Chapter 35
Interference
PowerPoint® Lectures for
University Physics, 14th Edition
Lectures by Jason Harlow
– Hugh D. Young and Roger A. Freedman Amended by Scott Hildreth – Chabot College
© 2016 Pearson Education Inc.
Learning Goals for Chapter 35
• What happens when two waves combine (interfere) in space.
• Understand interference pattern formed by interference of
two coherent light waves.
• Calculate intensity at locations of interference pattern.
• Understand how interference occurs when light reflects from
two surfaces of a thin film.
• Interferometry, & how interference enables measurement of
extremely small distances.
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Introduction
• Why soap bubbles show
vibrant color patterns, even
though soapy water is
colorless!
• What causes the multicolored
reflections from CDs & DVDs?
• Look at optical effects that
depend on wave nature of light.
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Principle of superposition
• Interference occurs in any situation
where two or more waves overlap in space.
• Total wave at any point at any instant of time
is governed by principle of superposition:
When two or more waves overlap,
resultant displacement at any point & at any instant
is found by adding instantaneous displacements
that would be produced by individual waves
if each were present alone.
Wave fronts from a disturbance
Interference effects most easily seen when combine
sinusoidal waves with a single frequency & wavelength.
Constructive and destructive interference
• Concepts of
constructive
& destructive
interference apply to
water waves as well
as to light & sound
waves.
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Constructive and destructive interference
• Two identical sources of
monochromatic waves,
S1 and S2.
• Two sources permanently
in phase
vibrating in unison.
• Constructive interference
occurs at point a
equidistant from both
sources.
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Conditions for Constructive Interference
• Distance S2 to point b
is exactly two (2)
wavelengths greater
than distance S1 to b.
• Both waves arrive in
phase
• Waves “reinforce”
each other.
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Conditions for Destructive Interference
• Distance S1 to point c
is half-integral # of
wavelengths greater
than distance S2 to c.
• Both waves cancel or
partly cancel each
other.
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Constructive and Destructive interference
• Start with 2 identical sources of
monochromatic waves, S1 & S2,
in phase.
• Red curves show all positions
where constructive interference
occurs (antinodal curves).
• Nodal curves would show
where destructive interference
occurs.
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Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where
one part of a wave front can interact with another part.
One way to study this is to do a double-slit experiment:
So what do we see?
Is light a series of bullets?
Or a wave?
Coherent Light Source arriving at both slits at exactly
the same time…
Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where
one part of a wave front can interact with another part.
One way to study this is to do a double-slit experiment:
Is light a series of bullets?
Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where
one part of a wave front can interact with another part.
One way to study this is to do a double-slit experiment:
Or is light a wave that interferes?
Interference – Young’s Double-Slit Experiment
If light is a wave,
there should be an
interference pattern.
Two-source interference of light
• Young’s Experiment (1801)
• Interference of waves from slits S1 & S2 produces a pattern on
the screen. (Check out weblink!)
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Two-source interference of light
• Thomas Young
• “The last man who knew everything”
• Dr. of Medicine (haemodynamics)
• Wave Theory of Light
• Young’s Modulus (Stress/Strain)
• Founder of physiological optics
• Music (Young temperament)
• Translator of Rosetta Stone…
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Young’s Double-Slit Experiment
Interference occurs because each point on screen is NOT
same distance from both slits.
Depending on the path length difference, wave can interfere
constructively (bright spot) or destructively (dark spot).
Constructive Interference from two slits
• Constructive interference (reinforcement) occurs at points
where path difference is integral number of wavelengths
path difference d = mλ
• Bright regions on screen occur at angles θ for which
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Constructive Interference from two slits
Constructive Interference from two slits
Destructive Interference from two slits
• Destructive interference (cancellation) occurs, forming dark
regions on screen, at points for which path difference is
a half-integral number of wavelengths.
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Destructive Interference from two slits
Young’s Double-Slit Experiment
Interference pattern lines
(a) Will there be an infinite number of points on the
viewing screen where constructive and destructive
interference occur, or only a finite number of points?
(b) Are neighboring points of constructive
interference uniformly spaced, or is the spacing
between neighboring points of constructive
interference not uniform?
Young’s Double-Slit Experiment
Between the maxima and the minima, the
interference varies smoothly.
Young’s Double-Slit Experiment
But wait… what are these lines??? Stay tuned!
Line spacing for double-slit interference.
A screen containing two slits 0.100 mm apart is 1.20 m
from the viewing screen.
Light of wavelength λ = 500 nm falls on the slits from a
distant source. Approximately how far apart will adjacent
bright interference fringes be on the screen?
Example: Changing wavelength
•(a) What happens to interference pattern in
previous example if incident light (500 nm) is
replaced by light of wavelength 700 nm?
•(b) What happens if wavelength stays at 500 nm
but slits are moved farther apart?
•Check out http://www.walter-
fendt.de/html5/phen/doubleslit_en.htm
Young’s Double-Slit Experiment
Since position of maxima (except central one) depends on
wavelength, first- and higher-order fringes contain a
spectrum of colors.
Example :
Wavelengths from double-slit interference.
White light passes through two slits 0.50 mm apart, & an
interference pattern is observed on a screen 2.5 m away.
First-order fringe resembles a rainbow with violet and red
light at opposite ends. The violet light is about 2.0 mm and
the red 3.5 mm from the center of the central white fringe.
Estimate wavelengths for the violet and red light.
Two-source interference of light
• If distance R to screen is much greater than distance d
between slits….
• Use approximate geometry (c).
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Electric field in interference patterns
• Find intensity at any point in two-source interference pattern,
by combining two sinusoidally varying fields.
• If two sources are in phase, then waves arriving at point P
differ in phase by amount ϕ proportional to difference in
path lengths
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Electric field in interference patterns
• Assuming amplitudes of two waves are both approximately
equal to E at point P, combined amplitude is:
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Phasor diagram for superposition
• To add two sinusoidal
functions with a phase
difference, use phasor
representation for simple
harmonic motion &
voltages & currents in AC
circuits
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Phasor diagram for superposition
• Each sinusoidal function is
represented by rotating vector
(phasor)
• Projection on horizontal axis
at any instant represents
instantaneous value of
sinusoidal function.
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Intensity in Double-Slit Interference Pattern
The two waves can be
added using phasors,
to take the phase
difference into account:
Intensity in interference patterns
• Intensity at any point in two-source interference pattern:
• I0 is maximum intensity = 2 e0cE2 = four times larger than
intensity from each individual source.
• Averaged over all phase differences, I = I0/2, just twice
intensity of each source.
• Total energy isn’t changed – just redistributed in space.
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Intensity in Double-Slit Interference Pattern
The electric fields at the
point P from the two slits
are given by
where
Intensity in Double-Slit Interference Pattern
The time-averaged intensity is proportional to the
square of the field:
Intensity in Double-Slit Interference Pattern
Intensity as a function of angle.
Intensity in interference patterns
• Phase difference is:
• Must use wavelength in medium!
l = l0 / n
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Interference in thin films
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Phase shifts during reflection
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Phase shifts during reflection
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Interference in thin films: No relative shift
• For light of normal incidence on thin film wavelength λ in
film, where neither (or both) reflected waves have a half-
cycle phase shift:
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Interference in thin films: Half-wave shift
• For light of normal incidence on a thin film wavelength λ in
film, when only one of reflected waves has a half-cycle phase
shift:
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Nonreflective coatings
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Sketch of a Michelson Interferometer
End Mirror End Mirror
Beam Splitter
Viewing
Laser Screen
4
Michelson interferometer
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LIGO
4
LIGO sites
LIGO (Washington) LIGO (Louisiana)
(4km and 2km) (4km)
Funded by the National Science Foundation; operated by Caltech and MIT; the
research focus for more than 670 LIGO Scientific Collaboration members worldwide.
5
The LIGO Observatories
LIGO Hanford Observatory (LHO)
H1 : 4 km arms
H2 : 2 km arms
LIGO Livingston Observatory (LLO)
L1 : 4 km arms
• Adapted from “The Blue Marble: Land Surface, Ocean Color and Sea Ice” at visibleearth.nasa.gov
• NASA Goddard Space Flight Center Image by Reto Stöckli (land surface, shallow water, clouds). Enhancements by Robert Simmon (ocean color,
compositing, 3D globes, animation). Data and technical support: MODIS Land Group; MODIS Science Data Support Team; MODIS Atmosphere Group;
MODIS Ocean Group Additional data: USGS EROS Data Center (topography); USGS Terrestrial Remote Sensing Flagstaff Field Center (Antarctica); 5
Sensing the Effect of a Gravitational Wave
Gravitational
wave changes Change in arm length is
arm lengths 10-18 meters,
and amount of or about
light in signal 2/10,000,000,000,000,000
inches
Laser
signal
5
Gravitational Wave Detection
• Suspended Interferometers
Suspended mirrors in “free-fall”
g.w. output
port
power recycling
mirror
LIGO design length sensitivity: 10-18m
5
Core Optics Suspension and Control
Optics
suspended
as simple
pendulums
Shadow sensors & voice-coil
actuators provide
damping and control forces
Mirror is balanced on 30 micron
diameter wire to 1/100th degree of arc
5
Seismic Isolation – Springs and Masses
damped spring
cross section
5
How Small is 10-18 Meter?
One meter, about 40 inches
10,000 Human hair, about 100 microns
100 Wavelength of light, about 1 micron
10,000 Atomic diameter, 10-10 meter
100,000 Nuclear diameter, 10-15 meter
1,000 LIGO sensitivity, 10-18 meter
5
Gravitational Waves
Transverse distortions of spacetime due to
motion of massive astronomical bodies.
Expected sources:
• Inspiraling neutron stars/black holes
• (Asymmetric) supernovae
• Rotating pulsars
• Cosmic gravitational-wave background
Expected properties:
• Quadrupole polarization
• Propagating at speed of light
• Strains of ΔL/L = 10-21 or less
Strength of GWs
• For binary system shown in the
figure, strain h=2δl/l
becomes h~R1R2/Dd
• For neutron stars of 1.4M⊙out
at Virgo cluster (15Mpc):
h~10-21
• Crazy but within current reach!
Binary: a typical GW source
Hulse-Taylor Binary Pulsar
17 / sec
~ 8 hr
PSR 1913 + 16, measured in
1975
System should lose energy
through gravitational radiation
» Stars get closer together
» Orbital period gets shorter
Why Are We Looking?
“Chirp Signal”
We can use weak-field gravitational waves to study
strong-field general relativity.
Fabry-Perot Michelson
Interferometer
Uses light interference to
measure path length
difference between two
arms
Each arm is a Fabry-Perot
cavity, effectively
increasing arm length
Geometry ideally suited for
quadrupole radiation
Fabry-Perot Michelson
Interferometer
Each arm is Fabry-
Perot cavity,
effectively
increasing arm
length
FP cavities increase path length
by ~280 times, making
equivalent length 1120 km long!
Latest Detection: Neutron Star Collision
