Non-linear dispersive waves with a small dissipation

The general theory of Whitham for slowly-varying non-linear wavetrains is extended to the case where some of the defining partial differential equations cannot be put into conservation form. Typical examples are considered in plasma dynamics and water waves in which the lack of a conservation form is due to dissipation; an additional non-conservative element, the presence of an external force, is treated for the plasma dynamics example. Certain numerical solutions of the water waves problem (the Korteweg-de Vries equation with dissipation) are considered and compared with perturbation expansions about the linearized solution; it is found that the first correction term in the perturbation expansion is an excellent qualitative indicator of the deviation of the dissipative decay rate from linearity.

A method for deriving necessary and sufficient conditions for the existence of a general uniform wavetrain solution is presented and illustrated in the plasma dynamics problem. Peaking of the plasma wave is demonstrated, and it is shown that the necessary and sufficient existence conditions are essentially equivalent to the statement that no wave may have an amplitude larger than the peaked wave.

A new type of fully non-linear stability criterion is developed for the plasma uniform wavetrain. It is shown explicitly that this wavetrain is stable in the near-linear limit. The nature of this new type of stability is discussed.

Steady shock solutions are also considered. By a quite general method, it is demonstrated that the plasma equations studied here have no steady shock solutions whatsoever. A special type of steady shock is proposed, in which a uniform wavetrain joins across a jump discontinuity to a constant state. Such shocks may indeed exist for the Korteweg-de Vries equation, but are barred from the plasma problem because entropy would decrease across the shock front.

Finally, a way of including the Landau damping mechanism in the plasma equations is given. It involves putting in a dissipation term of convolution integral form, and parallels a similar approach of Whitham in water wave theory. An important application of this would be towards resolving long-standing difficulties about the "collisionless" shock.

Some approximate solutions of dynamic problems in the linear theory of thin elastic shells

Some aspects of wave propagation in thin elastic shells are considered. The governing equations are derived by a method which makes their relationship to the exact equations of linear elasticity quite clear. Finite wave propagation speeds are ensured by the inclusion of the appropriate physical effects.

The problem of a constant pressure front moving with constant velocity along a semi-infinite circular cylindrical shell is studied. The behavior of the solution immediately under the leading wave is found, as well as the short time solution behind the characteristic wavefronts. The main long time disturbance is found to travel with the velocity of very long longitudinal waves in a bar and an expression for this part of the solution is given.

When a constant moment is applied to the lip of an open spherical shell, there is an interesting effect due to the focusing of the waves. This phenomenon is studied and an expression is derived for the wavefront behavior for the first passage of the leading wave and its first reflection.

For the two problems mentioned, the method used involves reducing the governing partial differential equations to ordinary differential equations by means of a Laplace transform in time. The information sought is then extracted by doing the appropriate asymptotic expansion with the Laplace variable as parameter.

Evaluation of the Photometrics CH250 CCD Camera for use in the NOAA/MLML Marine Optics System

This report summarizes initial work to incorporate Photometries CH250 charge-coupled device (CCD) detectors in the NOAAIMLML Marine Optics System (MOS). The MOS spectroradiometer will be used primarily in the Marine Optics Buoy (MOBY) to surface truth the ocean color satellite, SeaWiFS, scheduled for launch later this year. This work was funded through Contract NAS5-31746 to NASA, Goddard Space Flight Center. (PDF contains 24 pages)

A singularly perturbed linear two-point boundary-value problem

We consider the following singularly perturbed linear two-point boundary-value problem:

Ly(x) ≡ Ω(ε)D_xy(x) - A(x,ε)y(x) = f(x,ε) 0≤x≤1 (1a)

By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)

Here Ω(ε) is a diagonal matrix whose first m diagonal elements are 1 and last m elements are ε. Aside from reasonable continuity conditions placed on A, L, R, f, g, we assume the lower right mxm principle submatrix of A has no eigenvalues whose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existence of a unique solution of (1). These sufficient conditions are used to define when (1) is a regular problem. It is then shown that as ε → 0^+ the solution of a regular problem exists and converges on every closed subinterval of (0,1) to a solution of the reduced problem. The reduced problem consists of the differential equation obtained by formally setting ε equal to zero in (1a) and initial conditions obtained from the boundary conditions (1b). Several examples of regular problems are also considered.

A similar technique is used to derive the properties of the solution of a particular difference scheme used to approximate (1). Under restrictions on the boundary conditions (1b) it is shown that for the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately represents the solution of the reduced problem.

Furthermore, the existence of a similarity transformation which block diagonalizes a matrix is presented as well as exponential bounds on certain fundamental solution matrices associated with the problem (1).

Deep tissue fluorescence imaging with time-reversed light

Advances in optical techniques have enabled many breakthroughs in biology and medicine. However, light scattering by biological tissues remains a great obstacle, restricting the use of optical methods to thin ex vivo sections or superficial layers in vivo. In this thesis, we present two related methods that overcome the optical depth limit—digital time reversal of ultrasound encoded light (digital TRUE) and time reversal of variance-encoded light (TROVE). These two techniques share the same principle of using acousto-optic beacons for time reversal optical focusing within highly scattering media, like biological tissues. Ultrasound, unlike light, is not significantly scattered in soft biological tissues, allowing for ultrasound focusing. In addition, a fraction of the scattered optical wavefront that passes through the ultrasound focus gets frequency-shifted via the acousto-optic effect, essentially creating a virtual source of frequency-shifted light within the tissue. The scattered ultrasound-tagged wavefront can be selectively measured outside the tissue and time-reversed to converge at the location of the ultrasound focus, enabling optical focusing within deep tissues. In digital TRUE, we time reverse ultrasound-tagged light with an optoelectronic time reversal device (the digital optical phase conjugate mirror, DOPC). The use of the DOPC enables high optical gain, allowing for high intensity optical focusing and focal fluorescence imaging in thick tissues at a lateral resolution of 36 µm by 52 µm. The resolution of the TRUE approach is fundamentally limited to that of the wavelength of ultrasound. The ultrasound focus (~ tens of microns wide) usually contains hundreds to thousands of optical modes, such that the scattered wavefront measured is a linear combination of the contributions of all these optical modes. In TROVE, we make use of our ability to digitally record, analyze and manipulate the scattered wavefront to demix the contributions of these spatial modes using variance encoding. In essence, we encode each spatial mode inside the scattering sample with a unique variance, allowing us to computationally derive the time reversal wavefront that corresponds to a single optical mode. In doing so, we uncouple the system resolution from the size of the ultrasound focus, demonstrating optical focusing and imaging between highly diffusing samples at an unprecedented, speckle-scale lateral resolution of ~ 5 µm. Our methods open up the possibility of fully exploiting the prowess and versatility of biomedical optics in deep tissues.

Coherent population transfer with chirped few-cycle laser pulses in an excited-doublet four-level system

The behavior of population transfer in an excited-doublet four-level system driven by linear polarized few-cycle ultrashort laser pulses is investigated numerically. It is shown that almost complete population transfer can be achieved even when the adiabatic criterion is not fulfilled. Moreover, the robustness of this scheme in terms of the Rabi frequencies and chirp rates of the pulses is explored.

Giant Kerr nonlinearity induced by interacting dark resonances

A scheme for giant enhancement of the Kerr nonlinearity in a four-level system with double dark resonances is proposed. Compared with that generated in a single-dark-resonance system, the Kerr nonlinearity can be enhanced by several orders of magnitude with vanishing linear absorption. We attribute this dramatic enhancement to the interaction of dark resonances. (c) 2005 Optical Society of America.

Nonlinear optics in planar silica-on-silicon disk resonators

Optical frequency combs (OFCs) provide direct phase-coherent link between optical and RF frequencies, and enable precision measurement of optical frequencies. In recent years, a new class of frequency combs (microcombs) have emerged based on parametric frequency conversions in dielectric microresonators. Micocombs have large line spacing from 10's to 100's GHz, allowing easy access to individual comb lines for arbitrary waveform synthesis. They also provide broadband parametric gain bandwidth, not limited by specific atomic or molecular transitions in conventional OFCs. The emerging applications of microcombs include low noise microwave generation, astronomical spectrograph calibration, direct comb spectroscopy, and high capacity telecommunications.

In this thesis, research is presented starting with the introduction of a new type of chemically etched, planar silica-on-silicon disk resonator. A record Q factor of 875 million is achieved for on-chip devices. A simple and accurate approach to characterize the FSR and dispersion of microcavities is demonstrated. Microresonator-based frequency combs (microcombs) are demonstrated with microwave repetition rate less than 80 GHz on a chip for the first time. Overall low threshold power (as low as 1 mW) of microcombs across a wide range of resonator FSRs from 2.6 to 220 GHz in surface-loss-limited disk resonators is demonstrated. The rich and complex dynamics of microcomb RF noise are studied. High-coherence, RF phase-locking of microcombs is demonstrated where injection locking of the subcomb offset frequencies are observed by pump-detuning-alignment. Moreover, temporal mode locking, featuring subpicosecond pulses from a parametric 22 GHz microcomb, is observed. We further demonstrated a shot-noise-limited white phase noise of microcomb for the first time. Finally, stabilization of the microcomb repetition rate is realized by phase lock loop control.

For another major nonlinear optical application of disk resonators, highly coherent, simulated Brillouin lasers (SBL) on silicon are also demonstrated, with record low Schawlow-Townes noise less than 0.1 Hz^2/Hz for any chip-based lasers and low technical noise comparable to commercial narrow-linewidth fiber lasers. The SBL devices are efficient, featuring more than 90% quantum efficiency and threshold as low as 60 microwatts. Moreover, novel properties of the SBL are studied, including cascaded operation, threshold tuning, and mode-pulling phenomena. Furthermore, high performance microwave generation using on-chip cascaded Brillouin oscillation is demonstrated. It is also robust enough to enable incorporation as the optical voltage-controlled-oscillator in the first demonstration of a photonic-based, microwave frequency synthesizer. Finally, applications of microresonators as frequency reference cavities and low-phase-noise optomechanical oscillators are presented.

Diagonal forms, linear algebraic methods and Ramsey-type problems

This thesis focuses mainly on linear algebraic aspects of combinatorics. Let N_t(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of N_t(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H.

As a continuation, the author determines the formula for diagonal forms of integer matrices obtained from other combinatorial structures, including incidence matrices for subgraphs of a complete bipartite graph and inclusion matrices for multisets.

One major application of diagonal forms is in zero-sum Ramsey theory. For instance, Caro's results in zero-sum Ramsey numbers for graphs and Caro and Yuster's results in zero-sum bipartite Ramsey numbers can be reproduced. These results are further generalized to t-uniform hypergraphs. Other applications include signed bipartite graph designs.

Research results on some other problems are also included in this thesis, such as a Ramsey-type problem on equipartitions, Hartman's conjecture on large sets of designs and a matroid theory problem proposed by Welsh.

Toxicity of linear alkylbenene sulphonate (LAS) detergent, to Clarias gariepinus fingerlings

The acute toxicity of Linear Alkylbenzene Sulphonate (LAS) detergent to Clarias gariepinus fingerlings was investigated using static bioassays and continous aeration over a period of 96h. The 96h LC sub(50) was determined as 24.00mgL super(-1). During the exposure period, the test fish exhibited several behavioural changes before death such as restlessness, rapid swimming, loss of balance, respiratory distress and haemorrhaging of gill filaments amongst others. Opercula ventilation rate as well as visual examination of dead fish indicates lethal effects of the detergent on the fish. Water quality examination showed increase in pH from 6.55 to the alkaline, death point of 10.55. There was also a remarkabel rise of alkalinity from 20.00mgL super(-1) to 52.50mgL super(-1)

Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Linear chirp control of infrared signal in a biased semiconductor thin film

Ultrashort light-matter interactions between a linear chirped pulse and a biased semiconductor thin film GaAs are investigated. Using different chirped pulses, the dependence of infrared spectra on chirp rate is demonstrated for a 5 fs pulse. It is found that the infrared spectra can be controlled by the linear chirp of the pulse. Furthermore, the infrared spectral intensity could be enhanced by two orders of magnitude via appropriately choosing values of the linear chirp rates. Our results suggest a possible scheme to control the infrared signal.