The Refractive Index of Air with a Michelson Interferometer · 06/11/2013 · The Refractive Index...

$: The Refractive Index of Air with a Michelson Interferometer · 06/11/2013 · The Refractive Index of Air with a Michelson Interferometer Joshua Webster Partners: Billy Day & Josh$
The Refractive Index of Air with a Michelson Interferometer

Joshua Webster

Partners: Billy Day & Josh Kendrick

PHY 3802L

11/06/2013

Webster Lab 3: Interferometry

1

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.


2

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


3

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.


4

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.


5

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 values we can use the formula for error propagation found in the

Appendix (A.1):

√(

)

(

)

(

)

√(

)

(

)

(

)

(

)


6

√(

)

(

)

(

)

(

)

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.


7

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 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

Diagram 1 Diagram 2


8

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.


9

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 Micrometer Reading

(μm) ∆(micrometer reading) (μm)

Wavelength (nm)

∆(Wavelength) (nm)

Initial Reading N/A 501 5 N/A N/A

1 20 507 5 600 50

2 20 514 5 700 50

3 20 520 5 600 50

4 20 526 5 600 50

5 20 533 5 700 50

6 20 540 5 700 50

7 20 546 5 600 50

Average St Dev

643 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) = 22.3

Room Pressure (in Hg) = 30.016

Cumulative Fringe Count

∆(Cumulative Fringes) Cell Pressure (in Hg)

Absolute Pressure (in Hg)

∆(Abs. Pressure)

(in Hg) Refractive Index (n-1)

1 0.25 1.75 28.3 0.5 0.000223152

2 0.25 3.50 26.5 0.5 0.000209336

3 0.25 5.00 25.0 0.5 0.000197494

4 0.25 6.50 23.5 0.5 0.000185652

5 0.25 7.75 22.3 0.5 0.000175784

6 0.25 9.50 20.5 0.5 0.000161968

7 0.25 10.50 19.5 0.5 0.000154073


10

Table 3: The second series of data using the air cell.






∆(Abs. Pressure)


1 0.25 2.75 27.3 0.5 0.000232286

2 0.25 4 26.0 0.5 0.000221637

3 0.25 5.5 24.5 0.5 0.000208858

4 0.25 6.5 23.5 0.5 0.000200339

5 0.25 8 22.0 0.5 0.00018756

6 0.25 9.75 20.3 0.5 0.000172651

7 0.25 10.75 19.3 0.5 0.000164132

Table 4: The third series of data using the air cell.






∆(Abs. Pressure)


1 0.25 2.5 27.5 0.5 0.000225841

2 0.25 4 26.0 0.5 0.000213529

3 0.25 5.5 24.5 0.5 0.000201218

4 0.25 7 23.0 0.5 0.000188907

5 0.25 8.5 21.5 0.5 0.000176595

6 0.25 9.5 20.5 0.5 0.000168388

7 0.25 11 19.0 0.5 0.000156076


11

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.



y = -0.6807x + 20.105 R² = 0.9967

0

1

2

3

4

5

6

7

8

18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Cu

mu

lati

ve F

rin

ge C

ou

nt


Cumulative Fringes Vs. Absolute Pressure (Series 1)

Series 1

Linear (Series 1)

y = -0.7343x + 21.084 R² = 0.9965

0

1

2

3

4

5

6

7

8

18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Cu

mu

lati

ve F

rin

ge C

ou

nt



Series 2

Linear (Series 2)


12



Graph 4: This graph depicts the values represented in Tables 2-4, and is a combination of

Graphs 1-3.

y = -0.7072x + 20.377 R² = 0.9976

0

1

2

3

4

5

6

7

8

18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Cu

mu

lati

ve F

rin

ge C

ou

nt



Series 3

Linear (Series 3)

y = -0.6807x + 20.105 R² = 0.9967

y = -0.7343x + 21.084 R² = 0.9965

y = -0.7072x + 20.377 R² = 0.9976

0

1

2

3

4

5

6

7

8

18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Cu

mu

lati

ve F

rin

ge C

ou

nt


Cumulative Fringes Vs. Absolute Pressure Series 1

Series 2

Series 3

Linear(Series 1)


13

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) ∆(n-1)

28.266 2.23E-04 2.77E-05

26.516 2.09E-04 2.67E-05

25.016 1.97E-04 2.59E-05

23.516 1.86E-04 2.50E-05

22.266 1.76E-04 2.43E-05

20.516 1.62E-04 2.33E-05

19.516 1.54E-04 2.28E-05

AVG St. Dev.

1.87E-04 2.50E-05

Series 2


27.266 2.32E-04 2.87E-05

26.016 2.22E-04 2.79E-05

24.516 2.09E-04 2.70E-05

23.516 2.00E-04 2.64E-05

22.016 1.88E-04 2.54E-05

20.266 1.73E-04 2.44E-05

19.266 1.64E-04 2.38E-05

AVG St. Dev.

1.98E-04 2.50E-05

Series 3


27.516 2.26E-04 2.80E-05

26.016 2.14E-04 2.72E-05

24.516 2.01E-04 2.63E-05

23.016 1.89E-04 2.54E-05

21.516 1.77E-04 2.45E-05

20.516 1.68E-04 2.39E-05

19.016 1.56E-04 2.30E-05

AVG ST. Dev.

1.90E-04 2.50E-05


14

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

linearly with the pressure of the gas. To begin with, there are ⁄ wavelengths of light

within the cell (taking into account both passes of the laser light beam). At the final pressure,

there are ⁄ wavelengths within the cell. The difference between these two values is

the number of fringes counted while evacuating the air cell, . So, and

⁄ .

Therefore, the slope of the graph of cumulative fringes versus pressure is:


15

( )

In the equation above, and are the initial and final pressures, and and are the index of

refraction that correspond to the pressure values.

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:

√(

)

(

)

(

)

(

)


16

This value for the uncertainty in seems to be consistent in relation to the value for .

To determine an accepted value for based on the measurements taken in this lab we

can use equation 9:

[ ]

Using the pressure value given by the CRC, 101.325 kPa, at a temperature of 22.4 (the

average temperature during our observations):

[ ]

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.


17

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.


18

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.


19

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.


20

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 AVG (n-1) St. Dev.

1 0.0001868 2.501E-05

2 0.0001982 2.502E-05

3 0.0001901 2.504E-05

Tot. AVG 0.0001917 2.503E-05


21

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.

√(

)

(

)

(

)


22

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/Pasco012-

07137b.pdf

Refractive Index. (n.d.). Retrieved October 27, 2013, from Wikipedia:

http://en.wikipedia.org/wiki/Refractive_index

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