Optical Propagation in Solids

This chapter emphasizes the linear optical properties of solids as a function of frequency and temperature. Such information is basic to understanding the performance of optical fibers, lenses, dielectric and metallic mirrors, window materials, thin films, and solid-state photonic devices in general. Optical properties are comprehensively covered in terms of mathematical models of the complex index of refraction based on those discussed in Chapters 4 and 5. Parameters for these models are listed in Appendix 4. A general review of solid-state properties precedes this development because the choice of an optical material requires consideration of thermal, mechanical, chemical, and physical properties as well. This section introduces the classification of optical materials and surveys other material properties that must be considered as part of total optical system design involving solidstate optics. Solid-state materials can be classified in several ways. The following are relevant to optical materials. Three general classes of solids are insulators, semiconductors, and metals. Insulators and semiconductors are used in a variety of ways, such as lenses, windows materials, fibers, and thin films. Semiconductors are used in electrooptic devices and optical detectors. Metals are used as reflectors and high-pass filters in the ultraviolet. This type of classification is a function of the material’s electronic bandgap. Materials with a large room-temperature bandgap (Eg > 3eV) are insulators. Materials with bandgaps between 0 and 3 eV are semiconductors. Metals have no observable bandgap because the conduction and valence bands overlap. Optical properties change drastically from below the bandgap, where the medium is transparent, to above the bandgap, where the medium is highly reflective and opaque. Thus, knowledge of its location is important. Appendix 4 lists the bandgaps of a wide variety of optical materials. To characterize a medium within the region of transparency requires an understanding of the mechanisms of low-level absorption and scattering. These mechanisms are classified as intrinsic or extrinsic. Intrinsic properties are the fundamental properties of a perfect material, caused by lattice vibrations, electronic transitions, and so on, of the atoms composing the material.

Optical Electromagnetics I

In this chapter, the optical spectrum is defined and subdivided into many sub-bands, which are traditionally determined by transparency in various media. Propagation of the electromagnetic field in vacuum, as based on Maxwell’s equations, and basic notions of geometrical and physical optics, are covered. The theoretical and conceptual foundation of the remaining chapters is established in this chapter and the next. Optical electromagnetic propagation is generally and often accurately described by classical geometrical optics or ray optics. When diffraction or wave interference is of concern, then the more complete field of physical optics is used. Geometrical optics requires precise knowledge of the spatial and spectral dependence of the index of refraction. This requires electrodynamics, which is most appropriately described by quantum optics. These topics are covered in the first five chapters. The definitions of the optical spectrum and the various models for describing propagation are introduced in the following. The optical electromagnetic field covers the range of frequencies from microwaves to the ultraviolet (UV) or wavelengths from 10 cm to 100 nm. This is a very liberal definition covering six orders of magnitude, yet the description of propagation is very similar over this entire band, and distinct from radio-wave propagation and x-ray propagation. A listing of the nomenclature for the different spectral bands within the range of optical wavelengths is given in Table 1.1. Other commonly used units of spectral measure such as wave number, frequency, and energy are also listed in the table. These various quantities are related to wavelength by the following formulas: where c is the speed of light (c = 2.99792458 × 108 m/sec), λ is wavelength, f is frequency in hertz, E is energy, h is Planck’s constant (h = 6.6260755(40) × 10−34 J sec), and ν is frequency in wave numbers (the number of wavelengths per centimeter). Although wavelength is commonly used by applied scientists and engineers, frequency is the most appropriate unit for the theoretical description of light–matter interactions. Because of the importance of spectroscopy in the discussion of optical propagation, the spectroscopic unit of wave number will be consistently used.

Optical Propagation in Water

From a basic physics perspective, liquids are the least understood state of matter. Yet this medium plays an important role in the process of life on this planet. The human body is largely composed of liquids, and three-quarters of the surface of the earth is covered by seawater. The main liquid of interest in this chapter, and to the applied scientist and engineer, is water. The importance of understanding the optical properties of water cannot be overemphasized. The chapter appropriately begins with a discussion of the optical properties of pure water, since it is the main ingredient in seawater and in biomedical fluids. Pure water is an insulator with a strong dipole moment and an effective electronic band edge in the ultraviolet near 0.16 μm (62,500 cm−1). Absorption near the band edge shows similar structure to that observed in solids. Water has extensive infrared vibrational bands just as in the gas phase. Dipoles in a liquid can partially rotate in response to the polarization of the incident microscopic field, and Debye relaxation bands occur in the microwave region. A permittivity model for Debye relaxation was presented in Chapter 4 by Eq. 4.60. This is an important mechanism that describes the optical properties of liquids at far-infrared and microwave frequencies.

Optical Propagation in Gases and the Atmosphere of the Earth

Propagation within the atmosphere is an important consideration concerning the performance of many electro-optical systems. An electro-optical system can be described as containing three basic components: source, detector, and propagation medium. Because of the quality of source and detection systems today, often the limiting factor in overall system performance is the propagation medium. Thus a thorough discussion of the atmosphere and various mechanisms of attenuation is required. Absorption, scattering, and turbulence are the dominant mechanisms of signal loss and distortion. This chapter covers gaseous absorption and scattering in the atmosphere of the earth. Turbulence is not covered, and the reader is referred to other texts (see Chapter 1, Refs. 1.10 and 1.11). The atmosphere surrounds and protects the earth in the form of a gaseous blanket that acts as the transition between the solid surface of the earth and the near-vacuum of the outer solar atmosphere. It acts as a shield against harmful particle radiation, meteors, and high-energy photons. The dynamics of the atmosphere drive the weather on the surface. It provides for life itself as part of the earth’s biosphere. Thus optical propagation in this medium has many important characteristics and consequences. These include meteorological optics, infrared and visible astronomy, remote sensing, and electro-optical systems performance in general. Therefore, it is appropriate to begin this chapter with an introduction to the nature of the atmosphere. The atmosphere is composed of gases and suspended particles or aerosols at various temperatures and concentrations as a function of altitude and azimuth. The variations in altitude show a marked structure. Six main horizontal layers form the stratified structure of the atmosphere, as shown in Fig. 7.1. The lowest is the troposphere, which extends from ground level to approximately 11 km (36,000 ft or 7 mi.). The temperature in this layer generally decreases with increasing altitude at the rate of 6.5 K/km. However, variations can exist on this rate, which creates interesting refractive effects. The pressure varies from one atmosphere at sea level to a few tenths of an atmosphere at the top of this layer.

Experimental Techniques

This chapter presents basic experimental techniques and various apparatus for measuring the complex index of refraction and related quantities. Generally, measurements of transmittance, reflectance, and emittance are made using spectrometers or lasers. Other important techniques, which measure directly the real refractive index, n, the absorption coefficient, βabs , and the scattering coefficient, βsca, such as interferometry, ellipsometers, calorimetry, and scatterometers, are also introduced. Ultimately, experimental procedures must be taught in the laboratory. Thus, devoting only one chapter to experimental technique and five to theory is not indicative of the importance of this fundamental topic. By discussing the measurement of basic optical parameters, it is intended that the concepts developed in the first five chapters will be reinforced. All of the theoretical models developed in the previous chapters contain measurable parameters. Basic theory often helps guide the design of a good experiment. Once data is available, it can be used to check the assumptions of the theory. This interplay between experiment and theory is an essential part of definitive work. The chapter has two main parts; the first covers measurements of the real and imaginary parts of the complex index of refraction and the second covers measurements of scattering. As established in Chapter 2, the characterization of bulk absorption mechanisms on optical propagation is accomplished by the complex index of refraction. Considerable effort was expended in Chapters 3, 4, and 5 to obtain models of the complex index. Thus, at this point, we wish to find ways to experimentally measure the complex index of refraction for various media. The broad-band spectral response of a medium is commonly measured by a spectrometer. There are two main types of spectrometers, dispersive and interferometric. Generally, spectrometers make broad-band transmission, emission, and reflection measurements, and therefore indirectly measure, n̄. Interferometric measurements, are the exception. Lasers, which feature narrow-band, high-intensity, highly directional light are often used to complement and calibrate broad-band spectrometer measurements. The highest accuracy measurements of the absorption coefficient are obtainable by laser techniques, which can directly measure the components of the complex index.

Electrodynamics II: Microscopic Interaction of Light and Matter

Although the primarily phenomenological theory of absorption and refraction of light by matter, based on classical models as presented in Chapter 4, is very useful, it is incomplete and often inadequate. A more complete and accurate picture of electrodynamics is given by the theory of quantum optics, and that is the topic of this chapter. The models developed in this chapter are more detailed and therefore more complicated than the phenomenological models of Chapter 4. The most robust models, which are applied in Part II, are presented in this chapter. The quantum models accurately represent experimental data and allow extrapolation and interpolation of such data. Many practical computer based models concerning optical propagation are based on this theory. The theory of elastic scatter as presented in Chapter 4 is consistent with quantum optics and is not presented again. (However, inelastic scatter must address the quantum nature of the scattering medium.) Quantum optics is not completely covered in this chapter. Entire textbooks are devoted to this diverse and comprehensive topic covering optics (see Refs. 5.1–5.3). The emphasis of this book is on absorption and reflection spectroscopy. Now details of internal structure of the medium impacting light–matter interaction are examined. The classical oscillator model is upgraded by semiclassical radiation theory and a quantum oscillator model is developed. Semiclassical radiation theory is based on a quantized medium coupled to a classical field. It is often applied to laser theory, where near-line-center stimulated emission dominates. The quantum oscillator model again utilizes the quantized medium and classical field, but with more attention to detailed balance between absorption and emission. It satisfies causality and the fundamental symmetry relationships established in Chapter 2. These quantum optics models are more complete formalisms and provide solutions to the shortcomings of classical electrodynamics. Of particular interest to propagation in gaseous media is the line shape in the far wing. To achieve long path lengths, propagation near line center of a resonance must be avoided. Line shape models in quantum optics accurately represent much of the frequency and temperature dependence observed in experimental data.

Optical Electromagnetics II

In this chapter the same basic topics are addressed as in the previous chapter, but now in the presence of matter. This greatly complicates the description of optical propagation and continues to be the primary topic of the remaining chapters. A formal structure is developed to handle absorption and scattering phenomena in general. The modeling of optical propagation is reduced to having to know the complex index of refraction of the medium. A macroscopic description represents the large-scale observable character of optical propagation. At this level, many models are phenomenological, but lead to important general properties, definitions, formulas, and the establishment of basic concepts. Because microscopic models to be presented in future chapters contain considerable detail, this section is an important prerequisite to the remaining text. Again, plane waves are a useful tool for the description of optical propagation. The Poynting vector, causality, and Poynting’s theorem are used to develop and derive quantities and relationships concerning radiometry and the flow of electromagnetic power at optical frequencies. Consider Maxwell’s equations again, but in the presence of linear isotropic matter. Now the constitutive relations will play a more important role and are the foundation of classical dispersion theory.

Propagation Background and Noise

Noise from the detector and source is always considered in a system design study. However, as an optical field propagates, it acquires additional noise and background radiance from the path defined by the source and by the field of view of the detector. This is typically in the form of propagation path emission and background emission, and bulk scattering within the propagation medium and surface scattering at the propagation medium boundaries of hot-object radiation (e.g., the sun) into the sensor field of view. In many cases this severely limits system performance. Also, in a passive system when no source is present, the background radiance is the signal of interest. Path emission can be modeled by the radiation transfer equation given by Eq. 2.85a. The source function, ℘+(s), must now be given an explicit representation. Figure 11.1 illustrates the incremental emittance per incremental length and bandwidth along an optical path in thermal equilibrium.

Electrodynamics I: Macroscopic Interaction of Light and Matter

Thus far, we have developed the properties of the electromagnetic field at optical frequencies, based on Maxwell’s equations. These equations further give a classical macroscopic perspective on the coupling of the propagation media to the field, as presented in Chapter 2. The macroscopic properties of a medium are based on averaged microscopic properties. The microscopic energy structure of matter was presented in Chapter 3, covering gases, solids, and liquids by employing mostly quantum models. We now proceed to the next level of development, the dynamic description of the interaction between the optical field and the propagation medium as a function of the field frequency and propagation media variables (e.g., energy structure, temperature, and pressure). In this chapter, the classical electromagnetic field is coupled to discrete frequency oscillators via Newton’s equation of motion. This approach leads to the popular classical oscillator model, often presented in introductory books on lasers. The classical oscillator model is an incomplete theory and can be only a semiempirical model. In the next chapter, a more detailed and comprehensive approach, which also includes statistical and quantum mechanics, is used leading to robust semiclassical and quantum oscillator models. This chapter and the next are the basis for the applied models presented in Part II of this book. Classical electrodynamics is based on Maxwell’s equations, as given in Chapter 2, and the Lorentz force relation, as given below: . . . F = q[E + (v × B)]. (4.1) . . . These equations cover the classical description of the interaction of light and matter. The first term in Eq. 4.1 represents coupling of the electric field to the medium. As discussed in Chapter 2 (Section 2.2), the leading mechanism for this is the electric dipole moment. To see that this is the coupling mechanism in the first term, consider the potential function driving this force, F = −∇V(r) = −∇(−qr · E). The above expression contains the dipole moment, as defined in Chapter 2. The second term in Eq. 4.1 represents coupling of the magnetic field to the medium.

Particle Absorption and Scatter

Particles are composed of solids and/or liquids, thus the bulk optical properties of these media must be known before propagation modeling within a medium of suspended particles (called aerosols when in air) can begin. We return to our discussion of propagation in the atmosphere and oceans of the earth that began in Chapters 7 and 9, and we now include attenuation by small particles. Particles vary in size, shape, concentration, and composition. Size and concentration distributions are described in the following two sections. The composition of the most common particles is presented in the last section. Unfortunately, a representation of shape variation does not exist. As mentioned in Chapter 4 (Section 4.4.2 on Mie scattering), a collection of real aerosols will have a range of different radii. This is called a polydisperse medium. Various models are used to represent particle size distributions. One commonly used model for particle number density as a function of radius is the modified gamma distribution function, as given by . . . ρp(r) = Arα exp(−brγ), (10.1) . . . where A, b, α, and γ are empirically determined parameters. This function represents the number of particles per unit volume and unit radius as a function of radius r. The total particle number density is obtained by integrating ρp(r ) over all r.