# Interference by dffhrtcv3

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```									Interference and Diffraction
Certain phenomena require the light
to be treated as waves.

Two relevant examples are:
interference and diffraction
Plane Electromagnetic Waves

Ey                                     ˆ
E(x, t) = EP sin (kx-t) j
ˆ
B(x, t) = BP sin (kx-t) z
Bz

Waves are in phase,                                c
but fields oriented at 900
Speed of wave is c                                     x
c  1 / 00  3 108 m / s
At all times E = c B
Plane Electromagnetic Waves

E and B are perpendicular
to each other, and to the
direction of propagation
of the wave.

The direction of propagation
is given by the right hand            Electromagnetic waves
rule: Curl the fingers from E         propagate in vacuum
to B, then the thumb points           with speed c, the speed
in the direction of propagation.      of light.
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

+              =
Combination of Waves
The combining of two waves to form a composite wave is called:
Interference

+              =

Constructive interference
(Waves almost in phase)

The interference is constructive
if the waves reinforce each other.
Combination of Waves
The combining of two waves to form a composite wave is called:
Interference

(Waves almost cancel.)
+               =
Destructive interference
(Close to p out of phase)

The interference is destructive
if the waves tend to cancel each other.
Interference of Waves

+                =

Constructive interference
(In phase)

+            =
(Waves cancel)

( p out of phase)           Destructive interference
Interference of Waves
When light waves travel different paths,
and are then recombined, they interfere.
1                           Each wave has an electric field
*   2
whose amplitude goes like:
E(s,t) = E0 sin(ks-t) î

Mirror                  Here s measures the distance
traveled along each wave’s path.

+              =

Constructive interference results when light paths differ
by an integer multiple of the wavelength: s = m 
Interference of Waves
When light waves travel different paths,
and are then recombined, they interfere.
1                           Each wave has an electric field
*   2
whose amplitude goes like:
E(s,t) = E0 sin(ks-t) î

Mirror                  Here s measures the distance
traveled along each wave’s path.

+          =

Destructive interference results when light paths differ
by an odd multiple of a half wavelength: s = (2m+1) /2
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. That’s
because most light comes in many short bursts strung
together.
Incoherent light: (light bulb)

random phase “jumps”
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. That’s
because most light comes in many short bursts strung
together.
Incoherent light: (light bulb)

random phase “jumps”

Laser light is an exception:     Coherent Light: (laser)
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
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
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
If the paths cause a phase
difference p, reflected waves
cancel out.
Thin Film Interference
Ray 1 has a phase change of phase of p upon reflection
Ray 2 travels an extra distance 2t (normal incidence approximation)

1
2      n=1
oil on water
t                     n>1     optical film on glass
soap bubble

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 p out of phase
 2 t = m n  2 n t = m 
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.

1
2      n=1
oil on water
t                    n>1      optical film on glass
soap bubble
Thin films work with even low
coherence light, as paths are short

When ray 2 is p out of phase, the rays interfere destructively.
This is how anti-reflection coatings work.
Diffraction
What happens when a planar wavefront
of light interacts with an aperture?

If the aperture is large
compared to the wavelength
this would be expected....

…Light propagating
in a straight path.
Diffraction

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

Not really…
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.

instead of following a straight path, is called: DIFFRACTION
Diffraction
Slit width = a, wavelength =                  I

q

Huygen’s Principle

Huygen first explained this in 1678 by proposing that all
planar wavefronts are made up of lots of spherical
wavefronts..
Huygen’s 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
Huygen’s 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.
Huygen’s 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”
Huygen’s 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.
Diffraction at Edges

what happens to the
shape of the field at this
point?
Diffraction at Edges

Light gets diffracted at the edge of an opaque barrier
 there is light in the region obstructed by the barrier

I
Double-Slit Interference

Because they
waves will
eventually
interfere with
one another and
produce
interference
fringes
Double-Slit Interference
Double-Slit Interference
Double-Slit Interference
Double-Slit Interference

Bright
fringes
Double-Slit Interference

screen

Bright
fringes
Double-Slit Interference

screen

Bright
fringes

Thomas Young (1802) used double-slit
interference to prove the wave nature of light.
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 it’s a half-multiple they cancel.

P

r1
y
r2

d
q
L
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 it’s a half-multiple they cancel.

d sin q = m   bright fringes
P
d sin q = ( m+1/2)   dark fringes
r1
Now use y = L tan q; and for
r2           y     small y  sin q  tan q = y / L

d                                   y bright = mL/d
q
L                      y dark = (m+ 1/2)L/d
Multiple Slit Interference

With more than two slits, things get a little more complicated

P

y

d

L
Multiple Slit Interference

With more than two slits, things get a little more complicated
Now to get a bright
fringe, many paths
P
must all be in phase.

y    The brightest fringes
become narrower but
brighter;
d
and extra lines show up
L
between them.
Such an array of slits is called a “Diffraction Grating”
Multiple Slit Interference

The most intense diffraction lines appear when:
d sin(q) = m 
Note that each wavelength  is diffracted
at a different angle q

S

q
All of the lines (more intense and less intense) show up at the
set of angles given by: d sinq = (n/N)  (N = number of slits).
Single Slit Diffraction
Each point in the slit acts as a source of spherical wavelets

Slit width a

q

For a particular direction q, wavelets will                               I
Intensity
interfere, either constructively or destructively,         distribution
resulting in the intensity distribution shown.
Angular Spread: q ~  / a
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           S1
sin q =  / a                                q      D
a = slit width              S2

Using sin q  q  qmin =  / a        Dmin / L   / a
Example: Double Slit Interference
Light of wavelength  = 500 nm is incident on a double
slit spaced by d = 50 m. What is the fringe spacing on
the screen, 50 cm away?

d
50 cm
Example: Double Slit Interference
Light of wavelength  = 500 nm is incident on a double
slit spaced by d = 50 m. What is the fringe spacing on
the screen, 50 cm away?

d
50 cm

y  L / d

 50  10 m 500  10 m 50  10 m
2            9              6

 5mm
Example: Single Slit Diffraction
Light of wavelength  =
500 nm is incident on a
slit a=50 m wide. How
a
wide is the intensity
distribution on the
screen, 50 cm away?                50 cm
Example: Single Slit Diffraction
Light of wavelength  =
500 nm is incident on a
slit a=50 m wide. How
a
wide is the intensity
distribution on the
screen, 50 cm away?                          50 cm
q   / a

y  L q  50  10 m 500  10 m 50  10 m  5mm
2            9                6

What happens if the slit width is doubled?
The spread gets cut in half.
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 q = m 

sin q  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

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