Standing Waves, Beats,
and Group Velocity
Superposition again
Standing waves: the
sum of two oppositely
traveling waves
Beats: the sum of
two different
frequencies
Group velocity: the speed of information
Group-velocity dispersion
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Leerdoelen
In dit college leer je:
• Dat lichtgolven kunnen interfereren (superpositie)
• Hoe lichtgolven bij verschillende golflengtes samen een puls
vormen
• Interferentiepatronen berekenen
• Wat de groepssnelheid is van een lichtpuls en hoe je die berekent
• Analyseren wat de invloed van dispersie is op lichtpulsen
• Een golf f = f (x,t). Bedenk steeds of in een plot x constant wordt
gehouden (“film”) of dat t constant is (“foto”).
• Hecht: 7.2
2
Sums of fields: Electromagnetism is linear,
so the principle of Superposition holds.
If E1(x,t) and E2(x,t) are solutions to the wave equation,
then E1(x,t) + E2(x,t) is also a solution.
Proof:
 2 ( E1  E2 )  2 E1  2 E2
 2 
2
x
x
x 2
and
 2 ( E1  E2 )  2 E1  2 E2
 2  2
2
t
t
t
 2 ( E1  E2 ) 1  2 ( E1  E2 )   2 E1 1  2 E1    2 E2 1  2 E2 
 2
 2  2
 2  2
0
2
2
2 
2 
x
c
t
c t   x
c t 
 x
This means that light beams can pass through each other.
It also means that waves can constructively or destructively interfere.
3
Superposition allows waves to pass
through each other.
Otherwise they'd get
deformed while overlapping
4
Intensity of multiple waves
When multiple light waves are present, the intensity is calculated using
the total field:
Suppose there are two light waves with fields E1 and E2: the resulting
intensity becomes:
in which:
( E1 + E2 )( E*1 + E*2 ) = E1E1* + E1E2* + E2 E1* + E2 E2*
So the intensity of the total field is:
I = I1 + I 2 + ce Re{E1 × E2* }
This cross-term (called the interference term) can be as large as the
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intensity of the individual beams!
Add waves with the same frequency
and k but different complex amplitude.
It's easy to add waves with the same complex exponentials:
where all initial phases are lumped into E1, E2, and E3.
~
~
~
Sum the amplitude: Phasor addition.
i 3
i1
ia 2
E

E
e
E

E
e
E
=
E
e
~3
~1 ~1
~2
~3
~2
E3
~
E
~ tot
Im
E1
~
1
2
E2
~
3
Re
6
But sometimes the complex exponentials will be different!
Now we’ll add waves with different
complex exponentials.
Note
the plus
sign!
7
Adding waves of the same frequency and
amplitude, but opposite direction, yields a
standing wave.
Waves propagating in opposite directions:
And the intensity of this light wave then becomes:
(E0 is assumed to be real)
8
l
A Standing Wave
The points where
the amplitude is
always zero are
called nodes.
Nodes
E
The points where
the amplitude
oscillates
maximally are
called antinodes.
x
Note that the
nodes and antinodes are each
separated by l/2.
Anti-nodes
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Beams Crossing at an Angle
x
q
z
= E0 expéëi(w t - kz cosq )ùûéëexp(-ikxsin q ) + exp(+ikxsin q )ùû
= 2E0 expéëi(wt - kz cosq )ùû cos(kxsin q )
Fringe spacing, L:
I(x,t) µ I0 cos (kxsin q ) = I éë1+ cos(2kxsin q )ùû
2
1
2 0
L = 2p/(2ksinq)
L = l/(2sinq)
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Big angle: small fringes.
Small angle: big fringes.
The fringe spacing, L:
Large angle:
L  l / (2sin q )
As the angle decreases to
zero, the fringes become
larger and larger, until finally, at
q = 0, the intensity pattern
becomes constant.
Small angle:
L = 0.1 mm is about the minimum
fringe spacing you can see:
q  sin q  l / (2L)
 q  0.5 m / 200  m  1/ 400 rad  0.15
11
Laser beams crossing at an angle
Finite (laser) beams yield fringes only where the beams overlap.
x
12
Two point sources
Interfering spherical waves also yield standing waves
and fringes (in 2D and 3D)
Examples with different separations. Note the different
node patterns.
13
Superposition of waves with different
frequencies [at one point x]
Etot (t) = Re{E0 exp(iw 1t) + E0 exp(iw 2t)}
Let
w ave =
w1 + w 2
2
and Dw =
Take E0 to be real.
w1 - w 2
2
1  ave  
2  ave  
So : Etot (t) = Re{E0 exp i(w avet + Dw t) + E0 exp i(w avet - Dw t)}
= Re{E0 exp(iw avet)[exp(iDw t) + exp(-iDw t)]}
= Re{2E0 exp(iw avet)cos(Dw t)}
= 2E0 cos(w avet)cos(Dw t)
We typically have that 1 and 2 are similar, this means ave >> 
Summing waves of two different frequencies yields the product
of a rapidly varying cosine (w ave ) and a slowly varying cosine (Dw ).
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When two cosines or sines of different
frequency interfere, the result is beats.
In phase
Out of phase In phase Out of phase In phase
Individual
Waves
Sum
Envelope
1
2p/
1 beat
Irradiance
time
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When two light waves of different frequency
interfere, they also produce beats.
Etot (x,t) = Re{E0 expi(k1x - w 1t) + E0 expi(k2 x - w 2t)}
Let
Similiarly,
Take E0 to be real.
k1 + k2
k1 - k2
kave =
and Dk =
2
2
w +w
w -w
w ave = 1 2 and Dw = 1 2
2
2
So :
Etot (x,t) = Re{E0 expi(kave x + Dkx - w avet - Dw t) + E0 expi(kave x - Dkx - w avet + Dw t)}
= Re{E0 expi(kave x - w avet) éëexpi(Dkx - Dw t) + exp[-i(Dkx - Dw t)]ùû}
= Re{2E0 expi(kave x - w avet)cos(Dkx - Dw t)}
= 2E0 cos(kave x - w avet)cos(Dkx - Dw t)
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Traveling-Wave Beats
In phase
Out of phase In phase Out of phase In phase
Individual
waves
Sum
Envelope
Intensity
z
17
Seeing Beats
However, a sum of many
frequencies will yield a
train of well-separated
pulses:
In phase
It’s usually very difficult to see optical
beats because they occur on a time
scale that’s too fast to detect.
This is why we said earlier that beams
of different colors don’t interfere, and we
only see the average intensity.
Out of phase
Individual
waves
Etot ( z, t )  2E0 cos(kave z  avet ) Pulse(kmin z  mint )
Sum
Pulse separation: 2p/min
2p/max
Irradiance
I tot ( z, t )  Pulse(kmin z  mint )
2
time
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Group velocity
Light-wave beats (continued):
Etot(x,t) = 2E0 cos(kave x–ave t) cos(kx–t)
This is a rapidly oscillating wave: [cos(kave x–ave t)]
carrier wave
with a slowly varying amplitude: [2E0 cos(kx–t)]
amplitude
The phase velocity comes from the rapidly varying part: v = ave / kave
What about the other velocity—the velocity of the amplitude?
Define the group velocity: vg
º  /k
In general, we define the group velocity as:
vg º ¶w ¶ k
19
Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies).
For our example,
Dw
vg º
Dk
c0 k1 - c0 k2
=
n1k1 - n2 k2
[ = kc0]
where k1 and k2 are the k - vector magnitudes in vacuum.
If n1 = n2 = n,
c0 k1 - k2
c0
vg =
=
= phase velocity
n k1 - k2
n
If n1 ¹ n2 ,
vg ¹ c
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When the group and phase velocities
are different…
More generally, vg ≠ vf, and the carrier wave (phase fronts)
propagates at the phase velocity, and the pulse (irradiance)
propagates at the group velocity (usually slower).
The carrier wave:
The envelope (irradiance):
Now we must multiply together these two quantities.
21
Phase and Group
Velocities
The phase velocity, vf, is that of the
high-frequency oscillations. The group
velocity, vg, is that of the pulse envelope.
In vacuum
Most
common
case
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The group velocity is the velocity of
the envelope or irradiance: the math.
The carrier wave propagates at the phase velocity.
And the envelope propagates at the group velocity:
E(t) = E0 (z - v g t) exp[ik(z - vf t)]
Or, equivalently, the irradiance propagates at the group velocity:
E(t) µ
I(z - v g t) exp[ik(z - vf t)]
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Calculating the Group Velocity, vg  d /dk
Now,  is the same in or out of the medium, but k = k0 n, where k0 is
the k-vector in vacuum, and n is what depends on the medium.
So it's easier to think of  as the independent variable:
v g  1/  dk / d 
Using k =  n() / c0, calculate: dk/d = (n +  dn/d) / c0
vg  c0 / n   dn/d) = (c0 /n) / (1 +  /n dn/d)
Finally:
  dn 
v g  vf / 1 

n
d



So the group velocity equals the phase velocity when dn/d = 0, such
as in vacuum. Otherwise, since n [usually] increases with , dn/d >
0, and:
vg < vf
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Calculating group velocity vs. wavelength
We more often think of the refractive index in terms of wavelength, so
let's write the group velocity in terms of the vacuum wavelength l0.
Use the chain rule :
dn
dn d l0
=
dw d l0 dw
Now, l0 = 2p c0 / w , so :
Recalling that :
we have :
or :
d l0 -2p c0
-2p c0
- l02
=
=
=
2
2
dw
w
(2p c0 / l0 ) 2p c0
æ c0 ö é w dn ù
v g = ç ÷ / ê1+
è n ø ë n dw úû
æ c0 ö éê 2p c0 ìï dn æ - l02 ö üï ùú
v g = ç ÷ / 1+
ç
÷ý
í
ç
÷ ú
ê
n
n
l
d
l
è ø
0 ï
0 è 2p c0 ø ï
î
þû
ë
æ c0 ö æ l0 dn ö
v g = ç ÷ / ç 1n d l0 ÷ø
è nø è
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The group velocity is less than the phase
velocity in non-absorbing regions.
vg = c0 / (n +  dn/d)
In regions of normal dispersion, dn/d is positive. So vg < c for these
frequencies.

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Group velocity dispersion is the variation
of group velocity with wavelength
GVD means that the group velocity will be different for different
wavelengths in the pulse. So GVD will lengthen a pulse in time.
vg(blue) < vg(red)
Because short pulses have such large ranges of wavelengths,
GVD is a bigger issue than for nearly monochromatic light.
28
Group-velocity dispersion is undesirable
in telecommunications systems.
Train of input telecom pulses
Dispersion causes short
pulses to spread in time and
to become long pulses.
Many km of fiber
Dispersion dictates the
wavelengths at which telecom
systems must operate and
requires fiber to be very
carefully designed to
compensate for dispersion.
Train of output telecom pulses
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Tot slot
Wat hebben we vandaag gezien:
• Superpositie en interferentie van licht
• Lichtpulsen zijn een superpositie van lichtgolven met
verschillende golflengtes
• Fase- en groepssnelheid van lichtpulsen
• De invloed van dispersie op lichtpulsen
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