ABSTRACT
Several wave scattering phenomena in optics are modeled by the method of fictitious sources (MFS). Despite its interesting features, the effectiveness of the MFS and its applicability are restricted by open issues, including the placement of the fictitious sources (FS) and the fields' convergence. Concerning these issues, we investigate here the MFS convergence and study oscillations in its solutions for a representative scattering problem of a dielectric cylinder illuminated by a current filament. It is shown analytically that, when the FS radii lie in the interior and exterior of two disks with certain critical radii, the MFS currents' series diverge while the respective fields converge, as the FS number N tends to infinity. Asymptotic formulas of the divergent currents are established, exhibiting that they increase exponentially with N and oscillate. Numerical simulations are included, demonstrating that (i) the divergent currents oscillate for sufficiently large N, (ii) the oscillating values are fairly approximated by the derived asymptotic expressions, and (iii) these oscillations are inherent in MFS and are not due to ill-conditioning; hence, they cannot be overcome by improving the hardware or software. The possibility of obtaining convergent and correct fields from divergent intermediary currents may lead to a potential significant advance of the applicability of the MFS.
ABSTRACT
Traveling waves in two coupled parallel infinite linear point-scatterer arrays are studied analytically for the first time to our knowledge. The two arrays are considered to be generally offset in the axial direction. It is found that slow quasi-even/odd supermodes are supported, as a result of the coupling-induced splitting of the modes of the single array, in direct analogy to standard optical waveguide couplers. Exactly even/odd supermodes are supported when the axial offset is zero. Mode splitting, dispersion curves, and coupling length are numerically investigated versus the inter-element spacing, the inter-array distance, and the axial offset. Potential applications of the concept are in directional optical couplers made of metallic or dielectric nanoparticle chains.
ABSTRACT
The evaluation of acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources is based on well-known volume integral equations; this is particularly true for bodies of arbitrary shape and/or composition, for which separation of variables fails. In this paper the investigation focuses on acoustic (scalar fields) in inhomogeneous spheres of arbitrary composition, i.e., with r-, theta- or even phi-dependent medium parameters. The volume integral equation is solved by a hybrid (analytical-numerical) method, which takes advantage of the orthogonal properties of spherical harmonics, and, in particular, of the so-called Dini's expansions of the radial functions, whose convergence is optimized. The numerical part comes at the end; it involves the evaluation of certain definite integrals and the matrix inversion for the expansion coefficients of the solution. The scalar case treated here serves as a steppingstone for the solution of the more difficult electromagnetic problem.
