Evanescent electromagnetic waves abound in the vicinity of luminous objects. These waves, which consist of oscillating electric and magnetic fields in regions of space immediately surrounding an object, do not transfer their stored energy to other regions and, therefore, remain localized in space. Like all electromagnetic waves, the behavior of evanescent waves is governed by Maxwell's equations, and their presence in the vicinity of an object helps to satisfy the requirements of field continuity at the object's boundaries. Evanescent fields decay exponentially with distance away from the object's surface, making them exceedingly difficult to detect at distances much greater than a wavelength.

When a beam of light shines on a diffraction grating, for example, various diffracted orders partake of the energy of the incident beam and carry it away in different directions. At the same time, evanescent waves are created around the grating, which ensure the continuity of the field at the grating's corrugated surface. Similarly, a beam of light shining on an aperture or on a small particle sets up evanescent fields around the boundaries of these objects. Perhaps the best-known example of evanescence, however, is provided by total internal reflection (TIR) from an internal facet of a prism (see Figure 28.1). Here the evanescent field is formed in the free-space region behind the prism, and remains distinct and isolated from the propagating (i.e., incident and reflected) beams; this phenomenon was discussed briefly in Chapter 27.

Photolithography is the technology of reproducing patterns using light. Developed originally for reproducing engravings and photographs and later used to make printing plates, photolithography was found ideal in the 1960s for mass-producing integrated circuits. Projection exposure tools, which are now used routinely in the semiconductor industry, have continually improved over the past several decades in order to satisfy the insatiable demand for reduced feature size, increased chip size, improved reliability and production yield, and lower overall cost. High-numerical-aperture lenses, short-wavelength light sources, and complex photoresist chemistry have been developed to achieve fabrication of fine patterns over fairly large areas. Research and development efforts in recent years have been directed at improving the resolution and depth of focus of the photolithographic process by using phase-shifting masks (PSMs) in place of the conventional binary intensity masks (BIMs). In this chapter we describe briefly the principles of projection photolithography and explore the range of possibilities opened up by the introduction of PSMs.

Basic principles

Figure 42.1 is a diagram of a typical projection system used in optical lithography. A quasi-monochromatic, spatially incoherent light source (wavelength λ) is used to illuminate the mask. Steps are usually taken to homogenize the source, thus ensuring a highly uniform intensity distribution at the plane of the mask. The condenser stop may be controlled to adjust the degree of coherence of the illuminating beam; this control of partial coherence is especially important when PSMs are used to improve the performance of optical lithography beyond what is achievable with the traditional BIMs.

The state of polarization of a given beam of light is modified upon reflection from (or transmission through) an object. The resulting change in polarization state conveys information about the structure and certain physical properties of the illuminated region. Polarization microscopy is a variant of conventional optical microscopy that enables one to monitor these changes over a small area of a specimen. Such observations then allow the user to identify and analyze the specimen's structural and other physical features.

Traditionally, observations with a polarization microscope have been categorized “orthoscopic” or “conoscopic.” orthoscopic observations involve direct imaging of the sample itself, thus allowing one to view the indentations, striations, variations of optical activity and birefringence, etc., over the sample's surface. conoscopic observations, however, involve illuminating a crystalline surface with a cone of light and then imaging the exit pupil of the objective lens. This mode of observation is used in characterizing the crystal's ellipsoid of birefringence and identifying its optical axes.

The polarization microscope

Figure 39.1 is a simplified diagram of a polarization microscope. The light source is typically an extended white light source, such as a halogen lamp or an arc lamp. The collected and collimated beam from the source is linearly polarized as a result of passage through a polarizer. In metallurgical microscopes, such as the one shown here, the objective lens is used both for illuminating the sample and for collecting the reflected light.

Following the publication of the first edition of this book, I wrote (or co-wrote) nine additional columns for Optics & Photonics News (OPN), which appeared between April 2001 and April 2007. Some of these columns were included in the Japanese enlarged edition of the book, published in 2006; all nine columns are now included in this second English edition. Throughout the years, I also wrote four columns which were not submitted to OPN, because they ended up being somewhat lengthy and perhaps too mathematical for the general readership of the OPN; these appear here for the first time as Chapters 9, 14, 18, and 25.

The selection of topics and the exposition style of the thirteen new chapters of the present edition follow the same principles and guidelines as did the original thirty-seven chapters of the first edition. In each case a topic is chosen either for its intrinsic value as a foundational contribution to the science of optics (e.g., the Sagnac effect, second-order coherence, the Doppler shift), or because of its technological significance (e.g., optical pulse compression, semiconductor diode lasers, diffractive optical elements). To a large extent, the fifty chapters of the present book are independent of each other and can be read in any desired sequence. Occasionally, when the information in one chapter could benefit the understanding of the material in another, cross references are provided. The presentation style is pedagogic and informal, with mathematics used sparingly unless it is deemed essential and unavoidable.

Holography dates from 1947, when the Hungarian-born British scientist Dennis Gabor (1900–1979) developed the theory of holography while working to improve electron microscopy. Gabor coined the term “hologram” from the Greek words holos, meaning whole, and gramma, meaning message. The 1971 Nobel prize in physics was awarded to Gabor for his invention of holography.

Further progress in the field was prevented during the following decade because the light sources available at the time were not truly coherent. This barrier was overcome in 1960, with the invention of the laser. In 1962 Emmett Leith and Juris Upatnieks of the University of Michigan recognized, from their work in side-looking radar, that holography could be used as a three-dimensional visual medium. They improved upon Gabor's original idea by using a laser and an off-axis technique. The result was the first laser transmission hologram of three-dimensional objects. The basic off-axis technique of Leith and Upatnieks is still the staple of holographic methodology. These transmission holograms produce images with clarity and realistic depth, but require laser light to view the holographic image.

The Russian physicist Uri Denisyuk combined holography with Lippmann's method of color photography. In 1962 Denisyuk's approach produced a white-light reflection hologram, which could be viewed in the light from an ordinary light bulb. In 1968 Stephen Benton, then at Polaroid corporation, invented white-light transmission holography. This type of hologram can be viewed in ordinary white light and is commonly known as the rainbow hologram.

In the classical electromagnetic theory the wave-vector k = (2π/λ)σ underlies the Fourier space of propagating (or radiative) fields. The k-vector combines into a single entity the wavelength λ and the unit vector σ that signifies the beam's propagation direction. The Fourier transform relation between the three-dimensional space of everyday experience and the space of the wave-vectors (the so-called k-space) gives rise to relationships between the two domains analogous to Heisenberg's uncertainty relations.

Considering that in quantum theory the electromagnetic k-vector is proportional to the photon's momentum (p = ħk, where ħ = h/2π, h being the Planck constant), one should not be surprised to find relationships between dimensions of a beam in the X Y Z-space and its momentum spread in the k-space. Such relationships impose fundamental limits on the ability of measurement systems to determine the various properties of electromagnetic fields.

In this chapter we address two problems that have widespread applications in optical metrology, spectroscopy, telecommunications, etc., and discuss the constraints imposed by the uncertainty principle on these problems. The first topic of discussion is the separation of two overlapping beams of identical wavelength having slightly different propagation directions. This will be followed by an analysis of the limits of separating co-propagating beams having slightly different wavelengths.

Angular separation and the limit of resolvability

Figure 19.1 shows an aperture of diameter D, which transmits two plane waves of the same wavelength λ propagating in slightly different directions.

Diffraction gratings have been used in spectroscopy and other studies of electromagnetic phenomena for nearly two centuries. Josef Fraunhofer (1787–1826), the discoverer of the dark lines in the solar spectrum, built the first gratings in 1819 by winding fine wires around two parallel screws. Henry Rowland made significant contributions to the fabrication of precise, large-area, high-frequency ruled gratings in the 1880s. Robert Wood, who succeeded Rowland in the chair of experimental physics at Johns Hopkins University in 1901, used these ruled gratings extensively in his researches and discovered, among other things, the “anomalous” behavior of metallic gratings, which he first published in 1902. John William Strutt (Lord Rayleigh) developed a theoretical model of these gratings around 1907 and was successful in explaining certain features of Wood's anomalies. However, it is only during the past thirty years or so that a thorough understanding of nearly all aspects of the behavior of diffraction gratings has been achieved through the consistent application of Maxwell's equations with the help of advanced analytical and numerical techniques.

Modern gratings having a few thousand lines per millimeter with near-perfect periodicity are fabricated over fairly large areas (grating diameters of around one meter or so are possible). The groove shapes can be controlled to be sinusoidal, rectangular, triangular, or trapezoidal, and one can obtain shallow or deep grooves (relative to the groove width) by current manufacturing techniques.

When thinking about traditional optical materials one invokes a notion of homogeneous media, where imperfections or variations in the material properties are minimal on the length scale of the wavelength of light λ (Fig. 1.1 (a)). Although built from discrete scatterers, such as atoms, material domains, etc., the optical response of discrete materials is typically “homogenized” or “averaged out” as long as scatterer sizes are significantly smaller than the wavelength of propagating light. Optical properties of such homogeneous isotropic materials can be simply characterized by the complex dielectric constant ε. Electromagnetic radiation of frequency ω in such a medium propagates in the form of plane waves E,H ̴ ei(k·r− ωt) with the vectors of electric field E(r,t), magnetic field H(r,t), and a wave vector k forming an orthogonal triplet. In such materials, the dispersion relation connecting wave vector and frequency is given by εω2 = c2k2, where c is the speed of light. In the case of a complex-valued dielectric constant ε, one typically considers frequency to be purely real, while allowing the wave vector to be complex. In this case, the complex dielectric constant defines an electromagnetic wave decaying in space, |E|, |H| ̴ e−Im(k)·r, thus accounting for various radiation loss mechanisms, such as material absorption, radiation scattering, etc.

Another common scattering regime is a regime of geometrical optics. In this case, radiation is incoherently scattered by the structural features with sizes considerably larger than the wavelength of light λ (Fig. 1.1(b)).

In this section we discuss in more detail the guiding properties of photonic bandgap waveguides, where light is confined in a low refractive index core. We first describe guidance of TE and TM waves in a waveguide featuring an infinite periodic reflector operating at a frequency in the center of a bandgap. In this case, radiation loss from the waveguide core is completely suppressed. We then use perturbation theory to characterize modal propagation loss due to absorption losses of the constitutive materials. Finally, we characterize radiation losses when the confining reflector contains a finite number of layers.

Figure 4.1 presents a schematic of a waveguide with a low refractive index core surrounded by a periodic multilayer reflector. The analysis of guided states in such a waveguide is similar to the analysis of defect states presented at the end of Section 3. As demonstrated in that section, a core of low refractive index nc surrounded by a quarter-wave reflector can support guided modes with propagation constants above the light line of the core refractive index kx ⊂ [0, ωnc]. Note that a core state with kx = 0 defines electromagnetic oscillations perpendicular to a multilayer plane, therefore such a state is a Fabry–Perot resonance rather than a guided mode. In the opposite extreme, a core state with kx ̴ ωnc defines a mode propagating at grazing angles with respect to the walls of a waveguide core, which is typical of the lowest-loss leaky mode of a large-core photonic bandgap waveguide.

In Chapter 4 we have derived perturbation theory for Maxwell's equations to find corrections to the electromagnetic state eigenfrequency ω due to small changes in the material dielectric constant. Applied to the case of systems incorporating absorbing materials, we have concluded that absorption introduces an imaginary part to the modal frequency, thus resulting in decay of the modal power in time. While this result is intuitive for resonator states localized in all spatial directions, it is somewhat not straightforward to interpret for the case of waveguides in which energy travels freely along the waveguide direction. In the case of waveguides, a more natural description of the phenomenon of energy dissipation would be in terms of a characteristic modal decay length, or, in other words, in terms of the imaginary contribution to the modal propagation constant. The Hamiltonian formulation of Maxwell's equations in the form (2.70) is an eigenvalue problem with respect to ω2, thus perturbation theory formalism based on this Hamiltonian form is most naturally applicable for finding frequency corrections. In the following sections we develop the Hamiltonian formulation of Maxwell's equations in terms of the modal propagation constant, which allows, naturally, perturbative formulation with respect to the modal propagation constant.

Eigenstates of a waveguide in Hamiltonian formulation

In what follows, we introduce the Hamiltonian formulation of Maxwell's equations for waveguides, [1] which is an eigenvalue problem with respect to the modal propagation constant β. A waveguide is considered to possess continuous translational symmetry in the longitudinal ẑ direction.

In this section we investigate photonic bandgaps in two-dimensional photonic crystal lattices. We start by plotting a band diagram for a periodic lattice with negligible refractive-index-contrast. We then introduce a plane-wave expansion method for calculating the eigenmodes of a general 2D photonic crystal, and then develop a perturbation approach to describe bandgap formation in the case of photonic crystal lattices with small refractive index contrast. Next, we introduce a modified plane-wave expansion method to treat line and point defects in photonic crystal lattices. [1,2] Finally, we introduce perturbation formulation to describe bifurcation of the defect states from the bandgap edges in lattices with weak defects.

The two-dimensional dielectric profiles considered in this section exhibit discrete translational symmetry in the plane of a photonic crystal, and continuous translational symmetry perpendicular to the photonic crystal plane direction (Fig. 6.1). The mirror symmetry described in Section 2.4.7 suggests that the eigenmodes propagating strictly in the plane of a crystal can be classified as either TE or TM, depending on whether the vector of a modal magnetic or electric field is directed along the ẑ axis.

Two-dimensional photonic crystals with diminishingly small index contrast

In the case of a 2D discrete translational symmetry, the dielectric profile transforms into itself ε(r + δr) = ε(r) for any translation along the lattice vector δr defined as δr = ā1N1 + ā2N2,(N1, N2) ⊂ integer.

This chapter is dedicated to periodic structures that are geometrically more complex than 2D photonic crystals, but not as complex as full 3D photonic crystals. In particular, we consider the optical properties of photonic crystal fibers, optically induced photonic lattices, and photonic crystal slabs.

Photonic crystal fibers

First, we consider electromagnetic modes that propagate along the direction of continuous translational symmetry ẑ of a 2D photonic crystal (Fig. 7.1(a)). In this case, modes carry electromagnetic energy along the ẑ direction, and, therefore, can be considered as modes of an optical fiber extended in the ẑ direction and having a periodic dielectric profile in its cross-section. Fibers of this type are called photonic crystal fibers. From Section 2.4.5, it follows that fiber modes can be labeled with a conserved wave vector of the form k = kt + ẑkz, where kt is a transverse Bloch wave vector, and kz ≠ 0. To analyze the modes of a photonic crystal fiber with periodic cross-section, we will employ the general form of a plane-wave expansion method presented in Section 6.2.

Furthermore, if a defect that is continuous along the ẑ direction is introduced into a 2D photonic crystal lattice (Fig. 7.1(b)), such a defect could support a localized state (see Section 6.6), thus, effectively, becoming the core of a photonic crystal fiber.

In this chapter, we will consider reflective properties of planar multilayers, and guidance by multilayer waveguides. We will first introduce a transfer-matrix method to find electromagnetic solutions for a system with an arbitrary number of planar dielectric layers. We will then investigate the reflection properties of a single dielectric interface. Next, we will solve the problems of reflection from a multilayer stack, guidance inside a dielectric stack (planar waveguides), and finally, propagation perpendicular to an infinitely periodic multilayer stack. We will then describe omnidirectional reflectors that reflect radiation completely for all angles of incidence and all states of polarization. Next, we will discuss bulk and surface defect states of a multilayer. We will conclude by describing guidance in the low-refractive-index core waveguides.

Figure 3.1 presents a schematic of a planar multilayer. Each stack j = [1 …N] is characterized by its thickness dj and an index of refraction nj. The indices of the first and last half spaces (claddings) are denoted n0 and nN+1. The positions of the interfaces (except for j = 0) along the axis ẑ are labeled zj,j = [1 …N + 1], whereas z0 can be chosen arbitrarily inside of a first half space. In the following, we assume that the incoming plane wave has a propagation vector k confined to the xz plane. The planar multilayer possesses mirror symmetry with respect to the mirror plane xz.