Localized Patterns and Solitons in Cylindrical Microresonators with Anomalous GVD

Optical frequency combs (OFCs) are spectra characterized by a set of regular discrete lines equally spaced. These combs have numerous practical applications that include spectroscopy, telecommunications, and metrology[1]. Traditionally, generating wideband OFCs using integrated devices involves a microring resonator driven with a continuous wave laser. OFC in this context form thanks to the interplay between nonlinearity, dispersion, driving and losses [2]. Recently, an alternative method for OFC generation is attracting attention [3]–[5]. The device is a cylinder pumped with a spatially extended pump. The finite extension of the pump plays a fundamental role in explaining the complex physics of the system [4], [5]. This aspect had been typically neglected in the seminal analysis of 2D dissipative Kerr solitons [6] and pattern formation [7]. The system is described by a two-dimensional equation with a localized pump: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\partial_{t}E=-(1+i\Delta)E+\frac{i}{2}\nabla^{2}E+i\vert E\vert ^{2}E+S\exp(-x^{2}/\sigma^{2})$</tex> (see [4] for a derivation), where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${E}$</tex> is the electric field envelope, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\nabla$</tex> is the transverse Laplacian, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${S}$</tex> is the pump strength, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${\Delta}$</tex> is the detuning, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${\sigma}$</tex> is the spatial width of the driving field. We consider anomalous GVD and self-focusing regime (elliptic case).