The Precision Interferometer

Transcription

1 Learning Outcomes (the skills you will acquire Proficiency in making fine adjustments of optical components Understand how to use an interferometer Obtain a practical understanding of the interference of light waves Measure the wavelength difference between the sodium D lines Observe the beating of these wavelengths by means of the interferometer Practice in taking clear and intelligible laboratory notes Preparatory Task: Work through the interactive screen experiment version of this experiment on and also the micrometer tutorial interactive screen experiment in the same location, describing the reading of micrometer scales. Read the entire script and write a short paragraph in your lab book summarizing the experiment and its aims. Make sure that you can follow the derivation in Task 5. Write a brief outline of the theory behind the experiment and perform Tasks 1 and 2 below. When performing this experiment, you must avoid touching any of the optical surfaces, particularly the mirrors, which are uniformly flat to a fraction of a micron. Touching these surfaces would leave deposits which deform the surface and hence distort the interference pattern that you are to observe. 1. Introduction Using a prism spectrometer, it is possible to measure the wavelength ( λ = nm of the characteristic yellow emission lines of a sodium lamp. However, the spectral resolution of the prism is insufficient to separate the two component emission lines, denoted D 1 and D 2 ; this can be achieved by means of a precision interferometer. In this experiment, you will measure the wavelength difference, ( λ 1 - λ 2, between the sodium D lines using the interferometer in its Fabry-Perot mode of operation. The Fabry-Perot interferometer consists of two half-silvered mirrors, which reflect 50% of the light incident upon them and transmit the remaining 50%. One of these mirrors is adjustable, by means of the two black tilt screws which are mounted diagonally on the rear of the mirror holder; they must not be confused with the tension screw, which incorporates a metal spring and which should not be touched. Locate the adjustable mirror on the interferometer mounting and check that you identify correctly the two black tilt screws (if in any doubt, ask the demonstrator. Facing the adjustable mirror is the second half-silvered mirror, which is movable by means of the micrometer mounted on the side of the interferometer base; you will not

2 need to touch the movable mirror itself. Light from a sodium lamp enters on the far side of the interferometer through a diffusing plate, and the interference pattern is viewed through the hole in the component holder which is located on the near side of the interferometer. The interference fringes appear to be located at infinity and so should be observed with the eye relaxed. Tasks 1 and 2: Determine the effect of interference between two coherent beams of monochromatic light. The equation for a light wave of amplitude a propagating in the x- direction is ϕ(x,t = asin(ωt kx where k = 2π /λ is the wave number, ω = 2πf is the angular frequency, and t is the time. If we choose x = 0 as the location of the observer, then ϕ(x = 0,t = asinωt. In the case of the precision interferometer, the wave of amplitude a from mirror M 1 arrives at the observer with phase δ 1, and the wave with the same amplitude a from M 2 arrives with phase δ 2. The resulting disturbance is ϕ(x = 0,t = asin(ωt + δ 1 + asin(ωt + δ 2 = 2asin(ωt + δ 1 + δ 2 2 cos( δ δ What is the relationship between δ 1 and δ 2 such that the resulting light intensity, I = ϕ 2, is a maximum? Express the phases, δ 1 and δ 2, in terms of the corresponding distances travelled by the light, d 1 and d 2, and the wavelength of the light, λ. 2. Preparation The initial adjustment of the tilt of the mirror is by means of the two black screws. If the two mirrors are not parallel, you will see multiple images of the diffusing plate (which appears yellow when the sodium lamp is on when you look through the hole in the component holder. You can superpose all these images by changing the tilt of the rear mirror, by means of successive small adjustments of the tilt screws. When a single image of the illuminated source is obtained, fringes arising from the interference of the light reflected from and transmitted by the mirrors come into view. The fringes are concentric circles, which can be centred in the field of view by making fine adjustments to the tilt of the rear mirror; the surfaces of the two mirrors will then be parallel. [Note that other faint circular fringes may be visible when the mirrors are incorrectly adjusted. These fringes can be identified by the fact that they do not move when the separation of the mirrors is varied by means of the micrometer.]

3 Task 3: Adjust the mirror by means of the tilt screws, as described above, until a circular fringe pattern is obtained, centred in the field of view. Show this circular fringe pattern to the laboratory demonstrator. Change the separation of the two mirrors by means of the micrometer. Each rotation of the micrometer corresponds to a displacement of 25 microns of the movable mirror. When the micrometer is turned clockwise, the separation of the mirrors increases and fringes appear to emerge from the centre of the pattern. Turning the micrometer anti-clockwise decreases the mirrors separation and causes the fringe pattern to move towards the centre. The maximum displacement of the movable mirror, over the entire course of the micrometer, is approximately 1 mm. Note that changing the sense of rotation of the micrometer gives rise to a mechanical effect known as backlash, which occurs because the micrometer is operated by means of a screw thread. Its effects on your measurements can be almost eliminated by turning the micrometer through one complete turn before taking any readings, and by turning always in the same sense (clockwise or anti-clockwise during any given set of measurements. Task 4: Quote the precision to which you can measure distance from the micrometer scale. 3. Measurements As mentioned earlier, the yellow light of the lamp is due to the two sodium D lines, D 1 and D 2, which have slightly different wavelengths; their intensity ratio is 1:2. As you adjust the distance between the mirrors by means of the micrometer, the fringe pattern from line D 1 moves, relative to the fringe pattern from D 2. When the bright fringes from both patterns coincide, there is maximum visibility, as the bright fringes reinforce each other. On the other hand, when the bright fringes of D 1 lie between the bright fringes of D 2, the field of view is almost uniformly yellow: there is minimum visibility of the fringes. As you increase or decrease the distance between the mirrors, you will run through successive positions of maximum and minimum fringe visibility. A few trial runs may be necessary in order to be able to measure consistently the positions of, in particular, minimum fringe visibility.

4 Task 5: Turn the micrometer anti-clockwise until it is fully extended (highest reading. Starting from this position, rotate the micrometer clockwise, taking measurements of the positions of successive maximum and minimum fringe visibility until the micrometer reaches the end of its course. Repeat the measurements for the opposite sense of rotation of the micrometer. If you are working as a pair, your partner should make the same measurements. Compare your results. Useful tip: successive positions of maximum and minimum fringe visibility are separated by roughly five full turns of the micrometer screw. These measurements can be used to determine the wavelength difference between the D 1 and D 2 lines, as follows. Let the wavelength of the D 1 line be λ 1 and of the D 2 line be λ 2, and suppose that a position of maximum fringe visibility occurs at a micrometer reading s. At this position, the condition of constructive interference, corresponding to a bright fringe, is 2s = mλ 1 = nλ 2 where m and n are integers. Let the next position of maximum fringe visibility occur at a micrometer reading of (s+t. At this point, the fringe patterns corresponding to the two wavelengths, λ 1 and λ 2, have slipped by one fringe, relative to each other, and hence 2(s + t = (m + pλ 1 = (n + p +1λ 2 where p is the number of fringes of λ 1 between the two successive positions of maximum fringe visibility. Eliminating m, n and p between these equations gives t = λ 1 λ 2 2(λ 1 λ 2 λ 2 2(λ 1 λ 2 and λ is the mean wavelength of the D 1 and D 2 lines ( λ = nm. The value of t is given by the gradient of the graph, determined under Task 6.

5 Tasks 6 and 7: Tabulate your measurements of the successive positions of maximum and minimum fringe visibility, using Excel. Plot your results, with the number of the reading along the x-axis, denoting maxima by successive integers and minima by successive half-integers (i.e. 1, 1.5, 2, 2.5,, and the micrometer readings along the y-axis, with their error bars. Determine the gradient of this curve, with its error bar. Determine the difference in the wavelengths of the sodium D lines, ( λ 1 - λ 2 nm, from the previous equation, together with the associated error bar. 5. Exploration: white light fringes The Michelson interferometer is probably best known because of its use in the Michelson-Morley experiment, which was designed to demonstrate the motion of the Earth through the aether. In fact, no such motion was detected, and this led to the abandonment of the concept of the aether and the adoption, by Einstein, in his theory of special relativity, of the postulate that the velocity of light is independent of the motion of its source. The principle of operation of the Michelson interferometer is illustrated in Fig. 1. The aim is to measure the path difference between two coherent beams of light, which travel along the two arms of the interferometer. Light from a source of radiation (a sodium lamp or a tungsten filament lamp, in this experiment is split into two beams of equal intensity by the beam-splitter, A, one surface of which is a half-silvered mirror, which transmits half of the incident light and reflects the other half. One beam is reflected towards the mirror M 1, reflected back from M 1 and transmitted through the beam-splitter to the observer. The other beam is transmitted through the beamsplitter to the mirror M 2, reflected back and then reflected from the half-silvered surface of the beam-splitter towards the observer. When the mirrors are adjusted so that the two beams are parallel as they approach the observer, interference fringes are seen. The compensating plate, C, is identical in size and construction to the beamsplitter, A, except that it is not silvered and hence transmits the light incident on it. Its function is to ensure that the distance travelled in glass along both arms of the interferometer is identical. Figure 1: The light paths through a Michelson interferometer. M 1 and M 2 are the mirrors, and A is the beam-splitter; C is the compensating plate.

6 Let the distance from A to M 1 be denoted d 1, and the distance from A to M 2 be denoted d 2. Then, the difference between the path lengths of the two coherent beams of light is 2(d 1 where the factor of 2 arises because each beam of light travels twice the length of the respective arm of the interferometer. If the wavelength of the light is λ, then 2(d 1 /λ is the path difference expressed as a multiple of the wavelength. When 2(d 1 /λ is an integer, the two beams of light seen by the observer are exactly in phase and interfere constructively; a bright fringe is observed. When 2(d 1 /λ is a halfinteger, the two beams interfere destructively; a dark fringe is observed. Note that, when d 1 = d 2 exactly, constructive interference occurs, whatever the value of λ. One can then observe white light fringes. Slightly adjust the mirror M 2 so that the fringe pattern is slightly offcentre and hyperbolic fringes can be seen in the field of view. Using the micrometer, vary the distance d 1 of the mirror arm. You will once again observe the change from maximum to minimum fringe visibility. In addition, you will see that the curvature of the fringes tends either to increase or to decrease. You need to adjust d 1 in the sense of decreasing fringe curvature. Eventually, the fringes become straight, and if you continue to change d 1 in the same direction, the fringes begin to curve in the opposite sense. Adjust d 1 so that the fringes are, as near as you can judge, parallel lines. You are now close to the position at which d 1 = d 2, and white light fringes can be observed. Turn off the sodium lamp and use instead a tungsten filament (white light source. Careful adjustment of d 1 by means of the micrometer will bring the white light fringes into view; they have a characteristic appearance, being coloured towards the edges of the pattern, but dark near the centre. If the adjustment of the micrometer in one direction fails to reveal the fringes, turn it carefully in the opposite sense. It may prove necessary to repeat the adjustment, to the position of straight, parallel fringes, using the sodium lamp, and then switch to the white light source once more. When you obtain the white light fringe pattern, show it to the laboratory demonstrator. Sketch the fringe pattern in your lab book, noting the colour gradation. At the centre of this fringe pattern, d 1 and d 2 are equal to within one wavelength. Measure d 1 approximately, using a ruler, and hence estimate the precision to which d 1 = d 2, assuming that you can locate the centre of the white light fringe pattern to within one wavelength (taken to be at the centre of the visible spectrum, i.e. at approximately 600 nm.

7 6. To conclude Make sure your lab note book contains enough guidance for the marker to be able to follow the presentation of your results. State in words what each numerical result corresponds to and say how it was obtained. For your extended report Give some background to the Michelson-Morley experiment. List a few other applications of precision interferometry.