Index of Refraction of Air (last edited 12/20/2006 )

Dr. Larry Bortner

To find the index of refraction of air at 15 ̊C and 101.325 kPa using an interferometer.

Air, or any other low density gas, retards a light wave going through it, but it does not slow it down very much. The quantity that we use to indicate the speed of light in a substance is called the index of refraction n and is defined as the ratio of the speed of light in vacuum to the speed of light in the material. It is inversely proportional to the speed, so the higher n is, the slower the wave. If the light wave speed in air is just a little slower than the speed in vacuum, we expect the index of refraction to be just a little bit larger than one.

One explanation of how light travels through air is that the electrons of the constituent atoms are induced to oscillate by the electromagnetic wave that is light. This takes energy from the wave. But an oscillating charge emits its own wave at the same frequency. There is a finite time lag between when the energy is absorbed and when it is re-emitted. This interaction of the wave with the charge slows down the progress of the wave.

**Index of Refraction and Wavelength**

In the *Lens Equation *and *Dispersion *experiments, we
investigated the effects of light slowing down when it
goes through glass. In dispersion, we know that n
varies in a material according to the wavelength;
recall that *normal dispersion *is when n increases
when λ increases. (For λ=5800Å, at T=15°C,
and
P=standard atmospheric pressure, n =1.0002773.)

Another way to express the index of refraction is by the ratio of wavelengths, since the frequency of the wave will not change when it enters another medium:

(1)

where
λ is the wavelength in vacuum and
λ_{air} is the
wavelength in air.

This means that any method where we can measure the difference between the two wavelengths is a way we can calculate n. Thus we have an experimental method of finding the speed of light in air.

Define the *optical path length *
L_{op} of a light beam
between two points to be the number of waves w_{#} in
that path, times the wavelength of the light in
vacuum:

(2)

One way to detect the expected small difference in
wavelengths is to use a very sensitive instrument
known as a *Michelson interferometer. *This apparatus
splits monochromatic light into two beams, the
*sample *and the *reference *beam. The sample beam
undergoes a physical process that changes its optical
path length. It is then recombined with the reference
beam that has traveled the same physical distance.
From the superposition principle, this recombination
results in areas of constructive and destructive
interference, or light and dark fringes. Analyzing this
interference pattern of the two waves gives
information on the perturbing agent.

The interferometer design (Fig. 1) is representative
of a large group of devices utilizing wave phenomena
such as electromagnetic radiation or sound. A filter
at position A selects a single wavelength from an
extended source. Light from a particular point S on
the diffusing screen strikes a partially-silvered mirror
(the *beam splitter*) which transmits half of the
original beam (This half is the reference beam*.*)
through a glass compensator toward mirror M_{1}_{} and
reflects the other half (the sample beam) toward
mirror M_{2}. The mirrors reflect both beams back to
the beam splitter where a portion of the light gets
recombined on route to the observer at D. The
compensator at C (virtually identical to the beam
splitter except it’s not silvered) insures that the
distances traveled by the reference and sample
beams through glass can be made equal. In practice,
the total path lengths of the sample and reference
beams do not have be exactly equal, just
comparable.

Figure 1 Diagram of an interferometer.

When the mirrors are perpendicular to the incoming beams and the optical path lengths of the two beams are equal, the interference pattern will be a series of concentric circles as in Fig. 2. By making one of the mirrors slightly off perpendicular or by having different path lengths, the fringe pattern can be straight lines (portions of very large circles).

Figure 2 The resulting interference pattern when the reference beam and the sample beam path lengths are the same.

**Measuring Small Path Differences**

The difference in the path lengths of successive rings
or fringes is one wavelength. This means that if M_{2}
moves forward or back by
λ/2, the optical path of
the sample beam changes by that twice much, or
λ,
and that there is a *movement *in the field of view of
one fringe. In practice, we set the mirrors such that
the pattern is one of straight lines. A sharp object
like a pin point placed in the observer’s field of view
provides a reference point so that we can count
fringes moving past it as the physical conditions
change.

To measure the index of refraction of a gas such as air, we put a gas cell with transparent glass ends in the path of the interferometer sample beam (Fig. 3). Identical glass plates are put in the reference beam path to ensure nearly equal path lengths. Air is pumped out of the chamber and slowly bled back in as the pressure is monitored. The greater the number of gas molecules in the chamber, the slower light travels. Thus the optical path length increases as the pressure increases.

Figure 3 Interferometer set up for this experiment, with a gas cell in the beam path.

If the physical distance light travels through the gas cell chamber one way is L, the optical path length of the sample beam when the cell is under a vacuum is

(3)

(Remember that it goes through twice.) As noted, if there is air in the chamber, the wavelength changes, hence the number of wavelengths and the optical path length change:

(4)

**Counting Fringes (Wavelengths)**

The number of fringes i that reflects the optical path difference between these two conditions is the difference in the number of wavelengths:

(5)

We expect the number of wavelengths in air to depend on the pressure and temperature of the gas. Here the index of refraction is written as a function of those quantities.

**nair as a Function of Pressure, Temperature, and
Wavelength**

To find the functional form of this dependence, we assume that the perturbation from the ideal index of refraction (n=1) is linear with the number density of gas molecules. That is, the perturbation takes the form of

(6)

where N is the number of molecules in a volume V
and C_{1}_{} is an arbitrary constant. The ideal gas law is

(7)

where k is Boltzmann's constant. Solving for the number density N/V and substituting into Eq. 6 gives

(8)

where C_{2} incorporates k.

Many labs have done precise measurements of the
index of refraction of gases and empirical fits have
been made to this data. Let n=n(P_{0},T_{0}). For dry air
containing 0.03% by volume CO_{2} at a reference
temperature T_{0}=15°C and standard atmospheric
pressure P_{0}=101.325 kPa, the *CRC Handbook for
Chemistry and Physics *lists the following wavelength-dependent fit:

(9)

where the wavelength λ is in mm. Using ratios, we can express the deviation of n from unity at any pressure and temperature, relative to the reference values:

(10)

**Fringes as a Function of Pressure**

Substituting Eq. 10 into Eq. 5 then gives the number of fringes that indicates the optical path difference between the sample beam with air in the cell at any pressure P and temperature T and the sample beam with air in the cell at 15°C and one atmosphere pressure:

(11)

The quantity in braces is the extra sample beam optical path length that is greater than 2L, measured in the number of vacuum wavelengths. Assuming the temperature doesn't change, plotting i vs. this number gives a straight line with slope n-1 that we can compare to the empirical value from Eq. 9.

The single-traverse path length L in the cell is not
directly measurable. From Fig. 4, what we *can
*measure is the total length L_{T} of the cell, the depth h
of the lip between the glass and the outer extent of
the metal, and the thickness t of the two identical
glass plates used as compensators for the glass
plates in the cell.

Figure 4 Gas cell detail, showing dimensions of interest. L is a calculated value, the others are measured.

The length is then

(12)

The following equipment and accessories are needed:

• Ealing interferometer set up in Michelson mode

• hand vacuum pump

• mercury light

• absolute temperature sensor, u{T}=0.5 K

• pressure sensor, u{P}=0.1 kPa

• needle valve

• Science Workshop interface

To minimize distortion, the mirrors used in the
interferometer are all *front-surface *optical elements,
meaning that the reflective coatings are on the front
surface of the glass instead of the rear surface. This
means that even *touching *the mirrors can damage or
destroy them. When working with the
interferometer, observe this general rule:

Never touch the active surfaces of optical elements. (Don’t touch the mirror!)

If your interferometer needs adjustment, ask your instructor for assistance.

The canister that houses the mercury bulb becomes very hot after a short time. If you have to reposition the lamp, do so by moving its support base. These bulbs have an important operational characteristic:

When the bulb is turned off after warming up, it won’t turn on again until it has cooled down, a 10- 15 minute process.

To avoid this delay, be sure that you have taken all of your data before you turn off your lamp.

1. Click on Start> Science Workshop> Third Quarter> Speed of Light in Air, then click on the program Start button. Record the ambient temperature in °C.

2. Record the following dimensions in cm for the gas cell:

L_{T}=6.540

h=0.254

t=1.000

Assume an uncertainty of 0.003 cm for each.

3. Turn on the mercury light source. Check to see that an interference pattern of dark, fairly straight lines is visible in the field of view of your interferometer. If not, have your instructor make the necessary adjustments.

4. Use the hand vacuum pump to reduce the cell
pressure to P_{start}~20 kPa. The fringes will move
across the field of view as you pump, indicating
the change in path length as noted in the
Background. The previous user may have left the
needle valve slightly open. Be sure it is closed.

5. While observing the fringe pattern, adjust the needle valve so that the pressure increases by 1 kPa every two or three seconds.

6. Position the mouse cursor over the Keep button.

7. Pump out air until P_{start}~10 kPa, or as low as you
can go. Using long strokes is more effective than
quick, short ones.

8. *One *person needs to observe the fringes and
indicate when there's been a movement of a
single fringe, and to do this for a total of ten
fringes. This entails looking at the fringe pattern
with one eye closed and your head in the same
position for the duration of the ten-fringe
movement. The fringe observer needs to do two
things:

®
1. click on Keep every time a fringe passes the
reference point *and *

® 2. count the total number of fringes (the number of times Keep was clicked).

a. Continue this for a total of 10 fringes.

b. Press the red Stop button.

c. Record the approximate starting pressure and the corresponding Run #.

d. Each partner should be the fringe observer at least five times.

9. Repeat Steps 7 and 8 for values of P_{start} in kPa of
the following:

15

20

25

30

35

40

45

This gives you a total of eight (8) data sets of 10 points each.

Do not quit DataStudio until the end of class.

The wavelength of the green line in the mercury spectrum is λ=5461 Å.

1. Click on Start> Templates> Third Quarter> Speed of Light in Air to start Excel. Enter in your names.

2. Enter in the measured cell dimensions, the temperature, and the necessary reference values.

3. Click on the data staging tab at the bottom of the spreadsheet to go to that sheet.

4. Return to the DataStudio window.

a. Position the mouse cursor over the Pressure (kPa) heading.

b. Click on this heading to highlight all of the data of this run.

c. Copy this (Ctrl C).

d. Switch back to Excel, select an empty green cell, and paste (Ctrl V).

5. Repeat Step 4 until all data sets have been transferred.

6. Return to the calculation section by clicking on the main tab at the bottom of the Excel spread sheet.

7. Do the required calculations in the top section:

a. L (Eq.11).

b. λ in cm.

c. T in K.

d. T_{0} in K.

e. (n-1) x 10^{8} at T_{0} and P_{0}_{} (Eq.9)

f. n-1 at T_{0}_{} and P_{0}.

g. n at T_{0}_{} and P_{0}.

8. For each of the eight data sets:

a. Express the experimental pressures as the number of vacuum wavelengths in the cell,

b. Do a least squares fit to find n-1 and its uncertainty.

9. Plot all eight data sets on the same graph, with a legend.

10. Your experimental value of n-1 is the average of these slopes. Find this, as well as the standard error.

11. Find the average error of the slopes.

12. Compare the experimental n-1 with the book value, using the maximum of the standard error and the average error as the uncertainty in n-1.

1. The book value of n-1 (Eq. 9) is for dry air. How would humidity affect your results?

2. How could you use a Michelson interferometer to measure the temperature coefficient of expansion of a material? Explain.