Different time scales in plasmonically enhanced high-order-harmonic generation

We investigate high-order-harmonic generation in inhomogeneous media for reduced dimensionality models. We perform a phase-space analysis, in which we identify specific features caused by the field inhomogeneity. We compute high-order-harmonic spectra using the numerical solution of the time-dependent Schrödinger equation, and provide an interpretation in terms of classical electron trajectories. We show that the dynamics of the system can be described by the interplay of high-frequency and slow-frequency oscillations, which are given by Mathieu's equations. The latter oscillations lead to an increase in the cutoff energy, and, for small values of the inhomogeneity parameter, take place over many driving-field cycles. In this case, the two processes can be decoupled and the oscillations can be described analytically.

Figure 1

Effective potentials $V_{\mathrm{eff}}(x,t)$ for spatially inhomogeneous fields, considering several values of $\beta , {\mathcal{E}}_t=0.07\mathrm{a}.\mathrm{u}.$ and ${\mathcal{E}}_t=-0.07\mathrm{a}.\mathrm{u}.$ [(a) and (b), respectively], together with the phase portraits for the Hamiltonian (1) and $\beta =0.02$ (bottom panels). In (c) and (d), we consider the field to be ${\mathcal{E}}_t=0.07\mathrm{a}.\mathrm{u}.$, and ${\mathcal{E}}_t=-0.07$ a.u., respectively. The separatrices are given by the red lines in the figure, and the numbers near each contour denote the corresponding total energy of the system. The Stark saddle coordinate $x_S$ and the fixed point $x_{\beta }$ due to the inhomogeneity are indicated in the figure. The contours in blue are related to the energies lower than that of the Stark saddle. The red dashed lines give the separatrices for the homogeneous case $\beta =0$, which occur at at energy $E_{\mathrm{sep}}=-0.52\mathrm{a}.\mathrm{u}.$ The black dashed lines give the phase-space trajectory for $E=-0.5\mathrm{a}.\mathrm{u}.$ and $\beta =0$.

Figure 2

Equilibrium curves as functions of the instantaneous field value ${\mathcal{E}}_t$, for different values of the inhomogeneity parameter $\beta$. The dashed lines correspond to the Stark saddle, while the dot-dashed and solid lines give the saddle and center caused by the inhomogeneity, respectively. The fine short-dashed lines give the asymptotic value $x_{\beta }=-1/\left(2\beta \right)$, which will be reached for ${\mathcal{E}}_t\to \infty$.

Figure 3

HHG spectra computed from the dipole acceleration (9) for different values of the inhomogeneity parameter $\beta$ (solid line) and for the homogeneous case (gray dots). The external field is given by Eq. (3), and its temporal part by Eq. (4), with frequency $\omega =0.05\mathrm{a}.\mathrm{u}.$, amplitude $E_0=0.075\mathrm{a}.\mathrm{u}.$ and phase $\varphi =-\pi /2$. (a), (b), and (c) correspond to inhomogeneity parameters $\beta =0.002, \beta =0.005$, and $\beta =0.01$, respectively. The pulse duration is six cycles for $\beta =0.002$ and $\beta =0.005$, and five cycles for $\beta =0.01$. The cutoff harmonics are indicated by the arrows in the figure, and have been obtained from classical-trajectory computations. Notice the absence of the cutoff in the panel corresponding to $\beta =0.01$, and that the set of harmonics between 230 and 300 in (b) are damped following a ramplike structure instead of a sharp cutoff.

Figure 4

Time-frequency maps and classical returning times (superimposed dots) as functions of the harmonic order and the field cycles, obtained with inhomogeneity parameters (a) $\beta =0.002$, (b) $\beta =0.005$, and (c) $\beta =0.01$. The kinetic energy of the electron upon return is related to the harmonic frequency by $E_{\mathrm{kin}}\left(t_R\right)=\mathrm{\Omega }-I_p$, where $I_p$ is the ionization potential. The remaining field and atomic parameters are the same as in the previous figure.

Figure 5

Blowups of the time-frequency maps in Fig. 4 and 4, together with the classical returning times (superimposed dots). (a) and (b) correspond to $\beta =0.002$, while (c) and (d) have been computed for $\beta =0.005$. The left and right panels give the Gabor spectra for $0.5T\le t\le 2.5T$ and $2T\le t\le 5T$, respectively, where $T=2\pi /\omega$ is the field cycle. The arrows in the figure, whose styles have been matched with those in Fig. 3, indicate the cutoff energies.

Figure 6

Fourier spectra and time frequency maps (top and bottom panels, respectively) computed for $\beta =0.002$, and the same field parameters in Fig. 3, but using the two symmetric potentials defined in Eq. (16). (a) and (c) correspond to the two-saddle potential ${\tilde{V}}_l^{(-)}(x,t)$, while (b) and (d) have been calculated using the two-center potential ${\tilde{V}}_l^{(+)}(x,t)$.

Figure 7

Numerical solutions of Mathieu's equation (dashed line) in (a) phase space and as a position-time plot for (b) $\beta =0.002$, (c) $\beta =0.001$, and (d) $\beta =0.0005$. The initial positions and momenta in the three cases are $x\left(0\right)=0$ and $p\left(0\right)=0$, respectively. For reference, the continuous line in (a) shows a closed orbit resulting from the propagation under a homogeneous laser field. Continuous lines in (b), (c), and (d) represent Dehmelt's approximation to Mathieu's equation.

Figure 8

Phase-space trajectories starting at $x\left(0\right)=-3.65\mathrm{a}.\mathrm{u}.$ and $p\left(0\right)=0.05\mathrm{a}.\mathrm{u}.$ for a field of amplitude $E_0=0.075\mathrm{a}.\mathrm{u}.$, and frequency $\omega =0.05\mathrm{a}.\mathrm{u}.$ (a) and (b) correspond to inhomogeneity parameters $\beta =0.002$ and $\beta =0.012$, respectively. The propagation time employed for $\beta =0.002$ was $t=11T$, while for $\beta =0.012$ we used $t=2T$, where $T=2\pi /\omega$ is a cycle of the laser field. The dashed (blue) lines have been computed without the atomic potentials, while in the solid (gray) lines $V_a\left(x\right)$ has been included.

Figure 9

Time-frequency maps computed for $\beta =0.001$ (top) and $\beta =0.002$ (bottom), together with the kinetic energies of trajectories returning to the core for $\beta =0.001$. Here, the dotted (white) line indicates the analytical result obtained by inserting (24) into the time derivative of Dehmelt's approximation in (22) whereas the black dots show the results obtained by evolving the ensemble of classical trajectories placed initially at the core. In both, the analytical approximation and the ensemble of trajectories the initial conditions, at the initial time $t_0=0$, are sampled from a Gaussian distribution with parameter width $\gamma =0.5$.