Introduction

The dielectric coatings typically used in high precision optical measurement consist of alternating layers of materials with different refractive indices (see Figure 9.1). For highreflectivity, the coating layers form pairs of high and low refractive index materials, referred to as doublets, with each component layer a quarter wave thick. In most cases, it is safe to assume that the coating reflects incident light from a mirror surface; a conceptually perfect plane at the boundary between the coating and the vacuum. However, a more precise picture is that of a reflection from each interface between coating layers, all of which add at the mirror surface to form the total reflected wave.

The displacement of the mirror surface, as interferometrically measured with the reflected wave, is given by the simple relation Δz = Δϕλ/4π, where Δϕ is the change in reflection phase of a field with wavelength λ, and Δz is the apparent change in position of the surface. In this chapter we will consider thermo-optic noise, which manifests itself as a change in the reflection phase of the coating resulting from thermal fluctuations in the coating materials.

How to change the reflection phase of a coating

The thermoelastic mechanism

Thermal expansion of the coating materials causes the thickness of a coating to change as a function of temperature. These temperature changes can be driven, or can result from statistical fluctuations in the coating material.

As Lord Kelvin was renowned for saying – “to measure is to know” – and indeed precision measurement is one of the most challenging and fundamentally important areas of experimental physics.

Over the past century technology has advanced to a level where limitations to precision measurement systems due to thermal and quantum effects are becoming increasingly important. We see this in experiments to test aspects of relativity, the development of more precise clocks, the measurement of the Gravitational Constant, experiments to set limits on the polarisation of the vacuum, and the ground based instruments developed to search for gravitational radiation.

Many of these experimental areas use laser interferometry with resonant optical cavities as short term length or frequency references, and thermal fluctuations of cavity length present a real limitation to performance. This has received particular attention from the community working on the upgrades to the long baseline gravitational wave detectors, LIGO, Virgo and GEO 600, the signals from all likely sources being at a level where very high strain sensitivity – of the order of one part in 1023 over relevant timescales – is required to allow a full range of observations. Research towards achieving such levels of strain measurement has shown that the thermal fluctuations in the length of a well designed resonant cavity are currently dominated by those due to mechanical losses in the dielectric materials used to form the multi-layer mirror coating used, with the fluctuations of the mirror substrate materials also playing an important part.

In recent years, intense laser fields have become available, over a wide frequency range, in the form of short pulses. Such laser fields are strong enough to compete with the Coulomb forces in controlling the dynamics of atomic systems. As a result, atoms in intense laser fields exhibit new properties that have been discovered via the study of multiphoton processes. After some introductory remarks in Section 1.1, we discuss in Section 1.2 how intense laser fields can be obtained by using the “chirped pulse amplification” method. In the remaining sections of this chapter, we give a survey of the new phenomena discovered by studying three important multiphoton processes in atoms: multiphoton ionization, harmonic generation and laser-assisted electron–atom collisions.

Introduction

If radiation fields of sufficient intensity interact with atoms, processes of higher order than the single-photon absorption or emission play a significant role. These higher-order processes, called multiphoton processes, correspond to the net absorption or emission of more than one photon in an atomic transition. It is interesting to note that, in the first paper he published in Annalen der Physik in the year 1905, his “Annus mirabilis,” Einstein [1] not only introduced the concept of “energy quantum of light” – named “photon” by Lewis [2] in 1926 – but also mentioned the possibility of multiphoton processes occurring when the intensity of the radiation is high enough, namely “if the number of energy quanta per unit volume simultaneously being transformed is so large that an energy quantum of emitted light can obtain its energy from several incident energy quanta.”

In this chapter we turn to the formulation of the theory of the interaction of intense laser fields with atoms in the important case where the laser photon energy is much smaller than the ionization potential of the initial atomic state. When the intensity is sufficiently high and the frequency sufficiently low, ionization proceeds as if the laser electric field were quasi-static. In this regime, it is appropriate to make the “strong-field approximation,” or SFA, in which one assumes that an active electron, after having been ionized, interacts only with the laser field and not with its parent core. Using this approximation, Keldysh [1] showed that analytical expressions for the rate of ionization can be obtained when the electric-field amplitude, the laser frequency and the binding energy of the initial state are such that the Keldysh parameter γK defined by Equation (1.8) is much less than unity and the photoelectron does not escape by over-the-barrier ionization (OBI). However, the applicability of the SFA extends beyond this regime and, more importantly, it can be used to investigate high-order ATI and high-order harmonic generation. The SFA also provides a framework in which the physical origin of these processes, embodied in the semi-classical three-step recollision model introduced in Section 1.3, can be understood.

We begin in Section 6.1 by examining the low-frequency limit of the Floquet theory and showing how the total ionization rate of the atom can be obtained using the adiabatic approximation.

In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively.

We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies, which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.

The availability of intense laser fields over a wide frequency range, in the form of short pulses of coherent radiation, has opened a new domain in the study of light–matter interactions. The peak intensities of these laser pulses are so high that the corresponding laser fields can compete with the Coulomb forces in controlling the dynamics of atomic systems. Atoms interacting with such intense laser fields are therefore exposed to extreme conditions, and new phenomena occur which are known as multiphoton processes. These phenomena generate in turn new behaviors of bulk matter in strong laser fields, with wide-ranging applications.

The purpose of this book is to give a self-contained and unified presentation of high- intensity laser–atom physics. It is primarily aimed at physicists studying the interaction of laser light with matter at the microscopic level, although it is hoped that any scientist interested in laser–matter interactions will find it useful.

The book is divided into three parts. The first one contains two chapters, in which the basic concepts are presented. In Chapter 1, we give a general overview of the new phenomena discovered by studying atomic multiphoton processes in intense laser fields. In Chapter 2, the theory of laser–atom interactions is expounded, using a semi-classical approach in which the laser field is treated classically, while the atom is described quantum mechanically.

In this chapter, we shall discuss the theory of laser–atom interactions, using a semi-classical method in which the laser field is treated classically, while the atom is studied by using quantum mechanics. This semi-classical approach constitutes an excellent approximation for intense laser fields, since in that case the number of photons per laser mode is very large [1, 2]. In addition, spontaneous emission can be neglected. We begin therefore by giving in Section 2.1 a classical description of the laser field in terms of electric- and magnetic-field vectors satisfying Maxwell's equations. We start by considering plane wave solutions of these equations. Then general solutions describing laser pulses are introduced. The dynamics of a classical electron in the laser field, and in particular the ponderomotive energy and force, are discussed in Section 2.2. Neglecting first relativistic effects, we write down in Section 2.3 the time-dependent Schrödinger equation (TDSE), which is the starting point of the theoretical study of atoms in intense laser fields, and introduce the dipole approximation. In the subsequent two sections, we study the behavior of the TDSE under gauge transformations and the Kramers frame transformation. In view of the central role that the time evolution operator plays in the development of the theory of laser–atom interactions, some general properties of this operator are reviewed in Section 2.6.

One of the major non-perturbative methods used to study atoms in intense laser fields is the direct numerical integration of the wave equations describing atoms interacting with laser fields [1]. This is an attractive alternative to the methods discussed in the two preceding chapters, since solutions of the wave equations can be obtained by numerical integration for a wide range of laser intensities and frequencies. In addition, no restrictions need to be imposed on the type of laser pulses which are used, making the numerical integration of wave equations particularly useful for the study of interaction of atoms with short laser pulses.

However, the numerical integration of the wave equations is computationally very intensive, for the following reasons. Firstly, at high laser intensities, and especially for low frequencies, the ionized electrons can acquire quite high velocities and their quiver motion becomes much larger than the size of the initial atomic orbit. The corresponding wave packets can therefore travel large distances in short time intervals. As a result, the spatial grids used to follow the motion of these wave packets must be large and have small spatial separations. Secondly, the discretization of time, used in all numerical integration schemes, requires a large number of small steps in order to obtain accurate results. Thirdly, the direct numerical integration of the wave equations becomes extremely demanding for atoms with more than one active electron.

In this chapter, we focus our attention on the multiphoton ionization (MPI) of atoms interacting with intense laser fields with wavelengths in the infra-red to visible part of the spectrum. Section 8.1 is devoted to multiphoton single ionization. We begin by giving an overview of key early experiments, in particular those exploring the phenomenon of “above-threshold ionization” (ATI). We then discuss general features of ATI spectra and consider how these features can be understood within the framework of the semi-classical model. We conclude the section by examining how two-color processes can be used to study MPI. In Section 8.2, we analyze multiphoton double ionization, a process which has attracted considerable attention due to the prominent role played by electron correlation effects. Detailed reviews of atomic multiphoton ionization and ATI have been given by Joachain [1], DiMauro and Agostini [2], Protopapas, Keitel and Knight [3], Joachain, Dörr and Kylstra [4], Kylstra, Joachain and Dörr [5], Dörner et al. [6], Becker et al. [7] and Lewenstein and L'Huillier [8].

Multiphoton single ionization

As noted in Section 1.3, multiphoton ionization (MPI) was first observed in 1963 by Damon and Tomlinson [9] and also investigated in 1965 by Voronov and Delone [10] and Hall, Robinson and Branscomb [11]. In the following two decades, a number of experiments were performed to study various aspects of MPI, and results were obtained concerning the dependence of the ionization yields on the laser intensity, absolute MPI cross sections and the resonantly enhanced multiphoton ionization (REMPI) phenomenon.