Michelson Interferometer • May 2015

Michelson InterferometerScott McIntosh

University of California, Davis

Abstract

The Michelson Interferometer is a very simple device consisting of a few mirrors and a light source. In thisexperiment we observed the behavior of three kinds of light in just such a device and used our findings todetermine various wavelength relationships. We found that the interferometer is a very precise instrumentfor being something so basic. Our measurements and calculations were found to be in very close agreementwith commonly accepted values.

I. Introduction

The Michelson Interfereometer was in-vented by Albert Michelson in an at-tempt to prove the existence of a luminif-

erous ether, which at the time was thought tobe the medium through which electromagneticwaves propagated. The experiment eventuallyproved there was no ether, and laid the ground-work for special relativity.

The device itself consists of three mirrors(one partially reflecting and two totally reflect-ing), a source, and a detector (in our case ascreen). A diagram is given in figure 1 below:

Figure 1: diagram of a Michelson interferometer

As shown in the diagram, one mirror is afixed distance from the beam splitter, while theother is adjustable by means of a micrometer.The source laser emits light which gets splitinto two beams and sent to the two mirrors.One beam gets reflected back to the splitterand reflected again to the screen. The secondbeam gets reflected and sent back through the

splitter and projected onto the screen. Sincethese are now two separate beams, an interfer-ence pattern is observed.

If the mirrors are properly aligned a "bulls-eye" pattern will be seen. This is due to thefact that the beams are superimposed on topof each other. If on the other hand, the mirrorsare not properly aligned, the beams will beprojected next to each other, creating a linearinterference pattern.

The objective of this experiment is to em-ploy the Michelson interferomteter to measurethe wavelength of monochromatic and dichro-matic light and observe the interference patterncreated by broad spectrum light.

II. Methods

In measuring the wavelength of monochro-matic light, a Helium-Neon laser is used asthe source and the interferometer is calibrated.Once the "bullseye" pattern is observed the mi-crometer is used to move the non-stationarymirror. As the mirror moves, the interferencepattern changes, and the position of the max-ima and minima will move inward or outward,depending on which way the micrometer isturned.

The intensity of the interference pattern forthis light is given by equation 1 below:

I(x) =Iinc2

[1 + cos

(2πx

λ

)](1)

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Michelson Interferometer • May 2015

where x is the path difference between thetwo mirrors.

We can see that the maxima and minimaoccur when x is equal to an integer multipleof λ. By turning the micrometer and cyclingthe position of the maxima N times, and mea-suring ∆x we can calculate the wavelength byemploying the following formula:

∆x = Nλ (2)

Due to the mechanical action of the inter-nal parts of the interferometer, we must add ascaling factor of 2

5 . Solving for λ we get:

25

∆xN

= λ (3)

In measuring the wavelength of dichro-matic light, a sodium lamp is used as thesource and the interferometer is calibrated.Measurements are taken in the same manneras monochromatic light.

The intensity of dichromatic light is givenby the sum of equations 4 and 5 below:

I1(x) =Iinc(λ1)

2

[1 + cos

(2πxλ1

)](4)

I2(x) =Iinc(λ2)

2

[1 + cos

(2πxλ2

)](5)

Ordinarily an intensity equation would re-quire an interference cross-term, but since theseare two different wavelengths of light, no inter-ference term is necessary. The intensity can beapproximated by the following equation:

I(x) ≈ Iinc(λ̄)

[1+ cos

(2πx

λ̄

)cos

(π(∆λ)x

λ̄2

)](6)

where we define λ̄ and ∆λ as follows:

λ̄ =λ1 + λ2

s(7)

∆λ =| λ1 − λ2 | (8)

By modifying equation 3 we can derive aresult for λ̄:

25

∆xN

= λ̄ (9)

Next we must determine the path differencebetween the envelope maxima and minima. Wedo this by finding how far the mirror must bemoved between points where the overall inten-sity of the fringes drops to a minimum. Weshall refer to this distance as ∆x′. By employ-ing the following equation we can derive thevalue for ∆λ

∆λ =5λ̄2

∆x′(10)

In observing the interference pattern cre-ated by broad spectrum light, we simply swapout our previous sources for a white light andcalibrate the interferometer.

III. Results

For monochromatic light we measured ∆x tobe approximately 0.0044cm after N = 25 turns.Using equation 3, we found the wavelength tobe approximately 704nm.

For dichromatic light we measured ∆x tobe approximately 0.0039cm after N = 25 turns,and ∆x′ to be approximately 0.0152cm. Usingequations 9 and 10 we found λ̄ to be 624nmand ∆λ to be 13nm.

In observing broad spectrum light, we sawthat there was no "bullseye" pattern, but ratherlong fringes. In the middle of the pattern thefringes were almost solid black, and at theedges the fringes spread out to where we couldobserve the entire visible light spectrum. Thismakes sense because white light contains allvisible frequencies of light.

IV. Discussion

The purpose of this experiment was to observethe behavior of monochromatic, dichromatic,and broad spectrum light in a Michelson inter-ferometer. We employed the interferometer todetermine the wavelength of monochromatic

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Michelson Interferometer • May 2015

light, and the average wavelength and absolutedifference of wavelengths of dichromatic light.

Our calculated value for the wavelengthof the monochromatic Helium-Neon laser was704nm. This measurement is only 10.6% differ-ent from the accepted value of 633nm.

Our calculated value for the average wave-length of dichromatic Sodium light was 624nm,which is 5.7% different from the accepted valueof 589.3nm. Our calculated value for the abso-lute difference of the dichromatic wavelengthswas 13nm, which is 182% different from theaccepted value of 0.6nm.

Since our monochromatic wavelength anddichromatic average wavelength are in suchclose agreement, we can easily see the utilityand precision of the Michelson interferome-

ter. Even though our value for the absolutedifference of dichromatic wavelengths is twoorders of magnitude higher than the acceptedvalue, this is not unexpected, considering theextremely small scale of what we were workingwith.

The thermal expansion of the micrometeris one example of a possible source of error.Another possible source of error is impropercounting of the number of maxima/minimacycles (N). Because even very small changes inmirror displacement correspond to very largechanges in phase difference, it is difficult toproperly ascertain the exact number of cycles.Regarless of this, our results were still wellwithin the acceptable range of error.

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