Huygens-Fresnel Picture for High Harmonic Generation in Solids

High harmonic generation (HHG) is usually described by the laser-induced recollision of particlelike electrons, which lies at the heart of attosecond physics and also inspires numerous attosecond spectroscopic methods. Here, we demonstrate that the wavelike behavior of electrons plays an important role in solid HHG. By taking an analogy to the Huygens-Fresnel principle, an electron wave perspective on solid HHG is proposed by using the wavelet stationary-phase method. From this perspective, we have explained the deviation between the cutoff law predicted by the particlelike recollision model and the numerical simulation of semiconductor Bloch equations. Moreover, the emission times of HHG can be well predicted with our method involving the wave property of electrons. However, in contrast, the prediction with the particlelike recollision model shows obvious deviations compared to the semiconductor Bloch equations simulation. The wavelike properties of the electron motion can also be revealed by the HHG in a two-color field.

Figure 1

Sketch for HHG analogy to the Huygens-Fresnel principle.

Figure 2

Time frequency property of HHG for the channel $k_l=0$. (a) The absolute value of partial derivative of quasiclassical action $|{\partial }_{t^{\prime }}S(k_l,t_r,t^{\prime })|$ for the different emission times $t_r=0.0450{\mathrm{T}}_0$, $0.665{\mathrm{T}}_0$, and $0.900{\mathrm{T}}_0$. (b) The values $|{\partial }_{t^{\prime }}S(k_l,t_r,t_i)|$ (black line) and effective electron-hole displacement $|x(k_l,t_r,t_i)/a_x|$ (red line) with $t_i$ determined by Fig. 2. (c) The relation between the most probable emission energy and emission (ionization) times. The gray and black lines are obtained with our WSPM method. The light blue and blue lines are obtained with the CSPM method. (d) The time frequency property of HHG obtained with the numerical integration of Eq. (2). The black line and blue line are the results obtained with the WSPM and CSPM.

Figure 3

The time frequency property of HHG from the numerical results for channels (a) $k_l=0.02$ ${\mathrm{L}}_{\mathrm{kx}}$, and (b) $k_l=0.04$ ${\mathrm{L}}_{\mathrm{kx}}$. (c) The result for SBEs. The WSPM results are marked by black, purple, and green dots that are copies from Figs. 2, 3, and 3. The CSPM result is marked by blue points. (d) The simulated cutoff law (red circles); the prediction from WSPM (solid black line) and CSPM (blue line).

Figure 4

(a) The value of $\mathrm{\Sigma }(t_{2N})$ for different channels. (b) Simulated spectrum for HHG. Each harmonic order has been normalized to simplify comparison. The laser parameters are $\omega =0.0152\mathrm{a}.\mathrm{u}.$, $F_0=0.003\mathrm{a}.\mathrm{u}.$, and $F_{2{\omega }_0}=0.01{\mathrm{F}}_0$.