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