The traditional Beer law for absorption in an optical medium is precise only at incident power densities lower than few kW.At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission.In previous models that considered those processes,analytical expression for the absorption law could not obtained.In order to solving this problem,the two-level energy rate equations is solved analytically and it leads to a generalized absorption law in this paper and then the applied criterion of the generalized Beer law is obtained and analyzed.The generalized absorption law reduces to Beer law at low power densities.By making comparison between the experimental result and the calculated result of the generalized Beer law,it shows that this generalized law is fast and exact because it accounts for absorption,stimulated and spontaneous emissions.
The traditional Beer law for absorption in an optical medium is precise only at incident power densities lower than few kW.At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission.In previous models that considered those processes,analytical expression for the absorption law could not obtained.In order to solving this problem,the two-level energy rate equations is solved analytically and it leads to a generalized absorption law in this paper and then the applied criterion of the generalized Beer law is obtained and analyzed.The generalized absorption law reduces to Beer law at low power densities.By making comparison between the experimental result and the calculated result of the generalized Beer law,it shows that this generalized law is fast and exact because it accounts for absorption,stimulated and spontaneous emissions.
The traditional Beer law for absorption in an optical medium is precise only at incident power densities lower than a few kW.At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission.In previous models that considering those processes,analytical expression for the absorption law could not be obtained.In order to solving this problem,the two-level energy rate equations is solved analytically and it leads to a generalized absorption law in this paper and then the applied criterion of the generalized Beer law is obtained and analyzed.The generalized absorption law reduces to Beer law at low power densities.By making comparison between the experimental result and the calculated result of the generalized Beer law,it shows that this generalized law is fast and exact because it accounts for absorption,stimulated and spontaneous emissions.
Relations described by Einstein connecting the rates of spontaneous emission, stimulated emission, and absorption of radiation by an atomic system in free space are generalized to apply to broadband spectra of quantized systems dilutely distributed in a dielectric medium. Although the gross features of the broadband emission and absorption spectra can be qualitatively different (especially at low temperatures), the various spectra are connected at any specific frequency by simple expressions. For a two-level system imbedded in a medium at temperature T, typical equations connecting stimulated-emission and absorption cross sections $\ensuremath{\sigma}(\ensuremath{\omega})$ to the rate $f(\ensuremath{\omega})$ of spontaneous emission of photons per unit solid angle per unit frequency interval are: ${\ensuremath{\sigma}}_{e}(\ensuremath{\omega})={\ensuremath{\sigma}}_{a}(\ensuremath{\omega})\mathrm{exp}[\frac{\ensuremath{\hbar}(\ensuremath{\mu}\ensuremath{-}\ensuremath{\omega})}{\mathrm{kT}}]=f(\ensuremath{\omega}){(\frac{2\ensuremath{\pi}c}{\ensuremath{\omega}n})}^{2}$, where $\ensuremath{\hbar}\ensuremath{\mu}$ is a temperature-dependent excitation potential and n is the index of refraction of the host material.
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.
A dynamic extended Beer–Lambert law with photodissociation and recombination is proposed, which is built to describe the interactions between light and molecular gases. The model comprises two spatiotemporally coupled first-order nonlinear partial differential equations. Numerical solutions of these equations systematically reveal the nonlinear transmission properties and steady-state dissociation behavior of the medium from the weak- to intense-light regime. In the weak-light limit, the model naturally reduces to the classic Beer–Lambert law, with the system reaching a steady state characterized by high molecular density and exponential light attenuation. Conversely, in the intense-light regime, low molecular concentration and pronounced photoinduced transparency occurs, and light attenuates linearly instead of exponentially. A method for measuring the recombination rate constant under steady-state conditions is presented, which imposes significantly lower demands on both the light intensity and the temporal resolution of the measurement equipment compared to conventional approaches.
A theoretical study has been made of the transmission of a monochromatic, well-collimated beam of radiation normally incident on idealized material containing only two energy levels. The nonlinear partial differential equations governing the variation of population and photon density in space and time have been solved exactly for arbitrary initial conditions under the assumption that spontaneous emission and thermal relaxation from the excited state can be neglected. These conditions are satisfied in practice when the radiation is in the form of a pulse whose duration is short compared with the characteristic times of these relaxation processes. Two cases are considered in detail: (a) when all atoms are in the ground state and the material absorbs; and (b) when there is a population inversion and the material amplifies. In case (a), the radiation, with a characteristic velocity, ``bores'' its way through an optically dense substance emerging delayed in time and leaving the material in a perfectly transparent or saturated state. In case (b), an incoming pulse is amplified and sharpened—to a degree determined by the gain of the medium. In addition to these cases, the dependence of the apparent spontaneous emission lifetime on the size and population distribution of the material is mentioned.
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.