Background
My project this summer stemmed from what was originally a very lofty project idea. When Dr. Noé first introduced
us to the concept of acousto-optics on our first day in the LTC, I immediately started thinking about the real
world applications of such a field. I wanted to develop a sensor that would be able detect brain
infections by evaluating the change in pressure in the spinal cord due to a fluid influx. After a week or so of
reading into this, I unfortunately (but fortunately) came to the conclusion that using a Fiber-Bragg grating (FBG)
was almost out of the question. Although I wouldn't be able to pursue making a pressure sensor, the idea of
real-world applications to whatever research I ended up condcting really stuck with me.
Coming into this summer, I
had a background in polymers because of my research in the chemical engineering department at UC Santa Barbara the
year before. After stumbling upon a paper by Velasquez et. al., I decided to explore the overlap between polymers
and optics. Within a week, I was introduced to the many applications of birefringent materials. If properly
used, they could create bandpass filters to allow certain wavelengths of light through, polarizing them onto
certain planes. I soon came to learn that these filters were used in display applications everywhere and
decided to look more into this.
Introduction
Birefringent materials have two orthogonal axes with slightly different indices of refraction through which
light can travel. Depending on the angle of the incident light, its polarization state can be altered. There are
three types of polarized light: eliptical, circular, and linear. For the purposes of this project, I mainly focused
on linearly polarized light, which propogates along one given field, and circularly polarized light, which maintains the same amplitude but changes
direction azimuthally as it propogates. The x- and y-components of the electric field vector of linearly polarized
light maintain a 180-degree phase shift, while these components maintain a 90-degree phase shift in circularly
polarized light. The relative phase shift between the x- and y-components is known as retardance, which is often dependent upon
wavelength.
Birefringent filters utilise changes in the state of polarized light in anisotropic materials to transmit
different intensities of light at certain wavelengths. Such filters are employed in display and color filtering
technology, as well as wavelength division multiplexing systems for optical communications.Although generally considered to be extremely
intricate and expensive, birefringent materials can be as simple as sheets of polymer film. For my project I used the optical
properties of
cellophane tape to explore and create such devices.
Polymer birefringence is predominantly the result of manufacturing processes. When these
materials are subjected to stressed from stretching and molding processes, the induced
stress shows up as birefringence in the finished materials. When the monomers chains form
parallel strands, one refractive index is parallel to the main chain consisting of carbon
bonds, while the other is
perpendicular. Light travelling perpendicular to the main chain will encounter double
bonds, retarding the light more.
My project came in three steps. First, I was to characterize the birefringence of the material I was using ("High Performance" Scotch brand
clear cellophane packaging tape). Then, with those parameters, I was able to mathematically model using Jones calculus birefringent filters that
could polarize the incoming light in a certain manner at particular wavelengths. The third and final step was building the filters and observing
their
effects on transmitted light.
Characterizing Birefringence
In order to determine the retardance of the tape at particular wavelengths, I had to
simulate a set-up that could measure the intensity of light coming through the tape at
certain wavelengths, thicknesses of the tape, and angles of polarization. I created samples
of parallel tape that would act as retarders at specific wavelengths for the incident
light. The set-up consisted of a halogen bulb and two linear polarizers. The polymer
samples were placed in between either the crossed of parallel polarizers. Due to the
brifringent properties of the thin films, at different oritentations of the polarizers,
different wavelenths of light were transmitted. The films displayed different colors with
each oritentation of the polarizers.
Using a ThorLabs CSS100 Spectrometer, I recorded the intensity of the light transmitted
at different wavelengths. The samples I created produced a series of periodic oscillations,
with complete extinctions at the minima and maxima transmission at the maxima. Below is an
example of the transmission data of a ten layer sample between the two parallel
polarizers.
I then had to normalize the data to determine the retardance of light at particular
wavelengths and eventually calculate the order of rotation for each sample. I did this by
dividing the intensity of the output light by the incident light between two parallel
polarizers to characterize the retardance of the samples at particular wavelengths. At the
minima for the parallel polarizers and the maxima for the crossed polarizers, the filters
behaved like half-wave plates, due to the 180-degree rotation of the light. The
intersection of the crossed and parallel transmission graphs represent quarter-wave plates
because their behavior doesn't change, despite the rotation of the linear analyzer. The
graph below is the normalized data for a six layer sample.
After gathering the data, I realized that beyond just understanding at which wavelengths
the polymer acted like a half- and quarter-waveplate, I could also determine the order of
retardance. After discussion with Marty and Dr. Noé, I used a technique which I've coined
as the "k-plot", to determine the total retardance of the extinction points of the samples.
I plotted hte inverse-wavelength using the following equation:
Using a trial-and-error method, I graphed several orders of odd multiples of pi. A line
drawn through those points that passes through the origin indicates the proper k-values.
The ten layers of tape, for example, was determine to have an 8
th order of
retardance. Its k-plot is shown below.
Modeling Filters
Jones calculus is a powerful tool that I really devoted a lot of my summer to learning.
I used this branch of mathematics to theoretically design the filters. In theory, setting
up a series of equations would allow me to find an angle theta at which I could rotate two
samples of cellophane tape relative to each otherto produce circularly polarized light.
After modeling, I discovered that it is impossible to produce circularly polarized light
from linearly polarized light at the wavelength it behaves like a half-wave plate. There
does exist quarter-wave plate retardation in the cellophane samples I create, as indicated
in the earlier graph, but it can only be produced at those wavelengths.
Unexplained Results
In principle, as demonstrated through Jones calculus, it is impossible to produce
circularly polarized light from linearly polarized light with only half-wave plates. Yet,
when two six layer samples were rotated 165 degrees relative to each other, the filter
yielded no visible maxima. Instead, the intensities leveled out to the intensities of the
previously established quarter-wave retarder wavelengths, like we would expect for
circularly polarized light.
Future Work
In the future, I really hope to replicated and look further into my unexplained results.
Clearly there is something going on in the graph above, even if it isn't circularly
polarized light.
I also would like to further examine the retardance caused by the cellophane tape by
using an interferometer to observe the change in the fringe patterns induced by the
birefringence of the material.
Studying achromatic waveplates and figuring out how to make them with polymers (as I did
with wavelength-dependent birefringent filters) would also be a great way to continue this
project in the future.
References
[1] P. Velasquez et. al., “Interference Birefringent Filters Fabricated with Low Cost
Commercial Polymers,” Am. J. Phys. 73, 357 (2005).
[2] F.L. Pedrotti & S.J. Pedrotti, Introduction to Optics (Second Edition).
[3] Special Optics, "Retardation Plate Theory".