Diffraction

• How do we know light is a wave?

• Waves undergo diffraction

• if a wave encounters an object that has an opening of dimensions similar to its , part of the wave will flare out through the opening

• can be understood using Huygen’s argument

• true for all waves e.g ripple tank

Water waves flare out when passing through opening of width a

a

e.g. sound v=f=v/f = (340m/s)/1000Hz = .34 m a of door ~ 1 m => a~ 3

e.g. light ~ 500 nm = 5 x 10-7 m => need smaller opening

tangent towavelets

a =4

Fresnel Bright Spot

Geometrical Optics

Wave Optics

Shine monochromaticlight on a solid sphere.

What image is producedbehind it?

1. C close to Bno diffraction=>geometrical shadow

2. C very far from B => Fraunhoffer diffraction

3. Intermediate case- rays not parallel => Fresnel diffraction

Fraunhoffer diffraction is the easiest to handle

Single Slit Diffraction

Single Slit Diffraction

z

Assume screen is far enough away that red rays are parallel

Path difference between neighbouring rays is z sin

Total electric field due to r1 and r2 isE(r1,t)=Em[sin(kr1-t) + sin(kr1-t+k z sin)]

1 1

1

cos( ) cos( sin )sin

2sin sin

s 2in 2

m

m

Ekr t kr t k

k

Ekr

k

a

t

Single Slit• E(r1,t)=Em[sin(kr1-t) + sin(kr1- t+k z sin)]

• where range on z is 0 z a

1 1

0

( , ) sin( sin )a

mE r t E kdz zr t k

1 0cos( sin ) |sin

m aEkr t kz

k

1sin( / 2)A kr t sinka Phase difference between top and bottom

Single Slit• Amplitude =[2(Em /ksin)sin(kasin/2)]

=[2(Ema /)sin(/2)]

• = (Ema)sin(/2)/(/2)

• I = I0 (sin(/2)/(/2))2

• where I0 = (Ema)2 is the maximum intensity

• note: lim sin(x)/x => 1 x => 0

• intensity is maximum at = 0

sinka

Single Slit• I = I0 (sin(/2)/(/2))2 = 0 when /2 = m= asin/• asin = m for a dark fringe ( m0)• note : m=0 is a maximum!• where are the other maxima?• maximize sin(x)/x with respect to x• (d/dx) [sin(x)/x ] = cos(x)/x - sin(x)/x2 = 0• or x = tan(x) => x =0 is a solution• plot x and tan(x ) versus x and look for

intersections

x ~ (m+ 1/2) , m=1,2,3

x 2 3

tan(x)

sin(x)/x

[sin(x)/x]2