NORMAL MODES OF PROPAGATION

The propagation of an optical wave is governed by Maxwell's equations. The propagation characteristics depend on the optical property and the physical structure of the medium. They also depend on the makeup of the optical wave, such as its frequency content and its temporal characteristics. In this chapter, we discuss the basic propagation characteristics of a monochromatic optical wave in three basic categories of medium: an infinite homogeneous medium, two semi-infinite homogeneous media separated by an interface, and an optical waveguide defined by a transverse structure. Some basic effects of dispersion and attenuation on the propagation of an optical wave are discussed in Sections 3.6 and 3.7.

The optical property of a medium at a frequency of ω is fully described by its permittivity ε(ω), which is a tensor for an anisotropic medium but reduces to a scalar for an isotropic medium. For a homogeneous medium, ε(ω) is a constant of space; for an optical structure, it is a function of space variables. Without loss of generality, we designate the z coordinate axis to be the direction of optical wave propagation in an isotropic medium; thus the longitudinal axis of an optical waveguide that is fabricated in an isotropic medium is the z axis. For this reason, ε(ω) has only transverse spatial variations that are functions of the transverse coordinates, which are x and y in the rectilinear coordinate system, or Φ and r in the cylindrical coordinate system. We use the rectilinear coordinates for our general discussion. The exception is optical wave propagation in an anisotropic crystal, for which the natural coordinate system is that defined by its principal axes but an optical wave does not have to propagate along its principal z axis.

For the following discussion in this section, we consider propagation in an isotropic medium, which is not necessarily homogeneous in space. The wave propagates in the z direction, and the possible inhomogeneity characterizing the optical structure is described by a scalar permittivity ε(x, y), as illustrated in Fig. 3.1. If the medium is homogeneous, then ε(x, y) = ε is a constant of space, as shown in Fig. 3.1(a).

OPTICAL RESONATOR

As discussed in Section 5.3, multiple reflections take place between the two reflective surfaces of a Fabry–Pérot interferometer, resulting in multiple transmitted fields. A transmittance peak occurs when the round-trip phase shift φRT between the two reflective surfaces is an integral multiple of 2π so that all of the transmitted fields are in phase. From the viewpoint of the field inside the interferometer, this condition results in optical resonance between the two reflective surfaces. Thus a Fabry–Pérot interferometer behaves as an optical resonator, also called a resonant optical cavity. At resonance, the field amplitude inside an optical resonator reaches a peak value due to constructive interference of multiple reflections. The optical energy stored in an optical cavity peaks at its resonance frequencies.

An optical cavity can take a variety of forms. Figure 6.1 shows the schematic structures of a few different forms of optical cavities. Though an optical cavity has a clearly defined longitudinal axis, the axis can lie on a straight line, as in Fig. 6.1(a), or it can be defined by a folded path, as in Figs. 6.1(b), (c), and (d). A linear cavity defined by two end mirrors, as in Fig. 6.1(a), is known as a Fabry–Pérot cavity because it takes the form of the Fabry–Pérot interferometer. A folded cavity can simply be a folded Fabry–Pérot cavity that supports a standing intracavity field, as in Fig. 6.1(b). A folded cavity can also be a non-Fabry–Pérot ring cavity that supports two independent, contrapropagating intracavity fields, as in Figs. 6.1(c) and (d).

An optical cavity provides optical feedback to the optical field in the cavity. Optical resonance occurs when the optical feedback is in phase with the intracavity optical field. The optical feedback in a Fabry–Pérot cavity is provided simply by the two end mirrors that have the reflective surfaces perpendicular to the longitudinal axis, as in Figs. 6.1(a) and (b). In a ring cavity, it is provided by the circulation of the laser field along a ring path defined by mirrors, as in Fig. 6.1(c), or a ring path defined by an optical fiber, as in Fig. 6.1(d). The cavity can also be constructed with an optical waveguide, as in the case of a semiconductor laser or a fiber laser.

TYPES OF OPTICAL MODULATION

Optical modulation allows one to control an optical wave or to encode information on a carrier optical wave. The inverse process that recovers the encoded information is demodulation. There are many types of optical modulation, which can be categorized in several different ways.

1. 1. According to the particular optical-field parameter being modulated, optical modulation can be categorized into different modulation schemes: phase modulation, frequency modulation, polarization modulation, amplitude modulation, spatial modulation, and diffraction modulation.

2. 2. Depending on whether the information is encoded in the analog or digital form, optical modulation can be either analog modulation or digital modulation.

3. 3. Optical modulation can be categorized as direct modulation or external modulation. Direct modulation is directly performed on an optical source, which is usually a light-emitting diode (LED) or a laser, without using a separate optical modulator. External modulation is performed on an optical wave using a separate optical modulator to change one or more characteristics of the wave.

4. 4. Optical modulation is accomplished by varying the optical susceptibility of the modulator material. Depending on whether the real or imaginary part of the susceptibility is responsible for the functioning of the modulator, optical modulation can be categorized as refractive modulation or absorptive modulation. Refractive modulation is performed by varying the real part of the susceptibility, thus varying the refractive index of the material; absorptive modulation is performed by varying the imaginary part of the susceptibility, thus varying the absorption coefficient of the material.

5. 5. Optical modulation can be categorized according to the physical mechanism behind the change of the optical susceptibility, such as electro-optic modulation, acousto-optic modulation, magneto-optic modulation, all-optical modulation, and so forth.

6. 6. Depending on the geometric relation between the modulating signal and the modulated optical wave, optical modulation can be transverse modulation or longitudinal modulation. In transverse modulation, the signal is applied in a direction perpendicular to the propagation direction of the optical wave. In longitudinal modulation, the signal is applied along the propagation direction of the optical wave.

7. 7. Optical modulation can be performed on unguided or guided optical waves. Correspondingly, the structure of an optical modulator can take the form of a bulk or waveguide device. A bulk modulator is used to modulate an unguided optical wave. A waveguide modulator is used to modulate a guided optical wave.

PHYSICAL PRINCIPLES OF PHOTODETECTION

Photodetection converts an optical signal into a signal of another form. Most photodetectors convert optical signals into electrical signals that can be further processed or stored. All photodetectors are square-law detectors that respond to the power or intensity, rather than the field amplitude, of an optical signal. The electrical signal generated by an optical signal is either a photocurrent or a photovoltage that is proportional to the power of the optical signal. Based on the difference in the conversion mechanisms, there are two classes of photodetectors: photon detectors and thermal detectors. Photon detectors are quantum detectors based on the photoelectric effect, which converts a photon into an emitted electron or an electron–hole pair; a photon detector responds to the number of photons absorbed by the detector. Thermal detectors are based on the photothermal effect, which converts optical energy into heat; a thermal detector responds to the optical energy, rather than the number of photons, absorbed by the detector. Because of this fundamental difference, the general characteristics of these two classes of photodetectors have a number of important differences.

The response of a photon detector is a function of the optical wavelength with a long-wavelength cutoff, whereas that of a thermal detector is wavelength independent. A photon detector can be much more responsive than a thermal detector in a particular spectral region, which typically falls somewhere within the range from the near ultraviolet to the near infrared. By comparison, a thermal detector normally covers a wide spectral range from the deep ultraviolet to the far infrared with a nearly constant response. Photon detectors can be made extremely sensitive. Some of them have a photon-counting capability that is not possible for a thermal detector. A photon detector can be designed to have a high response speed capable of following very fast optical signals. Most thermal detectors are relatively slow in response because the speed of a thermal detector is limited by thermalization through heat diffusion and by heat dissipation when the power of an optical signal varies. For these reasons, photon detectors are suitable for detecting optical signals in photonic systems, whereas thermal detectors are most often used for optical power measurement or infrared imaging. In this chapter, only the basic principles of photon detectors are discussed because our major concern is photodetection for photonics applications.

POPULATION RATE EQUATIONS

From the discussion in the preceding chapter, it is clear that population inversion is the basic condition for an optical gain. For any system in its normal state in thermal equilibrium, a low-energy level is always more populated than a high-energy level, hence there is no population inversion. Population inversion in a system can only be accomplished through a process called pumping by actively exciting the atoms in a low-energy level to a high-energy level. If left alone, the atoms in a system relax to thermal equilibrium. Therefore, population inversion is a nonequilibrium state that cannot be sustained without active pumping. To keep a constant optical gain, continuous pumping is required to maintain population inversion. This condition is clearly consistent with the law of conservation of energy: amplification of an optical wave leads to an increase in optical energy, which is possible only if the required energy is supplied by a source.

Pumping is the process that supplies energy to the gain medium for the amplification of an optical wave. There are many different pumping techniques, including optical excitation, electric current injection, electric discharge, chemical reaction, and excitation with particle beams. The use of a specific pumping technique depends on the properties of the gain medium being pumped. The lasers and optical amplifiers of particular interest in photonic systems are made of either dielectric solid-state media doped with active ions, such as Nd:YAG and Er:glass fiber, or direct-gap semiconductors, such as GaAs and InP. For a dielectric gain medium, the most commonly used pumping technique is optical pumping using either an incoherent light source, such as a flashlamp or a light-emitting diode, or a coherent light source from another laser. A semiconductor gain medium can also be optically pumped, but it is usually pumped by electric current injection. In this section, we consider the general conditions for pumping to achieve population inversion. Detailed pumping mechanisms and physical setups are not addressed here because they depend on the specific gain medium used in a particular application.

The net rate of increase of the population density in a given energy level is described by a rate equation. As we shall see below, pumping for population inversion in any practical gain medium always requires the participation of more than two energy levels.

The field of photonics has matured into an important discipline of modern engineering and technology. Its core principles have become essential knowledge for all undergraduate students in many engineering and scientific fields. This fact is fully recognized in the new curriculum of the Electrical Engineering Department at UCLA, which makes the principles of photonics a required course for all electrical engineering undergraduate students. Graduate students studying in areas related to photonics also need this foundation.

The most fundamental concepts in photonics are the nature of optical fields and the properties of optical materials because the entire field of photonics is based on the interplay between optical fields and optical materials. Any photonic device or system, no matter how simple or sophisticated it might be, consists of some or all of these functions: the generation, propagation, coupling, interference, amplification, modulation, and detection of optical waves or signals. The properties of optical fields and optical materials are addressed in the first two chapters of this book. The remaining nine chapters cover the principles of the major photonic functions.

This book is written for a one-quarter or one-semester undergraduate course for electrical engineering or physics students. Only some of these students might continue to study advanced courses in photonics, but at UCLA we believe that all electrical engineering students need to have a basic understanding of the core knowledge in photonics because it has become an established key area of modern technology. Many universities already have departments that are entirely devoted to the field of photonics. For the students in such photonics-specific departments or institutions, the subject matter in this book is simply the essential foundation that they must master before advancing to other photonics courses. Based on this consideration, this book emphasizes the principles, not the devices or the systems, nor the applications. Nevertheless, it serves as a foundation for follow-up courses on photonic devices, optical communication systems, biophotonics, and various subjects related to photonics technology. Because this book is meant for a one-quarter or one-semester course, it is kept to a length that can be completed in a quarter or a semester. Because it likely serves the only required undergraduate photonics course in the typical electrical engineering curriculum, it has to cover most of the essential principles.