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-Pérot 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 90o in lead of the driving field, and the phase of absorption lags 90o behind the transmitted field. Also spontaneous emission must obey energy conservation, hence it occurs with 90o 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.
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-Pérot 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 90o in lead of the driving field, and the phase of absorption lags 90o behind the transmitted field. Also spontaneous emission must obey energy conservation, hence it occurs with 90o 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.
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.
The laser linewidth manifests the spectral coherence of a lasing resonator mode. It is of fundamental importance, theoretically as much as practically. For more than half a century we have been told that the laser linewidth is a result of quantum fluctuations due to spontaneous emission, which induce amplitude and phase fluctuations, whereas stimulated emission occurs in phase with the incident field. Unfortunately, in such a scenario each emission process would violate the law of energy conservation. We show that a (spontaneous or stimulated) emission process occurs 90 degrees in lead of the incident field, converts one atomic excitation into one photon in the lasing mode and, therefore, conserves the energy, whereas the mentioned fluctuations are the result of vacuum energy and occur at an orders-of-magnitude faster time scale than an emission process.
The processes of absorption and stimulated emission are investigated via (i) energy conservation in a Fabry-Perot resonator, (ii) the Lorentz-oscillator model, and (iii) the Kramers-Kronig relations. Vacuum fluctuations are added and the relation between stimulated and spontaneous emission is clarified. It is shown that previous models start from incorrect assumptions about the physical foundations of spontaneous emission. The non-orthogonality of longitudinal optical modes in a linear resonator is investigated and its consequences for the laser linewidth are discussed. Based on these results, a semi-classical approach to the laser linewidth below, around, and above the laser threshold which is consistent with resonator theory is presented. These results also shed new light on the usually considered linewidth-narrowing and -broadening factors.
It is generally believed that absorption and stimulated emission are inverse processes, as both are driven by an external field, their strength is quantified by the same Einstein B coefficient, and they occur with a defined phase, opposite to each other, namely in phase and in anti-phase with the driving field, whereas spontaneous emission is a different process that occurs with an arbitrary phase with respect to a potential incident field. Recently, the phase relation in absorption and emission was shown to differ from this believe. Here it is verified via the amplitude–phase diagram and via the interference of sine waves that, precisely speaking, only the absorption process, in which a number φ + 1 of incident photons is decreased by one photon, and the emission process, in which a number φ of incident photons is increased by one photon, are truly inverse processes also in their phase. Particularly, this implies that absorption of a single incident photon and spontaneous emission of a photon into an empty mode are inverse processes in the amplitude–phase diagram.
This chapter discusses the details of what stimulated emission of radiation is, why it is the major physical phenomenon at the heart of the laser, and how it is exploited in ways to create laser light beams. It describes the Bohr model of the atom and how electrons can only make quantized leaps in energy levels from one atomic orbital or shell to the next. The chapter considers Pauli’s Exclusion Principle and singlet, doublet, and triplet states of the atom. It also describes the interaction of light with atoms through absorption, spontaneous emission, and stimulated emission. The chapter shows that the quantum mechanical harmonic oscillator and the ground state of the universe is a nonzero electromagnetic field and therefore is the trigger for spontaneous emission. It looks at the models for absorption, spontaneous and stimulated emission, and vibrational states.
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.