under other parameter conditions, we have calculated for the periodic state the Stokes resonant frequency (or the frequency spacing between the harmonic components) at different pump powers, frequency detuning, and fiber lengths. The calculated results are presented in Fig. 10.

As shown in Fig. 10, the Stokes resonant frequency component in the periodic state still keeps a little difference with the frequency detuning ΔΩ/2π for different pump powers, frequency detuning, and fiber lengths, and is also determined by the integral multiple of the relaxation-oscillation frequency to make sure its magnitude the most close to the frequency detuning ΔΩ/2π in each case. Moreover, it can be found evidently from Fig. 10 that the Stokes resonant frequency component for the periodic state in the case of a certain frequency detuning and fiber length increases with the pump power increasing. It means that the corresponding relaxation-oscillation frequency generated in the evolution process to the periodic state increases with the increasing of the pump power, which is contrary to the trend presented in the CW state (shown in Fig. 4). Actually, as discussed above, there exists the frequency mixing effect in the evolution process which means that the harmonic frequency components of the pump wave, except the initially injected one, are generated through the interaction between the Stokes and acoustic waves, namely the anti-Stokes scattering process. Thus, there exists energy reflux from the Stokes wave to the pump wave, which means that at this moment the Stokes experiences a loss and consequently is accelerated instead of being slow down. Although the Stokes wave is slow down overall, the extent of being slow down for the Stokes wave will decrease with the enhancement of the frequency mixing effect, and hence the relaxation-oscillation frequency increases with the pump power.

#### 3.3 Chaos

In Subsection 3.2, we have analyzed that when the injected pump power increases above ${P}_{1}^{cr}$, the output Stokes wave eventually evolves to a periodic state with a series of separate frequency components. Moreover, relaxation oscillation is only observed during the evolution process to the periodic state but vanishes finally. Similar to the CW state, the periodic state may be not always stable in the whole unstable region of the CW state, which means that there may also exist transition points distinguishing the stable and unstable regions of the periodic state. Evidently, ${P}_{2}^{cr}$ (2.899W) represents the threshold input pump power at which the periodic state begins to be unstable. When the injected pump power increases above ${P}_{2}^{cr}$, relaxation oscillation does not weaken with time but sustains all the while. As a result, the output Stokes wave eventually evolves to the periodic state no longer but presents quasi-random behaviors, as presented in Fig. 2(c). Actually, the SBS process described by Eq. (1) belongs to the kind of deterministic systems, and as a result, the aforementioned quasi-random behaviors should be treated as chaos induced by the SBS nonlinearity. In this subsection, we will analyze the chaotic features of the output Stokes wave and its corresponding physical mechanism.

In order to judge whether the output Stokes behavior belongs to chaos when the injected pump power exceeds ${P}_{2}^{cr}$, we study the temporal characteristics of the output Stokes wave from two aspects including the largest Lyapunov exponent [33] and phase portrait. Generally, the Lyapunov exponents quantify the average rate of convergence or divergence of nearby trajectories in the phase space. A positive exponent implies divergence, and a negative one convergence. For time series produced by a dynamical system, the presence of a positive Lyapunov exponent indicates chaos. Therefore, in many applications it is sufficient to calculate only the largest Lyapunov exponent to judge whether the system presents as chaos attractor.

First, we calculate the largest Lyapunov exponent of the output Stokes wave for different injected pump powers which are all larger than ${P}_{2}^{cr}$. The calculated results are presented in Fig. 11. It should be noted that the all the simulation parameters except the injected pump power are kept unchanged with those employed in Fig. 2(c).

As shown in Fig. 11, the largest Lyapunov exponent of the output Stokes wave is positive when the injected pump power increases above ${P}_{2}^{cr}$, which indicates that the output Stokes wave eventually evolves to chaos after the injected pump power exceeds the second critical power. Moreover, it is found clearly from Fig. 11 that the largest Lyapunov exponent increases with the injected pump power. Therefore, we can conclude that the chaotic extent of the output Stokes wave increases with the pump power.

Then, we construct the phase portrait for the output Stokes wave shown in Fig. 2(b) and (c). As shown in Fig. 12, we can see a typical periodic trajectory when the injected pump power *P _{p0}* = 0.6W, and a typical chaotic one when

*P*= 3W. As a result, the traces presented provide evidences of chaos attractor and are indicative of the breakdown of the limit-cycle behavior corresponding to the periodic state described in Subsection 3.2, which suggests a Ruelle-Takens route (meaning that the system output is successively stable, periodic or quasi-periodic and eventually chaotic when a controlled system parameter is adjusted monotonously) to chaos [34].

_{p0}Next, we analyze what the underlying physical mechanism responsible for the chaos is. As mentioned above, relaxation oscillation sustains all the time when the injected pump power increases above ${P}_{2}^{cr}$, and as a result, the output Stokes wave eventually presents chaotic behaviors. Thus, it seems to indicate that relaxation oscillation triggers directly the generation of chaos. Through careful observation on the optical spectrum corresponding to the chaotic behaviors, it is found that there exists the characteristic relaxation-oscillation frequency. However, this frequency does not remain constant but distributes in a certain range, which results in that there are too many new frequencies emerged so that the whole spectrum is mixed together and contains very complicated fine structures. Figure 13 presents a detailed part of the whole optical spectrum given in Fig. 2(c).

In Subsection 3.1, we have analyzed that the actual relaxation-oscillation period at the Stokes resonant frequency component exceeds 2T_{r} due to the SBS-induced group velocity change. Therefore, we believe that the qualitative process of the chaotic behaviors observed can be elucidated by the following model. For each one of the three waves, when the relaxation oscillation keeps existing in the SBS process, every single frequency component within the set of $\left\{\mathrm{...},{f}_{-1}^{s},{f}_{0}^{s},{f}_{1}^{s},\mathrm{...}\right\}$ induced by the mixing effect is turned into a frequency cluster with the interval of relaxation-oscillation frequency. However, the SBS-induced group velocity change is different for each component in the cluster. As a result, it can be learned from Eq. (2) that the SBS-induced relaxation-oscillation frequency change for each component in the cluster is also different with each other. Thus, during the following relaxation-oscillation process, each component in the cluster will generate different relaxation-oscillation frequency corresponding to its own frequency value, and will excite a new frequency cluster with its corresponding relaxation-oscillation frequency around itself. Because of the different frequency intervals, the newly generated frequency clusters do not overlap with each other, which means that there are new frequency components emerged. Moreover, due to the mixing effect, more and more new frequency components will be generated during the SBS process. As the relaxation oscillation remains sustaining the aforementioned process will go on continually, eventually resulting in that the whole spectrum is mixed together and contains very complicated fine structures. Consequently, in the time domain, the output Stokes wave presents chaotic behavior instead of the periodic one. On the basis of the aforementioned analysis, it can be concluded that the chaotic behaviors presented by the output Stokes wave are resulted from interaction among the relaxation oscillation, SBS-induced group velocity change and frequency mixing effect.

## 4. Conclusion

In this paper, we have discovered numerically and analytically that frequency detuning from resonance can give rise to rich dynamic behaviors of stimulated Brillouin scattering in optical fibers without external feedback. With the frequency detuning from resonance, SBS presents a variety of classifiable dynamical features, including steady state, periodic state and chaos. We have analyzed that the main mechanisms responsible for these dynamic behaviors include the transient response of acoustic wave, relaxation oscillation, three-wave coherent coupling-based frequency mixing effect and SBS-induced group velocity change. Our work thus sheds new light on the SBS dynamics in optical fibers, and can provide meaningful instructions in various application areas of SBS, such as Brillouin amplification, pulse compression, optical fiber sensing, and signal-processing techniques. Moreover, in the regime of non-resonance, simple generation of chaos in optical fibers without providing end-face feedback may find applied potentials in chaos-based cryptography and long-distance chaotic enciphering communication. Furthermore, these findings may have a relevance to the dynamics of other similar stimulated scattering phenomena and nonlinear processes.

## References and links

**1. **R. V. Johnson and J. H. Marburger, “Relaxation oscillations in stimulated Raman and Brillouin scattering,” Phys. Rev. A **4**(3), 1175–1182 (1971). [CrossRef]

**2. **I. Bar-Joseph, A. A. Friesem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of stimulated Brillouin scattering in single-mode optical fibers,” J. Opt. Soc. Am. B **2**(10), 1606–1611 (1985). [CrossRef]

**3. **E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. **66**(11), 1454–1457 (1991). [CrossRef] [PubMed]

**4. **W. Lu, A. Johnstone, and R. G. Harrison, “Deterministic dynamics of stimulated scattering phenomena with external feedback,” Phys. Rev. A **46**(7), 4114–4122 (1992). [CrossRef] [PubMed]

**5. **D. Yu, W. Lu, and R. G. Harrison, “Physical origin of dynamical stimulated Brillouin scattering in optical fibers with feedback,” Phys. Rev. A **51**(1), 669–674 (1995). [CrossRef] [PubMed]

**6. **V. Lecoeuche, B. Ségard, and J. Zemmouri, “Modes of destabilization of Brillouin fiber ring lasers,” Opt. Commun. **134**(1-6), 547–558 (1997). [CrossRef]

**7. **V. Lecœuche, B. Ségard, and J. Zemmouri, “On route to chaos in stimulated Brillouin scattering with feedback,” Opt. Commun. **172**(1-6), 335–345 (1999). [CrossRef]

**8. **C. Montes, A. Mamhoud, and E. Picholle, “Bifurcation in a cw-pumped Brillouin fiber-ring laser: Coherent soliton morphogenesis,” Phys. Rev. A **49**(2), 1344–1349 (1994). [CrossRef] [PubMed]

**9. **C. Montes, D. Bahloul, I. Bongrand, J. Botineau, G. Cheval, A. Mamhoud, E. Picholle, and A. Picozzi, “Self-pulsing and dynamic bistability in cw-pumped Brillouin fiber ring lasers,” J. Opt. Soc. Am. B **16**(6), 932 (1999). [CrossRef]

**10. **J. Gao, Y. Ding, Z. Chen, and C. Lin, “Extended chaotic domain in the long optical fibers based on SBS process,” Physica B **442**, 1–5 (2014). [CrossRef]

**11. **S. V. Chernikov, Y. Zhu, J. R. Taylor, and V. P. Gapontsev, “Supercontinuum self-Q-switched ytterbium fiber laser,” Opt. Lett. **22**(5), 298–300 (1997). [CrossRef] [PubMed]

**12. **Y. X. Fan, F. Y. Lu, S. L. Hu, K. C. Lu, H. J. Wang, X. Y. Dong, J. L. He, and H. T. Wang, “Tunable high-peak-power, high-energy hybrid Q-switched double-clad fiber laser,” Opt. Lett. **29**(7), 724–726 (2004). [CrossRef] [PubMed]

**13. **A. A. Fotiadi, P. Mégret, and M. Blondel, “Dynamics of a self-Q-switched fiber laser with a Rayleigh-stimulated Brillouin scattering ring mirror,” Opt. Lett. **29**(10), 1078 (2004). [CrossRef] [PubMed]

**14. **M. Djouher, K. Abdelamid, L. Hervé, and S. François, “Brillouin instabilities in continuously pumped high power fiber lasers,” J. Nonlinear Opt. Phys. Mater. **18**(01), 111–120 (2009). [CrossRef]

**15. **H. Lü, P. Zhou, X. Wang, and Z. Jiang, “Hybrid ytterbium/Brillouin gain assisted partial mode locking in Yb-doped fiber laser,” IEEE Photonics J. **7**(3), 1501611 (2015). [CrossRef]

**16. **C. G. Atkins, D. Cotter, D. W. Smith, and R. Wyatt, “Application of Brillouin amplification in coherent optical transmission,” Electron. Lett. **22**(10), 556–558 (1986). [CrossRef]

**17. **A. R. Chraplyvy and R. W. Tkach, “Narrowband tunable optical filter for channel selection in densely packed WDM systems,” Electron. Lett. **22**(20), 1084–1085 (1986). [CrossRef]

**18. **X. Bao, D. J. Webb, and D. A. Jackson, “32-km distributed temperature sensor based on Brillouin loss in an optical fiber,” Opt. Lett. **18**(18), 1561–1563 (1993). [CrossRef] [PubMed]

**19. **D. Garus, K. Krebber, F. Schliep, and T. Gogolla, “Distributed sensing technique based on Brillouin optical-fiber frequency-domain analysis,” Opt. Lett. **21**(17), 1402–1404 (1996). [CrossRef] [PubMed]

**20. **T. Schneider, A. Wiatrek, and R. Henker, “Zero-broadening and pulse compression slow light in an optical fiber at high pulse delays,” Opt. Express **16**(20), 15617–15622 (2008). [CrossRef] [PubMed]

**21. **G. Qin, T. Sakamoto, N. Yamamoto, T. Kawanishi, H. Sotobayashi, T. Suzuki, and Y. Ohishi, “Tunable all-optical pulse compression and stretching via doublet Brillouin gain lines in an optical fiber,” Opt. Lett. **34**(8), 1192–1194 (2009). [CrossRef] [PubMed]

**22. **M. Laroche, H. Gilles, and S. Girard, “High-peak-power nanosecond pulse generation by stimulated Brillouin scattering pulse compression in a seeded Yb-doped fiber amplifier,” Opt. Lett. **36**(2), 241–243 (2011). [CrossRef] [PubMed]

**23. **Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science **318**(5857), 1748–1750 (2007). [CrossRef] [PubMed]

**24. **Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. **94**(15), 153902 (2005). [CrossRef] [PubMed]

**25. **K. Y. Song, M. Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express **13**(1), 82 (2005). [CrossRef] [PubMed]

**26. **G. K. W. Gan, Y. G. Shee, K. S. Yeo, G. A. Madhiraji, F. R. Adikan, and M. A. Mahdi, “Brillouin slow light: substantial optical delay in the second-order Brillouin gain spectrum,” Opt. Lett. **39**(17), 5118–5121 (2014). [CrossRef] [PubMed]

**27. **K. Y. Song, M. González Herráez, and L. Thévenaz, “Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers,” Opt. Express **13**(24), 9758–9765 (2005). [CrossRef] [PubMed]

**28. **L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics **2**(8), 474–481 (2008). [CrossRef]

**29. **J. O. White, A. Vasilyev, J. P. Cahill, N. Satyan, O. Okusaga, G. Rakuljic, C. E. Mungan, and A. Yariv, “Suppression of stimulated Brillouin scattering in optical fibers using a linearly chirped diode laser,” Opt. Express **20**(14), 15872–15881 (2012). [CrossRef] [PubMed]

**30. **E. Petersen, Z. Yang, N. Satyan, A. Vasilyev, G. Rakuljic, A. Yariv, and J. O. White, “Stimulated Brillouin scattering suppression with a chirped laser seed: comparison of dynamical model to experimental data,” IEEE J. Quantum Electron. **49**(12), 1040–1044 (2013). [CrossRef]

**31. **G. P. Agawal, *Nonlinear Fiber Optics* (Academic, 2001), Chap. 9.

**32. **G. L. Keaton, M. J. Leonardo, M. W. Byer, and D. J. Richard, “Stimulated Brillouin scattering of pulses in optical fibers,” Opt. Express **22**(11), 13351–13365 (2014). [CrossRef] [PubMed]

**33. **M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D **65**(1-2), 117–134 (1993). [CrossRef]

**34. **S. Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange Axiom *A* attractors near quasi periodic flows on *T ^{m}, m*≥3,” Commun. Math. Phys.

**64**(1), 35–40 (1978). [CrossRef]