published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Acoustic metamaterials are sub-wavelength structures designed to overcome limitations in the material properties of conventional materials. The present research focuses on the nonlinear acoustic and dynamic response of a specific type of engineered microstructure called a snapping acoustic metamaterial (SAMM). Snapping of these elements is defined as large, rapid deformations induced by infinitesimal perturbations in the time-varying external pressure. Snapping behavior in SAMM elements results from their non-monotonic stress-strain response, which displays regimes of positive and negative stiffness. This work presents a modeling study of the nonlinear behavior of both individual SAMM elements and a heterogeneous material containing a dilute concentration of SAMM elements embedded in a nearly incompressible viscoelastic solid. Two different scenarios are considered: (i) nonlinear wave propagation in the heterogeneous medium, and (ii) forced nonlinear dynamics of inclusions embedded in a viscoelastic medium. The nonlinearity of the SAMM elements is represented by a cubic pressure-volumetric strain relationship based on finite element model results from previous work. The effective nonlinear response of a heterogeneous mixture of SAMM elements embedded in a matrix, characterized by the parameters B/A and C/A, is then determined using both a nonlinear mixture law and a nonlinear Hashin-Shtrikman approach. The former estimate is limited to matrix materials with zero shear modulus, which cannot stabilize SAMM inclusions in regimes of negative stiffness. The augmented Hashin-Shtrikman method, however, includes nonlinear elasticity and the shear modulus of the matrix material. It therefore provides accurate estimates of the homogenized material when SAMM elements display negative stiffness and enhanced acoustical nonlinearity. The distortion of an acoustic wave propagating through the effective medium is studied through numerical solution of a nonlinear evolution equation that includes both quadratic and cubic nonlinearity. Finally, the forced nonlinear dynamic response of both a single SAMM element in a matrix and a domain of effective medium material embedded in matrix is considered. This behavior is of interest for generating enhanced absorption of acoustic wave energy because snapping leads to large hysteresis in the stress-strain response. A generalized Rayleigh-Plesset analysis is adapted to model the large-deformation dynamics associated with the system. / text

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This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.

The fundamental limitations on the operation of optical bistable devices in a ring cavity and in GaAs etalons are investigated. Experimental results of spontaneous transitions due to shot noise fluctuations are found in good agreement with various "ladder" models, if one allows the counting rates to vary accordingly. Stability analysis for two-photon homogeneously broadened media reveals single-wavelength instabilities for the laser but not for absorptive optical bistability. Appreciable regions of sidemode gain exist for both problems allowing for multiwavelength instabilities to occur. GaAs bistable devices show attractive features such as low power and high speed at room temperature for optical processing. However, experimental evidence in GaAs confirm the computer simulations of bistability that cavity losses, due to unsaturable background absorption, limit the switching power at room temperature. Methods to overcome the different limitations in GaAs devices toward parallel computation are then addressed from a system approach.

Nonlinear optical transverse effects, which arise from the interaction of a Gaussian beam with a nonlinear medium, are discussed. They are cw on-resonance enhancement (CORE), cw off-resonance rings, crosstalk, intracavity phase switching, self-defocusing of a bistable etalon, and thin-sample-encoding (TSE) self-lensing (i.e., self-focusing or self-defocusing) bistability and instabilities. CORE is the enhancement of the on-axis intensity of an on-resonance laser beam in a two-level medium. We report the first observation of CORE using sodium vapor. Cw off-resonance rings, resulting from a pure phase encoding or from simultaneous phase and amplitude encodings, are observed using sodium vapor. These rings have been observed many time before, but we have made the first careful comparison with numerical calculations. Parallel operation and external switching are two crucial themes for the application of bistable optical devices (BODs). For parallel operation, crosstalk between nearby BODs on the same etalon due to diffraction coupling may need to be avoided. Minimum separations needed to ensure independent operation are computed. For external switching, intracavity phase switching using a plane-wave input can switch on (or off) a BOD using an external pulse. The validity of this concept for a Gaussian input is numerically simulated. Self-defocusing effects, resulting in some interesting effects in the transverse bistable loops, are observed using a GaAs-GaAlAs bistable etalon. The results are confirmed by numerical simulations. TSE self-lensing bistability is observed using a short sodium cell and a single mirror. TSE self-lensing bistability results from an enhancement in the feedback via self-lensing of the beam. The lack of an optical resonator and the use of a thin medium make TSE setup a good candidate to study the Ikeda instability. We report the first observation of the Ikeda instability in this all-optical passive system using a cw input. Numerical simulations show a rich bifurcation sequence not yet fully studied experimentally.

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The effect of linear absorption on TE0 nonlinear guided waves and the effect of linear absorption, input-beam misalignment and nonlinear saturation on soliton emission from a nonlinear waveguide have been numerically investigated using the beam propagation method. In the first case the distribution of the absorption is found to have a dramatic effect on the propagation of the nonlinear guided waves. In the second case results reminiscent of the lossless case are found to survive in the presence of these complications.

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