Constructive vs. destructive interference; Coherent vs ...

Constructive vs. destructive interference;
Coherent vs. incoherent interference

Waves that combine Constructive
in phase add up to
relatively high irradiance. = interference
(coherent)

Waves that combine 180° Destructive
interference
=out of phase cancel out
and yield zero irradiance. (coherent)

Waves that combine with = Incoherent
lots of different phases addition
nearly cancel out and Source: Tribino, Georgia Tech
yield very low irradiance.

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:

or: I = 12 cε E~!0 2

where: " 2
E! 0
= E! 0x E! 0*x + E! 0 y E! 0*y + E! 0z E! 0*z

This formula only works when the wave is of the form:

! k! ⋅ r! −ω t &'
( )( )E!r!,t E" 0
= Re exp$%i

The relative phases are the key.

The irradiance (or intensity) of the sum of two waves is:

{ }I = I1 + I2 + cε Re E! 1 ⋅ E! 2* E~ 1 and ~E2 are complex amplitudes.

If we write the amplitudes in E! i ∝ Ii exp[−iθi ] Im E! i
terms of their intensities, Ii, A
and absolute phases, θi, θi
Re

I = I1 + I2 + 2 I1 I2 Re{exp[−i(θ1 −θ2 )]} Im

Imagine adding many such fields. 2 I1 I2
In coherent interference, the θi – θj
will all be known. I1 + I2 θ1 – θ2Re

In incoherent interference, the θi – θj 0 I
will all be random.

Adding many fields with random phases

We find: " r"
exp[i(k
E! total = [E! 1 + E! 2 + ...+ E! N ] ⋅ −ω t )]

{ }Itotal = I1 + I2 + ...+ IN + cε Re E! 1E! 2* + E! 1E! 3* + ...+ E! N−1E! 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!

Itotal = I1 + I2 + … + In All the Im
relative Re
phases
exp[i(θk −θl )]
The intensities simply add! I1+I2+…+IN

Two 20W light bulbs yield 40W. exp[i(θi −θ j )]

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
interference (possibly 2).
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

x2 + (R − d )2 = 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 E! 1 ⋅ E! 2* + I2

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 = 2I0 + cε Re E! 0 exp[iωt]⋅ E! 0* exp[−iω(t − τ )]
{ }= 2I0 + cε Re E! 0 2 exp[iωτ ]
= 2I0 + cε E! 0 2 cos[ωτ ]
“Dark fringe” I

“Bright fringe”

I = 2I0 + 2I0 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:!

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

ωτ = 2ωL/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!

{ }Iout = I 1+ I2 + cε Re E0 exp[i(ωt − kz − 2kL1)]E0* exp[−i(ωt − kz − 2kL2)]

{ }= I + I + 2I Re exp[2ik(L2 − L1)] since I ≡ I1 = I2 = (cε0 / 2) E0 2

= 2I {1+ cos(kΔL)} “Dark fringe” I

“Bright fringe”

where: ΔL = 2(L2 – L1)

Fringes (in delay): ΔL = 2(L2 – L1)

The Michelson Input!
Interferometer beam!

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

Delay!
Mirror!

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

Fringes (in delay)
I

λ



ΔL = 2(L2 – L1)

Crossed Beams x

r
! r!)] k+
kk!!kk!−++−⋅⋅r!==r!r!=kk==ccxkkooxˆcssc+θoθossyzzθˆˆθyˆ+−zz+kk−+sszkikizˆnnssθθiinnxxθˆθˆ
⇒ x θ

r z


x exp[−i(ω k− ⋅ r!

! E0* !
{ }I = 2I0 + cε Re E0 exp[i(ωt − k ⋅ t − k )]
−
+

Cross term is proportional to:

{ }Re E0 exp[i(ωt − kz cosθ − kx sinθ ]E0* exp[−i(ωt − kz cosθ + kx sinθ ]

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

∝ E0 2 cos(2kx sinθ ) I


Fringe spacing: Λ = 2π /(2k sinθ )
x
= λ /(2sinθ )

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 /(2sinθ )

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 /(2sinθ )

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

θ ≈ sinθ = λ /(2Λ)
⇒ θ ≈ 0.5µm / 200µm

≈ 1/ 400 rad = 0.15o

The Michelson x Input!
Interferometer beam!
and Spatial Fringes z
Fringes
Mirror! Mirror!

Suppose we misalign the mirrors Beam-!
so the beams cross at an angle splitter!
when they recombine at the beam
splitter. And we won't scan the delay.

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

{ }Re E0 exp[i(ωt − kz cosθ − kx sinθ ] E0* exp[−i(ω t − kz cosθ + kx sinθ ]

∝ Re{exp[−2ikx sinθ ]} Fringes (in position)

∝ cos(2kx sinθ ) I



Crossing beams maps x
delay onto position.

The Michelson x Input!
Interferometer beam!
and Spatial Fringes z

Mirror!

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

Mirror!

{ }Re E0 exp[i(ωt − kz cosθ − kx sinθ + 2kd ] E0* exp[−i(ω t − kz cosθ + kx sinθ ]

∝ Re{exp[−2ikx sinθ + 2kd ]}

∝ cos(2kx sinθ + 2kd )

The fringes will shift in Fringes (in position)
phase by 2kd.
I



x

The Unbalanced Misalign mirrors, so
Michelson Interferometer beams cross at an angle.

Input! x

beam!

Now, suppose an object is θ

z

placed in one arm. In addition Mirror!

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
will distort the spatial fringes.
one path

Input!
beam!

Mirror! θ



Beam-!
splitter!

Mirror!

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

Technical point about Michelson

interferometers:

the compensator Input! Beam-!
plate beam! splitter!

Output!
beam!

Mirror!

So a compensator Mirror! If reflection occurs off
plate (identical to the the front surface of
beam splitter) is beam splitter, the
usually added to transmitted beam
equalize the path passes through beam
length through glass. splitter three times;
the reflected beam
passes through only
once.

The Mach-Zehnder Interferometer

Mirror! Beam-! Output
splitter! beam

Object

Input! Mirror!
beam!

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!

Borel, et al., Yellow
Opt. Expr. 12, indicates
1996 (2004) peak field
regions.

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 Light source
small object, it not only re-
emits the wave in the
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 L1 Potential
constant along a L2 wave-front
wave-front, we
compute the phase L3 φi = k Li
delay from one wave- L4
front to another
potential wave-front.

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 θi θr
perpendicular to

the k-vectors.

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
a wave front

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.