Anomalous Dispersion in Gases Derived from the Optical Depth. Theoretical Treatment: Line by Line Calculations
NORWEGIAN DEFENCE RESEARCH ESTABLISHMENT KJELLER
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This report describes a theoretical development of the anomalous dispersion in a mixture of gases, from the optical depth. The dispersion is determined by a Hilbert transform of the optical depth. An optical depth formulation as defined by Van Vleck and Huber is fundamental to this treatment. The Hilbert transform is performed without approximations giving an expression for the dispersion which is valid for all wavenumber subject to the validity of the dipole approximation and a classical description of the electromagnetic field. The derived expression is adequate for the Voigt, Lorentz and Doppler line shape functions. The effect of line coupling is not taken into account. An approximate expression for the dispersion is implemented for line by line calculations of dispersion in a modified FASCOD2. This approximation is valid when homogenous Lorentz broadning is dominant and the resonance wave number is greater than 1cm-1 30 GHz. Atmospheric dispersion for a specific atmospheric model is calculated throughout the entire wave number region 0-17900 cm-1. Examples demonstrate the correctness of the line by line calculated dispersion in relation to theoretical results and measurements.