Physics 262

Lab #4: Interferometer

John Yamrick

Abstract

This experiment revolved around the construction and use of a Michelson Interferometer. The apparatus was used to measure the wavelength of a given green laser diode to be 548.0 nm ± 5.2 nm and the wavelength of a given red laser diode to be 650.7 nm ± 6.0 nm. A lower bound for the coherence length of the green laser diode was measured to be 175 cm ± 2 cm. Finally, the index of refraction was measured for light passing through a plane of glass (ng = 1.56 ± 0.15) and through a chamber of pressurized air (results plotted in the experimental results section).

Introduction

A Michelson Interferometer is a device used to measure very small distances by observing the interference pattern created by recombining a split light beam whose parts have followed different optical paths. In most of this experiment, one of the optical paths was held steady while a perturbation was introduced into the other path. Comparing the changes in the interference pattern compared to baseline or calibration data allowed measurements of the beam’s wavelength or calculation of the index of refraction through one of the optical paths. The coherence length, taken to be the greatest distance for which an interference pattern is still visible, required no perturbation to obtain.

Theoretical Background

When two beams of light are superimposed onto each other, the net result is the sum of their constituent electromagnetic fields. If the two beams have the same wavelength and amplitude, then the difference in their phases at any given point will determine whether the sinusoidal waves interfere in a constructive or destructive manner.

A Michelson Interferometer operates by aiming two portions of a split light beam through a diverging lens and onto an observation screen. By aiming the beams at slightly different locations, the relative path length between the two beams will vary for different points in the plane of observation. This translates into a difference of phase between the two waveforms incident upon a particular point. This causes interference.

The interference manifests itself in a circular pattern with rings of constructive and destructive interference. The rings near the center are largest, because they are the areas where the two incident rays have diverged least from their direction of propagation. Away from the center, smaller changes in angle start producing larger changes in path length to the screen, and consequently the rings grow narrower. Depending on where the beams are aimed, various portions of this pattern may be visible.

As stated earlier, it is the phase difference between the two beams that is significant. Thus, any perturbation in the optical path of one of the beams prior to recombination will be carried through to the final interference pattern. In this experiment, path length is changed directly (via an electrostrictive actuator) and indirectly by altering the index of refraction of a portion of the path (via pressurization of the air in a portion of the path, or the inclusion of a variable width of glass).

Experimental Procedure

Apparatus

Polarized He-Ne laser (632.8 nm), polarizing cube (beam-splitter/recombiner), electrostrictive actuator with mounted mirror and control box, signal generator, photodiode, oscilloscope, mirrors, diverging lens, green and red diode lasers, glass plate on rotating mount, air cell with pressure gauge and hand pump

Procedure

The first task of the experiment was to determine the wavelengths of the green and red diode lasers. Before this could be done, calibration data needed to be collected using a laser with a known wavelength.

The apparatus was set up such that the beam was immediately split using the polarizing cube. One part of the beam went a short distance and then reflected back on itself off of a stationary mirror. The other part reflected back off of the mirror mounted on the electrostrictive actuator. When the beams recombined, the phase difference between them could be controlled by adjusting the position of the electrostrictive actuator.

In order to see the interference this produced, the recombined beam was sent through a diverging lens and aimed at the wall. By using the fine position adjustments on the two mirrors, the two portions of the split beam could be aimed slightly apart from each other in order to produce a clear interference pattern.

The signal generator was fed to the control box for the actuator so that it moved forward and backward at a steady pace. This was accomplished by sending it a very low frequency triangle wave. Data was collected over part of a single cycle of the waveform where the control voltage went from 30V to 70V.

The photodetector was set up so that, as the actuator moved, the fringes would move across the front of it. The output was made visible on the oscilloscope. Each fringe that passed showed that the optical path difference between the two beams had increased by one wavelength. Thus, the total distance which the actuator moved over the control voltage range could be determined using the wavelength of the known laser, and this could be used to calculate the wavelengths of the unknown diode lasers. In the next part of the experiment, the coherence length of the lasers was measured. The coherence length represents the greatest path length difference possible before the two beams stop creating interference. It gives an indication of the spectral density of the laser. To measure it, the path of one of the split beams was held constant while the other was gradually increased.

When computing optical path length, the index of refraction of the medium is just as important as the dimensional length. In the last two parts of the experiment, the index of refraction of a medium was calculated by comparing its effect on path length relative to a baseline. First, the index of refraction of glass was calculated by observing the number of wavelength changes when a beam crossed it at an angle instead of normal to the plane. Second, the index of refraction of air as a function of pressure was calculated by observing the path length change as an air cell is pressurized.

Experimental Results and Discussion

Wavelengths: In the initial experimental setup (with one of the split paths stationary and the other hitting the electrostrictive actuator) the following data was obtained for each of the three lasers:

Wavelengths Counted Trial #: He-Ne Laser Green Diode Red Diode 1 137 159 133 2 137 155 133 3 134 157 132 4 135 156 130 Average: 135.75 156.75 132.00 Std Dev: 1.30 1.48 1.22

The He-Ne results provides the total distance the electrostrictive actuator travels:

Where N is the number of counts. This d is the same for all of the lasers, so we can determine the other wavelengths relative to the He-Ne:

To determine error, we simply multiply by the percentage error in the count total. That is: (1.48/156.75) = 0.00944 for the green diode and

(1.22/132.00) = 0.00924 for the red diode.

This corresponds to:

Coherence Length: Attempts were made to determine the coherence length of each of the three lasers, but with limited success. The He-Ne exhibited a very long coherence length (consequent of its sharp frequency spectrum) and measuring it on the length of the lab table did not do it justice. The coherence length of the red diode laser seemed to fluctuate dramatically (possibly due to the laser entering different modes of radiation) and no measurement was possible. The green diode laser at least seemed to be in a stable mode, and while the interference began to diminish, there was not quite enough length on the table to find a distance where it actually disappeared. Instead, the distance of 175 cm ± 2 cm was measured as a lower bound on the coherence length as the largest distance for which the existence of an interference pattern was confirmed.

Index of Refraction (glass): The index of refraction for the glass plate can be calculated by rotating the plate from an orientation normal to the beam to one at an angle θ. Two sets of measurements were taken, one going from normal to 17° and the other from normal to 7°.

Angle 17° 7° 46 8 47 8 46 7 Counts 53 8 50 53 Average 49.2 7.75 Standard Deviation 3.02 0.43

As the glass is rotated, the optical path changes. It goes through a greater amount of glass and a lesser amount of air. Both effects need to be accounted for. The equation derived geometrically in the lab notebook gives such an expression:

Where d is the thickness of the glass, b is the x-coordinate of the optical path through the glass, ∆L is the path difference measured by counting fringes, and h is the actual optical path through the glass

To solve this, plug in the definitions of h and b:

(Snell’s Law)

Because the index of refraction appears inside the cosine term (this is a consequence of Snell’s Law) this leads to a transcendental equation. An iterative approach in Excel can provide a solution for ng. (The excel file will be e-mailed along with the lab report). At heart, ng is a function of θ, d, and the number of counts. To determine the total error, the error for each of these three real physical quantities is determined, and the effects of their max and min values on the value of ng are examined. The effects are then combined into a worst case scenario, and a new range of error is determined. d = 1 mm ± 0.05 mm

θ = 17° ± 0.5° cavg = 49.2 ± 3.0

The extremes of these results result in an ng between 1.4133 and 1.7069. ng = 1.56 ± 0.15

Index of Refraction (pressure cell): The final portion of the experiment was to measure the index of refraction within an air cell as a function of pressure. The apparatus was arranged in the same way as for measuring the index of refraction of glass with the air cell being placed in the position of the glass plate. Now, instead of rotating the plate, the cell was held stationary and evacuated. The pressure change was read off the dial as a certain number of fringe changes were counted.

Fringes Reduction in Pressure (kPa) ∆n n Counted 0 0 0 0 0 0.000000 1.000293 5 7 7 6 7 -0.000019 1.000274 10 16 16 15 15 -0.000038 1.000255 15 23 23 23 23 -0.000057 1.000236 20 31 31 30 32 -0.000076 1.000217 25 39 39 39 39 -0.000095 1.000198 30 45 47 46 47 -0.000114 1.000179 35 51 53 53 55 -0.000134 1.000159 40 58 61 60 64 -0.000153 1.000140

The length of the air cell was measured to be 8.295 cm ± 0.0005 cm.

In this plot, a linear trend is seen from the recorded value for the index of refraction in air (taken to be 1.000293) towards a value of 1 in vacuum.

Summary

Interferometry allows precise measurement of physical distances down to the wavelength of light being used by the apparatus. Such measurements can be used to determine the wavelengths of other light beams or the index of refraction of materials. In this experiment, a 632.8 nm He-Ne laser was used to determine the wavelengths of a green diode laser to be 548.0 nm ± 5.2 nm and a red diode laser to be 650.7 nm ± 6.0 nm. The index of refraction of a pane of glass was determined to be 1.56 ± 0.15, and the index of refraction of air in a pressure cell was plotted. Direct measurement showed that the coherence length of the green diode laser exceeds 175 cm ± 2 cm. min θ max θ min d max d min c max c max n min n Starting Value for n: 1.5875 1.4873 1.5787 1.4949 1.4844 1.5861 1.7069 1.4133 θ (degrees): 16.5 17.5 17 17 17 17 16.5 17.5 # of counts: 49.2 49.2 49.2 49.2 46.2 52.2 52.2 46.2 d (m): 0.001 0.001 0.00095 0.00105 0.001 0.001 0.00095 0.00105 wavelength (m): 6.33E‐07 6.33E‐07 6.33E‐07 6.3E‐07 6E‐07 6.33E‐07 6.328E‐07 6.33E‐07 path difference ‐ ∆L (m): 1.56E‐05 1.56E‐05 1.56E‐05 1.6E‐05 1E‐05 1.65E‐05 1.6516E‐05 1.46E‐05 ф (radians): 0.179876 0.203586 0.186273 0.19685 0.1983 0.185394 0.16717002 0.214408 calculated h (m): 0.001016 0.001021 0.000967 0.00107 0.001 0.001017 0.00096343 0.001075 calculated b (m): 0.00101 0.001016 0.000961 0.00107 0.001 0.001011 0.00095641 0.00107 calculated n: 1.5875 1.4873 1.5787 1.4949 1.4844 1.5861 1.7069 1.4133

Results show that in order to max n, we want to minimize θ and d, but max c To calculated the minimal n, we do the reverse