Several methods exist for designing photonic crystal cavities. Gradient search can lead to fast convergence on the optimal fitness design, but requires a convex search space, which is in general not guaranteed. The systematic parameter search is reliable in finding the optimum solution within a very small parameter space. But the method still takes a long time and the parameter space can be overly restrictive.
We investigated the genetic algorithm search for finding un-intuitive structures[1]. A flexible design methodology allows us to optimize PC structures for specific objectives. In this paper, we report the results of several such GA-based PC optimizations. We show that the GA performs well even in very complex design spaces, and therefore has great potential as a robust design tool in a range of PC applications.
The inverse-field approach derives a structure starting from a given electromagnetic mode to [2-4]. Though the analytical approach is general, we demonstrated its utility on the design of cavities with very large Q> 10^6 and near-minimal mode volume ~ (lambda/n)^3. The results are obtained within a single computational step. We first derive a simple expression of the modal out-of-plane radiative loss and demonstrate its utility by the straightforward calculation of Q factors on several cavity designs. Based on this radiation expression, the recipe begins with choosing the FT mode pattern that gives the desired radiation losses. For high-Q cavities with minimal radiative loss inside the light cone, we show that the transform of the mode should be centered at the extremes of the Brillouin Zone, as far removed from the light cone as possible. Next we proved that for a cavity mode with a Gaussian envelope, Q/V grows exponentially with mode volume V while the cavity with the sinc field envelope should lead to even higher Q's by completely eliminating the Fourier components in the light cone. Finally, we derived approximate solutions to the inverse problem of designing a cavity that supports a desired cavity mode. This approach yields very simple design guides that lead to very large Q/V. Since it eliminates the need for lengthy trial-and-error optimization, our recipe enables rapid and efficient design of a wide range of PC cavities. We also showed that the analytical approach may be extended to two dimensions.
The FDTD analysis is not only useful in finding the optimal design, but also in understanding discrepancies between that design and the actual fabricated structure. We have demonstrated a novel method for analyzing discrepancies between designed and fabricated photonic structures[5]. The method makes few idealizations; the planar profile is derived directly from SEM images of the structure, and is translated in the third dimension by the known depth of the structure. With more refined imaging techniques, the third dimension may be scanned separately as well. The calibration from SEM image to dielectric structure in the computer is straightforward and repeatable, and we showed that discretization on the order of 10\unit{nm} gives good results for the PC structures considered here. The analysis of real structures allows direct examination of fabrication errors and their effects on the electromagnetic modes. In addition, by linking the experimentally observed results with simulations of the same structure, this method allows precise evaluation of analytical error models.
Building on the analysis of cavity perturbations, we next considered if intentional perturbations can actually improve certain aspects of a given photonic crystal design[6]. We found that small perturbations can be used to dramatically change the radiated mode of a photonic crystal cavity. This makes it possible to design structures that have both high Q, small mode volume, and directional emission. The latter property was thought to be reserved for large-volume modes. We also found that a similar perturbative approach can be used to couple waveguides modes to radiative modes with desired field distribution.
[1] Joel Goh, Ilya Fushman, D. Englund, and J. Vuckovic. Genetic optimization
of photonic bandgap structures. Opt. Express, 15(13):8218–8230, 2007.
[2] J. M. Geremia, J. Williams, and H. Mabuchi. An inverse-problem ap-
proach to designing photonic crystals for cavity QED. Physical Review E,
66(066606), June 2002.
[3] K. Srinivasan and O. Painter. Momentum Space design of high-Q photonic crystal cavities. Opt. Express,
10(15):670–684, June 2002.
[4] D. Englund, I. Fushman, and J. Vuckovic. General Recipe for Designing
Photonic Crystal Cavities. Opt. Express, 12(16):5961–75, August 2005.
[5] D. Englund and J. Vuckovic. A direct analysis of photonic nanostructures.
Opt. Express, 14(8):3472–83, April 2006.
[6] Dirk Englund, Mitsuru Toishi, and Jelena Vuckovic. Controlled coupling
between photonic crystal and radiative modes by small index perturbations.
in preparation, 2008.