Resonant scattering states in 2D nanostructured waveguides: a boundary wall approach

Abstract

We determine resonant scattering states of two-dimensional photonic crystal nanostructures with defects. To do so, we use the boundary-wall method originally introduced to obtain the scattering eigenstates of one electron moving in a medium with arbitrary boundaries. We investigate geometries including beam bending and interferometer-like waveguides, as well as waveguides connected by resonant cavities. We are able to identify the electromagnetic modes that, due to the special resonance condition attained in the vicinity of defects, provide optimal transmission of an incoming plane wave. Based on the generality of the boundary wall technique and its numerical simplicity and efficiency to identify resonant scattering modes, we briefly discuss further possible applications of this method in analysing the performance of general photonic band-gap devices.

Scalar wave scattering in spherical cavity resonator with conical channels

“…Then the profiles are obtained by numerically integrating jψrj 2 over the volume of the input and output channels, respectively. Appendix A gives a brief description of the boundary wall method, and for a more detailed definition we refer the reader to the works by da Luz et al [25] and Zanetti et al [26,27].…”

The flexibility in choosing distinct Green’s functions for the boundary wall method: waveguides and billiards

“…To facilitate finding the billiard systems eigenstates, it is appropriate to choose a 'seed' state φ k [12] with a large number of modes in the y direction [25]. Thus, we assume the following steady state for the waveguide in the absence of C…”

“…Also, the BWM is valid in any spatial dimension (see, e.g., [12,19]). The BWM has been employed in many distinct applications, as for the investigation of matter waves [20][21][22], analysis of diverse optical processes [23][24][25][26] and description of certain nanostructure properties [18,27,28].…”

Scattering in the Poincaré disk and in the Poincaré upper half-plane

“…Our starting point is the Lippmann-Schwinger (LS) equation [16,17,19,25] which is a Fredholm integral equation of the second kind, and as such, it is a difficult task to overcome. Recently, we presented several exact solutions of the LS equation for two, and three-dimensional barriers [26][27][28] modeled as boundary walls-a name coined by da Luz and collaborators [29] who introduced this method and carried out applications of this technique in condensed matter systems as photonic crystals [30,31]. These boundary wall potentials are composed by singular distributions running along curves [32,33], just as introduced in the original work [29], or defined over surfaces as we studied [34,35].…”