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Experiment 4 – The Michelson Interferometer 1Experiment 4The MichelsonInterferometer1 IntroductionThere are, in general, a number of types of optical instruments thatproduce optical interference. These instruments are grouped under thegeneric name of interferometers. The Michelson interferometer causesinterference by splitting a beam of light into two parts. Each part ismade to travel a different path and brought back together where theyinterfere according to their path length difference.You will use the Michelson interferometer to observe the interferenceof two light sources: a HeNe laser and a sodium lamp. You will studyinterference patterns quantitatively to determine the wavelengths andsplitting of the Na D lines empirically. You will use the HeNe laserinterference spectrum to calibrate the interferometer.2 Background - see Pedrotti3, Sections 8-1 to 8-3, aswell as Chapters 5 and 7.2.1 The Michelson InterferometerThe Michelson interferometer is a device that produces interferencebetween two beams of light. A diagram of the apparatus is shown inFig. 1. The basic operation of the interferometer is as follows. Lightfrom a light source is split into two parts. One part of the light travelsa different path length than the other. After traversing these differentpath lengths, the two parts of the light are brought together to interferewith each other. The interference pattern can be seen on a screen.Light from the source s trikes the beam splitter (designated by S).The beam splitter allows 50% of the radiation to be transmitted to thetranslatable mirror M1. The other 50% of the radiation is reflected toExperiment 4 – The Michelson Interferometer 2Figure 1: Schematic illustration of a Michelson interferometer.the fixed mirror M2. The compensator plate C is introduced along thispath to make each path have the same optical path length when M1and M2are the same distance from the beam splitter. After returningfrom M1, 50% of the light is reflected toward the frosted glass screen.Likewise, 50% of the light returning from M2is transmitted to theglass screen. At the screen, the two beams are superposed and one canobserve the interference between them.2.2 Interference of Waves With a Single FrequencyIf two waves simultaneously propagate through the same region ofspace, the resultant electric field at any point in that region is thevector sum of the electric field of each wave. This is the principle ofsuperposition. (We assume all waves have the same polarization).If two beams emanate from a common source, but travel over twodifferent paths to a detector, the field at the detector will be determinedby the optical path difference, which we will denote by ∆x = x2− x1.A related quantity is the phase difference, ∆φ, given by∆φ =2πλ∆x = k∆x, (1)where k is the wavenumber. Constructive interference occurs when∆φ = 2mπ, m = 0, ±1, ±2, ±3, . . . . (2)Destructive interference occurs when∆φ = ± (2m + 1) π, m = 0, 1, 2, 3, . . . . (3)Experiment 4 – The Michelson Interferometer 3Figure 2: Beat signal from two input frequencies into a Michelson in-terferometer2.3 Interference of Waves with Two FrequenciesWe will now consider the case of two frequencies with wavenumbers k1and k2that together follow two different paths with a difference of ∆x.The sum of the waves with different amplitudes at point x along thex-axis is given by:ET=eixk1+ ei(x+∆x)k1E1+eixk2+ ei(x+∆x)k2E2(4)If we let a = E2/E1and define δk = (k1− k2) /2, after a lot of algebra,we can write the intensity (E∗TET) as:21 + a + a2+ a cos 2δk∆x + (1 + a) (cos k1∆x + a cos k2∆x)(5)Figure 2 shows the exp ec ted signal, which consists of a fast oscillationas well as a slow oscillation characteristic of δk.Experiment 4 – The Michelson Interferometer 43 ExperimentIn the following experiments, you will calibrate the movement of M1with the HeNe laser and use the interferometer to accurately measurethe wavelengths of the fine structure doublet of the sodium D line, aconsequence of the spin of the electron.3.1 Calibration with HeNe Laser LightInject the laser beam into the Michelson intererometer. Make sure thebeam is properly retro-reflected. Initially, you will see two bright spotson the screen. Adjust the angle of the fixed mirror until these two spotsoverlap. You can use lenses to expand the beam if necessary.Note, take care when moving M2as the interference is verysensitive to its alignment. As you translate mirror M1, you will seefringes appearing and disappearing on the screen. The wavelength ofthe light can be found usingλ =2dM 1m(6)where dM 1is the distance mirror M1was moved and m is the numberof rings that disappeared (or appeared) while M1was being moved.Note that the interferometer has a lever arm reduction factor of five,so that the distance that M1moves is one-fifth the distance that themicrometer moves.Use the synchronous motor to facilitate the turning of the microm-eter. As the micrometer is turning, record the interference data withthe computer. The motor runs at 0.5 rpm and the micrometer moves5 × 10−4m/rev. This can give you a check of things, but we will usethe HeNe data (look up the HeNe wavelength on the web) to accuratelycalibrate the speed.3.2 Sodium LightNow use the sodium lamp to produce an interference pattern. Sincethe spectrum of this light consists primarily of two closely spaced lines(a doublet), each wavelength will produce its own set of fringes. Yourgoal will be to e mpirically determine λ1and λ2by measuring the finelyspaced fringes and the beat pattern.It is much more challenging to get good interference patterns withthe lamps, so take your time and play with alignment, lamp placement,and possible lens placement. You will need to increase the gain in yourdetection system.You should observe both the finely spaced pattern as well as a mod-ulation in the contrast at the differenc e frequency of the two lines. Yourgoal is to as accurately as possible measure the wavelength of the twoExperiment 4 – The Michelson Interferometer 5sodium lines. Be sure to carefully estimate uncertainties.Each of the doublet lines of the sodium lamp are not monochromaticdue to broadening from pressure effects and motion of the atoms in thelamp (Doppler effect). This means the coherence length is not thatlarge. If the path length difference is too large, you will not see anyfringes. Me asure the coherence length of your


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UMD PHYS 375 - The Michelson Interferometer

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