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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 Hecht, Chap. 9, Pedrotti Chap. 7and 82.1 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 strikes 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 3 2.3 Fringe Visibility for Interferograms Made With Two Frequencies We will now consider the case of two frequencies with numbers 1kand 2k that together follow two different paths with a common path difference ofx!. The two frequencies cannot show interference for times longer than their periods. Thus each frequency produces its own two beam interference intensity pattern: 1 2 1 22 cosI I I I I!= + + and (4) ' ' ' ' ' '1 2 1 22 cosI I I I I!= + + with 1k x!= " and '2k x!= " being the corresponding phase differences for the two beam paths. Both frequency light beams travel the same paths and have the same path difference x!. Write the ratio ' '1 2 1 2/ /I I I I r= = because the beam splitter ratio is the same for both frequencies and the intensity ratio for the two frequencies as '1 1/I I a=. Then the total intensity is given by: '1 1(1 )(1 ) 2 (cos cos )TI I r a I r a! != + + + +. (5) The first term is the average intensity and the second term with the two cosine functions is the sum of the two interference terms. The interference terms don’t simplify much further but the dominant behavior for 1a  has the dependence on ! and '!of the form '1 2 1 22cos( ) cos( ) 2cos(( ) ) cos(( ) )2 2 2 2k k k kx x! ! ! !+ "+ "= # #. This product of cosines shows a fast oscillation at the mean wavenumber frequency and a slow envelope at the wavenumber difference frequency. An example plot of Eq. (5) is shown in Figure 2 for the case I1 = 1, r = 0.7 and a = 0.8. Difference in distance of the two interferometer arms (Arb. Units)Intensity (Arb. Units)Figure 2: Intensity beat signal from two input frequencies into a Michel-son interferometer. The function shown is for the case I1= 1, r = 0.7and a = 0.8..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 interferometerlever arm reduction factor is 5X, so that the wavelength of the lightcan be found usingλ =152dm(4)where d is the distance the micrometer moves and m is the number ofrings that dissapeared (or appeared) while mirror M1moves. Note thatfor a given d, M1moves d/5. Use the synchronous motor to facilitatethe turning of the micrometer. As the micrometer is turning, recordthe interference data with the computer. The motor runs at 0.5 rpmand the micrometer moves 5 × 10−4m/rev. This can give you a checkof things, but we will use the HeNe data (look up the HeNe wavelengthon the web) to accurately calibrate 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 empirically determine λ1and λ2by measuring the finelyspaced fringes


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UMD PHYS 375 - The Michelson Interferometer

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