Introduction

Parametric four-wave mixing (FWM) and stimulated Raman scattering (SRS) are two fundamental processes of third-order nonlinear optical materials, and silica optical fibers in particular (Agrawal, 2006). As will be clear from many of the preceding chapters in this book, they also play a central role in supercontinuum generation. On the one hand, FWM involves the elastic interaction between two pump photons as well as a Stokes and an anti-Stokes photon symmetrically located around the pump frequency (Stolen, 1975; Stolen and Bjorkholm, 1982). It is of tremendous interest for large bandwidth optical amplification (Hansryd et al., 2002), including ultra-low-noise phase-sensitive amplification (McKinstrie and Radic, 2004; Oo-Kaw et al., 2008), but also for parametric oscillators (Coen and Haelterman, 2001; Sharping, 2008), wavelength converters (Islam and Boyraz, 2002) including noise-free schemes (Gnauck et al., 2006; Méchin et al., 2006) or optical signal processing in general (Salem et al., 2008). Its efficiency critically depends on a phase-matching relation, which is related to momentum conservation. On the other hand, SRS is characterized by the stimulated inelastic conversion of photons from an intense optical pump wave into lower frequency Stokes photons through the resonant excitation of a vibration in the transmission medium (i.e., optical phonons) (Bloembergen, 1967; Stolen et al., 1972). Here phase-matching is automatically satisfied. The SRS gain, in contrast to the FWM gain, exhibits an asymmetric profile (Stolen and Ippen, 1973): only the down-shifted Stokes waves are amplified while the up-shifted anti-Stokes waves are exponentially absorbed.

Introduction

Supercontinuum (SC) generation, the creation of broadband spectral components from an intense light pulse passing through a nonlinear medium, is of great theoretical interest as well as having numerous applications in optical frequency metrology, bio-imaging and spectroscopy (Dudley et al., 2006). In particular, the demonstration of efficient SC generation in a silica photonic crystal fibre (PCF) and silica fibre tapers using a Ti:Sapphire laser (Birks et al., 2000, Ranka et al., 2000) had a striking impact on this research field. The advent of this new class of waveguide, capable of engineered dispersion and strong confinement of light, facilitated research on the fundamental study of the evolution of ultra-fast pulses in highly nonlinear wave-guides, as well as the development of practical broadband light sources using the proper combination of fibres and laser pulses. Although the successful demonstration of ultra-broadband light generation often spanning more than an octave has been made in silica fibre, the small Kerr nonlinear coefficient of silica still limits its practicality. The ideal SC light source would use a compact, low power pulsed laser. This goal has motivated the study of SC generation in waveguides with higher nonlinear coefficients and lower energy thresholds or decreased device length to initiate the nonlinear process. Several approaches based on this idea have been reported utilising highly nonlinear material in a fibre geometry such as lead-silicate, bismuth and chalcogenide fibres (Brambilla et al., 2005, Leong et al., 2006, Mägi et al., 2007), and in a planar waveguide geometry including silicon (Boyraz et al., 2004), AlGaAs (Siviloglou et al., 2006) and chalcogenide waveguides (Psaila et al., 2007, Lamont et al., 2008).

Introduction

It is perhaps not surprising that using extremely high power and short pulse duration pump sources leads to dramatic nonlinear processes in optical fibres; in contrast, the generation of a supercontinuum from a continuous pump wave of relatively meagre power is at first sight, astounding. Yet supercontinua spanning over 1000 nm have been generated with pump powers of a few tens of watts – orders of magnitude lower than pulse pumped systems.

The key to continuous wave (CW) supercontinuum generation is the utilisation of modulation instability (MI). This is inherent to any anomalously dispersive, nonlinear medium, and has been observed in a wide range of systems. This instability can enable the creation of the extremes of peak power and pulse duration necessary for dramatic nonlinear processes to occur, even from very low power CW pump lasers. But although MI from CW pump lasers was observed in the 1980s by Itoh et al. (1989), other factors required for efficient continuum generation were missing, causing another decade to pass before such results were obtained.

A full review of experimental results will be presented later, but for reference, some examples of continuous wave supercontinua are shown in Fig. 8.1, which illustrate the high spectral power and the spectral smoothness and flatness which are characteristic of CW continuum generation.

Although significantly different from the physical mechanisms involved in ultra-short – femtosecond based – supercontinuum generation, the basic physical processes underpinning CW continuum generation are the same as for longer pump pulses (greater than a few picoseconds) and the observations and conclusions developed in the 1980s surrounding these type of sources apply to the CW regime.

Spectral broadening and the generation of new frequency components is an inherent feature of nonlinear optics, and has been studied in both bulk media and optical fiber waveguides since the 1960s. However, it was not until the early 1970s that the mechanism was widely applied to provide an extended “white-light” source for time resolved spectroscopy, which was later coined “a supercontinuum” by the Alfano group. Subsequent developments in the late 1970s in low-loss optical fibers with conventional structures for telecommunications led to the introduction of fiber as an ideal platform for supercontinuum generation. At the same time, the development of optical soliton physics throughout the late 1980s and early 1990s laid the theoretical foundation and established all the experimental mechanisms required for the production of this versatile source. Despite this progress, however, extensive laboratory deployment remained inhibited by unwieldy pump sources and unreliable system integration.

The advent of photonic crystal fiber in the late 1990s, together with developments in efficient high power and short pulse fiber lasers, fuelled a revolution in the generation of ultrabroadband high brightness optical spectra through the process of supercontinuum generation. Experiments using photonic crystal fiber in 1999–2000 attracted widespread interest and excitement because of the combination of high power, high coherence and the possibility to generate spectra spanning more than an octave. Moreover, the design freedom of photonic crystal fiber allowed supercontinuum generation to be optimized to the wider range of available pump sources, and experiments reported broadband spectra covering the complete window of transmission of silica based fiber using input pulses with durations ranging from several nanoseconds to several tens of femtoseconds, as well as high power continuous wave sources.

With the invention of the laser (Maiman 1960), rapid technological development of Q-switching (McClung and Hellwarth 1962) and mode locking techniques (Mocker and Collins 1965, DeMaria et al. 1966) allowed the achievement of the shortest, controllable, man-made pulse durations, and, consequently, for even modest pulse energies, unprecedented optical peak powers were achievable with ever-decreasing pulse durations, establishing a trend which continues to the present day. The enormous optical field strengths generated at the focal point of a pulsed laser ensured that the corresponding electronic polarization response of a transparent medium was nonlinear, in that higher order terms of the expansion describing the polarization needed to be considered despite the then insignificance of the magnitude of the second and third order susceptibilities and as a consequence ushered in the era of nonlinear optics. The first nonlinear optical process to be reported was second harmonic generation (Franken et al. 1961), which although observable, is of little importance in relation to the subject matter of this book, supercontinuum generation in optical fibres. However, this was followed by reports of frequency mixing (Bass et al. 1962) and parametric generation (Giordmaine and Miller 1965, Akhmanov et al. 1965). Essential for supercontinuum generation are the processes that result from the third order nonlinear term (Maker and Terhune 1965).

Introduction

This chapter provides a succinct overview of the various nonlinear effects that can occur when a light field propagates in an optical fibre. Given that nonlinear fibre optics is a very mature research field, it has been covered in much detail by many previous reviews and monographs. The reader is particularly referred to Agrawal (2007) for a treatment that combines a review of both theory and experiments in a way that is simultaneously accessible and technically comprehensive. Other monographs that contain valuable material and references to the original literature include Taylor (2005) and Alfano (2006). In this treatment we provide only a brief introduction to the major concepts, placing particular emphasis on effects that play an important role in supercontinuum generation. Where appropriate, these effects will be discussed in more detail in other chapters. We do, however, treat the numerical modelling of nonlinear pulse propagation in more depth than is usually found in the literature.

Modelling nonlinear pulse propagation

The propagation of an electromagnetic wave or pulse depends on the medium in which it propagates. In vacuum a pulse can propagate unchanged. When propagating in a medium, however, an electromagnetic field interacts with the atoms, which generally means that the pulse experiences loss and dispersion. The latter effect occurs because the different wavelength components of the pulse travel at different velocities due to the wavelength dependence of the refractive index. In an optical waveguide, the total dispersion has an additional component due to the light confinement called waveguide dispersion, which cannot be neglected.

Introduction

The generic single-mode optical fibre is a fibre drawn from fused silica. Silica fibres are available at very low cost, being produced by sophisticated processes in massive quantity for the global telecommunications network. They are very transparent, and much optical equipment is designed around their transparency windows, making research at these wavelengths particularly convenient. Devices or effects using silica fibres are therefore compatible with a wide range of existing optical fibre technologies, and so easy to incorporate into existing systems should the need or opportunity arise. Silica fibres are available in a wide range of different forms, especially with the development and commercial availability of silica photonic crystal fibres (PCF) and microstructured fibres.

Many of the above arguments rely for their worth on the already widespread deployment of silica optical fibres. However, there are also fundamental reasons to work with silica fibres, many of them the same reasons that they are already so popular. Silica – glassy SiO2 – can be synthesised with remarkably high levels of purity at very low cost: silicon is an abundant and easily-obtained element in the earth's crust, while the chemical processes used to synthesise silica are efficient and simple. Of course, the fact that these processes have been scaled up to produce large volumes of material further reduces the cost of production. The transparency window of silica is broad, spanning both visible and near-infra-red wavelengths. Practically, silica optical fibres are usually used in the spectral range 300–2500 nm.

Introduction

As discussed in Chapter 1, nonlinear fibre optics is a very mature field of research, and nonlinear propagation effects have been studied under a wide range of conditions. The basic physical processes that cause spectral broadening in fibres were outlined in Chapter 3, and reference was made to particular mechanisms that contribute to supercontinuum generation.

Whilst it is often convenient to discuss nonlinear spectral broadening in terms of effects that occur in either the normal or anomalous dispersion regime of a fibre, supercontinuum generation is more complex as it involves interactions that generate new spectral components on both sides of the zero dispersion wavelength (ZDW). The aim in this chapter is to extend the description of these interactions given in the preceding chapter in order to illustrate in more detail some commonly observed features of fibre supercontinuum generation. We do not intend to reproduce the comprehensive reviews that already exist in the literature (see the references given in Chapter 3); our objective is rather to provide a succinct overview of supercontinuum broadening mechanisms, and to introduce some of the subject matter that will be considered in more detail in subsequent chapters of the book.

The different regimes of supercontinuum generation can be broadly distinguished by considering short (femtosecond) versus long (picosecond, nanosecond and continuous wave) pump pulses. In this chapter, we consider the general features of supercontinuum generation for three particular commonly-observed cases.

Casimir's 1948 calculation (Casimir, 1948) of the vacuum force between two dielectric bodies is restricted to an idealized situation: two infinitely conducting parallel plates of infinite extension. Real dielectrics are neither infinitely conducting nor infinitely extended though. Moreover, in most experimental tests of the Casimir effect (Lamoreaux, 1997, 1999; Bordag et al., 2001; Lamoreaux, 2005; Munday et al., 2009) the force between a plate and a sphere was measured, because it is very difficult in practice to keep microscopic plates exactly parallel, a problem avoided by using a sphere above a plate instead. The most comprehensive general theory of the Casimir effect is known as Lifshitz theory(Landau and Lifshitz, Vol. IX, 1980). This theory was pioneered by Evgeny M. Lifshitz in 1955 (Lifshitz, 1955) and further developed by Igor E. Dzyaloshinskii, Lifshitz and Lev P. Pitaevskii in 1961 (Dzyaloshinskii et al., 1961). Here we explain the central concepts of Lifshitz theory. We test the theory on Casimir's case and mention some of the results beyond it. Lifshitz theory avoids the artefacts of Casimir's simple calculation (Casimir, 1948) and it is significantly more flexible and general, but it is also technically complicated and sometimes not very intuitive. Although we try to elucidate Lifshitz theory as much as possible, it still remains a rather heavy theoretical machinery à la russe. It seems remarkable that such a complicated theory may give such simple results as Casimir's formula (2.50) and the generalizations we are going to discuss.

A note to the reader

Quantum optics has grown from a sub-discipline in atomic, molecular, and optical physics to a broad research area that bridges several branches of physics and that captures the imagination of the public. Quantum information science has put quantum optics into the spotlight of modern physics, as has the physics of ultracold quantum gases with its many spectacular connections to condensed-matter physics. Yet through all these exciting developments quantum optics has maintained a characteristic core of ideas that I try to explain in this slim volume.

Quantum optics focuses on the simplest quantum objects, usually light and few-level atoms, where quantum mechanics appears in its purest form without the complications of more complex systems, often demonstrating in the laboratory the thought experiments that the founders of quantum mechanics dreamed of. Quantum optics has been, and will be for the foreseeable future, quintessential quantum mechanics, the quantum mechanics of simple systems, based on a core of simple yet subtle ideas and experiments.

One of the strengths of quantum optics is the close connection between theory and experiment. Although this book necessarily is theoretical, many of the theoretical ideas I describe are guided by experiments or, in turn, have inspired experiments themselves. Another strength of quantum optics is that it is done by individuals or small teams. One single person or a small group can build and perform an entire experiment.

Lindblad's theorem

Although the fundamental laws of physics explain a great deal of the material world, they often seem to contradict our immediate experience. For instance, as physicists we believe that all elementary interactions are reversible, but as people we are immersed in irreversibility. The collision of two atoms may well be reversible and predictable, but for us time flows in a definite direction and life is sometimes unpredictable. “What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know. ” (Attributed to Augustine.) Nevertheless, as we try to explain in this section, one can include irreversibility, the true arrow of time, into quantum mechanics in a concise and elegant form put on solid mathematical ground in Lindblad's theorem (Lindblad, 1976). Having then understood the basics of irreversible quantum dynamics, life may not be quite the same any more.

Irreversibility

A beam splitter is a reversible optical instrument: one could collect the outgoing light beams and bring them together again in a second beam splitter where they constructively interfere to re-establish the light in its original state. But now imagine a light beam is scattered many times. For example, light entering biological tissue is interacting with each of the cells along its path. Some of the light is scattered and only a part of the light continues in the original direction.

Parametric amplifier

In 1935 Erwin Schrödinger (Schrödinger, 1935a) coined the scientific term entanglement:

In a related paper (Schrödinger, 1935b) where Schrödinger also mentioned the famous cat, written in his native German, Schrödinger used the word Verschränkungfor entanglement, which is probably better. Verschränkung is a joiner's term for dovetailing pieces of furniture, Schränke in German. The word entanglement conjures up images of things entangled in nets, whereas entangled quantum states rather resemble neat bridges between different quantum objects, possibly at different places, dovetails across space. Entangled states have turned from posing a theoretical problem into becoming an experimental tool, first for testing the nonlocality of quantum mechanics and later for applying nonlocality and the related massive parallelism of the quantum world in quantum information science. The workhorse of most quantum optical applications of entanglement is the parametric amplifier.

Beam splitter

Compared with other fundamental experiments in physics, optical tests of quantum mechanics are often distinguished by their simplicity. Most quantum optical experiments do not require a whole industry - an optical table of equipment and a few people are often sufficient. Good ideas, good research problems are more important. “Research is to see what everybody has seen and to think what nobody has thought” (Jammer, 1989). A simple optical beam splitter, for instance, is already a nice device to demonstrate the quantum nature of light. Quite a number of puzzling quantum effects have been seen by splitting or recombining photons at a small cube of glass. Additionally, the beam splitter serves as a theoretical paradigm for other linear optical devices. Interferometers, semitransparent mirrors, dielectric interfaces, wave guide couplers, and polarizers are all described sufficiently well by a simple beam splitter model. The quantum effects of almost all passive optical devices can be understood assuming appropriate beam splitter models. (It's all done with mirrors.)

Heisenberg picture

An ideal beam splitter is a reversible, lossless device in which two incident beams may interfere to produce two outgoing beams. For instance, a dielectric interface inside a cube or plate of glass splits a light beam into two, see Fig. 5.1. We may reverse this situation by sending the two beams back to the cube, where they interfere constructively to restore the original beam.

Light in media

The backbone of theoretical quantum optics is the quantum field theory of light, the quantum theory of the electromagnetic field. Quantum electrodynamics becomes surprisingly complicated if one insists on formulating it in a relativistically and gauge-invariant form, as one surely should do in elementary particle physics. Moreover, in quantum optics, we are concerned with light quanta in materials such as glass, with quantum electromagnetism in media, which is even more complicated. To cut a long story short, here we follow a minimalistic approach where we develop the essentials of quantum electrodynamics with as little technical effort as possible, sailing around the cliffs of relativistic quantum field theories. We do not assume any prior knowledge of classical field theory; we only borrow a few ideas from classical electromagnetism (Jackson, 1998). Complications and problems still remain, but hopefully these are mostly no longer formal difficulties, but rather conceptual problems of the quantum nature of light. As Dr Samuel Johnson said on poetry “We all know what light is; but it is not easy to tell what it is. ”

Maxwell's equations

What is light? Light is a quantum object- we describe its properties by quantum observables, Hermitian operators, and its state by a state vector | ψ _ or density matrix ρ. We use the Heisenberg picture of quantum mechanics where the operators evolve in time, but the quantum state does not change.