`color{blue} ✍️`If we look clearly at the shadow cast by an opaque object, close to the region of geometrical shadow, there are alternate dark and bright regions just like in interference.
`color{blue} ✍️`This happens due to the phenomenon of diffraction. Diffraction is a general characteristic exhibited by all types of waves, be it sound waves, light waves, water waves or matter waves.
`color{blue} ✍️`Since the wavelength of light is much smaller than the dimensions of most obstacles; we do not encounter diffraction effects of light in everyday observations.
`color{blue} ✍️`However, the finite resolution of our eye or of optical instruments such as telescopes or microscopes is limited due to the phenomenon of diffraction. Indeed the colours that you see when a CD is viewed is due to diffraction effects. We will now discuss the phenomenon of diffraction.
`color{brown} {bbul{"The single slit"}}`
`color{blue} ✍️`In the discussion of Young’s experiment, we stated that a single narrow slit acts as a new source from which light spreads out. Even before Young, early experimenters – including Newton – had noticed that light spreads out from narrow holes and slits.
`color{blue} ✍️`It seems to turn around corners and enter regions where we would expect a shadow. These effects, known as diffraction, can only be properly understood using wave ideas. After all, you are hardly surprised to hear sound waves from someone talking around a corner.
`color{blue} ✍️`When the double slit in Young’s experiment is replaced by a single narrow slit (illuminated by a monochromatic source), a broad pattern with a central bright region is seen. On both sides, there are alternate dark and bright regions, the intensity becoming weaker away from the centre (Fig. 10.16).
`color{blue} ✍️`To understand this, go to Fig. 10.15, which shows a parallel beam of light falling normally on a single slit LN of width a. The diffracted light goes on to meet a screen. The midpoint of the slit is M. A straight line through M perpendicular to the slit plane meets the screen at C.
`color{blue} ✍️`We want the intensity at any point P on the screen. As before, straight lines joining P to the different points L,M,N, etc., can be treated as parallel, making an angle `θ` with the normal MC. The basic idea is to divide the slit into much smaller parts, and add their contributions at P with the proper phase differences.
`color{blue} ✍️`We are treating different parts of the wavefront at the slit as secondary sources. Because the incoming wavefront is parallel to the plane of the slit, these sources are in phase. The path difference NP – LP between the two edges of the slit can be calculated exactly as for Young’s experiment. From Fig. 10.15,
`color{purple}{NP – LP = NQ}`
`color{purple}{= a sin θ}`
`color {blue}{≈ aθ}`
.............(10.21)
`color{blue} ✍️`Similarly, if two points `M_1` and `M_2` in the slit plane are separated by y, the path difference `M_2P – M_1P ≈ yθ.`
`color{blue} ✍️`We now have to sum up equal, coherent contributions from a large number of sources, each with a different phase. This calculation was made by Fresnel using integral calculus, so we omit it here. The main features of the diffraction pattern can be understood by simple arguments.
`color{blue} ✍️`At the central point C on the screen, the angle θ is zero. All path differences are zero and hence all the parts of the slit contribute in phase. This gives maximum intensity at C. Experimental observation shown in Fig. 10.15 indicates that the intensity has a central maximum at `θ = 0` and other secondary maxima at `θ l (n+1/2) λ//a,` and has minima (zero intensity) at `θ approx nλ//a,` `n = ±1, ±2, ±3, ....` It is easy to see why it has minima at these values of angle. Consider first the angle θ where the path difference aθ is λ. Then,
`color {blue}{θ ≈ λ /a}`
............(10.22)
`color{blue} ✍️`, divide the slit into two equal halves LM and MN each of size a/2. For every point `M_1` in LM, there is a point `M_2` in MN such that `M_1M_2 = a//2`. The path difference between `M_1` and `M_2` at P = `M_2P – M_1P` `= θ a//2 = λ//2` for the angle chosen.
`color{blue} ✍️`This means that the contributions from `M_1` and `M_2` are 180º out of phase and cancel in the direction θ = λ/a. Contributions from the two halves of the slit LM and MN, therefore, cancel each other. Equation (10.22) gives the angle at which the intensity falls to zero.
`color{blue} ✍️`One can similarly show that the intensity is zero for `θ = nλ/a,` with n being any integer (except zero!). Notice that the angular size of the central maximum increases when the slit width a decreases.
`color{blue} ✍️`It is also easy to see why there are maxima at `θ = (n + 1/2) λ//a` and why they go on becoming weaker and weaker with increasing n. Consider an angle `θ = 3λ//2a` which is midway between two of the dark fringes. Divide the slit into three equal parts. If we take the first two thirds of the slit, the path difference between the two ends would be
`color {blue}{2/3 a xx theta = (2a)/3 xx (3λ)/(2a)=λ}`
............(10.23)
`color{blue} ✍️`The first two-thirds of the slit can therefore be divided into two halves which have a `λ/2` path difference.
The contributions of these two halves cancel in the same manner as described earlier. Only the remaining one-third of the slit contributes to the intensity at a point between the two minima.
`color{blue} ✍️`Clearly, this will be much weaker than the central maximum (where the entire slit contributes in phase). One can similarly show that there are maxima at `color{purple}{(n + 1/2) θ//a}` with `color{purple}{n = 2, 3,}` etc.
`color{blue} ✍️`These become weaker with increasing n, since only one-fifth, one-seventh, etc., of the slit contributes in these cases. The photograph and intensity pattern corresponding to it is shown in Fig. 10.16.
`color{blue} ✍️`There has been prolonged discussion about difference between intereference and diffraction among scientists since the discovery of these phenomena. In this context, it is interesting to note what Richard Feynman* has said in his famous Feynman Lectures on Physics:
`color{blue} ✍️`No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them.
`color{blue} ✍️`The best we can do is, roughly speaking, is to say that when there are only a few sources, say two interfering sources, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.
`color{blue} ✍️`In the double-slit experiment, we must note that the pattern on the screen is actually a superposition of single-slit diffraction from each slit or hole, and the double-slit interference pattern. This is shown in Fig. 10.17.
`color{blue} ✍️` It shows a broader diffraction peak in which there appear several fringes of smaller width due to double-slit interference. The number of interference fringes occuring in the broad diffraction peak depends on the ratio `d/a,` that is the ratio of the distance between the two slits to the width of a slit. In the limit of a becoming very small, the diffraction pattern will become very flat and we will obsrve the two-slit interference pattern [see Fig. 10.13(b)].
`color{blue} ✍️`In the double-slit interference experiment of Fig. 10.12, what happens if we close one slit? You will see that it now amounts to a single slit. But you will have to take care of some shift in the pattern.
`color{blue} ✍️`We now have a source at S, and only one hole (or slit) `S_1` or `S_2.` This will produce a single- slit diffraction pattern on the screen.
`color{blue} ✍️`The centre of the central bright fringe will appear at a point which lies on the straight line `SS_1` or `SS_2,` as the case may be. We now compare and contrast the interference pattern with that seen for a coherently illuminated single slit (usually called the single slit diffraction pattern).
`color{blue} {(i)}` The interference pattern has a number of equally spaced bright and dark bands. The diffraction pattern has a central bright maximum which is twice as wide as the other maxima. The intensity falls as we go to successive maxima away from the centre, on either side.
`color{blue} {(ii)}` We calculate the interference pattern by superposing two waves originating from the two narrow slits. The diffraction pattern is a superposition of a continuous family of waves originating from each point on a single slit.
`color{blue} {(iii)}` For a single slit of width a, the first null of the interference pattern occurs at an angle of `λ/a.` At the same angle of `λ/a,` we get a maximum (not a null) for two narrow slits separated by a distance a.
`color{blue} ✍️`One must understand that both d and a have to be quite small, to be able to observe good interference and diffraction patterns. For example, the separation d between the two slits must be of the order of a milimetre or so.
`color{blue} ✍️`The width a of each slit must be even smaller, of the order of 0.1 or 0.2 mm. In our discussion of Young’s experiment and the single-slit diffraction, we have assumed that the screen on which the fringes are formed is at a large distance.
`color{blue} ✍️`The two or more paths from the slits to the screen were treated as parallel. This situation also occurs when we place a converging lens after the slits and place the screen at the focus. Parallel paths from the slit are combined at a single point on the screen.
`color{brown} {"Note"}` that the lens does not introduce any extra path differences in a parallel beam. This arrangement is often used since it gives more intensity than placing the screen far away. If f is the focal length of the lens, then we can easily work out the size of the central bright maximum. In terms of angles, the separation of the central maximum from the first null of the diffraction pattern is `λ//a`. Hence, the size on the screen will be `fλ/a.`
`color{blue} ✍️`If we look clearly at the shadow cast by an opaque object, close to the region of geometrical shadow, there are alternate dark and bright regions just like in interference.
`color{blue} ✍️`This happens due to the phenomenon of diffraction. Diffraction is a general characteristic exhibited by all types of waves, be it sound waves, light waves, water waves or matter waves.
`color{blue} ✍️`Since the wavelength of light is much smaller than the dimensions of most obstacles; we do not encounter diffraction effects of light in everyday observations.
`color{blue} ✍️`However, the finite resolution of our eye or of optical instruments such as telescopes or microscopes is limited due to the phenomenon of diffraction. Indeed the colours that you see when a CD is viewed is due to diffraction effects. We will now discuss the phenomenon of diffraction.
`color{brown} {bbul{"The single slit"}}`
`color{blue} ✍️`In the discussion of Young’s experiment, we stated that a single narrow slit acts as a new source from which light spreads out. Even before Young, early experimenters – including Newton – had noticed that light spreads out from narrow holes and slits.
`color{blue} ✍️`It seems to turn around corners and enter regions where we would expect a shadow. These effects, known as diffraction, can only be properly understood using wave ideas. After all, you are hardly surprised to hear sound waves from someone talking around a corner.
`color{blue} ✍️`When the double slit in Young’s experiment is replaced by a single narrow slit (illuminated by a monochromatic source), a broad pattern with a central bright region is seen. On both sides, there are alternate dark and bright regions, the intensity becoming weaker away from the centre (Fig. 10.16).
`color{blue} ✍️`To understand this, go to Fig. 10.15, which shows a parallel beam of light falling normally on a single slit LN of width a. The diffracted light goes on to meet a screen. The midpoint of the slit is M. A straight line through M perpendicular to the slit plane meets the screen at C.
`color{blue} ✍️`We want the intensity at any point P on the screen. As before, straight lines joining P to the different points L,M,N, etc., can be treated as parallel, making an angle `θ` with the normal MC. The basic idea is to divide the slit into much smaller parts, and add their contributions at P with the proper phase differences.
`color{blue} ✍️`We are treating different parts of the wavefront at the slit as secondary sources. Because the incoming wavefront is parallel to the plane of the slit, these sources are in phase. The path difference NP – LP between the two edges of the slit can be calculated exactly as for Young’s experiment. From Fig. 10.15,
`color{purple}{NP – LP = NQ}`
`color{purple}{= a sin θ}`
`color {blue}{≈ aθ}`
.............(10.21)
`color{blue} ✍️`Similarly, if two points `M_1` and `M_2` in the slit plane are separated by y, the path difference `M_2P – M_1P ≈ yθ.`
`color{blue} ✍️`We now have to sum up equal, coherent contributions from a large number of sources, each with a different phase. This calculation was made by Fresnel using integral calculus, so we omit it here. The main features of the diffraction pattern can be understood by simple arguments.
`color{blue} ✍️`At the central point C on the screen, the angle θ is zero. All path differences are zero and hence all the parts of the slit contribute in phase. This gives maximum intensity at C. Experimental observation shown in Fig. 10.15 indicates that the intensity has a central maximum at `θ = 0` and other secondary maxima at `θ l (n+1/2) λ//a,` and has minima (zero intensity) at `θ approx nλ//a,` `n = ±1, ±2, ±3, ....` It is easy to see why it has minima at these values of angle. Consider first the angle θ where the path difference aθ is λ. Then,
`color {blue}{θ ≈ λ /a}`
............(10.22)
`color{blue} ✍️`, divide the slit into two equal halves LM and MN each of size a/2. For every point `M_1` in LM, there is a point `M_2` in MN such that `M_1M_2 = a//2`. The path difference between `M_1` and `M_2` at P = `M_2P – M_1P` `= θ a//2 = λ//2` for the angle chosen.
`color{blue} ✍️`This means that the contributions from `M_1` and `M_2` are 180º out of phase and cancel in the direction θ = λ/a. Contributions from the two halves of the slit LM and MN, therefore, cancel each other. Equation (10.22) gives the angle at which the intensity falls to zero.
`color{blue} ✍️`One can similarly show that the intensity is zero for `θ = nλ/a,` with n being any integer (except zero!). Notice that the angular size of the central maximum increases when the slit width a decreases.
`color{blue} ✍️`It is also easy to see why there are maxima at `θ = (n + 1/2) λ//a` and why they go on becoming weaker and weaker with increasing n. Consider an angle `θ = 3λ//2a` which is midway between two of the dark fringes. Divide the slit into three equal parts. If we take the first two thirds of the slit, the path difference between the two ends would be
`color {blue}{2/3 a xx theta = (2a)/3 xx (3λ)/(2a)=λ}`
............(10.23)
`color{blue} ✍️`The first two-thirds of the slit can therefore be divided into two halves which have a `λ/2` path difference.
The contributions of these two halves cancel in the same manner as described earlier. Only the remaining one-third of the slit contributes to the intensity at a point between the two minima.
`color{blue} ✍️`Clearly, this will be much weaker than the central maximum (where the entire slit contributes in phase). One can similarly show that there are maxima at `color{purple}{(n + 1/2) θ//a}` with `color{purple}{n = 2, 3,}` etc.
`color{blue} ✍️`These become weaker with increasing n, since only one-fifth, one-seventh, etc., of the slit contributes in these cases. The photograph and intensity pattern corresponding to it is shown in Fig. 10.16.
`color{blue} ✍️`There has been prolonged discussion about difference between intereference and diffraction among scientists since the discovery of these phenomena. In this context, it is interesting to note what Richard Feynman* has said in his famous Feynman Lectures on Physics:
`color{blue} ✍️`No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them.
`color{blue} ✍️`The best we can do is, roughly speaking, is to say that when there are only a few sources, say two interfering sources, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.
`color{blue} ✍️`In the double-slit experiment, we must note that the pattern on the screen is actually a superposition of single-slit diffraction from each slit or hole, and the double-slit interference pattern. This is shown in Fig. 10.17.
`color{blue} ✍️` It shows a broader diffraction peak in which there appear several fringes of smaller width due to double-slit interference. The number of interference fringes occuring in the broad diffraction peak depends on the ratio `d/a,` that is the ratio of the distance between the two slits to the width of a slit. In the limit of a becoming very small, the diffraction pattern will become very flat and we will obsrve the two-slit interference pattern [see Fig. 10.13(b)].
`color{blue} ✍️`In the double-slit interference experiment of Fig. 10.12, what happens if we close one slit? You will see that it now amounts to a single slit. But you will have to take care of some shift in the pattern.
`color{blue} ✍️`We now have a source at S, and only one hole (or slit) `S_1` or `S_2.` This will produce a single- slit diffraction pattern on the screen.
`color{blue} ✍️`The centre of the central bright fringe will appear at a point which lies on the straight line `SS_1` or `SS_2,` as the case may be. We now compare and contrast the interference pattern with that seen for a coherently illuminated single slit (usually called the single slit diffraction pattern).
`color{blue} {(i)}` The interference pattern has a number of equally spaced bright and dark bands. The diffraction pattern has a central bright maximum which is twice as wide as the other maxima. The intensity falls as we go to successive maxima away from the centre, on either side.
`color{blue} {(ii)}` We calculate the interference pattern by superposing two waves originating from the two narrow slits. The diffraction pattern is a superposition of a continuous family of waves originating from each point on a single slit.
`color{blue} {(iii)}` For a single slit of width a, the first null of the interference pattern occurs at an angle of `λ/a.` At the same angle of `λ/a,` we get a maximum (not a null) for two narrow slits separated by a distance a.
`color{blue} ✍️`One must understand that both d and a have to be quite small, to be able to observe good interference and diffraction patterns. For example, the separation d between the two slits must be of the order of a milimetre or so.
`color{blue} ✍️`The width a of each slit must be even smaller, of the order of 0.1 or 0.2 mm. In our discussion of Young’s experiment and the single-slit diffraction, we have assumed that the screen on which the fringes are formed is at a large distance.
`color{blue} ✍️`The two or more paths from the slits to the screen were treated as parallel. This situation also occurs when we place a converging lens after the slits and place the screen at the focus. Parallel paths from the slit are combined at a single point on the screen.
`color{brown} {"Note"}` that the lens does not introduce any extra path differences in a parallel beam. This arrangement is often used since it gives more intensity than placing the screen far away. If f is the focal length of the lens, then we can easily work out the size of the central bright maximum. In terms of angles, the separation of the central maximum from the first null of the diffraction pattern is `λ//a`. Hence, the size on the screen will be `fλ/a.`