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Journal of the Optical Society of America

General Rate Equations for the Two-Level Pulsed Laser

T. T. Kikuchi

Published Aug 1, 1964• 1 citation
Publisher

Abstract

A new set of general rate equations has been derived by second quantization, which describe the operation of Q-switched lasers in terms of parameters of the system. The relative importance of induced emission, correlation, spatial cross relaxation, and interference on energy build-up are estimated near the initial time, when the active ions are predominantly in the upper state. At this period, only induced emission contributes significantly to the radiant energy growth. Because of the random location of the active ions in the laser material and because of the choice of nonlossy form of vector potential, the full effects of coherence and correlation are not manifest. Also, during this period, the spectral width for the most probable axial mode varies inversely as the square root of time. On the other hand, the linewidth due to mode selection varies inversely as the square of the time. Furthermore, for a laser with maximum inversion and with gain equal to 5×109 sec−1, we can consider all the active ions to be behaving identically only for times much less than 4 nsec.

DOI
10.1364/josa.54.000983

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A new set of general rate equations has been derived by second quantization, which describe the operation of Q-switched lasers in terms of parameters of the system. The relative importance of induced emission, correlation, spatial cross relaxation, and interference on energy build-up are estimated near the initial time, when the active ions are predominantly in the upper state. At this period, only induced emission contributes significantly to the radiant energy growth. Because of the random location of the active ions in the laser material and because of the choice of nonlossy form of vector potential, the full effects of coherence and correlation are not manifest. Also, during this period, the spectral width for the most probable axial mode varies inversely as the square root of time. On the other hand, the linewidth due to mode selection varies inversely as the square of the time. Furthermore, for a laser with maximum inversion and with gain equal to 5×109 sec−1, we can consider all the active ions to be behaving identically only for times much less than 4 nsec.
The conventional rate equations for a Q-switched laser are augmented to explicitly include the effects of time- and level-dependent pumping, thermalization among the sublevels in the upper and lower multiplets, and multiplet relaxation in a homogeneously broadened four-level laser medium. To make the numerical computations more generally valid, we introduce a number of dimensionless variables. We show that the initial set of five coupled differential equations can be reduced to a simple set of two coupled equations for the inversion density and photon flux. Via numerical modeling, we have investigated the manner in which both thermalization and lower multiplet relaxation affect Nd:YAG laser characteristics such as output energy and temporal waveform. Our numerical results confirm earlier predictions that the Q-switched Nd:YAG laser output energy increases monotonically by a factor of 3.33 as one progresses from the assumption of slow to rapid thermalization and by an additional factor of 1.46 if one further assumes a terminal multiplet relaxation which is fast relative to the resonator photon decay time. We also find that the laser pulsewidth is substantially broadened when the resonator photon decay time is comparable to the thermalization, and to a lesser extent, the terminal multiplet relaxation times.
We have undertaken the complete temporal description of pulsed emission by a homogeneously broadened laser, including the effects of spontaneous emission, the detailed geometry of the laser cavity, and the variation of atomic polarization and level populations over wavelength distances. The model is based on traveling-wave equations which are derived from Maxwell's equations and solved in conjunction with boundary conditions imposed at the cavity mirrors. Thus, any direct assumptions concerning the nature of the laser's longitudinal mode structure is avoided. Variations in polarization and population over wavelength distances are treated by means of expansions in spatial Fourier series, having as fundamental a half optical wavelength. The Fourier series are truncated after the first harmonic. The treatment differs from earlier work in that the dephasing of the dipole moment is treated exactly without a rate-equation approximation. Spontaneous emission is simulated both as to spectrum and Gaussian character by including in the dipole equations stochastic shot-noise sources. The model equations are solved numerically, and results include the details of QQQ-switched pulse evolution from noise for both passive and active switching. In the case of an actively switched laser, the two-photon fluorescence intensity pattern has been calculated. It reveals a well-defined structure of subsidiary intensity maxima, even though subcavities are not assumed in the calculation. The pattern can be correlated directly with the emission pulse structure, and should vary from shot to shot. No single point in the pattern is suitable for a peak-to-background ratio determination. However, if the background is averaged over a distance in the fluorescing medium equal to twice the separation between cavity mirrors, the peak-to-background ratio would be \ensuremath{\approx} 1.6, indicating a highly uncorrelated spectrum.
Traditionally, the active material in a laser is modelled as independent emitters, but in recent years it has become increasingly clear that radiative coupling between emitters can significantly change the characteristics of small lasers. Collective effects in free space such as superradiance have been studied extensively [1,2], but the effects of inter-emitter correlation in micro- and nano-cavities need further examination to be put on firm theoretical ground. Several studies of collective effects in nano-cavities have been made [3-6], but the theoretical models employed are intricate, and numerical methods are needed both to generate the dynamic equations and to solve them. We propose a model where the complexity is strongly reduced, allowing analytical solutions [7]. We consider a collection of identical two-level emitters interacting with a single cavity mode. We start from Maxwell-Bloch equations, but instead of making the typical adiabatic elimination of the polarization, we allow the polarization decay rate to be of the same magnitude or smaller than other decay rates. Hence, the traditional laser rate equations for the photon number and the population inversion must be supplemented by equations for the emitter-field correlation and the emitter-emitter correlation. This gives us four generalized laser rate equations, which we solve analytically in steady state. Comparing with the steady state results obtained from the traditional laser rate equations we see that inclusion of collective effects leads to a reduction of the photon number for small pump rates, similarly to what is found in [4]. From the generalized laser rate equations, we derive a measure of the strength of collective effects in terms of laser parameters: This describes the difference between results with and without inter-emitter correlations, and it goes smoothly to zero as we approach parameter values where the traditional laser rate equations become valid. To gain insight into the photon statistics of the laser, we construct dynamic equations for higher order correlations of operators. We derive an analytical expression for the zero-delay photon auto-correlation function, and for low pump rates we find that the interaction of emitters results in super-thermal values of the auto-correlation. This feature is observed in experiments and numerical models [4-5], and with our analytical expressions, we are able to pinpoint the parameter combinations for which the collective effects have the largest impact. Considering the same model in terms of the Fourier components of the operators, we find results for the photon number that agree well with the previous approach, while allowing computation of the linewidth. Thus, we can examine how emitter-emitter correlation affects the line broadening of the laser.
In this chapter, laser rate equations were used to derive another set of simpler rate equations that couple cavity photon number and the population of excited active atoms as the dynamical variables describing the laser. These equations were used to explore the buildup of laser oscillations from noise in a laser cavity and the remarkable growth of cavity photon number as the laser crosses threshold of oscillation. These modified rate equations were used to derive a Master equation for the cavity photon number probabilities. The steady-state solution of this equation was used to discuss equally remarkable transformation of photon statistics of laser light near threshold. The effect of spontaneous emission on setting the ultimate limit on laser linewidth was also discussed.
We present a general theory that connects the pump process in multilevel lasers with the statistics of the laser field. The key ingredient in our approach is the derivation of an effective master equation that involves only the two laser levels and that contains a non-Markovian pump term. This pump term gives rise to a narrowing of the photon-number distribution in steady state. The qualitative features of this dynamic noise reduction can be inferred from the eigenvalues of the pump term, which are known analytically. As the mechanism that is responsible for the noise reduction, we identify a correlated excitation process in which the effective excitation rate from the lower to the upper laser level depends on the photon number. This correlated excitation process is very different from an excitation at equidistant times, such as in lasers with periodic external injection. We stress that our approach is general and the treatment nonlinear because we do not resort to approximations such as an adiabatic elimination of atomic variables or the linearization of a Fokker-Planck equation. It is therefore particularly relevant for the description of systems where the active medium consists of a few atoms only and where the field losses are not negligible on the time scale of the atomic relaxations. This would be the case in an ion-trap laser. For such a situation our results are substantially different from those obtained by a linearized treatment. \textcopyright{} 1996 The American Physical Society.
The presence of an internal layer with an active medium in a hot radiation source is considered, which can be described by a quantum two-level system located near equilibrium. The population of the upper and lower levels is approximately equal. It is shown that during convection from deeper hot layers, which supports the inversion of the populations of the active system, generation of induced radiation pulses is possible, the intensity of which is comparable to or greater than the intensity of the background spontaneous radiation of the source. With a sufficient thickness of the surface layers due to the effects of radiation scattering in them, the emission spectrum of a completely black body may well form there. Pulse generation near a previously detected new threshold of induced radiation can lead to a periodic change in the radiation intensity of the source as a whole. This threshold is determined by the equality of the squared population inversion to the total number of states. The generation of pulses of induced radiation is considered both in Einstein's representation, on the basis of balanced equations, and using a semiclassical description for small values of population inversion and for low levels of electric field intensity, when the Rabi frequency is less than the line width. The description of the induced radiation process is reduced to a one-parameter system of equations. Periodic solutions are represented by closed trajectories on the phase plane (relative density of quanta, relative density of population inversion). A similar layer with an active medium, which can be described by a quantum two-level system located near equilibrium, can exist in stars and is most likely localized in the photosphere. If there is significant convection in the star’s atmosphere, conditions can be realized for generating pulses of induced radiation. It turns out that one can see the similarity of the obtained solutions with known observations of changes in the luminosity of Cepheid stars (Cepheus delta and the North Star).