A theory for a gas laser with a Gaussian radial field distribution is considered. Rotating wave approximation is adopted to calculate the atomic polarization of the two−level system. An asymmetry of the shape of the Lamb dip and a shift of its center frequency are numerically obtained in the self−consistent laser field.
Theoretical studies of standing and traveling wave lasers are presented. The spectral behavior of the amplified spontaneous emission (superradiance) in a pencil-shaped column is investigated theoretically and experimentally. Spectral narrowing followed by rebroadening has been observed as predicted by the theory. A geometrical representation of an interacting two-level atomic system has been developed and applied to an optical laser. A semi-classical model for a standing wave laser has been examined using the rotating wave approximation, and compared to the exact solution. Depth and width of the Lamb dip are determined theoretically and experimentally. The effect of radial profile of the laser field has been studied using the self-consistent-field approach, and the radius of the beam was measured at various positions from the window.
In Chaps. 8 and 9, we considered a laser medium consisting of homo geneously broadened, stationary atoms. When the lasing atoms move as in a gas, they see an electric field with shifted frequency due to the Doppler effect, as shown in Fig. 10-1. One might argue that it is merely necessary to average the complex polarization or the susceptibility Xn over the frequency range corresponding to the velocity distribution, that is, calculate a new ^ n , for which Here the frequency co = coo + Kv, where coo is the frequency at atomic line (1) center, v is the component of velocity along the laser (z) axis, and the frequen cy distribution is determined by a Maxwell-Boltzmann velocity distribution W(y) = ( w)-1 exp [ - (v/w)2]. (2) Here u is the most probable speed of the atom, and K is the wave number. The simple recipe (1) is, in fact, valid for an inhomogeneously broadened mediumt consisting of stationary atoms such as ruby at low temperatures. However, in the standing-wave laser, the atoms not only see Doppler-shifted frequencies, but also move through the standing-wave electric field, effectively seeing an amplitude-modulated field. Equivalently, each atom sees two frequencies, for the standing-wave field is the sum of oppositely directed running waves, one of which appears to be Doppler upshifted, one down shifted. Hence it is necessary to consider both the frequency shift and the time dependence of the atomic z coordinate when calculating the polarization of a gaseous medium.
We show that in a gas laser the inhomogeneity in the distribution of amplifying atoms and the transverse Gaussian distribution of the saturating electromagnetic field give rise to different lens effects which are generally opposite to each other and influence the line shapes in different ways, shifting the Lamb dip either to the blue or to the red. For those laser phenomena which increase one or the other of the lens effects, the role of a diaphragm is of particular importance, since it determines the spatial distribution of the resonant field. We present here a new method for studying the resonant diffracted field in a laser. Also, we give a unified description of the various inhomogeneities. Doing away with the usual method of mean-field approximation, we numerically calculate, for a diaphragmed, monomode weak-gain gas laser, the intensity, variation of the beam diameter, and the radius of curvature as functions of the frequency.
The self-consistent field approximation (SCFA)[1]-[4] is used to calculate the steady-state mode energies and frequency shifts for a gas laser. The SCFA is a statistical approximation which was shown[3] to be accurate for small values of a parameter (\gamma^{2}N), which is always small for gas lasers. The SCFA is simpler and more convenient for studying the complicated dynamical effects that occur in gas lasers than the more recent[5]-[8] quantum theoretical models. It does not, of course, give any new description of the laser photon statistics, since the approximation treats the field as a pure coherent state[9] and is, therefore, equivalent to the semiclassical theory. The main results of this paper are to show how dynamical effects can be calculated nonperturbatively for arbitrary values of the field strength, and to analyze the different roles played by different kinds of inhomogeneous effects in gas lasers. These inhomogeneous effects are due to the dependence of the atomic inversion density on the atomic positions and velocities. Some numerical results are given which show that the nonperturbative theory can explain some effects not previously explained by perturbation theories. [1], [2].
We derive the equation of motion for the quantum-mechanical radiation density matrix of a gas laser to lowest order in the dimensionless coupling constant. Our derivation is fully quantum mechanical and we can calculate the coherence properties from the radiation density matrix. Our model consists of N two-level systems interacting with radiation in a cavity in the presence of dissipation, pumping, and collisions. The method we use is a generalization of the Bogoliubov derivation of the kinetic equation for a small parameter. Our derivation holds for any physically realizable pump power, and near threshold reduces to Lamb's near-threshold theory. With the equation of motion for the radiation density matrix, we obtain solutions both for when the average field is nonzero and for when it is zero. The steady-state electromagnetic density is the same in both cases except for a small spontaneous-emission term. We show that the reason gas lasers do not satisfy rate equations is the existence of zeroth-order correlations between the internal atomic variables and atomic center-of-mass variables. It is these same zeroth-order correlations which are responsible for the Lamb dip. Our derivation includes collisions and reduces the calculations of their effect to quadrature.
The Lamb theory of the optical maser is applied to circularly polarized atomic transitions, and used to consider the beat frequencies and the coherence properties of such orthogonal fields when axial magnetic fields are applied to the gaseous laser. The beat frequency approaches zero in near-zero magnetic fields and synchronization can then occur between the right- and left-handed circularly polarized oscillations. For a resonator with no undue polarization constraint, such a strong coupling gives rise to a linearly polarized output, and to a rotation of the plane of polarization with increasing magnetic field. A self-consistent expression is derived for this rotation under steady-state conditions, and a maximum rotation of $\ifmmode\pm\else\textpm\fi{}\frac{1}{4}\ensuremath{\pi}$ with magnetic field is indicated before the synchronization breaks down and circularly polarized beat phenomena appear. The rotation with magnetic field depends on the laser intensity, on the anisotropy in the cavity losses, and on the position of the cavity resonance within the Doppler linewidth. Also, the angle of rotation is indeterminate unless such anisotropy is present. Other regions of such coherence can occur at higher magnetic fields, where the beat frequency again approaches zero. These depend on the detailed shape of the various dispersion curves of the laser medium. The results derived from the theory used are in general agreement with experimental observations on the 1.153-\ensuremath{\mu} He-Ne laser transition.