Introduction In this chapter we will present the theory of spectral gain in optical amplifiers and the related optical power distribution. The concepts are general, as they apply to every optical amplifier configuration, independent of the specific architecture and implementation technology. We will configure the optical amplifying medium as a traveling-wave device, where the optical field undergoes optical amplification as it propagates from the input to the output sections. Generally speaking, optical amplification is achieved by a net stimulated emission process. Referring to a generic section z of infinitesimal thickness dz , optical gain is determined by the imbalance between net stimulated emission and cavity losses, including extrinsic absorption, reflection and radiated fields. As we know from the theory of light amplification in the three-level atomic system presented in Chapter 5, stimulated absorption of photons is governed by the same transition rate as stimulated emission, and is included in the net stimulated emission process. With this in mind, we note the identity between stimulated emission and the stimulated absorption rates, Γ 12 = Γ 21 , and the net stimulated photon term Γ[ j 2 ( t ) – j 1 ( t )] p ( t ) in the rate equation. In order to develop the traveling-wave model, the photon density becomes a function of the longitudinal position 0 ≤ z ≤ L along the amplifier medium. Accordingly, we will set p ( t , z ) as the relative density of stimulated photons per unit volume (0 ≤ p ( t , z ) ≤ 1) at time t and section z . When photons with appropriate energy proceed along the amplifying medium, they are amplified by means of net stimulated emission. At some section z = z 0 , the steady-state photon density P ( z ) can approach the saturation density P sat ( z ).
Introduction In this chapter we will present the theory of spectral gain in optical amplifiers and the related optical power distribution. The concepts are general, as they apply to every optical amplifier configuration, independent of the specific architecture and implementation technology. We will configure the optical amplifying medium as a traveling-wave device, where the optical field undergoes optical amplification as it propagates from the input to the output sections. Generally speaking, optical amplification is achieved by a net stimulated emission process. Referring to a generic section z of infinitesimal thickness dz , optical gain is determined by the imbalance between net stimulated emission and cavity losses, including extrinsic absorption, reflection and radiated fields. As we know from the theory of light amplification in the three-level atomic system presented in Chapter 5, stimulated absorption of photons is governed by the same transition rate as stimulated emission, and is included in the net stimulated emission process. With this in mind, we note the identity between stimulated emission and the stimulated absorption rates, Γ 12 = Γ 21 , and the net stimulated photon term Γ[ j 2 ( t ) – j 1 ( t )] p ( t ) in the rate equation. In order to develop the traveling-wave model, the photon density becomes a function of the longitudinal position 0 ≤ z ≤ L along the amplifier medium. Accordingly, we will set p ( t , z ) as the relative density of stimulated photons per unit volume (0 ≤ p ( t , z ) ≤ 1) at time t and section z . When photons with appropriate energy proceed along the amplifying medium, they are amplified by means of net stimulated emission. At some section z = z 0 , the steady-state photon density P ( z ) can approach the saturation density P sat ( z ).
Introduction Most of the lightwave transmission systems operating today benefit fully from extensive deployment of optical amplifiers (OA). These components perform the amplification of optical intensity directly in the optical domain, without resorting to any electrical-domain regeneration. Optical amplifiers increase the average number of output photons per unit time by several orders of magnitude compared with the incoming signal intensity. However, as with every amplifying process, optical amplification cannot be achieved without paying back for the unavoidable noise power added to the signal intensity. As we will see in detail in Chapter 6, the optical amplification process comes with optical noise generated by spontaneous emission inside the active medium. The intensity of the optical signal is represented by the flux of the signal photon population. Conversely, the photons generated by the spontaneous emission process inside the optical cavity represent the noise photon population. The fundamental phenomenon supporting optical amplification is stimulated emission. The basic property of stimulated emission, making the great difference from spontaneous emission and enabling optical amplification, is the coherent nature of the photon generation process. Accordingly, both signal and noise photons go through the same multiplication process by means of stimulated emission. Unless the optical cavity where the stimulated emission takes place (the active cavity) presents selective features for enhancing the signal photon population relative to the spontaneous one, there is no way to avoid noise photons going through stimulated emission, filling most of the cavity population. As a consequence, both photon populations increase, leading respectively to amplified signal intensity and to amplified spontaneous emission intensity. It should be apparent that every noise photon can create a new identical noise photon as a result of stimulated emission, effectively increasing the noise intensity in the active optical cavity. This is the well-known mechanism of amplified spontaneous emission (ASE) noise in optical amplification. Any improvement in the optical amplifier structure devoted to increasing stimulated over spontaneous emission will probably increase the optical signal-to-noise ratio (OSNR) at the cavity output. The spectrum of gain characteristic of the optical active medium is usually wide enough to support an efficient amplification of the broadband spontaneous emission spectrum, leading to the broadband ASE noise power spectral density.
Introduction Previous discussions (see Section 10.4) suggested that stimulated emission can be used to generate optical gain, that is, to amplify radiation. The reader certainly has experience in the amplification of electronic signal. For example, radio receivers capture faint radio waves and turn them into a signal that is powerful enough to drive large speakers. This electronic amplification increases the amplitude of the signal while faithfully preserving its acoustic frequencies and modulation characteristics. Similarly, optical amplification is expected to increase the amplitude of an optical signal while preserving its frequency, its modulation characteristics, and its coherence. The latter requirement is of particular significance for optical radiation, where the coherence of naturally occurring radiation rarely exceeds one micrometer. Lasers are the primary source for coherent radiation. They depend on stimulated emission for amplification and for the generation of coherent radiation (the word laser is the acronym of Light Amplification by Stimulated Emission of Radiation). For amplification, an atomic system that is part of the laser medium must be prepared with a sufficiently large number of particles in the excited state. (The term atomic system is used here to describe all microscopic systems including molecules and free electrons.) Radiation passing through that excited medium encounters multiple events of stimulated emission, each event contributing one photon that is added coherently to the propagating beam. When the number of events of stimulated emission exceed all losses by absorption or scattering, the incident radiation is amplified.
A photon, unlike an electron, can stimulate an excited atom to emit a second photon into the same electromagnetic mode. This stimulated emission process is the foundation for light amplification and oscillation (i.e., self-generation). Initially, the term LASER referred to Light Amplification by Stimulated Emission of Radiation. Nowadays laser often means Light Oscillation by Stimulated Emission, which should literally be called "LOSER" instead of "LASER." To distinguish the above two devices, the former is called a laser amplifier, the latter a laser oscillator (Siegman 1986). Laser oscillation occurs when the photon generation rate exceeds the photon loss rate in a system. If gain saturation were absent, the photon number in a laser oscillator would diverge in time. In other words, the rate equation for the photon number would acquire an unstable solution above the oscillation threshold. In reality, gain saturation reduces the photon generation rate to the photon loss rate so that the number of photons in the oscillator remains at a finite value.
Introduction When an electromagnetic wave propagates through amedium stimulated emissions increase the intensity of the wave, while absorptions diminish it. The overall intensity will increase if the number of stimulated emissions can be made larger than the number of absorptions. If we can create such a situation then we have built an amplifier that operates through the mechanism of stimulated emission. This laser amplifier , in common with electronic amplifiers, has useful gain only over a particular frequency bandwidth. Its operating frequency range will be determined by the lineshape of the transition, and we expect the frequency width of its useful operating range to be of the same order as the width of the lineshape. It is very important to consider how this frequencywidth is related to the various mechanisms by which transitions between different energy states of a particle are smeared out over a range of frequencies. This line broadening affects in a fundamental way not only the frequency bandwidth of the amplifier, but also its gain. A laser amplifier can be turned into an oscillator by supplying an appropriate amount of positive feedback. The level of oscillation will stabilize because the amplifier saturates. Laser amplifiers fall into two categories, which saturate in different ways. A homogeneously broadened amplifier consists of a number of amplifying particles that are essentially equivalent, whereas an inhomogeneously broadened amplifier contains amplifying particles with a distribution of amplification characteristics. Homogeneous line broadening All energy states, except the lowest energy state of a particle (the ground state) cover a range of possible energies.
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.
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.