Creating High-Harmonic Beams with Controlled Orbital Angular Momentum

A beam with an angular-dependant phase $\mathrm{\Phi }=\ell \varphi$ about the beam axis carries an orbital angular momentum of $\ell \hslash$ per photon. Such beams are exploited to provide superresolution in microscopy. Creating extreme ultraviolet or soft-x-ray beams with controllable orbital angular momentum is a critical step towards extending superresolution to much higher spatial resolution. We show that orbital angular momentum is conserved during high-harmonic generation. Experimentally, we use a fundamental beam with $|\ell |=1$ and interferometrically determine that the harmonics each have orbital angular momentum equal to their harmonic number. Theoretically, we show how any small value of orbital angular momentum can be coupled to any harmonic in a controlled manner. Our results open a route to microscopy on the molecular, or even submolecular, scale.

Figure 1

(a) OAM is imparted to the pump with a spatial light modulator. (b) High-order harmonics are generated in argon and (c) measured in a spectrometer. The grating allows us to resolve each harmonic. Moving the slit and grating horizontally allows us to recover the full transverse profile of the harmonics. (d) Intensity profile for the 13th harmonic. All harmonics show the same characteristic intensity profile.

Figure 2

(a) Generating harmonics from two distinct beams at focus creates an OAM beam and a Gaussian reference beam. (b) The interference, in the far field, of the harmonic beam containing OAM and a Gaussian harmonic beam results in a $n$-fork interference pattern as shown in the upper left-hand image, where $n$ is the OAM per photon of the $n$th harmonic. The vertical Fourier transform of such a pattern exhibits increasing frequency from one side of the beam to the other, with $n+1$ steps, as shown in the lower left-hand image. Experimentally, the interference patterns and their Fourier analysis are similar to those predicted. For harmonic 11, we count a difference of 11 fringes between left and right for harmonics produced from a $\ell =\pm 1$ fundamental beam. The charge $\ell$ of the harmonic is reversed when we inverse the charge of the fundamental beam. These results confirm that OAM is conserved in high-harmonic generation.

Figure 3

In the two top figures, we approximately trace the fringes on top and bottom that are linked from left to right. We measure the beam size between these fringes at two $x$ positions (with uncertainty $\pm 0.1\mathrm{mrad}$) and measure the frequency at these positions with the bottom figures (uncertainty $\pm 0.1{\mathrm{mrad}}^{-1}$). (a) For harmonic 13: on the left (${\theta }_x=-3.6\mathrm{mrad}$), the beam size is $\mathrm{\Delta }{\theta }_y=(5.6+5.3)10.90\pm 0.14\mathrm{mrad}$; on the right (${\theta }_x=5\mathrm{mrad}$), the size is $\mathrm{\Delta }{\theta }_y=(4.8+4.3)9.10\pm 0.14\mathrm{mrad}$. The frequency on the left is $2.2{\mathrm{mrad}}^{-1}$ so that the number of fringes is $23.98\pm 1.13$. The frequency on the right is $1.2{\mathrm{mrad}}^{-1}$ so that the number of fringes is $10.92\pm 0.93$. The difference is $13.1\pm 1.5$ fringes. (b) For harmonic 15: on the left (${\theta }_x=-4\mathrm{mrad}$), the beam size is $\mathrm{\Delta }{\theta }_y=(5.7+5.7)11.40\pm 0.14\mathrm{mrad}$; on the right (${\theta }_x=3\mathrm{mrad}$), the size is $\mathrm{\Delta }{\theta }_y=(5.2+4.4)9.60\pm 0.14\mathrm{mrad}$. The frequency on the left is $2.4{\mathrm{mrad}}^{-1}$ so that the number of fringes is $27.36\pm 1.19$. The frequency on the right is $1.3{\mathrm{mrad}}^{-1}$ so that the number of fringes is $12.48\pm 0.98$. The difference is $14.9\pm 1.5$ fringes. In both cases, the results show that OAM is conserved.

Figure 4

(a) Controlling high-harmonic generation with a second-harmonic, $\ell =1$ beam incident at an angle creates a rapidly moving grating at the focal plane, shown here at a given instant. (b) Harmonics are created in the different diffraction orders, from a combination of ($m_{\omega }$, $m_{2\omega }$) photons, where $m_{\omega }$ is the number of fundamental photons and $m_{2\omega }$ is the number of control photons involved in the process. The position at the far field is determined by conservation of linear momentum, and the orbital angular momentum of each harmonic beam is given by $m_{2\omega }$ times the OAM of the control beam.