3.38. Diffraction

1. 3.38. Diffraction • Diffraction is a scale phenomenon that can only be described by use of waves. • Consider a plane wave front of wavelength l incident on an aperture of width a. • How does the wave change when it propagates through the aperture? • We can understand the changes by using Huygens’ principle. • There are two cases to consider • Aperture width a greater than wavelength • Aperture width comparable to the wavelength.

2. 3.38. Diffraction • Aperture width a greater than wavelength l. • Here plane wave front of wavelength l incident the aperture of width a. • In the aperture we can set up many secondary sources. • These produce secondary wavelets. • In the centre of the aperture the leading edges of the wavelets produces a plane wave front. • At the edges, the wavelets form a curved wavefront. • Hence the wave front appears to propagate through the aperture as if it were a plane wave in the centre but is only disrupted at the edges.

3. a > l

4. 3.38. Diffraction • Aperture width a comparable to wavelength l. • Here plane wave front of wavelength l incident the aperture of width a. • In the aperture we can set up many secondary sources. • These produce secondary wavelets. • In the centre of the aperture the leading edges of the wavelets produces a plane wave front. • At the edges, the wavelets form a curved wavefront. • Hence the wave front appears to propagate through the aperture as if it were a plane wave in the centre but is only disrupted at the edges.

5. a ~ l

6. Source 1 Detector , x Source 2 3.39 Interference Consider two light sources that produce harmonic waves of equal amplitude A but equal frequency f. A detector is placed a distance x1 from source 1 and a distance x2 from source 2. x1 x2 How does the signal recorded by a detector vary as we vary the distances x1 and x2? Need to use the principle of linear superposition to obtain the answer.

7. Let the wave function for source 1 be • y1(x,t) = Asin(kx1-wt + f1(t)) • and for source 2 • y2(x,t) = Asin(kx2-wt + f2(t)) • From principle of linear superposition we have • yt(x,t) = y1(x,t) + y2(x,t) • yt(x,t) = Asin(kx1-wt + f1(t)) + Asin(kx2-wt + f2(t)) • Which becomes

8. Term that depends on separation of sources Harmonic wave at ave distance • Thus at x we have a harmonic wave with phase that depends on the average distance of the detector from the sources multiplied by a term that depends solely on the difference in distance between the two sources. • Let us assume that the initial phase of each wave is zero. We will discuss the significance of the initial phase later. • The displacement at the point x is given by

9. The detector responds to the intensity of the signal. Hence we find If we use the relationship We find

10. When k(x1-x2) = 2nπ (n = integer) the intensity is a maximum When k(x1-x2) = (2n+1)π (n = integer) the intensity is zero Thus there is a periodic modulation of the intensity that depends on the difference in distance between the two sources and the detector. The term (x1-x2) is known as the path difference.

11. The term (x1-x2) is known as the path difference. For an intensity maximum k(x1-x2) = 2nπ But k = 2π/l So (x1-x2) = nl Thus when the path difference is an integer number of wavelengths there is an intensity maximum For an intensity minimum k(x1-x2) = (2n+1)π But k = 2π/l So (x1-x2) = (n+1/2)l Thus when the path difference is a half integer number of wavelengths there is an intensity minimum.

12. If we now include the initial phase for each of the waves how is the intensity pattern affected? • We find that the displacement is given by • The intensity becomes

13. We can understand the effect of the initial phase with reference to the cosine term in the intensity • The difference between f1(t) and f2(t) remains constant for all time then the interference pattern remains constant. • If the difference between f1(t) and f2(t) varies in a random fashion then we find that the value of the cosine term fluctuates randomly between +1 and -1. • As a result the intensity pattern varies in an unpredictable fashion and the fringes are “washed out”.

14. If f1(t) - f2(t) remains constant for all time the sources are said to be coherent. • Example of a coherent source - loud speakers connected to the same frequency source • For coherent sources the intensity I(x,t) is given by • If f1(t) - f2(t) varies randomly as a function of time the sources are said to be incoherent. • Example of incoherent source - car headlights • For incoherent sources the intensity I(x,t) is given by

15. 3.40 Young’s Slits Experiment: Interference from two sources Two slits s1 and s2 are separated by a distance d. A screen is placed a distance L from the slits (L>>d). The nth bright interference fringe is seen a B a height yn from the centre of the screen. The slits are illuminated by a white light source located behind the pinhole p. This ensures that the slits are illuminated by coherent radiation.

16. 3.40 Young’s Slits Experiment: Interference from two sources S1B = x1 S2B = x2 S2B - S1B = x2 - x1 x2 - x1 = r r = dsina yn = Ltana For the nth bright fringe the path difference between S2B and S1B must be an integer number of wavelengths. x2 - x1 = nl r = nl

17. 3.40 Young’s Slits Experiment: Interference from two sources S1B = x1 S2B = x2 S2B - S1B = x2 - x1 x2 - x1 = r r = dsina yn = Ltana Now the angle a is small and so sina = tana. Hence r= ynd/L so ynd/L = nl Thus yn = Lnl/d The spacing between successive fringes is ∆ = Ll/d

19. 3.41 Single slit diffraction Consider a slit of width a. A plane wave is incident on the slit. A set of secondary sources are set up in the slit. We find that at an angle b the source at 1 is p ( l/2) out of phase with the source at 5. The source at 5 is p ( l/2) out of phase with the source at 9. Also the source at 5 is at the centre of the slit. So when the condition asinb = ml is satisfied destructive interference occurs.