The evolution of the first and second moments of the photon density distribution for a two-mode laser system was studied by proposing appropriate transition rate expressions and using the Kubo equations developed for a Chapman-Kolmogorov system. A numerical analysis technique was used in solving the rate equations and computer plots were generated for various combinations of laser parameter values. Anomalously large fluctuations, similar to those occurring in single-mode lasers, were observed during photon buildup of the modes. When only one mode was oscillating, a photon buildup in the non-lasing mode was seen to cause similar fluctuations in the lasing mode. In all cases, these fluctuations were accompanied by a strong linear relationship between the modes. (Author)
The evolution of the first and second moments of the photon density distribution for a two-mode laser system was studied by proposing appropriate transition rate expressions and using the Kubo equations developed for a Chapman-Kolmogorov system. A numerical analysis technique was used in solving the rate equations and computer plots were generated for various combinations of laser parameter values. Anomalously large fluctuations, similar to those occurring in single-mode lasers, were observed during photon buildup of the modes. When only one mode was oscillating, a photon buildup in the non-lasing mode was seen to cause similar fluctuations in the lasing mode. In all cases, these fluctuations were accompanied by a strong linear relationship between the modes. (Author)
A quantitative analysis is made of quantum fluctuations in the number of photons in the resonator of a single-mode laser considered in a quasi-linear approximation. An expression is obtained for a general four-level system, which is exact above the lasing threshold, but can also be used to calculate the fluctuations in the case of three- and two-level lasers. Above the threshold when certain relationships between the spontaneous decay constants are satisfied, the photon statistics are of the sub-Poisson type in four- and three-level systems. However, for other relationships between the decay constants, pumping rate, and average number of photons, fluctuations in the resonator obey both the Poisson and super-Poisson statistics with variances much greater than the average number of photons. Both Poisson and super-Poisson statistics with large values of the variance are found for a two-level system. The results of calculations indicate that, in general, there is no coherent Glauber state of the field in a laser resonator when the threshold is significantly exceeded.
Using a generalized version of the Volterra-Wiener functional technique, we study a model for a many-mode laser action in which the coupling among modes takes place through the intensities. It is shown that the coupled Langevin equations for the field amplitudes can be solved avoiding the infinities at the mode thresholds. In particular, mean photon numbers, variances, and correlation and cross-correlation functions of the modes can be computed. In order actually to carry out the computations, it is necessary to evaluate the solution of a set of coupled deterministic first-order differential equations and to diagonalize a matrix whose dimension is given by the number of modes envisaged. As an application, the two-mode case is treated in detail and, besides a derivation of the stationary properties, in part already known, the transient photon statistics, during the buildup of the two modes, is thoroughly studied. The advantage of this approach is due to its simplicity and to the possibility of dealing with a large number of interacting modes.
We derive the equation of motion for the field density matrix of the degenerate two-photon laser under conditions of two-photon resonance starting from the full microscopic Hamiltonian. Our results are compared with the corresponding quantities obtained from the standard effective Hamiltonian. As we have shown for the nondegenerate case, the full diagonal density-matrix equations tend to the effective Hamiltonian density-matrix equations in an appropriate limit, but the equations of motion for the off-diagonal elements do not coincide. The equations obtained using a more accurate form of the effective Hamiltonian, in which Stark shifts are included, do agree. In this paper we concentrate on the photon-number distribution and the nature of the phase transition that takes place in the neighborhood of the two-photon lasing transition threshold. We determine the steady-state mean photon numbers and consider the stability of the solutions. A solution is found where the mean photon number is zero, but this is found to be a stable solution only for sufficiently weak pumping, whereas in the standard effective Hamiltonian approach it is always stable. As the pumping strength is increased, a range of detunings is reached in which there are two stable solutions. The amplitude of the zero-photon peak diminishes very rapidly as the pumping strength increases. Finally, for sufficiently large detunings, a single, stable, steady-state solution is obtained. The nature of the phase transition is illustrated by presenting plots of the photon-number distribution against detuning and pumping rate. The photon-number fluctuations about the mean are also discussed.
Buildup of laser oscillation is expressed in terms of the probabilities p(n)(t) that there are n photons in the laser cavity. These probabilities obey a coupled set of difference-differential equations of motion. The method and results of numerical integration of these equations are given here for the initial condition that no photons exist (radiation field vacuum). Corresponding equations of motion for the moments of the photon distribution are derived, and their time dependences are given. A movie showing the time development of the p(n) from an initial vacuum is given on the corners of the November 1970 issue of Applied Optics and can be seen by flipping through the pages.
Single-mode cavity laser light statistics is considered within the framework of rate equations. According to this approach, fluctuations are caused by jumps in active and detecting atoms. The algebra is simple, allowing analytical expressions for the intracavity Fano factor and the photocurrent spectral density to be obtained. Poissonian, quiet, and optical pumps are considered. The results are verified by comparison with Monte Carlo simulations. An essentially exhaustive investigation of sub-Poissonian light generation by classical laser schemes, two-mode lasers, and semiconductor lasers is proposed.
The probability distribution p(n) for the detection of n photons emitted by a single-mode dye laser in a short time at various excitations is measured, together with the distribution of time intervals between pairs of detected photons. The first measurement yields the probability density p(I) and the moments of the light intensity I, and the second one the two-time intensity-correlation function. Measurements are also performed at three different wavelengths. p(I) is found to have a two-component structure, and the relative intensity fluctuations $\frac{〈{(\ensuremath{\Delta}I)}^{2}〉}{{〈I〉}^{2}}$ grow, apparently without limit, as the excitation tends to zero, so that no thermal state is reached. A possible explanation in terms of pumping fluctuations is discussed. The correlation function has the form of a sum of three exponential functions, and the amplitudes and decay constants of the three components are derived. The effective decay-time constant exhibits a thirteenth-power-law dependence on frequency of the laser light.