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.

/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

Optical impulse-response characterization of diffusive media can be of importance in various applications, among them optical imaging in the security and medical fields. We present results of an experimental technique that we developed for acquiring the impulse response, based upon the Kramers–Kronig algorithm, and have been applied for optical imaging of objects hidden behind clothing. We demonstrate three-dimensional imaging with 5mm depth resolution between diffusive layers.

/pdf/optical-imaging-of-hidden-objects-behind-clothing-363hfb0m1w.pdf

Optical imaging of hidden objects behind clothing

The dependence of the transition between the ballistic and the diffusive regimes of turbid media on the experimental solid angle of the detection system is analyzed theoretically and experimentally. A simple model is developed which shows the significance of experimental conditions on the location of the ballistic–diffusive transition. It is demonstrated that decreasing the solid angle expands the ballistic regime; however, this benefit is bounded by the initial Gaussian beam diffraction. In addition, choosing the appropriate wavelength according to the model’s principles provides another means of expanding the ballistic regime. Consequently, by optimizing the experimental conditions, it should be possible to extract the ballistic image of a tissue with a thickness of 1 cm.

https://www.spiedigitallibrary.org/journals/journal-of-biomedical-optics/volume-18/issue-10/106006/Effect-of-measurement-on-the-ballisticdiffusive-transition-in-turbid-media/10.1117/1.JBO.18.10.106006.pdf

Effect of measurement on the ballistic-diffusive transition in turbid media.

We investigate the fundamental limitations of dispersion mitigation filters. By analyzing the dispersion compensating process from basic principles, we demonstrate how a digital filter can mitigate arbitrarily weak dispersion without oversampling. We calculate the maximum distance the signal can pass with and without dispersion compensation, beyond which no data decoding is possible. Furthermore, we show the exact mathematical relation between this maximum distance and the length of the compensating filter - with and without a Forward Error Correction (FEC).

Fundamental limitations of dispersion mitigation filters.

The real spectrum of bound states produced by PT-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. On the other hand, it is known that the PT-symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) PT symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose PT symmetry remains unbroken for arbitrarily large values of the gain-loss coeffcient. Further, we introduce an extended 2D model with the imaginary part of potential ~ xy in the Cartesian coordinates. The latter model is not a PT-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the PT symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable PT symmetry, while generic soliton families are found in a numerical form. The PT-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with winding numbers m = 1, 2, and 3. In the model with imaginary potential ~ xy, families of single-and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.

Robust PT symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

We apply the multiply subtractive Kramers-Kronig (MSKK) method to the derivative of a medium's optical transfer function. That is, the phase “difference” or derivative Δθ (instead of the phase) can be evaluated from the measurements of the relative derivative of the intensity Δln[I(ω)]=ΔI(ω)/I(ω) with the aid of a few Δθ measurements. As a result, we obtain a method that integrates two different techniques, MSKK and spectral ballistic imaging. We show that the transfer function can be evaluated with great accuracy without the need to measure the phases at all but rather its derivative, which is a much simpler process.

Er'el Granot

Papers

Optical imaging of hidden objects behind clothing

Effect of measurement on the ballistic-diffusive transition in turbid media.

Fundamental limitations of dispersion mitigation filters.

Robust PT symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

Differential multiply subtractive Kramers-Kronig relations