Angular momentum of light

The angular momentum of light is a vector quantity that expresses the amount of dynamical rotation present in the electromagnetic field of the 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. The angular momentum of light is a vector quantity that expresses the amount of dynamical rotation present in the electromagnetic field of the 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. These two forms of rotation are therefore associated with two distinct forms of angular momentum, respectively named light spin angular momentum (SAM) and light orbital angular momentum (OAM). The total angular momentum of light (or, more generally, of the electromagnetic field and the other force fields) and matter is conserved in time. It is well known that 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. Less widely known is the fact that light may also carry angular momentum, which is a property of all objects in rotational motion. For example, a light beam can be rotating around its own axis while it propagates forward. 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. For a light beam, one can usually distinguish two “forms of rotation”, the first associated with the dynamical rotation of the electric and magnetic fields around the propagation direction, and the second with the dynamical rotation of light rays around the main beam axis. 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. An important limiting case in which the distinction is instead clear and unambiguous is that of a “paraxial” light beam, that is a well collimated beam in which all light rays (or, more precisely, all Fourier components of the optical field) only form small angles with the beam axis. For such a beam, SAM is strictly related with the optical polarization, and in particular with the so-called circular polarization. OAM is related with the spatial field distribution, and in particular with the wavefront helical shape. In addition to these two terms, if the origin of coordinates is located outside the beam axis, there is a third angular momentum contribution obtained as the cross-product of the beam position and its total momentum. This third term is also called “orbital”, because it depends on the spatial distribution of the field. 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: