Attosecond lighthouses in gases: A theoretical and numerical study

We present an extensive theoretical and numerical study of the attosecond lighthouse effect in gases. We study how this scheme impacts the spatiotemporal structure of the driving laser field all along the generation medium, and show that this can modify the phase matching relation governing high-harmonic generation (HHG) in gases. We then present a set of numerical simulations performed to test the robustness of the effect against variations of HHG parameters, and to identify possible solutions for relaxing the constraint on the driving laser pulse duration. We thus demonstrate that the lighthouse effect can actually be achieved with laser pulses consisting of up to $\sim 8$ optical periods available from current lasers without postcompression, for instance by using an appropriate combination of $800\text{−}$ and $1600\text{-nm}$ wavelength fields.

Figure 1

(a) WFR velocity as a function of STC parameter $\xi$ calculated for ${\tau }_L=5\text{fs}$ (black squares), $10\text{fs}$ (empty circles), and $20\text{fs}$ (black stars). (b) Normalized WFR velocity as a function of the distance to the focus $z_1=z-z_f$ (in units of $z_R$), for ${\tau }_L=5\text{-fs}$ and for $\xi =2/3\times {\xi }_{\text{max}}$ (dashed green line), ${\xi }_{\text{max}}$ (solid red line) and $3/2\times {\xi }_{\text{max}}$ (dotted blue line). The corresponding WFR velocities in $z_1=0$ are indicated by the vertical lines in (a). (c) Variation of the phase with the distance to the focus in $x=w_0$, with the same parameters and same color codes and symbols as in (b). The curve with black circles depicts the phase of a focused regular Gaussian beam.

Figure 2

Space-time laser electric-field distribution calculated for ${\tau }_L=5\text{fs}$, $\xi =0.17\mathrm{fs}/\mathrm{mm}$, and $z_R=400\mu \text{m}$. (a) Distributions at focus and (b) after propagation on distances of $0.5\times z_R$, (c) $1.0\times z_R$, and (d) $2.5\times z_R$. Time is normalized to the laser period $T_L$.

Figure 3

Far-field space-time square modulus of the attosecond pulse electric fields generated by a laser beam (a) without WFR $(\xi =0)$ and (b) using pulse parameters of Fig. 2. In both (a) and (b), the peak intensity in the middle of the jet is $8.65\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$. (c) Temporal intensity profile of the attosecond pulses obtained by integrating the signal between the white lines in (a) and (b). The pulse duration of individual attosecond pulses is $160\text{as}$ (FWHM) without WFR (dashed line) and $240\text{as}$ with WFR (solid line).

Figure 4

Comparison of the HH time-integrated far-field angular intensity profile (black line), and the spatially integrated attosecond pulse train temporal intensity profile (gray line). Negative $\theta$ (upper scale) values correspond to early times in the pulse. The correspondence between $\theta$ and $t$ is given by $\theta =v_r^{\text{max}}t$, with $v_r^{\text{max}}=3\times {10}^{-3}\mathrm{rad}/\mathrm{fs}$.

Figure 5

Evolution of HH (relative) far-field intensity profile with focus position for (a) $z_f/z_R=-1.25$, (b) $-0.5$, (c) 0.0, and (d) $+0.5$. Negative values of $z_f/z_R$ correspond to focus positions in front of the gas target. Normalization is performed with respect to our reference case, displayed in (b). See text for details.

Figure 6

Influence of the medium length on HH far-field intensity profile for $I_L(x=z=0)=I_{\text{sat}}$, ${\tau }_L=5\text{fs}$, $P=10\text{mbar}$, and (a) $L_{\text{med}}/z_R=0.25$ (reference case), (b) 1.25, and (c) 2.5.

Figure 7

Evolution with focal length of HH far-field intensity profile. $I_L(x=z=0)=I_{\text{sat}}$, ${\tau }_L=5\text{fs}$, and $P=10\text{mbar}$. Black curve: $f=1\text{m}$ (reference case). Gray curve: $f=2\text{m}$.

Figure 8

HH far-field intensity profile calculated for $P=10\text{mbar}$, ${\tau }_L=5\text{fs}$, and (a) $I_L(x=z=0)=0.6\times I_{\text{sat}}$, (b) $I_{\text{sat}}$ (reference case), (c) $2\times I_{\text{sat}}$, and (d) $4\times I_{\text{sat}}$. The arrow indicates the time direction. Panels (e) and (f) display the spatiotemporal intensity distributions of the attosecond pulses in the middle of the gas medium (in $z=0$) for $I_L(x=z=0)=I_{\text{sat}}$ [case of panel (b)] and $2\times I_{\text{sat}}$ [case of panel (c)], respectively.

Figure 9

Evolution with pressure of HH far-field intensity profile. $I_L(x=z=0)=I_{\text{sat}}$ and ${\tau }_L=5\text{fs}$. (a) $P=10\text{mbar}$ (reference case). (b) $P=20\text{mbar}$. (c) $P=80\text{mbar}$.

Figure 10

Dependence of HH far-field intensity profile on STC parameter. $I_L(x=z=0)=I_{\text{sat}}$ and $P=10\text{mbar}$. (a) $\xi =\frac{2}{3}\times {\xi }_{\text{max}}$. (b) $\xi ={\xi }_{\text{max}}$ (reference case). (c) $\xi =\frac{3}{2}\times {\xi }_{\text{max}}$.

Figure 11

HH far-field intensity profiles calculated for ${\tau }_L=5\text{fs}$, $\xi =0.17\mathrm{fs}/\mathrm{mm}$ (black line, reference case), and ${\tau }_L=10\text{fs}$, $\xi =0.34\mathrm{fs}/\mathrm{mm}$ (gray line). $I_L(x=z=0)=I_{\text{sat}}$ and $P=10\text{mbar}$.

Figure 12

Influence of spatial laser beam shape on HH far-field intensity profile. $I_L(x=z=0)=I_{\text{sat}}$ and $P=10\text{mbar}$. Black line: Gaussian (reference case). Gray line: flat-top.

Figure 13

Comparison between the HH intensity far-field profile obtained by mixing 400 and $800\text{-nm}$ wavelength fields of $10\text{-fs}$ duration $(\xi =0.34\mathrm{fs}/\mathrm{mm})$ (gray line) and our reference case (black line). $I_L({\lambda }_L=800\text{nm})=I_{\text{sat}}$ and $I_L({\lambda }_L=400\text{nm})=0.1\times I_{\text{sat}}$ in $x=z=0$.

Figure 14

HH intensity far-field profiles calculated for a ${\tau }_L=5\text{-fs}$, ${\lambda }_L=800\text{-nm}$ laser field (a) $(f=2\text{m})$, (b) a ${\tau }_L=10\text{-fs}$, ${\lambda }_L=1600\text{-nm}$ field $(\xi =0.34\mathrm{fs}/\mathrm{mm})$, and (c) a $20\text{-fs}$ duration bichromatic field obtained by mixing 800 and $1600\text{-nm}$ wavelength $(\xi =0.68\mathrm{fs}/\mathrm{mm})$. In the first two cases, $I_L=I_{\text{sat}}$ in $x=z=0$, while in the latter one, $I_L({\lambda }_L=800\text{nm})=0.1\times I_{\text{sat}}$.