EOP3056 Optical Metrology and Testing Experiment OM2: The Mach-Zehnder Interferometer

Transcription

1 EOP3056 Optical Metrology and Testing Experiment OM2: The Mach-Zehnder Interferometer 1.0 Objectives To construct a Mach-Zehnder interferometer from discrete optical components. To explain how Mach-Zehnder interferometer produce an interference patterns. To analyze the Mach-Zehnder interferometer behaviours and characteristics. To measure the index of refraction of a gas. To observe the effect on fringe pattern when a parallel glass plate is inserted into one of the beams. Practice in taking clear and intelligible laboratory notes. Proficiency in making fine adjustments of optical components 2.0 Apparatus (number in the brackets is the number of sets) 1. HeNe laser Laser L (LLL-2) 2. Laser Holder (LEPO-44) 3. Optical Rail with carriers (LEPO-54) 4. Beam Expander (f =15mm) 5. 3-D Adjustable Post Holder (LEPO-17) 6. Lens Holder (LEPO-9) 7. Beam Splitter BS1 (5:5) 8. Lens Holder (LEPO-9) 9. Magnetic Base (LEPO-4) 10. Two-axis Tilt Holder (LEPO-8) 11. Flat Mirror M1 12. Magnetic Base (LEPO-4) 13. White Screen (LEPO-14) 14. Magnetic Base (LEPO-4) 15. Two-axis Tilt Holder (LEPO-8) 16. Beam Splitter BS2 (5:5) 17. Magnetic Base (LEPO-4) 18. Two-axis Tilt Holder (LEPO-8) 19. Flat Mirror M2 20. Magnetic Base (LEPO-4) *He-Ne Laser (LLL-2), beam expander, beam splitter BS1 and mirror M1 are aligned on an optical rail (LEPO-54) 3.0 Introduction The Mach-Zehnder interferometer represents another topology for a two-beam interferometer; its relevance to theoretical insight, and its usefulness in optical testing and usage. It is another two-beam interferometer, but one in which the two beams are entirely unidirectional and non-overlapping, and capable of wide separation. The ability to pass one of its two separated beams through test media of bulky proportions helps to account for its popularity and use. 3.1 Introduction Faculty of Engineering, Multimedia University Page 1

2 Generally, an interferometer starts with a coherent beam of light, splits it into two beams that are coherent with each other, and then passes the two beams through different paths before recombining them. Very small changes in the path length difference between the two paths can be measured by observing the pattern of the interference fringes. A Mach-Zehnder Interferometer (MZI) recombines the beams on a different beam splitter than the one that originally split them. A Michelson Interferometer uses only one beam splitter for both purposes. The MZI arrangement is often used because it allows the experimenter to access and modify the individual beams. The figure shows the basic configuration for the MZI. A beamsplitter BS1 made from a partially silvered mirror reflects part of the incident laser beam and allows another part to pass. Each of these beams is reflected by another mirror M1 or M2 and the beams come together on BS2. The recombined beams can be viewed on Screen 1 or Screen 2. If they arrive out of phase, they will cancel and cause a dark fringe, and if they arrive in phase, they cause a bright fringe. The phase differences are caused by slightly different path lengths in the two arms (BS1-M1-BS2 and BS1-M2-BS2). Figure 1: The Mach-Zehnder Interferometer If the beams consist of plane waves, and are almost but not quite perfectly parallel, then the light and dark fringes appear as parallel lines on the screen. If the path length in one of the arms changes, this causes the fringes to move. If the path length difference between the two arms changes by one wavelength, the fringe pattern on the screen will advance by one complete cycle from dark to light and back to a dark fringe. If one of the beams converges or diverges slightly, then its wavefronts are spherical, and the fringe pattern is circular. Nachman explains how the fringe pattern shape and the direction of fringe movement can be predicted. Even if the paths in the arms are the same length, a phase difference between the beams can be Mach-Zehnder Interferometer caused by changing the index of refraction in one of the arms. For example, a transparent chamber containing air in one beam can be evacuated to measure air s index of refraction nair, which is very close to 1. Or, if a thin glass plate placed in one beam is tilted so that the beam has to cross a greater thickness of glass, that beam will be retarded and the fringes will shift. Plane wave light from the source strikes a semi-reflective mirror at A, a beam splitter where it divides into two beams. The first reflects from the splitter towards B, where it reflects towards another beam splitter at D. From there half of its energy is transmitted towards E and half reflected towards F. The second is transmitted towards C and then reflected towards D where half is transmitted towards F and half reflected towards E. Faculty of Engineering, Multimedia University Page 2

3 There are therefore four different beams to be considered, and in order to calculate the wave field at either of the detectors you need to know the way the amplitudes and phases of these four beams have changed since they were all a single beam at the point A. The following considerations apply. Since the beams are plane waves, their amplitudes do not decrease with distance. However at each reflection/transmission at a beam splitter, the energy of the beam divides into two. The ratio of the intensity of the reflected wave over the intensity of the incident wave is called the reflectivity or reflectance (R) of the semi-reflective surface. Note that the ratio of the amplitude of the reflected wave over the amplitude of the incident wave is usually defined as the coefficient of reflection (r) and the reflectivity is the square of this. The phase of an electromagnetic wave at any point in space changes linearly with the length of the path along which the wave has travelled. If the wave has gone a distance L after leaving some reference point, the change in phase is proportional to how many wavelengths this distance is equivalent to, that is where λ0 is the wavelength in vacuum and n is the refractive index, since there is 2π phase for every wavelength of extra path. For many dielectric materials, when a wave reflects from a surface of greater refractive index, it suffers a change of phase equal to π radians. But when it reflects from a surface of smaller refractive index, there is no change of phase. Reflection from a mirror usually involves a change of phase π. But at reflection from a semireflective mirror it is critical whether reflection takes place at the front or the back of the mirror, which is whether or not the beam travels through the glass being reflected. In the present application, the reflection at A involves a π phase change, as does the reflection CDE. But for the reflection BDF there is no phase change. Assume the electric field of the initial beam at the point A, at time t, is given by: Then the electric fields of the two beams reaching detector 1 at the point E, the first which travels along the path ACDE, and the second along ABDE are: Faculty of Engineering, Multimedia University Page 3

4 Likewise the electric fields of the two beams reaching detector 2 at the point F, travelling along ACDF and ABDF are: E 3,F (t) = A 0 (1 R) cos(ωt 2πnL ACDF / λ 0 2π ) Eq. 7 E 4,F (t) = A 0 Rcos(ωt 2πnL ABDF / λ 0 2π ) Eq. 8 Note, in the ideal case where the reflectivity of each beam splitters is 50% and the path lengths around both sides of the interferometer are equal, all the energy goes into one detector only. If the refractive index is different in the two paths, whether there is destructive or constructive interference at the detectors will depend sensitively on that difference. It is not surprising therefore that the Mach-Zehnder interferometer finds its greatest application in measuring refractive indexes. 3.2 Application The Mach Zehnder modulator has two spatially separated equivalent paths. It is typically used to measure an optical phase shift, i.e. a path-length change, due to an object in one arm while the other arm serves as a reference beam to which the phase shift is compared. You can for instance use this technique to measure the index of refraction of gas in a cell by measuring the phase shift as a function of the gas pressure. Alternatively one can use such an interferometer to visualize object that result in a phase shift but are otherwise transparent, e.g. a gas flow such as hot air, where the shape of the optical fringes reects the spatial index profile of the object. 3.3 Measuring the refractive index of a gas with a Mach-Zehnder interferometer One of the simplest experiments involves the measurement of the index of refraction of a gas with an airtight test cell placed in one of the optical arms of the Mach-Zehnder interferometer. It should be clear that the outputs of the detectors of a Mach-Zehnder interferometer are exquisitely sensitive to the optical path length its beams travel. If the length of either arm changes by just half a wavelength (a change less than 10 6 m in 10 1 m) the intensity can change from maximum to minimum. By the same token, it is also sensitive to changes in refractive index along the beam paths. That is why this instrument is often used to measure refractive indexes, particularly of gases. Figure 2 illustrates the experimental setup. The interferometer is aligned as described in the previous experiment. Next, the test cell is evacuated and positioned in one optical arm. If the fringe spacing is too small, a positive lens may be placed between plate P2 and the screen. By adjusting the position of the screen, the fringe pattern can be magnified several times. This permits a more clearly visible pattern and reduces eye strain when it becomes necessary to count the moving fringes. Faculty of Engineering, Multimedia University Page 4

5 Figure 2: Arrangement for measuring index changes in a gas cell The measurement begins by closing off the valve connecting the vacuum pump to the evacuated test cell and slowly opening the needle valve which permits the gas, whose index of refraction is to be measured, to enter the cell. As soon as the gas enters the cell, the fringes begin to move and must be counted as they go past a reference mark on the screen. With some care, even fractional parts of a fringe can be measured quite accurately. The refractive index n at the particular wavelength λ is then computed using Equation 9. ( m)λ n = Eq. 9 L where: m = The number of fringes moving past the reference mark on the screen. L = The length of the test cell Evaluation and results In gases, the refractive index is linearly dependent on the pressure p. n(p) = n(p = 0) + n/ p p with n(p = 0) = Eq. 10 Thus, in the following evaluation we will determine the differential quotient n n / p = [n(p + p) - n(p)]/ p Eq. 11, p from the measurement data. The optical path length d in the evacuable chamber is the product of the geometric length s of the chamber and the pressure-dependent refractive index n(p) of the gas in the chamber. By changing the pressure in the chamber from p to the value p + p, we change the optical path length by d = n(p + p) s n(p) s Eq. 12 During evacuation, we may observe motion in the interference lines on the translucent screen. Starting from the ambient air pressure p0, we can count Z(p) shifts in the chamber until pressure p is reached. A shift of the maxima by exactly one position corresponds to a change of l in the optical path length. Thus, the optical path length changes between pressure p and p + p by d = (Z(p) Z(p + p)) λ Eq. 13 From Eq. 12 and Eq. 13, we can conclude that: n(p + p) n(p) = (Z(p + p) Z(p)) λ/s and, on the basis of Eq. 11, n/ p = - Z/ p λ/s Faculty of Engineering, Multimedia University Page 5

6 4.0 Warnings and precautions Students are responsible to be careful the below warnings and precautions. Students are responsible to own and other personal safety. 4.1 Laser safety The helium-neon (HeNe) laser used is a class IIIa laser which can cause permanent damage to your vision (retina). Never look at a direct laser beam or a direct reflection of a laser beam from a specular (mirror, glass, metal, etc.) surface. Never put your eyes at the plane where a laser beam is guided to traverse by optical components. Do not wear rings, watches or other shiny jewellery when working with lasers. (All these objects could send laser beams towards your eyes or those other persons nearby). Never insert an optical component directly into a laser beam (to avoid any possible beam reflections from the component, e.g. from the chamfers of the component). Never simply flip an optical component in a laser beam (to avoid any possible beam reflections from other specular objects located within the same workspace). Use only diffuse reflectors (e.g. rough surface white papers) for viewing or tracing HeNe laser beam. Always block laser beam close to the laser when the experiment is left unattended. 4.2 Partial and diffuse reflections of laser beam In a darkened room, our pupils will be expanded and will let in 60 times more light than in a lighted room. This experiment has many partial reflections (from lens, transparent apertures, anti-reflection surface of a beam splitter) and diffuse reflections (from various objects: viewing screen, holders, mounts, posts, etc.) Hence, this experiment will be performed in a lighted room. Furthermore, the light intensity of the fringes on the viewing screen is sufficiently high to be viewed in lighted room. 4.3 Tracing laser beam An experiment normally involves more than one optical component and mechanical part which can give total or partial reflections of laser beam. It is always required to know a laser beam direction and position. Tracing technique is always used. To do this tracing, put a beam stop (a rough surface white paper for HeNe laser) at a position where a laser beam direction and position are known and move the beam stop away in the laser beam direction until to the desired distance or location. 4.4 Handling optical components The optical components used are expensive. Never touch the optical surfaces of lenses, mirrors, beam splitters, etc with your skin (finger, nose, etc.) or any objects (except lens tissues). The coatings on the surfaces can be degraded by the fatty acids of human grease or scratched by the objects. It is the same of the air blown out from human mouth which contains acidic moisture. In this experiment, all the optical components have been mounted on their holders with mounting posts, always carry the optical components at the mounting posts. Never remove the optical components from their holders. 4.5 Adjustment knobs of adjustable mirror mounts Never turn an adjustment knob of a mirror mount more than a few turns. It should never be far from its medium position. The spring of the mirror mount could be damaged if it is over stretched. Faculty of Engineering, Multimedia University Page 6

7 4.6 Clamping screws There are clamping screws on the post holder and the laser mounting assembly. Do not over tighten these screws. This may damage the screw thread or break the mechanical clamping parts. Instead, tighten the screws until the holders are sufficient to hold the required parts without moving. E.g. tighten the clamping screw of a post holder until it is just sufficient to hold its mounting post without sliding down. Note that the required strength for tightening a clamping screw depends on the load to be held without moving. 4.7 No rush work You are advised not to carry out this experiment in rush to avoid any mistakes which could cause the damages as mentioned previously, especially your eyes. As an example, a cutting of a mounting post across a laser beam may send a reflected laser beam towards your or your co-worker s eyes. Although the laser beam sweeps across your eyes in a short instant, it may temporarily cause a dark line existing in your vision. Faculty of Engineering, Multimedia University Page 7

8 5.1 Experiment Figure 3: Schematic view of Mach-Zehnder Interferometer The experimental procedures below only include the important steps (including the safety steps) for carrying out this experiment. They do not contain all the details on the adjustment and alignment of the laser beam. You need to think and feel on them, e.g. how much and how light to turn an adjustment knob of a mirror mount for a small beam movement in the required direction. The below are mechanical parts for optical alignment. Note that optical alignment needs patience and time. You must make sure not to knock down any optical component along the optical alignment. This experiment is carried out in a lighted room. Never switch off the room lights. If necessary, you may block the lights from the room lamps to reach to the screen. Faculty of Engineering, Multimedia University Page 8

9 5.2 Procedures for setting up a Mach-Zehnder Interferometer Figure 4: Setup of the Mach-Zehnder interferometer on the laser optics base plate with evacuable chamber a laser optics base plate b, c beamsplitters d, e planar mirrors with fine adjustment f spherical lens g translucent screen h evacuable chamber k hose connection for vacuum pump (Cognitive Analysing, Level 4) [15 marks] Procedures for observation of interferometer behaviors and characteristics In this lab a Mach-Zehnder interferometer will be used to observe interference between two plane waves, a plane wave and a spherical wave, and a plane wave and a cylindrical wave. An important part of this lab is determining whether a wavefront is diverging or converging by looking at the interference fringes and lightly pushing on one mirror in the system. You will use this procedure a lot during the semester. Preparation i. Adjust the mirrors (d and e) so that the two beams are superimposed at the reflective surface of the beamsplitter (c). ii. To get the two plane waves nearly parallel, put a lens in the interferometer output and adjust the second beamsplitter so both beams are superimposed in the focal plane of the lens. There should be no shear between the two beams. iii. Repeat i) and ii) until there is no shear between the beams. Faculty of Engineering, Multimedia University Page 9

10 iv. Adjust the tip and tilt of the second beamsplitter to get interference fringes. Faculty of Engineering, Multimedia University Page 10

11 5.1.2 Mach-Zehnder Interferometer alignment i. Aligning the Mach Zehnder interferometer is not easy. As a general rule we force the beam to go square with the holes in the optical breadboard/table ii. Aim the beam as well as we can parallel to the row of holes. iii. Position the laser and the metal beams for the complete set up, also place the beam splitters, polariser and mirrors iv. Align the laser such that the laser beam is horizontal and well positioned on the optical breadboard. v. Adjust the beam splitter at an angle of 45 with respect to the beam axis and adjust its tilt so that the two beams (transmission and reflection) parallel to the table. vi. Place the planar mirror (d) in the partial beam reflected by the beam divider (b) so that the laser beam strikes it in the center. vii. By turning the optics base on the interferometer base plate, align the planar mirror so that the beam is deflected by 90 and travels on a path parallel to the transmitted partial beam. viii. Place planar mirror (e) in the transmitted partial beam opposite planar mirror (d) in the assembly as shown in Fig. 1 so that the laser beam strikes it in the center. ix. By turning the optics base on the interferometer base plate, also align this planar mirror so that the partial beam is deflected by 90. x. Fasten the translucent screen (g) in the base and set it up behind the laser optics base plate as shown in Fig. 1 so that the partial beam reflected by the planar mirror (e) strikes it in the center. xi. Set up beam divider (c) antiparallel to beam divider (b) so that it is struck by both partial beams at an angle of 45 ; make sure that the partially transparent layer is facing the screen (g) Subsequent adjustment: i. The components are correctly arranged when the beam path from beam divider to beam divider forms a rectangle. Keep path lengths in the two arms as similar as possible. ii. Keep the beams parallel to the table top and at the same height. iii. Correct the beam path if necessary. iv. Make sure all the optical elements are rigidly supported. v. Amplitudes and polarizations of the beams should be the same when they recombine. vi. Readjust the planar mirrors and beam dividers so that the most intensive beams of the two reflex groups coincide on the screen (g). vii. Change the distance between the screen (g) and the second beam divider (c) and check whether the reflexes of the two partial beams remain virtually coincident, i.e. sufficiently parallel Readjusting the vertical beam path: i. If the partial beams diverge from the horizontal plane: ii. Measure the heights of the partial beams over the laser optics base plate behind each optical component precisely using the wooden ruler, and correct the inclinations of the planar mirrors and beam dividers as necessary. iii. Adjust the optical components so that the most intensive beams of the two reflex groups coincide on the transparent screen. Faculty of Engineering, Multimedia University Page 11

12 iv. Change the distance between the screen (g) and the second beam divider (c) again and check whether the reflexes of the two partial beams are parallel. v. Repeat the readjustment as necessary Correcting the horizontal beam path: i. Ideally, the partial beams exit the beam divider at virtually the same point and recombine on translucent screen. ii. If the partial beams diverge in the horizontal plane: iii. Check the paths of the partial beams from beam divider (b) to beam divider (c) and correct the alignment of the corresponding components if the beam paths do not describe a rectangle. iv. Shift the planar mirror (e) parallel to the long side of the laser optics base plate and align it so that the partial beam it reflects coincides with that reflected from planar mirror (d) both on beam divider (c) and on the translucent screen (g) Spherical lens: i. Place the spherical lens (f) on the laser optics base plate between beam divider (c) and the translucent screen (g) (the small opening of the lens holder must face toward the beam divider). ii. Adjust the height and lateral position of the spherical lens so that the two partial beams pass through it axially. iii. If necessary, correct the beam path by readjusting one of the planar mirrors Fine adjustment: i. If you do not yet see a pattern of lines on the translucent screen: ii. Change the beam path by slightly changing the alignment of the beam dividers or the planar mirrors; readjust the spherical lens as necessary. iii. The more the partial beams run in parallel between the beam divider (c) and the screen (g), the wider and farther apart the interference lines are. iv. Adjust the interference pattern so that it is easy to observe by slightly changing the alignment of the beam dividers or the planar mirrors. v. If you cannot achieve a satisfactory image by fine adjustment, repeat the interferometer adjustment procedure from the beginning. vi. The interference pattern is much brighter and easier to observe when the laser is switched to an output power of 1 mw. As this can change the beam path slightly, you may need to adjust the beam path or the position of the spherical lens. vii. Observe the changes of fringes to observe the interference patterns. Record down your observations. viii. Write down the conditions for to obtain bright and dark fringes in terms of optical path difference (OPD) in wavelengths and in terms of phase. Remember that phase equals OPD wavenumber. ix. What is the phase shift on reflection? What is the phase shift when going from n 1 to n 2 when (a) n 1>n 2 and (b) n 1<n 2. Explain the reason in both cases. Faculty of Engineering, Multimedia University Page 12

13 5.1.8 Evacuable chamber and hand vacuum and pressure pump: Note: reflections of the laser beam occur at the glass surfaces of the evacuable chamber. In some cases, these may even strike the emission aperture of the laser beam and affect the quality of the laser beam. If this happens, turn the chamber somewhat. i. Seal one of the hose connections of the evacuable chamber tightly using a stopper (included in the scope of supply). ii. Mount the evacuable chamber on an optics base and place it in the beam path e.g. between beam divider (b) and planar mirror (e) so that the partial beam passes iii. through it axially. Do not change the positions of the other optical components. Connect the vacuum pump to the other hose connection of the evacuable chamber using the tubing, without pulling the chamber off the laser optics base plate by the tube; connect a suitable hose adapter to the hose connection. iv. Set up a strain-relief fitting using the small stand base and the universal clamp as and attach the tube next to the laser optics base plate so that the measurement cannot be falsified by twisting or shifting the evacuable chamber Lens alignment Lens alignment: It is important to know how a lens is aligned properly. There are two partial reflections from a lens, one from each surface, which form two spots on an aperture screen. In this experiment, the alignment sequences are: i. Slide up/down the lens mounting post until the spots centers are at the same height of the aperture. ii. Slide left/right the post holder until the two spots are overlapping. iii. Rotate the post holder until the two spots are centered about the aperture. (Note that the lens mounting assembly does not allow vertical tilting. Hence, the spots may not be coincident at the top or bottom of the aperture.) Note on the movements of the spots with respect to each of the adjustments. Repeating up/down, left/right, rotate movements may be required to align the lens properly. This alignment consumes time, depending on individual alignment skill. Consider geometrical optics of ray reflection and refraction. 5.2 Measurement of the Refractive Index of a Gas. i. In the second experiment, the refractive index of air is determined. To achieve this, an evacuable chamber is placed in the path of one component beam of the Mach-Zehnder interferometer. Slowly evacuating the chamber alters the optical path length of the respective component beam ii. Avoid mechanical shocks to the laser optics base plate (e.g. do not shake or bump the table). iii. Avoid air streaking in the setup, e.g. through breathing or drafts. iv. Mark the position of an intensity maximum on the translucent screen (g) at which the passing interference lines can be counted. v. Evacuate the chamber (h) slowly, until the next intensity maximum has moved to exactly the marked point. vi. Read off the corresponding underpressure on the manometer of the hand vacuum and pressure pump and write this value down. vii. Repeat this process until the maximum possible underpressure is reached. Recommended, but not absolutely necessary: Faculty of Engineering, Multimedia University Page 13

14 viii. Using the valve on the hand vacuum and pressure pump, let air into the vessel slowly until the previous intensity maximum is exactly at the marked position. ix. Read off the corresponding underpressure on the manometer of the hand vacuum and pressure pump and write this value down. x. Repeat this process until the normal air pressure is reached. (Cognitive Analysing, Level 4) [5 marks] Fringe motion due to thermal air currents. i. Insert your hand underneath one arm of the interferometer. Observe the interferometer pattern as the air above your hand warms up and a plume rises. Explain the reasons. ii. Put your hand underneath the beam before the first beamsplitter. Any fringe motion, why? iii. Is the interferometer sensitive to vibration? iv. Explain why aberrations in the collimated laser beam affect the shape of the interference fringes. v. If a positive spherical lens is placed in one beam, do both output beams have the same sign, ie. Are they both contracting, both expanding, or one of each? Record your observation. (Cognitive Analysing, Level 4) [5 marks] 6.0 Discussion i. If a mirror s surface flatness has to be measured, where would you insert the mirror in the setup? How would you calculate the surface profile (small surface deviations (from flatness) in terms of bumps and holes)? (Cognitive Analysing, Level 4) [5 marks] ii. Explain how the above measurements would differ with respect to measurements using Michelson Interferometer. (Cognitive Evaluating, Level 5) [5 marks] iii. The following interferogram was obtained testing a window of refractive index 1.5 in a Mach-Zehnder interferometer using a helium-neon laser. When a hot tip of a soldering iron is placed in the arm of the interferometer containing the window the fringes bend toward the right in the interferogram. [5 marks] iv. What is the peak-valley error, in units of microns, in the thickness of the window? Is the center of the window too thick or too thin? Explain. Faculty of Engineering, Multimedia University Page 14

15 (Cognitive Evaluating, Level 5) [5 marks] Faculty of Engineering, Multimedia University Page 15

16 7.0 Conclusion Conclude based on the discussed matters. (Cognitive Evaluating, Level 5) [3 marks] MARKING SCHEME 1. Experiment objectives 2% 2. Procedures, results, answers and discussions for all questions and assignments 45% 3. Conclusion 3% LABORATORY REPORT Date of submission: within 14 days after performing the experiment Place of submission: submit to the laboratory where you conducted the experiment Length of report: Your definition. Write the necessary things. Report contents: Report must include the following: i. Experiment observations ii. Discussion iii. Conclusion End of Lab Sheet Faculty of Engineering, Multimedia University Page 16