Advanced Mathematical Methods for Scientists and Engineers

We review recent progress in inducing and harnessing stimulated Brillouin scattering (SBS) in integrated photonic circuits. Exciting SBS in a chip-scale device is challenging due to the stringent requirements on materials and device geometry. We discuss these requirements, which include material parameters, such as optical refractive index and acoustic velocity, and device properties, such as acousto-optic confinement. Recent work on SBS in nano-photonic waveguides and micro-resonators is presented, with special attention paid to photonic integration of applications such as narrow-linewidth lasers, slow- and fast-light, microwave signal processing, Brillouin dynamic gratings, and nonreciprocal devices.

Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits

This article reviews two types of ultrasound-mediated biophotonic imaging–acousto-optical tomography (AOT, also called ultrasound-modulated optical tomography) and photo-acoustic tomography (PAT, also called opto-acoustic or thermo-acoustic tomography)–both of which are based on non-ionizing optical and ultrasonic waves. The goal of these technologies is to combine the contrast advantage of the optical properties and the resolution advantage of ultrasound. In these two technologies, the imaging contrast is based primarily on the optical properties of biological tissues, and the imaging resolution is based primarily on the ultrasonic waves that either are provided externally or produced internally, within the biological tissues. In fact, ultrasonic mediation overcomes both the resolution disadvantage of pure optical imaging in thick tissues and the contrast and speckle disadvantages of pure ultrasonic imaging. In our discussion of AOT, the relationship between modulation depth and acoustic amplitude is clarified. Potential clinical applications of ultrasound-mediated biophotonic imaging include early cancer detection, functional imaging, and molecular imaging.

/pdf/ultrasound-mediated-biophotonic-imaging-a-review-of-acousto-3eaqlzes3c.pdf

Ultrasound-mediated biophotonic imaging: A review of acousto-optical tomography and photo-acoustic tomography

We investigate the effect of nonlinearity in novel parity-time (PT) symmetric potentials. We show that new types of nonlinear self-trapped modes can exist in optical PT synthetic lattices.

/pdf/optical-solitons-in-pt-periodic-potentials-gr30h4uehw.pdf

Optical solitons in PT periodic potentials

Optical nonlinearities give rise to many ubiquitous effects in optical fibers. These effects are interesting in themselves and can be detrimental in optical communications, but they also have many useful applications, especially for the implementation of all-optical functionalities in optical networks. In the present paper, we briefly review the different kinds of optical nonlinearities encountered in fibers, pointing out the essential material and fiber parameters that determine them. We describe the effects produced by each kind of nonlinearity, emphasizing their variations for different values of essential parameters. Throughout the paper, we refer to recent systems applications in which these effects have been dealt with or exploited.

Optical nonlinearities in fibers: review, recent examples, and systems applications

It is shown that when the initial particles probability density is discontinuous the emerging currents appear instantaneously, and although the density beyond the discontinuity is initially negligible the currents there have a finite value. It is shown that this non-equilibrium effect can be measured in real experiments (such as cooled Rubidium atoms), where the discontinuity is replaced with finite width (hundreds of nanometers) gradient.

Transients with time-independent currents

Despite the high prevalence of singular potential in general, and rectangular potentials in particular, in applied scattering models, to date little is known about their short time effects. The reason is that singular potentials cause a mixture of complicated local as well as non-local effects. The object of this work is to derive a generic method to calculate analytically the short-time impact of any singular potential. In this paper it is shown that the scattering of a smooth wavefunction on a singular potential is totally equivalent, in the short-time regime, to the free propagation of a singular wavefunction. However, the latter problem was totally addressed analytically in Ref. [7] . Therefore, this equivalency can be utilized in solving analytically the short time dynamics of any smooth wavefunction at the presence of a singular potentials. In particular, with this method the short-time dynamics of any problem where a sharp boundaries potential (e.g., a rectangular barrier) is turned on instantaneously can easily be solved analytically.

Short-time quantum dynamics of sharp boundaries potentials

Recently, many evidences demonstrate that partial Differential Phase Shift Keying (i.e., when the delay inside the Delay Interferometer is shorter than the symbol period) can partially compensate the signal deformation caused by spectrally narrowing the optical channel (by interleavers, add-drop elements, WDM filters, etc.). In this paper the source of this effect is investigated with numerical simulations and, to the best of our knowledge for the first time, analytically. We found that our analytical analysis matched the simulation results with high accuracy. Furthermore, a phenomenological relation, which relates the optimum Free Spectral Range to the channel bandwidth, was derived.

Partial Differential Phase Shift Keying - Theory and Motivation

The existence of Fano resonances in dynamic resonant tunneling (RT) systems has been investigated. Fano resonances are characterized by the appearance of a 100% reflection coefficient in proximity to a high transmission coefficient. For a Fano resonance to appear, a bound state must exist. On the other hand, a resonant tunneling process is characterized by a high transmission and the existence of a quasi-bound state (QBS) instead of a bound one. It has been shown that, by narrowing the width of the barrier, the resonance energy of the QBS gradually decreases and eventually turns into a bound state. Consequently, in a dynamic RT process, there are two scenarios: either a bound state exists, in which case, Fano resonances exist for any barrier width, or a QBS exists, and the barrier should be narrow enough for the Fano resonance to appear. In both cases, the incoming particle’s frequency must be lower than the oscillating well’s frequency. In this work, these resonances are investigated in detail, and both exactly numerically and approximated analytical expressions are derived for both the weak and strong oscillating amplitude regimes. One of the conclusions is that, when the oscillating frequency is low enough, multiple Fano resonances can appear by varying the barrier’s width. Since these resonances are very sharp and zero transmission can easily be detected, this property can be used as a very accurate method for measuring the barrier’s width, even when the particle’s de-Broglie wavelength is much larger than the barrier’s width.

/pdf/multiple-fano-resonances-in-dynamic-resonant-tunneling-3eyp8f83.pdf

Multiple Fano Resonances in Dynamic Resonant Tunneling Processes

The conductivity of an amorphous sample at low temperatures is calculated. While Mott's variable range hopping theory considers infinite samples, the proposed formalism treats finite ones. It turns out that this is a crucial difference. The model predicts a transition temperature (Tc) between two conductivity behaviours: for T Tc (the transition temperature, Tc, depends on the Fermi energy and on the sample's characteristics). The former resembles the simple two-dimensional Mott conductivity behaviour, while the latter resembles the Efros and Shklovskii conductivity theory. We also show a simple connection between these temperatures: .

Er'el Granot

Papers

Transients with time-independent currents

Short-time quantum dynamics of sharp boundaries potentials

Partial Differential Phase Shift Keying - Theory and Motivation

Multiple Fano Resonances in Dynamic Resonant Tunneling Processes

Resonant-tunnelling conductance of a finite-size amorphous sample