EOP4056 Optical Metrology and Testing: Experiment OM1
EOP4056 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 @ = 480 – 700 nm, AOI = 45o, 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 90o 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. M1 and M2 are highly polished plane mirrors. C is a compensating plate which is identical to BS except the partially reflecting coating.
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 M1 and M2 are made exactly or closely perpendicular to each other. Depending on the bandwidth (coherency) of the source, the optical distance of M1 and M2 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 M2 formed by reflection in BS. Assume the distance of M2 to BS is larger than that of M1 to BS and M1 and M2’ are parallel. L is swung over about BS so that it is in line with M1 and BS. L1’ and L2’ are the images of L in M1 and M2’, 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.
Consider a single point source P on L emitting light in all directions (the same from P1’ and P2’). Let a ray from P reflected at an angle by M1 and M2’ 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 P1’ and P2’ with an angle to the optical axis is 2dCos.
Hence, it can be generalized that split parallel rays will reinforce each other to produce maximum intensity (constructive interference) fringes for those angles satisfying the relation
2dCos = m ------Eq. 1
where m is an integer and is the wavelength. For a given d, m and , is constant, constructive interference will lie in the form of circles with their centers on the optical axis, which each is corresponding to an m value.
The intensity distribution across the rings is given by
I A2 = 4a2cos2(/2) ----- Eq. 2
where a is the amplitude of the split waves, A is that of their resultant and is the phase difference given by
= (2/)*2dCos ------Eq. 3
Fringes of this kind, where parallel beams are brought to interference with phase difference determined by the angle of inclination , 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 increases and therefore decreases. Hence, the rings shrink toward the center, with the highest-order one (Cos = 1) disappearing whenever d decreases by /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 M2’ is further moved, fringes reappear at the center and move outward.
3.2 Fringes of equal thickness
If the mirror M2’ and M1 are not exactly parallel, fringes will still be seen with monochromatic light. In this case the space between the mirrors is wedge-shaped.
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 /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 m = 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,
= o/n ------Eq. 4
, where o is the wavelength of the light in vacuum and n is the refractive index of the material which is a function of wavelength, n(o). If a material has thickness d = m, where m is a positive integer, then d = m*o/n and hence nd = mo, where nd is the optical path length of the light with 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
mo = 2nad ------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.
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 0o to an angle , 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 0o to . b1 is the beam path when the sample at 0o while b2 is the beam path when the sample at 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.
To derive an equation for calculating the refractive index of a transparent solid with the measured and the number of fringes shifted (N), consider the plate is rotated about the point O in Figure 1.
When the plate is at 0o:
Base on equation Eq. 2, ------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. is the optical path length difference between the interference beams.
When the plate is at degrees (This rotation shifts a number of fringes):
------Eq. 7
From Eq. 6 and Eq. 7:
------Eq. 8
, where m2 – m1 = N is the number of fringes shifted, is the path length change in air and is the path length change in the sample.
To get , consider the triangular OPQ, where OP = t = lg1, OQ = a = lg2 and PQ = b,
= a – t ------Eq. 9
To get , consider the triangular OQR, where OR = c,
= – (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 na = 1, the refractive index equation is
---- Eq. 11
, where the term N2o2/4t is negligible for visible wavelengths and may be ignored.
Note that na = 1.000293 at standard temperature (0oC) and pressure (1 atm) for o = 589.29 nm (Sodium D light). It is reasonable to take n = 1.0003 at room temperature and 1 atm for o = 632.8 nm (HeNe laser red light).
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 alighted 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
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.
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.
- Laser mounting assembly:
- Laser tube height: slide post clamp up/down along its mounting post
- Laser tube horizontal tilting: rotate post clamp about its mounting post
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.