Ultrashort, Strongly Focused Laser Pulses in Free Space

Technological advances in ultrafast optics now allow the generation of laser pulses whose duration is as short as a few optical cycles of the electric field; furthermore, these pulses can be focused to a spot size comparable to the wavelength. These strongly focused, ultrashort laser pulses have found applications, for instance, in high-resolution microscopy, particle trapping and electron acceleration. In order to characterize the spatiotemporal behavior of such ultrashort, tightly focused pulses, one needs the expressions of their electromagnetic fields. Ultrafast nonparaxial pulsed beams must be modeled as exact solutions to Maxwell's equations. Many studies on the propagation of a pulsed beam are based on a scalar paraxial theory, which provides an accurate description of the pulsed beam propagation when the beam divergence angle is small and the beam spot size is much larger than the wavelength for each spectral component. However, the analysis of tightly focused laser beams requires expressions of optical beams that extend beyond the paraxial approximation. Moreover, the vector nature of light cannot be neglected to properly describe tightly focused beams. Also, the appropriate spectrum amplitude must be employed in order to model ultrashort pulses. Many authors have proposed expressions for the electromagnetic fields of laser pulsed beams, but most of these models are incomplete. For example, Wang and co-workers presented scalar paraxial pulsed Gaussian beams that have a Gaussian spectrum (Wang et al., 1997), but their expressions are not suitable to describe ultrashort pulses, as reported by Porras (Porras, 1998). Caron and Potvliege suggested forms of spectra, which are appropriate to characterize pulses of very small duration, but the expressions for their vectorial nonparaxial ultrashort pulses are written in terms of numerically calculated angular spectra (Caron & Potvliege, 1999). Lin et al. presented closed-form expressions for subcycle pulsed focused vector beams that are exact solutions to Maxwell’s equations obtained in the context of the so-called complex-source point method, but they used an unsuitable Gaussian spectrum (Lin et al., 2006). Recently, an der Brügge and Pukhov have provided solutions for ultrashort focused electromagnetic pulses found with a more appropriate spectral amplitude, but the expressions hold true only in the paraxial regime (an der Brügge & Pukhov, 2009). The aim of this chapter is to provide a simple and complete strategy to correctly model strongly focused, ultrashort laser pulses. Three main tools are employed to find the expressions for the fields of such pulsed beam. First, the Hertz potential method is used in