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Journal of Applied Physics

Analytical Line Shapes for Lorentzian Signals Broadened by Modulation

Rolf Arndt

Published Aug 1, 1965• 271 citations
Publisher

Abstract

General analytical expressions are given for the line shapes of Lorentzian signals broadened by modulation. The results are discussed for the cases of the first and second harmonic. The linewidths and the signal intensities have been calculated analytically as a function of the modulation amplitude.
DOI
10.1063/1.1714523

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According to the standard Karplus-Schwinger theory of saturation broadening, saturated line shapes are Lorentzians with linewidths that increase linearly with the perturbing field strength. This, however, is not what is observed experimentally when the saturating field is inhomogeneous. If the saturating field strength varies significantly over the experimental signal volume, we find that the saturated line shapes are strikingly non-Lorentzian. The ``sharpness'' of the experimental line shapes is quantified by an ``effective linewidth,'' which is the half width at half maximum of a Lorentzian that approximates the experimental line shape near line center. For certain classes of inhomogeneous fields, we find that this effective linewidth increases approximately as the square root of the saturating field strength, rather than linearly. We show that this class of inhomogeneous fields is distinguished by the presence of a node in the field geometry, and that the effect arises because the line shape near line center is dominated by the atomic population in the vicinity of the node. These results indicate the importance of understanding and accounting for inhomogeneous field effects when extracting physical information from experimental line shapes.
It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δ ν = 1 + I ν 1 / I s Δ ν H for condition 2). Here, I v 1 and Is are the intensity of the large signal and the saturation intensity at the center frequency Vo, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.
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A general theory of line broadening is used to study the effects of collisions on absorption line shapes when self-broadening is important. Special attention is focused on the role of processes in which an excited atom transfers its electronic excitation to an unexcited atom; it is this process which distinguishes self-broadening from foreign-gas broadening. Several forms for the absorption rate are presented corresponding to particular assumptions about the amplitudes associated with this phenomenon. For example, one approximation leads to a generalized Lorentz-Lorenz law in which the one-atom polarizability is replaced by one that accounts for line broadening. Finally, the importance of many-body effects in the calculation of the linewidth is briefly discussed to determine when simple impact theories are applicable.
A theory of line shape for a Doppler broadened probe in the presence of a strong infrared pump is presented. The optical analog of Bloch equations has been developed; the T1, T2 relaxations are introduced phenomenologically in these equations. Doppler broadening of both the pump and signal transitions are taken into account to obtain the absorption coefficient for the signal. This leads to a Doppler broadened Gaussian function weighted by a pump induced Lorentzian function whose position and linewidth are controlled by the detuning and power respectively of the pump radiation. The absorption coefficient is further modulated by a Lorentzian function containing the fourth power of the radiation. The off-resonance pumping leads to two symmetrically located ‘‘Lamb dips’’ or ‘‘peaks’’ around the Gaussian pedestal for co- and counter propagating laser radiations. A calculation of the line shape of the central sub-Doppler feature for the on-resonance pumping is in good agreement with the observed variation of linewidth with pressure. Numerical results are presented graphically to demonstrate the pump and signal power dependence of the central hole or peak for the bent and cascade-type double resonances. The results obtained confirm the expected behavior based on level populations.