Angular momentum of light
While traveling approximately in a straight line, a beam of light can also be rotating (or "spinning", or "twisting") around its own axis.
This rotation, while not visible to the naked eye, can be revealed by the interaction of the light beam with matter.
There are two distinct forms of rotation of a light beam, one involving its polarization and the other its wavefront shape.
Light, or more generally an electromagnetic wave, carries not only energy but also momentum, which is a characteristic property of all objects in translational motion.
The existence of this momentum becomes apparent in the "radiation pressure" phenomenon, in which a light beam transfers its momentum to an absorbing or scattering object, generating a mechanical pressure on it in the process.
Light may also carry angular momentum, which is a property of all objects in rotational motion.
Again, the existence of this angular momentum can be made evident by transferring it to small absorbing or scattering particles, which are thus subject to an optical torque.
These two rotations are associated with two forms of angular momentum, namely SAM and OAM.
However this distinction becomes blurred for strongly focused or diverging beams, and in the general case only the total angular momentum of a light field can be defined.
OAM is related with the spatial field distribution, and in particular with the wavefront helical shape.
However, since its value is dependent from the choice of the origin, it is termed "external" orbital angular momentum, as opposed to the "internal" OAM appearing for helical beams.
One commonly used expression for the total angular momentum of an electromagnetic field is the following one, in which there is no explicit distinction between the two forms of rotation:
However, another expression of the angular momentum naturally arising from Noether’s theorem is the following one, in which there are two separate terms that may be associated with SAM (
These two expressions can be proved to be equivalent to each other for any electromagnetic field that satisfies Maxwell’s equations with no source charges and vanishes fast enough outside a finite region of space.
A gauge-invariant version can be obtained by replacing the vector potential A and the electric field E with their “transverse” or radiative component
The latter expression has further problems, as it can be shown that the two terms are not true angular momenta as they do not obey the correct quantum commutation rules.
[citation needed] An equivalent but simpler expression for a monochromatic wave of frequency ω, using the complex notation for the fields, is the following:[2]
In the case of transparent media, in the paraxial limit, the optical SAM is mainly exchanged with anisotropic systems, for example birefringent crystals.
Indeed, thin slabs of birefringent crystals are commonly used to manipulate the light polarization.
Whenever the polarization ellipticity is changed, in the process, there is an exchange of SAM between light and the crystal.
In the paraxial limit, the OAM of a light beam can be exchanged with material media that have a transverse spatial inhomogeneity.
For example, a light beam can acquire OAM by crossing a spiral phase plate, with an inhomogeneous thickness (see figure).
[6] A more convenient approach for generating OAM is based on using diffraction on a fork-like or pitchfork hologram (see figure).
[7][8][9][10] Holograms can be also generated dynamically under the control of a computer by using a spatial light modulator.
[11] As a result, this allows one to obtain arbitrary values of the orbital angular momentum.
Another method for generating OAM is based on the SAM-OAM coupling that may occur in a medium which is both anisotropic and inhomogeneous.
In particular, the so-called q-plate is a device, currently realized using liquid crystals, polymers or sub-wavelength gratings, which can generate OAM by exploiting a SAM sign-change.
[12][13][14] OAM can also be generated by converting a Hermite-Gaussian beam into a Laguerre-Gaussian one by using an astigmatic system with two well-aligned cylindrical lenses placed at a specific distance (see figure) in order to introduce a well-defined relative phase between horizontal and vertical Hermite-Gaussian beams.
The possible applications of the orbital angular momentum of light are instead currently the subject of research.
In particular, the following applications have been already demonstrated in research laboratories, although they have not yet reached the stage of commercialization:
