Realization of a Continuously Phase-Locked Few-Cycle Deep-UV/XUV Pump-Probe Beamline with Attosecond Precision for Ultrafast Spectroscopy

Chemical and physical processes in molecules can be controlled through the manipulation of quantum interferences between rotational, vibrational, and electronic degrees of freedom. Most of the past efforts have been focused on the control of nuclear dynamics. Even though electronic coherence and its coupling to nuclear degrees of freedom may profoundly affect the outcome of these processes, electron dynamics have received less attention. Proper investigation of electron dynamics in materials demands ultrafast sources in the visible, ultraviolet (UV), and extreme ultraviolet (XUV) spectral region. For this purpose, a few-cycle deep-UV and XUV beamlines have been constructed for studying ultrafast electron dynamics in molecules. To ensure the required high temporal resolution on the attosecond time scale, vibration isolation from environmental mechanical noise and active stabilization have been implemented to achieve attosecond timing control between pump and probe pulses with excellent stability. This is achieved with an actively phase-stabilized double-layer Mach-Zehnder interferometer system capable of continuous time-delay scans over a range of 200 fs with a root-mean-square timing jitter of only 13 as over a few seconds and ~80 as of peak-to-peak drift over several hours.

FORCED DIFFUSION EQUATION WITH PIECEWISE CONTINUOUS TIME DELAY

In this article, we study the boundary value problem (BVP) for forced diffusion equation with piecewise constant arguments. We give explicit formula for solving BVP. The problem of finding periodic solutions of some differential equations with piecewise constant arguments (DEPCA) is reduced to solving a system of algebraic equations. Using the method of finding periodic solutions of DEPCA, the solutions of BVP are given in several examples which are periodic in time.

New Stability Analysis and Design of Discrete Time Delay Control for Nonaffine Nonlinear Systems

Time Delay Control (TDC) for linear and nonlinear systems with uncertain dynamics has been widely discussed in the literature, as it has a very simple and compact form. It uses time-delayed signals for estimating unknown dynamics at an instant, and uses feedback linearization for cancellation of known and estimated unknown dynamics. While the original formulation and most of the analyses have been focused on the continuous version of the controller, its implementation is more natural in digital form. This paper extends the recently improved sufficient Bounded Input - Bounded Output (BIBO) stability conditions for continuous Time Delay Control of nonaffine nonlinear systems to discrete Time Delay Control, including simplified approximate conditions under various assumptions. Additionally. asymptotic stability is established for similar conditions. The derived conditions are contrasted with earlier results for continuous Time Delay Control. Examples are used to illustrate the differences in continuous and discrete TDC, related to performance as well as sufficient and actual conditions for stability.