Polaroid sheets look superficially like some kind of ordinary gray plastic, but they have a very peculiar property. They have a built-in axis, and light that passes normally through a sheet of Polaroid vibrates only along that axis. Such light is said to be linearly polarized in the direction of the polarizer's axis. If linearly polarized light goes through a second polaroid sheet (called the "analyzer" when it comes second), only the component of the amplitude along the new axis is transmitted. Here "component" means in the sense of linear algebra: project the polarization direction of the light perpendicularly onto the new axis. The projection amounts to a multiplicative factor cos(A), if A is the angle between the polarizer axis and the analyzer axis. In particular, if A is 0 degrees, meaning the polarizer and analyzer have their axes in the same direction, then all of the polarized light goes through the analyzer, and if A is 90 degrees, meaing the polarizer and analyzer are "crossed", then none of the polarized light goes through the analyzer. (The foregoing is an idealization, of course, describing an ideal polaroid sheet. Perfect transparency and perfect extinction are not to be expected.) At intermediate angles A, the amplitude of the light is reduced by the factor cos(A), but the intensity of the light, i.e., its brightness, or, more technically, its energy current density, is proportional to the square of the amplitude, and so it is reduced by the factor cos²(A). It is this energy current density which is measured by any kind of brightness meter. In the experiment below, we measure brightness with two photodiodes, PD1 and PD2.

Experiment:

The experimental setup corresponds almost exactly to the polarizer-analyzer situation described in the introduction. You can rotate the analyzer and thus measure the intensity seen by PD1 as a function of angle A. The main difficulty is that the light source, a laser, is not at all steady! You will see, in fact, that its intensity has very large variations on a time scale of only a few seconds, so that any naive attempt to measure what is seen by PD1 would mainly see the variability of the light source, which is not at all what we want. For this reason a beam splitter is part of the design, to pick off some (presumably constant) fraction of the incident beam, and send it to PD2. When the intensity seen by PD2 diminishes, we know that is because the beam intensity diminished. That same beam intensity is incident on the analyzer, where it is diminished by the further factor cos²(A). Thus the ratio of intensity seen by PD1 to that seen by PD2 should be constant at fixed A, and should show the cosine-squared dependence when A is varied. So that is the strategy: we normalize the intensity seen by PD1 using that seen by PD2.

Thus we must read two outputs at once, both DC voltages (the photocurrents generated by light falling on the photodiodes produce voltage drops across a resistor proportional to the current, and hence proportional to brightness). The photodiodes need a +/-15V power supply. There is a junction box to make all circuit connections easy, including the output connection. It would work to use two voltmeters, and have two people ready to read them at the same moment. Instead we have a Keithley 2000 multimeter with two channels interfaced to a computer. The multimeter reads both voltages, and sends both values to the computer, where we can also do the necessary arithmetic, graph results, etc. The leads for connecting the multimeter to the photodiode outputs are found at the back, in pairs, for channel 1 and channel 2. There is also a button on the front panel of the multimeter which must be pushed in to select the input connections at the back of the instrument.

The computer interface to the instrument uses Matlab, a versatile scientific computation and visualization program, which you may have used elsewhere. Start Matlab by double-clicking its icon on the desktop. Start the interface by typing in the command "keithley2000", followed by a return. The interface is graphical and runs in its own window, which should appear. Set it to 2 channels. In order to understand better what is happening, choose 10 samples per scan, and choose 1 second as the time per sample. Now when you push the StartScan button, you can literally hear what happens. Each second there is an audible CLICK as little relays switch the instrument from one channel to the other. It takes noticeable time to make the switch, about 0.1 second, but fortunately the laser intensity doesn't change much over that time. (This extra time is the reason it takes over 11 seconds to scan both channels 10 times, as you can see in the command window.) You can visualize the fluctuating laser intensity with the PlotData button. You will probably find that in 11 seconds the intensity wandered quite a bit. But let us see if the normalization scheme worked. The acquired data are in a 10-by-2 array called "data1" (unless you changed the name). Each row is one of the samples, in order of acquisition. Each column is one of the channels. Channel 1 is in the first column, which in Matlab notation is "data1(:,1)", and Channel 2 is in the second column, "data1(:,2)". The colon (:) means the first index, the row index, goes through all its values, but the second index, the column index, is fixed, indicating which column. Matlab computes the ratio of these two readings, row by row, with "data1(:,1)./data1(:,2)". The symbol for division here is "./", not the usual "/", since strictly speaking division of a column by a column makes no sense. The "./" means do it elementwise. Try this -- is the ratio fairly constant, even though the intensities themselves are varying?

Before starting serious data taking, be sure you know which channel is which. Do a 10-second scan and block the light to PD1 for at least one "click" and then block the light to PD2 for at least one "click". When you graph the result you can see which trace is which, and identify which is Channel 2 by noting the last value read (which will still be visible on the face of the instrument). This way you will know which channel is looking at PD1 and which at PD2. Quite possibly the ratio we tried in the paragraph above was the inverse of the one we actually want!

When you know how to take good data, find the normalized intensity transmitted through the analyzer as a function of analyzer angle, taking a measurement every 10 degrees for a full 180 degrees, i.e. a full period of this periodic phenomenon. Then see how well it fits the prediction that it is proportional to cos²(A).