# Orbital angular momentum of light

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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.

## Introduction

Different columns show the beam helical structures, phase fronts, and corresponding intensity distributions.

A beam of light carries a linear momentum $\mathbf{P}$, and hence it can be also attributed an external angular momentum $\mathbf{L}_e=\mathbf{r}\times\mathbf{P}$. This external angular momentum depends on the choice of the origin of the coordinate system. If one chooses the origin at the beam axis and the beam is cylindrically symmetric (at least in its momentum distribution), the external angular momentum will vanish. The external angular momentum is a form of OAM, because it is unrelated to polarization and depends on the spatial distribution of the optical field (E).

A more interesting example of OAM is the internal OAM appearing when a paraxial light beam is in a so-called "helical mode". Helical modes of the electromagnetic field are characterized by a wavefront that is shaped as a helix, with an optical vortex in the center, at the beam axis (see figure). The helical modes are characterized by an integer number $m$, positive or negative. If $m=0$, the mode is not helical and the wavefronts are multiple disconnected surfaces, for example, a sequence of parallel planes (from which the name "plane wave"). If $m=\pm 1$, the handedness determined by the sign of $m$, the wavefront is shaped as a single helical surface, with a step length equal to the wavelength $\lambda$. If $|m|\geqslant 2$, the wavefront is composed of $|m|$ distinct but intertwined helices, with the step length of each helix surface equal to $|m|\lambda$, and a handedness given by the sign of $m$. The integer $m$ is also the so-called "topological charge" of the optical vortex. Light beams that are in a helical mode carry nonzero OAM.

In the figure to the right, the first column shows the beam wavefront shape. The second column is the optical phase distribution in a beam cross-section, shown in false colors. The third column is the light intensity distribution in a beam cross-section (with a dark vortex core at the center).

The beam photons in this case have an OAM of $m\hbar$ directed along the beam axis. This OAM is origin-independent.

An example of optical modes having a helical wavefront is provided by the set of Laguerre-Gaussian modes.[1]

## Mathematical expressions for the orbital angular momentum of light

The classical expression of the orbital angular momentum in the paraxial limit is the following:[2]

$\mathbf{L}=\epsilon_0\sum_{i=x,y,z}\int \left(E^i\left(\mathbf{r}\times\mathbf{\nabla}\right)A^i\right)d^{3}\mathbf{r} ,$

where $\mathbf{E}$ and $\mathbf{A}$ are the electric field and the vector potential, respectively, $\epsilon_0$ is the vacuum permittivity and we are using SI units. The $i$-superscripted symbols denote the cartesian components of the corresponding vectors.

For a monochromatic wave this expression can be transformed into the following one:[3]

$\mathbf{L}=\frac{\epsilon_0}{2i\omega}\sum_{i=x,y,z}\int \left({E^i}^{\ast}\left(\mathbf{r}\times\mathbf{\nabla}\right)E^{i}\right)d^{3}\mathbf{r} .$

This expression is generally nonvanishing when the wave is not cylindrically symmetric. In particular, in a quantum theory, individual photons may have the following values of the OAM:

$\mathbf{L}_z=m\hbar .$

The corresponding wave functions (eigenfunctions of OAM operator) have the following general expression:

$\langle \mathbf{r}|m\rangle\propto e^{i m \phi} .$

where $\phi$ is the cylindrical coordinate. As mentioned in the Introduction, this expression corresponds to waves having a helical wavefront (see figure above), with an optical vortex in the center, at the beam axis.

## Production of OAM states

Orbital angular momentum states with $l = \pm 1$ occur naturally, there are a number of ways of creating OAM states of arbitrary $l$ in the lab including via spiral phase plates, spatial light modulators and q-plates.

Spiral wave plates, made of plastic or glass, are plates where the thickness of the material increases in a spiral pattern in order to imprint a phase gradient on light passing through it. For a given wavelength, an OAM state of a given $l$ requires that the step height, the height between the thinest and thickest parts of the plate be given by $s = l\lambda/(n-1)$ where $n$ is an integer. Although the wave plates themselves are efficient, they are relatively expensive and are, in general, not adjustable to different wavelengths of light.[4]

Another way to modify the phase of the light is with a diffraction grating. For an $l=0$ state, the diffraction grating would consist of parallel lines. However, for an $l=1$ state, there will be a "fork" dislocation, and the number of lines above the dislocation will be one larger than below. An OAM state with $l>1$ can be created by increasing the difference in the number of lines above and below the dislocation.[5] As with the spiral wave plates, these diffraction gratings are fixed for $l$, but are not restricted to a particular wavelength. A spatial light modulator operates in a similar way to the diffraction grating, but can be controlled by computer to dynamically generate a wide range of OAM states.

## Potential use in telecommunications

Research into OAM has suggested that light waves could carry hitherto unprecedented quantities of data through optical fibres. According to preliminary tests, data streams travelling along a beam of light split into 8 different circular polarities have demonstrated the capacity to transfer up to 2.5 terabits of data (equivalent to 66 DVDs or 320 Gigabytes) per second.[6] Further research into OAM multiplexing in the radio and mm wavelength frequencies has been shown in preliminary tests to be able to transmit 32 gigabits of data per second over the air.[7] There is an ongoing discussion if this will add any capacity on top of other schemes, such as MIMO.

## Quantum Information Applications

OAM states can be generated in coherent superpositions and they can be entangled, which is an integral element of schemes for quantum information protocols. These states can be generated using parametric down conversion, and correlations measured using spatial light modulators (SLM).[8] Using qudits (with d levels, as opposed to a qubit's 2 levels) has been shown to improve the robustness of quantum key distribution schemes. OAM states provide a suitable physical realisation of such a system, and a proof of principle experiment (with 7 OAM modes from $l=-3$ to $l=3$) has been demonstrated.[9]

## References

1. ^ Siegmam, Anthony E. (1986). Lasers. University Science Books. p. 1283. ISBN 0-935702-11-3.
2. ^ Belinfante, 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. doi:10.1016/S0031-8914(40)90091-X.
3. ^ Humblet, J. (1943). "Sur le moment d'impulsion d'une onde electromagnetique". Physica (Utrecht) 10 (7): 585–603. Bibcode:1943Phy....10..585H. doi:10.1016/S0031-8914(43)90626-3.
4. ^ Beijersbergen, M.W.; Coerwinkel, R.P.C.; Kristensen, M.; Woerdman, J.P. (December 1994). "Helical-wavefront laser beams produced with a spiral phaseplate". Optics Communications 112 (5-6): 321–327. doi:10.1016/0030-4018(94)90638-6.
5. ^ Bazhenov, V.Yu.; Soskin, M.S.; Vasnetsov, M.V. (May 1992). "Screw Dislocations in Light Wavefronts". Journal of Modern Optics 39 (5): 985–990. doi:10.1080/09500349214551011.
6. ^ "'Twisted light' carries 2.5 terabits of data per second". BBC. 25 June 2012. Retrieved 25 June 2012.
7. ^ Yan, Yan (16 September 2014). "High-capacity millimetre-wave communications with orbital angular momentum multiplexing". Nature Communications 5: 4876. Bibcode:2014NatCo...5E4876Y. doi:10.1038/ncomms5876. PMC 4175588. PMID 25224763. Retrieved 18 September 2014.
8. ^ B. Jack, a. M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. G. Ireland, S. M. Barnett, and M. J. Padgett, Phys. Rev. A - At. Mol. Opt. Phys. 81, 1 (2010).
9. ^ M. Mirhosseini, arXiv 1402.7113 (2014).

## Further reading

• Allen, L.; Barnett, 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.
• Andrews, David L. & Babiker, Mohamed (2012). The Angular Momentum of Light. Cambridge: Cambridge University Press. p. 448. ISBN 9781107006348.