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Journal of Physics A Mathematical and General

An amplifying solution of the Maxwell-Bloch equations with atomic relaxation and field losses

J.A. Hermann

Published Nov 1, 1980• 4 citations
Publisher

Abstract

Linear analytic solutions of the homogeneously broadened, unidirectional Maxwell-Bloch equations with field losses are found in circumstances where amplification can occur. Particular stress is laid upon the similarity-variable solution. A number of inequalities are derived which suffice to differentiate between the physical regimes of superfluorescence, amplified spontaneous emission and swept-gain amplification.

DOI
10.1088/0305-4470/13/11/024

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Linear analytic solutions of the homogeneously broadened, unidirectional Maxwell-Bloch equations with field losses are found in circumstances where amplification can occur. Particular stress is laid upon the similarity-variable solution. A number of inequalities are derived which suffice to differentiate between the physical regimes of superfluorescence, amplified spontaneous emission and swept-gain amplification.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.
The linear and nonlinear properties of propagation of an electromagnetic field through a medium may be derived by the effective elimination of the medium from the coupled Maxwell-Bloch equations. In this paper we suggest, on the other hand, the elimination of the field from the Maxwell-Bloch equations as a means to derive a nonlinear master equation for, e.g., an atomic sample in a laser field. This presents a straightforward and physically intuitive derivation of equations for ``nonlinear atom optics,'' obtained recently by other approaches.
The analysis in Chapter 3 used a phenomenological form of the permittivity to describe active materials. A proper understanding of optical amplification requires a quantum-mechanical approach for describing the interaction of light with atoms of an active medium [1]. However, even a relatively simple atom such as hydrogen or helium allows so many energy transitions that its full description is intractable even with modern computing machinery [2, 3]. The only solution is to look for idealized models that contain the most essential features of a realistic system. The semiclassical two-level-atom model has proven to be quite successful in this respect [4]. Even though a real atom has infinitely many energy levels, two energy levels whose energy difference nearly matches the photon energy suffice to understand the interaction dynamics when the atom interacts with nearly monochromatic radiation. Moreover, if the optical field contains a sufficiently large number of photons (> 100), it can be treated classically using a set of optical Bloch equations. In this chapter, we learn the underlying physical concepts behind the optical Bloch equations. We apply these equations in subsequent chapters to actual optical amplifiers and show that they can be solved analytically under certain conditions to provide a realistic description of optical amplifiers. It is essential to have a thorough understanding of the concept of a quantum state [5]. To effectively use the modern machinery of quantum mechanics, physical states need to be represented as vectors in so-called Hilbert space [6].
In a related study the equations governing a high-gain laser amplifier have been solved numerically without making the usual slowly-varying-amplitude derivative approximation in Maxwell's equations, and thus the field amplitudes are not restricted to vary negligibly in a distance of one wavelength. The results reveal an instability that involves reflections and oscillatory growth of perturbations away from steady state. In the present study steady-state solutions of the field equations are obtained analytically, and transformations are described for converting the equations to alternative intensity-equation sets. These intensity equations are linearized and solved for the growth rate and oscillation period of the perturbations.
For a pencil-shaped extended medium with Fresnel number equal to 1, we have quantum mechanically derived a description of the initiation of superfluorescence in terms of Maxwell-Bloch equations with a fluctuating source due to the zero-point fluctuations of the vacuum field. By the introduction of classical behavior, these equations are extended to include nonlinear behavior due to decreasing atomic inversion. The principal assumption in the derivation is that the main features of superfluorescence are governed by the interaction of atoms with field modes inside two small solid angles around the pencil axis. The delay ${\ensuremath{\tau}}_{D}$, defined as the time at which the mean-squared tipping angle of the collective Bloch vector attains the value 1, turns out to be given by ${\ensuremath{\tau}}_{D}=(\frac{{\ensuremath{\tau}}_{R}}{4}){[\mathrm{ln}{(2\ensuremath{\pi}N)}^{\frac{1}{2}}]}^{2}$, where ${\ensuremath{\tau}}_{R}$ is the radiation time for collective decay and NNN is the number of atoms. The corresponding effective initial tipping angle roughly equals 2(N)12\frac{2}{{(N)}^{\frac{1}{2}}}(N)21​2​. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}$ of the delay of the superfluorescence pulse satisfies approximately $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}=\frac{2.3}{\mathrm{ln}N}$. A brief comparison with previous treatments of superfluorescence is given.