Agenda

1
Agenda

• This Week
• Interference in waves
• Today
• Phasors
• Tuesday
• Lab, Quiz on Lenses/Mirrors/Geo Optics
• WedFri
• 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 D
6
At Intersection
Wave
Upper Path
Lower 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
• f is phase offset Also depends on light source
• Same light source, same f.

8
Wave dependencies
Wave Equation Generic

• w k depend on frequency wavelength Same for
  both paths
• Time is just a measurement of time As in its
  215
• 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

• w 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, 2p radians
proceed for wave
10
Wave AppearanceRepeats on 2p Time or distance
2p
E
11
Wave dependencies
Wave Equation Generic
Equivalence over period
12
Wave dependencies (k)
Wave Equation Generic

• w k depend on frequency wavelength Same for
  both paths
• When time x l (wavelength) has passed, then one
  cycle has occurred. Wave equation looks same for
  any distance l or.

This is true as when distance l is traveled, 2p
radians proceed for wave k 2p/l (like w
2p/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 D
14
At Intersection
Wave
Upper Path
Lower 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 f. Single light source, so
f same for each Laser so frequency and w same for
each. Trig anyone? Can add using trig identities,
math handbooks, math software. Or old school
(1st time?)
15
Examine Wave Addition
Stuff inside is angle Amplitude same for both,
but doesnt have to be. Show generic method for
addition Works for waves, complex s, AC
circuits, and quantum mechanics
16
Graphical Representation
Lower Path
E
q (wt f kL)
YLX Ecos(wt f kL)
Hypotenuse is just ELE
17
Graphical Representation
Lower Path
Upper Path
E
E
qL (wt f kL)
qU (wt f kL2)
YLX Ecos(wt f kL)
YUX Ecos(wt f kL2)
Hypotenuse is just ELE
Hypotenuse is just EUE
18
To Add add x y parts
Lower Path
Upper Path
E
E
qL (wt f kL)
qU (wt f kL2)
YLX Ecos(wt f kL)
YUX Ecos(wt f kL2)
19
To Add add x y partsGood? Horrible?
Combined
Dq
E
ET
Lets find Dq Angle (phase) difference
E
qL (wt f kL)
YUX Ecos(wt f kL2)
YLX Ecos(wt f kL)
20
Interference View

• Single Light Source
• Laser Coherent (phase set)

Intersection of Light Bottom Path travels L Top
Path travels L2 L D
21
Phase Difference

• Lower Path
• Angle qL wt kL f
• Upper Path
• Angle qU wt kL2 f
• L2 L D
• Angle qU wt kL f kD
• Dq qU qL kD

22
To Add add x y partsGood? Horrible?
Combined
DqkD
E
ET
Lets find Dq Angle (phase) difference
E
qL (wt f kL)
YUX Ecos(wt f kL2)
YLX Ecos(wt f kL)
23
To Add add x y partsLaw of Cosines?
Combined
DqkD
E
ET
Lets find Dq Angle (phase) difference
E
qL (wt f kL)
YUX Ecos(wt f kL2)
YLX Ecos(wt f kL)
24
Law of Cosines?Interior angle is p kD.
Lets find Dq Angle (phase) difference
Combined
DqkD
E
ET
p-kD
E
qL (wt f kL)
ET2 EU2 EL2 2EUELcos(p-kD)
25
Law of Cosines?Interior angle is p kD.
Lets find Dq Angle (phase) difference
Combined
DqkD
E
ET
ET2 EU2 EL2 2EUELcos(p-kD) ET2 2E2
2E2cos(p-kD) ET2 2E2 2E2cos(kD) Trig fun
part 2 ET2 2E2 (1cos(kD))
p-kD
E
qL (wt f kL)
Notice Final wave only depends on phase
difference!
26
Law of Cosines?Interior angle is p kD.
Lets find Dq Angle (phase) difference
Combined
DqkD
E
ET
ET2 EU2 EL2 2EUELcos(p-kD) ET2 2E2
2E2cos(p-kD) ET2 2E2 2E2cos(kD) Trig fun
part 2 ET2 2E2 (1cos(kD)) 1 cosq
2cos2(q/2) Trig fun part 3 ET2 2E2
(2cos2(kD/2)) ET2 4E2 (cos2(kD/2)) ET
2Ecos(kD/2) Theorists find that
enjoyable. Experimentalists find that in a book
p-kD
E
qL (wt f kL)
27
Interference Implications

• Single Coherent Light Source
• Split paths

Final Amplitude ET 2E cos(kD/2) D is path
length difference k 2p/l. Max when cos(kD/2)
/-1 Min when cos(kD/2) 0
28
Interference ImplicationsMax ? constructive
Final Amplitude ET 2E cos(kD/2) D is path
length difference k 2p/l. Max when cos(kD/2)
/-1 kD/2 np, n 0,1,2.
Maxima
Check at home Intensity I 0.5e0cET2 Minima
when D(n1/2)l
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