In 1992, Les Allen and coworkers within Han Woerdman's group realized that light beams with an azimuthal phase dependence described by an additional exp(iâθ) term, where â can take any integer value and θ is the angular position within the beam, carry an orbital angular momentum. This orbital angular momentum is additional to the spin angular momentum associated with circular polarization. These azimuthally phased beams comprise â intertwined helical phase fronts, skewed with respect to the beam axis, meaning that at all positions within the beam cross section, the wavefront normal and associated momentum density has a well-defined azimuthal component. The insight of Allen and coworkers was that for all beams described in this way, when integrated over the beam cross section, the orbital angular momentum (OAM) is ââ per photon. Laguerre-Gaussian and high-order Bessel beams are both examples of mode families having these properties. A further feature of these beams is that the phase singularity on the beam axis precludes a nonzero on-axis intensity and therefore all such beams with â â 0 have an annular intensity cross section (see Fig. 4.1). The circulation of the optical momentum and energy around the singularity leads them also to be called optical vortices. The further recognition that beams carrying defined amounts of OAM could be created within the laboratory means that over the last 15 years they have formed the basis of many investigations, ranging from the transfer of OAM to microscopic particles (in optical spanners or optical wrenches) to studies of the fundamental properties of light in both the classical and quantum regimes. However, although OAM is an immensely useful concept, one must emphasize that all its properties are simply those predicted by standard electromagnetic theory. The purpose of this chapter is to consider how sets of helical laser modes carrying OAM can be used for information transfer.
In 1992, Les Allen and coworkers within Han Woerdman's group realized that light beams with an azimuthal phase dependence described by an additional exp(iâθ) term, where â can take any integer value and θ is the angular position within the beam, carry an orbital angular momentum. This orbital angular momentum is additional to the spin angular momentum associated with circular polarization. These azimuthally phased beams comprise â intertwined helical phase fronts, skewed with respect to the beam axis, meaning that at all positions within the beam cross section, the wavefront normal and associated momentum density has a well-defined azimuthal component. The insight of Allen and coworkers was that for all beams described in this way, when integrated over the beam cross section, the orbital angular momentum (OAM) is ââ per photon. Laguerre-Gaussian and high-order Bessel beams are both examples of mode families having these properties. A further feature of these beams is that the phase singularity on the beam axis precludes a nonzero on-axis intensity and therefore all such beams with â â 0 have an annular intensity cross section (see Fig. 4.1). The circulation of the optical momentum and energy around the singularity leads them also to be called optical vortices. The further recognition that beams carrying defined amounts of OAM could be created within the laboratory means that over the last 15 years they have formed the basis of many investigations, ranging from the transfer of OAM to microscopic particles (in optical spanners or optical wrenches) to studies of the fundamental properties of light in both the classical and quantum regimes. However, although OAM is an immensely useful concept, one must emphasize that all its properties are simply those predicted by standard electromagnetic theory. The purpose of this chapter is to consider how sets of helical laser modes carrying OAM can be used for information transfer.
Photons are an ideal candidate for encoding both classical and quantum information. Besides spin angular momentum associated with circular polarization, single photon can also carry other fundamentally new degree of freedom of orbital angular momentum related to the spiral phase structure of light. The key significance of orbital angular momentum lies in its potential in realizing a high-dimensional Hilbert space and in encoding a high-dimensional quantum information. Since Allen et al. [Allen L, Beijersbergen M W, Spreeuw R J C, Woerdman J P 1992 Phys. Rev. A 45 8185] recognized the physical reality of photon orbital angular momentum in 1992, rapidly growing interest has been aroused in orbital angular momentum (OAM) from both classical and quantum points of view. Here we present an overall review on the high-order orbital angular momentum of photon, including its preparation and manipulation based on some specific techniques and also its applications. The spatial light modulator is a commercial device that has been widely employed to generate the OAM beams. We make and identify the optical OAM superposition with very high quantum numbers up to l=360. Recently, the metallic spiral phase mirrors were also developed to produce high-order OAM beams up to l=5050. In addition, the Q-plates made of anisotropic and inhomogeneous liquid crystals were invented to generate high-order OAM beams in a polarization-controllable manner, and the OAM superposition of l=± 50 were achieved. Owing to high rotational symmetry, these high OAM beams have been found to have more and more important applications in the fields of high-sensitivity sensing and high-precision measurements. Two fascinating examples are discussed in detail. The first example is that the research group led by Prof. Zeilinger has prepared and observed the quantum entanglement of high orbital angular momenta up to l=±300 by the technique of polarization-OAM entanglement swapping, and they demonstrated that the angular resolution could be significantly improved by a factor of l. Their result was the first step for entangling and twisting even macroscopic, spatially separated objects in two different directions. The second example is that the research group led by Prof. Padgett has demonstrated an elegant experiment of rotational Doppler effects for visible light with l=±20 OAM superposition. They showed that a spinning object with an optically rough surface might induce a Doppler effect in light reflected from the direction parallel to the rotation axis, and the frequency shift was proportional to both the disk's angular speed and the optical OAM. The potential applications in noncontact measurement of angular speed and in significant improvement of angular resolution for remote sensing will be particularly fascinating.
Optical vortices became a hot topic since almost two decades ago, when it was recognized that Laguerre-Gaussian laser modes carry orbital angular momentum [Allen et al. Phys Rev A 45, 8185 (1992)] related with a screw phase dislocation, and different from the spin angular momentum associated to circular polarization. In 1995, this dynamical quantity was transferred to matter in an optical micromanipulation system for the first time [H. He, et al., Phys. Rev. Lett. 75, 826 (1995)], and since then, a number of studies on angular momentum of light have unveiled different interesting aspects on the subject. However, there are still open questions, which have arisen together with the generation of novel light beams, such as vector vortices, for instance. In contrast with scalar vortices, with usual polarization states (linear, circular, elliptical), the orientation and magnitude of the electric field of vector vortices (solutions of the vector wave equation) is a function of space and time. In this work, we present an experimental study of the local angular momentum density of a Bessel vector vortex of first order by means of an optical trap. For this purpose, we used different probe particles in order to sense the local contribution to the optical angular momentum in each region of the beam. But optical fields are not the only wave fields that may exhibit phase dislocations or singularities. There are close analogies between light and sound fields that can be exploited in order to get a better understanding of common phenomena and study new aspects in both branches of physics. Here we also present the first experimental demonstration and theoretical analysis of acoustical vortices in free field, with similar properties to those of the optical vortices, including the angular momentum that can be transferred to matter. The corresponding analogies and differences with the optical case turn out to be very enlightening for the understanding of the phenomenon of angular momentum in wave fields.
Introduction Although the orbital angular momentum (OAM) of light is often quantified in terms of ℓℏ per photon, one should recognise that this result can be derived directly from Maxwell's equations and is most certainly not an exclusively quantum property. Following the recognition of optical OAM in 1992 [1], the immediate thrust of experimental work was to observe OAM transfer to microscopic objects. These exciting experiments by Rubinsztein- Dunlop [2, 3] and others established that the predicted angular momentum was truly mechanical, but did not provide insight on OAM at the level of single photons. Laguerre- Gaussian modes were identified as being a natural basis set for describing beams with OAM, where their phase terms describe the helical phasefronts corresponding to an OAM of ℓℏ per photon. The polarisation of light has played an important historical role in the experimental investigation of quantum entanglement, quantum gate operations and quantum cryptography. A potential advantage of OAM is that it resides in a higher-dimensional state space which, as follows from the theory, is infinite-dimensional. One of the experimental challenges is to access a large number of these dimensions. In quantum communication, the most obvious advantage of high n -dimensional systems is the increased bandwidth [4], but qunits also lead to stronger security [5, 6]. An increase in the number of entangled quantum modes has also been shown to improve measures of nonlocality [7], to allow quantum computation with better fault tolerance [8] and to simplify quantum logic structures [9].
Twisted light has received a lot of attention in recent years. And there are justified reasons for this. The properties of twisted light are unusual and differ fundamentally from the plane-wave properties. The phase fronts of these light beams have a helical structure, giving them the name twisted. This helical phase structure originates from the well-defined orbital angular momentum (OAM) that they carry along their propagation direction and in addition to the spin angular momentum (SAM). Moreover, they have a characteristic intensity profile, which consists of concentric rings with a central minimum, and a phase singularity in the beam center. This thesis is based on two main parts, each dealing with a different aspect of twisted light. These investigations are carried out using Bessel beams, one type of twisted light. Twisted light beams are cylindrically symmetrical. Therefore, they are defined on the basis of circularly polarized light. This fact leads to the first question: Is it possible to generate twisted light in other polarization states? In this work, we derive how this can be achieved by using the superposition principle and how this can be used to construct linearly, radially, or azimuthally polarized Bessel beams. The focus in the second part is on excitations of atoms by twisted Bessel beams. In particular, we analyze how the use of twisted light can modify individual atomic multipole transitions. The obtained results show that for an efficient modification a precise localization of the target atom is required. In the last chapter, both aspects are combined. We analyze transitions between magnetic hyperfine states in a single trapped atom (or ion), which is exposed to an additional external magnetic field. The presented calculations for these multipole excitations are based on a formalism that accounts for the alignment of the applied external magnetic field with respect to the light propagation direction, and additionally different polarization states of the radiation field are considered. The obtained results indicate that twisted light could be of particular interest for precision experiments. One possible application is the use in single-ion clocks. The work shows that high multipole orders of the vortex beams allow excitations in the intensity minimum in the center of these beams without causing large disturbances, and thus may enable a significant reduction of the light shift in atomic transitions.
More than a century ago, pioneeringworks carried out by Poynting (1909) and other physicists have provided the evidence of the validity of Marxwell equations for an electromagnetic field, showing how a beam of light carries energy and momentum, both in the linear and angular components. In particular the angular momentum of light, related to the generator of rotations in quantum mechanics, has been typically associated with its polarization, and more specifically with its circular polarization components. An optical beam traveling in the positive direction of the z axis that is circularly polarized, carries a z-component angular momentum content σ = ±h per photon, which is positive if the circular polarization is left-handed and negative if it is right-handed. This angular momentum content is not just a formal property, but a very concrete one that can have significant mechanical effects as for example an observable induced rotation by absorption to amaterial particle Beth (1936); Friese et al. (1998)
The orbital angular momentum density of Bessel beams is calculated explicitly within a rigorous vectorial treatment. This allows us to investigate some aspects that have not been analysed previously, such as the angular momentum content of azimuthally and radially polarized beams. Furthermore, we demonstrate experimentally the mechanical transfer of orbital angular momentum to trapped particles in optical tweezers using a high-order Bessel beam. We set transparent particles of known dimensions into rotation, where the sense of rotation can be reversed by changing the sign of the singularity. Quantitative results are obtained for rotation rates. This paper's animations are available from the Multimedia Enhancements page.