Quantenmechanische Behandlung des optischen Masers.

In the present paper we give a fully quantummechanical treatment of the self-sustained oscillation of one mode in solid-state lasers. The total laser system consists of various subsystems: The lasing mode is coupled to the atoms of the active material and to a loss mechanism. It is assumed to be in complete resonance with the homogeneously broadened atomic transition. The pump of the active atoms, which are assumed to have only two levels, is brought about by their interaction with a large system of negative temperature. The active atoms decay not only by induced and spontaneous emission into the lasing mode, but also by spontaneous emission into the continuum of nonlasing modes (and possibly by nonradiative transitions). This process is fully taken into account. The pumping process and the spontaneous emission into the continuum of nonlasing modes are treated as in a preceding paper. There we have shown that the coordinates of these fields can be eliminated in some sense and give rise to a mean dissipative motion of the atoms and to fluctuations. Using the Heisenberg picture we obtain a system of coupled nonlinear equations of motion for the atomic operators and for the creation operator of the oscillating mode. We then eliminate the atomic operators by the iteration procedure of the semiclassical laser theory. This leaves us with a nonlinear differential equation of the van-der-Pol type for the creation operator of the laser mode, which contains the fluctuations of the pumping process, the spontaneous emission into the continuum and the loss mechanism as inhomogeneities of operator character. Such an operator equation has previously been obtained by Haken, who has shown, that in the neighbourhood of the stationary saturated level of oscillation the amplitude is highly stabilized, whereas the phase undergoes an undamped diffusion process. This process takes the phase in the course of time arbitrarily far from any given initial value. We use Haken's method of solution and demonstrate that the correct commutation rules for the oscillating mode [ b( t), b( t)]=1 are preserved for all times. Besides these quantum mechanical properties our solution contains all the well known results of the semiclassical theory. Our main result is the expression for the linewidth, which is caused by phase diffusion. The half width at half maximum power is in circular frequencies given by κ is the half width of the cavity, Γ the half width of the atomic transition (we have assumed κ ≪ Γ), σ the critical inversion per atom, P the energy radiated per sec and $$n_{TH} = \left[ {e^{\frac{{\hbar \omega }}{{kT}}} - 1} \right]^{ - 1}$$ the number of thermal noise quanta. We prove that $$\frac{1}{{2\sigma _k }} = n_{SP}$$ , where n is the number of spontaneous noise quanta in the mode, n is determined in complete analogy to n+1/2. It arises from the spontaneous emission of the active atoms into the laser mode and is for optical frequencies much more important than n+1/2, which stems from the finite temperature of the cavity walls and the vacuum fluctuations of the cavity. Both noise sources, however, enter our formula entirely symmetrical. The two terms containing the cavity fluctuations represent the old Townes formula corrected by a factor 1/2. The spontaneous noise term on the other hand agrees in the limit κ≪ Γ with Haken's expression, if we account for the different description of the pumping process. Before we study the nonlinear oscillation of the mode above threshold, we investigate its behaviour in the linear or amplification region below threshold. The laser line is shown to grow out of a broad background of spontaneous emission noise. [ABSTRACT FROM AUTHOR]