In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.

In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.

In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.

Die Weiterentwicklung von Ultrakurzimpulslaserquellen hat die Horizonte für Wissenschaft, Medizin und Industrie stetig erweitert. Ultrakurze Impulsdauern und hohe Energien erzeugen Spitzenleistungen auf der Gigawatt-Skala, deren zeitliche und spektrale Charakteristik ideale Voraussetzungen für nichtlineare zeitaufgelöste Spektroskopie und ultraschnelle nichtlineare Optik bieten. Die Untersuchung von Molekülschwingungen im sogenannten Fingerabdrucksbereich (engl. fingerprint region) und die effiziente lasergetriebene Erzeugung von Hohen-Harmonischen- und Röntgenimpulsen benötigen Laserquellen im mittleren bis langwelligen Infrarot. Da oberhalb einer Wellenlänge von 4 μm keine Festkörperlaserquellen existieren, hat sich optische parametrische Verstärkung zur Schlüsseltechnik in diesem Wellenlängenbereich entwickelt. In dieser Arbeit werden Laserimpulse oberhalb von 4 μm Wellenlänge mittels optischer parametrischer Verstärkung gestreckter Impulse erzeugt, deren Energien den Micro- bis Millijoule Bereich bei einer Kilohertz-Wiederholrate erreichen. Die Pumpwellenlänge von 2 μm ist vorteilhaft gegenüber den üblicherweise verwendeten Pumpen im nahen Infraroten und erlaubt zur Generation der Eingangsspektren besonders innovative kompakte Laserarchitekturen. Es werden zwei Systeme im mittleren und langwelligen Infrarot entwickelt basierend auf einem Cr:ZnS Eingangslaser, die bisherigen Systemen in Energie und Spitzenleistungen überlegen sind. Während sich die Laserquelle im mittleren Infraroten durch seine Durchstimmbarkeit auszeichnet, wird mit den langwelligen infraroten Impulsen erstmals einen nichtlineare Absorptionsmessung an Wasser durchgeführt. / The progress in the development of ultrafast laser sources has opened up new horizons in science, medicine and industry. Pulses of ultrashort duration and high energy reach gigawatt peak power which offer ideal conditions for time-resolved nonlinear absorption spectroscopy and ultrafast nonlinear optics. The investigation of vibrational states of biomolecules in the so-called fingerprint region and strong-field experiments aiming for the generation of high-harmonics or x-rays quest for such laser sources in the mid- to long-infrared spectral range. Due to the lack of existing solid state lasers beyond 4 μm, optical parametric amplification has emerged as the key technique to generate adequate infrared pulses. In this work, optical parametric chirped pulse amplification (OPCPA) is the key technique used to generate 100 μJ-level energy pulses at kHz repetition rate beyond 4 μm. In this context, novel front-end architectures are designed, tailored to compactness and to exploit the advantages of 2 μm pumped OPCPA over the typically used near-infrared drivers around 1 μm. The novel front-end based on a femtosecond Cr:ZnS oscillator emitting 30 fs pulses at 2.4 μm provides the necessary spectral components for the 2 μm pump and the signal. Two OPCPA systems in the mid-wave infrared (MWIR) and long-wave infrared (LWIR) spectral region, superior in terms of pulse energy and peak power compared with existing systems, are developed. While the tunability of the first system is unique, the second system is used to for the first time demonstrate a nonlinear transmission experiment in water by direct excitation of the L2 libration.

<p>Nonlinear transmission lines (NLTLs) are of great interest for high power microwave (HPM) generation because they can sharpen pulses to create an electromagnetic shockwave to produce oscillations from 100 MHz to low GHz. NLTLs provide frequency agility, compactness, durability and reliability, providing a solid-state radiofrequency (RF) source for producing HPM. The essential component of NLTLs is the nonlinear material, typically a dielectric that varies with voltage or a magnetic material whose permeability varies with current, incorporated in the transmission line in various topologies. This thesis presents an alternative approach involving designing composites comprised of nonlinear dielectric inclusions (barium strontium titanate (BST)) and/or nonlinear inductive inclusions (nickel zinc ferrites (NZF)) in a polymer base host material, analogous to electromagnetic interference designs that incorporate stainless steel inclusions of various shapes in a plastic to tune the composite’s electromagnetic properties at GHz. Appropriately designing NLTL composites requires predicting these effective properties both in linear (for a fixed and low voltage and current) and nonlinear regions (permittivity and permeability become voltage dependent and current dependent, respectively) prior to designing HPM systems comprised of them. As a first step, this thesis evaluates and benchmarks composites models in the commercial software CST Microwave Studios (CST MWS) to various effective medium theories (EMTs) to predict the permittivity and permeability of composites of BST and/or NZF inclusions in the linear regime, compared with experimental measurements. The manufacturing and measurement of the nonlinear composites will be briefly discussed with an analysis of the homogeneity of a composite sample using 3D X-ray scan. Long-term application of these approaches to predicting the effective nonlinear composite permittivity and permeability and future work will be discussed.</p>

<p>Nonlinear transmission lines (NLTLs) provide a solid state alternative to conventional vacuum based high power microwave (HPM) sources. The three most common NLTL implementations are the lumped element, split ring resonator (SRR), and the nonlinear bulk material based NLTLs. The nonlinear bulk material implementation provides the highest power output of the three configurations, though they are limited to pulse voltages less than 50 kV; higher voltages are possible when an additional insulator is used, typically SF₆ or dielectric oil, between the nonlinear material and the outer conductor. The additional insulator poses a risk of leaking if structural integrity of the outer conductor is compromised. The desire to provide a fieldable NLTL based HPM system makes the possibility of a leak problematic. The work reported here develops a composite based NLTL system that can withstand voltages higher than 50 kV and not pose a risk of catastrophic failure due to a leak while also decreasing the size and weight of the device and increasing the output power.</p> <p>Composites with barium strontium titanate (BST) or nickel zinc ferrite (NZF) spherical inclusions mixed in a silicone matrix were manufactured at volume fractions ranging from 5% to 25%. The dielectric and magnetic parameters were measured from 1-4 GHz using a coaxial airline. The relative permittivity increased from 2.74±0.01 for the polydimethylsiloxane (PDMS) host material to 7.45±0.33 after combining PDMS with a 25% volume fraction of BST inclusions. The relative permittivity of BST and NZF composites was relatively constant across all measured frequencies. The relative permeability of the composites increased from 1.001±0.001 for PDMS to 1.43±0.04 for a 25% NZF composite at 1 GHz. The relative permeability of the 25% NZF composite decreased from 1.43±0.05 at 1 GHz to 1.17±0.01 at 4 GHz. The NZF samples also exhibited low dielectric and magnetic loss tangents from 0.005±0.01 to 0.091±0.015 and 0.037±0.001 to 0.20±0.038, respectively, for all volume fractions, although the dielectric loss tangent did increase with volume fraction. For BST composites, all volume fraction changes of at least 5% yielded statistically significant changes in permittivity; no changes in BST volume fraction yielded statistically significant changes in permeability. For NZF composites, the change in permittivity was statistically significant when the volume fraction varied by more than 5% and the change in permeability was statistically significant for variations in volume fraction greater than 10%. The DC electrical breakdown strength of NZF composites decreased exponentially with increasing volume fraction of NZF, while BST composites exhibited no statistically significant variation with volume fraction. </p> <p>For composites containing both BST and NZF, increasing the volume fraction of either inclusion increased the permittivity with a stronger dependence on BST volume fraction. Increasing NZF volume fraction increased the magnetic permeability, while changing BST volume fraction had no effect on the composite permeability. The DC dielectric breakdown voltage decreased exponentially with increased NZF volume fraction. Adding as little as 5% BST to an NZF composite more than doubled the breakdown threshold compared to a composite containing NZF alone. For example, adding 10% BST to a 15% NZF composite increased the breakdown strength by over 800%. The combination of tunability of permittivity and permeability by managing BST and NZF volume fractions with the increased dielectric breakdown strength by introducing BST make this a promising approach for designing high power nonlinear transmission lines with input pulses of hundreds of kilovolts.</p> <p>Coaxial nonlinear transmission lines are produced using composites with NZF inclusions and BST inclusions and driven by a Blumlein pulse generator with a 10 ns pulse duration and 1.5 ns risetime. Applying a 30 kV pulse using the Blumlein pulse generator resulted in frequencies ranging from 1.1 to 1.3 GHz with an output power over 20 kW from the nonlinear transmission line. The output frequencies increased with increasing volume fraction of BST, but the high power oscillations characteristic of an NLTL did not occur. Simulations using LT Spice demonstrated that an NLTL driven with a Blumlein modulator did not induce high power oscillations while driving the same NLTL with a pulse forming network did. </p> <p>Finally, a composite-based NLTL could be driven directly by a high voltage power supply without a power modulator to produce oscillations both during and after the formed pulse upon reaching a critical threshold. The output frequency of the NLTLs is 1 GHz after the pulse and ranged from 950 MHz to 2.2 GHz during the pulse. These results demonstrate that the NLTL may be used as both a pulse forming line and high power microwave source, providing a novel way to reduce device size and weight, while the use of composites could provide additional flexibility in pulse output tuning. </p>

No description available.

<p dir="ltr">High-power microwave (HPM) devices are generally vacuum-based devices that transform electron beam energy into microwaves with peak powers above 100 MW from 1-300 GHz. Solid-state HPM devices provide more compactness and greater reliability while consuming less power. Nonlinear transmission lines (NLTLs) provide a solid-state alternative to HPM generation by sharpening the input pulses from a pulse forming network to create output oscillations.</p><p dir="ltr">The first section of this dissertation evaluates and explores the feasibility of using nonlinear composites containing ferroelectric (e.g., Ba_2/3Sr_1/3TiO₃, BST) and/or ferromagnetic (e.g., Ni_1/2Zn_1/2Fe₂O₄, NZF) inclusions in a linear polymer host (polydimethylsiloxane, PDMS) to tune NLTL properties for HPM applications. Appropriately modelling and designing NLTLs using nonlinear composites require accurately characterizing their linear and nonlinear electromagnetic properties. We first studied the electromagnetic properties of the composites using theoretical, numerical, and experimental approaches. Incorporating these composite models and characterizations into NLTL simulations will be discussed.</p><p dir="ltr">Vacuum-based HPM devices, such as magnetrons and crossed-field amplifiers, generally operate in the space-charge-limited region, which corresponds to the maximum current possible for insertion into the device. This motivated studying the space-charge-limited current and electron flow in a two-dimensional (2D) planar diode with various crossed-magnetic fields using particle-in-cell (PIC) simulations. For non-magnetically insulated diodes (electrons emitted from the cathode can reach the anode), analytical and/or semi-empirical solutions are derived for electrons with nonzero monoenergetic initial velocity that agree well with PIC simulations. For magnetically insulated conditions, we developed new metrics using simulations and analytic theories to assess electron cycloidal and Brillouin flow to understand the implications of increasing injection current for 2D diodes. These analyses provide details on the operation of these devices at high currents, particularly virtual cathode operation, that may elucidate behavior near their limits of operation.</p>