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Agenda
• This Week– Interference in waves
• Today– Phasors
• Tuesday– Lab, Quiz on Lenses/Mirrors/Geo Optics
• Wed&Fri– Finish Chapter 35
Interference
• Wave Phenomenon
• Speakers
• Fundamentals– Multiple “sources”– Correlated – “Coherence”– Waves interfere at some location – Usual: Spatial without time dependence
Waves from 2 Point Sources
• Waves Interfere with each other
• Notice Lines – Time independent
Waves from 2 Point Sources
• Path dependence for each source• Set up mathematical method - waves
, cosx t A t kx
Set Up Interference• Single Light Source
• Laser – Coherent (phase set)
Intersection of LightBottom Path travels LTop Path travels L2 = L+
At Intersection
2
, cos
, cos , , cosL U
x t E t kx
x t E t kL x t E t kL
Wave
Lower Path Upper Path
E is electric field amplitudeBefore we continue….What is “k”? What is w?
Wave Parameters
, cosx t E t kx 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 sourceSame light source, same f.
Wave dependencies
, cosx t E t kx Wave Equation: Generic
& k depend on frequency & wavelength: Same for both pathsTime is just a measurement of time: As in it’s 2:15“x” is a measure of how far wave has traveled in spaceTime is same for both paths, x is different
Wave dependencies
, cosx t E t kx
Wave Equation: Generic
& k depend on frequency & wavelength: Same for both pathsWhen time t = T (period) has passed, then one cycle has occurred. Wave equation looks same for any period T: or….
, cos cosx t E t T kx E t kx
This is true as when time T passes, 2 radians proceed for wave
Wave AppearanceRepeats on 2
Time or distance
E
Wave dependencies
, cosx t E t kx
Wave Equation: Generic
, cos cos
22
cos cos
2cos cos
2cos cos 2
cos 2 cos
x t E t T kx E t kx
fT
E t T kx E t T kx
E t T kx E t T kxT
E t T kx E t kxT
E t kx E t kx
Equivalence over period
Wave dependencies (k)
, cosx t A t kx
Wave Equation: Generic
& k depend on frequency & wavelength: Same for both pathsWhen time x = (wavelength) has passed, then one cycle has occurred. Wave equation looks same for any distance : or….
, cos cosx t A t k x A t kx
This is true as when distance is traveled, 2 radians proceed for wavek = 2 (like w = 2/T)k utilized often in wave mechanics (i.e. quantum)
Interference View• Single Light Source• Laser – Coherent (phase set)
Intersection of LightBottom Path travels LTop Path travels L2 = L+
At Intersection
2
, cos
, cos , , cosL U
x t E t kx
x t E t kL x t E t kL
Wave
Lower Path Upper Path
2, cos cosT x t A t kL A t kL
For electromagnetic waves: Amplitude E (B as well)Both waves, time is same. t refers to a “time” not how long wave has traveledCoherent means phase is set by . Single light source, so same for eachLaser so frequency and same for each.Trig anyone?Can add using trig identities, math handbooks, math software….Or old school (1st time?)
Examine Wave Addition
2, , , cos cosT L Ux t x t x t E t kL E t kL
Stuff inside is angleAmplitude same for both, but doesn’t have to be.Show generic method for additionWorks for waves, complex #’s, AC circuits, and quantum mechanics
Graphical Representation
Lower Path
LX = Ecos(t + + kL)
Hypotenuse is just EL=E
= (t + + kL)
E
Graphical Representation
Lower Path
LX = Ecos(t + + kL)
Hypotenuse is just EL=E
L = (t + + kL)
Upper Path
UX = Ecos(t + + kL2)
Hypotenuse is just EU=E
U = (t + + kL2)
EE
To Add: add x & y parts…
Lower Path
LX = Ecos(t + + kL)
L = (t + + kL)
Upper Path
UX = Ecos(t + + kL2)
U = (t + + kL2)
EE
To Add: add x & y parts…Good? Horrible?
Combined
LX = Ecos(t + + kL)
UX = Ecos(t + + kL2)L = (t + + kL)
Let’s find …Angle (phase) difference
E
EET
Interference View
• Single Light Source
• Laser – Coherent (phase set)
Intersection of LightBottom Path travels LTop Path travels L2 = L+
Phase Difference
• Lower Path– Angle L = t + kL +
• Upper Path– Angle U = t + kL2 +
– L2 = L +
– Angle U = t + kL + k
• = U – L = k
To Add: add x & y parts…Good? Horrible?
Combined
LX = Ecos(t + + kL)
UX = Ecos(t + + kL2)L = (t + + kL)
k
Let’s find …Angle (phase) difference
E
EET
To Add: add x & y parts…Law of Cosines?
Combined
LX = Ecos(t + + kL)
UX = Ecos(t + + kL2)L = (t + + kL)
k
Let’s find …Angle (phase) difference
E
EET
Law of Cosines?Interior angle is – k.
Combined
L = (t + + kL)
kLet’s find …Angle (phase) difference
-kE
EET
ET2 = EU
2 + EL2 – 2EUELcos(-kD)
Law of Cosines?Interior angle is – k.
Combined
L = (t + + kL)
kLet’s find …Angle (phase) difference
-kE
EET
ET2 = EU
2 + EL2 – 2EUELcos(-k)
ET2 = 2E2 – 2E2cos(-k)
ET2 = 2E2 + 2E2cos(k) [Trig fun part 2]
ET2 = 2E2 (1+cos(k))
Notice Final wave only depends on phase difference!
Law of Cosines?Interior angle is – k.
Combined
L = (t + + kL)
kLet’s find …Angle (phase) difference
-kE
EET
ET2 = EU
2 + EL2 – 2EUELcos(-k)
ET2 = 2E2 – 2E2cos(-k)
ET2 = 2E2 + 2E2cos(k) [Trig fun part 2]
ET2 = 2E2 (1+cos(k))
1 + cos = 2cos2(/2) [Trig fun part 3]ET
2 = 2E2 (2cos2(k/2))ET
2 = 4E2 (cos2(k/2))ET = 2Ecos(k/2)Theorists find that… enjoyable….Experimentalists find that … in a book
Interference Implications• Single Coherent Light Source
• Split paths
Final AmplitudeET = 2E cos(k/2) is path length differencek = 2/.Max when cos(k/2) = +/-1Min when cos(k/2) = 0
Interference ImplicationsMax constructive
Final AmplitudeET = 2E cos(k/2) is path length differencek = 2/.Max when cos(k/2) = +/-1
k = n, n =0,1,2….
22 2
2
kn
n n
kn
Check at home:Intensity: I = 0.50cET
2 Minima when =(n+1/2)
Maxima
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