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].

Introduction and preview

While everything in Chapter 6 came under the umbrella of linear optics, in this chapter we will be looking at a range of nonlinear phenomena associated with optical pulses. We will consider the process of self-phase modulation in considerable detail, and see how it can be used, in combination with dispersion, to stretch and compress optical pulses, and to generate optical solitons. We will examine the adverse effects of group velocity dispersion on second harmonic generation, and examine various ways of optimising the bandwidth in optical parametric chirped pulse amplification. Finally, we will discover how nonlinear optical techniques can be used in the diagnosis of ultrashort pulses and for the stabilisation of the carrier-envelope phase.

Wave equation for short pulses

A detailed derivation of the differential equation governing the propagation of short optical pulses under nonlinear conditions is long and intricate; see for example [59]. To avoid this, we will take a series of reasonable steps that lead to the correct conclusion.

We start, as in Section 2.2, by substituting Eqs (2.4) and (2.5) into Eq. (2.3), but we now retain three time-dependent terms that were previously discarded.

The analysis in Chapter 3 used a phenomenological form of the permittivity to describe active materials. A proper understanding of optical amplification requires a quantum-mechanical approach for describing the interaction of light with atoms of an active medium [1]. However, even a relatively simple atom such as hydrogen or helium allows so many energy transitions that its full description is intractable even with modern computing machinery [2, 3]. The only solution is to look for idealized models that contain the most essential features of a realistic system. The semiclassical two-level-atom model has proven to be quite successful in this respect [4]. Even though a real atom has infinitely many energy levels, two energy levels whose energy difference nearly matches the photon energy suffice to understand the interaction dynamics when the atom interacts with nearly monochromatic radiation. Moreover, if the optical field contains a sufficiently large number of photons (> 100), it can be treated classically using a set of optical Bloch equations. In this chapter, we learn the underlying physical concepts behind the optical Bloch equations. We apply these equations in subsequent chapters to actual optical amplifiers and show that they can be solved analytically under certain conditions to provide a realistic description of optical amplifiers.

It is essential to have a thorough understanding of the concept of a quantum state [5]. To effectively use the modern machinery of quantum mechanics, physical states need to be represented as vectors in so-called Hilbert space [6].

The study of optical effects in active media is rich with unexpected consequences. For instance, one may imagine that absorption and amplification in a dielectric medium will exhibit some sort of symmetry because both are related to the same imaginary part of the dielectric constant, except for a sign change. It turns out that such a symmetry does not exist [1]. This issue has also been investigated in detail for random or disordered gain media, with varying viewpoints [2]. Traditionally, much of the research on amplifying media has considered the interaction of light within the entire volume of such a medium. Recent interest in metamaterials and other esoteric structures in which plasmons are used to manipulate optical signals has brought attention to the role of surface waves in active dielectrics [3, 4].

From a fundamental perspective, the main difference between active and passive media is that spontaneous emission cannot be avoided in gain media. As a result, a rigorous analysis of gain media demands a quantum-mechanical treatment. Spontaneous emission in a gain medium depends not only on the material properties of that medium but also on the optical modes supported by the structure containing that material [5]. By a clever design of this structure (e.g., photonic crystals or microdisk resonators with a metallic cladding), it is possible to control the local density of optical modes and the spontaneous emission process itself. In Chapter 4, we discuss how to model a gain medium under such conditions by deploying optical Bloch equations.

Optical parametric amplifiers constitute a category of amplifiers whose operation is based on a physical process that is quite distinct from other amplifiers discussed in the preceding chapters. The major difference is that an atomic population is not transferred to any excited state of the system, in the sense that the initial and final quantum-mechanical states of the atoms or molecules of the medium remain unchanged [1]. In contrast, molecules of the gain medium end up in an excited vibrational state, in the case of Raman amplifiers. Similarly, atoms are transferred from an excited electronic state to a lower energy state in the case of fiber amplifiers and semiconductor optical amplifiers.

Optical parametric amplification is a nonlinear process in which energy is transferred from a pump wave to the signal being amplified. This process was first used to make optical amplifiers during the 1960s and has proved quite useful for practical applications. In Section 8.1 we present the basic physics behind parametric amplification, and then focus on the phase-matching requirement in Section 8.2. In Section 8.3 fiber-based parametric amplifiers are covered in detail, because of the technological importance of such amplifiers. Section 8.4 is devoted to parametric amplification in birefringent crystals, while Section 8.5 focuses on how phase matching is accomplished in birefringent fibers.

Modern optical fibers exhibit very low losses (≈ 0.2 dB/km) in the 1.55 μm wavelength region that is of interest for telecommunications applications. Even though light at wavelengths in this region can be transmitted over more than 100 km before its power degrades considerably, an optical amplifier is eventually needed for any telecommunications system to restore the signal power to its original level. Since a fiber-based amplifier is preferred for practical reasons, such amplifiers were developed during the 1980s by doping standard optical fibers with rare-earth elements (known as lanthanides), a group of 14 elements with atomic numbers in the range from 58 to 71. The term rare appears to be a historical misnomer because rare-earth elements are relatively abundant in nature. When these elements are doped into silica or other glass fibers, they become triply ionized. Many different rare-earth elements, such as erbium, holmium, neodymium, samarium, thulium, and ytterbium, can be used to make fiber amplifiers that operate at wavelengths covering a wide range from visible to infrared. Amplifier characteristics, such as the operating wavelength and the gain bandwidth, are determined by the dopants rather than by the fiber, which plays the role of a host medium. However, because of the tight confinement of light provided by guided modes, fiber amplifiers can provide high optical gains at moderate pump power levels over relatively large spectral band-widths, making them suitable for many telecommunications and signal-processing applications [1–3].

The recent development of artificially structured optical materials—termed optical metamaterials—has led to a variety of interesting optical effects that cannot be observed in naturally occurring materials. Indeed, the prefix “meta” means “beyond” in the Greek language, and thus a metamaterial is a material with properties beyond those of naturally occurring materials. Examples of the novel optical phenomena made possible by the advent of metamaterials include optical magnetism [1, 2], negative refractive index [3, 4], and hyperbolic dispersion [5, 6]. Metamaterials constitute a 21st-century area of engineering science that is not only expanding fundamental knowledge about electromagnetic wave propagation but is also providing new solutions to complex problems in a wide range of disciplines, from data networking to biological imaging. Although metamaterials have attracted public attention, most people see them only in devices such as Harry Potter's cloak of invisibility, or machines like StarCraft's Arbiter, with the ability to make things invisible. Indeed, the research on metamaterials indicates that the invisibility cloak is a real possibility, and might find applications in advanced defence technologies. However, it is worth mentioning other opportunities where such advanced materials can find practical applications. A very important one is the transformation of evanescent waves into propagating waves, enabling one to view subwavelength-scale objects with an optical microscope, thereby surpassing the diffraction limit [7–9].

Light gets scattered when it encounters an obstacle or inhomogeneity even on a microscopic scale. A well-known example is the blue color of the sky, resulting from Rayleigh scattering of light by molecules in the air. Such redirection of energy can be used to amplify signals by taking power from a “pump” wave co-propagating with the signal in an appropriate optical medium. An example of this is provided by Raman scattering. Having said that, it is important to realize that scattering does not always occur when light interacts with a material [1]. In some cases, photons get absorbed in the medium, and their energy is eventually dissipated as heat. In other cases, the absorbed light may be re-emitted after a relatively short time delay in the form of a less energetic photon [2], a process known as fluorescence. If fluorescence takes place after a considerable delay, the same process is called phosphorescence [3].

For a photon to get absorbed by a material, its energy must correspond to the energy required by the atoms or molecules of that material to make a transition from one energy level to a higher energy level. In contrast, the scattering of photons from a material can take place without such a requirement. However, if the energy of the incident photon is close to an allowed energy transition, significant enhancement of scattering can occur.

An integral feature of any optical amplifier is the interaction of light with the material used to extract the energy supplied to it by an external pumping source. In nearly all cases, the medium in which such interaction takes place can be classified as a dielectric medium. Therefore, a clear understanding of how light interacts with active and passive dielectric media of finite dimensions is essential for analyzing the operation of optical amplifiers. When light enters such a finite medium, its behavior depends on the global properties of the entire medium because of a discontinuous change in the refractive index at its boundaries. For example, the transmissive and reflective properties of a dielectric slab depend on its thickness and vary remarkably for two slabs of different thicknesses even when their material properties are the same [1].

In this chapter we focus on propagation of light through a dispersive dielectric slab, exhibiting chromatic dispersion through its frequency-dependent refractive index. Even though this situation has been considered in several standard textbooks [2, 3], the results of this chapter are more general than found there. We begin by discussing the state of polarization of optical waves in Section 2.1, followed with the concept of impedance in Section 2.2. We then devote Section 2.3 to a thorough discussion of the transmission and reflection coefficients of a dispersive dielectric slab in the case of a CW plane wave. Propagation of optical pulses through a passive dispersive slab is considered in Section 2.4, where we also provide simple numerical algorithms.

Semiconductor optical amplifiers (SOAs) are increasingly used for optical signal processing applications in all-optical integrated circuitry [1, 2]. Research on SOAs started just after the invention of semiconductor lasers in 1962 [3]. However, it was only after the 1980s that SOAs found widespread applications [4, 5]. The effectiveness of SOAs in photonic integrated circuits results from their high gain coefficient and a relatively low saturation power [6, 7]. In addition, SOAs are often used for constructing functional devices such as nonlinear optical loop mirrors [8, 9], clock-recovery circuits [10, 11], pulse-delay discriminators [12–14], and logic elements [15, 16].

A semiconductor, as its name implies, has a conductivity in between that of a conductor and an insulator. Some examples of elemental semiconductors include silicon, germanium, selenium, and tellurium. Such group-IV semiconductors have a crystal structure similar to that of diamond (a unit cell with tetrahedral geometry) and the same average number of valence electrons per atom as the atoms in diamond. Compound semiconductors can be made by combining elements from groups III and V or groups II and VI in the periodic table. Two group III–V semiconductors commonly used for making SOAs are gallium arsenide (GaAs) and indium phosphide (InP). These semiconductors enable one to manipulate properties such as conductivity by doping them with impurities, and allow the formation of the p–n junctions required for the electrical pumping of SOAs.