High-order-harmonic generation driven by pulses with angular spatial chirp

We present and analyze a technique to drive high-order harmonics by laser pulses with an angular spatial chirp. Results of our numerical simulations show that each harmonic is emitted with an angular chirp which scales inversely with the harmonic order and leads to additional control of the spatial and temporal resolution of the spectrum. In particular, the use of angular chirp leads to separation of the harmonics in two dimensions where (i) high spectral resolution can be achieved and (ii) the temporal periodicity of the harmonic pulse trains can be controlled. We show that this technique does not require carrier-envelope-phase stabilization when using few-cycle laser pulses.

Figure 1

(a) Schematic of a beam with transverse chirp at the focal plane, with the transverse spectrum, ${\left|E(x,\lambda )\right|}^2$ (b) and temporal profile, $\text{Re}{\left[E(x,t)\right]}^2$ (c) of the electric field $E$ at the focal plane. The variation of the local frequency across the focal spot leads to wave fronts with surface normal directions that vary from one cycle to the next, which has been applied to obtain the so-called lighthouse effect. (d) Schematic of a beam with angular chirp at the focal plane proposed in this work to drive HHG: A transverse spatially chirped beam is focused, so each frequency beam propagates along a different direction, overlapping at the focal plane. The transverse spectrum ${\left|E(x,\lambda )\right|}^2$ (e) and temporal profile $\text{Re}{\left[E(x,t)\right]}^2$ (f) of the electric field $E$ are shown both at the focal plane, where the beam exhibits a pulse front tilt with wave fronts all perpendicular to the optical axis.

Figure 2

Angular distribution of high-order harmonics driven by a nonchirped (a) and an angularly chirped beam with (b) ${\gamma }_0=16.4$ as and (c) ${\gamma }_0=35.0$ as (other parameters: $\lambda =800$ nm, ${\tau }_p=7.7$ fs, ${\varphi }_{CEO}=0,E_0^2=2\times {10}^{14} {\text{W/cm}}^2$). The dashed pink line represents the fitted angular chirp rate ${\gamma }_q={\gamma }_0/q$. On the right of each panel is shown the angularly integrated (blue line) and the on-axis (dashed gray line) spectrum.

Figure 3

Angular distribution of the high-order harmonics driven by a ${\tau }_p=3.8$ fs pulse, (a) without angular chirp and (b) angularly chirped with ${\gamma }_0=16.4$ as. All other parameters of the beam are as in Fig. 2. On the right of panels (a) and (b1) are shown the angularly integrated (blue line) and the on-axis (dashed gray line) spectrum. For the angularly chirped case, an additional spectrum at 1.75 mrad is shown (orange dashed line). The dashed pink line in panel (b1) represents the model through the expected angular chirp rate ${\gamma }_q={\gamma }_0/q$ for each odd harmonic order. In panel (b2) we present the angular distribution of the attosecond pulse trains after Fourier transform of the spectra in panel (b1) and filtering the low-order harmonics $q\le 9$. Note, that the intensity yield is shown in logarithmic scale in panel (b1) but in linear scale in panel (b2). The periodicity of the attosecond pulse trains detected on axis $\left(T_1\right)$ and at 1.75 mrad off axis $\left(T_2\right)$ is modified by $T_2-T_1=61$ as.

Figure 4

CEP invariance of angularly chirped HHG. Panels (a) and (b) show the electric field at focus $(z=0)$ at three different transversal positions ($x=0$, 5.2, and $10.4 \mu \text{m}$), with ${\varphi }_{CEO}=0$ and ${\varphi }_{CEO}=\pi /2$. (c) HHG spectrum (linear scale) detected at two angles (0 mrad, solid lines; 1.7 mrad, dashed lines), driven by an input angularly chirped beam with ${\tau }_p=3.8$ fs, ${\gamma }_0=16.4$ as, and ${\varphi }_{CEO}=0$ (green) and ${\varphi }_{CEO}=\pi /2$ (pink).