Generation of high order optical harmonics from solid surfaces

Abstract

An intense ultrashort laser pulse impinging on a solid target can create a thin surface layer of plasma in which the density drops from solid density to the vacuum level in a distance much shorter than the wavelength. The plasma performs an oscillatory motion in response to the electromagnetic forces of the intense laser light. It is shown that the reflected light experiences a phase modulation which leads to odd and even harmonics of the fundamental laser frequency. For a very strongly driven surface plasma, the velocity of the reflecting boundary can approach the speed of light. Under these conditions retardation effects become important and are expected to lead to efficient generation of very high order harmonics. PACS: 42.65.Ky; 52.40.Nk; 52.50.Jm Generation of optical harmonics of very high order is a subject of great current interest. Harmonics exceeding order 300 have recently been observed during the interaction of an intense laser pulse of only several femtoseconds with a jet of helium gas [1, 2]. In gases, harmonic generation is due to the strong nonlinear response of the atomic electrons when the laser field approaches the field ionization limit. High order harmonic generation from solids was observed for the first time many years ago by Carman et al. [3, 4] who used nanosecond laser pulses. A key point of the theoretical explanation [5, 6] was the assumption of a step-like plasma density gradient. Electrons driven across this steep gradient by the laser field perform a strongly anharmonic motion which leads to the generation of odd and even harmonics. The formation of a steep density gradient was attributed to the action of the ponderomotive force, which counteracts the plasma expansion. Creation of a plasma on the surface of a solid using femtosecond laser pulses leads quite naturally to the formation of a steep plasma density gradient, because there is no time for significant plasma expansion during the interaction. A thin layer of plasma is formed in which the plasma density drops from solid density to vacuum in a very short distance. Specularly reflected coherent harmonics from solid surfaces up to the seventh order have been observed by Kohlweyer et al. [7], and up to the eighteenth by the author and coworkers [8]. In these experiments, intense femtosecond laser pulses from a titanium sapphire CPA laser were used. Norreys et al. [9] reported generation of harmonics as high as 75 with picosecond laser pulses. In this case the harmonic radiation was not confined to the specular direction and was spread out over a large solid angle. Recent particle-in-cell (PIC) simulations by Gibbons [10] and by Lichters et al. [11] provided new insight into high order harmonic generation from a plasma–vacuum boundary. In particular, they demonstrated the importance of relativistic effects and showed that harmonic generation could be extended well beyond the high frequency limit of the earlier theories [5, 6] by increasing the laser intensity to the relativistic regime. Lichters et al. [11] showed that their detailed numerical simulations of the complex collective electron dynamics are in excellent agreement with a simple model in which harmonic generation is interpreted as an anharmonic distortion of the laser field upon reflection from a rapidly oscillating surface. This moving mirror model originally due to Bulanov et al. [12] turns out to be extremely useful for understanding high order harmonic generation from solid targets. 1 The moving mirror model 1.1 General considerations The threshold of plasma formation in typical dielectric solids is about 1013 W/cm2 for a 100 fs laser pulse [13]. When the laser intensity exceeds this threshold, condensed matter is very rapidly turned into a plasma. During the short interaction time with the laser pulse, the ions can be regarded as fixed positive background charges. Electrons in a skin depth layer experience strong electromagnetic forces and are driven back and force across the vacuum boundary. The basic approximation of the moving mirror model [11, 12, 14] is to neglect the details of the electron spatial distribution and to represent the