OAM of Light: Origins and Applications

Definition

Orbital Angular Momentum (OAM) of light is generating growing interest within the scientific community. This entry reviews the origins and applications of OAM. It is the counterpart of linear momentum for systems in rotation. The general expression of OAM is discussed, followed by its implications in terms of phase distribution and donut-shaped intensity profiles. Applications described include the generation of optical torque, telecommunications enhancement, and the rotational Doppler effect, emphasizing the role and consequences of angular momentum. In particular, its use to manipulate systems or to detect rotations is described. Finally, further developments and technological barriers are considered.

1. Introduction

The concept of linear momentum has been known since 1619, when Johannes Kepler attempted to explain why the tail of a comet always points away from the Sun [1]. The prediction that light carries linear momentum and can exert pressure on any surface it encounters was made by James Clerk Maxwell in 1862 [2], and was experimentally verified by Piotr Lebedev in 1900 [3], as well as independently by Edward Leamington Nichols and Gordon Ferrie Hull in 1901 [4,5]. Linear momentum has also been proposed for use in solar sails [6,7], following the ideas of Jules Verne in his 1865 book From the Earth to the Moon [8]. More recently, laser cooling [9,10,11] and optical trapping [12] have been based on the exchange of linear momentum.

Curiously, light can also carry angular momentum, either Spin Angular Momentum (SAM) or Orbital Angular Momentum (OAM), and can generate torques. Initial experiments involving SAM have been performed on macroscopic [13,14,15] and microscopic [16] systems. More recently, exchanges involving OAM have also been reported [17,18,19]. While OAM and SAM may be linked in tightly focused beams, they can be clearly separated within the paraxial approximation [20,21,22]. OAM may have further applications, for example, by offering new diversity in telecommunications [23,24]. The aim of this entry is to provide a comprehensive explanation of OAM and to explore several potential applications.

2. Physical Origin of the OAM

Electromagnetic fields carrying OAM are solutions of Maxwell’s equations. Several general studies have addressed OAM fields, not limited to optical frequencies (see, e.g., [25,26,27]). The purpose of this section is not to provide a complete survey of OAM field theory, but rather to give insights into the physical origin of OAM fields.

The existence of OAM was already described in Poynting’s early work [28,29]. It gained renewed interest in the 1990s [30,31] and is now a well-recognized field of study [32,33,34,35]. Electromagnetic fields carrying OAM typically exhibit a non-uniform phase distribution, $\phi$ (see Figure 1). It varies as

$$\phi =\mathsf{\ell }\theta$$

(1)

$\theta$ is the polar coordinate and ℓ is the so-called topological charge of the beam. The phase varies by $2\pi \mathsf{\ell }$ around the beam axis on a plane perpendicular to its propagation direction. This type of beam is sometimes called a vortex beam, or a twisted beam. Due to a phase singularity, there is no intensity at the center of the beam, resulting in a donut-shaped intensity distribution.

One of the most commonly used beams carrying OAM is the Laguerre-Gaussian (LG) beam [36]. The electric field can be expressed as

$$E^{\mathsf{\ell }}(r,\varphi )=E_0{\left({\displaystyle \frac{2}{\left|\mathsf{\ell }\right|!\pi }}\right)}^{1/2}{\left(r\sqrt{2}\right)}^{\left|\mathsf{\ell }\right|}{e}^{-r^2}{e}^{-\mathsf{ı}\mathsf{\ell }\varphi }$$

(2)

r being the radius normalized to the beam waist $w_0$ ($r=1$ corresponds to a radius equal to $w_0$), and $\varphi$ being the azimuthal angle. Its phase obviously verifies Equation (1). It corresponds to a vortex beam whose amplitude tends toward zero at the center, except for $\mathsf{\ell }=0$, which represents the fundamental Gaussian beam. $E_0$ is a constant characteristic of the field. For the fundamental beam, it is the beam intensity at the center. LG beams are solutions of the equation of propagation of electromagnetic fields. They form an orthonormal basis.

Most importantly, such LG beams carry OAM. In fact, each photon carries an angular momentum $\mathsf{\ell }\hslash$ ($\hslash =h/2\pi$ is the reduced Planck constant) that can induce a torque. The available torque for $N=P/\left(h\nu \right)=P\lambda /\left(hc\right)$ photons per second (where P is the optical power, $\nu$ is the light frequency and $\lambda$ its wavelength; c is the velocity of light) is given by

$$\Gamma =N\mathsf{\ell }\hslash ={\displaystyle \frac{P\lambda }{2\pi c}}\mathsf{\ell }$$

(3)

It is worth noting that this equation is a direct consequence of angular momentum conservation. The angular momentum of photons is converted into a torque.

Although it is not the purpose of this review, let us briefly summarize the general methods used to generate OAM beams, highlighting the main advantages of each technique. A more complete review can be found, for example, in references [34,37,38]. The easiest way to generate OAM beams is to use a specially designed forked grating [39], or alternatively, to use holograms [40] or Spatial Light Modulators [41]. This technique is easy to handle and implement. However, when high purity is needed, the beam has to be filtered. Another way to easily generate OAM beams is to transform a Hermite-Gaussian (HG) beam into a LG beam using cylindrical lenses [42], or, in an equivalent manner, using tilted usual lenses. In fact, in this case, the HG beam generation replaces the LG beam generation.

An elegant way to create OAM beams is to employ a spiral phase plate [43]. Such a plate acts as an azimuthally variable phase retardation that generates the phase of Equation (1). Alternatively, one can use a transverse phase profile to generate any type of beam, which can also be multiplexed [44] This spatial mode multiplexer is based on Multi-Plane Light Conversion and achieves a very high mode selectivity with low crosstalks. They are particularly well adapted to a given wavelength. However, these systems are rather bulky and thus restrict their use in integrated systems. To that purpose, although they are still based on a transverse phase profile, metamaterials could pave the way for the development of ultrathin optical devices [45,46,47]. Thin nanostructured holographic plates or compact planar metamaterial structures can generate structured beams within a broadband frequency range. They could then be used for integrated systems that directly generate OAM beams.

One can also mention the direct generation of LG beams with high topological charges inside a laser cavity, achieved by annular beam pumping [48,49]. A circular diaphragm produces a hollow-shaped intensity distribution in the near field of a pump laser beam. The intensity distribution of the lasing beam reflects the intensity distribution of the pumping beam. The order of the mode can be varied by changing the geometry of the cavity. It has to be noted that, in the terahertz range of the electromagnetic spectrum, metamaterials have also been used also to produce OAM beams [50,51,52], with the same principle as in optics. Several metasurfaces have been designed to demonstrate. High purity beams have been produced. In the radio domain, arrays of dipole emitters have been arranged for OAM generation [53]. A given phase difference between the emitters at a given location leads to an OAM carrying beam. The higher the number of dipoles, the higher the purity and the directivity of the generated beam.

3. Applications of OAM

3.1. Atom Guiding, Telecommunications and Interferometry

Although the term OAM refers to orbital angular momentum, some applications exploit the intensity or phase distribution of OAM beams. Strictly speaking, they are not based on the OAM of light. This section briefly mentions some of them.

3.1.1. Atom Guiding

One can benefit from the intensity distribution of LG beams. For $\mathsf{\ell }\ne 0$, this distribution has a donut shape (see Figure 1) with no intensity at its center. It can thus be used to trap or guide atoms that are repelled by light (dark traps) [54,55,56]. For example, the trapping of laser cooled atoms has been demonstrated in the core of a doughnut beam made of LG modes, enclosed by two additional beams [57]. Alternatively, three-dimensional geometries can trap atoms in a superposition of suitably phased LG modes in so-called bottle beams [58,59,60]. Such traps result from the superposition of LG modes that creates a ring-shaped intensity null bounded harmonically in all directions. For example, spin-relaxation lifetimes of up to 1.5 s have been observed for trapped rubidium atoms. LG beams can also be used as a funnel to guide atoms at their center either in free space [61], or in hollow core fibers [62]. In the case of free space propagation, cold atoms have been channeled up to a distance of 30 cm using high purity LG beams, whereas cold rubidium atoms have been guided in a 100 μm-diameter hollow-core silica fiber. This could be very helpful for atom manipulation.

On the other hand, LG beams and their superpositions are well suited for generating optical ring traps or ring lattices, where the atoms are trapped at locations of maximum intensity [63,64]. The optical ring lattice is formed by superposing two LG modes with opposite topological charges. The ring lattice trap could be rotated and the atoms can then be manipulated or rearranged. Such traps could be valuable in atom optics or atom interferometry. They may also simulate condensed matter effects such as persistent currents [65,66] or Mott insulators [67], which may pave the way towards atomtronics, i.e., performing the same functions as electronics but with atoms instead of electrons. Atoms could even be transferred between bright and dark ring traps, simply by modifying the laser detuning [68], or during propagation [69]. Depending on the detuning between the atomic resonance and the trapping laser, the atoms are confined in a dark or bright region. This approach may be valuable in microstructure or chiral material fabrication, since one can adapt the trap parameters when needed.

However, these applications rely on the beam’s intensity distribution and are not directly related to OAM.

3.1.2. Telecommunications

Since the LG beams considered in Equation (2) form an orthonormal basis, it was early recognized that LG beams may offer new diversity in telecommunications [23,24,70,71]. Moreover, since for a given frequency, the topological charge can vary from $-\infty$ to $+\infty$, there is, in principle, an infinite number of modes that could be used, with the same frequency. Since the different modes can be easily multiplexed and demultiplexed, as already mentioned, this leads to a superposition of independent modes without crosstalk. They propagate in the same direction in the case of free space, or within the same support in the case of guided propagation. However, the typical size of the beam increases with the charge [72,73], which limits the number of usable modes in practice.

Nevertheless, for a given optical frequency, such diversity with an infinite number of orthogonal modes can be obtained in any orthogonal basis, such as Hermite-Gaussian beams [74]. This is a form of Multiple Input-Multiple Output (MIMO) technique [75]. MIMO is an antenna technology used in wireless communications, where multiple antennas are employed at both the transmitter and receiver. The main advantage of using LG beams is that they maintain cylindrical symmetry throughout the telecommunication system [76,77]. Another point is that OAM carrying beams could be more robust against propagation turbulence [78]. This is particularly true when dealing with multiplexed beams [79], since the crosstalk between channels may be low. In particular, it was found both theoretically and experimentally, that the relative phase of the superposition of modes is not affected by the atmosphere. They remain orthogonal during propagation. This also establishes the feasibility for performing long-distance quantum experiments with the OAM of photons. However, it is also applicable to all spatially structured modes.

There is also another point linked to telecommunication that uses OAM beams. It deals with quantum key distribution and quantum entanglement. It has been shown that the use of spatial encoding, such as with OAM beams, dramatically increases the quantum key distribution transmission rate [80,81]. Specifically, The use of OAM beams allows for the transfer of more than 1 bit per photon, since OAM modes form a basis of infinite dimension. It has also been shown that quantum key distribution systems based on spatial-mode encoding can be more resilient against intercept-resend eavesdropping attacks. Although it is not, strictly speaking, restricted to OAM beams, it may strongly improve quantum cryptography communications. Quantum entanglement of the OAM state [82] enables high-dimensional entanglement, allowing for quantum cloning [83] and quantum teleportation [84]. In particular, multiple degrees of freedom can be simultaneously teleported. This leads a full quantum description of the teleported particle. It demonstrates an increase in technical control of scalable quantum technologies. It should be noted that these achievements could have been realized with other kinds of spatially structured light [85].

However, the very promising applications of OAM in telecommunications rely on the orthogonality properties of LG beams rather than on the angular momentum of the electromagnetic field. They are not directly related to OAM.

3.1.3. Interferences

OAM beams can also interfere. While interference may serve as a tool to fully characterize the OAM beam and to measure its topological charge [86,87,88], they can also be used in metrology [89,90,91]. Due to their unique spiral phase structure, vortex beams offer advantages for optical metrology applications, enabling highly sensitive detection of chiral interactions between light and matter. They effectively probe complex media, with applications in environmental monitoring, deep tissue imaging, and noisy channel communication. Figure 2 shows examples of interference patterns between ℓ and −ℓ OAM beams. These produce a daisy-like interference pattern, where the number of petals is twice the topological charge. Typically, to observe interference fringes in a standard experiment using plane waves, one must modulate the phase in one arm of the interferometer (for example, by changing the optical path). By contrast, in our case, the dark and bright fringes already exist in the pattern, from one petal to another. The phase modulation in one arm leads to a rotation of the interference pattern. The sensitivity of such an interferometer, which uses OAM, could be higher than that of conventional interferometers.

However, once again, this effect is not directly linked to the OAM. It only takes advantage of the phase distribution of the OAM beam to produce the daisy-like interference pattern.

3.2. OAM Transfer

As already mentioned, the applications of OAM beams described previously do not rely on the properties of the orbital angular momentum, but rather on the phase or intensity distribution of these fields.

One of the first major property of OAM beams is that angular momentum can be transferred to matter. Indeed, these beams that carry OAM that can be converted to a torque (see Equation (3)). This torque may then induce rotation in the system to which it has been applied. For this to occur, the system must either absorb the electromagnetic wave (see Figure 3 or change the topological charge of the beam. Due to OAM conservation, the system must then acquire OAM. However, this torque is quite weak. For a visible wavelength $\lambda =500$ nm and for a power of $P=1$ W, the torque equals $\Gamma =2.6\times {10}^{-16}$ Nm. At higher wavelengths, in the centimeter range ($\lambda =15$ cm, corresponding to a frequency $\nu =2$ GHz), and with the same power, the torque increases to $\Gamma =8.0\times {10}^{-11}$ Nm. One might think that using centimeter waves could make it easier to rotate systems. However, the characteristic size of the systems has to be of the order of or larger than the wavelength, and therefore increases accordingly. It thus also increases with it. Nevertheless, it is worth noting that, in the case of torques, systems could be suspended or floating. Gravity acts along the axis of the torque, which allows the elimination of gravitational effects on the rotational dynamics.

3.2.1. OAM Transfer to Atoms and Molecules

Observing atom rotation due to OAM transfer is quite challenging, especially for atoms at room temperature, where thermal agitation may mask the azimuthal movement. Additionally, the typical size of a focused OAM beam is on the order of the wavelength at minimum, while atomic sizes are fractions of a nanometer. As a result, the atom does not experience the vortex nature of the beam. This difficulty is reduced at low temperatures and in condensates.

The first direct observation of OAM-transfer was performed in 2006 on a Na condensate [92]. The transfer mechanism relies on a stimulated Raman process with a very ingenious scheme. The initial and final states of the two-photon transition differ in their linear momentum. The atoms absorb a photon from a Gaussian fundamental beam and emit it again into a counter-propagating OAM beam. There is thus both OAM transfer and a linear momentum transfer of $2\hslash k$, causing atoms that have undergone the Raman transition to be separated from the original condensate. In another experiment [93], the linear momentum was canceled using nearly co-propagating Raman beams.

One challenge could be to generate OAM beams with smaller sizes, thus surpassing the diffraction limit for OAM beams. This could be achieved, for example, using the Arago Poisson spot [94]. The diffraction of a plane wave by a circular object leads to a beam whose transverse dimension goes bellow dimensions of usual beams, in the nanometer range. An alternative approach could be to use plasmonics with metallic nanoscale resonant optical antennas [95,96], which significantly increase the intensity of the OAM beam, while still maintaining a beam of very small dimensions. This originates from the evanescent field in the vicinity of the nanoscale antennas. The goal of such beams is to induce transitions other than the dipolar transition associated with $|△J_z|⩾2$ such as dipolar magnetic, quadrupolar, or higher-order transitions [97], which are forbidden transitions within the usual dipolar approximation. Here, $△J_z$ refers to the change in angular momentum involved in the selection rules for atomic transitions. OAM transfer to a bound electron can also be realized with the help of polarization [98]. It should be noted that researchers have also succeeded in demonstrating interactions of OAM beams with large atoms, such as Rydberg atoms [99,100].

3.2.2. OAM Transfer to Micro/Meso-Scopic Particles

Since microscopic or mesoscopic particles have characteristic scales on the order of the typical size of an optical beam, it is much easier to transfer OAM to such particles. The first reported rotation was achieved using an absorbing micrometer-size particle with a few milliwatts of laser power, resulting in rotation frequencies from 1 to 10 Hz [17]. Biological cells have also been trapped and rotated [101]. Silica spheres of different diameters in the micrometer range have been set into rotation using high order LG beams [102]. The rotation detection has been performed following defects on the spheres. Furthermore, depending on the OAM beam structure, trapped objects could have been experimentally separated spatially depending on their size. This could have applications in the filtration of liquids. The main advantage of using OAM, compared with the transfer of spin angular momentum, lies in the possibility of transferring more than ℏ per photon. A discussion on the use of OAM beams to rotate microscopic particles can be found in [19,35,103].

One research direction would be to use OAM in optofluidics, i.e., creating microfluidic functions with light. It would be practically beneficial to rotate micromotors using OAM and to incorporate them into liquid flows [104,105]. It should be noted that the diffraction pattern behind an asymmetric object corresponds to an OAM beam [106,107], which can induce rotation of the object due to the conservation of angular momentum.

3.2.3. OAM Transfer to Macroscopic Objects

The first experiment conducted with a macroscopic object was performed in the radio domain [18]. An OAM beam generated by a so-called turnstile antenna was reflected (with its topological charge inverted [108,109]) in a copper ring that was set into rotation. A similar experiment was carried out in the optical domain with an absorbing object. This enabled the precise measurement of the torque, which can be considered a torque meter [110]. In fact, this can provide evidence of the quantization of light. The ratio of the torque $\Gamma$ with the optical power P divided by the optical pulsation $\omega$ equals

$${\displaystyle \frac{\Gamma \omega }{P}}=\mathsf{\ell }$$

(4)

Experimentally, it is found that this value is always an integer, indicating that the topological charge is quantized. Dividing the torque by the quantum of action ℏ times the topological charge yields the number of particles involved in the torque per second, each carrying an angular momentum equal to $\mathsf{\ell }\hslash$. These particles could be associated with photons. This number also equals the power divided by $\hslash \omega$, which is usually assumed to be the number of photons in the light beam. This kind of reasoning is very similar to that of Raman and Bhagavantam [111], used to precisely demonstrate the quantization of light by considering polarization.

We anticipate that transferring OAM to macroscopic objects could be a valuable tool for systems that experience little friction, such as those in vacuum, like satellites in space. Satellite positioning is a real challenge. Currently, it is usually achieved using heavy systems inside the satellite that are set into rotation. Due to the conservation of angular momentum, the satellite also rotates, but in the opposite direction. However, these systems also generate harmful vibrations [112]. The emission of OAM beams with long wavelengths by the satellite itself could be a promising alternative.

3.3. Rotational Doppler Effect

The rotational Doppler effect is the counterpart of the usual linear Doppler effect, but for systems in rotation. A helpful illustration has been proposed by Miles Padgett to provide an intuitive understanding of this effect [113]: “Place a watch at the centre of a rotating turntable, and, viewed from above, its hands will seem to rotate more quickly. This classical effect applies to all rotating vectors, for example to the spatial pattern of the electric field of any light beam carrying angular momentum. The electric-field vector rotates at the frequency of the light, but an additional rotation of the beam about its axis of propagation will speed up or slow down the field rotation, resulting in a frequency shift proportional to the rate of rotation of the beam.” This idea is depicted in Figure 4. It is very similar to the one-day gain achieved by Phileas Fogg during his eastward around-the-world circumnavigation in J. Verne’s famous novel “Around the World in Eighty Days” [114].

3.3.1. Rotational Doppler Effect in Terms of Energy Conservation

Let us draw a parallel with the usual Doppler effect [115,116]. Consider the simple case of the normal reflection of a wave on a moving surface (velocity $\overrightarrow{v}$ in the direction of propagation). During reflection, for a single photon, the change in optical linear momentum during the interaction is $\overrightarrow{p}=2\hslash \overrightarrow{k}$. The work done by the force then equals $2\hslash \overrightarrow{k}·\overrightarrow{v}$. Due to energy conservation, this work implies that the energy of the photon must change accordingly

$$△E=\hslash △\omega =2\hslash \overrightarrow{k}·\overrightarrow{v},\mathrm{leading}\mathrm{to},△\omega =2\overrightarrow{k}·\overrightarrow{v}$$

(5)

which is the usual linear Doppler effect formula. It can be adapted for non-normal incidence.

Consider now the reflection of a single photon carrying OAM on a surface rotating at angular frequency $\Omega$, with a change in the topological charge $△\mathsf{\ell }$ (see Figure 5). Due to the conservation of total angular momentum, the rotating surface experiences a torque during the interaction. The work done by this torque equals $△\mathsf{\ell }\hslash \Omega$. Because of energy conservation, the energy of this single photon must change according to

$$△E=\hslash △\omega =\hslash \Omega △\mathsf{\ell },\mathrm{leading}\mathrm{to},△\omega =\Omega △\mathsf{\ell }$$

(6)

It should be noted here that the rotational Doppler effect is associated with a change in the topological charge. Therefore, there is no rotational Doppler shift upon simple reflection on a standard mirror.

3.3.2. Use of the Rotational Doppler Effect

The most common application of the rotational Doppler effect is in determining the rotation frequency of a diffusing object [117,118,119,120], even if the object cannot be detected using imaging techniques [121]. A light beam carrying OAM (or a superposition of OAM beams) impinges normally on a rotating, diffusing object, with the rotation axis aligned with the optical axis. After diffusion, the light no longer carries OAM. The rotational Doppler effect is detected via the beat frequency between the scattered light and a reference beam or another diffused beam. Such a technique can also be applied to detect non-uniform rotations, as occur in vortices in liquids [122,123,124] or gases [125]. The angular distribution of the rotation is inferred from the frequency-shifted diffused light at a given distance from the axis. This allows mapping of the entire vortex and identifying its type by varying the beam size or the topological charge.

Rotational Doppler effects can also be observed with a fundamental Gaussian beam incident on rotating objects with specific symmetries [126,127,128]. In this case, the reflected or transmitted light acquires this symmetry. After interaction, the beam can be decomposed into a Laguerre-Gaussian basis. Consequently, several peaks appear in the frequency spectrum, corresponding to rotationally Doppler-shifted components. In particular, an object with a $2\pi /\mathsf{\ell }$ symmetry results in a non-zero contribution from the LG mode with topological charge ℓ. This approach could, for example, be used to detect a rotating airplane propeller.

Actually, the rotational Doppler effect is still in its early stages of development. It aims to be, for rotating objects, what the linear Doppler effect is for objects moving with a longitudinal velocity. It could then be used to detect or characterize any form of rotation, including vortices with a distribution of rotational velocities, such as eddies, wake turbulences, or tornadoes. However, several challenges remain to be addressed. The first is the issue of misalignment [129]. Misalignment—whether a lateral displacement or a skew angle—leads to the appearance of multiple peaks in the frequency spectrum. From this spectrum, one must then determine both the rotation frequency of the object and the position of the rotation axis. Machine learning could be a valuable tool in this process [130].

4. Conclusions

In this entry, we have discussed the origin of OAM and its potential applications. Some popular uses, such as guiding or trapping, rely on the intensity distribution of the beam. Others, like telecommunications, are based on the orthogonality between different modes, which do not directly involve the angular momentum characteristics of the beam. Similarly, interferometry using OAM also leverages the phase distribution rather than the angular momentum itself. We have highlighted two applications that are truly rooted in OAM, namely angular momentum transfer and the rotational Doppler effect. Finally, we discussed their potential and current limitations.

OAM beams are not limited to the visible part of the spectrum. UV and X-ray OAM beams have been reported, which could be used to manipulate nanostructures, molecules, or atoms. This would not be feasible with visible beams because the wavelength and the ultimate size of the beam are too large compared to the characteristic scales of these systems. Conversely, radio OAM beams have also been demonstrated. Thanks to their long wavelength and the fact that the torque depends linearly on it, these beams can significantly increase the torque and be used to manipulate heavier objects. They could also be employed to detect larger vortices in meteorology, oceanography, or astronomy.

OAM is not limited to electromagnetic waves. It has also been observed in acoustic waves, which can propagate through liquids and matter with quite high angular momentum. Additionally, OAM has been demonstrated with matter waves, which can be used to rotate nanoparticles or to probe the chirality of systems and molecules.