Resonant-state expansion applied to three-dimensional open optical systems: Complete set of static modes

Abstract

We present two alternative complete sets of static modes of a homogeneous dielectric sphere, for their use in the resonant-state expansion (RSE), a rigorous perturbative method in electrodynamics. Physically, these modes are needed to correctly describe the static electric field of a charge redistribution within the optical system due to a perturbation of the permittivity. We demonstrate the convergence of the RSE towards the exact result for a perturbation describing a size reduction of the basis sphere. We then revisit the quarter-sphere perturbation treated in [Doost et al., Phys. Rev. A 90, 013834 (2014)], where only a single static mode per each angular momentum was introduced, and show that using a complete set of static modes leads to a small, though non-negligible correction of the RSE result, improving the agreement with finite-element simulations. As another example of applying the RSE with a complete set of static modes, we calculate the resonant states of a dielectric cylinder, also comparing the result with a finite-element simulation.

Full electromagnetic Green's dyadic of spherically symmetric open optical systems and elimination of static modes from the resonant-state expansion

“…We then introduce in Sec. II C a new version of the RSE with static modes entirely eliminated from the basis and provide its illustration for a size perturbation of a dielectric sphere in vacuum, demonstrating in particular a slow convergence, very similar to the standard version of the RSE [43].…”

“…As a result, a ML representation of the dyadic GF of an arbitrary optical system was obtained [35]. This form contains a summation over all the RSs of the system, supplemented with a proper set of static modes required for completeness [43].…”

“…In three dimensions, a homogeneous sphere in vacuum is obviously the simplest basis system allowing an exact analytic solution. It is important to note that the RSE is not limited to small perturbations but is capable of treating perturbations of arbitrary strength, and can be superior to existing computational methods in electrodynamics, such as finite difference in time domain and finite element methods, in terms of accuracy and efficiency, as demonstrated in [29,34,43]. Another significant advantage of the RSE compared to other methods is that it calculates an asymptotically complete set of the RSs of the target system within a wide spectral range; no RSs are missing and no spurious solutions are produced.…”

“…While applying the RSE to 3D open optical systems, it turned out that in addition to the RSs of the basis system, one has to include in the basis for completeness also an additional sets of static modes, in this way representing the static pole of the GF discussed above. In spite of the fact that the problem of static modes in the RSE has been addressed in [29,43], the RSE method still has an unsolved fundamental problem of correct and efficient inclusion of static modes, or even their partial or complete elimination. Indeed, there is presently available either a quick but incomplete static mode inclusion [29], or a complete inclusion of static modes which however suffers from a too slow convergence to the exact solution [43].…”

“…However, as required for verification, illustrations of the new versions of the RSE are provided for the exactly solvable case of an ideal sphere in vacuum. Comparisons with available commercial solvers treating non-spherical cases numerically, similar to those provided in [29,43], will be published elsewhere. Studying the convergence of the RSE towards the available exact solutions, we find the optimal versions of the method which can further be tested and used for non-spherical perturbations.…”