Modeling Frequency Micro-Combs with Second Kind Fredholm Equations

Current theoretical models describing the dynamics of frequency combs in passive Kerr micro-cavities are divided into two main categories. The first is based on the Ikeda map [1], originally conceived for large cavity loops [2], and provides an accurate description of the system as long as the wave functions remain localised and do not span all over the cavity. The second, more recent approach, is based on the modal decomposition approach [3], [4] and is particularly well suited for small microcavities, since the propagation variable is the physical time and the cavity field is described along a spatial and naturally periodic coordinate, which is also known as Lugiato-Lefever model [5], which is written in the standard form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\partial_t \Psi=\hat{\mathscr{O}}_1(\Psi, S)$</tex>, with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Psi(t, \phi)$</tex> representing the intracavity field and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S(t, \bar{z})$</tex> the driving field. Likewise, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\phi$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\bar{z}$</tex> represent the cavity and bus coordinates, respectively.