Liquids play an important role both in society and in many technologies. Although the viscosity, optical absorption, and scattering properties of chocolate make this particular liquid unsuitable for application in optics, the use of other fluids has resulted in fluidic micro-optics developing into an innovative new discipline. The use of fluids for optics is not new; macro-optical examples include rotating pools of mercury used as reflecting telescope mirrors (Borra, 1982), and we have already seen micro-optical examples in the previous chapters.

Fluidic micro-optics, for which the term optofluidics has attained common usage, is based on the combination of microfluidics and micro-optics, and has evolved in the last decade. Microfluidics, as we mentioned in Chapter 12, is a branch of microsystems engineering that employs microfabricated fluidic channels, micropumps, and microvalves to realize complete fluidic systems (Srinivasan et al., 2004; Kedzierski et al., 2009). Many microfluidic systems have been developed for biochemical analysis, in which optical components already play a role; laser diodes, LEDs, photodetectors, and other micro-optical devices have been integrated with fluidic structures to yield micro-total-analysis (so called μTAS) systems.

Optofluidics, however, concerns the use of fluidic effects and systems for the realization of the optical component itself (Psaltis et al., 2006; Monat et al., 2007).

Most optical systems benefit from tunability. The ability to controllably change the focal length, magnification, or orientation of a single component or optical assembly is essential for many applications, and is typically accomplished macroscopically using mechanical movement of parts of the system with respect to each other. If we think of focusing a microscope, zooming a camera lens, or rotating a mirror for beam alignment, tunability is usually accomplished by twisting a dial, pushing a knob, or shifting a fixture.

In the micro-world a host of effects provides a number of novel means to tune micro-optical components, such that tunable micro-optics has developed into a dynamic research area (Friese et al., 2007; Krogmann et al., 2007). On the one hand, new materials, such as liquids, distensible membranes, liquid crystals, and deformable structures, may be used for the fabrication of micro-optical components, and on the other, phenomena such as surface tension, pressure, mechanical deformation, and the application of electric or magnetic fields may be employed to tune the optical characteristics. Most of these do not apply to macroscopic systems, so that we have a uniquely micro-optical topic at hand.

We consider in this chapter some of the techniques and devices that have been used to realize tunable micro-optical components and systems. A sizable fraction of the concepts employ liquids, either as droplet lenses or in microfluidic systems, and other softmatter elements, such that we study a number of these in detail.

One of the distinguishing characteristics of micro-optics is the means by which micro-optical components and systems are fabricated. Classical optical elements, predominantly lenses and prisms, have been manufactured for centuries using well-established and highly refined grinding and polishing processes. As the size of lenses decreases below the millimeter range, however, these techniques reach their limits: grinding a lens with a diameter below 1 mm is difficult, and economically impracticable. As a result, micro-optics uses completely different fabrication techniques, namely those based on semiconductor manufacturing.

Semiconductor fabrication technology is extremely highly developed, and the factories that produce acres of processed silicon likely represent the most advanced manufacturing technologies on the planet. Whereas most of this industrial capacity is still used for generating the electronic chips on which much of human civilization relies, the past two to three decades have seen an expansion of these technologies into other realms, notably microelectromechanical systems (MEMS) (Menz et al., 2001; Senturia, 2001; Korvink and Paul, 2006; Liu, 2006; Gianchandani et al., 2007), a field that has diversified to include micromechanics, microfluidics, microacoustics and micro-optics. These sub-fields have benefited from semiconductor technology, not only in that extremely small structures may be defined but also that these may be mass-fabricated, in marked contrast to the fabrication of individual components that characterizes classical optics.

Much, if not most, interesting optics happens at interfaces, the transition from one material to another. The phenomena of reflection and refraction are both interface effects, and since these are the basis for the functional behavior of almost all optical components, we must address the question of what happens when an electromagnetic wave impinges on the boundary between two materials.

We begin this chapter with a discussion of refraction, which occurs when a beam of light strikes an interface and is deflected in its propagation direction. Based on consideration of the electromagnetic fields and their polarization across the material transition, we will then derive relationships between the reflected and transmitted parts of the optical field: both of these considerations, we will see, are to first order a function of the refractive indices of the materials on either side of the interface. After looking at the reflection and transmission of power, we will look at the important difference between external and internal incidence, and for the latter consider total internal reflection and the evanescent field.

Reflection and refraction

When a light beam is incident from a material with refractive index ni onto an interface formed with a material of refractive index nt, as shown in Figure 4.1, part of the light is reflected back into ni and a portion is transmitted into nt.

Of all subdisciplines of micro-optics, reflective micro-optics is perhaps the simplest from the optical point of view, yet the most successful commercially. Micromirrors are manufactured by the billions, and their simple optical function, namely to reflect an incident light beam, is the basis for a significant sector in the home entertainment industry on the one hand, and forms the backbone for the fiber networks on which the planet relies on the other.

We will see in the sections below how micromirrors have revolutionized video beamers and have become the mainstay for optical telecommunications. Beginning with a basic analysis of reflection for planar, parabolic, and spherical mirrors, we will subsequently look at the basic technology of micromirrors, their structure, optical behavior, and means for actuation. A special case for reflective micro-optics is that of adaptive optics, which relies on mirror arrays or deformable single mirrors to correct a distorted wavefront, and has become an essential component in medical and astronomical imaging systems. Finally, as micromirrors represent advanced technological products, we will examine a few case studies, showing where you are likely to have already encountered one of these components.

Reflection

Reflective micro-optics relies on reflection of an optical field from a mirror surface, which is, at least in its basics, relatively easy to understand. In striving for high reflectivity for ever larger wavelength ranges, engineers continue to study reflection both theoretically and experimentally, and this seemingly well-developed topic is still the subject of a fair amount of research activity.

Interference is one of the most important phenomena in optics, and forms the basis for diffractive optics, one of the most relevant subdisciplines of micro-optics. Interference effects result from the superposition of electromagnetic fields with closely matched frequency, and explain many naturally observed phenomena, such as the colorful rings generated by thin oil films on wet surfaces. For the optician interference is a tool useful both for the development of novel optical components and for high-accuracy measurement tasks.

This chapter begins by considering coherence and interference in general, and then focuses on interferometers, optical systems that take advantage of interference effects for determination of distances, wavelengths, refractive indices, and a host of related parameters. Interferometers are not only essential characterization tools for micro-optics, but are also fabricated using the technologies of micro- or integrated optics to realize a wide variety of measurement tasks. We then turn to the optics of thin-film multilayers, optical coatings on surfaces that rely on interference to generate customizable transmission or reflection properties. By depositing layers of precisely controlled refractive index in layers at the correct thicknesses, the interference of partially transmitted and partially reflected fields can generate high or low reflection, or, for example, filters that transmit only a certain band of wavelengths. The design and fabrication of anti-reflection coatings are important process steps in the manufacture of both macro and micro-optical components.

The optical components and systems we have considered to this point belong to the realm of free-space optics, meaning the optical field is transmitted through a homogeneous medium (such as vacuum or air) without being guided in any manner. The lenses, mirrors, and diffractive structures we have studied are intended to modulate the unguided beams of free-space optics.

A significant branch of optics, however, uses structures to guide an optical field using reflections at an optical interface. The optical structures that accomplish this, waveguides and optical fibers, may be considered to be a form of “optical wire,” in which the field is confined to a small area in two dimensions and transmitted along the third. Waveguides are micro-optical components for light transmission that are typically fabricated on glass, plastic or semiconductor substrates, giving rise to the concept of integrated optics, analogously to integrated electronics (S. Miller, 1969). When combined with integrated active optical devices, such as lasers, modulators or detectors, also waveguide-based, such guided wave chips form the basis for the field of photonics. Finally, optical fibers span several orders of magnitude in their dimensions, with micrometer-sized guiding regions and hundred kilometer lengths; they are also useful as light guides in hybrid micro-optical systems.

We consider guided-wave optics in this chapter, beginning with an overview of how light is transmitted in a waveguide, and introduce the discrete propagation modes that result.

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.