 In previous lectures, I discussed the basic requirement for the observation of interference fringes in the laboratory and discuss some important experimental arrangement to observe the interference fringes in the laboratory. In this lecture, I am going to describe some possible engineering applications of interference phenomena of light. We have already seen that neutral rings experiment can be used to determine the wavelength of monochromatic light, refractive index of liquids and to study the flatness of glass plate. The Michelson interferometer may be used to measure accurately wavelength of light and wavelength difference between the two closely spaced spectral line of certain sources like sodium. In addition to this, it is also used for evaluation of a standard meter in terms of wavelength of light. Similarly, Fabry-Parrot interferometer can be used for similar applications. Here, I am going to describe few other possible engineering applications of interference of light. One important application of interference of thin film is the coating of non-reflecting films. When these films are coated on surfaces of optical elements like lens or prism, the reflectivity of surfaces is very much reduced. Therefore, the loss of light by reflection at various surfaces of a system of lenses or prisms is appreciably reduced. The main idea behind the non-reflecting film is the destructive interference between the waves reflected from the air film interface and film glass interface by properly adjusting the thickness and refractive index of the film material. Suppose, the refractive index of the film material mu r is less than the refractive index of glass mu g and greater than the refractive index of air mu a. In this case, if a beam of light of wavelength lambda is incident at air film interface, then phase change of pi occurs on reflections at both air film and film glass interfaces. Therefore, if the film thickness t is adjusted such that the condition to mu f t is equal to lambda by 2 is satisfied for normal incidence, then the condition for destructive interference is satisfied for the reflected light. Thus, if film thickness t is lambda upon 4 mu f, the light of wavelength lambda is not reflected. As we have discussed earlier, there is no loss of energy in interference phenomena. Therefore, there is merely a redistribution of energy. So, in this case, the energy mostly appears in the transmitted beam. In case of white light, lambda is generally chosen near the middle of the visible spectrum that is lambda around 5000 angstrom. At angles other than normal incidence, the part difference will change because of the factor cosine theta. However, since cos theta does not change very much near 0 degree, so reflection remains quite low over a fair range of small angle about the normal. Furthermore, the film is non-reflecting only for a particular value of lambda. For a polychromatic light, the film's non-reflecting property will be falling off when wavelength is less or greater than 5000 angstrom. However, there is not much increase in reflectivity when we go towards red and violet ends of the visible spectrum. Here, it should be noted that although the destructive interference will be observed for film thickness lambda upon 4 mu f or 3 lambda upon 4 mu f or 5 lambda upon 4 mu f, the lowest thickness lambda upon 4 mu f is preferred. For this thickness, reflectivity is small for entire range of visible spectrum. Above, we have seen that a film of thickness lambda upon 4 mu f where lambda is the wavelength of light and mu f is the film refractive index which lies between the refractive indices of the two surrounding media acts like an anti-reflecting layer. This happens due to destructive interference occurring between the waves reflected from the top and the bottom interfaces. However, if the refractive index of the film is smaller or greater than both the surrounding media, then in such a case in addition to the phase difference due to the additional part travelled by the wave reflected from the lower interface, there would also be an extra phase difference of pi between the two reflected waves. In this case, a film of thickness lambda upon 4 mu f therefore, would increase the reflectivity rather than reducing it. Now, if you consider a medium consisting of alternate layers of high and low refractive indices of mu naught plus delta mu and mu naught minus delta mu of equal thickness T, such medium is called a periodic medium and the special period of variation of the refractive index is 2 T. Suppose, delta mu is much smaller than mu naught and we choose the thickness of each layer to be T equal to lambda upon 4 mu naught which we can take approximately equal to lambda upon 4 mu naught plus delta mu or approximately equal to lambda upon 4 mu naught minus delta mu. In this case, the reflection arising out of individual reflections from the various interfaces would all be in phase and should result in a strong reflection. Thus, for a strong reflection at a chosen wavelength lambda b, the period of the refractive index variation should be 2 T is equal to lambda b upon 2 mu naught. This condition is similar to the Bragg's diffraction of x-rays from various atomic layer for normal incidence and therefore, is referred to as the Bragg's condition. The quantity lambda b is often referred to as the Bragg wavelength. As we move away from the wavelength lambda b, the reflectivity of the periodic medium falls off sharply. We can obtain an approximate expression for the wavelength deviation delta lambda from lambda b which will produce a zero reflectivity. In order to do this, we first note that at lambda b, the waves reflected from each of the n individual layer are all in phase leading to a strong reflection. If we move away from lambda b, then the individual waves reflected from the various layers will not be in phase and thus reflectivity reduces. If we choose a wavelength lambda b plus delta lambda such that the reflection from layer 1 and layer n by 2 plus 1 from layer 2 and n plus 2 plus 2 and so on up to reflection from layer n by 2 and n are out of phase, then the reflectivity will be zero. For reflection from each of the top n by 2 layer, there is a reflection from a corresponding lower n by 2 layer which is out of phase. Thus when we move from lambda b to lambda b plus delta lambda, the waves reflected from the first and n by 2 plus 1 th layer should have an additional phase difference of pi. Thus we have 2 pi by lambda b into mu naught into n by n into 2 t by 2 minus 2 pi by lambda b plus delta lambda into mu naught n into 2 t upon 2 is equal to pi. Here the first term on the LHS left hand side is simply the phase difference at lambda b between reflection first and n by 2 plus 1 due to the extra part travelled by the later wave and second term is that at lambda b plus delta lambda. Assuming delta lambda much smaller than lambda b, we have 2 pi divided by lambda b square into mu naught into n into 2 t by 2 into delta lambda is equal to pi and from this we get delta lambda upon lambda b is equal to lambda b upon mu naught n into 2 t which we can write equal to 2 t upon L where L is equal to t into n is the total thickness of the periodic medium. Thus if the incident wave is polychromatic like white light, the reflected light may have a high degree of monochromatic city. This principle is used in white light holographic. Thus the periodic medium discussed above finds wide application in high reflectivity multi layers coating. So, now I am going to describe the working of interference filter. So, working of this interference filter can be understood with the help of Fabry-Parrot interferometer. I have already discussed the construction of this interferometer. If this interferometer is placed in a parallel beam of white light, interference will occur for all the monochromatic component of such light, but this will not manifest itself until the transmitted beam is dispersed by an auxiliary spectroscope. We then observe a series of bright fringes in the spectrum each formed by a wavelength somewhat different from the next. The maxima will occur at wavelength given by lambda is equal to 2 t cos theta divided by m where m is any whole number. If t is a separation of a few millimeter there will be very narrow fringes and high dispersion is necessary in order to separate them. Such fringes are known as channel spectrum or as itself butler bands and have been used for example in calibration of spectroscopes for the infrared and in accurate measurements of wavelength of the absorption lines in the solar spectrum. An application of these fringes having considerable practical importance uses the situation where t is extremely small so that only 1 or 2 maxima occur within the visible range of wavelengths. With white light incident only 1 or 2 narrow bands of wavelength will then be transmitted the rest of the light being reflected. So, a pair of semi transparent metallic film can act as a filter passing nearly monochromatic light. For maxima to be widely separated m must be small number this is attained only by having the reflecting surfaces very close together. If we wish to have the maxima for m is equal to 2 occur at a given wavelength lambda the metal films would have to be a distance lambda apart. The maxima m is equal to 1 will then appear at a wavelength of 2 lambda such minute separation can be attained however with modern techniques of thin film deposition for this a semi transparent metal is first deposited on a glass plate next a thin layer of some dielectric material such as cryolite is deposited on the top of this and then the dielectric layer is in turn coated with another similar film of metal. Finally, another plate of glass is placed over the film for mechanical protection. The completed filter then has the cross section as shown in this figure since the part difference is now in the dielectric of index and the wavelength of maximum transmission for normal incidence are given by lambda is equal to 2 n t upon m. If there are 2 maxima in the visible spectrum one of these can easily be eliminated by using colored glass for protecting cover plate. So, up till now we have discussed some important engineering application of interference now I am going to discuss about the coherence of light wave. So, first I will discuss about the temporal coherence and then I will discuss about the spatial coherence. In all the experimental arrangement for the formation of interference fringes discussed earlier it was assumed that the displacement that is the electric field associated with light wave remained sinusoidal for all values of time. Thus the displacement was assumed to be given by e is equal to a cos k x minus omega t plus 5 this equation predicts that at any position x the displacement is sinusoidal for all time varying from minus infinity to plus infinity. For example, at x is equal to 0 we have e is equal to a cos omega t minus 5 and variation of e bit time forms infinite wave trends as shown in this figure. However, this will result only from perfectly monochromatic source and this corresponds to an idealized situation because the radiation from an ordinary light source consist of finite size wave trend. The average duration of the wave trend is tau c that is the electric field remains sinusoidal for times of the order of tau c. Thus at a given point the electric field at a time t and t plus delta t will in general have a definite phase relationship if delta t is much smaller than tau c and do not have any phase relationship if delta t is much greater than tau c. The time duration tau c is known as the quarence time of the source and the field is set to remain quarence for time tau c. The length l of the wave trend is given by l is equal to c into tau c where c is the velocity of light. This is referred to as quarence length for example, for neon light tau c is of the order of 10 to the power minus 10 second and for the red cadmium line it is of the order of 10 to the power minus 9 second and so the corresponding quarence length will be 3 centimeter and 30 centimeter respectively. The correlation between a wave at a given time and a certain time later is known as the temporal quarence. Again let us consider here the Young's double slit experiment. In this experiment the interference pattern observed around the point p at time t is due to the superposition of waves emanating from slits S 1 and S 2 at time t minus r 1 upon c and t minus r 2 upon c respectively where r 1 and r 2 are the distances S 1 p and S 2 p respectively. If r 2 minus r 1 upon c is much smaller than tau c then the wave arriving at p from S 1 and S 2 will have a definite phase relationship and an interference pattern of good controls will be observed. On the other hand if the power difference r 2 minus r 1 is large enough such that r 2 minus r 1 upon c is much greater than tau c then the waves arriving at p from S 1 and S 2 will have no fixed phase relationship and no interference pattern will be observed. Thus the central fringe for which r 1 is equal to r 2 will in general have a good contrast and as we move towards higher order fringes the contrast of the fringes will gradually become poorer. Similarly, in the Michelson interferometer experiment if d is the distance between mirror m 1 and m 2 prime virtual image of mirror m 2 then for definite phase relationship between the beam reflected from m 1 and m 2 that is 2 d upon c much less than 2 d upon c than tau c and well defined fringes will be observed in this case. On the other hand if 2 d upon c is much greater than tau c then in general there is no definite phase relationship between the two beams and no interference pattern is observed. It may be mentioned that there is no definite distance at which the interference pattern disappears. As the distance increases the contrast of the fringes becomes gradually poorer and eventually the fringe system disappears. The coherence time for a laser beam is usually much larger than in comparison to ordinary light sources. For example, for a helium neon laser coherence time as large as 50 milliseconds have been obtained this would imply a coherence length of 15,000 kilometer. Commercially available helium neon laser have coherence time of the order of 15 nanosecond this means the coherence length is about of the order of 15 meter. Thus, using such as laser beam high contrast interference fringes can be obtained even for a part difference of few meters. In the Michelson interferometer experiment the decrease of degrees in contrast of the fringes can also be interpreted as being due to the fact that the source is not emitting at a single frequency. But over a narrow band of frequency as shown here when the part difference between the two interfering beams is 0 or very small the different wavelength components produce fringes superimposed on one another and the fringe contrast is good. On the other hand when the part difference is increased different wavelength component produce fringes fringe patterns which are is slightly displaced with respect to one another and the fringe contrast becomes poorer. We can equally well say that the poor fringe visibility for a large optical part difference is due to the non monochromaticity of the light source. It can be shown that the temporal coherence tau c of the beam is directly related to the spectral with delta lambda. The relation is given by this equation delta lambda is equal to lambda square upon c tau c or delta lambda is equal to lambda square upon l. Since frequency is related to wavelength by equation nu is equal to c upon lambda we can write spectral width in terms of frequency delta nu is equal to c by l which is approximately equal to 1 over tau c. Thus the frequency spread of a spectral line is inverse of the coherence length. If the coherence time tau c is large then the value of delta nu will be small that is the wave will be more monochromatic. So, now I am going to describe the spatial coherence that is the correlation between the two points at a certain distance away. So, earlier we considered the coherence of the field arriving at a particular point in space from a point source through two different optical paths. Now, let us discuss the coherence property properties of the field associated with the finite dimension of the source. We consider the young double hole experiment with the point source S being equidistant from S 1 and S 2. We assume S to be nearly monochromatic so that it produces interference fringes of good contrast on the screen P p prime. The point O on the screen is such that S 1 O is equal to S 2 O. Clearly the point source S will produce an intensity maxima around the point O. We next consider another similar source S prime at a distance l from S. We assume that the waves from S and S dash have no definite phase relationship. Thus the interference pattern observed on the screen P p prime will be superposition of the intensity distribution of the interference pattern formed due to S and S dash. If the separation l is slowly increased from 0, the contrast of the fringes on the screen P p prime becomes poorer because of the fact that interference pattern produced by S dash is slightly shifted from that produced by S. Clearly if S dash S 2 minus S dash S 1 equal to lambda by 2 the minima of the interference pattern produced by S will fall on the maxima of the fringe pattern produced by S dash and no fringe pattern will be observed. With the help of this figure it can be shown that the fringe pattern will disappear when l is of the order of lambda a upon 2 d. Now, if we have an extended incoherent source whose linear dimension is of the order of lambda a upon d. Then for every point on the source there is a point at a distance of lambda a upon 2 d which produces fringes which are shifted by half a fringe width. Therefore the interference pattern will not be observed. Thus for an extended incoherent source interference fringes of good contrast will be observed only when l is much smaller than lambda a upon d or d is much smaller than lambda a upon l. Equivalently for a given source of width l interference fringes of good contrast will be formed by interference of light from 2 point S 1 and S 2 separated by a distance d much smaller than lambda a upon l. Now, if theta is the angle subtended by the source at the slit then theta approximately lambda upon a and above condition for obtaining the fringe of good contrast takes the form d much less than lambda a upon l or d is much smaller than lambda upon theta. On the other hand if d is approximately lambda by theta the fringes will be of poor contrast. The distance lambda by theta gives the distance over which the beam may be assumed to be specially coherent and is referred to as the lateral coherence width l w.