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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Laser beam coherence

Alexandru Popa

Published Jun 29, 2001• 1 citation
Publisher

Abstract

In recent papers we presented a wave model for closed conservative bounded systems, whose behavior is described by the Schrodinger equation. For such a system, the Schrodinger equation is equivalent to the wave equation. The analysis of the wave properties reveals an intrinsic coherence of the electron motions. We approximate the fundamental and metastable states of laser active medium by closed conservative bounded systems. We show that the coherence of the laser radiation could be explained by a transition between these coherent states.

DOI
10.1117/12.432882

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In recent papers we presented a wave model for closed conservative bounded systems, whose behavior is described by the Schrodinger equation. For such a system, the Schrodinger equation is equivalent to the wave equation. The analysis of the wave properties reveals an intrinsic coherence of the electron motions. We approximate the fundamental and metastable states of laser active medium by closed conservative bounded systems. We show that the coherence of the laser radiation could be explained by a transition between these coherent states.
Some preliminary results of a theoretical investigation of the origin of laser coherence properties are presented. A model consisting of NNN two-level atoms in a perfect cavity interacting through their dipole moments with a single mode of the radiation field is adopted, and the resulting many-body problem is made solvable through the introduction of the self-consistent-field approximation (SCFA). It is shown that the SCFA is consistent with Glauber's suggestion that an ideal laser might be in a "coherent state," or pure eigenstate of the positive-frequency part of the electric-field operator. A conservation law is derived which assures the positive definiteness of the field-density operator, and it is shown that Glauber's coherent state is that solution of the SCFA equations which minimizes the corresponding constant of the motion. When the SCFA is modified by the introduction of particle-field correlations, the conservation law is destroyed, and the coherent state is no longer and allowed solution.
The quantum-mechanical state of laser radiation near the generation threshold is studied. It is shown that there is a second threshold in laser generation which lies above the first one. At the second threshold coherence appears in laser radiation. In a small region between the two thresholds the laser radiation is in an essentially quantum state, in which the average field strength is close to zero, but the average energy keeps a macroscopic value. Active atoms in superposition states are the sources of coherence in lasers. The laser active medium sufficiently enlarges the characteristic time of coherence.
The basic phenomena of interaction between atoms and e.m. fields can be discussed in some detail without the need of a detailed model of the atom. Absorption and stimulated emission of radiation can be explained satisfactorily by the theory, where a classical electromagnetic wave provides a time-dependent perturbation to the energy of the atomic electrons. Laser radiation appears to a very good approximation to be a classical travelling wave of well-defined amplitude and phase. The radiation produced by a continuous laser is a good approximation to a coherent state. For conventional sources of radiation, the interaction energy is small compared with the energy an electron possesses by virtue of the attraction to the atomic nucleus, so perturbation theory produces quite acceptable results. The quantum mechanical analogy to the distortion of the classical charge cloud is given by the writing down of a general state function for the system, Ψ.
We predict a variety of photonic coherence phenomena in passive and active coupled ring resonators. Specifically, the effective dispersive and absorptive steady-state response of coupled resonators is derived, and used to determine the conditions for coupled-resonator-induced transparency and absorption, lasing without gain, and cooperative cavity emission. These effects rely on coherent photon trapping, in direct analogy with coherent population trapping phenomena in atomic systems. We also demonstrate that the coupled-mode equations are formally identical to the two-level atom Schrödinger equation in the rotating-wave approximation, and use this result for the analysis of coupled-resonator photon dynamics. Notably, because these effects are predicted directly from coupled-mode theory, they are not unique to atoms, but rather are fundamental to systems of coherently coupled resonators.
We extend the existing theory of continuous-wave lasers by systematically considering spontaneous emission. In a simple rate-equation approach, the laser eigenvalue, defined as the ratio of coherent photons coupled out of the resonator divided by the number of photons coupled in via spontaneous emission, emerges as the fundamental parameter describing a continuous-wave laser. We derive a general equation for the laser eigenvalue and confirm the point at which coherence manifests itself in the resonator. The theory describes all types of lasers of three-level, four-level, or any intermediate nature.
We have demonstrated in previous papers the following property of closed, conservative and bounded systems: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. In the present article, we show that this property is connected to the intrinsic wave properties of the system. This results from the equivalence between the Schrödinger equation and the wave equation, valid for conservative systems. As a consequence of the wave properties of the system, we show that the Hamilton-Jacobi equation has always periodical solutions, whose constants of motion are identical to the eigenvalues of the Schrödinger equation, written for the same system. It results that the calculation model presented in previous papers is generally valid in the case of closed, conservative and bounded systems. We present the applications of the model to the nitrogen and oxygen atoms, to the ions with the same structure, and to the He 2 , Be 2 and B 2 molecules.