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Wave Optics
Chapter 27
Wave Optics
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Linear Superposition and Interference
The Double-Slit Experiment
Interference in Thin Films
Interferometers
Interference and Diffraction
Diffraction and Spatial Resolution
Diffractive Spectral Dispersion: Gratings
Linear Superposition and Interference
From Chapter 17:
The Principle of Linear Superposition
When two or more waves are present simultaneously at
the same place, the resultant disturbance is the sum
of the disturbances from the individual waves.
In the chapter 17 context, we were talking about
mechanical waves. But the principle holds for
electromagnetic waves as well.
Linear Superposition and Interference
A few terms:
Coherent sources of light emit waves that have a constant phase
relationship.
Constructive interference occurs when the field vectors have the
same condition at the same place at the same time.
Destructive interference occurs when the field vectors are
exactly opposite in condition at the same place at the same
time.
Linear Superposition and Interference
If two waves from a coherent source travel two different paths to
arrive at a common point, their phase relationship at that point
depends on the difference in lengths between their paths.
Linear Superposition and Interference
Condition for destructive interference:
1

 path   m  
2

Condition for constructive interference:
 path  m
(where m is any integer)
Young’s Double-Slit Experiment
Image courtesy of
University of Toronto
physics department
Thomas Young, 1773 – 1829
English physicist, medical doctor, and Egyptologist
Also the inventor of Young’s modulus (strength of materials)
Young’s Double-Slit Experiment
Young’s experimental setup:
Young’s Double-Slit Experiment
Young’s result:
Young’s Double-Slit Experiment
Path difference geometry:
Young’s Double-Slit Experiment
Conditions for dark and bright fringes:
m
bright fringes: sin  
d
1

 m  
2

sin


dark fringes:
d
Thin-Film Interference
For light, as for any electromagnetic wave, wavelength,
frequency, and speed are related by
c

f
This applies in vacuum, where the speed is c. In a material with
refractive index n,
c
c 
n
n 


f
nf n
The wavelength has been reduced by a factor of n.
Thin-Film Interference
Two glass plates in air:
2
1
t
Wave 1 is reflected from the lower surface of the upper plate
Wave 2 is reflected from the upper surface of the lower plate
Wave 2 travels 2t farther than does wave 1
Thin-Film Interference
But we also find out …
2
1
t
… that when a wave is reflected while trying to pass from a
lower index into a higher one, it experiences a change of
phase at the interface that is equivalent to 1/2  in the higherindex material: n / 2.
Thin-Film Interference
How many “extra” wavelengths …
2
1
t
… does wave 2 experience, relative to wave 1?
2t
# of extra wavelengths:
extra distance
1

 2
phase change
Thin-Film Interference
Condition for constructive
interference:
# of wavelengths = m
2
1
t
2t
1
 m
 2

1
1
t   m  
2
2
Thin-Film Interference
2
1
1
t   m  
2
2
1
t
We see another bright fringe (constructive interference) for
every half-wavelength change in the thickness of the air
wedge.
Thin-Film Interference
For destructive interference:
1
# of extra wavelengths = m 
2
2
1
t
2t
1
1
1
 m
 t  m
 2
2
2
A dark fringe also occurs at half-wavelength intervals of air
wedge thickness.
Thin-Film Interference
Consider a soap bubble. Condition for
constructive interference between ray
that reflects from the outer surface and
ray that reflects from the inner surface:
1
2t   film  m film
2
1
1
1 
1
t   m   film 
 m  
2
2
2n film 
2
Destructive interference:
m
t
2n film
t
Thin-Film Interference
Film between two different materials:
t
n = n1
Both waves experience
/2 phase change at
reflection.
n = n2
n = n3
n3 > n2 > n1
Constructive interference:
2t  m film  m
m
t
2n2

n2
Destructive interference:
1
1 


2t   m   film   m  
2
2  n2


1

m



2
t
2n2
Thin-Film Interference
1
Film between two different materials:
Wave 1 experiences
/2 phase change at
reflection. Wave 2 does not.
2
t
n = n1
n = n2
n = n3
n2 > n3 > n1
Constructive interference:


2t   m film  m
2
1

m



2
t
2n2
n2
Destructive interference:
1
1 


  m   film   m  
2 
2
2  n2

m
t
2n2
2t 

Interferometers
An interferometer separates a light wave into two parts and then
recombines them, so that they interfere. How they interfere
depends on what happened to them while they were separate.
Many uses:
 measure wavelength of light
 measure displacements
 measure wavefront errors
caused by optics under test
 fundamental experiments in physics
Diffraction: Huygens’ Principle
Consider a plane wavefront.
Each point can be considered a source
of spherical “wavelets” that move
forward at the speed of the wave.
The next wavefront is a plane
tangent to the wavelets.
Diffraction: Huygens’ Principle
Christiaan Huygens
1629 – 1695
Dutch physicist and
astronomer
Early telescope maker
First observed Saturn’s rings
and one of Saturn’s moons
Image from National Center for Atmospheric Research
Diffraction: Huygens’ Principle
For every direction except
“straight ahead”:
For each wavelet source
point, we can find
another one that will
interfere destructively.
That is, as long as the
wavefront goes on
forever …
/2
Diffraction: Huygens’ Principle
If something limits or truncates
the wavefront, such as
passing through an opening,
points at and near the end
will lack “partners in
destruction” for some
directions.
The light spreads out:
diffraction.
Diffraction: Huygens’ Principle
For each point in the upper half of the
opening, in the direction , there is
a point in the lower half that
interferes destructively with it.
A dark fringe occurs in this direction.
Condition for dark fringes:
sin   m

W
Diffraction: Huygens’ Principle
Resulting far-field diffraction pattern from a single slit:
Diffraction: Circular Aperture
If light passes through a circular, rather than rectangular,
opening, a circular diffraction pattern results.
The angular radius of the first dark ring:
angle in radians
  1.22

D
aperture diameter
Diffraction and Spatial Resolution
What is the smallest angular distance between two objects, so
that we can look at an image of them and tell that there are
two instead of one? (Resolution)
Diffraction and Spatial Resolution
John William Strutt,
Lord Rayleigh (1842 – 1919)
English physicist
Nobel Prize in physics, 1904
(gas theory and discovery of argon)
Image from Nobel e-Museum
Diffraction and Spatial Resolution
Rayliegh’s Criterion for Resolution
Two point objects are resolved if their angular separation is at
least the radius of the first dark ring in their diffraction
patterns.
 min  1.22

D
Diffraction and Spatial Resolution
What does this mean to someone who designs optical systems?
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Increase aperture size: improve diffraction performance (min
gets smaller)
Increase aperture size: geometric aberrations become worse
(blur circle due to spherical, coma, etc.)
While geometric blur is large compared to diffraction pattern,
reducing aberrations strongly improves image quality
When geometric blur has been made small compared to
diffraction … diminishing returns for correction
improvements. System is “diffraction-limited.”
Diffractive Spectral Dispersion
Diffraction grating: a large number of parallel slits or lines, each
of which acts as a diffracting aperture.
Condition for
bright fringes:
m
sin  
d
Diffractive Spectral Dispersion
Condition for bright fringes:
m
sin  
d
grating “pitch”
Diffractive Spectral Dispersion
Gratings can be either transmissive or reflective.
They are used to separate light by wavelength (color) in optical
instruments called spectrometers.
Many natural processes can be identified and characterized from
the discrete wavelengths of the light that they emit, or absorb,
or both.
The atomic structures of crystals can act as gratings and produce
X-ray diffraction patterns. (Also useful in analytic science.)