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Chapter 5: Superposition of waves
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
in a linear world, disturbances coexist without causing further disturbance
Superposition of waves
 2  2  2
1  2
 2  2  2 2
2
x
y
z
v t
If 1 and 2 are solutions to the wave equation,
then the linear combination
  a1  b 2
is also a solution.
where a and b are constants
Superposition of light waves
1
  
E  E1  E 2
  
B  B1  B 2
2
-in general, must consider orientation of vectors (Chapter 7—next week)
-today, we’ll 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
Light side of life
Nonlinear optics is another story
for another course, perhaps
What happens when two plane waves overlap?
Superposition of waves of same frequency
E1  E01 cos( ks1  t  1 )
E2  E02 cos( ks2  t   2 )
initial phase (at t=0)
propagation distance
(measured from reference plane)
Superposition of waves of same frequency
E1  E01 cos( ks1  t  1 )
E2  E02 cos( ks2  t   2 )
simplify by intoducing constant phases:
 1  ks1  1
 2  ks2   2
thus
E1  E01 cos(1  t )
E2  E02 cos( 2  t ) .
At point P, phase difference is
 2  1  k (s2  s1 )  (2  1 )
and the resultant electric field at P is
ER  E1  E2  E01 cos(1  t )  E02 cos( 2  t )
Superposition of waves of same frequency
ER  E1  E2  E01 cos(1  t )  E02 cos( 2  t )
constructive
interference
destructive
interference
E1
E2
ER
“in step”
“out of step”
In between the extremes:
constructive
general superposition
destructive
notice the amplitudes can vary; it’s all about the phase
General case of superposition (same )
E1  E01 cos(1  t )
E2  E02 cos( 2  t )
ER  E1  E2  E01 cos(1  t )  E02 cos( 2  t )
where
1  ks1  1
and
 2  ks2  2
Expressed in complex form:
ER  Re( E01ei (1 t )  E02ei ( 2 t ) )  Re( e it ( E01ei1  E02ei 2 )
Simplify with phasors
Phasors, not phasers
Phasor diagrams
magnitude E0
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
Phasors in motion
http://resonanceswavesandfields.blogspot.com
Phasor diagrams
complex space representation; vector addition
ER  Re( E01ei (1 t )  E02ei ( 2 t ) )  Re( e it ( E01ei1  E02ei 2 )
ER  Re( E0 ei ( t ) )  E0 cos(  t )
E 0 e i
p  [21]
from law of cosines we get the amplitude of the resultant field:
2
2
E02  E01
 E02
 2 E01E02 cos( 2  1 )
Phasor diagrams
E0 sin   E01 sin 1  E02 sin  2
E0 cos   E01 cos 1  E02 cos  2
taking the tangent we get the phase of the resultant field
E01 sin 1  E02 sin  2
tan  
E01 cos 1  E02 cos  2
Works for 2 waves, works for N waves
-harmonic waves
-same frequency
N
N
N
i 1
j i i 1
E02   E02i  2 E0i E0 j cos( j   i )
N
tan  
E
i 1
N
E
i 1
0i
sin  i
0i
cos  i
Two important cases
for waves of equal amplitude and frequency
N
N
N
E   E  2 E0i E0 j cos( j   i )
2
0
i 1
2
0i
j i i 1
randomly phased
phase differences random
hence as N  
hencecos(
as  j   i )  0
N
2
E02   E02i  NE01
i 1
E  NE
2
0
2
01
coherent
in phase; all i are equal
N
N
N
i 1
j i i 1
E02   E02i  2 E0i E0 j
2


E    E0i   ( NE0i ) 2
 i 1

N
2
0
E N E
2
0
2
2
01
Lightbulb
Light from a light bulb
is very complicated!
1 It has many colors (it’s white), so we have to add waves of
many different values of  (and hence k-magnitudes).
2 It’s not a point source, so for each color, we have to add
waves with many different k directions.
3 Even for a single color along one direction, many different
atoms are emitting light with random relative phases.
Coherent vs. Incoherent light
Coherent light:
Incoherent light:
- strong
- relatively weak
- uni-directional
- omni-directional
 N2
- irradiance 
- irradiance  N
Coherence is a continuum
1
0
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
more on coherence next week
Color mixing intermezzo
Mixing the colors of light
Mixing colors to make a pulse of light
1. Single mode
Supress all modes except one
Intensity
Broadband laser operating regimes
2. Multi-mode
Statistical phase relation
amongst modes
Intensity
Time
IN
Time
T = 2L/c
Constant or linear phase
amongst modes
I  N2
Intensity
3. Modelocked
Time
Modelocking laser cavity
Boundary condition:
Allowed modes:
= const.
Mode distance:
Pulse duration:
Peak intensity:
DT  1 / (N
)
 N2 (coherent addition of waves)
Intermezzo: Femtowelt
http://www.physik.uni-wuerzburg.de/femto-welt/
Standing waves
- occur when wave exists in both forward and reverse directions
- if phase shift = p, standing wave is created
ER  (2E0 sin kx) cos t
A(x)
- when A(x) = 0, ER=0 for all t; these points are called nodes
- displacement at nodes is always zero
Standing wave anatomy
ER  A( x) cos t
where
A( x)  2E0 sin kx
- 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 cost = ±1
- hence, peaks occur at t = ½ mT (T is the period)
Standing waves in action
light
water
http://www.youtube.com/watch?v=0M21_zCo6UM
sound
http://www.youtube.com/watch?v=EQPMhwuYMy4
Superposition of waves of different frequencies
E1  E0 cos( k1 x  1t )
E2  E0 cos( k2 x  2t )
ER  E1  E2  E0 [cos( k1x  1t )  E0 cos(k 2  2t )]

b
1
1
cos   cos b  2 cos (  b ) cos (  b )
2
2
(1  2 ) 
(1  2 ) 
 ( k1  k 2 )
 ( k1  k 2 )
ER  2 E0 cos 
x
t  cos 
x
t
2
2
 2

 2

kp
p
kg
ER  2 E0 cos( k p x   p t ) cos( k g x   g t )
g
Beats
ER  2 E0 cos( k p x   p t ) cos( k g x   g t )
Here, two cosine waves, with p >> g
Beats
ER  2 E0 cos( k p x   p t ) cos( k g x   g t )
The product of the two waves is depicted as:
beat frequency:
b  2 g
 (1  2 ) 
 2

2


b  1  2
Acoustic analogy
2 frequencies
1
0
Time [s]
1
0
200
300
Frequency [Hz]
Intensity [A.U.]
Intensity [A.U.]
-1
-1
1
Amplitude [A.U.]
1
Amplitude [A.U.]
1
4 frequencies
-1
-1
400
0
Time [s]
1
0
Time [s]
1
0
200
300
Frequency [Hz]
400
400
1
Intensity [A.U.]
Amplitude [A.U.]
1
Intensity [A.U.]
Amplitude [A.U.]
-1
-1
300
Frequency [Hz]
Many frequencies
16 frequencies
1
0
200
1
-1
-1
0
Time [s]
1
0
200
300
Frequency [Hz]
400
Phase and group velocity
phase velocity:
vp 
p
kp

1  2
k1  k 2


k
group velocity:
g
d
vg 


kg
k1  k 2
dk
1  2
envelope moves with group velocity
carrier wave moves with phase velocity
Here, phase velocity = group velocity (the medium is non-dispersive).
In a dispersive medium, the phase velocity ≠ group velocity.
Superposition and dispersion
of a waveform made of 100 cosines with different frequencies
non-dispersive medium
dispersive medium
And the beat goes on
http://www.youtube.com/watch?v=umrp1tIBY8Q
Exercises
You are encouraged to solve
all problems in the textbook
(Pedrotti3).
The following may be
covered in the werkcollege
on 28 September 2011:
Chapter 5:
2, 6, 8, 9, 14, 18
(not part of your homework)