Transcript Lec-17_Strachan

Physics 213
General Physics
Lecture 17
Last Meeting: Mirrors and Lenses
Today: Interference
1
Huygen’s Principle


Huygen’s Principle is a geometric construction for determining the
position of a new wave at some point based on the knowledge of the
wave front that preceded it
All points on a given wave front are taken as point sources for the
production of spherical secondary waves, called wavelets, which
propagate in the forward direction with speeds characteristic of
waves in that medium
 After some time has elapsed, the new position of the wave front
is the surface tangent to the wavelets
Wave fronts
Huygen’s Construction for a
Plane Wave
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
At t = 0, the wave front
is indicated by the plane
AA’
The points are
representative sources
for the wavelets
After the wavelets have
moved a distance cΔt, a
new plane BB’ can be
drawn tangent to the
wavefronts
Huygen’s Construction for a
Spherical Wave

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
The inner arc
represents part of the
spherical wave
The points are
representative points
where wavelets are
propagated
The new wavefront is
tangent at each point to
the wavelet
5
Huygen’s Principle and the Law
of Reflection
• The Law of
Reflection can be
derived from
Huygen’s Principle
• AA’ is a wave front
of incident light
• The reflected wave
front is CD
Huygen’s Principle and the Law
of Reflection, cont
• Triangle ADC is
congruent to triangle
AA’C
• θ 1 = θ 1’
• This is the Law of
Reflection
8
9
10
Brightness~Intensity=E2max/2m0c
E=cB
11
Young’s Double Slit Experiment,
Diagram


The narrow slits, S1
and S2 act as
sources of waves
The waves
emerging from the
slits originate from
the same wave front
and therefore are
always in phase
Resulting Interference Pattern
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The light from the two slits form a visible pattern on a screen
The pattern consists of a series of bright and dark parallel bands
called fringes
Constructive interference occurs where a bright fringe appears
Destructive interference results in a dark fringe
Interference Patterns

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Constructive
interference occurs
at the center point
The two waves
travel the same
distance
 Therefore,
they
arrive in phase
Interference Patterns, 2
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
The upper wave has to
travel farther than the
lower wave
The upper wave travels
one wavelength farther


Therefore, the waves
arrive in phase
A bright fringe occurs
Interference Patterns, 3
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The upper wave travels
one-half of a
wavelength farther than
the lower wave
The trough of the
bottom wave overlaps
the crest of the upper
wave
This is destructive
interference

A dark fringe occurs
Interference Equations
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The path difference, δ,
is found from the tan
triangle
δ = r2 – r1 = d sin θ


This assumes the paths
are parallel
Not exactly parallel, but a
very good approximation
since L is much greater
than d
Interference Equations, 2
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For a bright fringe, produced by constructive
interference, the path difference must be either
zero or some integral multiple of the wavelength
δ = d sin θbright = m λ
= 0, ±1, ±2, …
 m is called the order number
m


When m = 0, it is the zeroth order maximum
When m = ±1, it is called the first order maximum
Interference Equations, 3
When destructive interference occurs, a
dark fringe is observed
 This needs a path difference of an odd half
wavelength
 δ = d sin θdark = (m + ½) λ

m
= 0, ±1, ±2, …
Interference Equations, 4
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The positions of the fringes can be measured
vertically from the zeroth order maximum
y = L tan θ  L sin θ
y=Lmλ/d
Assumptions
 L>>d
 d>>λ

Approximation

θ is small and therefore the approximation tan θ  sin θ can be
used
Interference Equations, final
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For bright fringes
ybright 

L
d
m
m  0,  1,  2
For dark fringes
ydark
L 
1

m  
d 
2
m  0,  1,  2
Phase Changes Due To
Reflection

An electromagnetic
wave undergoes a
phase change of 180°
upon reflection from a
medium of higher
index of refraction
than the one in which
it was traveling

Analogous to a reflected
pulse on a string
Phase Changes Due To
Reflection, cont

There is no phase
change when the wave
is reflected from a
boundary leading to a
medium of lower index
of refraction

Analogous to a pulse in a
string reflecting from a
free support
Interference in Thin Films
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Ray 1 undergoes a
phase change of 180°
with respect to the
incident ray
Ray 2, which is
reflected from the lower
surface, undergoes no
phase change with
respect to the incident
wave
Interference in Thin Films, Interference
Conditions – Normal Incidence
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Ray 2 also travels an additional
distance of 2t before the waves
recombine
For constructive interference
 2 n t = (m + ½ ) λ m = 0, 1, 2
…
 This takes into account both
the difference in optical
path length for the two rays
and the 180° phase change
For destruction interference
 2 n t = m λ m = 0, 1, 2 …