PHYS 415: OPTICS
Review of Interference and Diffraction
F. ÖMER ILDAY
Department of Physics,
Bilkent University, Ankara, Turkey
I used the following resources in the preparation of almost all these lectures:
Trebino’s Modern Optics lectures from Gatech (quite heavily used), and
various textbooks by Pedrotti & Pedrotti, Hecht, Guenther, Verdeyen, Fowles and Das
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Interference vs. Diffraction
• Interference is when we add up multiple but a finite number
of E&M waves
• Diffraction is when we add up a continuum of E&M waves.
• Fundamentally, there is no difference.
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Interference and Interferometers
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Varying the delay on purpose
Simply moving a mirror can vary the delay of a beam by many
wavelengths.
Input
beam E(t)
Mirror
Output
beam E(t–)
Translation stage
Moving a mirror backward by a distance L yields a delay of:
  2 L /c
Do not forget the factor of 2!
Light must travel the extra distance
to the mirror—and back!
Since light travels 300 µm per ps, 300 µm of mirror displacement
yields a delay of 2 ps. Such delays can come about naturally, too.
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We can also vary the delay using
a mirror pair or corner cube.
Mirror pairs involve two
reflections and displace
the return beam in space:
But out-of-plane tilt yields
a nonparallel return beam.
E(t)
Input
beam
Mirrors
E(t–)
Output
beam
Translation stage
Corner cubes involve three reflections and also displace the return
beam in space. Even better, they always yield a parallel return beam:
[Edmund
Scientific]
“Hollow corner cubes” avoid propagation through glass.
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The Michelson Interferometer
Input
beam
The Michelson Interferometer splits a beam
into two and then recombines them at the same
Mirror
beam splitter.
L2
Beamsplitter
Suppose the input beam is a plane wave:
Output
beam
L1
Delay
Mirror
I out  I 1  I 2  c Re E0 exp i ( t  kz  2kL1 ) E0* exp  i ( t  kz  2kL2 ) 
 I  I  2 I Re exp  2ik ( L2  L1 )
 2 I 1  cos(k L)
where: L = 2(L2 – L1)
Fringes (in delay):
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since I  I1  I 2  (c 0 / 2) E0
2
“Bright fringe”
“Dark fringe”
Iout
L = 2(L2 – L1)
Input
beam
The Michelson Interferometer
L2
Output
beam
Mirror
The most obvious application of the
Michelson Interferometer is to measure
the wavelength of monochromatic light.
Beamsplitter
L1
Delay
Mirror
Iout  2I 1  cos(k L)  2I 1  cos(2 L / )
Iout

L = 2(L2 – L1)
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Huge Michelson Interferometers may
someday detect gravity waves.
Gravity waves (emitted by all massive objects) ever so slightly warp
space-time. Relativity predicts them, but they’ve never been detected.
Supernovae and colliding black holes emit gravity waves that may be
detectable.
Gravity waves are “quadrupole”
waves, which stretch space in
one direction and shrink it in
Mirror
another. They should cause
one arm of a Michelson
interferometer to stretch and the
other to shrink.
L2
Beamsplitter
L1
L1 and L2 = 4 km!
Mirror
Unfortunately, the relative distance (L1-L2 ~ 10-16 cm) is less than the
width of a nucleus! So such measurements are very very difficult!
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The building
containing an arm
The LIGO project
CalTech LIGO
A small fraction
of one arm of the
CalTech LIGO
interferometer…
Hanford LIGO
The control center
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The LIGO folks think big…
The longer the interferometer
arms, the better the
sensitivity.
So put one in space,
of course.
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The Michelson Interferometer is a
Fourier Transform Spectrometer
L2
Suppose the input beam is not monochromatic
(but is perfectly spatially coherent):
 Iout =
Beamsplitter
2I + c  Re{E(t+2L1 /c) E*(t+2L2 /c)}
Now, Iout will vary rapidly in time, and most detectors will simply integrate
over a relatively long time, T :
T /2
U
I
L1
Delay
Mirror
T /2
out
(t )dt
 U  2 IT  c Re
T / 2
 E (t  2L / c)E *(t  2L / c) dt
1
2
T / 2
Changing variables: t' = t + 2L1 /c and letting = 2(L2 - L1)/c and T 

U  2 IT  c Re
 E(t ')E *(t ' dt '
The Field Autocorrelation!

Recall that the Fourier Transform of the Field Autocorrelation is the spectrum!!
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Fourier Transform Spectrometer Interferogram
A Fourier Transform Spectrometer's detected light energy vs. delay is called an
interferogram.
Integrated irradiance
Michelson interferometer
integrated irradiance
Spectrum
2/0
0
Intensity
1/
Delay

0
Frequency
The Michelson interferometer output—the interferogram—Fourier
transforms to the spectrum.
The spectral phase plays no role! (The temporal phase does, however.)
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Fourier Transform Spectrometer Data
Actual interferogram from a Fourier Transform Spectrometer
Interferogram
This interferogram
is very narrow, so
the spectrum
is very broad.
Fourier Transform Spectrometers are most commonly used 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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Michelson-Morley experiment
19th-century physicists thought that light was a vibration of a
medium, like sound. So they postulated the existence of a medium
whose vibrations were light: aether.
Michelson and Morley
realized that the earth could
not always be stationary
with respect to the aether.
And light would have a
different path length and
phase shift depending on
whether it propagated
parallel and anti-parallel or
perpendicular to the aether.
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Parallel and
anti-parallel
propagation
Mirror
Perpendicular
propagation
Beamsplitter
Mirror
Supposed velocity of
earth through the aether
Michelson-Morley Experiment: Results
The Michelson interferometer was
(and may still be) the most sensitive
measure of distance (or time) ever
invented and should’ve revealed a
fringe shift as it was rotated with
respect to the aether velocity.
Interference fringes showed no
change as the interferometer
was rotated.
Their apparatus
Michelson and Morley's results
from A. A. Michelson, Studies in
Optics
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The “Unbalanced” Michelson
Interferometer
Misalign mirrors, so
beams cross at an angle.
Input
beam
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[2if(x,y)].
x
q
Mirror
z
Beamsplitter
Now the cross term becomes:
Mirror
Place an
object in
this path
exp[if(x,y)]
Re{ exp[2if(x,y)] exp[-2ikx sinq] }
Iout(x)
Distorted fringes
(in position)
x
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The "Unbalanced" Michelson Interferometer
can sensitively measure phase vs. position.
Placing an object in one arm of a misaligned Michelson
interferometer will distort the spatial fringes.
Spatial fringes distorted
by a soldering iron tip in
one path
Input
beam
q
Mirror
Beamsplitter
Mirror
Phase variations of a small fraction of a wavelength can be measured.
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The Mach-Zehnder Interferometer
Beamsplitter
Mirror
Output
beam
Object
Input
beam
Beamsplitter
Mirror
The Mach-Zehnder interferometer is usually operated “misaligned” and
with something of interest in one arm.
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Mach-Zehnder Interferogram
Nothing in either path
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Plasma in one path
The Sagnac Interferometer
The two beams take the same path around the interferometer and the output light
can either exit or return to the source.
Mirror
Mirror
Mirror
Beamsplitter
Input
beam
Beamsplitter
Mirror
Input
beam
It turns out that no light exits! It all returns to the source!
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Mirror
Why all the light returns to the source in a Sagnac interferometer
Reflection off a
front-surface
mirror yields a
phase shift of p
(180 degrees).
Mirror
For the exit beam:
Clockwise path has phase
shifts of p, p, p, and 0.
Counterclockwise path
Input
has phase shifts of
beam
0, p, p, and 0.
Return
Perfect
cancellation!!
Beamsplitter
beam
Reflective
surface
Mirror
Exit
beam
Reflection off a backsurface mirror yields
no phase shift.
For the return beam: Clockwise path has phase shifts of p, p, p, and 0.
Counterclockwise path has phase shifts of 0, p, p, and p.
Constructive interference!
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The Sagnac Interferometer senses rotation
Suppose that the beam splitter moves by a distance, d, in the time, T, it takes light to
circumnavigate the Sagnac interferometer.
As a result, one beam will travel more, and the other less distance.
I out  E0 exp(ikd )  E0 exp(ikd )
2

I
sin
(kdradius,
)
0
If R = the interferometer
and W = its angular velocity:
2
R
d
d = Rq
q
Sagnac
Interferometer
q WT
d  R  T  R(2 R / c)  2( R 2 ) / c  2 Area / c
Thus, the Sagnac Interferometer's sensitivity to rotation depends on its area. And it
2
need not be round!
out
0
I
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 I sin (2k  Area / c)
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).
You see the
color l when:
2L = ml
L
You only see
bold colors when
m = 1 or 2.
Otherwise the
variation with l is
too fast for the
eye to resolve.
This effect also causes the colors in bubbles and oil films on puddles.
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Newton's Rings
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Multiple-beam interference:
The Fabry-Perot Interferometer or Etalon
A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and
forth. An etalon is a type of Fabry-Perot interferometer, and is a piece of glass with parallel
sides.
The transmitted wave is an infinite series of multiply reflected beams.
r, t = reflection, transmission coefficients from glass to air
L
Transmitted
wave: E0t
Incident wave: E0
Reflected
wave: E0r
nair = 1
Transmitted wave:
n
nair = 1
d = round-trip phase delay
inside medium = 2kL
t 2 E0
t 2 r 2 e  i E0
t 2 (r 2 ei ) 2 E0
t 2 (r 2 e  i )3 E0
E0t  t 2 E0  t 2 r 2e i E0  t 2 (r 2e i )2 E0  t 2 (r 2e i )3 E0  ...
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The Etalon (cont'd)
The transmitted wave field is:
E0t  t 2 E0  t 2 r 2e i E0  t 2 (r 2e i )2 E0  t 2 (r 2e i )3 E0  ...
 t 2 E0 1  (r 2ei )  (r 2ei )2  ...
E0t  t 2 E0 / 1  r 2e i 
The transmittance is:
E
T  0t
E0
2
2
t

1  r 2 e  i
2


t4

2  i
2  i 
(1

r
e
)(1

r
e )


 
 

t4
(1  r 2 )2
(1  r 2 )2



 {1  r 4  2r 2 [1  2sin 2 ( / 2)]}  {1  2r 2  r 4  4r 2 sin 2 ( / 2)]} 
4
{1

r

2cos(

)}

 
 

Dividing numerator and denominator by (1  r 2 ) 2
1
T
1  F sin 2  / 2 
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 2r 
2
 2r 

where: F  
2
1  r  1  R 
2
Etalon Transmittance vs. Thickness, Wavelength, or Angle
Transmittance
1
T
1  F sin 2  / 2 
Transmission
maxima occur
when d / 2 = mp:
2pL/l = mp
or:
  2kL  4 L / 
  L / 2m
The transmittance varies significantly with thickness or wavelength.
We can also vary the incidence angle, which also affects d (via L).
As the reflectance of each surface (r2) approaches 1 (F increases), the widths of the
high-transmission regions become very narrow.
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Does this look familiar?
Recall that a finite train of identical pulses can be written:
E (t )  {III(t / T ) g (t )}  f (t )
where g(t) is a Gaussian envelope over the pulse train.
The light field transmitted by the etalon!
g(t) = exp(-t/t)
E ( )  {III(T / 2 )  G( )}F ( )
The peaks become
Lorentzians.
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The Etalon Free Spectral Range
lFSR =
Free Spectral
Range
Transmittance
The Free Spectral Range is the wavelength range between
transmission maxima.
lFSR
  2kL  4 L / 
4 L
4 L

 2

  FSR

4 L


4 L[1  FSR /  ]
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
 2

4 L
4 L

 2

[1  FSR /  ]
1 1  FSR / 
1
2



 FSR 


2L
2L
T
Etalon Linewidth
1
1  F sin 2  / 2 
  4 L / 
The Linewidth dLW is a transmittance peak's full-width-half-max (FWHM).
1  F sin 2   LW / 2 / 2  2 or sin 2  LW / 4   1/ F
For d << 1, we can make the small argument approx:
 LW / 4 
2
 1/ F
Transmittance
Setting d equal to dLW/2 should yield T = 1/2:
lLW
l
  LW  4 / F
4 L
4 L
1  R 
 2r 


2



Substituting F  
and
we
have:
LW
LW
LW
2
2
 r 



1

R


Or:
 2 1  R  The linewidth is the etalon’s
LW 
wavelength-measurement


2 L  r  accuracy.
2
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The Interferometer or Etalon Finesse
The Finesse, F , is the ratio of the Free Spectral Range and the
Linewidth:
2
2L
F 
 2 1  R 
2 L  r 
Taking
r 1
 r 
 
1  R 
F   /[1  R]
The Finesse is the number of wavelengths the interferometer can
resolve.
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How to use an interferometer to measure wavelength
1. Measure the wavelength to within one Free Spectral Range
using a grating or prism spectrometer to avoid the interferometer’s
inherent ambiguities.
2. Scan the spacing of the two mirrors and record the spacing
when a transmission maximum occurs.
3. If greater accuracy is required, use another (longer)
interferometer with a FSR ~ the above accuracy (line-width) and
with an even smaller line-width (i.e., better accuracy).
Interferometers are the most accurate measures of wavelength available.
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Anti-reflection Coating
Notice that the center of
the round glass plate
looks like it’s missing.
It’s not! There’s an
“anti-reflection coating”
there (on both the front
and back of the glass).
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Anti-reflection Coating Math
Consider a beam incident on a piece of glass (n = ns) with a layer of material (n = nl) if
thickness, h, on its surface.
It can be shown that the Reflectance is (for such thin media, we need to go back to
Maxwell’s equations):
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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nl2  n0 ns
Multilayer coatings
Typical laser mirrors and camera
lenses use many layers.
The reflectance and transmittance
can be tailored to taste!
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Stellar interferometry
Stars are too small to resolve using normal
telescopes, but interferometry can see them.
Stellar
interferometers
operate in the radio
(when the signals
are combined
electronically) and
visible (where the
beams are
combined optically).
Taken from von der Luhe, of
Kiepenheuer-Institut fur
Sonnenphysik, Freiburg,
Germany.
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“Photonic crystals” use interference to guide light—
sometimes around corners!
Borel, et al.,
Opt. Expr. 12,
1996 (2004)
Yellow
indicates
peak field
regions.
Augustin, et al.,
Opt. Expr., 11,
3284, 2003.
Interference controls the path of light. Constructive interference
occurs along the desired path.
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Convolution
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The Convolution
The convolution allows one function to smear or broaden another.

f (t )  g (t ) 

f ( x) g (t  x) dx





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f (t – x) g ( x) dx
changing variables:
xt-x
The convolution can be
performed visually.
Here, rect(x) * rect(x) =
(x)
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Convolution with a delta function
f (t )   t  a) 


f (t  u )  (u – a) du

 f (t  a)
Convolution with a delta function simply centers the
function on the delta-function.
This convolution does not smear out f(t). Since a device’s
performance can usually be described as a convolution of
the quantity it’s trying to measure and some instrument
response, a perfect device has a delta-function instrument
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The Convolution Theorem
The Convolution Theorem turns a convolution into the inverse
FT of the product of the Fourier Transforms:
F {f (t )  g (t )} = F ( w
) G ( )
Proof:


F { f (t )  g (t )}     f ( x) g (t – x) dx  exp( i t ) dt
  




  f ( x)   g (t  x) exp(– it ) dt  dx

 





f ( x){G (  exp(– i x)} dx





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f ( x) exp(– i x) dx G (   F ( G ( 
The Convolution Theorem in action
rect( x)  rect( x)  ( x)
F {( x)} 
F {rect( x)} 
sinc(k / 2)
sinc 2 (k / 2)
sinc(k / 2)  sinc(k / 2)  sinc2 (k / 2)
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The Shah Function
The Shah function, III(t), is an infinitely long train of
equally spaced delta-functions.
t

III(t ) 
  (t  m)
m 
The symbol III is pronounced shah after the Cyrillic character III, which is
said to have been modeled on the Hebrew letter
(shin) which, in turn,
may derive from the Egyptian
a hieroglyph depicting papyrus plants
along the Nile.
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The Fourier Transform of the Shah Function
III(t)



   t  m) exp(i t )dt
 m 




t
 t  m) exp(i t )dt
If  = 2n, where n is an

integer, the sum diverges;
  exp( i m) otherwise, cancellation occurs.
m 
So:
F {III(t )}  III( 
F {III(t)}
m  

2
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Fraunhofer Diffraction
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Diffraction
Light does not always
travel in a straight line.
It tends to bend around
objects. This tendency
is called diffraction.
Shadow of a
hand
illuminated
by a
HeliumNeon laser
Any wave will do this,
including matter waves
and acoustic waves.
Shadow of
a zinc oxide
crystal
illuminated
by a
electrons
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Why it’s hard to see diffraction
Diffraction tends to cause ripples at edges. But poor source
temporal or spatial coherence masks them.
Example: a large spatially incoherent source (like the sun) casts
blurry shadows, masking the diffraction ripples.
Screen
with hole
A point source is required.
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Untilted rays
yield a perfect
shadow of the
hole, but off-axis
rays blur the
shadow.
Diffraction of a wave by a slit
Whether waves in water or electromagnetic
radiation in air, passage through a slit yields
a diffraction pattern that will appear more
dramatic as the size of the slit approaches
the wavelength of the wave.

slit size
  slit size
  slit size
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Diffraction of ocean water waves
Ocean waves passing through slits in Tel Aviv, Israel
Diffraction occurs for all waves, whatever the phenomenon.
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Diffraction Geometry
We wish to find the light electric field after a screen with a hole in it.
This is a very general problem with far-reaching applications.
y0
A(x0,y0)
y1
x0
P1
0
Incident
wave
x1
This region is assumed to be
much smaller than this one.
What is E(x1,y1) at a distance z from the plane of the aperture?
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Diffraction Solution
The field in the observation plane, E(x1,y1), at a distance z from the aperture plane is given
by a convolution:
E ( x1 , y1 ) 

h( x1  x0 , y1  y0 ) E ( x0 , y0 ) dx0 dy0
A (x0 , y0 )
where :
1 exp(ikr01 )
h( x1  x0 , y1  y0 ) 
i
r01
and :
r01  z   x0  x1    y0  y1 
2
2
2
A very complicated result! And we cannot approximate r01 in the exp
by z because it gets multiplied by k, which is big, so relatively small
changes in r01 can make a big difference!
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Fraunhofer Diffraction: The Far Field
Recall the Fresnel diffraction result:
 x12  y12 
exp(ikz )
E  x1 , y1  
exp ik

i z
2z 


  (2 x0 x1  2 y0 y1 ) ( x02  y02 )  
exp ik 

  E  x0 , y0  dx0 dy0
2
z
2
z
 
 
A (x0 , y0 )
Let D be the size of the aperture: D 2 = x02 + y02.
When kD2/2z << 1, the quadratic terms << 1, so we can neglect them:
 x12  y12 
exp(ikz )
E  x1 , y1  
exp ik

i z
2
z



 ik

exp   x0 x1  y0 y1   E  x0 , y0  dx0 dy0
 z

A ( x0 , y0 )
This condition corresponds to going far away: z >> kD2/2 = D2/
If D = 100 microns and  = 1 micron, then z >> 30 meters, which is far!
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Fraunhofer Diffraction Conventions
As in Fresnel diffraction, we’ll neglect the phase factors, and we’ll explicitly write the
aperture function in the integral:
 
E  x1 , y1  

 ik

exp   x0 x1  y0 y1   A( x0 , y0 ) E ( x0 , y0 ) dx0 dy0
 z

 
This is just a Fourier Transform!
E(x0,y0) = constant if a plane wave
Interestingly, it’s a Fourier Transform from position, x0, to another
position variable, x1 (in another plane). Usually, the Fourier “conjugate
variables” have reciprocal units (e.g., t & , or x & k). The conjugate
variables here are really x0 and kx = kx1/z, which have reciprocal units.
So the far-field light field is the Fourier Transform of the apertured field!
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The Fraunhofer Diffraction formula
We can write this result in terms of the off-axis k-vector components:
 
E  kx , k y  

E(x,y) = const if a plane wave
exp  i  k x x  k y y  A( x, y ) E ( x, y ) dx dy
Aperture function
 
where we’ve dropped
the subscripts, 0 and 1,
E  kx , k y   F
kx = kx1/z
and:
and
 A( x, y) E ( x, y)
ky = ky1/z
kx
kz
or:
qx = kx /k = x1/z
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and qy
= ky /k = y1/z
ky
Fraunhofer Diffraction from a slit
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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Fraunhofer Diffraction from a Square Aperture
The diffracted field is a sinc function
in both x1 and y1 because the Fourier
transform of a rect function is sinc.
Diffracted
irradiance
Diffracted
field
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Diffraction from a Circular Aperture
A circular aperture
yields a diffracted
"Airy Pattern,"
which involves a
Bessel function.
Diffracted Irradiance
Diffracted field
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Diffraction from small and large circular apertures
Far-field
intensity pattern
from a small
aperture
Recall the Fourier scaling!
Far-field
intensity pattern
from a large
aperture
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Fraunhofer diffraction from two slits
w
-a
w
0
a
x0
A(x0) = rect[(x0+a)/w] + rect[(x0-a)/w]
E ( x1 )  F { A( x0 )}
 sinc[w(kx1 / z ) / 2]exp[ia(kx1 / z )] 
sinc[w(kx1 / z ) / 2]exp[ia(kx1 / z )]
E ( x1 )  sinc( wkx1 / 2 z) cos(akx1 / z)
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kx1/z
Diffraction from one- and two-slit screens
Fraunhofer diffraction patterns
One slit
Two slits
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Diffraction from multiple slits
Slit
Pattern
Infinitely many equally spaced
slits yields a far-field pattern
that’s the Fourier transform
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Diffraction
Pattern
Young’s Two Slit Experiment and Quantum Mechanics
Imagine using a beam so weak that only one photon passes through the screen at a
time. In this case, the photon would seem to pass through only one slit at a time,
yielding a one-slit pattern.
Which pattern occurs?
Possible Fraunhofer diffraction patterns
Each photon
passes
through only
one slit
Each photon
passes
through
both slits
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Fresnel Diffraction
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Fresnel Diffraction: Approximations
But, in the denominator, we can approximate r01 by z.
And, in the numerator, we can write:
2
2
x x   y y 
r01  z 2   x0  x1    y0  y1   z 1   0 1    0 1 
 z   z 
2
But if 
1,
2
1   1  / 2
2
2
 1  x0  x1 2 1  y0  y1 2 
x

x
y

y
 0 1   0 1
 z 1  


z



 
2z
2z
 2  z  2  z  
This yields:
E ( x1 , y1 ) 

A ( x0 , y0 )
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2
2



x

x
y

y


1


0
1
0
1
exp ik  z 

  E ( x0 , y0 ) dx0 dy0
i z
2z
2z
 

 

Fresnel Diffraction: Approximations
Multiplying out the squares:
E  x1 , y1  

  ( x02  2 x0 x1  x12 ) ( y02  2 y0 y1  y12 )  
1
exp ik  z 

  E ( x0 , y0 ) dx0 dy0
i z
2z
2z
 
 
A ( x0 , y0 )
Factoring out the quantities independent of x0 and y0:
 x12  y12 
exp(ikz )
E  x1 , y1  
exp ik

i z
2
z



  (2 x0 x1  2 y0 y1 ) ( x02  y02 )  
exp ik 

 E  x0 , y0  dx0 dy0
2z
2 z  
 
A (x0 , y0 )
This is the Fresnel integral. It's complicated!
It yields the light wave field at the distance z from the screen.
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Diffraction Conventions
We’ll typically assume that a plane wave is incident on the aperture.
E ( x0 , y0 )  const
It still has an exp[i( t – k z)], but it’s
constant with respect to x0 and y0.
And we’ll explicitly write the aperture function in the integral:
 x12  y12 
exp(ikz )
E  x1 , y1  
exp ik

i z
2
z



  (2 x0 x1  2 y0 y1 ) ( x02  y02 )  


exp ik 

  A(x0 , y0 ) dx0 dy0
2
z
2
z


 

And we’ll usually ignore the various factors in front:
 
E  x1 , y1  

 
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2
2

  (2 x0 x1  2 y0 y1 ) ( x0  y0 )  

exp ik 

  A(x0 , y0 ) dx0 dy0
2z
2z 

 

Fresnel Diffraction: Example
Fresnel Diffraction from a single slit:
Close
to the
slit
Slit
Incident
plane wave
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z
Far
from
the
slit
Fresnel Diffraction from a Slit
Irradiance
This irradiance vs. position emerges from a slit illuminated by a laser.
x1
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Diffraction Approximated
The approximate intensity vs.
position from an edge:
Such effects can be
modeled by measuring the
distance on a “Cornu Spiral”
But most useful diffraction effects do not occur in the Fresnel
diffraction regime because it’s too complex.
For a cool Java applet that computes Fresnel diffraction patterns, try
http://falstad.com/diffraction/
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