1. PHY 102: Waves & Quanta Topic 7 Diffraction John Cockburn (j.cockburn@... Room E15)

2. Interference re-cap Phasors Single slit diffraction Intensity distribution for single slit

3. Electromagnetic Waves Where E 0 and B 0 are related by: E 0 = cB 0 INTENSITY of an EM wave  E 0 2 NB. we will see later that EM radiation sometimes behaves like a stream of particles (Photons) rather than a wave………………

4. Interference First, consider case for sound waves, emitted by 2 loudspeakers: Path difference =nλ Constructive Interference Path difference =(n+1/2)λ Destructive Interference (n = any integer, m = odd integer)

5. Interference

6. Young’s Double Slit Experiment Demonstrates wave nature of light Each slit S 1 and S 2 acts as a separate source of coherent light (like the loudspeakers for sound waves)

7. Young’s Double Slit Experiment Constructive interference: Destructive interference:

8. Young’s Double Slit Experiment Y-position of bright fringe on screen: y m = Rtan  m Small , ie r 1, r 2 ≈ R, so tan  ≈ sin  So, get bright fringe when: (small  only)

9. Young’s Double Slit Experiment: Intensity Distribution For some general point P, the 2 arriving waves will have a path difference which is some fraction of a wavelength. This corresponds to a difference  in the phases of the electric field oscillations arriving at P:

10. Young’s Double Slit Experiment: Intensity Distribution Total Electric field at point P: Trig. Identity: With  = (  t +  ),  =  t, get:

11. So, E TOT has an “oscillating” amplitude: Since intensity is proportional to amplitude squared: Or, since I 0  E 0 2, and proportionality constant the same in both cases:

12. For the case where y<<R, sin  ≈ y/R:

13. Young’s Double Slit Experiment: Intensity Distribution

14. 2-slit intensity distribution: “phasor” treatment Remember from Lecture 1, harmonic oscillation with amplitude A and angular frequency  can be represented as projection on x or y axis of a rotating vector (phasor) of magnitude (length) A rotating about origin. Light We can use this concept to add oscillations with the same frequency, but different phase constant  by “freezing” this rotation in time and treating the 2 oscillations as fixed vectors…… So called “phasor method”

15. 2-slit intensity distribution: “phasor” treatment Use phasor diagram to do the addition E 1 + E 2 Using cosine rule:

16. Another way: Complex exponentials

18. Single Slit Diffraction “geometrical” picture breaks down when slit width becomes comparable with wavelength

19. Single Slit Diffraction observed for all types of wave motion eg water waves in ripple tank

20. Single Slit Diffraction

21. Explain/analyse by treating the single slit as a linear array of coherent point sources that interfere with one another (Huygen’s principle)……………………. All “straight ahead” wavelets in phase → central bright maximum Destructive interference of light from sources within slit for certain angles Fraunhofer (“far-field”) case

22. Single Slit Diffraction From diagram, can see that for slit of width A, we will get destructive interference (dark band on screen) at angles  which satisfy…..: Choice of a/2 and a/4 in diagram is entirely arbitrary, so in general we have a dark band whenever; (m=±1, ±2, ±3………..)

23. Position of dark fringes in single-slit diffraction If, like the 2-slit treatment we assume small angles, sin  ≈ tan  =y min /R, then Positions of intensity MINIMA of diffraction pattern on screen, measured from central position. Very similar to expression derived for 2-slit experiment: But remember, in this case y m are positions of MAXIMA In interference pattern

24. Width of central maximum We can define the width of the central maximum to be the distance between the m = +1 minimum and the m=-1 minimum: Ie, the narrower the slit, the more the diffraction pattern “spreads out” image of diffraction pattern Intensity distribution

25. Single-slit diffraction: intensity distribution To calculate this, we treat the slit as a continuous array of infinitesimal sources: Can be done algebraically, but more nicely with phasors………………..

26. Single-slit diffraction: intensity distribution E 0 is E-field amplitude at central maximum  = total phase difference for “wavelets” from top and bottom of slit

27. Single-slit diffraction: intensity distribution How is  related to our slit/screen setup? Path difference between light rays from top and bottom of slit is From earlier (2-slit)

28. Single-slit diffraction: intensity distribution