High-order harmonic generation in an active grating

We study theoretically and experimentally high-order harmonic generation (HHG) using two noncollinear driving fields focused in gases. We show that these two fields form a nonstationary blazed active grating in the generation medium. The intensity and phase structure of this grating rule the far-field properties of the emission, such as the relative amplitude of the diffraction orders. Full macroscopic calculations and experiments support this wave-based picture of the process, complementing and extending its standard “photon” picture. This insight into the HHG process allows us to envision structuration schemes to convert femtosecond lasers to attosecond pulses with increased efficiency.

Figure 1

(a) Schematic of HHG with two noncollinear beams. (b) Intensity map close to focus, when the second beam has an amplitude of 20% compared to the main beam. (c) Corresponding electric field, with a zoom over the white rectangle plotted in the inset. The first beam is propagating along $z$, and the second $z^{\prime }$.

Figure 2

(a) Left column: electric field at focus resulting from the superposition of two beams making an angle of $2^{\circ }$ with relative amplitude of 1:0.05 (top line), 1:0.2 (middle line), and 1:0.8 (bottom line). One beam is horizontal; the second, less intense is coming from the top left of the figure. Middle and right columns: modulus and phase of the associated complex analytical signal. The color bars are the same for all columns.

Figure 3

Wave-vector modulus (a) and angle with respect to horizontal (b) across the transverse direction ($x$) for different ratios of the two fields: 1:0.05 (blue), 1:0.2 (orange), 1:0.4 (green), 1:0.6 (red), 1:0.8 (purple), and 1:1 (brown). The data corresponding to amplitudes of the analytical signal below 10% of its maximum has been discarded. (c),(d) Histograms of the distributions of $\frac{|k_s|-|k_1|}{|k_1|}$ and ${\theta }_s$ for the case of $\alpha =0.6$. Each vertical cut is a histogram for a given field amplitude $\tilde{E}$ giving finally these 2D histograms. The amplitude of the analytical signal ${\tilde{E}}_s$ has been normalized to 1.

Figure 4

(a) Electric field at focus resulting from the superposition of two beams making an angle of $2^{\circ }$ with relative amplitude of 1:0.4. The two waists are $100\mu \text{m}$, durations of 30 fs, $\lambda =800\mathrm{nm}$. (b) Amplitude and (c) phase of the associated analytical signal. (d) Relative amplitude $\left[\right|k_s\left(\vec{r}\right)|-k_1$ in ${10}^4{\mathrm{m}}^{-1}]$ and (e) inclination angle (in degrees) of the local wave vector. For maps (c)–(e) all data corresponding to magnitudes of the analytical signal below 10% of its maximum have been discarded.

Figure 5

(a),(b) Cuts through the maps displayed in Figs. 4 and 4 at $z=0$ (orange, solid), $z=5\lambda$ (blue dotted), and $z=-5\lambda$ (green, dash-dotted) along the transverse direction. For ${\theta }_s$, negative inclination angle means propagation toward the bottom right of the figure, i.e., SFG processes with our convention. (c),(d) Two-dimensional histograms corresponding to Figs. 4 and 4, like in Fig. 3.

Figure 6

Same histograms as in Fig. 5 for different ratios ($\alpha =0,0.2,0.4,0.6,0.8,1.0$ from left to right). The light-blue dashed line is at $\theta ·\alpha /(1+\alpha )$.

Figure 7

Intensities at $z=0$ (a)–(d) and in the far field (e)–(h) for $\alpha =0$ (blue), $\alpha =0.2$ (orange), $\alpha =0.6$ (green), and $\alpha =1.0$ (red). We consider H15. For the sake of visibility the curves are all normalized to 1 and shifted respectively by 0, 0.4, 0.8, and 1.2. (a) and (b) correspond to cuts analogous to the orange curve in Fig. 5. (c) is the amplitude of the ionization rate computed with the ADK model. (d) is a toy model corresponding to a series of Gaussian amplitude with periodicity $\mathrm{\Lambda }=1/\mathrm{\Delta }k$, of width $\mathrm{\Lambda }/5$ with a Gaussian global envelope of width $60\mu \text{m}$. The phase is here set according to Eq. (22). (e)–(h) Amplitudes of the Fourier transforms of (a)–(d). In (e), the phase of the signal is set to zero; in (f) and (g) the phases computed in Fig. 5 are plugged; in (h) we took the analytical formula of Eq. (24). The results are offset by the same quantities and the color code is the same on all plots. The distance is $z=0.3$ m. In (g) the black dotted line corresponds to the same as the red, but setting the phase to 0.

Figure 8

Top line: Amplitude (orange) and phase (purple) of harmonic 15 (a) and harmonic 27 (b) at the exit of the generating medium calculated with the model described in Sec. 4 versus the transverse direction $x$. The left (respectively, right) scale is used for amplitude (respectively, phase) for both panels. In panel (b), the amplitude of the driving field is displayed as a dashed line. It is offset by $2\times {10}^8\text{V}/\text{cm}$ and divided by ${10}^4$. Bottom line: Values of the derivative of the phase against $x$. In both plots various colors are results taken for successive maxima of the XUV field (blue: $x\simeq -40\mu \text{m}$; orange: $x\simeq 0$; green: $x\simeq 40\mu \text{m}$; red: $x\simeq 80\mu \text{m}$). The mean value is displayed in purple. (c) Variation of the blaze angle against the field ratio for harmonic $q=15$. The dashed line is the analytical formula given by Eq. (24). (d) Variation of the blaze angle against the harmonic number for $\alpha =0.25$.

Figure 9

Color maps of the intensities in the far field of harmonics 15 [(a),(c)] and 27 [(b),(d)] for perturbations $\alpha =0.18$ [(a),(b)] and $\alpha =0.5$ [(c),(d)]. All maps are individually normalized and the color maps are set equal. Cuts of the harmonic intensities 15 (e) and 27 (f) at ${\theta }_y=0$ for $\alpha =0.18$ (blue, no offset), $\alpha =0.25$ (orange, $\text{offset}=0.2$), $\alpha =0.31$ (green, $\text{offset}=0.4$), $\alpha =0.35$ (red, $\text{offset}=0.6$), $\alpha =0.43$ (purple, $\text{offset}=0.8$), and $\alpha =0.5$ (brown, $\text{offset}=1.0$). The successive cuts are offset by 0.2 for the sake of visibility. The intensities of H15 are divided by 2 to share the $y$ axis with H27.

Figure 10

Left column: Experimental images on the detector for five levels of perturbation $[\alpha \lesssim 0.15, \alpha =0.25, \alpha =0.38, \alpha =0.50$, and $\alpha =0.66$ from (a) to (e)]. Here $\theta =1^{\circ }$. All images are normalized to 1, and share the same color map. The vertical axis is the direction of the dispersion of the detector grating, while the horizontal axis is the direction of dispersion of the “active” grating. Right column: Lineouts corresponding to the sums between the dashed lines displayed in panel (a), for H9 (red, $\text{offset}=0.9$), H11 (green, $\text{offset}=0.6$), H13 (orange, $\text{offset}=0.3$), and H15 (blue, no offset). All curves are normalized and offset by 0.3 for the sake of visibility.

Figure 11

Lineouts of experimental images similar to those of Fig. 10, but for an angle $\theta =1.2^{\circ }$. Each panel corresponds to a given harmonic and all curves are normalized to 1 and progressively offset by 0.3. Different colors correspond to perturbation levels of $\alpha \lesssim 0.15$ (blue, no offset), $\alpha =0.25$ (orange,$\text{offset}=0.3$), $\alpha =0.38$ (green, $\text{offset}=0.6$), $\alpha =0.50$ (red, $\text{offset}=0.9$), and $\alpha =0.66$ (mauve, $\text{offset}=1.2$). The dashed vertical line is set at the location of the direct harmonics.

Figure 12

Normalized position of the dominant diffraction order as a function of the perturbation energy normalized to the energy in the main beam $[{\alpha }^2={\left(E_2/E_1\right)}^2]$. Experimental values for angles $\theta =13$ mrad (a) and $\theta =17$ mrad (b), and theoretical values for $\theta =20$ mrad (c). Each color/symbol corresponds to one harmonic [for (a),(b) H9: blue circles, H11: orange squares, H13: green diamonds, H15: red crosses; for (c) H15: brown squares, H27: light-green circles]. The dashed line is the analytical prediction according to Eq. (31), taking into account all magnification coefficients of our grating.