Degenerate four-wave mixing in triply resonant Kerr cavities

Abstract

We demonstrate theoretical conditions for highly efficient degenerate four-wave mixing in triply resonant nonlinear (Kerr) cavities. We employ a general and accurate temporal coupled-mode analysis in which the interaction of light in arbitrary microcavities is expressed in terms of a set of coupling coefficients that we rigorously derive from the full Maxwell equations. Using the coupled-mode theory, we show that light consisting of an input signal of frequency ω 0 − ω can, in the presence of pump light at ω 0 , be converted with quantumlimited efficiency into an output shifted signal of frequency ω 0 + ω, and we derive expressions for the critical input powers at which this occurs. We find the critical powers in the order of 10 mW, assuming very conservative cavity parameters (modal volumes ∼10 cubic wavelengths and quality factors ∼1000). The standard Manley-Rowe efficiency limits are obtained from the solution of the classical coupled-mode equations, although we also derive them from simple photon-counting "quantum" arguments. Finally, using a linear stability analysis, we demonstrate that maximal conversion efficiency can be retained even in the presence of self-and cross-phase modulation effects that generally act to disrupt the resonance condition.

High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities

“…In this paper, we explore realistic microcavity designs that enable highly efficient degenerate four-wave mixing (DFWM) beyond the undepleted pump regime. In particular, we extend the results of our previous work [41], which focused on the theoretical description of DFWM in triply resonant systems via the temporal coupled-mode theory (TCMT) framework, to account for various realistic and important effects, including linear losses, self-phase and cross-phase modulation, and frequency mismatch. Specifically, we consider the nonlinear process depicted in Fig.…”

“…[41] to include important effects associated with the presence of losses, selfphase and cross-phase modulation, and imperfect frequency matching. We consider the DFWM process depicted in Fig.…”

“…1, in which incident light from some input or output channel (e.g., a waveguide) at frequencies ω 0 and ω m is converted to output light at a different frequency ω p = 2ω 0 − ω m inside a triply resonant χ (3) cavity. The fundamental assumption of TCMT (accurate for weak nonlinearities) is that any such system, regardless of geometry, can be accurately described by a few sets of geometry-specific parameters [41]. These include the frequencies ω ck and corresponding lifetimes τ k (or quality factors Q k = ω ck τ k /2) of the cavity modes, as well as nonlinear coupling coefficients α kk and β k , determined by overlap integrals between the cavity modes (and often derived from perturbation theory [23]).…”

“…II, we revisit the TCMT framework introduced in Ref. [41], and extend it to include additional effects arising from cavity losses (Sec. II A), self-phase and cross-phase modulation (Sec.…”

“…These include studies of χ (2) processes such as second harmonic generation [26,[47][48][49], sum and difference frequency generation [50], and optical parametric amplification [27,28,51], as well as χ (3) processes such as third harmonic generation [47,52], four-wave mixing [53][54][55], and optical parametric oscillators [22,[56][57][58]. Studies that go beyond the undepleted regime and/or employ resonant cavities reveal complex nonlinear dynamics in addition to highefficiency conversion [23,[37][38][39]41,59,60], but have primarily focused on ring-resonator geometries due to their simplicity and high degree of tunability [60]. Significant efforts are underway to explore similar functionality in wavelength-scale photonic components (e.g., photonic crystal cavities) [49,50], although high-efficiency conversion has yet to be experimentally demonstrated.…”