Spheroidal Eigenvectors Research Articles

All–Anisotropic Spheroidal Photonic Antennas: Theory and Modeling

We introduce the concept of all-anisotropic spheroidal photonic antennas, where the nanoantenna is formed by a uniaxial anisotropic crystal, and study their modeling and optimization. For this purpose, we develop, for the first time, a rigorous theory for the electromagnetic (EM) scattering by anisotropic spheroids, using a full-wave method which features spheroidal eigenvectors with a set of discrete wavenumbers for the expansion of the fields in the region of anisotropy. We fully validate our theory by comparisons with the commercial HFSS software, as well as with a general purpose discrete dipole approximation method. Our implementation turns out to be fast, and serves as an efficient tool for modeling the resonant properties of spheroidal photonic antennas. In this context, we show that geometrical tailoring, by employing high-aspect ratio prolate and oblate spheroids, allows for optimizing the nanoantenna and achieving maximal forward-to-back scattering or maximal forward-scattering with reduced back-scattering. In addition, we demonstrate how the use of anisotropic materials enables us to set this optimum state at smaller wavelengths, even close to ultraviolet (UV), as compared to isotropic dielectrics, and we reveal the key role of the incident field's polarization in achieving optimized operation for both prolate and oblate configurations.

Efficient Calculation of the Electromagnetic Scattering by Lossless or Lossy, Prolate or Oblate Dielectric Spheroids

In this paper, we study the electromagnetic scattering of a plane wave by a prolate or oblate dielectric spheroid, which can be lossless or lossy. The presented efficient solution is obtained by applying a perturbation technique to the problem of the sphere using the spherical eigenvectors. This method allows a closed-form solution for the fields and the scattering cross sections, which is valid for small eccentricities of the spheroid. Alternatively, we construct the exact solution of the problem using the separation of variables in terms of the spheroidal eigenvectors and updated spheroidal algorithms that allow complex arguments. We compare the closed-form solution versus the exact solution and we conclude about its accuracy. Both polarizations are studied and numerical results are given for various values of the parameters.

Exact Eigenfrequencies in Concentric Prolate Spheroidal-Spherical Metallic Cavities

In this letter the eigenfrequencies in prolate spheroidal-spherical and spherical-spheroidal cavities are calculated. The metallic boundaries of the cavities are perfect conductors. The matrix equations are constructed using expansions connecting the spherical and the spheroidal eigenvectors. Then, by truncating the resulting matrices, the exact eigenfrequencies are obtained as the converged roots of determinantal equations. This method allows to study elongated spheroidal cavities that existing methods fail to handle. The method is validated by comparison with published results from the literature. Original results for spheroids having large eccentricities are given for various values of the parameters.

Electromagnetic Scattering From a Metallic Prolate or Oblate Spheroid Using Asymptotic Expansions on Spheroidal Eigenvectors

The electromagnetic scattering of a plane wave by a metallic spheroid is considered in this work. The solution is obtained using two alternative methods: the first method is based on applying asymptotic expansions on the spheroidal eigenvectors, allowing a closed-form solution for the fields and the scattering cross sections, which is valid for small eccentricities of the spheroid. The second method is based on the full wave analysis using the separation of variables in terms of the spheroidal eigenvectors, which allows to obtain the exact solution of the problem. Both prolate and oblate spheroids are considered, and both polarizations are studied. Numerical results are given for various values of the parameters.

ELECTROMAGNETIC SCATTERING BY A METALLIC SPHEROID USING SHAPE PERTURBATION METHOD

The scattering of a plane electromagnetic wave by a perfectly conducting prolate or oblate spheroid is considered analytically by a shape perturbation method. The electromagnetic field is expressed in terms of spherical eigenvectors only, while the equation of the spheroidal boundary is given in spherical coordinates. There is no need for using any spheroidal eigenvectors in our solution. Analytical expressions are obtained for the scattered field and the scattering cross-sections, when the solution is specialized to small values of the eccentricity h = d/(2a), (h 1), where d is the interfocal distance of the spheroid and 2a the length of its rotation axis. In this case exact, closed-form expressions, valid for each small h, are obtained for the expansion coefficients g(2) and g(4) in the relation S(h) = S(0)[1 + g(2)h2 + g(4)h4 + O(h6)] expressing the scattering cross-sections. Numerical results are given for various values of the parameters.

Electromagnetic Eigenfrequencies in Concentric Spheroidal-Spherical Cavities

The electromagnetic eigenfrequencies fnsm in perfectly conducting concentric spheroidal-spherical cavities are determined analytically. Two types of cavities are examined, one with spheroidal outer and spherical inner boundary and inversely for the other. The problem is solved by two different methods. In the first, the electromagnetic field is expressed in terms of both spherical and spheroidal eigenvectors, connected with one another by well-known expansion formulas. In the second, a shape perturbation method, the field is expressed in terms of spherical eigenvectors only, while the equation of the spheroidal boundary is given in spherical coordinates. The analytical determination of the eigenfrequencies is possible for small values of h = d/(2R2), (h « 1), with d the interfocal distance of the spheroidal boundary and 2R2 the length of its rotation axis. In this case exact, closed-form expressions are obtained for the expansion coefficients 9(2)nsm and 9(4)nsm in the resulting relation fnsm (h) = fns (0) [1 + h2 9(2)nsm + h4 9(4)nsm + O(h6)] . Analogous expressions are obtained by using the parameter v = 1 - (R2/R'2)2 (for lvl « 1) , with 2R'2 the length of the other axis of the spheroidal boundary. Numerical results are given for various values of the parameters.

On the solution of boundary value problems using spheroidal eigenvectors

In the present work an attempt is made to construct the Navier vector functions in spheroidal geometry in a form convenient for the solution of boundary value problems. The complexity of the problem is moderated by exploiting general properties of the vector operators, the underlying Helmholtz equation kernel functions as well as the symmetries of the problem. Finally, the proposed analysis is illustrated in the case of electromagnetic and elastic fields.

Electromagnetic Eigenfrequencies in a Spheroidal Cavity (Calculation By Spheroidal Eigenvectors)

The electromagnetic eigenfrequencies fnsm in a perfectly conducting spheroidal cavity are determined analytically. The analytical determination is possible in the case of small values of h = d/(2a), (h « 1), where d is the interfocal distance of the spheroidal cavity and 2a the length of its rotation axis. In this case exact, closed-form expressions are obtained for the expansion coefficients g(2) nsm and g(4) nsm in the resulting relation fnsm(h) = fns (0) [1 + h2g(2) nsm + h4g(4) nsm + O(h6)]. Analogous expressions are obtained with the use of the parameter v = 1- a2/b2 (for |v| << 1) , where 2b is the length of the other axis of the spheroidal cavity. The electromagnetic field is expressed in terms of spheroidal eigenvectors. Numerical results are given for the lower-order modes.

Electromagnetic Eigenfrequencies in a Spheroidal Cavity

The electromagnetic eigenfrequencies fnsm in a perfectly conducting spheroidal cavity are determined analytically, by a shape perturbation method. The analytical determination is possible in the case of small values of the quantity = 1- a2/b2, (|1), where 2a and 2b are the lengths of the rotation axis and the other axis of the spheroidal cavity, respectively. In this case, exact, closed-form expressions are obtained for the expansion coefficients and in the resulting relation fns (0) [1 + + There is no need for using any spheroidal eigenvectors in our solution. The electromagnetic field is expressed in a series of spherical eigenvectors, while the equation of the spheroidal boundary is given in terms of the spherical coordinates. Numerical results are given for the eigenfrequencies of the lower-order magnetic and electric modes.