The interaction of N atoms each with 3 levels at random lattice sites with a set of cavity modes is considered. The optical transition between the lowest two atomic levels is taken into account explicitely assuming a Lorentzian line shape, whereas the third level just serves for the pumping process. If homogeneous inversion of the atoms is assumed, only one coherent mode oscillates in the steady state. It is the one being closest to the atomic resonance and having highest r If, however, in the next approximation a mode-dependent depletion of the excited atomic states is taken into account, with increasing pumping rate several modes may oscillate simultaneously. The behaviour of two such modes is treated in detail and it is shown, that one obtains a stable configuration. Using a higher approximation the nonlinear interaction between these two modes brought about by the amplifying material is studied in detail. As a special result one obtains a repulsion of the frequencies of the modes as a function of pumping power in accordance with gaslaser experiments. Quantum noise effects are neglected throughout the present paper.

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Spatially homogeneous solutions of the Landau-Lifshitz-Gilbert equation are analysed. The conservative as well as the dissipative case is considered explicitly. For the linearly polarized driven Hamiltonian system we apply canonical perturbation theory to uncover the main resonances as well as the global phase space structure. In the case of circularly polarized driven dissipative motion we present the complete bifurcation diagram including bifurcations up to codimension three.

We start from quantum mechanical laser equations which were derived in a previous paper for an inhomogeneously broadened laser and which contain in particular the noise sources due to cavity losses, vacuum fluctuations, interaction with phonons and nonlasing photons and the pump. For the example of frequency locking caused by the nonlinear polarization we derive a quantum mechanical Langevin equation for the relative phase angle ~,= 2q/2-q/i--N3, where q/l, ~,//2, 1[I3 are the total phases of three axial modes which would be equally spaced in the unloaded cavity. In the resulting equation = 6 --t7 sin ~ +f (t) (1) the fluctuating forcef(t) is Markoffian and Gaussian, the second moment being given by d (t --t') 2 (F 1 + F 3 + 4/'2), where Fj is the linewidth of the individual unlocked mode, j.Eq. (1) is solved by Fokker-Planck techniques and numerical results are represented for the characteristic function of order 1. Furthermore the mean locking time is represented in a graphical plot. The results are also applicable to two modes which are coupled to each other, e.g. by loss modulation. In this case the second moment off(t) is given by 8(t

We present a quantum mechanical nonlinear treatment of the phase and amplitude flucutations of gas lasers, i.e. lasers with moving atoms, and of solid state lasers with an inhomogeneously broadened line. The atoms may possess an arbitrary number of levels. As in our preceding papers the noise due to the pump, incoherent decay, lattice vibrations or atomic collisions, as well as due to the thermal and zero point fluctuations of the cavity is completely taken into account. The linewidth (due to phase diffusion), and the intensity fluctuations (due to amplitude noise) are essentially expressed by the threshold inversion, the unsaturated inversion and the saturated population numbers of the two atomic levels, which support the laser modes. Our results apply to the whole threshold region and above up to essentially the same photon number, to which the previous semiclassical theories of inhomogeneously broadened lasers were applicable. For the example of a two-level system we also demonstrate the application of a new technique which allows us to eliminate rigorously the atomic variables (operators), yielding a set of nonlinear coupled equations for the lightfield operators alone. If the elimination procedure is carried out only partially and additional approximations are made, we find essentially the rate equations of McCtJMB~R, in a form derived by LAx. When we neglect noise, the nonlinear equation may be solved exactly in the case of single mode operation. By a suitable expansion of the exact multimode equations we find a convenient set of equations, which reduce in the noiseless case to those derived and used previously by HAKEN and SAU~RMANN as well as LAMB.