Scaling law from orbital angular momentum conservation in harmonic and high-order harmonic generation driven by spatiotemporal light fields

Abstract

Nonlinear photon upconversion processes driven by diverse forms of structured light are receiving increasing attention. In harmonic and high-order harmonic generation (HG and HHG) with Laguerre-Gauss (LG) beams, linear scaling the driver topological charge (TC) with the harmonic order is equivalent to driver orbital angular momentum (OAM) per photon scaling, and constitutes a proof of OAM conservation. However, with generic driving fields, such as non-LG vortices or spatiotemporal optical vortices, TC and OAM per photon may scale or not in a process in which the OAM is conserved. We find the physical magnitude that scales with generality when the OAM, either longitudinal or transverse, or its intrinsic part, is conserved. This new rule allows for the wealth of phenomena observed in HHG that are unintelligible from the rigid LG rule.

Figures (4)

• Figure 1: (a1) Left: Iso-intensity surface and phase of fundamental pulsed LG $\psi_1(x,y,t',0) = A(\frac{x}{x_0} + i\eta \frac{y}{y_0})e^{-\frac{x^2}{x_0^2}-\frac{y^2}{y_0^2}-\frac{t'^2}{t_0^2}}$ (left) at $\omega_1=2.35$ rad/fs, with $\ell=1$, $x_0 = 0.5$ mm, $y_0 = 0.5$ mm, $t_0 = 200$ fs, $\eta= 1$ and $A$ adjusted to obtain $W = 6.5$ mJ. Right: Iso-intensity surface and phase of the generated second harmonic (right) at $\omega_2 = 2\omega_1 = 4.7$ rad/fs at $z=20$$\mu$m. The effective nonlinear coefficient is $\kappa = 1.466\times10^{-4}$ V$^{-1}$, and $n_1 = n_2 = 1.661$ is assumed. (b1) The same but with fundamental distorted LG by setting the distotion parameter $\eta= 0.2$, and $A$ adjusted to obtain $W = 6.5$ mJ. (a2, b2) l-OAM (solid) and number of photons (dashed) for the fundamental and the harmonic fields as functions of the medium thickness, for the LG and distorted LG. (a3, b3) ratio of converted l-OAM to converted photon number (solid) and l-OAM per photon (dashed) for the fundamental (red) and the harmonic (blue) as functions of the medium thickness, for the LG and distorted LG.
• Figure 2: (a) Left: Iso-intensity surface and phase of the non-elliptical STOV $\psi_1(x,y,t',0) \propto(t'/\eta t_0 +ix/x_{0,f})\exp(t'^2/t_0^2-x^2/x_{0,f}^2-y^2/y_{0,f}^2)\exp[ik_1(x_0^2+y_0^2)/2f]$ at $\omega_1=2.35$ rad/fs, of charge $\ell=1$ and distortion parameter $\eta= 0.4$ at the entrance $z=0$ of the nonlinear medium, obtained by focusing with focal length $f=250$ mm the tilted lobed field $A (t/\eta t_0 + x/x_0)\exp(-x^2/x_0^2-y^2/y_0^2-t'^2/t_0^2)$ with $x_0 = y_0 = 1.0$ mm, $t_0 = 200$ fs, and $A$ adjusted to obtain $W = 6.5$ mJ. The focused STOV width is $x_{0,f}=y_{0,f} =2f/k_1x_0$. Right: Iso-intensity surface and phase of the generated second harmonic at $\omega_2 = 2\omega_1 = 4.7$ rad/fs at $z=20$$\mu$m. The nonlinear coefficient is $\kappa = 1.18\times10^{-5}$ V$^{-1}$ and $n_1 = n_2 = 1.661$. (b, c) Ratio of converted intrinsic t-OAM per converted photon number (solid) and intrinsic t-OAM per photon (dashed) for the fundamental (red) and the second harmonic (blue) as functions of the medium thickness, for the respective choices of the EC and the PC to extract the intrinsic t-OAM.
• Figure 3: Harmonic l-OAM per photon scaling in HHG fitting the scaling law Eq. (\ref{['undepleted2']}), as evaluated from the TSM (blue) and macroscopic SFA approximation (orange). HHG is driven by a 800 nm wavelength pulsed longitudinal vortex $\psi_1(x,y,t') = A(\frac{x}{x_0} + i\eta \frac{y}{y_0})e^{-\frac{x^2}{x_0^2}-\frac{y^2}{y_0^2}-\frac{t'^2}{t_0^2}}$ of TC $\ell =1$, with $A$ such that the peak field intensity is $1.4\times10^{14}\:\textrm{W/cm}^2$, $x_0=y_0=30\:\mu$m and $t_0=26.11$ fs. The vortex is deformed through the parameter $\eta$ as indicated. The intensity of the driving fields are shown in the insets. The driver TC and OAM per photon scaling are shown for comparison.
• Figure 4: Harmonic intrinsic t-OAM per photon scaling in HHG fitting Eq. (\ref{['undepleted2']}), as evaluated from the TSM (blue) and macroscopic SFA approximation (orange). In (a) HHG is driven by the 800 nm tilted-lobed field $\psi(x,y,t')= A (t'/t_0 - x/x_0)e^{-\frac{x^2}{x_0^2}-\frac{y^2}{y_0}-\frac{t'^2}{t_0^2}}e^{\frac{ik_1x^2}{2f}}$, with $x_0=y_0=30\:\mu$m and $t_0=26.11$ fs, and $A$ such that the peak intensity is $1.4\times10^{14}\:\textrm{W/cm}^2$, obtained by focusing a STOV of TC $\ell=1$ with a focal length $f=250$ mm. In (b) and (c) the driver is back- and forward- propagated from the focus distances $z_R/2=k_1x_0^2/4$. See Porras_nano_2025 for analytical expressions. These driving fields are the same as those used in porras2025arxivPRL. The intensity of the driving fields are shown in the insets. The driver TC and OAM per photon scaling are shown for comparison.