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...
Abstract-We studied the efficiency of different implementations of the split-step Fourier method for solving the nonlinear Schrödinger equation that employ different step-size selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems, including higher order solitons, collisions of soliton pulses, a single-channel periodically stationary dispersion-managed soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally third-order accurate split-step scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used step-size selection criteria, and it is robust for a wide range of systems providing a system-independent rule for choosing the step sizes. We find that a step-size selection method based on limiting the nonlinear phase rotation of each step is not efficient for many optical-fiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic step-size distribution to bound the amount of spurious four-wave mixing. This method is as efficient as other second-order schemes in the single-channel dispersion-managed soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations.
We report the first experimental demonstration of two-dimensional spatial solitary waves in secondorder nonlinear optical material. When an intense optical beam is focused into a phase-matchable second-order nonlinear material, the fundamental and generated second-harmonic fields are mutually trapped as a result of the strong nonlinear coupling which counteracts both diffraction and beam walkoff.
Nonlinear pulse propagation is investigated in the neighborhood of the zero-dispersion wavelength in monomode fibers. When the amplitude is sufficiently large to generate breathers (N > 1 solitons), it is found that the pulses break apart if lambda - lambda(0) is sufficiently small, owing to the third-order dispersion. Here lambda(0) denotes the zero-dispersion wavelength. By contrast, the solitary-wave (N = 1) solution appears well behaved for arbitrary lambda - lambda(0). Implications for communication systems and pulse compression are discussed.
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