Overcoming the detection bandwidth limit in precision spectroscopy: The analytical apparatus function for a stepped frequency scan

In a previous paper [Rohart et al., Phys Rev A 2014;90(042506)], the influence of detection-bandwidth properties on observed line-shapes in precision spectroscopy was theoretically modeled for the first time using the basic model of a continuous sweeping of the laser frequency. Specific experiments confirmed general theoretical trends but also revealed several insufficiencies of the model in case of stepped frequency scans. As a consequence in as much as up-to-date experiments use step-by-step frequency-swept lasers, a new model of the influence of the detection-bandwidth is developed, including a realistic timing of signal sampling and frequency changes. Using Fourier transform techniques, the resulting time domain apparatus function gets a simple analytical form that can be easily implemented in line-shape fitting codes without any significant increase of computation durations. This new model is then considered in details for detection systems characterized by 1st and 2nd order bandwidths, underlining the importance of the ratio of detection time constant to frequency step duration, namely for the measurement of line frequencies. It also allows a straightforward analysis of corresponding systematic deviations on retrieved line frequencies and broadenings. Finally, a special attention is paid to consequences of a finite detection-bandwidth in Doppler Broadening Thermometry, namely to experimental adjustments required for a spectroscopic determination of the Boltzmann constant at the 1-ppm level of accuracy. In this respect, the interest of implementing a Butterworth 2nd order filter is emphasized.