# How does optical “self-torque” make sense in terms of Maxwell's equations?

I saw an article in Science (arXiv:1901.10942) that describes "self-torque" of light.

My understanding is that a series of ultrafast optical pulses with different angular momenta combine in some gaseous medium to give a a new pulse with a time-dependent angular momentum. This phenomena is termed self-torque in the paper.

I have a lot of questions about this, most of which are probably due to the cartoon biasing my understanding of the work.

1. In what sense is it meaningful to call this phenomena self-torque? I always imagine self-torque as a nonlinear phenomenon in interacting systems, but Maxwell's equations in vacuum don't really provide this non-linearity. Is the self-torque aspect really happening in the gaseous medium?

2. Why is there a need to use ultrafast pulses to make a time-dependent orbital angular momentum of light? Or is it that ordinary light can have a time-dependent angular momentum but it is only useful if the time-dependence is on the ultrafast time scale?

3. How do we understand conservation of angular momentum in this scenario? Is it that the atoms in the gas have an equal and opposite "self-torque" after the light pulse has already left?

(Full disclosure: I'm an author in the paper in question.)

1. The name self-torque was chosen because this is a time derivative of an angular momentum, $$\xi = \frac{\mathrm{d} ⟨\ell⟩}{\mathrm d t},$$ which has the dimensions of a torque, but which is already imprinted into the light pulse, so it made a lot of sense. If nothing else, there's definitely no clear contender for how to name this!

It's probably not my place to comment on how strongly this can be identified with previous notions of self-torque in other domains, as I'm not familiar enough with those domains to speak to them beyond what's in the paper $$-$$ this is probably something for domain experts in those fields to comment on.

1. There is no intrinsic need for ultrafast pulses $$-$$ it is perfectly reasonable to think about implementing this in other spatial domains, from quasi-CW lasers in the visible domain all the way down to radio frequencies. (We're hoping that the OAM community will try this, of course, and that they'll help us figure out cool ways to use this in other spectral domains.)

[Edit: they did! Beams with self-torque have now been re-implemented in a separate spectral domain (at the telecomms wavelength, 1.5 $$\mu\rm m$$, with self-torque timescales of the order of $$1\:\rm ns$$, some five orders of magnitude slower than our original work), as reported in Nanophotonics 9, 2957 (2020).]

The simple reason for why we used ultrafast pulses is that the concept developed naturally out of work in this area. Specifically, the configuration that produces self-torqued high-order harmonic emission is a natural extension of the previous work by Laura Rego and the Salamanca team [Phys. Rev. Lett 117, 163202 (2016)] (itself a natural extension of previous OAM work in HHG), where they looked at the HHG produced by a superposition of $$\ell=1$$ and $$\ell=2$$ beams, and showed cool nonlinear effects. To go from there to the self-torque configuration, you just need to add a time delay between the two, which is one of the standard tweaks in ultrafast science.

(That said, once you add in that tweak, you still have to figure out what it is that you've made, and what it means, and how you could possibly measure it. And the experimental campaign to measure it was no walk in the park, either. These were major breakthroughs by the Salamanca and JILA teams, respectively $$-$$ I helped analyze the results of the simulations.)

$$\$$

Still, I would argue that there is a deeper reason why this new perspective came up within HHG and not elsewhere: nonlinear optics, and particularly extreme nonlinear optics, is different. It's bigger. It's harder. It forces you to think in new ways. And it's definitely cooler, of course!

More to the point, the habit of thinking about optics exclusively in terms of monochromatic light very often makes you lose sight, completely, of the time-domain aspect of the system; this gets you some powerful tools, but it also filters out useful information. HHG does not afford us this luxury: we have to think constantly in terms of the time-domain behaviour of light, and this brings to the surface a number of interesting features that are just completely lost by working with monochromatic light or in the frequency domain. Optical self-torque is an excellent example of this.

(And, since I'm in shameless-plug mode: this other recent paper (arXiv) is a similar example, where the time-domain perspective on multi-colour combinations, as developed in HHG for the 'bicircular' fields used to produce circularly-polarized harmonics, helped us uncover a completely new optical singularity with cool new symmetries $$-$$ and, in the process, to break the results that sparked this previous question. And this, again, plays nicely with HHG and its conservation laws (arXiv).)

1. HHG is a messy business. You have a lot of absorption, you have a lot of ionization, and you have a lot of nonlinear optical activity on top of the harmonic-generation process $$-$$ and all of those processes entail their own flows of energy, momentum, and spin and orbital angular momentum.

And yet, despite all that, time and time again we've found that as far as conservation laws go, HHG can be understood perfectly well as a parametric optical process: that is, as a process where the light does its own business and leaves the matter unchanged (ionization notwithstanding!), so that all the conserved-quantity flows happen from the driving field(s) to the harmonics. (See the last paper linked above for an up-to-date review of this aspect.)

In other words: the OAM comes directly from the infrared pulses that drive the harmonic generation. At the start of the process, you have the initial pulse with $$\ell_1=1$$ driving the show, which means that the $$q$$th harmonic comes out with an OAM of $$\ell^{(q)} = q\times \ell_1 = q$$; heuristically, you have $$q$$ photons of frequency $$\omega$$ and OAM $$\ell_1$$ combining to make one photon of frequency $$q\omega$$ and OAM $$q\ell_1$$ (though note that this is a heuristic that doesn't represent a rigorous perturbation-theory backing). Similarly, at the end of the process, the harmonics are driven by the second pulse, with $$\ell_2=2$$, giving the $$q$$th harmonic can OAM of $$\ell^{(q)} = q\times \ell_2 = 2q$$. And in the middle of the pulse, you have a mixture of those processes, which results in the smooth ramp-up of OAM in the emitted harmonics by mixing in photons from both drivers.