An introduction to twisted light

Twisted light is light carrying orbital angular momentum. At first sight, the idea of light carrying orbital angular momentum sounds intriguing. Perhaps, it is because the words trigger in us memories of very different physical entities, taught to us many times. Who can deny the ubiquity and importance of "orbital angular momentum" and "light"? However, at the same time these concepts may seem irreconcilable.

But, can you remember the ideas of multipolar transitions in nuclear physics? Despite the fact that those memories may lie far in our student past, we were told that atoms may emit waves having angular momentum.

Also, we know that light is made of photons, and photons possess spin, and spin is an angular momentum...though not an orbital one!

So, the idea of light having orbital angular momentum is actually not that strange to us. However, what was shown in the 90s was that light carrying orbital angular momentum, can be produced at will in the lab using conventional lasers and optics [1].

What is then light having orbital angular momentum? Besides definitions and complicated mathematics, we can make sense of the most important feature of twisted light by appealing to its effect on particles. A microscopic particle shined by light having only spin angular momentum (a conventional light field) will produce, needless to say, the spinning of the particle. That reflects the transfer of intrinsic/spins angular momentum from light to matter. On the other hand, if twisted light is used, the particle will also move around the beam axis, signaling the transfer of orbital angular momentum [2]. This is actually one of the uses one can give to twisted light: under certain conditions, the orbital and spin angular momenta can be manipulated separately, and use to move particles; these are the optical tweezers I mentioned before.

Let us just go a little into math, and first recall how a conventional plane wave looks like. Its electric field can be written by

\begin{eqnarray}
 \large \bf{E}(\bf{r},t)=\bf{E}_0(r) e^{i(q z-\omega t)}+c.c.
\end{eqnarray}
where the wave travels in the direction z with angular frequency ω and with amplitude E0. In the case of twisted light, the orbital angular momentum shows up as an extra complex phase

\begin{eqnarray}
 \large \bf{E}(\bf{r},t)=\bf{E}_0(r) e^{i(q z-\omega t)} e^{i \ell \varphi} + c.c.
\end{eqnarray}

where φ is the angle of the position vector in cylindrical coordinates [3]. The constant ℓ signals the amount of orbital angular momentum carried by the electromagnetic field. This angle-dependent phase is responsible for the screw-like wave-front, see the figure.

Besides the orbital angular momentum, twisted light has another important feature. It exhibits a phase singularity at its axis, where the electric and/or magnetic field can be zero. In future posts we will comments on new phenomena arising from this vortex.

References

[1] For a nice introduction to the general topic of twisted light, I recommend the article "Light´s orbital angular momentum", by Padgett et al, Phys. Today 57(5), 35 (2004).
[2] A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[3] The expression is somewhat more complicated, but I show the essential features.