1. Agenda This Week –Interference in waves Today –Phasors Tuesday –Lab, Quiz on Lenses/Mirrors/Geo Optics Wed&Fri –Finish Chapter 35

2. Interference Wave Phenomenon Speakers Fundamentals –Multiple “sources” –Correlated – “Coherence” –Waves interfere at some location –Usual: Spatial without time dependence

3. Waves from 2 Point Sources Waves Interfere with each other Notice Lines – Time independent

4. Waves from 2 Point Sources Path dependence for each source Set up mathematical method - waves

5. Set Up Interference Single Light Source Laser – Coherent (phase set) Intersection of Light Bottom Path travels L Top Path travels L2 = L+ 

6. At Intersection Wave Lower Path Upper Path E is electric field amplitude Before we continue…. What is “k”? What is w?

7. Wave Parameters Wave Equation: Generic E is amplitude: Same for both paths  & k depend on frequency & wavelength: Same for both paths  is phase offset: Also depends on light source Same light source, same f.

8. Wave dependencies Wave Equation: Generic  & k depend on frequency & wavelength: Same for both paths Time is just a measurement of time: As in it’s 2:15 “x” is a measure of how far wave has traveled in space Time is same for both paths, x is different

9. Wave dependencies Wave Equation: Generic  & k depend on frequency & wavelength: Same for both paths When time t = T (period) has passed, then one cycle has occurred. Wave equation looks same for any period T: or…. This is true as when time T passes, 2  radians proceed for wave

10. Wave Appearance Repeats on 2  Time or distance  E

11. Wave dependencies Wave Equation: Generic Equivalence over period

12. Wave dependencies (k) Wave Equation: Generic  & k depend on frequency & wavelength: Same for both paths When time x = (wavelength) has passed, then one cycle has occurred. Wave equation looks same for any distance : or…. This is true as when distance is traveled, 2  radians proceed for wave k = 2  (like w = 2  /T) k utilized often in wave mechanics (i.e. quantum)

13. Interference View Single Light Source Laser – Coherent (phase set) Intersection of Light Bottom Path travels L Top Path travels L2 = L+ 

14. At Intersection Wave Lower Path Upper Path For electromagnetic waves: Amplitude  E (B as well) Both waves, time is same. t refers to a “time” not how long wave has traveled Coherent means phase is set by . Single light source, so  same for each Laser so frequency and  same for each. Trig anyone? Can add using trig identities, math handbooks, math software…. Or old school (1 st time?)

15. Examine Wave Addition Stuff inside is angle Amplitude same for both, but doesn’t have to be. Show generic method for addition Works for waves, complex #’s, AC circuits, and quantum mechanics

16. Graphical Representation Lower Path  L X = Ecos(  t +  + kL) Hypotenuse is just E L =E  = (  t +  + kL) E

17. Graphical Representation Lower Path  L X = Ecos(  t +  + kL) Hypotenuse is just E L =E  L = (  t +  + kL) Upper Path  U X = Ecos(  t +  + kL 2 ) Hypotenuse is just E U =E  U = (  t +  + kL 2 ) E E

18. To Add: add x & y parts… Lower Path  L X = Ecos(  t +  + kL)  L = (  t +  + kL) Upper Path  U X = Ecos(  t +  + kL 2 )  U = (  t +  + kL 2 ) E E

19. To Add: add x & y parts… Good? Horrible? Combined  L X = Ecos(  t +  + kL)  U X = Ecos(  t +  + kL 2 )  L = (  t +  + kL)  Let’s find  … Angle (phase) difference E E ETET

20. Interference View Single Light Source Laser – Coherent (phase set) Intersection of Light Bottom Path travels L Top Path travels L2 = L+ 

21. Phase Difference Lower Path –Angle  L =  t + kL +  Upper Path –Angle  U =  t + kL 2 +  –L 2 = L +  –Angle  U =  t + kL +  k   =  U –  L = k 

22. To Add: add x & y parts… Good? Horrible? Combined  L X = Ecos(  t +  + kL)  U X = Ecos(  t +  + kL 2 )  L = (  t +  + kL)  k  Let’s find  … Angle (phase) difference E E ETET

23. To Add: add x & y parts… Law of Cosines? Combined  L X = Ecos(  t +  + kL)  U X = Ecos(  t +  + kL 2 )  L = (  t +  + kL)  k  Let’s find  … Angle (phase) difference E E ETET

24. Law of Cosines? Interior angle is  – k . Combined  L = (  t +  + kL)  k  Let’s find  … Angle (phase) difference  -k  E E ETET E T 2 = E U 2 + E L 2 – 2E U E L cos(  -kD)

25. Law of Cosines? Interior angle is  – k . Combined  L = (  t +  + kL)  k  Let’s find  … Angle (phase) difference  -k  E E ETET E T 2 = E U 2 + E L 2 – 2E U E L cos(  -k  ) E T 2 = 2E 2 – 2E 2 cos(  -k  ) E T 2 = 2E 2 + 2E 2 cos(k  ) [Trig fun part 2] E T 2 = 2E 2 (1+cos(k  )) Notice Final wave only depends on phase difference!

26. Law of Cosines? Interior angle is  – k . Combined  L = (  t +  + kL)  k  Let’s find  … Angle (phase) difference  -k  E E ETET E T 2 = E U 2 + E L 2 – 2E U E L cos(  -k  ) E T 2 = 2E 2 – 2E 2 cos(  -k  ) E T 2 = 2E 2 + 2E 2 cos(k  ) [Trig fun part 2] E T 2 = 2E 2 (1+cos(k  )) 1 + cos  = 2cos 2 (  /2) [Trig fun part 3] E T 2 = 2E 2 (2cos 2 (k  /2)) E T 2 = 4E 2 (cos 2 (k  /2)) E T = 2Ecos(k  /2) Theorists find that… enjoyable…. Experimentalists find that … in a book

27. Interference Implications Single Coherent Light Source Split paths Final Amplitude E T = 2E cos(k  /2)  is path length difference k = 2  /. Max when cos(k  /2) = +/-1 Min when cos(k  /2) = 0

28. Interference Implications Max  constructive Final Amplitude E T = 2E cos(k  /2)  is path length difference k = 2  /. Max when cos(k  /2) = +/-1 k  = n , n =0,1,2…. Check at home: Intensity: I = 0.5  0 cE T 2 Minima when  =(n+1/2) Maxima

29. What did we learn? Light is electromagnetic wave –Electric Field part most important –All you need for intensity –Varies in time & space Interference –Defined by path LENGTH difference –time independent –Path length referenced to wavelength –Coherent, linked sources