ComplexABCDtransformations for optical ring cavities with losses and gain

Abstract

We propose a new design concept and a fabrication process for three-dimensional (3D) silicon photonic crystals on a 100 nm scale that does not require any alignment processes. The elemental technique used in this process is two directional electrochemical etching processes at a particular magnetic field. First, we have performed photonic band calculation and estimated device parameters to obtain the maximum photonic band gap in the visible range centered at around 800 nm. Next, we have experimentally observed the formation of a two-dimensional periodic pore with a diameter of 80 nm and an aspect ratio above 80 on an n þ (100) silicon substrate. Finally, we have fabricated 3D microstructures by two directional etching processes. A clear directionality for the pore formation was observed in two directions, showing the possibility of projecting the patterns formed on the slope to the side of the wafer. These fundamental etching processes can be applied to the fabrication of 3D photonic crystals in the visible range without any alignment processes.

New type of unstable optical resonators

“…In the presence of losses, the elements of matrices T (4) become complex; however, the matrices themselves remain unimodal, as in the case of perfect resonators [1]. A more general analysis of complex transformations in the mechanics of narrow beams has been performed by Maslov [5], who has shown that, if we have a beam, field asymptotic (1) remains correct in the complex case.…”

“…(5) must have solutions with a negative (positive) imaginary part. For passive resonators, this condition is sufficient because, in such resonators, the sign of parameter Im H is retained along the beam [1]. In order to solve Eq.…”

“…The coincidence of the ABCD matrices corresponding to the forward and backward waves follows from the initial equations in which k is replaced with −k [1]. In the presence of losses, the elements of matrices T (4) become complex; however, the matrices themselves remain unimodal, as in the case of perfect resonators [1].…”

“…In our recent studies [1]- [3], [6] we have shown that active ring resonators (the resonators containing an amplifying medium) can posses unidirectional stability. The purpose of this study is to demonstrate that unidirectional stability is not necessary related to the processes occurring in the amplifying medium.…”