Controlling the orbital angular momentum of high harmonic vortices

Optical vortices, which carry orbital angular momentum (OAM), can be flexibly produced and measured with infrared and visible light. Their application is an important research topic for super-resolution imaging, optical communications and quantum optics. However, only a few methods can produce OAM beams in the extreme ultraviolet (XUV) or X-ray, and controlling the OAM on these beams remains challenging. Here we apply wave mixing to a tabletop high-harmonic source, as proposed in our previous work, and control the topological charge (OAM value) of XUV beams. Our technique enables us to produce first-order OAM beams with the smallest possible central intensity null at XUV wavelengths. This work opens a route for carrier-injected laser machining and lithography, which may reach nanometre or even angstrom resolution. Such a light source is also ideal for space communications, both in the classical and quantum regimes.

We make the following assumptions to describe the interaction of a single perturbing photon.
 We consider a single perturbing photon in a mode that optimally overlaps in space (S) and time (t) the driving laser pulse.  We use E=h to estimate a classical energy of a single photon.  We estimate the classical intensity by E/St.  We use the known characteristics of diffraction gratings to estimate the diffracted light, although the diffraction efficiency is insufficient to describe the behavior of a single photon at specific frequency.
Our intension with the estimate below is to motivate the requirement for a quantum theory of extreme nonlinear optics. We are motivated by the potential to use one photon of an entangled pair as the perturbation beam. Of course, quantum optics can be important under much less extreme circumstances 1 .
For our estimate, we will use an 800nm fundamental pulse with 1mJ energy per pulse or N=4×10 15 photons per pulse, 5×10 14 W cm -2 intensity at focus and 10 -4 energy conversion to the sum of all harmonics in neon gas. For convenience we will assume that 100eV (65 th harmonic) can represent the properties of all diffracted photons. These choices define the threshold diffraction efficiency η TXUV =1.67×10 -10 that we will need for the grating to diffract one photon.
The diffraction efficiency of a sinusoidal phase grating is given by, where δ XUV is the phase modulation depth of the emission dipole at XUV wavelength in near field.
Knowing the required modulation, we can estimate the energy ratio between the strong and weak IR beams needed to create it. We define a perturbation parameter ε= E p /E d , where E p and E d are the perturbing and driving electric field respectively.
The phase estimate can be broken into two parts: a. the phase modification due to changes of the electron trajectory caused by amplitude variation of the driving laser field; b. the phase modification due to change of recombination time due to the phase variation of the driving field.
a. For the nonlinear phase that arises due to changes to the electron trajectory, the unperturbed phase of q th harmonics at frequency Ω is 2 The change of action caused by a perturbed field can be evaluated by where t c is time of recombination and t b is time of birth, p is the canonical momentum, I p is the ionization potential of the gas medium, A is the unperturbed vector potential of the driving laser field, A p is the vector potential of the perturbing laser field, E 0 is the amplitude of the driving laser field, ω is the frequency of driving laser field and Δφ is the phase difference between the driving and perturbing laser fields.
b. For the nonlinear phase influenced by the phase variation of the driving field, The total phase shift of the dipole emission equals to the sum of the above two contributions. Since the perturbation is small we only keep the first order term. The coefficient dδ/dε is plotting in Supplementary  Figure 3. The phase modulation is stronger for electrons which recombined later (the higher order harmonics).

Supplementary Figure 3 Phase modulation coefficient dδ/dε in radians for different electron trajectories
Taking the trajectory corresponding to the 65 th order (the dashed line), we can find that the ratio between modulation depth (dδ) and the perturbation parameter (dε) is 95. Therefore the ratio between the weak and strong IR field is