Tailored orbital angular momentum in high-order harmonic generation with bicircular Laguerre-Gaussian beams

We report on a method to generate extreme ultraviolet vortices from high-order harmonic generation with two-color counter-rotating Laguerre-Gaussian ($\text{LG}$) beams that carry a well-defined orbital angular momentum (OAM). Our calculations show that the OAM of each harmonic can be directly controlled by the OAM of the incident $\text{LG}$ modes. Furthermore, we show how the incoming $\text{LG}$ modes have to be tailored, in order to generate every possible value of OAM in the emitted harmonics. In addition, we analyze the emitted harmonics with respect to their divergence and find that it decreases with the harmonic order and increases with the OAM of the emitted harmonic.

Figure 1

Scheme of HHG with bicircular $\text{LG}$ beams. The setup is shown in the center of (a). The incident beam is a superposition of a circularly polarized ${\text{LG}}^{\omega ⥁}$ mode with its counter-rotating second harmonic ${\text{LG}}^{2\omega ⥀}$ of the same intensity. The gas target is approximated by a thin layer at the focus of the beam extended over the whole beam, which is indicated by the black dots in the near field. At each point in the gas target harmonics are emitted. The emitted harmonics contribute to the far field. The intensity and phase profile of a ${\text{LG}}_{1,0}$ and a ${\text{LG}}_{2,0}$ mode are displayed in (b), while (c) shows the electric field (dark green) and the vector potential (light green) of a $\omega \text{−}2\omega$ bicircular field. Finally, a sketch of the emitted harmonics in a bicircular field is shown in (d). The harmonics exhibit the polarization of the incident beam with the fundamental frequency $\omega$ (red) followed by a harmonic with the polarization of the incident second harmonic (blue). Every third harmonic is suppressed for a $\omega \text{−}2\omega$ bicircular field.

Figure 2

Far-field phase (columns 1 and 2), intensity (columns 3 and 4) profiles, and a lineout through the intensity profiles (column 5) for $13\text{th}$ and $14\text{th}$ harmonic generated by a superposition of two counter-rotating LG modes ${\text{LG}}_{{\ell }_1,0}^{\omega ⥁}\oplus {\text{LG}}_{{\ell }_2,0}^{2\omega ⥀}$ with different combinations of $({\ell }_1,{\ell }_2)$. Upper row $({\ell }_1,{\ell }_2)=(1,1)$, middle row (1,2), lower row (2,1). For a better comparison of the intensity distributions, the maximum of each intensity profile was normalized to 1.