Our laser is designed to sweep back and forth over a small range of photon frequencies that correspond the the frequencies of the Rb transitions we wish to study.
You will determine the calibration factor by building a Michelson Interferometer on your optical board. By directing the laser beam into the interferometer, and recording the intensity of the output on the scope, you can determine the calibration factor needed. Ultimately what we want to measure are the energy differences between hyperfine levels, so all we need are frequency differences between features in the spectra. We can accomplish this by using a Michelson interferometer to measure the change in frequency of the laser as a function of time.
The geometry of the Michelson interferometer is shown in Fig. 10. The beams from the two arms of the interferometer will combine at the photodetector with varying degrees of constructive interference depending on their phase difference $\Delta\phi$. It can be shown that the phase difference depends on the difference in lengths of the two arms of the interferometer and the frequency of the light as
| $\phi_1 - \phi_2 = \Delta \phi = \dfrac{4\pi f}{c}(L_1 - L_2)$ | (7) |
where $f$ is the frequency of the light, $L_1$ and $L_2$ are the path lengths of the two arms of the interferometer, and $c$ is the speed of light. From this relation, it can be shown that the frequency spacing of the interference maxima at the output of the interferometer is
| $\Delta f = \dfrac{c}{2(L_1 - L_2)}$ | (8) |
Try to make $L_1 - L_2$ as large as possible. This results in closer fringe spacing which allows you to better characterize the frequency sweep rate.
Some tips for aligning the interferometer.
- Always build your optical layouts front to back. Meaning begin with the first optical component the beam will interact with, carefully align it, then move on to the next component leaving the previous one alone.
- In order to have the two output beams come together properly at the photo detector:
- Choose one arm of the interferometer to build first.
- Place an iris approximately halfway between the beam splitter and the mirror as shown.
- Adjust the beam splitter to send the beam through the center of the iris.
- Then setup the mirror to reflect the beam back on itself and through the iris. This will ensure that the beam is well aligned.
- This return beam determines where you place the detector, do not make any further adjustments to the mirror until you get to the fine tuning stage.
- Now when you setup the second mirror, all you have to do is adjust it so that its return beam overlaps the first one at the detector. Do not make adjustments to the beam splitter or the first mirror, work only with the second mirror. This will ensure that the two beams are nearly perfectly square to one another which is what you need to see the interference fringes.
Aligning a Michelson interferometer can be tedious. Be patient and you will succeed!
Once you have aligned the interferometer with the alignment laser you can switch to the infrared laser and view the signal on the scope. If you did the initial alignment well you should see at least a hint of an interference pattern on the signal. At this point it is a matter of making very minute adjustments to the two mirrors, one at a time, in order to maximize the amplitude of the interference pattern.
- Make a tiny adjustment to one of the mirrors and take your hand off the mount(your finger pressure is enough to change the path length of the light at this point).
- If the interference pattern improved, make another adjustment in the same direction, if it got worse adjust the mirror in the other direction.
- Once you have maximized the amplitude of the interference signal by adjusting one mirror axis, repeat the process for the other axis of that mirror.
- Repeat for the other mirror, going back and forth between the two mirrors until you can no longer improve the interference pattern.
In order to receive credit for this part of the lab you need to do the following.
- Capture the digitized scope trace of the interference pattern.
- Measure and record the lengths of the two arms of the interferometer. Include uncertainties.
- Measure the position of the interference maxima using the scope cursor feature. Alternatively you can do this outside of lab with the digitized data you saved from the scope.
- Calculate the conversion coefficient to convert time differences on the scope into laser frequency differences. Show your calculations.
- Apply your calibration to the FWHM measurements that you made in the earlier check point exercise, in place of the approximate value we provided for you.
Note that you will have multiple interference maxima to work with. This gives you the opportunity to take multiple readings and average, or check for linearity of the sweep. You do not necessarily need to do this for the purpose of this check point exercise, but you will want to more carefully consider these factors for your measurement of the hyperfine energy splitting.