Spin angular momentum of light
An electromagnetic wave is said to have circular polarization when its electric and magnetic fields rotate continuously around the beam axis during the propagation. The circular polarization is left () or right () depending on the field rotation direction (but be careful that both conventions are used in science, depending on the subfield).
When a light beam is circularly polarized, each of its photons carries a spin angular momentum of , where is the reduced Planck constant and the sign is positive for left and negative for right circular polarizations (this is adopting the convention most commonly used in optics). This SAM is directed along the beam axis (parallel if positive, antiparallel if negative). The above figure shows the instantaneous structure of the electric field of left () and right () circularly polarized light in space. The green arrows indicate the propagation direction.
The mathematical expressions reported under the figures give the three electric-field components of circularly polarized plane wave propagating in the direction, in complex notation.
In particular, this expression shows that the SAM is nonzero when the light polarization is elliptical or circular, while it vanishes if the light polarization is linear.
In the quantum theory of the electromagnetic field, the SAM is a quantum observable, described by a corresponding operator:
where is the unit vector in the propagation direction, and are respectively the creation and annihilation operators for photons in the mode k and polarization state .
The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves:
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- Born, M. & Wolf, E. (1999). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.). Cambridge: Cambridge University Press. ISBN 0-521-64222-1.
- Allen, L.; Barnnet, Stephen M. & Padgett, Miles J. (2003). Optical Angular Momentum. Bristol: Institute of Physics. ISBN 978-0-7503-0901-1.
- Torres, Juan P. & Torner, Lluis (2011). Twisted Photons: Applications of Light with Orbital Angular Momentum. Bristol: Wiley-VCH. ISBN 978-3-527-40907-5.