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.

Introduction

Third-order nonlinear processes are based on the term ε0χ(3)E in the polarisation expansion of Eq. (1.24). Just as the second-order processes of the previous chapter coupled three waves together, so third-order processes couple four waves together, and hence are sometimes called four-wave processes.

Third-order processes are in some ways simpler and in other ways more complicated than their second-order counterparts. Perhaps the most important difference is that third-order interactions can occur in centrosymmetric media. This means that, while crystals can be used if desired, one is often dealing with optically isotropic materials in which the complexity of crystal optics is absent. However, the tensor nature of the third-order coefficients is still not entirely straightforward, even in isotropic media.

A second issue is that phase matching is automatic for many third-order processes and, even when it is not, the existence of four waves, potentially travelling in four different directions, makes phase matching easier to achieve. Of course, phase matching was also automatic at second order for the Pockels effect and optical rectification, but those were somewhat special cases involving DC fields.

At first sight, automatic phase matching sounds like a good thing, but the benefit (if it is such) comes at a serious price. When phase matching was critical, and one had to work to achieve it for a particular combination of frequencies, it did at least provide a way of promoting one particular nonlinear process, while discriminating against all the others.

Introduction to high harmonic generation

The treatment of frequency mixing in the early chapters of this book was based on the assumption that the applied EM fields were weak compared to the internal fields within the nonlinear media. This enabled the polarisation to be expanded as a power series in the field, and perturbation theory to be used to calculate the nonlinear coefficients.

In this chapter, we consider the generation of optical harmonics in the strong-field regime, where the fields are sufficiently intense to ionise the atoms in a gaseous medium, and the electrons released then move freely in the field, at least to a good approximation. As we shall discover, an enormous number of harmonics can be generated in these circumstances, extending across a broad spectral plateau that extends deep into the soft X-ray region. High harmonic generation (HHG) therefore creates a table-top source of coherent X-rays, which has already been applied in lensless diffraction imaging [107]. Moreover, if the huge spectral bandwidth is suitably organised, HHG enables pulses as short as 100 attoseconds or even less to be generated [108]. These can then serve as diagnostic tools on unprecedented time-scales. They have already been used to probe proton dynamics in molecules with a 100-as time resolution [109], and to measure the time delay of electron emission in the photoelectric effect for the first time [110]. Other applications are described in [111].

Table D1 contains information about which d elements are non-zero for each of the 21 non-centrosymmetric crystal classes. Column 3 lists the number of independent elements for each class and (in brackets) the lower number of independents when the Kleinmann symmetry condition (KSC) applies.

Detailed information is given in column 4 where semicolons separate independent elements. Elements that are equal to others under all circumstances (in value if not in sign) are so indicated, while elements grouped within brackets are identical if the KSC applies. In a few cases (e.g. class 24) d14 = −d25, but both are zero under Kleinmann symmetry as indicated by the ‘= 0’. The rare class 29 is non-centrosymmetric, but other aspects of the symmetry force all elements of the d matrix to be zero.

Note that, in biaxial media, the mapping from the crystallographic (abc) to the physical (xyz) axes is not necessarily straightforward. It cannot be assumed that xyz → 123, which many people would take for granted. For a detailed discussion, see Sections 4.4 and 4.6, as well as Chapter 2 of Dmitriev et al. [26].