A truly monochromatic beam of light, if it ever existed, would be perfectly coherent. Suppose that such a beam is split into two parts and each part propagated over an arbitrary distance. When the parts are finally brought together and mixed, no matter how different the two path lengths may have been, the resulting waveform will exhibit constructive and destructive interference in the form of bright and dark fringes. The coherence length of a monochromatic beam is therefore infinite, in the sense that the path-length difference can be as large as desired without hampering one's ability to create interference patterns.

Real sources of light, of course, are never monochromatic. white light restricted to the visible range of wavelengths from 400 nm to 700 nm, for example, has a coherence length of only a couple of micrometers. A green filter passing sunlight at λ0 = 550 nm with a 10 nm bandwidth produces a beam with a coherence length of about 50 μm. The red line of cadmium (λ0 = 643.8 nm) has a nearly Gaussian spectrum with a 0.0013 nm width at half peak intensity, leading to a coherence length of nearly 30 cm. This is similar to the coherence length of a short, inexpensive HeNe laser (λ0 = 632.8 nm) with a few longitudinal modes and a typical bandwidth of Δf ≈ 1 GHz. A stabilized HeNe laser operating in a single longitudinal mode (Δf ≈ 100 kHz) has a coherence length of several kilometers.

Diffractive optical elements (DOEs), which are relatively new additions to the toolbox of optical engineering, can function as lenses, gratings, prisms, aspherics, and many other types of optical element. Typically formed in a film of only a few microns thickness, a DOE may be fabricated on an arbitrarily-shaped substrate. Flexible functionality, wide range of available optical aperture, light weight, and low manufacturing cost are among the advantages of DOEs. They can be fabricated in a broad range of materials such as aluminum, silicon, silica, and plastics, thus providing flexibility in selecting the base material for specific applications. The effects of temperature change, thermal gradients, shock, and stress in thin film optical devices, however, can cause deformation of the substrate and ultimately alter the behavior of a DOE.

DOEs are wavelength sensitive; for instance, the focal length and aberration characteristics of a diffractive lens can vary substantially if the wavelength of the incident light is changed. DOEs can duplicate most of the functions provided by conventional glass optics provided that the optical system operates over a narrow spectral bandwidth, or the operation of the system requires chromatic dispersion. To date, DOEs have found widespread application in beam-combiners, head-mounted displays, beam-shaping optics, laser collimators, spectral filters, compact spectrometers, diode laser couplers, projection displays, compact disk (CD) and digital versatile disk (DVD) players, laser resonators, computer interconnects, solar concentrators, laser material processing, and wavelength division multiplexers/demultiplexers.

Ernst Abbe (1840–1905), professor of physics and mathematics at the University of Jena, Germany, and major partner in the Carl Zeiss company, made important contributions to the theory and practice of optical microscopy. His compound microscope was a superb optical design based on a theoretical understanding of diffraction and minimization of the effects of aberrations. Abbe enunciated his famous sine condition regarding the axial point in the object plane of a centered image-forming system such as a microscope or a telescope. When this condition is satisfied, “aberration-free” imaging of the object points located in the vicinity of the optical axis is assured. This chapter provides an heuristic description of the sine condition, which, in the words of Conrady, is “one of the most remarkable and labor-saving theorems in the whole realm of applied optics”.

As the chapter follows a rather unconventional approach towards explaining the sine condition, it is worthwhile to highlight its main features at the outset. An introduction of the necessary geometric-optical concepts provides the basis for defining the sine condition. This is followed by establishing, for an axial object point, a one-to-one mapping between the principal planes of the imaging system. The wavefront entering the system at the first principal plane (p.p.) is thus related to that emerging from the second p.p.

To describe the imaging of near-axis regions, we switch to a wave-optical viewpoint.

Readers are undoubtedly familiar with the phenomenon of total internal reflection (TIR), which occurs when a beam of light within a high-index medium arrives with a sufficiently great angle of incidence at an interface with a lower-index medium. What is generally not appreciated is the complexity of phenomena that accompany TIR. For instance, consider the simple optical setup shown in Figure 27.1, where a uniform beam of light is brought to focus by a positive lens, being reflected, somewhere along the way, at the rear facet of a glass prism. Assuming a refractive index n = 1.65 for the prism material, the critical angle of incidence is readily found to be θcrit = sin−1(1/n) = 37.3°. Let the lens have numerical aperture NA = 0.2 (i.e., f-number = 2.5). Then the range of angles of incidence on the prism's rear facet will be (33.5°, 56.5°). The majority of the rays thus suffer total internal reflection and converge, as depicted in Figure 27.1, towards a common focus in the observation plane.

Figure 27.2 shows computed plots of intensity and phase at the observation plane, indicating that the focused spot essentially has the Airy pattern, albeit with minor deviations from the ideal. The diameter of the first dark ring, for example, is approximately 6λ, which is close to the theoretical value of 1.22λ /NA for the Airy disk.

The principles of operation of Fabry–Pérot interferometers are well known, and their application in spectroscopy has established their status as one of the most sensitive instruments ever invented. However, the behavior of a Fabry–Pérot device in polarized light, especially when birefringence and optical activity are present within the mirrors or in the cavity, is less well known. We devote this chapter to a description of some of these phenomena, in the hope of clarifying their physical origins and perhaps suggesting some new applications.

The dielectric mirror

A multilayer stack mirror is shown schematically in Figure 15.1. The substrate is a transparent slab of glass, and the layers are made of high- and low-index dielectric materials. In the examples used in this chapter the low-index layers will have (n, k) = (1.5, 0) and thickness d = 105.5 nm, and the high-index layers will have (n, k) = (2, 0) and d = 79.125 nm. (At the operating wavelength, 633 nm, both these layers will be one quarter-wave thick.) Figure 15.2 shows computed plots of amplitude and phase for the reflection coefficients of a 10-layer mirror. At normal incidence (θ = 0) both p- and s-components of polarization have an amplitude reflectivity |ρ| = 0.844. The mirror, therefore, reflects about 71% of the incident optical power and transmits the remaining 29%. Ignoring any loss of light at the uncoated facet of the substrate, the amplitude transmission coefficient (outside the substrate) turns out to be |τ| = 0.536.

Introduction

The degree of first-order temporal coherence, a function denoted by g(1)(τ), provides information about the coherence length and the power spectral density of a light source. However, without additional information, g(1)(τ) has no bearing on intensity fluctuations and higher-order statistics of the emitted light. A quasi-monochromatic laser beam and the beam of light from an incandescent light bulb, provided that the latter is properly filtered to match the spectral line-shape of the former, will have identical degrees of first-order coherence. Any interferometric experiment involving the splitting and superposition of amplitudes would yield identical results for the laser beam and the (properly filtered) thermal light. Therefore, on the basis of such experiments alone, there is no way to distinguish the two light sources. It turns out, however, that the intensity fluctuations of laser light are fundamentally different from those of thermal light. The two sources can, therefore, be distinguished based on their second-order coherence properties.

An ideal photodetector produces an electrical signal proportional to the “cycle-averaged intensity” of the E-field (or B-field) of the light beam at the location of the detector. Assuming that the electrical bandwidth of the detector (including all associated circuitry) is greater than the bandwidth of the incident light wave by at least a factor of 2, the output of the detector should accurately represent the intensity fluctuations of the light beam as a function of time.

The diffraction-limited focusing of a laser beam to either explore or modify a surface is the basis of several important technologies. Examples include scanning optical microscopy, optical disk data storage, and laser printing. The size of the focused spot and the corresponding depth of focus are important factors in determining the performance characteristics of these systems. In this chapter we examine methods of forming the focused spot, and clarify the relation between spot size and depth of focus.

Principle of operation

The essential features of a scanning optical microscope are shown in Figure 37.1. A laser beam is sent through an objective lens to form a focused spot on the sample. Ideally, the objective is corrected for all aberrations, yielding a diffraction-limited focused spot. The light reflected from the sample returns through the objective and is redirected by the beam-splitter to a detection module. The detection module may be designed to monitor the power, the phase, or the polarization state of the returning beam. The electrical signal S(x, y) produced by the detector is thus representative of the small area of the sample illuminated by the focused spot at and around the point (x, y). The sample is moved to different locations by the XY stage on which it is mounted; the signal S(x, y), plotted against the sample's position, yields an image of the sample's surface over the desired area.

An omni-directional dielectric mirror (also known as a one-dimensional photonic bandgap crystal) exhibits 100% reflectivity at all angles of incidence and for all states of incident polarization. Unlike metallic mirrors, which absorb a small fraction of the incident optical power, dielectric reflectors are lossless. These properties make omni-directional dielectric mirrors ideal candidates for applications in which a beam of light in an unknown or unpredictable polarization state is likely to arrive at the mirror from any direction, and in which loss of light at the mirror, no matter how small, is deemed intolerable. A good example is provided by the walls of an optical waveguide. Since there are numerous reflections from the wall as a beam of light travels through, even small losses at each encounter with a wall rapidly deplete the beam's energy.

A typical omni-directional reflector is a periodic stack of bilayers, each bilayer consisting of a high-index and a low-index dielectric layer. The larger the refractive indices of the available dielectrics (and also the larger the difference between these indices), the easier it is to design the reflector. For example, if the two materials available for fabricating a stack of bilayers have indices n1 = 1.5 and n2 = 2.0, it is impossible to obtain omni-directionality for both p- and s-polarized light. However, with n1 = 1.5 and n2 ≥ 2.3, an omni-directional reflector can be designed.

The Scottish physicist John Kerr (1824–1907) discovered the magneto-optical effect named after him in 1888. When linearly polarized light is reflected from the polished surface of a magnetized medium its polarization vector rotates and becomes somewhat elliptical. The direction of rotation and the sense of ellipticity are reversed when the direction of magnetization M of the sample is reversed, thus providing a powerful tool for optically monitoring the state of magnetization of the sample under investigation.

The physical mechanism of the Kerr effect is identical to that of the Faraday effect and, in fact, the same theoretical model can be used to describe both phenomena, one in reflection, the other in transmission (see Chapter 12, “The Faraday effect”).

The Kerr effect can be analyzed under quite general conditions, with the direction of magnetization of the sample oriented arbitrarily relative to the plane of incidence of the light beam. However, the three geometries shown in Figure 13.1 are of particular importance and will be analyzed separately in the present chapter. When the magnetization M is perpendicular to the sample's surface, the observed phenomenon is referred to as the polar Kerr effect. When M is parallel to the surface and in the plane of incidence, the Kerr effect is longitudinal. Finally, when M is parallel to the surface but perpendicular to the plane of incidence, the observed phenomenon is known as the transverse Kerr effect.

Zernike invented the phase-contrast microscope in 1935, and was awarded the 1953 Nobel prize in physics for this achievement. In an ordinary optical microscope, an object that imparts a phase modulation to the incident light will produce only a faint image. This faint image may be attributed to the diffraction of a small amount of the light out of the entrance pupil of the objective lens. To improve this image, Zernike in effect extracted a reference beam from the light collected by the objective lens and produced an interferogram of the object at the image plane of the microscope, thus converting phase information into amplitude (or intensity) modulation.

The principles of operation of the phase-contrast microscope are by now fully understood. Both spatially coherent and spatially incoherent light may be used in this type of microscopy. For best results, a quasi-monochromatic light source with a reasonable coherence time must be employed. Our goal in the present chapter is to give a simple explanation of the main ideas behind the method and to provide a pictorial survey of this important branch of modern optical microscopy.

The phase-contrast microscope

The diagram in Figure 38.1 shows the main elements of a phase-contrast microscope. The light source may be a coherent source (e.g., a laser) or an incoherent one (e.g., a tungsten lamp or an arc lamp); monochromaticity may be achieved by means of a colored glass filter. The condenser lens projects the source onto the object, whose image is formed by the objective lens.

The apertures of classical optics simply block those parts of an incident wavefront that fall outside the aperture, allowing everything else to go through intact. Moreover, multiple apertures act upon an incident beam independently of each other, polarization effects are usually negligible (i.e., scalar diffraction), and it is not necessary to keep track of both the electric- and the magnetic-field components of the beam.

All of the above assumptions break down when apertures shrink to dimensions comparable to or smaller than a wavelength. For example, transmission through two small adjacent apertures cannot be treated by assuming that only one aperture is open at a time, then adding the fields transmitted by the individual apertures. (This is because the electric charge and current distributions in the vicinity of one aperture are influenced by the radiation pattern of the other aperture.) Polarization effects are extremely important for small apertures, as exemplified by the case of a normally incident beam going through an elliptical aperture in a thin metal film; whereas in the case of polarization (i.e., E-field) parallel to the long axis of the ellipse there is negligible transmission, when the incident polarization is rotated 90° to point along the ellipse's minor axis, the aperture transmits a substantial fraction of the incident light. Finally, to analyze the interaction of light with small apertures, it is generally necessary to keep track of both E and B components of the electromagnetic wave, as the modification of one of these fields produces non-trivial changes in the other field's distribution.

An informal survey of some colleagues and students revealed that the notion of reciprocity in optics is not widely appreciated. One colleague even justified the prevailing ignorance by drawing a parallel between reciprocity in optics and complementarity in quantum mechanics: “Both are true statements which have little, if any, practical value in their respective domains.” This chapter is an attempt at explaining the concept of reciprocity, clarifying some associated misconceptions, and pointing out its practical applications.

Non-reciprocity of Faraday rotators

No one disputes that a Faraday rotator is a non-reciprocal element. The usual argument goes as follows. Let a linearly polarized beam of light be fully transmitted through a polarizing beam-splitter (PBS) before being directed through a 45° Faraday rotator, as shown in Figure 17.1. If the beam is reflected back (by an ordinary mirror, for example), it retraces its path through the rotator and emerges with its polarization vector rotated by a full 90°. At the PBS, therefore, the returning beam will be deflected away from its original path. (This, in fact, is a well-known method of isolating laser diodes from spurious reflections within a given system.) Since the reflected light does not return on its original path, and since the PBS is believed to be reciprocal, the argument is taken as proof of the non-reciprocity of the Faraday rotator.

Although it is true that Faraday rotators are non-reciprocal, there is a flaw in the above argument, which will become clear upon inspection of the system of Figure 17.2.

Robert N. Hall, born in New Haven, Connecticut in 1919, joined General Electric's Research and Development Center after graduating from the California Institute of Technology. In 1962, having realized that a semiconductor junction could support population inversion, Hall built the first semiconductor injection laser. This device, based on a specially designed p-n junction, operated when an electric current injected the electrons directly into the junction, thus allowing for highly efficient generation of coherent light from a compact source. Today, diode lasers based on Hall's original idea are used, among other places, in CD and DVD players, laser printers, and fiber-optic communication systems.

In this chapter we describe the basic features of the beam of light emitted by a diode laser, and discuss methods to analyze and manipulate this beam. Collimation and beam-shaping with a pair of cylindrical lenses will be shown to be a simple and flexible method that may be applied not only to diode lasers but also to beams emerging from optical fibers.

Characteristics of diode lasers

A semiconductor diode laser shown schematically in Figure 34.1 consists of a gain layer (only a few ten nanometers thick), surrounded by guiding layers for confining the laser mode. The guiding layers' index of refraction is somewhat greater than that of the surrounding regions (substrate and cladding), thus permitting confinement by total internal reflection.