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:
2pL/l = 2mp
or:
l  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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The Etalon Free Spectral Range
The Free Spectral Range is the wavelength range between transmission maxima.
lFSR =
Free Spectral
Range
lFSR
2p L
2p L

 2p
l
l  lFSR

2p L
l

2p L[1  lFSR / l ]
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l
 2p

2p L
2p L

 2p
l
l[1  lFSR / l ]
1 1  lFSR / l 1
l2



 lFSR 
l
l
L
L
Etalon Linewidth and Finesse
The Linewidth dLW is a transmittance peak's full-width-half-max (FWHM).
1
T
1  F sin 2 d / 2 
Setting d equal to dLW/2 should yield T = 1/2:
1  F sin 2  d LW / 2  / 2   2 or sin 2 d LW / 4   1/ F
For d << 1, we can make the small argument approx:
d LW / 4 
2
 1/ F  d LW  4 / F
The Finesse, F, is the ratio of the
Free Spectral Range and the Linewidth:
2
 2r  we have:
Substituting F  
2
1  r 
d = 2p corresponds to
one FSR
2p
p F
I 

2
4/ F
I  p /[1  r 2 ]
taking r  1
The Finesse is the number of wavelengths the interferometer can resolve.
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Simplest Laser Cavity is a Fabry-Perot!
LASER: Light Amplification by Stimulated Emission of Radiation
Required: gain medium + pump source + cavity
(reflectors or recirculating loop)
Example:
Fabry-Perot cavity
(two mirrors)
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The Amazing Wave – Laser
Optical
Communication
Laser
Machining
Bar code
scanner
Cosmic MW
background
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http://www.laserfest.org
Laser lithography
Medical
Imaging
Laser wake field
particle accelerator
An Fabry-Perot cavity (Etalon)
with an inverted medium
Ei
Er
R
Er
Ei
Et
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2
2
T
Et
Ei
Oscillation from photon noise?
2
2
An Passive Fabry-Perot Etalon
E(1)
Ei
r
E(2)
Two parallel mirrors
r
E0t
T
E0
2
t2

1  r 2e  id
2


t4

2  id
2  id 
(1

r
e
)(1

r
e )

l
T
(tt ' ) 2
1
(1  rr ' )  4rr ' sin ( d )
2
2
E(1)t E(2)t
d 
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2pn
l
2l cos 
2
An FP with an Inverted Medium
Then the propagation vector is
k
k

c

c
 
 

c

c
Passive
n
with no gain medium
(n  dn(k ))  i / 2  i
dn
Gain
2
Loss
(tt ' e (  )l ) 2
T  1 R 
(1  rr ' e
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
d 
(   ) l 2
2p (n  dn)
l
)  4rr ' e
2l cos 
(   ) l
1
sin ( d )
2
2
An FP with an Inverted Medium
An optical gain over certain spectral region
(tt ' ) 2 e (  )l
T  1 R 
(1  rr ' e
(   ) l 2
)  4rr ' e
Oscillation from photon noise?
what is the condition for
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(  ) l
1
sin ( d )
2
2
Here comes laser!
T 
An FP Etalon with an Inverted Medium
An optical gain over certain spectral region k
T  1 R 
1
(1  rr ' e (  )l ) 2  4rr ' e (  )l sin 2 ( d )
2
c
( 2)  p 
p
lp
2nc l cos 
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l
(tt ' ) 2 e (  )l
1  1 
  
(1)   ln
2l  R1 R2 
c
d 
2p (n  dn)
Gain threshold
Cavity L modes
2l cos 
Ch7. Interference and Diffraction
1. Interference and fringes
2. Fabry–Perot interferometer
3. Division of amplitude & wavefront division
4. Diffraction (I)
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Interference & Diffraction
Division of amplitude
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Division of wavefront
Matter of scale
Fraunhofer Diffraction
Fraunhofer Diffraction from
a slit is simply the Fourier
Transform of a rect function,
which is a sinc function.
The irradiance is then sinc2
.
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Example: the Fourier Transform of a
rectangle function: rect(t)
1/ 2
1
F ( )   exp( i t )dt 
[exp( i t )]1/1/2 2
i
1/ 2
1
[exp(i / 2)  exp(i
i
 exp(i / 2)  exp(i sin(



2i


F (   sinc(
Imaginary
Component = 0
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More About Sinc(x)
Sinc(x/2)
transform
function.
is
of
the Fourier
a rectangle
Sinc2(x/2)
transform
function.
is
of
the
a
Fourier
triangle
Sinc2(ax) is the diffraction
pattern from a slit.
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Fraunhofer Diffraction
Fraunhofer Diffraction from
a slit is simply the Fourier
Transform of a rect function,
which is a sinc function.
The irradiance is then sinc2
.
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Diffraction and Fourier Transform
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Interference from Two Equal
Sources of Separation f
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Young’s Double Slit Experiment
The distance on the
screen between two bright fringes
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