diffraction how do we know light is a wave? waves undergo diffraction if a wave encounters an object...
Post on 19-Dec-2015
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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