Coherent beams and cross terms
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)
Temporal crossed terms
Combining a Beam with a Delayed Replica of Itself Has “Fringes”
I  2 I 0  c Re E0 exp[it ]  E0* exp[i (t   )]


 2 I 0  c Re E0 exp[i ]
2
 2 I 0  c E0 cos[ ]
2
Fringes (in delay)
I  2 I 0  2 I 0 cos[ ]
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I
- 
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 L)
Fringes (in delay)
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-
The Michelson Interferometer is a
"Fourier Transform Spectrometer"
Suppose the input beam is not monochromatic
(but still has constant amplitude throughout space):
Þ
Iout =
2I + c e Re{E(t+L1/c) E*(t+L2 /c)}
Now, Iout will vary rapidly in time, and most detectors will simply
integrate over a relatively long time, T:
T /2
U

I Out (t )dt  U  2 IT  c Re
T / 2
T /2

E (t  L1 / c ) E *(t  L2 / c ) dt
T / 2
t' = t + L1/c &  = (L2 - L1)/c & T 
U  2 IT  c Re

 E (t ') E *(t '  dt '

The Field
Autocorrelation!
The Fourier Transform of the Field Autocorrelation is the spectrum!!
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Fourier Transform Spectrometer
A Fourier Transform Spectrometer's detected light energy vs. delay is
called an interferogram.
Interferogram
This interferogram
is very narrow, so
the spectrum
is very broad.
Fourier Transform Spectrometers find use in the infrared where the fringes
in delay are most easily generated. As a result, they are often called
FTIR's.
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Time domain interference detection
Fourier Transform Infrared (FTIR) Spectrometer
Soukoulis’ group
Wang’s group
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FTIR Data Acquisition
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Example: why is water is blue?
Colors from vibrations: A FTIR study
Crater lake, Oregon, USA
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Multilayer coatings
Typical laser mirrors and camera
lenses use many layers.
The reflectance and transmittance
can be tailored to taste!
Dr. Pain’s book
PP. 350-353
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Examples: high reflection & anti-reflection
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Laser mirrors, camera and microscope lens
Anti-Reflection Coating
R=0
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n  n0 ns
2
l
Anti-reflection Coating Math
Consider a beam incident on a piece of glass (n=ns) with a layer of
material (n=nl) of thickness, h, on its surface.
It can be shown that the Reflectance is:
nl2 (n0  ns ) 2 cos 2 (kh)  (n0 ns  nl2 ) 2 sin 2 (kh)
R 2
nl (n0  ns ) 2 cos 2 (kh)  (n0 ns  nl2 ) 2 sin 2 (kh)
At normal incidence, and if kh   / 2 (i.e., h   / 4)
(n0 ns  nl2 ) 2
R
(n0 ns  nl2 ) 2
Notice that R=0 if:
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n  n0 ns
2
l
An Fabry-Perot Interferometer (Etalon)
Ei
Er
R
Er
Ei
Et
2
2
T
Et
Ei
2
2
A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and
forth. An etalon is a piece of glass with parallel sides.
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Multiple-beam interference: The FabryPerot Interferometer or Etalon
The transmitted wave is an infinite series of multiply reflected beams.
r, t = reflection, transmission coefficients from glass to air
Transmitted
wave: E0t
Incident wave: E0
Reflected
wave: E0r
Transmitted wave:
n=1
n
n=1
d = round-trip phase delay
inside medium
t 2 E0
t 2 r 2 e  id E0
t 2 (r 2 e  id ) 2 E0
t 2 (r 2 e  id )3 E0
E0t  t 2 E0  t 2 r 2e  id E0  t 2 (r 2e  id ) 2 E0  t 2 (r 2e  id )3 E0  ...
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The Etalon (continued)
The transmitted wave field is:
E0t  t 2 E0  t 2 r 2e  id E0  t 2 (r 2e  id ) 2 E0  t 2 (r 2e  id )3 E0  ...
 t 2 E0 1  (r 2 e  id )  (r 2 e  id ) 2  ...
E0t  t 2 E0 / 1  r 2eid 
E
The transmittance is: T  0t
E0
2
2
t

1  r 2e  id
2


t4

2  id
2  id 
(1

r
e
)(1

r
e )


 
 

t4
(1  r 2 ) 2
(1  r 2 ) 2



 
 

4
4
2
2
2
4
2
2
{1  r  2 cos(d )}  {1  r  2r [1  2sin (d / 2)]}  {1  2r  r  4r sin (d / 2)]} 
2 2
Dividing numerator and denominator by (1  r )
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1
T
2
1  F sin d / 2 
where:
 2r 
F 
2
1

r


2
Etalon Transmittance vs.
Thickness, Wavelength, or Angle
Transmission maxima
occur when:
2L/ = 2m
or:
  L/m
The transmittance varies significantly with thickness or wavelength.
We can also vary the incidence angle, which also affects d.
As the reflectance of each surface (r2) approaches 1, the widths of the
high-transmission regions become very narrow.
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