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