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Journal of the Optical Society of America B

Semiclassical theory of dye lasers: the single-frequency and multifrequency steady states of operation

Hong Fu, Hermann Haken

Published May 1, 1988• 23 citations
Publisher

Abstract

A semiclassical theory of dye lasers is presented in which the relevant energy-level diagram of a dye molecule is assumed to consist of a bandlike ground state with many sublevels and an excited single state. This theory not only describes the single-frequency operation, which has a low instability threshold, but also describes the two-frequency and multifrequency steady states of operation and the transitions between the different steady states. The general solution of a multifrequency operation is given explicitly and differs essentially from the well-known Rabi oscillation. The theoretical predictions are in good agreement with recent experiments done by Hillman et al., [ Phys. Rev. Lett.52, 1605 ( 1984)], which cannot be explained by the conventional Maxwell–Bloch laser theory derived from two-level atoms.

DOI
10.1364/josab.5.000899

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•Dec 1, 1987•2 citations
A semiclassical theory of dye lasers is presented in which the relevant energy-level diagram of a dye molecule is assumed to consist of a bandlike ground state with many sublevels and an excited single state. This theory not only describes the single-frequency operation, which has a low instability threshold, but also describes the two-frequency and multifrequency steady states of operation and the transitions between the different steady states. The general solution of a multifrequency operation is given explicitly and differs essentially from the well-known Rabi oscillation. The theoretical predictions are in good agreement with recent experiments done by Hillman et al., [ Phys. Rev. Lett.52, 1605 ( 1984)], which cannot be explained by the conventional Maxwell–Bloch laser theory derived from two-level atoms.
A semiclassical description for dye lasers is proposed, where the energy-level diagram of the dye molecule is assumed to consist of a continuous bandlike ground state and an excited singlet state. Unidirectional single-frequency (s.f.) operation is discussed. The linear-stability analysis for this operation reveals a very low threshold instability, which may appear generally in practical lasers. The ratio of the instability threshold to the lasing threshold may be of any value greater than 1, depending mainly on the bandwidth and the distribution of the dipole moments on the band, but it is independent of the cavity loss. This instability may account for that observed in recent experiments by Hillman, Krasinki, Boyd, and Stroud [Phys. Rev. Lett. 52, 1605 (1984)]. A general approach to analyzing the linear stability of the s.f. operation of the Maxwell-Bloch equations is also proposed, which states that only the eigenvalues of a 2\ifmmode\times\else\texttimes\fi{}2 matrix are relevant: one concerns the stability of the s.f. operation near the lasing threshold, the other determines the instability threshold of this operation.
We develop a fully quantum-mechanical theory for the organic-dye-solution laser, obtain density-matrix equations of motion for the single-mode radiation-density operator and the matter-density operator, and solve and investigate the steady-state case. We generalize the usual Born-Markoff approximation master equation for two matter states to include four matter states, each one of which interacts with the laser radiation field. This allows us to treat exactly the organic-dye molecular triplet-state levels which participate in the laser operation in an essential way. For experimentally realizable conditions the steady-state solution contains features which are qualitatively different from nondye lasers. These effects are directly attributable to intensity-dependent triplet-state radiation absorption losses. At threshold the diagonal matrix elements of the radiation photon distribution (Rn{R}_{n}Rn​) can be a decreasing function of the photon number nnn with an inflection point rather than the usual truncated Gaussian. This necessitates redefinition of threshold. For pumping just above threshold there may be both a maximum and a minimum in Rn{R}_{n}Rn​ rather than just a maximum as in usual laser theories. We also specify the effect of triplets on the narrowing of Rn{R}_{n}Rn​ for pumping above threshold and the subsequent widening of Rn{R}_{n}Rn​ for pumping well above threshold. Many of our results require photon-counting experiments for verification. Also, our equations are easily reducible to a semiclassical theory, where our treatment of the triplets variables is an improvement over the usual rate equations.
In an attempt to explain the instability observed by Hillman et al.,1 we have analyzed the domain of parameters in which their dye laser operates. This leads to the consideration of an asymptotic limit for which the slowly varying envelope approximation no longer holds. As a result, the electric field equation must retain its second-order space and time derivatives. This opens new possibilities for the steady states and their stability properties, (i) Even in the case of perfect matching between atomic and cavity frequencies, the laser operating frequency is displaced from the cavity resonance, (ii) The single-mode stable steady state is not necessarily the mode whose frequency is nearest to the atomic transition, (iii) Instability of the steady state can occur without restriction on the laser cavity quality.
The steady-state properties of a dye laser model with white quantum noise and strongly colored pump noise are investigated. An effective diffusion coefficient in the steady state is presented. Our coefficient differs from that of Fox and Roy [Phys. Rev. A 35, 1838 (1987)]. We compare our results with the measurements and simulations of Lett, Short, and Mandel [Phys. Rev. Lett. 52, 341 (1984)], and the results of Fox and Roy, respectively. We find that the quantum noise plays an important role in the steady-state analysis of laser fluctuations below and near threshold.
Abstract : A semiclassical analysis of injected lasers with stable and unstable resonators and arbitrary outcoupling fractions was performed. Homogeneously broadened lasers with Fabry-Perot and with positive branch, confocal unstable resonators were modeled. Both time-dependent and steady-state analyses of free-running and injected lasers were performed. Because of deficiencies in the standing wave semiclassical and the simple saturable gain models, a new semiclassical theory of homogeneously broadened lasers were developed. The theory is predicted upon the following assumptions: the atoms are two-level, a single longitudinal mode exists in the cavity, the transverse mode structure can be adequately modeled by the geometric optics mode, and the laser electric fields can be described by a pair of counterpropagating planar, cylindrical, or spherical traveling waves with different amplitudes. A general semiclassical theory was first developed, and then specialized to the Fabry-Perot and positive branch, confocal unstable resonator cases. Both the Maxwell-Bloch and rate equation approximation forms of the theory were derived. The general theory is sufficiently broad to allow the modeling of the geometric optics mode of any standing wave resonator configuration. The theory was shown to reduce to the simple saturable gain and standing wave semiclassical models in the appropriate limits. The analysis of the injected lasers were performed with the new semiclassical theory. Both time-dependent and steady-state analyses were performed.