Ch7. Interference and Diffraction
1. Spatial and temporal coherence
2. Cross terms and fringes
3. Examples
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The Temporal Coherence Time and
the Spatial Coherence Length
tc
The temporal coherence time is the time over which the beam wavefronts remain
equally spaced. Or, equivalently, over which the beam remains sinusoidal.
Lc
The spatial coherence length is the distance over which the beam wavefronts
remain flat.
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Spatial and Temporal Coherence
Spatial and
Temporal
Incoherence
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Spatial
Coherence;
Temporal
Incoherence
Temporal
Coherence;
Spatial
Incoherence
Spatial and
Temporal
Coherence:
Lasers Emit Temporally Coherent Light
The coherence time is given by:
t c  1/ Dv
where Dn is the light bandwidth (the width of the spectrum).
Sunlight is temporally very incoherent because its bandwidth is
very large (the entire visible spectrum).
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The Spatial Coherence
The van Cittert-Zernike Theorem states that the spatial
coherence area Ac is given by:
D
c 
2
d
2
2
where d is the diameter of the light source and D is the distance away.
Basically, wavefronts smooth
out as they propagate away
from the source.
Starlight is spatially very coherent because stars are very far away.
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The Irradiance Revisited
The most general plane-wave electric field is:
 


E r , t  Re E0 exp i (k  r   t ) 
where the amplitude is both complex and a vector:
E0   E0 x , E0 y , E0 z 
The irradiance is:
c
c
*
*
*
*


I
E0  E0 
E

E

E

E

E

E
0
x
0
x
0
y
0
y
0
z
0
z

2
2 
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Polarization Dependence
Because the irradiance is given by:
c
c
*
 E0 x E0 x*  E0 y E0 y *  E0 z E0 z * 
I
E0  E0 

2
2 
combining two waves of different polarizations is different from combining
waves of the same polarization.
Different polarizations (say x and y):
c
 E0 x  E0 x*  E0 y  E0 y *   I1  I 2
I

2 
Same polarizations (say x and x, so we'll omit the x-subscripts):
I
Therefore:
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c 
*
*
*
E1  E1  2 Re E1  E2  E2  E2 

2 




I  I1  c Re E1  E2  I 2
*
Cross term!
Spatial Crossed Terms
x
k  k cos  zˆ  k sin  xˆ
k  k cos  zˆ  k sin  xˆ
k

z
k  r  k cos  z  k sin  x
k  r  k cos  z  k sin  x
k


I  2 I 0  c Re E0 exp[i(t  k  r )]E0* exp[ i(t  k  r )]
Cross term is proportional to:


Re E0 exp i ( t  kz cos   kx sin   E0 exp  i ( t  kz cos   kx sin  
 Re exp  2ikx sin  
 cos(2kx sin  )
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*
Fringes (in position)
Examples: Fresnel's Biprism
A prism with an apex angle of
about 179° refracts the left
half of the beam to the right
and the right half of the beam
to the left.
Fringe pattern
observed by interfering
two beams created by
Fresnel's biprism
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Angle Dependence of Fringes
The fringe spacing, :
Large angle:
  2 /(2k sin  )
  /(2sin  )
  sin    /(2)
   0.5 m / 200 m
Small angle:
 1/ 400 rad  0.15
0.1 mm is about the minimum fringe spacing seen by eye
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A Misaligned Michelson Interferometer
If the input beam is a plane wave,
crossing beams maps
delay onto position.


Re E0 exp i ( t  kz cos   kx sin   E0 exp  i ( t  kz cos   kx sin  
 Re exp  2ikx sin  
 cos(2kx sin  )
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*
Fringes (in position)
A Misaligned Michelson Interferometer
Now, suppose an object is
placed in one arm. In addition
to the usual spatial factor,
one beam will have a spatially
varying phase, exp[if(x,y)].
Now the cross term becomes:
Re{ exp[if(x,y)] exp[-2ikxsin]}
Distorted fringes
(in position)
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A Misaligned Michelson Interferometer
Placing an object in one arm of a
misaligned Michelson interferometer will
distort the spatial fringes.
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Mach-Zehnder Interferometer
The Mach-Zehnder interferometer is usually operated “misaligned” and with
something of interest in one arm.
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Newton's Rings
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Quiz
Q: He-Ne lasers can have coherence times as
long as about a second.
This is amazing; how many cycles that have to
be locked in the beam?
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Newton's Rings
Get constructive interference when an integral number of half
wavelengths occur between the two surfaces (that is, when an
integral number of full wavelengths occur between the path of the
transmitted beam and the twice reflected beam).
This effect also causes the colors in bubbles
and oil films on puddles.
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Combining a Beam with a Delayed
Replica of Itself Has “Fringes”
The irradiance is given by:


I  I1  c Re E1  E2  I 2
*
Suppose the two beams are E0exp(it) and E0exp[i(t-t)], that is, a beam
and itself delayed by some time t:
I  2 I 0  c Re E0 exp[it ]  E0* exp[i (t  t )]


 2 I 0  c Re E0 exp[it ]
2
 2 I 0  c E0 cos[t ]
2
Fringes (in delay)
I
I  2 I 0  2 I 0 cos[t ]
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- t
The Michelson Interferometer
The Michelson Interferometer
splits a beam into two and then
recombines them at the same
beam splitter.
Fringes (in delay)
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The Michelson Interferometer


I out  I 1  I 2  c Re E0 exp i (t  kz  kL1 ) E0 exp  i (t  kz  kL2 ) 
 I  I  2 I Re exp ik ( L2  L1 ) 
*
since I  I1  I 2  (c 0 / 2) E0
2
 2 I 1  cos(k DL)
Fringes (in delay)
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