No abstract

We establish an exact partial differential equation to model Kerr comb generation in whisperinggallery mode resonators. This equation is a variant of the Lugiato-Lefever equation that includes higher-order dispersion and nonlinearity. This spatio-temporal model, whose main variable is the total intracavity field, is significantly more suitable than the modal expansion approach for the theoretical understanding and the numerical simulation of wide-span combs. It allows us to explore pulse formation in which a large number of modes interact cooperatively. This versatile approach can be straightforwardly extended to include higher-order dispersion, as well as other phenomena like Raman, Brillouin and Rayleigh scattering. We demonstrate for the first time that when the dispersion is anomalous, Kerr comb generation can arise as the spectral signature of dissipative cavity solitons, leading to wide-span combs with low pumping.PACS numbers: 42.62. Eh, 42.65.Hw, 42.65.Sf, 42.65.Tg The development of frequency combs -equidistant frequency lines from a short-pulse laser -revolutionized the measurement of frequencies [1] and has opened up a host of potential applications in fundamental and applied physics, including the measurement of physical constants, the detection of earth-like planets, chemical sensing, the generation, measurement, and distribution of highly accurate time, and the generation of low-phase-noise microwave radiation [2]. Ti:sapphire lasers were used as the original source of frequency combs, but in the past seven years, alternative fiber laser sources have developed [2]. Recently, Del 'Haye, et al. [3] demonstrated that it is possible to use the whispering gallery modes in microresonators in combination with the Kerr effect to generate an equidistant frequency comb that is also referred to as a Kerr comb. Since many applications in both fundamental and applied science would benefit from the small size, simplicity, robustness, and low power consumption of these whispering gallery mode sources, a considerable worldwide effort has gone into understanding and controlling them [4]. In particular, there have been several efforts to develop mathematical models of these sources [5][6][7][8], but all the efforts to date have serious drawbacks.A complete modal expansion has been derived to describe the growth of the Kerr combs from noise [5,6]. This model predicts a cascaded growth in which a primary comb is first generated, which then generates a secondary comb, and later higher order combs. This model is in complete agreement with experiments [5]. However, * E-mail: yanne.chembo@femto-st.fr it is difficult and computationally expensive to use this mode expansion beyond the primary comb generation because the number of modes that must be kept in a calculation grows like the third power of time. Moreover, it is difficult to study pulse formation in this model, since a large number of modes interact cooperatively. Pulse formation plays a critical role in comb generation. It is desirable to find a spatio-tempora...

We propose a detailed stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs in whispering gallery mode resonators pumped in the normal dispersion regime. We analyze the spatial bifurcation structure of the stationary states depending on two parameters that are experimentally tunable, namely the pump power and the cavity detuning. Our study demonstrates that the non-trivial equilibria play an important role in this bifurcation map, as their associated eigenvalues undergo critical bifurcations that are foreshadowing the existence of localized spatial structures. In particular, we show that in the normal dispersion regime, dark cavity solitons can emerge in the system, and thereby generate a Kerr comb. We also show how these solitons can coexist in the resonator as long as they do not interact with each other. The Kerr combs created by these (sets of) dark solitons are also analyzed, and their stability is discussed as well.

International audienceWe describe a general framework based on modal expansion for the study of optical-frequency combs generated with monolithic whispering-gallery-mode resonators. We obtain a set of time-domain rate equations describing the dynamics of each mode as a function of the main characteristics of the cavity, namely, Kerr nonlinearity, absorption, coupling losses, and cavity dispersion (geometrical and material). A stability analysis of the various side modes is performed, which finds analytically the threshold power needed for comb generation. We show that the various whispering gallery modes are excited in a nontrivial way, strongly dependent on the value of the overall cavity dispersion. We demonstrate that the combs are not simply generated through a direct transfer of energy from the pumped mode to all their neighbors but rather through complex intermediate interactions. Anomalous cavity dispersion is also demonstrated to be critical for these cascading processes, and comb generation is thereby unambiguously linked to modulational instability. This theory accurately describes the emergence of spectral modulation and free spectral-range tunability in the comb. It also enables a clear understanding of the various phenomena responsible for the spectral span limitation. Our theoretical predictions are in excellent agreement with the numerical simulations, and they successfully explain the internal mechanisms responsible for the generation of hundreds of Kerr modes in monolithic whispering-gallery-mode resonators

Reservoir computing, originally referred to as an echo state network or a liquid state machine, is a braininspired paradigm for processing temporal information. It involves learning a "read-out" interpretation for nonlinear transients developed by high-dimensional dynamics when the latter is excited by the information signal to be processed. This novel computational paradigm is derived from recurrent neural network and machine learning techniques. It has recently been implemented in photonic hardware for a dynamical system, which opens the path to ultrafast brain-inspired computing. We report on a novel implementation involving an electro-optic phase-delay dynamics designed with off-the-shelf optoelectronic telecom devices, thus providing the targeted wide bandwidth. Computational efficiency is demonstrated experimentally with speech-recognition tasks. State-of-the-art speed performances reach one million words per second, with very low word error rate. Additionally, to record speed processing, our investigations have revealed computing-efficiency improvements through yet-unexplored temporalinformation-processing techniques, such as simultaneous multisample injection and pitched sampling at the read-out compared to information "write-in".

Optical frequency comb generation in whispering gallery mode resonators has been demonstrated in several experiments. The spectra of the combs exhibit a wide variety of complex profiles that are not fully understood. We report a detailed study on frequency comb generation in whispering gallery mode resonators including a complete stability analysis and numerical simulations. We show that the interaction of dispersion and nonlinearity is the key in determining the stability of the comb, the complex characteristics of its spectral profile, and its frequency span. The results will be important for understanding the essential physical processes leading to efficient comb generation.

No abstract

We report on the experimental demonstration of a hybrid optoelectronic neuromorphic computer based on a complex nonlinear wavelength dynamics including multiple delayed feedbacks with randomly defined weights. This neuromorphic approach is based on a new paradigm of a brain-inspired computational unit, intrinsically differing from Turing machines. This recent paradigm consists in expanding the input information to be processed into a higher dimensional phase space, through the nonlinear transient response of a complex dynamics excited by the input information. The computed output is then extracted via a linear separation of the transient trajectory in the complex phase space. The hyperplane separation is derived from a learning phase consisting of the resolution of a regression problem. The processing capability originates from the nonlinear transient, resulting in nonlinear transient computing. The computational performance is successfully evaluated on a standard benchmark test, namely, a spoken digit recognition task.