lab 6 - interferometry experiment
TRANSCRIPT
Experimentally Finding the Index of Refraction of a Glass Prism via a Twyman-
Green Interferometer
CREATED AND PERFORMED BY BRIAN HALLEE
Performed Tuesday, November 16, 2010
Historical Background
The specific set-up used in our experiment is known as the Twyman-Green
Interferometer named after its discoverers: Frank Twyman and Arthur Green. The two
scientists (an engineer and chemist, respectively) introduced the apparatus in 1916 in order to
expand what is known as a Michelson interferometer to measure the refraction properties of
lenses and prisms.1 Hence, it is important to first touch this foundational experiment before
moving to our own apparatus. While the mathematics and physics of the contraption will be
explained thoroughly in the following theory section, it is worthwhile to note that this optical
device has made advances and tested predictions made in a slew of areas outside the realms of
strict optics. Albert Michelson devised the experiment in 1881 in order to observe fringes in
light waves theoretically known to occur.2 At this point in the 19th century, the scientific masses
were still largely, if not entirely, convinced that light was strictly a wave. Thomas Young had
successfully proved this (to a point) utilizing his “double-slit experiment”. While in reality
Young simply used a slip of card and a light beam to demonstrate light interference, a slightly
more involved technique eventually won the honor of wielding his name. This was a primitive
interferometry experiment that allowed light to enter in though two different slits, interfere,
and hit a photo-sensitive plate that displayed the differing levels of intensity along a horizontal
axis.3 Thus, Michelson expanded on this idea and developed an apparatus that was to be
dynamic in that the fringes could be moved or altered in some controlled manner. It uses a mix
of perfect mirrors and half-silvered mirrors to achieve a merging to two separate light beams
right at the point of detection on a screen. As such, the “interferometer” is actually a broad
term for a slew of lenses and mirrors (with a light-source included) that together generates an
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interference pattern on a screen. With such a relatively simple set of lenses, physicists have
been able to use it to experimentally test special relativity, discover the hyperfine structure,
measure lunar tidal effects, and most importantly standardize the meter based upon the speed
of light. Perhaps the most interesting use of his interferometry included the partnership of
Edward Morely (1887) in attempting to prove or disprove the “aluminiferous aether” thought to
be the medium for allowing light to pass through space (very much analogous to air’s role in
sound waves). The upper limit on light’s speed was still hotly debated at this time. Thus,
Michelson and Morley applied interferometry to the issue by attempting to measure a delay in
light beams meeting at a point. This was accomplished by assessing fringe shifts where, if the
“aether” was stationary relative to the sun, the Earth’s motion should produce a shift of roughly
4% of a single fringe. While the initial experiments the two men underwent utilized equipment
with far too much inaccuracy to ever measure the aether, the procedure was quickly improved
by Michelson and other scientists who inadvertently led the experiment to be labeled “The
Most Famous Failed Experiment”. This was naturally due to the fact that no aether need exist
to propagate electromagnetic waves. Taking the Michelson-interferometer notion one step
further, the Twyman-Green interferometer replaces the adjustable mirror with a glass prism
that can be tuned to force the light to pass through more of its glass. After passing through the
glass, the light returns via a stationary mirror. Frank Twyman, an electrical engineer by trade,
met up with Alfred Green who was the lead foreman at the optical shop at the University of
Liverpool.4 Together they expanded the Michelson interferometer to measure properties such
as the index of refraction for different materials (We use a BK7 glass prism in our own
experiment). Another difference between the two was Twyman’s strict use of a point source of
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Figure 1: The left side depicts a collimated beam of light.
Source: http://en.wikipedia.org/wiki/File:Collimator.jpg
light (e.g. laser) while the Michelson apparatus is able to be used with either a point source or
extended source of light. Overall, these pioneering interferometry scientists have introduced a
whole new standard of accuracy in measurements and have applied that accuracy to many
diverse fields.
Theoretical Basis
The physics behind the Twyman Green interferometer lies entirely in introductory and
modern optics. As previously noted, the “interferometer” is, in fact, a blanket-term for the
arrangement of lenses, apertures, the light source and screen (See figure 2 on the following
page). While our specific equipment will be more deeply described in the apparatus section of
this report, it is fitting to briefly describe the theory of how these individual pieces form a very
precise tool when brought together. Firstly, the light source must be in a special form denoted
as “collimated”. It need not necessarily take the form of a laser. However, this is typically the
more convenient and cost-effective option, and it is the source of choice in our own
experiment. The word collimated simply denotes the “rays” (or
paths of photons) of light are parallel and facing in the same
direction, and the process is depicted in figure 1. The
collimation is achieved by passing the rays of light through a
converging lens that causes the light to converge to a point.
After meeting at this point, the rays will begin to diverge and enter a projection lens that is
strategically placed a known distance from the point depending on its focal length. After
passing through this second lens, the beams will be absolutely parallel to each other and can be
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justly classified as collimated. As depicted in figure 2, once formed, this collimated beam of
light is split apart by a special device set at the center of the apparatus. This is a carefully
constructed perfectly flat piece of glass with a thin film of silver on the right side. This silver
acts to send the beam in two separate directions by reflecting any rays that interact with a
silver particle. Silver acts as a perfect mirror.
Consequently, considering the glass is
positioned 45° with respect to the incoming
beam, the beam is reflected 90° upward
toward a mirror. The remaining beam passes
through the splitter unimpeded towards the
prism. If the silvered glass is manufactured
correctly, this will split the beam almost exactly
in half. The ultimate purpose of this separation
is to have the two beams interfere again at the
detector (or screen) shown in figure 2 in order to observe the constructive and destructive
interference via fringe forming. Our Twyman-Green interferometer introduces more
complexity than seen in the figure above by passing the unimpeded beam through a glass prism
before reaching the mirror. The prism itself deserves a distinctive amount of attention in order
to understand how its index of refraction can be determined from this experiment, and how it
determines the interference effects detected after the beams have been amalgamated. As
depicted in figure 3, the unimpeded beam actually enters into a triangular prism at a 45° angle
relative to the normal of the face. The figure gives a depiction of the prism and its effects when
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unimpeded beam actually enters into a triangular prism at a 45° angle relative to the normal of
the face. The figure gives a depiction of
the prism and its effects when placed at
two different depths in the beam. The
separation distance is denoted by the
variable∆ x. You can easily infer from
figure 3 that when a beam is forced to
travel through more glass, it travels a lesser
distance overall than one that travels
through less. The double arrows in the figure depict the fact that once the beam hits the mirror
it will undergo perfect reflection and travel the exact same path in the opposite direction. We
have, thus far, treated the fact that light is perfectly reflected by a mirror as an axiom. This is
due to the fact that it simply makes classical mechanical sense, and there is little mathematical
theory to be applied to a flat mirror. However, it is not exactly clear how light undergoes a
change in direction (as depicted in figure 3) upon entering and exiting solid glass. Thus, we
have arrived at the point where mathematical rigor need be introduced to our interferometer.
The law governing the behavior of light at the junction of two materials is known as Snell’s Law,
and it was derived by Willebrord Snellius in 1621 who was the first to place the phenomenon in
strict mathematical terms. It is written as follows:
n1∗sin(θ¿¿1)=n2∗sin(θ¿¿2)¿¿ (eqn. 1)
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Figure 4: The constructive and destructive interference of ripples in a water tank.
Source: http://en.wikipedia.org/wiki/File:Two-point-interference-ripple-tank.JPG
The thetas in Snell’s Law represent the angles between the light beam and the normal of the
appropriate medium. The n-values represent what is dubbed the index of refraction of the
medium. It is a fundamental constant that differs with every material. Therefore, referring to
figure 3, light passing through the prism in the top-state would have a θ1 = θ and a θ2 = Φ.
Naturally, if the indexes of refraction and/or incidence angles are known, equation 1 allows you
to isolate a single variable and observe how the light will respond to the medium change. Thus,
Snell’s Law suggests that path-change is the norm for light at junctions of two mediums of
different indexes of refraction (Which is the case for glass/air), and gives reasoning as to why
this is the case in figure 3. Now that a basic theory has been applied to the prism arrangement,
it is imperative to bestow mathematical order to the interference of the beams once they meet
again at the detector (or screen, in our case). The beam that
was simply reflected 90° and reflected again to the
screen will have traveled a vastly different distance (in
terms of wavelengths) than the beam that enters the
prism. Thus, holistically, it would seem that physics
would require us to have our interferometer calibrated
and positioned to accuracy on the order of nanometers.
This would be an arduous, if not impossible, task for
undergraduates to partake in. However, if we exploit
the wave-like property of light we can arrive at a
solution that rests on what is known as constructive and
destructive interference. This phenomenon, depicted in
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figure 4, occurs in wave propagation when two waves interfere at particular phase-differences.
If both waves are in phase they will exhibit constructive interference shown by the light peaks
in figure 4. Waves become out of phase when they meet after having traveled paths that lead
them to become half a wavelength out of step with each other. Thus, peaks meet troughs and
the entire wave is canceled perfectly. Consolidating this notion mathematically, we arrive at
the formula below:
r2−r1=mλ (eqn. 2)
R1 & R2 = The distance light travels in that specific path before being detectedm = The total number of fringes displayed or detectedλ = The wavelength of the laser
Equation 2 describes the fact that when two waves meet, if their traveled-path difference is
exactly equal to an integer number of wavelengths (denoted by m) the resulting wave is one of
constructive interference. In our experiment, we are focused on observing how an increased
amount of prism glass will affect the fringes on the detector. It has been qualitatively shown via
figure 2 that this causes the path to increase in distance traveled. Thus, we are actually
concerned with the change in path difference after the initial path difference. Therefore, we
can take equation 2 one step further as
∆ L=∆m∗λ (eqn. 3)
where ΔL represents the change in path difference over the course of our experiment, and Δm
denotes the number of integer wavelengths the path difference spans. This, of course, means
∆m is the exact amount of bright fringes that are detected or displayed over the procedure,
and that fact will come in handy when it comes time to gather data from our experiment. After
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deriving equation 3, we no longer care about the beam that doesn’t pass through the prism, as
its path length is of no significance and it simply serves as a control beam to interfere with our
“prism beam”. Armed with this equation, we can apply it solely to the geometry of figure 3,
and use Snell’s law to come up with the index of refraction of our prism. We have mentioned
the fact that the prism can be placed further into the beam, but we have yet to give any
inclination as to how this is achieved. In our experiment, as with most Twyman-Green
interferometers, the prism is clamped onto a table that is coupled to a micrometer-driven, 1-
direction translation arm that can move the prism with a precision of approximately 0.25 μm.
Considering the wavelength of red laser light is in the range of 600-700 nm, adjustments on the
order of micrometers is absolutely necessary. Using figure 3 as our reference, we can find the
distance light travels in the two instances of the prism by using the variables given in the
diagram as follows:
Path Distance (Top )=2∗(∆ x+n∗L1+δ+d ) (eqn. 4)
PathDistance (Bottom )=2∗(n∗L2+d ) (eqn. 5)
∆x = Distance from the defined origin of the prismn = Index of refraction of the glassL1 = The distance light travels in the top instance in the prismL2 = The distance light travels in the original instance in the prismδ = The slightly additional distance light travels from the top-instance of the prism to the reflecting mirrord = The distance from the prism to the reflecting mirror
The reason for the multiplication by two in the above two equations is due to the fact that after
reflection the light travels the exact same path through the prism. Thus, in order to fully
account for the path change this “doubling-back” must be accounted for. Now considering the
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prism is the only dynamic piece in our experiment, it is also the only segment that contributes
to the path-length change of the light. Thus, we can substitute equations 4 and 5 into equation
3 by subtracting them as follows:
2∗(∆ x+n∗L1+δ+d )−2∗(n∗L2+d )=2∗(∆ x+n∗L1+δ+d−n∗L2−d )
→2∗(∆ x+n∗(L1−L2 )−δ )=∆ L=∆m∗λ (eqn. 6)
Using the geometry of the prism in figure 2 and some trigonometric identities, we can achieve
the following equation for ΔL:
∆ L= n∗∆ xcos (θ−φ)
−∆x−n∗∆ x∗sin2(θ−φ)
cos (θ−φ) (eqn. 7)
Finally, after substituting equation 7 into equation 6 and rearranging terms to solve for n, we
achieve a relatively simple equation governing the index of refraction for our prism:
n=√22
∗√( ∆m∆ x∗λ+1)
2
+1 (eqn. 8)
Thus, the measurement of the index of refraction boils down to two variables which are all very
easy to measure with a proper Twyman-Green apparatus. The wavelength (λ) is treated as a
given considering practically all laser manufacturers print the wavelength direction on the
device to roughly 4-5 significant figures. The distance the prism moved from the defined origin
(Δx) is measured by keeping track of the number of micrometers moved on the micrometer.
For equation 8 to work correctly, both Δx and λ must be in meters. Finally, perhaps the most
troublesome aspect of this experiment is achieving a value for Δm. In our experiment, the
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detector in figure 2 is replaced by a paper screen where we are able to visibly depict the
interference fringes with the naked eye. The value for Δm is acquired by observing how many
bright fringes move across a reference point over the course of the prism movement.
Equation 8 was entirely developed resting on our own geometry and experimental assumptions
(such as the assumption that the prism is truly 45 degrees to the incoming beam). While there
is nothing inherently wrong with our derivation, it is fitting to compare it with empirical data
that is long-accepted in the community for the borosilicate crown glass (BK7) used to form our
prism. This reference takes the form of an equation known as the Sellmeier Equation, and it
was developed as an empirical relationship between the refractive index and wavelength for a
transparent medium by W. Sellmeier in 1871.5 The equation is shown below:
n2=1+B1 λ
2
λ2−C1
+B2 λ
2
λ2−C2
+B3 λ
2
λ2−C3 (eqn. 9)
The constants B# and C# are experimentally determined coefficients that are unique to the
transparent medium. Luckily, large databases have been compiled on the coefficients of
numerous mediums, and BK7 is one of the most defined and accurately studied mediums of
them all. Therefore, due to the faith we hold in the accuracy of our laser’s wavelength and the
ideal nature of the Sellmeier Equation, we can comfortably use this as the standard for
refraction index in order to determine if our procedure or calculations are faulty using our
theoretically-derived equation 8.
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Figure 5: The schematic for our own Twyman-Green Interferometer
Source: Lab Handout Packet
Apparatus
The Twyman-Green interferometer we
utilized deviated slightly from the “ideal”
interferometer described in the previous
section. The actual set-up used in our
experiment can be viewed in figure 5, and
we will describe each component starting
from the top. First, the entire
interferometer was slot-mounted onto a
shock-absorbing, air-equipped, auto-
leveling table. This ensured that all the
components were directly in line with
one another and slight bumps or vibrations
would not disturb the interference to any
significant degree. The laser device we utilized was mounted to the table, and emitted a strong
collimated red laser beam with a frequency of 623.8nm. This light was fed into a steering
mirror that was placed at a 45° angle with respect to the incoming beam. This mirror was a
perfectly flat reflecting mirror that re-routed all incoming light 90° to the right. In order to
ensure near-perfect collimation, the beam was then fed into an objective lens with a small
enough focal length to be utilized for a microscope. This caused any non-parallel beams to
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diverge to a point that was exactly a focal length away from the projection lens. Once the rays
reached the projection lens they were perfectly collimated. The key difference in our apparatus
with respect to the ones previously mentioned is the angle at which the half-silvered beam
splitter was oriented. While the exact angle with respect to the incoming beam was unknown
to us, the splitter did not deflect the beam in a right angle. Fortunately, the apparatus was
constructed prior to our arrival, as this saved us from guessing how the beam reflected off of
this splitter. This also caused the screen to be positioned at an odd angle (roughly 45°) as
depicted in figure 5. Unlike the “detector” shown in figure 2, we utilized a paper screen which
was hoisted using an arm that clamed directly to the table. The fringes observed with this
screen were roughly 1-2 mm in size. Thus, no magnification or extra lenses were needed in
order to count the fringes over the course of the experiment. As mentioned earlier, the prism
seen in figure 5 was mounted on a table that was adjustable to move in a translational direction
using a micrometer arm. After setting some arbitrary origin as zero on the micrometer, the
knob on the device was twisted in order to cause the light beam to pass through more glass.
The device was pre-constructed to ensure the face of the prism met the beam at a perfect 45°
angle. Following the prism, a flat reflecting mirror is required to contain the beam and pass it
back through the prism and return it to the splitter. All of the individual components make up
the interferometer as a whole, and cause the beam to be near-perfectly contained within the
device after exciting the light source.
Procedure
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Considering the device was fully constructed an operational before arrival, the
procedure required to undergo the experiment was relatively simple and uninvolved. The first
step included making sure that all the components seen in figure 5 were indeed in their proper
arrangement on the table. This does not, however, infer that any adjustments were to be
made to the components if they were skewed as the alignments and positions are extremely
precise. Consequently, an unjust perturbation of a single lens or mirror could potentially
destroy the delicate fringing effects needed to analyze the prism. After ensuring the quality of
the apparatus, the laser device was switched on and a few seconds passed to allow for the
device to warm up. Once the beam was visible on the screen, the micrometer was twisted until
the arm was locked at zero. At this point, the experiment could begin. Located on the screen
was a tiny, but easily discernable, black dot that was used as a reference point for the passage
of bright fringes. The screen was adjusted slightly so that once the micrometer was set to zero,
the dot rested on a bright fringe. After this was completed, the micrometer was turned slowly
so that the number of fringes that passed by the dot could be tallied in the head of the
experimenter. After reaching a certain amount, the procedure was stopped, and the number of
fringes and distance on the micrometer were both recorded in a notebook. The micrometer’s
knob repeated every 25μm. Thus, four complete turns on the device signified a 100μm change
in distance of the prism from its origin. The meter could be read to an accuracy of roughly
0.25μm (250 nm). Thus, the prism distance garnered accuracy very close to the order of the
beam’s wavelength. This procedure was followed four times in order to reach fringes of 100,
150, 200, and 250. A simple time-saving trick after reaching the 100th fringe is to record the
number displayed on the micrometer and how many turns after that number it takes for the
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meter to return to the origin. Thus, the experimenter is able to resume counting where they left
off by returning to this point instead of repeating the fringes already counted. Once the
distances were logged for each fringe count, the experiment was completed and the laser was
powered down. In order to resolve the refraction index for the prism, calculations using the
Sellmeier equation and equation 8 were used as demonstrated in the following section.
Sample Calculations
Although there are very few intermediate steps to undertake in order to achieve an
experimental value for the refraction index, equation 8 is delicate in that all the units must be in
their fundamental form. Thus, we will work towards this via a sample calculation using the data
from our first run (100 fringes). The raw data from all four of our runs can be viewed in either
the appendix of this report, or the Excel spreadsheet on the accompanying disc. After reaching
100 fringes, the micrometer read 22.0μm. After turning clockwise and reaching the first zero,
the knob made 2 complete cycles before reaching the origin. Thus, the total distance for this
run (Δx) was
22μm+¿ 2 turns∗¿ μmturn
¿=22 μm+50μm=72 μm (1)
As mentioned, the units must be in fundamental form for equation 8 to work. We convert the
answer found in (1) to meters as follows:
72μm∗( 10−6m1μm )=7.2 x10−5m (2)
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Obviously, Δm is 100 considering that was the limit of our procedure. Therefore, the only
remaining variable is the wavelength of the beam. As mentioned in the apparatus section, the
manufacturer of our laser light source has concluded that the emitted beam’s wavelength is
632.8 nm. Utilizing a similar procedure as (2), we achieve this value in meters as follows:
623.8nm∗( 10−9m1nm )=6.238 x10−7m (3)
We now have all the variables in their necessary form to apply equation 8 to our first run. The
procedure is shown below:
n=√22
∗√( 100 fringes7.2 x10−5m∗(6.238 x 10−7m )+1)
2
+1=¿1.505029 (4)
Now, considering that natural air has a refraction index of roughly 1, this seems reasonable.
However, as stated, the truest comparison against known empirical data comes with the
Sellmeier equation. The company Schott glass maintains a database that includes the Sellmeier
coefficients for BK7 glass. These coefficients are located in the appendix of this report, and they
will be utilized to perform the calculation needed to derive the accepted value of BK7’s
refraction index. Because of the operations performed in each term of the equation, it will be
broken up into three steps below, and recombined to finally solve for the index. A final note of
interest is the fact that the Sellmeier equation was constructed assuming the wavelength to be
in μm. Thus, a quick calculation used to convert (3) into micrometers is shown below:
6.238 x10−7m∗( 106 μm1m )=0.6238 μm (5)
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Using (5) and the first two Sellmeier coefficients, we solve for the first term:
B1 λ2
λ2−C1
=(1.039612 )∗(0.6238μm)2
(0.6238μm)2−0.006001=1.055428154 (6)
The second term is found similarly:
B2 λ2
λ2−C2
=(0.231792 )∗(0.6238μm)2
(0.6238μm)2−0.020018=0.243989454 (7)
And finally the third term:
B3 λ2
λ2−C3
=(1.010469 )∗(0.6238μm)2
(0.6238μm)2−103.5607=−.003922328 (8)
Combining (6), (7), and (8), and adding 1 we achieve the accepted Sellmeier value (equation 9)
for BK7 as follows:
nsellmeier2=1+1.055428154+0.243989454+−.003922328=2.29549528 (9)
Finally, the refraction index is found by taking the square root of (9):
nsellmeier=√2.29549528=1.515089 (10)
This comes very close to the value we found in (4) using just one run. In the following section,
we will be concerned with the deviation of our run values from (10). Thus, it is fitting to
introduce the equations used to find these deviations and to perform a sample calculation using
run 1. The equation for % deviation is shown below:
%Deviation=nrun−nsellmeier
nrun
∗100% (11)
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Applying (11) to run 1, we achieve a % deviation as follows:
%Deviationrun1=1.505029−1.515089
1.505029∗100%=0.664013% (12)
Lastly, it will be important to average all the runs together in order to negate any minor
observation errors when reading the micrometer. The average of any set of values is found
using the following formula:
Average=x=∑0
n
xn
n
(13)
In (13), x represents a value in the series and n is the total number of values taken into
consideration. Thus, using the four n-values found (see Appendix – Raw Data) and applying (13)
to them, we find the average refraction index of our experiment as such:
nexp=14∗(1.505029+1.505029+1.505983+1.505792 )=1.5054581 (14)
Discussion
Overall, the four experimental runs produced a very precise set of results for the
refraction index of our prism. All the indexes were within one thousandth of each other, so we
can be sure that our procedure was, at the very least, uniform. Using (11) from the previous
section we can quickly find the percent deviation of each run in order to potentially develop
trends or factors that lead us to the values we garnered.
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%Deviationrun1=1.505029−1.515089
1.505029∗100%=0.664013%
%Deviationrun2=1.505029−1.515089
1.505029∗100%=0.664013%
%Deviationrun3=1.505983−1.515089
1.505983∗100%=0.601034%
%Deviationrun4=1.505792−1.515089
1.505792∗100%=0.613648%
Although the deviations are relatively small, it is important to remember the precision of the
average interferometer. Thus, all error must be duly accounted for. Firstly, I remain convinced
that a single fringe was missed while counting the first 100 fringes. Assuming this is true, the
true Δm for the first run is 101. I feel this is the case due for two reasons. First, I did
subconsciously feel as though I missed a fringe over the course of my first run. Likewise, if you
repeat (4) using Δm = 101, the refraction index more closely resembles that of the Sellmeier
result:
nrun 1=√22
∗√( 101 fringes7.2 x10−5m∗(6.238x 10−7m )+1)
2
+1=1.510518
This value only deviates roughly 0.3% from the Sellmeier value, effectively cutting the previous
deviation in half. Considering that I marked the location of each stopping point and returned to
it after plotting the data, one missed fringe early on is automatically resonated throughout all
the data points. Thus, 50% of the deviation of each experimental run is likely due to this one
missed fringe.
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When observing the deviation between different runs, we can observe a small trend as the
number of fringes is increased. The refraction index increases by roughly a thousandth upon
reaching 200 fringes. My hypothesis is that the further the prism intruded on the beam, the
more possibility there was for the beam to pass through glass imperfections and impurities.
These may have caused the actual refraction index to increase, which also caused the
experimental index to more closely approach the Sellmeier value. We can observe a more
accurate deviation value by taking the average refraction index found in (14) and applying (11)
to it as follows:
%Deviation=1.505481−1.5150891.505481
∗100%=0.64%
The reason a problem exists with such a small deviation is the fact that the index of refraction is
inherently a small varying number. In fact, almost all refractive materials have indices that lie
between 1 and 2.5. Likewise, some materials such as oxygen, helium, and nitrogen exhibit
indices that vary only in the ten-thousandth place! This is also the case for some of the organic
alcohols such as Acetone and ethanol.6 Consequently, the accuracy of measuring the refractive
index is of utmost importance as it requires many significant figures to separate one material
from another.
Nonetheless, while the Sellmeier equation is empirically based, some experts continue to argue
as to what the true value is for Borosilicate crown glass. For example, a center for occupational
research in Texas gives a value of 1.50917 for a red laser with wavelength of 640nm7. While our
value still undershoots this index by roughly four hundredths, it exhibits less than half the
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deviation of the Sellmeier value, and suggests that not all BK7 glass is made exactly the same
and that experts still argue over its true value.
Referring back to equations 2 and 3, we have yet to mention the implications if such properties
such as the laser’s wavelength was not treated as a given at the start of the experiment. If this
was the case, we could use equation three to experimentally arrive at the wavelength by
counting the number of fringes and the distance it took to achieve them. Naturally, the
wavelength would only be as precise as the tool used to measure ∆L. This is due to the fact
that Δm is an integer with an infinite set of significant figures. Thus, we would not be at a total
loss if the manufacturer had not stated the wavelength, so long as we were equipped with a
precise micrometer. Theoretically, we could also reverse this situation by utilizing equation 2 to
solve for a distance change if the wavelength is known. If we removed the prism from our
apparatus, we would again be dealing with a Michelson interferometer. With this, we could
measure a distance by moving the mirror in the direction of the beam and counting the fringes.
In this way, using equation 2, distances that are comparable to a wavelength of light can be
measured with ease, and this is precisely how the interferometer plays such a crucial role in
bestowing a large degree of accuracy to distances and their subsequent applications.
In summary, due to the properties of light being so delicately small and easily perturbed, the
Michelson and Twyman-Green interferometers are able to exploit interference to shed light on
these properties, and apply their precision to other seemingly unrelated variables. This
experiment proved that light, even at its abhorrently high speed, is able to be easily
manipulated through refractive and reflective materials, and the properties of alteration (the
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index of refraction) are able to be solved for using simple theoretical geometry and derived
fundamental laws such as Snell’s law. Lastly, we proved that equations governing this
experiment can be rearranged to indirectly measure infinitesimal properties such as the
wavelength of light or distances on the order of nanometers.
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Works CitedEncyclopaedia Britannica. (n.d.). Tywman-Green Interferometer. Retrieved November 18, 2010, from
Encyclopaedia Britannica: http://www.britannica.com/EBchecked/topic/611419/Twyman-Green-interferometer
Hugh D. Young, R. A. (2007). University Physics. Pearson Addison-Wesley.
Menzies, A. (1960). Frank Twyman 1876-1959. In T. R. Society, Biographical Memoirs (pp. 269-279). Royal Society Publishing.
Polyanskiy, M. (2008-2010). Refractive Index Database. Retrieved November 28, 2010, from RefractiveIndex.info: http://refractiveindex.info/
Scheider, W. (1986). Do the "Double Slit" Experiment the Way it Was Originally Done. Retrieved November 18, 2010, from CavendishScience: http://cavendishscience.org/phys/tyoung/tyoung.htm
University of Tennessee: Knoxville. (n.d.). Michelson Interferometer. Retrieved November 18, 2010, from University of Tennessee: http://electron9.phys.utk.edu/optics421/modules/m5/Interferometers.htm
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Endnotes
Page | 23
Appendix – Raw Data
Δm
Δx (Raw)
Δx (μm)Measurement (μm)
# of Turns n - exp % Dev
100 22 2 721.50502
90.66401
3
150 8 4 1081.50502
90.66401
3
200 18.75 5 143.751.50598
30.60103
4
250 4.75 7 179.751.50579
20.61364
8
n - sell B1: 1.03961212
term 1 Term 2 Term 3 n^2 n-sell B2:0.231792344
1.055428154 0.243989454 -0.003922328 2.29549528 1.515089199 B3: 1.01046945
C1:0.006000699
C2:0.020017914
C3: 103.560653
n - exp (AVG)1.505458
1Final % Dev 0.64 %
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Appendix – Disc Contents
Root Directory
Lab 6 – Inteferometry Experiment.doc
o The official Microsoft Word 2003 format lab report concerning the Interferometry Experiment
Interferometry Lab Data.xls
o The Microsoft Excel worksheet that contains the raw data taken during the course of the experiment. The spreadsheet also contains operated-on data such as the final values, deviations, and Sellmeier values found in the Appendix (Raw Data) section of this report
Fig1.gif
o The collimated beam diagram found on page 3
Fig2.gif
o The Michelson Interferometer diagram found on page 4
Fig3.jpg
o The prism diagram found on page 5
Fig4.jpg
o The interference ripples picture found on page 6
Fig5.gif
o The Twyman-Green diagram found on page 11
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1 Encyclopaedia Britannica. (n.d.). Tywman-Green Interferometer. Retrieved November 18, 2010, from
Encyclopaedia Britannica: http://www.britannica.com/EBchecked/topic/611419/Twyman-Green-
interferometer
2 University of Tennessee: Knoxville. (n.d.). Michelson Interferometer. Retrieved November 18, 2010,
from University of Tennessee:
http://electron9.phys.utk.edu/optics421/modules/m5/Interferometers.htm
3 Scheider, W. (1986). Do the "Double Slit" Experiment the Way it Was Originally Done. Retrieved
November 18, 2010, from CavendishScience: http://cavendishscience.org/phys/tyoung/tyoung.htm
4 Menzies, A. (1960). Frank Twyman 1876-1959. In T. R. Society, Biographical Memoirs (pp. 269-279).
Royal Society Publishing.
5 W. Sellmeier, Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen, Annalen der
Physik und Chemie 219, 272-282 (1871).
6 Polyanskiy, M. (2008-2010). Refractive Index Database. Retrieved November 28, 2010, from
RefractiveIndex.info: http://refractiveindex.info/
7 Pedrotti, Leno. Prisms to Deviate Light by Refraction. Prisms. Texas: The Center for Occupational
Research and Development, 1987.
Hugh D. Young, R. A. (2007). University Physics. Pearson Addison-Wesley.