Group delay dispersion
Phase velocity (1)
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.
Direction of propagation
Phase velocity (2)
In a medium with relative dielectric constant er , the phase velocity in free space is equal to:
r
p
cv
e
For a wave of frequency f, we can define the wavelength in the medium as:
f
vp
m
Since er > 1, m is smaller than the wavelength in vacuum.
Propagation constant
)( ztjAe
For an electromagnetic wave we can define the wave propagation constant as
m
2
The wave can be expressed as where =2f is the angular frequency. The propagation constant measures the amount of phase accumulated per unit length.
The phase velocity can be calculated as
fv mp
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:
100 )()(
Then we can express a wavepacket as:
)()()( 1001 )()(),(ztjjztj eAdeeAdtzA
d
dvg
1
1
The group velocity is:
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
ω=2πf
ω0=2πf0 (f0 : signal central frequency)
Z0,
vg : Group velocity
Group delay dispersion [s2/rad.m]
)( tzjAe
22
Group delay variation
Frequency x Length
gvd
d 11
Dcd
vd
cd
d g
2
)/1(
2
22
2
2
2
...)(2
1)()( 2
2
0100
Examples
Standard single-mode optical fibers (D=16ps/nm.km at =1.55mm or 22=0.128ps/GHz.km)
Microstripline on Alumina substrate, (22=0.7ps/GHz.cm)
18cm microstripline
Pulse propagation
100 )()(
2
2
0100 )(2
1)()(
0 100 200 300 400 500 600 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200 14000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3
02
2
0100 )(6
1)(
2
1)()(
1st order
0 100 200 300 400 500 600 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd order
0 100 200 300 400 500 600 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3rd order
Material vs waveguide dispersion
Material dispersion: dispersion due to frequency-dependent response of a material to waves.
Waveguide dispersion: mode dispersion due to frequency-dependent properties of the guiding structure.
)()( e
c
2
1
ck
Waveguide modes
zxy
yz
x
xyz
ejy
h
x
h
ejx
hhj
ejhjy
h
e
e
e
Using Maxwell equations , EH ej HE mj
And assuming , )(),( ztjeyx eE )(),( ztjeyx hH
zxy
yz
x
xyz
hjy
e
x
e
hjx
eej
hjejy
e
m
m
m
We obtain:
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
and E=H=0 unless TEM modes are not dispersive (except for material dispersion).
me
me
In the case of TE and TM modes is a function of frequency. For example in a rectangular waveguide:
2
1
ck
yx
xy
eh
eh
e
e
yx
xy
he
he
m
m
0)(
0)(
,
22
,
22
yx
yx
h
e
me
me
If (TE,TM modes), the transverse components of E and H are uniquely determined by the longitudinal ones. For a TEM mode:
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:
In the presence of second order dispersion 2:
dttzA
dttzAtzT
2
22
2
|),(|
|),(|)(
zj
eH2
02 )(
2)(
A(0,t) A(z,t)
Impact of dispersion on pulse width (2)
(B is the pulse bandwidth)
2
2
22 )()0()( BzTzT
We can compute the pulse width in the Fourier domain:
dzA
dzAzA
zT2
2
2*
2
|),(ˆ|
),(ˆ),(ˆ
)( zj
eAzA2
02 )(
2),0(ˆ),(ˆ
in odd terms)(),(ˆ),0(ˆ),(ˆ 22
0
2
22
2
2
2
zzAAzA
The derivative inside the integrand is:
Resulting in:
dAB 22
0
2 |),0(ˆ|)(
( > 0)
Examples
2
2
22 )()0()( BzTzT
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
2
2
22 )()0()( BzTzT
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.
1 0 1 0 0 0 1 0 1 0 10Gb/s
+
mod 2
Delay T
Delay T
+
d(k) {0,1}
b(k)
b(k-1)
b(k)=d(k) d(k-1)
b(k-1)
c(k) {0,1,2}
c(k)=b(k)+ d(k-1)
+
-1
e(k) {-1,0,1}
e(k)=c(k)-1
Single sideband modulation
The spectrum of an amplitude modulated signal has two sidebands next to the carrier component.
|S()|
)cos()()( 0ttAts
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 Single sideband
Single sideband modulation (2)
Single sideband modulation can be obtained in different ways
Low pass/high pass filtering Hartley modulator
m(t)
90⁰ 90⁰
cos(0t)
S +
-
)cos( tA m )cos()cos( 0ttA m
)sin()sin( 0ttA m
)cos( 0ttA m
|S()|
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):
effA
Pnnn 20
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.
dztPA
nd
eff
NL )(2 2
Solitons (2)
In the presence of dispersion and nonlinearities, the envelope of a pulse A(z,t) evolves according to the Nonlinear Schrodinger Equation:
Dispersion Kerr effect
0||2
1 2
2
2
2
AA
t
A
z
Ai
The equation admits special solutions, called solitons, that propagate undistorted:
2/
2
sech),( ziettzA
Phase conjugation
ω0 ω0
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.
signal conjugated
signal
Digital Equalization
Various digital filter designs (FIR,IIR...)
Maximul likelihood sequence estimators offer the best performance Very useful for dynamically evolving channels.
The received signal can be digitally processed to correct for dispersion.
Digital Equalization
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.
Photodiode
Data
E/O modulator
Laser LO
Coupler
Optical Link
Predistortion
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.
Photodiode
Predistortion
I/Q modulator
Laser
Data
I Q
Optical Link
Nonuniform Lines
Z(x),(x)
)()(
)()(
)()()()(
xVxZ
xj
dx
xdI
xIxZxjdx
xdV
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
At microwave frequencies: nonuniform microstripline with varying width.
At optical frequencies: nonuniform fiber with varying core refractive index.
13-26GHz
=1.55mm
Nonuniform Lines (3)
)()(
)()(
)()()()(
xVxZ
xj
dx
xdI
xIxZxjdx
xdV
ZIVb
ZIVa
axKbxjxKdx
db
bxKaxjxKdx
da
)())()((
)())()((
We can write a system of coupled-mode equations for the propagating wave a and the counterpropagating wave b:
Z(x),(x)
a
b
dx
xZdxK
))(ln()(
The coefficient K(x) controls the amount of coupling between the modes.
Periodic Grating
x
dZxZ
2sinexp)( 0
For a periodic perturbation, the forward and backward waves achieve maximum coupling when the period of the perturbation is /2
Z1 Z2 Z1 Z2 Z1 Z2
d=/2
To avoid harmonics, the best impedance distribution is
d=/2
r
cd
cf
e2
19 19.5 20 20.5 21
4 3 2 1 0
GHz
Gro
up d
elay
(ns)
Phas
e (D
eg)
20 10 0 -10 -20
Chirped delay lines
4
2π
50
2
0
sinL
CxCx+a
e=Z(x)
W(x)
2L)4)L4(xAe=W(x) //
a0 : Modulation spatial period
C : Chirping factor
L : Total length (-L/2<x<L/2)
0
2π4π1
af
c
ε
Cc
ε=τ(f)
rr
Group delay
Impedance