Symphony on strong field approximation

Abstract

In the last three decades, we have witnessed incredible advances in laser technology and in the understanding of nonlinear laser-matter interactions, crowned recently by the award of the Nobel prize to Gérard Mourou and Donna Strickland [1,2]. It is now routinely possible to produce few-cycle femtosecond (1 fs = 10 −15 s) laser pulses in the visible and mid-infrared regimes [3,4]. By focusing such ultrashort laser pulses on gas or solid targets, possibly in a presence of nano-structures [5], the targets are subjected to an ultra-intense electric field, with peak field strengths approaching the binding field inside the atoms themselves. Such fields permit the exploration of the interaction between strong electromagnetic coherent radiation and an atomic or molecular system with unprecedented spatial and temporal resolution [6]. On one hand, HHG nowadays can be used to generate attosecond pulses in the extreme ultraviolet [7,8], or even in the soft X-ray regime [9]. Such pulses themselves may be used for dynamical spectroscopy of matter; despite carrying modest pulse energies, they exhibit excellent coherence properties [10,11]. Combined with femtosecond pulses they can also be used to extract information about the laser pulse electric field itself [12]. HHG sources therefore offer an important alternative to other sources of XUV and X-ray radiation: synchrotrons, free electron lasers, X-ray lasers, and laser plasma sources. Moreover, HHG pulses can provide information about the structure of the target atom, molecule or solid [13][14][15]. Of course, to decode such information from a highly nonlinear HHG signal is a challenge, and that is why a possibly perfect, and possibly "as analytical as possible" theoretical understanding of these processes is in high demand. Here is the first instance where SFA offers its basic services.Since electronic motion is governed by the waveform of the laser electric field, an important quantity to describe the electric field shape is the so-called absolute phase or carrier-envelope phase (CEP). Control over the CEP is paramount for extracting information about electron dynamics, and to retrieve structural information from atoms and molecules [13,16,17]. For instance, in HHG an electron is liberated from an atom or molecule through ionization, which occurs close to the maximum of the electric field. Within the oscillating field, the electron can thus accelerate along oscillating trajectories, which may result in recollision with the parent ion, roughly when the laser field approaches a zero value. Control over the CEP is particularly important for HHG, when targets are driven by laser pulses comprising only one or two optical cycles. In that situation the CEP determines the relevant electron trajectories, i.e. the CEP determines whether emission results in a single or in multiple attosecond bursts of radiation [16,18].The influence of the CEP on electron emission is also extremely important. It was demonstrated for instance in an anti-correlation experiment, in which the number of AT...

Time-independent approximations for time-dependent optical potentials

“…When the laser field term involving E(t) dominates the potential term and κ(t) =const several well known and successful approach have been developed. For instance, the strong field approximation is a mixture of perturbative expansions based on the Du-Hamel formula carried out on the level of the time-evolution operator [15,16,17,11].…”

“…In more concrete settings less general approximation methods have been developed to account for the specifics of the system. For instance, a very active field of research involving timedependent Hamiltonians is the area of strong laser fields [11]. The systems considered in this context involve Stark potentials of the form V (x) + xE(t), with E(t) describing a laser field and the term xE(t) dominating or being comparable in strength to the potential V (x).…”

“…In the high intensity regime the two perturbative series, one in V (x) and the other in xE(t), are mixed and terminated after the first iteration. The expressions obtained in this manner are commonly referred to as the strong field approximation [15,16,17,11].…”

Crystal-momentum-resolved contributions to multiple plateaus of high-order harmonic generation from band-gap materials

“…Although focusing on the response of a single Bloch electron (e.g., at the point) has provided useful insights into the HHG mechanisms [26,27], the contributions from multiple k points and their quantum interference have turned out to be important for both intraband and interband harmonics, attracting more and more attention (see, e.g., recent works [43,45,46]). In addition to the methods mentioned above, the strong-field approximation (SFA) method was used to investigate HHG in solids [58]. A comparison study of the SFA method and the TDSE for a one-dimensional (1D) model was conducted in Ref.…”

Perspective on Petahertz Electronics and Attosecond Nanoscopy

“…While the lightwave-induced strong-field processes in gas-phase atoms and molecules have been under intensive investigations, leading to the birth of attosecond physics (see e.g. [3,[8][9][10][11][12][13][14]), the exploration of lightwave-driven petahertz electronics in condensed matter at conditions of extreme nonlinearity is still in its early phase and has become an emerging field of research. Lightwave electronics in solids, i.e.…”