Introduction

This chapter presents the electromagnetic theory that describes the main characteristics of surface electromagnetic modes in general and surface plasmons (SPs) in particular that propagate along single- and double-interface planar guiding structures. We begin with an introduction to electromagnetic theory that discusses Maxwell's equations, the constitutive equations and the boundary conditions. Next, Maxwell's equations in terms of time-harmonic fields, electric and magnetic fields in terms of each other, and the resultant wave equations are presented. Group velocity and phase velocity, surface charge at a metal/dielectric interface and the perfect electric conductor conclude this introduction. Following this introduction are sections that describe the properties of electromagnetic modes that single- and double-interface planar guiding structures can support in terms of the media they are composed of. These media will be presented in terms of their permittivity and permeability whose real part can be either positive or negative. A new formalism will be developed to treat such media in the context of natural materials such as metals and dielectrics and in terms of a collection of subwavelength nanostructures dubbed metamaterials. Finally, the power flow along and across the guiding structures is presented, and the reflectivity from the base of a coupling prism and the accompanied Goos–Hänchen shift are treated. The material covered in this chapter draws heavily from Refs. [1] to [3] for the theory of electromagnetic fields and from Refs. [4] and [5] for the theory of optical waveguides.

Introduction

List of topics investigated in this chapter

Surface modes propagating along a single-interface structure composed of combinations of lossless, nondispersive DPS-, ENG-, DNG- and MNG-type media have been explored in the previous chapter. In this chapter we expand the investigation to double-interface structures and treat the properties depicted in Table 5.1. Here, however, the structures are restricted to those whose cover and substrate are composed of the same lossles DPS-type medium, and to guides composed of lossless DPS-, ENG-, DNG- and MNG-type media with real-valued ϵr and μr. The propagation constant, β, is calculated for a freely propagating mode, and the electric (E) and magnetic (H) fields are evaluated together with the local power flow, sz. The mode solutions were adapted from Refs. [1] and [2].

System

Geometry of the system

Figure 5.1 is a schematic diagram of a double-interface structure composed of a substrate, a guide with thickness dg and a cover. We assume the existence of a surface mode at this double-interface structure whose propagation constant, β, points along the z-direction, and the normal to the interface is along the x-direction. The thickness of the guide adds another parameter when solving for the modes that this structure can support. In particular, for a symmetric structure where the guide is a DNG-type, there is a critical guide thickness, dcr, below which the modes exhibit unique properties.

Introduction

List of topics investigated in this chapter

The previous two chapters dealt with single- and double-interface structures composed of combinations of DPS-, ENG-, MNG- and MNG-type media that support freely propagating modes. In these chapters ϵr and μr were real positive or real negative. The topics covered in this chapter and summarized in Table 6.1 deal with a single-interface structure composed of a metallic (ENG-type) substrate having a complex ϵr and with μr = 1 and a dielectric (DPS-type) cover that can support a SP mode. The propagation constant, β, is calculated for a freely propagating mode and for a mode excited and loaded by a prism coupler using the Otto (O) and Kretschmann (K) configurations. Next, the electric and magnetic fields, local power flow, wave impedance and charge density wave at the substrate–cover interface are evaluated. Finally, the reflectivity of an incident electromagnetic wave off the base of the prism, ℛ, is calculated for both configurations. The theory of singleinterface surface plasmons was adapted from Refs. [1] to [4] and recent reviews from Refs. [5] and [6].

System

Geometry of the system

The single-interface structure considered in this chapter, shown in Fig. 6.1, consists of a thick planar metallic silver (ENG-type) substrate and a thick planar dielectric (DPS-type) cover.

Introduction

The purpose of this chapter is to highlight some of the applications of SP physics that have either already been demonstrated or that hold promise for the future. For example, chemical and biological sensing using SPs is a solid commercial success with demonstrable advantages in certain areas over competitive technologies. There are some applications that are just now being introduced to the market, such as medical diagnostics and treatments with gold nanoparticles. Other applications of SPs discussed in this chapter will never become commercially successful, but nevertheless were included because they illustrate the wide range of areas in which SP physics has been applied. A fourth category of applications has already received a certain degree of success in the laboratory and may have an enormous commercial potential, but only time will tell. This category would include most of the nanophotonics applications and heat-assisted magnetic recording. Finally, there is a fifth category of potential applications that would still have to be considered highly speculative, but may one day become the most marvelous of all, including such things as “invisibility cloaks” and “perfect lenses.” Although these applications are not considered in any particular order, among the first applications of SP physics was the study of the optical properties of metals.

Measuring the optical constants of metals

Standard techniques for measuring the optical properties of thin films include ellipsometry and reflection/transmission measurements.

Nanoparticles

Introduction

In Chapters 2 to 8 we considered two-dimensional, planar surfaces that support propagating SP modes, and quasi-one-dimensional surfaces like nanowires or nanogrooves that support both propagating and nonpropagating SP modes. In this chapter we consider the quasi-zero-dimensional surfaces for nanoparticles (NPs) and nanoholes. Obviously these surfaces support only localized SP modes. Nevertheless, the richness of the SP phenomena found in these structures has generated a great deal of interest and is finding applications in a variety of areas. A beautiful microscopic image of silver nanoparticles is shown in Fig. 9.1 (see the color figure in the online supplement at www.cambridge.org/9780521767170). The different sizes and shapes of the particles determine the resonant frequencies of the SP modes, resulting in the wide range of colors.

We begin this chapter by studying spherical NPs. Both near-field and far-field properties can be calculated in several different ways, including the quasistatic approximation, Mie theory and a variety of numerical methods such as finitedifference time-domain (FDTD). Ellipsoidal particles and the lightning-rod effect are studied in the quasistatic approximation. Nanovoids are the complement to NPs and also exhibit localized SP modes. A sum rule is derived which connects resonant SP frequencies of a particle and its complementary void. Nanoshells, nanodisks, nanorods and nanotriangles are a few of the NP shapes that have been synthesized and studied in the literature and their SP properties are considered here.

When deciding how to organize a book on surface plasmons, it seemed natural to consider the dimensionality of the surfaces on which they exist. On planar surfaces, which include both semi-infinite surfaces as well as multilayer thin films, there is a rich body of phenomena related to propagating surface plasmons. The same is true for surfaces of nanoparticles having a rich variety of phenomena for localized surface plasmons. Surfaces of nanowires and nanogrooves lie in between these two regimes, and these surfaces support both propagating and nonpropagating surface plasmons. In this book, therefore, we have initially categorized the chapters by surface dimensionality, trying to point out both the differences and similarities of the surface plasmon phenomena in these three regimes.

This book does not hesitate to include mathematical derivations of the equations that describe the basic surface-plasmon properties. After all, it was our desire to base the book on Mathematica precisely so that these equations could be explored in detail. Our derivations of the properties of surface plasmons are based on Maxwell's equations in SI units. In Chapter 2, Maxwell's equations are introduced for dense media, i.e., media which can be described by frequency dependent permittivity, permeability and conductivity. Because interfaces are essential to surface plasmons, the electromagnetic boundary conditions are required. Practically all of the results in this book are based on time-harmonic fields that can be most simply represented in complex notation. Unfortunately, in the literature there is no standard definition for the complex functional dependence on time of the electric and magnetic fields.

Introduction

A wide variety of optical techniques have been developed for exciting SPs. As seen in Chapters 8 and 9, it is possible to excite localized SPs on nanowires and nanoparticles simply by shining a beam of light on these surfaces. On the other hand, it is not as simple to excite the nonradiative SPs on planar surfaces that do not directly couple to an incident plane wave, because the momentum of the incident photon cannot be matched to that of the SP. As described in Section 2.16, there are several approaches for overcoming this difficulty, including focusing a beam of light on the edge of a metallic film (end-fire coupling), using a diffraction grating to directly match the wave vector of the incident photon to that of the SP, or using attenuated total reflection via prism coupling. The prism-coupling equations were derived in Chapter 2 and the important aspects of resonance angle and line width were discussed and illustrated in detail. In this chapter we do not rederive those results, but rather, discuss some of the practical issues involved in using these configurations, and especially make use of those results for comparison to the grating coupler. The mathematics of vector diffraction, which is necessary for studying the grating coupler, is described in detail in the appendix of this chapter, which can be found online in the supplementary material at www.cambridge.org/9780521767170.

Introduction

List of topics investigated in this chapter

The topics covered in this chapter and summarized in Table 7 are similar to those covered in Chapter 6. Here, however, we deal with a double- rather than a singleinterface structure which consists of identical lossless dielectric (DPS-type) cover and substrate bounding a lossy metallic (ENG-type) guide, as shown in Fig. 7.1. Such a symmetric structure produces two solutions to the mode equation that are distinct from each other for a finite guide thickness. As the thickness of the guide increases, the two solutions converge to that of a single-interface DPS-ENG-type structure. The propagation constants of these modes are calculated for the freely propagating case and for the case where the modes are excited and loaded by a prism coupler. For each mode the electric and magnetic fields are evaluated, together with the local power flow, wave impedance and surface charge density at each guide interface. In Chapter 6 we dealt with a single-interface structure and used the Otto (O) or Kretschmann (K) configurations to excite a mode. Here, because we have a double-interface structure, we use the general prism coupling configuration (G). The reflectivity of an incident electromagnetic wave off the base of the prism, ℛ, is calculated for this configuration. The theory of double-interface surface plasmons was adapted from Refs. [1] to [4] and the concept of short-range (SR) and long-range (LR) SPs and recent reviews are from Refs. [5] to [11].