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.

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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.

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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 emphasized that the bunching parameter β ≡
p
 B
 /p
 D
 ,
i.e. the ratio between the probability to measure two bosons and two distinguishable
particles at the same state, is a constant of motion and depends only on the overlap
between the initial wavefunctions. This ratio is equal to β = 2 / (1 +
I
 2), where I is the overlap integral
between the initial wavefunctions. That is, only when the initial wavefunctions are
orthogonal this ratio is equal to 2, however, this bunching ratio can be reduced to 1,
when the two wavefunctions are identical. This simple equation explains the experimental
evidences of a beam splitter. A straightforward conclusion is that by measuring the
local bunching parameter β (at any point in space and time) it is possible
to evaluate a global parameter I (the overlap between the initial wavefunctions).
The bunching parameter is then generalized to arbitrary number of particles, and in an
analogy to the two-particles scenario, the well-known bosonic enhancement appears only
when all states are orthogonal.

State orthogonality, boson bunching parameter and bosonic enhancement factor

Gamma-ray-irradiated lead-silicate glass shows enhanced photoinduced quasi-phase-matched second-harmonic generation This is due to the formation of color centers, which serve as traps for space charge, resulting in an internal dc field due to the coherent photovoltaic effect The color center formation also enhances the χ(3) nonlinearity, which contributes to the photoinduced quasi-phase-matched second-harmonic generation as well We separate these two contributions and show their respective roles

Optical nonlinearity in gamma-ray-irradiated lead-silicate glass

A generic formalism for the propagation of a pulse with sharp boundaries in any linear medium is derived. It is shown that such a pulse experiences generic deformations. For any given linear medium, the pulse deformation is expressed as a generic differential operator, which characterizes the medium and which operates on the pulse at the singular points (the sharp boundaries). The theory is then applied to a Fabry–Perot etalon and to dispersive media with second order dispersion, third order dispersion, and a combination of both. Simple approximate expressions are also derived for a relatively short, i.e., low dispersive, medium and compared with exact numerical solutions.

Generic propagation of sharp boundaries electromagnetic signals in any linear dispersive medium

In this chapter, a simple market model is presented to illustrate how random entrepreneurial activity can be responsible for the establishment of economic equilibria without the assumption of perfect knowledge. In this model it is assumed that the entrepreneurs (both traders and producers) have no information regarding the other entrepreneurs’ preferences, wealth, or production skills. The only information they have is the past transaction prices, and yet this information is sufficient for the market to reach equilibrium price. Equilibrium is not a stationary process on the microscopic level. It is a process, which consists of interactions between entrepreneurs, who act randomly without insight. Consequently, the market price continuously oscillates randomly around the equilibrium values. The higher the risk the producers are willing to take, the more stable is the equilibrium. When entrepreneurial actions are depressed, the market may drift from its optimal point. This model also investigates the more realistic scenario, in which, due to specialization, the production boundary frontiers are convex (instead of linear). It is shown that in this case, the drifts are suppressed and the optimal equilibrium is more stable. Moreover, the amount of risk aversion has a clear effect on the production growth of the economy. The lower the risk aversion is, the higher is the growth rate of the economy.

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The Roll of the Entrepreneur in the Establishment of Economic Equilibria

We investigate the dynamics of the eigenstate of an infinite well under an abrupt shift of the well's wall. It is shown that when the shift is small compared to the initial well's dimensions, the short-time behavior changes from the well-known ${t}^{3/2}$ behavior to ${t}^{1/2}$. It is also shown that the complete dynamical picture converges to a universal function, which has fractal structure with dimensionality $D=1.25$.

Er'el Granot

Papers

State orthogonality, boson bunching parameter and bosonic enhancement factor

Optical nonlinearity in gamma-ray-irradiated lead-silicate glass

Generic propagation of sharp boundaries electromagnetic signals in any linear dispersive medium

The Roll of the Entrepreneur in the Establishment of Economic Equilibria

Bound eigenstate dynamics under a sudden shift of the well’s wall