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 recom- bined to producing an interference pattern. The struc- ture 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 mea- FIG. 1: Your Michelson interferometer setup. The interferometer is in the lower left, the sodium lamp power source in surements of the interference pattern for the accurate the upper left, and sodium lamp in the upper right portion of comparison of wavelengths, to measure the refractive in- the picture. dex 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. Fi- nally 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 FIG. 2: Typical Michelson interferometer. The important will likely NOT be adequate to do well on the quiz. parts of a Michelson interferometer include a sturdy base, a diffusing glass, a beam splitter, a movable mirror with a The optical circuit of a Michelson interferometer is micrometer screw for measuring distance of movement, a fixed shown schematically in Figs. 2 and 3. Light from a source mirror, and compensating glass. The light source can be a S passes through a ground glass plate DB (optional) and spectral lamp, a collimated laser beam, or even a white light strikes the beam splitter P. The beam splitter P is a par- source. tially silvered mirror (50% reflecting). Half of the incident light amplitude toward mirror M1 and transmits half of the incident amplitude toward mirror M2. A mi- same integrated thickness of glass. (Note that otherwise crometer adjuster screw is attached to the movable mirror the beam that travels along path P-M1-P-O would pass M1, permitting it to be moved toward or away from the through a thickness of glass three times while the beam beam splitter in small, precise steps. The two mirrors, that travels along the other path would pass through the beam splitters, and compensating glass are flat to about same thickness of glass only once. The compensating a 1/4 of an optical wavelength. The compensating glass, glass is not necessary to produce fringes using laser light, CG, of identical composition and thickness to the beam but it is essential for producing interference fringes with splitter, is included so that each of the two beams (paths white light, such as those shown in Figure 9. P-M1-P-O and P-M2-P-O in Figure 3) passes through the 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 diﬀer- FIG. 6: Circular fringes (equal inclination) seen in Michelson ence is an integer number m = 0, 1, 2... of wavelengths, interferometer destructive interference occurs (dark fringes).

mλ = 2d cos θ, m = 1, 2, 3.... (1) by the observer situated at point “O” in Fig. 3, consising of a series of concentric rings. Each ring corresponds to a where m is the “order” of the interference. Note that different angle φ, as illustrated in Figure 6. In this case, the beam in one arm undergoes an additional external when M1 is translated a distance δz along the optic axis, reflection, and thus incurs one additional π phase shift, the number of fringes N that will appear (or disappear) relative to the beam in the other arm, which is why the at the center of the bulls-eye pattern is: above condition produces a dark, rather than a bright, fringe. N = 2 δz/λ If the two mirrors M1 and M2 are not aligned precisely perpendicular to one another the path difference will de- Thus, if you can measure the displacement of M1 which pend on the particular region of mirror M1 (and the cor- causes a known number of fringes to to appear (or disap- responding region of M2) which we are observing from pear) from the center of the pattern, an unknown wave- the position O. The field of view, then, seen by looking length can be measured. Conversely, you can use a known at mirror M1 from position O will be made up of a series wavelength to calibrate the micrometer screw; i.e., con- of alternately bright and dark fringes, nearly straight and vert microns of travel of the screw to microns of travel of parallel, as shown in Figure 5. If the path difference is the mirror (which are not necessarily equal!). 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 II. ALIGNMENT OF THE INTERFEROMETER closely spaced, so much so that they may be unresolv- USING A LASER able by the naked eye. Such fringes are shown in Figure 5. 1. Place and orient your steering mirror to direct the ex- If the two mirrors are precisely aligned exactly parallel pended beam from a HeNe laser into the the input to one another, a “bulls-eye” fringe pattern will be seen 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 super- imposing 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 guid- ance.

III. CALIBRATING THE MICROMETER SCREW USING A HENE LASER AS A WAVELENGTH REFERENCE. FIG. 7: The right (upper) and wrong (lower) appearance of M1 can be translated without disturbing the alignment the fringes. In the wrong case, the strong fringes die out abruptly on the left side of the disc, here demarcated with of the interferometer. Each tick on the thimble of the the blue dashed line. In the right case, the strong fringes go micrometer adjuster of M1 corresponds to 1 micron of all the way to the left edge. The difference is more obvious movement of the spindle. One complete revolution of the when viewed in person. 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 smaller readings). Record the reading of the mi- is 5mm of movement of the spindle. Make SURE you crometer. are clear on how to read the scales on the the micrometer adjuster before you start taking calibration measure- 3. Count the number of fringes that pass through the ments. (See http://en.wikipedia.org/wiki/File: center of the field of view as the micrometer screw is Micrometer_caliper_parts_0001.png if you need a turned slowly in the direction of decreasing reading. picture.) After counting 10 fringes, record the micrometer reading again. When you stop turning the screw 1. Set the micrometer screw to approximately 5 mm. at the end of 10 fringes, be very careful to NOT accidentally slip a fringe while you are recording 2. Turn the micrometer screw a quarter-turn in the the micrometer reading. direction of smaller reading. This is done to avoid backlash since all readings will be taken with 4. Continue this process for 20 groups of 10 fringes. the screw moving in the same direction (towards You will find that this procedure requires a cer- 4

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 λa, λb. Let δd be the path length difference between the two interferometer arms at FIG. 8: Successive fields of view in interferometer alignment 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 tain amount of technique (and patience), since the θ = 0. slightest movement of the screw will gain or lose a fringe. maλa = 2 δd (3)

mbλb = 2 δd (4) IV. ANALYSIS

for orders ma and mb which, as integers, must be related Enter these points into an Excel spreadsheet, export by as CSV, import the data to python, and perform a fit of the data to a linear model. Consider carefully what free mb = ma + M, (5) parameters you want to include in your model. Don’t where M is the “order” of the coincidence, or its number worry about including error bars in this fit. You know of sharpening coincidences which would have been ob- that each group of 10 fringes actually moves the mirror by served if one had started observing from the white light 5 wavelengths of the HeNe laser. Plot the actual mirror condition δd = 0, in which case both interference pat- displacement vs reading of the micrometer. Fit this data terns would have had dark central fringes, as there would to a straight line. The slope of the line is the calibration be no path length difference for either (any!) wavelength. constant K that you are seeking Substituting (2) and (3) into (4) we have: microns of actual mirror travel K = (2) 2 δd/λb = 2 δd/λa + M (6) microns of travel of the screw threads. or The uncertainty in the slope as reported by the fitting routine will be a useful estimate of your uncertainty in λ λ λ¯2 δd = a b M ≈ M (7) the calibration procedure, and you will use this informa- 2∆λ 2∆λ tion to estimate a systematic uncertainty in your sodium wavelength measurements of the next section. 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 V. MEASUREMENT OF CLOSELY SPACED slope of a linear fit to the data will give us ∆λ. Note SPECTRAL LINES VIA MICHELSON that this is true EVEN if we are off in our reckoning of INTERFEROMETERY. 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 In the next part of the lab you will use your calibrated horizontal axis M 7→ M + X Therefore it is not critical Michelson interferometer to measure a small difference to begin the measurement at the white light condition, in wavelength between two closely spaced spectral lines although it does help to make the sharpening coincidence of a sodium lamp. The 589nm “yellow” line of sodium more obvious. Also note that the same equations apply actually consists of two distinct lines, separated by a few for “wash-outs”, which are typically easier to identify. In tenths of a nanometer. When a sodium lamp is used as a this lab you will look for wash-outs. 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 VI. PROCEDURE the two sets of fringes coincide (bright regions overlap- ping bright regions), and the total intensity pattern ob- served is a bulls-eye pattern of moderately high contrast 1. Turn on the sodium lamp and wait at least 5 minutes (a “sharpening coincidence”). When the M1 is moved, for the light to reach full intensity. the two sets of fringes evolve slightly differently, and at 2. Direct the light from the lamp into the interferometer some setting will anti-coincide (bright regions overlap- using the steering mirror. ping dark regions) so that a total intensity pattern dis- plays no fringes (a “wash-out anticoincidence”). We can 3. Position a ground glass diffuser at the output port. 5

4. Spin the micrometer screw to nearly its maximum- 8. Continue recording the positions of washouts until reading. you run reach the minimum reading of the micrometer screw. 5. 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 (un- luckily) landed on a wash-out — try spinning the VII. ANALYSIS 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 Use these data, along with your measurement of mean bumped. wavelength, and Eq. (5), to determine the sodium D-line splitting. Propagate all your uncertainties, including the 6. Use the same fringe counting procedure you did for systematic uncertainty in the micrometer screw calibra- the HeNe calibration to measure the (mean) wave- tion, to get a total uncertainty in your D-line splitting length of the (two) sodium D-line(s). For this cal- measurement. culation, utilize the micrometer screw calibration you measired earlier with the HeNe laser.

Now you will measure the Sodium D-Line doublet splitting VIII. EXTRA CREDIT – WHITE LIGHT FRINGES 7. Spin the micrometer screw to nearly its maximum- reading. Now you will adjust the interferometer to the “white 8. Turn the micrometer toward smaller readings until light position,” when the two arms are exactly equal in you see the first sharpening event. Record the mi- length. Tilt your work lamp down so that you can see crometer position. the lightbulb when you look into the output of the interferometer. Put a ground glass diffuser on the input to 9. Continue turing the micrometer toward smaller the interferometer. Position the screw at approximately readings and record its value for every subsequent at 1/2 its full range. Look into the interferometer, and, washout you see. It is ok, and indeed recom- with patience and care, slowly turn the micrometer screw mended, to scan back and forth across a washout towards smaller readings. White light fringes will only position to determine its location, but remember to exist for approximately 1/8 of a turn of the micrometer always “finish up” your screw turning by slightly screw – otherwise you see nothing but the frosted glass. advancing the screw in the direction of smaller Fig. 9 proves it can be done. Take turns looking for the reading BEFORE recording its value. (This is es- fringes if you get tired. If you do not see any fringes, go sential to eliminate errors due to backlash in the back to your original position and turn the screw towards screw threads.) Estimate an uncertainty in deter- increasing reading. When you see the fringes, shout “Eu- mining the location of each washout. Be sure to reka!” Count how many fringes of each color can be seen describe the procedure for making this uncertainty on either side of the central maximum, and report this estimate in your manuscript. in your write-up. Take a picture with your cell phone! 6

FIG. 9: White light fringes in a Michelson interferometer from this lab. Photograph taken by your instructor with an iPhone.