Constructive vs.

destructive interference;
Coherent vs. incoherent interference

Waves that combine Constructive

in phase add up to interference
relatively high irradiance.
= (coherent)

Waves that combine 180° Destructive

out of phase cancel out = interference
and yield zero irradiance. (coherent)

Waves that combine with

Incoherent
lots of different phases = addition
nearly cancel out and
yield very low irradiance.

Source: Tribino, Georgia Tech

Interfering many waves: in phase, out of
phase, or with random phase…
Im

Re
If we plot the
Waves adding exactly
complex
in phase (coherent
amplitudes: constructive addition)

Waves adding exactly Waves adding with

out of phase, adding to random phase,
zero (coherent partially canceling
destructive addition) (incoherent
addition)
The Irradiance (intensity) of a light wave!
The irradiance of a light wave is proportional to the square of the
electric field:
! 2

or: I = 12 c ε E
~0

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 − ω t &'
$
( )
"
The relative phases are the key.
The irradiance (or intensity) of the sum of two waves is:

I = I1 + I 2 + cε Re { E1 ⋅ E2* } E
~1
and ~E2 are complex amplitudes.
! !
Im
If we write the amplitudes in Ei
A
!
terms of their intensities, Ii,
and absolute phases, θi,
Ei ∝ I i exp[−iθ i ] θi
! Re

I = I1 + I 2 + 2 I1 I 2 Re {exp[−i(θ1 − θ 2 )]} Im

Imagine adding many such fields. 2 I1 I 2

In coherent interference, the θi – θj I1 + I 2 θ1 – θ2
will all be known. Re
0
In incoherent interference, the θi – θj I
will all be random.
Adding many fields with random phases
We find:
" "
Etotal = [ E1 + E2 + ...+ E N ] exp[i( k ⋅ r − ω t)]
! ! ! !
I total = I1 + I 2 + ...+ I N + cε Re { E1 E2* + E1 E3* + ...+ E N −1 E N* }
! ! ! ! ! !
I1, I2, … In are the irradiances of the Ei Ej* are cross terms, which have the
various beamlets. They’re all phase factors: exp[i(θi-θj)]. When the
positive real numbers and they add. θ’s are random, they cancel out!

All the
Itotal = I1 + I2 + … + In relative Im
phases
Re
I1+I2+…+IN
The intensities simply add!
Two 20W light bulbs yield 40W.
exp[i (θi − θ j )] exp[i(θ k − θl )]
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

You only see bold
color λ when
colors when m = 1
constructive
(possibly 2).
interference
Otherwise the
occurs.

variation with λ is
too fast for the
L eye to resolve.

This effect also causes the colors in bubbles and oil films on puddles.
Newton's Rings Animation - http://extraphysics.com/java/models/newtRings.html

2
x2 + ( R − d ) = R2
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 AR-
coated!
The irradiance when combining a beam with
a delayed replica of itself has fringes.
The irradiance is given by:

I = I1 + cε Re { E1 ⋅ E2* } + I 2
! !
Suppose the two beams are E0 exp(iωt) and E0 exp[iω(t-τ)], that is,
a beam and itself delayed by some time τ :

I = 2I 0 + cε Re { E0 exp[iωt]⋅ E0* exp[−iω (t − τ )]}

! !
2
{
= 2I 0 + cε Re E0 exp[iωτ ]
2
! }
= 2I 0 + cε E0 cos[ωτ ] “Dark fringe” I
“Bright fringe”
!
I = 2I 0 + 2I 0 cos[ωτ ]
τ
Varying the delay on purpose
Simply moving a mirror can vary the delay of a beam by many
wavelengths.!
Input !
Mirror! beam!E(t)

Output !
beam!E(t–τ)

Translation stage!

Moving a mirror backward by a distance L yields a delay of:!

Do not forget the factor of 2!!
τ = 2 L /c Light must travel the extra distance !
to the mirror—and back!!

ωτ = 2 ωL / 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 Input!
beam!

The Michelson Interferometer splits a L2 Output!

beam into two and then recombines Mirror!
beam!
them at the same beam splitter.
Beam-! L1
splitter!
Suppose the input beam is a plane Delay!
wave: Mirror!

I out = I 1 + I 2 + cε Re {E0 exp [i(ω t − kz − 2kL1 )] E0* exp [−i(ω t − kz − 2kL2 )]}
2
= I + I + 2 I Re {exp [2ik ( L2 − L1 )]} since I ≡ I1 = I 2 = (cε 0 / 2) E0
= 2 I {1 + cos(k ΔL)} “Dark fringe” I
“Bright fringe”

where: ΔL = 2(L2 – L1)

Fringes (in delay): ΔL = 2(L2 – L1)

 Input!
The Michelson beam!
Interferometer L2 Output!
Mirror! beam!
The most obvious application of Beam-! L1
the Michelson Interferometer is splitter!
to measure the wavelength of Delay!
monochromatic light. Mirror!

Iout = 2I {1 + cos(k ΔL)} = 2I {1 + cos(2π ΔL / λ )}

Fringes (in delay)

I
λ

ΔL = 2(L2 – L1)
 Crossed Beams x
r
! k+
k!+ = k cosθ ẑ + k sin θ x̂
k− = k cosθ ẑ − k sin θ x̂ θ

z

!
r = xx̂ + yŷ + zẑ
! ! r
⇒ k!+ ⋅ r = k cosθ z + k sin θ x k−
!
k− ⋅ r = k cosθ z − k sin θ x
! ! * ! !
I = 2I 0 + cε Re { E0 exp[i(ω t − k+ ⋅ r )]E0 exp[−i(ω t − k− ⋅ r )]}
Cross term is proportional to:
Re {E0 exp [i (ωt − kz cos θ − kx sin θ ] E0* exp [ −i(ωt − kz cos θ + kx sin θ ]}

{ 2
∝ Re E0 exp [ −2ikx sin θ ] } Fringes (in position)
2
I

∝ E0 cos(2kx sin θ )
Fringe spacing: Λ = 2π /(2k sin θ )
x
= λ /(2sin θ )
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, Λ: Large angle:

Λ = λ /(2sin θ )

As the angle decreases to Small angle:

zero, the fringes become
larger and larger, until finally, at
θ = 0, the intensity pattern
becomes constant.
You can't see the spatial fringes unless
the beam angle is very small!
The fringe spacing is:

Λ = λ /(2sin θ )

Λ = 0.1 mm is about the minimum fringe spacing you can see:

θ ≈ sin θ = λ /(2Λ)
⇒ θ ≈ 0.5µ m / 200µ m
≈ 1/ 400 rad = 0.15o
 The Michelson x
Input!
Interferometer z
beam!

and Spatial Fringes Mirror!

Suppose we misalign the mirrors Beam-! Fringes

so the beams cross at an angle splitter!
when they recombine at the beam
splitter. And we won't scan the delay. Mirror!

If the input beam is a plane wave, the cross term becomes:

Re {E0 exp [i (ω t − kz cos θ − kx sin θ ] E0* exp [ −i (ω t − kz cos θ + kx sin θ ]}

∝ Re {exp [ −2ikx sin θ ]}
Fringes (in position)
∝ cos(2kx sin θ ) I

Crossing beams maps
delay onto position. x
 The Michelson x
Input!
beam!
Interferometer z

and Spatial Fringes Mirror!

Beam-! Fringes
Suppose we change one arm’s splitter!
path length.
Mirror!

Re {E0 exp [i (ω t − kz cos θ − kx sin θ + 2kd ] E0* exp [ −i (ω t − kz cos θ + kx sin θ ]}

∝ Re {exp [ −2ikx sin θ + 2kd ]}
∝ cos(2kx sin θ + 2kd )
Fringes (in position)
I

The fringes will shift in
phase by 2kd.
x
The Unbalanced Misalign mirrors, so
beams cross at an angle.
Michelson Interferometer Input!
beam! x
Now, suppose an object is
placed in one arm. In addition Mirror! θ
z
to the usual spatial factor,
Beam-!
one beam will have a spatially splitter!
varying phase, exp[2iφ(x,y)]. Place an
Mirror! object in
this path
Now the cross term becomes:
exp[iφ(x,y)]
Re{ exp[2iφ(x,y)] exp[-2ikx sinθ] }
Iout(x)
Distorted
fringes
(in position)
x
The Unbalanced Michelson Interferometer
can sensitively measure phase vs. position.

Spatial fringes distorted Placing an object in one arm of a

by a soldering iron tip in misaligned Michelson interferometer
one path will distort the spatial fringes.

Input!
beam!

θ

Mirror!

Beam-!
splitter!

Mirror!

Phase variations of a small fraction of a wavelength can be measured.

Technical point about Michelson
interferometers:
the compensator Input!
beam! Beam-!
plate splitter!

Output!
beam!
Mirror!

If reflection occurs off

the front surface of
So a compensator beam splitter, the
plate (identical to the transmitted beam
beam splitter) is passes through beam
usually added to splitter three times;
equalize the path Mirror! the reflected beam
length through glass. passes through only
once.
The Mach-Zehnder Interferometer

Beam-!
splitter!
Mirror!
Output
beam
Object

Input!
beam!
Mirror!
Beam-!
splitter!

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

Photonic crystals use interference to
guide light—sometimes around corners!
Yellow
indicates
Borel, et al., peak field
Opt. Expr. 12, regions.
1996 (2004)

Augustin, et al.,
Opt. Expr., 11,
3284, 2003.

Interference controls the path of light. Constructive interference occurs

along the desired path.
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.
Scattering
Molecule
When a wave encounters a
small object, it not only re-
emits the wave in the Light source
forward direction, but it
also re-emits the wave in
all other directions.

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 − ω 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
L1
wave-front, we
compute the phase L2
delay from one wave-
L3 Potential
front to another wave-front
potential wave-front. L4
φi = k Li
Scatterer
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. θi θr

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: θi = θr.
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.

φ = ka sin(θtoo big) θi θtoo big φ = ka sin(θi)

Potential
wave front
a

Coherent destructive interference occurs for a reflected beam direction

if the angle of incidence ≠ the angle of reflection: θi ≠ θr.
 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.