The value of the Einstein $B$ coefficient for stimulated emission for a collection of atoms interacting with a radiation field is shown to depend upon the velocity distribution of the excited atoms and also upon the variation of field intensity with frequency. An emission line narrowing phenomenon for radiating matter enclosed in a nonresonant cavity with reflecting walls is predicted. It is shown that a loss mechanism must be introduced if significant line narrowing is to result from a single pass of radiation through an amplifying medium. An additional contributing effect to the mode pushing and mode pulling phenomena in a gas laser is demonstrated. A method of setting a gas laser to a predetermined frequency is described.
The value of the Einstein B coefficient for stimulated emission for a collection of atoms interacting with a radiation field is shown to depend upon the velocity distribution of the excited atoms and also upon the variation of field intensity with frequency. An emission line narrowing phenomenon for radiating matter enclosed in a nonresonant cavity with reflecting walls is predicted. It is shown that a loss mechanism must be introduced if significant line narrowing is to result from a single pass of radiation through an amplifying medium. An additional contributing effect to the mode pushing and mode pulling phenomena in a gas laser is demonstrated. A method of setting a gas laser to a predetermined frequency is described.
The phenomena which are principally responsible for the interaction of radiation with matter in everyday life are absorption and spontaneous emis sion. Light falling on matter is absorbed, leaving atoms in excited states which spontaneously emit radiation with a spread of frequency inversely proportional to the decay time. The situation is modified when the atomic decay times are sufficiently long and the radiation sufficiently intense, for then radiation may fall on excited atoms, resulting in stimulated emission rather than absorption. We saw in Chap. 2 that this third process was required to obtain the Planck blackbody radiation law in Einstein’s derivation. It is stimulated emission which enables an atomic medium to amplify incident radiation, and therefore is an essential ingredient in light amplifiers and oscillators. In fact, the word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.
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.
Introduction The idea of making use of the phenomenon of stimulated emission from an atom or molecule for amplification of the electromagnetic field, and then combining the amplifier with a resonator to make an oscillator, is due to Townes and his co-workers (Gordon, Zeiger and Townes, 1954, 1955) and independently to Basov and Prokhorov (1954, 1955). The former constructed the first MASER, or microwave amplifier by stimulated emission of radiation . In 1958 Schawlow and Townes proposed an application of the same principle to the optical domain (Schawlow and Townes, 1958), in which a two-mirror Fabry–Perot interferometer serves as the optical resonator and an excited group of atoms as the gain medium. The first LASER, or light amplifier by stimulated emission of radiation , was constructed by Maiman (1960). The gain medium was ruby, which was excited by a bright flash of light from a gas discharge tube, whereupon the laser delivered a short optical pulse. The first continuously operating – or CW – laser was developed by Javan and his co-workers (Javan, Bennett and Herriott, 1961); it made use of a He:Ne gas mixture, that was continuously excited by a discharge, as the gain medium. This type of laser is still widely used today. Since that time many different kinds of laser have been developed, ranging in wavelength from the infrared to the ultraviolet. Some produce light at several different frequencies at once, and some are tunable over a wide range.
Suppose that the gas is in equilibrium with a radiation field B(?, T). In a steady state, the number atoms leaving the lower state must equal the number leaving the upper state. Combining the last two equations, The probability that an atom will be in a state with energy Eu is exp{- Eu/kT) . This is proved in any book on statistical mechanics, but it is beyond the scope of this chapter. The relative populations of the two states is Using equation (33), Therefore, combining equations (37) and (38), Since the constants and depend only on the structure of the atom, and not on temperature or other aspects of the environment,25 Then So, using the Plank black-body law [equation (2)], These coefficients are called the Einstein coefficients; they depend only on the nature of the atom and not on the temperature, the ambient radiant intensity, or any other aspect of the atom’s environment. Therefore, equations (42) and (44) are true in general. In particular, note that this argument shows that stimulated emission must exist; i.e., . This proved the existence of stimulated emission long before it had been observed in the laboratory; today, of course, it is the basis of the laser (Light Amplification by Stimulated Emission of Radiation) industry. Stimulated emission is necessary, as I have already mentioned, to keep the Second Law of thermodynamics from being violated, because without it, atomic absorption and emission processes could not be in balance with emission from a black body. If there is an enclosure with perfectly black walls with temperature T, then the radiation field inside the enclosure must be equal to the Planck function B(?, T). But now introduce a gas into that enclosure. Without stimulated emission, there would be a continuous transfer of energy either to or from the gas. This, as we know, is impossible.
We extend the investigation of the long-time behavior of a model consisting of N two-level atoms interacting with a single electromagnetic cavity mode to include interaction with many cavity modes. We show that, as a consequence of the coupling between radiation modes produced by spatial density variations of the population inversion, there is no strictly stationary state possible for multimode behavior. However we obtain a stationary state by neglecting the rapidly oscillating terms. The steady-state population inversion is then a solution of an eigenvalue problem. The eigenvalue determinate is a function of the number of modes and the coupling between modes in addition to the usual dependence on frequencies and relaxation times. We explicitly solve for the unique eigenvalue in special cases. The corresponding eigenvector gives the steady-state mode intensity ratios. The absolute values of the steady-state intensities are determined by the energy conservation equation generalized to include pumping and dissipation. We also calculate the steady-state frequency shifts for each mode. The mode frequency shifts are practically independent of each other and have the same functional form as the single-mode frequency shifts.
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.