Optical Gas-Phase Frequency References Based on Photonic Crystal Technology:Impact of Slow Light on Molecular Absorption

Optical frequency references are devices providing well-defined and stable optical frequency responses to incoming radiation for applications such as high-precision spectroscopy and optical fibre communications. To stabilise the emission frequency of lasers, which drifts with time mostly because of fluctuations in temperature and mechanical vibrations, atomic and molecular optical transitions can be used since they show precise and well-defined frequency responses to incoming radiation. However from a practical standpoint conventional gas cell devices cannot be easily integrated into existing optical systems because of their bulky dimensions. To replace conventional absorption cells, photonic crystal fibres filled with gas-phase material are promising devices owing to their robustness, reliability, and portable characteristics. In addition they can be directly embedded into existing optical systems and they perform well in harsh environments. In this experimental study, optical gas-phase frequency references based on photonic crystal technology are realised. The gas-sensing properties of different photonic crystal fibre samples are studied and the long-term stability and reliability of fibre gas cells are demonstrated. In addition an analytical model predicting the gas-filling time in photonic crystal fibres (PCF) is developed and can be applied to any type of fibre, fibre geometry, or length. Then fibre gas cells filled with acetylene gas, a recognised frequency reference gas-phase material, have been prepared to conduct fundamental research on slow & fast light generation in optical fibres to verify the possibilities of slow light in enhancing light-matter interaction. The group velocity of light is controlled by modifying the material and structural dispersive properties of the PCF absorption cells through stimulated Brillouin scattering and cavity ring resonators, respectively. We could demonstrate that material slow light has no impact on the molecular absorption effect whereas structural slow light has an impact on the absorption efficiency scaling linearly with the group index. Such radically different responses to slow light suggest that group velocity is not the universal physical quantity scaling light-matter interaction, and that the optical absorption of molecules is more closely related to the velocity of the electromagnetic energy. Finally the impact of slow light on the molecular absorption efficiency is also evaluated in dispersion-engineered photonic crystal waveguides. We demonstrate that in planar photonic crystal waveguides the field enhancement and its evanescent fraction have more impact on the absorption efficiency than the reduction of the group velocity of light

Stabilization of cascaded nonlinear systems under sampling and delays

Over the last decades, the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model. Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system

Delayed point control of a reaction–diffusion PDE under discrete-time point measurements

We consider stabilization problem for reaction–diffusion PDEs with point actuations subject to a known constant delay. The point measurements are sampled in time and transmitted through a communication network with a time-varying delay. To compensate the input delay, we construct an observer for the future value of the state. Using a time-varying observer gain, we ensure that the estimation error vanishes exponentially with a desired decay rate if the delays and sampling intervals are small enough while the number of sensors is large enough. The convergence conditions are obtained using a Lyapunov–Krasovskii functional, which leads to linear matrix inequalities (LMIs). We design output-feedback point controllers in the presence of input delays using the above observer. The boundary controller is constructed using the backstepping transformation, which leads to a target system containing the exponentially decaying estimation error. The in-domain point controller is designed and analysed using an improved Wirtinger-based inequality. We show that both controllers can guarantee the exponential stability of the closed-loop system with an arbitrary decay rate smaller than that of the observer’s estimation error

Delayed boundary control of a heat equation under discrete-time point measurements

We consider a reaction-diffusion PDE under continuously applied boundary control that contains a constant delay. The point measurements are sampled in time and transmitted through a network with a time-varying delay. We construct an observer that predicts the value of the state allowing to compensate for the constant boundary delay. Using a time-varying injection gain, we ensure that the estimation error vanishes exponentially with a desired decay rate if the delays and sampling intervals are small enough while the number of sensors is large enough. The stability conditions, obtained via a Lyapunov-Krasovskii functional, are formulated in terms of linear matrix inequalities. By applying the backstepping transformation to the future state estimation, we derive a boundary controller that guarantees the exponential stability of the closed-loop system with an arbitrary decay rate smaller than that of the observer. The results are demonstrated by an example

Observer-based input-to-state stabilization of networked control systems with large uncertain delays

We consider output-feedback predictor-based stabilization of networked control systems with large unknown time-varying communication delays. For systems with two networks (sensors-to-controller and controller-to-actuators), we design a sampled-data observer that gives an estimate of the system state. This estimate is used in a predictor that partially compensates unknown network delays. We emphasize the purely sampled-data nature of the measurement delays in the observer dynamics. This allows an efficient analysis via the Wirtinger inequality, which is extended here to obtain exponential stability. To reduce the number of sent control signals, we incorporate the event-triggering mechanism. For systems with only a controller-to-actuators network, we take advantage of continuously available measurements by using a continuous-time predictor and employing a recently proposed switching approach to event-triggered control. For systems with only a sensors-to-controller network, we construct a continuous observer that better estimates the system state and increases the maximum output sampling, therefore, reducing the number of required measurements. A numerical example illustrates that the predictor-based control allows one to significantly increase the network-induced delays, whereas the event-triggering mechanism significantly reduces the network workload

Regularisation and Long-Time Behaviour of Random Systems

Schenke A. Regularisation and Long-Time Behaviour of Random Systems. Bielefeld: Universität Bielefeld; 2020.In this work, we study several different aspects of systems modelled by partial differential equations (PDEs), both deterministic and stochastically perturbed. The thesis is structured as follows: Chapter I gives a summary of the contents of this work and illustrates the main results and ideas of the rest of the thesis. Chapter II is devoted to a new model for the flow of an electrically conducting fluid through a porous medium, the tamed magnetohydrodynamics (TMHD) equations. After a survey of regularisation schemes of fluid dynamical equations, we give a physical motivation for our system. We then proceed to prove existence and uniqueness of a strong solution to the TMHD equations, prove that smooth data lead to smooth solutions and finally show that if the onset of the effect of the taming term is deferred indefinitely, the solutions to the tamed equations converge to a weak solution of the MHD equations. In Chapter III we investigate a stochastically perturbed tamed MHD (STMHD) equation as a model for turbulent flows of electrically conducting fluids through porous media. We consider both the problem posed on the full space $\R^{3}$ as well as the problem with periodic boundary conditions. We prove existence of a unique strong solution to these equations as well as the Feller property for the associated semigroup. In the case of periodic boundary conditions, we also prove existence of an invariant measure for the semigroup. The last chapter deals with the long-time behaviour of solutions to SPDEs with locally monotone coefficients with additive L\'{e}vy noise. Under quite general assumptions, we prove existence of a random dynamical system as well as a random attractor. This serves as a unifying framework for a large class of examples, including stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$\alpha$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard-type equations, stochastic Kuramoto-Sivashinsky-type equations, stochastic porous media equations and stochastic $p$-Laplace equations

Generation and metrology of ultrashort pulses and their application in attosecond science

This thesis deals with the dynamical processes in atoms and small molecules initiated by the absorption of ultrashort, coherent light pulses. The studied phenomena takeplace on the femtosecond (1 fs = 10−15 s) and attosecond (1 as = 10−18 s) timescales and critically depend on the properties of the light fields that drive them. Wework with infrared (IR) femtosecond laser pulses, which we manipulate through nonlinear interactions with matter to either study these interactions themselves or applythem to investigate other light-induced processes.One part of this thesis focuses on the generation and characterisation of IR pulses spectrally broadened through the Kerr effect. We use a technique called dispersion scanto temporally compress and at the same time measure pulses broadened in gas-filled hollow-core fibres. We propose multiple improvements to this well-established characterisation technique. Further, we investigate femtosecond filamentation in gases, a process with highly complex dynamics involving several non-linear processes including the Kerr effect and ionisation. We develop a method that allows us to measure the electric field of a laser pulse undergoing filamentation in three dimensions, whilealso scanning along the filament length. Our technique provides access to pulses with desirable characteristics that may be generated at a point inside the filament, simultaneously enabling their measurement and extraction for applications. In addition, this technique opens up the possibility to explore intricate filament dynamics.In the other part of this work, we up-convert the IR laser pulses into trains of extreme ultraviolet (XUV) attosecond pulses through a non-linear process called high-orderharmonic generation. We combine the IR and XUV pulses to study the photoionisation dynamics in different species using a method known as RABBIT (Reconstructionof Attosecond Beating By Interference of Two-photon transitions). In this technique, a target gas is ionised by the XUV field, creating an electron wave-paket (EWP) in thecontinuum, while a weak IR pulse probes the system. The EWP scatters off the ionic potential, acquiring an additional phase as it propagates. Recording the photoelectronspectrum as a function of the IR-XUV time delay allows us to infer time-resolved information about the ionic potential. We apply this method to investigate the dynamicsof different ionisation processes in noble gases (He, Ar, and Xe) and the N2 molecule. The high spectral resolution of our electron spectrometer allows us to disentanglethe contributions from different ionisation channels. In addition, we perform angle-resolved measurements, investigating the coherent superposition of final stateswith different angular momenta