# Interference-Induced Phenomena in High-Order Harmonic Generation from Bulk Solids
## 1 Introduction  
The first observation of photoemission spectra with clear high-order harmonics was reported using gaseous target media [1; 2]. In noble gas samples, the atoms driven by intense laser fields can be sources of secondary radiation ranging from visible to extreme ultraviolet regimes. Besides the wide bandwidth, this radiation has excellent temporal and spatial coherence. Additionally, the process of high-order harmonic generation (HHG) can be applied for the creation of attosecond pulses [3; 4; 5], which are ideal tools for the time-resolved investigation of electronic processes. These are the first experiments that successfully generated such short pulses and triggered the fast development of measurement and control techniques, as well as that of theoretical models [6; 7; 8; 9; 10; 11; 12; 13]. Typical intensities of the HHG signals are 5 to 10 orders of magnitude lower than that of the applied exciting field, which is partially due to the low particle density in gas samples. One possible way to increase the conversion efficiency is using solid targets [14]. For the case of surfaces, there are two mechanisms, the appearance of which strongly depends on the laser intensity: coherent wake emission (CWE) [15] and relativistic oscillating mirror (ROM) [16]. Besides surfaces, bulk solids have also been demonstrated to produce high-order harmonics, such as using few-cycle mid infrared (0.34-0.38 eV) pulses impinging on a wide-bandgap zinc oxide (ZnO) crystal [17; 18; 19; 20]. The gallium selenide (GaSe) single crystal has also been used as a target when the control of terahertz high harmonic generation was studied [21]. Considering real-time observation of crystal electrons, a non-perturbative quantum interference was identified between interband transitions as a salient HH generation mechanism [22]. In Reference [23], a direct comparison of the HHG signal was given in the solid and gas phases ofargon and krypton. Properties of the HHG radiation itself have also been investigated in various samples [24; 25; 26; 27; 28; 29]. For a recent review, see [30]. Besides the experimental results, several theoretical methods were developed in order to describe the interaction of intense electromagnetic fields and matter. The quasi-classical method, used to explain high-order harmonics, is a well-known three step model [31]. One of the most successful quantum-mechanical treatments of the process was introduced by [PERSON] et al. [32], known as the strong field approximation model of HHG, which has also been applied for solid samples [33; 34]. Saddle-point approximation can also be used to evaluate the integrals that appear in the description of both gases and solids [35; 36; 37]. The effective Bloch equations that were already developed in Reference [38], and the density matrix formalism in the velocity gauge [39] are closely related to the method that we are going to use in the following. As a different approach, ab initio calculations can provide effective methods for solving the time-dependent quantum mechanical problem of driving many electron systems. For example, the nonlinear response to strong laser fields was studied by the three-dimensional, time-dependent, two-particle reduced-density-matrix theory (TD-2 RDM) [40]. In Reference [41], the time-dependent density functional theory (TDDFT) was used to investigate the dependence of the process on the ellipticity of the laser field. Other TDDFT simulations revealed how the inhomogeneity of the electron-nuclei potential affects the process of HHG in solids [42]. In the current work, we focus on the case of bulk solids. As we shall see, a one-dimensional model [43] (similar to the one used in [44]) for the crystalline solid with a bandgap having the order of a few eV can reproduce the qualitative features of the HHG spectra (plateau region and cutoff, including its intensity dependence). The most important positive feature of this simple model is its transparency and low computational costs.