This chapter reviews the main concepts of electromagnetic theory relevant for the understanding of this textbook. Based on Maxwell’s equations, we derive the wave equation and discuss homogeneous solutions, such as plane waves and evanescent waves. We derive the boundary conditions at interfaces between homogeneous media and the Fresnel reflection and transmission coefficients. We discuss energy conservation, causality, and reciprocity of electromagnetic fields. Point response functions are introduced (Green functions) in order to derive the inhomogeneous solution of the wave equation. The chapter concludes with the angular spectrum representation, a framework that allows arbitrary fields to be described as a superposition of plane and evanescent waves.

In practice, a radiating source is most commonly placed close to a surface or multiple interfaces. Examples are antennas mounted over ground or molecules placed on dielectric surfaces or waveguides. The topic has a long history dating back to Arnold Sommerfeld’s paper in 1909. To derive the fields of an arbitrary oriented dipole over a layered medium we have to find the corresponding Green function. We start by decomposing the free-space dyadic Green function into $s$ and $p$ polarized parts and then evaluate reflection and transmission for the two polarizations separately. Once the Green function of the layered reference system is found, we proceed to derive the radiated power and the far-fields of the dipole. We analyze the radiation patterns and the modes into which the energy is most effectively coupled. For dipoles over dielectric half-spaces, we find that evanescent field components couple predominantly into supercritical angles, giving rise to what is termed “forbidden light.” We discuss how recorded radiation patterns can be used to determine the orientation of the radiating dipole, such as the orientation of molecules fluorescing near a dielectric surface. The chapter concludes by reviewing the image dipole method and discussing its validity.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The properties of optical materials are determined by the fundamental constituents of matter. In this chapter we discuss how to design materials with unusual optical properties by arranging meta-atoms, smaller than the wavelength of light, fabricated using nanotechnology. Meta-atoms arranged in periodic arrays with lattice constants of the order of the wavelength lead to photonic crystals. Photons in photonic crystals behave analogous to electrons in regular crystals, allowing principles from solid-state physics, such as doping, to be carried over. This results in fascinating effects such as photonic bandgaps and localized states of light. Metamaterials arise if meta-atoms are densely packed such that light propagates as if it were in a homogeneous medium. By tuning the properties of the meta-atoms, these materials can be tailored to exhibit exotic optical properties such as negative or near-zero refractive index. Finally, we introduce metasurfaces which directly mold the flow of light – a property that can be used to create ultraflat optical elements.

This introductory chapter sets the stage for the research field of nano-optics. It introduces the fundamental concept of localizing light beyond the diffraction limit through the superposition of propagating and evanescent waves, emphasizing the critical role of evanescent waves. Additionally, it provides a historical overview of the key developments that have shaped nano-optics, and outlines the scope of the book.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

In contrast to photonic crystals, random media lack spatial symmetry but are characterized by spatial and temporal correlations in the dielectric function. This results in unique effects on light propagation through such media. We begin by deriving the transport equation for light in random media, which, under specific conditions, leads to the diffusive behavior of photons. Following this, we explore phenomena such as Anderson localization and coherent backscattering, which can be attributed to time-reversed scattering pathways and the associated interference effects of photons. Lastly, we examine the application of random media as linear optical elements capable of focusing light to subwavelength spots, introducing the concept of singular-value decomposition in this context.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

Based on the angular spectrum representation, we discuss the focusing and localization of electromagnetic fields. In the paraxial limit (weak focusing) we derive the Gaussian beam and discuss its key properties, including its collimation range and divergence. Using the method of stationary phase, we show how the far-field of any known field distribution can be derived and how these far-fields can be embedded in the angular spectrum representation in order to rigorously calculate strongly focused wave fields. Higher-order modes, such as Hermite–Gaussian beams, radially /azimuthally polarized beams, and orbital angular momentum (OAM) beams are introduced and the calculation of focused fields at interfaces is discussed. The chapter concludes with a derivation of the image of a point source, the so-called point-spread function, and a discussion of how it limits the resolution in optical microscopy.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

In this chapter we discuss semianalytical methods for calculating optical fields in arbitrary geometries. Semianalytical methods rely on numerical procedures to derive analytical solutions for the problem at hand. Examples are the multiple-multipole method (MMP), the coupled-dipole method (CDM), or the method of moments (MoM). Based on the volume integral equation we show the equivalence of the CDM and the MoM. The comparison allows us to derive the most general form of the polarizability $\alpha$ of a small scatterer. We show that it reproduces the dynamic and quasi-static polarizabilities derived in previous chapters. We derive an equation for calculating the Green function of an arbitrary system, known as the Dyson equation, and discuss how it can be used to iteratively determine the electromagnetic field in an arbitrary geometry.