Group delay dispersion
Phase velocity (1)
Direction of propagation
For a sinusoidal electromagnetic wave (a single frequency component)
the speed of propagation of the points of constant phase is called
phase velocity
If you pick a particular phase of the wave, it will appear moving at the
phase velocity.
Phase velocity (2)
In a medium with relative dielectric constant er , the phase velocity in
free space is equal to:
vp 
c
er
For a wave of frequency f, we can define the wavelength in the
medium as:
m 
vp
f
Since er > 1, m is smaller than the wavelength in vacuum.
Propagation constant
For an electromagnetic wave we can define the wave propagation
constant as

The wave can be expressed as
angular frequency.
2
m
Ae j (t  z ) where =2f is the
The propagation constant measures the amount of phase accumulated
per unit length.
The phase velocity can be calculated as

v p   m f

Wavepacket
In practice every electromagnetic signal contains multiple frequency
components.
The superposition of a group of plane waves is called a wavepacket.
A modulated signal is also represented by a wavepacket.
f1
f2
f1+f2
Group velocity
What is the propagation velocity of a wavepacket?
If the phase velocity vp is a function of frequency, the velocity of a
wave packet is different.
The propagation velocity of a wavepacket is called group velocity vg
Group velocity (2)
If a wavepacket is narrowband (its frequency components are
concentrated near a single frequency 0), then we can approximate the
propagation constant near 0 as:
 ( )  0  (  0 )1
Then we can express a wavepacket as:
A( z, t )   dA( )e j (t z )  e j ( 10 0 )  dA( )e j (t 1z )
The group velocity is:
d
vg 

1 d
1
Group delay dispersion
Group delay dispersion is the variation of group
delay with the signal frequency.
Different frequency components travel at different
speed: pulses spread in time!
With dispersion
Without dispersion
Examples
Optical prism (the refractive index varies
with frequency)
Optical fibers
Microwave
rectangular waveguides
Quantify dispersion
Ae j ( z t )
Z0, 
1
 ( )   0  (  0 ) 1  (  0 ) 2  2  ...
2
1 
d
1

d vg
ω=2πf
ω0=2πf0 (f0 : signal central frequency)
vg : Group velocity
d 2
2 d (1 / vg )
2
2 


D
2
d
2c d
2c
22 
Group delay dispersion [s2/rad.m]
Group delay variation
Frequency x Length
Examples
Standard single-mode optical fibers
(D=16ps/nm.km at =1.55mm
or 22=0.128ps/GHz.km)
18cm microstripline
Microstripline on Alumina substrate,
(22=0.7ps/GHz.cm)
Pulse propagation
1
0.9
 ( )  0  (  0 )1
1st order
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
700
1
1
0. 9
0.9
0. 8
1
 ( )   0  (  0 ) 1  (  0 ) 2  2
2
2nd order
0.8
0. 7
0.7
0. 6
0.6
0.
0.55
0.
0.44
0.
0.33
0.22
0.
0.11
0.
0
0
0
0
100
200
200
300
400
400
500
600
600
700
1
0.9
1
1
 ( )   0  (  0 ) 1  (  0 ) 2  2  (  0 )3
2
6
3rd order
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
700
800
Material vs waveguide dispersion
Material dispersion: dispersion due to frequency-dependent
response of a material to waves.

 ( ) 
e ( )
c
Waveguide dispersion: mode dispersion due to frequencydependent properties of the guiding structure.
 c 
  k 1  

2
Waveguide modes
Using Maxwell equations
And assuming E  e( x, y)e
We obtain:
  H  jeE ,   E   jmH
j (t  z )
,
hz
 jhy  jeex
y
h
 j hx  z  jee y
x
hy hx

 jeez
x y
H  h( x, y)e
j (t  z )
ez
 je y   jmhx
y
e
 jex  z   jmhy
x
e y ex

  jmhz
x y
We can calculate ex,y and hx,y from ez and hz .
Therefore we classify modes based on their components along z:
TE (ez=0),TM (hz=0) and TEM (ez=hz=0) modes.
Waveguide dispersion
In the case of TE and TM modes  is a function of frequency. For
example in a rectangular waveguide:
 
  k 1  c 

2
If    me (TE,TM modes), the transverse components of E and H
are uniquely determined by the longitudinal ones.
For a TEM mode:
hy  eex
e y  mhx
(  2   2 me )ex , y  0
 hx  ee y
ex  mhy
(  2   2 me )hx , y  0
and E=H=0 unless    me
TEM modes are not dispersive (except for material dispersion).
Impact of dispersion on pulse width
Dispersion is relevant especially relevant in high data rate digital
communication systems. How much is the widening of a pulse subject to
group delay dispersion?
We can define the r.m.s. width of a pulse A(t) as:
T 2 ( z) 
2
2
t
|
A
(
z
,
t
)
|
dt

2
|
A
(
z
,
t
)
|
dt

In the presence of second order dispersion 2:
A(0,t)
H ( )  e
j
2
2
( 0 ) 2 z
A(z,t)
Impact of dispersion on pulse width (2)
We can compute the pulse width in the Fourier domain:
2

ˆ * ( z,  )
A
Aˆ ( z,  )d
2


2

T ( z)  

ˆ ( z,  ) |2 d
|
A


Aˆ ( z,  )  Aˆ (0,  )e
j
2
2
( 0 ) 2 z

The derivative inside the integrand is:
2 ˆ
2 ˆ
ˆ ( z,  )  2 (   ) 2 z 2  terms odd in 
A
(
z
,

)

A
(
0
,

)

A
2
0
 2
 2
Resulting in:
T 2 ( z )  T 2 (0)  ( 2 Bz ) 2

B 2   (  0 ) 2 | Aˆ (0,  ) |2 d

(B is the pulse bandwidth)
( > 0)
Examples
T 2 ( z )  T 2 (0)  ( 2 Bz ) 2
The problem is particularly important in optical fiber links, characterized by extremely high bit-rates
and long distances. For example:
•
For a 10Gb/s link over standard single/mode fiber (D=16ps/nm.km), DT=10%T over 100km
•
For a 20Gb/s link over standard single/mode fiber (D=16ps/nm.km), DT=180%T over 100km
By comparison microwave circuits are much more dispersive. If we send a 10Gb/s signal over
microstripline we obtain DT=10%T over 20cm.
Dispersion has been mainly studied at optical frequencies, because of the long propagation distances.
However it is becoming more important at wireless frequencies due to the increase in data rates.
Overcoming dispersion

Canceling dispersion

Pulse shaping

Modulation formats insensitive to dispersion

Solitons

Digital filtering and predistortion
Canceling dispersion
Dispersion can be corrected by introducing an equal but
opposite amount of dispersion in the channel
In optical communication systems this can be realized with
dispersion compensating fibers or chirped fiber gratings.
Dispersion compensating fibers have negative dispersion
coefficient D , which is typically ten times higher than in
standard single mode fibers.
Drawback: DCFs high loss.
Pulse shaping
T 2 ( z )  T 2 (0)  ( 2 Bz ) 2
By reducing the pulse bandwidth B we can reduce T(z) (the
signal is narrowband and the frequency components
propagation velocities are similar)
However: to reduce the bandwidth we need to increase the
period T(0)
What is the optimum compromise
between bandwidth and pulse width?
Pulse shaping (2)
The pulse shape achieving the best possible simultaneous time
and frequency concentration (maximum dispersion tolerance) is
the prolate spheroidal wave function.
Duobinary
3-level encoding. Zeros are encoded as either 1 or -1. Ones are encoded as 0.
Transmit R bit/s in less than R/2 Hz bandwidth.
b(k)=d(k)  d(k-1)
mod 2
b(k)
d(k)
{0,1}
Delay
T
+
b(k-1)
c(k)=b(k)+ d(k-1)
e(k)=c(k)-1
-1
b(k-1)
1
0
1
0
0
Delay
T
0
+
+
c(k)
{0,1,2}
1
0
1
0
e(k)
{-1,0,1}
10Gb/s
Single sideband modulation
The spectrum of an amplitude modulated signal has two
sidebands next to the carrier component.
s(t )  A(t ) cos(0t )
One sideband contains all the signal information. If we suppress
the upper sideband, we can transmit the same information in
half the bandwidth and lower the dispersion.
|S()|
Double sideband

|S()|
Single sideband

Single sideband modulation (2)
Single sideband modulation can be obtained in different ways
Low pass/high pass filtering
|S()|

Hartley modulator
A cos(mt ) cos(0t )
A cos(mt )
m(t)
cos(0t)
90⁰
90⁰
+
S
A sin(mt ) sin(0t )
A cos(mt  0t )
OFDM
Orthogonal Frequency Division Multiplexing consists in splitting a
wideband signal into multiple independent narrowband frequency
channels.
Easy to perform digitally with Fast Fourier Transform.
The channels are orthogonal (the cross-correlation is zero), therefore they
can be easily separated even if tightly spaced.
Narrowband channels are less sensitive to dispersion and multi-path
interference.
Disadvantages: sensitive to Doppler shift
multiple transmitters required.
Solitons (1)
In optical fibers for example the refractive index increases with optical power
(Kerr effect):
P
n  n0  n2
Aeff
Aeff : Fiber effective area
n2 : Refractive index nonlinearity coefficient
The Kerr effect generates self-phase modulation in optical fibers that can be
used to cancel the phase distortion due to dispersion.
d NL
2 n2

P(t )dz
 Aeff
Solitons (2)
In the presence of dispersion and nonlinearities, the envelope of a pulse
A(z,t) evolves according to the Nonlinear Schrodinger Equation:
A 1  A
i
  2 2   | A |2 A  0
z 2
t
2
Dispersion
Kerr effect
The equation admits special solutions, called solitons, that propagate
undistorted:
   iz / 2
A( z, t )  sech 
t e
 2 
Phase conjugation
signal
ω0
ω0
conjugated
signal
A phase conjugator converts mirrors the spectral components of a signal (it
is equivalent to invert the sign of the signal phase).
Employed mainly at optical frequencies.
Can be implemented at optical frequencies using nonlinear crystals.
Digital Equalization
The received signal can be digitally processed to correct for dispersion.
Various digital filter designs (FIR,IIR...)
Maximul likelihood sequence estimators offer the best performance
Very useful for dynamically evolving channels.
Digital Equalization
Data
Optical
Link
Coupler
E/O
modulator
Laser
LO
Photodiode
At optical frequencies digital equalizers require a coherent receiver to
achieve optimum performance.
Coherent detection detects both the phase and amplitude information of the
optical signal, while a photodiode alone would detect only power.
Predistortion
Data
Predistortion
I
Q
I/Q
modulator
Laser
Optical
Link
Photodiode
An alternative approach consists in predistorting the transmitted signal
(with digital or analog techniques) to compensate for dispersion in the
communication channel.
Dispersion is a linear effect, therefore we need to invert the transfer
function of the channel. This can be done either at the transmitter or the
receiver.
Requires amplitude and phase modulation.
Useful in optical communication channels to avoid coherent detection.
Nonuniform Lines
Z(x),(x)
dV ( x)
  j ( x ) Z ( x ) I ( x )
dx
dI ( x)
 ( x)
j
V ( x)
dx
Z ( x)
Nonuniform lines (transmission lines having nonuniform characteristic
impedance Z(x) and/or propagation constant (x) ) can be used to generate
a desired amount of dispersion.
Impedance modulation is commonly used at microwave frequencies while
modulation of the propagation constant is more common in optics.
Nonuniform Lines - Examples
13-26GHz
At microwave frequencies:
nonuniform microstripline with
varying width.
=1.55mm
At optical frequencies: nonuniform
fiber with varying core refractive
index.
Nonuniform Lines (3)
dV ( x)
  j ( x ) Z ( x ) I ( x )
dx
dI ( x)
 ( x)
j
V ( x)
dx
Z ( x)
a
b
a  V  ZI
b  V  ZI
Z(x),(x)
We can write a system of coupled-mode equations for the propagating
wave a and the counterpropagating wave b:
da
 ( K ( x)  j ( x))a  K ( x)b
dx
db
 ( K ( x)  j ( x))b  K ( x)a
dx
K ( x) 
d ln( Z ( x))
dx
The coefficient K(x) controls the amount of coupling between the modes.
Periodic Grating
For a periodic perturbation, the forward and backward waves achieve
maximum coupling when the period of the perturbation is /2
fc 
c
d=/2
2d e r
Z1
Z2
Z1
Z2
4
20
3
10
2
0
1
-10
0
19
-20
19.5
20 20.5
GHz
21
d=/2
Phase (Deg)
Group delay (ns)
To avoid harmonics, the best impedance distribution is
Z1
Z2
  2
Z ( x)  Z 0 exp  sin
  d

x  

Chirped delay lines
  2π

L2 
W(x)sin 
+Cx  xC 
 a
4 


0

Z(x) = 50e
a0 : Modulation spatial period
C : Chirping factor
L : Total length (-L/2<x<L/2)‫‏‬
W(x) = Ae
4(x L / 4)/ L)2
Impedance
Group delay
τ(f) =
εr 1  4π εr
2π 

f 
c C  c
a0 