Role of Radial Charges on the Angular Momentum of Electromagnetic Fields: Spin-$3/2$ Light

Electromagnetic fields carry a linear and an angular momentum, the first being responsible for the existence of the radiation pressure and the second for the transfer of torque from electromagnetic radiation to matter. The angular momentum is considered to have two components, one due to the polarization state of the field, usually called spin angular momentum (SAM), and one due to the existence of topological azimuthal charges in the field phase profile, which leads to the orbital angular momentum (OAM). These two contributions to the total angular momentum of an electromagnetic field appear, however, to not be independent of each other, something which is described as spin-orbit coupling. Understanding the physics of this coupling has kept scientists busy for decades. Very recently it has been shown that electromagnetic fields necessarily carry also invariant radial charges that, as discussed in this Letter, play a key role in the angular momentum. Here we show that the total angular momentum consists in fact of three components: one component only dependent on the spin of the field, another dependent on the azimuthal charges carried by the field, and a third component dependent on the spin and the radial charges contained in the field. By properly controlling the number and coupling among these radial charges it is possible to design electromagnetic fields with a desired total angular momentum. Remarkably, we also discover fields with no orbital angular momentum and a spin angular momentum typical of spin-$3/2$ objects, irrespective of the fact that photons are spin-1 particles.

Figure 1

Panel (a): Absolute value of the coupling function $f(m,m^{\prime })=1/[2i\pi (m-m^{\prime })]+1/[2{\pi }^2(m-m^{\prime }{)}^2]$ among neighboring radial modes. The curve refers to the coupling of the zeroth order radial charge (shown close to the origin of the $x$ axis in the graph) with the neighboring radial charges, of order $m\in [-5,5]$. On the $x$ axis the order of each charge is reported, while the phase profile of each charge is used as marker on the curve. Only radial charges corresponding to HNMs of same azimuthal charge $n$ can couple to each other. Panels (b), (c), (d), and (e): Example of spin-$3/2$ electromagnetic field. The field is derived by a scalar angular spectrum $A(\rho ,\phi )=\sin [2\pi m\lambda \sqrt{(1/{\lambda }^2)-{\rho }^2}+n\phi ]/{[(1/{\lambda }^2)-{\rho }^2]}^{1/4}$. The absolute value, and phase, of the corresponding scalar field $U(x,y,z)$, at $z=0$, is given in panels (b) and (c), respectively. Panels (d) and (e) show the amplitude and phase of the corresponding scalar spectrum, $A$, within the circle of radius $1/\lambda$ in Fourier space. The example corresponds to $m=3$ and $n=2$. The full electromagnetic field $\mathbf{E}$ associated to the angular spectrum $A$ is obtained through Eq. (7). More generally, any electromagnetic field carrying HNMs with no azimuthal charge (i.e., $n=0$) will give a $\gamma =3{\sigma }_z/(2\omega )$.