EOP3056 Optical Metrology and Testing Experiment OM1: Introduction to Michelson Interferometer

EOP3056 Optical Metrology and Testing Experiment OM1: Introduction to Michelson Interferometer 1.0 Objectives To construct a Michelson interferometer from discrete optical components To explain how Michelson's interferometer produced an interference pattern. To analyze the Michelson interferometer behaviors and characteristics 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) Optical breadboard (1) 1.5mW HeNe laser with mounting assembly (1) Beam steering mirror with mounting assembly (4) Bi-concave lens, f = -25.4 mm, with mounting assembly (1) Bi-convex lens, f = +200 mm, with mounting assembly (1) Broad band beam splitter, R/T 50/50 @ λ = 480 700 nm, AOI = 45 o, with mounting assembly (1) Beam stop (2) and viewing screen (1) 2 mm plastic aperture (1) and 2 mm paper apertures (2) Plastic ruler (1) and cardboard marked with 90 o vertical line (1) Rotation stage mounted with a microscope slide (1) Aluminium coated microscope slide with mounting assembly (1) 3.0 Introduction The Michelson interferometer is an important example of interferometers based on division of amplitude. A partially reflecting mirror is used to divide a wave (beam) into two resulting waves (beams) which wave fronts (beam sizes) maintain the original width but have reduced amplitudes. These two beams are sent in quite different directions against plane mirrors, from where they are brought together again to form interference fringes. Figure 1: The Michelson interferometer. S is a light source. L is a diffusing ground-grass plate or a lens to extend the small light source S. BS is a partially reflecting mirror or beam splitter. M 1 and M2 are highly polished plane mirrors. C is a compensating plate which is identical to BS except the partially reflecting coating. Faculty of Engineering, Multimedia University Page 1

The Michelson interferometer had been invented before the first ruby laser was built on 1960. An extended light source is required to extend the field of view of an observer. For small light source, an extender L is required. The light coming from the source S is divided into (1) a reflected and (2) a transmitted beam of equal intensity by the beam splitter. These two beams are reflected by mirrors M1 and M2 and return to the beam splitter. Part of the beam coming from M1 passes through the beam splitter and part of the beam coming from M2 is reflected by the beam splitter. Thus, the two beams are brought together to form interference fringes. Since one beam passes through BS 3 times, a compensating plate C is required for another beam so that both beams pass through equal thickness of glass. The inclusion of a compensating plate negates the effect of dispersion (optical path varying with wavelength) due to the glass medium of the beam splitter. Hence, a very board bandwidth source can generate fringes. However, C is not required if laser source is used. To obtain fringes, the mirrors M 1 and M 2 are made exactly or closely perpendicular to each other. Depending on the bandwidth (coherency) of the source, the optical distance of M 1 and M 2 to the partially reflecting coating must be closely equal. However, if a laser source is used, optical path difference up to 10 cm can still produce fringes. 3.1 Circular fringes To understand the origin of the fringes formed, Figure 1 is redrawn with all the elements in a straight line. Figure 2: A conceptual rearrangement of the Michelson interferometer Ob is the observer viewing at BS where all the elements can be seen in BS. M2 is the image of M 2 formed by reflection in BS. Assume the distance of M 2 to BS is larger than that of M 1 to BS and M 1 and M 2 are parallel. L is swung over about BS so that it is in line with M 1 and BS. L 1 and L 2 are the images of L in M 1 and M 2, respectively. A large collecting lens can be put at De position to form fringes on a screen located at the focal plane of the lens. From the figure, the two virtual sources L1 and L2 are coherent in that the phases of corresponding points in the two virtual sources are exactly the same at all instants. If d is the separation of M1 and M2, the separation of L1 and L2 is 2d. When d is exactly an integral number of half wavelengths, all rays of light reflected normal to the mirrors will be in phase. However, rays of light reflected at an angle will not in phase in general. Faculty of Engineering, Multimedia University Page 2

Consider a single point source P on L emitting light in all directions (the same from P 1 and P 2 ). Let a ray from P reflected at an angle θ by M 1 and M 2 to form two rays (actually formed by BS). Since M1 and M2 are parallel, the two rays are also parallel. The path difference between the two rays coming to the observer from corresponding points P 1 and P2 with an angle θ to the optical axis is 2dCosθ. Hence, it can be generalized that split parallel rays will reinforce each other to produce maximum intensity (constructive interference) fringes for those angles θ satisfying the relation 2dCosθ = mλ ------ Eq. 1 where m is an integer and λ is the wavelength. For a given d, m and λ, θ is constant, constructive interference will lie in the form of circles with their centers on the optical axis, which each θ is corresponding to an m value. The intensity distribution across the rings is given by I A 2 = 4a 2 cos 2 (δ/2) ----- Eq. 2 where a is the amplitude of the split waves, A is that of their resultant and δ is the phase difference given by δ = (2π/λ)*2dCosθ ------ Eq. 3 Fringes of this kind, where parallel beams are brought to interference with phase difference determined by the angle of inclination θ, are often referred to as fringes of equal inclination. A particular ring corresponds to a fixed order m. As M2 is moved toward M1, d decreases, from Eq. 1, Cos θ increases and therefore θ decreases. Hence, the rings shrink toward the center, with the highest-order one (Cosθ = 1) disappearing whenever d decreases by λ/2. Each remaining ring broadens as more and more fringes vanish at the center, until only a few fringes fill the whole field of view. By the time d is equal to zero, the center fringe will have spread out, filling the entire field of view. When M 2 is further moved, fringes reappear at the center and move outward. 3.2 Fringes of equal thickness If the mirror M 2 and M 1 are not exactly parallel, fringes will still be seen with monochromatic light. In this case the space between the mirrors is wedge-shaped. Faculty of Engineering, Multimedia University Page 3

Figure 3: The formation of fringes with inclined mirrors in the Michelson interferometer The two rays reaching the observer Ob from a point P on the source are not no longer parallel, but appear to diverge from a point P near the mirrors (note the schematic is drawn not in scale). For various positions of P on the extended source, it can be showed that the path difference between the two rays remains constant but that the distance of P from the mirrors changes. If the angle between the mirrors is not too small, the distance of P from the mirrors is not great. If the distance between the mirrors, d = 0, the fringes are straight because the variation of the path difference across the field of view is due primarily to the variation of the thickness of the air-film between the mirrors. With a wedge-shaped film, the locus of points of equal thickness is a straight line parallel to the edge of the wedge. If d has an appreciable value, the fringes are not exactly straight because there is also some variation of the path difference with angle as mentioned in section 3.1. They are in general curved and are always convex toward the thin edge of the wedge. If d is decreased (M2 moves towards M1 without changing the inclination of M2 ), the fringes will move to curve-in side, a new fringe crossing the center each time d changes by λ/2. When d approaches 0, the fringes become straighter until when d = 0, M2 intersects M1, the fringes are perfectly straight. When M2 moves further, the fringes curve in the opposite direction. For large path differences, large d, the fringes cannot be seen. Because the principal variation of path difference results from a change of the thickness d, these fringes have been termed fringes of equal thickness. 3.3 Application The Michelson interferometer can be used for many applications. It is a non-contact measuring technique via the observation of the interference/fringe pattern or the fringe transition. The equation mλ = 2d is often used for various types of measurements. In most cases, d is to be measured. Measurement examples are wavelength of light source, displacement of object, thickness of thin film, refractive index of transparent solid/liquid/gas, surface flatness/quality of optical components, thermal expansion coefficient of solid, pulse duration of pico/femto-second light pulse, and vibration frequency of object. 3.4 Measurement of refractive index of transparent solid The wavelength of a light is shortened when it propagates in a transparent material, λ = λo/n ------ Eq. 4, where λo is the wavelength of the light in vacuum and n is the refractive index of the material which is a function of wavelength, n(λo). If a material has thickness d = mλ, where m is a positive integer, then d = m*λo/n and hence nd = mλo, where nd is the optical path length of the light with λ o, i.e. the path length passed by the light in vacuum. In Michelson interferometer, both interference beams propagate in their interferometer arms twice, hence mλ o = 2n ad ------ Eq. 5, where na is the refractive index of air and d is the distance between the two interferometer arm mirrors. 2nad is the optical path length difference between the two interference beams. Faculty of Engineering, Multimedia University Page 4

To measure the refractive index of a transparent solid with a Michelson interferometer, it is necessary to slowly vary the length of the solid through which the interference beam passes so that the number of fringes shifted on the viewing screen can be counted. A technique is used for such measurement. Figure 1 shows the arrangement of the Michelson interferometer with a transparent solid located at one arm of the interferometer. As the plate is rotated from 0 o to an angle θ, the light passes through a greater path length in the transparent material. This shifts a number of fringes on the viewing screen. Figure 4: Michelson interferometer arrangement for the measurement of the refractive index of a transparent solid. The notations S, L, BS, M1 and M2 are the same as given in Experiment OM1, Figure 1. The sample is rotated about the point O from 0 o to θ. b1 is the beam path when the sample at 0 o while b2 is the beam path when the sample at θ angle. The path length of the beam in the sample is increased from t to a while its path length in air is decreased by (c t), where t is the thickness of the sample. Faculty of Engineering, Multimedia University Page 5

To derive an equation for calculating the refractive index of a transparent solid with the measured θ and the number of fringes shifted (N), consider the plate is rotated about the point O in Figure 1. When the plate is at 0 o : Base on equation Eq. 2, m 1 λ o = 2n a l a1 + 2n g l g1 ------ Eq. 6, where ng is the refractive index of the sample, la1 is the path length in air and lg1 is the path length in the sample. 2n a l a1 + 2n g l g1 is the optical path length difference between the interference beams. When the plate is at θ degrees (This rotation shifts a number of fringes): m 2 λ o = 2n a l a2 + 2n g l g 2 ------ Eq. 7 From Eq. 6 and Eq. 7: (m 2 m 1 )λ o = Nλ o = 2n a (l a2 l a1 ) + 2n g (l g 2 l g1 ) ------ Eq. 8, where m 2 m 1 = N is the number of fringes shifted, l a2 l a1 is the path length change in air and l g 2 l g1 is the path length change in the sample. To get l g 2 l g1, consider the triangular OPQ, where OP = t = lg1, OQ = a = lg2 and PQ = b, l g 2 l g1 = a t ------ Eq. 9 To get l a2 l a1, consider the triangular OQR, where OR = c, l a2 l a1 = (c t) = t c ------ Eq. 10 The negative sign for (c t) is included because the path length passed by the beam through air is reduced by (c t) as shown in Figure 1. After the derivation of equations for a and c, substitution into Eq. 9 and Eq. 10 and then into Eq. 8 with n a = 1, the refractive index equation is (2t Nλ )(1 Cosθ) + N 2 2 λ o o n g = 4t 2t(1 Cosθ) Nλ o ---- Eq. 11 2, where the term N 2 λo /4t is negligible for visible wavelengths and may be ignored. Note that na = 1.000293 at standard temperature (0 o C) and pressure (1 atm) for λo = 589.29 nm (Sodium D light). It is reasonable to take n = 1.0003 at room temperature and 1 atm for λ o = 632.8 nm (HeNe laser red light). Faculty of Engineering, Multimedia University Page 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 jewelry 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. 4.6 Clamping screws Faculty of Engineering, Multimedia University Page 7

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 8

5.0 Experiment Figure 5: Schematic view of Michelson Interferometer. The mounting assemblies of the components are not shown in the schematic diagram. M1, M2, M3 and M4 are the beam steering mirrors. L1 is the bi-concave lens and L2 is the bi-convex lens. BS is the beam splitter, where S1 is the beam splitting surface and S2 is the anti-reflection surface. BSB is the beam stop for blocking laser beam close to the laser. A1, A2, A3, A4, A5 and A6 are the aperture or beam-stop locations along the setup of the Michelson interferometer. The numbers at the outside of the optical breadboard indicate the distances in inches. The distance between adjacent screw holes is one inch. The adjustable mirror mount has two fine thread screws for adjustment, a steel ball and two pulling springs. 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. i. Laser mounting assembly: a. Laser tube height: slide post clamp up/down along its mounting post b. Laser tube horizontal tilting: rotate post clamp about its mounting post Faculty of Engineering, Multimedia University Page 9

Loose the post clamping screw a little bit (don t loose too much) for movement. Do not over-tighten the clamping screw. Laser tube vertical tilting is not allowed. ii. Adjustable mirror mount: a. Laser beam horizontal fine steering (X): turn the horizontal fine adjustment screw b. Laser beam vertical fine steering (Y): turn the vertical fine adjustment screw iii. Mounting post and its post holder a. Laser beam horizontal coarse steering (X): rotate the mounting post in its post holder b. Optical component height: slide the mounting post up/down in its post holder Loose the post holder clamping screw for movement. Do not over-tighten the clamping screw. Laser beam vertical coarse steering (Y) is not allowed. The optical component here can be mirror, lens, beam splitter, etc. iv. 2 mm aperture: It is always useful for optical alignment to mark laser beam center or the optical axes of optical components and to act as a screen to monitor the reflections from optical components that are inserted in the beam after the aperture. Student must read and understand sections 4.0 to 4.7 and 5.0 before performing the experiment below. 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. 5.1 Procedures for setting up a Michelson Interferometer 5.1.1 U-shaped beam alignment i. The laser beam will traverse in a U-shaped path as shown in Figure 4 and in a plane which is parallel with the surface of the optical breadboard. Hence, you should never put your eyes at this laser traversing plane level. ii. Make sure the laser is not turned on. Remove L1, L2, BS and all the apertures from the laser beam path. iii. Laser is still off. Check the directions of the laser output, M1 and M2 so that when the laser is turned on, the laser beam will traverse approximately in a U-shaped path. Using a plastic ruler, check whether the center-heights of the laser output, M1 and M2 are about 12 cm. If necessary, correspondingly adjust the mounting assemblies of the laser (do not loose the clamping screw too much, else the clamp will slide down), M1 and M2. (Note: normally, they are in the correct positions unless someone has moved their positions.) iv. Laser is still off. Put a beam stop (BST or a rough surface white paper) close to the laser. Turn on the laser. Trace the laser beam until a position close to M1. (Note that if you are sure the approximate position of the laser beam close to M1, you may put BST directly close to M1.) With the laser beam falling on BST, make sure M1 can capture the laser beam if BST is removed. Adjust the laser mounting assembly if Faculty of Engineering, Multimedia University Page 10

necessary. Put another beam stop (BSB) close to the laser (to block the laser beam). Move BST to A2 position. Remove BSB (to unblock the laser beam). Check whether the laser beam intersects M1 at its center. Adjust the laser mounting assembly if necessary. v. Note that from this onwards, procedures will be mentioned in more simple forms, including precaution steps. It is your responsibility to make sure you can understand the meanings of the simple forms. With BST at A2, trace the beam until M2. Make sure M2 can capture the beam. Adjust M1 mounting post if necessary (do not accidentally knock down BST). Move BST to A5 (block the beam first). Using a plastic ruler, record the center-height h of the beam just after M1. (Caution: Never touch the mirror surface! The value h may not be the actual value. Make sure you use the same end of the ruler touching the breadboard surface for other centerheight measurements). Check whether the beam in the M1-M2 arm intersects M2 at its center and has the same center-height h just before M2. If necessary, correspondingly adjust the adjustment knobs and mounting post of M1 and the mounting post of M2 (don t knock down BST). (For easy alignments for the rest of experiments, make the M1-M2-arm beam in parallel with the right edge of the breadboard. You may use the cardboard marked with 90 o vertical line and a line of screw holes on the surface of the breadboard for this parallel alignment. Note that the beam is not on top of a line of screw holes but offset about 3 mm from the center of the screw hole.) vi. With BST at A5, trace the beam until A6 position. Using the cardboard marked with 90 o vertical line, check whether the beam in M2-M4 arm is parallel to the front edge of the breadboard (or a line of screw holes on the surface of the breadboard, offset about 3mm from screw hole center). The beam at A6 must have the same centerheight h. (This parallel arrangement is important for easy alignment for the rest of experiments.) If necessary, correspondingly adjust the adjustment knobs and mounting post of M2 (don t let the beam shoot outside BST). Now, the beam has been aligned to traverse in a U-shaped path and in a plane parallel to the breadboard surface. 5.1.2 Laser beam expander alignment i. An inverted telescope is used to expand a laser beam. In this experiment, Galilean telescope is used. ii. 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: (1) Slide up/down the lens mounting post until the spots centers are at the same height of the aperture. (2) Slide left/right the post holder until the two spots are overlapping. (3) 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. iii. Put the 2 mm plastic aperture at A1 position and align it so that the beam passes its center. (For easy viewing the beam passing the aperture center, put a rough surface white paper close to the aperture at the exit side, the back-scattered laser light will help you to view the aperture opening area.) Put a 2 mm paper aperture Faculty of Engineering, Multimedia University Page 11

at A4 position and align its center at the beam center. Put another 2 mm paper aperture at A2 position and align its center at the beam center. iv. Block the beam. (Caution: Never turn the lens chuck screws! The lens can easily drop out from the lens chuck.) Put in L1 (Bi-concave lens) at about 4 screw-holes distance from M1 and align roughly its center at the beam path. (Use the centers of A2 and A4 apertures or 5.1.1 v as a guide and use a ruler for L1 center-height.) Unblock the beam. Use the lens alignment guides, align L1 so that the transmitted beam is centered on A4 aperture and the two reflected beams are centered about A2 aperture. (They may be at the top or bottom of the aperture. Remove the aperture at A2, you can see the two large-diameter, low-intensity spots at A1. One of the spot may have concentric circular bright and dark rings). v. Put the paper aperture (initially at A2) at A3 position and align its center at the beam center. (Note that the diffraction of the 2 mm aperture may affect the beam direction after the aperture. Hence, the aperture must be properly center at the beam center. The aperture at A4 is a useful reference for this aperture centering.) Block the beam. Put in L2 (Bi-convex lens) at about 7 screw-holes distance from L1 and align roughly its center at the beam path (including its centerheight). Unblock the beam. Use the lens alignment guides, align L2 so that the transmitted beam is centered on A4 aperture and the two reflected beams are centered about A3 aperture. With the information in 5.1.1 vi, the beam center at A6 should be the same position (height = h, offset about 3mm from screw hole center). vi. Make sure M4 can reflect back the beam to M2. Remove BST at A6. Finely turn the M4 adjustment knobs until the reflected (returning) beam passes through A4 aperture and then A3 aperture, and lastly its center falls at about 1 mm from the edge of either side of A1 aperture. Remove A3 and A4 aperture. Carefully move L2 back and forth along the beam until the returning beam size at A1 is the same size as the incoming beam passing through the aperture. (Note that the returning beam must be located about 1 mm from the edge as that before moving the lens. You can use a white paper to determine the incoming beam size that passes the aperture.) The beam after L2 is well collimated. Measure and record the beam sizes at A4, d A4 = mm and A6, d A6 = mm. 5.1.3 Michelson interferometer alignment i. The beam splitting surface of the beam splitter has reflectance, R 50% and transmittance, T 50% at wavelengths, λ = 480 700 nm (broad band) and angle of incidence, AOI = 45 o. The anti-reflection surface is not totally 0% reflectance. Hence, it is tilted about 30 arc minutes to the beam splitting surface to avoid interference due to multiple internal reflections. ii. Put a paper aperture at A5 position and align its center at the beam center. Use the information in 5.1.1 vi to confirm the aperture centering (diffraction effect). iii. Block the beam. Put in BS with the surface marked with > facing to M4. Roughly align S1 surface center (Use the information in 5.1.1 vi. Make sure the centerheight of BS is equal to h) and its orientation (45 o between S1 surface and the beam path) so that when the beam is unblocked, part of the incoming beam will be reflected to M3 by S1 surface. The angle between the reflected beam to M3 and the incoming beam from M2 is roughly 90 o. iv. Unblock the beam. Perform repeating adjustment of moving the BS base plate back and forth along the incoming beam from M2 and rotate the base plate until the angle between the incoming beam from M2 and the reflected beam to M3 is close or equal to 90 o. At the same time, the reflected beam falls on about the center of M3. (Use the Faculty of Engineering, Multimedia University Page 12

cardboard marked with 90 o vertical line. The holes on the breadboard surface can help to determine 90 o angle. This alignment consumes time!) After the 90 o has been achieved, note on the position and orientation of the base plate relative to the screw holes in the incoming beam direction. (This noting is important for following adjustment.) Adjust M3 height if necessary. (Note that the center-height of the beam on M3 may not be exactly equal to h and the transmitted beam to M4 is not centered. All these are fine.) v. Rotate the mounting post of M3 if necessary. Turn the adjustment screws of M3 so that the beam reflected from M3 passes through the aperture at A5, then the aperture at A2 and lastly falls on the same position (about 1 mm from the same edge). (Note that the beam initially may not fall on the paper of the aperture at A4. Hence, you need to trace the beam with a rough surface white paper to know its position outside the aperture paper. The alignment consumes time and requires very fine adjustment to get the final position.) Remove the aperture at A5. The coincident position of the two spots on A1 aperture should not be changed. If they are changed, adjust accordingly to get the same position (about 1 mm from the same edge). At this coincidence of the two spots, a series of bright and dark fringes appears on the viewing screen. vi. The S1 surface may not be at the center of the beam at M3-BS direction after A5 aperture is removed. Put BST (or a paper aperture) at the front of M4, move the base plate along M3-BS direction until the spot on BST is the best without blocking by the lens chucks. (Note the position of the base plate in M2-BS direction must be the same as noted in 5.1.3 iv). (The BS has been aligned in all X, Y and Z directions). The interferometer alignment has been completed. Note that aligning the two beams on A1 aperture is good for a quick rough alignment to get fringes appearing on the screen. However, this is only applied for a collimated or closely collimated beam. 5.2 Procedures for observation of interferometer behaviors and characteristics i. After the interferometer alignment is completed, one of the interferometer mirrors will not be moved anymore. Let M3 is fixed all the time. This section will observe how the orientation and separation of the fringes changed with respect to the orientation and tilting angle of the M4 mirror movements. Finely turn (repeat to turn a small step and release) the adjustment screws of M4 so that approximately five bright fringes appear across the beam on the screen. Perform the following experiment in sequence. You may sketch the fringes change after every movement of M4 for easy discussion. ii. Use a rough surface white paper. View the fringes on the white paper from the screen position to a position close to BS. Record this observation in words. iii. Finely turn the horizontal adjustment screw of M4 clockwise and observe the movement of the fringes on the screen until the number of bright fringes is equal to 2 or 10 (depending on the number decreasing or increasing). Record this observation in words (for fringes changed with respect to mirror movements). iv. Finely turn the horizontal adjustment screw of M4 counter-clockwise and observe the movement of the fringes on the screen until the number of bright fringes is equal to 2 or 10. Record this observation in words. v. The observations are the same for turning the vertical adjustment screw of M4. Now, finely turn these two screws until approximately five horizontal bright fringes. Faculty of Engineering, Multimedia University Page 13

Draw the fringes on a graph paper in details (relative separation, thickness and intensity of the bright and dark fringes). Ask your instructor to verify your drawn fringes and the fringes appearing on the screen. vi. By selecting the correct adjustments screw so that only the number of horizontal bright fringes changes without changing the orientation of the fringes. Finely turn this adjustment screw clockwise and counter-clockwise and observe the number of the bright fringes. Make sure you turn the screw for the whole range until the number of fringes becomes very large or equals to 1 (or 0) and then increases again. Record this observation in words. (Let 0 o tilting angle is assigned for 1 or 0 number of fringes, clockwise produces positive tilting angles and counter-clockwise produces negative tilting angles. Record only comparative, no quantitative.) vii. Turn the correct adjustment screw so that the number of fringes becomes 1 (or 0) and then turn another adjustment screw to get approximately 5 vertical bright fringes. Record this observation with respect to the mirror tilting angle and direction in this sequence of screw adjustments. viii. Turn the correct adjustment screw so that the number of fringes becomes 1 (or 0). At this position, let assign the horizontal adjustment screw angle as θx = 0 o and the vertical adjustment screw angle as θ y = 0 o. Turn the horizontal adjustment screws and observe θ x until there are about 10 bright fringes on the screen, let assign θ x = θ x1. Turn the vertical adjustment screw with the same turning angle, i.e. θ y = θ x1. Record what you observe on the fringes changed. Then, turn the vertical adjustment screw so that θ y = 0 o and then further turn it until θ y = -θ x1. Record what you observe on the fringes changed. ix. Turn the correct adjustment screw to get approximately 5 vertical bright fringes. Move L2 closer to L1 by 4 screw-holes distance. Align L2 as mentioned in 5.1.2. Record the fringes changed in words. Turn the correct adjustment screw to bring to one or two bright fringes. Record the fringes changed in words. Further turn the adjustment screw and also another adjustment screw until a single spot (either bright or dark) with side ring. Draw the fringes. x. Move L2 closest to L1 but the base plates of L2 and L1 are not touching each other. Put BST or a paper aperture at A6 to align L2. Draw the fringes. Turn one of the adjustment screws and record the fringes changed. Turn back the adjustment screw to the previous fringes before go next step. xi. Remove L2 from the laser beam path (block the beam first). Draw the fringes. Turn one of the adjustment screws and record the fringes changed. Turn back the adjustment screw to the previous fringes before go next step. xii. Remove L1 from the laser beam path (block the beam first). Draw the fringes. Turn one of the adjustment screws and record the fringes changed. Now, you should be able to control the number of fringes and their orientation, etc. 6.1 Procedures for refractive index of glass measurement The Michelson interferometer is to be set up to produce several (about 5) concentric circular fringes on a viewing screen. The conditions to produce this fringe pattern are (1) an expanded and diverging laser beam and (2) sufficiently large BS-M4 arm length comparing with BS- M3 arm length (refer to Figure 4 of Experiment OM1). This fringe pattern aims for ease of counting the number of fringes appearing/disappearing at the center of the concentric circular fringes. Faculty of Engineering, Multimedia University Page 14

i. Figure 4 of Experiment OM1 is always referred in the following alignment procedures. ii. If necessary, refer to Section 5.1 of Experiment OM1 for the detailed alignment procedures. iii. Take precaution on the laser beam during the alignments when the laser beam is not expanded and diverged. iv. Remove the bi-concave lens L1 and bi-convex lens L2. v. Align the laser beam on the center of the mirror M2. vi. Align the laser beam in M2-BS arm so that it is parallel to the front edge of the optical breadboard. Use the optical breadboard screw holes as a guide. vii. Align the beam splitter BS so that (1) the laser beam falls on the BS center and (2) the reflected beam to M3 is 90 o to the M2-BS beam. Use the optical breadboard screw holes to locate BS position before the 90 o alignment. Note S1 surface is on the M2 side, not on the M4 side. viii. Align M3 so that the reflected beam from M3 is closely coincident with the incoming beam. It should fall on ~1 mm from edge of either side of the aperture A1. ix. Locate M4 so that BS-M4 arm length is 1 inch longer than BS-M3 arm length. Align M4 so that the reflected beam from M4 is coincident with that from M3 on the aperture A1 paper and the viewing screen. At this point, you can see some fringes on the viewing screen and the aperture A1 paper. x. Put the bi-concave lens L1 in M1-M2 arm at a position 4 inches from M1 and align L1 (up/down, left/right, rotate) so that the laser beam falls on the screen with about symmetrical intensity profile. You can see at about 5 concentric circular fringes with their centers at the center of the intensity profile. If necessary, align M4 to get the center. xi. Mount the rotation stage with a microscope glass plate in between BS and M4 with two clamping screws (hand tightening is enough). Rotate the rotation stage adjustment knob so that the glass plate surface is 90 o to the laser beam in horizontal direction (vertical direction cannot be aligned). For accurate alignment, remove L1, put a rough surface white paper at the front of M4 (do not touch the M4 surface with the paper), if the 90 o is achieved, all the secondary dots are aligned vertically with the primary bright dot. Put in L1 and align it. The same fringe pattern can be seen on the screen. Note that there may have a faint secondary fringe pattern visible at the center of the screen. This is formed by the Fabry-Perot interference between the front and back surfaces of the glass plate. xii. Record the angle reading on the rotation stage (r 0). This is the 0 o position. For accurate refractive index measurement, the glass plate must be as 90 o as possible and the angle must be read as precise as possible. xiii. Slowly rotate the rotation stage adjustment knob and count 20 fringe transitions (appearing/disappearing at the center). Read the angle as precise as possible (r20). Continue to count another 20 fringe transitions and read the angle (r40). xiv. Repeat procedure 11, 12 and 13 to make sure the measurements are correct. If these two sets of reading are different too much. Repeat again. Data recording: r 0 = angle reading at the 0 o position r 20 = angle reading after 20 fringe transitions r 40 = angle reading after another 20 fringe transitions Set 1: r 0 = o, r 20 = o, r 40 = o, θ 1 = r 20 r 0 = o, θ 2 = r 20 r 0 = o Faculty of Engineering, Multimedia University Page 15

Set 2: r0 = o, r20 = o, r40 = o, θ1 = r20 r0 = o, θ2 = r20 r0 = o Set 3: Calculations: 1. Calculate the average value of θ 1 and θ 2. From Eq. 8 with N 2 λ o 2 /4t ignored, since n g is constant, the thickness of the glass plate can be derived and the resultant equation is given below. t = 1 λ o N 1 N 2 (Cosθ 1 Cosθ 2 ) 2 N 1 Cosθ 1 (1 Cosθ 2 ) N 2 (1 Cosθ 1 )Cosθ 2 ----- Eq. 12 where λo = 632.8 nm, N1 = 20, N2 = 40, θ1 = the angle from the 0 o position to make 20 fringe transitions and θ 2 = the angle from the 0 o position to make 40 fringe transitions. (Cognitive Analysing, Level 4) [5 marks] 2. Determine the thickness of the glass plate with Eq. 12. Compare with the theoretical value. (Cognitive Evaluating, Level 5) [5 marks] 3. Determine the refractive index of the glass plate with Eq. 11 (ignore N 2 λ o 2 /4t term). Compare with the theoretical value. (Cognitive Evaluating, Level 5) [5 marks] Faculty of Engineering, Multimedia University Page 16

Discussions Analyse and discuss on the matters along with the experiments Setup of the interferometer Experiment observation related to theory (Cognitive Understanding, Level 2) 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 17