Chapter 5: Superposition of waves

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Chapter 5 Superposition of waves
Chapter 5 Superposition of waves
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Superposition principle
At a given place and time, the net response
caused by two or more stimuli is the sum of the
responses which would have been caused by each
stimulus individually.
applies to any linear system
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in a linear world, disturbances coexist without
causing further disturbance
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Superposition of waves
If y1 and y2 are solutions to the wave equation,
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Superposition of light waves
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-in general, must consider orientation of vectors
(Chapter 7next week) -today, well treat
electric fields as scalars -strictly valid only
when individual E vectors are parallel -good
approximation for nearly parallel E
vectors -also works for unpolarized light
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Light side of life
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Nonlinear optics is another story
for another course, perhaps
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What happens when two plane waves overlap?
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Superposition of waves of same frequency
initial phase (at t0)
propagation distance (measured from reference
plane)
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Superposition of waves of same frequency
simplify by intoducing constant phases
thus
.
At point P, phase difference is
hence the resultant electric field at P is
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Superposition of waves of same frequency
constructive interference
destructive interference
in step
out of step
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In between the extremes
constructive
destructive
general superposition
notice the amplitudes can vary its all about
the phase
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General case of superposition
where
and
Expressed in complex form
Simplify with phasors
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Phasors, not phasers
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Phasor diagrams
E0
magnitude
a
angle
projection onto x-axis
clock analogy - time is a line - but time has
repeating nature - use circular, rotating
representation to track time
phasors -represent harmonic motion -complex
plane representation - use to track
waves -simplifies computational manipulations
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Phasors in motion
http//resonanceswavesandfields.blogspot.com/2007/
08/phasors.html
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Phasor diagrams
complex space representation vector addition
p - a2-a1
from law of cosines we get the amplitude of the
resultant field
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Phasor diagrams
taking the tangent we get the phase of the
resultant field
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Works for 2 waves, works for N waves
- harmonic waves -same frequency
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Two important cases
for waves of equal amplitude and frequency
hence as
randomly phased
coherent
phase differences random
in phase all ai are equal
hence as
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Lightbulb
Light from a light bulb is very complicated!
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Coherent vs. Incoherent light
Incoherent light - relatively weak -
omni-directional - irradiance ? N
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Coherence is a continuum
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Coherent fixed phase relationship between the
electric field values at different locations or
at different times Partially coherent some
(although not perfect) correlation between phase
values Incoherent no correlation between
electric field values at different times or
locations
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more on coherence next week
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Mixing the colors of light
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How to make a pulse of light
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Laser operating regimes
1. Single frequency
Intensity
Time
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Standing waves

• occur when wave exists in both forward and
  reverse directions
• if phase shift p, standing wave is created
• when A(x) 0, ER0 for all t these points are
  called nodes
• displacemeent at nodes is always zero

A(x)
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Standing wave anatomy

• nodes occur when A(x) 0
• - A(x) 0 when sinkx 0, or kx mp (for m
  0, 1, 2, ...)
• - since k 2p/l, x ½ ml
• - ER has maxima when coswt 1
• - hence, peaks occur at t ½ mT (T is the
  period)

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Standing waves in action
light
water
http//www.youtube.com/watch?v0M21_zCo6UM
sound
http//www.youtube.com/watch?vEQPMhwuYMy4
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Superposition of waves of different frequency
a
b
wp
wg
kp
kg
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Beats
Here, two cosine waves, with wp gtgt wg
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Beats
The product of the two waves is depicted as
beat frequency
Phasor representation http//www.compadre.org/osp
/items/detail.cfm?ID8174
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Acoustic analogy
2 frequencies
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Phase and group velocity
phase velocity
group velocity
envelope moves with group velocity carrier wave
moves with phase velocity
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Superposition and dispersion
of a waveform made of 100 cosines with different
frequencies
non-dispersive medium
dispersive medium
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And the beat goes on
http//www.youtube.com/watch?vumrp1tIBY8Q
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Exercises
You are encouraged to solve all problems in the
textbook (Pedrotti3). The following may be
covered in the werkcollege on 22 September
2010 Chapter 5 2, 6, 8, 9, 14, 18
(not part of your homework)