# Spin angular momentum of light

The spin angular momentum of light (SAM) is the component of angular momentum of light that is associated with the quantum spin and the wave's circular or elliptical polarization.

## Introduction

Left and right circular polarization and their associate angular momenta

An electromagnetic wave is said to have circular polarization when its electric and magnetic fields rotate continuously around the beam axis during propagation. The circular polarization is left (${\displaystyle L}$) or right (${\displaystyle R}$) depending on the field rotation direction and, according to the convention used: either from the point of view of the source, or the receiver. 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 (SAM) of ${\displaystyle \pm \hbar }$, where ${\displaystyle \hbar }$ is the reduced Planck constant and the ${\displaystyle \pm }$ sign is positive for left and negative for right circular polarizations (this is adopting the convention from the point of view of the receiver 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 (${\displaystyle L}$) and right (${\displaystyle R}$) 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 ${\displaystyle z}$ direction, in complex notation.

## Mathematical expression

General expression for the spin angular momentum in the paraxial limit is[1]

${\displaystyle \mathbf {S} =\epsilon _{0}\int \left(\mathbf {E} \times \mathbf {A} \right)\,d^{3}\mathbf {r} ,}$

where ${\displaystyle \mathbf {E} }$ and ${\displaystyle \mathbf {A} }$ are the electric field and magnetic vector potential respectively, ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity, and we are using SI units.

Monochromatic-wave case:[2]

${\displaystyle \mathbf {S} ={\frac {\epsilon _{0}}{2i\omega }}\int \left(\mathbf {E} ^{*}\times \mathbf {E} \right)\,d^{3}\mathbf {r} .}$

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:

${\displaystyle \mathbf {S} =\sum _{\mathbf {k} }\hbar \mathbf {u} _{\mathbf {k} }\left({\hat {a}}_{\mathbf {k} ,L}^{\dagger }{\hat {a}}_{\mathbf {k} ,L}-{\hat {a}}_{\mathbf {k} ,R}^{\dagger }{\hat {a}}_{\mathbf {k} ,R}\right),}$

where ${\displaystyle \mathbf {u} _{\mathbf {k} }}$ is the unit vector in the propagation direction, ${\displaystyle {\hat {a}}_{\mathbf {k} ,\pi }^{\dagger }}$ and ${\displaystyle {\hat {a}}_{\mathbf {k} ,\pi }}$ are respectively the creation and annihilation operators for photons in the mode k and polarization state ${\displaystyle \pi }$.

In this case, for a single photon the SAM can only have two values (eigenvalues of the SAM operator):

${\displaystyle \mathbf {S} _{z}=\pm \hbar .}$

The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves:

${\displaystyle |\pm \rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\\pm i\end{pmatrix}}.}$

## References

1. ^ Belintante, F. J. (1940). "On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields". Physica. 7 (5): 449–474. Bibcode:1940Phy.....7..449B. CiteSeerX 10.1.1.205.8093. doi:10.1016/S0031-8914(40)90091-X.
2. ^ Humblet, J. (1943). "Sur le moment d'impulsion d'une onde electromagnetique". Physica. 10 (7): 585–603. Bibcode:1943Phy....10..585H. doi:10.1016/S0031-8914(43)90626-3.

• Born, M. & Wolf, E. (1999). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-64222-4.
• 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.
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.

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.

Index of physics articles (S)

The index of physics articles is split into multiple pages due to its size.

Orbital angular momentum multiplexing

Orbital angular momentum (OAM) multiplexing is a physical layer method for multiplexing signals carried on electromagnetic waves using the orbital angular momentum of the electromagnetic waves to distinguish between the different orthogonal signals.Orbital angular momentum is one of two forms of angular momentum of light. OAM is distinct from, and should not be confused with, light spin angular momentum. The spin angular momentum of light offers only two orthogonal quantum states corresponding to the two states of circular polarization, and can be demonstrated to be equivalent to a combination of polarization multiplexing and phase shifting. OAM on the other hand relies on an extended beam of light, and the higher quantum degrees of freedom which come with the extension. OAM multiplexing can thus access a potentially unbounded set of states, and as such offer a much larger number of channels, subject only to the constraints of real-world optics.

As of 2013, although OAM multiplexing promises very significant improvements in bandwidth when used in concert with other existing modulation and multiplexing schemes, it is still an experimental technique, and has so far only been demonstrated in the laboratory. Following the early claim that OAM exploits a new quantum mode of information propagation, the technique has become controversial; however nowadays it can be understood to be a particular form of tightly modulated MIMO multiplexing strategy, obeying classical information theoretic bounds.

Orbital angular momentum of light

The orbital angular momentum of light (OAM) is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. It can be further split into an internal and an external OAM. The internal OAM is an origin-independent angular momentum of a light beam that can be associated with a helical or twisted wavefront. The external OAM is the origin-dependent angular momentum that can be obtained as cross product of the light beam position (center of the beam) and its total linear momentum.

Photon

The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force (even when static via virtual particles). The photon has zero rest mass and always moves at the speed of light within a vacuum.

Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of both waves and particles. For example, a single photon may be refracted by a lens and exhibit wave interference with itself, and it can behave as a particle with definite and finite measurable position or momentum, though not both at the same time as per the Heisenberg's uncertainty principle. The photon's wave and quantum qualities are two observable aspects of a single phenomenon—they cannot be described by any mechanical model; a representation of this dual property of light that assumes certain points on the wavefront to be the seat of the energy is not possible. The quanta in a light wave are not spatially localized.

The modern concept of the photon was developed gradually by Albert Einstein in the early 20th century to explain experimental observations that did not fit the classical wave model of light. The benefit of the photon model is that it accounts for the frequency dependence of light's energy, and explains the ability of matter and electromagnetic radiation to be in thermal equilibrium. The photon model accounts for anomalous observations, including the properties of black-body radiation, that others (notably Max Planck) had tried to explain using semiclassical models. In that model, light is described by Maxwell's equations, but material objects emit and absorb light in quantized amounts (i.e., they change energy only by certain particular discrete amounts). Although these semiclassical models contributed to the development of quantum mechanics, many further experiments beginning with the phenomenon of Compton scattering of single photons by electrons, validated Einstein's hypothesis that light itself is quantized. In 1926 the optical physicist Frithiof Wolfers and the chemist Gilbert N. Lewis coined the name "photon" for these particles. After Arthur H. Compton won the Nobel Prize in 1927 for his scattering studies, most scientists accepted that light quanta have an independent existence, and the term "photon" was accepted.

In the Standard Model of particle physics, photons and other elementary particles are described as a necessary consequence of physical laws having a certain symmetry at every point in spacetime. The intrinsic properties of particles, such as charge, mass, and spin, are determined by this gauge symmetry. The photon concept has led to momentous advances in experimental and theoretical physics, including lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. It has been applied to photochemistry, high-resolution microscopy, and measurements of molecular distances. Recently, photons have been studied as elements of quantum computers, and for applications in optical imaging and optical communication such as quantum cryptography.

Photon polarization

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon

can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave. Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.

Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with two pairs (or one broken pair) of polaroid sunglasses.

The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.

Q-plate

A q-plate is an optical device which can generate light beams with orbital angular momentum of light (OAM) from a beam with well-defined Spin angular momentum of light (SAM). It is currently realized using liquid crystals, polymers or sub-wavelength gratings.

A method for generating orbital angular momentum of light (OAM) is based on the SAM-OAM coupling that may occur in a medium which is both anisotropic and inhomogeneous. In the case of the q-plate, the OAM sign is controlled by the input polarization.

Spin angular momentum

Spin angular momentum may refer to:

Spin angular momentum of light, a property of electromagnetic waves

A type of quantum mechanics angular momentum operator

Spin polarization

Spin polarization is the degree to which the spin, i.e., the intrinsic angular momentum of elementary particles, is aligned with a given direction. This property may pertain to the spin, hence to the magnetic moment, of conduction electrons in ferromagnetic metals, such as iron, giving rise to spin-polarized currents. It may refer to (static) spin waves, preferential correlation

of spin orientation with ordered lattices (semiconductors or insulators).

It may also pertain to beams of particles, produced for particular aims, such as polarized neutron scattering or muon spin spectroscopy. Spin polarization of electrons or of nuclei, often called simply magnetization, is also produced by the application of a magnetic field. Curie law is used to produce an induction signal in Electron spin resonance (ESR or EPR) and in Nuclear magnetic resonance (NMR).

Spin polarization is also important for spintronics, a branch of electronics. Magnetic semiconductors are being researched as possible spintronic materials.

The spin of free electrons is measured either by a LEED image from a clean wolfram-crystal (SPLEED) or by an electron microscope composed purely of electrostatic lenses and a gold foil as a sample. Back scattered electrons are decelerated by annular optics and focused onto a ring shaped electron multiplier at about 15°. The position on the ring is recorded. This whole device is called a Mott-detector. Depending on their spin the electrons have the chance to hit the ring at different positions. 1% of the electrons are scattered in the foil. Of these 1% are collected by the detector and then about 30% of the electrons hit the detector at the wrong position. Both devices work due to spin orbit coupling.

The circular polarization of electromagnetic fields is due to spin polarization of their constituent photons.

In the most generic context, spin polarization is any alignment of the components of a non-scalar

(vectorial, tensorial, spinor) field with its arguments, i.e., with the nonrelativistic three spatial or

relativistic four spatiotemporal regions over which it is defined. In this sense, it also includes

gravitational waves and any field theory that couples its constituents with the differential

operators of vector analysis.

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