 Interference by division of wave front, by M. K. Srivastav, Department of Physics, University of Technology, Erurki, Uttarakhand. In the last three lectures on interference, we have seen that the principle of superposition leads to an interference pattern when we consider light of the same frequency. However, a very important difference exists between monochromatic light source and the source of sound. The sound source emits a sinusoidal wave of infinite longitudinal extent. On the other hand, the monochromatic light source really consists of a very large number of independent atomic sources which emit for a finite period of time. The atoms also keep moving randomly and colliding with each other. The result is that the emission consists of wavelengths of finite length and random initial phase. Two interfering sources must therefore be obtained from a single source so that this random variation of phase gets cancelled and the phase difference between the two sources so obtained is steady and does not depend on time. It depends only on the path difference between the two beams when they reach the screen where the pattern is observed. It leads to a steady and stationary pattern. Such sources as you know are called coherent sources. You have seen that there are two basic procedures to obtain such a pair of sources. These are division of wave front and division of amplitude. We have gone through various practical setups like Young's double hole arrangement, Lloyd's mirror, personal by prism, personal double mirror, parallel sided film, wedge shaped film, Newton's rings which is a very simple laboratory setup and Michelson's interferometer which is a precision measuring device. In the present and the next two lectures, we shall go through various examples and problems and make comments to help fix up your ideas and illustrate the principles. Let us begin with division of wave front in this lecture. Consider Young's double hole arrangement. It is S1 and S2 are the two sources which are illuminated by the basic source S. They form a coherent pair. The interference is observed on the screen. We shall consider really how the intensity varies as we move along the screen parallel to line of sources. The illumination at any point P depends on the path difference S2 P minus S1 P of the distances from the sources S1 and S2. For a fixed value of the path difference, the locus of the point P on the screen is a hyperbola. You see hyperbola is the locus of a point. The difference of whose distances from two fixed points is fixed. Those two fixed points here are S1 and S2 and the fixed difference is the path difference. This gives you an idea of the shape of the fringes. Hyperbolic for a small values of X compared to the distance D of the screen from the line of sources, the loci are state lines parallel to the X axis. Thus, we observe approximately the state line fringes on the screen. Note that the fringes are state lines even though the sources S1 and S2 are point sources. If we had instead of these point sources, we would have obtained again the state line fringes naturally with increased intensities. Now, these fringes are non-localized. They can be photographed by just placing a photographic film where we have got the screen. The dark and bright fringes are equally spaced. This is the basic characteristic of any interference pattern. And the distance between any two consecutive bright or consecutive dark fringes that is the fringe width is given by lambda capital D upon small d. Lambda is the wavelength of the light. Capital D is the distance of the screen from the line of sources and small d is the distance between the pair of sources. We have assumed in this derivation that small d is very very small compared to the capital D which is usually always the case. If we are interested in the angular width, then the angular width of the fringes measured from the midpoint of the silits is naturally given by lambda upon small d. At the nth bright fringe, counted from the central one where the path difference is 0, the path difference for the nth bright fringe is n lambda. At a point distance y from the central point, that is the central fringe, 0th order fringe, the path difference and the phase difference are y small d upon capital D and the phase difference 2 pi by lambda times y small d upon capital D. If a convex lens is placed immediately after the silits, the fringe width on the screen placed at the focal plane of the lens is given by lambda f upon d, d has been replaced by the focal length. Now, then the interesting thing, the energy distribution on the screen, this is given by I as a function of y, the distance of the point from the central point, central fringe is equal to 4 I naught cos square of pi lambda d upon lambda capital D. I 0 is the intensity of the individual sources which has been taken to be equal to each other. You see the density on the screen that varies from 0 at a minimum, naturally where cos factor is 0 or the maximum value is 4 times I naught at a maximum. The average intensity is just equal to twice I naught, which is simply the sum of the intensities. The main idea here is just to show that there is no loss of energy, the interference simply leads to a redistribution of the energy on the screen. Let us consider another arrangement, the interference pattern produced by two point sources S 1 and S 2 on a plane P P prime, which is perpendicular to the line joining S 1 and S 2, this is typical. S 1 and S 2, this line, line of sources is perpendicular to the screen. Capital D is the distance of the screen from one of the sources S 2 and for the other source, the distance is small d plus capital D. We are interested on the pattern on the screen P P. In order to determine the shape of the interference pattern, let us find out the locus of the points P on the screen for a fixed part difference delta equal to S 1 P minus S 2 P. The x axis is taken to be perpendicular to the plane of the figure, y axis is along the line P P and the z axis is along the line line of sources S 1 S 2 with the screen placed at the origin, which is at z is equal to 0. The coordinates of the point P are x, y, 0, z is 0 and those of S 1 and S 2 are 0, x and y are both 0. So, 0, 0 minus capital D minus a small d and for S 2, 0, 0 and minus capital D respectively. Now, for the path difference, we consider the distances capital D plus a small d square plus x square plus y square whole under root. This is one of the distance minus capital D square plus x square plus y square whole under root. This is the other distance and the difference of these two is equal to the path difference, which is capital delta. Now, we transfer one term to the right hand side and then a square. This is the usual method of solving such equations. Some of the terms cancel out. The result is we get x square plus y square is equal to whole of this expression depending on a small d capital D and capital delta. Some constant value and this shows that differences will be circular. That is the interesting part. The figure next we have that shows them for two different values of b. The first one when the d is 20 centimeters and the second figure where the rings are finer is when the distance is 10 centimeters. Now, in the plane of the figure, if we put x is equal to 0 and consider the variation with respect to y for any given value of delta, then the expression is given by y is equal to all this complicated expression. But the interesting thing is when capital D is very large compared to small d, it becomes y equal to some value like here, which is given d upon capital delta and the whole square root of the product d minus delta into d plus delta. If there is an mth order ring at the point p, which means delta is equal to m lambda, which is the path difference, then the ym for that is given by this expression d upon m lambda multiplied by a square root of b square minus m square lambda square. One thing is interesting here and that is as we move away from the point o, which was the point in the line of sources on the screen, the order of differences decreases. Highest order is at the point o and it is given by d by lambda. You see in the basic Young's two-hole arrangement at the central point, the path difference was 0 and as we move away from the central point, the path difference goes on increasing. One lambda for the first bright fringe, two lambda for the second bright fringe and like that. Here the highest order is at the central point. The path difference here is not 0. It cannot be 0 and as we move away from it, the path difference decreases. The first bright fringe will be one order less, next bright fringe two order less and like this. Now, let us consider what should be the width of the sources and how does it affect the pattern. You see in all the arrangements like Lloyd's mirror or the Fresnel bi prism or the Fresnel double mirror, which depend on the division of wave front. The sources S1 and S2, they must be point sources or they should be thin slits. That is very important. This is the situation we like for a good interference pattern. If they are wide, not the point sources, not the thin slits, the fringe system becomes blurred. In the case of Fresnel bi prism and Fresnel double mirror, the two virtual images are similarly placed. We shall see what is the meaning of similarly placed and the result is that the various coherent pairs of point sources they are in, in the wide sources, they are displaced with respect to each other. So, this is the arrangement one has in the Fresnel bi prism. S is the basic source. When the light passes through the bi prism, the usual reflection causes the creation of the two sources, S1 virtual sources, S1 and S2, but S1 and S2 are similarly placed with respect to S. This is the situation. If S is wide, we have shown it say points 1 to 7. 1 to 7 does not mean anything. Just they are wide sources and S1 and S2 have been shown. Now, you see corresponding to the point 1, the two virtual sources are 1 prime and 1 double prime. Similarly, corresponding to the point 2, the two virtual sources are 2 prime and 2 double prime. You see the distance between the corresponding pairs is same. So, the small d in the expression remains the same, but the midpoint of the two sources that is getting shifted for 1-1 pair it is 1, for 2-2 pair again 2, 3-3 pair it is 3. So, it is shifted. The fringe there is resulting from them are also similarly displaced making the whole pattern less distinct. In the case of Lloyd's mirror, the original source and its virtual image are symmetric with respect to the line of the mirror and the result is that individual coherent pairs here have varying distance between them. You see that is the arrangement in the Lloyd's mirror. S1 is the basic source. The light directly reaches to the point A on the screen. Then light reaching via reflection as if coming from the virtual source S2 and these two superpose and cause interference. Let us consider S1 and S2. There should be point sources or the silits parallel to the plane of the mirror, but if they are wide as shown here S1 is the basic source. S2 is the virtual image corresponding to the point 1 prime. You have the point 1 double prime corresponding to the point 2 prime. You have 2 double prime, 3 prime, 3 double prime. You see the midpoint of the pair is same for 1-1 pair or 2-2 pair or 3-3 pair, but the distance between the sources keeps changing. 1-1 pair or the nearest then the 2-2 pair then the 3-3 pair then the 4-4 pair. So here the distance between the sources is varying. The resulting fringes produced by them have varying fringe width in the overall pattern, the central fringe and the first few fringes are okay and then the pattern becomes blurred. Let us consider what happens if white light is used in place of a monochromatic light, white light fringes. If white light is used the wavelengths therein vary from about 4000 x times in the violet region to about 7000 for the red side. The central fringe now here in this case will be white because all the wavelengths will constructively interference here. You see the path difference for the central fringe is 0, 0 for all wavelengths. No problem. As the fringe width depends on the wavelength following the white central fringe we will have colored fringes, but only few of them. The fringes will soon disappear because at points far away from the central fringe there will be so many wavelengths in the white light which will constructively interfere that we will observe uniform white illumination. The white light fringes are sometimes though they are very useful as the central fringe can be identified being distinct from all other fringes. Only the central fringe is white all other are colored. If the light is monochromatic the whole patterns looks alike. You cannot identify if a particular fringe is the central fringe or not. Let us consider another interesting thing a displacement of fringes. If a plate of thickness t and refractive index n is introduced in the path of light from one of the sources it introduces an additional optical path given by n minus 1 times t. It is shown here s 1 and s 2 are those two sources this is small plate is introduced in the path of one of them. So, we consider the light reaching the point p from the source s 1 and s 2 is small d as before the distance between the two sources. We want to see how the fringe pattern gets shifted gets changed originally the central fringe is at the point o and now it has got shifted to the point o prime. The result is that the fringes on the screen get shifted as we sign the figure by an amount y which is given by n minus 1 times t capital D upon small d. Now, if monochromatic light is being used this shift will not be observed as all the fringes look alike. So, whether you put the plate or you remove it you would not find any change in the pattern you would not be able to observe the shift. However, the white light is being used only the central fringe zero order is white and is then different from other fringes which are colored. The shift of the central fringe can be observed and measured this method can thus be used to measure thickness of thin transparent sheets. And the interesting thing is the contrast in the fringe pattern the visibility factor. So, that it is defined as the difference of the maximum to minus minimum intensity divided by their sum multiplied by 100 to expresses as a percentage. Now, the maximum intensity is naturally the depends on the sum of the amplitudes which is squared. So, if i 1 and i 2 are the intensities of the two sources the square root of i 1 is proportional to the amplitude. So, the amplitude from one of the sources plus the amplitude from the other source whole thing is squared is proportional to the maximum intensity. Similarly, for the minimum when they are in opposite phase square root of i 1 is the amplitude of 1 square root of i 2 is the amplitude of the other whole thing is squared proportional to the intensity at a minimum where the two sources I mean their disturbance reaches in opposite phase. So, we have got this expression these two factors difference between them and some of these factors multiplied by 100. Now, naturally you can see here if the two sources are of equal intensity i 1 is equal to i 2 you will find that the difference factor will be 0 and then in that case the visibility will be 100 percent the pattern will be very bright crisp sharp this is the ideal situation. But if for example, let us take a situation where i 2 is only one tenth of i 1. Now, again we have calculated the visibility factor 1 plus square root of 0.1 whole square that is for the maximum and 1 minus square root of 0.1 whole square that is for the minimum. So, i maximum minus i minimum divided by i maximum plus i minimum in this case multiplied by 100 makes it only 57 percent. So, the contrast has fallen quite a bit. I think this is all what we plan to do in this lecture. So, we come to the end of this. Thank you.