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Acta Physica Sinica

LANGEVINIAN THEORY OF TWO-PHOTON LASERS BY OPTICAL PUMPING

Xiangming Hu, Wang Dexin, 华中师范大学物理系,武汉,430070

Published Jan 1, 1992• 0 citations
Publisher

Abstract

In this paper, a two-photon laser by optical pumping, with two-photon losses, is studied by means of the Langevinian theory, the langevinian equation is obtained. Then the steady-state average photon number, the frequency pulling, the linewidth and the dynamical Stark effect are obtained. From the analysis of the gain, it is shown that, for the incoherent pumping, only the two-step process of single-and two-photon gives contribution. And for the coherent pumping, in addition, the one-step process of single and two-photon also contributes. And two processes are confined by the negative-feedback and saturation. The linewidth is independent of the steady state average photon number.

DOI
10.7498/aps.41.437

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In this paper, a two-photon laser by optical pumping, with two-photon losses, is studied by means of the Langevinian theory, the langevinian equation is obtained. Then the steady-state average photon number, the frequency pulling, the linewidth and the dynamical Stark effect are obtained. From the analysis of the gain, it is shown that, for the incoherent pumping, only the two-step process of single-and two-photon gives contribution. And for the coherent pumping, in addition, the one-step process of single and two-photon also contributes. And two processes are confined by the negative-feedback and saturation. The linewidth is independent of the steady state average photon number.
The Langevin quantum theory has been used to treat the optically pumped three-level laser system. We derive an equation of motion for laser field in the type of Van der Pol equation when the photon number of the laser field is not too high. It includes not only both incoherent and coherent pumping, but also both strong and weak optical pumping. If an average over reservoir is taken, the equation will give all the semi-classical results. Our results show that the laser gain comes from two basic processes, i.e., a two-photon process and a two-step process. When the pumping strength is larger than some value, the dynamic Stark effect will appear. The laser linewidth is mainly contributed by the thermal noise, the vaecum fluctuation, the spontaneous emissions between the laser levels and the spontaneous Raman scattering.
The Langevin quantum theory has been used to treat the optically pumped three-level laser system. We derive an equation of motion for laser field in a type of Van der Pol equation when the photon number of the laser field is not too high. It includes not only both incoherent and coherent pumping, but also both strong and weak optical pumping. If an average over the reservoir is taken, the equation will give all the semi-classical results. Our results show that the laser gain comes from two basic processes, i.e., a two-photon process and a two-step process. When the pumping strength is larger than a certain value, the dynamic Stark effect will appear. The laser linewidth is mainly contributed by the thermal noise, the vacuum fluctuation, the spontaneous emissions between the laser levels and the spontaneous Raman scattering.
A quantum theory of a two-photon laser is developed by making use of the antinormal-ordering Q function. Starting from the microscopic atom-field interaction Hamiltonian for cascade three-level atoms and including saturable absorbers, we first present a master equation for the field-density operator, and then transform the master equation into a Fokker-Planck equation for the Q function. The Q-function approach enables us to study the two-photon laser analytically, obtaining simple expressions for nonlinear gain, mean photon number, frequency pulling, natural linewidth, and photon-number variance in a unified method. We find that the field in a two-photon laser will build up from a vacuum without triggering if its linear gain is larger than the cavity loss. Also hysteresis can occur in the two-photon laser even without triggering. With an overall two-photon resonance, the normalized photon-number variance approaches a common value 11/8 well above ``threshold,'' independent of the one-photon detuning. We compare these results with previous results obtained from an effective interaction Hamiltonian. We find that the domain of validity of the effective Hamiltonian for the photon-number variance is much smaller than that for the mean photon number. Well above ``threshold'' the effective Hamiltonian overestimates both the natural linewidth and the photon-number variance.
Starting from the microscopic Hamiltonian of the three-level model and making use of the slowly varying amplitude approximation and the two-photon rotating wave approximation, we derive an effective Hamiltonian for the two-level two photon model under conditions where the two-photon process dominates the other radiartive processes, so that an exact relation between the three-level model and the two-level two-photon model is established. The effective Hamiltonian includes a term come from the optical Stark effect which was neglected in the conventional treatment. The coefficient of this term may have the same order of magnitude as the effective coupling constant. Based on this effective Hamiltonian, the standard method of Haken's laser theory is applied to study the threshold, the mean photon number, the self-induced frequency shift, the photon statistics and the fluctuation in steady states. Results are compared with previous ones which neglect the Strk effect.
We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit. As a by-product we also study the ordinary two-photon laser and find, e.g., photon-number variance and laser linewidth.