Predicting the whispering gallery mode spectra of microresonators

The whispering gallery modes (WGMs) of optical resonators have prompted intensive research efforts due to their usefulness in the field of biological sensing, and their employment in nonlinear optics. While much information is available in the literature on numerical modeling of WGMs in microspheres, it remains a challenging task to be able to predict the emitted spectra of spherical microresonators. Here, we establish a customizable Finite- Difference Time-Domain (FDTD)-based approach to investigate the WGM spectrum of microspheres. The simulations are carried out in the vicinity of a dipole source rather than a typical plane-wave beam excitation, thus providing an effective analogue of the fluorescent dye or nanoparticle coatings used in experiment. The analysis of a single dipole source at different positions on the surface or inside a microsphere, serves to assess the relative efficiency of nearby radiating TE and TM modes, characterizing the profile of the spectrum. By varying the number, positions and alignments of the dipole sources, different excitation scenarios can be compared to analytic models, and to experimental results. The energy flux is collected via a nearby disk-shaped region. The resultant spectral profile shows a dependence on the configuration of the dipole sources. The power outcoupling can then be optimized for specific modes and wavelength regions. The development of such a computational tool can aid the preparation of optical sensors prior to fabrication, by preselecting desired the optical properties of the resonator.Comment: Approved version for SPIE Photonics West, LASE, Laser Resonators, Microresonators and Beam Control XV

Comparative study of density functional theories of the exchange-correlation hole and energy in silicon

We present a detailed study of the exchange-correlation hole and exchange-correlation energy per particle in the Si crystal as calculated by the Variational Monte Carlo method and predicted by various density functional models. Nonlocal density averaging methods prove to be successful in correcting severe errors in the local density approximation (LDA) at low densities where the density changes dramatically over the correlation length of the LDA hole, but fail to provide systematic improvements at higher densities where the effects of density inhomogeneity are more subtle. Exchange and correlation considered separately show a sensitivity to the nonlocal semiconductor crystal environment, particularly within the Si bond, which is not predicted by the nonlocal approaches based on density averaging. The exchange hole is well described by a bonding orbital picture, while the correlation hole has a significant component due to the polarization of the nearby bonds, which partially screens out the anisotropy in the exchange hole.Comment: 16 pages, 5 figures, RevTeX, added conten

Quantum Computing and Communications

This book explains the concepts and basic mathematics of quantum computing and communication. Chapters cover such topics as quantum algorithms, photonic implementations of discrete-time quantum walks, how to build a quantum computer, and quantum key distribution and teleportation, among others

Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\Delta + E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure