Introduction

Extracellular electric potentials, such as local field potentials (LFPs) or the electroencephalogram (EEG), are routinely measured in electrophysiological experiments. LFPs are recorded using micrometer-size electrodes, and sample relatively localized populations of neurons, as these signals can be very different for electrodes separated by 1 mm (Destexhe et al., 1999a) or by a few hundred micrometers (Katzner et al., 2009). In contrast, the EEG is recorded from the surface of the scalp using millimeter-scale electrodes and samples much larger populations of neurons (Niedermeyer and Lopes da Silva, 1998). LFPs are subject to much less filtering compared to EEG, because EEG signals must propagate through various media, such as cerebrospinal fluid, dura mater, cranium, muscle and skin. LFP signals are also filtered, because the recording electrode is separated from the neuronal sources by portions of cortical tissue. Besides these differences, EEG and LFP signals display the same characteristics during wake and sleep states (Steriade, 2003).

The observation that action potentials have a limited participation in the genesis of the EEG or LFPs dates from early studies. Bremer (1938, 1949) was the first to propose that the EEG is not generated by action potentials, based on the mismatch of the time course of EEG waves with action potentials. Eccles (1951) proposed that LFP and EEG activities are generated by summated postsynaptic potentials arising from the synchronized excitation of cortical neurons. Intracellular recordings from cortical neurons later demonstrated a close correspondence between EEG/LFP activity and synaptic potentials (Klee et al., 1965; Creutzfeldt et al., 1966a, 1966b).

This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω0 equal to the experimentally determined absorption frequency.

Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave.

The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.

The difference between laser light and the light emitted by an incoherent source can only be fully appreciated with reference to certain notions of energetic photometry, which are spelt out in the first part of this complement. It will be shown in Section 3C.2 how the laws of photometry drastically reduce the energy density that can be obtained from a conventional incoherent source (such as a heated filament, or a discharge lamp) in comparison with a laser source (Section 3C.3). Far from being merely circumstantial, these laws for classical sources are of a fundamental kind that can be deduced from the basic principles of thermodynamics. Another way to relate these properties of light to the fundamental principles of physics is to examine them in the context provided by the statistical physics of photons, as will be discussed in Sections 3C.4 and 3C.5.

Conservation of radiance for an incoherent source

Étendue and radiance

An incoherent source comprises a large number of independent, elementary emitters, emitting electromagnetic waves with a random distribution of uncorrelated phases. It emits light in every direction. A light beam produced by this source can be decomposed into elementary pencils of light. Since the light is incoherent, the total power carried by the beam is the sum of the powers carried by the elementary pencils.

An elementary pencil is defined by the element dS of the source from which it originates, and a second surface element dS′, as shown in Figure 3C.1.

Introduction

In 1961, just a few months after Maiman invented the ruby laser, Franken focused the pulses emitted from such a laser, of wavelength 694 nm, on a quartz plate, and examined the spectrum of the light transmitted using a simple prism (see Figure 7.1). He thus discovered that ultra-violet light of wavelength 347 nm was emerging from the quartz plate. Clearly, as it propagated through the quartz, the light of frequency ω had generated the second harmonic, of frequency 2ω.

It thus transpires that in optics, as in any other part of physics, a system subjected to a strong enough sinusoidal excitation will leave the linear response regime. Nonlinearities cause harmonics of the excitation frequency to appear.

But what intensity is needed before nonlinear effects will appear? One might think that a natural scale would be the electric field of the nucleus at the location of an atomic electron. In the case of the hydrogen atom in its ground state, this field is about, or 3 × 1011 V.m−1. (Here e is the charge of the electron, and a0 the Bohr radius, of the order of 5 × 10−11 m). In fact, experiment shows that, in the transparency zone of a dielectric material like quartz, a field of just 107 V.m−1 (corresponding to a light intensity of 2.5 kW.cm−2) is sufficient for nonlinear effects to appear perturbatively.

One-photon sources are important elements in quantum optics. The archetypal example is an atom raised to an excited state at time t = 0, then de-exciting with emission of a single photon. The development of this kind of source depends on progress with experimental techniques, e.g. the possibility of isolating a single atom, molecule or quantum well. In this complement, we present the formalism for describing the corresponding radiation, and use it to discuss some spectacular experiments which bring out properties quite incompatible with a classical description of the electromagnetic field. We begin in Section 5B.2 by describing the anti-correlation between detections on either side of a semi-reflecting mirror, establishing the quantitative difference with a classical field. Section 5B.3 discusses a quantum optical effect that was only demonstrated at the beginning of the twenty-first century, namely the quantum coalescence of two one-photon wave packets on a semi-reflecting mirror, which occurs even when the two photons were emitted by independent atoms. An analogous effect, the Hong–Hou–Mandel effect, is discussed in Chapter 7. These effects exemplify quantum interference involving two photons. Finally, Section 5B.4 is concerned with quantum calculations involving quasi-classical states. As we now know, this leads to results that are identical to the predictions of semi-classical theory.

In this complement we discuss several examples of optical phenomena in media where the refractive index depends nonlinearly on the intensity, known as optical Kerr media. This nonlinear effect exists in all materials, even isotropic ones, like glass or fused silica, but it is particularly marked in certain physical systems to be exemplified in Section 7B.1. After investigating the propagation of light through such media in Section 7B.2, we shall discuss three applications of the optical Kerr effect (which can be studied in any order). We begin by describing a bistable optical system, when this nonlinear medium is inserted in a Fabry–Perot cavity (Section 7B.3). We then study phase conjugate mirrors and examine their potential applications in adaptive optics (Section 7B.4). Finally, we discuss certain effects occurring during the propagation of an isolated wave, bounded either transversely or temporally, in a Kerr medium, and describe self-focusing effects (Section 7B.5) and self-phase-modulation effects (Section 7B.6). In particular, we shall show that nonlinear effects and dispersion effects can compensate to produce stable structures known as solitons, which maintain their shape during propagation.

Examples of third-order nonlinearities

Nonlinear response of two-level atoms

We begin by studying a simple case of a nonlinear interaction, namely a two-level quantum system under the effects of a plane wave.

The spectral width of the output of most single-mode lasers is determined by technical limitations associated with the stability of the optical length of the laser cavity (see Section 3.3.3). However, in the absence of these, there is a more fundamental limit to the degree of monochromaticity that can be achieved. This limit, known as the Schawlow–Townes limit is, in fact, rather narrower than the passive bandwidth of the laser cavity or the width of the gain curve of the active medium it contains. We calculate in a heuristic fashion in this complement the Schawlow–Townes limit for a laser operating far above threshold.

The fundamental mechanism for the spectral broadening of a laser output beam is the spontaneous emission by the gain medium of photons into the laser mode. Spontaneous emission adds to the complex field of the laser mode εL a fluctuating field, εsp corresponding to the addition of a single photon with a random phase. The total field therefore undergoes amplitude and phase fluctuations. The fluctuations of the amplitude are damped by the gain saturation of the amplifying medium and only the phase fluctuations persist, because the mechanism responsible for laser oscillation does not impose any phase to the generated field. Thus, in the course of successive spontaneous emission events, the phase of the laser field undergoes a random walk. After a time τc (the field correlation time) the phase of the laser field can no longer be predicted; it has lost all memory of its initial value.

In this chapter we shall describe the principle of the operation of lasers, their common features and the properties of the light they emit. Our aim is not to provide an exhaustive catalogue of the types of laser available at the time of writing. Such an account would, in any case, soon be obsolete. Rather, we shall use concrete examples of existing systems to illustrate important features or general principles. We do not want either to give an extensive theoretical description of a laser's properties and of its dynamics. We restrict ourselves here to a rather simplified approach to its main features and refer the reader to more specialized handbooks for further information (see the further reading section at the end of the main chapter).

The physical principles accounting for laser operation can appear quite straightforward. This impression stems from the fact that the essential concepts are now well understood, whilst the detail and some incorrect notions are passed over in silence. It is interesting to note, however, how painstaking our progress in understanding lasers has been. It is usually considered that the prehistory of the laser commenced in 1917 when Einstein introduced the notion of stimulated emission. In fact, Einstein was led to the conclusion that such a phenomenon must occur from considerations of the thermodynamic equilibrium of the radiation field and a sample of atoms at a finite temperature T.