What is color imaging science?

Color imaging science is the study of the formation, manipulation, display, and evaluation of color images. Image formation includes the optical imaging process and the image sensing and recording processes. The manipulation of images is most easily done through computers in digital form or electronic circuits in analog form. Conventional image manipulation in darkrooms accounts only for a very small fraction of the total images manipulated daily. The display of color images can use many different media, such as CRT monitors, photographic prints, half-tone printing, and thermal dye-transfer prints, etc. The complete imaging chain from capture, through image processing, to display involves many steps of degradation, correction, enhancement, and compromise. The quality of the final reproduced images has to be evaluated by the very subjective human observers. Sometimes, the evaluation process can be automated with a few objectively computable, quantitative measurements.

The complexity of color imaging science stems from the need to understand many diverse fields of engineering, optics, physics, chemistry, and mathematics. Although it is not required for us to be familiar with every part of the process in detail before we can work in and contribute to the color imaging science field, it is often necessary for us to have a general understanding of the entire imaging chain in order to avoid making unrealistic assumptions in our work. For example, in digital image processing, a frequently used technique is histogram-equalization enhancement, in which an input image is mapped through a tonal transformation curve such that the output image has a uniformly distributed histogram of image values.

Within our domain of interest, images are formed by light and its interaction with matter. The spatial and spectral distribution of light is focused on the sensor and recorded as an image. It is therefore important for us to first understand the nature and the properties of light. After a brief description of the nature of light, we will discuss some of its basic properties: energy, frequency, coherence, and polarization. The energy flow of light and the characterization of the frequency/wavelength distribution are the subjects of radiometry, colorimetry, and photometry, which will be covered in later chapters. The coherence and the polarization properties of light are also essential for understanding many aspects of the image formation process, but they are not as important for most color imaging applications because most natural light sources are incoherent and unpolarized, and most imaging sensors (including our eyes) are not sensitive to polarization. Therefore, we will discuss these two properties only briefly. They are presented in this chapter. Fortunately there are excellent books [208, 631, 871] covering these two topics (also, see the bibliography in Handbook of Optics [84]). From time to time later in the book, we will need to use the concepts we develop here to help us understand some of the more subtle issues in light–matter interaction (such as scattering and interference), and in the image formation process (such as the OTFs).

What is light?

The nature of light has been one of the most intensively studied subjects in physics. Its research has led to several major discoveries in human history.

The interaction between light and matter is often very complicated. The general description of the resulting phenomena often uses empirical measurement functions, such as the bidirectional spectral reflectance distribution function (BSRDF) to be discussed in the next chapter. However, the optical properties of a homogeneous material in its simple form (such as gas or crystal) can be calculated from physical principles. Understanding the basic optical properties of material is important because it serves as a foundation for understanding more complex phenomena. In this chapter, we will first discuss the physical properties of light, matter, and their interaction for simple cases. We will then derive the optical “constants” of material that characterize the propagation of light in the material.

Light, energy, and electromagnetic waves

For color imaging applications, light can be defined as the radiant electromagnetic energy that is visible either to our visual system, or to the image capture devices of interest. (When discussing visual systems of different species, we have to vary its range accordingly.) In optics, the scope of definition of light is larger, including other wavelengths for which the behavior of optical elements (such as lenses) can be described by the same laws as used for the visible spectrum. In physical chemistry, light is sometimes used to denote electromagnetic waves of all frequencies.

The electromagnetic spectrum that is visible to our eyes is from about 360 nm to about 830 nm in the air (according to the CIE specifications), corresponding to the frequency range of 3.61 × 1014-8.33 × 1014 Hz.

Although our interest in the human visual system is mainly in the role it plays as the observer in a color imaging system, we cannot completely treat it as a big black box, because the system is nonlinear and far too complicated to be characterized as such. There have been many attempts to apply linear system theory to the human visual system and characterize it with a system transfer function. That approach may serve some purposes in a few applications, but it is inadequate for most color imaging applications. Another possible approach is to treat it as many medium-sized black boxes, one for each special aspect in image perception. This is a more practical approach and it has been used very well for many applications. For example, we can measure the human contrast sensitivity as a function of luminance, field size, viewing distance, noise level, and chromatic contents, and the results can be used to design the DCT quantization tables for color image compression. However, the medium-sized black box approach does not give us much insight or guidance when our problem becomes more complicated or when we have a new problem. The size of the black boxes has to be reduced. An extreme limit is when each box corresponds to a single neuron in the visual pathway. Even then, some details inside the neuron may still be important to know. In general, how much detail we need to know is highly application-dependent, but the more we know the better we are equipped to deal with image perception related questions.

Rate equation and population inversion

In a material which has only two energy levels, χ″ is always positive because ρ11 > ρ22 for E1 < E2 at thermal equilibrium. Prior to the invention of lasers, there was no known method to achieve ρ22 > ρ11. However, we now know that a negative χ″ in Eq. (5.41a) (i.e. ρ22 > ρ11) can be achieved by pumping processes that are available in materials that have multiple energy levels, as described in the following.

Let there be many energy levels in the material under consideration, as shown in Fig. 6.1. Let there be a mechanism in which the populations at E1 and E2, i.e. N1 = Nρ11 and N2 = Nρ22, are increased by pumping from the ground state at pump rates R1 and R2. In solid state lasers, the pumping action may be provided by an intense optical radiation causing stimulated transition between the ground state and other higher energy states, where the particles in the higher energy states relax preferentially into the E2 state. In gas lasers, the molecules in the ground state may be excited into higher energy states within a plasma discharge; particles in those higher energy states then relax preferentially to the E2 state. Alternatively, collisions with particles of other gases may be utilized to increase the number of particles in the E2 state. Various schemes to pump different lasers are reviewed in ref. In order to obtain amplification, it is necessary to have R2 ≫ R1.

In Chapters 1 and 2 we discussed the propagation of laser radiation and the cavity modes as TEM waves. The amplitude and phase variations of these waves are very slow in the transverse directions. However, in applications involving single-mode optical fibers and optical waveguides, the assumption of slow variation in the transverse directions is no longer valid. Therefore, for electromagnetic analysis of such structures, we must go back to Maxwell's vector equations. Fortunately, the transverse dimensions of the components in these applications are now comparable to or smaller than the optical wavelength; solving Maxwell's equations is no longer a monumental task.

Many of the theoretical methods used in the analysis of optical guided waves are very similar to those used in microwave analysis. For example, modal analysis is again a powerful mathematical tool for analyzing many devices and systems. However, there are also important differences between optical and microwave waveguides. In microwaves, we usually analyze closed waveguides inside metallic boundaries. Metals are considered as perfect conductors at most microwave frequencies. In these closed structures, we have only a discrete set of waveguide modes that have an electric field terminating at the metallic boundary. We must avoid the use of metallic boundaries at the optical wavelength because of their strong absorption of radiation. Thus, we use open dielectric waveguides and fibers in optics, with boundaries extending theoretically to infinity. These are open waveguides. There are three important differences between optical and microwave waveguide modes and their utilization.

Introduction

Radiation from lasers is different from conventional optical light because, like microwave radiation, it is approximately monochromatic. Although each laser has its own fine spectral distribution and noise properties, the electric and magnetic fields from lasers are considered to have precise phase and amplitude variations in the first-order approximation. Like microwaves, electromagnetic radiation with a precise phase and amplitude is described most accurately by Maxwell's wave equations. For analysis of optical fields in structures such as optical waveguides and single-mode fibers, Maxwell's vector wave equations with appropriate boundary conditions are used. Such analyses are important and necessary for applications in which we need to know the detailed characteristics of the vector fields known as the modes of these structures. They will be discussed in Chapters 3 and 4.

For devices with structures that have dimensions very much larger than the wavelength, e.g. in a multimode fiber or in an optical system consisting of lenses, prisms or mirrors, the rigorous analysis of Maxwell's vector wave equations becomes very complex and tedious: there are too many modes in such a large space. It is difficult to solve Maxwell's vector wave equations for such cases, even with large computers. Even if we find the solution, it would contain fine features (such as the fringe fields near the lens) which are often of little or no significance to practical applications.

All optical oscillators and amplifiers can be analyzed as an electromagnetic structure, such as a cavity or a transmission line, which contains an amplifying medium. The operating characteristics of lasers are governed by both the electromagnetic properties of the structures and the properties of the amplifying medium.

Stimulated emission and absorption of radiation from energy states are the physical basis of amplification in all laser materials. In order to study lasers, it is necessary to understand the energy levels of the amplifying medium and the stimulated transitions involving them. Quantum mechanical analysis of the energy states and the stimulated emission and absorption is presented in most books on lasers. However, quantum mechanical analyses cover only the analysis of individual atoms. We need to relate the effects of the quantum mechanical interactions to the macroscopic properties of the materials so that properties of the lasers may be analyzed. Therefore, the macroscopic susceptibility of materials related to stimulated emission and absorption is the focus of discussion in this chapter.

It is not the purpose of this book to teach quantum mechanics. There are already many excellent books on this topic. The readers are assumed to have a fundamental knowledge of quantum mechanics. However, it is necessary first to review some of the major steps in order to understand precisely the notation used and the meaning of the results. This is presented in Section 5.1.

It is well known that basic solid state and gas laser cavities consist of two end reflectors that have a certain transverse (or lateral) shape such as a flat surface or a part of a large sphere. The reflectors are separated longitudinally by distances varying from centimeters to meters. The size of the end reflectors is small compared with the separation distance. All cavities for gaseous and solid state lasers have slow lateral variations within a distance of a few wavelengths (such as the variation of refractive index and gain of the material and the variation of the shape of the reflector). Therefore these cavity modes are analyzed using the scalar wave equation. Laser cavities are also sometimes called Fabry–Perot cavities because of their similarity to Fabry–Perot interferometers. However, Fabry–Perot interferometers have distances of separation much smaller than the size of the end reflectors. The diffraction properties of the modes in Fabry–Perot interferometers are quite different from the properties of modes in laser cavities.

The analysis of the resonant modes is fundamental to the understanding of lasers. Modes of solid state and gas lasers are solutions of Eqs. (1.28a) and (1.28b), known as Gaussian modes. They are TEM modes. The analysis of laser modes and Gaussian beam optics constitutes a nice demonstration of the mathematical techniques presented in Chapter 1.

When I look back at my time as a graduate student, I realize that the most valuable knowledge that I acquired concerned fundamental concepts in physics and mathematics, quantum mechanics and electromagnetic theory, with specific emphasis on their use in electronic and electro-optical devices. Today, many students acquire such information as well as analytical techniques from studies and analysis of the laser and its light in devices, components and systems. When teaching a graduate course at the University of California San Diego on this topic, I emphasize the understanding of basic principles of the laser and the properties of its radiation.

In this book I present a unified approach to all lasers, including gas, solid state and semiconductor lasers, in terms of “classical” devices, with gain and material susceptibility derived from their quantum mechanical interactions. For example, the properties of laser oscillators are derived from optical feedback analysis of different cavities. Moreover, since applications of laser radiation often involve its well defined phase and amplitude, the analysis of such radiation in components and systems requires special care in optical procedures as well as microwave techniques. In order to demonstrate the applications of these fundamental principles, analytical techniques and specific examples are presented. I used the notes for my course because I was unable to find a textbook that provided such a compact approach, although many excellent books are already available which provide comprehensive treatments of quantum electronics, lasers and optics.

The general principles of amplification and oscillation in semiconductor lasers are the same as those in solid state and gas lasers, as discussed in Chapter 6. A negative χ″ is obtained in an active region via induced transitions of the electrons. When the gain per unit distance is larger than the propagation loss, laser amplification is obtained. In order to achieve laser oscillation, the active material is enclosed in a cavity. Laser oscillation begins when the gain exceeds the losses, including the output. However, the details are quite different. In this chapter, the discussion on semiconductor lasers will use much of the analyses already developed in Chapters 5 and 6; however, the differences will be emphasized.

In semiconductor lasers, free electrons and holes are the particles that undertake stimulated emission and absorption. How such free carriers are generated, transported and recombined has been discussed extensively in the literature. We note here, in particular, that free electrons and holes are in a periodic crystalline material. The energy levels of electrons and holes in such a material are distributed within conduction and valence bands. The distribution of energy states within each band depends on the specific semiconductor material and its confinement within a given structure. For example, it is different for a bulk material (a three-dimensional periodic structure) and for a quantum well (a two-dimensional periodic structure).

In order to understand optical fiber communication components and systems, we need to know how laser radiation functions in photonic devices. The operation of many important photonic devices is based on the interactions of several guided waves. We have already discussed the electromagnetic analysis of the individual modes in planar and channel waveguides in Chapter 3. From that discussion, it is clear that solving Maxwell's equations simultaneously for several modes or waveguides is too difficult. There are only approximate and numerical solutions. In this chapter, we will first learn special electromagnetic techniques for analyzing the interactions of guided waves. Based on these techniques, practical devices such as the grating filter, the directional coupler, the acousto-optical deflector, the Mach– Zehnder modulator and the multimode interference coupler will be discussed. The analysis techniques are very similar to those techniques used in microwaves, except we do not have metallic boundaries in optical waveguides, only open dielectric structures.

The special mathematical techniques to be presented here include the perturbation method, the coupled mode analysis and the super-mode analysis (see also ref.). In guided wave devices, the amplitude of radiation modes is usually negligible at any reasonable distance from the discontinuity. Thus, in these analyses, the radiation modes such as the substrate and air modes in waveguides and the cladding modes in fibers are neglected. They are important only when radiation loss must be accounted for in the vicinity of any dielectric discontinuity.

Experiments with single photons

Central to the entire discipline of quantum optics, as should be evident from the preceding chapters, is the concept of the photon. Yet it is perhaps worthwhile to pause and ask: what is the evidence for the existence of photons? Most of us first encounter the photon concept in the context of the photo-electric effect. As we showed in Chapter 5, the photo-electric effect is, in fact, used to indirectly detect the presence of photons – the photo-electrons being the entities counted. But it turns out that some aspects of the photo-electric effect can be explained without introducing the concept of the photon. In fact, one can go quite far with a semiclassical theory in which only the atoms are quantized with the field treated classically. But we hasten to say that, for a satisfactory explanation of all aspects of the photo-electric effect, the field must be quantized. As it happens, the other venerable “proof” of the existence of photons, the Compton effect, can also be explained without quantized fields.

In an attempt to obtain quantum effects with light, Taylor, in 1909, obtained interference fringes in an experiment with an extremely weak source of light. His source was a gas flame and the emitted light was attenuated by means of screens made of smoked glass.