This paper describes a model and provides the mathematical formulation for describing the spike pulsing phenomenon observed in optically pumped three-level lasers. The model is based on the pumping of electrons from a ground state to an excited pump band from which they relax very rapidly to the excited laser level. With population inversion then achieved, laser action starts and repetitive pulses with decreasing amplitude are generated; the pulses then damp out and the laser output reaches a steady state as long as sufficient pump radiation energy is available. The equations describing the system in terms of the population inversion and photon density are nonlinear. However, approximations are made which permit the required relations between the transition rates to be established for such action to progress. The time variation of the population inversion and the photon density at the laser frequency are obtained for both the pumping time interval and the stimulated emission laser region.
This paper describes a model and provides the mathematical formulation for describing the spike pulsing phenomenon observed in optically pumped three-level lasers. The model is based on the pumping of electrons from a ground state to an excited pump band from which they relax very rapidly to the excited laser level. With population inversion then achieved, laser action starts and repetitive pulses with decreasing amplitude are generated; the pulses then damp out and the laser output reaches a steady state as long as sufficient pump radiation energy is available. The equations describing the system in terms of the population inversion and photon density are nonlinear. However, approximations are made which permit the required relations between the transition rates to be established for such action to progress. The time variation of the population inversion and the photon density at the laser frequency are obtained for both the pumping time interval and the stimulated emission laser region.
The discrete nature of active media atoms, laser light photons and their Poisson distribution law allow us to apply correctly Markov processes theory with discrete states and continuous time to the description of laser generation process. The equations of photons quantity in laser resonator and population inversion are deduced. The results coincide with known equations. Equations of photons quantity dispersion and population inversion dispersion are deduced also. The analysis of these equations permits us to explain the operation of spiking pulsed multimode lasers.
Introduction We were able to show in Chapter 3 that a medium in which we can obtain a population inversion (i.e. a situation in which the population density in the excited state is greater than that in the fundamental level) allows for optical gain of an electromagnetic wave having a frequency near to the resonant frequency of the system. By introducing feedback of the amplified signal into the medium, the system can be made to oscillate naturally, resulting in laser oscillations . To obtain this population inversion, we must introduce at least a third (and perhaps even a fourth) energy level into the system. (We saw how a two-level system under the influence of an intense pump beam will saturate with no resulting population inversion.) The aim then of this chapter is to introduce the concepts necessary to extend our two-level system into a working model capable of illustrating the phenomenon of laser oscillation. We will not spend too much time discussing atomic transition lasers as they do not figure readily in our treatment of quantum electronic properties of semiconductors. An exception will be made, however; we brush upon the particular topics of a diode pumped laser in Complement 4.E and a quantum cascade laser in Complement 13.H. Population inversion and optical amplification Population inversion We will show how population inversion can be achieved by carrier transfer from higher lying levels to the upper level of a two-level subsystem of interest.
Solutions of the rate equations for a two-level system describing spectral properties in the presence of stimulated emission under picosecond-pulse pumping conditions are examined. The materials of interest are broad band tunable, vibronic laser crystals whose lasing properties under normal pumping conditions have been related to spectral properties through the McCumber theory1. The population inversion at threshold can then be determined by three methods: lifetime shortening, band narrowing, and peak gain measurements.
The nonlinear theory of stimulated Raman scattering of high-energy long laser pulses in two-level media is constructed and provides a unified global interpretation of the experiments of Dr\"uhl, Wenzel, and Carlsten [Phys. Rev. Lett. 51, 1171 (1983)]. It is proved that the model, resulting from the slowly varying envelope approximation and including phase mismatch and detuning from Raman resonance, is integrable by the spectral transform method for arbitrary boundary values. The essential variable in the theory is the phase of the Stokes pulse relative to the pump. For long duration input pump pulses and when pump and Stokes pulses experience a rapid \ensuremath{\pi} phase shift, an anomalous spike of pump radiation is created in the pump depletion zone. It is shown in particular that the decay of Raman spikes is related to the velocity of this \ensuremath{\pi} phase shift. For fast flips, the possibility of observing spike overflow is established and, with multiple phase flips, a multiple-spike phenomenon is produced, suggesting new experiments. Lastly, the number of optical phonons participating to the interaction is explicitly calculated.
POPULATION RATE EQUATIONS From the discussion in the preceding chapter, it is clear that population inversion is the basic condition for an optical gain. For any system in its normal state in thermal equilibrium, a low-energy level is always more populated than a high-energy level, hence there is no population inversion. Population inversion in a system can only be accomplished through a process called pumping by actively exciting the atoms in a low-energy level to a high-energy level. If left alone, the atoms in a system relax to thermal equilibrium. Therefore, population inversion is a nonequilibrium state that cannot be sustained without active pumping. To keep a constant optical gain, continuous pumping is required to maintain population inversion. This condition is clearly consistent with the law of conservation of energy: amplification of an optical wave leads to an increase in optical energy, which is possible only if the required energy is supplied by a source. Pumping is the process that supplies energy to the gain medium for the amplification of an optical wave. There are many different pumping techniques, including optical excitation, electric current injection, electric discharge, chemical reaction, and excitation with particle beams. The use of a specific pumping technique depends on the properties of the gain medium being pumped. The lasers and optical amplifiers of particular interest in photonic systems are made of either dielectric solid-state media doped with active ions, such as Nd:YAG and Er:glass fiber, or direct-gap semiconductors, such as GaAs and InP. For a dielectric gain medium, the most commonly used pumping technique is optical pumping using either an incoherent light source, such as a flashlamp or a light-emitting diode, or a coherent light source from another laser. A semiconductor gain medium can also be optically pumped, but it is usually pumped by electric current injection. In this section, we consider the general conditions for pumping to achieve population inversion. Detailed pumping mechanisms and physical setups are not addressed here because they depend on the specific gain medium used in a particular application. The net rate of increase of the population density in a given energy level is described by a rate equation. As we shall see below, pumping for population inversion in any practical gain medium always requires the participation of more than two energy levels.
A three level model for optically pumped alkali metal vapor lasers is developed by considering the steady state rate equations for the longitudinally averaged number densities of the ground 2 S1/2 and first excited 2 P3/2, and 2 P1/2 states. For efficient operation, the collisional relaxation between the two upper levels should be fast relative to stimulated emission. By assuming a statistical distribution between the upper two levels, the limiting analytic solution for the quasi-two level system is achieved. A second limiting solution is identified for strongly bleached conditions where the atom recycle rate, limited by spin-orbit relaxation, fully specifies the output power. Performance in the intermediate regime depends significantly on pump laser bandwidth and requires numerical simulation. The ratio of populations for the two excited, 2 P3/2,1/2 states completes an analytic solution and depends primarily on pump laser bandwidth, threshold and alkali concentration.