THE ABSORPTION AND STIMULATED EMISSION OF RADIATION

So far in this book we have emphasized the spontaneous emission of radiation. This occurs at a constant rate, independent of any external influences. Now we wish to turn our attention to the processes of stimulated emission and absorption of radiation which are induced by the presence of an external electromagnetic wave. In this chapter we prepare the foundations for an understanding of both the formation of spectral lines, which is considered in detail in Chapter 10, and the physics of gas lasers which is discussed in Chapters 11-13. 

We commence by deriving the absorption cross-section of a classical electric dipole oscillator.. The result should be similar to that obtained on the basis of the quantum theory and is of further interest since the frequency dependence of the cross-section is predicted in a simple way. Next we obtain the relations between the spontaneous emission transition probability, and the 

Einstein coefficients for absorption and stimulated emission, denoted by and respectively. The expressions for B j, and Bj are then confirmed by means of quantum mechanics using time-dependent perturbation theory. This enables the probability of stimulated emission and absorption of radiation to be given in terms of the oscillator strengths of spectral lines. Finally we show that there is close agreement between the classical and quantum-mechanical expressions for the total absorption cross-section and explain how the atomic frequency response may be introduced into the quantum-mechanical results. 

Claseiaal desar-iption of absorption by electrio dipole osoiIlator 

We consider first the absorption of radiation by a classical atom, represented by an harmonically oscillating 

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THE ABSORPTION AND STIMULATED EMISSION OF RADIATION 273 we can transform equation (9.6) to (Problem 9.1)... 

Sources. Laser sources make use of population inversion. When radiation enters a medium, both absorption and stimulated emission of radiation may occur and the change in flux at the exit is ... 

So far we have treated absorption and stimulated emission of radiation. However, it is well known that an atom can emit radiation even when it is not externally perturbed, i.e. spontaneous emission. It is not possible to treat this process fully here, since consideration of the quantization of the electromagnetic field as described by Quantum ElectroDynamics (QED) is necessary. According to QED a coupling between the atom and the "vacuum state" of the field is responsible for the emission. 

Absorption and stimulated emission of 10.6- um radiation by C02 at the P(20) rotational line,... 

In the above rather simplified analysis of the interaction of light and matter, it was assumed that the process involved was the absorption of light due to a transition m - n. However, the same result is obtained for the case of light emission stimulated by the electromagnetic radiation, which is the result of a transition m -> n. Then the Einstein coefficients for absorption and stimulated emission are identical, viz. fiOT< n = m rt. 

The word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation. The physical process upon which lasers depend, stimulated emission, was first elucidated by Einstein in 1917 (1). Einstein showed that in quantized systems three processes involving photons must exist absorption, spontaneous emission, and stimulated emission. These may be represented as follows ... 

Equation (A3.7) shows the equality between the probabilities of absorption and stimulated emission that we have already established for monochromatic radiation in Equation (5.15). Equation (A3.8) gives the ratio of tlie spontaneous to the induced transition probability. It allows us to calculate the probability A of spontaneous emission once the Einstein B coefficient is known. 

Considerable attention has been paid in the past few years to the study of both the absorption and emission spectra of the rare earths. This has been boosted further by the development of the new branch of physics, the Laser (light amplification through stimulated emission of radiation). The study of the optical spectra of ions yields valuable information about the energy levels of normal configurations and of excited states, and also about the nature of their environment. However, a detailed analysis of optical spectra demands a considerable knowledge of theoretical techniques. Recent advances in paramagnetic resonance techniques [479] have enabled us to understand the nature of the ground states of the rare earth ions in crystalline environments. 

These equations are similar to those of first- and second-order chemical reactions, I being a photon concentration. This applies only to isotropic radiation. The coefficients A and B are known as the Einstein coefficients for spontaneous emission and for absorption and stimulated emission, respectively. These coefficients play the roles of rate constants in the similar equations of chemical kinetics and they give the transition probabilities. 

In addition to absorption and stimulated emission, a third process, spontaneous emission, is required in the theory of radiation. In this process, an excited species may lose energy in the absence of a radiation field to reach a lower energy state. Spontaneous emission is a random process, and the rate of loss of excited species by spontaneous emission (from a statistically large number of excited species) is kinetically first-order. A first-order rate constant may therefore be used to describe the intensity of spontaneous emission this constant is the Einstein A factor, Ami, which corresponds for the spontaneous process to the second-order B constant of the induced processes. The rate of spontaneous emission is equal to Aminm, and intensities of spontaneous emission can be used to calculate nm if Am is known. Most of the emission phenomena with which we are concerned in photochemistry—fluorescence, phosphorescence, and chemiluminescence—are spontaneous, and the descriptive adjective will be dropped henceforth. Where emission is stimulated, the fact will be stated. 

The probability of stimulated emission by an excited molecule, Bpv, is the same as that of the reverse process, absorption by the ground-state molecule. This is a consequence of the law of microscopic reversibility if the number of excited molecules Nm is equal to the number of ground-state molecules N , then the rates of stimulated absorption and emission must be equal. If by some means a population inversion can be produced, Nm > Nn, the net effect of interaction with electromagnetic radiation of frequency v will be stimulated emission (Figure 2.4). This is the operating principle of the loser. LASER is an acronym for light amplification by stimulated emission of radiation (Section 3.1). 

The acronym LASER (Light Amplification via the Stimulated Emission of Radiation) defines the process of amplification. For all intents and purposes this method was elegantly outlined by Einstein in 1917 [11] wherein he derived a treatment of the dynamic equilibrium of a material in a electromagnetic field absorbing and emitting photons. Key here is the insight that, in addition to absorption and spontaneous emission processes, in an excited system one can stimulate the emission of a photon by interaction with the electromagnetic field. It is this stimulated emission process which lays the conceptual foimdation of the laser. 

LASER is an acronym that stands for Light Amplification by Stimulated Emission of Radiation. The laser produces an intense, highly directional beam of light. The most common cause of laser-induced tissue damage is thermal in nature, where the tissue proteins are denatured due to the temperature rise following absorption of laser energy. The human body is vulnerable to the output of certain lasers, and under certain circumstances, exposure can result in damage to the eye and skin. 

The emitted photon travels in phase with and in the same direction as the initial photon. These two photons can similarly interact with additional excited states, stimulating the emission of further photons, resulting in light amplification (laser is an acronym for light amplification by stimulated emission of radiation), For stimulated emission to occur, the probability (or rate) of stimulated emission must exceed that of absorption. At thermal equilibrium, the relative population of the ground and excited state in a two-level system has A n > Nm- As discussed in Chap. 1, for stimulated emission to dominate absorption, a population inversion is required, such that To create this... 

The first relation shows that the probabilities for absorption and stimulated emission are the same for a transition between states 1 and 2. This is in accordance with the result obtained above using first-order perturbation theory. Note, that the result (4.26) is independent of the strength of the radiation field. It was in discussions of this kind that A. Einstein, in 1917, found it necessary to introduce the concept of stimulated emission in order to obtain agreement with the statistical laws known at that time [4.12]. 

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