Saddle point approaches in strong field physics and generation of attosecond pulses
Attoscience is the emerging field that accesses the fastest electronic processes occurring at the atomic and molecular length scales with attosecond (1 as = 10 –18 s) time resolution having wide ranging physical, chemical, material science and biological applications. The quintessential and one of the most fundamental processes in this domain is the generation of phase locked XUV attosecond pulses. The theoretical approach to understand the process incorporates a fully quantum or semi classical or relativistic description of coherent charge dynamics in intense ultrashort electromagnetic fields driving a quantum system (an atom, a molecule, solid band gap materials or surface plasmas). Modelling of such physical and dynamical systems in science and also in many other branches often leads to equations represented in terms of complex multi-dimensional integrals. These integrals can often be solved using the stationary phase approximation, which leads to a series of equations identifying the points in the multi-dimensional space, having most significant contributions in their evaluation. These points are usually indicated as saddle points. The description of the dynamics of quantum mechanical or relativistic systems that results from such an approach enables near to classical physics intuitive perceptions of the processes under investigation. Thus, the saddle point methods are very powerful and valuable general theoretical tools to obtain asymptotic expressions of such solutions and help also to gain physical insights on the underlying phenomena. Such techniques developed in the past have been adapted to study the emission of as pulses by different physical systems and have been widely employed in calculating and estimating the response of matter to intense electromagnetic pulses on ultrafast time scales. Here we provide an extensive disposition of the saddle point approaches unifying their ubiquitous applications within the domain of attoscience valid for simple atomic to more complex condensed matter systems undergoing ultrafast dynamics and present current trends and advancements in the field. In this review we would delineate the methodology, present a synthesis of seminal works and describe the state of the art applications. Finally we also address ultrashort time dynamics of novel materials that have gained much attention recently, namely lower dimensional material systems and micro-plasma systems.