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

Simple quantum theory of bi-compound laser

S. V. Petrushkin, Самарцев Виталий Владимирович

Published Apr 20, 2004• 0 citations
Publisher

Abstract

The loss mechanism to account for the loss of laser photos from the cavity due to absorption by the second dopant inside the laser rod is investigated. We introduced a system of three-level ions as a dissipative subsystem and considered that photons can be lost through their coupling to this quantum subsystem. The conventional quantum laser theory is then expanded for this special case.

DOI
10.1117/12.562196

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The loss mechanism to account for the loss of laser photos from the cavity due to absorption by the second dopant inside the laser rod is investigated. We introduced a system of three-level ions as a dissipative subsystem and considered that photons can be lost through their coupling to this quantum subsystem. The conventional quantum laser theory is then expanded for this special case.
A semiclassical treatment of the radiation loss in a laser, previously investigated by Baseia and Nussenzveig [Opt. Acta 31, 39 (1984)], is extended to the quantum-theory level. The model employs an optical cavity having an output coupling that includes the radiation loss due to the beam extraction through appropriate boundary conditions, instead of the unrealistic loss reservoir of the conventional treatment. This allows one to work entirely within the continuous spectrum and obtain the equation for the radiation field both inside and outside the cavity. Inside the cavity the results agree, in the coherent representation and absence of quantum noise, with those of the previous semiclassical theory. Also, the results of both treatments for the field inside the cavity coincide with those of the conventional theory, since we reinterpret its ``mode amplitude'' as a collective variable. Outside the cavity (an inaccessible region for a standard approach) the explicit solution for the laser field exhibits a natural relationship between the intensities inside and outside the cavity. Further considerations, such as those concerning the role played by the noise term in the field buildup, are also discussed.
In this chapter we treat single-mode laser operation in which both the atoms and the field obey the laws of quantum mechanics. For simplicity, we suppose for most of the discussion that the field is tuned to atomic line center and that the active atoms are two-level systems which pass through the cavity in a time r, as in Sec. 16-1. To simulate a homogeneously broadened medium, we suppose that the transit times are distributed with the probability y exp(— yr). The more realistic (and complicated) calculation underlying the theory of Chap. 8 is given in Appendix I. We also suppose that excitation occurs only to the upper level; Probs. 17-13 and 17-14 consider excitation to the lower level as well. We describe cavity losses by an ensemble of nonresonant atoms injected into the lower level, as discussed in Sec. 16-1. The response of these “loss” atoms does not saturate (i.e., small coupling constant), as does that of the active atoms. The loss atoms here play the same role as the fictitious con­ ductivity in the semiclassical laser theory of Chap. 8. As shown in Sec. 16-5, an equivalent description is obtained by a reservoir of simple harmonic os­ cillators with a spread of frequencies. Our discussion depends on the reservoir theory of Sec. 16-1, in which the reduced density operator represents the ra­ diation field. In Chap. 20 we give a related development in which the Langevin (Heisenberg picture) approach is used.
Laser actions and photon statistics for a nondegenerate two–photon laser in an effective three–level atomic system with arbitrary detuning are investigated both semiclassically and quantum mechanically, Due to the two–photon effect, the radiation field easily displays a sub–Foissonlan statistical distribution. The laser actions for a two–photon and a single–photon loss mechanisms are compared. It Is found that the detailed balance in the single–photon loss mechanism can still be preserved in the case that a first–order phase transtion takes place.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
In this chapter, we turn our considerations to the simplest model of a laser that includes all the essential features of any practical laser. The one-photon losses (due to coupling to the environment), linear gain (due to inverted atoms) and saturation (due to nonlinear loss), that are experienced by a single-mode (laser) cavity are all included in the calculations and their role in the laser properties is fully explored.We point out that all lasersmust have these three elements to operate as a laser. If there is only a linear gain, then of course there can be no output, since an output causes a loss. With both gain and loss, but not saturation, the laser intensity is either near zero (well below threshold), or else it rises infinitely (well above threshold). Neither case is very useful as a model of a real laser, which has a finite output above threshold. This is why a saturation mechanism is needed, such as an intensity dependent nonlinear loss. This simple model used here is then obtainable in the limit of a high-Q laser operating not too far above threshold.