Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators

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...

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

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Raman Self-Frequency Shift of Dissipative Kerr Solitons in an Optical Microresonator

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Detuning-dependent properties and dispersion-induced instabilities of temporal dissipative Kerr solitons in optical microresonators

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