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].

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.

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.

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.

Introduction

In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics.

Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes, which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11].

There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m, which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.

Artificial optical materials and structures have enabled the observation of various new optical effects. For example, photonic crystals are able to inhibit the propagation of certain light frequencies and provide the unique ability to guide light around very tight bends and along narrow channels. With metamaterials, on the other hand, one can achieve negative refraction. The high field strengths in optical microresonators lead to nonlinear optical effects that are important for future integrated optical networks, and the coupling between optical and mechanical degrees of freedom opens up the possibility of cooling macroscopic systems down to the quantum ground state. This chapter explains the basic underlying principles of these novel optical structures.

Photonic crystals

Photonic crystals are materials with a spatial periodicity in their dielectric constant, a system that was first analyzed by Lord Rayleigh in 1887 [1]. Under certain conditions, photonic crystals can create a photonic bandgap, i.e. a frequency window within which propagation of light through the crystal is inhibited. Light propagation in a photonic crystal is similar to the propagation of electrons and holes in a semiconductor. An electron passing through a semiconductor experiences a periodic potential due to the ordered atomic lattice. The interaction between the electron and the periodic potential results in the formation of energy bandgaps. It is not possible for the electron to pass through the crystal if its energy falls within the range of the bandgap.

Why should we care about nano-optics? For the same reason we care about optics! The foundations of many fields of the contemporary sciences have been established using optical experiments. To give an example, think of quantum mechanics. Blackbody radiation, hydrogen lines, or the photoelectric effect were key experiments that nurtured the quantum idea. Today, optical spectroscopy is a powerful means to identify the atomic and chemical structure of different materials. The power of optics is based on the simple fact that the energy of light quanta lies in the energy range of electronic and vibrational transitions in matter. This fact is at the core of our abilities for visual perception and is the reason why experiments with light are very close to our intuition. Optics, and in particular optical imaging, helps us to consciously and logically connect complicated concepts. Therefore, pushing optical interactions to the nanometer scale opens up new perspectives, properties and phenomena in the emerging century of the nanoworld.

Nano-optics aims at the understanding of optical phenomena on the nanometer scale, i.e. near or beyond the diffraction limit of light. It is an emerging new field of study, motivated by the rapid advance of nanoscience and nanotechnology and by their need for adequate tools and strategies for fabrication, manipulation and characterization at the nanometer scale. Interestingly, nano-optics predates the trend of nanotechnology by more than a decade. An optical counterpart to the scanning tunneling microscope (STM) was demonstrated in 1984 and optical resolutions had been achieved that were significantly beyond the diffraction limit of light.

We are very pleased that this textbook found wide use and reasonably high demand. Since the first printing of the first edition in 2006, the field of nano-optics has gained considerable momentum and new research directions have been established. Among the new topics are metamaterials, optical antennas, and cavity optomechanics, to name but a few. The high field localization associated with metals at optical frequencies has given rise to the demonstration of truly nanoscale lasers and the high nonlinearity of metals is being used for frequency conversion in subwavelength volumes. These new trends define a clear motivation for a second edition of Principles of Nano-Optics.

The overall structure of the book has been left unchanged with the exception of a new chapter on optical antennas (Chapter 13). Chapter 2 (Theoretical foundations) has been adjusted to include topics such as reciprocity and energy density in lossy media, and Chapter 4 (Resolution and localization) has been extended by including new microscopy techniques, such as structured illumination and localization microscopy. Chapter 5 received a major polish: optical microscopy is now classified in terms of interaction orders between probe and sample. On the other hand, Chapter 6 has been condensed since some near- field techniques are no longer of general interest. Several new topics have been included in Chapter 8, which covers the theory of localized light-matter interactions. Among the new sections is a discussion of Fano interference, strong coupling between modes, and level crossing.

In order to measure localized fields one needs to bring a local probe into close proximity to a sample surface. Typically, the probe-sample distance is required to be smaller than the size of lateral field confinement and thus smaller than the spatial resolution to be achieved. An active feedback loop is required in order to maintain a constant distance during the experiment. However, the successful implementation of a feedback loop requires a sufficiently short-ranged interaction between the optical probe and the sample. The dependence of this interaction on the probe-sample distance should be monotonic in order to ensure a unique distance assignment. A typical block diagram of a feedback loop applied to scanning probe microscopy is shown in Fig. 7.1. A piezoelectric element P(ω) is used to transform an electric signal into a displacement, whilst the interaction measurement I(ω) takes care of the reverse transformation. The controller G(ω) is used to optimize the speed of the feedback loop and to ensure stability according to well-established design rules. Most commonly, a so-called PI controller is used, which is a combination of a proportional gain (P) and an integrator stage (I).

Using the (near-field) optical signal itself as a distance-dependent feedback signal seems to be an attractive solution at first glance. However, it turns out that this is problematic. (1) In the presence of a sample of unknown and inhomogeneous composition, unpredictable variations in the near-field distribution give rise to a non-monotonic distance dependence.

Optical measurement techniques, and near-field optical microscopy in particular, exist in a broad variety of configurations. In the following we will derive an interaction series to understand and categorize different experimental configurations. The interaction series describes multiple scattering events between an optical probe and a sample and is similar to the Born series in light scattering. We will start out by discussing far-field microscopy first and then proceed with selected configurations encountered in near-field optical microscopy.

The interaction series

The interaction of light with matter can be discussed in terms of light scattering events [1, 2]. Figure 5.1 is a sketch of a generic geometry considered in the following. The sample and – in the case of near-field optical microscopy – also an optical probe, which is positioned in close proximity, are assumed to be described by dielectric susceptibilities η(r) and χ(r), respectively. An incident light field Ei is illuminating the probe-sample region. Ei is assumed to be a solution of the homogeneous Helmholtz equation (2.35). The incoming field causes a scattered wave Es, which is detectable in the far-field. The total field is then given by E = Ei + Es. In a qualitative picture, there are several processes that can convert an incoming photon into a scattered photon. For example, the incoming photon may be scattered only at the probe or only at the sample before traveling into the far-field.

Near-field optical probes, such as laser-irradiated apertures or metal tips, are the key components of the near-field microscopes discussed in the previous chapter. No matter in which configuration a probe is used, the achievable resolution depends on how well the probe is able to confine the optical energy. This chapter discusses light propagation and light confinement in different probes. Fundamental properties are discussed and an overview of fabrication methods is provided. The most common optical probes are (1) uncoated tapered glass fibers, (2) aperture probes, and (3) pointed metal/semiconductor structures and resonant-particle probes. The reciprocity theorem of electromagnetism states that a signal remains unchanged upon exchange of source and detector (see Chapter 2.13). We therefore consider all probes as localized sources of light.

Light propagation in a conical transparent dielectric probe

Transparent dielectric probes can be modeled as infinitely long glass rods with a conical and pointed end. The analytically known HE11 waveguide mode, incident from the infinite cylindrical glass rod and polarized in the x-direction, excites the field in the conical probe. For weakly guiding fibers, the modes are usually designated LP (linearly polarized). In this case, the fundamental LP01 mode corresponds to the HE11 mode. The tapered, conical part of the probe may be represented as a series of disks with decreasing diameters and infinitesimal thicknesses. At each intersection, the HE11 field distribution adapts to the distribution appropriate for the next slimmer section.

The thermal and zero-point motion of electrically charged particles inside materials gives rise to a fluctuating electromagnetic field. Quantum theory tells us that the fluctuating particles can only assume discrete energy states and, as a consequence, the emitted fluctuating radiation takes on the spectral form of blackbody radiation. However, while the familiar blackbody radiation formula is strictly correct at thermal equilibrium, it is only an approximation for non-equilibrium situations. This approximation is reasonable at large distances from the emitting material (far-field) but it can strongly deviate from the true behavior close to material surfaces (near-field).

Because fluctuations of charge and current in materials lead to dissipation via radiation, no object at finite temperature can be in thermal equilibrium in free space. Equilibrium with the radiation field can be achieved only by confining the radiation to a finite space. However, in most cases the object can be considered to be close to equilibrium and the non-equilibrium behavior can be described by linear-response theory. In this regime, the most important theorem is the fluctuation-dissipation theorem. It relates the rate of energy dissipation in a non-equilibrium system to the fluctuations that occur spontaneously at different times in equilibrium systems.

The fluctuation-dissipation theorem is of relevance for the understanding of fluctuating fields near nanoscale objects and optical interactions at nanoscale distances (e.g. the van der Waals force). This chapter is intended to provide a detailed derivation of important aspects in fluctuational electrodynamics.