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IEEE Journal of Quantum Electronics

Effects of thermalization on Q-switched laser properties

John J. Degnan, D. Barry Coyle, Richard B. Kay

Published May 1, 1998• 72 citations
Publisher

Abstract

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.

DOI
10.1109/3.668777

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Modeling the Dynamic Emission of a Polarized Cr4+:YAG Crystal Investigated by Double-Pulse Pumping Generated by a Nd-Laser

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A mathematical model describing the dynamic emission of intracavity polarized isotropic Cr 4+ :YAG solid--state saturable absorber was developed.This model considers double pumping laser pulses to simulate the actions of Cr 4+ :YAG as a dual Q-switched crystal (1.06 µm) and lasing medium (1.4 µm).The model describes the time evolution of interaction between the pumping laser pulse, partial polarizer and the polarized saturable absorber.The analysis of the polarization process is based on the assumption that at each moment of lasing evolution, the state of polarization represented by an eigenstate corresponds to the lowest radiation losses state.The model offers a simple mechanism for studying the kinetics of the pulsed lasers and the influence of the variations of the pumping laser power and nonlinear anisotropy parameter on the characteristics of the output laser pulses 1.06 µm and 1.4 µm.The angular rotation of the passive switch Cr 4+ :YAG reveals that the transmission of the polarized 1.06 µm laser radiation is strongly anisotropic in the saturation regime.The suggested model estimates the transmission of pumping laser density of the polarized light 1.06 µm radiation as a function of nonlinear anisotropy parameter, temporal behavior of the relative population inversion of Nd-laser, population inversion of the different polarized levels of Cr 4+ :YAG pulsed laser and the output laser pulse densities under impact of different values of the nonlinear anisotropy which is due to the self-induced anisotropy of its saturated absorption.

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Instabilities in a coherently pumped laser

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We solved in steady state in the rotating wave limit a five-level model which is fully coherent on the coupled pump–laser transition and includes relaxation of diagonal elements through, two rotational reservoirs connected by vibrational relaxation. The equations were solved in steady state for χ P ″ and χ S ″, the imaginary susceptibilities on the pump and laser transitions. These results were velocity integrated to generate susceptibilities for the inhomogeneously broadened system with arbitrary field strengths and relaxation rates.1 The effects of pump absorption were included by numerically integrating the spatial differential equation for the pump intensity down the laser axis.

•Apr 26, 1987•1 citation

General Rate Equations for the Two-Level Pulsed Laser

T. T. Kikuchi

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.

Journal of the Optical Society of America•Aug 1, 1964•1 citation
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.
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.
A mathematical model describing the dynamic emission of intracavity polarized isotropic Cr 4+ :YAG solid--state saturable absorber was developed.This model considers double pumping laser pulses to simulate the actions of Cr 4+ :YAG as a dual Q-switched crystal (1.06 µm) and lasing medium (1.4 µm).The model describes the time evolution of interaction between the pumping laser pulse, partial polarizer and the polarized saturable absorber.The analysis of the polarization process is based on the assumption that at each moment of lasing evolution, the state of polarization represented by an eigenstate corresponds to the lowest radiation losses state.The model offers a simple mechanism for studying the kinetics of the pulsed lasers and the influence of the variations of the pumping laser power and nonlinear anisotropy parameter on the characteristics of the output laser pulses 1.06 µm and 1.4 µm.The angular rotation of the passive switch Cr 4+ :YAG reveals that the transmission of the polarized 1.06 µm laser radiation is strongly anisotropic in the saturation regime.The suggested model estimates the transmission of pumping laser density of the polarized light 1.06 µm radiation as a function of nonlinear anisotropy parameter, temporal behavior of the relative population inversion of Nd-laser, population inversion of the different polarized levels of Cr 4+ :YAG pulsed laser and the output laser pulse densities under impact of different values of the nonlinear anisotropy which is due to the self-induced anisotropy of its saturated absorption.
The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.
We solved in steady state in the rotating wave limit a five-level model which is fully coherent on the coupled pump–laser transition and includes relaxation of diagonal elements through, two rotational reservoirs connected by vibrational relaxation. The equations were solved in steady state for χ P ″ and χ S ″, the imaginary susceptibilities on the pump and laser transitions. These results were velocity integrated to generate susceptibilities for the inhomogeneously broadened system with arbitrary field strengths and relaxation rates.1 The effects of pump absorption were included by numerically integrating the spatial differential equation for the pump intensity down the laser axis.
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.