Author: James BarnesLab Partner: Jon TimcheckThe Ohio State University - Physics Dept.Physics 5700

The Michelson Interferometer(Dated: April 29, 2015)

This experiment studies the applications of the famous Michelson interferometer. First the in-terferometer is used to measure the wavelengths of three sources: a HeNe laser, a thallium vaporlamp, and a cadmium vapor lamp. We measured the wavelength of the HeNe to be 633.4 ± 2.0nm,which is within one standard deviation of the accepted value. The thallium and cadmium sourceswere used to measure the wavelength of the green line and red line that appear in the emissionspectrum of thallium and cadmium respectively. We measured the wavelength of thallium’s greenline to be 538.8±20nm and the wavelength of cadmium’s red line to be 742.5±23nm, which are bothwithin one standard deviation of the accepted values. Next, the interferometer was used to measurethe separation of wavelengths that in both the mercury and sodium doublets. The separation ofmercury’s doublet was found to be 1.95 ± 0.14nm and sodium’s doublet was measured to have aseparation of 0.55±0.02nm. Both of these measurements were within 2 standard deviations of theiraccepted values. Finally, the interferometer was used to measure the index of refraction of a pieceof mica. The index of refraction measured in this experiment was 1.543± 0.018 which was within 2standard deviations of the accepted value.

PACS numbers:

Prior to the Michelson-Morley experiment conductedin 1887 by Albert A. Michelson and Edward Morley, itwas believed that both light and sound both requireda medium to propagate through. The Dutch physicistChristiaan Huygens proposed the existence of a luminif-erous aether as light’s transmission medium. The ideawas that Earth was moving through this aether, whichallowed light from celestial bodies to reach us. However,this idea becomes problematic when one considers therelative motion of the Earth’s orbit with respect to thismedium. Depending on the time of year, the Earth’s or-bit would either increase or decrease the speed of light.Undertaking the task of describing this aether, Michelsonbuilt a device that could be used to compare the speedof light in perpendicular directions. Michelson’s exper-iment provided compelling evidence against the aethertheory. His results influenced the interest and develop-ment of special relativity. The Michelson interferometerended up being more useful than originally thought, withnumerous applications in chemistry, optics, and physics.More recently, the interferometer has been modified inthe hopes of measuring gravitational waves [2], whichwere predicted by Einstein’s general relativity and arehypothesized to transport energy as gravitational radia-tion.

In this experiment we used a Michelson Interferometerto measure the wavelength of various sources, the wave-length separation in emission spectra doublets, and theindex of refraction of mica. All of these measurementsare dependent on the production of interference fringes.These interference patterns are produced when the twobeams are recombined at the beam splitter at the samefrequency. Due to the phase difference between the two

Figure 1: The basic structure of the Michelson interferometerused in this experiment.

beams, both constructive and destructive interference areobserved. This phase difference is due to the differencein the length each beam’s path traveled.

As shown in Figure 1 [9], the Michelson interferome-ter consists of a beam splitter, and both a movable andfixed mirror, denoted as M1 and M2 respectively. Asthe name implies, a beam splitter is a half-silvered mir-ror that splits a beam of light into two parts, these twonew beams are equal in intensity. A compensating glasswas added to the interferometer because if one tracesthe paths of the beams, it becomes clear that the beamfrom M1 passes through the beam splitter 3 times andthe beam from M2 would only pass through the splitteronce. The compensating glass is used to negate the effectof dispersion, therefore a source with a large bandwidthcould still produce fringes. Dispersion is caused by thevarying speeds of light in a given material for differentwavelengths [7]. So if a white light was sent through anoptical fiber, the beam will slowly separate into its var-ious color components [7]. Therefore, adding the com-

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Figure 2: One can see that circular fringes are produced whenM1 and M2 are perpendicular to each other. Figure from thewebsite hyperphysics.

pensating lens ensures that any optical differences in thebeams are purely a product of the actual path length dif-ference [4].

The two beams are then brought back together pro-ducing the desired interference pattern. The movementof the M1 is controlled by a micrometer with a mechan-ical advantage of 5:1. M2 is not entirely stationary, itstilt can manipulated by two dials on the back of the mir-ror’s frame. This tilt governs the geometry of the fringesproduced as shown in Figure 2 and can produce eitherhyperbolic, circular, or straight fringes. In this experi-ment two methods were used to measure the wavelengthof the source. Both methods utilized the basic principleof comparing the displacement of M1 to the number offringe repetitions which occured over said displacement[8]:

λ =2L

n(1)

such that L is the path length difference between M1 andM2, and n is the number of fringe repetitions. The keydifference between the two methods is how the fringe rep-etitions are counted.

The primary method requires the use a Thorlabs modelDCC1545M camera as the detector, and would act as theobserver, as shown in Figure 1. Due to the various typesand intensities of the sources used, one would have tooften manipulate the photographic settings of the cam-era. To find the number of fringe repetitions for equation(1), we picked a fixed spot on the computer screen andcounted the number of bright fringes that crossed thefixed point.

The secondary method utilized a photodetector con-nected to a universal counter to count the number offringe repetitions, albeit the HeNe laser was the onlysource that was bright enough to employ this method.Before the photodetector could be used to count thefringe repetitions, the trigger on the counter had to beset. The trigger was set so that when a dark fringe tran-sitioned into a bright fringe the trigger would be tripped,

Figure 3: The circular fringes obtained with the thalliumsource.

and would be turned off once the intensity of the fringewent below the set value, thus completing one fringe rep-etition. The shape of the fringe was irrelevant in the pri-mary method, but the secondary method required the useof the circular fringes. If the straight fringes were usedon the photodetector, multiple fringes would fall uponthe iris of the photodetector, which would increase themeasured intensity and tamper with the collected data.However, if one were to use circular fringes, the centerfringe could be centered on the iris of the detector sothat only one fringe was being measured at a given time.If for some reason one could only use straight fringes,a single slit with a small enough width could be placedin front of the photodetector so that only one fringe ismeasured at a given time. This is not the case for ourexperiment. For the HeNe laser, we are interested inobserving the real fringes, which means the fringes can bedisplayed on screen without the use of a condensing lens[4]. However, we did expand the laser beam with a f=-0.5cm lens because the beam was too narrow initially. Thenas mentioned earlier, the tilt of M2 was manipulated un-til the fringe pattern appeared on a white viewing card.The camera was then aligned so that the fringes werecentered on the display. Six trials were conducted overa displacement of M1 of 0.2 mm, which can be seen inTable 1. We can find δλ by using the general formula forerror propagation[5]:

δλ =

√(∂λ

∂nδn)2 + (

∂λ

∂LδL)2.

δn was then found to be 2.26 counts by calculating thestandard deviation of n [5]:

δn =

√√√√1

6

6∑i=1

(ni − n̄)2

δL is the error in the micrometer reading, which is 0.005mm, because the micrometer can be read to the nearest0.01 mm. The weighted average was then calculated tobe 633.4 ± 2.0nm, which is within 1 standard deviationof the theoretical value of 632.8nm [1].

Using the camera method, we used the interferometerto study the emission lines produced by thallium and cad-

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TABLE I: Data collected using the photodetector to measurethe wavelength of the HeNe laser.

L(mm) n(#counts) λ(nm)0.2 630 634.9- 633 631.9- 632 632.9- 630 634.9- 629 635.9- 625 629.9

mium vapor lamps. These emission spectra are frequen-cies emitted as electromagnetic radiation when an atomtransititions from a high energy state to a low energystate. The produced photon is generated from the poten-tial difference between the two states, and is unique foreach atom [3]. The vapor lamps used in this experimentgenerate light by sending an electric discharge through agas of the desired element which then ionzies the gas.

The HeNe laser was then replaced by a green filteredthallium source, which required the primary methodmentioned earlier due to its low intensity. The filterplaced on the thallium lamp absorbed all of the non-green light rays, and only allowed the passage of greenlight. Unlike the HeNe laser, we were interested in ob-serving the virtual fringes, which the system requires acondensing lens to produce discernible fringes [4]. A lenswith a f=0.5 cm was placed just before the camera sothat the virtual fringes will converge on the camera len[1]. Six trials were conducted over a displacement of M1of 0.2 mm. Using the same data analysis structure out-lined for the HeNe laser, the weighted average wavelengthwas calculated to be 538.8±20nm for the thallium greenline. The theoretical wavelength for the thallium greenline is 535.8 nm[6], therefore our measured wavelengthwas within 1 standard deviation of the theoretical value.

The thallium source was then replaced by a red fil-tered cadmium vapor lamp. The cadmium source alsorequired the condensing lens that was used on the thal-lium source to observe its virtual fringes. Again, 6 trialswere conducted over a displacement of 0.2 mm, whichgave us a weighted average wavelength of 742.5± 23nm.The theoretical wavelength of the red line produced bythe cadmium source is 734.5 nm[6], therefore our mea-sured wavelength is within 1 standard deviation of theaccepted value.

Although the process of measuring a wavelength for theHeNe laser and the two vapor lamps are practically iden-tical, it should be known that the two types of sourcesare very different. The HeNe laser is monochromatic,and therefore variations of the electric field at any twopoints in space are completely correlated, implying thatthe laser is a coherent source [2]. However, this cannot besaid about the two vapor lamp sources. Both the ampli-tude and the phase of the electric field produced by the

source exhibit random fluctuations, this is because thesource is not monochromatic. So if we compare two waveswith different points of orgin on the lamp, the randomfluctuations would be uncorrelated, meaning our vaporlamps are only partially coherent. This creates a prob-lem because interference patterns must be produced bya coherent source [4]. The filter on the two vapor lampsallow us to focus on one wavelength, however the inter-ferometer can still exhibit the temporal coherence of thetwo beams by simply varying the difference in the pathlengths traveled by the beams [2]. When the path lengthdifference is increased the fringes become indistinguish-able because increasing the difference only increases thetravel time for the beam reflected by M1. Therefore theinterferometer can be used to increase the interferencevisibility by adjusting the path length difference. Thisinterference visibility describes the quality of the fringesproduced by an interferometric system [4]

V =Imax − Imin

Imax + Imin

As mentioned earlier, interferometers can also be usedto measure the wavelength separation in emission spectradoublets. The two doublets analyzed in this experimentare the sodium and mercury yellow doublets. Doubletsare defined as two dengenerate wavefunctions occurringin the emission spectra of an atom with the such thatthe two wavefunctions only differ by their angular spinmomenta [3]. These doublets are produced by the finestructure of the atoms, which describes the splitting ofemission lines due to the spin of an electron [3].

Now consider the two different wavelengths that occurin a doublet, and for notation’s sake, let the shorter wave-length in the doublet be denoted as λ1 and the longer asλ2. Due to the low intensity of the vapor lamps used,we could look into the beam splitter and see reflectedimages of the source. This greatly simplified the processof obtaining interference fringes because we were able toview the two reflected images and align them as shown inFigure 2(a). Once the interference pattern was obtained,one would notice that by varying the micrometer, the res-olution of the fringes also varied. As seen in Figure 3, thedifference between low contrast and high contrast fringesis noticable. The low contrast fringes are produced whenthe dark fringes of λ1 coincide with the bright fringesof λ2. The high contrast fringes are produced when thebright fringes of λ1 coincide with the bright fringes ofλ2. With this in mind, we then dialed the micrometer sothat low contrast fringes appeared and then increased thedisplacement of M1 while counting the number of com-pleted low contrast to low contrast cycles. We calculatedthe average displacement per cycle, which will be denotedas Lavg. Since λ2 > λ1, we know that λ2 = λ1 +∆λ suchthat ∆λ is the separation of the doublet. We also knowthat since this is a doublet that ∆λ << λ1. So, if we let

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Figure 4: The low contrast circular fringes compared to thehigh constrast circular fringes.

d1 be the difference between the arm lengths to M1 andM2 such that a high contrast fringes appears, and that adark fringe appears at θ for both wavelengths [1]

2d1 cos θ = m1λ1 = m2λ2

We then increased the difference between the two armsuntil the next set high contrast fringes appear, and wecan denote this new difference in displacement of the mir-rors as d2 [1]:

2d2 cos θ = m′1λ1 = m′2λ2

such that m′1 = m1 + k + 1 and that m′2 = m2 + k. Wecan then substract these two equations to find [1]:

2∆d cos θ = kλ2 = (λ1∆λ

)λ2

∆λ =λ1λ2

2∆d cos θ' λ̄2

2∆d cos θ(2)

such that λ̄ is average wavelength of the doublet. We canalso count near the center of the circular fringes so thatthe term cos θ = 1.

So for the mercury doublet, we see that λ1 occurs at576.9 nm and λ2 occurs at 579.1 nm [6]. Therefore, theaverage wavelength of the 578.0 nm for λ̄ was used forthis experiment. Five trials were conducted over 3 com-plete low to low contrast cycles with the results shownin Table II. We then found the weighted average of ∆λto be 1.95 ± 0.14nm. Comparing this to the theoreticalwavelength separation of the mercury doublet of 2.106nm [6], we see that our measured value is within 2 stan-dard deviations of the accepted value.

The mercury vapor lamp was then replaced by a yel-low filtered sodium vapor lamp. This time λ1 occursat 588.99 nm and λ2 occurs at 589.59 nm, giving us aλ̄ that is equal to 589.3 nm [6]. So following the sameprocess used to find the mercury doublet, we found theweighted average of ∆λ to be 0.55 ± 0.02nm. Thus, ourmeasured separation is within two standard deviations ofthe accept value of 0.59 nm [6]. Finally, the interfer-ometer was set up so that we could measure the indexof refraction of the mica. We were able to find the in-dex of refraction because like with the compensating lens,

Figure 5: (a) Shows the setup of the interferometer and (b)shows the path of the beam through the mica. [9]

any change in a beams path creates an optical difference.Shown in Figure 5(a), the beam that is reflected off ofM1 must pass through our slab of mica (two times dueto the return trip) before the recombination of the twobeams. Figure 5(b) shows the beam’s path through themica. The mica was mounted on a rotational stage whicha needle that allowed us to measure the angle of rotationfor the mica. From Figure 5 and Snell’s Law, we knowthat sinφi = nmica sinφr, note that the index of refrac-tion for air is 1. To find when the mica was normal tothe beam, we watched the movement of the fringes onthe computer screen. When the mica passes through thenormal angle the fringes change their direction of move-ment on the computer screen. The HeNe laser was usedas the source for this portion of the experiment. We thencan calculate the index of refraction using the equationgiven in reference [10]:

nmica =(2t−Nλ)(1− cos θ)

2t(1− cos θ)−Nλ(3)

such that nmica is the index of refraction for the mica,λ is the wavelength of the laser, t is the thickness of themica, and N is the number of fringes repetitions thatoccur while the mica was rotated an angle of θ. Withequation 3 in mind, one can see that once the thicknessis found, we can find N for a set θ. Using a caliper, wemeasured the thickness of the mica to be 0.07±0.005mm.We then conducted 16 trials, over 3 set values of θ whichwere 0.528, 0.779, and 0.905 rads. See Table II for asample of the data collected.

TABLE II: Sample data for the index of refraction

θ(rads) δθ(rads) N δN nmica δn0.905 0.0089 39 0.8165 1.529 0.0769- - 40 - 1.555 0.0823- - 40 - 1.555 0.0823- - 41 - 1.582 0.0881

We then found the δN by finding the standard devia-

5

tion for each set value of θ, so:

δN =

√√√√1

4

4∑i=1

(Ni − N̄)2.

Then we used the general formula for error propagationfor nmica to calculate δn

δn =

√(∂n

∂tδt)2 + (

∂n

∂θδθ)2 + (

∂n

∂NδN)2

δθ was found to be 0.0089 rad, this was calculated byconverting the angle read off of the rotating stage to thenearest 0.071 units into radians. The units of the ro-tating stage are 360 degrees divided into 50 units. Thecaliper could read the thickness of the mica to the near-est 10−5m, therefore the δt = 5 ∗ 10−6m. Thus, after our16 trials we found the weighted average of nmica to be1.543±0.018. It can then be infered that our mica samplewas muscovite which has a theoretical index of refractionof 1.563 [6]. Therefore we see that our measured value iswithin 2σ of the accepted value.

Even though most of our measurements were within atleast two standard deviations of their corresponding the-oretical values, there are still several factors to considerwhen dealing with an interferometer. One must con-sider the interferometer’s sensitivity to movement. Eventhough the entire experiment was carried out on an op-tics table, simply grabbing the micrometer would causea noticable sway in the fringes. The backlash on themicrometer must also be consider. However, preventivesteps were taken to reduce any effect backlash could haveimposed. To counter backlash, we would overshoot theinitial position and then would slowly approach the de-sired value from the same direction. Finally, I think thecounting of fringes imposed the greatest threat to accu-racy. Even though the fringe repetitions were recordedon a video, there were occasions that keeping track of the

fringes was near impossible. Also deciding on the errorin the count of fringes was a little arbitrary at first, butI think our approach of fixing the length difference andthen finding the standard deviation in the counts pro-vides a reasonable error in fringe repetitions.

This experiment shows some of the nontrivial appli-cations of the famous Michelson interferometer. WithMichelson’s tool, physicists were able to develope a bet-ter understanding of the behavior of light. Not only hasthis instrument given us advances in the field of physics,but also in both chemistry and optics.

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[2] Hariharan, P. (2007). Basics of interferometry (2nd ed.).Amsterdam: Elsevier Academic Press.

[3] McQuarrie, D. (1983). Quantum chemistry. Mill Valley,Calif.: University Science Books.

[4] Hecht, E. (2002). Optics (4th ed.). Reading, Mass.:Addison-Wesley.

[5] Taylor, J. (1997). An introduction to error analysis: Thestudy of uncertainties in physical measurements (2nded.). Sausalito, Calif.: University Science Books.

[6] Weast, R. (2014). CRC handbook of chemistry andphysics (95th ed.). Boca Raton, FL: CRC Press.

[7] Fincham, W., Freeman, M. (2003). Optics (11th ed.).London: Butterworths.

[8] Harris, R. W. (1974). Theory and Experiments forTwo Configurations of the Modular Interferometer: TheMichelson and Fabry-Perot Interferometers. GaertnerScientific Corp.

[9] Fendley, J. (1982). Measurement of refractive index usinga Michelson interferometer. Physics Education, 209-211.

[10] Department of Physics. (2010). Michelson Interferome-try. LUMS School of Science and Engineering.