Existence theorems for a nonlinear second-order distributional differential equation

In this work, we are concerned with existence of solutions for a nonlinear second-order distributional differential equation, which contains measure differential equations and stochastic differential equations as special cases. The proof is based on the Leray--Schauder nonlinear alternative and Kurzweil--Henstock--Stieltjes integrals. Meanwhile, examples are worked out to demonstrate that the main results are sharp.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of King Saud University - Science', ISSN 1018-3647. Submitted 05-March-2017; revised 24-April-2017; accepted for publication 26-April-201

Optical bandgap engineering in nonlinear silicon nitride waveguides

Silicon nitride is awell-established material for photonic devices and integrated circuits. It displays a broad transparency window spanning from the visible to the mid-IR and waveguides can be manufactured with low losses. An absence of nonlinear multi-photon absorption in the erbium lightwave communications band has enabled various nonlinear optic applications in the past decade. Silicon nitride is a dielectric material whose optical and mechanical properties strongly depend on the deposition conditions. In particular, the optical bandgap can be modified with the gas flow ratio during low-pressure chemical vapor deposition (LPCVD). Here we show that this parameter can be controlled in a highly reproducible manner, providing an approach to synthesize the nonlinear Kerr coefficient of the material. This holistic empirical study provides relevant guidelines to optimize the properties of LPCVD silicon nitride waveguides for nonlinear optics applications that rely on the Kerr effect

Thermal noise reduction in soliton microcombs via laser self-cooling

Thermal noise usually dominates the low-frequency region of the optical phase noise of soliton microcombs, which leads to decoherence that limits many aspects of applications. In this work, we demonstrate a simple and reliable way to mitigate this noise by laser cooling with a pump laser. The key is rendering the pump laser to simultaneously excite two neighboring cavity modes from different families that are respectively red and blue detuned, one for soliton generation and the other for laser cooling

The Proportion of Weierstrass Semigroups

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.Comment: 15 pages. Corrected typos, some minors mathematical changes, added some discussion. To appear in J. Algebr

High-Q Si3N4 microresonators based on a subtractive processing for Kerr nonlinear optics

Microresonator frequency combs (microcombs) are enabling new applications in frequency synthesis and metrology – from high-speed laser ranging to coherent optical communications. One critical parameter that dictates the performance of the microcomb is the optical quality factor (Q) of the microresonator. Microresonators fabricated in planar structures such as silicon nitride (Si3N4) allow for dispersion engineering and the possibility to monolithically integrate the microcomb with other photonic devices. However, the relatively large refractive index contrast and the tight optical confinement required for dispersion engineering make it challenging to attain Si3N4 microresonators with Qs > 107 using standard subtractive processing methods – i.e. photonic devices are patterned directly on the as-deposited Si3N4 film. In this work, we achieve ultra-smooth Si3N4 microresonators featuring mean intrinsic Qs around 11 million. The cross-section geometry can be precisely engineered in the telecommunications band to achieve either normal or anomalous dispersion, and we demonstrate the generation of mode-locked dark-pulse Kerr combs as well as soliton microcombs. Such high-Qs allow us to generate 100 GHz soliton microcombs, demonstrated here for the first time in Si3N4 microresonators fabricated using a subtractive processing method. These results enhance the possibilities for co-integration of microcombs with high-performance photonic devices, such as narrow-linewidth external-cavity diode lasers, ultra-narrow filters and demultiplexers

When Does Size Matter? An Empirical Study of Consumer Demographics and Product Package Choices

From the Washington University Senior Honors Thesis Abstracts (WUSHTA), 2017. Published by the Office of Undergraduate Research. Joy Zalis Kiefer, Director of Undergraduate Research and Associate Dean in the College of Arts & Sciences; Lindsey Paunovich, Editor; Helen Human, Programs Manager and Assistant Dean in the College of Arts and Sciences Mentor: Tat Cha

Bidirectional initiation of dissipative solitons in photonic molecules

Dissipative solitons (DSs) can be generated in microresonators featuring Kerr nonlinearities via continuous wave (CW) pumping, forming a frequency comb in the spectral domain. While single cavity DSs have been thoroughly investigated in the last years, recent efforts have moved towards photonic molecules (linearly coupled cavities). These arrangements give rise to exotic physical phenomena and practical improvements in terms of conversion efficiency and tuneable comb dynamics. In a recent study of normal dispersion photonic molecules, we found that DSs can be generated in absence of intracavity CW bistability. Here, we show that this feature enables the CW initiation of DSs, tuning the laser into resonance either from the blue side or the red side. While DS initation from the red side has been demonstrated with the photorefractive effect, this is the first demonstration of bidirectional initiation that only requires a Kerr nonlinear medium

Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed

Frequency-Comb-Assisted Swept-Wavelength Interferometry

Swept-wavelength interferometry (SWI) is a highly sensitive and versatile technique implemented in a diverse array of industrial and scientific applications. SWI uses a continuously tunable laser to capture the magnitude and phase response of a device under test (DUT). The prevalent non-linear tuning of the laser calls for an auxiliary interferometer for the calibration of the laser frequency on the fly [1]. However, this approach is susceptible to environmental perturbations, and the inherent dispersion of the interferometer introduces systematic errors. Laser frequency combs can be used as optical rulers against which to calibrate tunable lasers with high- precision and, when self-referenced, with high accuracy [2]. Here, we apply this comb-based calibration approach in the context of SWI for the first time and illustrate its relevance for the characterization of high-Q microresonators

Optical transmission in triple-film hetero-opals

Angle-resolved transmission of s-polarized light in triple-film hetero-opals has been investigated in the spectral range including high-order photonic band gaps, and compared to the transmission of its constituent single-film opals. The interfaces do not destroy the predominantly ballistic light propagation over the studied frequency and angular ranges, but heterostructuring leads to a smoothed angular distribution of intensity of the transmitted light and to the reconstruction of the transmission minima dispersion. The interface transmission function has been extracted by comparing the transmission of the hetero-opal and its components in order to demonstrate the difference. This deviation from the superposition principle was provisionally assigned to light refraction and reflection at the photonic crystal interfaces and to the mismatch between mode group velocities in hetero-opal components