Proposal for High-Energy Cutoff Extension of Optical Harmonics of Solid Materials Using the Example of a One-Dimensional ZnO Crystal

We propose a novel approach based on the subcycle injection of carriers to extend the high-energy cutoff in solid-state high harmonics. The mechanism is first examined by employing the standard single-cell semiconductor Bloch equation (SC SBE) method for one-dimensional (1D) Mathieu potential model for ZnO subjected to the intense linearly polarized midinfrared laser field and extreme-ultraviolet pulse. Then, we use coupled solution of Maxwell propagation equation and SC SBE to propagate the fundamental laser field through the sample, and find that the high-harmonics pulse train from the entrance section of the sample can inject carriers to the conduction bands with attosecond timing, subsequently leading to a dramatic extension of high-energy cutoff in harmonics from the backside. We predict that for a peak intensity at $2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, as a result of the self-seeding, the high-energy cutoff shifts from 20th (7.75 eV) order to around 50th (19.38 eV) order harmonics.

Figure 1

(a) Vector potential of the incident IR (red) and EUV (purple) pulses. (b) Electron trajectory in conventional three-step model for HHG in solids. (c) Electron trajectory in extended three-step model, i.e., acceleration of electrons initially locating far from $\mathrm{\Gamma }$ on highest valence band $V1$, transition assisted by one-photon absorption, acceleration on lowest conduction band $C1$, tunneling to higher conduction band $C2$ near the BZ boundary, acceleration, and recombination. $t_n$ in (b) and (c) represents the moment marked in (a).

Figure 2

(a) Band structure of model ZnO along $\mathrm{\Gamma }\text{−}M$, where the three bands strongly coupled are highlighted, including the highest valence band $V1$ and two lowest conduction bands $C1$ and $C2$, to get an intuitive understanding of the extended three-step model in following analysis. (b) MTM between each pair of highlighted bands. Only real values are presented since 1D real and symmetric potential results in real ${\mathit{p}}_{mn}^{\mathit{k}}$ values. Note that the Fermi energy has been reset to 0 eV and two valence bands and three bands are used in simulation to get converged results.

Figure 3

(a) Temporal profiles of the driving IR field (red) and the injecting UV pulse at 0 (blue, referred to as case 1) and $\frac{1}{4}{T}_L$ delay (green, case 2). Note the UV pulse is injected at the zero crossing (negative maximum) of the vector potential of the IR pulse for case 1 (case 2). The field amplitudes have been scaled up separately. (b) HHG spectra for IR only (red), case 1 (blue), and case 2 (green). Dashed lines mark the minimal and maximal gaps between $V1$ and $C1$, and $V1$ and $C2$.

Figure 4

Evolution of the carrier population growth on different bands for case 1 (left) and case 2 (right), relative to the case when only the IR field is presented. The red and black dashed lines show the central crystal momentum of excited carriers on $C2$ and $C1$, respectively.

Figure 5

(a) Emission spectra from SC SBE (black solid line), reflected (blue solid line), and transmitted (red solid line) emissions by MP SBE, respectively. (b) Emission spectra from a single cell subjected to the fundamental pulse and additional components of $11\text{−}27{\omega }_L$ harmonics (green solid line). Averaged intensity of odd harmonics in the forward radiation (red circles) is plotted as a guide to the eye. Dotted lines in (a) and (b) depict the minimal and maximal energy difference between the two lowest conduction bands and the highest valence band. Time-frequency spectrogram of backward (c) and forward (d) field in logarithmic scale. The square of the vector field potential $A$ is illustrated in white dashed lines.