Carrier-wave steepened pulses and gradient-gated high-order harmonic generation

We show how to optimize the process of high-order harmonic generation (HHG) by gating the interaction using the field gradient of the driving pulse. Since maximized field gradients are efficiently generated by self-steepening processes, we first present a generalized theory of optical carrier-wave self-steepened (CSS) pulses. This goes beyond existing treatments, which only consider third-order nonlinearity, and has the advantage of describing pulses whose wave forms have a range of symmetry properties. Although a fertile field for theoretical work, CSS pulses are difficult to realize experimentally because of the deleterious effect of dispersion. We therefore consider synthesizing CSS-like profiles using a suitably phased subset of the harmonics present in a true CSS wave form. Using standard theoretical models of HHG, we show that the presence of gradient-maximized regions on the wave forms can raise the spectral cutoff and so yield shorter attosecond pulses. We study how the quality of the attosecond bursts created by spectral filtering depends on the number of harmonics included in the driving pulse.

Figure 1

Self-steepening at the LDD shocking distance for (a) ${\chi }^{\left(2\right)}$, (b) ${\chi }^{\left(3\right)}$, and (c) ${\chi }^{\left(4\right)}$ nonlinearities. Although a material with only a ${\chi }^{\left(4\right)}$ nonlinearity is not realistic, its inclusion helps illustrate some important features. We can test inversion symmetry by replacing $E$ with $-E$. The initial pulse has $E=E_0\text{sin}\left(\omega t+{\varphi }_1\right)\text{sech}\left(0.28\omega t/\tau \right)$, with a wavelength of 800 nm, ${\varphi }_1=0$, and $\tau =1$; $n_0=1$. The nonlinear strength is ${\chi }^{\left(n\right)}{E}^{n-1}=0.02$.

Figure 2

(Color online) Logarithmic plot of $E{\left(\omega \right)}^2$ every half micron, approaching $S_2$ in a ${\chi }^{\left(2\right)}$ material. The spectra flatten out as increasing amounts of ever higher-order harmonics are generated as the pulse approaches the shock. The pulse and material parameters are as for Fig. 1. Each ordinate division represents five orders of magnitude.

Figure 3

MOC shocking distances $S_2$ as a function of CEP $\varphi$, allowing for different pulse widths. In the long pulse $\left(\tau =28\right)$ case, a shallow minimum centered around ${\varphi }_1=\pi /2$ is just visible, caused by the term in the square root of Eq. (2). Apart from the varying pulse length, the pulse and material parameters are as for Fig. 1.

Figure 4

(Color online) Wavelet transform of the HHG from a normal driving pulse with $\lambda =800\text{nm}$, an 8-fs FWHM with peak intensity $5\times {10}^{14}\text{W}/{\text{cm}}^2$. Since the driving pulse retains inversion symmetry, high harmonics with a similar distribution are generated every half-cycle. Note that the center of the pulse produces the bulk of the HHG, which [following Eq. (5)] depends on $E^2$. The 1D TDSE model of HHG was used to generate this figure.

Figure 5

(Color online) Wavelet transform of the HHG from a synthesized ${\chi }^{\left(2\right)}$ CSS-like driving pulse using amplitudes and phases from Eqs. (3, 4) with $F=4$ and spectral contributions above the sixth harmonic removed. The energy and other relevant characteristics of the driving pulse are same as for Fig. 4. It clearly shows the alternating emission from each half cycle of the steepened pulse. The 1D TDSE model of the HHG process was used to generate this figure.

Figure 6

(Color online) HHG spectra for different driving pulses, offset vertically for clarity. Each ordinate division represents four orders of magnitude. The pulse energy and other relevant characteristics are the same as for Fig. 4. The synthesized ${\chi }^{\left(2\right)}$ FMS and CSS pulses contain second, third, and fourth harmonics; the synthesized ${\chi }^{\left(3\right)}$ CSS pulse contains third and fifth harmonics. Bottom curve to top curve: normal (red), ${\chi }^{\left(2\right)}$ FMS (blue), ${\chi }^{\left(2\right)}$ CSS (black), and ${\chi }^{\left(3\right)}$ CSS (magenta).

Figure 7

(Color online) xuv burst intensity profiles and FWHM resulting from high-pass-filtered HHG spectra for the various CSS driving pulses seen in Fig. 6. Bottom curve to top curve: normal (red), ${\chi }^{\left(2\right)}$ FMS (blue), ${\chi }^{\left(2\right)}$ CSS (black), and ${\chi }^{\left(3\right)}$ CSS (magenta). The curves have been individually scaled to aid visibility.

Figure 8

(Color online) xuv burst durations and intensities resulting from high-pass-filtered HHG spectra for synthesized ${\chi }^{\left(2\right)}$ CSS-like driving pulses, as a function of the maximum harmonic component included. The burst durations are denoted with $\times$ and use the left-hand scale; the intensities are denoted with ○ and use the right-hand scale. The 1D TDSE model was used.

Figure 9

(Color online) xuv burst durations and intensities resulting from high-pass-filtered HHG spectra for synthesized ${\chi }^{\left(3\right)}$ CSS-like driving pulses, as a function of the maximum harmonic component included. The burst durations are denoted with $\times$ and use the left-hand scale; the intensities are denoted with ○ and use the right-hand scale. The 1D TDSE model was used.

Figure 10

(Color online) Upper frame: the vector potential $A\left(t\right)$ corresponding to the field profile shown in the lower frame. Note how the peaks in $A\left(t\right)$ correspond to the regions of high field gradient. Lower frame: the trajectories of only those high-energy ionized electrons that recollide with the nucleus to emit high-order harmonic radiation, calculated using the SFA model [27]. The dashed (red) line shows the CSS field profile, in this case for the center of a three-cycle pulse with peak intensity $5\times {10}^{14}\text{W}/{\text{cm}}^2$. The CSS-enhanced trajectories (starting at $A$, on the left) recollide over a short period of time at high field gradients, giving rise to a brief xuv emission with larger cutoff energy. The latter set of recolliding trajectories (starting at $B$, right) have a lower energy and recollide over a longer time interval at low field gradients.