Wave Optics
Chapter 25
Wave OpticsThe field of wave optics studies the properties of
light that depend on its wave natureOriginally light was thought to be a particle and that
model successfully explained the phenomena discussed in geometric options
Other experiments revealed properties of light that could only be explained with a wave theory
Maxwell’s theory of electromagnetism convinced physicists that light was a wave
Wave vs. Geometric OpticsThe wavelength of light plays a key role in determining
when geometric optics can or cannot be usedWhen discussing image characteristics over distances
much greater than the wavelength, geometric optics is extremely accurate
When dealing with sizes comparable to or smaller than the wavelength, wave optics is requiredExamples include interference effects and propagation
through small openingsEven more experiments led to the quantum theory of
lightLight has properties of both waves and particles
InterferenceOne property unique to
waves is interferenceInterference of sound
waves can be produced by two speakers
When the waves are in phase, their maxima occur at the same time at a given point in space
Section 25.1
Interference, cont.The total wave displacement at the listener’s
location is the sum of the displacements of the two individual waves
If two waves are in phase, the sum of their displacements is largeThe wave interfere constructivelyThis produces a large amplitude and a large intensity
Section 25.1
Interference, finalThe maximum of one wave can coincide with the
minimum of the other waveThese waves are out of phase
The interference is destructive when the waves are out of phaseIf the waves are 180° out of phase, the sum of the
displacements of the two waves is zero
Section 25.1
Conditions for InterferenceTwo or more interfering waves travel through
different regions of space over at least part of their propagation from source to destination
The waves are brought together at a common pointThe waves must have the same frequency and must
also have a fixed phase relationshipThis means that over a given distance or time interval
the phase difference between the waves remains constant
Such waves are called coherent
Section 25.1
CoherenceThe eye cannot follow variations of every cycle of
the wave, so it averages the light intensityFor waves to interfere constructively, they must stay
in phase during the time the eye is averaging the intensity
For waves to interfere destructively, they must stay out of phase during the averaging time
Both of these possibilities involve the wave having precisely the same frequency
Section 25.1
Coherence, cont.
With slightly different frequencies, the interference changes from constructive to destructive and back
Over a large number of cycles, the waves average no interference
Section 25.1
Michelson InterferometerThe Michelson
interferometer is based on the interference of reflected waves
Two reflecting mirrors are mounted at right angles
A third mirror is partially reflectingCalled a beam splitter
Section 25.2
Michelson Interferometer, cont.The incident light hits the beam splitter and is
divided into two wavesThe waves reflect from the mirrors at the top and
right and recombine at the beam splitterThe only difference between the two waves is that
they travel different distances between their respective mirrors and the beam splitter
The path length difference is ΔL = 2L2 – 2L1
Section 25.2
Michelson Interferometer, finalThe path length
difference is related to the wavelength of the light
If N is an integer, the two waves are in phase and produce constructive interference
If N is a half-integer the waves will produce destructive interference
LN
Section 25.2
Interference ConditionsFor constructive interference, ΔL = m λFor destructive interference, ΔL = (m + ½) λ
m is an integer in both casesIf the interference is constructive, the light intensity
at the detector is largeCalled a bright fringe
If the interference is destructive, the light intensity at the detector is zeroCalled a dark fringe
Section 25.2
Measuring Length with a Michelson InterferometerUse the light from a laser and adjust the mirror to
give constructive interferenceThis corresponds to one of the bright fringes
The mirror is then moved, changing the path lengthThe intensity changes from high to zero and back to
high every time the path length changes by one wavelength
If the mirror moves through N bright fringes, the distance d traveled by the mirror is
2N
d Section 25.2
Measuring Length, cont.The accuracy of the measurement depends on the
accuracy with which the wavelength is knownMany laboratories use helium-neon lasers to make
very precise length measurementsA similar type of interference effect is used to read
information from CDs and DVDs
Section 25.2
LIGOLIGO – Laser Interferometer Gravitational Wave
ObservatoryDesigned to detect very small vibrations associated
with gravitational waves that arrive at the Earth from distant galaxies
By using a long distance between the beam splitter and the mirrors, the LIGO interferometers are sensitive to very small percentage changes in that distance
Section 25.2
Thin-Film Interference
Assume a thin soap film rests on a flat glass surfaceThe upper surface of the soap film is similar to the
beam splitter in the interferometerIt reflects part of the incoming light and allows the
rest to be transmitted into the soap layer after refraction at the air-soap interface
Section 25.3
Thin-Film Interference, cont.The transmitted ray is partially reflected at the
bottom surfaceThe two outgoing rays meet the conditions for
interferenceThey travel through different regions
One travels the extra distance through the soap film
They recombine when they leave the filmThey are coherent because they originated from the
same source and initial ray
Section 25.3
Thin-Film Interference, finalThe index of refraction of the film also needs to be
accounted forFrom the speed of the wave inside the film
The wavelength changes as the light wave travels from a vacuum into the filmThe frequency does not change
The number of extra wavelengths is
ƒfilm filmfilm
cn
Section 25.3
2
film
dN
Frequency of a Wave at an InterfaceWhen a light wave
passes from one medium to another, the waves must stay in phase at the interface
The frequency must be the same on both sides of the interface
Section 25.3
Phase Change and ReflectionWhen a light wave reflects from a surface it may be
invertedInversion corresponds to a phase change of 180°
There is a phase change whenever the index of refraction on the incident side is less than the index of refraction of the opposite side
If the index of refraction is larger on the incident side the reflected ray in not inverted and there is no phase change
Section 25.3
Phase Change and Reflection, Diagram
Section 25.3
Phase Changes in a Thin FilmThe total phase change in a thin film must be
accounted forThe phase difference due to the extra distance
traveled by the rayAny phase change due to reflection
For a soap film on glass, nair < nfilm < nglass
There are phase changes for both reflections at the soap film interfacesThe reflections at both the top and bottom surfaces
undergo a 180° phase changeSection 25.3
Thin-Film Interference, Case 1Both waves reflected by a thin film undergo a phase
changeThe number of extra cycles traveled by the ray
inside the film completely determines the nature of the interference
If the number of extra cycles, N, is an integer, there is constructive interference
If the number of extra cycles is a half-integer, there is destructive interference
Section 25.3
Case 1, cont.Equations are
These equations apply whenever nair < nfilm < n(substance
below the film)
2
12
2
film
film
md constructive interference
n
md destructive interference
n
Section 25.3
Thin-Film Interference, Case 2Assume the soap bubble is surrounded by airThere is a phase change at the top of the bubbleThere is no phase change at the bottom of the
bubbleSince only one wave undergoes a phase change,
the interference conditions are12
2
2
film
film
md constructive interference
n
md destructive interference
n
Section 25.3
Thin-Film Interference: White LightEach color can interfere
constructively, but at different anglesBlue will interfere
constructively at a different angle than red
When you look at the soap film the white light illuminates the film over a range of angles
Section 25.3
Antireflection CoatingsNearly any flat piece of
glass may act like a partially reflecting mirror
To avoid reductions in intensity due to this reflection, antireflective coatings may be used
The coating makes a lens appear slightly dark in color when viewed in reflected light
Section 25.3
Antireflective Coatings, cont.Many coatings are
made from MgF2
nMgF2 = 1.38
There is a 180° phase change at both interfaces
Destructive interference occurs when
2
12
2MgF
md
n
Section 25.3
Antireflective Coatings, finalThe smallest possible value of d that gives
destructive interference corresponds to m = 0MgF2 is a popular material for antireflective coatings
because it can be made into very uniform films with small thicknesses
Although an antireflective coating will work best only at one wavelength, it will give partially destructive interference at nearby wavelengths
To function over the entire range of visible wavelengths, the coatings are made using multiple layers that give perfect destructive interference at different wavelengths
Section 25.3
Thin-Film Interference
Assume a thin soap film rests on a flat glass surfaceThe upper surface of the soap film is similar to the
beam splitter in the interferometerIt reflects part of the incoming light and allows the
rest to be transmitted into the soap layer after refraction at the air-soap interface
Section 25.3
Phase Change and Reflection, Diagram
Section 25.3
Case 1, cont.Equations are
These equations apply whenever nair < nfilm < n(substance
below the film)
2
12
2
film
film
md constructive interference
n
md destructive interference
n
Section 25.3
Thin-Film Interference, Case 2Assume the soap bubble is surrounded by airThere is a phase change at the top of the bubbleThere is no phase change at the bottom of the
bubbleSince only one wave undergoes a phase change,
the interference conditions are12
2
2
film
film
md constructive interference
n
md destructive interference
n
Section 25.3
Thin-Film Interference
Section 25.3
2
12
2
film
film
md constructive interference
n
md destructive interference
n
12
2
2
film
film
md constructive interference
n
md destructive interference
n
m=0,1,2..
Coherent Interference IntensityI = utotal x c = ½ εo c
Eo2
Electric Fields add linearly
Intensity units W/m2
21max III
21min III
Light Through a Single Slit
Light passes through a slit or opening and then illuminates a screen
As the width of the slit becomes closer to the wavelength of the light, the intensity pattern on the screen and additional maxima become noticeable
Section 25.4
Single-Slit DiffractionWater wave example
of single-slit diffractionAll types of waves
undergo single-slit diffraction
Water waves have a wavelength easily visible
Diffraction is the bending or spreading of a wave when it passes through an opening
Section 25.4
Huygens’ PrincipleIt is useful to draw the
wave fronts and rays for the incident and diffracting waves
Huygen’s Principle can be stated as all points on a wave front can be thought of as new sources of spherical waves
Section 25.4
Double-Slit InterferenceLight passes through two very narrow slitsWhen the two slits are both very narrow, each slit
acts as a simple point source of new wavesThe outgoing waves from each slit are like simple
spherical wavesThe double slit experiment showed conclusively that
light is a waveExperiment was first carried out by Thomas Young
around 1800
Section 25.5
Young’s Double-Slit ExperimentLight is incident onto
two slits and after passing through them strikes a screen
The incident light shown is a plane waveThis is easy to achieve
with a laserCould also be achieved
with a lens
Section 25.5
Young’s Experiment, cont.The experiment satisfies the
general requirements for interference The interfering waves travel through
different regions of space as they travel through different slits
The waves come together at a common point on the screen where they interfere
The waves are coherent because they come from the same source
Interference will determine how the intensity of light on the screen varies with position
Section 25.5
Young’s Experiment, finalAssume the slits are very narrowAccording the Huygen’s principle, each slit acts as a
simple source with circular wave fronts as viewed from above
The light intensity on the screen alternates between bright and dark as you move along the screenThese areas correspond to regions of constructive
interference and destructive interference
Section 25.5
Double Slit AnalysisDetermine the path
length between each slit and the screen
Assume W is very largeIf the slits are separated
by a distance d, then the difference in length between the paths of the two rays is
ΔL = d sin θ
Section 25.5
Double Slit Analysis, cont.If ΔL is equal to an integral number of complete
wavelengths, then the waves will be in phase when they strike the screenThe interference will be constructiveThe light intensity will be large
If ΔL is equal to a half number of complete wavelengths, then the waves will not be in phase when they strike the screenThe interference will be destructiveThe light intensity will be zero
Section 25.5
Conditions for InterferenceFor constructive interference, d sin θ = m λ
m = 0, ±1, ±2, … Will observe a bright fringe
For destructive interference, d sin θ = (m + ½) λm = 0, ±1, ±2, … Will observe a dark fringe
Section 25.5
Double-Slit Intensity PatternThe angle θ varies as
you move along the screenEach bright fringe
corresponds to a different value of m
Negative values of m indicate that the path to those points on the screen from the lower slit is shorter than the path from the upper slit
Section 25.5
Spacing Between SlitsNotation:
d is the distance between the slitsW is the distance between the slits and the screenh is the separation between the slits
For m = 1,
Since the angle is very small, sin θ ~ θ and θ ~ λ/dBetween m = 0 and m = 1, h = W tan θ
sind
Section 25.5
ApproximationsSince the wavelength of light is small, the angles
involved in the double-slit analysis are also smallFor small angles, sin θ ~ θ and tan θ ~ θUsing the approximations, h = W θ = W λ / d
Section 25.5
Interference with Monochromatic LightThe conditions for interference state the interfering
waves must have the same frequencyThis means they must have the same wavelength
Light with a single frequency is called monochromaticConsists of one color
Light sources with a variety of wavelengths are generally not useful for double-slit interference experimentsThe bright and dark fringes may overlap or the total
pattern may be a “washed out” sum of bright and dark regions
No bright or dark fringes will be visible
Section 25.5
Single-Slit Interference
Slits may be narrow enough to exhibit diffraction but not so narrow that they can be treated as a single point source of waves
Assume the single slit has a width, wLight is diffracted as it passes through the slit
and then propagates to the screen
Section 25.6
Single-Slit AnalysisThe key to the
calculation of where the fringes occur is Huygen’s principle
All points across the slit act as wave sources
These different waves interfere at the screen
For analysis, divide the slit into two parts
Section 25.6
Single-Slit Fringe LocationsIf one point in each part of the slit satisfies the
conditions for destructive interference, the waves from all similar sets of points will also interfere destructively
Destructive interference will produce a dark fringeConditions for destructive interference are
w sin θ = ±m λm = 1, 2, 3, … The negative sign will correspond to a fringe below the
center of the screenSection 25.6
Single-Slit Analysis – Bright FringesThere is no simple formula for the angles at which
the bright fringes occurThe intensity on the screen can be calculated by
adding up all the Huygens wavesThere is a central bright fringe with other bright
fringes that are lower in intensityThe central fringe is called the central maximumThe central fringe is about 20 times more intense than
the bright fringes on either side
Section 25.6
Single-Slit – Central MaximumThe width of the
central bright fringe is approximately the angular separation of the first dark fringes on either side
The full angular width of the central bright fringe = 2 λ / w
If the slit is much wider than the wavelength, the light beam essentially passes straight through the slit with almost no effect from diffraction
Section 25.6
Double-Slit Interference with Wide SlitsWhen the slits of a double-slit experiment are not
extremely narrow, the single-slit diffraction pattern produced by each sit is combined with the sinusoidal double-slit interference pattern
A full calculation of the intensity pattern is very complicated
Section 25.6
Diffraction GratingAn arrangement of
many slits is called a diffraction grating
AssumptionsThe slits are narrow
Each one produces a single outgoing wave
The screen is very far away
Section 25.7
Diffraction Grating, cont.Since the screen is far away, the rays striking
the screen are approximately parallelAll make an angle θ with the horizontal axis
If the slit-to-slit spacing is d, then the path length difference for the rays from two adjacent slits isΔL = d sin θ
If ΔL is equal to an integral number of wavelengths, constructive interference occurs
For a bright fringe, ΔL = d sin θ = m λm = 0, ±1, ±2, …
Section 25.7
Diffraction Grating, finalThe condition for bright
fringes from a diffraction grating is identical to the condition for constructive interference from a double slit
The overall intensity pattern depends on the number of slits
The larger the number of slits, the narrower the peaks
Section 25.7
Grating and Color Separation
A diffraction grating will produce an intensity pattern on the screen for each color
The different colors will have different angles and different places on the screen
Diffraction gratings are widely used to analyze the colors in a beam of light
Section 25.7
Diffraction and CDsLight reflected from
the arcs in a CD acts as sources of Huygens waves
The reflected waves exhibit constructive interference at certain angles
Light reflected from a CD has the colors “separated”
Section 25.7
Optical ResolutionFor a circular opening
of diameter D, the angle between the central bright maximum and the first minimum is
The circular geometry leads to the additional numerical factor of 1.22
D1.22
Section 25.8
Telescope Example
Assume you are looking at a star through a telescopeDiffraction at the opening produces a circular
diffraction spotAssume there are actually two starsThe two waves are incoherent and do not interfereEach source produces its own different pattern
Section 25.8
Rayleigh Criterion
If the two sources are sufficiently far apart, they can be seen as two separate diffraction spots (A)
If the sources are too close together, their diffraction spots will overlap so much that they appear as a single spot (C)
Section 25.8
Rayleigh Criterion, cont.Two sources will be resolved as two distinct sources
of light if their angular separation is greater than the angular spread of a single diffraction spot
This result is called the Rayleigh criterionFor a circular opening, the Rayleigh criterion for
the angular resolution is
Two objects will be resolved when viewed through an opening of diameter D if the light rays from the two objects are separated by an angle at least as large as θmin
Dmin
1.22
Section 25.8
ScatteringWhen the wavelength is
larger than the reflecting object, the reflected waves travel away in all direction and are called scattered waves
The amplitude of the scattered wave depends on the size of the scattering object compared to the wavelength
Blue light is scattered more than redCalled Rayleigh
scatteringSection 25.9
Blue SkyThe light we see from the
sky is sunlight scattered by the molecules in the atmosphere
The molecules are much smaller than the wavelength of visible lightThey scatter blue light
more strongly than redThis gives the
atmosphere its blue color
Section 25.9
Scattering, Sky, and SunBlue sky
Although violet scatters more than blue, the sky appears blue
The Sun emits more strongly in blue than violetOur eyes are more sensitive to blueThe sky appears blue even though the violet light is
scattered moreSun near horizon
There are more molecules to scatter the lightMost of the blue is scattered away, leaving the red
Section 25.9
Nature of LightInterference and diffraction show convincingly that
light has wave propertiesCertain properties of light can only be explained with
a particle theory of lightColor vision is one effect that can be correctly
explained by the particle theoryHave strong evidence that light is both a particle and
a waveCalled wave-particle dualityQuantum theory tries to reconcile these ideas
Section 25.10
Quantum Mechanics?!Single electrons fired
through double slitInterference? With…?Quantum Computing
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