Short pulse laser beam beyond paraxial approximation

Abstract

Nonparaxial perturbative equations are derived from the scalar wave equation by taking into account spatiotemporal couplings. General solutions are obtained in Fourier space and further transformed back in direct space. They depend on parameters that can be used to match various boundary conditions and the perturbative expansion of any nonparaxial exact solutions. This parametrization is used to study the sensitivity of direct electron acceleration off an ultrashort tightly focused laser pulse to nonparaxial corrections of radially polarized electromagnetic fields.

Maxwell-consistent, symmetry- and energy-preserving solutions for ultrashort-laser-pulse propagation beyond the paraxial approximation

“…II H). This is a fundamental difference with respect to previous works, where, for example, in order to determine the high-order corrections, some authors had considered ad hoc assumptions such that they are zero at the beam focal point [21,27], they follow the structure of a spherical wave emanating from the beam focal point [23] or they must match some known nonparaxial solutions [31]. Indeed, in the particular solutions proposed by most of these works dealing with Hermite-Gaussian and Laguerre-Gaussian paraxial families, spurious homogeneous solutions are found when a Gram-Schmidt orthogonalization process is applied in the focal plane [37,38].…”

“…A Taylor expansion of Eq. ( 13) in powers of κ ⊥ (around κ ⊥ = 0) and ξ (around ξ = 0), reveals that the general solution of the wave equation depends on powers of ε [31]. Motivated by this fact, in order to solve Eq.…”

“…Later, Porras et al [29,30] proposed a similar time-domain perturbative approach, based on a different expansion parameter, in order to study the propagation of vectorial few-cycle light pulses. More recently, Favier et al took into account spatiotemporal couplings in the wave equation in order to ex-tend Lax perturbative equations to few-cycle pulses [31]. In the transverse-spatial and temporal Fourier domain, they linked the Lax series with a Taylor expansion of the exact solution of the wave equation, but their proposed high-order corrections hinged on an arbitrary number of integration constants, which were chosen to match some known nonparaxial solutions.…”

Linearly polarized laser beam with generalized boundary condition and non-paraxial corrections

“…They can, in principle, be obtained by means of a fit to a given numerical description of the electromagnetic field numerically derived or experimentally measured, up to a certain precision. The procedure that is described here is strictly valid for monochromatic pulses only, which is sufficient in the case studied in this article, though it can be extended to generally astigmatic beams [20] and electromagnetic fields involving spatio-temporal couplings [11]. Extreme cases where a very broad angular spectrum as those considered for instance in [21] are unlikely described by such series expansion, owing to large values of the expansion parameter ε.…”

“…These effects are indeed of importance, especially when dealing with longitudinal chromatism [10]. Such spatio-temporal couplings can also be included within perturbation series expansions [11] for few cycle fields. Solving the Maxwell equations in a consistent way at a given order of the series expansion in momentum space including such effects was demonstrated to match numerical calculations with good precision [12], though requiring numerical integration to obtain position space field values.…”