All-optical diode in an asymmetrically apodized Kerr nonlinear microresonator system

We show numerically that a Kerr nonlinear system composed of two channel waveguides coupled periodically by circular microresonators can be used as an all optical diode. The diode has low intensity requirements (50 MW/cm²) and compact dimensions (100  m).     

A Simple Proof of the Matrix-Valued Fejér-Riesz Theorem

A very short proof of the Fejér-Riesz lemma is presented in the matrix case.      

Nonlinear wave interaction in photonic band gap materials

We present detailed analytical and numerical studies of nonlinear wave interaction processes in one-dimensional (1D) photonic band gap (PBG) materials with a Kerr nonlinearity. We demonstrate that some of these processes provide efficient mechanisms for dynamically controlling so-called gap-solitons. We derive analytical expressions that accurately determine the phase shifts experienced by nonlinear waves for a large class of non-resonant interaction processes. We also present comprehensive numerical studies of inelastic interactions, and show that rather distinct regimes of interaction exist. The predicted effects should be experimentally observable, and can be utilized for probing the existence and parameters of gap solitons. Our results are directly applicable to other nonlinear periodic structures such as Bose–Einstein condensates in optical lattices.     

On the integrability of the ergodic hilbert transform for a class of functions with equal absolute values

It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable functionf such that the ergodic Hilbert transform of any function equal in absolute values tof is non-integrable.     

Discrete optical soliton scattering by local inhomogeneities

The nonlinear wave scattering by local inhomogeneities in discrete optical systems is studied both analytically and numerically. The presented theory describes the reflection and transmission of discrete optical solitons at a point defect. In particular, the derived expressions determine the reflected and transmitted pulses from the incident one. In the range of validity, the analytical results are in excellent agreement with the numerical simulations. It is demonstrated that the point defects in structured optical materials represent effective tool for controlling and manipulation of the nonlinear light pulses.     

Justification of the nonlinear Schrödinger equation in spatially periodic media

The dynamics of the envelopes of spatially and temporarily oscillating wave packets advancing in spatially periodic media can approximately be described by solutions of a Nonlinear Schr?dinger equation. Here we prove estimates for the error made by this formal approximation using Bloch wave analysis, normal form transformations, and Gronwall’s inequality.     

Matrix spectral factorization and wavelets

In this paper, recently published results on matrix spectral factorization is reviewed, and their connection to wavelet matrices is revealed.     

On a relationship between the integrabilities of various maximal functions

It is shown that the right-sided, left-sided, and symmetric maximal functions of any measurable function can be integrable only simultaneously. The analogous statement is proved for the ergodic maximal functions.     

Forty years of theory-inspired experiments on charge-transfer via solutions and electrodes: the Georgian accents

Journal of Solid State Electrochemistry - This scholarly review, which also contains some specific historical details, is written by a few of researchers representing two generations succeeding the...