Introduction

High-quality optical cavities – and hence optical coatings – have emerged as an indispensable tool in physics. One of their key uses is in precision sensing and spectroscopy, where high-performance coatings bridge a broad spectrum of applications from large-scale gravitational wave interferometers (see Chapter 14) to atomic optical clocks (see Chapter 15). Radiation pressure forces inside of such cavities, i.e. the momentum transfer of photons onto the cavity boundaries, impose a fundamental limit on their performance, the so-called standard quantum limit (see Section 1.4), when combined with unavoidable quantum fluctuations in a laser beam. Another limitation arises from random phase fluctuations that are induced by the thermal noise of the coating's mechanical motion (see Chapter 3). With the improved performance of optical coatings, optical cavities are gradually becoming sensitive enough to observe these optomechanical effects. Curiously, what is being considered a nuisance by some fields of research has led within only a few years to the development of a completely new field of optical physics – cavity optomechanics – that exploits optomechanical interactions in high-finesse optical cavities as a new interface between light and matter. A common goal in this field is to exploit the quantum regime of mechanical devices through quantum optical control. This work has immediate implications with respect to fundamental questions, such as the quantum-classical transition of massive objects, and new applications including quantum limited mechanical sensors or mechanical interfaces for quantum information processing.

Introduction

Mechanical and optical thermal noises play an important role in many precise optomechanical experiments, in which positions of test bodies are monitored by laser beams. Much of the initial experimental and theoretical research in this area was driven by the physics of gravitational-wave interferometers, where thermal fluctuations are expected to be the dominant source of noise in the frequency band between about 10 and 100 Hz (Harry et al., 2002) (see Chapter 14). Recently, it has become clear that controlling thermal noise will be key in several other fields, notably in designing laser cavities with higher frequency stability (Numata et al., 2004) (see Chapter 15), in reaching the quantum limit in macroscopic opto-mechanical experiments (Kippenberg and Vahala, 2008) (see Chapter 16), and in cavity QED experiments (Miller et al., 2005) (see Chapter 17). In this chapter we review the statistical-mechanics formalism which is used to theoretically calculate mechanical and optical thermal noise. For completeness, we also add a discussion of another important limitation in mechanical measurements, the so-called Standard Quantum Limit.

Theory of mechanical thermal noise

The theory of time-dependent thermodynamical fluctuations has been extensively developed for the past century. One of the fundamental results in this field is the Fluctuation–Dissipation Theorem, which was originally formulated by Callen and Welton (1951). Callen and Welton's insight was that the intensity of random fluctuation in some macroscopic degree of freedom of the thermodynamic system was proportional to the strength with which was coupled to the microscopic degrees of freedom of the heat bath.

Introduction

In this chapter we review recent research on using alternative beam shapes for reducing thermal noise and other thermal effects in mirrors. High-precision laser interferometry experiments typically make use of the fundamental Gaussian beam, which can be generated with the great spatial stability that is important for achieving low-noise signal readouts. However, the Gaussian beam might not be the optimal choice for all high-precision measurements. The idea of using an alternative beam geometry, specifically a flat-top intensity profile, to reduce mirror thermal noise in optical cavities was discussed first by Kip Thorne and his research group at Caltech in 2000 (Thorne et al., 2000). Since then several groups have made progress in taking this idea closer to reality. Despite these efforts, no highprecision interferometric measurement actually showing lower thermal noise has so far been undertaken with alternative beam shapes.

Alternative beam shapes improve thermal noise, and specifically coating thermal noise, by effectively increasing the wm parameter of Equation 4.9 and subsequent related equations. Straightforward increasing of the beam width wm of a Gaussian beam will work up to a point, but the optical loss from light spilling over the mirror edge generally will be unacceptable at some level. Thus, with purely Gaussian beams, there is a tradeoff between lower thermal noise (larger wm) and lower shot noise (higher optical power, thus smaller wm). Using beam shapes other than Gaussian is a way to change this tradeoff, because alternative beam shapes can be effectively larger while keeping the light mostly contained on the face of the optic.

Overview

Light incident on high quality optical surfaces and substrates can be lost from the beam in two possible ways. The light can be scattered from the beam due to imperfections such as microroughness or point defects, which is discussed in Chapter 11. Alternatively, the light can be absorbed by the coating or the substrates. Experiments over many decades have shown that for high quality mirrors reflecting radiation in the IR range, the dominant loss mechanism is scatter by often more than an order of magnitude. This large bias towards loss due to scatter makes it very challenging to measure the absorption in such optics. The optical absorption of the mirror coatings in high precision applications like gravitational wave interferometers (see Chapter 14) typically ranges from a fewtenths of parts per million (ppm) to several ppm. Given that the loss in mirrors is completely dominated by scatter, one may ask the question: why then is such a small level of absorption important? The reason is thermal aberrations.

Clearly, absorption at far larger than ppm levels is widely tolerated in many other types of optical instruments – metallic mirror coatings typically absorb a few percent of light incident upon them, for example. But most optical instruments are either of low power, or low precision, or both. Absorption does play a central role in setting the damage threshold intensity in high power optics. Precision measurements are typically not at such high optical power that laser damage is an issue.

Introduction

Many high precision optical measurements require high laser power, low optical loss or a combination of both to achieve their sensitivity requirements. Optical loss in an apparatus is often dominated by light scattering from coatings while optical absorption sets limits on the advances that can be made with higher optical power. In some cases, the achievable sensitivity given these limits is not sufficient and experiments require more advanced techniques such as a manipulation of the quantum behavior of light or avoiding any transmissive optics. In this chapter we give an overview of our current understanding of scattered light from high performance coatings. We also present simple overviews on squeezed light, in Section 11.3, and diffractive optics, in Section 11.4.

Optical loss

Optical power loss in experiments can result from a variety of processes; scattering, absorption, transmission, polarization effects, and imperfect quantum efficiency in detection. Some precision optical measurements are limited by the standard quantum limit, discussed in Section 1.4 and below in Section 11.3, where optical power becomes a key parameter in sensitivity. When position of a test mass is measured interferometrically, the signal will scale linearly with the laser power while the quantum shot noise increases as the square-root of power. In these situations, the signal-to-shot-noise ratio will increase proportionally to the square-root of the power. Optical loss can limit the achievable power in a system, and degrade this signal-to-noise ratio. It is also a limiting factor in the quantum noise reduction achievable with squeezed states of light, as discussed below in Section 11.3.