14 Chapter. Interference and Diffraction

Transcription

1 14 Chapter Interference and Diffraction 14.1 Superposition of Waves Interference Conditions for Light Sources Young s Double-Slit Experiment Example 14.1: Double-Slit Experiment Intensity Distribution Example 14.: Intensity of Three-Slit Interference Diffraction Single-Slit Diffraction Example 14.3: Single-Slit Diffraction Intensity of Single-Slit Diffraction Intensity of Double-Slit Diffraction Patterns Diffraction Grating Summary Appendix: Computing the Total Electric Field Solved Problems Double-Slit Experiment Phase Difference Constructive Interference Intensity in Double-Slit Interference Second-Order Bright Fringe Intensity in Double-Slit Diffraction Conceptual Questions Additional Problems Double-Slit Interference Interference-Diffraction Pattern Three-Slit Interference Intensity of Double-Slit Interference Secondary Maxima

2 Interference-Diffraction Pattern

3 Interference and Diffraction 14.1 Superposition of Waves Consider a region in space where two or more waves pass through at the same time. According to the superposition principle, the net displacement is simply given by the vector or the algebraic sum of the individual displacements. Interference is the combination of two or more waves to form a composite wave, based on such principle. The idea of the superposition principle is illustrated in Figure (a) (b) (c) (d) Figure Superposition of waves. (a) Traveling wave pulses approach each other, (b) constructive interference, (c) destructive interference, (d) waves move apart. Suppose we are given two waves, ψ 1 (x,t) = ψ 1 sin(k 1 x ± ω 1 t + φ 1 ), ψ (x,t) = ψ sin(k x ± ω t + φ ). (14.1.1) The resulting sum of the two waves is ψ (x,t) = ψ 1 sin(k 1 x ± ω 1 t + φ 1 ) + ψ sin(k x ± ω t + φ ). (14.1.) The interference is constructive if the amplitude of ψ ( x, t) is greater than the individual ones (Figure b), and destructive if smaller (Figure c). As an example, consider the superposition of the following two waves at t = : The resultant wave is given by ψ 1 (x) = sin x, ψ (x) = sin(x + π / 4) (14.1.3) 14-

4 ψ (x) = ψ 1 (x) + ψ (x) = sin x + sin(x + π / 4) = (1+ )sin x + cos x, (14.1.4) where we have used the trigonometric identity,sin( α + β ) = sinα cos β + cosα sin β and sin( π / 4) = cos( π / 4) = /. Further use of the identity a asin x + bcos x = a + b sin x + a + b = a + b cosφ sin x + sinφ cos x b cos x a + b (14.1.5) with then leads to = a + b sin(x + φ), 1 b φ = tan a. (14.1.6) ψ (x) = (5 + ) 1/ sin(x + φ), (14.1.7) 1 where φ = tan ( /(1 + )) = 3.4 =.53 rad. The superposition of the waves is depicted in Figure Figure Superposition of two sinusoidal waves. We see that the wave has a maximum amplitude when sin( x + φ) = 1, or x = π / φ. The interference there is constructive. On the other hand, destructive interference occurs at x = π φ =.61 rad, where sin( π ) =. 14-3

5 Interference Conditions for Light Sources Suppose we are considering two light waves. In order to form an interference pattern, the incident light must satisfy two conditions: (i) The light sources must be coherent. This means that the waves from the sources must maintain a constant phase relation. For example, if two waves are phase shifted by φ = π, this phase shift must not change with time. (ii) The light must be monochromatic. This means that the light consists of just one wavelength λ = π / k. Light emitted from an incandescent light bulb is incoherent because the light consists of waves of different wavelengths and they do not maintain a constant phase relationship. Thus, no interference pattern is observed. 14. Young s Double-Slit Experiment Figure Incoherent light source In 181 Thomas Young carried out an experiment in which the wave nature of light was demonstrated. The schematic diagram of the double-slit experiment is shown in Figure Figure Young s double-slit experiment. 14-4

6 A monochromatic light source is incident on the first screen that contains a slit S. The emerging light then arrives at the second screen which has two parallel slits S 1 and S. which serve as the sources of coherent light. The light waves emerging from the two slits then interfere and form an interference pattern on the viewing screen. The bright bands (fringes) correspond to interference maxima, and the dark band interference minima. Figure 14.. illustrates ways in which the waves could combine to interfere constructively or destructively. Figure 14.. Constructive interference (a) at P, and (b) at P 1. (c) Destructive interference at P. The geometry of the double-slit interference is shown in the Figure Figure Double-slit experiment Consider light that falls on the screen at a point P a distance y from the point O that lies on the screen a perpendicular distance L from the double-slit system. A distance d separates the two slits. The light from slit will travel an extra distance δ = r r1 to the 14-5

7 point P than the light from slit 1. This extra distance is called the path difference. From Figure 14..3, we have, using the law of cosines, and d d r1 = r + dr cos π θ = r + dr sin θ, (14..1) d d r = r + dr cos π + θ = r + + dr sin θ. (14..) Subtracting Eq. (14.1) from Eq. (14..) yields r r = ( r + r )( r r ) = dr sinθ. (14..3) In the limit L >> d, where the distance to the screen is much greater than the distance between the slits, the sum of r and 1 r may be approximated by r 1 + r r, and the path difference becomes δ = r r1 d sinθ. (14..4) In this limit, the two rays r 1 and r are essentially treated as parallel (see Figure 14..4). Figure Path difference between the two rays, assuming L >> d. Whether the two waves are in phase or out of phase is determined by the value of δ. Constructive interference occurs when δ is zero or an integer multiple of the wavelength λ, δ m = d sinθ m = mλ, m =, ± 1, ±, ± 3, (constructive interference), (14..5) 14-6

8 where m is called the order number. The zeroth-order ( m = ) maximum corresponds to the central bright fringe at θ =, and the first-order maxima ( m = ± 1) are the bright fringes on either side of the central fringe. When δ is equal to an odd integer multiple of λ /, the waves will be 18 out of phase at P, resulting in destructive interference with a dark fringe on the screen. The condition for destructive interference is given by δ m = d sinθ m = m + 1 λ, m =, ± 1, ±, ± 3, (destructive interference).(14..6) In Figure 14..5, we show how a path difference of δ = λ / ( m = in Eq. (14..6)) results in destructive interference and δ = λ ( m = 1 in Eq. (14..5)) leads to constructive interference. Figure (a) Destructive interference. (b) Constructive interference. To locate the positions of the fringes as measured vertically from the central point O, in addition to L >> d, we shall also assume that the distance between the slits is much greater than the wavelength of the monochromatic light, d >> λ. The conditions imply that the angle θ is very small, so that y sinθ tanθ =. (14..7) L Substituting Eq. (14..7) into the constructive and destructive interference conditions given in Eqs. (14..5) and (14..6), the positions of the bright and dark fringes are, respectively, and y m = m λl, m =, ±1, ±, ± 3, (constructive interference), (14..8) d y m = (m +1/ ) λl, m =, ±1, ±, ± 3, (destructive interference). (14..9) d 14-7

9 Example 14.1: Double-Slit Experiment Suppose in the double-slit arrangement, y =. cm. d =.15mm, L = 1cm, λ = 833nm, and (a) What is the path difference δ for the rays from the two slits arriving at point P? (b) Express this path difference in terms of λ. (c) Does point P correspond to a maximum, a minimum, or an intermediate condition? Solutions: (a) The path difference is given by δ = d sinθ. When L >> y, θ is small and we can make the approximation sinθ tan θ = y / L. Thus, δ d y L = ( m). 1 m 1.m = m. (b) From the answer in part (a), we have hence δ = 3.λ. δ λ = m m 3., (c) Because the path difference is an integer multiple of the wavelength, the intensity at point P is a maximum Intensity Distribution Consider the double-slit experiment shown in Figure Figure Double-slit interference 14-8

10 The total instantaneous electric field E at the point P on the screen is equal to the vector sum of the two sources: E = E1 + E. The magnitude of the Poynting vector flux S is proportional to the square of the total field, S E = ( E + E ) = E + E + E E. (14..1) Taking the time-average of S, the intensity I of the light at P is I = S E + E + E E 1 1. (14..11) The term E1 E represents the correlation between the two light waves. For incoherent light sources, since there is no definite phase relation between E 1 and E, the cross term vanishes, and the intensity due to the incoherent source is simply the sum of the two individual intensities, Iinc = I1 + I. (14..1) For coherent sources, the term E1 E is non-zero. In fact, for constructive interference, E1 = E, and the resulting intensity is I = 4I, (14..13) 1 which is four times greater than the intensity due to a single source. When destructive interference takes place, E1 = E, and E1 E I1, and the total intensity becomes as expected. I = I I + I =, (14..14) Suppose that the waves emerged from the slits are coherent sinusoidal plane waves. Let the electric field components of the wave from slits 1 and at P be given by and E 1 = E sin(ωt), (14..15) E E sin( (14..16) respectively, where the waves from both slits are assumed to have the same amplitude E. For simplicity, we have chosen the point P to be the origin, so that the kx dependence in the wave function is eliminated. Because the wave from slit has traveled an extra 14-9

11 distance δ to P, E has an extra phase constant φ relative to E 1, which traveled from slit 1. For constructive interference, a path difference of δ = λ would correspond to a phase shift of φ = π. This implies that the ratio of path difference to wavelength is equal to the ratio of phase shift to π, Therefore the phase shift is δ φ =. λ π (14..17) π π φ δ d sinθ λ λ (14..18) Assuming that both fields point in the same direction, the total electric field may be obtained by using the superposition principle discussed in Section 13.5, E = E 1 + E = E sin(ωt) + sin(ωt + φ) = E cos(φ / )sin(ωt + φ / ) (14..19) where we have used the trigonometric identity sinα + sinβ = sin α + β cos α β. (14..) The intensity I is proportional to the time-average of the square of the total electric field, I E = 4E cos (φ / ) sin (ωt + φ / ) = E cos (φ / ). (14..1) Therefore the intensity is I = I cos (φ / ), (14..) where I is the maximum intensity on the screen. Substitute Eq. (14..18) into Eq. (14..) yielding I = I cos (πd sinθ / λ). (14..3) The plot of the ratio I / I as a function of d sinθ / λ is shown in Figure

12 Figure Intensity as a function of d sin θ / λ For small angle θ, using Eq. (14..7), the intensity can be rewritten as Example 14.: Intensity of Three-Slit Interference πd I = I cos λl y. (14..4) Suppose a monochromatic coherent source of light passes through three parallel slits, each separated by a distance d from its neighbor, as shown in Figure Figure Three-slit interference. The waves have the same amplitude E and angular frequency ω, but a constant phase difference φ = π d sin θ / λ. (a) Show that the intensity is I = I 9 πd sinθ 1+ cos λ, (14..5) 14-11

13 where I is the maximum intensity associated with the primary maxima. (b) What is the ratio of the intensities of the primary and secondary maxima? Solutions: (a) Let the three waves emerging from the slits be described by the functions E 1 = E sin(ωt), E = E sin(ωt + φ), E 3 = E sin(ωt + φ). (14..6) Using the trigonometric identity sinα + sinβ = cos α β sin α + β, (14..7) the sum of E 1 and E 3 is E 1 + E 3 = E sin(ωt) + sin(ωt + φ) = E cos(φ)sin(ωt + φ). (14..8) The total electric field at the point P on the screen is then the sum E = E 1 + E 3 + E = E sin(ωt + φ)(cos(φ) + 1). (14..9) The intensity is proportional to E, I E = E (cos(φ) + 1) sin (ωt + φ) = 1 E (cos(φ) + 1), (14..3) where we have used sin (ωt + φ) = 1/. The maximum intensity I is attained when cosφ = 1. Thus, I I = ( 1+ cos(φ) ) Substitute φ = π d sin θ / λ into Eq. (14..31). Then the intensity is 9. (14..31) I I sin ( 1 cos ) I π 1 cos d θ = + φ = λ. (14..3) (b) The interference pattern is shown in Figure

14 Figure Intensity pattern for triple slit interference. From the figure, we see that the minimum intensity is zero, and occurs when cosφ = 1/. The condition for primary maxima is cosφ = + 1, which gives I / I = 1. In addition, there are also secondary maxima that are located at cosφ = 1. The condition implies φ = (m + 1)π, which implies that d sinθ / λ = (m + 1 / ), m =,± 1,±,. The intensity ratio is I / I = 1/ Diffraction In addition to interference, waves also exhibit another property diffraction, which is the bending of waves as they pass by some objects or through an aperture. The phenomenon of diffraction can be understood using the Huygens-Fresnel principle that states that Every unobstructed point on a wavefront will act a source of secondary spherical waves. The new wavefront is the surface tangent to all the secondary spherical waves. Figure illustrates the propagation of the wave based on the Huygens-Fresnel principle. Figure Huygens-Fresnel principle. According to the Huygens-Fresnel principle, light waves incident on two slits will spread out and exhibit an interference pattern in the region beyond (Figure 14.4.a). The pattern is called a diffraction pattern. On the other hand, if no bending occurs and the light wave 14-13

15 continue to travel in straight lines, then no diffraction pattern would be observed (Figure 14.4.b). Figure (a) Spreading of light leading to a diffraction pattern. (b) Absence of diffraction pattern if the paths of the light wave are straight lines. We shall restrict ourselves to a special case of diffraction called Fraunhofer diffraction. In this case, all light rays that emerge from the slit are approximately parallel to each other. For a diffraction pattern to appear on the screen, a convex lens is placed between the slit and screen to provide convergence of the light rays Single-Slit Diffraction In our consideration of Young s double-slit experiments, we have assumed the width of the slits to be so small that each slit is a point source. In this section we shall take the width of slit to be finite and see how Fraunhofer diffraction arises. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Figure

16 Figure Diffraction of light by a slit of width a. In diffraction of Fraunhofer type, all rays passing through the slit are approximately parallel. In addition, each portion of the slit will act as a source of light waves according to the Huygens-Fresnel principle. We start by dividing the slit into two halves. At the first minimum, each ray from the upper half will be exactly 18 out of phase with a corresponding ray form the lower half. For example, suppose there are 1 point sources, with the first 5 in the lower half, and 51 to 1 in the upper half. Source 1 and source 51 are separated by a distance a / and are out of phase with a path difference δ = λ /. Similar observation applies to source and source 5, as well as any pair that are a distance a / apart. Thus, the condition for the first minimum is Therefore a λ sinθ =. (14.4.1) λ sinθ =. (14.4.) a Applying the same reasoning to the wavefronts from four equally spaced points a distance a / 4 apart, the path difference would be δ = a sin θ / 4, and the condition for destructive interference is λ sinθ =. (14.4.3) a The argument can be generalized to show that destructive interference will occur when asinθ m = mλ, m = ±1, ±, ± 3, (destructive interference) (14.4.4) Figure illustrates the intensity distribution for a single-slit diffraction. Note that θ = is a maximum

17 Figure Intensity distribution for a single-slit diffraction. By comparing Eq. (14.4.4) with Eq. (14..5) we see that the condition for minima of a single-slit diffraction becomes the condition for maxima of a double-slit interference when the width of a single slit a is replaced by the separation between the two slits d The reason is that in the double-slit case, the slits are taken to be so small that each one is considered as a single light source, and the interference of waves originating within the same slit can be neglected. On the other hand, the minimum condition for the single-slit diffraction is obtained precisely by taking into consideration the interference of waves that originate within the same slit. Example 14.3: Single-Slit Diffraction A monochromatic light with a wavelength of λ = 6 nm passes through a single slit which has a width of.8 mm. (a) What is the distance between the slit and the screen be located if the first minimum in the diffraction pattern is at a distance 1. mm from the center of the screen? (b) Calculate the width of the central maximum. Solutions: (a) The general condition for diffraction destructive interference is sin θ m = m λ a m = ±1, ±, ± 3, For small θ, we employ the approximation sinθ tan θ = y / L, which yields 14-16

18 y λ m. L a The first minimum corresponds to m = 1. If y 1 = 1. mm, then L = ay 1 mλ = ( m)( m) = 1.33 m. 1(6 1 9 m) (b) The width of the central maximum is (see Figure 14.5.) w = y 1 = ( m) =. mm Intensity of Single-Slit Diffraction How do we determine the intensity distribution for the pattern produced by single-slit diffraction? To calculate this, we must find the electric field by adding the field contributions from each point. Let s divide the single slit into N small zones each of width Δ y = a / N, as shown in Figure The convex lens is used to bring parallel light rays to a focal point P on the screen. We shall assume that Δy << λ so that all the light from a given zone is in phase. Two adjacent zones have a relative path difference, δ = Δ y sinθ. The relative phase shift Δ β is given by the ratio Δβ δ Δy sinθ π = =, Δ β = Δ y sinθ. (14.5.1) π λ λ λ Figure Single-slit Fraunhofer diffraction 14-17

19 Suppose the wavefront from the first point (counting from the top) arrives at the point P on the screen with an electric field given by E 1 = E 1 sin(ωt). (14.5.) The electric field from point adjacent to point 1 will have a phase shift Δ β, and the field is E = E 1 sin(ωt + Δβ). (14.5.3) Because each successive component has the same phase shift relative to the previous one, the electric field from point N is The electric field is the sum of each individual contribution, E = E 1 + E + E N = E 1 The total phase shift between the point N and the point 1 is E N = E 1 sin(ωt + (N 1)Δβ). (14.5.4) sin(ωt) + sin(ωt + Δβ) + + sin(ωt + (N 1)Δβ).(14.5.5) π π β = NΔ β = NΔ y sinθ = a sinθ, (14.5.6) λ λ where NΔ y = a. The expression for the total field given in Eq. (14.5.5) can be simplified as follows. Apply the trigonometric relation cos( α β ) cos( α + β ) = sinα sin β : cos( ωt Δβ / ) cos( ωt + Δ β / ) = sinωt sin( Δβ / ) cos( ωt + Δβ / ) cos( ωt + 3 Δ β / ) = sin( ωt + Δβ )sin( Δβ / ) cos( ωt + 3 Δβ / ) cos( ωt + 5 Δ β / ) = sin( ωt + Δβ )sin( Δβ / ) cos[ ωt + ( N 1/ ) Δβ ] cos[ ωt + ( N 3/ ) Δ β ] = sin[ ωt + ( N 1) Δβ ]sin( Δβ / ) Adding all the terms individual equations together in Eq. (14.5.7) yields cos(ωt Δβ / ) cos[ωt (N 1/ )Δβ] = sin(δβ / ) sinωt + sin ωt + Δβ ( ) + + sin( ωt + (N 1)Δβ ) The two terms on the left-hand-side in Eq. (14.5.8) combine to yield (14.5.7). (14.5.8) cos(ωt Δβ / ) cos[ωt (N 1/ )Δβ] = sin(ωt + (N 1)Δβ / )sin(nδβ / ). (14.5.9) 14-18

20 Substitute Eq. (14.5.9) into Eq. (14.5.8), yielding sinωt + sin(ωt + Δβ) + + sin(ωt + (N 1)Δβ) sin[ωt + (N 1)Δβ / ]sin(β / ) =. sin(δβ / ) (14.5.1) [See Appendix for alternative approaches to simplifying Eq. (14.5.5).] We can determine the electric field by substituting Eq. (14.5.1) into Eq. (14.5.5), sin(β / ) E = E 1 sin(ωt + (N 1)Δβ / ). ( ) sin(δβ / ) The intensity I is proportional to the time average of E, E sin(β / ) = E 1 sin(δβ / ) sin (ωt + (N 1)Δβ / ) = 1 E sin(β / ) 1.(14.5.1) sin(δβ / ) We can express the intensity as I I N = sin( β / ), ( ) sin( Δβ / ) where the extra factor N has been inserted to ensure that I corresponds to the intensity at the central maximum β = ( θ = ). In the limit where Δβ, In this limit, the intensity is N sin( Δβ / ) NΔ β / = β /, ( ) sin(β ) I = I β / sin(πasinθ / λ) = I. ( ) πasinθ / λ In Figure 14.6., we plot the ratio of the intensity I / I as a function of β /

21 Figure Intensity of the single-slit Fraunhofer diffraction pattern. From Eq. ( ), we determine that the condition for minimum intensity is π λ asinθ m = mπ, m = ±1, ±, ± 3,. We can then determine the various angles θ m such that the intensity is minimum, sinθ m = m λ, m = ±1, ±, ± 3,. ( ) a In Figure the intensity is plotted as a function of the angle θ, for a = λ and a = λ. We see that as the ratio a / λ grows, the peak becomes narrower, and more light is concentrated in the central peak. In this case, the variation of I with the width a is not shown. Figure Intensity of single-slit diffraction as a function of θ for a = λ and a = λ. 14-

22 14.7 Intensity of Double-Slit Diffraction Patterns In the previous sections, we have seen that the intensities of the single-slit diffraction and the double-slit interference are given by, sin(πasinθ / λ) I = I, single-slit diffraction, πasinθ / λ I = I cos (φ / ) = I cos (πd sinθ / λ), double-slit interference. Suppose we now have two slits, each having a width a, and separated by a distance d. The resulting interference pattern for the double-slit will also include a diffraction pattern due to the individual slit. The intensity of the total pattern is the product of the two functions, sin(πasinθ / λ) I = I cos (πd sinθ / λ). (14.6.1) πasinθ / λ The first and the second terms in the above equation are referred to as the interference factor and the diffraction factor, respectively. While the former yields the interference substructure, the latter acts as an envelope that sets limits on the number of the interference peaks (see Figure ).. Figure Double-slit interference with diffraction. We have seen that the m interference maximum occurs when d sinθ = mλ, while the condition for the first diffraction minimum is a sinθ = λ. A particular interference maximum with order number m may coincide with the first diffraction minimum. The value of m depends only on the spacing between slits and the size of opening. We first form the ratio 14-1

23 d sinθ asinθ = mλ λ (14.6.) We now solve Eq. (14.6.) for the value of m, d m =. (14.6.3) a Since the mth fringe is not seen, the number of fringes on each side of the central fringe is m 1. Thus, the total number of fringes in the central diffraction maximum is 14.8 Diffraction Grating N = ( m + 1) + 1 = m 1. (14.6.4) A diffraction grating consists of a large number N of slits each of width a and separated from the next by a distance d, as shown in Figure Figure Diffraction grating If we assume that the incident light is planar and diffraction spreads the light from each slit over a wide angle so that the light from all the slits will interfere with each other. The relative path difference between each pair of adjacent slits is δ = d sinθ, similar to the calculation we made for the double-slit case. If this path difference is equal to an integral multiple of wavelengths then all the slits will constructively interfere with each other and a bright spot will appear on the screen at an angle θ. Thus, the condition for the principal maxima is given by d sinθ m = mλ, m =, ± 1, ±, ± 3,. (14.7.1) If the wavelength of the light and the location of the m-order maximum are known, the distance d between slits may be readily deduced. 14-

24 (a) (b) Figure Intensity distribution for a diffraction grating for (a) N = 1 and (b) N = 3. The location of the maxima does not depend on the number of slits, N. However, the maxima become sharper and more intense as N is increased. The width of the maxima can be shown to be inversely proportional to N. In Figure 14.8., we show the intensity distribution as a function of β / for diffraction grating with N = 1 and N = 3. Notice that the principal maxima become sharper and narrower as N increases. The observation can be explained as follows: suppose an angle θ (recall that β = π a sin θ / λ ) which initially gives a principal maximum is increased slightly, if there were only two slits, then the two waves will still be nearly in phase and produce maxima which are broad. However, in a diffraction grating with a large number of slits, even though θ may only be slightly deviated from the value that produces a maximum, it will be exactly out of phase with a light wave emerging from some other slit. Because a diffraction grating produces peaks that are much sharper than the two-slit system, it gives a more precise measurement of the wavelength Summary Interference is the combination of two or more waves to form a composite wave based on the superposition principle. In Young s double-slit experiment, where a coherent monochromatic light source with wavelength λ emerges from two slits that are separated by a distance d, the condition for constructive interference is δ m = d sinθ m = mλ, m =, ± 1, ±, ± 3, (constructive interference), where m is called the order number. The condition for destructive interference is 14-3

25 d sinθ m = (m + 1/ )λ, m =, ± 1, ±, ± 3, (destructive interference). The intensity in the double-slit interference pattern is I = I cos (πd sinθ / λ), where I is the maximum intensity on the screen. Huygens-Fresnel principle that states that every unobstructed point on a wavefront will act a source of secondary spherical waves. The new wavefront is the surface tangent to all the secondary spherical waves. Diffraction is the bending of waves as they pass by an object or through an aperture. In a single-slit Fraunhofer diffraction, the condition for destructive interference is sinθ m = m λ, m = ±1,±,± 3, (destructive interference), a where a is the width of the slit. The intensity of the interference pattern is sin(β ) I = I β / sin(πasinθ / λ) = I, πasinθ / λ where β = π asin θ / λ is the phase difference between rays from the upper end and the lower end of the slit, and I is the intensity at θ =. For two slits each having a width a and separated by a distance d, the interference pattern will also include a diffraction pattern due to the single slit, and the intensity is sin(πasinθ / λ) I = I cos ( πd sinθ / λ). πasinθ / λ 14.1 Appendix: Computing the Total Electric Field In Section 14.6 we used a trigonometric relation and obtained the total electric field for a single-slit diffraction. Below we show two alternative approaches of how Eq. (14.5.5) can be simplified. (1) Complex representation: 14-4

26 The total field E may be regarded as a geometric series. We begin with the Euler formula n n n n n+ 1 ix ( ix) ( 1) x ( 1) x e = = + i = cos x + i sin x. (14.9.1) n! ( n)! (n + 1)! n= n= n= In the Euler formula, sin x = Im(e ix ), where the notation Im stands for the imaginary part. We can write the sum of terms that appears on the right-hand-side of Eq. (14.5.5) as sin(ωt) + sin(ωt + Δβ) sin(ωt + (N 1)Δβ) = Im e iωt + e i(ωt + Δβ ) i(ωt +( N 1)Δβ ) e = Im eiωt (1+ e iδβ e i( N 1)Δβ ) in Δβ 1 e e in Δβ / (e in Δβ / e in Δβ / ) = Im e iωt = Im e iωt (14.9.) 1 e iδβ e iδβ / (e iδβ / e iδβ / ) sin(β / ) sin(β / ) = Im e i(ωt +( N 1)Δβ / ) = sin(ωt + (N 1)Δβ / ) sin(δβ / ) sin(δβ / ), where we have used two mathematical results, and where β = NΔβ. N n+ 1 n 1 a a = 1 + a + a + =, a < 1, (14.9.3) 1 a n= (e in Δβ / e in Δβ / ) sin(β / ) = (e iδβ / e iδβ / ) sin(δβ / ), (14.9.4) The total electric field then becomes sin(β / ) E = E 1 sin(ωt + (N 1)Δβ / ), (14.9.5) sin(δβ / ) which is the same as that given in Eq. (14.5.1). () Phasor diagram: Alternatively, we may also use phasor diagrams to obtain the time-independent portion of the resultant field. Before doing this, let s first see how phasor addition works for two wave functions. 14-5

27 Figure Addition of two phasors. Let E 1 = E 1 sin(α) and E = E sin(α + φ), with the sum of the two fields given by E = E 1 + E = E 1 sin(α) + E sin(α + φ) = E sin(α + φ '), (14.9.6) where φ ' is a phase constant that we will determine. Using the phasor approach, the fields E 1 and E are represented by two-dimensional vectors E 1 and E, respectively. The addition of E = E1 + E is shown in Figure The idea of this geometric approach is based on the fact that when adding two vectors, the components of the resultant vector are equal to the sum of the individual components. The vertical component of E is the resultant field E and is the sum of the vertical projections of E 1 and E. If the two fields have the same amplitude E1 = E, the phasor diagram becomes 14-6

28 Figure Addition of two phasors with the same amplitude. From the diagram, we see that η + φ = π and φ ' + η = π. We can then solve for the new phase constant, φ ' = π η = π 1 (π φ) = φ. (14.9.7) From Figure 14.1., we have that cos E / E φ ' =. (14.9.8) 1 Combining Eqs. (14.9.7) and (14.9.8), we can solve for the amplitude E = E 1 cosφ ' = E 1 cos(φ / ). (14.9.9) The resultant field is then E = E 1 cos(φ / )sin(α + φ / ). (14.9.1) Alternatively the sum of the fields can be determined using the trigonometric identity given in Eq. (14..7). Now let s turn to the situation where there are N sources, as in our calculation of the single-slit diffraction intensity in Section By setting t = in Eq. (14.5.5), the timeindependent part of the total field is E = E 1 + E + E N = E 1 sin(δβ) + + sin((n 1)Δβ). ( ) The corresponding phasor diagram is shown in Figure Notice that all the phasors lie on a circular arc of radius R, with each successive phasor differed in phase by Δ β. 14-7

29 Figure Phasor diagram for determining the time-independent portion of E. From the Figure , we determine that Because the length of the arc is NE1 β E sin / =. (14.9.1) R = Rβ, we have where β NE1 β sin( β / ) E = Rsin = sin = E1 β Δβ /, ( ) β = NΔ β. The result is completely consistent with that obtained in Eq. ( ). The intensity is proportional to E, I sin( β / ) sin( β / ) I = I N = Δβ / β /. ( ) which reproduces the result in Eq. ( ) Solved Problems Double-Slit Experiment In Young s double-slit experiment, suppose the separation between the two slits is d =.3 mm. If a beam of 5-nm light strikes the slits and produces an interference pattern. How many maxima will there be in the angular range 45. < θ < 45.? 14-8

30 Solution: On the viewing screen, light intensity is a maximum when the two waves interfere constructively. This occurs when d sinθ m = mλ, m =, ± 1, ±,, (14.1.1) 4 where λ is the wavelength of the light. At θ = 45., d = 3. 1 m, and 9 λ = 5 1 m, we obtain d sinθ m = = (14.1.) λ Thus, there are 45 maxima in the range < θ < 45.. By symmetry, there are also 45 maxima in the range 45. < θ <. Including the one for m = straight ahead, the total number of maxima is N = = 95. (14.1.3) Phase Difference In the double-slit interference experiment shown in Figure 14..3, suppose d =.1 mm and L = 1. m, and the incident light is monochromatic with a wavelength λ = 5 nm. (a) What is the phase difference between the two waves arriving at a point P on the screen when θ =.8? (b) What is the phase difference between the two waves arriving at a point P on the screen when y = 4. mm? (c) If φ = 1/ 3 rad, what is the value of θ? (d) If the path difference is δ = λ / 4, what is the value of θ? Solutions: (a) The phase difference φ between the two wavefronts is given by With θ =.8, we have π π φ = δ d sinθ λ = λ. (14.1.4) π φ = = 7 (5. 1 m) 4 (1. 1 m) sin(.8 ) 17.5 rad. (14.1.5) 14-9

31 (b) When θ is small, we make use of the approximation sinθ tanθ = y / L. Thus, the phase difference becomes π y φ d λ L. (14.1.6) For y = 4. mm, we have (c) For φ = 1/ 3 rad π φ = ( m) ( m), we have Therefore θ = m 1.m = 5.3 rad. (14.1.7) 1 3 rad = π λ d sinθ = π ( m) ( m)sinθ. (14.1.8) (d) For δ = d sin θ = λ / 4, we have 7 1 λ m θ = sin = sin d = 4(1. 1 m). (14.1.9) Constructive Interference Coherent light rays of wavelength λ are illuminated on a pair of slits separated by distance d at an angle θ 1, as shown in Figure If an interference maximum is formed at an angle θ at a screen far from the slits, determine the relationship between θ 1, θ, d and λ. 14-3

32 Figure Solution: The path difference between the two rays is δ = d sinθ d sinθ. (14.1.1) 1 The condition for constructive interference is δ = mλ, where m =, ±1, ±, is the order number. Thus, we have d(sinθ 1 sinθ ) = mλ. ( ) Intensity in Double-Slit Interference Let the intensity on the screen at a point P in a double-slit interference pattern be 6.% of the maximum value. (a) What is the minimum phase difference (in radians) between sources? (b) For the minimum phase difference, what is the corresponding path difference if the wavelength of the light is λ = 5 nm? Solution: (a) The average intensity is given by where I is the maximum light intensity. Thus, I = I cos (φ / ), (14.1.1).6 = cos (φ / ). ( ) 14-31

33 Therefore the phase difference is φ = cos 1 ( I / I ) = cos 1 (.6) = 78.5 = 1.37 rad. ( ) (b) The phase difference φ is related to the path difference δ and the wavelength λ by δ = λφ (5 nm)(1.37 rad) = = 19 nm. ( ) π π Second-Order Bright Fringe A monochromatic light is incident on a single slit of width.8 mm, and a diffraction pattern is formed at a screen that is.8 m away from the slit. The second-order bright fringe is at a distance 1.6 mm from the center of the central maximum. What is the wavelength of the incident light? Solution: The general condition for destructive interference is sin θ m = m λ a y L, ( ) where the small-angle approximation has been made. Thus, the position of the mth order dark fringe measured from the central axis is y m = m λl a. ( ) Let the second bright fringe be located halfway between the second and the third dark fringes. Therefore, y b = 1 ( y + y ) = 1 λl ( + 3) 3 a = 5λL a. ( ) The approximate wavelength of the incident light is then λ a y b 5L = ( m)( m) = m. ( ) 5(.8 m) Intensity in Double-Slit Diffraction Coherent light with a wavelength of λ = 5 nm is sent through two parallel slits, each having a width a =.7 µm. The distance between the centers of the slits is 14-3

34 d =.8 µ m. The screen has a semi-cylindrical shape, with its axis at the midline between the slits. (a) Find the angles of the interference maxima on the screen. Express your answers in terms of the angles the location of the maxima make with respect to the bisector of the line joining the slits. (b) How many bright fringes appear on the screen? (c) For each bright fringe, find the intensity, measured relative to the intensity I, associated with the central maximum. Solutions: (a) The condition for double-slit interference maxima is given by Therefore d sin θ m = mλ m =, ±1, ±,. (14.1.) 1 mλ θm = sin d. (14.1.1) With λ 7 = 5. 1 m and d 6 =.8 1 m, Eq. (14.1.1) becomes θ m = sin 1 m m m = sin 1 (.179 m). (14.1.) The solutions are θ = θ ±1 = sin 1 (±.179) = ±1.3 θ ± = sin 1 (±.357) = ±.9 θ ±3 = sin 1 (±.536) = ±3.4 θ ±4 = sin 1 (±.714) = ±45.6 θ ±5 = sin 1 (±.893) = ±63. θ ±6 = sin 1 (±1.7) = no solution. (14.1.3) Thus, we see that there are a total of 11 interference maxima. (b) The general condition for single-slit diffraction minima is asinθ m = mλ, thus θ m = sin 1 (mλ / a) m = ±1, ±,. (14.1.4) With λ 7 = 5. 1 m and a 7 = 7. 1 m, Eq. (14.1.4) becomes 14-33

35 The solutions are θ m = sin 1 m m m (.714 m). (14.1.5) θ ±1 = sin 1 (±.714) = ± 45.6 (14.1.6) θ ± = sin 1 (± 1.43) = no solution. Since these angles correspond to dark fringes, the total number of bright fringes is N = 11 = 9. (c) The intensity on the screen is given by where I is the intensity at θ =. sin (πa sin θ / λ) I = I πa sin θ / λ (i) At θ =, we have the central maximum and I / I = 1.. (ii) At θ = ± 1.3, we have that, (14.1.7) Therefore π (.7 µm)sin 1.3 πa sin θ / λ = ± = ±.785 rad = ±45.. (14.1.8).5 µm I sin 45. =.811 I ± =.785. (14.1.9) (iii) At θ = ±.9, we have πa sin θ / λ = ±1.57 rad = ±9.. (14.1.3) The intensity ratio is (iv) At θ = ± 3.4, we have I I sin 9. = ± = ( ) πa sin θ / λ = ±.36 rad = ±135 (14.1.3) The intensity ratio is 14-34

36 I I sin 135 = ± = ( ) (v) At θ = ± 63., we have πa sin θ / λ = ±3.93 rad = ±5. ( ) The intensity ratio is I sin 5 =.34 I ± = ( ) 14.1 Conceptual Questions 1. In Young s double-slit experiment, what happens to the spacing between the fringes if (a) the slit separation is increased? (b) the wavelength of the incident light is decreased? (c) the distance between the slits and the viewing screen is increased?. In Young s double-slit experiment, how would the interference pattern change if white light were used? 3. Explain why the light from the two headlights of a distant car does not produce an interference pattern. 4. What happens to the width of the central maximum in a single-slit diffraction if the slit width is increased? 5. In a single-slit diffraction, what happens to the intensity pattern if the slit width becomes narrower and narrower? 6. In calculating the intensity in double-slit interference, can we simply add the intensities from each of the two slits? Additional Problems Double-Slit Interference In the double-slit interference experiment, suppose the slits are separated by d = 1. cm and the viewing screen is located at a distance L = 1. m from the slits. Let the incident light be monochromatic with a wavelength λ = 5 nm

37 (a) Calculate the spacing between the adjacent bright fringes. (b) What is the distance between the third-order fringe and the centerline? Interference-Diffraction Pattern In the double-slit Fraunhofer interference-diffraction experiment,. mm separates the slits, which are.1 mm wide. The incident light is monochromatic with a wavelength λ = 6 nm. How many bright fringes are there in the central diffraction maximum? Three-Slit Interference Suppose a monochromatic coherent light source of wavelength λ passes through three parallel slits, each separated by a distance d from its neighbor. (a) Show that the positions of the interference minima on a viewing screen a distance L >> d away is approximately given by where n is not a multiple of 3. y n = n λl 3d, n = 1,,4,5,7,8,1,, (b) Let L = 1. m, λ = 45 nm, and d =.1 mm. What is the spacing between the successive minima? Intensity of Double-Slit Interference In the double-slit interference experiment, suppose the slits are of different size, and the fields at a point P on the viewing screen are E 1 = E 1 sin(ωt), E = E sin(ωt + φ) Show that the intensity at P is I = I + I + I I cosφ. 1 1 where I 1 and I are the intensities due to the light from each slit Secondary Maxima In a single-slit diffraction pattern, we have shown in 14.6 that the intensity is 14-36

38 sin(β / ) I = I β / sin(πasinθ / λ) = I. πasinθ / λ (a) Explain why the condition for the secondary maxima is not given by β m / = (m + 1 / )π, m = 1,,3,. (b) By differentiating the expression above for I, show that the condition for secondary maxima is β β = tan. (c) Plot the curves y = β / and y = tan( β / ). Using a calculator which has a graphing function, or mathematical software, find the values of β at which the two curves intersect, and hence, the values of β for the first and second secondary maxima. Compare your results with β m / = (m + 1/ )π Interference-Diffraction Pattern If there are 7 fringes in the central diffraction maximum in a double-slit interference pattern, what can you conclude about the slit width and separation? 14-37