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Journal of Applied Physics

Effects of Drift and Diffusion of Excited States on Spatial Hole Burning and Laser Oscillation

Hans Günter Danielmeyer

Published Jul 1, 1971• 69 citations

Abstract

The rate equation for the population inversion of homogeneously broadened 4-level lasers is solved for the cases of small and large mobility of excited states. Considered are drift and diffusion, i.e., collective and random motions along the laser axis. Both drift and diffusion smooth spatial holes which are burned into the population inversion by standing-wave fields. A drift results also in a drag effect, i.e., a phase lag between the population inversion and the standing-wave field. For homogeneously broadened lasers, sufficient smoothing of spatial holes will result in spontaneous full-power single-frequency operation. This is examined by calculating the unsaturated gains of secondary modes as a function of the excitation, drift, and diffusion parameters. Spatial hole burning is insignificant for gas lasers because of their large carrier diffusion. Furthermore, ionic drift in gas-ion lasers alone could effectively quench spatial hole burning. This is in contrast to dye lasers where drifts, i.e., longitudinal flow rates of the order of the sound velocity, would be required to obtain a single-frequency output. In solid-state lasers like ruby and Nd: YAG, energy diffusion between lattice sites is shown to significantly reduce spatial hole depths although the matrix elements for the exchange are quite small. An experiment is described which confirms the theory and allows the measurement of small diffusion constants of excited laser states. For the 4F3/2 level in Nd: YAG a value of 5×10−7 cm2/sec is obtained. This energy exchange among Nd3+ sites contributes 0.3 GHz to the homogeneous linewidth. Implications for axial mode selection and single-frequency oscillation of various lasers are discussed.

DOI: 10.1063/1.1660694

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The rate equation for the population inversion of homogeneously broadened 4-level lasers is solved for the cases of small and large mobility of excited states. Considered are drift and diffusion, i.e., collective and random motions along the laser axis. Both drift and diffusion smooth spatial holes which are burned into the population inversion by standing-wave fields. A drift results also in a drag effect, i.e., a phase lag between the population inversion and the standing-wave field. For homogeneously broadened lasers, sufficient smoothing of spatial holes will result in spontaneous full-power single-frequency operation. This is examined by calculating the unsaturated gains of secondary modes as a function of the excitation, drift, and diffusion parameters. Spatial hole burning is insignificant for gas lasers because of their large carrier diffusion. Furthermore, ionic drift in gas-ion lasers alone could effectively quench spatial hole burning. This is in contrast to dye lasers where drifts, i.e., longitudinal flow rates of the order of the sound velocity, would be required to obtain a single-frequency output. In solid-state lasers like ruby and Nd: YAG, energy diffusion between lattice sites is shown to significantly reduce spatial hole depths although the matrix elements for the exchange are quite small. An experiment is described which confirms the theory and allows the measurement of small diffusion constants of excited laser states. For the 4F3/2 level in Nd: YAG a value of 5×10−7 cm2/sec is obtained. This energy exchange among Nd3+ sites contributes 0.3 GHz to the homogeneous linewidth. Implications for axial mode selection and single-frequency oscillation of various lasers are discussed.

A theory is presented for the effect of cross relaxation (CR) on the saturation with driving-signal intensity of the steady-state gain of a laser amplifier with an inhomogeneously broadened linewidth. The condition of steady-state gain results when the amplifier excitation level W and the driving-signal intensity are constant for about 3τ or longer, where τ is the fluorescence lifetime of the upper laser level. The driving signal is derived from an oscillator operated in the normal mode and which does not saturate the amplifier inversion density over times short compared with τ. We assume that CR, at a rate F, restores to thermal equilibrium any distortion of the spectral inversion density caused by the driving signal. Formulas are derived for steady-state gain saturation with arbitrary F for two cases: the spectral width of the driving signal is (1) narrow and (2) broad relative to the ionic or homogeneous line. The ionic linewidth Δvi is assumed to be much less than the inhomogeneous linewidth Δva and refers to the minimum frequency interval for which infinitely fast CR is operative. The theory also yields the CR coefficients f=(Δvi/Δva) [F/(τ−1+W)] for the narrow-band case and f*=(Δvs/Δva) [F/(τ−1+W)] for the broadband case, where Δvs is the driving-signal bandwidth. These coefficients significantly parametrize the CR effect: for f, f*>1, hole-burning is not severe, whereas for f, f*<1, considerable hole-burning takes place and the gain saturates faster. Finally, the theory is applied to experimental results obtained for Nd3+:glass broadband and narrow-band amplification; an approximate lower bound of 4×105 sec−1 is determined for F.

A stochastic, semiclassical model is developed for a multimode, homogeneously broadened laser with rapid-dipole dephasing (e.g., semiconductor or dye lasers). The theory is valid for systems with a free spectral range that is large compared to the decay rates of the population inversion and the field. Population dynamics including temporal beating effects and relaxation oscillations, spatial-hole burning, coherent-wave mixing, and quantum noise are all included self consistently. A principal restriction of the model, when applied to coupled-cavity systems, is that the modes satisfying the boundary conditions of the gain medium must form a subset of the modes defined by the cavity and that only modes in this subset are above threshold. For typical systems, the model is valid for pump rates up to several times threshold. The effects on the emission linewidth of coherent-wave mixing and the number of modes included in the simulation at various pump rates are discussed.

We report the observation of a spectral feature in the gain and absorption profiles of a multilevel system. The gain experienced by the sidebands of a weakly modulated laser field Is measured as a function of modulation frequency. For small modulation frequencies the sidebands experience diminished absorption or gain depending on the time-averaged inversion of the system. The width of the spectral feature depends on spontaneous as well as stimulated population transition rates. This paper Is an extension of single-field modulation experiments In collisionally broadened media.1,2 Two lasers are used in our experiment. One laser is used as an incoherent pump field that provides population for the upper level of the other optical transition. The second laser stimulates this population to the lower or ground level depending on the system configuration. In the experiment we modulate the second laser field. Using alexandrite as an active medium we are able to study both three- and four-level atomic systems.3 The stability of a single-mode laser to the growth of neighboring modes is related to the gain experienced by the sidebands. The competition between modes is strongest when their frequency separation is less than the width of this spectral feature.

This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.

Homogeneously broadened bidirectional ring lasers show alternation of the lasing direction induced dynamically by a long-lived population grating and slight detuning of the laser cavity from the atomic resonance1. We show that the mode switching is accompanied by abrupt jumps by pi in the total relative phase of the system2. Our analysis reveals that the carrier frequencies of the two modes are not equal even though the two unidirectional steady state solutions have the same frequency2. When one mode dominates and the other is suppressed, the dominant mode has the steady state frequency. However, a mode growing in strength has a frequency closer to the atomic resonance while a mode falling in strength has a frequency further from the atomic resonance. The dynamically induced mode-pulling and mode-pushing can be as much as 25 % of the steady state frequency detuning. The shifting frequencies of the modes indicate that the standing wave interference pattern rotates in the laser ring cavity, alternately moving clockwise and counterclockwise with nearly equal and constant speeds. The longitudinal spatial hole-burning pattern consequently also rotates in the ring tending to wash out the population grating and keeping it from developing an amplitude of much more than one percent of the spatially averaged population inversion.

We have studied microscopic effects leading to mode competition in an inhomogeneously broadened medium of a gas laser with a weakly anisotropic cavity. The polarization bistability observed in some lasers, which is believed to be caused by hole burning, we attribute to microscopic coherent effects induced by two light fields. Due to these effects one field can be coupled with another. Hole burning is found to be important, but not crucial. The exact solution of the equation of motion for field amplitudes is compared with the third order approximation of perturbation theory. The role of four wave mixing, important for bistability of axial modes in a homogeneously broadened medium, is discussed.