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

Page | 1

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

Page | 2

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

Page | 3

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

Page | 4

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)

Page | 5

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

Page | 6

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

Page | 7

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

Page | 8

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

Page | 9

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.

Page | 10

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

Page | 11

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

Page | 12

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

Page | 13

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)

Page | 14

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)

Page | 15

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)

Page | 16

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.

Page | 17

%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.

Page | 18

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

Page | 19

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

Page | 20

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.

Page | 21

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

Page | 22

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 %

Page | 24

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

Page | 25

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.