Introduction

Helical phase fronts are commonly associated with Laguerre-Gaussian (LG) beams and optical vortices. These beams are characterized by an input field normally expressed as exp(−ilϕ), where l is an integer called the topological charge, which describes the magnitude and handedness of the helical phase profile. In quantum theory, the topological charge relates to a quantized orbital angular momentum (OAM) of lℏ per photon [1–2]. When projected in the far field, the azimuthal components of the phase create destructive interference giving rise to a dark centre surrounded by a high-intensity ring of light via constructive interference. As the topological charge is increased, the steeper phase gradients deflect light farther off-axis, thereby enlarging the light ring surrounding the dark centre.

When focused using a lens, the propagation of an LG beam along the optical axis follows a conical ray until it reaches a minimum ring radius at the focus and then conically increases after the focus. A high concentration of photons is maintained at the outskirts of the conical beam forming a ring at the transverse plane. Increasing the topological charge (and consequently the OAM) disperses the distribution of photons around a larger ring. Aside from circularly symmetric beams, higher-order LG beams bring about unique laser transverse modes, which display unique three-dimensional (3D) light patterns. Hence, the interplay of helical phase fronts results in far field beam profiles that can be uniquely utilized in optical tweezers [3–7] or as means of engineering the point spread function in laser microscopy [8].

Introduction

Higher-order modes of optical beams have been the subject of many studies over the past 20 years [1, 2]. Because laser resonators deliver in principle quite complex spatial patterns, spatial modes of beams were studied intensively when lasers were first developed [3]. Beams in higher-order spatial modes are solutions of the paraxial wave equation, with Hermite-Gauss and Laguerre-Gauss beams being the most important families of beams. These are solutions of the wave equation in Cartesian and cylindrical coordinates, respectively. The latter beams have been at the heart of a revival of research on higher-order modes due to the orbital angular momentum that they carry [4]. They have also received much attention due to the phase singularities present in their transverse amplitude [5]. Higher-order modes have stimulated much research in the application of forces and torques to objects in optical tweezers [1, 2]. Their usefulness has carried higher-order spatial modes further into new paths, in studies with non-classical sources of light and applications in quantum information [6].

The beams mentioned above are scalar solutions of the wave equation and thus independent of the polarization of the light. Vector beams are formed by the non-separable combinations of spatial and polarization modes. This enhanced modal space produces an interesting set of beams that offer new effects and applications. The origin of these beams is not recent either, as the possibility of combining higher-order spatial modes and polarization started with those early studies of modes as well.

For optical fields the notion of a total angular momentum has long been known. The concept of a light beam carrying orbital angular momentum, however, was unfamiliar until it was discovered that Laguerre-Gaussian beams, within the paraxial approximation, carry a well-defined orbital angular momentum [1, 2]. This discovery started the modern interest in orbital angular momentum of light. In this chapter we discuss the theoretical framework of orbital angular momentum of light in terms of fields and light beams and how to generate these. The material in this chapter is based in parts on the PhD thesis of Götte [3].

Introduction

A quantitative treatment of the mechanical effects of light became possible only after light had been integrated into Maxwell's dynamical theory of electromagnetic waves. With this theory Poynting [4] derived a continuity equation for the energy in the electromagnetic field. After Heaviside [5, 6] introduced the vectorial notation for the Maxwell equations this continuity equation could be written in its modern form using the Poynting vector. Interestingly, the linear momentum density in the electromagnetic field is also given by the Poynting vector apart from constant factors depending on the chosen system of units. Poynting [7] also derived an expression for the angular momentum of circularly polarised light by means of a mechanical analogue in the form of a rotating shaft. Later, Poynting's expression was verified by measuring the torque on a quarter wave-plate due to circularly polarised light [8].

Elementary particle physics addresses the question, “What is matter made of?” on the most fundamental level – which is to say, on the smallest scale of size.

It's a remarkable fact that matter, at this scale, comes in tiny chunks, separated by vast empty spaces. This is radically different from our everyday experience, where matter appears to fill everything. Solids seem … well … solid, liquids are smooth and continuous, and even gases, such as the air in a room, occupy every nook and cranny. But the world of the very small is much more like the night sky, with pinpoints of light here and there, but mostly just nothing.

Even more surprising, these “chunks” come in a relatively small number of different types – there are protons and electrons, pi mesons and neutrinos, … only a few dozen in all. Again, this is totally different from our macroscopic world, where there are rocks and dirt clods, bananas and chimpanzees, trees, books, and people – variety without limit.

Most astonishing of all, the chunks of any particular type are not just “pretty similar,” like two Fords coming off the same assembly line, but absolutely utterly identical. There aren't fat electrons and skinny ones, or young electrons and old electrons, or happy electrons and sad electrons – if you've seen one, you've seen them all. There is nothing remotely like this absolute identicalness in everyday life – it is a quantum phenomenon, inconceivable in classical physics, where you could always, if necessary, paint a red spot on the object, or stamp a serial number on it.

Introduction

Vortices and vorticity are common objects of study in the field of fluid dynamics [1]. In a fluid, vortices appear as spinning, often turbulent, flows, giving rise to common phenomena such as tornadoes and whirlpools. In their more discreet manifestations, fluid vortices are thought to be responsible for the destruction of bridges due to the formation of vortex streets and structural resonance [2] and for the flight of insects [3], ending a long debate in the community about the reversibility of the movement of insect wings. Because of the importance of all those natural phenomena, the study of vortex dynamics and the creation and annihilation of hydrodynamical vortices is a matter of intense study.

When the first properties of superfluids were found, the question of whether vortices could appear in such irrotational fluids was addressed by prominent scientists [4, 5]. It was found that vortices could also appear in such systems, but they could only appear as point vortices, i.e. singular locations where the vorticity of the fluid was infinite, whereas everywhere else the vorticity was strictly zero. This property was also found later on in Bose-Einstein condensates (BECs), where the quest for exciting and controlling the point vortices of quantum systems is still a hot topic. The dynamics of and processes related to the birth and destruction of vortex-antivortex pairs play an important role in the physics of these exotic systems.

Introduction

There exist two well-established methods to trap charged particles: the Penning trap [1] and the Paul trap [2]. In the Penning trap the particle is confined in space by a combination of static magnetic and electric fields. In the Paul trap the trapping is caused by a high frequency electric quadrupole field. The subject of this article is to present a third mechanism for trapping charged particles – trapping by beams of electromagnetic radiation. It was discovered some time ago [3–6] that a properly prepared beam acts as a “waveguide” for particles, confining their motion in the transverse directions. Similar phenomena occur in RF fields [7]. In all these cases, the essential role is played by the electric field configuration in the plane perpendicular to the beam axis (for nonrelativistic electrons, the magnetic field is less important). Particles are confined to the vicinity of the minimum-energy points. In particular, for beams of electromagnetic radiation carrying orbital angular momentum such points lie on the beam axis. One beam may confine particles only in the transverse direction. Two or three crossing beams may fully confine particles, acting as a substitute for the Paul trap.

Trapping of charged particles by beams of electromagnetic radiation is based on the same general principles as the Paul trap. In the Paul trap the quadrupole radio-frequency field is produced by a system of electrodes. In our case, the field is that of freely propagating electromagnetic beams. In both cases the essential role is played by two factors: the proper shape of the force field and fast oscillation or rotation. These features are clearly seen when we invoke the notion of the ponderomotive potential.

The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure.

The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found.

Introduction

The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2π radian range or whether it evolves continuously without bound.

Cosmology is the study of the structure and evolution of the universe as a whole. Prior to the twentieth century this was the domain of religion and speculative philosophy, but it is now a serious branch of physics, and one with spectacular implications. The transition from myth to science was brought about, in large measure, by two empirical findings: Hubble's observation (in 1929) that the Universe is expanding, and the discovery by Penzias and Wilson of Cosmic Microwave Background radiation (in 1965). Taken together, they indicate that the Universe as we know it began with a gigantic explosion 13.7 billion years ago – the Big Bang. In the next two sections we will explore these developments.

But first I would like to alert you to a fundamental assumption that informs almost all modern thinking about cosmology. Before Copernicus, most people assumed that the Earth is at the center of the Universe. The Sun, the Moon, the planets, and the stars orbit around us. Copernicus showed, to the contrary, that while the Moon orbits the Earth, the planets (including the Earth) orbit the Sun. And we now know that the Sun itself is just one of billions and billions of stars. The Earth is not at the center of the Universe – nor is the Sun. Modern cosmology takes the Copernican revolution to its logical conclusion: The Universe has no center, or any kind of “preferred” location – at a given time, it is the same in all places and in all directions. This is known as the Cosmological Principle.

Introduction

When light engages with matter, the interactions that take place at the photon level are subject to the operation of powerful underlying symmetry laws. Such principles underpin the physics of even the simplest photon interactions, as for instance in the familiar Planck- Einstein relation E = hν for the absorption or emission of radiation. As emerged from Noether's work [1], this manifestation of overall conservation of energy is a direct consequence of a system invariance under temporal translation [2]. In connection with specific atomic photophysics, the term ‘selection rule’ is often used in connection with other space or time symmetries, these frequently being manifest as constraints over the conservation of quantized angular momentum. Obvious examples are the rules that govern the ‘allowed’ and ‘forbidden’ lines in atomic spectra, where the associated conditions over the geometric disposition and flow of charge emerge in the form of transition multipoles [3, 4].

The angular momentum attributes of light are most familiar in connection with the integer spin of the photon. Circularly polarized states have well-defined spin angular momentum along the direction of propagation [5], and numerous chiral or gyrotropic interactions exploit the differences in behaviour – observed in a material that is itself chirally constituted – between light beams of left- and right-handedness [6–8]. The principles are well known, and their applications have a surprisingly wide compass.

Classical physics, some aspects of which we discussed in Chapter 1, is – for the most part – comforting to our intuitions. You probably wouldn't have come up with Newton's second law (F = ma) on your own (after all, nobody did before Newton), but once it is on the table it feels right. It seems consistent with our everyday experience. Classical physics refines and perfects our intuitions, but it doesn't upset them. By contrast, the four revolutions in twentieth-century physics are wildly counterintuitive; they seem to contradict everything we thought we understood – everything we took for granted about the world. That is, in part, what makes them so interesting. But it also raises a recurring question: “If this is really true, how come I never noticed it before?” I hope you will keep a skeptical eye on that subtext, as we go along.

Einstein's postulates

Einstein published his Special Theory of Relativity in 1905. The special theory is not an account of any particular physical phenomenon; rather, it is a description of the arena in which all phenomena occur. It is a theory of space and time themselves. As such, it takes precedence over all other theories. If you were to propose a new model of elementary particles, say, the first thing to ask would be, “Is it consistent with special relativity?” If not, you have some fast talking to do. As Kant would say, special relativity is a prolegomenon to any future physics.

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].

Introduction

The orbital angular momentum (OAM) carried by light is widely seen as an extremely useful optical characteristic, with applications in many areas of optics. It was Allen et al. [1] who recognised that a helically phased light beam with a phase cross-section of exp(iℓϕ) carries an OAM, with a value of ℓℏ per photon. Such a light beam contains an optical vortex line of ℓ on its axis. One issue that is yet to be completely resolved is the development of a simple and 100% efficient method for the measurement of OAM.

A better known case of optical angular momentum is spin angular momentum (SAM). SAM is associated with the polarisation state of the light; the spin angular momentum in a left and right circularly polarised beam is σℏ=±1, per photon, respectively [2]. The SAM can be easily determined through the use of a polarising beam splitter, where a π/4 waveplate converts circular polarised light into a p- or s-polarised state which is then transmitted or reflected to give one of two outputs, as shown in Fig. 13.1(a).

OAM arises from the amplitude cross-section of the beam and is therefore independent of the spin angular momentum. One key characteristic of beams carrying OAM is that whereas SAM has only two orthogonal states, the OAM is described by an unbounded state space, i.e. ℓ (as in exp(iℓϕ) can take any integer value [3].

Introduction

The theoretical foundations for describing the influence of twisted light on atoms, ions and molecules are essentially the same as those leading to their cooling and trapping by ordinary plane wave laser light [1–4]. A third of a century has passed since the Doppler mechanism was first put forward by Wineland and Dehmelt as a mechanism for cooling ions [5], and then by Hänsch and Schawlaw for neutral atoms [6]. The key role played by the Doppler mechanism is best illustrated by the simplest case of one-dimensional optical molasses, where a two-level atom is subject to two identical counter-propagating plane waves of a frequency near to resonance with the atomic transition frequency. In terms of a classical description, at small velocities the atom experiences a friction force F= −αv, where v is the velocity vector of the atomic centre of mass parallel to the common axis of the light beams, and α is a friction coefficient. The friction force gradually cools the atoms to progressively smaller, albeit limited, velocities and can be made to operate in all three dimensions by use of multiple beams. The cooling technique based on the Doppler mechanism has been superseded by more effective cooling mechanisms, especially evaporative and Sisyphus cooling mechanisms, culminating, with the achievement of super-cooling, in the realisation of Bose-Einstein condensates (BECs) [7]. Nevertheless, the Doppler mechanism remains important as a mechanism for the early stages of cooling and still plays a key role in the interaction of atomic systems with laser light.

Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.”

I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world).

Photons

Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c, and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid.

Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.