Waveform-Controlled Relativistic High-Order-Harmonic Generation

We consider the efficiency limit of relativistic high-order-harmonic emission from solid targets achievable with tailored light fields. Using one-dimensional particle-in-cell simulations, the maximum energy conversion efficiency is shown to reach as high as 10% for the harmonics in the range of 80–200 eV and is largely independent of laser intensity and plasma density. The waveforms most effective at driving harmonics have a broad spectrum with a lower-frequency limit set by the width of the incident pulse envelope and an upper limit set by the relativistic plasma frequency.

Figure 1

(a) Attosecond pulse generation efficiency $(I_a/I_L)$ for driving pulses with 10% of incident energy in the $n$th harmonic with phase ${\varphi }_n$. (b) Maximum and minimum generation efficiency achieved for varied ${\varphi }_n$ at fixed $n$. Attosecond pulses are found by filtering the reflected spectrum so that $20<\omega /{\omega }_L<100$. For all points, $N=400$, $a_0=40$, and ${\theta }_L=30°$.

Figure 2

(a) Optimal waveform at $N=400$, $a_0=40$, and ${\theta }_L=30°$ for $n=1$, 3, 5. The $n=5$ solution (fundamental and first four harmonics) is the global optimum, with no additional enhancements seen for including six or more frequencies. Inset: Fraction of incident energy contained in each frequency. (b) Corresponding spectra of incident and reflected fields, with ${\omega }^{-8/3}$ scaling indicated for reference.

Figure 3

Attosecond pulse generation efficiency ($I_a/I$) achieved with optimization over $n$ frequencies. Where not otherwise noted, the simulation parameters are $N=400$, $a_0=40$, ${\theta }_L=30$, $L/\lambda =0$, and $\tau =3\mathrm{fs}$, and attosecond pulse intensity is calculated by filtering out frequencies lower than $20{\omega }_L$. (a)–(d) The effect of varying (a) $N$, (b) $a_0$, (c) ${\theta }_L$, and (d) $L/\lambda$. In (d), $a_0=20$ and $N_{\max }=200$.

Figure 4

Relativistic plasma frequency (dashed line) compared to the effective laser frequency (${\omega }_{LE}$) of optimal incident waveform for varied target density to laser amplitude ratio. The effective laser frequency is calculated by taking the time between the zero crossing of the field at $t=0$ (near $t_2$) and the maximum field intensity (at $t_3$) as one-quarter of the effective laser period. Incident waveforms are shown for selected points at $N/a_0=0.5$, 10, and 40. The error bars indicate the range of effective frequencies calculated from waveforms producing within 5% of the maximum efficiency at those parameters. Inset: The trajectory of an emitting electron bunch for normal incidence geometry, with the ion positions marked by the vertical line and the laser incident from the left. Electrons begin near the center of the schematic, are pushed away from the ions between $t_1$ and $t_2$, and then turn before emitting at $t_3$.

Figure 5

Efficiency of attosecond pulse generation (filtered by $\omega /{\omega }_L>20$) for optimized (black squares) and single frequency [$n=1$] (circles) driving waveforms with varied $a_0$ and $N$ at $\theta =30°$. The maximum efficiencies achieved with a restricted number ($n=2–5$) of available frequencies (triangles) illustrates the number of harmonics required to build an optimal waveform as a function of $a_0$ and $N$.

Figure 6

Attosecond pulse trains (filtered by $10<\omega /{\omega }_L<100$) produced by 30 fs (FWHM, field) incident pulses at either (a) $\lambda =0.8\mu \mathrm{m}$ or (b) a mixture of 0.8, 1.6, and $3.2\mu \mathrm{m}$ light with $N=400$, $a_0=40$, and ${\theta }_L=30°$. The left vertical axis is for the incident electric field (light gray region) and its envelope, and the right axis is the scale for the attosecond pulse intensity.