Quardrature Michelson Alignment

Introduction

This interferometer takes a linearly polarized input signal and rotates it with the first half wave-plate (HWP) so that it lays entirely in the horizontal plane. After passing through a non-polarizing beam splitter (NBS), half of the signal is reflected by a mirror back into the non-polarizing beamsplitter; this signal is still linearly polarized in the horizontal plane. The other half of the signal is sent through a quarter wave plate (QWP) (with its fast axis at pi/8 to the horizontal plane) to a mirror where it is reflected and then passes once more through the quarter waveplate (QWP) back into the non-polarizing beamsplitter (NBS). The effect of passing the quarter wave plate twice, is that it rotates the linearly polarized beam from the horizontal plane into a linearly polarized beam at 45 degrees with the horizontal plane. (Actually, the linearly polarized signal is converted by the QWP into a elliptically polarized signal and after being reflected by the mirror into the opposite elliptical polarization direction, it will be converted by the QWP back into a linearly polarized signal, albeit one which is at 45 degrees with respect to original one.)

The two linearly polarized beams are then passed through a second quarter wave plate which is at 45 degrees with the horizontal plane. Since the reference beam is already at 45 degrees, it is not affected by this second quarter wave plate. However, since the measurement beam is at 45 degrees with this second quarter waveplate's fast axis, it will be converted into a circularly polarized signal.

A second, polarizing beam splitter (PBS) breaks the linearly polarized beam into its horizontal and vertical polarization components. The (average) intensity for each polarization component is then measured by two photo diodes and its ratio corresponds to the phase between the measurement and the reference arm. In other words, it represents to the optical path length difference between the two interferometer arms. Furthermore, by observing the change in the phase difference, we can at any time unambiguously determine the direction of the change.

For a more mathematical description see the Setup3 Theory (Mathematica & PDF Files Attached) section.

Alignment

The alignment procedure for the Quadrature Michelson Interferometer is relatively straightforward. For this setup, you will need:

3x mirrors

2x irises

1x HeNe laser

1x laser mount

1x neutral density filter

1x polarizing beamsplitter (optional)

1x translation stage

1x Quadrature Michelson Interferometer (this is already assembled for you)

1x Tektronix 4034 Digital Phosphor Oscilloscope

Step 1 - Laser

Mount the laser to a post and screw it onto the optics table. Rotate the post so that if you were to look at the laser from a top-down view, the beam would be roughly parallel to a line of holes in the table.

Step 2 - Mirrors

Place one mirror in front of the laser such that it reflects the beam by roughly 90 degrees. Now place a second mirror some distance away from the first so that it reflects the beam by another 90 degrees. The direction of the beam after reflecting off both mirrors should be opposite what it was after leaving the laser. There is no required distance between the two mirrors or the first mirror and the laser; you are free to choose whatever distances are most conducive to your experiment. Figure 1 shows the placements of the two mirrors.

Height align the two irises with the laser. Use the irises to walk the mirrors so that the beam is parallel to both the surface of the table and the grid of holes on the table.

Figure 1: The placements o f the two mirrors and the neutral density filter. The polarizing beam splitter is not necessary for this setup; it was only used for testing purposes. Note that the red line denotes the path of the HeNe beam.

Step 3 - Neutral Density Filter

We will use a neutral density filter to adjust the beam intensity. Place the neutral density filter somewhere between the mirrors. Place it so that beam hits near the outer edge of the filter. Rotate the filter so that the reflected beam hits the mirror mount of the mirror close to the laser.

Step 4 - Rough Alignment of the Quadrature Michelson Interferometer

Place the interferometer after the second mirror so that the beam strikes the half-wave plate (HWP) at nearly normal incidence. After passing through the HWP, the beam is incident on the first non-polarizing beamsplitter, so you should see that the part of the beam passes straight through the splitter and hits the mirror attached to the interferometer and the other part is ejected from the interferometer all together. To get a rough alignment of the setup, you will make adjustments to the interferometer in order to steer that beam that is emitted from the interferometer.

Block any mirrors in the interferometer with lens tissue and retro-reflect beam (from the front surface of the non-polarizing beam splitter) back to laser but not into laser while carefully moving and adjusting the entire interferometer assembly. (Note this beam might be weak since only a small percentage is retro reflected. Use the trick with a note card with a hold to locate the retro beam.)

Step 5 - Adjust Mirror

You now will need to make sure that beam hits the mirror attached to the interferometer with normal incidence. Get a piece of paper and poke a small hole in it. Place the paper in the path of the beam near the laser so that the beam passes through the hole that you made. When you do this, you should see at least one red dot appear on the paper. Adjust the knobs on the interferometer mirror and see if any of the dots move. If none of the dots move, try moving the paper down the path of the beam until you find a dot that does move when you adjust the mirror's knobs. The one that moves is the retroreflection of the mirror and that is the one we are interested in. Once you have located it, adjust the mirror's knobs to get the dot close to hole in the paper. The end goal of this step is to move the dot to a position where it can be seen on the front of the laser. You will want the dot to be as close to the opening on the front of the laser as possible without actually reentering the laser. See Figure 2.

Figure 2: An image depicting the front of the laser. The bright dot near the center is the retroreflection from the mirror.

Step 6 - Translation Stage and Mirror

Note that the instructions given for this step may not apply to your particular experiment. The mirror and translation stage placed in this step are often replaced by a setup specific to the experiment being performed. Nevertheless, the general ideas found in this step will still be true for your setup.

Mount a mirror to a translation stage and mount the translation stage to the table so that the mirror reflects the beam that leaves the interferometer back into the setup. Place a KimTech wipe in front of the interferometer mirror to block the beam. Repeat step 6 for this new mirror. It is important that the retroreflections of the two mirrors overlap. When they overlap, you will see an interference pattern on the front of the laser. Continue adjusting this new mirror to make the interference fringes as large as possible. This interference pattern can be seen in Figure 3.

Figure 3: The front of the laser is again depicted. The retroreflections from the two mirrors are causing an interference pattern.

Note that this is not the largest the fringes can be; you will be able to get them even larger than is depicted here.

Step 7 - Understanding How the Interferometer Works and Exercises with Polarization Optics

Before proceeding with the alignment of the interferometer, this is a good time to experiment with the setup to make sure you understand what is going on. Note that for this section, I will adopt the naming convention outlined in Figure 4.

Note: this section in italics is purely for educational not alignment purposes and you may skip directly to step 8 "Detailed Instructions for the Optimal Alignment of the Retarder Plates"

Figure 4: Diagram of the Michelson Quadrature interferometer with naming convention.

1) Turn on the Tektronix Oscilloscope. Connect Detector X to channel 1 and Detector Y to channel 2 of the oscilloscope. Initially, the scope displays voltage vs time, but you will want it to display the voltage in Y vs the voltage in X. To get this, press the button labelled "Acquire", then change "XY Display" to "on".

2) WHILE WEARING NITRILE GLOVES, remove QWP 2. Do not remove the HWP or the QWP 1 but block QWP1 by placing a piece of lens paper between it and mirror 2.

3) Confirm that the signal is entirely in the x-axis by rotating the HWP. Rotate the HWP through 360 degrees while blocking the beam through the QWP1. What you will find is that the HWP rotates the polarization of the beam, so different percentages of the beam's intensity are sent to Detector X and Detector Y. The result is that by rotating the HWP, the signal sweeps out a quarter circle on the screen. Rotating the HWP by 45 degrees changes the signal by 90 degrees. For example, when the HWP is at (~352 +/- n*90) degrees, the signal is entirely along the x-axis and when the HWP is at (~307 +/- n*90) degrees, the signal is entirely along the y-axis.

Before moving onto the next step, adjust the HWP so that the signal is entirely in the x-axis.

4) Now put QWP 2 back in while leaving QWP 1 blocked. Rotate QWP 2. You should find that some of the signal will now be in the y-axis, but never more than half. That is to say, the line connecting the dot on the screen and the origin will never make an angle with the x-axis greater than 45 degrees. If you keep turning the QWP after that angle is 45 degrees, the signal regresses back to the x-axis.

What is happening is that when you insert a QWP is that you are turning the linearly polarized signal into a circularly polarized signal. However, to obtain circularly polarized light, the QWP must be at the proper angle to retard the signal in one axis by exactly one quarter wave. For any other angle, you will get elliptically polarized light, (i.e., light where the x and y components are never identical), or simply linearly polarized light. Since the original signal entering the QWP is entirely in the x-direction, there are three possibility: if the QWP is aligned that its fast or slow axis is parallel to the x-axis, you will not see any changes, i.e., the light from the QWP remains linearly polarized. When the fast and slow axes are at 45 degrees with the x-axis, you will get circularly polarized light and the x and y components are identical, creating a point at 45 degrees with respect to the origin on the scope. At any other angle, the resulting beam will be elliptically polarized, with most of the signal in the x-axis and some in the y-axis (retarded.)

You will notice that when you hit the table with something, the signal on the screen does not respond because there is no interference happening within the setup. This may seem like a strange comment to make at the moment, but you will see shortly that the Quadrature Michelson Interferometer is extremely sensitive and when it is aligned you will see the signal respond to hitting the table.)

5) Unblock QWP 1 with QWP 2 removed. Rotate QWP 1. Like before, you will see that the y-component of the signal increases, but never exceeds the x-component. This phenomenon happens for a different reason this time. When the beam enters the interferometer, a beam splitter breaks it into two arms: one that goes through QWP 1 before reflecting off a mirror and one that reflects off a mirror without passing through a wave plate at all. Because the beam is initially polarized along the x-axis, the component that does not get sent through QWP 1 is guaranteed to also be in the x-axis. The component that does go through QWP 1, however, can have a y-component to its polarization. Moreover, because the light in that arm has to go through QWP 1 twice, that arm can be entirely converted to a y-polarized beam. Hence, as you rotate QWP 1, the signal traces out 1/8th of a circle on the screen from the x-axis to line with a 45 degree angle.

However, you will notice that this time that there is noise in the signal when you hit the table. Now that both mirrors of the interferometer are unblocked, there is interference occurring within the setup, which causes noise in the signal. When QWP 1 is rotated so that the signal is still entirely in the x-axis you will find that there are huge fluctuations in the amplitude. This is because the beams in both interference arms are polarized in the x-axis, which means that you can see complete constructive and destructive interference.

When you rotate the QWP 1 so that the signal is not along the x-axis, you will find smaller fluctuations because the QWP causes light in that arm to be partially polarized in the y-axis. The portion of the light that is polarized along the y-axis cannot interfere with light polarized along the x-axis. This has two important results: there are smaller fluctuations in the signal's noise, as previously mentioned, and the fluctuations can only happen along the x-axis because the component of the light along the y-axis has nothing to interfere with.

Briefly block QWP 1 and rotate the HWP so that the signal is at a 45 degree angle. Unblock QWP 1 and hit the table. Now that both arms have x and y components to their polarizations, the noise created by hitting the table traces out an ellipse. Rotating QWP 1 causes the ellipse to move about the screen.

Put the return the HWP back to its original position so that the signal entering the first beam splitter has only an x-axis component.

Step 8 - Detailed Instructions for the Optimal Alignment of the Retarder Plates

You are now ready to align the interferometer.

1) Rotate the HWP so that laser signal hitting the first (non-polarized) beam splitter is entirely in the x-direction. (Note you do not need an external polarizing beam splitter discussed above to obtain this result.) To do so be sure to, first, remove QWP 2 and, second, block the beam path between QWP 1 and Mirror 2 with a piece of paper. Rotate the HWP so that the signal displayed on the scope is entirely along the x-axis.

2) Adjust QWP 1 so that the signal in this arm is at 45 degrees: with QWP 2 still removed and the beam path between QWP 1 and Mirror 2 unblocked, rotate till QWP 1 until the dot on the scope is at 45 degrees with respect to the origin. (Alternatively, briefly block the path for Mirror 1 and you should see a signal entirely in the y-direction.)

3) Put QWP 2 back in. Create some small vibrations on the optical table by hitting it with a soft object and meanwhile rotate QWP 2 to make the ellipse traced out by the signal as circular as possible.

4) Make small adjustments to the mirrors so that the retroflected beams interfere and adjust QWP2 more, if necessary. You may also adjust the irises that are built into the detectors by finding at what point they diminish the signal and then slightly backing off from this point.

Once you are able to hit the table and have the signal trace out a circle, like in Figure 5 you are done!

Figure 5: An image of the oscilloscope screen when the signal traces out a circular path.

Data Analysis

This page describes how to convert the voltage from the X and Y detectors into path length changes and velocity.