Introduction

This chapter is focused on design strategies for minimizing Brownian (see Chapter 4) and, more generally, thermal noises (see Chapters 3 and 9) in high-reflectivity optical coatings. It is organized as follows: in Section 12.2 we review the basic formulas needed to describe the optical properties of dielectric coatings (an ab-initio derivation of these formulas is included in the Appendix). Brownian noise formulas are the subject of Section 12.3. Section 12.4 presents the key ideas of coating thickness optimization. Thermo-optic noise issues are reviewed in Section 12.5, together with a discussion of pertinent minimization criteria. Section 12.6 contains a few comments on material characterization, and touches the important topic of glassy mixture modeling and optimization.

Coating formulas

In this section we summarize the basic coating formulas on which the subsequent analysis is based. A compact ab-initio derivation of these results is given in the Appendix.

Optical coatings are modeled as stacks of planar layers terminated on both sides by homogeneous halfspaces; the relevant geometry and notation is sketched in Figure 12.1. Layers are identified by an index i = 1, 2, …, NL. It is understood that i = 0 and i = NL + 1 correspond to the left halfspace and the substrate, respectively. It is convenient to introduce a local coordinate system (x, y, zi) for each layer, so that the internal layers i = 1, 2, …, NL correspond to -di ≤ zi ≤ 0, the left halfspace is defined by - ∞ < z0 ≤ 0, and the substrate by 0 ≤ zNL + 1 < ∞.

Cavity quantum electrodynamics (cavity QED) describes the interactions of single atoms and single photons in a resonator. In the laboratory, state-of-the-art experiments are able to observe the dynamics of these simple yet fundamental quantum systems. The recent development of quantum information science has also introduced a new role for cavity QED systems as quantum interfaces, in which information can be coherently mapped between photonic and atomic quantum bits.

Groundbreaking optical cavity QED experiments in the past twenty years would not have been possible without the technological development of mirror coatings with extremely low scatter and absorption losses. Thermal noise has typically played a smaller role in cavity QED experiments but in certain instances is nevertheless important to consider.

In this chapter, we begin with an overview of optical cavity QED experiments and some of the central questions that they address. It is hoped that this description, while by no means exhaustive, will provide a window for the nonspecialist into basic techniques and motivations in the field. We then turn to focus on the roles played by scattering and absorption loss and mechanical loss. We discuss how these losses have set limits on feasible experiments and the extent to which improvements may be possible.

Experimental realizations of cavity QED systems

An idealized cavity QED system consists of one atom placed between two highly reflective mirrors (Figure 17.1). In this model, a single atomic transition interacts with photons in a single cavity mode; the atom is often pictured at an antinode of the cavity standing wave, where it has a maximal dipole interaction with the cavity field.

Phase noise and shot noise are often the fundamental limiting factors of sensitivity in precision optical systems. These noises determine the so-called standard quantum limit (Braginsky et al., 2003), see Section 1.4. At the same time the fundamental frequency stability in high-finesse optical resonators may also be determined by other fundamental effects originating in mechanical, thermodynamical, and quantum properties of solid boundaries (Braginsky et al., 1979). Many of these effects were initially identified and calculated on the forefront of laser gravitational wave antenna research (see Chapter 14) but are becoming increasingly important in other optical systems (Numata et al., 2004; Webster et al., 2008; Matsko et al., 2007; Savchenkov et al., 2007) (see Chapters 15, 16, and 17).

Excess optical phase noise is added to a probe optical wave reflected from mirrors forming an optical cavity due to variation of boundary conditions produced by fluctuations of the surface and optical thickness of the multilayer coating. We characterize below different effects leading to phase noise starting from fluctuations originating in the mirror's substrate and in its interferometric multi-layer reflective coating. In addition to intrinsic noises produced by internal properties of optical mirrors, there are also extrinsic noises imprinted onto the output phase due to different nonlinear effects in the bulk and coating of the mirrors. As, for example, fluctuations of input power producing fluctuations of thickness and refractive index in the coating due to local heating (the photothermal effect, see Sections 3.4 and 3.10).

Introduction

Some of the most energetic and intriguing phenomena in our universe, including black hole coalescence and the first moments of the Big Bang, cannot be observed with traditional “light telescopes”. These events can only be “seen” with gravitational wave observatories that detect minute ripples in the fabric of space-time. Gravitational wave detection has the potential to open a new window for humanity to view and understand the universe. However the experimental requirements for detection are extreme and meeting the sensitivity limits has spurred research in noise reduction. The study of thermal noise in optical coatings was driven by the need formirror displacement sensitivities of 10-19 min advanced gravitational wave interferometers (Levin, 1998; Harry and The LIGO Scientific Collaboration, 2010; Harry et al., 2002).

Einstein's General Theory of Relativity states that an accelerating mass will produce a gravitational wave. The acceleration produces a distortion in space-time that propagates at the speed of light (see Einstein (1915), Einstein (1916), and Barish and Weiss (1999)). Gravitational waves interact extremely weakly with matter and as a result are very difficult to detect. Thus, an astrophysical event is required to produce gravitational waves that are strong enough to be detected on Earth. These weakly interacting waves are rarely scattered or absorbed, therefore the information they carry is virtually unaltered regardless of the time and distance they have traveled to us or the intervening objects and conditions they have encountered.

Introduction

The use of cryogenic mirrors can be an attractive way of minimizing thermal noise due to optical coatings and substrates. As long as special care is taken in choosing materials, thermally induced fluctuations should be “frozen” at low temperatures. In this chapter, we introduce the current understanding of temperature dependence of mirror thermal noise, some practical design issues, and examples of cryogenic experiments.We will mainly focus on off-resonant thermal noise and physical cooling issues of macroscopic optics. For more on cooling of resonant thermal noise, see Chapter 16.

Temperature dependence of mirror thermal noise

In sensitive optical experiments, thermal fluctuations of coatings and substrates are important sources of noise and both tend to be reduced at cryogenic temperatures. The gains from cryogenics could come from several directions: the direct reduction of temperature itself, the decrease in mechanical losses (occurring in some materials), and the precipitous drop in specific heat and the increase in the mean free path of phonons. Appropriate materials and beam parameters must be adopted so that every kind of thermal noise is minimized by cryogenics simultaneously. Typically, Brownian thermal noise in coatings improves very slowly from cryogenics. This is due to a weak temperature dependence of mechanical loss. Coating thermal noise is also usually dominant over the substrate Brownian noise, so improvements of coating thermal noise directly impact measurement sensitivity. Substrate thermoelastic noise (see Chapter 7), coating thermo-optic noise (see Chapter 9), and other thermal noises (see Chapter 3) generally have less importance at cryogenic temperatures and at low frequencies.

Introduction

Mirror thermal noise has been directly measured in optical cavities with high displacement sensitivity. Direct comparisons between theory and measurement were done in dedicated setups which directly measure mirror thermal noise. In addition, frequency stabilities achieved by fixed-spacer cavities are limited by mirror thermal noise, ultimately by coating thermal noise. In this chapter, we will give an overview of such verification experiments and the experimental techniques which are used.

General considerations

There are two ways to form Fabry–Perot cavities to measure mirror thermal noise (see Figure 5.1).

• Free-mirror configuration. Two mirrors are mechanically isolated from each other, like in a gravitational-wave interferometer (see Chapter 14). This configuration more easily allows isolation of the mirror thermal noise from other thermal noises, but is more challenging for suppression of seismic noise. As a result, its measurement frequency range is limited to ≿10 Hz. The displacement measurement is obtained from a control (error) signal which keeps the free-mirror cavity on resonance (Black et al., 2004a). Two identical test cavities allow for common-mode noise rejection. This has typically been done electronically. An optical recombination is also possible, as a Fabry–Perot Michelson interferometer, for the common-mode noise rejection. However, it adds another degree of freedom to be controlled.

• […]

Introduction

Optical cavities are extremely useful devices in laser-based research. Within the context of precision measurement, they enable tests of the laws that govern the macroscopic structure of the universe, embodied in the search for gravitational waves (Abbott and The LIGO Scientific Collaboration, 2009) (see also Chapter 14). At the other end of the length scale, cavity-stabilized lasers are powerful tools for precision spectroscopy that probe nature at the quantum mechanical level, through tests of quantum electrodynamics (QED) (Kolachevsky et al., 2009). Furthermore, cavities enable high-sensitivity broadband spectroscopy (Thorpe et al., 2006), which has practical applications in trace gas sensing; exploration of new light – matter interaction regimes in cavity QED (Miller et al., 2005) (see also Chapter 17); tests of fundamental physical principles including relativity (Brillet and Hall, 1979; Hils and Hall, 1990; Eisele et al., 2009), local position invariance (Blatt et al., 2008), and the time invariance of the fundamental constants of nature (Fortier et al., 2007b); and nonlinear optics, including coherent light build-up for studies of extremely nonlinear effects (Gohle et al., 2005; Yost et al., 2009). In general, optical cavities have become indispensable tools at the heart of many modern experiments.

In conjunction with optical frequency combs, cavity-stabilized laser systems have enabled the development of highly accurate frequency standards based on neutral atoms (Sterr et al., 2004; Takamoto et al., 2005; Ludlow et al., 2006; Le Targat et al., 2006; Ludlow et al., 2008; Lemke et al., 2009) and trapped ions (Diddams et al., 2001; Madej et al., 2004; Margolis et al., 2004; Rosenband et al., 2008).

Dedicated to Robert Kirk Burrows

In 1999, I was a young postdoc moving to Syracuse University to work on LIGO, which had been a dream of mine since I was first introduced to gravitational wave detection as an undergraduate in Kip Thorne's class at Caltech. I had done my PhD in gravitational wave detection, but using the older technology of resonant masses rather than LIGO's laser inteferometry. I was concerned that my background would not prove appropriate. I soon found a common issue, thermal noise, that I was able to focus on. Beyond just a good fit for me, thermal noise was actually a topic in flux within LIGO at the time. A talented young theorist at Caltech named Yuri Levin had just shown that the optical coatings on the LIGO mirrors could well contribute much more thermal noise than anyone had anticipated. What was missing were realistic numbers to plug into Yuri's formulas to see just how big of an impact coating thermal noise might have. This became one of my principal roles in LIGO, as part of a group of experimentalists interested in this question at Stanford, Glasgow, as well as Syracuse and other collaborating institutions.

Since then, we in LIGO have found that coating thermal noise is a very important limit to sensitivity, and we have engaged in over a decade of theoretical, experimental, and modeling work to better understand and reduce it.

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.”