The propagation of waves through nonlinear media is of interest here, namely as it pertains to two specific examples, a nonlinear dielectric and a hyperelastic solid. In both cases, we examine the propagation of two-dimensional, weakly nonlinear, quasi-planar waves. It is found that such systems will have a nonlinearity that is intrinsically cubic, and therefore, a classical Zabolotskaya-Khokhlov equation cannot give an accurate description of the wave evolution. To determine the general evolution equation in such systems, a multi-timing technique developed by Kluwick and Cox (1998) and Cramer and Webb (1998) will be employed. The resultant evolution equations are seen to involve only one new nonlinearity coefficient rather than the three coefficients found in other studies of cubically nonlinear systems. After determining the general evolution equation, inclusion of relaxation, dispersion and dissipation effects can be easily incorporated. / Master of Science

Intensive demands have arisen to characterize nonlinear optical properties of materials for applications involving optical limiters, waveguide switches and bistable light switches. The technique of Pulse Delay Modulation is described which can monitor nonlinear changes in transmission with shot noise limited signal-to-noise ratios even in the presence of large background signals. The theoretical foundations of the experiment are presented followed by actual measurements of beam depletion due to second harmonic generation in a LiIO3 crystal and two-photon absorption in the semiconductor ZnSe. Sensitivity to polarization rotation arising from the Kerr Effect in carbon disulfide, saturable absorber relaxation in modelocking dyes and photorefractive effects in ZnSe are demonstrated. The sensitivity of Pulse Delay Modulation is combined with Fabry-Perot enhancement to allow the measurement of two-photon absorption in a 0.46pm thick interference filter spacer layer. Also included is a study of nonlinear optical limiting arising from dielectric breakdown in gases.

Thulium (Tm) based high repetition rate compact optical frequency comb sources operating in the 2 µm regime with femtosecond pulse durations enable a wide range of applications such as precise micro-machining, spectroscopy and metrology. Applications such as metrology and spectroscopy rely on the stability of mode-locked lasers (MLLs) which provide extreme precision, yet, the complex dynamics of such highly nonlinear systems result in unstable events which could hinder the normal operation of a MLL. MLL as a nonlinear system inherently exists a wide variety of complex attractors, which are sets of states that the system tends to evolve toward, exhibiting unique behaviors. Complex phenomena including pulsating solitons, chaotic solitons, period-doubling, soliton explosion, etc., have been predicted theoretically and observed experimentally in the past decade. However, most experimental observations rely on conventional characterization methods, which are limited to the scanning speed of the spectrometer and the electronic speed of photodetector and digitizer, so that the details of the non-repetitive events can be buried. In recent years, a technique called dispersive Fourier transform (DFT) has been developed and allows consecutive recordings of the pulse-to-pulse spectral evolution of a femtosecond pulse train, opening a whole new world of nonlinear dynamics in MLL. In this dissertation, we first demonstrate the ability of scaling the repetition rate of a Tm MLL to repetition rate as high as 1.25 GHz through miniaturizing the cavity. Our approach of maintaining comparable pulse energies while scaling the repetition rates allows a high-quality femtosecond mode-locking performance with low noise performance in Tm soliton lasers. Then we experimentally study the transition dynamics between consecutive multi-pulsing states through adjusting pump power with a constant rate in an erbium-doped fiber laser, specifically the build-up and annihilation of soliton pulses between a double pulsing and a three-pulse state utilizing DFT. To investigate real-time laser dynamics in Tm based laser systems, we propose and develop a DFT system that up-converts the signal to the 1 µm regime via second harmonics generation (SHG) and stretches the signal in a long spool of single-mode fiber to realize DFT. This approach overcomes the limitation of bandwidth of 2 µm photodetector and high intrinsic absorption of 2 µm light in fused silica fibers. The SHG-DFT system is used to study dynamics of both explosions in a chaotic state between stable single-pulsing and double-pulsing state, and explosions induced by soliton collision in a dual-wavelength vector soliton state. We also study dynamics of transient regimes in a Tm-doped fiber ring laser that can be switched between conventional soliton and dissipative soliton, revealing how spectral filtering plays a role in obtaining stable stationary states. / 2022-11-23T00:00:00Z

It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait. In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas] The xi may generally be considered to be concentrations of species in a chemical reaction, in which case the k's are rate constants. In some cases the xi may be considered to be position and momentum variables in a mechanical system. We will divide the equations into two classes: those in which the evolution can be carried out by the action of one of Lie's transformation groups of the plane, and those for which this is not possible. Members of the first class can be integrated by quadrature either directly or by use of an integrating factor; those in the second class cannot. Of those in the first class the most interesting evolve by transformations of the projective group, and these, as well as the equations that cannot be integrated by quadrature, we study in some detail. We seek a qualitative analysis of systems which have no linear terms in their evolution equations when the origin from which the xi are measured is a critical point. The standard, linear, phase plane analysis is of course not adequate for our purposes.

By nature discrete solitons represent self-trapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity. In optics, this class of self-localized states has been successfully observed in both one-and two-dimensional nonlinear waveguide arrays. In recent years such lattice structures have been implemented or induced in a variety of material systems including those with cubic (Kerr), quadratic, photorefractive, and liquid-crystal nonlinearities. In all cases the underlying periodicity or discreteness leads to new families of optical solitons that have no counterpart whatsoever in continuous systems. In the first part of this dissertation, a theoretical investigation of linear and nonlinear optical wave propagation in semi-infinite waveguide arrays is presented. In particular, the properties and the stability of surface solitons at the edge of Kerr (AlGaAs) and quadratic (LiNbO3) lattices are examined. Hetero-structures of two dissimilar semi-infinite arrays are also considered. The existence of hybrid solitons in these latter types of structures is demonstrated. Rabi-type optical transitions in z-modulated waveguide arrays are theoretically demonstrated. The corresponding coupled mode equations, that govern the energy oscillations between two different transmission bands, are derived. The results are compared with direct beam propagation simulations and are found to be in excellent agreement with coupled mode theory formulations. In the second part of this thesis, the concept of parity-time-symmetry is introduced in the context of optics. More specifically, periodic potentials associated with PT-symmetric Hamiltonians are numerically explored. These new optical structures are found to exhibit surprising characteristics. These include the possibility of abrupt phase transitions, band merging, non-orthogonality, non-reciprocity, double refraction, secondary emissions, as well as power oscillations. Even though gain/loss is present in this class of periodic potentials, the propagation eigenvalues are entirely real. This is a direct outcome of the PT-symmetry. Finally, discrete solitons in PT-symmetric optical lattices are examined in detail.

This work is divided into two main parts. In the first part (chapters 2-7) we consider the nonlinear response of nano-particle colloidal systems. Starting from the Nernst-Planck and Smoluchowski equations, we demonstrate that in these arrangements the underlying nonlinearities as well as the nonlinear Rayleigh losses depend exponentially on optical intensity. Two different nonlinear regimes are identified depending on the refractive index contrast of the nanoparticles involved and the interesting prospect of self-induced transparency is demonstrated. Soliton stability is systematically analyzed for both 1D and 2D configurations and their propagation dynamics in the presence of Rayleigh losses is examined. We also investigate the modulation instability of plane waves and the transverse instabilities of soliton stripe beams propagating in nonlinear nano-suspensions. We show that in these systems, the process of modulational instability depends on the boundary conditions. On the other hand, the transverse instability of soliton stripes can exhibit new features as a result of 1D collapse caused by the exponential nonlinearity. Many-body effects on the systems' nonlinear response are also examined. Mayer cluster expansions are used in order to investigate particle-particle interactions. We show that the optical nonlinearity of these nano-suspensions can range anywhere from exponential to polynomial depending on the initial concentration and the chemistry of the electrolyte solution. The consequence of these inter-particle interactions on the soliton dynamics and their stability properties are also studied. The second part deals with linear and nonlinear properties of optical nano-wires and the coupled mode formalism of parity-time (PT) symmetric waveguides. Dispersion properties of AlGaAs nano-wires are studied and it is shown that the group velocity dispersion in such waveguides can be negative, thus enabling temporal solitons. We have also studied power flow in nano-waveguides and we have shown that under certain conditions, optical pulses propagating in such structures will exhibit power circulations. Finally PT symmetric waveguides were investigated and a suitable coupled mode theory to describe these systems was developed.

There is a need for a nonlinear flutter analysis method capable of predicting limit cycle oscillation in aeroelastic systems. A review is conducted of analysis methods and experiments that have attempted to better understand and model limit cycle oscillation (LCO). The recently developed method of nonlinear normal modes (NNM) is investigated for LCO calculation. Nonlinear normal modes were used to analyze a spring-mass-damper system with nonlinear damping and stiffness to demonstrate the ability and limitations of the method to identify limit cycle oscillation. The nonlinear normal modes method was then applied to an aeroelastic model of a pitch-plunge airfoil with nonlinear pitch stiffness and quasi-steady aerodynamics. The asymptotic coefficient solution method successfully captured LCO at a low relative velocity. LCO was also successfully modeled for the same airfoil with an unsteady aerodynamics model with the use of a first order formulation of NNM. A linear beam model of the Goland wing with a nonlinear aerodynamic model was also studied. LCO was successfully modeled using various numbers of assumed modes for the beam. The concept of modal truncation was shown to extend to NNM. The modal coefficients were shown to identify the importance of each mode to the solution and give insight into the physical nature of the motion. The quasi-steady airfoil model was used to conduct a study on the effect of the nonlinear normal mode's master coordinate. The pitch degree of freedom, plunge degree of freedom, both linear structural mode shapes with apparent mass, and the linear flutter mode were all used as master coordinates. The master coordinates were found to have a significant influence on the accuracy of the solution and the linear flutter mode was identified as the preferred option. Galerkin and collocation coefficient solution methods were used to improve the results of the asymptotic solution method. The Galerkin method reduced the error of the solution if the correct region of integration was selected, but had very high computational cost. The collocation method improved the accuracy of the solution significantly. The computational time was low and a simple convergent iteration method was found. Thus, the collocation method was found to be the preferred method of solving for the modal coefficients. / Ph. D.

This dissertation reports a sensitive single beam experimental technique for measuring nonlinear refraction and nonlinear absorption in a wide variety of materials. The experimental setup is described and a comprehensive theoretical analysis including cases where nonlinear refraction and nonlinear absorption are also presented.

The objective of this research is to develop large signal modeling and optimization methods for Capacitive Micromachined Ultrasonic Transducers (CMUTs), especially when they are used in an array configuration. General modeling and optimization methods that cover a large domain of CMUT designs are crucial, as many membrane and array geometry combinations are possible using existing microfabrication technologies. Currently, large signal modeling methods for CMUTs are not well established and nonlinear imaging techniques utilizing linear piezoelectric transducers are not applicable to CMUTs because of their strong nonlinearity. In this work, the nonlinear CMUT behavior is studied, and a feedback linearization method is proposed to reduce the CMUT nonlinearity. This method is shown to improve the CMUT performance for continuous wave applications, such as high-intensity focused ultrasound or harmonic imaging, where transducer linearity is crucial. In the second part of this dissertation, a large signal model is developed that is capable of transient modeling of CMUT arrays with arbitrary electrical terminations. The developed model is suitable for iterative design optimization of CMUTs and CMUT based imaging systems with arbitrary membrane and array geometries for a variety of applications. Finally, a novel multi-pulse method for nonlinear tissue and contrast agent imaging with CMUTs is presented. It is shown that the nonlinear content can be successfully extracted from echo signals in a CMUT based imaging system using a multiple pulse scheme. The proposed method is independent of the CMUT geometry and valid for large signal operation. Experimental results verifying the developed large signal CMUT array model, proposed gap feedback and multi-pulse techniques are also presented.

published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy