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arXiv:0808.3122 · Physical Review A

Nonlinear harmonic generation and devices in doubly resonant Kerr cavities

Hila Hashemi, Alejandro W. Rodríguez, John D. Joannopoulos, Marin Soljacic A, Steven G. Johnson

Published Jan 13, 2009• 41 citations
arXivHTMLPDF

Abstract

We describe a theoretical analysis of the nonlinear dynamics of third-harmonic generation $(\ensuremath{\omega}\ensuremath{\rightarrow}3\ensuremath{\omega})$ via Kerr $({\ensuremath{\chi}}^{(3)})$ nonlinearities in a resonant cavity with resonances at both $\ensuremath{\omega}$ and $3\ensuremath{\omega}$. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of $\ensuremath{\sim}V∕{Q}^{2}$, where $V$ is the modal volume and $Q$ is the lifetime, and can even exhibit 100% harmonic conversion efficiency at a critical input power. However, we show that it also exhibits a rich variety of nonlinear dynamics, such as multistable solutions and long-period limit cycles. We describe how to compensate for self- and cross-phase modulation (which otherwise shifts the cavity frequencies out of resonance), and how to excite the different stable solutions (and especially the high-efficiency solutions) by specially modulated input pulses.

DOI
10.1103/physreva.79.013812

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We describe a theoretical analysis of the nonlinear dynamics of third-harmonic generation (\ensuremathω\ensuremath→3\ensuremathω)(\ensuremath{\omega}\ensuremath{\rightarrow}3\ensuremath{\omega})(\ensuremathω\ensuremath→3\ensuremathω) via Kerr (\ensuremathχ(3))({\ensuremath{\chi}}^{(3)})(\ensuremathχ(3)) nonlinearities in a resonant cavity with resonances at both \ensuremathω\ensuremath{\omega}\ensuremathω and 3\ensuremathω3\ensuremath{\omega}3\ensuremathω. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of \ensuremath∼V∕Q2\ensuremath{\sim}V∕{Q}^{2}\ensuremath∼V∕Q2, where VVV is the modal volume and QQQ is the lifetime, and can even exhibit 100% harmonic conversion efficiency at a critical input power. However, we show that it also exhibits a rich variety of nonlinear dynamics, such as multistable solutions and long-period limit cycles. We describe how to compensate for self- and cross-phase modulation (which otherwise shifts the cavity frequencies out of resonance), and how to excite the different stable solutions (and especially the high-efficiency solutions) by specially modulated input pulses.
In this work, we derive general conditions to achieve high efficiency cascaded third harmonic generation and three photon parametric down conversion in Kerr nonlinear resonant cavities. We employ the general yet rapid temporal coupled-mode method, previously shown to accurately predict electromagnetic conversion processes in the time domain. In our study, we find that high-efficiency cascaded third harmonic generation can be achieved in a triply resonant cavity. In contrast, high-efficiency cascaded three-photon parametric down conversion cannot be achieved directly in a triply resonant cavity, although a combination of two doubly resonant cavities and three waveguides is an effective alternative. The stabilities of the calculated steady-state solutions for both processes are revealed by applying Jacobian matrices. Finally, we find that the inclusion of self- and cross- phase modulation introduces multi-stable solutions. Further study is required to find a simple way to reliably achieve stable conversion at the highest possible efficiency.
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 ${\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\Delta}\ensuremath{\omega}$ can, in the presence of pump light at ${\ensuremath{\omega}}_{0}$, be converted with quantum-limited efficiency into an output shifted signal of frequency ${\ensuremath{\omega}}_{0}+\ensuremath{\Delta}\ensuremath{\omega}$, 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  10~10 10 cubic wavelengths and quality factors  1000~1000 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.
Second-harmonic generation (SHG) in doubly resonant optical resonators (resonant for the pump and for the harmonic wave) is a subject rich in outstanding challenges: the demonstration of self-pulsing, in which the intracavity field amplitudes undergo a Hopf bifurcation to a stable limit cycle,1 and chaos; the generation of near-perfect squeezing near but below the self-pulsing threshold; the generation of squeezed radiation in the self-pulsing regime; and the demonstration of quantum nondemolition measurements.
It is known1 that in type-I second-harmonic generation in a doubly resonant cavity where a subharmonic mode is coupled to a harmonio mode through a χ(2) process, self-pulsing may occur for high input fundamental power. Squeezing2 and QND measurements3 are possible near the self-pulsing threshold. For type-II harmonic generation, however, there are two sub-harmonic modes coupled to the harmonic mode through the same χ(2) process but with different arrangements of polarizations. The extra subharmonic mode will make the dynamics of the system totally different.
We consider second-harmonic generation by a non-linear crystal placed in an empty resonant cavity and pumped by an external coherent field. It was recently predicted that the stationary state loses its stability through a Hopf bifurcation when the input intensity reaches a critical value. A new amplitude self-modulated state then arises. We construct explicitly this new time-periodic solution and prove its stability in the vicinity of the transition point. As the input intensity is further increased, numerical investigations indicate a new transition to another time-periodic solution; this transition involves an hysteresis cycle. The two periodic solutions differ by the number of maxima in one period: at low input intensity, there is one maximum per period, whereas at high input intensity there are two.
We calculate the spectrum of third-harmonic generation (THG) for multimode fundamental input fields. We consider input fields from (1) a laser with gain on several cavity modes, (2) pulses with short temporal duration, and (3) frequency-modulated input fields. The effects of phase modulation in the nonlinear medium on the spectrum of the THG field are considered. Using the estimates by Miles and Harris [IEEE J. Quantum Electron. QE-9, 470 (1973)] for the nonlinear susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-3\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) in alkali-metal vapors, we find that phase modulation can profoundly affect the efficiency and the spectrum of the THG. Generally, the self-phase modulation of the fundamental will tend to reduce the phase matching of an intense fundamental field and the third harmonic and thereby lower the conversion efficiency. New frequencies not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{k}}$, where {${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$} are frequencies in the fundamental laser spectrum, are created even at relatively low conversion due to phase modulation. Effects of cross phase modulation are also studied.