Existence of different, frequently incommensurate scales is a common phenomenon in nature. Interactions between processes characterized by different scales can result in a multitude of emergent phenomena when a system cannot be described as a scale-separated hierarchy of underlying processes but presents a substantially new entity with qualitatively new properties and behavior. Striking examples are life, fractals, and chaos. Here, we shall demonstrate the quite nontrivial phenomena: chaotic and stochastic resonances and anti-resonance on examples of laser systems. The phenomena of resonant stochastization (stochastic anti-resonance), self-ordering (stochastic resonance), and resonant chaotization of coherent structures (dissipative solitons) are considered on the examples of mode-locked lasers and Raman fiber amplifiers. Despite a well-known effect of noise suppression and global regularization of dynamics due to the resonant interaction of noise and regular external periodic perturbation, here we report about the reverse situation when the regular and noise-like perturbations result in the emergent phenomena ranging from the coherent structure formation to the fine-grained chaotic/noisy dynamics. We guess that the nonlinear optical systems can be considered in this context as an ideal test-bed for “metaphorical modeling” in the area of deterministic and stochastic dynamics of resonance systems.
- chaotic and stochastic resonance/anti-resonance
- soliton-emergence phenomena
- resonant soliton–linear wave interaction
- noise-assisted coherence
- “metaphorical” optical modeling
- resonance vector mode-locking
Is a noise so destructive? This question is not only philosophical because it is directly addressable. We live in a noisy environment, and who knows, would such environment be extremely constructive, namely constructive? Why not? For instance, the growth of initial quantum gravitational perturbations gives birth to our Universe as a quite-ordered structure, and our brains are very flowing but constructive, isn’t that so? The key point is a resonant interaction of noise with a nonlinear system [1, 2]. The resonance phenomena in nonlinear systems are fraught with counterintuitive consequences. Noise can enhance a system’s internal coherence [stochastic resonance, (SR)] or damage it [stochastic anti-resonance, (SAR)]. Both effects are resonantly sensible to the system parameters that allow naming both phenomena as a resonance with taking into account a principal difference between the linear and nonlinear systems far from an equilibrium state .
The notion of SR occurred unexpectedly from the studies regarding the long-term climatic changes (i.e., the ice ages) when the short-term (1-year scaled) climatic noise enhances resonantly an incommensurable weak variation (~105 years) of the Earth ecliptic [4, 5]. The excellent surveys expose a further progress forwarding this direction [3, 6–11]. A development of the SR ideology in the fields of neuroscience, biology, and information processes was especially exciting. A noise-induced resonant enhancement of neural sensibility, adaptivity, and learning capability was demonstrated and analyzed [10, 12–18].
The classical theory of SR was based on the resonant transitions in noisy bi-stable nonlinear systems [19–21]. Further studies revealed that both SR and SAR cover an extremely broad range of phenomena including escape from the metastable state, threshold “firing” dynamics, dynamics assisted by deterministic chaos, regularization induced by coherent periodic or continuous structures without a noise assistance, etc [3, 22, 23]. Therefore, the terms of SR and SAR can be misleading in some respects, and it is better to speak about a broad range of phenomena in the nonlinear systems far from equilibrium, which is caused by the resonant-like interaction between processes with incommensurable characteristic scales [3, 24].
As a classical illustration of SR, one may consider the so-called FitzHugh-Nagumo (FN) model (e.g., see  and references ibidem), which describes a noise excitable evolution in a very simple two-dimensional form:
where a potential function is defined as f(t) = x − ax3, typically ϵ defines a ratio of evolutional scales between x and y variables, γ is a coupling parameter, β is a friction coefficient, and s(t) is a periodic external force (s(t) = α cos[ωt], usually). The last term in Eq. (1) describes a Wiener stochastic process with volatility 2D. The stochastic Eq. (1) is treated in the Stratonovich’s sense. Evolution of dynamical variables in the absence of noise and periodic modulation is shown in Figure 1, which demonstrates a relaxation to local minimum of potential.
Separated effects of small harmonic modulation and noise are shown in Figure 2. One can see that they have a “perturbative” character and induce the small oscillations/fluctuations around the potential local minimum.
However, the situation changes drastically under the common action of noise and external modulation (Figure 3). Extremal and almost regular spikes appear at a frequency, which is lower than the modulational one and incommensurable with the noise scale. One may consider this example based on the FN-model as an impressive and quite simple illustration of SR.
At this moment, there is a huge amount of work concerning the SR and SAR as well as their variations and modifications. We refer a reader to the above-cited books and surveys (the reference list is not exhaustive, of course). A spectacular demonstration of SR in a ring dye laser  gave impetus to an intensive exploration of this field. Therefore, we intend to discuss some aspects of SR, SAR, coherent resonant, and multi-scale phenomena regarding laser optics and solitonics.
This paper is organized as follows. In the next section, we expose the SAR phenomenon in a Raman fiber amplifier. Then, the chaotic resonance of dissipative soliton with linear waves will be considered. Further, the SR and SAR as well as multi-scale resonant phenomena in mode-locked lasers will be exposed. Finally, the resonance vector mode-locking will be described in a nutshell.
2. SAR in a Raman fiber amplifier
A Raman amplifier can be considered as a test-bed for the study of SAR and multi-scale dynamics due to a comparative simplicity and realizability and, simultaneously, high practical significance. The latter is defined by the fact that Raman amplification provides an efficient tool for optical telecommunication lines with frequency multiplexing (for details see ). In such lines, there are very different scales: a length corresponding to the width of pulse carrying information (~10 – 100 mm), commensurable lengths of polarization beats and inherent stochastic distortions of a fiber (~10 – 100 m), attenuation length (~10 km), nonlinear and dispersion lengths (>100 km), and overall propagation length (>107 m) .
The Raman amplification is sensitive to the relative polarization of gain and signal—a gain is maximum for copolarized pump and signal but minimum for their mutually transverse polarizations. Since beat rates for signal (bs) and pump (bp) differ, it causes a periodical modulation of the Raman gain with fiber length . Simultaneously, the polarization properties (birefringence) of fiber are sensitive to the inevitable stochastic breakdowns of the fiber cylindrical symmetry . Thus, one has all necessary prerequisites for the manifestation of SR and SAR phenomena.
The extended vector theory of the stimulated Raman scattering with taking into account the random birefringence is presented in [26–28]. The system of stochastic differential equations describing an evolution of copropagating pump and signal states of polarization (SOP) under the action of random fiber birefringence can be written in the following form :
where and are the projections of signal and pump powers with the corresponding unit vectors and (, ) in the Stokes representation. The Raman gain coefficient is gR, and the pump/signal frequencies are ωp, s, respectively. The attenuation constants for the pump and signal are αp, s. The most interesting parameters are bp, s = 2π/Lp, s (Lp, s are the pump/signal beat lengths, respectively) and the Wiener stochastic term with the zero drift and the volatility σ2 = 1/Lc: 〈β(z), β(z′)〉 = σ2δ(z − z′) (Lc is a correlation length of the stochastic material birefringence).
The variation of Lc (correlation length defining a noise “strength”) relatively Lp, s (periods of the deterministic polarization beatings) causes a transition between the different regimes.
A strong polarization pump/signal coupling (Figure 4, left) corresponds to a case of Lp, s ≫ Lc when a noise is too “fine-grained” and cannot distort nonlinear trapping of signal by pump. As a result, the mutual polarizations of pump and signal are highly correlated, and the signal fluctuations are small (≈1% in the case under consideration; see Figure 5, left).
Further decrease of Lp, s relatively Lc causes almost complete decoupling of pump and signal (so-called diffusion limit; Figure 4, right) when a fiber behaves like an isotropic medium. Noise plays important but diminishing role (Figure 5, right).
Such a resonant-like enhancement of irregularity that depends on the relative strengths of noise and regular oscillations is an example of SAR. Figure 6 is a spectacular illustration of this phenomenon based on the model of Eqs. (2a) and (2b) . We can see here the resonant enhancement of a Raman gain standard deviation defined as in dependence on a fiber length L and a polarization mode-dispersion parameter defining relative contribution of stochastic and deterministic polarization effects (λs and c are the signal wavelength and the speed of light, respectively).
The phenomenon of SAR can be explained as an escape from a metastable state corresponding to pump-signal pulling with an effective potential barrier ΔU and an “intra-well relaxation time” (or length in our case) τi (Figure 7, left) [27, 31]. The random fluctuations can cause an escape from this metastable state with an escape rate defined by the so-called Kramers time (length) τk (D is an effective “temperature” defining a noise strength) [6, 32]. The periodic (T) modulation of potential barrier caused by the polarization beatings can enlarge this effective temperature and, thereby, increase the escape rate (Figure 7, right) [27, 31].
Thus, a Raman fiber amplifier can be considered as a simple and practically valuable test-bed for a demonstration of SAR that is a phenomenon of noise-induced escape from the metastable state. Practical control of this phenomenon is especially important for the development of modern high-speed optical communication lines that promise to exceed the limits of existing broadband information infrastructure.
3. Chaotic resonance between a dissipative soliton and linear waves
Dissipative soliton (DS) is a well-localized structure self-emergent in dissipative systems. Such structures appear in different areas ranging from physics to biology, medicine, and even economy and sociology [33, 34]. The simplest equation regarding the DS modeling is the so-called generalized complex nonlinear Ginzburg-Landau equation (CNGLE) [33–35]:
where a field of amplitude A, slowly-varying with a local time t, propagates along a coordinate z under action of dissipative (first braces) and nondissipative (second braces) factors. Then, σ-parameter corresponds to energy-dependent net-loss, α-parameter defines spectral losses, κ- and ζ-parameters describe effective nonlinear gain and its depletion, respectively. Nondissipative factors are self-phase modulation (SPM, γ), group-delay dispersion (GDD, β2) with low-order correction to the latter (third-order dispersion or TOD, β3). s(t) describes a complex white noise.
The general-form of DS solutions of Eq. (3) is unknown, and the extensive numerical simulations are required to investigate the complexity of DS dynamics. However, there are some very simple considerations based on resonance/balance relations, which allow understanding some basic properties of DS.
Indeed, a steady-state solution of Eq. (3) has a form A(t, z) = E(t) exp(−iqz), where the soliton wave number q is related to the carrier-envelope offset , which results from nonlinear phase shift caused by SPM: q = γP0 (P0 is a DS peak power) . The dispersion relation for linear waves is k(ω) = β2ω2/2. Hence, to be stable (i.e., nonradiating), the DS spectrum has to be truncated at the frequencies ±Δ : k(±Δ) = q, where (Figure 8) .
Simultaneously, the spectral loss ~αΔ2 has to be compensated by the nonlinear gain ~κP0. This condition plus the resonant condition give the rough stability criterion for DS:
which interrelates dissipative and nondissipative factors contributing to DS formation (more precise analysis can be found in ).
TOD modifies the dispersion condition for a linear wave so that the resonant condition becomes: (black/red online/curve in Figure 9c). It can cause an appearance of additional resonant frequency which proximity to DS spectrum (i.e., to one of the other resonant frequencies) can initiate chaotic dynamics [39, 40]. This conjecture was confirmed in  both experimentally and numerically.
Figure 9 demonstrates an example of such chaotization obtained from numerical simulations of Eq. (3) (for details, see ). The Wigner function (time-frequency diagram, Figure 9a) consists of strongly distorted DS-part near 2.3 μm (dark-red region online) and long dispersive tale in spectral domain around 2.4 μm (yellow – light blue region online), which co-propagates with DS and, as an analysis shows, collides with it in time domain. As a result, the DS spectrum becomes modulated chaotically (Figure 9b), but the averaged spectrum looks quite smooth with the characteristic shape of “Boa constrictor digesting an elephant” (Figure 9d) . As was mentioned, these phenomena can be explained as a nonlinear resonance of three-coupled oscillators [41, 42] when the TOD-induced resonant point (DW in Figure 9c) approaches one of the other two (R2, in our case).
The last statement can be confirmed by a reconstruction of phase space corresponding to the chaotic dynamics in Figure 9. Such a reconstruction is based on the standard lag-delayed procedure when one tries to reconstruct an N-dimensional phase space from time-series data V(t) by the means of following discretization: [V(t), V(t + L), V(t + 2L), …, V(t + (N − 1)L)], where L is a time-lag . As a rule, an appropriative time-lag is defined from the first zero of autocorrelation function of time-series (peak powers in our case). The corresponding reconstruction is shown in Figure 10 . One can see, that the chaotic trajectory of P0(t) is completely embedded in the three-dimensional manifold, and the attracting manifold has a typical toroidal shape. Both facts validate a model of nonlinear resonance of three coupled oscillators.
4. Stochastic resonance and anti-resonance in mode-locked lasers
A laser, as a device locking electromagnetic waves, possesses a discrete set of longitudinal modes, i.e., set of standing waves, which interacts irregularly due to random mutual phases. Locking of a mutual phase between modes, namely mode-locking, results in the generation of a high-intensive ultra-short laser pulse circulating with the repetition rate multiple of the period of laser resonator (e.g., see [44, 45]). In the time domain, the ultra-short pulse formation can be described in the frameworks of the so-called fluctuation model [46, 47], which treats a pulse1 emerging as a process of amplification and selection of noise fluctuations (see Figure 11). Such a model demonstrates a crucial role of noise in ultra-short pulse dynamics. The noise is not only a source of pulse formation, but it can also affect the pulse dynamics at all stages. In particular, it is a source of “linear” (dispersive) waves, which can resonantly interact with a pulse and randomize its dynamics (see the previous section).
The typical equation with “distributed” laser parameters is Eq. (3). It describes a multitude of realistic phenomena intrinsic to the pulse dynamics. But in many real-world situations, the “discretized” models are more relevant. For instance, let us consider a system of laser resonantly coupled with an external resonator. In the dispersion-less case, the field evolution in a laser can be described as follows :
Here, both z and t are discretised so that z is a cavity transit number, and t is a local time discretised with a step Δt. The coefficient g(t) describes a local gain for a 4-level active medium with the maximal gain gm for full population inversion: σ14, σ32, and T31 are absorption, emission cross-section, and gain relaxation time, respectively. νp and ν are pump and generation wavelength, respectively.
is a noise term with the coherence time tcoh, a noise level s0, and a random phase ϕ(t) .
Spectral dissipation is provided by a Fabry-Pérot etalon with a group-delay tf :
where Rf = tf/(tf + Δt).
The field A is coupled with the field in external resonator B(z, t) :
where R, T, and θ are reflection, transmission, and coupling coefficients, respectively.
It is interesting to consider the question about disruptiveness of noise for pulse formation. Figure 12 shows the regions of formation of stable pulses from noise (so-called regions of mode-locking self-start) for different tcoh. One can see, that the decrease of noise coherence is destructive from the points of view of the mode-locking regions size and the threshold pump intensity providing mode-locking.
However, the simulations demonstrate that even very low-frequency external modulation (e.g., by moving resonator mirror inducing the Doppler shift of optical wave) can suppress noise (see Figure 13) and stabilize dynamics [52, 53]. This phenomenon can be interpreted as a manifestation of resonant interaction of scale-incommensurable processes. Moreover, exactly such a resonance provides mode-locking self-start in the majority of lasers (moving mirror technique  or even simple mirror knocking).
In close connection with the phenomenon mentioned above, one has to note that a nonlinear interconnection between the scale-incommensurate processes is a crucial factor defining all considered phenomena. An interesting example closely connected with previous one is a laser mode-locked by external phase modulation [55, 56]. This system can be described by following equation (compare with Eq. (3)):
where −σ is a saturated gain, ω is a modulation frequency normalized to modulation depth, and δ is a mismatch between the modulation and cavity periods. In the absence of SPM, the pulse width τ is . It increases with saturated net-gain −σ, is closely associated with the modulation frequency ω, and exceeds substantially the minimal value defined by spectral dissipation. However, the nonlinearity (namely, SPM) can modify a situation crucially [55, 56]. Firstly, the pulse width decreases (not increases) with a gain that allows generating high energy, and simultaneously, short pulses. Secondly, and it is a first nontrivial fact, pulse width can be extremely short () and reach scales incommensurable with the modulation frequency. Third impressive fact is that the modulation frequency providing stable mode-locking can be extremely small in comparison with the laser cavity period (ω is lower by approximately three orders of magnitude in comparison with a linear case). It seems that this effect is closely related to the above considered noise suppression due to the Doppler shift.
Returning to an effect of noise on the mode-locking self-start illustrated in Figure 12, one may consider another interesting manifestation of SR/SAR in mode-locked lasers. External resonator providing mode-locking can be considered as a Fabry-Pérot interferometer resonantly matched with a laser cavity (see above). This interferometer can contain a nonlinear medium, and such a system possesses rich dynamical properties, in particular, it can cause spontaneous formation of ultra-short pulses (mode-locking). Examples of regions of such mode-locking are shown in Figure 12. It is interesting that such regions are inhomogeneous. Figure 14 demonstrates the mean-square deviation σI of pulse peak intensity inside two such regions in dependence on pump.
One can see, that the pulse is highly stable inside the mode-locking region and destabilizes only on stability border in the case of (1). But in the case of (2), the behavior of σI becomes strongly nonmonotonic. Pronounced peaks in the σI-dependence is the classical SAR manifestation caused by the excitation of noise with subsequent formation of the pulse satellites whose interaction with main pulse perturbs strongly the latter. The regions of SAR alternate with the regions of regular dynamics. Thus, the mode-locking region can be granulated.
In all examples above considered, a mode-locking resulting in the pulse appearance was caused by either loss self-modulation or external periodical modulation. However, the mode-locking can appear spontaneously due to spontaneous multimode instability (so-called Risken-Nummedal-Graham-Haken effect, RNGH) [57, 58]. However, such self-mode-locking is unstable. Nevertheless, the stable self-mode-locking was obtained in Er-fiber laser due to beatings induced by the difference of intra-laser (fiber + polarization components) birefringence and that induced by polarization hole burning in active medium (Er-doped fiber). That is the so-called resonance vector mode-locking . The beatings generate the spectral satellites (sidebands) for each laser mode produced by multimode instability (see Figure 15). Adjusting of intra-laser birefringence by polarization controller shifts these sidebands to adjacent modes that cause a resonance between them with subsequent stable mode-locking.
But that is not all. The generated comb of locked modes can excite the transverse acoustic waves in a fiber through electrostriction effect . The resonance between the comb and these waves lock (trap) a pulse in time domain that provides an unprecedented stability of the pulse train. The last is highly required for metrology, high-resolution spectroscopy, etc .
The nonlinear resonance phenomena are illustrated as examples of fiber Raman amplifiers and mode-locked lasers. These systems proved their advantage as an ideal test-bed for “metaphoric modeling” of complex nonlinear systems due to comparative simplicity, high-speed statistic gathering, and precise controllability.
We considered the phenomenon of the so-called stochastic anti-resonance as examples of a fiber Raman amplifier and a laser mode-locked by resonant coupling with a nonlinear Fabry-Pérot interferometer.
In the first case, the regular polarization beatings between pump and Raman signal are coupled resonantly with the stochastic birefringence caused by material defects (stochastic changes of fiber symmetry). As a result, there is a region of parameters (first of all, so-called polarization mode-dispersion parameter) where the evolution of the state of polarization becomes highly irregular that manifests itself in resonance growth of relative standard deviation of Raman gain. This phenomenon was interpreted as a noise-induced escape from metastable state and quantitatively characterized by an abrupt decrease of the characteristic Kramers length.
In the second case, it is shown a crucial dependence of mode-locking ability on noise correlation time so that the growth of irregularity squeezes the mode-locking region and increases the mode-locking threshold. Nontrivial effect of noise manifests itself inside the region of parameters, where spontaneously born pulse exists. Namely, a monotonic variation of the pump causes the alternation of maximums and minimums of the pulse peak intensity mean-square deviation. Thus, the stochastic anti-resonances exist inside the mode-locking region, which, thereby, has a granular structure.
The interesting example of scale hierarchy in a mode-locked laser is demonstrated. The matter is that extremely slow (~1 kHz), external modulation can suppress noise in mode-locked laser through the Doppler effect. This effect is broadly known for experimenters using the resonator mirror knocking for the mode-locking self-start.
Active mode-locked lasers can demonstrate another aspect of scale hierarchy in the nonlinear resonance phenomena. The laser phase nonlinearity coupled with the external phase modulation can provide generation of pulses whose widths are not limited by modulation frequency but only by intra-laser spectral dissipation. Moreover, laser mode-locking can be reached at anomalously low (in comparison with laser resonator round-trip) modulation frequencies. One may bring the last effect into correlation with that mentioned in the previous paragraph, but this issue demands a further consideration.
The phenomenon, which is connected closely with the resonance in systems possessing a scale hierarchy, is a so-called resonance vector mode-locking. In this case, a spontaneous locking of laser modes emerging as a result of multimode instability (RNGH) is stabilized due to the polarization beating caused by intra-laser birefringence and birefringence induced by polarization hole burning in the active medium. That results in stable self-mode-locking, which is stabilized additionally through a resonant coupling with the acoustic waves excited by the mode-locking itself through the electrostriction effect.
The dissipative soliton resonance with linear waves originating from noise can be considered separately in some way. It is shown, that this resonance defines the dissipative soliton characteristics, namely, its spectral width. When the resonant conditions change due to the contribution of higher-order dispersions (third-order in our case), the dissipative soliton can emit a radiation, which interacts with soliton in turn. As a result, the dissipative soliton dynamics becomes chaotic, that can be classified as chaotic resonance in terms of nonlinear resonance of three coupled oscillators.
The unified viewpoint on the nonlinear stochastic and chaotic phenomena in the field of laser physics and solitonics remains undeveloped yet. Such a viewpoint would be a part of general thermodynamic and kinetic theory of dissipative systems promising a strong practical impact in different areas ranging from physics to biology, medicine, and sociology.
I acknowledge my colleagues Dr. E. Sorokin and Dr. S. Sergeyev for their valuable contribution to the presented research findings. The author acknowledges the support from Austrian Science Fund (FWF project P24916-N27). Part of computational results has been achieved using the Vienna Scientific Cluster (VSC).
- We use the term “pulse” instead of DS because it is more appropriate to “discretized” systems for which the notion “soliton” can be misleading.