We describe a theoretical analysis of the nonlinear dynamics of third-harmonic generation (ω→3ω) via Kerr (χ(3)) nonlinearities in a resonant cavity with resonances at both ω and 3ω. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of ∼V/Q2 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/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.
Using exact numerical methods for finite-size nonplanar objects, we demonstrate a stable mechanical suspension of a silica cylinder within a metallic cylinder separated by ethanol, via a repulsive Casimir force between the silica and the metal. We investigate cylinders with both circular and square cross sections, and show that the latter exhibit a stable orientation as well as a stable position, via a method to compute Casimir torques for finite objects. Furthermore, the stable orientation of the square cylinder is shown to undergo an unusual 45o transition as a function of the separation length-scale, which is explained as a consequence of material dispersion.
We compare several methods for the efficient generation of correlated random sequences (colored noise) by filtering white noise to achieve a desired correlation spectrum. We argue that a class of IIR filter-design techniques developed in the 1970s, which obtain the global Chebyshev-optimum minimum-phase filter with a desired magnitude and arbitrary phase, are uniquely suited for this problem but have seldom been used. The short filters that result from such techniques are crucial for applications of colored noise in physical simulations involving random processes, for which many long random sequences must be generated and computational time and memory are at a premium.
We consider the design of tapers for coupling power between uniform and slow-light waveguides in photonic crystals. We describe new optimization methods for designing robust tapers, which not only perform well under nominal conditions, but also over a given set of parameter variations. When the set of parameter variations models the inevitable variations typical in the manufacture or operation of the coupler, a robust design is one that will have a high yield, despite these parameter variations. We introduce the ideas of successive refinement, and robust optimization based on multi-scenario optimization with iterative sampling of uncertain parameters, using a fast method for approximately evaluating the reflection coefficient. We compare our robust design results to a linear taper, and to optimized tapers that do not take parameter variation into account. We verify the robust performance of our designs using an accurate, but much more expensive, method for evaluating the reflection coefficient.
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