Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides

Abstract

Abstract:We derive a sufficient condition for the existence of indexguided modes in a very general class of dielectric waveguides, including photonic-crystal fibers (arbitrary periodic claddings, such as "holey fibers"), anisotropic materials, and waveguides with periodicity along the propagation direction. This condition provides a rigorous guarantee of cutoff-free index-guided modes in any such structure where the core is formed by increasing the index of refraction (e.g. removing a hole). It also provides a weaker guarantee of guidance in cases where the refractive index is increased "on average" (precisely defined). The proof is based on a simple variational method, inspired by analogous proofs of localization for two-dimensional attractive potentials in quantum mechanics.

On the spectrum of waveguides in planar photonic bandgap structures

“…This technique has previously been extensively used in the study of waveguides (e.g. [18,[28][29][30]). …”

Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps

“…where the rightmost integrand of (26) vanishes for α → 0 because differentiating γ results in at least one α factor. Equation ( 28) is strictly negative under conditions (20) and (21). Namely, the V 2 term is smaller than the V term by (21), and (20) implies that the…”

“…13 to index guiding in the periodic Maxwell's equations. 20 (Indefinite-sign localization was also considered for discrete Schrödinger operators. 21 ) As for localization in arbitrary gaps, however, we are not aware of published rigorous results in 2d that are valid for weak V .…”

High-efficiency second-harmonic generation in doubly-resonant χ^(2) microring resonators

“…We solve this problem by designing the second waveguide to have a low-frequency cutoff > ω 1 , so that ω 1 cannot couple out via that channel. In 3d below, a low-frequency cutoff is introduced simply by the waveguide substrate [58][59][60]. In our 2d model system here we obtain a cutoff by the simple expedient of placing the ω 2 waveguide next to a perfect electric conductor (PEC).…”

“…10. (Note that the substrate causes a cutoff in both waveguides, but the cutoff is < ω 1 for the input waveguide [58][59][60].) Since we use separate input/output waveguides, it should be possible to independently control their coupling Q's by varying the waveguide-ring separations.…”