Simulation of terahertz generation in corrugated plasma waveguides

We simulate the response of a corrugated plasma channel to an ultrashort laser pulse in two dimensions with the goal of demonstrating the production of terahertz frequency electromagnetic modes. Corrugated channels support electromagnetic modes that have a Floquet-type dispersion relation and thus consist of a sum of spatial harmonics with subluminal phase velocities. This allows the possibility of phase matching between the ponderomotive potential associated with the laser pulse and the electromagnetic modes of the channel. Since the bandwidth of an ultrashort pulse includes terahertz frequencies, significant excitation of terahertz radiation is possible. Here we consider realistic density profiles to obtain predictions of the terahertz power output and mode structure for a channel with periodic boundary conditions. We then estimate pulse depletion effects from our simulation results. The fraction of laser energy converted to terahertz is independent of laser pulse energy in the linear regime, and we find it to be around 1%. Extrapolating to a pulse energy of 0.5 J gives a terahertz power output of 6 mJ with a pulse depletion length of less than 20 cm.

Figure 1

(a) Diagram of experimental setup for producing a corrugated plasma channel [14, 15]. As an alternative to a spatially modulated formation pulse, a modulated cluster density may be used in conjunction with a uniform formation pulse. (b) Dispersion plot of EM modes in the $\delta$-function corrugated channel (considered by Antonsen et al. [14]), which consists of a channel with a density profile that has a train of $\delta$ functions, thus creating a period system while allowing the use of analytic results from the axially uniform case. Here, $d$ is the distance between consecutive $\delta$ functions, while $\omega$ and $k_c$ are the frequency and wavenumber, respectively.

Figure 2

Plot of normalized band-gap size as a function of normalized central band-gap frequency, obtained from the numerical calculation of the dispersion relation, performed in the Appendix. Note that the band-gap size vanishes rapidly with increasing frequency, at a point far below the typical frequencies associated with an optical pulse.

Figure 3

Plots of the Fourier transforms of 50-fs laser pulses with wavelength 800 nm and spot size 15 $\mu$m, recorded initially (red, solid) and after (orange, circles) propagation for 40 Rayleigh lengths for normalized vector potential amplitudes of (a) $a_0=0.2$, (b) $a_0=0.4$, and (c) $a_0=0.8$. These potentials correspond to pulse energies of 0.03, 0.1, and 0.5 J, respectively. Plot (d) is for $a_0=0.8$ for a pulse propagating over one Rayleigh length. These plots were generated using the simulation WAKE [20].

Figure 4

Plots from a simulation of a laser pulse passing through a uniform plasma showing (a) density perturbation $\delta n/n_0$ calculated analytically (red, solid) and numerically (orange, circles) for a uniform plasma and (b) difference between power input and output as a function of normalized stepsize squared. Plots of ${\mathrm{Ẽ}}_z(k_z,\omega )$ from a simulation of a laser pulse passing through a parabolic plasma channel evaluated for (c) $v_g=c$ at $r=0$ and (d) for $v_g=2c$ at $r>r_0$. The dotted blue lines are the dispersion curves for plasma waves and the first three radial eigenmodes.

Figure 5

Axial and radial power flow density in frequency space for (a) uniform plasma, (b) $\delta =0.05$ axial corrugations but no radial density dependence, (c) a finite radius plasma channel with no axial corrugations, and (d) a finite radius plasma channel with $\delta =0.05$ axial corrugations.

Figure 6

Plots involving various quantities as a function of density for channel widths $w_{\mathrm{ch}}=15$ $\mu$m (blue, solid), $w_{\mathrm{ch}}=25\mu$m (red, dashed), $w_{\mathrm{ch}}=50$ $\mu$m (green, dash-dot), and $w_{\mathrm{ch}}=75$ $\mu$m (magenta, dotted). The quantities are (a) average power flow in the radial direction, (b) average power flow in the axial direction, (c) percentage of laser energy converted to terahertz, and (d) angle between the Poynting vector measured outside the channel and the axis.

Figure 7

Plots involving various quantities as a function of pulse duration for density $n_0=1.25\times {10}^{18}$ cm${}^{-3}$ and channel width $w_{\mathrm{ch}}=25$ $\mu$m. The quantities are (a) average power flow in the radial direction, (b) average power transferred from the laser pulse to the plasma, (c) percentage of laser energy converted to terahertz and (d) average power transferred from the laser pulse to the plasma rescaled for fixed peak ponderomotive potential. Note that at $\tau =50$ fs, these quantities do not match the results in Fig. 6 because the simulation length was shorter, and the damping rate $\nu$ was necessarily larger.

Figure 8

Average radial power spectral density for different cutoff radii and for corrugated channels with mode width (a) $w_{\mathrm{ch}}=15$ $\mu$m and (b) $w_{\mathrm{ch}}=50$ $\mu$m. The topmost plots have $r_c=1.5w_{\mathrm{ch}}$, and the cutoff radius increases by $0.5w_{\mathrm{ch}}$ for each successive plot. The subfigure to the right of each power plot displays a $z$-averaged radial density profile.

Figure 9

Two-dimensional Fourier transforms of $E_z$ taken at fixed radius outside the channel for (a) $w_{\mathrm{ch}}=15$ $\mu$m and (b) $w_{\mathrm{ch}}=50$ $\mu$m. The red (solid) line is the lightline of the laser pulse, and the blue (dotted) curves are the functions in Eq. (12) that constitute the approximate dispersion relation.

Figure 10

Plots of power density in frequency space as a function of density for channel widths (a) $w_{\mathrm{ch}}=15$ $\mu$m, (b) $w_{\mathrm{ch}}=25$ $\mu$m, (c) $w_{\mathrm{ch}}=50$ $\mu$m, and (d) $w_{\mathrm{ch}}=75$ $\mu$m. The orange (dashed) lines indicate the first five Floquet modes of the fundamental radial eigenmode, as predicted by the small $\delta$ theory.

Figure 11

Log plots of power density in frequency space as a function of density for channel widths (a) $w_{\mathrm{ch}}=50$ $\mu$m and (b) $w_{\mathrm{ch}}=75$ $\mu$m. Comparing this with Fig. 10 verifies that the frequency predictions are accurate for large mode widths.

Figure 12

Dispersion curves (blue, dotted) generated by evaluating a finite-sized version of the determinant shown in Eq. (A1). Fig. (a) contains the dispersion construction discussed in Sec. 2, which is the dispersion relation from a single-element determinant reproduced many times. The remaining figures contain dispersion curves calculated for (b) 3, (c) 9, and (d) 15 nonzero Fourier coefficients. The lightline (red, solid) is displayed also.

Figure 13

Frequencies of lightline intersections with the dispersion curves in Fig. 12a (orange, dashed) and Fig. 12b (red, solid) for (a) $\delta =0.05$ and (b) $\delta =0.9.$