Interference

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Interference and Diffraction
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Certain phenomena require the light (the
electromagnetic radiation) to be treated as
waves. Two relevant examples are interference
and diffraction
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Plane Electromagnetic Waves
Waves are in phase, but fields oriented at
900 Speed of wave is c At all times E c B
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Plane Electromagnetic Waves
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Combination of Waves
In general, when we combine two waves to form a
composite wave, the composite wave is the
algebraic sum of the two original waves, point by
point in space Superposition Principle.

• When we add the two waves we need to take into
  account their
• Direction
• Amplitude
• Phase

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Combination of Waves
The combining of two waves to form a composite
wave is called Interference
The interference is constructive if the waves
reinforce each other.
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Combination of Waves
The combining of two waves to form a composite
wave is called Interference
The interference is destructive if the waves
tend to cancel each other.
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Interference of Waves
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Interference of Waves
When light waves travel different paths, and are
then recombined, they interfere.
Each wave has an electric field whose amplitude
goes like E(s,t) E0 sin(ks-?t) î
Here s measures the distance traveled along each
waves path.
Constructive interference results when light
paths differ by an integer multiple of the
wavelength ?s m ?
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Interference of Waves
When light waves travel different paths, and are
then recombined, they interfere.
Each wave has an electric field whose amplitude
goes like E(s,t) E0 sin(ks-?t) î
Here s measures the distance traveled along each
waves path.
Destructive interference results when light paths
differ by an odd multiple of a half wavelength
?s (2m1) ?/2
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Interference of Waves
Coherence Most light will only have
interference for small optical path differences
(a few wavelengths), because the phase is not
well defined over a long distance. Thats
because most light comes in many short bursts
strung together.
Incoherent light (light bulb)
random phase jumps
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Interference of Waves
Coherence Most light will only have
interference for small optical path differences
(a few wavelengths), because the phase is not
well defined over a long distance. Thats
because most light comes in many short bursts
strung together.
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Thin Film Interference
We have all seen the effect of colored
reflections from thin oil films, or from soap
bubbles.
Film e.g. oil on water
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Thin Film Interference
We have all seen the effect of colored
reflections from thin oil films, or from soap
bubbles.
Rays reflected off the lower surface travel a
longer optical path than rays reflected off upper
surface.
Film e.g. oil on water
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Thin Film Interference
We have all seen the effect of colored
reflections from thin oil films, or from soap
bubbles.
Rays reflected off the lower surface travel a
longer optical path than rays reflected off upper
surface.
Film e.g. oil on water
If the optical paths differ by a multiple of ?,
the reflected waves add. If the paths cause a
phase difference ?, reflected waves cancel out.
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Thin Film Interference
Ray 1 has a phase change of phase of ? upon
reflection Ray 2 travels an extra distance 2t
(normal incidence approximation)
Constructive interference rays 1 and 2 are in
phase ? 2 t m ?n ½ ?n ? 2 n t (m ½) ?
?n ?/n
Destructive interference rays 1 and 2 are ? out
of phase ? 2 t m ?n ? 2 n t m ?
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Thin Film Interference
When ray 2 is in phase with ray 1, they add up
constructively and we see a bright region.
Different wavelengths will tend to add
constructively at different angles, and we see
bands of different colors.
Thin films work with even low coherence light, as
paths are short
When ray 2 is ? out of phase, the rays interfere
destructively. This is how anti-reflection
coatings work.
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Diffraction

• What happens when a planar wavefront
• of light interacts with an aperture?

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Diffraction
If the aperture is small compared to the
wavelength, would this be expected?
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Diffraction

If the aperture is small compared to the
wavelength, would the same straight propagation
be expected? Not really
In fact, what happens is that a spherical
wave propagates out from the aperture.
All waves behave this way.
This phenomenon of light spreading in a broad
pattern, instead of following a straight path,
is called DIFFRACTION
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Diffraction
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Huygens Principle

• Huygen first explained this in 1678 by proposing
  that all planar wavefronts are made up of lots of
  spherical wavefronts..

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Huygens Principle

• Huygen first explained this in 1678 by proposing
  that all planar wavefronts are made up of lots of
  spherical wavefronts..

That is, you see how light propagates by breaking
a wavefront into little bits
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Huygens Principle

• Huygen first explained this in 1678 by proposing
  that all planar wavefronts are made up of lots of
  spherical wavefronts..

That is, you see how light propagates by breaking
a wavefront into little bits, and then draw a
spherical wave emanating outward from each
little bit.
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Huygens Principle

• Huygen first explained this in 1678 by proposing
  that all planar wavefronts are made up of lots of
  spherical wavefronts..

That is, you see how light propagates by breaking
a wavefront into little bits, and then draw a
spherical wave emanating outward from each
little bit. You then can find the leading edge a
little later simply by summing all these little
wavelets
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Huygens Principle

• Huygen first explained this in 1678 by proposing
• that all planar wavefronts are made up of lots of
  
• spherical wavefronts..

That is, you see how light propagates by breaking
a wavefront into little bits, and then draw a
spherical wave emanating outward from each
little bit. You then can find the leading edge a
little later simply by summing all these little
wavelets
It is possible to explain reflection and
refraction this way too.
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Diffraction at Edges
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Diffraction at Edges
Light gets diffracted at the edge of an opaque
barrier ? there is light in the region obstructed
by the barrier
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Double-Slit Interference
Because they spread, these waves
will eventually interfere with one another
and produce interference fringes
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Double-Slit Interference
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Double-Slit Interference
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Double-Slit Interference
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Double-Slit Interference
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Double-Slit Interference
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Double-Slit Interference
screen
Bright fringes
Thomas Young (1802) used double-slit
interference to prove the wave nature of light.
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Double-Slit Interference
Light from the two slits travels different
distances to the screen. The difference r1 - r2
is very nearly d sinq. When the path
difference is a multiple of the wavelength these
add constructively, and when its a half-multiple
they cancel.
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Double-Slit Interference
Light from the two slits travels different
distances to the screen. The difference r1 - r2
is very nearly d sin q. When the path
difference is a multiple of the wavelength these
add constructively, and when its a half-multiple
they cancel.
Now use y L tan q and for small y ? sin ? ?
tan ? y / L
y
y bright mlL/d y dark (m 1/2)lL/d
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Multiple Slit Interference
With more than two slits, things get a little
more complicated
P
y
d
L
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Multiple Slit Interference
With more than two slits, things get a little
more complicated
Now to get a bright fringe, many paths must all
be in phase. The brightest fringes become
narrower but brighter and extra lines show
up between them.
P
y
d
L
Such an array of slits is called a Diffraction
Grating
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Multiple Slit Interference
The most intense diffraction lines appear
when d sin(q) m l Note that each wavelength
? is diffracted at a different angle ?
All of the lines (more intense and less intense)
show up at the set of angles given by d sinq
(n/N) l (N number of slits).
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Single Slit Diffraction
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The Diffraction Limit
Diffraction imposes a fundamental limit on the
resolution of optical systems
Suppose we want to image 2 distant points, S1
and S2, through an aperture of width a
Two points are resolved when the maximum of one
is at the minimum of the second
L
The minima occurs for sin ? ? / a
S1
D
q

a slit width
S2
Using sin ? ? ? ? ?min ? / a
Dmin / L ? ? / a
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Example Double Slit Interference
Light of wavelength l 500 nm is incident on a
double slit spaced by d 50 mm. What is the
fringe spacing on the screen, 50 cm away?
d
50 cm
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Example Double Slit Interference
Light of wavelength l 500 nm is incident on a
double slit spaced by d 50 mm. What is the
fringe spacing on the screen, 50 cm away?
d
50 cm
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Example Single Slit Diffraction
Light of wavelength l 500 nm is incident on a
slit a50 mm wide. How wide is the intensity
distribution on the screen, 50 cm away?
a
50 cm
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Example Single Slit Diffraction
Light of wavelength l 500 nm is incident on a
slit a50 mm wide. How wide is the intensity
distribution on the screen, 50 cm away?
a
50 cm
What happens if the slit width is doubled? The
spread gets cut in half.
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Beams of green (? 520 nm) and red (? 650 nm)
light impinge normally on a grating with 10000
lines per cm. What is the separation ?y of the
first order diffracted beams on a screen,
parallel to the grating, and located at a
distance L 50 cm away from it?
Grating equation ? d sin ? m ?
sin ? ? y/L for small angles, and m 1 for first
order 10000 lines/cm ? d 0.0001 d y / L ?
? y ? L /d
Y green 520x10-7x50 / 0.0001 26 cm Y red
650x10-7x50 / 0.0001 32.5 cm ?y 6.5 cm