The Refractive Index of Air with a Michelson Interferometer
Joshua Webster
Partners: Billy Day & Josh Kendrick
PHY 3802L
11/06/2013
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Lab 3: Interferometry
Abstract:
The purpose of the experiments performed in this report is to aid in the familiarity with a
Michelson interferometer. The first section deals with determining the wavelength of a laser light
using the geometrical relationship between the beam and the interferometer device. The laser
light was determined to have a wavelength of 643 ± 50 nm. The second section deals with
determining the index of refraction of air using a vacuum air cell. The refractive index of air was
found to be 0.000192 ± 0.000025.
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Lab 3: Interferometry
Table of Contents
Abstract: .......................................................................................................................................... 1
Introduction ..................................................................................................................................... 3
Background ..................................................................................................................................... 4
Experimental Techniques................................................................................................................ 7
Diagrams and Images .................................................................................................................. 7
Data ................................................................................................................................................. 9
Analysis......................................................................................................................................... 14
Discussion ..................................................................................................................................... 17
Conclusion .................................................................................................................................... 20
Appendix ....................................................................................................................................... 21
References ..................................................................................................................................... 22
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Lab 3: Interferometry
Introduction
The Michelson interferometer optical configuration was originally invented (and named
after) Albert Abraham Michelson who was the first American physicist to win the Nobel Prize in
sciences in 1907. The famous Michelson-Morley experiment, in which an attempt was made to
measure the effect of the “aether wind” on the speed of light, was conducted using a similar
apparatus as was used in the experiments performed in this laboratory report.
The Michelson interferometer setup used in this lab consists of two mirrors (one
stationary, one adjustable), a beam splitter, and a light source. Fine adjustments to the mirror (to
within one micron) are made in this experiment using a micrometer. The purpose for the
Michelson interferometer configuration is to produce noticeable interference patterns by splitting
a light source into two separate beams. One of the beams of light is reflected by the fixed mirror
into a device or in the case of this experiment, onto a surface (wall). The other beam is reflected
by the movable mirror onto the wall. A distinguishable interference pattern results when both of
the light beams are brought together. These interference patterns are characterized by fringes,
and they can be counted or measured for analysis as is done in this report.
Experiments were conducted in this lab to understand the uses and capabilities of the
Michelson interferometer, to measure the wavelength of a laser light, and to measure the
refractive index of air. The following sections of this report will consist of the Backround,
Experimental Techniques, Data, Analysis, Discussion, Conclusion, Appendix, and References.
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Lab 3: Interferometry
Background
The first section of the experiment was conducted to measure the laser light wavelength.
After the initial reading of the micrometer was recorded, a specific number of fringes were
counted by dialing the micrometer. This was repeated several times, and each micrometer
reading was recorded. Using the micrometer readings and a set number of fringes between each
reading, a formula can be derived to determine the wavelength of the laser light. The laser light
we were using was red, so one should theoretically expect to find a wavelength around 600 to
700 nanometers.
For the equation above: is the wavelength of the laser light, is the distance associated with
fringes which will be the initial micrometer reading subtracted from the final, and there is a
factor of two because the light beams traverses twice the distance.
An average value for the wavelengths can be obtained using the normal arithmetic average
formula:
̅
In the above formula:
∑
is the number of wavelengths, and
is the individual wavelength value.
The standard deviation was also calculated using the standard formula:
√
∑
̅
An air cell was used to determine the refractive index of air. Air was evacuated from the
cell until a certain number of fringes were counted. This was repeated several times, and the air
pressure inside the cell was recorded at each interval. The room pressure and temperature were
also recorded and a graph was created for the cumulative fringes vs. absolute pressure.
From equation 1 we can develop a relation to determine the index of refraction of air.
Because of the air cell, there is an index of refraction associated that needs to show up in the
equation. The equation accounting for the index of refraction, , is therefore:
In the equation above, L represents the geometrical length of the air cell.
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Lab 3: Interferometry
The uncertainty in the slope can be calculated using the error propagation formula found in the
Appendix (A.1):
√(
)
(
)
√(
)
√( )
The Lorenz-Lorentz Law:
(
(
(
)
)
)
for a gas is proportional to the density, so for an ideal gas,
Upon combining equations 4 and 5 the resultant equation becomes,
(
)(
)(
)
In the equation above:
is the refractive index at a certain pressure, , and temperature .
⁄ is the slope of the graph of cumulative counts versus pressure, and is the temperature
when the measurement was taken. is absolute Kelvin.
For the uncertainty in the
Appendix (A.1):
values we can use the formula for error propagation found in the
√(
√(
)
)
(
(
)
(
)
(
5
)
)
(
)
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Lab 3: Interferometry
√(
)
(
)
(
)
(
)
To determine an accepted value for
based on the measurements taken in this lab an
equation given by the CRC Handbook of Chemistry and Physics needs to be used:
[
]
For the equation above: P and T represent the pressure (in Pascals) and temperature (in
respectively.
6
),
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Lab 3: Interferometry
Experimental Techniques
Diagrams and Images
Diagram 1: Shown below is the Michelson interferometer configuration used during the
experiments performed in this lab. However, a compensator was not used in our setup.
Diagram 1
Diagram 2
Diagram 2: (Above, right) This diagram is the exact configuration used in this lab for measuring
the refractive index of air with the air cell in the uncompensated leg.
The first section of the lab was to correctly configure the Michelson interferometer and to
achieve proper alignment of the laser-interferometer setup so that the fringe pattern was easily
visible. With a fair amount of confidence in the alignment the experiment was advanced to
section 2. With the interferometer aligned to produce a circular fringe pattern on the wall, a
bright fringe on-axis was obtained using the micrometer, and the first micrometer reading was
recorded. The micrometer was then slowly adjusted in the direction of increasing size until 20
fringes moved across the wall. At each 20 fringe interval, the micrometer reading was recorded.
This was repeated a total of 7 times.
In the next section of the lab, an air cell was placed in between the stationary
mirror and the light beam. The air cell was evacuated using a hand pump with a pressure gauge.
At each successive fringe count the pressure in the air cell was recorded. This procedure was
done 7 times, and then repeated 3 times. Before each set of measurements the room pressure and
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Lab 3: Interferometry
temperature was recorded. A plot of the cumulative fringes versus the absolute pressure was
constructed. The rest of the lab was to understand the formulae that describe the relationships
dealing with the interferometry experiments conducted.
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Lab 3: Interferometry
Data
Table 1: The table below shows the micrometer readings for the first section of the lab, and also
shows the wavelengths calculated for each reading. Uncertainties are estimated.
Measurement # # Fringes
Initial Reading
N/A
1
20
2
20
3
20
4
20
5
20
6
20
7
20
Micrometer Reading ∆(micrometer Wavelength ∆(Wavelength)
(μm)
reading) (μm)
(nm)
(nm)
501
5
N/A
N/A
507
5
600
50
514
5
700
50
520
5
600
50
526
5
600
50
533
5
700
50
540
5
700
50
546
5
600
50
Average
643
St Dev
53
Table 2: This is the first series of data using the air cell. The absolute pressure is the cell
pressure subtracted from the room pressure. Uncertainties are estimated.
Room Temperature (ᵒC) =
Room Pressure (in Hg) =
Cumulative
Fringe
∆(Cumulative
Count
Fringes)
1
0.25
2
0.25
3
0.25
4
0.25
5
0.25
6
0.25
7
0.25
Cell Pressure (in Hg)
1.75
3.50
5.00
6.50
7.75
9.50
10.50
9
22.3
30.016
Absolute
∆(Abs.
Pressure Pressure) Refractive
(in Hg)
(in Hg)
Index (n-1)
28.3
0.5 0.000223152
26.5
0.5 0.000209336
25.0
0.5 0.000197494
23.5
0.5 0.000185652
22.3
0.5 0.000175784
20.5
0.5 0.000161968
19.5
0.5 0.000154073
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Lab 3: Interferometry
Table 3: The second series of data using the air cell.
Room Temperature (ᵒC) =
Room Pressure (in Hg) =
Cumulative
Fringe
∆(Cumulative
Count
Fringes)
1
0.25
2
0.25
3
0.25
4
0.25
5
0.25
6
0.25
7
0.25
Cell Pressure (in Hg)
2.75
4
5.5
6.5
8
9.75
10.75
22.4
30.016
Absolute
∆(Abs.
Pressure Pressure) Refractive
(in Hg)
(in Hg)
Index (n-1)
27.3
0.5 0.000232286
26.0
0.5 0.000221637
24.5
0.5 0.000208858
23.5
0.5 0.000200339
22.0
0.5 0.00018756
20.3
0.5 0.000172651
19.3
0.5 0.000164132
Table 4: The third series of data using the air cell.
Room Temperature (ᵒC) =
Room Pressure (in Hg) =
Cumulative
Fringe
∆(Cumulative
Count
Fringes)
1
0.25
2
0.25
3
0.25
4
0.25
5
0.25
6
0.25
7
0.25
Cell Pressure (in Hg)
2.5
4
5.5
7
8.5
9.5
11
10
22.5
30.016
Absolute
∆(Abs.
Pressure Pressure) Refractive
(in Hg)
(in Hg)
Index (n-1)
27.5
0.5 0.000225841
26.0
0.5 0.000213529
24.5
0.5 0.000201218
23.0
0.5 0.000188907
21.5
0.5 0.000176595
20.5
0.5 0.000168388
19.0
0.5 0.000156076
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Lab 3: Interferometry
Graph 1: This graph contains the data for Series 1 listed in Table 2. A linear fit line is shown.
The error bars are estimated and listed in the table.
Cumulative Fringes Vs. Absolute Pressure (Series 1)
8
Cumulative Fringe Count
7
6
5
4
Series 1
3
Linear (Series 1)
2
y = -0.6807x + 20.105
R² = 0.9967
1
0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
Absolute Pressure (in Hg)
Graph 2: This graph contains the data for Series 2 listed in Table 3. A linear fit line is shown.
The error bars are estimated and listed in the table.
Cumulative Fringes Vs. Absolute Pressure (Series 2)
8
Cumulative Fringe Count
7
6
5
4
Series 2
3
Linear (Series 2)
2
y = -0.7343x + 21.084
R² = 0.9965
1
0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
Absolute Pressure (in Hg)
11
26.0
27.0
28.0
29.0
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Lab 3: Interferometry
Graph 3: This graph contains the data for Series 3 listed in Table 4. A linear fit line is shown.
The error bars are estimated and listed in the table.
Cumulative Fringes Vs. Absolute Pressure (Series 3)
8
Cumulative Fringe Count
7
6
5
4
Series 3
3
Linear (Series 3)
2
y = -0.7072x + 20.377
R² = 0.9976
1
0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
Absolute Pressure (in Hg)
Graph 4: This graph depicts the values represented in Tables 2-4, and is a combination of
Graphs 1-3.
Cumulative Fringe Count
Cumulative Fringes Vs. Absolute Pressure
8
Series 1
7
Series 2
6
Series 3
5
Linear
(Series 1)
4
y = -0.6807x + 20.105
R² = 0.9967
3
2
y = -0.7343x + 21.084
R² = 0.9965
1
y = -0.7072x + 20.377
R² = 0.9976
0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
Absolute Pressure (in Hg)
12
26.0
27.0
28.0
29.0
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Lab 3: Interferometry
Table 5: This table shows the absolute pressures from the tables above with their respective
index of refraction and uncertainty associated. The data for each series (1-3) is listed. Also
shown are the average values for the index of refraction and the standard deviation.
Absolute Pressure (in Hg) Refractive Index (n-1)
28.266
2.23E-04
26.516
2.09E-04
25.016
1.97E-04
23.516
1.86E-04
22.266
1.76E-04
20.516
1.62E-04
19.516
1.54E-04
AVG
1.87E-04
Series 2
Absolute Pressure (in Hg) Refractive Index (n-1)
27.266
2.32E-04
26.016
2.22E-04
24.516
2.09E-04
23.516
2.00E-04
22.016
1.88E-04
20.266
1.73E-04
19.266
1.64E-04
AVG
1.98E-04
Series 3
Absolute Pressure (in Hg) Refractive Index (n-1)
27.516
2.26E-04
26.016
2.14E-04
24.516
2.01E-04
23.016
1.89E-04
21.516
1.77E-04
20.516
1.68E-04
19.016
1.56E-04
AVG
1.90E-04
13
∆(n-1)
2.77E-05
2.67E-05
2.59E-05
2.50E-05
2.43E-05
2.33E-05
2.28E-05
St. Dev.
2.50E-05
∆(n-1)
2.87E-05
2.79E-05
2.70E-05
2.64E-05
2.54E-05
2.44E-05
2.38E-05
St. Dev.
2.50E-05
∆(n-1)
2.80E-05
2.72E-05
2.63E-05
2.54E-05
2.45E-05
2.39E-05
2.30E-05
ST. Dev.
2.50E-05
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Lab 3: Interferometry
Analysis
The wavelength in the first section of the report was calculated using equation 1. This equation is
based off of geometry, and the fact that the light beam traverses the distance twice.
The average wavelength was calculated using the arithmetic average formula.
̅
̅
∑
∑
This wavelength measurements accuracy depends on the precision of knowing when a
new fringe begins. There is a bit of “gray” area in this measurement. The uncertainty in the
measurement of the fringe beginning and ending does account for something, but the micrometer
is so precise that slight human error is fairly insignificant.
The standard deviation of the wavelengths agrees fairly well with the estimated
uncertainty. Taking a look at the big picture, since the values were between 600 and 700
nanometers and there was almost an even amount of each of those numbers, it makes sense that
the uncertainty would be right around 50 which is half of the difference between the two values.
Deriving the equation for the index of refraction we need to look at the relation
⁄ , where
is the wavelength of the light in a vacuum, and
is the index of refraction for
the material. It can be assumed at low pressures, that the index of refraction for a gas varies
⁄ wavelengths of light
linearly with the pressure of the gas. To begin with, there are
within the cell (taking into account both passes of the laser light beam). At the final pressure,
⁄ wavelengths within the cell. The difference between these two values is
there are
the number of fringes counted while evacuating the air cell, . So,
and
⁄ .
Therefore, the slope of the graph of cumulative fringes versus pressure is:
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Lab 3: Interferometry
(
)
In the equation above, and are the initial and final pressures, and
refraction that correspond to the pressure values.
and
are the index of
However, we can just use the slope calculated from the graph and for the formula use
for
rise over run. This makes the error propagation formula much simpler. Using the equation for
error propagation found in the Appendix (A.1):
√( )
√(
Calculating the
(
)
)
(
)
values, we use equation 7:
(
(
)(
)(
)
)(
)
For values of
obtained in this lab, there are uncertainties associated with the fringe
count, wavelength, air cell length, pressure, and temperature. For the fringe count, a human
measurement error can be accounted for by estimating a
fringe count error. The wavelength
standard deviation was calculated to be
. The error in the air cell length can be estimated
based on measurement error to be
. Error in the pressure can be attributed to measurement
error of around
. Error in temperature can be estimated to be
based on possible device error (a digital thermometer was used).
Calculating the uncertainty associated in the
√(
)
(
value we can use equation 8:
)
(
15
)
(
)
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Lab 3: Interferometry
This value for the uncertainty in
seems to be consistent in relation to the value for .
To determine an accepted value for
can use equation 9:
based on the measurements taken in this lab we
[
]
Using the pressure value given by the CRC, 101.325 kPa, at a temperature of 22.4
average temperature during our observations):
[
(the
]
We can now multiply this number by the number stated in the CRC for the
value
for the index of refraction of air at 640 nm wavelength (which was closest to our determined
wavelength). The CRC gives a value of 27644, so multiplying by the above determined number
gives 26949. This would be the accepted value of the index of refraction at a wavelength and
temperature close to ours.
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Lab 3: Interferometry
Discussion
The values obtained in this lab seem to be quite acceptable. The value obtained for the
wavelength of the laser light was within the range of acceptable values (around 600-700 nm) for
light in the red spectrum, which was the color of the laser being used. The
values
determined for the refractive index of air were in fair agreement with each other, and were
acceptable (however, slightly off) when compared to the value for the refractive index found
using online resources. The accepted value for the refractive index of air,
value,
was determined to be 26949. Compared to our value of 19170 ± 2503, the accepted value is a
little past the high end of the uncertainty on our number. This could be due to a few associated
errors in the experiment that were not numerically accounted for. It should also be stated that the
accepted value obtained is not a concrete answer for our specific wavelength. It is used as an
accepted value for a wavelength “near” our actual wavelength.
Assuming that the error on the fringe count is 1, how many fringes would you need to count in
order to know the value of wavelength to within 0.l nm?
Assuming only error in the fringe count, using equation 1 from the Background and A.1
from the Appendix we can calculate the amount of fringes necessary to count in order to have a
0.1 nm uncertainty.
√(
√(
)
(
)
)
√
√
The fact that you are counting only integer numbers of fringes leads to a major contribution
to the uncertainty of your measurement of n-1. If you continued to use only integral values,
would a laser with a shorter or longer wavelength give more precise results? Why?
If only integer numbers of fringes are measured, then a longer wavelength laser would
give more precise results. This is because the length that the micrometer adjusts the mirror would
be greater for larger wavelengths. The increased wavelength would make counting the fringes
easier, and thus likely result in smaller contributions of measurement error.
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Lab 3: Interferometry
The cell length was measured between the inside surfaces of the glass ends. Is there an error
introduced by the glass plates? If so, what is it?
There is an error introduced by the glass plates. This error is due to the width of the plates
(approximately 0.1 cm width for each plate, two total), which does have some effect in error
propagation. Since the light traverses the glass plates twice, once going in and once coming back
out, the width would have to be multiplied by a factor of 4.
The list below lists some effects which could influence your measurement. Discuss their
importance (or non-importance?). For your result, and give suggestions on how to avoid these
effects or correct for them.

Influence of temperature on barometer reading
This should be negligible, because the temperature and pressure was recorded at the
beginning of each series, and there were only very slight deviations.

Variation of temperature during your measurement of S(T)
The variation of the temperature could result in some error contribution. Throughout the
course of this lab the temperature only very slightly changed, and most likely did not contribute
any significant amount of error.

Deviation from the ideal gas laws assumed in equation (3), equation (6) in this report
Deviation from the ideal gas law would contribute to the uncertainty, but only very slightly.
The ideal gas law is really only an approximation for a real gas that exhibits similar behavior to
an ideal gas. The ideal gas law also neglects molecular size and intermolecular attractions, which
could also lead to uncertainty. This could be accounted for by using a more elaborate equation
such as Van der Waals equation of state.

Change in cell length due to atmospheric pressure when the air cell is partially
evacuated
This would definitely influence measurements if not accounted for. Because of the small size
of the air cell, any physical change (no matter how minute) could show up in the data. An
experiment could minimize the possibility of something like this occurring by using a larger cell,
so that small effects would not contribute any significant amount. Extremely precise instruments
could also be used, so that small fluctuations in pressure could be measured.

Influence of the relative humidity of the air on the refractive index
Relative humidity is of significant importance when dealing with refractive index. In fact,
most published calculations account for this. An increase in humidity brings with it an increase
in the dispersion of kinetic energy resulting in a decrease in the measured index of refraction.
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
Lab 3: Interferometry
Drift in the interferometer during observations
This is probably the largest source of error. Even though the interferometer setup is fairly
stable and reliable, the potential for drift to occur is still very real. To minimize the risk the
experimenters should be very careful with the device after alignment. The interferometer could
also be mounted on a surface for added stability.
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Lab 3: Interferometry
Conclusion
The set of experiments performed in this lab were conducted to familiarize oneself with
the various uses of a Michelson interferometer. The wavelength of light source can be
determined using the Michelson interferometer configuration, and in the case of the experiments
performed in this lab, the laser light being utilized was determined to have a wavelength of 643 ±
50 nm. Also, the refractive index of gases can be determined with a Michelson interferometer.
The refractive index of air
–
was determined by the experiments performed in this lab to
be 0.000192 ± 0.000025. The table below shows the conclusive data found in the experiment for
the refractive index of air.
Table 6: This table shows the average n-1 values and standard deviations for each series. Also
shown are the total averages of both the average (n-1) values and the standard deviations.
Series
1
2
3
Tot. AVG
AVG (n-1)
0.0001868
0.0001982
0.0001901
0.0001917
20
St. Dev.
2.501E-05
2.502E-05
2.504E-05
2.503E-05
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Lab 3: Interferometry
Appendix
A.1 Formula for the propagation of errors:
Given a function, , with variables , , and . The uncertainty in is the square root of the sum
of the squares of the partial derivatives of with respect to each variable, and each partial
derivative is multiplied by the square of it’s uncertainty.
√(
)
(
)
(
21
)
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Lab 3: Interferometry
References
Lide, D. R., & ed. (2005). CRC Handbook of Chemistry and Physics. Boca Raton, FL: CRC
Press.
PASCO scientific. (1990). Precision Interferometer. Retrieved October 27, 2013, from PASCO
scientific:
http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/optics/Pasco01207137b.pdf
Refractive Index. (n.d.). Retrieved October 27, 2013, from Wikipedia:
http://en.wikipedia.org/wiki/Refractive_index
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