Abstract

In this experiment I studied the polarization of laser light passed through several polaroids configurations in order to determine the effects of the changing the polarization on the light, also circularly polarized and somewhat depolarized light was studied.  The precent polariztion of 1, 2 and 3, polarizer configurations were found to 18.0 +/- 1.67, 81.8 +/- 1.7, and 35.4 +/- 4.3.  The wax paper was 18.2 +/- 18.0.  The laser was also shined through a glass block in order to determine Brewster's angle for that glass.  This angle was found to be 57 +/- 2.1 degrees with a precent polarization of 92.6 +/- 2.07.

Introduction

This experiment dealt with examining the polarization of light under several differant configurations.  In the first portion of the experiment we were primarily studying polarization of a laser with regards to the effect of passing it through 1,2 or 3 polaroids.  Each time one of the polaroids was connected to a motor that allowed it to rotate at constant angular velocity thus giving us the chance to study the effect of the orientation of the polaroid on the intensity of the light it transmitted.  In the section portion of the experiment one or two quarter wave plates were introduced in order to give us the oppurtunity to study circularly polarized light and the effect of the orientation of the two quarter wave plates on each other.  Also in order to study our ability to depolarize, polarized light we used a wax screen as a scattering mechanism.  Finally we studied the Brewster's angle of a glass block by measuring the blocks ability to eliminate one directional component of a incident laser beam.

Experiment

Section 1:  As shown below (Fig. 1) an optic track was setup containing a laser, a beam adjuster, one to three polaroids, one of which connected to a motor along for rotation at a constant velocity, and a photodiode connected to a computer for collecting the measurements of the beams intensity.  In order to minimize the effect of ambiant light on the data, the light level in the room was reduced below the sensitivity of the photodiode.  This configuration was used in three forms, first with a single rotating polaroid placed on the track, then with a stationary polaroid placed between teh rotating polaroid and the laser and lastly with stationary polaroids both in front and behind the rotating polaroid.  The laser beam and polaroids were adjusted so that the laser passed parrallel to the optics track and through the center of each polaroid before reaching the photo diode.  The determination of the center of the polaroids was made visually and adjusted by hand.  For the setup containing all three polaroids the stationary polaroids were first positioned relative to each other so as to produce a minimum amount of transmitted light reaching the diode (This was also done by hand since at levels close to 0 intensity the photodiode was less sensitive than mere sight).  The rotating polaroid was then inserted between the two.  In all cases the computer was used to record the data for later analysis.

Section 2:  In this part of the experiment the properties of circularly polarized light were observed.  This was accomplished through the use of quarter wave plates.  The setup for this was essentially identical to Section 1 (see Fig 1), except that there was a stationary polarizer in the first position, followed by one or two quarter wave plates and then the rotating polaroid.  With only one quarter wave plate the orientation of the quarter wave plate relative to the stationary polaroid was adjusted so that intensity of the light did not vary (approximately) relative to the angle of the rotating polarizer.  This condition represents circularly polarized light and was recorded.  Furthermore with out adjusting the configuration of the first quarter wave plate, the second quarter wave plate was added.  The effect of changing the orientation by rotating the second quarter wave plate clockwise through 180 degrees was measred by recording data at 25 degree increments.

Section 3:  This section again used the optics track and works like Section 1 (see Fig 1) except that this time the laser passed first through a polaroid then a piece of wax paper, then the rotating polaroid then a bi-convex lens used to focus the somewhat dispersed light onto the photodiode.  This portion of the experiment was meant to measure the wax paper's ability to depolarize the light passing through it.

Section 4:  In the final section of this experiment was designed to measure the ability to polarize light by reflection at Brewster's angle.  Again the laser was mounted on the optics track and passed first through a beam adjuster and then through a stationary polaroid.  From there it travelled to and impacted a glass block positioned vertically on a rotating table with angular scale.  A cross beam was attached under the rotating table so as to allow the photodiode to be positioned off to the side of the optics track in order to record the intensity of the reflected beam (see Fig 2).  First the block was adjusted so as to retro-reflect the beam back into the laser, this angle on the scale was recorded as the bas angle, then the angle of the table and the orientation of the polarizer were simultaneously adjusted until the intensity of the reflected beam achieved a minimum as judged visually.  In order to measure the angular properties of this beam the first polarizer was removed and the rotating polarizer was placed on the cross beam so that the reflected laser first passed through the center of it and then onto the photodiode.

Additional notes on Data Acquisition : Unless otherwised noting in the analysis all samples were recorded at 1000, values per sec for a total of 5000 values.  The values recorded were in volts which is linearly proportional to the intensity of the beam, and have been scaled in the analysis for the range 0 to 1 such that 0 is no detectable beam and 1 is the maximal recorded beam for that data sample.  This is linearly proportional to the intensity of the beam and since all the analysis that follows deals in comparing relative magnitude of intensity the proportionality factors would cancel.  Furthermore simultaneously to the signal acquired from the photodiode the computer recorded a signal which indicated when the rotating diode had completed a full period this allows one to determine the relative position of the polarizer which with respect to the photodiode signal.

Results/Analysis

General Information regarding signal analysis:  In all cases the signals were analyzed by using Mathematica to automate the task of finding the maxima and minima of the signal (see Analysis, Section 3 for special notes on Wax).  Additionally it was found neccesary to devise a way of automaticly eliminating extraneous data which was caused by the equipment.  Since the extraneous points were all located significantly off the curve and occurred in <1% of the data, I decided to tabulate the lower 95% of the values for DI = I_N+1 - I_N, where I_N is the nth measure of the intesity value, and then computed the mean and standard deviation of this quantity.  I then set the maximum allowable change between two data points equal to the mean change plus 3 dtandard deviations, and any place where the data varied by more than that was adjusted down to this limit, and thus brought in line with the data around it.  Additionally when considering the case where the extreme value of one peak occured at more than one data point on that peak, the reported value will be the average of the positions.  Computed wavelength measures (unless otherwise noted) are taken as the differance in position of the second peak and next to last peak over the number of waves between the two.  Thus by using multiple wavelengths we can reduce the error in the wavelength.  This was then compared to the cycle rate as measured by the rotation of the polarizer to figure out the relative period.

Section 1, Part 1 : With only the rotating polarizer the intensity plot was found to be as follows :

One feature of noting in this plot is that there is a variation between the magnitude of the top even numbered peaks and the top odd numbered peaks of approximately 1%, this I attribute to not having the laser exactly centered on the rotating polaroid.  Additionally one can note a roughly triangular wave underlying this pattern, which is shown in detail here :

This may result either from fluctuations in the laser's inherent polarization or from the equipment used to measure the intensity, but it appears to occur in all the samples for which one can distinguish features this finely.  This is therefore taken as a significant component of the error.

One of the features we most desired to measure was the precent polarization of the light, which can be given by the formula :  % Polarization = (I_max - I_min)/(I_max + I_min) * 100 , where I_max and I_min are the maximum and minimum intensity values respectively.  In order to find the maximum and minimum values a straight average was preformed on the values of the minima and maxima shown in the above graph (Fig 3), with the following result :

When calculating this error there were a number of things that were considered, including, potential fluctuations in the laser, the discreteness of the measuring devices, the sensitivity limitations of the equipment, and fluctuations in the signal itself.  After some consideration it was decided that the only two signifcant contributors to error deviations in the measurements from peak to peak, and the buried triangular wave discussed above, therefore the reported error is the quadrature combination of the standard deviation of the peak measurements and the range of the small scale signal fluctuation near the extrema.

The other thing we wished to know is how the function varies with respect to rotation of the polarizer.

To do this I started by figuring out the wavelength of the pattern above by looking at the seperation of the peaks and then compared this to the data for the seperation of each full rotation of the polarizer.  With the results as follows:

This is the expected result, that the wave should repeat for every 180 degree rotation of the polarizer.  Both the error on wavelength and that on speed are based on how localized the data is versus how many periods we can consider it over.

Part 2:

The analysis from part 1 is repeated for this data set.

As one would expect the use of two polarizers interacting greatly reduced the baseline polarization of the light.

Part 3

In this case since the polarizers 1 and 3 are out of phase by Pi/2, we can see that the transmitted light never decreases to zero because as the rotating polarizer moves it brings them more in and out of phase but never totally either.

It should be noted that the value for waves is determined by counting the two peaks (higher and lower) as one wavelength because they are significantly differant in height.

Section 2:

This portion of the lab results will be somewhat qualitative.  To begin with when using the single quarter wave plate, while I was not able to totally elimanate angular dependance, I did get the variations down to less than a tenth of the amplitude as shown here:

As one can plainly see while there is still an obvious angular dependance here the % of the light which is polarized is very small as it should be.

When one inserts the second quarter wave plate the effect is essentially to bring the polarization back in phase but with an angular shift which depends on the differance in the orientation of the two plates.  It was found that rotating the orientation clockwise caused the waveform to systematicly shift to the right.

Section 3:

The attempt at producing unpolarized light by shining a laser through a sheet of wax paper was not wholly successful.

The apparent granularity was caused by the limit of the resolution of the equipment to pick up on the low levels of light transmitted by the wax paper.  As one can see from this image (well sort of see) there is a distinct waveform at the bottom of the graph where many of the data points lie, and if one just considers that, the light can be thought of as being significant unpolarized, however looking at the full spectrum we get the following results:

Unfortunately because the system is so poorly behaved it is very hard to determine precisely where the data falls, especially because of the large variations about the maximum values the error for this turns out to be very large and additionally one must consider that the dispersing and bringing back together means that it is not easy to determin how much of the signal actually reaches the photodiode.  It should also be noted that the 18.2% polarization is essentially the same as that found for Part 1 of Section 1 where we determined the polarization of the laser beam through only one polaroid.  So it may actually be the case that some signifcant portion fo the light coming through the wax paper is actually just that of the laser being unaffected, which may account for the data points not lying on the somewhat well-defined region at the bottom.

If we make a rough guess at what the maximum of this dominate lower function is, and set that to be I_max = 0.77 +/- 0.1, we can then get that % Polarization is 6.7 +/- 15.7, which while it lends some credence to tnotion that there might be a depolarizing effect involved here, it obviously shows that without a way to produce a clearer signal there will be no way to clearly whether the wax produces any real affect on the polarization of the laser.

Also note that for the data computation on this section no adjustment of the data was made to elimante randomness since the signal essentally swamped any of that.

Section 4:

Finding Brewster's Angle was not horribly difficult in lab, though the precision with which it is known is not great.  The retroreflection angle was found to 3° +/- 0.5°, where the error is a measure of how well one can determine when the laser is exactly passing back into its point of origin.  The adjusted angle at which the intensity of the light was minimized was found to 304° +/- 2°, where the 2° arrises from the serious difficulty of determining when everything is positioned perfectly.  Thus giving Brewsters angle to be 57° +/- 2.06°.
Since tan q_B = n/n₀, where q_B is the Brewster angle, n is the index of refraction of the object the beam is entering and n0 is the index of refraction medium it is leaving we can find the index of refraction of the glass by first assuming that the index of refraction of air is essentially that of vacuum, n₀ = 1.  Therefore we arrive at n=1.53 +/- 0.121.

The graph of the reflected beam looked as follows :

As is evident from the image the light reflected from the glass surface is nearly completely polarized, with the actual data going as follows :

This it should be noted is the most polarized smaple we created through this entire experiment.
 

Conclusion

After conducting multiple experiments on polarized light we have found that the use of polarizers can give rise to many interesting and unusual effects that can be studied.  In particular the ability to use ordinary glass to generate high quality polarization is interesting.  Though we did find that better equipment and/or a more constant medium might be useful for getting a clearer picture of what was actually occuring in some of these experiments.