The Lugiato-Lefever Equation (LLE), first developed to provide a description of
spatial dissipative structures in optical systems, has recently made a significant
impact in the integrated photonics community, where it has been adopted to help
understand and predict Kerr-mediated nonlinear optical phenomena such as parametric
frequency comb generation inside microresonators. The LLE is essentially an
application of the nonlinear Schrodinger equation (NLSE) to a damped, driven Kerr
nonlinear resonator, so that a periodic boundary condition is applied. Importantly,
a slow-varying time envelope is stipulated, resulting in a mean-field solution in
which the field does not vary within a round trip. This constraint, which
differentiates the LLE from the more general Ikeda map, significantly simplifies
calculations while still providing excellent physical representation for a wide
variety of systems. In particular, simulations based on the LLE formalism have
enabled modeling that quantitatively agrees with reported experimental results on
microcomb generation (e.g., in terms of spectral bandwidth), and have also been
central to theoretical studies that have provided better insight into novel
nonlinear dynamics that can be supported by Kerr nonlinear microresonators. The
great potential of microresonator frequency combs (microcombs) in a wide variety of
applications suggests the need for efficient and widely accessible computational
tools to more rapidly further their development. Although LLE simulations are
commonly performed by research groups working in the field, to our knowledge no free
software package for solving this equation in an easy and fast way is currently
available. Here, we introduce pyLLE, an open-source LLE solver for microcomb
modeling. It combines the user-friendliness of the Python programming language and
the computational power of the Julia programming language.
pyLLE: A Fast and User Friendly Lugiato-Lefever Equation Solver