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Chin. Phys.; (United States)

Rate equation theory of two-photon lasers

Fan-Yi Lin

Published Apr 1, 1983• 0 citations
OpenAlex

Abstract

Two-photon lasers and laser amplifiers are studied by using the rate equations. The steady solution is obtained and its stability is analyzed. The time evolution of large deviation from the steady solution is studied by using computer simulation. The population inversion and the trigger input must exceed certain thresholds to operate a two-photon laser. The method for producing giant pulse is discussed. The possibility of building a practical two-photon laser is analyzed based on the known data. A Doppler-free two-photon laser needs much lower threshold but much more serious frequency matching among the atom system, cavity, and the trigger input. Such a Doppler-free two-photon laser could find applications in laser spectroscopy and in the accurate metrology. Two-photon laser amplifiers distort the input waveform seriously. This distortion depends strongly on the input intensity.

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Rate equation theory of two-photon lasers

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Two-photon lasers and laser amplifiers are studied by using the rate equations. The steady solution is obtained and its stability is analysed. The time evolution of large deviation from the steady solution is studied by using computer simulation. The population inversion and the trigger input must exceed certain thresholds to operate a two-photon laser. The method for producing giant-pulse is discussed. The possibility of building a pratical two-photon laser is analysed basing on the known data. A Doppler-i'ree two-photon laser needs much lower threshold but much more serious frequency matching among the atom system, cavity and the trigger input. Such a Doppler-free two-photon laser could find applications in laser spectroscopy and in the accurate metrology. Two-photon laser amplifiers distort the input waveform seriously. This distortion depends strongly on the input intensity.

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Doppler-free population inversion induced by coherent two-photon resonance is calculated analytically for a two-photon analog of the hyperbolic-secant pulse and numerically for a Gaussian, linearly chirped pulse. The numerical calculations are carried out for a two-photon resonance of a chirped ruby laser line with the $6{S}_{\frac{1}{2}}\ensuremath{\rightarrow}9{D}_{\frac{3}{2}}$ transition in Cs. Complete inversion is found when the fractional change in frequency during the pulse lies between 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$. The coherent two-photon resonances are described by optical Bloch equations which are derived from a multiple-time-scale perturbation theory.

Physical Review A•Aug 1, 1976•14 citations
Two-photon lasers and laser amplifiers are studied by using the rate equations. The steady solution is obtained and its stability is analyzed. The time evolution of large deviation from the steady solution is studied by using computer simulation. The population inversion and the trigger input must exceed certain thresholds to operate a two-photon laser. The method for producing giant pulse is discussed. The possibility of building a practical two-photon laser is analyzed based on the known data. A Doppler-free two-photon laser needs much lower threshold but much more serious frequency matching among the atom system, cavity, and the trigger input. Such a Doppler-free two-photon laser could find applications in laser spectroscopy and in the accurate metrology. Two-photon laser amplifiers distort the input waveform seriously. This distortion depends strongly on the input intensity.
Two-photon lasers and laser amplifiers are studied by using the rate equations. The steady solution is obtained and its stability is analysed. The time evolution of large deviation from the steady solution is studied by using computer simulation. The population inversion and the trigger input must exceed certain thresholds to operate a two-photon laser. The method for producing giant-pulse is discussed. The possibility of building a pratical two-photon laser is analysed basing on the known data. A Doppler-i'ree two-photon laser needs much lower threshold but much more serious frequency matching among the atom system, cavity and the trigger input. Such a Doppler-free two-photon laser could find applications in laser spectroscopy and in the accurate metrology. Two-photon laser amplifiers distort the input waveform seriously. This distortion depends strongly on the input intensity.
In a recent article1to which we shall refer as [S&B], a two-photon laser was proposed and analyzed. This device consists of a cavity resonant at frequency vAand containing ions of type B with an inverted population NB/V between levels separated by an energy difference hvBsuch that vB= 2vA; it is necessary that the system not lase at vB, which criterion can be satisfied by low reflectivity of the cavity at frequency vB, by strong parasitic absorption in the laser material near frequency vB, or preferably by a choice of ion such that the transition vBis highly forbidden to a single-quantum process. The authors of [S&B] thus show that a certain priming density of photons of frequency vAwill provoke the simultaneous emission from the inverted population NBof pairs of photons vAat a rate exceeding the cavity loss,t he process diverging until the population inversion is eliminated.
We discuss the propagation of an electromagnetic pulse through an active medium prepared in a state of inversion between two levels of the same parity. Since no electric dipole transition is possible between the chosen atomic levels, we investigate the possibility of amplification of an injected signal having a carrier frequency equal to one-half the atomic-transition frequency. We show that under suitable conditions a nonlinear atomic polarization can be generated which oscillates at the same frequency as the incident electromagnetic pulse. The coupled atom-field evolution is described by the usual self-consistent approach. When atomic relaxation effects are negligible, we derive an equation describing the spatial evolution of the energy of the propagating pulse. From this equation we characterize the threshold condition for power amplification and classify the multiple steady-state solutions of the propagation problem. The evolution of the pulse envelope through the amplifier is analyzed with the help of a hybrid computer simulation. Pulse-envelope modulation and multiple-pulse formation even in the asymptotic limit of long amplifiers are displayed.
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.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
Doppler-free population inversion induced by coherent two-photon resonance is calculated analytically for a two-photon analog of the hyperbolic-secant pulse and numerically for a Gaussian, linearly chirped pulse. The numerical calculations are carried out for a two-photon resonance of a chirped ruby laser line with the $6{S}_{\frac{1}{2}}\ensuremath{\rightarrow}9{D}_{\frac{3}{2}}$ transition in Cs. Complete inversion is found when the fractional change in frequency during the pulse lies between 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$. The coherent two-photon resonances are described by optical Bloch equations which are derived from a multiple-time-scale perturbation theory.