PHY 102: Waves & Quanta

Topic 6

Interference

John Cockburn (j.cockburn@... Room E15)

•Electromagnetic Waves

•Interference of Sound Waves

•Young’s double slit experiment

•Intensity distribution for Young’s experiment

Electromagnetic Radiation

•Visible light is an example of ELECTROMAGNETIC RADIATION:

Electromagnetic Waves

•Existence predicted by James Clerk Maxwell (1865)

•Consist of “crossed” time-varying electric and magnetic fields

•Transverse wave, both electric and magnetic fields oscillate in a direction perpendicular to propagation direction

•No medium is necessary: Electromagnetic waves can propagate through a vacuum

•Constant speed of propagation through a vacuum: c ≈ 3 x 108 ms-1

Electromagnetic Waves

Electromagnetic Waves

•It can be shown from MAXWELL’S EQUATIONS of Electromagnetism (See second year course) that the electric and magnetic fields obey the wave equations:

2

2

002

2

t

E

x

E yy

2

2

002

2

t

B

x

B zz

2

2

22

2 ),(1),(

t

txy

vx

txy

“standard” linear wave equation

00

1

c

Electromagnetic Waves

)sin(0 tkxEEy

)sin(0 tkxBBz

Where E0 and B0 are related by: E0 = cB0

INTENSITY of an EM wave E02

NB. we will see later that EM radiation sometimes behaves like a stream of particles (Photons) rather than a wave………………

Speed of light in a material

•Constant speed of propagation through a vacuum: c ≈ 3 x 108 ms-1

•But, when travelling through a material, light “slows down”

n

cv

n is the “refractive index” of the material.

Frequency of the radiation is constant, so from v = fλ, wavelength must decrease by a factor of 1/n.

(NB refractive index depends on the wavelength of the light)

Interference

First, consider case for sound waves, emitted by 2 loudspeakers:

Path difference =nλConstructive Interference

Path difference =(n+1/2)λDestructive Interference

(n = any integer, m = odd integer)

Interference

Interference

For interference effects to be observed,

•sources must emit at a single frequency

•Sources must have the same phase OR have a FIXED phase difference between them. This is known as COHERENCE

Conditions apply to interference effects for both light and sound

Example calculation

For what frequencies does constructive/destructive interference occur at P?

Young’s Double Slit Experiment

•Demonstrates wave nature of light

•Each slit S1 and S2 acts as a separate source of coherent light (like the loudspeakers for sound waves)

Young’s Double Slit Experiment

Consider intensity distribution on screen as a function of (angle measured from central axis of apparatus)……………………….

If light behaves as a conventional wave, then we expect high intensity (bright line) at a position on the screen for which r2-r1 = nλ

Expect zero intensity (dark line) at a position on the screen for which r2-r1=(n+1/2)λ

Young’s Double Slit Experiment

Assuming (justifiably) that R>>d, then lines r2 and r1 are approximately parallel, and path difference for the light from the 2 slits given by:

sin12 drr

Young’s Double Slit Experiment

Constructive interference:

Destructive interference:

nd sin

2

1sin nd

Young’s Double Slit Experiment

Y-position of bright fringe on screen: ym = Rtanm

Small , ie r1, r2 ≈ R, so tan ≈ sin

So, get bright fringe when:

d

nRym

(small only)

Young’s Double Slit Experiment:Intensity Distribution

For some general point P, the 2 arriving waves will have a path difference which is some fraction of a wavelength.

This corresponds to a difference in the phases of the electric field oscillations arriving at P:

tEE sin01

tEE sin02

Young’s Double Slit Experiment:Intensity Distribution

Total Electric field at point P:

tEtEEEETOT sinsin 0021

Trig. Identity:

2

1sin

2

1cos2sinsin

With = (t + ), = t, get:

tEETOT sin

2cos2 0

tEETOT sin

2cos2 0

So, ETOT has an “oscillating” amplitude:

2cos2 0

E

Since intensity is proportional to amplitude squared:

2cos4 22

0

EITOT

Or, since I0E02, and proportionality constant the same in both cases:

2cos4 2

0

IITOT

differencepath

2

difference phase

sin

2

d

2cos4 2

0

IITOT

sin

cos4 20

dIITOT

For the case where y<<R, sin ≈ y/R:

R

dyIITOT

20 cos4

Young’s Double Slit Experiment:Intensity Distribution

R

dyIITOT

20 cos