2. Chapter 5: Superposition of waves

3. Superposition principle applies to any linear system At a given place and time, the net response caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.

4. in a linear world, disturbances coexist without causing further disturbance

5. then the linear combination where a and b are constants is also a solution. Superposition of waves If  1 and  2 are solutions to the wave equation,

6. Superposition of light waves 1 2 -in general, must consider orientation of vectors (Chapter 7—next week) -today, we’ll treat electric fields as scalars -strictly valid only when individual E vectors are parallel -good approximation for nearly parallel E vectors -also works for unpolarized light

7. Light side of life

8. Nonlinear optics is another story for another course, perhaps

9. What happens when two plane waves overlap?

10. Superposition of waves of same frequency propagation distance (measured from reference plane) initial phase (at t =0)

11. Superposition of waves of same frequency simplify by intoducing constant phases: thus At point P, phase difference is and the resultant electric field at P is.

12. “in step” Superposition of waves of same frequency “out of step” constructive interference destructive interference

15. In between the extremes: notice the amplitudes can vary; it’s all about the phase constructivedestructivegeneral superposition

16. Simplify with phasors where and Expressed in complex form: General case of superposition (same  )

17. Phasors, not phasers

18. Phasor diagrams projection onto x -axis magnitude angle   clock analogy: -time is a line -but time has repeating nature -use circular, rotating representation to track time phasors: -represent harmonic motion -complex plane representation -use to track waves -simplifies computational manipulations

19. Phasors in motion http://resonanceswavesandfields.blogspot.com

20. Phasor diagrams complex space representation; vector addition      from law of cosines we get the amplitude of the resultant field:

21. Phasor diagrams taking the tangent we get the phase of the resultant field

22. Works for 2 waves, works for N waves -harmonic waves -same frequency

23. hence as Two important cases for waves of equal amplitude and frequency randomly phasedcoherent phase differences random hence as in phase; all  i are equal

24. Light from a light bulb is very complicated! 1 It has many colors (it’s white), so we have to add waves of many different values of  (and hence k -magnitudes). 2It’s not a point source, so for each color, we have to add waves with many different k directions. 3Even for a single color along one direction, many different atoms are emitting light with random relative phases. Lightbulb

25. Coherent light: - strong - uni-directional - irradiance  N 2 Incoherent light: - relatively weak - omni-directional - irradiance  N Coherent vs. Incoherent light

26. Coherent fixed phase relationship between the electric field values at different locations or at different times Partially coherent some (although not perfect) correlation between phase values Incoherent no correlation between electric field values at different times or locations 1 0 Coherence is a continuum more on coherence next week

27. Color mixing intermezzo

28. Mixing the colors of light

29. Mixing colors to make a pulse of light

30. Time Intensity 1. Single mode Supress all modes except one Time Intensity 2. Multi-mode Statistical phase relation amongst modes I  N Time Intensity 3. Modelocked Constant or linear phase amongst modes I  N 2 T = 2L/c Broadband laser operating regimes

31. Boundary condition: Allowed modes: Mode distance: = const. Pulse duration:  T  (N  Peak intensity:  N 2 (coherent addition of waves) Modelocking laser cavity

32. http://www.physik.uni-wuerzburg.de/femto-welt/ Intermezzo: Femtowelt

33. Standing waves - occur when wave exists in both forward and reverse directions - if phase shift = , standing wave is created - when A ( x ) = 0, E R =0 for all t ; these points are called nodes - displacement at nodes is always zero A(x)A(x)

34. Standing wave anatomy - nodes occur when A ( x ) = 0 - A ( x ) = 0 when sin kx = 0, or kx = m  (for m = 0, ±1, ±2,...) - since k = 2  x = ½ m - E R has maxima when cos  t = ±1 - hence, peaks occur at t = ½ mT ( T is the period) where

35. Standing waves in action http://www.youtube.com/watch?v=0M21_zCo6UM light water sound http://www.youtube.com/watch?v=EQPMhwuYMy4

36. Superposition of waves of different frequencies  pp gg kpkp kgkg

37. Beats Here, two cosine waves, with  p >>  g

38. Beats beat frequency: The product of the two waves is depicted as:

39. 2 frequencies4 frequencies 16 frequencies Many frequencies Acoustic analogy

40. Here, phase velocity = group velocity (the medium is non-dispersive). In a dispersive medium, the phase velocity ≠ group velocity. Phase and group velocity phase velocity: group velocity: envelopemoves with group velocity carrier wave moves with phase velocity

41. non-dispersive mediumdispersive medium Superposition and dispersion of a waveform made of 100 cosines with different frequencies

42. http://www.youtube.com/watch?v=umrp1tIBY8Q And the beat goes on

43. You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 28 September 2011: Chapter 5: 2, 6, 8, 9, 14, 18 Exercises (not part of your homework)