Two-dimensional dissipative rogue waves due to time-delayed feedback in cavity nonlinear optics

TL;DR

This paper demonstrates that time-delayed feedback can induce and control the spontaneous formation of two-dimensional rogue waves in broad-area nonlinear optical systems, revealing a universal instability mechanism.

Contribution

It introduces a novel mechanism for rogue wave generation via time-delayed feedback in optical systems, applicable to both Lugiato-Lefever and laser models.

Findings

Rogue waves are induced by delayed feedback in optical systems.

The probability distribution of pulse heights shows long-tailed behavior.

The phenomenon is universal across different optical models.

Abstract

We demonstrate a way to generate a two-dimensional rogue waves in two types of broad area nonlinear optical systems subject to time-delayed feedback: in the generic Lugiato-Lefever model and in model of a broad-area surface-emitting laser with saturable absorber. The delayed feedback is found to induce a spontaneous formation of rogue waves. In the absence of delayed feedback, spatial pulses are stationary. The rogue waves are exited and controlled by the delay feedback. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical contribution which is often considered as a signature of the presence of rogue waves appears for sufficiently strong feedback. The generality of our analysis suggests that the feedback induced instability leading to the spontaneous formation of two-dimensional rogue waves is an universal phenomenon.

Equations

$$\frac{dE}{dt}=ia{\nabla }_{\perp }^{2}E-(\alpha +i\theta )E+i|E{|}^{2}E+E_i+$$

$$\eta e^{i\varphi }\left[E(t-\tau )-E(t)\right].$$

$$\frac{dE}{dt}$$

$$\frac{dN}{dt}$$

$$\frac{dn}{dt}$$

Full text

**Optical rogue waves may be generated by fibers and appear with a very large amplitude comparing to other surrounding pulses. They consist of optical pulses that appear and disappear suddenly. The long tail probability distribution is the fundamental characteristics accounting for the generation of rogue waves. The theory of rogue waves was mainly developed in the framework of focusing nonlinear Schroedinger equation. However, the nonlinear Schroedinger equation does not admit two-dimensional solutions due to collapse dynamics. In this contribution, we consider a two-dimensional rogue waves in cavity nonlinear optics described by the well known Lugiato-Lefever model equation. In particular, we discuss the effect of a delay feedback on the generation of two-dimensional rogue waves in the transverse plane of the cavity. In the absence of the delay effect, the two-dimensional pulses are stationary in time. When increasing the strength of the delay feedback, the pulse undergoes instabilities that lead to the formation of rogue waves. **

I Introduction

Spatial and/or temporal confinement of light leading to the formation of localized structures (LS’s) are drawing considerable attention both from fundamental as well as from applied point of views. These stationary solutions occur in a dissipative environment and belong to the class of dissipative structures. Their existence, due to the occurrence of a Turing (or modulational) instability, has been abundantly discussed and is by now fairly well understood (see recent overviews on this issue TTK_2007 ; Nail_lect ; Leblond-Mihalache ; Clerc ; Tlidi-PTRA ; Lugiatobook ; Tlidi_Clerc_sringer ). Along another line of research, theoretical and experimental studies have shown the possibility to generate rogue waves (RW) in fiber optics. Rogue waves are rare giant pulses or extreme events. They are also called dissipative rogue waves and have been generated in passively mode-locked lasers Akhmediev_2011 . The long tail probability distribution is the fundamental characteristics accounting for the generation of rogue waves. The modulational instability is an important mecanism for the creation of ocean waves as shown by Peregrine Peregrine . In addition, Peregrine solitons are considered as a prototype of rogue waves formation. Experimental confirmation of Peregrine solitons has been demonstrated in optical fiber Mussot_08 ; Kibler and in water wave tank Chabchoub ; chabchoub2 systems. Small amplitude pulses may grow to large amplitudes if their frequencies fall in the band of unstable mode with a positive gain. Nonlinear interaction between unstable frequencies may lead to a very complicated wave dynamics. Analytical study of the nonlinear interaction between two frequencies solutions of the nonlinear Schroedinger equation in the form of the collision of Akhmediev breathers has been reported in Nail .

The formation of RW in optics has been the subject of intense research since the pioneering work by Solli et al. Solli . Since then, the number of systems in which rogues waves are identified to appear has become important and can be witnessed by recent review papers Akhmediev_13 ; Onorato ; Dudley_14 ; Akhmediev_2016 . Recently, several studies have indicated that spatially extended systems exhibit rogue waves in the transverse section of the light Odent ; Arecchi ; Birkholz ; Montica ; Lushnikov ; Marshal ; Leonetti ; Pierangeli . Rogue wave may be generated in anisotropic inhomogeneous nonlinear medium Zhong .

The purpose of this paper is to report that the appearance of two-dimensional rogue waves in dissipative systems: driven optical cavities subjected to optical injection and broad-area surface-emitting lasers with saturable absorber. We propose a mechanism to generate two-dimensioal rogue waves formation by time delayed feedback scheme. This mechanism has been recently applied successfully in one-dimensional systems Tlidi_Intropy ; Akhmediev_2016 . In the first part of the present paper, we focus on two-dimensional Lugiato-Lefever model. We show that depending on the strength of the delay feedback, localized structures become unstable and rogue waves are formed. The rogue wave formation can occur in the regime where the transmitted intensity as a function of the input intensity is monostable. We provide a statistical analysis showing a non-Gaussian profile of the probability distribution with a long tail and pulse intensity height well beyond two times the significant wave height (SWH). The SWH is defined as the mean height of the highest third of waves. In the second part, we perform the same analysis by using a model describing broad-area surface-emitting laser with saturable absorber. The generality of our analysis suggests that the instability leading to the spontaneous formation of rogue waves is an universal phenomenon. It is worth to mention an important theoretical work on rogue waves by Akhmediev et al. Akmhediev_OL_09 in the framework on Schroedinger nonlinear equation. However, we choose two systems that cannot be described by the Nonlinear Schroedinger equation. This is because the nonlinear Schroedinger equation does not admit two-dimensional solutions due to collapse dynamics.

In the case of small area semiconductor laser where the diffraction is neglected, there exist a narrow parameter regions where the laser intensity exhibits high intensity pulses in the time domain Tredicci . When the delay feedback is taken into account temporal rogue waves have been also generated Reinoso . Rogue waves as extreme events have been reported in one dimensional systems such as all fiber cavities Conforti and whispering gallery mode resonators Chembo14 . More recently, an analysis of rogue waves supported by experimental data in a semiconductor microcavity laser with intracavity saturable absorber has been reported Selmi . In all these works, the generation of rogue waves have been studied in strictly one dimensional setting.

In this paper, after an introduction, we characterize in section II, the formation of rogue waves generated in two-dimensional Lugiato-Lefever Model with a delay feedback. In Section III, we consider a model describing broad-area surface-emitting laser with a saturable absorber with a time delay feedback and we show the occurrence of two-dimensional rogue waves in this system. We conclude in section VI.

II Lugiato-Lefever model

We propose for the first time a mechanism of the formation of two-dimensional spatial rogue waves based on the time delayed feedback control scheme. To illustrate this mechanism we consider a nonlinear passive cavity subjected to time-delayed feedback. This system is modeled by the well known Lugiato-Lefever (LL) equation that describes a ring cavity filled with a Kerr media and driven coherently by an injected signal LL . This model equation is valid under the following approximations: the cavity possesses a high Fresnel number, i.e. it is a large-aspect-ratio system and we assume that the cavity is much shorter than the diffraction and the nonlinearity spatial scales; and for the sake of simplicity, we assume a single longitudinal mode operation. We implement in the LL model an optical time-delayed feedback as a single round-trip delay term Tlidi_prl09 ; Panajotov_epjd10 . More precisely, we adopt the Rosanov rosanov75 and in the Lang and Kobayashi LK approach to model the time-delay feedback. The LL model with optical feedback reads:

[TABLE]

Here $E=E(x,y,t)$ is the normalized mean-field cavity electric field and $\theta$ is the frequency detuning parameter. $\alpha$ is the cavity losses, and $E_{i}$ is the input field amplitude assumed to be real, positive and independent of the transverse coordinates. Diffraction is modeled by the Laplace operator $\nabla^{2}_{\perp}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$ acting on the transverse plane $(x,y)$. The diffraction coefficient is $a$. For simplicity we normalized the losses to $\alpha=1$, like in classical Lugiato-Lefever equation. In this case, the homogeneous steady states are $I_{i}=I_{S}\left[1+(\theta-I_{S})^{2}\right]I_{S}$, with $I_{i}=E_{i}^{2}$ and $I_{S}=|E_{S}|^{2}$. Depending on choice of $\theta$ the curve $I_{S}(I_{i})$ is either monostable ($\theta<\sqrt{3}$) or bistable ($\theta>\sqrt{3}$) LL . The delay feedback is modelled by an external cavity operating in a self-imaging configuration Tlidi_prl09 ; Panajotov_epjd10 . This allows us to compensate diffraction in the external cavity. This feedback is characterized by the time-delay $\tau$, feedback strength $\eta$ and phase $\phi$. In the absence of delay feedback, the LL model has been derived for other systems such as all fiber cavity Healterman and whispering gallery mode resonators Chembo . In these two systems the diffraction term modeled by the Laplace operator is replaced by chromatic dispersion effect modeled by a second derivative with respect to the retarded time in the reference frame moving with the group velocity of light. The LL model has also been derived for a cavity with left-handed materials Kockaert.2006 ; Lendert07 where diffraction coefficient is negative $a<0$.

It is well known that in the absence of delay feedback, i.e. $\eta=0$, the Lugiato-Lefever equation admits stationary localized structures in one and two-dimensional settings Scroggie similar to those reported in TML94 . Experimental evidence of spatial LS in Kerr medium has been realized in Odent.2011 ; Odent-RC .

We fix all the parameters in Lugiato-Lefever model and only vary the strength of the delay feedback. In the absence of delay feedback, i.e. $\eta=0$, a single or multipeak stationary localized structures are formed (see Fig. 1). The occurrence of stationary localized structures does not require a bistable homogeneous response curve. The prerequisite condition for their formation is the coexistence between a single homogeneous flat solutions and the spatially periodic pattern. This coexistence occurs in the monostable regime Scroggie , i.e., for $\theta<\sqrt{3}$. A weakly nonlinear analysis in one LL and in two transverse dimensions TML96 have shown that this coexistence occurs when $\theta>41/30$. In this regime, the relative stability analysis has shown that the only stable periodic pattern in two-dimensions are hexagons, other symmetries are unstable TML96 . The interaction of well-separated localized structures has been also discussed VTZ_2012

When increasing the value of $\eta\tau$ above $1$, such localized structures exhibit a regular drift with a constant velocity (not shown) that has been reported in Krassimir_2016 . The delay feedback allows for the motion of stationary localized structure when the product $\eta\tau$ reaches the value of $+1$ for $\phi=\pi$ Tlidi_prl09 ; Panajotov_epjd10 ; Pimenov_pra13 . Optical delay feedback may also be at the origin of a drift bifurcation leading to the motion of localized strctures in the Lugiato-Lefever equation Krassimir_2016 .

The linear stability of the homogeneous steady states (HSS) is analyzed by considering small fluctuations around the steady-state that are modulated with transverse wavevector $k_{\bot}$. The optical feedback impacts the stability of the homogeneous solution by both its magnitude and phase. This is illustrated in Fig. 2 for the case of monostable HSS. Without optical feedback the HSS is stable with respect to spatially-homogeneous perturbations ($k_{\bot}=0$) [see Fig. 2 (b)] and Turing (modulationally) unstable above $E_{i}\sim 1.24$. The optical feedback drastically changes the stability creating a plead of Hopf bifurcations and thus making the HSS unstable in the whole range of $E_{i}$ as depicted in Fig. 2 (c). It also modifies the region of Turing instabilities [see Fig. 2 (d)].

When further increasing the value of $\eta$, spatial-temporal chaos appears which further develops so that very high amplitude pulses (extreme events) appear (see the snapshot of the optical intensity shown in Fig. 3 for $\eta=0.7$, $\tau=10$, and $\phi=3\pi/4$). A statistical analysis shows that the height of such extreme events is more than twice the significant wave height (SWH) - see Fig. 4. This figure shows a non Gaussian statistics of the wave intensity, with a long tail of the probability distribution typical for rogue waves formation.

III Broad-area surface-emitting laser with a saturable absorber

Another problem which produces spatial rogue waves is the broad-area surface-emitting laser with a saturable absorber. Recently, spatiotemporal chaos and extreme events have been demonstrated experimentally in an extended microcavity laser in 1D configuration Selmi . Here, we consider the control of two-dimensional rogue waves by time-delayed optical feedback. We consider the mean field model describing the space-time evolution of broad area vertical cavity surface emitting laser (VCSEL) with saturable absorption Bache_apb05 and modify it by adding a delay optical feedback from a distant mirror in a self-imaging configuration, i.e. light diffraction in the external cavity is compensated Panajotov_ol14 :

[TABLE]

Here $E$ is the slowly varying mean electric field envelope. $N$ ($n$) is related to the carrier density, $\alpha$ ($\beta$) is the linewidth enhancement factor and $b_{1}$ ($b_{2}$) is the ratio of photon lifetime to the carrier lifetime in the active layer (saturable absorber) (normalization is the same as in Bache_apb05 ). $\mu$ is the normalized injection current in the active material, $\gamma$ measures absorption in the passive material and $s=a_{2}b_{1}/(a_{1}b_{2})$ is the saturation parameter with $a_{1(2)}$ the differential gain of the active (absorptive) material. The diffraction of intracavity light $E$ is described by the Laplace operator $\nabla^{2}_{\perp}$ acting on the transverse plane $(x,y)$ and carrier diffusion and bimolecular recombination are neglected. Time and space are scaled to the photon lifetime $\tau_{p}$ and diffraction length, respectively. The feedback is characterized by the time-delay $\tau$, the feedback strength $\eta$ and phase $\phi$.

We consider the same laser parameters as in Panajotov_ol14 : $\alpha=2$, $\beta=0$, $b1=0.04$, $b2=0.02$, $\gamma=0.5$, $s=10$, $\mu=1.42$ and optical feedback with a time-delay of $\tau=100$ and phase $\phi=0$. The current is chosen such that the laser without optical feedback resides in a bistable region between the zero homogeneous solution ($E=0$, $N=\mu$, $n=-\gamma$) and the lasing solution ($E=\sqrt{I}e^{iwt}$, $N=\mu/(1+I)$, $n=-\gamma/(1+sI)$). For this choice of parameters the upper branch exhibits a subcritical Turing type of bifurcation allowing for the formation of LSs, which experiences a period-doubling bifurcation to spatially localized chaos Panajotov_ol14 . The time-delayed feedback parameters are: $\tau=100$, $\eta=0.75$ and $\phi=0$. We integrate numerically Eqs. (1)-(4) by using the standard split-step method with periodic boundary conditions.

Statistical analysis of pulse height distribution of spatial-temporal chaos in the model of a broad-area surface-emitting laser with a saturable absorber is presented in Fig. 5. The long-tailed statistical contribution serves as a signature of the presence of rogue waves: rogue waves with pulse heights more than twice the SWH appear in the system.

IV Conclusion

We demonstrate a way to generate rogue waves by time-delayed feedback in two generic nonlinear systems: a broad area nonlinear optical resonator subject to optical injection and a broad-area surface-emitting laser with a saturable absorber. While in the absence of delayed feedback the spatial pulses are stationary, for sufficiently strong feedback spontaneous formation of rogue waves is observed. These rogue waves are clearly exited and controlled by the feedback. The generality of our analysis suggests that the feedback induced instability leading to the spontaneous formation of rogue waves is an universal phenomenon.

Acknowledgements.

M.T. received support from the Fonds National de la Recherche Scientifique (Belgium). We aknowledge the support of the Interuniversity Attraction Poles program Photonics@be of the Belgian Science Policy Office (BelSPO), under Grant IAP 7-35. K.P. is grateful to the Methusalem foundation for financial support.