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Home , Interferometry, Michelson interferometer

Optical

MIT Department of Physics (Dated: August 27, 2013) The objective of this experiment is to observe the interference of by combining coherent monochromatic light beams using a . You will determine the of the light source from your measured interference pattern.

PREPARATORY QUESTIONS have perfectly constructive interference. Similarly, two waves of equal amplitude and but completely Please visit the Optical Interferometry chapter on the out of phase with one another, i.e. crests of one wave 8.13x website at mitx.mit.edu to review the background line up with troughs of another wave and vice versa, will material for this experiment. Answer all questions found result in completely destructive interference. in the chapter. Work out the solutions in your laboratory We can derive a more general equation for interfer- notebook; submit your answers on the web site. ence between two superimposed waves with the same frequency, but each with arbitrary amplitude and phase. Since this experiment is about optical interference with SAFETY Michelson interferometers, we will talk specifically about waves in the electric field, but the following discussion CAUTION: TAKE EXTREME CARE WHEN applies to waves in general. i(φ1−ωt) HANDLING OPTICAL COMPONENTS. Do Suppose we have one wave of the form E1e and i(φ2−ωt) not touch , lenses, beam splitters, or any another wave E2e where E1 and E2 are the mag- other optical surfaces with your bare hands. The nitudes of their respective fields, φ is the phase, ω is the oils on your fingers will significantly degrade the angular frequency, and t is time. Superimposing both performance of these , severely compromis- waves gives ing your data. E = E ei(φ1−ωt) + E ei(φ2−ωt). (1) WARNING: TAKE CARE WHEN ACTIVAT- T 1 2 ING A . The human eye does not take If φ1 − φ2 = 2nπ where n = {0, 1, 2,...}, then the waves kindly to laser light. Take whatever precautions are perfectly in phase and we recover the case of com- are necessary to ensure that no spurious or stray pletely constructive interference. Likewise, if φ1 − φ2 = beams leave your experimental area where they (2n + 1)π, then the two waves are out of phase and we may be absorbed by your, your partner’s, or some recover the case of completely destructive interference. other unsuspecting individual’s retina. This is illustrated in Figure1.

I. INTERFERENCE a) b)

When we superimpose two waves of the same or nearly ET the same frequency traveling in a medium, we can observe interference. Interference is, in fact, one of the corner- E1 stones of wave phenomena. Young’s double slit experi- ment, which showed interference fringes (bright and dark E2 bands) for light passing through two slits, was an early indicator of the wave nature of light. Since then, many FIG. 1. An illustration of how superimposing two waves re- variants have been used in thought experiments and real sults in an interference pattern. Figure 1(a) represents com- experiments that address the wave-particle duality that pletely constructive interference, whereas Figure 1(b) repre- is a building block of . sents completely destructive interference. The for waves states that when two or more waves interact in a region of space, the net However, we often measure intensity of the field rather amplitude at each point is the sum of the amplitudes than the field itself. The intensity is proportional to the of the individual waves. If the waves sum to a larger square of the field, amplitude than each individual wave, then we have con- structive interference. Likewise, if the waves add up to be ∗ 2 2 I ∝ hE ET i = E + E + 2E1E2 cos(φ1 − φ2), (2) smaller than the amplitude of the individual waves, then T 1 2 we have destructive interference. If two waves of the same where h·i indicates time averaging. We can also write I frequency and amplitude are in phase, i.e. crests line up in terms of√ the intensities of the individual fields, I = with crests and troughs line up with troughs, then we I1 + I2 + 2 I1I2 cos(φ1 − φ2). 2

How well two waves interfere with each other will de- pend on their . Mathematically, coherence is a measure of how well correlated the waves are, as quan- tified by the cross-correlation function. Physically, co- `1 herence is a measure of how constant the phase remains Half-Silvered between two waves. For this experiment, we will con- Mirror centrate on temporal coherence: the correlation between Laser Mirror the wave and a time-delayed version of itself. In essence, temporal coherence tells us how monochromatic a wave is. Interferometric visibility V, also known as contrast or `2 , is an experimentally observable measure of how coherent two waves are. Visibility is defined as I − I V = max min . (3) Imax + Imin Photodetector An example of an interferogram with Imin and Imax mea- surements can be seen in Figure2. Since the visibility FIG. 3. Michelson interferometer configuration. depends on the coherence of the two waves, any dissimi- larities between them will result in decreased visibility. splitter. The light beams from the two paths are aligned to impinge on the same point on the photodetector where they interfere and are detected. The amplitude of the de- tected light is determined by the total phase difference Imax between the combined beams. We now derive a general case for Michelson interfer- ence in a similar manner to that of SectionI. We aim iωt monochromatic light of the form E◦e on to the . Two beams of equal frequency and amplitude come out of the beam splitter. (Realistically, the am- plitudes will not be exactly equal, but they will be very I min close.) The beam intensity — not the electric field — 2 gets reduced by a factor of 2. Since√ I ∝ E , we di- vide the resulting electric fields by 2 rather than by FIG. 2. Visibility is a measure of how well two waves interfere. 2. The beam that transmits through will have the form E = √E◦ eiωt, whereas the reflected beam will obtain a π Higher visibility means more range between the maximum 1 2 and minimum intensity. phase shift and will have the form E = √E◦ ei(ωt+π) = 2 2 − √E◦ eiωt. Both waves will be reflected by the mir- 2 rors (both beams pick up another π phase shift, but this fact is not important because we care about rela- II. MICHELSON INTERFEROMETERS tive phase shift) and return to the beam splitter with a phase shift that depends on the length of the interferom- The Michelson interferometer is the most common con- eter arms. The superimposed electric field will therefore figuration for optical interferometry. Michelson interfer- E◦ iωt 2ik`1 2ik`2 2π be ET = 2 e (e − e ), where k = λ is the ometers can be used for astronomical interferometry, for and λ is the wavelength of the laser. optical coherence (a medical imaging tech- In practice, it is more useful to derive the equation nique), and for detection [1]. The for intensity since that is what the photodetector will Michelson interferometer is best known for its use in the actually measure: Michelson-Morley experiment, which disproved the exis- ∗ tence of the luminiferous ether and paved the way to the I ∝ ET ET development of the [2]. 2 E◦ 2 = (e2ik`1 − e2ik`2 ) Shown in Figure3, a Michelson interferometer uses op- 4 tical interference to detect the length difference between 2 E◦ two paths `1 and `2, however small they may be. It = [1 − cos(2k(`1 − `2))]. (4) does so by splitting a laser beam into two perpendicular 2 paths using a half-silvered mirror acting as a 50 % beam Using this, we can readily solve for the conditions of max- splitter. The beams then bounce off mirrors at the end imum and minimum intensity which should correspond of their respective paths and are returned to the beam to maximally constructive and destructive interference. 3

If 2(`1 − `2) = nλ, where `1 and `2 are the different path mirror will translate by 41.6 ± 1.3 nm. You may or may lengths, λ is the beam wavelength, and n = {0, 1, 2,...}, not find it useful to add a lens to focus the incident beam then the two beams are perfectly out of phase, resulting on the photodiode. Finally, the overall setup is kept as in completely destructive interference and a minimum compact as possible. The laser only remains coherent beam power reaching the photodetector. Conversely, if over a length scale of a couple tens of centimeters. 2(`1 − `2) = λ(2n + 1)/2, the two beams are perfectly in phase, resulting in completely constructive interference. N.b.: If you are trying to convince yourself that these re- IV. PROCEDURE lationships are true, keep in mind that the half-silvered beam splitter has a preferential direction and only im- 1. Begin by assembling the experiment as shown parts a π phase change on reflected laser light when in- in Figure4. Remember to keep the length cident from one direction. Also keep in mind that the of each arm of the interferometer approximately waves traverse the arms twice, so the difference between the same length. CAUTION: TAKE EX- the lengths of the arms is doubled. Further discussion TREME CARE WHEN HANDLING OP- can also be found in [3]. TICAL COMPONENTS. Do not touch mir- rors, lenses, beam splitters, or any other op- tical surfaces with your bare hands. The oils III. EXPERIMENTAL SETUP on your fingers will significantly degrade the performance of these optics, severely com- promising your data.(N.b.: The grid spacing of Function O’scope the optical breadboard is 1”.) PZT Generator Mirror 2. Make sure that the output of the photodiode is con- nected to CHANNEL 1 of the oscilloscope and the

`1 output of the function generator is connected to Half-Silvered CHANNEL 2. Also make sure that the output of the Mirror function generator is connected to the PZT. Laser Mirror 3. WARNING: YOU ARE ABOUT TO TURN ` ON A LASER. The human eye does not take 2 kindly to laser light. Take whatever precau- tions are necessary to ensure that no spuri- Lens ous or stray beams leave your experimental area where they may be absorbed by your, your partner’s, or some other unsuspecting Photodetector individual’s retina. Turn on the laser. Align the optics and follow the paths of each beam to make FIG. 4. A block diagram of the experimental setup. Recall sure that they both hit the photodiode at the same that the arm lengths `1 and `2 should be approximately equal. location. You may find it extremely helpful to align each piece of the optical setup individually rather A block diagram of the setup for this introductory ex- than the entire system at once. Consider the diffi- periment can be seen in Figure4 and is very similar to culty of minimizing a system of so many degrees of what was discussed in the previous section. We specify freedom simultaneously! the light source as a laser, which emits a light of a single 1 unknown wavelength. You will derive this wavelength in 4. Set the function generator to output a 10 Hz si- lab. We also specify the photodetector to be an ampli- nusoid or triangle waveform with a peak-to-peak fying photodiode, which translates the light incident on voltage of 10 V. (Your function generator may in- dicate peak-to-peak voltage by Vpp.) The output it into a voltage. The photodiode output voltage VPD is proportional to the intensity of the light incident on of the function generator will drive the PZT. Re- call that the amount that the mirror will translate the : VPD ∝ I. Most notable in the setup is the piezoelectric transducer (PZT) attached to one of the mirrors. The PZT will displace the mirror by an amount depending on the voltage applied to it. The calibration 1 As with all suggested apparatus settings in Junior Lab lab man- factor ∆`/∆V between mirror displacement and applied uals, the given value will be near a value which gives good signal, voltage has been determined for each PZT available in but you may find that a different value works better. In general, the lab to within an uncertainty of a few percent. Each do not use a value simply because the lab manual say so. Instead, device is labeled with its measured value, for example try to understand what experimental condition would cause one 41.6 ± 1.3 nm/V. That means that if 1 V is applied to to prefer one value over another, and then optimize your choice of settings for that condition. this example PZT using the function generator, then the 4

is determined by the measured ∆`/∆V as labeled description and sketches to allow evaluation of sys- on your device, which will be several tens of nm/V. tematic errors. 5. You should see some sort of fringing pattern from the photodiode output on the oscilloscope. Tweak V. ANALYSIS your alignment to obtain the largest possible sig- nal. Using the “cursors” or the grid on the os- 2 As with all experiments, you should perform at least cilloscope, measure the minimum and maximum a rough analysis before you leave the lab to confirm that voltage coming from the photodiode. You will use you have the necessary data to perform the analysis and these numbers to calculate the visibility. You may that the data you have obtained are of sufficient quality. find it useful to “RUN/STOP” or “SINGLE mode”the Never assume that an experiment has gone as planned display on the oscilloscope while making your mea- unless you have first taken action to assure that it has surement. done so, and then followed up with action to confirm 6. Measure the driving voltages from the function gen- success. Successful experiments require good planning: erator which correspond to the minima and max- plans which depend on everything going as planned are ima in the photodiode output. From these you will bad plans! 2 E◦ − use the relationships [1 − cos(2k(`1 − `2))] and 1. Calculate the visibility Imax Imin . Explain why the 2 Imax+Imin the value of ∆`/∆V to derive the wavelength of the visibility is less than ideal. laser. 2. From the calibration of the driving voltage to mir- 7. Review your measurements as they are recorded in ror displacement and from your measurements from your notebook. Be sure that you have recorded in Step6 of the previous section, calculate the wave- enough data to allow evaluation of the statistical length λ of the laser beam. (random) error in your measurements. Be sure that you have recorded enough information in narrative

[1] B. P. A. et al, Reports on Progress in Physics 72, 076901 (2009), http://stacks.iop.org/0034-4885/72/ i=7/a=076901. [2] A. A. Michelson and E. W. Morley, American Journal of Science 34, 333 (1887). [3] E. Hecht, Optics (Addison-Wesley, 1998).

2 Whenever experimental conditions feasibly allow, it is desirable senses and other mental pathways, bringing about a more active to use your own mind and hands to record data, rather than an awareness and understanding of the experiment. automated data logger, as this process more readily engages the