Interference
When you see interference
and when you don’t
The Michelson
Interferometer
“Fringes”
Prof. Rick Trebino
Georgia Tech
www.physics.gatech.edu/frog/lectures
Opals use interference
between tiny
structures to yield
bright colors.
Constructive vs. destructive interference;
Coherent vs. incoherent interference
Waves that combine
in phase add up to
relatively high irradiance.
Waves that combine 180°
out of phase cancel out
and yield zero irradiance.
Waves that combine with
lots of different phases
nearly cancel out and
yield very low irradiance.
=
Constructive
interference
(coherent)
=
Destructive
interference
(coherent)
=
Incoherent
addition
Interfering many waves: in phase, out of
phase, or with random phase…
Im
Re
If we plot the
complex
amplitudes:
Waves adding exactly
out of phase, adding to
zero (coherent
destructive addition)
Waves adding exactly
in phase (coherent
constructive addition)
Waves adding with
random phase,
partially canceling
(incoherent
addition)
The Irradiance (intensity) of a light wave
The irradiance of a light wave is proportional to the square of the
electric field:
I 
or:
1 c  E0
2
~
2
where:
2
E0  E0 x E0*x  E0 y E0* y  E0 z E0*z
This formula only works when the wave is of the form:


E  r , t   Re E0 exp i k  r   t 


The relative phases are the key.
The irradiance (or intensity) of the sum of two waves is:
I  I1  I 2  c Re E1  E2*
E
and E
are complex amplitudes.
~1
~2
Im
If we write the amplitudes in
terms of their intensities, Ii,
and absolute phases, qi,
Ei
Ei  Ii exp[iqi ]
In incoherent interference, the qi – qj
will all be random.
qi
Re
I  I1  I 2  2 I1 I 2 Re exp[i(q1  q2 )]
Imagine adding many such fields.
In coherent interference, the qi – qj
will all be known.
A
Im
2 I1 I 2
I1  I 2
0
q1 – q2
I
Re
Adding many fields with random phases
We find:
Etotal  [ E1  E2  ...  EN ] exp[i(k  r   t )]
I total  I1  I 2  ...  I N  c Re E1E2*  E1E3*  ...  EN 1EN* 
I1, I2, … In are the irradiances of the
various beamlets. They’re all
positive real numbers and they add.
Itotal = I1 + I2 + … + In
The intensities simply add!
Two 20W light bulbs yield 40W.
Ei Ej* are cross terms, which have the
phase factors: exp[i(qi-qj)]. When the
q’s are random, they cancel out!
All the
relative
phases
I1+I2+…+IN
exp[i(qi  q j )]
Im
Re
exp[i(q k  ql )]
Scattering
Molecule
When a wave encounters a
small object, it not only reemits the wave in the
forward direction, but it
also re-emits the wave in
all other directions.
Light source
This is called scattering.
Scattering is everywhere. All molecules scatter light. Surfaces
scatter light. Scattering causes milk and clouds to be white and
water to be blue. It is the basis of nearly all optical
phenomena.
Scattering can be coherent or incoherent.
Spherical waves
A spherical wave is also a solution to Maxwell's equations and is a
good model for the light scattered by a molecule.
Note that k and r are
not vectors here!
E(r , t )   E0 / r  Re{exp[i(kr   t )]}
where k is a scalar, and
r is the radial magnitude.
A spherical wave has spherical wave-fronts.
Unlike a plane wave, whose amplitude remains constant as it
propagates, a spherical wave weakens. Its irradiance goes as 1/r2.
Scattered spherical waves often
combine to form plane waves.
A plane wave impinging on a surface (that is, lots of very small
closely spaced scatterers!) will produce a reflected plane wave
because all the spherical wavelets interfere constructively along a
flat surface.
To determine interference in a given
situation, we compute phase delays.
Wave-fronts
Because the phase is
constant along a
wave-front, we
compute the phase
delay from one wavefront to another
potential wave-front.
L1
L2
L3
L4
Scatterer
Potential
wave-front
i  k Li
If the phase delay for all scattered waves is the same (modulo 2p),
then the scattering is constructive and coherent. If it varies uniformly
from 0 to 2p, then it’s destructive and coherent.
If it’s random (perhaps due to random motion), then it’s incoherent.
Coherent constructive scattering:
Reflection from a smooth surface when angle
of incidence equals angle of reflection
A beam can only remain a plane wave if there’s a direction for which
coherent constructive interference occurs.
The wave-fronts are
perpendicular to
the k-vectors.
qi qr
Consider the
different phase
delays for
different paths.
Coherent constructive interference occurs for a reflected beam if the
angle of incidence = the angle of reflection: qi = qr.
Coherent destructive scattering:
Reflection from a smooth surface when the
angle of incidence is not the angle of reflection
Imagine that the reflection angle is too big.
The symmetry is now gone, and the phases are now all different.
 = ka sin(qtoo big)
qi qtoo big
a
 = ka sin(qi)
Potential
wave front
Coherent destructive interference occurs for a reflected beam direction
if the angle of incidence ≠ the angle of reflection: qi ≠ qr.
Incoherent scattering: reflection from a
rough surface
No matter which direction we
look at it, each scattered wave
from a rough surface has a
different phase. So scattering is
incoherent, and we’ll see weak
light in all directions.
This is why rough surfaces look different from smooth surfaces and
mirrors.
Diffraction Gratings
Scattering ideas explain what
happens when light impinges
on a periodic array of grooves.
Constructive interference
occurs if the delay between
adjacent beamlets is an integral
number, m, of wavelengths.
Path difference: AB – CD = m
a sin(q m )  sin(qi )  m
Scatterer
a
D
qm
C
Incident qi
wave-front
qm
a
A
qi
Scatterer
B
Potential
diffracted
wave-front
AB = a sin(qm)
CD = a sin(qi)
where m is any integer.
A grating has solutions of zero, one, or many values of m, or orders.
Remember that m and qm can be negative, too.
Diffraction
orders
Because the diffraction angle depends
on , different wavelengths are
separated in the nonzero orders.
Diffraction angle, qm()
Incidence
angle, qi
First order
Zeroth order
Minus
first order
No
wavelength
dependence
occurs in
zero order.
The longer the wavelength, the larger its deflection in each
nonzero order.
Real diffraction gratings
m = -1
m=0
m =1
m=2
Diffracted white light
White light diffracted by a real grating.
Diffraction
gratings
The dots on a CD are
equally spaced (although
some are missing, of
course), so it acts like a
diffraction grating.
World’s largest diffraction grating
Lawrence Livermore National Lab
The irradiance when combining a beam with
a delayed replica of itself has fringes.
The irradiance is given by:

I  I1  c Re E1  E
*
2
 I
2
Suppose the two beams are E0 exp(it) and E0 exp[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 )]


 2I 0  c Re E0 exp[it ]
2
 2 I 0  c E0 cos[t ]
2
“Dark fringe”
I
“Bright fringe”
I  2 I 0  2 I 0 cos[t ]
t
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–t)
Translation stage
Moving a mirror backward by a distance L yields a delay of:
t  2 L /c
t
Do not forget the factor of 2!
Light must travel the extra distance
to the mirror—and back!
 2 L / c  2 k L
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.
The Michelson Interferometer
The Michelson Interferometer splits a
beam into two and then recombines Mirror
them at the same beam splitter.
Input
beam
L2
Output
beam
Beamsplitter
Suppose the input beam is a plane
wave:
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)
since I  I1  I 2  (c 0 / 2) E0
“Dark fringe”
I
“Bright fringe”
where: L = 2(L2 – L1)
Fringes (in delay):
2
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(2p L / )
Fringes (in delay)
I

L = 2(L2 – L1)
Crossed Beams
x
k  k cos q zˆ  k sin q xˆ
k  k cos q zˆ  k sin q xˆ
k
q
z
r  xxˆ  yyˆ  zzˆ
 k  r  k cosq z  k sin q x
k
k  r  k cos q z  k sin q x


I  2I 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 q  kx sin q  E0* exp  i (t  kz cos q  kx sin q 


 Re E0 exp  2ikx sin q 
2
 E0 cos(2kx sin q )
Fringe spacing:   2p /(2k sin q )
  /(2sin q )
2
Fringes (in position)
I
x
Irradiance vs. position for crossed
beams
Fringes occur where the beams overlap in space and time.
Big angle: small fringes.
Small angle: big fringes.
The fringe spacing, :
Large angle:
   /(2sin q )
As the angle decreases to
zero, the fringes become
larger and larger, until finally, at
q = 0, the intensity pattern
becomes constant.
Small angle:
You can't see the spatial fringes unless
the beam angle is very small!
The fringe spacing is:
   /(2sin q )
 = 0.1 mm is about the minimum fringe spacing you can see:
q  sin q   /(2)
 q  0.5 m / 200 m
 1/ 400 rad  0.15
The Michelson
Interferometer
and Spatial Fringes
Suppose we misalign the mirrors
so the beams cross at an angle
when they recombine at the beam
splitter. And we won't scan the delay.
x
z
Input
beam
Mirror
Beamsplitter
Fringes
Mirror
If the input beam is a plane wave, the cross term becomes:
Re E0 exp i( t  kz cos q  kx sin q  E0* exp  i ( t  kz cos q  kx sin q 
 Re exp  2ikx sin q 
 cos(2kx sin q )
Crossing beams maps
delay onto position.
Fringes (in position)
I
x
The Michelson
Interferometer
and Spatial Fringes
Suppose we change one arm’s
path length.
x
z
Input
beam
Mirror
Beamsplitter
Fringes
Mirror
Re  E0 exp i( t  kz cos q  kx sin q  2kd  E0* exp  i( t  kz cos q  kx sin q 
 Re exp  2ikx sin q  2kd 
 cos(2kx sin q  2kd )
Fringes (in position)
I
The fringes will shift in
phase by 2kd.
x
Misalign mirrors, so
beams cross at an angle.
The Unbalanced
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[2i(x,y)].
Input
beam
x
q
Mirror
z
Beamsplitter
Mirror
Now the cross term becomes:
Re{ exp[2i(x,y)] exp[-2ikx sinq] }
Place an
object in
this path
exp[i(x,y)]
Iout(x)
Distorted
fringes
(in position)
x
The Unbalanced Michelson Interferometer
can sensitively measure phase vs. position.
Spatial fringes distorted
by a soldering iron tip in
one path
Placing an object in one arm of a
misaligned Michelson interferometer
will distort the spatial fringes.
Input
beam
q
Mirror
Beamsplitter
Mirror
Phase variations of a small fraction of a wavelength can be measured.
Technical point about Michelson
interferometers:
Input
the compensator
beam
Beamsplitter
plate
Output
beam
Mirror
So a compensator
plate (identical to the
beam splitter) is
usually added to
equalize the path
length through glass.
Mirror
If reflection occurs off
the front surface of
beam splitter, the
transmitted beam
passes through beam
splitter three times;
the reflected beam
passes through only
once.
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.
Mach-Zehnder Interferogram
Nothing in either path
Plasma in one path
Newton's Rings
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  when
constructive
interference
occurs.
L
You only see bold
colors when m = 1
(possibly 2).
Otherwise the
variation with  is
too fast for the
eye to resolve.
This effect also causes the colors in bubbles and oil films on puddles.
Other applications of interferometers
To frequency filter a beam (this is often done inside a laser).
Money is now coated with interferometric inks to help foil
counterfeiters. Notice the shade of the “20,” which is shown from two
different angles.
Anti-reflection Coatings
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).
Such coatings have
been common on
photography lenses and
are now common on
eyeglasses. Even my
new watch is ARcoated!
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.