The so-called PT symmetric devices, which feature ε((-x)) = ε((x))* associated with parity-time symmetry, incorporate both gain and loss and can present a singular eigenvalue behaviour around a critical transition point. The scheme, typically based on co-directional coupled waveguides, is here transposed to the case of variable gain on one arm with fixed losses on the other arm. In this configuration, the scheme exploits the full potential of plasmonics by making a beneficial use of their losses to attain a critical regime that makes switching possible with much lowered gain excursions. Practical implementations are discussed based on existing attempts to elaborate coupled waveguide in plasmonics, and based also on the recently proposed hybrid plasmonics waveguide structure with a small low-index gap, the PIROW (Plasmonic Inverse-Rib Optical Waveguide).
We analyze the operation of 2 × 2 switches composed of two coupled waveguides operating on the basis of parity-time (PT) symmetry: the two waveguides differ through their gain or loss factors and not through the real part of their propagation constant. Plasmonics constitutes a preferred application for such systems, since combination of plasmonics with gain is increasingly mastered. The exact PT-symmetric case (gain and loss of identical absolute value) is considered as well as various unbalanced cases, thanks to their respective switching diagrams. Although perfect signal-conserving cross and bar states are not always possible in the latter cases, they can nevertheless form the basis of very good switches if precise design rules are followed. We draw from the analysis what the optimal configurations are in terms of, e.g., guide gain or gain-length product to operate the switch. Many analytical or semi-analytical results are pointed out. A practical example based on the coupling of a long-range surface-plasmon-polariton and a polymeric waveguide having gain is provided.
PT-symmetric structures, such as a pair of coupled waveguides with balanced loss/gain, exhibit a singularity of their eigenvalues around an exceptional point, hence a large apparent differential gain. In the case of fixed losses and variable gain, typical of plasmonic systems, a similar behavior emerges but the singularity is smoothened, especially in more confined structures. This reduces the differential gain around the singular point. Our analysis ascribes the origin of this behavior to a complex coupling between the waveguides once gain is present in an unsymmetrical fashion, even if guides feature the same modal gains in isolation. We demonstrate that adjunction of a real index variation to the variable waveguide heals the singularity nearly perfectly, as it restores real coupling. We illustrate the success of the approach with two geometries, planar or channel, and with different underlying physics, namely dielectric or plasmonic.
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