With the extreme miniaturization of NEMS, the role of nonlinear dynamics has become increasingly important -- even when the dynamics are driven by the Brownian force. This nonlinearity has imposed a mechanical noise floor on the linear frequency measurements made in sensing applications. Given that NEMS also become more sensitive as they become smaller, this floor has resulted in a complex interplay between the nonlinear regime and the linear sensitivity required to make continued advancements in exercising ultra-sensitive measurements. Recently, this has led to efforts to more accurately characterize the edge of the linear regime. Inside of the nonlinear regime, there are also ongoing fundamental studies in theory and experiment to partially characterize the nonlinear behavior of NEMS. Theoretically, these systems are frequently studied by decomposing the nonlinear continuous system into one or more nonlinear oscillators. However, in many of these works, the nonlinear spring constants are estimated at the lowest order. As such, there is a clear need to more accurately characterize and scale the nonlinear coefficients for NEMS. This work considers a long and slender NEMS resonator in the form of a doubly-clamped beam in tension. Using nonlinear Euler-Bernoulli beam theory, the geometric nonlinearity due to the stretching of the neutral axis is considered. We extensively explore simplifications in using a Galerkin discretization of the continuous system, where a single mode's dynamics are described as a damped, Duffing oscillator. We examine limitations of current approaches and find that using a tensioned and doubly-clamped mode shape for the trial function more accurately predicts experiment. Additionally, we find that doubly-clamped beams of finite tension may have their boundary conditions modified to that of a hinged-hinged beam in tension with little to no loss of generality. This modification allows for closed-form scaling of the critical amplitude and dynamic range at arbitrary mode number and tension. We extend this approach to scale the relative influence of bending, tension, and nonlinearity with increasing mode number n, finding that bending and nonlinear influences quickly outgrow the contributions of the intrinsic tension. Where applicable, these results are compared with experiment, and we obtain good agreement. To validate the approximate Galerkin formulations, we develop a finite element method to calculate the nonlinear coefficients of symmetric NEMS resonators. Unlike previous works, the present formulation may include all nonlinearity due to geometry, and the nonlinear amplitude-frequency backbone described by the Duffing oscillator is found as an excellent approximation for large amplitude beams. For beams of zero intrinsic tension, the finite element method obtains excellent agreement with the literature. For beams of finite tension and varying mode number, we find the error from the Galerkin discretization is small (≈5%). In addition, we theoretically explore the stochastic dynamics of a Duffing oscillator driven nonlinearly by the Brownian force. To access this regime experimentally with current nanomechanical systems, we motivate an experimental "synthetic noise" to approximate the Brownian force in the proximity of a single mode. As a measure of drive magnitude, we vary an effective temperature to explore the linear and nonlinear stochastic dynamics of a doubly-clamped nanoresonator. Using similitude and the Fluctuation-Dissipation theorem, we show that varying the effective temperature of the synthetic noise offers a window into the fundamental limits of thermally-driven nonlinearities. We compare theory, numerics, and experiment where applicable, obtaining good agreement for both limits of frequency shifts in the weakly nonlinear case. This research was supported by the National Science Foundation, grant number CMMI-2001559, and portions of the computations were conducted using the resources of Virginia Tech's Advanced Research Computing (ARC) center. / Master of Science / Nanoelectromechanical systems (NEMS) are nanoscale mechanical structures that convert physical stimuli (force, mass, acceleration, charge, etc.) to measurable electrical signals. Due to the extremely small size of NEMS, they offer an unprecedented level of sensitivity in a variety of measurement applications. However, as NEMS become smaller, the response of these mechanical structures begin to exhibit nonlinear behaviors. Said otherwise, proportional inputs (such as drive strength) do not result in proportional outputs. These nonlinear behaviors include a variety of undesired effects, such as multi-valued unstable/stable solutions and a noisy resonant frequency. In this work, we study a NEMS resonator in the form of a doubly-clamped beam, and we consider the stretching of the mid-plane as the nonlinearity. Here, the stretching of the mid-plane (an axial strain) is nonlinearly dependent on the amplitude of vibration, inducing a nonlinear tension force. With this model, it is typical to represent a singular nonlinear vibrational mode as a simple harmonic oscillator with an additional cubic term. In order to better characterize the edge of the nonlinear regime, the relative strength of the cubic term must be known. We thoroughly explore existing and new simplifications to obtain the nonlinear coefficient for the cubic term, demonstrating two possible approaches for better accuracy in beams of varying tension and mode number. These simplifications are validated by comparing with the present finite element method to determine the nonlinearity in symmetric NEMS resonators. Using these new insights, theory and numerics are used to explore the behavior of a doubly-clamped beam in a stochastic (random) force field. This force field is tailored to represent the collisions of surrounding molecules at the nanoscale, allowing exploration of nonlinear behavior at its fundamental limits. Where applicable, theory and numerics are compared to experiment, and we obtain good agreement. This research was supported by the National Science Foundation, grant number CMMI-2001559, and portions of the computations were conducted using the resources of Virginia Tech's Advanced Research Computing (ARC) center.

The objective of this research is to show that nonlinearity can be used to improve vibration absorption and suppression of unwanted vibrations in a main system due to external excitation. This was shown by investigating two systems a SDOF (with hardening nonlinearity) and a 2DOF (with softening nonlinearity). The aim of carrying out these investigations was to introduce a passive nonlinear system that can update itself and self-regulate to suppress undesired oscillations. To fulfil the desired gaol, various types of springs were considered and investigated. A commercially available spring called Mag-spring has been chosen and a function for its nonlinearity has been investigated. Mag-Spring is a newly invented spring which is designed to exhibit constant force at its operating range. However, this spring has a special non-linear behaviour before reaching to the constant force domain which is the main focus of the investigation presented in this thesis. The nonlinear behaviour of Mag-spring encouraged the idea that vibration design is possible by the advantages that can be gained from magnetic technology. The added benefit through this new Mag-spring, is that it solved some of the concerns assotiated with old available ones. The most concern norrowing the usage of magnetic springs, is the air gap between the two magnets, which make the attraction or repulsive force unstable through the spring’s working range. Linmot Company, introduced a solution to this concern by introducing a teflone that works as a bearing between the two magnets, which fixed the distance between them while they are sliding againest each other. In the first scenario of this study, a hardening nonlinear spring was added in parallel to a system with a single degree of freedom. The system will remain single degree of freedom as the spring was added in parallel without additional mass. The hardening spring shows low stiffness at low amplitude and high stiffness at higher amplitude. In this study, it was shown that nonlinearity affects the dynamic performance of a system and makes the natural frequency amplitude dependant. As the amplitude of vibration increases, consequently, stiffness increases and the natural frequency shifts away from the excitation frequency. For this investigation, a vibrating system with one degree of freedom has been built based on a mathematical model simulated and tested in Matlab software. Mag-spring was used to introduce the nonlinear stiffness to the system. Unbalance mass mounted to a disc fixed to a rotational machine has been used to create a forced vibration system with variable frequency. The response of the system with and without nonlinearity effect was monitored with an accelerometer. Simulation and experimental results showed that nonlinearity could shift the resonance frequency of the SDOF system by 10% (hardening of the system), without affecting the stiffness of the system at normal working condition. In the second scenario, a softening nonlinear spring was added as a vibration absorber to a system with a single degree of freedom, to make the system with two degree of freedom. The softening spring shows high stiffness at low amplitude and low stiffness at high amplitude. The rationale behind this is to introduce a spring which is hard at high frequency and soft at low frequency, which as a result will make the ratio √(k_a/m_a ) of the absorber follows the excitation frequency (ω) allowing the system to update itself and self-regulate providing vibration cancellation at more than one frequency value and widen the vibration cancellation range (ω_n2-ω_n1). It was shown that the Mag-spring could show a softening behaviour in a limited domain if its operating position is shifted. A program has been written to simulate the behaviour of all nonlinear system with two degree of freedom (nonlinear absorber). At this program, the maximum amplitude of each time domain was used to produce the frequency domain of the amplitude of the system. The amplitude of the vibration for a linear and a nonlinear absorber was compared. The results showed that the nonlinear absorber suppresses and reduces the vibration amplitude of the main system better than the linear absorbers with up to 60% reduction in magnification ratio and from 5% to 10% in widening the cancellation range (ω_n2-ω_n1). In the last scenario of this study, 4 different ideal softening stiffness curves were introduced based on theoretical methods. Their vibration response was calculated and compared to the nonlinear absorber (Mag-spring) and a linear absorber. This study shows that when nonlinearity is designed properly, it could provide a distinguished vibration cancellation response resulting more than 60% vibration cancellation improvement. This study demonstrated the possibility of developing a passive self-regulating tuned mass system involving the usage of nonlinearity. Nonlinearity will enhance the vibration cancellation by allowing the system to update itself and as a result will make the vibration absorption to be effective within a frequency range rather than single frequency unlike the classical tuned mass system. This study, to the best knowledge of the author, can be classified as an uncommon study in vibration systems investigations.

I have conducted detailed experimental and theoretical studies of the nonlinear optical properties of semiconductor materials useful for optical limiting. I have constructed optical limiters utilizing two-photon absorption along with photogenerated carrier defocusing as well as the bound electronic nonlinearity using the semiconducting material ZnSe. I have optimized the focusing geometry to achieve a large dynamic range while maintaining a low limiting energy for the device. The ZnSe monolithic optical limiter has achieved a limiting energy as low as 13 nJ (corresponding to 300W peak power) and a dynamic range as large as 105 at 532 nm using psec pulses. Theoretical analysis showed that the ZnSe device has a broad-band response covering the wavelength range from 550 nm to 800 nm. Moreover, I found that existing theoretical models (e.g. the Auston model and the band-resonant model using Boltzmann statistics) adequately describe the photo-generated carriers refractive nonlinearity in ZnSe. Material nonlinear optical parameters, such as the two-photon absorption coefficient β_2=5.5cm/GW, the refraction per unit carrier density σ_n=-0.8∗10^-21cm^3 and the bound electronic refraction n_2=-4∗10^-11esu, have been measured via time-integrated beam distortion experiments in the near field. A numerical code has been written to simulate the beam distortion in order to extract the previously mentioned material parameters. In addition, I have performed time-resolved distortion measurements that provide an intuitive picture of the carrier generation process via two-photon absorption. I also characterized the optical nonlinearities in a ZnSe Fabry-Perot thin film structure (an interference filter). I concluded that the nonlinear absorption alone in the thin film is insufficient to build an effective optical limiter, as it did not show a net change in refraction using psec pulses. An innovative numerical program was developed to simulate the nonlinear beam propagation inside the Fabry-Perot structure. For comparison, pump-probe experiments were performed using both thin film and bulk ZnSe. The results showed relatively long carrier lifetimes (>300 psec) in both samples. A numerical code was written to fit the pump-probe experimental results. The fitting yielded that carrier lifetimes (recombination through traps), radiative decay rate, two-photon absorption coefficient as well as the free carrier absorption coefficient for ZnSe bulk material.

No description available.

Nonlinear fiber devices have been attracting considerable attention in recent years, due to their inherent ultrafast response time and potential applications in optical communication systems. They usually require long fibers to generate sufficient nonlinear phase shifts, since nonlinearities of conventional silica-core silica-clad fibers are too low. These long devices, however, cause the serious problems of pulse walk-off, pulse broadening, and polarization fluctuation which are major limiting factors for response time, switching bandwidth, and maximum transmittable bit-rate. Therefore, short device length is indispensable for achieving ultrafast switching and higher bit-rate data transmission. To shorten the required device length, fiber nonlinearities should be increased. In this dissertation, as a way of increasing fiber nonlinearities, high-nonlinearity materials of Litharge, Bismite, Tellurite, and Chalcogenide glasses have been considered. Although they have high nonlinearities, they also have high group-velocity dispersion and high losses deteriorating the performance of nonlinear fiber devices seriously. The aim of this work is to investigate how these high-nonlinearity glasses affect the performance of nonlinear fiber devices, taking into consideration both the advantages and disadvantages. To achieve it, the critical properties of various nonlinear fiber devices constructed with the different types of high-nonlinearity glasses and different types of fibers have been evaluated. It turned out that the required device lengths of nonlinear fiber devices constructed with the high-nonlinearity glasses were significantly reduced and high group-velocity dispersions and losses could not be major problems due to the extremely short device length. As a result, it would be possible to suppress the problems of pulse walk-off, pulse broadening, and polarization fluctuation in nonlinear fiber devices by introducing high-nonlinearity glasses, thus enabling ultrafast switching and higher bit-rate data transmission. Furthermore, in this dissertation, a new scheme of wavelength-division demultiplexing based on the optical Kerr effect has been proposed for the first time. The new scheme can turn the disadvantage of the extremely high group-velocity dispersion of high-nonlinearity glasses into an advantage of wavelength-division demultiplexing. Finally, it now would be possible to greatly increase maximum transmittable bit-rate in optical communication systems by simultaneously demultiplexing optical time-division-multiplexed signals and wavelength-division-multiplexed signals with an optical Kerr effect-based demultiplexer. / Ph. D.

To help resolve certain practical issues with acoustical methods for landmine detection, experiments were performed using a pulsed, standoff source consisting of sixteen speakers mounted on a circular arc. This source, as well as a pair of 18-inch subwoofers, were used separately for acoustical excitation of the buried mine, and the response of the target site was examined as a function of source frequency, sound pressure level, and excitation signal type, with a particular focus on multitone signals. In addition, modeling was undertaken to investigate the effects of nonlinearity, including bimodular nonlinearity, on frequency generation. A numerical, time-domain solution based on a lumped-element model proposed by Donskoy et al. [J. Acoust. Soc. Am. 117, 690 (2005)] was developed and used to simulate pulsed excitation and the effects of bimodular nonlinearity, which allowed experimentally observed spectra to be compared with modeled results. / text

In order to be able to determine neutrino mass hierarchy in the neutrino oscillation experiment JUNO we need to understand the dependence of the response of the signal from the scintillator on the deposited energy inside the scintillator. We measure the nonlinearity of the signal response via the Compton scattering inside the scintillator and via the precision gamma spectroscopy inside the HPGe detector. We observe effects of different parameters on the experiment via the Monte Carlo simulations. We also improve the data processing of the measured data and discuss what improvements of the experiment we can use in the future.

This thesis introduces the linear and nonlinear properties of photonic crystal fibre (PCF), describes the fabrication and characterisation of different PCFs, and demonstrates their applications to supercontinuum (SC) generation and single-photon sources. The linear properties of PCF include endlessly single-mode transmission, highly controllable dispersion and birefringence. These unique properties have made PCFs the best media to demonstrate all kinds of nonlinear effects such as self-phase modulation (SPM), cross-phase modulation (XPM), Raman effects, four-wave mixing and modulation instability (FWM and MI), and soliton effects. The combination of these nonlinear effects has led to impressive spectral broadening known as SC generation in PCFs. The intrinsic correlation of signal and idler photons from FWM has brought PCF to the application of single-photon generation. Four projects about SC generation were demonstrated. The first was visible continuum generation in a monolithic PCF device, which gave a compact, bright (-20 dBm/nm), flat and single-mode visible continuum source extending to short wavelength at 400 nm. The second was polarised SC generation in a highly bire-fringent PCF. A well linearly polarised continuum source spanning 450-1750 nm was achieved with >99% power kept in a single linear polarisation. This polarised continuum source was then applied to tuneable visible/UV generation in a BIBO crystal. The third was residual pump peak removal for SC generation in PCFs. The fourth was to design an all-fibre dual-wavelength pumping for spectrally localised continuum generation. Two projects about photon pair generation using FWM were then demonstrated. One was an all-fibre photon pair source designed in the telecom band for quantum communication. This source achieved >50% heralding efficiency which is the highest in fibre photon pair sources reported so far. Another one was to design birefringent PCFs for naturally narrow band photon pair generation in the Si SPAD high detection efficiency range. 0.122 nm bandwidth signal photons at 596.8 nm were generated through cross polarisation phase matched FWM in a weakly birefringent PCF pumped by a picosecond Ti:Sapphire laser at 705 nm in the normal dispersion regime.

"As a part of an effective SETAR (self-exciting threshold autoregressive) mod- eling methodology, it is important to identify processes exhibiting SETAR-type non- linearity. A number of tests of nonlinearity have been developed in the literature, including those of Keenan (1985), Petruccelli and Davies (1986), Tsay (1986, 1989), Luukkonen (1988), and Chan and Tong (1990). However, it has recently been shown that all these tests perform poorly for SETAR-type nonlinearity detection in the presence of outliers. In this project we develop an improved test for SETAR-type nonlinearity in time series. The test is an outlier-robust variant of the Petruccelli and Davies (1986) test based on the cumulative sums of ordered weighted residuals from generalized maximum likelihood fits (which we call CUSUM-GM). The properties of the proposed CUSUM-GM test are illustrated by means of Monte Carlo simulations. The merits, in terms of size and power, of the proposed test are evaluated relative to the test based on ordered residuals from the ordinary least squares fit (which we call CUSUM-LS) and also to that of other tests for nonlinearity developed in literature. The simulations are run for uncontaminated data and for data contaminated with additive and innovational outliers. The simulation study strongly supports the validity of the proposed robust CUSUM-GM test, particularly in situations in which outliers might be a problem."

The main subject of this thesis is solitons due to degenerate parametric four-wave mixing. Derivation of the governing equations is carried out for both spatial solitons (slab waveguide) and temporal solitons (optical fibre). Higher-order effects that are ignored in the standard paraxial approximation are discussed and estimated. Detailed analysis of conventional solitons is carried out. This includes discovery of various solitons families, linear stability analysis of fundamental and higher-order solitons, development of theory describing nonlinear dynamics of higher-order solitons. The major findings related to the stationary problem are bifurcation of a two-frequency soliton family from an asymptotic family of infinitely separated one-frequency solitons, jump bifurcation and violation of the bound state principle. Linear stability analysis shows a rich variety of internal modes of the fundamental solitons and existence of a stability window for higher-order solitons. Theory for nonlinear dynamics of higher-order solitons successfully predicts the position and size of the stability window, and various instability scenarios. Equivalence between direct asymptotic approach and invariant based approach is demonstrated. A general analytic approach for description of localised solutions that are in resonance with linear waves (quasi-solitons and embedded solitons) is given. This includes normal form theory and approximation of interacting particles. The main results are an expression for the amplitude of the radiating tail of a quasi-soliton, and a two-fold criterion for existence of embedded solitons. Influence of nonparaxiality on soliton stability is investigated. Stationary instability threshold is derived. The major results are shift and decreasing of the size of the stability window for higher-order solitons. The latter is the first demonstration of the destabilizing influence of nonparaxiality on higher-order solitons. Analysis of different aspects of solitons is based on universal approaches and methods. This includes Hamiltonian formalism, consideration of symmetry properties of the model, development of asymptotic models, construction of perturbation theory, application of general theorems etc. Thus, the results obtained can be extended beyond the particular model of degenerate four-wave mixing. All theoretical predictions are in good agreement with the results of direct numerical modeling.