Orbital angular momentum of light

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

A beam of light carries a linear momentum ${\displaystyle \mathbf {P} }$, and hence it can be also attributed an external angular momentum ${\displaystyle \mathbf {L} _{e}=\mathbf {r} \times \mathbf {P} }$. 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). If the phase varies around the axis of such a wave, it carries orbital angular momentum.[1]

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 helical modes are characterized by an integer number ${\displaystyle m}$, positive or negative. If ${\displaystyle m=0}$, 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 ${\displaystyle m=\pm 1}$, the handedness determined by the sign of ${\displaystyle m}$, the wavefront is shaped as a single helical surface, with a step length equal to the wavelength ${\displaystyle \lambda }$. If ${\displaystyle |m|\geqslant 2}$, the wavefront is composed of ${\displaystyle |m|}$ distinct but intertwined helices, with the step length of each helix surface equal to ${\displaystyle |m|\lambda }$, and a handedness given by the sign of ${\displaystyle m}$. The integer ${\displaystyle m}$ is also the so-called "topological charge" of the optical vortex. Light beams that are in a helical mode carry nonzero OAM. As an example, any Laguerre-Gaussian mode with rotational mode number ${\displaystyle l\neq 0}$ has such a helical wavefront.[2]

The classical expression of the orbital angular momentum in the paraxial limit is the following:[3]

$${\displaystyle \mathbf {L} =\epsilon _{0}\sum _{i=x,y,z}\int \left(E^{i}\left(\mathbf {r} \times {\boldsymbol {\nabla }}\right)A^{i}\right)d^{3}\mathbf {r} ,}$$

where ${\displaystyle \mathbf {E} }$ and ${\displaystyle \mathbf {A} }$ are the electric field and the vector potential, respectively, ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity and we are using SI units. The ${\displaystyle i}$-superscripted symbols denote the cartesian components of the corresponding vectors.

For a monochromatic wave this expression can be transformed into the following one:[4]

$${\displaystyle \mathbf {L} ={\frac {\epsilon _{0}}{2i\omega }}\sum _{i=x,y,z}\int \left({E^{i}}^{\ast }\left(\mathbf {r} \times \mathbf {\nabla } \right)E^{i}\right)d^{3}\mathbf {r} .}$$

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:

$${\displaystyle \mathbf {L} _{z}=m\hbar .}$$

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

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

where ${\displaystyle \phi }$ 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.

A light beam with a given orbital angular momentum (OAM) can be generated by letting a standard Gaussian beam impinge on a display of a spatial light modulator (SLM). If the phase profile on SLM is flat, the SLM works effectively as a mirror. If the phase has a helical profile, the resulting beam is a Laguerre-Gaussian (LG) beam with a well-defined OAM. In real applications, there is a non-negligible admixture in the reflected beam in the form of a Gaussian beam. One can get rid of it by superposing the helical phase on the SLM with a diffraction grating.

Orbital angular momentum states with ${\displaystyle l=\pm 1}$ occur naturally. OAM states of arbitrary ${\displaystyle l}$ can be created artificially using a variety of tools, such as using 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 ${\displaystyle l}$ requires that the step height —the height between the thinnest and thickest parts of the plate— be given by ${\displaystyle s=l\lambda /(n-1)}$ where ${\displaystyle n}$ is an integer. Although the wave plates themselves are efficient, they are relatively expensive to produce, and are, in general, not adjustable to different wavelengths of light.[5]

Another way to modify the phase of the light is with a diffraction grating. For an ${\displaystyle l=0}$ state, the diffraction grating would consist of parallel lines. However, for an ${\displaystyle l=1}$ 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 ${\displaystyle l>1}$ can be created by increasing the difference in the number of lines above and below the dislocation.[6] As with the spiral wave plates, these diffraction gratings are fixed for ${\displaystyle l}$, but are not restricted to a particular wavelength.

A spatial light modulator operates in a similar way to diffraction gratings, but can be controlled by computer to dynamically generate a wide range of OAM states.

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.[18] 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. The fundamental communication limit of orbital-angular-momentum multiplexing is increasingly urgent for current multiple-input multiple-output (MIMO) research. The limit has been clarified in terms of independent scattering channels or the degrees of freedom (DoF) of scattered fields through angular-spectral analysis, in conjunction with a rigorous Green function method.[19] The DoF limit is universal for arbitrary spatial-mode multiplexing, which is launched by a planar electromagnetic device, such as antenna, metasurface, etc., with a predefined physical aperture.

Determining the spin angular momentum (SAM) of light is simple – SAM is related to the polarization state of the light: the AM is, per photon, in a left and right circularly polarized beam respectively. Thus the SAM can be measured by transforming the circular polarization of light into a p- or s-polarized state by means of a wave plate and then using a polarizing beam splitter that will transmit or reflect the state of light.[5]

The development of a simple and reliable method for the measurement of orbital angular momentum (OAM) of light, however, remains an important problem in the field of light manipulation. OAM (per photon) arises from the amplitude cross-section of the beam and is therefore independent of the spin angular momentum: whereas SAM has only two orthogonal states, the OAM is described by a state that can take any integer value N.[20] As the state of OAM of light is unbounded, any integer value of l is orthogonal to (independent from) all the others. Where a beam splitter could separate the two states of SAM, no device can separate the N (if greater than 2) modes of OAM, and, clearly, the perfect detection of all N potential states is required to finally resolve the issue of measuring OAM. Nevertheless, some methods have been investigated for the measurement of OAM.

OAM states can be generated in coherent superpositions and they can be entangled,[26][27] which is an integral element of schemes for quantum information protocols. Photon pairs generated by the process of parametric down-conversion are naturally entangled in OAM,[28][29] and correlations measured using spatial light modulators (SLM).[30]

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 ${\displaystyle l=-3}$ to ${\displaystyle l=3}$) has been demonstrated.[31]

In 2019, a letter published in the Monthly Notices of the Royal Astronomical Society presented evidence that OAM radio signals had been received from the vicinity of the M87* black hole, over 50 million light years distant, suggesting that optical angular momentum information can propagate over astronomical distances.[32]

Wikimedia Commons has media related to Orbital angular momentum of waves.

• Phorbitech

• Glasgow Optics Group
• Leiden Institute of Physics
• ICFO
• Università Di Napoli "Federico II"
• Università Di Roma "La Sapienza"
• University of Ottawa
• Elementary demonstration using a laser pointer