F. Seydou

Computation of the Helmholtz Eigenvalues in a Class of Chaotic Cavities Using the Multipole Expansion Technique

In this paper, we present a numerical computation of the energy levels and the corresponding wave functions in a microwave resonator using the multipole expansion technique. The approach permits closed form, fast, and robust solutions of the Helmholtz equation (and the Schrodinger equation for two-dimensional systems) in an important class of wave chaos problem. In particular, wave functions inside the billiard are expressed in terms of a simple expansion of Hankel functions. The implementation of the approach is described, and the classical bowtie cavity is considered as a case study to demonstrate the versatility and efficiency of the method. To validate the accuracy, the field distribution and the eigenvalues calculated using this approach are compared to the solution obtained by boundary integral method. The case when the cavity contains objects (perfect electric conductors and/or dielectrics) is also presented and discussed.

Numerical computation of the Green’s function for two-dimensional finite-size photonic crystals of infinite length

We develop a numerical algorithm that computes the Greens function of Maxwell equation for a 2D finite-size photonic crystal, composed of rods of arbitrary shape. The method is based on the boundary integral equation, and a Nyström discretization is used for the numerical solution. To provide an exact solution that validates our code we derive multipole expansions for circular cylinders using our integral equation approach. The numerical method performs very well on the test case. We then apply it to crystals of arbitrary shape and discuss the convergence.

Annular photonic crystals: computation and analysis of the green’s function

In this paper we discuss two-dimensional (2D) annular PC. In particular, we consider a finite size PC composed of an air-hole and dielectric rod array such that an air-hole is centered within each dielectric rod. The clusters we consider consist of parallel, disjoint two-layered cylinders of arbitrary shapes. We implement computational techniques and analyze the Greens function, using both the semi-analytical (for circular cylinders) and numerical-boundary element (for arbitrary cylinders) methods.

A multipole expansions method for acoustic wave propagation in vocal tract

The paper investigates the problem of wave motion in a vocal tract. We try to solve the problem using the Helmholtz equation, one of the most important mathematical models used to describe the time harmonic behavior of various vibration and wave propagation phenomena. The motivation of our work is to understand the main characteristics of wave propagation in the vocal tract, which is modeled as a series of 8 concatenated tubes. Some examples of research applications include human speech production, speech recognition and speaker identification. Some of the characteristic quantities to be calculated in these problems include scattering amplitudes, transmission and reflection coefficients, and resonance frequencies. We use multipole expansions to derive a closed form solution of the problem. For testing the method, we show the numerical results for the case of a disc where the solution is well known. Our approach shows excellent results. We intend to extend the method for the three dimensional case and to related inverse problems.

Quasi-Analytical Computation of Energy Levels and Wave Functions in a Class of Chaotic Cavities with Inserted Objects

A simple multipole expansion method for analytically calculating the energy levels and the corresponding wave functions in a class of chaotic cavities is presented in this work. We will present results for the case when objects, which might be perfect electric conductors and/or dielectrics, are located inside the cavity. This example is demonstrative of typical experiments used in chaotic cavities to study the probabilistic eigenvalue distribution when objects are inserted into the cavity.

TWO METHODS FOR PROFILE RECONSTRUCTION IN SCATTERING THEORY

In this paper we discuss the problem of finding a profile for the scattering of electromagnetic waves with fixed frequencies. This type of problem is one of the most important and interesting problems arising in mathematical physics and has applications in geophysics, technology, medicine and nondestructive testing (cf. Rfs. 1, 2, 3, 4, 5). The difficulty in solving such problems is due to two unpleasant facts: They are nonlinear and, more seriously, they fall in a group of problems called ill-posed that is the solution-if it exists at all-does not depend on the data. Many authors suggested methods for overcoming these difficulties. The methods can be broadly divided into two classes: Optimization and iterative methods. In the previous works most algorithms were for TM wave illuminations whereas much less was reported for the most complicated TE case. The TE incident field is complicated compared to the TM case because of the strong nonlinearity and singularity of the kernel of the integral equation. In our work we consider one optimization (the dual space) method and one iterative (the simplified Newton) method. We shall consider the two scattering problems: the TM and the TE polarizations. It is well known that a combination of the two cases improves the quality of reconstruction. Our objective is to implement numerically the two methods to both problems and discuss the advantages and disadvantages in each case. We shall emphasize on the frequency and the type of profiles for the reconstructions. The simplified Newton method we shall develop has many advantages over other iterative methods: It is computationally inexpensive and the forward problem does not need to be solved at each iteration step. We shall compare that results with the implementation of the dual space method for different profiles and frequencies.

Two methods for solving a 3D acoustic inverse scattering problem

We consider the problem of finding the refractive index of a buried object by using far‐field measurements in an inhomogeneous medium. We describe two methods for solving the inverse problem. Both methods are implemented in two steps in order to better deal with the ill‐posedness of the problem. In the first method an integral equation of the first kind is derived for the far‐field operator which is solved via least‐squares and Tikhonov regularization. We then use the solution of the integral equation to derive an over‐posed boundary value problem, i.e., the Helmholtz equation in a bounded domain with Cauchy data on the boundary. The index that satisfies this over‐posed problem most closely is obtained via the Levenberg–Marquardt algorithm. The second method is an iterative method and is based on the Lippmann–Schwinger equation. It is implemented via the Newton method. The first step consists, as for the other method. Here we use a Fourier integral approach and regularization via discretization. The secon...

Analysis of particular phononic structures using single integral equations

A fast method for determining phononic (and photonic) bandgaps in composite materials is developed. It is known that in the propagation of waves in a 3D medium containing N scatterers arranged periodically, there exist refractive indices for which such structures have bandgaps, i.e., frequencies for which no waves can propagate inside. Our task is to find the frequencies that generate these prohibited waves. This requires the solution of an eigenvalue problem for the Helmholtz operator. To solve this problem we choose an alternate route which uses boundary integral equations. We derive a single integral equation on each of the interfaces between the outer region and the scatterers, considering a general transmission boundary condition, by using a hybrid method using layer potentials and Green’s formula. This approach reduces the number of unknowns considerably in comparison to other methods, but requires the treatment of large dense matrices and many matrix vector multiplications. To remedy this, we use t...