Frequency Modulation in Microwave Spectroscopy

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Abstract
The absorption coefficient of a gas is related to the density matrix of that gas. The density matrix is obtained formally from its equation of motion, which takes into account a possible time-dependence of the resonant frequency of the molecule and of the frequency of the exciting radiation. The shape of the absorption line is calculated explicitly in a variety of cases. For low frequency modulation, it reduces to Lorentz's expression with a time-dependent frequency difference. For high frequency sinusoidal modulation, resolved sidebands to the resonant frequency are obtained. Finally, for square wave modulation, the line shape is again given by Lorentz's expression with a varying frequency difference, except that damped oscillations are superimposed on the expected square wave time-dependence. In this case, the deviations from the Lorentz expression are no greater than about ten percent for the zero frequency and fundamental frequency components, if the modulation frequency is less than one-half of the half-width of the absorption line.

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