Department of Physics, Middlebury College

1992-93

Modern Physics Laboratory



I. The Michelson Interferometer



Discussion

The Michelson interferometer is often introduced in the context of special relativity where its historic role in providing evidence against the existence of an absolute rest frame defined by the electromagnetic ether is emphasized. Although historians still debate the extent to which Michelson's ether-drift experiments influenced Einstein's creation of the special theory of relativity, an assessment of Michelson's pioneering contributions to interferometry is much less controversial, as evidenced by the Nobel Prize awarded to Michelson in 1907 for his many discoveries in optics. Michelson's optical inventions have had enduring value and his interferometer in particular has become an indispensable tool in scientific applications such as high resolution spectroscopy and atomic length standards, and in technical applications, where displacements as small as a fraction of the wavelength of visible light must be measured.

In this experiment you will assemble a Michelson interferometer that operates simultaneously with a red ( = 6328 Å) HeNe laser and a green ( = 543 nm; further precision is being withheld intentionally) so called "GreNe" laser. The GreNe laser uses a HeNe discharge tube, but the optical resonator forces lasing at the 3s2-->2p10, 543 nm transition, rather than at the more common 3s2-->2p4, 6328 Å transition. You will investigate the Michelson interferometer (a) as a means for determining the wavelength of a light beam and establishing an optical length standard, (b) as a wavemeter, to compare the wavelength of two light sources to an accuracy of 1 part in 104, and (c) to measure the index of refraction of air at two different wavelengths.

Procedure

While assembling and operating the interferometer it is important to remember that laser beams can cause severe eye damage. Keep your head well above the horizontal plane of the laser beams at all times. Use white index cards to locate beamspots along the various optical paths. When moving optical components or metal tools through the laser beams you may deflect laser light momentarily at your lab partner or yourself. If you see that you may deflect laser light accidentally during a particular step in the assembly procedure, temporarily block or attenuate the laser beam until all optical components are in their proper place. It is a good policy to be aware of any stray laser beam reflections and to warn your lab partner of any danger. If you are unsure of how to proceed safely with a given step in the assembly of the interferometer, please ask the instructor for assistance.

You will assemble the Michelson interferometer seen in the photograph of Fig. 1 and described in the diagram of Fig. 2. When you arrive in the laboratory the interferometer will be completely assembled and the various operational modes of the interferometer will be demonstrated to you. In your lab notebook you should note (a) the alignment procedure, (b) the rough placement of major optical components, (c) the various electronics connections and the settings on electronic instruments, and (d) the proper operation of the vacuum pumping system. You will then be asked to leave the laboratory so that the instructor can dismantle the optical, electrical, and vacuum components of the interferometer. You will then be invited to return to the laboratory to begin your investigations.

The alignment of the interferometer is the most difficult task you will face in this experiment, but it is relatively straightforward if you keep four general themes in mind.

(a) Add optical components to the interferometer one at a time, starting from a bare optical bench. When adding a new optical component ask yourself what its function is and be sure that it is adjusted correctly to perform that function. When you are sure that a given optical component is fulfilling its role in the interferometer, tighten down its screws and adjustment knobs, and proceed to add additional components.

(b) The Michelson interferometer is a planar device and throughout assembly and alignment it is essential that all laser beams are at the same vertical height above the table top.

(c) Although it is not important that laser beams strike the exact centers of the mirrors or pass through the central volume of the beamsplitter or gas cell, it is important that components are mounted initially so that a range of movement is possible.

(d) Do not overtighten the mounting screws used by the various optical mounts. You can easily cause permanent damage to the threads of the mounting holes in the optical table top.

The following steps will guide you in your investigations.

(1) Optical alignment.

(a) Mount the HeNe and GreNe lasers securely to the optical bench in the positions indicated in Fig. 2. Adjust the lasers so both laser beams (i) pass directly above the centers of a line of mounting holes in the optical table top, to within ±1 mm over the entire length of the table, and (ii) are at the same vertical height to within ±1 mm over the entire length of the table. It is important that these conditions be achieved before proceeding to the next step.

Steps (b) through (d) will deal entirely with the GreNe laser beam; less confusion will result if at this point you simply block the HeNe laser beam at the laser exit.

(b) Insert M1 and reflect the GreNe beam back to the GreNe laser so that the reflected spot on the GreNe laser is positioned 1 cm to the left of the GreNe laser exit hole and at the same vertical height, as indicated by the circle with fringes at the GreNe laser in Fig. 2. A white index card with a hole in it will allow the GreNe laser beam to pass through the card and the beam reflected from M1 to be observed clearly on the white card.

(c) Insert the beamsplitter BS. You will see unwanted reflections of the GreNe laser beam from the front surface of beamsplitter that are clearly visible near the exit hole of the GreNe laser. Identify these unwanted reflections and distinguish them from the beam that reflects from M1, as described in step (b). Adjust mirror M1 so that the beam reflected from M1 is again positioned 1 cm to the left of the GreNe laser exit hole, as described in step (b). Adjust beamsplitter BS so that unwanted reflections from the surfaces of the beamsplitter are at least a few mm from the spot made by the reflection from M1. At this time you should also adjust the beamsplitter so that the beam to be deflected to M2 has the correct vertical height.

(d) Install mirror M2 on the translation stage and mount the stage in the position shown in Fig. 2. Adjust M2 so that the reflected beam M2-BS-GreNe overlaps the reflection spot of the beam M1-BS-GreNe at the point 1 cm to the left of the exit hole of the GreNe laser, as set in part (c). At this point you have completed a two arm interferometer circuit for the GreNe laser beam and interference fringes should be clearly visible at the place where the beams reflected from M1 and M2 overlap (indicated by the circle with fringes at the GreNe laser in Fig. 2). Make small adjustments to mirrors M1 and M2 to obtain large, high contrast fringes.

(e) Block the GreNe laser beam and allow the HeNe beam to pass through the apparatus. Insert M3 and position it so that the red reflected beams M2-BS-GreNe and M1-BS-GreNe overlap at a point 1 cm to the left of the GreNe laser exit port at the same vertical height. Small adjustments to the beamsplitter BS and to the horizontal position of mirror M3 may be necessary; however, you should not adjust mirrors M1 and M2, otherwise their alignment for the GreNe beam may be destroyed. When proper alignment is achieved you should observe red interference fringes at the overlap point 1 cm to the left of the GreNe laser exit port. If you also look about 1 cm to the left of HeNe exit port you will also see red fringes (indicated by the circle with fringes at the HeNe laser in Fig. 2). Make small adjustments to M3 and BS to obtain high contrast, red fringes.

(f) Unblock the GreNe beam so that both laser beams pass through the interferometer. Make small adjustments to the mirrors and the beamsplitter so that high quality, red and green fringe patterns are simultaneously visible at the overlap points 1 cm to the left of both the HeNe and the GreNe laser exit ports.

(g) Insert mirror M4 to deflect both laser beams to the filter and photodiode area of the optical bench. Take care not to block the HeNe laser beam with the edge of M4.

(h) Insert red transmission filter F1 and photodiode detector PD1. Rotate F1 about a vertical axis to optimize the red interference fringes on the active area of PD1. Insert green transmission filter F2 and photodiode detector PD2. Rotate F2 about a vertical axis to optimize the green interference fringes on the active area of PD2.

(i) If red and green interference fringes are clearly visible on the faces of both PD1 and PD2, respectively, then optical alignment is correct and complete. Congratulations!

(2) Electronics.

A block diagram of the signal processing electronics used to amplify, count and display the fringe pattern signals at the two photodiode detectors is given in Fig. 3. The two SR560 Low-Noise Preamplifiers should be set for DC coupling and, initially, a bandpass filter set for the range 100-3000 Hz should be used. When set to the Start position, the HP 5326B and HP 5327B Timer-Counters will count fringes until each counter is set manually to the Stop position.

Use the Newport Translation Stage Motor Controller to set mirror M2 into translation at its highest speed. Adjust the preamplifier gains so that the photodiode signals at Ch. 1 and Ch. 2 of the oscilloscope are both at least 3 V peak-to-peak. The photodiode signal should be optimized by rotating filters F1 and F2 and by adjusting the positions of photodiode detectors PD1 and PD2. If you feel lucky, you might even try optimizing the detector signals by making small adjustments to the beamsplitter BS or to the mirrors.

When the detector signals for the red and green fringe patterns are both of good quality on the oscilloscope over the entire range of travel of the translation stage, make a record of the Ch. 1 and Ch. 2 displays using the plotter. Affix the plot to a page in your laboratory notebook.

(3) Measuring wavelength.

When the translation stage supporting M2 is moved a distance d away from the beamsplitter BS, the laser beams travelling along the path BS-M2-BS must traverse an additional distance 2d before combining and interfering with the reference beam along path BS-M1-BS. This means that N = 2d/air additional light waves fit into the path BS-M2-BS and that the same number of fringes will move across the face of the photodiodes. Thus, by counting the number of fringes N at the photodiodes for a given translation distance d, one can determine the wavelengths of the two laser beams.

Measure how many fringes move across the face of the two detectors when the translation stage is moved through a distance d = 2.00 cm. Be sure that the two HP Timer-Counters trigger properly over the entire translation distance d of your measurements. It is best to start the HP Timer-Counters after the translation motor is fully up to speed. Perform fringe counting measurements for d = 2.00 cm at least five times and record your results in a neat, labeled table in your lab notebook. For each of the five measurements, determine values for HeNe and GreNe, in units of Å. Determine the mean values and standard deviations for your sets of measurements for HeNe and GreNe. What are the fractional uncertainties in your wavelength determinations by this method? Compare your mean values for HeNe and GreNe to currently accepted values for these wavelengths. Are the accepted values compatible with your mean values for HeNe and GreNe and your uncertainty estimates?

(4) Optical length standards.

You probably noticed in step (3) that despite the careful use of a great deal of complicated and expensive optical and electronic instrumentation, the precision of your wavelength determination was ultimately limited by your ability to measure the translation distance d using a micrometer or ruler. The problem would not be solved even if the old platinum-irridium standard meter bar were available alongside the translation stage, because the standard meter bar is of an inappropriate length and the bar is not provided with the millimeter rulings necessary for the present experiment. Better rulers, standard length bars, and microscopes for reading rulings on them, do not provide a solution for measuring small displacements of the kind that can be determined with ease by interferometric devices.

The solution to the problem of measuring distances with greater accuracy is to define the unit of length in terms of interferometrically determined quantities. Michelson was the first to demonstrate that interferometry could provide a length standard that was more precise, more reproducible, more portable, and more secure against war or natural disaster, than any material length standard. As a result of his pioneering work, atomic length standards were adopted, and for the years 1960-1983, the meter was defined as exactly 1,650,763.73 wavelengths in vacuum of the orange 5d5-->2p10 transition of the 86Kr atom. With the 86Kr meter standard and an interferometer such as the one you have just assembled, it would be possible to measure a translation distance of d = 2.00 cm to an absolute accuracy of ±1 fringe or ± 6 x 10-5 cm. In current practice, fractions of a fringe can be counted to an uncertainty as small as ±0.001 fringe.

(5) Wavemeter.

The interferometer you have assembled is typical of a configuration called a wavemeter that is used in modern laser laboratories to measure the wavelength of a laser light source with high precision and high accuracy. A wavemeter uses a reference laser beam with a precisely determined wavelength (in our case, the HeNe laser with wavelength HeNe = 6328.1646 Å in air at STP2) to measure the unknown wavelength of another laser (in our case, the GreNe laser). You probably have a fairly good estimate of GreNe from part (3) and you can read from the label on the GreNe laser housing that it operates at 543 nm. But in this part of the experiment you will determine GreNe to five significant figures using the Michelson interferometer. The absolute accuracy of your determination of GreNe will actually exceed the absolute accuracy (±1 Å) attainable with the expensive, 1 m, high resolution spectrometer used later in the optical spectroscopy experiments of this laboratory notebook.

To make a wavemeter measurement, a single HP Timer-Counter is used in what is referred to as the Ratio Mode. The amplified signal from PD1 (red) is connected to Ch. A of the HP 5326B Timer-Counter and the amplified signal from PD2 (green) is connected to the OSC input on the rear panel, with the OSC INT-EXT switch on the rear panel set to EXT. On the front panel, set the Function switch to FREQ A and the display slide-switch to HOLD.

In order to obtain correct Ratio results for steps (a), (b), and (c) below, it is important that the translation stage motor runs smoothly over the entire translation distance d necessary to obtain a ratio measurement to the desired precision. It is a good practice to monitor the oscilloscope traces of PD1 and PD2 to ensure they maintain an amplitude of at least 3 V peak-to-peak at the HP 5326B Timer-Counter. The Trigger light on the HP Timer-Counter should glow continuously if smooth fringe counting is taking place.

(a) Set the Time Base/Multiplier switch to the position labeled 103 in blue lettering. With the translation stage moving, push the reset button and wait a few seconds until the counting stops. The counter display should read 8.58 or 8.59. It is easy to obtain a ratio from this displayed number. First, ignore the decimal point in the displayed number and ignore the units indicated on the display. Then divide the numerals on the display by the multiplier setting, 103, to find the ratio 858/103 = 0.858, which is the ratio of the number of red fringes counted by Ch. A to the number of green fringes counted at the EXT. oscillator input. When this ratio is combined with the accurately known value for HeNe in air, given above, one obtains GreNe = (6328 Å)(0.858) = 5430 Å to three significant figures, in agreement with the wavelength given on the label of the GreNe laser housing. You can repeat the ratio measurement by pushing the RESET button on the HP 5326B Timer-Counter.

Make a total of five measurements at the 103 Frequency/Multiplier switch setting and record your results in a table in your lab notebook. In your table, use your ratio measurement to compute five values for GreNe in air. Compute the mean value and the standard deviation of your values for GreNe. Be sure to report the proper number of significant figures in your computations.

(b) Set the Time Base/Multiplier switch to 104 (blue lettering). With the translation stage moving, you will now have to wait ten times longer than you did in part (a) to obtain a ratio, but the ratio measurement will include four significant figures on the HP counter display.

Make a total of five measurements at the 104 Time Base/Multiplier switch setting and record your results in a table in your lab notebook. In your table, use your ratio measurement to compute five values for GreNe. Compute the mean value and the standard deviation of your determinations of GreNe. Be sure to report the proper number of significant figures in your computations.

(c) Now set the Time Base/Multiplier switch to 104 (blue lettering). Make a total of five measurements at the 104 Time Base/Multiplier switch setting and record your results in a table in your lab notebook. In your table, use your ratio measurement to compute five values for GreNe. Compute the mean value and the standard deviation of your determinations of GreNe. Be sure to report the proper number of significant figures in your computations.

(6) Index of refraction of air.

A light beam has a shorter wavelength in a solid or gaseous medium than it does in vacuum. If the wavelengths in air and vacuum are respectively air and vac then they are related by air = vac/nair, where nair is the index of refraction of air. When the gas cell in the arm BS-M1-BS is evacuated fully, the number of waves along the evacuated length L of the cell is given by Nvac = 2L/vac, where waves travelling along both directions in the cell are being counted. When air is let into the gas cell to reestablish 1 atm of air pressure, the number of waves contained within the air volume is given by Nair = 2L/air. The difference N = Nair - Nvac can be written

N = 2L/air ( 1 - (1/nair) )   (1)

and N is the increase in the number of waves present in arm BS-M1-BS as 1 atm of air pressure is reestablished in the gas cell. This increase in the number of waves in the gas cell arm leads to NHeNe and NGreNe fringes moving over the detectors PD1 and PD2, respectively. Substituting NHeNe and NGreNe into Eq. (1) one can determine the index of refraction of air at the HeNe and GreNe wavelengths.

(a) Before making index of refraction measurements, make sure that good quality fringe signals can still be seen on the oscilloscope when the translation stage is in motion. Turn off the translation stage when the fringe signals are properly adjusted.

(b) Set the Tektronix 2211 Digital Storage Oscilloscope in the Store mode, and set the Sec/Division switch to 20 ms/div. At this setting, if the Sec/Div calibrate knob is rotated to the uncalibrated position, the Sec/Division scale of the oscilloscope is expanded by a factor of 100 and the Sec/Division scale is now 2 s/div. See Ref. 3 for further details of this useful method for slowing down the time base of this oscilloscope. The oscilloscope is now in Roll Mode and the 10 cm width of the CRT display will contain 20 s of data moving slowly to the left like a strip chart.

Set the low pass filter of the SR560 preamplifier to pass DC to 10 Hz to match the rather low frequency of the fringes produced as the gas cell leaks back up to atmospheric pressure.

(c) Pump the gas cell down for at least 30 s. If the oscilloscope is set properly you will observe many fringes passing by the photodiode detectors as the air is evacuated from the gas cell. When pumping is complete, close the valve leading from the vacuum pump to the gas manifold.

Now, open the valve leading to the precision leak valve, so that air leaks slowly into the gas cell. On the oscilloscope you should see fringe signals appear rapidly and then more slowly as the gas cell returns to atmospheric pressure. By repeating the pump-down/leak-up procedure several times it should be possible to find a setting of the precision leak valve such that all fringes produced during the leak up period can be contained within the 20 s store window of the oscilloscope. When you have obtained a satisfactory display of the fringe signals during the entire leak up period, save the oscilloscope display and plot it. A sample oscilloscope trace of the leak-up process is given in Fig. 4.

Make at least three measurements of the fringe patterns that occur during the leak-up process. Determine the number of fringes NHeNe and NGreNe that occur during each leak-up measurement and use these to determine the index of refraction of air at the HeNe and GreNe laser wavelengths using Eq. (1) for each measurement. By finding the mean and standard deviation of your index of refraction measurements, report final values and associated uncertainties for the index of refraction of air at the HeNe and GreNe laser wavelengths.

To make your index of refraction calculation you will need to remove the windows of the gas cell to measure the evacuated length L of the gas cell.

References
1. R.A. Serway, C.J. Moses, and C.A. Moyer, Modern Physics (Saunders College Publishing, Philadelphia, 1989), pp. 267-293.
2. American Institute of Physics Handbook, edited by D.E. Gray, 3rd ed. (McGraw-Hill, New York, 1972), p. 7-38.
3. Tektronix 2211 Digital Storage Oscilloscope, manual (Tektronix, Beaverton, OR, 1988), p. 3-10.

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