Download Polarization and Vector Dot Product

Polarization and Vector Dot Product
Developers
AB Overby
Objectives
The objectives of this experiment are to observe the effect of polarization on an electromagnetic
wave by measuring the amount of light passing through two polarizing filters.
Preparation
Background
Read sections regarding electromagnetic waves in textbook. Also, read sections concerning opamps in textbook and read description of LED’s in Section 3.13 of this text.
Electromagnetic waves exist as transverse waves. Meaning they travel, or propagate, in one
direction and consist of electrical and magnetic fields oscillating perpendicular to the direction of
propagation and to each other. A transverse wave can be polarized or un-polarized. An unpolarized wave has electrical and magnetic fields that oscillate in all directions while a polarized
wave has distinct directions, or orientations, of oscillation. Depending on the orientation of its
electric and magnetic fields, a transverse wave can be linearly, circularly, or elliptically polarized.
The equation for a transverse wave is given in eq. 1.
(1)
A wave is linear polarized when = 2n (n = 1,2,3….). Linearly polarized waves will be studied
during this experiment. A sketch of the amplitude of a linearly polarized transverse electric field
is shown in Figure 1.
Figure 1: Sketch of the amplitude of the electric field of a linearly polarized transverse wave
(black line) propagating in the z direction. The blue and red lines are the amplitudes of the
components in the x and y directions, respectively. (From
http://en.wikipedia.org/wiki/Polarization_%28waves%29)
Since polarized waves have direction and orientation, combining multiple waves causes
opposing orientations to cancel each other out. The resulting wave is actually the vector dot
product of the two combined waves. Both nature and modern technology take advantage of this
property of polarization.
Light is an electromagnetic wave and propagates as a transverse wave in free space and, as
such, is subject to the properties of polarization. At its source, like the sun or a lamp, light is unpolarized. It will remain un-polarized until running into a polarizing medium. Polarizing mediums
come in all forms, some absorb unwanted orientations while others reflect, refract, or scatter
differing orientations.
The Earth’s atmosphere actually partially polarizes entering sunlight. Both nature and modern
technology take advantage of this polarizing effect. Anti-glare coatings and polarizing sunglasses
work by filtering out light that is polarized from reflecting off of flat smooth surfaces, such as a
windshield or the surface of water. Some 3D viewing technologies work by polarizing the
projector light and having viewers wear special filtering glasses. Many insects are sensitive to
polarized sunlight. Some bees use polarization to navigate and map the way to food sources.
This experiment will use a set of two polarization filters that allow a specific orientation of
light, which is an electromagnetic (EM) wave, to pass through while absorbing all others. The
output of one filter is the linear polarization of un-polarized or randomly polarized light. Because
the output of one filter is linearly polarized, the amount of the light that passes through the
second filter will depend on the orientation of the second filter with respect to the first filter.
This is the same as taking the vector dot product of two EM-waves.
A common analogy is the idea of two picket fences with a rope running between the two.
When the rope is wiggled the first picket fence will block (absorb) all orientations except for the
one parallel to it the fence. The second fence will then absorb all orientations not parallel to it.
By controlling the angle of the second picket fence relative to the first, the wiggle (orientation)
and amplitude (magnitude) of the rope (EM-wave) can be manipulated as desired.
Figure 2: EM Wave Passing Through Polarizing Filter
The concept of placing a polarizer in a polarized beam of light (or EM wave) and measuring
the output intensity versus the input intensity is known as Malus’ Law (eq. 2) and is described as
such:
(2)
Where I is intensity of a plane wave as it exits the polarizer, Io is the intensity of the wave as it
enters the polarizer, and is angle of polarizer with respect to angle of polarization of wave
entering the polarizer. Note that intensity, which is expressed in terms of power/unit crosssectional area or power/unit emission angle, where the intensity of an EM field is related to the
amplitude of its electric field by the following equation:
Where is the phase velocity of the EM field, r is the relative dielectric constant of the medium
(air in this case) in which the light is propagating, o is the vacuum permittivity, r is the relative
permeability, c is the speed of light in vacuum, n is the index of refraction of the medium in
which the light is propagating, and E is the amplitude of the electric field of the EM wave.
Another measurement of polarization is the degree of polarization and it can be given by the
equation:
(3)
Imax and Imin are measured by finding the largest and smallest amplitude of the intensity of light
as it exits the polarizer as the angle of the polarizer is rotated with respect to the angle of
polarization of the plane wave of light and the intensity of light entering the polarizer is held
constant. I.e., Imax and Imin occur when in Malus’s Law is equal to 0and /2, respectively.
Since V is the ratio of the difference of between Imax and Imin divided by their sum, the closer the
degree of polarization is to 1.0 the more ideal the polarizer is. Both of these equations will need
to be modeled in MATLAB, along with a graph of Malus’ law. Figure 3 shows an expected output
of these two equations when the polarizer used is non-ideal and allows some light to pass
through when is equal to /2. The degree of polarization V printed below the graph in Figure 5
is calculated using the maximum and minimum values of the beam intensity in the plot.
The light source that will be used in this experiment, a light-emitting diode (LED), is a source
of un-polarized or randomly polarized light. The LED will be mounted above the ANDY board and
should be taped to the top of the two-piece fixture such that the emitting region is circled in the
middle of the two-piece fixture that holds two polarizers and is pointed down through the
fixture. It is important that the alignment of the position of the LED with respect to the upper
piece of the fixture remains constant during the experiment. To drive the LED, a 1 k resistor
should be used to limit the current through the LED to approximately 7mA. This resistor may be
decreased in value to no less than 270, if the signal measured at the output of the
photodetector package is very small. Figure 4 shows how the LED should be connected.
Figure 3: Plot of the intensity of the plane wave exiting a polarizer as a function of the angle
between the polarizer and the angle of polarization of the light as calculated using Malus’s Law
Figure 4: Circuit to drive the LED at a constant current
For this experiment you will need to configure the OPT101 photodiode with transimpedance
amplifier as displayed in Figure 4. Note that pin 2 is floating (i.e., will not be connected) and pin 5
is the output of the amplifier integrated in the same package as the photodetector.
Measurements of the intensity of light transmitted through the polarizing films will be taken at
pin 5. Please keep lengths of wire used to connect power to the OPT101 to a minimum as the
filter mount will need to fit over the photodiode. Whenever possible, make a connection to a pin
on OPT101 at a point outside of the area in which the fixture will sit as seen in figure 5.
The fixture holds two polarizing filters, one in each of the two pieces. The filter in the upper
fixture is used to select a specific polarization of light from the set of polarized photons emitted
by the LED. The second polarizer in the fixture, located in the lower half of the fixture, will be
rotated with respect to the upper filter. Each filter will absorb all orientations not matching
their orientation, therefore the maximum amount of light out of one filter will be 50% of the
total light intensity emitted by the LED. Io can be found by placing the two polarizing filters in a
0o orientation. Note that the filters are less than ideal and absorb or reflect more light than an
ideal case.
Figure 5: OPT101 with connections outside fixture
Figure 6: OPT101 Pinout
References
Polarization (waves). Wikipedia [Viewed on March 24, 2011]
http://en.wikipedia.org/wiki/Polarization_%28waves%29
How Do We Know Light Is A Wave? Light Waves and Color – Lesson 1. The Physics Classroom.
[Viewed on March 24, 2011] http://www.physicsclassroom.com/class/light/u12l1e.cfm#trans
Materials
The equipment and components required to perform this experiment are:
 ANDY Board
 DMM
 1 ea 1 kΩ resistor
 1 ea OPT101



1 ea red LED
2 ea Polarizing film
1 ea Polarizing film mount
Procedure
Analysis:
1. Write a MATLAB program to graph the intensity of light as a function of the difference in
angle of polarization between the first and second polarizing filters. The program should be
written such that the user can enter Imax and Imin. The output of the MATLAB program should
be a plot of Malus’ Law along with a printout of the values of the maximum and minimum
intensities and V, the calculated degree of polarization.
Measurements:
2. Build the photodiode and LED circuits shown in Figures 4 and 6.
3. Place the two-piece filter without the polarizing filters over the photodiode and tape the LED
in place on the upper piece of the fixture. Measure the voltage output of the photodetector
after aligning the position of the fixture to maximize the voltage from the
photodetector/amplifier assembly as measured at pin 5.
4. Calculate the intensity of the light that reaches the photodetector using information about
the area of the photodetector and responsivity of the detector at the LED emission
wavelength in V/W provided in the OPT101 datasheet. Assume that the light intensity is
uniform over the photodetector, an assumption that is valid if the angle of emission from the
LED is reasonably broad and/or the distance between the LED and photodetector is large.
5. Carefully insert one circular filter into the fixture that has the LED taped to it, making sure
that the face of the filter is perpendicular to the wall of the fixture and that you do not
disturb the location of the LED. If needed, place a small amount of glue along the edge of
the polarizing filter to hold the filter in place in the fixture.
6. Place the two-piece filter together. Measure the voltage output of the photodetector after
aligning the position of the fixture to maximize the voltage from the photodetector/amplifier
assembly as measured at pin 5. The measured voltages should be less than that measured in
Step 3.
7. Carefully insert the second circular filter into the lower piece of the fixture, making sure
that the filters are parallel with each other when the two fixtures are put together and that
you do not disturb the location of the LED.
8. Set the two piece fixture together such that the pin of one fixture is inserted into one of the
end holes in the face of the other fixture. This will be the 0° position. Place the two-piece
fixture over the photodetector, moving the fixture until you have maximized the voltage out
of the photodetector. Record the output voltage and calculate the intensity of the light
detected.
9. Rotate the two pieces of the fixture so that the pin in one of the pieces is aligned with the
next hole in the other fixture. This aligns the two filters such that the difference between
the angle of polarization of the two filters is now 30o from the original position. Align the
fixture over the photodetector so that the voltage signal is maximized. Record voltage
measurement and calculate the intensity of the light detected. The measured voltage should
be less than that measured in Step 6.
10. Repeat Step 9, moving the pin position to the other holes so that the difference in the angle
of polarization is increased to 60° and 90° of the original position. The measured voltages
should be less than that measured in Step 5.
11. Normalize all of the intensities calculated in Steps 6, 8, 9, and 10 to the intensity of light from
the LED measured in Step 3.
12. Using the values of normalized intensity of light at each angle (0°, 30°, 60° and 90°), generate
a graph of intensity vs. difference between angle of polarization of the two filters. Fit the
data in the graph to Malus’ Law and determine Imax or Imin. Note that it is unlikely that any of
the light intensities that you calculated will be equal to Imax or Imin because of the random
process used to position the two filters with respect to each other in Steps 6 and 7.
13. Calculate V the degree of polarization. Compare the value of V with that obtained by five
other teams in your class. Calculate the average and standard deviation for V.
14. Determine the difference in the angle of polarization between the two filters when the two
pieces of the fixture were at the 0° position (Step 7).
15. Ideally, intensity calculated in Step 6 is equal to 50% of the calculated intensity of light from
the LED without the filters in place, as measured in Step 3. Furthermore, Imax should be equal
to the square of the normalized value of the intensity calculated in Step 6. Calculate the
percent deviation of these two quantities and provide an explanation for the differences
observed.