Abstract Optical gain can be regarded as the growth of an electromagnetic wave propagating through a dielectric as described by Maxwell’s equations, with electric polarisation due to dipoles induced by the field of the wave in classical atomic oscillators. The real and imaginary parts of the complex susceptibility give the amplitude gain and refractive index change respectively. Over time the phase relation between oscillation of individual dipoles is lost by collisions; this loss of coherence causes the optical response of the ensemble to be broadened in energy or wavelength. The response is also broadened by gradual energy loss (damping) causing a reduction in the amplitude of the oscillating field. These effects are characterised by time constants T22 and T1 respectively, which combine to produce homogeneous broadening with Lorentzian lineshape. Absorption by an atomic ensemble is expressed as an optical cross section of each atom.
Abstract Optical gain can be regarded as the growth of an electromagnetic wave propagating through a dielectric as described by Maxwell’s equations, with electric polarisation due to dipoles induced by the field of the wave in classical atomic oscillators. The real and imaginary parts of the complex susceptibility give the amplitude gain and refractive index change respectively. Over time the phase relation between oscillation of individual dipoles is lost by collisions; this loss of coherence causes the optical response of the ensemble to be broadened in energy or wavelength. The response is also broadened by gradual energy loss (damping) causing a reduction in the amplitude of the oscillating field. These effects are characterised by time constants T22 and T1 respectively, which combine to produce homogeneous broadening with Lorentzian lineshape. Absorption by an atomic ensemble is expressed as an optical cross section of each atom.
Abstract The present chapter deals with the macroscopic theory of light-matter interaction. By using the Maxwell’s equations, the wave equation is developed first for free space, and then for a dielectric, and a conductor. The relationship between refractive index and absorption coefficient is then established. The phase and group velocities of the EM wave are defined. The well-known damped harmonic oscillator model is then used to find expressions for real and imaginary parts of the susceptibility. At the end, the two-level atomic system is used to illustrate fundamental processes of light-matter interaction, namely, absorption, spontaneous, and stimulated emissions, and the rates of these processes in terms of Einstein’s A and B coefficients. Absorption, gain, and self-sustained oscillation in a medium with a Fabry–Perot resonator are discussed using the two-level atomic model.
This chapter presents the electron oscillator model of light-matter interaction. It models atomic response to the fields in an electromagnetic wave after a classical driven charged (dipole) oscillator. It is used to calculate atomic polarization and dielectric susceptibility and derive an equation of motion for atomic polarization for a collection of atoms. By replacing the oscillator number density in this equation with the level population density difference and interpreting oscillator frequency with transition frequency and decay rate with spontaneous emission rate a quantum mechanically correct equation for atomic polarization is obtained. Poynting theorem on energy exchange between light and atoms in an EM wave then yields quantum mechanically correct rate equations for population densities. These equations together with the equation for atomic polarization provide a complete description of atomic response to light. Modifications to the atomic response due to finite atomic lifetimes, collisions, atomic motion, local variation of resonance frequency and other factors are also discussed in terms of lineshape function and cross-section.
Abstract The quantum mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.
The effects of light absorption and amplification on the transition rates of the electric-dipole emission of atoms or molecules embedded in micrometer-sized dielectric spheres are investigated in the framework of classical electrodynamics. The expression for the form factor of a resonant emission line is derived. Its dependence on the imaginary part of the light’s refractive index is investigated. It is discovered that the active sphere does not act as an optical resonator if light amplification is strong.
Introduction One of the most impressive accomplishments of wave optics is its success in providing a coherent and succinct description of the interactions between electromagnetic waves and matter (gases, solids, etc.). Maxwell's equations, which describe the propagation of light, and the Laplace–Lorentz equations, which describe the source terms of light, allow one to take into account the phenomena of refraction, diffusion, and diffraction of light by dense media. It is amazing – to say the least – that such a theory can account for the complex interactions of an electromagnetic wave with an immense ensemble of atoms (each approximated in terms of individual harmonic oscillators), by means of a simple optical index n op . Such an achievement is reminiscent and, indeed, on par with the level of concision achieved by the concept of effective mass in representing the interaction of a conduction electron with a crystalline lattice. In the description of all these effects, an electromagnetic wave with (angular) frequency ω forces free carriers into oscillatory motion at the same frequency, leading to radiative re-emission at this same frequency. This behaviour is a natural by-product of the linear equations we have employed up until now. In this chapter, we will see that non-linear media (i.e. materials whose response to external excitations contains non-linear terms), may by used to perform frequency conversion as evidenced, for example, by second harmonic light generation or optical parametric oscillations.
The laser gain equation is derived in two ways. First, the optical Bloch equations are derived, based upon a semiclassical treatment of a two-level atomic system and a radiation field. As a visualization aid, there follows a discussion of the formally identical classical Bloch equations, which describe the behavior of a magnetic moment in a combined static and rotating magnetic field. It is shown that laser gain is proportional to the imaginary part of the linear susceptibility, which is obtained from the semiclassical treatment. The second viewpoint uses a slightly modified form of Einstein’s A and B coefficients adapted to describing interactions between atoms and monochromatic fields. Various damping and broadening mechanisms are introduced in a reasonable but non-rigorous fashion, and the crucial distinction between homogeneous and inhomogeneous broadening is introduced.