Interference and Diffraction

Conditions for Interference

• Coherent Sources - The phase between waves from the multiple sources must be constant.

• Monochromatic - Wavelength from the sources must be of the same wavelength.

Interference• Remember the Superposition Principle

Constructive

Destructive),2,1,0( 12 nnrr

),2,1,0( 21

12 nnrr

#1

#2

r1

r2Source

Effect

Double Slit Interference

• Young’s Double-Slit Experiment• Two slits act like coherent sources• Assumption: d >> • Difference in path traveled is

• Constructive Interference sind

,...)2,1,0( sin mmd

d

Small Angle Approximation

• When << 1, then

and d L

ym = 1

m = 2

m = 0

m = -1

m = -2

tansin

mdLybright

21 m

dLydark

Intensity• Phase between waves from slits is

• Use a phasor diagram to predict the intensity

Ldy

22

E1

E2

tEE sin01

tEE sin02

2sin

2cos2 0

tEEsum

LdySSsum

20 cos4c

ES0

20

0 2

12

0-1

-2

Multiple Slits• 2 Slits

E1

E2

12

0-1

-2

12

0-1

-2

12

0-1

-2

• 4 Slits

• 3 Slits • Diffraction Grating

E1

E2

E3

E1

E2

E3

E4

mN

Resolving Power

Thin Films• Constructive interference between waves reflecting

off the top and bottom surfaces.

212 mtn

180° Phase Shift

0° Phase Shiftt

m - whole number

Single-Slit Diffraction

• Huygen’s Principle - Each point along the wave re-emits spherical waves.

m = -1

m = -2

d L

ym = 1

m = 2Destructive Interference

md sinIntensity

2

0 2/2/sin

SS

sin2 d

Resolution• Rayleigh Criterion - Can resolve two images if

the maximum of one image falls on the first minimum of another image.

a min

Slit resolution

Circular Aperture

D 22.1min

m

Resolving Detail• Detail at a distance can be

calculated by

• What detail can you see at 1 km (103m) with green light (500 nm)? (Let D=0.30cm)

rs minr

m s

Circular Aperture

D 22.1min

Interferometer• Comparing light from two separate

telescopes to increase the resolving power.• Very Large Array – Radio TelescopeVery Large Array – Radio Telescope

– 27 radio antennas in a Y-shaped 27 radio antennas in a Y-shaped – Fifty miles west of Socorro, New MexicoFifty miles west of Socorro, New Mexico– Each antenna is 25 meters in diameterEach antenna is 25 meters in diameter– Resolution of an antenna 36km acrossResolution of an antenna 36km across– Sensitivity of a dish 130 meters in diameter.Sensitivity of a dish 130 meters in diameter.

Diffraction and Interference

• Small regularly spaced openings result in bright spots.– Diffraction Grating

• Spacing of spots depend on distance between openings.– Crystallography