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 The cooling technique based on the Doppler mechanism has been superseded by more effective cooling mechanisms, especially evaporative and Sisyphus cooling mechanisms, culminating, with the achievement of super-cooling, in the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.

The interaction of light with nanoscale structures is at the core of nano-optics. As the structures become smaller and smaller the laws of quantum mechanics will become apparent. In this limit, the discrete nature of atomic states gives rise to resonant light-matter interactions. In atoms, molecules, and nanoparticles, such as semiconductor nanocrystals and other “quantum confined” systems, these resonances occur when the photon energy matches the energy difference of discrete internal (electronic) energy levels. Owing to the resonant character, light-matter interaction can often be approximated by treating these quantum emitters as effective two-level systems, i.e. by considering only those two (electronic) levels whose difference in energy is close to the interacting photon energy ħω0.

In this chapter we discuss quantum emitters that are used in optical experiments. We will discuss their use as single-photon sources and analyze their photon statistics. While the radiative properties of quantum emitters have been discussed in Chapter 8, this chapter focuses on the properties of the quantum emitters themselves. We adopt a rather practical perspective since more detailed accounts can be found elsewhere (see e.g. [1–4]).

Types of quantum emitters

The possibility of detecting single quantum emitters optically relies mostly on the fact that redshifted emission can be very efficiently discriminated against excitation light [5, 6]. This opens the road for experiments in which the properties of these emitters are studied or in which they are used as discrete light sources.

At the heart of nano-optics are light-matter interactions on the nanometer scale. For example, optically excited single molecules are used to probe local environments and metal nanostructures are exploited for extreme light localization and enhanced sensing. Furthermore, various nanoscale structures are used in near-field optics as local light sources.

The scope of this chapter is to discuss the interactions of light with nanoscale systems. The light-matter interaction depends on many parameters, such as the atomic composition of the materials, their geometry and size, and the frequency and intensity of the radiation field. Nevertheless, there are many issues that can be discussed from a more or less general point of view.

To rigorously understand light-matter interactions we need to invoke quantum electrodynamics (QED). There are many textbooks that provide a good understanding of optical interactions with atoms or molecules, and we especially recommend the books in Refs. [1–3]. Since nanometer-scale structures are often too complex to be solved rigorously by QED, one often needs to stick to classical theory and invoke the results of QED in a phenomenological way.

The multipole expansion

In this section we consider an arbitrary material system that is small compared with the wavelength of light. We call this material system a particle. Although it is small compared with the wavelength, this particle consists of many atoms or molecules. On a macroscopic scale the charge density ρ and current density j can be treated as continuous functions of position.

An optical antenna is a mesoscopic structure that enhances the local light-matter interaction. Similarly to their radiowave analogs, optical antennas mediate the information and energy transfer between the free radiation field and a localized receiver or transmitter. The degree of localization and the magnitude of transduced energy indicate how good an antenna is. We thus define an optical antenna as a device designed to efficiently convert freepropagating optical radiation to localized energy, and vice versa [1]. In this sense, even a standard lens is an antenna, but since the degree of localization is limited by diffraction, the lens is a poor antenna. To characterize the quality and the properties of an antenna, radio engineers have introduced antenna parameters, such as gain and directivity. Optical antennas hold promise for controllably enhancing the performance and efficiency of optoelectronic devices, such as photodetectors, light emitters, and sensors.

Although many of the properties and parameters of optical antennas are similar to those of their radiowave and microwave conuterparts, there are important differences resulting from their small size and the plasmon resonances of metal nanostructures. In this chapter we introduce the basic principles of optical antennas, building on the background of both radiowave antenna engineering and plasmonics.

Significance of optical antennas

The length scale of free radiation is determined by the wavelength λ, which is on the order of 500 nm. However, the characteristic size of the source generating this radiation is significantly smaller, typically sub-nanometer.