On the use of lenses to focus few-cycle pulses with controlled carrier–envelope phase

Abstract

From a model of focusing with lenses that includes the effects of the lens variable thickness, material dispersion, aperture, spherical and chromatic aberrations, we characterize the conditions under which a lens can focus to few-cycle, transform-limited pulses propagating without distortion along the focal region. A lens also allows to control the carrier-envelope phase shift along the focus. The carrier-envelope phase shift is drastically reduced by focusing with specific focal lengths and input spot sizes, which are of the same order as those typically used in experiments involving focusing for phase-sensitive, light-matter interactions.

Carrier-envelope phase shift of few-cycle pulses along the focus of lenses and mirrors beyond the nonreshaping pulse approximation: the effect of pulse chirp

“…Contrary to what is generally believed, chromatic aberration does not necessarily cause large pulse distortions in the focal region [9]. On the contrary, pulse broadening and distortion are negligible if the focal length variation ∼jf 0 0 jΔω within the pulse bandwidth Δω is much smaller than the focal depth 2L R;0 (prime signs in this paper denote d∕dω, and subscripts 0 denote evaluation at the carrier frequency ω 0 ) [9]. In terms of the dimensionless parameter…”

“…In this paper we evaluate the CEP evolution along the focal region of a focusing system taking into account the reshaping of the pulse during propagation through the focus due to its transversal localization, i.e., beyond the nonreshaping approximation implicit in all previous works [7][8][9]. We find that this reshaping does not alter previous results regarding the CEP variation curve in the focal region if the pulse is transform limited.…”

“…This is the only cause of focal CEP shift considered in all previous works [7][8][9], since the assumption of invariable envelope along the focal region is (more or less) implicit in all these works. Within this assumption, the most general expression found for the CEP shift from the focal point of a lens or mirror is [7][8][9] ΔΦz −tan −1 ζ gζ γζ 2 1 ζ 2 ;…”

“…The above simple formulas from the well known Gaussian beam propagation formalism [16] have recently been shown to provide an accurate description of the focused pulse in the case that the input spot size s is much smaller than the entrance aperture radius a L so that the effects of truncation and of spherical aberration are negligible [9]. Under these conditions, the simple dependence of the propagated spectrum in Eqs.…”

“…However, it is well known that Gouy phase causes a total CEP shift −π during propagation through a focal region and also that the specific CEP variation curve from the beginning to the end of the focal region depends on the focusing system and on the pulse in front of the focusing system. Within the approximation that the pulse temporal envelope remains unchanged during propagation through the focal region, the variation curve of the CEP has been evaluated for focusing mirrors [7,8] and for lenses [9] for general pulsed Gaussian beams in front of the focusing system. In the case of focusing with lenses, the conditions under which any vestige of the dispersion introduced by the lens material can be eliminated and under which the spherical and chromatic aberrations do not introduce any appreciable pulse broadening or distortion along the focal region, even when focusing few-cycle pulses have been recently described in [9].…”