Analysis and optimization of stress wave propagation in two-dimensional granular crystals with defects

Granular crystals are compact periodic assemblies of elastic particles in Hertzian contact whose dynamic response can be tuned from strongly nonlinear to linear by the addition of a static precompression force. This unique feature allows for a wide range of studies that include the investigation of new fundamental nonlinear phenomena in discrete systems such as solitary waves, shock waves, discrete breathers and other defect modes. In the absence of precompression, a particularly interesting property of these systems is their ability to support the formation and propagation of spatially localized soliton-like waves with highly tunable properties. The wealth of parameters one can modify (particle size, geometry and material properties, periodicity of the crystal, presence of a static force, type of excitation, etc.) makes them ideal candidates for the design of new materials for practical applications. This thesis describes several ways to optimally control and tailor the propagation of stress waves in granular crystals through the use of heterogeneities (interstitial defect particles and material heterogeneities) in otherwise perfectly ordered systems. We focus on uncompressed two-dimensional granular crystals with interstitial spherical intruders and composite hexagonal packings and study their dynamic response using a combination of experimental, numerical and analytical techniques. We first investigate the interaction of defect particles with a solitary wave and utilize this fundamental knowledge in the optimal design of novel composite wave guides, shock or vibration absorbers obtained using gradient-based optimization methods.

Phase conjugation via four-wave mixing in a resonant medium

This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process.

In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed.

In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile.

Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given.

Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10^-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.

Short-pulse laser absorption via J x B heating in ultrahigh intensity laser plasma interaction

An analytical fluid model for JxB heating during the normal incidence by a short ultraintense linearly polarized laser on a solid-density plasma is proposed. The steepening of an originally smooth electron density profile as the electrons are pushed inward by the laser is included self-consistently. It is shown that the JxB heating includes two distinct coupling processes depending on the initial laser and plasma conditions: for a moderate intensity (a <= 1), the ponderomotive force of the laser light can drive a large plasma wave at the point n(e)=4 gamma(0)n(c) resonantly. When this plasma wave is damped, the energy is transferred to the plasma. At higher intensity, the electron density is steepened to a high level by the time-independent ponderomotive force, n(e)> 4 gamma(0)n(c), so that no 2 omega resonance will occur, but the longitudinal component of the oscillating ponderomotive field can lead to an absorption mechanism similar to "vacuum heating." (c) 2006 American Institute of Physics.

Vacuum heating in the interaction of ultrashort, relativistically strong laser pulses with solid targets

An analytical fluid model for vacuum heating during the oblique incidence by an ultrashort ultraintense p-polarized laser on a solid-density plasma is proposed. The steepening of an originally smooth electron density profile as the electrons are pushed inward by the laser is included self-consistently. It is shown that the electrons being pulled out and then returned to the plasma at the interface layer by the wave field can lead to a phenomenon like wave breaking since the front part of the returning electrons always move slower than the trailing part. This can lead to heating of the plasma at the expense of the wave energy. An estimate for the efficiency of laser energy absorption by the vacuum heating is given. It is also found that for the incident laser intensity parameter, a(L)> 0.5, the absorption rate peaks at an incident angle 45 degrees-52 degrees and it reaches a maximum of 30% at a(L)approximate to 1.5.

Wave interactions in periodic structures and periodic dielectric waveguides

This work is concerned with a general analysis of wave interactions in periodic structures and particularly periodic thin film dielectric waveguides.

The electromagnetic wave propagation in an asymmetric dielectric waveguide with a periodically perturbed surface is analyzed in terms of a Floquet mode solution. First order approximate analytical expressions for the space harmonics are obtained. The solution is used to analyze various applications: (1) phase matched second harmonic generation in periodically perturbed optical waveguides; (2) grating couplers and thin film filters; (3) Bragg reflection devices; (4) the calculation of the traveling wave interaction impedance for solid state and vacuum tube optical traveling wave amplifiers which utilize periodic dielectric waveguides. Some of these applications are of interest in the field of integrated optics.

A special emphasis is put on the analysis of traveling wave interaction between electrons and electromagnetic waves in various operation regimes. Interactions with a finite temperature electron beam at the collision-dominated, collisionless, and quantum regimes are analyzed in detail assuming a one-dimensional model and longitudinal coupling.

The analysis is used to examine the possibility of solid state traveling wave devices (amplifiers, modulators), and some monolithic structures of these devices are suggested, designed to operate at the submillimeter-far infrared frequency regime. The estimates of attainable traveling wave interaction gain are quite low (on the order of a few inverse centimeters). However, the possibility of attaining net gain with different materials, structures and operation condition is not ruled out.

The developed model is used to discuss the possibility and the theoretical limitations of high frequency (optical) operation of vacuum electron beam tube; and the relation to other electron-electromagnetic wave interaction effects (Smith-Purcell and Cerenkov radiation and the free electron laser) are pointed out. Finally, the case where the periodic structure is the natural crystal lattice is briefly discussed. The longitudinal component of optical space harmonics in the crystal is calculated and found to be of the order of magnitude of the macroscopic wave, and some comments are made on the possibility of coherent bremsstrahlung and distributed feedback lasers in single crystals.

A study of nonlinear phenomena in the propagation of electromagnetic waves in a weakly ionized gas

This thesis is a study of nonlinear phenomena in the propagation of electromagnetic waves in a weakly ionized gas externally biased with a magnetostatic field. The present study is restricted to the nonlinear phenomena rising from the interaction of electromagnetic waves in the ionized gas. The important effects of nonlinearity are wave-form distortion leads to cross modulation of one wave by a second amplitude-modulated wave.

The nonlinear effects are assumed to be small so that a perturbation method can be used. Boltzmann’s kinetic equation with an appropriate expression for the collision term is solved by expanding the electron distribution function into spherical harmonics in velocity space. In turn, the electron convection current density and the conductivity tensors of the nonlinear ionized gas are found from the distribution function. Finally, the expression for the current density and Maxwell’s equations are employed to investigate the effects of nonlinearity on the propagation of electromagnetic waves in the ionized gas, and also on the reflection of waves from an ionized gas of semi-infinite extent.

I. Non-linear effects in a self-consistent calculation of SU_3 symmetry breaking in strong interaction coupling constants. II. Direct-channel reggeization of strong interaction scattering amplitudes

The thesis is divided into two parts. Part I generalizes a self-consistent calculation of residue shifts from SU₃ symmetry, originally performed by Dashen, Dothan, Frautschi, and Sharp, to include the effects of non-linear terms. Residue factorizability is used to transform an overdetermined set of equations into a variational problem, which is designed to take advantage of the redundancy of the mathematical system. The solution of this problem automatically satisfies the requirement of factorizability and comes close to satisfying all the original equations.

Part II investigates some consequences of direct channel Regge poles and treats the problem of relating Reggeized partial wave expansions made in different reaction channels. An analytic method is introduced which can be used to determine the crossed-channel discontinuity for a large class of direct-channel Regge representations, and this method is applied to some specific representations.

It is demonstrated that the multi-sheeted analytic structure of the Regge trajectory function can be used to resolve apparent difficulties arising from infinitely rising Regge trajectories. Also discussed are the implications of large collections of "daughter trajectories."

Two things are of particular interest: first, the threshold behavior in direct and crossed channels; second, the potentialities of Reggeized representations for us in self-consistent calculations. A new representation is introduced which surpasses previous formulations in these two areas, automatically satisfying direct-channel threshold constraints while being capable of reproducing a reasonable crossed channel discontinuity. A scalar model is investigated for low energies, and a relation is obtained between the mass of the lowest bound state and the slope of the Regge trajectory.

Spin wave resonance in ferromagnetic films

The objective of this investigation has been a theoretical and experimental understanding of ferromagnetic resonance phenomena in ferromagnetic thin films, and a consequent understanding of several important physical properties of these films. Significant results have been obtained by ferromagnetic resonance, hysteresis, torque magnetometer, He ion backscattering, and X-ray fluorescence measurements for nickel-iron alloy films.

Taking into account all relevant magnetic fields, including the applied, demagnetizing, effective anisotropy and exchange fields, the spin wave resonance condition applicable to the thin film geometry is presented. On the basis of the simple exchange interaction model it is concluded that the normal resonance modes of an ideal film are expected to be unpinned. The possibility of nonideality near the surface of a real film was considered by means of surface anisotropy field, inhomogeneity in demagnetizing field and inhomogeneity of magnetization models. Numerical results obtained for reasonable parameters in all cases show that they negligibly perturb the resonance fields and the higher order mode shapes from those of the unpinned modes of ideal films for thicknesses greater than 1000 Å. On the other hand for films thinner than 1000 Å the resonance field deviations can be significant even though the modes are very nearly unpinned. A previously unnoticed but important feature of all three models is that the interpretation of the first resonance mode as the uniform mode of an ideal film allows an accurate measurement of the average effective demagnetizing field over the film volume. Furthermore, it is demonstrated that it is possible to choose parameters which give indistinguishable predictions for all three models, making it difficult to uniquely ascertain the source of spin pinning in real films from resonance measurements alone.

Spin wave resonance measurements of 81% Ni-19% Fe coevaporated films 30 to 9000 Å thick, at frequencies from 1 to 8 GHz, at room temperature, and with the static magnetic field parallel and perpendicular to the film plane have been performed. A self-consistent analysis of the results for films thicker than 1000 Å, in which multiple excitations can be observed, shows for the first time that a unique value of exchange constant A can only be obtained by the use of unpinned mode assignments. This evidence and the resonance behavior of films thinner than 1000 Å strongly imply that the magnetization at the surfaces of permalloy films is very weakly pinned. However, resonance measurements alone cannot determine whether this pinning is due to a surface anisotropy, an inhomogeneous demagnetizing field or an inhomogeneous magnetization. The above analysis yields a value of 4πM=10,100 Oe and A = (1.03 ± .05) x 10^-6 erg/cm for this alloy. The ability to obtain a unique value of A suggests that spin wave resonance can be used to accurately characterize the exchange interaction in a ferromagnet.

In an effort to resolve the ambiguity of the source of pinning of the magnetization, a correlation of the ratio of magnetic moment and X-ray film thickness with the value of effective demagnetizing field 4πNM as determined from resonance, for films 45 to 300 Å has been performed. The remarkable agreement of both quantities and a comparison with the predictions of five distinct models, strongly imply that the thickness dependence of both quantities is related to a thickness dependent average saturation magnetization, which is far below 10,100 Oe for very thin films. However, a series of complementary experiments shows that this large decrease of average saturation magnetization cannot be simply explained by either oxidation or interdiffusion processes. It can only be satisfactorily explained by an intrinsic decrease of the average saturation magnetization for very thin films, an effect which cannot be justified by any simple physical considerations.

Recognizing that this decrease of average saturation magnetization could be due to an oxidation process, a correlation of resonance measurements, He ion backscattering, X-ray fluorescence and torque magnetometer measurements, for films 40 to 3500 Å thick has been performed. On basis of these measurements it is unambiguously established that the oxide layer on the surface of purposefully oxidized 81% Ni-19% Fe evaporated films is predominantly Fe-oxide, and that in the oxidation process Fe atoms are removed from the bulk of the film to depths of thousands of angstroms. Extrapolation of results for pure Fe films indicates that the oxide is most likely α-Fe₂O₃. These conclusions are in agreement with results from old metallurgical studies of high temperature oxidation of bulk Fe and Ni-Fe alloys. However, X-ray fluorescence results for films oxidized at room temperature, show that although the preferential oxidation of Fe also takes place in these films, the extent of this process is by far too small to explain the large variation of their average saturation magnetization with film thickness.

Regional and seasonal patterns of epipelagic fish assemblages from the central California Current

The coastal Pacific Ocean off northern and central California encompasses the strongest seasonal upwelling zone in the California Current ecosystem. Headlands and bays here generate complex circulation features and confer unusual oceanographic complexity. We sampled the coastal epipelagic fish community of this region with a surface trawl in the summer and fall of 2000–05 to assess patterns of spatial and temporal community structure. Fifty-three species of fish were captured in 218 hauls at 34 fixed stations, with clupeiform species dominating. To examine spatial patterns, samples were grouped by location relative to a prominent headland at Point Reyes and the resulting two regions, north coast and Gulf of the Farallones, were plotted by using nonmetric multidimensional scaling. Seasonal and interannual patterns were also examined, and representative species were identified for each distinct community. Seven oceanographic variables measured concurrently with trawling were plotted by principal components analysis and tested for correlation with biotic patterns. We found significant differences in community structure by region, year, and season, but no interaction among main effects. Significant differences in oceanographic conditions mirrored the biotic patterns, and a match between biotic and hydrographic structure was detected. Dissimilarity between assemblages was mostly the result of differences in abundance and frequency of occurrence of about twelve common species. Community patterns were best described by a subset of hydrographic variables, including water depth, distance from shore, and any one of several correlated variables associated with upwelling intensity. Rather than discrete communities with clear borders and distinct member species, we found gradients in community structure and identified stations with similar fish communities by region and by proximity to features such as the San Francisco Bay.

DEVELOPMENT OF THE WAVE CONTOURING RAFT.

The wave contouring raft system is the outcome of ideas initiated and developed by Sir Christopher Cockerell from 1972 onwards. His objective was to develop a wave energy device which is within the bounds of current technology. It should consist of simple, relatively small units, amenable to quantity production, which would enable a power generating system to be built up and commissioned in stages according to needs and production capability. This thinking led to the investigation of chains of pontoons, hinged together so that the passage of a wave down the chain causes the pontoons to oscillate relative to one another. Energy is extracted from the sea by applying a torque about the hinges to damp the motion. The work has involved extensive model testing in wave tanks and the building and testing of a 3-unit 1/10 scale power generating installation in the Solent, as well as design studies for a full size installation for Atlantic conditions.

Shock / boundary-layer interaction control using three-dimensional bumps for transonic wings

Three-dimensional bumps have been developed and investigated, aiming at the two major objectives of shock-wave / boundary-layer interaction control, i.e. drag reduction and suppression of separation, simultaneously. An experimental investigation has been conducted for a default rounded bump in channel now at University of Cambridge and a computational study has been performed for a spanwise series of rounded bumps mounted on a transonic aerofoil at University of Stuttgart. Observed in both cases are wave drag reduction owing to A-shock structures produced by three-dimensional surface bumps and mild control effects on the boundary layer. The effects of rough surface and tall extension have been investigated as well as several geometric variations and multiple bump configurations. A double configuration of narrow rounded bumps has been found to best perform amongst the tested, considerably reducing wave drag through a well-established A-shock structure with little viscous penalty and thus achieving substantial overall drag reduction. Counter-rotating streamwise vortex pairs have been produced by some configurations as a result of local flow separation, but they have been observed to be confined in relatively narrow wake regions, expected to be beneficial in suppressing large-scale separation under off-design condition despite increase of viscous drag. On the whole a large potential of three-dimensional control with discrete rounded bumps has been demonstrated both experimentally and numerically, and experimental investigation of bumps fitted on a transonic aerofoil or wing is suggested toward practical application.

An experimental study of unsteady separated shock wave boundary layer interactions

An experimental investigation of the unsteady interaction between a turbulent boundary layer and a normal shock wave of strength M∞ = 1.4 subject to periodic forcing in a parallel walled duct has been conducted. Emphasis has been placed on the mechanism by which changes in the global flow field influence the local interaction structure. Static pressure measurements and high speed Schlieren images of the unsteady interaction have been obtained. The pressure rise across the interaction and the appearance of the local SBLI structure have been observed to vary during the cycle of periodic shock wave motion. The magnitude of the pressure rise across the interaction is found to be related to the relative Mach number of the unsteady shock wave as it undergoes periodic motion. Variations in the upstream Influence of the interaction are sensitive to the magnitude and direction of shock wave velocity and acceleration and it is proposed that a viscous lag exists between the point of boundary layer separation and the shock wave position. Further work exploring the implications of these findings is proposed, including studies of the variation in position of the points of boundary layer separation and reattachment.

Experimental investigation of 3D shock / boundary layer interaction control in transonic flows

A novel supersonic wind tunnel setup is proposed to enable the investigation of control on a normal shock wave. Previous experimental arrangements were found to suffer from shock instability. Wind tunnel tests with and without control have confirmed the capability of the new setup to stabilise a shock structure at a target position without changing the nature of the shock wave / boundary layer interaction flow at M∞ = 1.3 and M ∞ = 1.5. Flow visualisation and pressure measurements with the new setup have revealed detailed characteristics of shock wave / boundary layer interactions and a λ-shock structure as well as benefits of control in total drag reduction in the presence of 3D bump control.

Micro-ramp control for oblique shock wave / boundary layer interactions

Supersonic engine intakes operating supercritically feature shock wave / boundary layer interactions (SBLIs), which are conventionally controlled using boundary layer bleed. The momentum loss of bleed flow causes high drag, compromising intake performance. Micro-ramp sub-boundary layer vortex generators (SBVGs) have been proposed as an alternative form of flow control for oblique SBLIs in order to reduce the bleed requirement. Experiments have been conducted at Mach 2.5 to characterise the flow details on such devices and investigate their ability to control the interaction between an oblique shock wave and the naturally grown turbulent boundary layer on the tunnel floor. Micro-ramps of four sizes with heights ranging from 25% to 75% of the uncontrolled boundary layer thickness were tested. The flow over all sizes of microramp was found to be similar, featuring streamwise counter-rotating vortices which entrain high momentum fluid, locally reducing the boundary layer displacement thickness. When installed ahead of the shock interaction it was found that the positioning of the micro-ramps is of limited importance. Micro-ramps did not eliminate flow separation. However, the previously two-dimensional separation was broken up into periodic three-dimensional separation zones. The interaction length was reduced and the pressure gradient across the interaction was increased.