Polarization properties of high harmonics generated on solid surfaces

Abstract

A detailed study of the electromagnetic field's configuration inside plasmas generated on solid surfaces by ultrashort laser pulses is given in the present work. The radiation due to the nonrelativistic motion of the electrons in the overdense region is considered the main source of high harmonics, whose polarization dependence on the fundamental beam is treated to explain the experimentally observed polarization properties of the harmonics.

Introduction

Several laboratories developed lasers capable of delivering very short pulses t<1 ps with high intensities I>10¹⁶ W/cm². High order harmonics were observed, resulting from these interactions of laser beams with solid targets. It was recently observed by using ultrashort laser pulses that even and odd harmonics occur for both laser polarization [1], [2], [3], [4], [5], [6]. Some theories [6], [7], [8], [9] were developed to explain the polarization dependence of harmonic generation. One of the theories referring to the polarization dependence of high harmonics generated on solid surfaces have been developed by von der Linde [7] using a perturbative treatment. This theory consider as the source of the high harmonics the coupling of different order velocity perturbations to the different order electron densities induced by the electric field of the incident laser light. Our earlier model [6] also consider the source of harmonics being the above mentioned coupling, but in this case the velocity and density components were calculated using the Fourier analysis of the Maxwell and continuity equation and of the equation of motion. A qualitative description was given by Refs. [8], [10] using the simple physical model of an oscillating plasma mirror and simulations with PIC codes were also carried out. The reflecting layer is considered to be the supercritical plasma, formed by the leading edge of the incident laser pulse, when the details of the density distribution are neglected. The phase modulation of this mirror gives rise to harmonic frequencies. All these theories predict that the even harmonics will be p-polarized no matter what the incident laser polarization is. The theories are in agreement with the experiments in case of odd harmonics which conserve the polarization of the fundamental laser beam. In the case of s-polarized incident beam there is a contradiction between the experiments and theory. The observation in this respect are also ambiguous. As we discussed in Ref. [6] the experiments carried out in this field can be divided into two categories: the experiments where the pulse length of the incident laser beam was shorter than 100 fs showed the dominance of p-polarized harmonics [7], [10], [11], [12], [13]. The second category, where the experiments were performed with pulse length longer than 500 fs both s- and p-polarized harmonics were observed, keeping the polarization of the original laser beam [2], [4], [5], [6].

In the present work we employ the theory of reflection and refraction, to understand how the electric and magnetic fields behave in plasmas where the index of refraction is complex. The role of the magnetic field can be of importance in such a medium, which means that the term $\text{(}\text{v}\text{→}\text{×}\text{B}\text{→}\text{)/c}$ in the equation of motion cannot be neglected. The nonrelativistic equation of motion of the electrons in the plasma governed by the complete Lorentz force due to the penetrating electromagnetic field has been solved exactly, and high harmonic components of the electron's velocity were obtained. Hence, these are the source of the radiation containing high harmonics.

In Section 2 the influence of different plasma regions on the propagation of the electromagnetic field is discussed. A subsection is dedicated to explain how the electromagnetic field could penetrate into the overdense region. In Section 3 the equation of motion is solved and the nonlinearities of the electron's motion are presented. In Section 4 the radiation field produced by the nonlinear motion of the electrons is depicted.