2.1 Wave propagation. Slides: Video Plane waves and interference Text reference: Quantum Mechanics for Scientists and Engineers Section B.

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1 2.1 Wave propagation Slides: Video Plane waves and interference Text reference: Quantum Mechanics for Scientists and Engineers Section B.4

2 Wave propagation Plane waves and interference Quantum mechanics for scientists and engineers David Miller

3 Wave equation in 3 dimensions Generalizing to 3 dimensions, the wave equation becomes where c t x y z or equivalently, with unit vectors xy ˆ, ˆ,andzˆ in the corresponding directions x ˆ ˆ ˆ ˆ ˆ ˆ x y y z z x x y y z z 2

4 Plane wave solutions We can check that a monochromatic (one frequency) plane wave of the form exp ikrt where r xxˆ yyˆ z zˆ and k k ˆ ˆ ˆ x xkyykzz is a solution when k / c First note that exp i krt xˆ yˆ zˆ exp ikxxkyy kz z t x y z ˆ ˆ ˆexp xx yy zz x y z i k k k ik xk yk z t ikexp i krt

5 Plane wave solutions So k r k r 2 exp i t exp i t ikexp i krt ikx ky kz exp i t x y z k r k x ky kz exp ik r t k 2 exp i krt

6 Plane wave solutions Since 2 2 then, with exp i krt k exp i krt and choosing k / c exp i t t 2 k r k / c expik r t 1 c 2 kr kr exp i t exp i t 2 So exp i krt is indeed a solution for any vector direction k provided k / c exp i krt t 2 2 k 2 k 2 ik r t exp 0

7 Wave interference k 1 One solution is the plane wave with wavevector k 1

8 Wave interference k 2 One solution is the plane wave with wavevector k 1 Another solution is the plane wave with wavevector k 2

9 Wave interference One solution is the plane wave with wavevector k 1 Another solution is the plane wave with wavevector k 2 Because the wave equation is linear the sum is also a solution showing interference

10

11 2.1 Wave propagation Slides: Video Diffraction Text reference: Quantum Mechanics for Scientists and Engineers Section B.4

12 Wave propagation Diffraction Quantum mechanics for scientists and engineers David Miller

13 Waves from a point source Here is a snapshot of a wave from a point source showing the waves are circular

14 Waves from a point source We can watch the wave from this point source as it propagates giving a circularly expanding wave

15 Waves from an aperture? Suppose we want to know the wave from an aperture a gap in a wall that is illuminated with a plane wave from the left

16 Waves from an aperture Suppose we want to know the wave from an aperture a gap in a wall that is illuminated with a plane wave from the left We could start by using just 3 point sources on this plane wave front to try to describe the transmitted wave

17 Waves from an aperture Suppose we want to know the wave from an aperture a gap in a wall that is illuminated with a plane wave from the left We could start by using just 3 point sources on this plane wave front to try to describe the transmitted wave

18 Waves from an aperture If we use 3 point sources on this plane wave front the result is not very accurate

19 Waves from an aperture If we use 5 point sources the result may not be accurate near the aperture but we see locally plane waves at larger distances

20 Waves from an aperture If we use 5 point sources the result may not be accurate near the aperture but we see locally plane waves at larger distances

21 Waves from an aperture With 9 point sources we see the circles are beginning to describe a straight line e.g., at their right edges and they generate plane phase fronts on the right as if the incident plane wave had propagated through the aperture

22 Waves from an aperture With 9 point sources we see the circles are beginning to describe a straight line e.g., at their right edges and they generate plane phase fronts on the right as if the incident plane wave had propagated through the aperture

23 Waves from an aperture With 17 point sources we have essentially converged on a good description in this model obtaining initially plane wave fronts from the original plane wave propagating through the aperture

24 Waves from an aperture With 17 point sources we have essentially converged on a good description in this model obtaining initially plane wave fronts from the original plane wave propagating through the aperture

25 Waves from an aperture We can also see that something is happening on the right the waves seem to be getting weaker in fact, they are actually spreading out beyond our picture

26 Diffraction angle d 2 For a source or aperture of width d the diffraction angle d where is the wavelength

27 Diffraction angle d 4 For a source or aperture of width d the diffraction angle d where is the wavelength Note the inverse relation larger aperture, smaller angle

28 Diffraction angle d For a source or aperture of width d the diffraction angle d where is the wavelength Note the inverse relation larger aperture, smaller angle smaller aperture, larger angle

29

30 2.1 Wave propagation Slides: Video Diffraction from periodic structures Text reference: Quantum Mechanics for Scientists and Engineers Section B.4

31 Wave propagation Diffraction from periodic structures Quantum mechanics for scientists and engineers David Miller

32 Scatterers..... Periodic structures Suppose we have a set of equally spaced identical scatterers and we shine a monochromatic wave at them from the right what does the backscattered light look like?

33 Periodic structures One scatterer will give a set of phase fronts concentric circles

34 Periodic structures One scatterer will give a set of phase fronts concentric circles Two will give two sets which will interfere

35 Periodic structures One scatterer will give a set of phase fronts concentric circles Two will give two sets which will interfere Multiple equally spaced scatterers give multiple sets

36 Periodic structures These scatterers all add in phase for particular directions, here straight ahead called a zeroth order diffraction

37 Periodic structures These scatterers all add in phase for particular directions, here straight ahead called a zeroth order diffraction an upward direction

38 Periodic structures These scatterers all add in phase for particular directions, here straight ahead called a zeroth order diffraction an upward direction a downward direction

39 .... Scatterer spacing in wavelengths 1.5 Periodic structures At large distances from the scatterer we get a multiple-beam diffraction pattern which looks like a set of points on a screen Larger scatterer separation gives beams closer in angle

40 ..... Scatterer spacing in wavelengths 2.0 Periodic structures At large distances from the scatterer we get a multiple-beam diffraction pattern which looks like a set of points on a screen Larger scatterer separation gives beams closer in angle

41 Scatterer spacing in wavelengths 2.5 Periodic structures At large distances from the scatterer we get a multiple-beam diffraction pattern which looks like a set of points on a screen Larger scatterer separation gives beams closer in angle

42 ..... Scatterer spacing in wavelengths 3.0 Periodic structures At large distances from the scatterer we get a multiple-beam diffraction pattern which looks like a set of points on a screen Larger scatterer separation gives beams closer in angle

43 .... Scatterer spacing in wavelengths 3.5 Periodic structures At large distances from the scatterer we get a multiple-beam diffraction pattern which looks like a set of points on a screen Larger scatterer separation gives beams closer in angle

44 .... Scatterer spacing in wavelengths 4.0 Periodic structures At large distances from the scatterer we get a multiple-beam diffraction pattern which looks like a set of points on a screen Larger scatterer separation gives beams closer in angle

45 Periodic structures The angle of these diffracted waves is given by simple geometry

46 s Periodic structures The angle of these diffracted waves is given by simple geometry sin s where is the wavelength s is the separation between scatterers

47 Periodic structures s 1 1 For larger spacings s multiple diffraction orders are possible sin1 s is a first order diffraction 2

48 Periodic structures s For larger spacings s multiple diffraction orders are possible 2 sin2 s is a second order diffraction

49 Periodic structures s For larger spacings s multiple diffraction orders are possible sin 1 s is a (negative) first order diffraction the sign is a matter of taste for this normal incidence

50 Periodic structures s For larger spacings s multiple diffraction orders are possible 2 sin 2 s is a (negative) second order diffraction the sign is a matter of taste for this normal incidence

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