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

Dr. Larry Bortner

Purpose

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


Background

Light Traveling Through a Gas

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=5800, 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:

ole.gif(1)


where λ is the wavelength in vacuum and λair 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 Lop 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:

ole1.gif(2)


Michelson Interferometer

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 M1 and reflects the other half (the sample beam) toward mirror M2. 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.

ay.gif

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).

?.gif

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 M2 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.

ole2.gif

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

ole3.gif(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:

ole4.gif(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:

ole5.gif(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

ole6.gif(6)

where N is the number of molecules in a volume V and C1 is an arbitrary constant. The ideal gas law is

ole7.gif(7)

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

ole8.gif(8)

where C2 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(P0,T0). For dry air containing 0.03% by volume CO2 at a reference temperature T0=15°C and standard atmospheric pressure P0=101.325 kPa, the CRC Handbook for Chemistry and Physics lists the following wavelength-dependent fit:

ole9.gif(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:

ole10.gif(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:

ole11.gif(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.


Gas Cell Path Length

The single-traverse path length L in the cell is not directly measurable. From Fig. 4, what we can measure is the total length LT 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.

ole12.gif

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

 

The length is then

ole13.gif(12)


Procedure

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:

         LT=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 Pstart~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 Pstart~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 Pstart 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.


Analysis

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.   T0 in K.

     e.   (n-1) x 108 at T0 and P0 (Eq.9)

     f.   n-1 at T0 and P0.

     g.   n at T0 and P0.

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.


 Questions

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.