If a metamaterial can be defined as a deliberately structured material that possesses physical properties that are not possible in naturally occurring materials, then deliberately structured surfaces that possess desirable optical properties that planar surfaces do not posses can surely be considered to be optical metamaterials. The surface structures displaying these properties can be periodic, deterministic but not periodic, or random.

In recent years interest has arisen in optical science in the study of such surfaces and the optical phenomena to which they give rise. A wide variety of these phenomena have been predicted theoretically and observed experimentally. They can be divided roughly into those in which volume electromagnetic waves participate and those in which surface electromagnetic waves participate. Both types of optical phenomena and the surface structures that produce them are described in this volume.

The first several chapters are devoted to optical interactions of volume electromagnetic waves with structured surfaces. One of the earliest examples of a structured surface that acts as an optical metamaterial, and the one that today is perhaps the best known and most widely studied, is a metal film pierced by a two-dimensional periodic array of holes with subwavelength diameters. It was shown experimentally by Ebbesen et al. [1] that the transmission of p-polarized light through this structure can be extraordinarily high at the wavelengths of the surface plasmon polaritons supported by the film.

Introduction

A key weakness of the simple approach to nonlinear optics adopted in Chapter 1 was that the physical origin of nonlinearity in the interaction of light and matter was hidden inside the χ(n) coefficients of the polarisation expansion. This is such a fundamental issue that it is difficult to avoid some mention of how nonlinearity arises within a quantum mechanical framework, even in an introductory text. Unfortunately, the standard technique for calculating the nonlinear coefficients is based on time-dependent perturbation theory, and the expressions that emerge begin to get large and unwieldy even at second order. While every effort has been made in this chapter to provide a gentle lead-in to this aspect of the subject, this is almost impossible to achieve given the inherent complexity of the mathematical machinery.

From a mathematical point of view, Schrödinger's equation is linear in the wave function, but nonlinear in its response to perturbations. At a fundamental level, this is where nonlinear optics comes from. The perturbations of atoms and molecules referred to here arise from external electromagnetic fields. When the fields are relatively weak, the perturbations are relatively small, and the theoretical machinery of time-dependent perturbation theory can be deployed to quantify the effects. This is the regime where the traditional polarisation expansion of Eq. (1.24) applies, indeed the terms in the expansion correspond to successive orders of perturbation theory.

I set out to write this book in the firm belief that a truly introductory text on nonlinear optics was not only needed, but would also be quite easy to write. Over the years, I have frequently been asked by new graduate students to recommend an introductory book on nonlinear optics, but have found myself at a loss. There are of course a number of truly excellent books on the subject – Robert Boyd's Nonlinear Optics, now in its 3rd edition [1], is particularly noteworthy – but none of them seems to me to provide the gentle lead-in that the absolute beginner would appreciate.

In the event, I found it a lot harder to maintain an introductory flavour than I had expected. I quickly discovered that there are aspects of the subject that are hard to write about at all without going into depth. One of my aims at the outset was to cover as much of the subject as possible without getting bogged down in crystallography, the tensor structure of the nonlinear coefficients, and the massive perturbation theory formulae that result when one tries to calculate the coefficients quantum mechanically. This at least I largely managed to achieve in the final outcome. As far as possible, I have fenced off the ‘difficult’ bits of the subject, so that six of the ten chapters are virtually ‘tensor-free’.

Introduction and preview

As noted in Section 1.8, the treatment presented in Chapter 1 was greatly over-simplified. The fact that the frequency dependence of the coefficients in the polarisation expansion was neglected gave the false impression that the coefficients governing all processes of a given order are the same, apart from simple factors. The tensor nature of the coefficients was completely ignored too.

Unfortunately, if one wants to understand nonlinear optical interactions in crystalline media, one cannot avoid getting to grips with the tensor nature of the nonlinear coefficients, a topic that is intricate and hard to simplify. At the very least, one needs to be able to interpret the numerical notation used to label the coefficients, and to know how to apply the data supplied in standard reference works in a given crystal geometry.

There is no disguising the fact that Section 4.3 in the present chapter is rather complicated. Readers who want to avoid the worst of the difficulties should skim it, always bearing in mind that the situation turns out to be far less alarming at the end of the journey than it seemed it might be at the beginning. Early on in that section, it looks as if there could be literally hundreds of separate nonlinear coefficients to deal with. But it soon emerges that the number is almost certainly no larger than 18, and perhaps only 10.

Preview

The central feature of this chapter is the phenomenon of birefringence, also known as double refraction, which occurs in crystals that are optically anisotropic. Given that birefringence is a linear optical effect, why is the whole of Chapter 3 being devoted to it? Firstly, most nonlinear crystals are birefringent, and so one naturally needs to know how light propagates in these media. Secondly, several important nonlinear optical techniques (the most obvious being phase matching) exploit birefringence to achieve their goal. Lastly, the material in this chapter provides essential background for the following chapter on the nonlinear optics of crystals.

Section 3.2 is a brief tutorial on crystal symmetry. Crystallography is something of a world on it own, and many people find it a complete mystery. Although the summary offered here is very basic, it should provide everything needed for what comes later.

Section 3.3 discusses the propagation of EM waves in optically anisotropic media, and contains a fairly detailed analysis of birefringence (double refraction), ordinary and extraordinary waves, and associated topics. The treatment is mainly centred on uniaxial media because of their relative simplicity and the fact that most nonlinear crystals are of this type.

Section 3.4 describes how birefringence can be exploited in the construction of wave plates, while Section 3.5 is reserved for a brief mention of biaxial media in which the propagation characteristics are considerably more complicated.

Introduction and preview

In this chapter, we will consider several of the basic frequency-mixing processes of nonlinear optics. The simplest is second harmonic generation (SHG), and we will take this as our basic example. In SHG, a second harmonic wave at 2ω grows at the expense of the fundamental wave at ω. As we will discover, whether energy flows from ω to 2ω or vice versa depends on the phase relationship between the second harmonic field and the nonlinear polarisation at 2ω. Maintaining the optimal phase relationship is therefore of crucial importance if efficient frequency conversion is to be achieved.

The SHG process is governed by a pair of coupled differential equations, and their derivation will be our first goal. The analysis in Section 2.2 is somewhat laborious, although the material is standard, and can be found in many other books, as well as in innumerable PhD theses. The field definitions of Eqs (2.4)–(2.5) are used repeatedly throughout the book, and are worth studying carefully.

In Section 2.3, the coupled-wave equations are solved for SHG in the simplest approximation. The results are readily extended to the slightly more complicated cases of sum and difference frequency generation, and optical parametric amplification, which we move on to in Section 2.4. The important case of Gaussian beams is treated in Section 2.5, where the effect of the Gouy phase shift in the waist region of a focused beam is highlighted.