We consider a system governed by the wave equation with index of refraction @math , taken to be variable within a bounded region @math and constant in @math . The solution of the time-dependent wave equation with initial data, which is localized in @math , spreads and decays with advancing time. This rate of decay can be measured (for @math , and more generally, @math odd) in terms of the eigenvalues of the scattering resonance problem, a non--self-adjoint eigenvalue problem governing the time-harmonic solutions of the wave (Helmholtz) equation which are outgoing at @math . Specifically, the rate of energy escape from @math is governed by the complex scattering eigenfrequency, which is closest to the real axis. We study the structural design problem: Find a refractive index profile @math within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact suppo...
Genetic algorithms have also been employed to minimize energy flux through the boundary @cite_3 @cite_24 . In @cite_1 @cite_32 @cite_43 the inverse method'' is employed, where a desired mode shape is chosen and then the material properties which produce that mode are found algebraically. In @cite_4 , the time-dependent problem is solved to steady state using a finite-difference method with perfectly matched layers to approximate the outgoing boundary conditions. The design problem is solved using a Nelder-Mead method.
@cite_4 @cite_1 @cite_32 @cite_3 @cite_24 @cite_43
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Optimization of a truncated, dielectric photonic crystal cavity leads to configurations that are far from truncated crystal cavities, and which have significantly better radiation confinement. Starting from a two-dimensional truncated photonic crystal cavity with optimal Q-factor, moving the rods from the lattice positions can increase the Q-factor by orders of magnitude, e.g., from 130 to 11 000 for a cavity constructed from 18 rods. In the process, parity symmetry breaking occurs. Achieving the same Q-factor with a regular lattice requires 60 rods. Therefore, using optimized irregular structures for photonic cavities can greatly reduce material requirements and device size. We describe a general recipe for designing high-quality factor (Q) photonic crystal cavities with small mode volumes. We first derive a simple expression for out-of-plane losses in terms of the k-space distribution of the cavity mode. Using this, we select a field that will result in a high Q. We then derive an analytical relation between the cavity field and the dielectric constant along a high symmetry direction, and use it to confine our desired mode. By employing this inverse problem approach, we are able to design photonic crystal cavities with Q > 4 ∙ 106 and mode volumes V (λ n)3. Our approach completely eliminates parameter space searches in photonic crystal cavity design, and allows rapid optimization of a large range of photonic crystal cavities. Finally, we study the limit of the out-of-plane cavity Q and mode volume ratio. Photonic band gap (PBG) materials are attractive for cavity QED experiments because they provide extremely small mode volumes and are monolithic, integratable structures. As such, PBG cavities are a promising alternative to Fabry-Perot resonators. However, the cavity requirements imposed by QED experiments, such as the need for high Q (low cavity damping) and small mode volumes, present significant design challenges for photonic band gap materials. Here, we pose the PBG design problem as a mathematical inversion and provide an analytical solution for a two-dimensional (2D) crystal. We then address a planar (2D crystal with finite thickness) structure using numerical techniques. We simulate an evolutionary process in the lab for designing a novel high confinement photonic structure, starting with a set of completely random patterns, with no insight on the initial geometrical pattern. We show a spontaneous emergence of periodical patterns as well as previously unseen high confinement subwavelength bowtie regions. The evolved structure has a Q of 300 and an ultrasmall modal volume of 0:112� �= 2n� 3 . The emergence of the periodic patterns in the structure indicates that periodicity is a principal condition for effective control of the distribution of light. Photonic structures consisting of periodic patterns of high and low index materials can alter the distribution of an electromagnetic field in space and frequency [1,2]. Applications include light emitters, modulators, switches, etc. [3‐7]. Periodic photonic structures have traditionally been hand designed with some insight from the extensive research of crystalline atomic lattice structures, where an analogy between electronic functions in crystalline structures and waves in periodic media with different dielectric functions is drawn. Based on these designs photonic structures that confine light, enhance and inhibit its propagation in specific directions have been demonstrated [8‐11]. It is not clear, however, if the periodicity of photonic structures is a necessary condition for controlling the distribution of light. This question is especially relevant since on one hand localization has been demonstrated in random media [12,13] while on other hand, recent discoveries of periodic photonic structure in biology [14,15] indicate that viable patterns can emerge through blind natural selection, suggesting that the periodicity of the structures is a principal condition for effective light manipulation. In order to address this question, we simulate an evolutionary process in the lab for designing novel photonic structures, starting with a set of completely random patterns, with no information on the initial geometrical pattern. We propose a novel geometry in a silicon planar resonator with an ultra-small modal volume of 0.01(λ 2n)3. The geometry induces strong electric field discontinuities to decrease the modal volume of the cavity below 1(λ 2n)3 The proposed structure and other common resonators such as 1D and 2D photonic crystal resonators are compared for tradeoffs in confinement and quality factors. Most current methods for the engineering of photonic crystal (PhC) cavities rely on cumbersome, computationally demanding trial-and-error procedures. In the present work, we take a different approach to the problem of cavity design, by seeking to establish a direct, semianalytic relationship between the target electromagnetic field distribution and the dielectric constant of the PhC structure supporting it. We find that such a relationship can be derived by expanding the modes of L-N-type cavities as a linear combination of the one-dimensional (1D) Bloch eigenmodes of a PhC W1 waveguide. Thanks to this expansion, we can also ascertain the presence of a well-defined 1D character in the modes of relatively short cavities (e.g., L9-15), thus confirming recent theoretical predictions and experimental findings. Finally, we test our method through the successful design of a cavity supporting a mode with Gaussian envelope function and ultralow radiative losses (quality factor of 17.5 x 10(6)).