Abstract In his celebrated 1916–17 papers in which he proposed the A and B coefficients for the spontaneous and stimulated emission of energy quanta from excited atoms, Einstein conjectured that stimulated emission involves the release of individual quanta (later dubbed “photons”) along the direction of an incident photon with the same energy, momentum, phase, and polarization state as that of the incident photon. According to classical electrodynamics, of course, an oscillating dipole must radiate an azimuthally symmetric electromagnetic field around its axis of oscillation. Nevertheless, Einstein suggested that the release of stored energy from excited atoms in the form of discrete quanta (photons) must be directional, and that, in the case of stimulated emission, the direction of the emitted photon must coincide with that of the incident photon. The goal of the present paper is to show that some of the prominent features of absorption and stimulated emission emerge from Maxwellian electrodynamics in conjunction with the simple mass-and-spring model of an atom known as the Lorentz oscillator model.
Abstract In his celebrated 1916–17 papers in which he proposed the A and B coefficients for the spontaneous and stimulated emission of energy quanta from excited atoms, Einstein conjectured that stimulated emission involves the release of individual quanta (later dubbed “photons”) along the direction of an incident photon with the same energy, momentum, phase, and polarization state as that of the incident photon. According to classical electrodynamics, of course, an oscillating dipole must radiate an azimuthally symmetric electromagnetic field around its axis of oscillation. Nevertheless, Einstein suggested that the release of stored energy from excited atoms in the form of discrete quanta (photons) must be directional, and that, in the case of stimulated emission, the direction of the emitted photon must coincide with that of the incident photon. The goal of the present paper is to show that some of the prominent features of absorption and stimulated emission emerge from Maxwellian electrodynamics in conjunction with the simple mass-and-spring model of an atom known as the Lorentz oscillator model.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
The coherence and directional properties of stimulated and spontaneous emission can be understood without a knowledge of quantum electrodynamics. This paper presents a semiclassical description of the interaction between a free atom and a weak optical field under conditions of either a stimulated emission or a stimulated absorption resonance. It is shown that, under certain conditions, the stimulated or spontaneous radiation fields emitted from a sample of atoms will propagate in the same direction as the incident beam, will be coherent with the incident beam, and will have the correct phase to provide amplification or attenuation.
We treat the interaction between an electromagnetic radiation bath and a material system in the semi-classical picture: the radiation is treated according to the Maxwell equations, but the process of emission/absorption is described through the concept of a photon; on the other hand, the material system is modelled according to quantum mechanics, assuming a discrete two-level energy spectrum. This very simple model allows one to elaborate the elementary theory of the absorption/emission coefficients, both for the spontaneous and for the stimulated processes which will be described within the electric dipole approximation. Selection rules for allowed transitions are eventually obtained and the nature of high-order transition is briefly outlined. Finally, a qualitative description of the light amplification by stimulated emission of radiation is presented.
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.
In this paper, four different ways are described in which homogeneous, isotropic, nonferromagnetic, Maxwellian media at rest may exhibit absorption and stimulated emission. These four ways may be identified by a positive or negative conductivity, a sink or source-type electric field, a complex dielectric constant, and a complex permeability. The forms of the electromagnetic waves, the refractive index, and the positive or negative extinction coefficient are determined for each process for both absorbing and active media. Spontaneous emission is associated with source-type electric and magnetic fields which generate conduction or displacement currents in the medium with improper phases to contribute to the radiant flux in the incident electromagnetic wave. Stimulated emission is characterized by exponentially increasing incident electric and magnetic fields and an excess of the time average of the outflow of radiant flux over the surfaces of an arbitrary volume. Stimulated emission appears when the time average of the rate at which the source-type electric and magnetic fields do work in carrying certain conduction or displacement currents in directions opposite to the incident electric and magnetic fields is greater than the time average of the rate at which the incident electric and magnetic fields dissipate energy by doing work in carrying other conduction or displacement currents in directions parallel to the incident electric and magnetic fields.
In the literature one finds several conflicting accounts of the phase difference of stimulated and spontaneous emission, as well as absorption, with respect to an existing (triggering) electromagnetic field. One of these approaches proposes that stimulated emission and absorption occur in phase and out of phase with their driving field, respectively, whereas spontaneous emission occurs under an arbitrary phase difference with respect to an existing field. It has served as a basis for explaining quantum-mechanically the laser linewidth, its narrowing by a factor of 2 around the laser threshold, as well as its broadening due to amplitude–phase coupling, resulting in Henry’s α-factor. Assuming the validity of Maxwell’s equations, all three processes would, thus, violate the law of energy conservation. In semi-classical approaches, we investigate stimulated emission in a Fabry–Perot resonator, analyze the Lorentz oscillator model, apply the Kramers–Kronig relations to the complex susceptibility, understand the summation of quantized electric fields, and quantitatively interpret emission and absorption in the amplitude–phase diagram. In all cases, we derive that the phase of stimulated emission is 90° in lead of the driving field, and the phase of absorption lags 90° behind the transmitted field. Also spontaneous emission must obey energy conservation, hence it occurs with 90° phase in lead of an existing field. These semi-classical findings agree with recent experimental investigations regarding the interaction of attosecond pulses with an atom, thereby questioning the physical explanation of the laser linewidth and its narrowing or broadening.