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.

The chapter provides an overview about superresolution microscopy techniques. We start out discussing the resolution limit and its origin and then review the principles of confocal microscopy in which the multiplication of illumination and detection point-spread function leads to enhanced resolution and contrast. Based on these concepts, resolution improvements due to nonlinear contrast mechanisms are discussed before introducing light-sheet microscopy with its superior axial resolution. The chapter proceeds by introducing structured illumination as a method to enhance the resolution in microscopy by optimizing the detectable bandwidth of spatial frequencies. Superresolution in microscopy is always based on prior information about the sample. In localization microscopy such prior information introduces additional dimensions to the spatial imaging problem, such as time or colour, that are then used to distinguish closely spaced single emitters. Several advanced superresolution microscopy techniques are discussed in that context, such as PALM and STORM as well as MINFLUX and SOFI. At the example of STED microscopy, we discuss how the nonlinearity associated with saturable transitions in conjunction with intensity zeros can in principle lead to unlimited spatial resolution.

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 chapter covers subwavelength-localized optical fields and their interaction with matter. Localized fields contain evanescent waves, which decay exponentially away from their source region. To study the interaction of localized fields with matter, we introduce field-confining structures known as optical probes. To interact effectively with the sample, these optical probes are placed within the range of the evanescent waves and raster-scanned across the sample, a technique known as near-field optical microscopy. Given that optical probes inevitably interact with the sample, we start out with a series expansion of these probe–sample interactions, gaining insights into their nature and strength. We then discuss fundamental aspects of light confinement concepts and the corresponding optical probes, such as subwavelength apertures and resonant scatterers. This includes an exploration of how different probe designs influence the probe performance. Finally, we address probe–sample distance control and categorize various realizations of near-field optical microscopes according to the leading terms of the interaction series. This categorization helps to differentiate between different types of microscopes and their specific applications, providing a comprehensive overview of the field.