Physics 18L: Modern Physics Lab
Lab 1: Malus’s Law
Jacob Schwartz
Partner: Samuel Jacobsen
University of California, Los Angeles
Professor Regan
Lab Section 1
April 7, 2010
Abstract
In this lab, we investigate the polarization of light and Malus’s Law. We project
light through a polarizer and analyzer and measure the resulting levels of light with
an analog light meter. By rotating the analyzer we can measure how polarized light
is transmitted or blocked by a second sheet of polaroid at an angle to the first in
accordance with Malus’s Law. After fitting our data to theory, I found that I had
underestimated our error bars (χ2 /DOF ≈ 1.8) and so I attempt to give an explanation
and to suggest improvements to our experiment. In the conclusion I propose that we
did indeed verify Malus’s Law.
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Introduction and Theory
Polarization is an important optical effect which exploits light’s nature as a transverse
wave. Polarization is important to science, for example as a way to view material stress,
but also in our daily lives in applications from LCD screens on digital watches to sunglasses,
photography, and 3D movies. When an unpolarized beam of light is passed through a first
polarizer sheet, only the electric field vector’s component in the direction of polarization
can pass through. This leads to a reduction in intensity of the initial beam by one half,
but now all the light is polarized in one plane. When this plane-polarized light is passed
through a second polarizer at an angle α to the first, only part of the electric field vector
E = E0 cos(α) can get through. Because the intensity of light I is proportional to the
square of the electric field vector E:
I(α) = I0 cos2 (α)
This equation is called Malus’s Law. In this experiment we will attempt to demonstrate
Malus’s Law by shining light through a pair of polarizers and recording the level of light
which is transmitted based on the angle between them. We will also investigate and attempt
to consider the effects of non-ideal polarizers, ie those which besides polarizing filter out
some additional light or which do not polarize the light completely.
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2.1
Experimental Arrangement and Procedure
Malus’s Law setup
To investigate Malus’s Law we used a setup with four main components: a projector, two
polaroid filters called the polarizer and the analyzer, and a light meter. Light shown from
the projector passes through the polarizer (our filter marked ”#2”), set at 0 degrees, to
become polarized. Then it passes through the analyzer (marked ”#8”), set at a variable
angle α where the light is reduced according to Malus’s law. Finally we measure the
remaining light using a fiber optic cable attached to a light meter.
0°
Projector
Polarizer “#2”
α°
Analyzer “#8”
Light Meter
Figure 1: Setup for Malus’s Law experiment
2.2
Measurement Procedures
Before starting we zeroed the light meter by covering up the receiving end of the meter
with a finger and adjusting the knob until the meter read zero. Removing the two polaroid
filters for now, we turned on the projector and took readings of the light meter every minute
to check its stability over time. Our second calibration measurement was to determine
whether the light coming from the projector was already somewhat polarized. With each
of our filters (called #2 and #8) in turn, we placed it in the path of the beam and rotated
it until we found the maxima and minima of light meter readings. We also took note of
the reduction in intensity due to the filter so that we could determine how much it blocked
light besides polarizing it.
Now we took data for Malus’s law. We placed both filters in the path of the beam
and set the polarizer to 0 degrees. With the analyzer also initially at 0 degrees, we took a
reading from the meter and turned the analyzer by 10 degrees until we covered the whole
circle.
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3.1
Data, Analysis, and Results
Malus’s Law Data
As specified in the procedure we took light meter readings for each α from 0 to 360 degrees.
From those values this plot was made:
Malus's Law Data and Best Fit
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Fit Function
Measured Values
Relative Intensity
5
4
3
2
1
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
Polarizer Angle (degrees)
Figure 2: Data from measurements and plot of best fit
I attempted to find the best fit of the graph with the function
I(α) = I0 + I1 cos2 (α − φ),
which describes a cos2 function with a raised floor I0 , amplitude I1 and phase shift φ. With
those three variables, I attempted to minimize the sum of the squares of the residuals.
Incorporating the errors present in the data, my lowest value for χ2 /DOF was about 1.8,
which tells that I did not take into account all the errors or different effects which were
present. The best value found for I0 was 0.11 ± 0.07, for I1 was 6.1 ± 0.1, and for φ was
−0.02 ± 0.04. The errors on those values were found by finding the difference between the
best value for a parameter and one which increased the χ2 by one.
In order to confirm that this best fit was reasonable, I made a plot to show the difference
between the data points and the points from the theoretical equation.
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Difference in Relative Intensity
Residuals
0.20
Data minus Theory
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
Polarizer Angle (Degrees)
Figure 3: Residual differences between data and theory
It is true that there is more area below the axis than above, but the way that I weighted
the errors (discussed in the next section) causes this to not be an issue. The errors in
absolute terms are optimized to be smaller around the minima and maxima of cos2 (0, 90,
180, 270 degrees) and larger on the regions with high slope.
3.2
Results of calibration measurements, sources of error
In our first check, of the stability of the meter over time, we found that it varied by roughly
±0.1 from the initial relative intensity value of 42.0 over the span of five minutes. To find
the error from meter fluctuations for the Malus’s Law data, we cannot simply extrapolate
downward by saying the error was 1/400 of the measured value because the meter itself has
a certain resolution, and we saw that at lower reading values it had a much higher jitter
proportionally. For the values that we read on the meter (which were in the neighborhood
of 5) I would estimate that the error from the meter’s instability was around δmeter = ±0.02.
I will use that number as a constant source of error over the whole 0 to 360 degree range.
The error in our ability to read the meter accurately varied between δread = ±0.02 and
±0.04 depending on where precisely the needle was in relation to the meter’s tick marks.
If the needle was directly over a tick mark it was generally easier to read, for example. I
recorded this error for each measurement and they are incorporated individually into the
χ2 value.
Another source of error was our ability to accurately measure the rotation of the analyzer. The analyzer could slip back and forth in its holder such that even when a certain
value’s tick mark was lined up with the reference mark the actual tilt could be several degrees different. I estimate that 2 degrees was the uncertainty. Since the measured intensity
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value’s rate of change depends on the angle, the error in measured value from this constant
2 degrees error changes with angle as well. It is larger for the angles where measured
intensity is changing rapidly and goes to zero for the extrema. The error from this source
is calculated by
δ
δα =
(I0 cos(α)2 ) = 2δα I0 cos(α) sin(α)
δα
The sum of each of these three errors was added in quadrature to find the total error.
q
2
2
+ δα2
δ = δmeter
+ δread
In our best fit model and in our data, the minimum value was not zero as it should have
been. This is because some ambient light was able to make it into the meter even when the
polarizer and analyzer were at right angles to each other. Perhaps we should have zeroed
the meter with ambient conditions instead of with a condition of total darkness.
Another possible source of error which we investigated was the polarization inherent in
the light from the projector. Using one of the polarizers (#8) we found that the maximum
light level measured by the detector was 11.6 ± 0.1 and the minimum was 10.5 ± 0.1. This
min
gives a degree of polarization (1 − max
) of 0.095 ± 0.012. It is likely that the polarization
from the projector slightly affected our data and caused a larger error, but in precisely
what way I do not know.
Finally, we measured the non-ideal nature of our polarizers, defined by the transmissioncorrection-factor F , defined as the observed maximum intensity divided by the expected
max intensity. Without any polarizers, our light meter read 41.8 ± 0.1. When we placed
the first polarizer in the beam, it should have read half of that, because
Z 2π
1
cos2 (α) dα = 1/2
2π 0
However the first polarizer, our #8, read a maximum of 11.6 ± 0.1 and the second, #2,
read a maximum of 11.2 ± 0.1. These correspond to F of 0.555 ± 0.005 and 0.533 ± 0.05,
respectively (an ideal polarizer would have F = 1). Multiplying 41.8 ∗ 0.5 ∗ .555 ∗ .533
gives a number very close to our observed maximum value when both polarizers were in
the beam of 6.2.
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Conclusion
When I first plotted the data for Malus’s law, I was quite worried to see that χ2 /DOF was
roughly 28. That was when I realized I had to incorporate the error from not knowing α
and from the meter’s instability. After optimizing again, I obtained a χ2 /DOF of 1.78,
which is still a bit too large to call the experiment a total success, but I believe this is a
very good value because it doesn’t yet take into account sources of errors and effects that
were not modeled, such as the ambient light in the room (which may have changed), and
polarized light from the projector.
If I were to do this experiment again, I would have been much more careful about
the error in α because that was by far our largest contribution to errors. It would have
helped to only turn the analyzer in one direction to lead up to a measurement rather than
turning it back and forth, so that this way it would always be in the same position inside
its housing when we took a measurement. It would have also been helpful to take more
readings more closely spaced, so that we make sure we hit the actual peak and also to have
a better resolution on the slopes of the graph, where data points are more sparse. However,
that would take much more time so I suppose every 10 degrees is a good compromise.
Looking at the graph, the plots of data and fit are almost on top of each other, which
leads me to believe that the problem with our data right now is that we need to incorporate
a few more sources of error, rather than Malus’s law actually being of a form having
dependancies other than cos2 (α). So, while our errors were not quite spot-on, we did show
that the intensity of light passing through a set of polarizers varies as cos2 (α), confirming
Malus’s law.
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