### Determining the Wavelength of an Unknown Laser

### The Challenges of Building a Travelling Mirror Michelson Interferometer

With the advent of tunable lasers, whether they be Dye pumped or Diode,
comes the problem of determining accurately at any instant the actual wavelength of light
emitted from the laser cavity. Therefore, there exists a need for a device, or
setup, that enables an experimentalist to determine the wavelength of light they are
dealing with very precisely. This apparatus would then have to possess the
capability to measure visible and infrared wavelengths to a few parts in 10^{7 }with
an update period of a few seconds or faster. This device is the Traveling Mirror
Michelson Interferometer (TMMI).

This device represents a scanning interferometer (see also my experiment with the Michelson Interferometer) in basic form, due to travelling retroreflector. As the carriage supporting the retroreflectors moves, the fringes at some observation point move over a photodetector of some sort so that the fringes are counted. It is this ratio of interference fringes that are simultaneously counted for each laser beam that produces the ratio of laser wavelengths. This aspect of the device renders it very useful when determining unknown laser wavelengths. If using a reference laser of known and constant wavelength, such as the HeNe laser, it becomes easy to determine the unknown using the fringe ratio.

Below is a diagram of the TMMI that I built in lab this semester:

One can follow the paths of the two lasers (red and blue paths) fairly easily. The beams are parallel and collinear for the entire trip, otherwise the wave nature of light would not allow us to acquire interference patterns. The difference in path length between when the beams get split up is also less than the coherence length of the HeNe laser. Otherwise, as mentioned on the basic theory page, no interference effects would occur. So, when the light reaches the beamsplitter, part of the beam is transmitted and part reflected. The two beams acquire a phase difference of pi, but in order to preserve conservation of energy, their intensities add to that of the laser light. It remains important that the beamsplitter be as close to 50% reflective-50%transmissive for the TMMI to operate properly. If this is the situation, then the resolution of the interference fringes will be close to maximized. If, however, that this is not the case, the more powerful beam, whether it be transmitted or reflected, will be to intense compared to the other causing poor interference fringes. This results because we want the amplitudes of the waves to combine (characteristic of coherent light) not the intensity! If the amplitude of one wave becomes significantly larger than the other, the principle of superposition becomes somewhat irrelevant because the smaller amplitude makes no difference to the larger one in either the case of constructive or destructive interference. After the beamsplitter, the light interacts with the retroreflector that sends it back to the beamsplitter where it recombines with itself to produce an interference pattern visible at the two photodiodes. Again, according to the wave nature of light, it remains imperative that the light be collinear and parallel so that the two beams don't interact with each other. Below is an actual picture of the device itself:

Here are the photodiodes onto which the interference patterns are projected:

Here is the Air Track with the Retroreflector Carriage on it:

Here is a picture of the electronic counter device, photomultiplier, and image of interference fringes on the oscilloscope:

Notice how the scope sees the fringes as a sine wave even though it is a DC signal from the Photodiodes. This tells us that the fringes aren't perfectly resolved as they pass over the face of the photodiode. Otherwise, we might see a square wave with just zero amplitude for destructive interference and some positive amplitude for constructive interference. The sine wave could also result from the background noise packed in with the fringes.

Like the Fabry-Perot and Michelson Interferometers, one of the most useful relationships in the wave theory describes the orders of constructive interference. Since the carriage travels twice the length of the air track, the interference fringe cycle becomes . This means that the length in-between adjacent orders of interference becomes

Just to put things into perspective, the carriage moves approximately 1.1 meters. This means that in one trip down the air track, there are approximately six million orders of interference that occur and each is separated by one-fourth the wavelength of light, or 158 nm for the HeNe laser. But how does this tie into determining the unknown wavelength of a laser? Well, using a reference laser, we know all the information needed to solve for an unknown wavelength using the order of interference relationship. This is done as follows:

This represents the ratio that will yield the unknown laser wavelength. The more accurate equation,

takes into account the indices of refraction. The number 200 comes from the phase lock multiplier, a device that multiplies the frequency of the fringes by a constant 200, and C is the counter reading which represents the ratio of the frequency of the fringes. Although I have not mentioned much about it up to this point, the phase lock frequency multiplier plays an integral role in the accuracy of the counter operation. Think of it this way, we are measuring the ratio of fringes from the known wavelength to the unknown wavelength. The stronger signal to the counter will definitely be that from the reference laser. Therefore, if we multiply its frequency by a large constant, a small change in the unknown fringe frequency will produce a large effect on the overall ratio since the smaller unamplified frequency of the unknown resides in the denominator. This addition of the frequency multiplier clearly boosts the accuracy of measurement of the instantaneous wavelength, as does the fact that the carriage travels twice the distance (down and back) giving us more data to collect and evaluate.

Of course, building the travelling mirror interferometer was no simple task. There were several aspects to be recognized that caused particular problems with this device:

- Getting the apparatus aligned initially takes some time. It helps to sit down and draw detailed ray diagrams that keep track of what light is going where. Multiple Mirrors were also a help. The more the mirrors you have, the more degrees of freedom that you have when it comes to alignment of the system.
- If the system gets out of alignment and one doesn't know which mirror got bumped or jostled to cause the malalignment. This takes patience, for when dealing with a very sensitive device that requires planar/collinear light that can't interfere anywhere along its path, except at the return trip to the beamsplitter, it can be frustrating. The best way to go about realignment, as I found, is to cover up the mirror on the track that corresponds to the beam reflected from the beamsplitter. Just deal with getting the transmitted beam correct first, because the beamsplitter acts as a partial mirror and can be easily adjusted, just not for the transmitted light beam. To make sure that the beam is aligned along the airtrack, you want it parallel and at the same height for every inch along it, look at the spots from the laser in the mirror at the end of the track. If the spot(s) is/are moving around, then the device isn't aligned properly. If the spot(s) move side to side horizontally on the mirror, then the beam isn't parallel to the air track. If the spot(s) move vertically on the mirror, then the beam isn't at the same height above the track at all times. Once the spots are stationary in the mirrors, using the scope and looking at the amplitude of the fringes on the photodiode represents the most accurate way to go about fine tuning the device. Simply fiddle with the mirrors until you see the strongest fringe pattern. The photodiode has much better eyes than yours to see the fringes!!!
- Let the HeNe reference laser warm up for a while, otherwise your fringes might be "jumpy". The HeNe laser is temperature dependant, and this becomes obvious if you view the modes with a spectrum analyzer. When freshly turned on, the modes tend to be mobile and the output isn't as stable when it has been on for a while.

- J.L. Hall and S.A. Lee, Appl. Phys. Lett.,
367 (1976).**29**

- Abstract
- History and Theory of Optics
- Michelson Interferometer
- Fraunhofer Diffraction
- Fabry-Perot Laser Cavity