Review of Waves

1. Review of Waves

2. Properties of electromagnetic waves in vacuum: Waves propagate through vacuum (no medium is required like sound waves) All frequencies have the same propagation speed, c in vacuum. Electric and magnetic fields are oriented transverse to the direction of propagation. (transverse waves) Waves carry both energy and momentum.

3. E and B fields in waves and Right Hand Rule: f+ solution Wave propagates in E x B direction f- solution

5. Special Case Sinusoidal Waves Wavenumber and wavelength These two contain the same information

6. Special Case Sinusoidal Waves Introduce Different ways of saying the same thing:

7. Polarizations We picked this combination of fields: Ey - Bz Could have picked this combination of fields: Ez - By These are called plane polarized. Fields lie in plane

8. Which of the following are valid EM waves? 1. Yes No A.Yes B. No 2.

9. 3. A.Yes B. No 4. What direction is this wave propagating in? Y Z X None of above

10. z Direction of propagation Ex . Wave Vector By , Bz y

11. Energy Density and Intensity of EM Waves Energy density associated with electric and magnetic fields For a wave: Thus: Units: J/m3 Energy density in electric and magnetic fields are equal for a wave in vacuum.

12. We now want to expand the picture in the following way: EM waves propagate in 3D not just 1D as we have considered. - Diffraction - waves coming from a finite source spread out. EM waves propagate through material and are modified. - Dispersion - waves are slowed down by media, different frequency waves travel with different speeds - Reflection - waves encounter boundaries between media. Some energy is reflected. - Refraction - wave trajectories are bent when crossing from one medium to another. EM waves can take multiple paths and arrive at the same point. - Interference - contributions from different paths add or cancel.

13. Waves emanating from a point source

15. Wave crests When can one consider waves to be like particles following a trajectory? Motion of crests Direction of power flow • Wave model: study solution of Maxwell equations. Most complete classical description. Called physical optics. • Ray model: approximate propagation of light as that of particles following specific paths or “rays”. Called geometric optics. • Quantum optics: Light actually comes in chunks called photons

16. What is the difference? Diffraction. Comparable to opening size Much smaller than opening size

17. Diffraction

18. Propagation of light through dielectric media In a dielectric the Electric field causes molecules/atoms to become polarized. Molecules align with field Medium becomes “polarized”

19. What happens when the electric field oscillates in time? A current is induced.

20. How is the Ampere-Maxwell Law modified? Dielectric For S2 there is both displacement current and real current

21. Real current given by dielectric constant Electric flux Dielectric constant In a dielectric material Bottom Line: Dielectric constant Replace by

22. Static dielectric constants To complicate matters, dielectric constant depends on frequency. Dielectric function depends on frequency

23. Consequences for EM Plane waves Propagation speed changes in a material Refraction Ratio of E to B changes Reflection For sinusoidal waves the following is still true

24. Index of refraction

25. For sinusoidal waves the following is still true Frequency is the same in both media Wavelength changes

26. A light wave travels through three transparent materials of equal thickness. Rank in order, from the largest to smallest, the indices of refraction n1, n2, and n3. • n1 > n2 > n3 • n2 > n1 > n3 • n3 > n1 > n2 • n3 > n2 > n1 • n1 = n2 = n3

27. Example: Optical fiber d Fused silica: n=1.48 Vacuum wavelength l = 1550 nm Wavelength in fiber l/n = 1045nm D = 10 mm P=1mW inside fiber I=P/A=? E=? In vacuum Rays bounce back and forth

28. Reflection from surface incident transmitted reflected Region 1 Region 2 Reflection coefficient What if “Index matched”

29. Ey, Bz How to calculate reflection At x=0 fields are continuous x region 1 0 region 2 For x<0 incident For x>0 reflected transmitted

30. Continuity of Ey at x=0 x<0 x>0 Continuity of Bz at x=0 x<0 x>0 Solve for reflected & transmitted pulses

31. Example: what fraction of power is reflected at boundary between glass and air? Reflection coefficient Air: n1=1.0003 Glass: n2 =1.5 ratio of reflected to incident electric field amplitude I=Power/Area ratio of reflected to incident intensity

32. Suppose the wave was incident on the boundary from the glass side incident transmitted reflected Region 1 Region 2 n1=1.5 n2=1.0 What is reflection coefficient?

33. incident transmitted reflected Region 1 Region 2 n1=1.5 n2=1.0 What is reflection coefficient? r=0 r =.2 C. r =-.2 D. r =5

34. Region 2 has higher velocity Region 2 has lower velocity

35. What if Same as for reflection from a conductor

36. Interference in 1 Dimension incident Incident and reflected fields add (superposition) reflected incident reflected Incident and reflected waves will interfere, changing the peak electric field at different points

37. Case #1: no reflection r=0. No interference, E plotted versus x for several values of t E plotted versus t for a single value of x. Same peak value independent of x Traveling wave

38. Case #2: total reflection r= -1. Anti-nodes E plotted versus x for several values of t nodes How far apart are the nodes? Anti-nodes? What is peak E? Case #3: total reflection r= +1. E plotted versus x for several values of t

39. = VSWR Emax Emin Case #4: partial reflection r= -0.5 E plotted versus x for several values of t How far apart are the minima? The Maxima? What is peak E? When reflection is not total there are still local maxima and minima. Voltage Standing Wave Ratio Pronounced “vizwarr”

40. Half Wavelength Window Eliminates Reflection incident incident transmitted reflected Region 1 Region 2 Region 1’ D When dielectric #2 is an integer number of half-wavelengths thick, no reflection

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42. Summary Waves are modified by dielectric constant of medium, k. All our Maxwell equations are valid provided we replace 3. Speed of waves is lowered. (Index of refraction - n) 4. Frequency of wave does not change in going from one medium to another. Wavelength does. 5. Waves are reflected at the boundary between two media. (Reflection coefficient)

43. = VSWR 6. Reflected waves interfere with incident waves. Distance between interference maxima/minima Ratio of maximum peak field to minimum peak field (Voltage standing wave ratio)

44. Solution of the Wave equation Where f+,- are any two functions you like, and is a property of space. Represent forward and backward propagating wave (pulses). They depend on how the waves were launched

45. Shape of pulse determined by source of wave. Speed of pulse determined by medium

46. What is the magnetic field of the wave? Notice minus sign

47. Ey (V/m) The graph at the top is the history graph at x = 4 m of a wave traveling to the right at a speed of 2 m/s. Which is the history graph of this wave at x = 0 m?

48. What is the frequency of this traveling wave? • 0.1 Hz • 0.2 Hz • 2 Hz • 5 Hz • 10 Hz

49. What is the phase difference between the crest of a wave and the adjacent trough? • 0 • π • π/4 • π/2 • 3 π/2