We next analyze our time dependent data to determine how linear our laser frequency scan is. To do this, we compare the time difference between any two hyperfine peaks and compare that to experimentally measured frequency differences. The experimental data is taken from \cite{Steck_2001} and from \cite{Steck_2008}, which measured the frequency difference between hyperfine peaks to kHz precision. If our frequency scan was linear in time, the ratio between any two hyperfine peaks in time should equal the actual frequency ratio. If the ratios are equal, we can use this calibration to plot our hyperfine data vs frequency. In order to measure transition frequency, we must first convert oscilloscope time to a frequency. We can do this by plotting a graph of \(\frac{\delta t}{\delta \omega}\) versus ratio number (Fig. \ref{fig:Ratios}). Since we expect the width of the peaks on the transmission graph to be the same, \[\label{eq3} \frac{t_{2}-t_{1}}{\omega_{2}-\omega_{1}}=\frac{t_{3}-t_{2}}{\omega_{3}-\omega_{2}}=\cdots\] Thus the plot of \(\frac{\delta t}{\delta \omega}\) versus ratio number should look like a horizontal line. From this line we can determine a conversion factor between frequency and time.

We then plot the ratios in a very specific way- they are plotted chronologically as we passed through our frequency scan. Ratio A corresponds to \(\frac{t_{2}-t_{1}}{\omega_{2}-\omega_{1}}\), B corresponds to \(\frac{t_{3}-t_{2}}{\omega_{3}-\omega_{2}}\), etc, so we have five ratios for our six peaks. By plotting the ratios in this way, any drift or non-linearity in the frequency scan can be inferred by observing the slope of the values of the ratios A, B, C, D, and E when plotted against the order they came in the frequency scan, since this way we can observe the frequency drift as time passed. A perfectly linear frequency scan would have all ratios be the same, so the slope of a line passing through them would be zero. There is clearly a slope to the ratios, an example of which is seen in Fig \ref{fig:Ratios}, indicating that our frequency scan was nonlinear. We could improve it in the future by doing a more time-intensive fit to the frequency scan.