A computational model study for complete frequency redistribution linear incoherent two-level atomic radiation trapping in optically dense media using the multiple scattering representation is presented. This model study discuss at length the influence of the spectral distributions, overall opacity and emission quantum yield to trapping distorted ensemble quantities stressing physical insight and with a non-specialist audience in mind. Macroscopic reemission yield, lifetime, steady state spectra and spatial distributions are calculated as a function of intrinsic emission yield, opacity and external excitation mode for Doppler, Lorentz and Voigt lineshapes. The work could constitute the basis for a final undergraduate or beginning graduate project in computational physics instruction and implements the analytical developments of the previous instalment of this contribution.
A computational model study for complete frequency redistribution linear incoherent two-level atomic radiation trapping in optically dense media using the multiple scattering representation is presented. This model study discuss at length the influence of the spectral distributions, overall opacity and emission quantum yield to trapping distorted ensemble quantities stressing physical insight and with a non-specialist audience in mind. Macroscopic reemission yield, lifetime, steady state spectra and spatial distributions are calculated as a function of intrinsic emission yield, opacity and external excitation mode for Doppler, Lorentz and Voigt lineshapes. The work could constitute the basis for a final undergraduate or beginning graduate project in computational physics instruction and implements the analytical developments of the previous instalment of this contribution.
A simple stochastic formulation of the multiple scattering representation solution of the classical linear incoherent trapping problem is presented for a broad audience. A clear connection with the alternative Holstein's solution ansatz is emphasized by the (re)interpretation of the fundamental mode as the one associated with a relaxed nonchanging spatial distribution of excitation. Expressions for overall relaxation parameters (ensemble emission yield and lifetime) as well as time-resolved (decay and spatial distribution) and steady-state quantities (spectra and spatial distribution) are given with the fundamental mode contribution singled out. The multiple scattering representation is advocated for final undergraduate and beginning graduate physics instruction based on physical insight and computation feasibility. This will be illustrated in the following instalment of this contribution.
We introduce an analytical method to investigate radiation trapping problems with Doppler frequency redistribution. The problem is formulated within the framework of the Holstein-Biberman-Payne equation. We interpret the basic integro-differential trapping equation as a generalized wave equation for a four-dimensional (4D) classical system (an associated quasiparticle). We then construct its analytical solution by a semiclassical approach, called the geometrical quantization technique (GQT). Within the GQT, it is shown that the spatial and frequency variables can be separated and that the frequency part of the excited atom distribution function obeys a stationary Schr\"odinger equation for a perturbed oscillator. We demonstrate that there is a noticeable deviation of the actual spectral emission profile from the Doppler line in the region of small opacities. The problem of calculating the spatial mode structure and the effective radiation trapping factors is reduced to the evaluation of wave functions and quantized energy values of the quasiparticle confined in the vapor cell. We formulate the quantization rules and derive the phase factors, which allow us to obtain analytically the complete spectrum of the trapping factors in 1D geometries (layer, cylinder, sphere) and other (2D and 3D) geometries when the separation of space variables is possible. Finally, we outline a possible extension of our method to treat radiation trapping effects for more general experimental situations including, for instance, a system of cold atoms.
Abstract It is already evident from the basic physical picture developed in Chapter l that mainly two parameters determine radiation trapping: the strength of the absorption and the spectral shape of the absorption. The strength of the absorption is characterized by the opacity of the atomic vapour, which is proportional to the absorption cross-section of the atoms and to the density of atoms. Hence, before we enter the mathematics of the trapping problem, we first have to determine the absorption cross-section and lineshape. We will not go into details (these are beyond the scope of this book), but only review the facts needed to approach our problem. In addition, there is another motivation for this chapter. It is not only nice but absolutely necessary to have all required formulas compiled in one consistent notation. One has to take really great care not to become completely lost by widely differing notation in the literature—the cgs-system and esu-units versus the SI-system, frequency versus angular frequency, atomic constants referenced to photon density, to energy density or to light intensity, the ubiquitous factor of two for polarization, and the like.
Generalized single-atom Maxwell-Bloch equations for optically dense media are derived taking into account noncooperative radiative atom-atom interactions. Applying a Gaussian approximation and formally eliminating the degrees of freedom of the quantized radiation field and of all but a probe atom leads to an effective time-evolution operator for the probe atom. The mean coherent amplitude of the local field seen by the atom is shown to be given by the classical Lorentz-Lorenz relation. The second-order correlations of the field lead to terms that describe relaxation or pump processes and level shifts due to multiple scattering or reabsorption of spontaneously emitted photons. In the Markov limit a nonlinear and nonlocal single-atom density matrix equation is derived. To illustrate the effects of the quantum corrections, we discuss amplified spontaneous emission and radiation trapping in a dense ensemble of initially inverted two-level atoms and the effects of radiative interactions on intrinsic optical bistability in coherently driven systems.
Abstract Radiation from spectral lines can be absorbed and re- emitted many times in atomic vapours before it reaches the boundaries of the container encasing the vapour. This effect is known as radiation trapping. It plays an important role practically everywhere where atomic vapours occur, e.g. in spectroscopy, in gas lasers, in atomic line filters, in the determination of atomic lifetimes, in measurements of atomic interaction potentials, and in electric discharge lamps. This book for the first time assembles all the information necessary for a treatment of practical problems, emphasizing both physical insights and mathematical methods. After an introduction that reviews resonance radiation and collisional processes in atomic vapours, physical effects and mathematical methods for various types of problems (e.g. with or without saturation, particle diffusion, reflecting cell walls, etc.) are explained in detail. The last part of the book describes the applications of these methods to a variety of practical problems like cross-section measurements or the design of discharge lamps.Atoms can both absorb and emit electromagnetic radiation like visible light In an assembly of many atoms, e.g., in a vapour or in a gas, radiation emitted by one atom may be reabsorbed by one of its neighbours, just to be reemitted a little later. The game can go on for quite some time—until the photon manages somehow to escape from this prison. At first glance, quite easy to understand. On the other hand, there must be some reason for this voluminous book full of sums, integrals and matrices. Our interest in this subject originally stemmed from investigations on atomic line filters we did for the European Space Agency, ESA. We believed that radiation trapping would have some influence on the performance of these filters and were looking for a simple means to take it into account We started out the same way as probably anyone new to the field would have done when faced with the problem of trapping. We read the classical paper of Holstein, and thought that was all there was to it. We soon had to realize, however, that the assumptions made by Holstein were not fulfilled in our problem, and decided to give the whole matter a closer look. That ‘closer look’ obtained a momentum of its own, and trapping became the centre of our own research in the following years. As our overview of the subject increased. we found that trapping plays a role in a staggering variety of situations. Trapping appears to be important in most experimental setups and in most engineering applications where atomic vapours are involved.
Advances in trap technology have made feasible experiments on single, or a few, atoms and ions. Photon counts from a single atom may exhibit new features that are washed out in ensembles. Wilser and I recently proposed a theoretical approach based on repeated gedanken measurements, which was then also independently proposed by Dalibard et al. and by Carmichael. The optical Bloch equations are not used but rather follow as a consequence. The approach is applied to a three-level V-system with small separation of the upper levels having parallel dipole moments. Macroscopic dark periods are predicted for such a laser-driven system, although none of the levels is metastable as in the Dehmelt system. The reason is a coherence effect. For particular values the dark period may become arbitrarily long. A similar result holds for A-systems. This explains the observed dark resonances in ensembles of such systems by increasingly long dark periods of individual atoms. The approach is then applied to atoms driven by incoherent light, and nonrate equations with off-diagonal terms are derived. It is shown that these may lead to coherence effects.