angular momentum of light
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The angular momentum of light is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
quantity that expresses the amount of dynamical rotation present in the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
of the
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
. 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 Eyes are organs of the visual system. They provide living organisms with vision, the ability to receive and process visual detail, as well as enabling several photo response functions that are independent of vision. Eyes detect light and conv ...
, 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 In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
shape. These two forms of rotation are therefore associated with two distinct forms of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, respectively named light spin angular momentum (SAM) and
light orbital angular momentum 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 in ...
(OAM). The total angular momentum of light (or, more generally, of the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
and the other
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
fields) and matter is conserved in time.


Introduction

Light, or more generally an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
, carries not only
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
but also
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
, which is a characteristic property of all objects in translational motion. The existence of this momentum becomes apparent in the "''
radiation pressure Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
'' phenomenon, in which a light beam transfers its momentum to an absorbing or scattering object, generating a mechanical
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
on it in the process. Light may also carry
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, 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 Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, 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 In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optic ...
'' light beam, that is a well
collimated A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction pr ...
beam in which all light rays (or, more precisely, all Fourier components of the
optical field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
) only form small angles with the
beam axis An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light propagat ...
. For such a beam, SAM is strictly related with the optical polarization, and in particular with the so-called
circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
. OAM is related with the spatial field distribution, and in particular with the
wavefront In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
contribution obtained as the cross-product of the beam position and its total
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
. 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.


Mathematical expressions for the angular momentum of light

One commonly used expression for the total
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of an
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
is the following one, in which there is no explicit distinction between the two forms of rotation: \mathbf = \varepsilon_0 \int \mathbf \times \left(\mathbf \times \mathbf\right) d^3\mathbf , where \mathbf and \mathbf are the electric and magnetic fields, respectively, \epsilon_0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
and we are using SI units. However, another expression of the angular momentum naturally arising from
Noether’s theorem Noether's theorem or Noether's first theorem states that every Differentiable function, differentiable Symmetry in physics, symmetry of the action (physics), action of a physical system with Conservative force, conservative forces has a corre ...
is the following one, in which there are two separate terms that may be associated with SAM (\mathbf) and OAM (\mathbf): \mathbf = \varepsilon_0 \int \left(\mathbf \times \mathbf\right) d^3\mathbf + \varepsilon_0 \sum_ \int \left(E^i \left(\mathbf \times \boldsymbol\right) A^i\right) d^3\mathbf = \mathbf + \mathbf , where \mathbf is the
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
of the magnetic field, and the ''i''-superscripted symbols denote the cartesian components of the corresponding vectors. These two expressions can be proved to be equivalent to each other for any electromagnetic field that vanishes fast enough outside a finite region of space. The two terms in the second expression however are physically ambiguous, as they are not
gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, es ...
-
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
. A gauge-invariant version can be obtained by replacing the vector potential A and the electric field E with their “transverse” or radiative component \mathbf_ and \mathbf_, thus obtaining the following expression: \mathbf_ = \varepsilon_0 \int \left(\mathbf_\times \mathbf_\right) d^3\mathbf + \varepsilon_0 \sum_ \int \left(_ \left(\mathbf \times \boldsymbol\right) A^i_\right) d^3\mathbf . A justification for taking this step is yet to be provided. 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. Their sum, that is the total angular momentum, instead does. An equivalent but simpler expression for a monochromatic wave of frequency ω, using the complex notation for the fields, is the following: \mathbf = \frac\int \left(\mathbf^\ast \times \mathbf\right) d^3\mathbf + \frac\sum_ \int \left(^ \left(\mathbf\times\boldsymbol\right) E^i\right) d^3\mathbf . Let us now consider the paraxial limit, with the beam axis assumed to coincide with the z axis of the coordinate system. In this limit the only significant component of the angular momentum is the z one, that is the angular momentum measuring the light beam rotation around its own axis, while the other two components are negligible. \mathbf \approx \frac\int \left(\left, E_\text\^2 - \left, E_\text\^2\right) d^3\mathbf +\frac \int \sum_ \left(^\ast \frac E^i\right) d^3\mathbf . where E_\text and E_\text denote the left and right circular polarization components, respectively.


Exchange of spin and orbital angular momentum with matter

When a light beam carrying nonzero angular momentum impinges on an absorbing particle, its angular momentum can be transferred on the particle, thus setting it in rotational motion. This occurs both with SAM and OAM. However, if the particle is not at the beam center the two angular momenta will give rise to different kinds of rotation of the particle. SAM will give rise to a rotation of the particle around its own center, i.e., to a particle spinning. OAM, instead, will generate a revolution of the particle around the beam axis. These phenomena are schematically illustrated in the figure. In the case of transparent media, in the paraxial limit, the optical SAM is mainly exchanged with anisotropic systems, for example
birefringent Birefringence is the optics, optical property of a material having a refractive index that depends on the Polarization (waves), polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or ...
crystals. Indeed, thin slabs of
birefringent Birefringence is the optics, optical property of a material having a refractive index that depends on the Polarization (waves), polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or ...
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. If the crystal is free to rotate, it will do so. Otherwise, the SAM is finally transferred to the holder and to the Earth.


Spiral Phase Plate (SPP)

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).


Pitch-Fork Hologram

A more convenient approach for generating OAM is based on using diffraction on a fork-like or pitchfork hologram (see figure). Holograms can be also generated dynamically under the control of a computer by using a
spatial light modulator A spatial light modulator (SLM) is an object that imposes some form of spatially varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the term SLM is used, it means that the transparency can ...
.


Q-Plate

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 A q-plate is an optical device that can form a light beam with orbital angular momentum (OAM) from a beam with well-defined spin angular momentum (SAM). Q-plates are based on the SAM-OAM coupling that may occur in media that are both anisotropic ...
is a device, currently realized using liquid crystals, polymers or sub-wavelength gratings, which can generate OAM by exploiting a SAM sign-change. In this case, the OAM sign is controlled by the input polarization.


Cylindrical Mode Converters

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 lens A cylindrical lens is a lens which focuses light into a line instead of a point, as a spherical lens would. The curved face or faces of a cylindrical lens are sections of a cylinder, and focus the image passing through it into a line parallel to i ...
es placed at a specific distance (see figure) in order to introduce a well-defined relative phase between horizontal and vertical Hermite-Gaussian beams.


Possible applications of the orbital angular momentum of light

The applications of the spin angular momentum of light are undistinguishable from the innumerable applications of the light polarization and will not be discussed here. 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: # Orientational manipulation of particles or particle aggregates in
optical tweezers Optical tweezers (originally called single-beam gradient force trap) are scientific instruments that use a highly focused laser beam to hold and move microscopic and sub-microscopic objects like atoms, nanoparticles and droplets, in a manner simila ...
# High-bandwidth information encoding in
free-space optical communication Free-space optical communication (FSO) is an optical communication technology that uses light propagating in free space to wirelessly transmit data for telecommunications or computer networking. "Free space" means air, outer space, vacuum, or ...
# Higher-dimensional quantum information encoding, for possible future
quantum cryptography Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
or
quantum computation Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
applications # Sensitive optical detection


See also

*
Angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
*
Circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
*
Electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
*
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...
*
Light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
*
Light orbital angular momentum 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 in ...
* Light spin angular momentum *
Optical vortices An optical vortex (also known as a photonic quantum vortex, screw dislocation or phase singularity) is a zero of an optical field; a point of zero Intensity (physics), intensity. The term is also used to describe a beam of light that has such a ze ...
*
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 signa ...
*
Polarization (waves) Polarization (also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the ...
*
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. Equi ...


References


External links


Phorbitech

Glasgow Optics Group

Leiden Institute of Physics

ICFO

Università Di Napoli "Federico II"



University of Ottawa


Further reading

* * * {{cite book, last1=Andrews, first1=David L., name-list-style=amp , last2=Babiker , first2= Mohamed, title=The Angular Momentum of Light, year=2012, publisher=Cambridge University Press, location=Cambridge, isbn=978-1-107-00634-8, pages=448, url=http://www.cambridge.org/de/knowledge/isbn/item6687744/The%20Angular%20Momentum%20of%20Light/ Angular momentum of light Light