Moule3:Interference-I Lecture 3: Interference-I Consier a situation where we superpose two waves. Naively, we woul expect the intensity (energy ensity or flux) of the resultant to be the sum of the iniviual intensities. For example, a room becomes twice as bright if we switch on two lamps instea of one. This actually oes not always hol. A wave, unlike the intensity, can have a negative value. If we a two waves whose values have opposite signs at the same point, the total intensity is less than the intensities of the iniviual waves. This is an example of a phenomena referre to as interference. 3. Young s Double Slit Experiment. We begin our iscussion of interference with a situation shown in Figure 3.. Light from a istant point source is incient on a screen with two thin slits. The separation between the two slits is. We are intereste in the image of the two slits on a screen which is at a large istance from the slits. Note that the point source is aligne with the center of the slits as shown in Figure 3.. Let us calculate the intensity at a point P locate at an angle θ on the screen. The raiation from the point source is well escribe by a plane wave by the time the raiation reaches the slits. The two slits lie on the same wavefront of this plane wave, thus the electric fiel oscillates with the same phase at both Point Source Slits Screen Figure 3.: Young s ouble slit experiment I 85
86 CHAPTER 3. INTERFERENCE-I P θ Source Slits Screen Figure 3.: Young s ouble slit experiment II Imaginary E ~ E ~ E ~ E ~ φ Imaginary φ φ φ ~ E Real ~ E Real Figure 3.3: Summation of two phasors the slits. If Ẽ (t) an Ẽ(t) be the contributions from slits an to the raiation at the point P on the screen, the total electric fiel will be Ẽ(t) = Ẽ(t) + Ẽ(t) (3.) Both waves originate from the same source an they have the same frequency. We can thus express the electric fiels as Ẽ(t) = Ẽe iωt, Ẽ (t) = Ẽe iωt an Ẽ(t) = Ẽeiωt. We then have a relations between the amplitues Ẽ = Ẽ + Ẽ. It is often convenient to represent this aition of complex amplitues graphically as shown in Figure 3.3. Each complex amplitue can be represente by a vector in the complex plane, such a vector is calle a phasor. The sum is now a vector sum of the phasors. The intensity of the wave is I = E(t)E(t) = ẼẼ (3.) where we have roppe the constant of proportionatily in this relation. It is clear that the square of the length of the resultant phasor gives the intensity. Geometrically, the resultant intensity I is the square of the vector sum of two vectors of length I an I with angle φ φ between them as shown in Figure 3.3. Consequently, the resulting intensity is I = I + I + I I cos(φ φ ). (3.3)
0. 0.05 0-0.05-0. -0.4-0. 0 0. 0.4 3.. YOUNG S DOUBLE SLIT EXPERIMENT. 87 Intensity λ/ 4 3-0.4-0. 0. 0.4 Θ Figure 3.4: Young ouble slit interference fringes with intensity profile Calculating the intensity algebraically, we see that it is I = [ẼẼ + ẼẼ + ẼẼ + Ẽ Ẽ] (3.4) = I + I + E E [ e i(φ φ ) + e i(φ φ ) ] (3.5) I = I + I + I I cos(φ φ ) (3.6) The intensity is maximum when the two waves have the same phase I = I + I + I I (3.7) an it is minimum when φ φ = π i.e the two waves are exactly out of phase I = I + I I I. (3.8) The intensity is the sum of the two intensities when the two waves are π/ out of phase. In the Young s ouble slit experiment the waves from the two slits arrive at P with a time elay because the two waves have to traverse ifferent paths. The resulting phase ifference is φ φ = π sin θ λ. (3.9) If the two slits are of the same size an are equiistant from the the original source, then I = I an the resultant intensity, I(θ) = I [ + cos ( )] π sin θ λ ( ) π sin θ = 4I cos λ (3.0) (3.)
k k 88 CHAPTER 3. INTERFERENCE-I Fresnel Biprism k B r A Θ / k Figure 3.5: Fresnel biprism to realise the Young s ouble slit For small θ we have I(θ) = I [ + cos ( )] πθ λ (3.) There will be a pattern of bright an ark lines, referre to as fringes, that will be seen on the screen as in Figure 3.4. The fringes are straight lines parallel to the slits, an the spacing between two successive bright fringes is λ/ raians. 3.. A ifferent metho of analysis. A Fresnel biprism is constructe by joining to ientical thin prisms as shown in Figure 3.5. Consier a plane wave from a istant point source incient on the Fresnel biprism. The part of the wave that passes through the upper half of the biprism propagates in a slightly ifferent irection from the part that passes through the lower half of the biprism. The light emanating from the biprism is equivalent to that from two exactly ientical sources, the sources being locate far away an there being a small separation between the sources. The Fresnel biprism provies a metho for implementing the Young s ouble slit experiment. The two waves emanating from the biprism will be coplanar an in ifferent irections with wave vectors k an k as shown in Figure 3.5. We are intereste in the intensity istribution on the screen shown in the figure. Let A be a point where both waves arrive at the same phase.ie φ(a) ie. Ẽ = Ẽ = Ee iφ(a). The intensity at this point will be a maximum. Next consier a point B at a isplacement r from the point A. The phase of the two waves are ifferent at this point. The phase of the first wave at the point B is given by φ (B) = φ(a) k r (3.3) an far the secon wave The phase ifference is φ (B) = φ(a) k r (3.4) φ φ = ( k k ) r (3.5)
3.. YOUNG S DOUBLE SLIT EXPERIMENT. 89 Using eq. (3.6), the intensity pattern on the screen is given by I( r) = I + I + I I cos[ ( k ) k r] (3.6) where I an I are the intensities of the waves from the upper an lower half of the biprism respectively. Assuming that the wave vectors make a small angle θ/ to the horizontal we have k = k[î + θ ĵ] an k = k[î θ ĵ] (3.7) where θ is the angle between the two waves. Using this an assuming that I = I we have ( )] πθ y I( r) = I [ + cos. (3.8) λ There will be straight line fringes on the screen, these fringes are perpenicular to the y axis an have a fringe spacing y = λ/θ. The analysis presente here is another way of analysing the Young s ouble slit experiment. It is left to the reaer to verify that eq. (3.) an eq. (3.8) are equivalent. Like Fresnel biprism one can also realise ouble slit experiment with Fresnel mirrors. Here one uses two plane mirrors an one of the mirrors is tilte slightly (θ < ) an glue with the other as shown in Figure 3.6. S 00 00 00 00 θ 00 θ S D S" Figure 3.6: Fresnel mirrors
90 CHAPTER 3. INTERFERENCE-I Problems. An electromagnetic plane wave with λ = mm is normally incient on a screen with two slits with spacing = 3 mm. a. How many maxima will be seen, at what angles to the normal? b. Consier the situation where the wave is incient at 30 to the normal.. Two raio antennas separate by a istance = 0 m emit the same signal at frequency ν with phase ifference φ. Determine the values of ν an φ so that the raiation intensity is maximum in one irection along the line joining the two antennas while it is minimum along exactly the opposite irection. How o the maxima an minima shift of φ is reuce to half the earlier value? 3. A lens of iameter 5.0 cm an focal length 0 cm is cut into two ientical halves. A layer mm in thickness is cut from each half an the two lenses joine again. The lens is illuminate by a point source locate at the focus an a fringe pattern is observe on a screen 50 cm away. What is the fringe spacing an the maximum number of fringes that will be observe? 4. Two coherent monochromatic point sources are separate by a small istance, fin the shape of the fringes observe on the screen when, a) the screen is at one sie of the sources an normal to the screen is along the line joining the two sources an b) when the normal to the screen is perpenicular to the line joining the sources. 5. The raiation from two very istant sources A an B shown in the Figure 3.7 is measure by the two antennas an also shown in the figure. The antennas operate at a wavelength λ. The antennas prouce voltage outputs Ṽ an Ṽ which have the same phase an amplitue as the electric fiel Ẽ an Ẽ incient on the respective antennas. The voltages from the two antennas are combine Ṽ = Ṽ + Ṽ an applie to a resistance. The average power P issipate across the resistance is measure. In this problem you can assume that θ (in raians). a. What is the minimum value of (separation between the two antennas) at which P = 0? b. Consier a situation when an extra phase φ is introuce in Ṽ before the signals are combine. For what value of φ is P inepenent of?
3.. YOUNG S DOUBLE SLIT EXPERIMENT. 9 A θ B Figure 3.7: 6. Lloy s mirror: This is one of the realisations of Young s ouble slit in the laboratory. Fin the conition for a ark fringe at P on the screen from the Figure 3.8. Also fin the number of fringes observe on the screen. Assume source wavelength to be λ. S S Q P 00 0 00 0 00000 00 000000 0 000000 O 000000 00000 00 0000 0000 00 0000 000000 00 0 A B 00 D Figure 3.8: Lloy s mirror 7. Calculate the separation between the seconary sources if the primary source is place at a istance r from the mirror-joint an the tilt angle is θ. 8. Two coherent plane waves with wave vectors k = k[cos 30 î + sin 30 ĵ] an k = k[sin 30 î + cos 30 ĵ] with k =. 0 6 m are incient on a screen which is perpenicular to the x axis to prouce straight line fringes. Determine the spacing between two successive ark lines in the fringe pattern.