Interference and Diffraction
Huygen’s Principle
•Any wave (including electromagnetic waves) is able to propagate because
the wave here affects nearby points there
•In a sense, the wave is the source for more of the wave
•A wave here creates waves in all the forward directions
•For a plane wave, the generated waves add up to make more plane waves
•Mathematically, this works, but for
plane waves, no one does it this way
Diffraction Through a Tiny Hole
•The waves come out in all directions
•It is only because the whole wave makes new waves that the
waves add up to only go forwards
•What if we let the wave pass through a tiny hole?
•Smaller than a wavelength
•Only one point acts as source
•Waves spread out in all directions
E0
E
sin  kr  t 
r
r
•What’s interesting is that oscillations
depend on distance from slit
E  E0 sin  kx  t 
E sin  kr  t 
Interference Through Two Slits
•Now imagine we have two slits, equally sized
•Each slit creates its own waves
•In some directions, crests add with
crests to make bigger “brighter”
crests
•In others, crests combine with
troughs to make minimum areas
•In the end, what you get is a
pattern of alternating light and dark
bands
•We’re about to need an
obscure math identity:
 A B 
 A B 
sin A  sin B  2sin 
cos



2
2




Interference Through Two Slits (2)
•What do the EM waves look like far away?
•Let the separation of the slits be d
•Let’s find total E-field at point P
E  E1  E2 ~ sin  kr1  t   sin  kr2  t 
d sin
 2sin  krave  t  cos  12 k  r2  r1  
d

rave  12  r1  r2 
r2
r2  r1  d sin 
r1
E
2sin  krave  t  cos  12 kd sin  
I
E2
sin 2  krave  t  cos2  12 kd sin  
P
Interference Through Two Slits (3)
I  cos
2

1
2
kd sin  
•Where is it bright?
•Where is it dark?
k
2

sin  bright  m d
  d sin  
I  I max cos 

 
2
m  0, 1, 2,
sin dark   m  12   d
Interference Through Four Slits
•What if we have more than two slits?
•Four slits, each spaced distance d apart
•Treat it as two double slits
E  E1  E2  E3  E4  E1,2  E3,4
 2sin  kr1,2  t  cos  12 kd sin    2sin  kr3,4  t  cos  12 kd sin  
r3,4
 4sin  krave  t  cos  d sin    cos  2 d sin   
I  E2
r1,2
P
•For four slits, every third band is bright
More Slits and Diffraction Gratings
•This process can be continued for more slits
•For N slits, every N – 1’th band is bright
•For large N, bands become very narrow
N=8
N = 16
N = 32
•A device called a diffraction
grating is just transparent with
closely spaced regular lines on it
•You already used it in lab
sin  bright
m

d
m  0, 1, 2,
•Diffraction gratings are another way to divide light into different colors
•More accurate way of measuring wavelength than a prism
•Commonly used by scientists
Resolution of Diffraction Gratings
m1
sin 1 
•Note that the angle depends on the wavelength
d
•With a finite number of slits, nearby wavelengths may overlap
m2
sin  2 
d
N=8

•The width of the peaks is about
1.1

  sin   
dN
•The difference between peaks is
m   
  sin   
d
•We can distinguish two peaks if:
m     N
•This quantity (mN) is called the

d
d
resolving power
•Even if N is very large, effectively N is how
many slits the light beam actually falls on
mN 


Diffraction Through a Single Slit
•What if our slit is NOT small compared to a wavelength?
•Treat it as a large number of closely spaced sources, by Huygen’s principle
•Let the slit size be a, and rave the distance to the center
•Let x be the distance of some point from the center
•The distance r will be slightly different from here to P
r  rave  x sin 
a x
E ~ sin  kr  t   sin  krave  kx sin   t 
r
1a
2
E   1 sin  krave  kx sin   t  dx
2a
rave
a /2
1

cos  krave  kx sin   t   a /2
k sin 
1


cos
kr

ka
sin



t
 ave 2

1
E


1
k sin    cos  krave  2 ka sin   t  
sin  12 ka sin  

sin  krave  t 
P
1
2 k sin 
Diffraction Through a Single Slit (2)
 sin  ka sin   
I  I max  1

 2 ka sin  
1
2
2
 sin  a sin    
I  I max 


a
sin




sin dark  m a
2
m  1, 2,
•Very similar to equation for multi-slit
diffraction, but . . .
•a is the size of the slit
•This equation is for dark, not light
•Note m= 0 is missing
•Central peak twice as wide
Screens and Small Angles
•Usually your slit size/separation is large compared to the wavelength
•Multi-slit:
Diffraction:
m
m
sin  dark 
sin  bright 
a
d
•When you project them onto a screen, you need to calculate locations of
these bright/dark lines
x
•For small angles, sin and tan are the same
 tan   sin 
L
x
L
xbright  m
xdark
L
d
L
m
a
Diffraction and Interference Together
a
•Now go through two finite sized slits
•Result is simply sum of each slit
•Resulting amplitude looks like:
d
a
E
sin  12 ka sin  
1
2
k sin 
sin  kr1  t   sin  kr2  t  
 sin  a sin    
2   d sin  
I  I max 
 sin 

 

  a sin   
2
a = d/5
•Resulting pattern has two kinds of variations:
•Fast fluctuations from separation d
•Slow fluctuations from slit size a
The Diffraction Limit
•When light goes through a “small” slit, its direction gets changed
•Can’t determine direction better than this

sin  min 
a

 min
a
a
•If we put light through rectangular (square) hole,
we get diffraction in both dimensions
•A circular hole of diameter D is a trifle smaller,
which causes a bit more spread in the outgoing wave
•For homework, use this formula; for tests, the
approximate formula is good enough

1.22
 min
 min 
D
D
a
D
Sample problem
If the pupil of your eye in good light is 2 mm in diameter, what’s
the smallest angle you can see using 500 nm visible light?
 min
1.22 1.22  5 107 m
4



3.05

10
rad  1 arc-min
3
D
2 10 m
•A degree is 1/360 of a circle, an arc-minute is 1/60 of
a degree, an arc-second is 1/60 of an arc minute
•Telescopes require large apertures to see small angles
Phases
•When you combine two (or more) waves, you need to know the phase shift
between them:
E  A sin  x   B sin  x   
•The angle is the phase shift
•When the phase shift is zero, the waves add constructively
•The result is bigger
•Same thing for any even multiple of 
•When the phase shift is , the waves add destructively
•The result is smaller
•Same thing for any odd multiple of 
•To find maximum/minimum effects, set phase shift to even/odd multiples of 
Spherical Mirrors: Finding the Image
•As a wave passes through any material, its phase shifts
•For a distance d, we have:
  kd

E  sin  kx  t 
2 d

•Recall, wavelength  changes inside a material
c
f 
n
f 0  c

0
n

2 dn
0
Reflection and Phase Shift
•When you reflect off of a mirror, the reflected
wave must cancel the incoming wave
•It has a  phase shift
•When you go from a low index of
refraction medium to a high one, some of
the wave is reflected
•It also has a  phase shift
 phase
shift
 phase shift
•When you go from a high index of
refraction medium to a low one, some of
the wave is reflected
•This has a 0 phase shift
0 phase shift
Interference From Thin Films
•Suppose we go through a thin soap film
•Index goes up then down
Front surface:
•Phase shift of  from reflection (low-high)
t
Back surface:
•Phase shift of 2t/ from traveling
•Phase shift of 0 from reflection
•Phase shift of 2t/ from traveling
4 t



Total phase shift between two reflected waves:

•Weak reflection when odd times :
4 tweak
•Strong reflection when even
  2m  1 


•Same results for index down then up
•Opposite for:
2tweak   m
•Index up, then up
•Index down, then down
2t
  m 1
strong

2

Applications of Thin Films Interference
•What if the light isn’t monochromatic?
2tweak  m
2tstrong    m  12 
•Some wavelengths are enhanced, others are not
•Soap bubbles
•Oil on water
•Newton’s rings: convex lens on flat glass plate
•Air gap changes thickness in circular pattern
•Alternating light/dark regions
d
narrow
air gap
Michelson Interferometer
•Interference easy to measure
•Can see much smaller than one wavelength
•LIGO, state of the art, can see 10-15 m!
Hanford, Washington
Laser
Detector
Mirrors
Crystal Scattering of X-rays
•Mysterious rays were discovered by Röntgen in 1895
•Suspected to be short-wavelength EM waves
•Order 1-0.1 nm wavelength
•Scattered very weakly off of atoms
•Bragg, 1912, measured wavelength accurately
2d cos  m
 
d
•Scattering strong only
if waves are in phase
•Must be integer
multiple of wavelength
Polarization
•Recall that light waves have electric and magnetic fields
perpendicular to the direction of motion
•But there are two independent ways of arranging this
•Called polarization
•Our eyes can’t tell these two polarizations apart
•But some instruments can measure or take advantage of
polarization
•We describe polarization by telling which direction the
electric field points, e.g. vertically or horizontally
B0
E0
E0
B0
Methods of Producing Polarization
Direct production
•Antennas produce waves that are automatically polarized
Scattering
•Light waves of all orientations hit small targets
•Target has vibrating charges, like an antenna
Reflection and Brewster’s Angle:
•When light hits a substance, some of it reflects and some refracts
•Fraction of each depends on polarization
n2
•There’s a special angle – Brewster’s angle –
tan  P 
n1
where reflected is completely polarized
E0
n1
P
n2
+
+
+
+
+
+
–
–
–
–
–
–
Methods of Producing Polarization (2)
Birefringent Crystals
•Index of refraction has to do with electric fields from the wave pushing atoms
around
•In some crystals, it is easier to push them one way than another
•Index of refraction depends on polarization
•You can use such birefringent crystals to sort light based on polarization
Selective absorption
•Similarly, some materials absorb one polarization better than another
E0
E0
E0
E0
E0
Some Uses for Polarization
Polarized Sun Glasses
•“Glare” comes mostly from light scattered in the atmosphere and
reflected from water
•Mostly polarized
•Sun glasses use selective absorption to eliminate it
Optical Activity
•Some materials are capable of rotating the plane of polarization
•These materials are not mirror-symmetric
•Enantiomers, especially biological molecules
•Studying rotation of polarized light detects presence of these molecules
•Someday use these to detect life on other planets?
E0
Sugar
water
E0