Introduction

In the first two chapters of this book we learned about the way a particle moves in a potential. Because in quantum mechanics particles have a wavy character, this modifies how they scatter from a change in potential compared with the classical case. In Section 3.8 we calculated transmission and reflection of an unbound particle from a one-dimensional potential step of energy V0. The particle was incident from the left and impinged on the potential barrier with energy E > V0. Significant quantum mechanical reflection probability for the particle occurred because the change in particle velocity at the potential step was large. This result is in stark contrast to the predictions of classical mechanics in which the particle velocity changes but there is no reflection.

In Section 3.10 we applied our knowledge of electron scattering from a step potential to the design of a new type of transistor. The analytic expressions developed were very successful in focusing our attention on the concept of matching electron velocities as a means of reducing quantum mechanical reflection that can occur at a semiconductor heterointerface. In this particular case, it is obvious that we could benefit from a model that is capable of taking into account more details of the potential. Such a model would be a next step in developing an accurate picture of transistor operation over a wide range of voltage bias conditions.

A major problem in nano-optics is the determination of electromagnetic field distributions near nanoscale structures and the associated radiation properties. A solid theoretical understanding of field distributions holds promise for new, optimized designs of near-field optical devices, in particular by exploitation of field enhancement effects and favorable detection schemes. Calculations of field distributions are also necessary for image reconstruction purposes. Fields near nanoscale structures have often to be reconstructed from experimentally accessible far-field data. However, most commonly the inverse scattering problem cannot be solved in a unique way and calculations of field distributions are needed to provide prior knowledge about source and scattering objects and to restrict the set of possible solutions.

Analytical solutions of Maxwell's equations provide a good theoretical understanding but can be obtained for simple problems only. Other problems have to be strongly simplified. A pure numerical analysis allows us to handle complex problems by discretization of space and time but computational requirements (usually given by cpu time and memory) limit the size of the problem and the accuracy of results is often unknown. The advantage of pure numerical methods, such as the finite-difference time-domain (FDTD) method or the finite-element (FE) method, is the ease of implementation. We do not review these pure numerical techniques since they are well documented in the literature. Instead we review two commonly used semi-analytical methods in nano-optics: the multiple multipole method (MMP) and the volume integral method.

Following the remarkable success of the first edition and not wanting to give up on a good thing, the second edition of this book continues to focus on three main themes: practicing manipulation of equations and analytic problem solving in quantum mechanics, utilizing the availability of modern compute power to numerically solve problems, and developing an intuition for applications of quantum mechanics. Of course there are many books which address the first of the three themes. However, the aim here is to go beyond that which is readily available and provide the reader with a richer experience of the possibilities of the Schrödinger equation and quantum phenomena.

Changes in the second edition include the addition of problems to each chapter. These also appear on the Cambridge University Press website. To make space for these problems and other additions, previously printed listing of MATLAB code has been removed from the text. Chapter 1 now has a section on harmonic oscillation of a diatomic molecule. Chapter 2 has a new section on quantum communication. In Chapter 3 the discussion of numerical solutions to the Schrödinger now includes periodic boundary conditions. The tight binding model of band structure has been added to Chapter 4 and the numerical evaluation of density of states from dispersion relation has been added to Chapter 5. The discussion of occupation number representation for electrons has been extended in Chapter 7.

The problem of dipole radiation in or near planar layered media is of significance to many fields of study. It is encountered in antenna theory, single molecule spectroscopy, cavity quantum electrodynamics, integrated optics, circuit design (microstrips), and surface contamination control. The relevant theory was also applied to explain the strongly enhanced Raman effect of adsorbed molecules on noble metal surfaces, and in surface science and electrochemistry for the study of optical properties of molecular systems adsorbed on solid surfaces. Detailed literature on the latter topic is given in Ref.[1]. In the context of near-field optics, dipoles close to a planar interface have been considered by various authors to simulate tiny light sources and small scattering particles. The acoustic analog is also applied to a number of problems such as seismic investigations or ultrasonic detection of defects in materials.

In his original paper, in 1909, Sommerfeld developed a theory for a radiating dipole oriented vertically above a planar and lossy ground. He found two different asymptotic solutions: space waves (spherical waves) and surface waves. The latter had already been investigated by Zenneck. Sommerfeld concluded that surface waves account for long-distance radio wave transmission because of their slower radial decay along the Earth's surface compared with space waves. Later, when space waves were found to reflect at the ionosphere, the contrary was confirmed. Nevertheless, Sommerfeld's theory formed the basis for all subsequent investigations. In 1911 Hörschelmann, a student of Sommerfeld, analyzed the horizontal dipole in his doctoral dissertation and likewise used expansions in cylindrical coordinates.

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.

The theory of quantum mechanics forms the basis for our present understanding of physical phenomena on an atomic and sometimes macroscopic scale. Today, quantum mechanics can be applied to most fields of science. Within engineering, important subjects of practical significance include semiconductor transistors, lasers, quantum optics, and molecular devices. As technology advances, an increasing number of new electronic and opto-electronic devices will operate in ways which can only be understood using quantum mechanics. Over the next thirty years, fundamentally quantum devices such as single-electron memory cells and photonic signal processing systems may well become commonplace. Applications will emerge in any discipline that has a need to understand, control, and modify entities on an atomic scale. As nano- and atomic-scale structures become easier to manufacture, increasing numbers of individuals will need to understand quantum mechanics in order to be able to exploit these new fabrication capabilities. Hence, one intent of this book is to provide the reader with a level of understanding and insight that will enable him or her to make contributions to such future applications, whatever they may be.

The book is intended for use in a one-semester introductory course in applied quantum mechanics for engineers, material scientists, and others interested in understanding the critical role of quantum mechanics in determining the behavior of practical devices. To help maintain interest in this subject, I felt it was important to encourage the reader to solve problems and to explore the possibilities of the Schrödinger equation.

Motivation

You may ask why one needs to know about quantum mechanics. Possibly the simplest answer is that we live in a quantum world! Engineers would like to make and control electronic, opto-electronic, and optical devices on an atomic scale. In biology there are molecules and cells we wish to understand and modify on an atomic scale. The same is true in chemistry, where an important goal is the synthesis of both organic and inorganic compounds with precise atomic composition and structure. Quantum mechanics gives the engineer, the biologist, and the chemist the tools with which to study and control objects on an atomic scale.

As an example, consider the deoxyribonucleic acid (DNA) molecule shown in Fig. 1.1. The number of atoms in DNA can be so great that it is impossible to track the position and activity of every atom. However, suppose we wish to know the effect a particular site (or neighborhood of an atom) in a single molecule has on a chemical reaction. Making use of quantum mechanics, engineers, biologists, and chemists can work together to solve this problem. In one approach, laser-induced fluorescence of a fluorophore attached to a specific site of a large molecule can be used to study the dynamics of that individual molecule. The light emitted from the fluorophore acts as a small beacon that provides information about the state of the molecule.

Near-field optical probes, such as laser-irradiated metal tips, are the key components of near-field optical microscopes discussed in the previous chapter. No matter whether the probe is used as a local illuminator, a local collector, or both, the optical spatial resolution solely depends on the confinement of the optical energy at the apex of the probe. This chapter discusses light propagation and light confinement in different probes used in near-field optical microscopy. Where applicable we study fundamental properties using electromagnetic theories (see Chapter 15) and provide an overview of current methods used for the fabrication of optical probes. We hope to provide the basic knowledge to develop a clear sense of the potentials and the technical limitations of the respective probes. The most common optical probes are (1) uncoated fiber probes, (2) aperture probes, (3) pointed metal and semiconductor probes, and (4) nano-emitters, such as single molecules or nanocrystals. The reciprocity theorem of electromagnetism states that a signal remains unchanged upon exchange of source and detector. Therefore, it suffices to investigate a given probe in only one mode of operation. In the majority of applications it is undesirable to expose the sample surface on a large scale due to the risk of photo-damage or long-range interference effects complicating image reconstruction. Therefore, we will preferentially consider the local illumination configuration.

Dielectric probes

Dielectric, i.e. transparent, tips are an important class of near-field optical probes and are the key components for the fabrication of more complex probes, e.g. aperture probes.

In recent years, artificial optical materials and structures have enabled the observation of various new optical effects and experiments. 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. The high field strengths in optical microresonators lead to nonlinear optical effects that are important for future integrated optical networks. This chapter explains the basic underlying principles of these novel optical structures. For a more detailed overview the reader is referred to review articles and books listed in the references.

Photonic crystals

Photonic crystals are materials with a spatial periodicity in their dielectric constant. Under certain conditions, photonic crystals can create a photonic bandgap, i.e. a frequency window in 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 in the range of the bandgap. However, defects in the periodicity of the lattice can locally destroy the bandgap and give rise to interesting electronic properties.