Michelson Interferometry and Measurement of the Sodium Doublet Splitting PHYS 3330: Experiments in Optics Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602 (Dated: Revised October 2011) In this lab we will use a Michelson interferometer to measure a the small difference in wavelength between two closely spaced spectral lines of a Sodium lamp. I. INTRODUCTION TO MICHELSON INTERFEROMETERY In the Michelson interferometer, an incident beam of light falls on a beam splitter which reflects roughly half of the incident light amplitude. Reflected and transmitted beams follow different paths, are reflected, and recombined to producing an interference pattern. The structure of the interference pattern depends upon differences in the length and alignment of the two arms of the interferometer, and also upon the surface smoothness of the optical components. One can make quantitative measurements of the interference pattern for the accurate comparison of wavelengths, to measure the refractive index of unknown substances, and measure the quality of optical components. The LIGO gravity wave detector is a Michelson interferometer with 2.5 mile long arms. You will need to complete some background reading before your first meeting for this lab. Please carefully study the following sections of the Newport Projects in Optics document (found in the Reference Materials section of the course website): 0.4 Interference Also, the excerpt from Melissinos Experiments in Modern Physics, included as an appendix to this manual. Finally read section 8.1 of your textbook Physics of Light and Optics, by Peatross and Ware. Your pre-lab quiz cover concepts presented in these materials AND in the body of this write-up. Don t worry about memorizing equations the quiz should be elementary IF you read these materials carefully. Please note that taking a quick look at these materials 5 minutes before lab begins will likely NOT be adequate to do well on the quiz. The optical circuit of a Michelson interferometer is shown schematically in Figs. 2 and 3. Light from a source S passes through a ground glass plate DB (optional) and strikes the beam splitter P. The beam splitter P is a partially silvered mirror (50% reflecting). Half of the incident light amplitude toward mirror M1 and transmits half of the incident amplitude toward mirror M2. A micrometer adjuster screw is attached to the movable mirror M1, permitting it to be moved toward or away from the beam splitter in small, precise steps. The two mirrors, beam splitters, and compensating glass are flat to about a 1/4 of an optical wavelength. The compensating glass, CG, of identical composition and thickness to the beam splitter, is included so that each of the two beams (paths P-M1-P-O and P-M2-P-O in Figure 3) passes through the FIG. 1: Your Michelson interferometer setup. The interferometer is in the lower left, the sodium lamp power source in the upper left, and sodium lamp in the upper right portion of the picture. FIG. 2: Typical Michelson interferometer. The important parts of a Michelson interferometer include a sturdy base, a diffusing glass, a beam splitter, a movable mirror with a micrometer screw for measuring distance of movement, a fixed mirror, and compensating glass. The light source can be a spectral lamp, a collimated laser beam, or even a white light source. same integrated thickness of glass. (Note that otherwise the beam that travels along path P-M1-P-O would pass through a thickness of glass three times while the beam that travels along the other path would pass through the same thickness of glass only once. The compensating glass is not necessary to produce fringes using laser light, but it is essential for producing interference fringes with white light, such as those shown in Figure 9. Light traveling along trajectories making an angle φ
2 FIG. 3: Optical arrangement and light path in Michelson interferometer FIG. 5: HeNe fringes in a Michelson interferometer from this lab. Photograph taking by your instructor with an iphone. FIG. 4: Condition for interference with respect to the optic axis accumulate a path length difference of 2d cos θ between the arms. When this difference is an integer number m = 0, 1, 2... of wavelengths, destructive interference occurs (dark fringes). mλ = 2d cos θ, m = 1, 2, 3... (1) where m is the order of the interference. Note that the beam in one arm undergoes an additional external reflection, and thus incurs one additional π phase shift, relative to the beam in the other arm, which is why the above condition produces a dark, rather than a bright, fringe. If the two mirrors M1 and M2 are not aligned precisely perpendicular to one another the path difference will depend on the particular region of mirror M1 (and the corresponding region of M2) which we are observing from the position O. The field of view, then, seen by looking at mirror M1 from position O will be made up of a series of alternately bright and dark fringes, nearly straight and parallel, as shown in Figure 5. If the path difference is near zero, the fringes will be broad and widely spaced in the field of view; if the path difference is on the order of 40 or 50 wavelengths the fringes will be narrow and closely spaced, so much so that they may be unresolvable by the naked eye. Such fringes are shown in Figure 5. If the two mirrors are precisely aligned exactly parallel to one another, a bulls-eye fringe pattern will be seen FIG. 6: Circular fringes (equal inclination) seen in Michelson interferometer by the observer situated at point O in Fig. 3, consising of a series of concentric rings. Each ring corresponds to a different angle φ, as illustrated in Figure 6. In this case, when M1 is translated a distance δz along the optic axis, the number of fringes N that will appear (or disappear) at the center of the bulls-eye pattern is: N = 2 δz/λ Thus, if you can measure the displacement of M1 which causes a known number of fringes to to appear (or disappear) from the center of the pattern, an unknown wavelength can be measured. Conversely, you can use a known wavelength to calibrate the micrometer screw; i.e., convert microns of travel of the screw to microns of travel of the mirror (which are not necessarily equal!). II. ALIGNMENT OF THE INTERFEROMETER USING A LASER 1. Place and orient your steering mirror to direct the expended beam from a HeNe laser into the the input port of the interferometer.
3 2. Observe three discs of light emerging from the output side. Two of the copies will lie almost on top of each other, but the third will likely be far to the side (or even absent), if the mirror M2 is misaligned. M2 is equipped with two screws on the back side that tilt the plane of the mirror. A slight adjustment of the mirror tilt screws will cause one of the three images to move. You can achieve the proper alignment of the mirrors by using the screws to superimposing the (movable) image onto the rightmost of the two stationary images. You will see interference fringes appear, though initially they may be very finely space. As you adjust M2 you must momentarily STOP turning the screws to look for fringes; you will not see fringes if you are turning the screws even if the mirrors are perfectly aligned, as the movement of the mirror blurs the pattern. 4. It turns out there are two orientations of the M2 which produce fringes with a HeNe laser in your interferometer. It is important for later stages of this lab that you now pick the correct orientation. To do this you must carefully observe the output and compare to Figure 7. In the wrong case, the strong fringes die out abruptly on the left side of the disc, when looking at a ground glass plate installed on the output port. In the correct case the strong fringes extend all the way to the left edge of the pattern. The difference is subtle. 5. While observing the fringes, carefully adjust both screws on mirror M2 so that the fringes take a circular bulls eye pattern. See Figure 8 for guidance. III. CALIBRATING THE MICROMETER SCREW USING A HENE LASER AS A WAVELENGTH REFERENCE. M1 can be translated without disturbing the alignment of the interferometer. Each tick on the thimble of the micrometer adjuster of M1 corresponds to 1 micron of movement of the spindle. One complete revolution of the thimble advances the the spindle through 50 microns, and moves the edge of the thimble across the distance of one tick-mark on the barrel. Thus, 10 ticks on the barrel is 5mm of movement of the spindle. Make SURE you are clear on how to read the scales on the the micrometer adjuster before you start taking calibration measurements. (See http://en.wikipedia.org/wiki/file: Micrometer_caliper_parts_0001.png if you need a picture.) 1. Set the micrometer screw to approximately 5 mm. 2. Turn the micrometer screw a quarter-turn in the direction of smaller reading. This is done to avoid backlash since all readings will be taken with the screw moving in the same direction (towards FIG. 7: The right (upper) and wrong (lower) appearance of the fringes. In the wrong case, the strong fringes die out abruptly on the left side of the disc, here demarcated with the blue dashed line. In the right case, the strong fringes go all the way to the left edge. The difference is more obvious when viewed in person. Record the reading of the mi- smaller readings). crometer. 3. Count the number of fringes that pass through the center of the field of view as the micrometer screw is turned slowly in the direction of decreasing reading. After counting 10 fringes, record the micrometer reading again. When you stop turning the screw at the end of 10 fringes, be very careful to NOT accidentally slip a fringe while you are recording the micrometer reading. 4. Continue this process for 20 groups of 10 fringes. You will find that this procedure requires a cer-
4 FIG. 8: Successive fields of view in interferometer alignment tain amount of technique (and patience), since the slightest movement of the screw will gain or lose a fringe. IV. ANALYSIS Enter these points into an Excel spreadsheet, export as CSV, import the data to python, and perform a fit of the data to a linear model. Consider carefully what free parameters you want to include in your model. Don t worry about including error bars in this fit. The slope of the best fit line is the calibration constant you need: microns of mirror displacement per microns of travel of the micrometer screw. The uncertainty in the slope as reported by the fitting routine will be a useful estimate of your uncertainty in the calibration procedure, and you will use this information to estimate a systematic uncertainty in your sodium wavelength measurements of the next section. V. MEASUREMENT OF CLOSELY SPACED SPECTRAL LINES VIA MICHELSON INTERFEROMETERY. In the next part of the lab you will use your calibrated Michelson interferometer to measure a small difference in wavelength between two closely spaced spectral lines of a sodium lamp. The 589nm yellow line of sodium actually consists of two distinct lines, separated by a few tenths of a nanometer. When a sodium lamp is used as a source for a Michelson interferometer, each line will produce its own set of fringes with a slightly shifted pattern relative to the other. At certain positions of mirror M1 the two sets of fringes coincide (bright regions overlapping bright regions), and the total intensity pattern observed is a bulls-eye pattern of moderately high contrast (a sharpening coincidence ). When the M1 is moved, the two sets of fringes evolve slightly differently, and at some setting will anti-coincide (bright regions overlapping dark regions) so that a total intensity pattern displays no fringes (a wash-out anticoincidence ). We can use the periodicity of the wash-out phenomenon to measure the sodium line spacing. The theory is described next. The two spectral lines whose difference is to be measured are at wavelengths λ a, λ b. Let δd be the path length difference between the two interferometer arms at some sharpening coincidence. At this coincidence each set of fringes satisfies a dark fringe criterion for the central fringe of each bullseye pattern (Equation (1) with θ = 0. m a λ a = 2 δd (2) m b λ b = 2 δd (3) for orders m a and m b which, as integers, must be related by m b = m a + M, (4) where M is the order of the coincidence, or its number of sharpening coincidences which would have been observed if one had started observing from the white light condition δd = 0, in which case both interference patterns would have had dark central fringes, as there would be no path length difference for either (any!) wavelength. Substituting (2) and (3) into (4) we have: or 2 δd/λ b = 2 δd/λ a + M (5) δd = λ aλ b 2 λ M λ 2 2 λ M (6) where λ is the mean wavelength of the two closely spaced lines. Thus, if we measure the mirror position for several sharpening coincidence orders M, M + 1, M + 2,... the slope of a linear fit to the data will give us λ. Note that this is true EVEN if we are off in our reckoning of the absolute order M by some an unknown integer X, as the slope we infer from the linear fit to the data is (of course) independent of arbitrary translations of the horizontal axis M M + X Therefore it is not critical to begin the measurement at the white light condition, although it does help to make the sharpening coincidence more obvious. Also note that the same equations apply for wash-outs, which are typically easier to identify. In this lab you will look for wash-outs. VI. PROCEDURE 1. Turn on the sodium lamp and wait at least 5 minutes for the light to reach full intensity. 2. Direct the light from the lamp into the interferometer using the steering mirror. 3. Position a ground glass diffuser at the output port. 4. If you have already achieved fringes with the HeNe laser you should immediately see fringes. If you do not see fringes it is possible that you have (unluckily) landed on a wash-out try spinning the micrometer screw a turn or two. If you still have no fringes, put the HeNe light back into the interferometer to check to see if something has been bumped.
5 5. Use the same fringe counting procedure you did for the HeNe calibration to measure the (mean) wavelength of the (two) sodium D-line(s). For this calculation, utilize the micrometer screw calibration you measired earlier with the HeNe laser. Now you will measure the Sodium D-Line doublet splitting 6. Spin the micrometer screw to nearly its maximumreading. 7. Turn the micrometer toward smaller readings until you see the first sharpening event. Record the micrometer position. 8. Continue turing the micrometer toward smaller readings and record its value for every subsequent washout you see. It is ok, and indeed recommended, to scan back and forth across a washout position to determine its location, but remember to always finish up your screw turning by slightly advancing the screw in the direction of smaller reading BEFORE recording its value. (This is essential to eliminate errors due to backlash in the screw threads.) Estimate an uncertainty in determining the location of each washout. Be sure to describe the procedure for making this uncertainty estimate in your manuscript. 8. Continue recording the positions of washouts until you run reach the minimum reading of the micrometer screw. VII. ANALYSIS Use these data, along with your measurement of mean wavelength, and Eq. (5), to determine the sodium D-line splitting. Propagate all your uncertainties, including the systematic uncertainty in the micrometer screw calibration, to get a total uncertainty in your D-line splitting measurement. VIII. EXTRA CREDIT WHITE LIGHT FRINGES Now you will adjust the interferometer to the white light position, when the two arms are exactly equal in length. Re-introduce the HeNe into the interferometer. Set the micrometer screw at approximately 90% of its full value. When mirror M1 is moved so as to approach the condition of equal path length of both arms, the fringes move appear to collapse into the center of the pattern, the center fringe growing ever larger as M1 is moved. Carefully turn the micrometer screw in a direction that causes the fringes to move toward the center. As you progress, the central fringe will become wider, eventually growing to fill the entire field of view when the white light condition is reached. If you pass the white light condition, the fringes will start to emerge from the center, rather than collapse into it. Now, since the mirrors in your interferometer are not perfectly flat, even at the white light condition the field of view can never be made perfectly uniform. Do the best you can with the HeNe. Tilt your work lamp down so that you can see the lightbulb when you look into the output of the interferometer. Put a ground glass diffuser on the input to the interferometer. Look into the interferometer, and, with patience and care, slowly turn the micrometer screw towards smaller readings. White light fringes will only exist for approximately 1/8 of a turn of the micrometer screw otherwise you see nothing but the frosted glass. Fig. 9 proves it can be done. Take turns looking for the fringes if you get tired. If you do not see any fringes, go back to your original position and turn the screw towards increasing reading. When you see the fringes, shout Eureka! The physics of white light fringes will be discussed in lecture in October.
FIG. 9: White light fringes in a Michelson interferometer from this lab. Photograph taken by your instructor with an iphone. 6