Hierarchy of coupled mode and envelope models for bi-directional microresonators with Kerr nonlinearity

Abstract

We consider interaction of counter-propagating waves in a bi-directionally pumped ring microresonator with Kerr nonlinearity. We introduce a hierarchy of the mode expansions and envelope functions evolving on different time scales set by the cavity linewidth and nonlinearity, dispersion, and repetition rate, and provide a detailed derivation of the corresponding hierarchy of the coupled mode and of the Lugiato-Lefever-like equations. An effect of the washout of the repetition rate frequencies from the equations governing dynamics of the counter-propagating waves is elaborated in details.

Controlling microresonator solitons with the counter-propagating pump

“…play a pivotal role in the theory of the microresonators with the counter-propagating waves [13]. Taking the discrete Fourier transforms of ψ ± allows reconstructing E (cf.…”

Coupled-mode theory for microresonators with quadratic nonlinearity

“…[61] uses an example a bidirectionally pumped Kerr resonator to provide a refreshed outlook on this formulation and includes a literature overview. Coupled mode equations, under quite general assumptions, are formally equivalent to an LL equation [61]. This formulation of LL is, however, different from the one derived from the Ikeda map.…”

Soliton blockade in bidirectional microresonators

“…One of the bottlenecks in studies of the bidirectional microresonators has been the absence of a sufficiently rigorous mathematical model to describe their operation in the multimode regime. A variety of qualitative models [7,8] have been used to treat problems of the nonlinear crosscoupling between the modes, backscattering, and also of the opposing group velocities of the two waves in different ways, see [16] for a comparative overview. These issues have been reconciled by demonstrating an equivalence of the coupled-mode model derived from the ab-initio Maxwell system to a certain form of the envelope equations [16][17][18].…”

“…Following Ref. [16], we define total electric field, E , in a microresonator as a superposition of the clockwise, '+', and counterclockwise, '−', rotating modes. If ψ ± µ (t) are the complex mode amplitudes, µ = −N/2…”

“…ω + is the laser frequency pumping the '+' wave, which is detuned from ω 0 by δ = ω 0 − ω + , and by ε = ω + − ω − from the pump frequency of the '−' wave, ω − . If ψ ± (t, θ) = ∑ µ ψ ± µ (t)e iµθ , then coupled mode equations for ψ ± µ are equivalent to the following equations for the ψ ± envelopes [16],…”