Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices

A typical photonic crystal (PhC) device has only a small number of distinct unit cells. The Dirichlet-to-Neumann (DtN) map of a unit cell is an operator that maps the wave field to its normal derivative on the boundary of the cell. Based on the DtN maps of the unit cells, a PhC device can be efficiently analyzed by solving the wave field only on edges of the unit cells. In this paper, the DtN map method is further improved by an operator marching method assuming that a main propagation direction can be identified in at least part of the device. A Bloch mode expansion method is also developed for structures exhibiting partial periodicity. Both methods are formulated on a set of curves for maximum flexibility. Numerical examples are used to illustrate the efficiency of the improved DtN map method.     

On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators describing nonlocal interactions in ${L^2(\Omega; d^n x)}$ , where ${\Omega \subset \mathbb{R}^n}$ , ${n\in\mathbb{N}}$ , ${n\geq 2}$ , are open sets with a compact, nonempty boundary ${\partial\Omega}$ satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in ${L^2(\Omega; d^{n} x)}$ to Fredholm perturbation determinants associated with operators in ${L^2(\partial\Omega; d^{n-1} \sigma)}$ , ${n\in\mathbb{N}}$ , ${n\geq 2}$ . This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrödinger operator on the half-line ${(0,\infty)}$ , in the case of local interactions, to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation.     

Arithmetic,Zeros, and Nodal Domains on the Sphere

We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelöf hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelöf.     

Nodal lines,ergodicity and complex numbers

This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is, the zeros of the analytic continuation of the eigenfunctions to the complexification of the underlying space. Such a complexification of the problem is analogous to studying the complex zeros of polynomials with real coefficients. The first result determines the limit distribution of complex zeros of `ergodic eigenfunctions' such as eigenfunctions of classically chaotic systems. The second result determines the expected distribution of complex zeros for complexifications of Gaussian random waves adapted to the Riemannian manifold. The resulting distribution is the same in both cases. It is singular along the set of real points.     

Core states and spectral flow in vortex dynamics

I review the semi-classical picture of how states bound in the core of a vortex in an S-wave superconductor respond to relative motion between the vortex and the condensate. I show how the momentum absorbed as a result of the Magnus force acting on the core leads to a change in the distribution of occupied states (“spectral flow”). In the simplest relaxation time approximation this modified distribution gives rise to the Kopnin–Kravtsov force on the vortex.     

Some properties of the spectral flow in semiriemannian geometry

Let be the action integral on a semiriemannian manifold ( , g) defined on the space of the curves z : [0, 1] → joining two given points z₀ and z₁. The critical points of ƒ are the geodesics joining z₀ and z₁. Let s ε [0, 1]. We study the behavior, in dependence of s, of the eigenvalues of the Hessian form of ƒ evaluated at z, restricted to the interval [0, s]. A formula for the derivative of the eigenvalues is given and some applications are shown.     

The integrable harmonic map problem versus Ricci flow

We construct a zero-curvature representation for a four-parameter family of non-linear sigma models with a Kalb–Ramond term. The one-loop renormalization is performed that gives rise to a new set of ancient and eternal solutions to the Ricci flow with torsion. Our analysis provides an explicit illustration of the role of the dilaton field for the renormalization of the non-linear sigma model.     

Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps

Efficient numerical methods for analyzing photonic crystals (PhCs) can be developed using the Dirichlet-to-Neumann (DtN) maps of the unit cells. The DtN map is an operator that takes the wave field on the boundary of a unit cell to its normal derivative. In frequency domain calculations for band structures and transmission spectra of finite PhCs, the DtN maps allow us to reduce the computation to the boundaries of the unit cells. For two-dimensional (2D) PhCs with unit cells containing circular cylinders, the DtN maps can be constructed from analytic solutions (the cylindrical waves). In this paper, we develop a boundary integral equation method for computing DtN maps of general unit cells containing cylinders with arbitrary cross sections. The DtN map method is used to analyze band structures for 2D PhCs with elliptic and other cylinders.     

On the Arnold cat map and periodic boundary conditions for planar elongational flow

In this paper we show that the periodic boundary conditions used to simulate planar elongational flow are closely related to the Arnold cat map. In particular the relationship between the Arnold cat map and the periodic boundary conditions devised by Kraynik and Reinelt [1992, Int. J. multiphase Flow, 18, 1045], the so-called K-R map, is demonstrated. It is shown that the family of lattices found by Kraynik and Reinelt corresponds to a subset of hyperbolic toral automorphisms. These lattices were previously found to be sufficient to enable molecular dynamics simulations of steady-state planar elongational flow of unrestricted duration. Within the frame of the cat map we provide a re-derivation for the set of eigenvalues, eigenvectors and orientation angles of the K-R map and find it to be considerably simpler than the original derivation provided by Kraynik and Reinelt.     

Time-delayed map as a model for open fluid flow

It is shown that a time-delayed map for just one (chaotic) element whose feedback is periodically interrupted can be exactly mapped to a coupled map lattice model for open fluid flow.     

Stochastic quantization,supersymmetry and the Nicolai map

We present a formulation of the method of stochastic quantization of Parisi and W that reveals its intimate connection with supersymmetry. The crucial ingredient of this analysis is the Nicolai map. By using supersymmetric Ward identities, we derive relations between two Fokker-Planck-type Hamiltonians which arise naturally in this formalism.     

Nodal domain statistics for quantum maps, percolation, and stochastic Loewner evolution

We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general Hamiltonian systems. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by stochastic Loewner evolution with diffusion constant close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.     

Phase diagrams in unidirectionally coupled map lattice for open traffic flow

The jamming transition from the free traffic to the oscillatory traffic is investigated with the unidirectionally coupled map lattice model which has the hyperbolic tangent local map. Spatio-temporal structures in the jamming transition are found with the use of numerical simulation. The traffic states are studied for both constant and noisy boundary conditions. We show the phase diagrams of different kinds of congested traffic. It is found that the noise at the boundary has an important effect on the traffic states. The traffic behavior in the coupled map lattice model exhibits a jamming transition similar to that found in the car-following model.     

Studies on the spectral width,monochromatic degree and spectral resolving power of prism monochromator

The angular width and geometric line width in the spectral plane for the output incoherent imaging spectral line are discussed respectively on the basis of structure of prism monochromator, the relationship between the spectral width and line width of output incoherent imaging spectral line in the spectral plane is presented. The expression of monochromatic degree of output light in the exit slit plane is presented. The spectral resolving power of incoherent imaging spectral line and coherent imaging spectral line of the diffractive waves are discussed respectively, and the expressions of the spectral resolving power are presented respectively.     

Fermionic path integrals,the nicolai map and the witten index

We present an explicit evaluation of the coherent-state fermion path integral and discuss our results in the light of supersymmetric quantum mechanics, the Nicolai map and the Witten index.     

Supertracks,supertrack functions and chaos in the quadratic map

Chaotic behavior for the quadratic map is conjectured to be a bounding phenomenon based on state-space trajectories starting from the maximum in the map. These trajectories, called supertracks, have no sensitivity to initial conditions. The chaotic regime is found to be characterized recursively by these trajectories as functions of the map parameter.     

Supersymmetry, PT-symmetry and spectral bifurcation

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.