Optical Communication Engineering Prof Derek Abbott Email: dabbott@eleceng.adelaide.edu.au

Phone: (+61 8) 8303-5748 Room location: Innova 3.47

How to pass this course: o Come to all lectures o Do all exercises

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o Work through text book examples

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o Focus and good study habits

o Do everything I say

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Pass

Best study habits

Poorest studyhabits

Highest IQ

Lowest IQ

TThhee BBiigg SSeeccrreett

2

Outline

1. Introduction

2. Fundamentals of Optics and Lightware Prop-agation

3. Optical Waveguides

4. Light Sources

5. Light Detectors

6. Fibre Components

7. Modulation

8. System Design

3

Introduction

A historical view:

1880 Bell’s Photophone

1900s Signal Lamps

1960s Lasers

1970s Low-loss Optical Fibres

1980s Analogue and Digital Communications

1990s Fibre Networks

4

The basic communications system (simplex):

TRANSMITTER RECEIVER

MessageOrigin

Modulator

CarrierSource

InputChannelCoupler

InformationChannel

OutputChannelCouple

Detector

SignalProcessing

MessageOutput

5

Message Origin

• Transducer (eg. microphone or video camera)

• Electrical Origin (eg. computer data)

• Some Combination of Signals

Modulator

• Two main functions

1. Converts message into correct format

2. Impresses the data onto the carrier

• Analogue or digital format

6

Carrier Source

• Laser Diode (LD) or Light Emitting Diode (LED) –

both small, lightweight and low power consumption.

• Ideally:

1. Single frequency of operation

2. High power output

• In practice these are not achieved, so

1. Spectral width =⇒ limited information capacity

2. Low power =⇒ limited path length

• Output power proportional to input current – infor-

mation contained in variation of optical power (In-

tensity Modulation)

• Operating frequency of LDs and LEDs comparable

with frequencies where fibres have low attenuation.

7

Input Channel Coupler

• Interface between light source and the information

channel

• Power lost due to transfer from light source to the

small fibre core

Information Channel

• Glass or plastic optical fibre

• Require:

1. low attenuation and large acceptance of light from

source

2. low distortion due to dispersion to maximize in-

formation rate

• Often a compromise between 1 and 2

8

Output Channel Coupler

• Delivers light from the fibre to the detector

• Usually efficient

Detector

• Converts modulated light power to an electrical sig-

nal

• Semiconductor photodiode

Signal Processor

• As for any other communication system - amplifica-

tion and filtering

• Aim for high SNR (analogue system) or low BER

(digital system)

Message Output

• As for any other communication system - provide

appropriate format for output message

9

Optical System Capacity

Bandwidth for:

telephone 4 kHz

AM broadcast 10 kHz

FM broadcast 200 kHz

commercial TV 6 MHz

Bit rate for:

telephone 64 kbps (8 bits×2×4000)

video 81 Mbps (9 bits×2×4.5×106)

Multiplexed systems available up to several Gbps.

10

Complexity and Cost of FibreSystems

< 100 kbps cheap, readily available components

100 kbps-10 Mbps moderate increase in expense

10 Mbps-100 Mbps need improved circuits and design

100 Mbps-1 Gbps costly and specialized components

Quality of Service

eg. Video transmission:

• Signal to Noise Ratio (SNR) > 104

• Bit Error Rate (BER) 10−9

11

Optical Spectrum

Useful wavelengths for optical communications:

0.2-0.4 µm UV

0.4-0.7 µm Visible

0.7-2.0 µm IR

0.7-2.0 µm most widely used for fibre communications

since losses are low in windows within this region.

12

Wave Nature of Light

• Used in the analysis of propagation and guiding of

light

• Wavelength λ = v/f

• Wavelength of operation always quoted as a free-

space wavelength (ie. v = c)

Particle Nature of Light

• Used in the analysis of light generation by sources

• Photons, having energyWp = hf , where h is Planck’s

constant (6.626× 10−24 Js)

13

Advantages and Disadvantagesof Fibres

Comparison with cable systems.

1. Economics - cost per unit of information transfer.

Materials: Readily available (SiO2, plastics)

Installation: Light weight, easy to transport

Operation: Same as for cable

Maintenance: Fibres require greater skill to repair

Strength: Fibres are rugged and flexible

2. Performance

(i) Power losses:

cable RG-19/U (100 MHz b/w) 23 dB/km

125 µm fibre at 0.82 µm (100 MHz b/w) 5 dB/km

fibres at 1.3 or 1.5 µm have < 0.5 dB/km

=⇒ fibres require fewer repeaters.

(ii) Information rate:

distortion limited at high bandwidths for fibres

loss limited for coax

5-10 times higher rates possible with fibres

14

(iii) Isolation: rejection of RFI and EMI

no crosstalk

nuclear EMP hardended

no need for common ground

(iv) Security: no radiation from fibre

(v) Transparency: compatible with existing systems

upgradable with planning

(vi) Environmental: fibres are corrosion resistant

fibres can stand thermal stress

(vii) Interconnections: fibre connections are lossy

and costly

15

Applications of OpticalCommunications

• Telephone Links

1. trunk lines between exchanges

2. trans-oceanic links

3. fibre to the home

• Cable TV

• Remote monitoring

• Surveillance Systems

• Data Networks (eg. LANs)

• Military Command and Control

• Sensors (eg. fibre gyroscope, hydrophone etc.)

16

Review of Optics

Ray Theory

Rules for ray tracing based on Geomatric Optics (GO):

1. Velocity of ray v = c/n, where n is the index of

reflection of the media in which the ray travels.

2. Rays travel in straight paths unless deflected by a

change in the medium.

3. At a plane boundary, rays are reflected at an angle

θr equal to the angle of incidence θi, ie. θr = θi

4. Any power crossing the boundary is represented by

another ray whose direction θt is given by Snell’s

Law:sin θtsin θi

=n1

n2

17

Reference system for reflection and transmission at a

plane boundary:

Boundary

reflected

incident

transmitted

Note that ray theory cannot predict the intensity of

the reflected or transmitted rays.

18

Lenses

Can be used to focus light onto a fibre. Eg. a

cylindrical lens of diameter D and made from

material with refractive index n:

D

f

Fibre

Spherical faces defined by radii R1 and R2, and the

focal length is

1

f= (n− 1)

(1

R1+

1

R2

)

19

• The ratio f/D is defined as the f -number of the

lens.

• Large f-numbers are readily obtained.

• Small f-numbers require thick lenses, and often result

in spherical aberrations which blur the focus.

• Parallel off-axis rays are focussed in the focal plane

at a position determined by the central (undeflected)

ray.

• The lens can also be used to collimate a point source

located in the focal plane.

20

Ray tracing through lenses:

1. Rays travelling through the centre are undeviated.

2. Rays travelling parallel to the axis pass through the

focal point after emerging from the lens.

3. Incident rays parallel to a central ray intersect the

central ray in the focal plane after emerging from

the lens.

4. Incident rays passing through the focal point travel

parallel to the lens axis after emerging from the lens.

ff

4

231

21

Cylindrical Lenses

Useful for improving the symmetry of radiation from

line sources (eg. LED).

Planar sourceFocal line

Cylindricallens

Top View

Side View

22

Graded Index (GRIN) Rod Lens

• Dielectric rod (or fibre) having refractive index de-

creasing with radial distance from centre.

• Rays travel sinusoidal paths with one cycle being the

pitch, P .

• Quarter-wave pitch can be used as a lens with short

focal length.

• GRIN Rod, GRIN Rod Lenses and a fibre coupling

system shown below.

P

Grade Index Rod:

Quarter Pitch Lenses:

Collimating

Focussing

Application to fibre coupling:

lens

fibre

23

Imaging with Lenses

f f

di

O

I

do

Thin lens equation:

1

do+

1

di=

1

f

Magnification:

M =dido

=1

do/f − 1

24

Angular range:

f f

dido

1

M=

tanαi/2

tanαo/2

• Magnification M > 1 implies a decrease in beam

spreading.

• Fibres accept small cone angles whereas LEDs and

LDs radiate angles =⇒ can use a lens to improve

coupling.

25

Numerical Aperture

• Measure of the ability of an optical system to collect

light over a wide range of angles.

Consider the lens and detector system:

d

f

d

f

• Maximum acceptance angle θ

tan =d

2f(1)

• Define Numerical Aperture, NA as

NA = n0 sin θ (2)

where n0 is the refractive index of the medium be-

tween the lens and the detector.

26

Fibre Numerical Aperture

Fibre NA = sin θ

Fibre

• Typical glass fibres for high bandwidth have NA in

the range 0.1 to 0.3 =⇒1. difficult to couple to (sensitive to alignment)

2. low efficiency (rays outside acceptance angle θ)

• Plastic fibres may be lossy, but can be made with NA

in the range 0.4 to 0.5 to improve coupling efficiency

27

Diffraction

Geometric optics may not be accurate when effects

due to finite dimensions become significant =⇒ require

Physical Optics (PO).

Some definitions:

• Transverse Plane is the plane perpendicular to the

direction of ray travel.

• Intensity is equal to the light power.

• Uniform Beam has the same intensity at all points

in transverse plane.

28

Example: A spherical lens focussing a collimated

light beam

• GO predicts that the beam will be focussed to a sin-

gle point (the focal point of the lens).

• Experiment and PO shows that a spot of diameter

d = 2.44λ× (f/D)

is formed, together with rings of diminishing intensity

around the spot.

• The pattern is formed due to diffraction caused by

the finite size of the lens.

Significance:

d may not be negligible when coupling to a small

diameter fibre

29

Gaussian Beams

A number of significant source (eg. gas lasers, some

LDs) radiate beams that have a Gaussian distribution

in the transverse plane.

.135

1

2w w w 2wr

I/I 0

where the intensity

I = I0e−2r2/w2

(3)

and w is the spot size - the radial distance at

which the intensity drop to I0/e2.

• Focussing a Gaussian beam with a lens yields a distri-

bution of light in the focal plane which is also Gaus-

sian:

I = I ′0e−2r2/w2

0

where the spot size in the focal plane is

w0 =λf

πw

30

Collimating a Gaussian Beam:

f

2w 2w0

z

• Close to the lens

I = I0e−2r2/w2

0

• At least distance from the lens, the beam diverges at

a constant angle θ radians, where

θ =2λ

πw

• The beam retains its Gaussian distribution, and at a

distance z has a spot size

w0 =λz

πw

• Small divergence angles are obtained when w λ.

31

Lightwave Fundamentals

Electromagnetic Waves

Light progates as waves of oscillating electric and

magnetic fields

E

z

t1 t3t2

t1 t3t2< <

where, for example, the electric field is described by

E = E0 sin(ωt− kz)

and

λ = wavelength

k =2π

λ=ω

v= propagation constant

32

• In free space, the notation λ0 and k0 are used. Note

that λ0/λ = n, hence wavelength in a dielectric

medium is shorter that in free-space.

• A plane wave has fields with the same phase over a

planar surface normal to the direction of propagation.

• The orientation in space of the electric field vector is

the polarization of the wave.

• A given distribution and polarization of fields is re-

ferred to as a mode.

• In optical fibres, many modes may exist and the wave

is said to be unpolarized.

33

• Attenuation due to energy loss is accounted for by

an attenuation constant α such that

E = E0e−αz sin(ωt− kz)

• Power (or intensity) is proportional to the square of

the field, so power loss (in dB) over length L is given

by

10 log(e−αL

)2

or power loss per meter is −8.685α where α is in

Np/m.

34

Dispersion

• Optical sojurce do not emit light at a single fre-

quency.

• Define

∆λ = line width or spectral width

∆f = optical bandwidth

which are related by

∆f

f=

∆λ

λ

• The smaller the line width, the more coherent is said

to be the source. Typical examples are as follows:

Soure ∆λ

LED 20-100 nm

LD 1-5 nm

NdYAG laser 0.1 nm

HeNe laser 0.002 nm

35

• Dispersion is the variation of wave velocity with

wavelength.

• Dispersion due to changes in the refractive index is

called Material dispersion.

• Dispersion due to the properties of the waveguide

modes in fibres is called waveguide dispersion.

• Dispersion due to modes having differing velocities is

called modal dispersion.

• All forms of dispersion have the same effect, and

cause pulse spreading in digital signals or amplitude

reduction in analogue signals.

36

Distortion due to pulse spreading in digital signals:

INPUT

time

optic

al p

ower

!#"%$&!OUTPUT

time

optic

al p

ower

'()*+,

Distortion due to amplitude reduction in analogue

signals:

INPUT

time

optic

al p

ower

-./012

OUTPUT

time

optic

al p

ower

345678

37

Analysis of Dispersion

Refractive Index Variation

9:

;<

=>

n 1.45

n’

n’’ 0

λ ≈ 1.3 µm for pure SiO2.

38

• Let τ be the time for a pulse to travel length L.

• τ/L will be function of wavelength for a dispersive

medium.

• Let λ1 and λ2 be the shortest and longest wave-

lengths of the pulse and hence:

∆λ = λ2 − λ1

∆(τ/L) = τ/L|λ2 − τ/L|λ1

where ∆(τ/L) is the pulse spread.

• Define the duration, ∆τ , of the pulse as the dura-

tion the instantaneous pulse power exceeds half

the peak power.

• From characteristics of τ/L it may be shown that

(τ/L)′ =∆(τ/L)

∆λ

= −λc

d2n

dλ2

= −λcn′′

39

• Define material dispersion, M , as

M =λn′′

c

usually in units of ps (delay) per nm (of spectral

width) per km (length).

• Pulse spread per unit length is therefore

∆(τ/L) = −M∆λ

where M > 0 implies that the shorter wavelengths

travel slower

• For SiO2, zero dispersion (M = 0) occurs around

λ0 = 1.3 µm. Doping with other materials may

change the zero dispersion point up to ±0.1 µm.

• Near the zero dispersion point pulse spreading can-

not be entirely neglected due to finite source spectral

width. For 1.2-1.6 µm, an approximate expression

for material dispersion is

M ≈ Mo

4

(λ− λ4

o

λ3

)

where Mo = −0.095 ps/(nm2×km)

40

Information Rate

• Pulse spreading limits the information rate.

• Consider a modulation frequency f = 1/T , where T

is the period.

• Source radiates between optical wavelengths λ1 and

λ2.

• Maximum allowable pulse spreading occurs when the

signals at λ1 and λ2 are time shifted so that they

cancel, ie.

∆τ =T

2and hence the modulation frequency is limited to

f <1

2∆τ

• Wavelengths between λ1 and λ2 do not cancel. The

limiting modulation frequency defined above is ap-

proximately the 3 dB optical bandwidth.

• The frequency-length limit is thus

f3dB × L =1

2∆(τ/L)

41

• For a Gaussian response, the actual analytic result is

f3dB × L =0.44

∆(τ/L)

• The total electrical loss for propagation along a path

may be written as La + Lf , where

La = loss due to attenuation and scattering

= loss due to pulse spreading

= −10 log e−0.693(f/f3dB)2

from which it can be determined that the loss f is

1.5 dB at a frequency corresponding to 0.707f3dB.

• THE OPTICAL 1.5 dB BANDWIDTH = THE ELEC-

TRICAL 3 dB BANDWIDTH

• The electrical frequency-length limit is therefore

f3dB(elec.)× L =0.35

∆(τ/L)

42

Consider a return-to-zero (RZ) digital signal:

z

T

1 1 0 1

timepo

wer

1/T 2/T

freq.

with a data rate RRZ = 1/T bps.

• The limit on the data rate imposed by the 3 dB dis-

persion limited bandwidth is

RRZ = f3dB(elec.) =0.35

∆τ

• Rate-length limit is therefore

RRZ × L =0.35

∆(τ/L)

• The allowable pulse spread is therefore 70% of the

pulse duration (or 35% of the time slot) to prevent

intersymbol interference.

43

Consider a non-return-to-zero (RZ) digital signal:

z

T

1 1 0 1

time

pow

er1/2T 1/T

freq.

with a data rate RNRZ = 1/T bps.

• The limit on the data rate imposed by the 3 dB dis-

persion limited bandwidth is

RNRZ = 2f3dB(elec.) =0.7

∆τ

• Rate-length limit is therefore

RNRZ × L =0.7

∆(τ/L)

• The allowable pulse spread is again 70% of the pulse

duration (here 70% of the time slot).

44

Information Capacity Examples

Limited by material dispersion in SiO2.

Source λ ∆λ M ∆(τ/L)

(µm) (nm) (ps/nm/km) (ns/km)

(i) LED 0.82 20 110 2.2

(ii) LED 1.5 50 (-)15 0.75

(iii) LD 0.82 1 110 0.11

(iv) LD 1.5 1 (-)15 0.015

OPTIC ELECTRICAL

f3dB × L RNRZ × L f3dB × L RRZ × L(GHz×km) (Gbps×km) (GHz×km) (Gbps×km)

(i) 0.23 0.32 0.16 0.16

(ii) 0.67 0.94 0.47 0.47

(iii) 4.55 6.4 3.2 3.2

(iv) 33.33 46.7 23.3 23.3

45

Resonant CavitiesFound in the construction of lasers: eg. the Fabry

Perot resonator below.Mirrors

Output

Amplifying Medium

A standing wave pattern set up between two mirrors:

L

• The cavity will support a longitudinal mode if

L =mλ

2where m is an integer

• The cavity resonants at

λ =2L

mand if the medium between the mirrors has a refrac-

tive index n, at a resonant frequency

f =mc

2nL

46

• Each value of m corresponds to a different longitudi-

nal mode.

• Spacing between adjacent modes is given by

∆fc =c

2Ln

• The corresponding separation in free-space wavelength

is

∆λc =λ2

0

c∆fc

where λ0 is the mean free-space wavelength.

47

Reflection at Plane Boundaries

Key reflecting surfaces in fibres include:

1. Input (air-glass) and output (glass-air) coupling.

2. Interface between fibre core and cladding.

3. Glass-air-glass boundary at fibre junctions.

• Reflection coefficient at a boundary between two

media for normal incidence is

ρ =n1 − n2

n1 + n2

where n1 is the media from which the wave is inci-

dent.

• A negative value of ρ indicates a 180 degree phase

shift of the reflected wave.

• Reflectance is the ratio of reflected to incident beam

intensity : ie. R = |ρ|2.

48

Fresnel’s Laws of Reflection

Parallel Polarization

?@ABCDEF GIH Et

Er

Ei

ρ|| =−n2

2 cos θi + n1

√n2

2 − n21 sin2 θi

n22 cos θi + n1

√n2

2 − n21 sin2 θi

Perpendicular Polarization

JLKMLNOLPQSR TVU

Et

Er

Ei

ρ⊥ =n1 cos θi −

√n2

2 − n21 sin2 θi

n1 cos θi +√n2

2 − n21 sin2 θi

49

• Note that the analysis only applies for specular re-

flection from perfectly smooth surfaces.

• Analysis does not apply to diffuse reflection from

rough surfaces.

• Zero reflection only occurs for parallel polarization at

the Brewster angle θB satisfied by

tan θB =n2

n1

• A quarter-wave antireflection coating may be used

to reduce reflections. From a quarter-wave layer with

refractive index n2, the normal incidence reflectance

is

R =

(n1n3 − n2

2

n1n3 + n22

)2

for which R = 0 when the layer has n2 =√n1n3.

50

Critical Angle Reflection

• Total internal reflection occurs for both polarizations

at an angle θc satisfied by

sin θc =n2

n1

only when n1 > n2.

• For incident angles θi ≥ θc, |ρ| = 1.

• Total internal reflection is the mechanism by which

waves are guided by an optical fibre:

1. Interference between the incident and reflected

waves gives a standing wave in the incident medium.

2. Outside the incident medium the field is an evanes-

cent wave, diminishing with distance as e−αz

where

α = k0

√n2

1 sin2 θi − n22

Deeper penetration is obtained for angles close to

θc.

z

Electric Field

n2n1

51

Summary of Useful Results

1. Pulse spread for material dispersion:

∆(τ/L) = −M∆λ

2. 3 dB optic bandwidth-length product:

f3dB × L =1

2∆(τ/L)

3. 3 dB electrical bandwidth-length product:

f3dB(elec.)× L =0.35

∆(τ/L)

4. Rate-length product (RZ):

RRZ × L =0.35

∆(τ/L)

5. Rate-length product (NRZ):

RNRZ × L =0.7

∆(τ/L)

52

6. Longitudinal mode separation in cavity:

∆fc =c

2Ln

7. Reflectance for normal incidence:

R =

(n1 − n2

n1 + n2

)2

8. Critical angle for total reflection:

sin θc =n2

n1

53

Optical Waveguides

1. Dielectric Slab Waveguide

y

x

z W Xn2

n1

n3

• For a ray to propagate in the central film n3 < n1

and n2 < n1.

• Smooth boundaries required for specular reflection.

• Suitable integrated optic material have low losses:

eg. Lithium Niobate (LiNbO3) 1 dB/cm and Gallium

Arsenide (GaAs) 2.2 dB/cm.

54

• Light rays are trapped in the film by total internal

reflection, hence critical angles for ray propagation

are

sin θc =n2

n1at lower boundary

sin θc =n3

n1at lower boundary

θ must be greater than the largest critical angle.

• When n2 = n3 we have a symmetrical slab waveg-

uide.

55

• The total field is the sum of two plane waves - one

travelling upward at angle θ and one travelling down-

ward at angle θ, both with a propagation constant

k = k0n1:

YZh hk

[

\Upward wave Downward wave

• The longitudinal propagation constant β along the

direction of wave propagation (z-direction) is

β = k sin θ = k0n1 sin θ

• Standing wave set up in the transverse direction (y-

direction), so the electric field for even modes (sym-

metrical about y = 0) is

E = E1 coshy sin(ωt− βz)

and for odd modes (asymmetrical about y = 0) is

E = E1 sinhy sin(ωt− βz)

where E1 is the peak value of the electric field and

h = k cos θ is the vertical component of k.

56

• The phase velocity is given by

vp =ω

β

• We define an effective refractive index neff as

neff =c

vp=β

k0= n1 sin θ

which plays the same role as n does in unguided

wave propagation. Hence in the waveguide, the wave-

length in the direction of propagation is

λg =λ0

neff

• Outside the film the fields decay with an attenuation

factor

α = k0

√n2

1 sin2 θ − n22

so the fields outside the film are

E = E2e−α(y−d/2) sin(ωt− βz) for y > d/2

E = E2e−α(y+d/2) sin(ωt− βz) for y < −d/2

for symmetrical slab waveguide.

57

Modes in Symmetrical Slab Waveguide

• Total internal reflection occurs for rays between the

critical angle θ = θc and θ = 90.

• At θ = 90, neff = n1, and at θ = θc, neff = n2, so

n2 ≤ neff ≤ n1

• Only certain ray directions, however, are permitted.

Each allowed direction corresponds to a mode.

• The condition for a particular mode is that the phase

shift ∆φ for a round trip as shown below must satisfy

∆φ = 2πm

where m is an integer.

] ^

• ∆φ is the sum of the phase shift along the path and

the phase shift due to the reflections.

• Waves that do not satisfy the condition on ∆φ will

attenuate rapidly due to destructive interference (ie.

they are evanescent.)

58

Transverse Electric (TE) Modes

• Waves polarized in the x-direction (perpendicular

polarization) give rise to TE modes

_E

• Solution to the round trip phase condition yields for

even symmetry in the transverse plane

tanhd

2=

1

n1 cos θ

√n2

1 sin2 θ − n22

• and for odd symmetry in the transverse plane

cothd

2=

1

n1 cos θ

√n2

1 sin2 θ − n22

where h = k cos θ = 2πn1λ0

cos θ.

59

Construction of mode charts:

For trapped rays in the range θc ≤ θ ≤ 90 determine

the lowest order soultion:

1. Choose θ

2. Determine neff = n1 sin θ

3. Solve for the lowest order solution of hd

4. Obtain (d

λ0

)

0

=hd

2πn1 cos θ

Higher order solution for TEm modes are noted from

the periodic nature of the tan and cot functions, and

are found from(d

λ0

)

m

=

(d

λ0

)

0

+m

2n1 cos θ

60

TE 0

TE 1

TE 2

TE 3

TE 4

TE 5

n1

n eff

n2

Eff

ectiv

e R

efra

ctiv

e In

dex

0 1 2 3 4 5

90o (Axial Rays)

`

acb

Propagation A

ngle

(Cut off)

dfehgji

• Cut-off occurs when θ = θc at(d

λ0

)

m,c

=m

2√n2

1 − n22

• The highest order mode that may propagate in a slab

waveguide with film thickness d has a mode number

N = int

1 +

2d√n2

1 − n22

λ0

• Minimise number of modes by making d small, or by

making n1 ≈ n2.

61

Transverse Mode Patterns

n2n1n2

TE0

TE1

TE2

ie. the mode index refers to the number of nulls in

the transverse electric field pattern.

62

Transverse Magnetic (TM) Modes

• Wave having parallel polarization give rise to TM

modes.

kE

• Solution to round trip phase condition yields for even

symmetry

tanhd

2=

n1

n22 cos θ

√n2

1 sin2 θ − n21

and for odd symmetry

cothd

2=

n1

n22 cos θ

√n2

1 sin2 θ − n21

• If n1 ≈ n2 then the mode chart is nearly the same

for TE modes.

• Cut-off values for TEm and TMm modes are the same

even if n1 6= n2.

• TM0 and TE0 modes have no cut-off, therefore a sin-

gle mode film cannot be obtained since discontinu-

ities may depolarize the wave.

63

Asymmetric Dielectric Slab Waveguide

• Occur when n2 6= n3, and are used in Optoelectronic

Integrated Circuits (OEICs).

• An example of an asymmetric slab waveguide is a

thin ZnS film (n1 = 2.29) on a glass substrate (n2 =

1.5). Here n3 = 1 (ie. air).

• Rays must exceed the larger of the two critical angles

to avoid leakage into the substrate and consequent

losses.

• TE and TM modes exist, but are no longer degener-

ate even for n1 ≈ n2.

• All modes, including the TE0 and TM0 modes, have

a non-zero cut-off condition.

• Transverse field distributions are asymmetrical.

64

Edge Coupling to Slab Waveguide

• Used here to demonstrate the potential difficulties

and inefficiencies.

• For practical purpose, prism coupling is commonly

used.

1. Numerical Aperture Considerations

ln0

mon p qn2

n2

n1

• From Snell’s law:

n0 sinα0 = n1 sinα1

= n1 cos θ

65

• If α0 is large, θ may drop below θc and the wave will

not propagate along the film.

• α0 is called the waveguide acceptance angle.

• At θ = θc

sin θ =n2

n1and hence

cos θ =

√n2

1 − n22

n1

giving a numerical aperture

NA = n0 sinα0 =√n2

1 − n22

• We define a fractional refrective index change as

∆ =n2 − n1

n1

which for n2 ≈ n1 yields

NA = n1

√2∆

66

The consequences:

For thin films, where only a few modes are supported:

• Propagation angles of different modes are widely spaced.

• Incident rays must match these angles for efficient

coupling.

For thick films, where many modes propagate:

• Propagation angles of different modes are closely spaced

in the range between θc and 90.

• All incident rays are captured within the acceptance

angle.

67

Other coupling considerations:

• Radiation modes: Untrapped rays do not reflect

100% of the light, but may still give rise to a wave

which attenuates along the length of the film as the

energy is transmitted across the boundaries.

• Cladding modes: Critical angle reflections may oc-

cur at the outside boundaries:

n0

n2

n2

n1

n0

68

Radiation from the end of the film:

• For thick films: the film will radiate over the accep-

tance angle.

• For thin films: the film will radiate according to the

distribution of each mode as determined by diffrac-

tion theory.

Note that a large acceptance angle implies the need

for large ∆, which in turn suggests that many modes

must be propagating.

69

Other losses:

• Reflection losses : Light is reflected from the end of

the film due to the change in refractive index. Antire-

flection coatings may be used to improve efficiency.

• Alignment losses : A lens can be used to focus light

onto the film if the numerical aperture of the source

and the film are mismatched.

• Mode mismatch losses : The transverse distribution

of the incident wave from the source must match that

of the mode in order to couple to it. This is especially

true if its is desired to couple to a single mode. (eg.

Gas laser radiation matches the TE0 mode distribu-

tion, whereas LED radiation is less well ordered.)

70

Dispersion and Distortion in Slab Waveguide

1. Material Dispersion : As discussed previously.

2. Waveguide Dispersion :

• neff is a function of wavelength for any given mode,

and hence

∆(τ/L) = −λcn′′eff∆λ = −Mg∆λ

• Mg can be obtained from the mode charts by

plotting the slopes of neff and n′eff.

3. Modal Distortion :

• Different modes travel with different net veloci-

ties, thus the power associated with each mode

arrives at the output at different times.

• This is not dependent on the source linewidth.

71

Modal Distortion

• Determine the time of travel for the axial ray:

ta =L

cn1

and for the ray at the critical angle (distance travelled

= Ln1/n2, velocity = c/n1):

tc =L

cn2n2

1

so we have

∆(τ/L) =n1(n1 − n2)

cn2

=n1∆

c

=NA2

2cn1

Note that all three mechanisms exist simultaneously

for multimode propagation.

72

Optical Fibres

Step Index (SI) Fibres

n2

n1

a

z core

cladding

n

Refractive index of core is n1 and of the cladding is

n2, where n1 > n2.

• Critical angle for trapped rays in the core is given by

sin θc =n2

n1

• Fractional refractive index change is

∆ =n2 − n1

n1

73

Features of the cladding:

• Light still travels in the caldding, but decays rapidly

away from the interface.

• The cladding should be non-absorbing.

• The cladding assists the support and handling of the

fibre.

• The cladding protects the core.

• Cladding modes are possible, but usually attenuate

rapidly away from the excitation. Buffer layers are

sometimes used eliminate cladding modes.

Fibre sizes (denoted core-size/cladding-size diameter

in µm):

common size 50/125, 100/140, 200/230.

74

Three common forms of SI fibre:

1. All glass fibres

• Smallest range of ∆ due to limited range of re-

fractive indices available with glasses.

• Low attenuation (eg. 2 dB/km).

• Smallest intermodal pulse spreading.

• Small NA, and hence inefficient coupling.

2. Plastic Cladding Silica (PCS) fibres

• Moderately larger range of ∆.

• Moderate attenuation (eg. 8 dB/km).

• Moderate pulse spreading.

• Higher NA, and generally larger core diameter

(eg. 200 µm).

3. All-plastic fibres

• Large range of ∆ due to larger range of refractive

indices with plastics.

• Very high attenuation (eg. 200 dB/km).

• Dispersion not usually an issue due to small path

lengths available.

• Highest NA, and large core diameter (eg. 1 mm).

75

Modes in SI Fibres

Plot a Mode Chart in terms of normalized frequency

V =2πa

λ0

√n2

1 − n22

where a is the core radius.

n1

n eff

n2

0 1 2 3 4 5 6V

HE11

TM01

TE01

HE21

EH11

HE12

HE31

76

• TE and TM modes exist.

• HE and EH are hybrid modes having both electric

and magnetic field components along the fibre axis.

• For a conventional SI fibre each curve of the mode

chart actually represents two orthogonal modes, both

travelling with the same velocity.

• Energy is easily coupled between the orthogonal modes

at discontinuities or inhomegeneities.

• Longitudinal propagation factor for a given mode is

found from

β = k0neff

77

• For large values of V , many modes will propagate

(multimode operation).

• The closer each mode is to cut-off, the deeper the

evanescent fields penetrate the cladding.

• For V > 10, the number of modes in all polarizations

is approximately

N =V 2

2

• Lowest order mode is the HE11 mode, having a Gaus-

sian transverse field pattern. For 1.2 < V < 2.4 the

spot size (or mode field radius) is given by

w

a= 0.65 + 1.619V −

32 + 2.879V −6

78

For Single Mode Operation in SI Fibre:

• A single mode will propagate if V < 2.405, and hence

a

λ0=

2.405

2π√n2

1 − n22

=2.405

2πNA

• Note that actually two orthogonal HE11 modes may

exist, travelling with the same velocity (ie. having

the same neff).

• A true single mode fibre can be obtained in a po-

larization preserving fibre which exhibits birefrin-

gence, where the effective refreactive index depends

on the polarization.

• Birefringence may be obtained by designing asym-

metries into the fibre as shown below.

elipticalcore

Borondoping

(a) (b)

• The mode polarization will depend on the polariza-

tion of the excitation.

79

Graded Index (GRIN) Fibre

n2 a

z core

cladding

nn(r)

n2 n1

• Index variation

n(r) =

n1

√1− 2(r/a)α∆ for r ≤ a

n1

√1− 2∆ = n2 for r > a

where

n1 = refractive index along fibre axis

n2 = refractive index of cladding

a = core radius

α = profile index

∆ =n2

1 − n22

2n21

≈ n1 − n2

n1for n1 ≈ n2

80

• Ray paths through GRIN fibre:

2a

escaping ray

• Acceptance angle and numerical aperture decrease

away from the fibre axis, since rays beyond the crit-

ical angle are not reflected at the outer boundary.

2a rtsouvtwox

Hence coupling efficiency is lower than SI fibres with

comparable core radius and ∆.

81

• A Parabolic Profile (α = 2) is commonly used:

n(r) = n1

√1− 2(r/a)2∆ for r ≤ a

which for ∆ 1 becomes

n(r) =

n1

[1− (r/a)2∆

]for r ≤ a

n1(1−∆) for r > a

• Numerical aperture of parabolic profile GRIN fibre

is

NA =√n(r)2 − n2

2

= n1(2∆)12

√1− (r/a)2

which, on-axis, is noted to be the same as for a SI

fibre.

82

• For paraxial rays, the ray path is given by

d2r

dz2=

1

n

dn

dr

which for a parabolic profile yields the ray positions

r(z) = r0 cos(√

Az)

+1√Ar′0 sin

(√Az)

and trajectories

r′(z) = −√Ar0 sin

(√Az)

+ r′0 cos(√

Az)

where A = 2∆/a2 and r0 is the initial position and

r′0 the initial trajectory (slope) of the ray.

• Typical fibre sizes are 50/125, 62.5/125 and 85/125.

Sometimes the axial NA is appended, eg. 50/125/0.2,

62.5/125/0.275, 85/125/0.26.

83

Modes in GRIN Fibres

• Explicit formula available for neff for a parabolic pro-

file:

neff =βpqk0

= n1 − (p + q + 1)

√2∆

k0a

• Lowest order mode is p = q = 0.

• Total number of modes in a multimode fibre is given

by

N =V 2

4ie. half the number found in a SI fibre for the same

V .

• Cut-off condition occurs when neff = n2. Approxi-

mate expression for single mode operation is

a

λ0<

1.4

π√n1(n1 − n2)

• Max. value of a/λ0 60% higher than for SI fibre.

84

Mode distributions for parabolic profile:

• Lowest order mode is circularly symmetrical and Gaus-

sian

E00 = E0e−α2r2/2 sin(ωt− β00z)

-2 0 2 yz

• Higher order modes not circularly symmetrical.

E10 = E1αxe−α2r2/2 sin(ωt− β00z)

0 |~-3 3

E20 = E2

[2(αx)2 − 1

]e−α

2r2/2 sin(ωt− β00z)

0 ~-3 3

85

Attenuation in Optical Fibres

Plastic fibres have high attenuation and are not used

for long distance communications.

Glass fibres are widely used for long distance

communications. Glasses are used with the following

properties:

• Formed by fusing molecules of Silica (SiO2).

• Product is a mixture having variations in molecular

locations (unlike a crystal).

• Glass may be doped with titanium, thallium, germa-

nium, boron etc. to vary the refractive index.

• High chemical purities are obtained.

86

1. Absorption Losses

(i) Intrinsic Absorption

• Natural property of the glass itself.

• Absorption due to electronic and molecular transi-

tion bands - has peak in UV and decreases with wave-

length.

• Absorption due to vibration of chemical bonds - has

peak in IR and increases with decreasing wavelength.

• Intrinsic losses are quite small in the wavelength re-

gion of interest.

87

(ii) Impurities

• Transition metals (Fe, Cu, V, Co, Ni, Mn, Cr) absorb

strongly in the wavelength region of interest.

• Transition metal atoms have complete inner shells.

The absorption of light elevates energy level to a

higher shell. Transition energy corresponds to pho-

ton energies in the useful region.

• OH ion has a stretching vibration resonance at 2.73

µm, with overtones at 1.37 µm, 1.23 µm and 0.95

µm when embedded in SiO2.

• OH impurities must be kept below 1 ppm by special

manufacturing precautions.

88

(iii) Atomic Defects

• Example: Titanium (Ti4+) dopant does not absorb,

but Ti3+ may be formed during fibrization, and in

this state absorbs heavily.

• Irradiation by X-rays, gamma-rays, neutrons etc. also

causes atomic defects – such defects are more signif-

icant at shorter wavelengths.

• Atomic defects also caused by contamination by met-

als (e.g. platinum from crucibles) during manufac-

ture.

89

2. Rayleigh Scattering

• Random molecular variations within the fibre are

caused when the glass cools after production.

• Concentration fluctuation also occur when fibre has

more than one oxide.

• These variations cause localized variations in the re-

fractive index, which are modelled as small (< wave-

length) scatterers embedded in an otherwise homo-

geneous medium.

• The scattering of photons in this manner is called

Rayleigh scattering:

and the loss is given by

L = 1.7

(0.85

λ0

)4

dB/km

90

• Rayleigh scattering is the fundamental limiting loss

for fibres.

• 1/λ4 variation suggests the need to work at longer

wavelength for lowest possible loss.

Inhomogeneities:

• Larger discontinuities can also cause scattering:

1. Discontinuities caused by a rough core-cladding

interface.

2. Material inhomogeneities on a large scale.

91

3. Geometric Effects

(i) Macroscopic Bending

• Bending a fibre onto a spool or around a corner:

R

at some radius R, the condition θ2 < θc is met, and

total internal reflection no longer occurs.

• Some light will leak from the bend.

• Higher order modes are most susceptible to loss at

bends.

92

(ii) Macroscopic Bending

• Local stresses casued by outer sheath protecting the

cable may cause small axial distortions randomly

along the fibre.

• Axial distortions cause losses as well as coupling be-

tween modes.

93

Total Attenuation

800 900 1000 1100 1200 1300 1400 1500 1600 17000

0.5

1.0

1.5

2.0

2.5

3.0

Wavelength (nm)

FirstWindow

TotalLoss

RayleighScattering

SecondWindow

OH AbsorptionPeak

ThirdWindow

Att

enat

ion

(dB

/km

)

Minimum loss in glass fibres occurs around 1.55 µm.

94

Total Attenuation

Optical Time Domain Reflectometry is often used

to measure attenuation.

• Access only needed at one end of the cable.

• Reflection from discontinuities can be localized, whereas

Rayleigh Scattering provides a continuous return.

Rayleigh Scattering

Fresnel Reflections

splice

Connector

End1 Distance End2

Opt

ical

Pow

er

95

Pulse Distortion and Information Rate

• A fibre link is said to be

– Power Limited : if the attenuation reduces the

signal level below that capable of reliable detec-

tion.

– Bandwidth Limited : if distortion precludes the

correct recovery of the signal at the receiver.

• The attenuation is the total of all the mechanisms

described above.

• The distortion is the total effect of

– Material Dispersion.

– Waveguide Dispersion.

– Modal Distortion.

96

Distortion in SI Fibres

• For multimode fibres, modal distortion (which is not

dependent on the source linewidth) is often the key

mechanism.

∆(τ/L)mod =n1

c∆ =

NA2

2cn1

• This gives an overestimate of the effect due to two

phenomena:

1. Mode Mixing.

2. Preferential Attenuation.

97

1. Mode Mixing:

• Energy is exchanged continuously between modes

along fibre.

• On average path travelled by each ray is nearly

the same.

• Mode mixing is not perfect, so multimode distor-

tion is still present.

• Also increases attenuation since some rays will be

deflected into paths at less than θc.

2. Preferential Attenuation:

• Higher order modes have longer path lengths and

higher attenuation.

• Contribution of higher order modes to pulse re-

construction is lower, and hence there is lower

distortion.

• Overall attenuation is higher if power is in higher

order modes.

• For short links, fewer higher order modes are ex-

cited.

98

Dispersion:

• Total dispersion is due to material dispersion (M)

and waveguide dispersion (Mg):

∆(τ/L)dis = −(M + Mg)∆λ

• Mg < M for short wavelengths.

• Mg significant in the range 1.2-1.6 µm where M is

smaller.

Total pulse spreading given by:

(∆τ )2 = (∆τ )2mod + (∆τ )2

dis

99

Distortion in Single Mode Fibres

• Only material and waveguide dispersion.

• Material dispersion most significant at short wave-

lengths (800-900 nm).

• Near 1.3 µm waveguide dispersion must be consid-

ered:

– just beyond zero matrial dispersion M < 0

– at this point Mg > 0

– the two phenomena can be used to counteract

one another.

• Long, high data-rate fibres can be constructed with

single mode operation.

100

• Since lowest attenuation occurs at 1.55 µm, meth-

ods to make dispersion small at this wavelength are

needed:

1. Dispersion shifted fibres have a triangular re-

fractive index profile to make M + Mg = 0 at

1.55 µm.

2. Dispersion flattened fibre have the core surrounded

first by a low refractive index layer to make M +

Mg small across a range of wavelengths.

0

1.2 1.3 1.4 1.5 1.6 1.7

Conventional

Dispersion Flattened

Dispersion Shifted

30

20

10

-10

-20

-30

Wavelength (um)

Dis

pers

ion

(ps/

nm/k

m)

101

Distortion in GRIN Fibres

• GRIN fibres have much lower modal distortion than

SI fibres, since non-axial rays passing through a lower

refracive index catch up with the axial rays. (α = 2

close to optimal.)

• Modal pulse spread per unit length given by

∆(τ/L) =n1∆2

2c

• Material and waveguide dispersion much the same as

for SI fibres.

102

Length Dependence of Pulse Spread

• For multimode fibres

– pulse broadening proportional to length over short

paths (< 1 km)

– over long paths pulses do not broaden as quickly.

• Over short lengths, mode mixing is incomplete.

• Over long paths an equilibrium is reached, the equi-

librium length Le being the transition between the

two conditions.

• Pulse spread can be written

∆τ = L∆(τ/L) for L ≤ Le

∆τ =√LLe∆(τ/L) for L > Le

• May be a trade off between a “poor” fibre encourag-

ing mode mixing but having high attenuation, and a

“good” fibre with lower attenuation but suppressing

mode mixing.

103

Connectors and Couplers

Sources of loss in fibre-to-fibre connections:

• Lateral Misalignment

Fibre CoreFibre Core ad

For multimode SI fibre:

L(dB) = −10 log

2

π

cos−1 d

2a− d

2a

√1−

(d

2a

)2

• Angular Misalignment

Fibre Core

Fibre Core

For multimode SI fibre:

L(dB) = −10 log

(1− n0θ

πNA

)

104

• Fibre Separation Loss

Fibre Core Fibre Core

x

Due to

– Reflections from the ends

– Acceptance angle limitation

L(dB) = −10 log

(1− xNA

4an0

)

NB. An index matching fluid may be used to decrease

fibre separation loss.

• Surface Roughness Loss

Fibre Core Fibre Core

Need to ensure that the ends are scribed and cut

smooth.

105

Connecting Different Fibres

2a1 2a2

Power loss L

No power loss

• Radii Difference (a2 < a1)

L(dB) = −10 log

(a2

a1

)2

• NA Difference (NA2 < NA1)

L(dB) = −10 log

(NA2

NA1

)2

• GRIN to SI Fibre: L = 0 dB

• SI to GRIN Fibre: L = 3 dB

106

Splices

• Fusion Splice – welding two fibres

• Adhesion Splice – bonding with epoxy

Connectors

• Low loss

• Repeatable

• Predictable

• Long life

• High strength

• Ease of assembly

• Ease of use

• Economical

• Compatible with environment

107

Source Coupling

• Reflection Loss

Fibre Core

• Area Mismatch Loss

Fibre Core

• Numerical Aperture Loss

Fibre Core

Lambertian Source to SI Fibre

L(dB) = −10 log(NA)2

Lambertian Source to GRIN Fibre

L(dB) = −10 log

(NA)2

[1− 0.5

(aeaf

)2]

where NA = axial NA, ae = radius of emit-ter and af = radius of fibre core.

108

Light Sources

Light Emitting Diodes

A pn semiconductor junction that emit light when

forward biased.

p n

Heavily DopedSemiconductor

I

V

109

ZERO BIAS

Conduction Band

Valence Band

WF

Wg

FORWARD BIASConduction Band

eVWF2

Valence Band

hfWF1

• When a free electron meets a free hole in the junction

region, they combine to give a photon.

• Radiation from a LED is caused by the recombina-

tion of holes and electrons injected into the junction

under forward bias conditions.

110

• The wavelength of operation is related to the bandgap

energy by

λ =hc

Wg

=1.24

Wg(eV )

• Bandgap energy (and hence operating wavelength)

varies depending on the proportion of the constituent

atoms:

Material λ (µm)

GaAs 0.9

AlGaAs 0.8-0.9

InGaAs 1.0-1.3

InGaAsP 0.9-1.7

• The homojunction LED described does not confine

the light well since

– charge carriers extend over large area

– after photons are created they diverge over unre-

stricted paths.

111

Heterojunction LED

• LED formed by the junction between dissimilar semi-

conductors: eg. the double heterojunction LED shown

below.

1.35 eVhf1.0 eV

InjectedElectrons

InjectedHoles

n-InP n-InGaAsP p-InP

Electron Barrier

Hole Barrier

ElectronEnergy

RefractiveIndex

3.53.2

• Free charges can only recombine in the narrow active

layer.

• Active layer has highest refractive index, thereby con-

fining propagation by total internal reflection (cf. slab

waveguide).

112

Surface emitting LED (Etched Well construction):

50 um

Metalisation

n-GaAs Substrate

n-AlGaAs Window

p-AlGaAs Active Layer

p-AlGaAs Confinement

p-AlGaAs Contact

InsulationSiO2

Metalisation

Epoxy

Fibre

113

Edge emitting LED:

Metalisation

Confinement

Active Layer

Confinement

Metalisation

Radiation

114

LED Characteristics

• The optical output power is (to a first approxima-

tion) proportional to the current i through the LED.

If N = i/e is the number of charge per unit time

available for recombination and η is the fraction of

charges that recombine to produce photons,

P = ηNWg Watts

= ηiWg(eV) eV/sec

• For digital modulation the LED is simply turned on

and off using current pulses (typically 50-100 mA).

• For analogue modulation a dc bias current is re-

quired to eliminate clipping of the output optical

power, so

i = Idc + ISP sinωt

P = Pdc + PSP sinωt

115

Bandwidth Considerations:

• For low modulation frequencies

PSP = a1ISP

where a1 = ∆P/∆i.

• For high modulation frequencies

– Junction and parasitic capacitances short circuit

the current, thereby reducing the optical power.

– Carrier lifetime τ is major limitation to high fre-

quency operation:

PSP =a1ISP√

1 + ω2τ 2

giving a 3 dB modulation bandwidth of

f3dB(elec.) =1

2πτ

116

• The rise time tr of a LED is defined as the time

taken for the optical power to rise from 10 to 90%

of its maximum in response to a step input current

(typically 5-200 ns). The electical bandwidth can be

obtained from

f3dB(elec.) =0.35

tr

(cf. an optical pulse having an impulse response of

width σ = 2tr.)

Spectral width:

• Carriers have energies distributed close to the Fermi

levels, and this distribution allows a range of photon

energies and hence wavelengths.

• Typical LED spectral widths are 20-50 nm for LEDs

operating in 800-900 nm range, and 50-100 nm for

LED operating in the longer wavelength regions.

117

Radiation:

• Surface emitters radiate in a Lambertian pattern

having a cos θ power distribution. The half-power

beamwidth is 120 degrees. Low coupling efficiency

to fibres may result.

Beam Intensity 120 degrees

-90 0 90

Beam Angle

118

• Edge emitters radiate an asymmetrical power pattern

due to the beam confinement caused by the modes

in the slab waveguide forming the active region.

Beam Intensity

120 degrees

-90 0 90

Beam Angle

30 degreesParallelPlane

PerpendicularPlane

119

Lifetime and Operating Conditions:

• Lifetime of 105 hours are common if the LED is oper-

ated within its power, current and temperature lim-

its.

• Temperatures between -70 and 120 degrees Celcius

can be tolerated, although output power decreases

with increasing junction temperature.

Packages:

• Packages with glass covers are used.

• An external (or integral) lens can improve coupling

efficiency to a fibre.

• A pigtail construction allows a fibre to be bonded to

the chip during manufacture.

120

Lasers

1. Solid state (eg. Nd:YAG laser)

LEDs

MirrorNd:YAG Rod

PartialMirror

Output

• Fairly narrow linewidths available (eg. 0.1 nm)

• Costly and complex for use in communications sys-

tems.

• Requires an external modulator.

121

External Modulation:

• An external device is used to modulate the output

rather than modulating the input current.

• External modulators make use of electro-optic ma-

terial (eg. LiNbO3) – refractive index changes in

proportion to applied electric field.

• Eg. Mach-Zehnder Interferometer Modulator:

V O/PI/P

OpticalWaveguide

• Optical length of each path can be varied to effect

addition or cancellation of waves at the output.

122

2. HeNe Gas Laser

Mirror PartialMirror

OutputHeNe Gas

Power Supply

electrodes

L

• Operating wavelength λ0 = 633 nm.

• Linewidth ∆ = 0.002 nm.

• Used to test fibre systems for defects and to measure

NA.

123

Some terms used when considering lasers:

1. Pumping threshold : input power level above which

the laser will start to emit.

2. Output spectrum : optical frequency (wavelength)

response of the laser of the laser. Usually as series of

closely spaced peaks.

3. Radiation pattern : range of angles over which light

is emitted. Dependent on emitting area and modes

of oscillation.

124

Laser Operation

• Allowed energy levels for gas atoms are distinct lines

(cf. bands in semiconductors).

• Atoms usually in the ground state (lowest energy

level).

• Atoms absorb energy to rise to excited states.

– For Nd:YAG laser by absorbing an incoming pho-

ton.

– For HeNe laser by absorbing energy from ionized

electrons.

• The energy difference between two excited states of

the Ne atom corresponds to the 633 nm radiation

from HeNe laser.

125

• Various possibilities exist for photon interaction:

1. An incoming photon at 633 nm is absorbed, rais-

ing the Ne atom to the upper excited level.

2. A Ne atom in the upper excited level drops to the

lower level: ie. spontaneous photon emission.

3. A Ne atom in the upper level may be induced by

an incoming photon at 633 nm to emit another

photon and drop to the lower level: ie. stimu-

lated photon emission.

• If there are more Ne atoms in the lower level absorp-

tion of photons dominates.

• If there are more Ne atoms in the upper level (a

population inversion) then the number of photons

increases – giving rise to gain.

126

• For an oscillator we require:

– Amplification-provided by the medium.

– Frequency governing structure-provided by the

medium.

– Feedback-provided by the mirrors in a Fabry Perot

cavity configuration.

• Oscillations occurs when the gain exceeds all the

losses, including that due to output through the pae-

tial mirror.

127

• Finite linewidth occurs due to

– Thermal effects on energy levels.

– Doppler effect on sources in motion.

• Output spectrum is dependent on the linewidth and

the spacing between longitudinal cavity modes:

c2L

Cavity resonances

f

Gain of medium

• Output radiation pattern is dependent on the trans-

verse distribution of the modes in the cavity.

– Multimode beams tend to be large.

– Single mode beams have Gaussian distribution.

128

3. Laser Diodes

• Construction similar to edge emitting LED

Metalisation

GaAs Substrate

n-AlGaAs, W =1.8eV, Confinement

n-AlGaAs,

p-AlGaAs, p-GaAs Contact

InsulationSiO2

Metalisation

Emitting Edge

g

Wg =1.55eV, Active Layer

Wg =1.83eV, Confinement

Stripe Contact

1 um

0.2 um1 um

1 um

Operation:

• Forward bias junction to junction ro inject charges

giving spontaneous photon emission (low intensity).

• With high current density, stimulated emission oc-

curs and the optic gain is large.

• Threshold current reached when gain is large enough

to overcome losses −→ laser operation.

129

• High power with low threshold is achieved by confin-

ing injected charges and light using a heterojunction

structure.

• Laser cavity is formed by reflections from the ends

of the slab waveguide (ie. AlGaAs-air interface R =

0.32.)

• Several longitudinal modes may occur, together with

several transverse modes.

• Single transverse mdoe (HE11) operation may occur

for large currents – a Gaussian beam results.

130

Laser Diode Characteristics

• Typical threshold currents 50-200 mA, below which

the laser diode does not emit significantly.

• Typical output powers 1-10 mW (CW operation) for

20-40 mA drive current above threshold.

• DC bias needed for both digital and analogue mod-

ulation (due to threshold).

• Linearity of the output power/drive current function

dependent on operating point.

• Laser diodes are thermally sensitive:

– threshold varies with temperature.

– output power varies with temperature.

– wavelength varies with temperature.

131

• Laser diodes have shorter rise times than LEDs.

– Spontaneous emission lifetime, τsp, is the av-

erage time free carriers exist before recombining

spontaneously (as in LEDs).

– Stimulated emission lifetime, τst, is the aver-

age time free carriers exist before being forced to

recombine by stimulation.

– For a laser diode τst < τsp.

• Rise times of 0.1-1 ns are common, allowing internal

modulation up to several GHz.

132

• Linewidth of 1-5 nm are achievable. Linewidth is

often dependent on drive current, since at sufficiently

high current a single cavity mode can be excited.

• Radiation patterns of laser diodes differ from LEDs:

– laser diodes radiate asymmmetrically with smaller

cone angles than LEDs (hence potential for im-

proved fibre coupling).

– since the laser diode emission is more coherent

than the LED, the narrower pattern cut is due

to the wide aperture as expected from diffraction

theory.

133

4. Optical Amplifiers

Two general types:

1. Semiconductor Optical Amplifier : use the gain as-

sociated with semiconductor matrials (cf. laser oper-

ation).

2. Erbium Doped Fibre Amplifier (EDFA): use exter-

nal optical pumping source.

Erbium Doped Fibre Amplifiers:

W/M W/MInput

Erbium Doped Fibre

(1.55 um)

Laser DiodePump Source

(1.48 um)

(1.55 um)

(1.48 um)

Output

134

Operation:

• Erbium doped silica fibre has optical gain near 1.55 µm.

• Efficient pumping bands are 980 and 1480 nm.

• Couple input and pumping signals to Erbium doped

fibre by wavelength division multiplexing (eg. Prism

or reflection grating).

• Pumping light absorbed by Erbium atoms causing

population inversion.

• Stimulated emission by the signal photons then oc-

curs, increasing the signal power along the fibre.

• Any remaining pumping signal removed by wave-

length division demultiplexing.

• If pumping signal strength insufficient for complete

fibre length, absorption then ocurrs.

Characteristics:

• For 10 mW pumping power, 30 dB gain available

with about 3 dB noise figure over a 20 nm band.

135

Optical Detectors

Two photoelectric effects can be used:

• External photoelectric effect : electrons freed from

surface of a metal by energy absorbed from incident

photons. (eg. vacuum photodiode).

• Efficient pumping bands are 980 and 1480 nm.

• Internal photoelectric effect : free charge carriers

generated in semiconductor junction by incoming pho-

tons (eg. semiconductor photodiode).

Some terms that are used:

• Responsivity: ρ = iP

(A/w)

• Spectral response: responsivity as a function of wave-

length.

• Rise time: as for light sources (f3dB = 0.35tr

)

136

Vacuum Photodiode

Cathode(cesium)

1.9 eVemittedelectrons

i

RL

V

Anode

incomingphotons

Vb

• A single electron is liberated from the cathode if a

photon has

hf ≥ Φ

where Φ is the work function of the metal.

• f = Φ/h defines the lowest optical frequency that

can be detected, or in terms of wavelength

λ0(µm) =1.24

Φ(eV)

137

• Quantum efficiency is defined as

η =number of emitted electrons

number of incident photons

• Responsivity is therefore calculated as

– number of photons per second striking cathode

= P/(hf)

– number of emitted electrons per second = ηP/(hf)

– current i = ηP/(hf)× e– hence

ρ =i

P=ηe

hf=ηeλ

hc

• Output voltage v = ρPRL, ie. output voltage is

proportional to incident optic power.

138

Semiconductor Photodiode

Formed by reverse biassing a pn-junction

hfi

RLVb

p nhf

Conduction Band

hf free electroncreated

free holecreated

depletion region

Valence Band

139

• A depletion region is formed at the junction.

• When a photon is absorbed in the depletion region,

the free carriers produced are accelerated across the

junction potential −→ is produced.

• When a photon is absorbed outside the depletion re-

gion, the forces are small on the free carriers pro-

duced and they often recombine−→ reduced respon-

sivity.

• Rise time is often long due to delays in current transit

time across depletion region.

140

PIN Photodiode

hfi

RLVb

p ni

V

• Solves problem of low responsivity and long rise time

by placing intrinsic semiconductor material (no dop-

ing) between the p and n region.

• High probability photons will be absorbed in the

large intrinsic region −→ high responsivity.

• Electrical forces strong across intrinsic region −→fast rise time.

141

An incoming photon must have an energy geater

than the material bandgap energy for electron-hole

pair creration, hence there is a cutoff wavelength

defined by

λ(µm) =1.24

Wg(eV)

Some material properties are:

η λ range Peak response Peak responsivity

µm µm (A/W)

Si 0.8 0.3-1.1 0.8 0.5

Ge 0.55 0.5-1.8 1.55 0.7

InGaAs 0.8 1.0-1.7 1.7 1.1

142

I-V Characteristics

-20 -15 -10 -5 0 0.50

-5

-10

-15

-20

Load LineP = 10 uW

20

30

40

Vb( )Photodiode Voltage (V)

Photoconductive Region(reverse biased)

Photovoltaic Region(forward biased)

Photodiode C

urrent (uA)

• I-V characteristic is the function of photodiode cur-

rent vs. voltage across the photodiode.

• The dark current Id is due to thermal generation of

free carriers (0.1 nA–100 nA depending on material

and temperature).

• Minimum detectable power is approximately

P = Id/ρ.

143

• Using the circuit below and the load line on the char-

acteristics we obtain

Vb + vd + RLid = 0

hf

RLVb V

+ -Vd

id

Hence can determine the output voltage for a given

optical input power

v = ρRLP

which is a linear relationship (until saturation is reached

at high optical powers).

144

Speed of response limited by

1. Carrier Transit Time:

• time for free charges to traverse the depletion re-

gion (approx. the width of the intrinsic layer for

PIN diodes).

• higher bias voltages reduce transit time.

• values of 1 ns or shorter are achievable.

2. Capacitance Effects:

RL VCdid

• Cd is the junction capacitance of the diode.

• Time constant isRLCd, giving a 10-90% rise time

of tr = 2.19CdRL, and a bandwidth of

f3dB =1

2πRLCd

145

Design trade-offs

• Design of diode: small junction area gives small ca-

pacitance and high bandwith, at the expense of re-

duced area for light capture.

• Choice of RL:

Eqn. Tradeoff

v = ρPRL RL large for large output voltage

Pmax = Vb/ρRL RL small for large dynamic range

f3dB = (2πRLCd)−1 RL small for large bandwidth

i2T = 4kT∆f/RL RL large to reduce thermal noise

current

146

I-V Converter:

• In previois circuit, voltage across diode decreases with

increasing optic power−→ nonlinearity as diode volt-

age drops to zero (ie. saturation).

• Without reducing RL we can use an I-V converter

circuit so vd = −Vb for all id (or optic powers).

Vb V

Rfid

• Output voltage v = −idRf .

147

Packaging

• Similar to that for LEDs and LDs, but less critical.

• Since the active area for detection is large, alignment

with the fibre is simpler.

• Photodetectors accept light over a wide angular range,

so NA mismatch between detector and fibre is not a

severe problem.

148

Avalanche Photodiode

• Employ a multiplying effect to provide internal gain.

• Used when incident optic power is less than a few

microwatts.

• Avalanche effect multiplication:

– Photon absorbed in depletion region yielding a

free electron-hole pair.

– Large electrical force provided by reverse bias ac-

celerates the charges

– Charges collide with neutral atoms freeing addi-

tional electron-hole pairs

– These additional electron-hole pairs also acceler-

ate and collide to free further charges.

149

• A large reverse bias is needed to stimulate the avalanche

effect.

• Gain

M =1

1−(

vdVBR

)n

where n > 1 and VBR is the reverse breakdown volt-

age of the diode.

• The current generated is thus

i =MηeλP

hc

and responsivity is

ρ =Mηeλ

hc

150

• An example is the Reach Through Diode construc-

tion shown below.

hf

RLVb

i

V

p+ n+p

electron-holecreation

avalanchemultiplication

• Only the electrons created take part in the avalanche

effect to improve the noise characteristic of the de-

vice.

151

Modulation

Baseband Modulation

LED Modulation:

Baseband Analogue Techniques

• Photocurrent and optical power given by

i = Idc + ISP sinωt

P = Pdc + PSP sinωt

• Modulation factor is defined as

m′ =ISPIdc

and the optic modulation factor as

m =PSPPdc

152

• Recalling the bandwidth limitation on the LED due

to carrier life time τ

PSP =a1ISP√1 + ω2τ 2

we can relate the modulation factors by

m =m′√

1 + ω2τ 2

which for ωτ 1 gives a linear transfer function

with m ≈ m′.

153

• An example of an analogue modulation circuit for a

LED:

Vdc

CR1

Vin Rin R2 Re

(LED)

154

LED Nonlinearities:

• Nonlinearities in the system are due mostly to those

of the source.

• Model the optic power as

P = Pdc + a1is + a2i2s

where is is the signal current and Pdc is the constant

power produced by the bias current.

• If is = I sinωt then

P = Pdc +1

2a2I

2 + a1I sinωt +1

2a2I

2 cos 2ωt

155

• Define total harmonic distortion (THD) as

THD =electrical power in harmonics

electrical power in fundamental

=

(electrical power in harmonics

electrical power in fundamental

)2

and THD (dB) = -10log THD.

• For a signal is = I sinωt we have

THD =1

4

(a2I

a1

)2

• Typical THD is 30-60 dB for LEDs.

156

• If the signal has two frequency components such that

is = I1 sinω1t + I2 sinω2t then the optic power is

P = Pdc +1

2a2

(I2

1 + I22

)

+a1(I1 sinω1t + I2 sinω2t)

−1

2

(I2

1 cos 2ω1t + I22 cos 2ω2t

)

+a2I1I2cos(ω1 − ω2)t− cos(ω1 + ω2)t

• Note, the last term is called intermodulation dis-

tortion −→ coupling of power between channels is

possible.

157

Baseband Digital Modulation

• No dc bias current is needed for digital modulation.

• Distortion is not a significant effect since only on/off

states are detected.

• Example of a digital modulation circuit for a LED:

Vdc

C

R1

Vin

R

R2

(LED)

158

Laser Diode Modulation

• Design problems arise from:

– Existence of theshold current

– Age dependence of threshold

– Temperature dependence of threshold

– Temperature dependence of wavelength

• Digital systems are usually biassed near threshold

(Idc ≈ ITH).

• Analogue circuits require bias well above threshold

for linearity.

• Wavelength shifts may be important for multichannel

systems or systems operating near zero dispersion.

• Feedback control can be used to compensate for tem-

perature and age variations (but does not affect the

wavelength variantion problem).

159

LD Analogue Modulation Circuits:

• As for LED but requiring much higher collector cur-

rent.

• THD > 30 dB for a good LD.

LD Digitial Modulation Circuits:

Example – Note the dc bias arrangement:

Vdc

C

R1

Vin

R

(LD)

160

Analogue Modulation Formats

• Notation:

i = I0 + Is cosωmt

P = P0 + Ps cosωmt

where ωm = 2πfm is the modulation frequency, I0

is the total dc current and P0 is the average optical

power.

1. AM/IM Subcarrier Modulation

• AM shifts the baseband to a different part of the

spectrum.

• Different channels can be received by filters tuned to

the appropriate subcarrier frequency.

161

• For a single sinusoidal modulation signal

i = I0 + Is(1 + m cosωmt) cosωsct

where ωsc = 2πfsc is the subcarrier frequency and

m ≤ 1. Hence the optic power is

P = P0 + Ps(1 + m cosωmt) cosωsct

• The spectrum of the AM signal is

SS ¡S¢

from which the bandwidth requirement B = 2fm is

obtained if fm is the highest modulating frequency.

162

Frequency Division Multiplex (FDM)

• Each channels has a different subcarrier frequency.

• Channels must be separated by more than 2ωm.

• Filters are used to separate channels.

• Nonlinearities create distortion and crosstalk between

channels.

• Total source power must be divided between the chan-

nels to prevent the peak drive current being exceeded.

• Used for cable TV because of the advantages:

– no additional conversion needed in modulation

format.

– smaller bandwidth than FM

163

2. FM/IM Subcarrier Modulation

• The modulating current is

i = I0 + Is cos[ωsct + θ(t)]

• If the modulating signal is a single sinusoid at fre-

quency fm = ωm/2π

i = I0 + Is cos[ωsct + β sinωt]

where β is the modulation index.

• If B is the basebandwith (= fm) and ∆f = βfm the

maximum frequency deviation, the total bandwidth

of the FM signal is

BT = 2B + 2∆f = 2fm(1 + β)

164

• The optic power is thus

P = P0 + Ps cos[ωsct + β sinωt]

• Since the information is extracted from the phase of

the subcarrier signal, effects of nonlinearities can be

minimised.

• FDM can be used with subcarrier frequencies sepa-

rated by at least BT .

• The FM format normally occupies a larger band-

width than the AM format.

165

Digital Modulation Formats

Why use digital anyway?

• Sources can be switched rapidly −→ large band-

widths achievable.

• Nonlinearities are not as important to accurate signal

reconstruction.

• Error checking can be used.

• Compatibility with non-optic digital data.

• Digital pulse are easily reconstructed by repeaters in

long links.

• Signal quality is often better than that in analogue

links.

166

1. Pulse Code Modulation

• Subcarrier switched on and off by the digital signal-

On/Off Keying (OOK).

• RZ and NRZ codes can be applied directly.

• For NRZ codes the spectrum contains a large and

important dc component, the magnitude of which

depends on the data −→ average optical power at

receiver varies.

• For RZ codes, the dc component is not significant

and can be filtered by capacitive coupling.

NRZ

RZ

0 1 0 1 1 1 0

167

• Where clock recovery is required a Manchester code

can be used. Data is represented by transitions.

(1→ 0 = downward transition).

NRZ

0 1 0 1 1 1 0

Manchester

• Where an Automatic Gain Control is used at the

receiver, a Bipolar Code can be used. Only changes

in data are transmitted. Here the average optical

power is constant regardless of the data.

NRZ

0 1 0 1 1 1 0

Bipolar

168

2. Other Digital Formats

• Pulse Position Modulation (PPM): The position of

a narrow pulse within a time frame gives the ampli-

tude of the sampled signal.

• Pulse Width Modulation (PWM): The width of the

pulse gives the amplitude of the sampled signal.

• Frequency Shift Keying (FSK): Subcarrier frequency

determines the data bit.

• Phase Shift Keying (PSK): Subcarrier phase deter-

mines the data bit.

169

Multiplexing Digital Signals

• Subcarrier frequency division multiplexing can be used

to provide several channels of digital data.

• Time division multiplexing can be used to inter-

leave several data streams onto the same channel.

• Trade-off between bandwidth for multiple subcarriers

or bandwidth for a single high data rate channel.

170

Heterodyne Receivers

Heterodyne Receivers make use of coherent detection

to use frequency modulated optic carriers, since phase

is recovered by downconversion.

Block diagram of an optic heterodyne receiver:

+Fibre+

LocalOscillator

ESIG ELO

ELO

ESIG

Photodetector

SignalProcessor

where ESIG is the signal electric field and ELO is the

local oscillator electric field, given by

ESIG = ES cos[ωct + θ(t)]

ELO = EL cos[(ωc + ωIF)t]

171

• ωc is the optic carrier frequency, ωIF is the optic lo-

cal oscillator frequency (called the intermediate fre-

quency, and θ(t) = β sinωmt is the optic frequency

modulation.

• The intensity of the signal received at the photode-

tector is

I = (ESIG + ELO)2

Eliminating all terms near 2ωc because they are be-

yond the bandwidth of the photodetector yields

I =1

2

(E2S + E2

L

)+ ESEL cos[ωIFt− θ(t)]

172

• The detected optical power is therefore

P = PS + PL + 2√PSPL cos[ωIFt− θ(t)]

• The average photocurrent is

idc =ηe

hf(PS + PL)

and the signal photocurrent is

iIF =2ηe

hf

√(PS + PL) cos[ωIF − θ(t)]

• It should be noted that the heterodyne process has

introduced signal gain by way of the local oscillator

power.

173

Component consideration for heterodyne

detection

Optical Mixer

Local Oscillator(LD)

PhotodetectorFibre

Beamsplitter

• Mixer operation relies on the interaction between the

local oscillator source and the input signal.

• The local oscillator source must have well defined,

signal polarized radiation −→ only single mode laser

diodes suffice.

174

Frequency modulation of LD transmitter

• The LD oscillation frequency depends on the instan-

taneous amplitude of the injected current. The cur-

rent determines the carrier density and temperature

in the active layer.

• The refractive index of the active layer is dependent

on the carrier density and temperature −→ so is the

resonant frequency of the cavity.

• ∆f = βfm is now the optic frequency deviation re-

quired.

• An alternative is to use external frequency modula-

tors using electro-optic or acousto-optic devices.

175

Optic Frequency Division Multiplexing

• Channels multiplexed onto different optic frequency

carriers:

CH.1

CH.2

CH.3

LD1

LD2

LDN

£¤¥¦

§¨

Cou

pler ©«ª­¬®©¯¬°±°±°¬®©²

+

LO

Fibre Link

Mixer

Photo-detector IF

Filt

ers

³´µ¶·¸¹º»¼½¾

• This is similar to wavelength division multiplexing.

• Subcarrier frequency division multiplexing can also

be used on each optic channel to further increase sig-

nal capacity.

176

Advantages and Disadvantages of Heterodyne

Detection

Advantages:

• Optical FM possible (phase detection less prone to

errors due to nonlinearities).

• Receiver has increased sensitivity.

Disadvantage:

• Complexity and cost of components.

177

Noise and Detection

Noise

1. Thermal Noise

• Originates in photodetector load resistance RL.

• Due to random thermally induced nature of elec-

trons.

• Mean square thermal noise current is given by

⟨i2NT⟩

=4kT∆f

RL

• This is superimposed on the detector current

i = ηeP/hf .

178

2. Shot Noise

• Originates in the photodetector itself.

• Due to discrete nature of electrons: ie. the total cur-

rent is due to a sum of individual electron move-

ments.

• Mean square shot noise current is given by⟨i2NS⟩

= 2eI∆f = 2e(ID + 〈is〉)∆f

where I is the average detector current, 〈is〉 is the

average value of the signal current, ID is the dark

current, and e is the electronic charge.

NOTE: up to about 10 GHz, both thermal and shot

noise may be assumed to have uniform spectra.

179

Derivation of 4kTR∆f

R

en2

C

in2

vn2

• Assumption 1: This circuit is modeled by a random

voltage generator en(t), a pure capacitorC, a lumped

resistor R, and resistance-free wires.

• Note that the capacitor is pure, so there is space be-

tween the plates, and we therefore expect the thermal

noise formula to be independent of C.

180

1. The Windowed Fourier Transform

By definition the Fourier transform In(ω) of the

fluctuating current in(t) through the circuit, is

given by

In(ω) =

∫ ∞

−∞in(t)e−jωtdt (4)

However, In(ω) can only exist if in(t) is absolutely

integrable, i.e.∫ ∞

−∞|in(t)|dt < +∞

• Unfortunately, this condition is not satisfied as in(t)

is a randomly varying function of time and does not

decay to zero as t→ ±∞.

• The instantaneous values of in(t) cannot be predicted

and this type of function represents an example of a

stochastic process.

181

• In practice, the random process can only be observed

for a finite window of time τ , so a dimensionless time

window function W (t, τ ) is defined. Provided a large

τ is chosen to minimize statistical sampling error, the

windowed version of Eqn. 4 becomes

In(ω, τ ) =

∫ ∞

−∞in(t, τ )e−jωtdt (5)

1

W

¿ÁÀÃÂÅÄ ÆÈÇÅÉ

182

• Assumption 2: The stochastic process is indepen-

dent of where the origin of W (t, τ ) is placed on the

time axis. This condition is referred as stationarity

and is reasonable in view of the observed nature of

thermal noise.

• This useful property means that Eqn. 5 is invariant

to the position of W (t, τ ).

• Although the presence of W in Eqn. 5 solves the in-

tegrability problem, it introduces the artifact of spec-

trum leakage.

• The leakage occurs essentially because the transform

of W , itself contains a continuous range of nonzero

frequency components.

• This can be ignored as we will eventually be consid-

ering just the limiting case as τ →∞.

183

Consider a windowed version of en(t) as

en(t, τ ) = en(t)W (t, τ ).

We now define the signals vn(t, τ ) and in(t, τ ) as the

response of the circuit to en(t, τ ). Note that as

τ →∞, vn(t, τ )→ vn(t) and in(t, τ )→ in(t).

We have in(t, τ ) = ddt[Cvn(t, τ )], thus

In(ω, τ ) =

∫ ∞

−∞

d

dt[Cvn(t, τ )]e−jωtdt

and integrating by parts

In(ω, τ ) = C[vn(t, τ )e−jωt

]+∞−∞+jωC

∫ ∞

−∞vn(t, τ )e−jωtdt

which reduces to

In(ω, τ ) = jωCVn(ω, τ ). (6)

• Although this result is trivial, it was important to

show that there were no window artifact problems.

• As dmWdtm = 0

∣∣t=±∞ it can be shown for the general

case that

dmvn(t, τ )

dtm←→ (jω)mVn(ω, τ )

is a windowed or time-limited Fourier pair.

184

2. The Johnson Thermal Noise Formula

• We now proceed to use the result in Eqn. 6 to find the

power spectrum in the capacitor and then produce

the celebrated Johnson formula.

• Consider the voltages around the loop, by Kirchhoff

vn(t) + in(t)R = en(t)

and viewed from the window W (t, τ ) this becomes

vn(t, τ ) + in(t, τ )R = en(t, τ ).

• Notice that by definition, en(t, τ ) does not contain

any delta function terms and therefore in(t, τ ) and

vn(t, τ ) must also be free of spikes.

• This can simply be demonstrated by reductio ad ab-

surdum : if in(t, τ ) contained a delta function pair,

due to windowing, then vn(t, τ ) would need an iden-

tical pair of opposing sign to balance the above equa-

tion. This would be impossible, however, as in(t, τ )

would then contain the second derivative and reason-

ing continues inductively ad infinitum.

185

• Now, taking Fourier transforms we have

Vn(ω, τ ) + In(ω, τ ) = En(ω, τ ).

substituting in Eqn. 6 gives

Vn =En

1 + jωRC(7)

multiplying by complex conjugates

|Vn|2 =|En|2

1 + (ωRC)2. (8)

• By conservation of energy, the total energy in the

time domain must equal that in the frequency do-

main, therefore∫ +∞

−∞e2n(t, τ )dt =

1

2π

∫ +∞

−∞|En(ω, τ )|2dω.

• This is known as the energy theorem or Plancherel’s

theorem (a special case of Parseval’s theorem).

• Each side of the equation represents total energy and

therefore |En|2 represents the energy density with

units of V2s/Hz.

186

• Due to the Hermitian property of the Fourier trans-

form, |En|2 is always even therefore, we can write the

one-sided form∫ +∞

−∞e2n(t, τ )dt =

1

2π

∫ +∞

0

2|En(ω, τ )|2dω.

• By definition of temporal average⟨e2n

⟩= lim

τ→∞

⟨e2n

⟩τ

= limτ→∞

1

τ

∫ +∞

−∞e2n(t, τ )dt

therefore⟨e2n

⟩= lim

τ→∞1

2π

∫ +∞

0

2|En(ω, τ )|2

τdω

where |En|2

τ is called the sample spectrum or peri-

odogram.

• It is permissible to move the limτ→∞ inside the in-

tegral provided the ensemble average is performed

first,

e2n =

1

2π

∫ +∞

0

limτ→∞

2|En|2τ

dω.

• As we have a random process, the limit would be in-

determinate had the ensemble average not been per-

formed first.

187

• Assumption 3: The process is ergodic, so temporal

and ensemble averages are equivalent, i.e.,

limτ→∞⟨e2n

⟩τ

= e2n.

• Thus

⟨e2n

⟩=

1

2π

∫ +∞

0

limτ→∞

2|En|2τ

dω. (9)

• By definition, the one-sided power spectral density

of en is

S(ω) = limτ→∞

2|En|2τ

,

therefore we can write Eqn. 9 as

⟨e2n

⟩=

1

2π

∫ +∞

0

S(ω)dω. (10)

188

• Assumption 4: The noise spectrum is white, there-

fore S(ω) = S0, a constant.

• For a practical measuring instrument bandwidth of

∆ω, Eqn. 10 becomes

⟨e2n

⟩=

1

2πS0∆ω. (11)

• Using identical arguments for the capacitor voltage,

vn, we have

⟨v2n

⟩=

1

2π

∫ +∞

0

limτ→∞

2|Vn|2τ

dω

and substituting in Eqn. 8

⟨v2n

⟩=

1

2π

∫ +∞

0

1

1 + (ωRC)2

limτ→∞

2|En|2τ

dω

=S0

2π

∫ +∞

0

dω

1 + (ωRC)2

=1

2π

S0

RC[arctan(ωRC)]∞0 =

1

2πS0

π

2RC.

• Putting this into Eqn. 11, to eliminate S0, gives

⟨e2n

⟩=

2

πC⟨v2n

⟩R∆ω. (12)

189

• Assumption 5: The system is in equilibrium with

surroundings.

• According to equipartition theory, a general dynam-

ical system in thermal equilibrium has on average a

potential energy of kT/2 for each degree of freedom.

• One short hand method for counting up the degrees

of freedom in a linear system is to count the num-

ber of independent quadratic variables in the energy

expression.

• By inspection of Eqn. 12, we see that our system

takes up energy as

1

2C⟨v2n

⟩

and, therefore, it has one degree of freedom, hence

1

2C⟨v2n

⟩=

1

2kT. (13)

190

• Assumption 6: Let us assume the system is classi-

cal (i.e., no quantum effects) so that the Maxwell-

Boltzmann kT term holds.

• Substituting Eqn. 13 into Eqn. 12 finally yields John-

son’s formula for open circuit noise voltage⟨e2n

⟩=

2

πkTR∆ω

= 4kTR∆f. (14)

• If the capacitor is replaced by an inductor L, the

analysis can be repeated in the current domain and

the generated short circuit noise current can be shown

to be ⟨i2sc⟩

=2

π

L

R

⟨i2n⟩

∆ω,

where⟨i2n⟩

is the observed noise current.

• The system now takes up energy as

1

2L⟨i2n⟩

=1

2kT

therefore ⟨i2sc⟩

=4kT∆f

R. (15)

which is the familiar current form for the Johnson

noise formula.

191

Signal-to-Noise Ratio

The photodetector circuit may be approximated as:

RLis iNT

2 iNS2

1. For the case of constant incident optic

power

• This may occur for the duration of a “1” in a digital

signal.

• The signal current is given as

is =ηeP

hf

192

• Average electrical signal power is given by

〈PES〉 = i2sRL =

(ηeP

hf

)2

RL

• Thermal noise power is given by

〈PNT 〉 = 4kT∆f

• Shot noise power is given by

〈PNS〉 = 2e∆f

(ηeP

hf+ ID

)RL

• Defining the signal-to-noise ratio (S/N) as the av-

erage signal power divided by the average noise power

we have

S/N =(ηeP/hf)2RL

2eRL∆f(ID + ηeP

hf

)+ 4kT∆f

193

• If the optic power is large, then shot noise dominates.

Neglecting the dark current yields

S/N =ηP

2hf∆f

for this case, which is referred to as shot noise lim-

ited or quantum limited.

• For quantum limited systems, as the optic power

increases by ∆P , the S/N increased by ∆P .

• Usually the optic power is small and thermal noise

dominates, giving

S/N =RL(ηeP/hf)2

4kT∆f

for this case, which is referred to as thermal noise

limited.

• For thermal noise limited systems, as the optic power

increases by ∆P , the S/N increases by 2∆P .

194

The S/N needs to be modified when the

photodetector has intenal gain. For an internal gain

M ,

• Shot noise power increases as Mn, where

– n = 2 for a photomultiplier tube

– n > 2 for an avalanche photodetector due to

excess noise.

• Hence,

S/N =(MηeP/hf)2RL

Mn2eRL∆f(ID + ηeP/hf) + 4kT∆f

195

2. For the case of a sinusoidally modulated

optic signal

• At the photodiode, the optic power is

Pi = P (1 + m cosωt)

which gives a photocurrent

i =ηeP

hf(1 + m cosωt)

• After amplification the signal current rises to

is =MηeP

hfm cosωt = ip cosωt

• The signal power can therefore be written

〈PES〉 =1

2RLi

2p =

1

2RL

(mMηeP

hf

)2

and hence the signal-to-noise ratio

S/N =(m2/2)(MηeP/hf)2RL

Mn2eRL∆f(ID + ηeP/hf) + 4kT∆f

196

3. For a Heterodyne System

• Average photodetector current is

idc =ηe

hf(PL + PS)

and the IF current is

iIF =2ηe

hf

√PSPL cos[ωIFt− θ(t)]

= iIF,p cos[ωIFt− θ(t)]

• Average signal power is therefore

〈PES〉 =1

2RLi

2IF,p

= 2RLPSPL

(ηe

hf

)2

197

• Shot noise is

〈PNS〉 = 2eRLI∆f

where ∆f is the IF bandwidth (double that of the

baseband signal) and the current I is

I = ID + Idc

= ID +ηe

hfPL

(1 +

PSPL

)

• Hence the signal-to-noise ratio for a heterodyne sys-

tem is

S/N =2(ηe/hf)2RLPSPL

2eRL∆f [ID + (ηePL/hf)(1 + PS/PL)] + 4kT∆f

198

Error Rates in Digital Systems

• Bit-error-rate is equivalent to the Probability of Error

Pe in determining the binary value of each bit.

• Errors are caused by noise on the received signal.

1. Thermal Noise Limited Systems

• The decision on “0” or “1” is made by comparing

the signal current amplitude with a predetermined

threshold.

• For “0”, the noise current may increase the level be-

yond the threshold. For “1”, the noise current may

add out of phase to drop the level below the thresh-

old.

199

• With the threshold set at half the peak signal cur-

rent (ie. at 0.5is), the probability of error and the

required signal-to-noise ratio are realted by

Pe =1

2− 1

2

(0.354

√S

N

)

where the error function erf(x) is given for x ≥ 3 by

erf(x) = 1− e−x2

x√π

• For a Pe, we can obtain the required S/N and hence

the required optic power for adequate receiver sensi-

tivity

P =hf

ηe

√4kT∆f

RL

√S

N

• The number of photons received per bit for a bit

interval τ is

np = (P/hf)τ

200

2. Shot Noise Limited Systems

• Also referred to as Quantum Limited, and requires

very high receiver sensitivity.

• Post detection processing counts the number of elec-

trons produced in the photodetector. A threshold on

the count is used to determine a “0” or “1”.

• The number of photonelectrons produced in a bit

interval τ when a “1” is received is

ns =ηPτ

hf=isτ

e

• The number of photoelectrons produced when a “0”

is received is due to the dark current, and is

nn =IDτ

e

201

• Optimal threshold when “1”s and “0”s are equally

likely is set at

kT = int

1 +

nsln(1 + ns/nn)

• The disadvantage is that the optic power must be

known to set the threshold.

• For small dark current (ie. nn ≈ 0), kT = 1 and

Pe = e−ns

202

Other Sources of Noise

Modal Noise

• Modal noise causes a random variation in optic power

in multimode fibres.

• Usually experimentally determined.

• Caused by the speckle pattern in the fibre croass-

section – ie. bright spots where mode interference is

additive.

• Speckle pattern changes with wavelength changes due

to temperature, modulation etc.

• Mechanical fibre movement also affects speckle pat-

tern.

• Connectors are sensitive to changes in the speckle

pattern and randomly varying transmitter power may

result.

203

Electronic Amplifier Noise

• An electronic amplifier normally follows the photode-

tector to increase receiver gain.

• Amplifier multiply the noise and also create their own

thermal noise.

• For a given amplifier noise figure F ,

(S/N)out =1

F(S/N)in

and the signal-to-noise ratio is degraded.

• For shot noise limited systems, APDs or heterodyne

detection provides noiseless amplification.

204

Optical Amplifier Noise

• Repeaters may be implemented using optical ampli-

tiers.

• Each amplifier has a gain Gk and a noise figure

Fk =(S/N)k,in(S/N)k,out

• If the transmission loss between the kth and k+ 1th

amplifier is αk, the total noise figure of a chain of N

repeaters is

F =F1

α1+

F2

α1G1α2+ · · · + FN∏N−1

i=1 (αiGi)αN

205

Laser Noise

• Laser output will fluctuate randomly even when driven

with a constant −→ may be significant in high speed

systems.

• Described by Random Intensity Noise (RIN) which

is minimised by operating well above the laser thresh-

old current.

• Mean square RIN current is

〈i2NL〉 = RIN(ρP )2∆f

or noise electrical power

〈PNL〉 =⟨i2NL⟩RL

• RIN is often specified in dB/Hz as

RINdB/Hz = 10 log

(⟨P 2NL

⟩

P 2∆f

)

where P is the average optic power.

206

• The RIN contribution to the signal-to-noise ratio can

be included as

S/N =(ρP )2RL

2eRL∆f(ID + ρP ) + 4kT∆f + RIN(ρP )2∆fRL

Jitter

• In bandwidth limited systems the digital pulses are

spread by dispersion.

• Where timing errors are present, pulse edges may

randomly overlap causing bit errors.

• Such overlap is referred to as jitter.

207

Receiver Circuit Design

Front end circuitry: photodetector and first amplifier.

BJT and FET Amplifiers

• Circuits:

R

Vb

Vcc

RL

Vo

R

VbRL

Vo

Vdd

208

• Denoting RT = RL||R||Rin and C = Cd+Cin where

Rin and Cin are the input resistance and capacitance

of the BJT or FET and Cd is the photodiode junction

capacitance, the circuit limited electrical bandwidth

is

f3dB =1

2πRTCT

• The amplifiers will have a noise figure F that de-

grades the signal-to-noise ratio.

• At low modulation frequencies (below 50 MHz) FETs

have superior noise performance. Above 50 MHz

BJTs are preferred.

209

High Impedance Amplifier

• In the BJT and FET amplifiers it would be desir-

able to make RL large to improve noise performance.

However, this degrades the dynamic range and band-

width of the receiver.

• A transimpedance amplifier is used, the noise prop-

erties of which approach that of a high impedance

amplifier.

Vb V

Rfid

• The electrical bandwidth is here

f3dB =1

2πRfCf

where Cf is the feedback capacitance of the amplifier.

210

Fibre Distribution Systems

Directional Couplers

1 2

34

• Throughput Loss :

LTHP = −10 logP2

P1

• Tap Loss (coupling factor):

LTAP = −10 logP3

P1

• Directionality (isolation):

LD = −10 logP4

P1

• Excess Loss :

LE = −10 logP2 + P3

P1

211

• Directional couplers are used as power splitters, where

the splitting ratio is defined as P2/P3. For example,

a 3 dB power splitter has LTHP = LTAP = 3 dB giving

equal power from ports 2 and 3.

• An example of directional coupler construction is the

fused biconically tapered section shown below. The

length of the fused section determines LTAP.

Cladding

Cladding

Cladding

2 Core

3 Core

L

212

Duplexing Network

1 2

34

T

R

1 2

34

T

R

DC DC

Loss due to the duplexers is 6 dB.

Tree Network

Bus Bus

DC

DC

T R

1

2 3 4

NTDC T T T DC

Network loss is

L = (N − 1)LTHP + LTAP + 2NLC

where LC is the connector loss.

213

Star Network

Ring Coupler

1 2 N

1’ 2’ N’

Network Loss is

L = −10 log

(1

N

)+ LE + 2LC

Ring Network

1

2 3

4

56

• No sharing of optical power −→ large networks can

be constructed.

• Reconfigurable when nodes fail.

• Fibre Distributed Data Interface (FDDI) LAN archi-

tecture.

214

Optical Isolators

• Used to prevent optical power being reflected back

or passed to a source.

• A Faraday rotation isolator can be used.

• Non-reciprocal materials such as Yttrium Iron Gar-

net (YIG) are used to rotate the direction of polariza-

tion in a manner dependent on propagation direction.

InputPolarizer

OutputPolarizer

45 degreeFaraday Rotator

215

Analog Example

• Specification:

Transmit a 6 MHz bandwidth signal from a surveil-

lance system over a distance up to 500 m achieving

a SNR at the receiver of 50 dB.

• Proposal:

Use baseband analogue modulation, multimode fibre,

LED source, silicon PIN photodetector short wave-

length (800-900 nm) window.

Options to improve performance if needed:

– Use single mode fibre

– Use an APD

– Use s LD

– Use longer wavelength

216

Design

1. Receiver:

• Use a BJT/load resistor circuit.

• Assume 100% modulation

• Typical PIN diode characteristies at 850 nm:

Cd = 5 pF

ρ = 0.5 A/W

ID = 1 nA

=⇒ for a 6 MHz bandwidth

RL ≤ (2πCdf3dB)−1

≤ 5.3 kΩ

∴ Choose RL = 5 kΩ. Hence ∆f = 6.4 MHz. Some

lattitude for other component limitations.

• Assume the amplifier has a 3 dB noise figure at 300 K.

217

2. Power Budget:

• Assume thermal noise limited:

S/N =12RL(ρP )2

4kT∆fF

If S/N = 50 dB, optic power required at photode-

tector is P = 6 µW.

∴ average photocurrent I = ρP = 3 µA

• Since I ID, ignore dark current.

• Check rms noise currents:

– Thermal noise current√〈i2NT 〉 =

√4kT∆fF

RL= 6.5 nA

– Shot noise current√〈i2NS〉 =

√2eI∆f = 2.5 nA

∴ Thermal noise limit OK.

• Check detector linearity:

Assume 5V bias =⇒ Imax =5 V

5000 Ω= 1 mA I(= 3 µA)

=⇒ no saturation.

218

Choose components:

• LED

Pavg = 1 mW @ 850 nm

tr = 12 ns

∆λ = 35 nm

emitting surface diameter 50 µm.

• Multimode SI Fibre

NA = 0.24

optic f3dB × L = 33 MHz× km

Loss = 5 dB/km

Core diameter 50 µm.

• Multimode GRIN Fibre

axia NA = 0.24

optic f3dB × L = 500 MHz× km (with LD)

Loss = 5 dB/km

Core diameter 50 µm.

219

Hence:

Source power 0 dBm

Receiver power > -22.2 dBm (6 µW)

SI fibre −→ LED coupling loss η = (NA)2

• SI coupling loss: 12.4 dB

GRIN fibre −→ LED coupling loss 3 dB worse

• GRIN coupling loss: 15.4 dB

Loss due to reflectance at glass/air interface

• Reflectance loss: 0.4 dB (total)

Connector loss – assume 2 connectors

• Connector loss: 2 dB (total)

∴ SI fibre can have loss 22.2−12.4−0.4−2 = 7.4 dB.

=⇒ Max. SI fibre length 7.45 = 1.48 km

GRIN fibre can have loss 4.4 dB

=⇒ Max. GRIN fibre length 4.45 = 880 m

220

3. Bandwidth Budget.

Work in rise times:

System tslight source tLS

fibre tFphotodetector tPD

t2s = t2LS + t2F + t2PD

Assume for the system that

ts =0.35

f3dB

= 58 ns

Photodiode tPD = 2.19 RLCd = 55 ns.

[Note tPD transmit time of ≈ 1 ns

=⇒ detector is circuit limited.]

LED tLS = 12 ns.

=⇒ Allowable fibre rise time tF ≤ 14 ns.

221

Recall:

f3dB(elec.) = 0.71 f3dB(opt.)

and tF ≈ ∆τ

• For SI fibre f3dB(elec.)× L = 23.4 MHz×km

=⇒ tF/L =0.35

23.4× 106= 15 ns/km

=⇒ max. fibre length is14

15= 930 m

∗ SI fibre system is bandwidth limited.

• For GRIN fibre f3dB(elec.)× L = 355 MHz×km

∴ tmod

L= 1 ns/km with LD

But need to include material dispersion due to LED

linewidth. ( M= 90 ps/nm/km)

∴ tdis

L= M∆λ = 3.2 ns/km

tF/L =

√(tmod

K

)2

+tdis

L= 3.2 ns/km

=⇒ max. fibre length is14

3.4= 4.1 km

∗ GRIN fibre system is power limited.

∴ Either system works for 500 m path!

222

Digital Example

• Specification:

Transmit a 400 Mbps NRZ data over 100 km with a

BER of 10−9 or better.

• Proposal:

Need a fibre with very high rate-length product and

very low attenuation

=⇒ Single mode fibre at 1550 nm

+ Very fast source =⇒ Laser diode

+ Very fast detector with gain

=⇒ InGaAs APD

Try a conventional BJT receiver circuit.

223

• Rise time budget:

RNRZ = 400× 106 =⇒ τ ≈ 1

RNRZ

= 2.5 ns

0 ÊË Ì

Allow the system rise time to be 70% of τ

=⇒ ts = 0.7τ = 1.75 ns

• Fibre selection: single mode @ 1550 nm

loss = 0.25 dB/km

Material dispersion M = −20 ps/nm/km

waveguide dispersion Mg = 4.5 ps/nm/km

• Source selection: LD @ 1550 nm (single mode)

∆λ = 0.15 nm

tLS = 1 ns

3 dB coupling loss to s.m. fibre

5 dBm average output power (3 mW output)

224

Letting

tF ≈ ∆τ = L(M + Mg)∆λ

= 0.23 ns

we then have for the photodetector circuit:

t2PD = t2S − t2F − t2LS

=⇒ tPD ≤ 1.4 ns

With an InGaAs APD:

Cd = 1 pF

tTR = 0.5 ns

where tTR is the transit time rise time.

Circuit limited rise time tRC = 2.19RLCd.

∴ t2PD = t2RC + T 2TR

=⇒ tRC ≤ 1.3 ns

hence RL ≤ 590 Ω

* Unless a transimpedance amplifier is used and the

circuit limited by feedback components.

225

Optical Storage

Data Channel

• To be useful, data is either

– Transmitted and used elsewhere, or

– Stored and retrieved later

Channel Errors

• Data will always be corrupted by errors in the chan-

nel

– CD-ROM scratches

– Radio interference

– Random noise on a phone line

• Is it possible to detect and correct errors?

– YES!

– Well, almost always.

226

OSI Communication Reference Model

227

Simple Error Detection

• Suppose our channel has a uniform bit error proba-

bility of 10−3

• Triple repetition

– 000 or 111

• Probability of a single or double bit error

≈ 3× 10−3 (Detected errors)

• Probability of a triple bit error

– 10−9 (Undetected error)

• Using triple repetition, use majority rule

– Correction rate 3× 10−3

– Residual error rate 3× 10−6 + 10−9

• This simple example is very inefficient

– Each channel bit represents 1/3 of a bit of useful

information

228

Hamming Distance

• Difference measure of binary vectors of length N

d(X, Y ) =

N∑

n=1

(Xn 6= Yn)

dmin = minX,Y

d(X, Y )

• Error detection possible (ε is number of error bits):

dmin ≥ ε + 1

• Error correction possible (τ is number of correctable

bits):

dmin ≥ 2τ + 1

• With k message bits and (n− k) check bits:

dmin ≤ n− k + 1

• Coding efficiency given by

R =k

n

229

Eight-to-Fourteen Modulation

• The first-line of error detection/corection codes on

CD or CD-ROM

– 8 bit codeword represented by 14 channel bits

– Choose 256 codewords with at least 4 bits differ-

ent from all other valid codewords

∗ 00011010001100

∗ 11001010101100, etc

∗ dmin = 4

∗ 512 valid codewords available

– Can detect up to 3 bit errors

– can correct 1 bit error

230

Error Correction Example

• 16 data bits with 8 parity bits

• Even parity forced in x and y

– 0,2 or 4 bits set to 1

• Any single message-bit error correctable

m1 m2 m3 m4 c1

m5 m6 m7 m8 c2

m9 m10 m11 m12 c3

m13 m14 m15 m16 c4

c5 c6 c7 c8

231

Convolutional Coding

• Send multiple copies of the same data

– D4 + D3 + 1, along with

– D4 + D3 + D + 1

232

Interleaving

• Error usually occur in a burst

MY**** HAS NO NOSE

• Since convolution relies on consecutive bits, we use

interleaving to break up bursty errors

233

Communication without Resend

• Broadcast data

– Digital TV

• Stored data

– CDROM or DVD

• Resend is not possible

– Require very low probability of uncorrectable er-

ror

– Requires very powerful error correction codes

Graceful Degradation

• Some data requires perfect transmission

– Text

– Binary executable programs

• In the case of multimedia, can “guess” missing con-

tent

– Repeat previous video frame section

– Repeat or silence audio

– Approximate image etc

234

CD-ROM

• Capacity 654.74 MBytes

• One spiral track of reflective pits and land

– Read by reflection of focussed laser

• Constant Linear Velocity

– Spins quickly for inner tracks, slowly for outer

tracks

• Mass produced or writable

• Data format standards, including audio

• Very high levels of error correction

235

Low Level Data Encoding Requirement

• 1. High information density

– Efficient use of medium

• Minimum intersymbol interference

– Maximise run length

• Self-clocking

– Minimise run length

• Low digital sum

– Uniform distribution of ones and zeros

Low-level Coding

• NRZ coding

– 1 is a transition between land and pit

– 0 remains as a constant land or pit

• Eight-to-Fourteen modulation

– 8 bits encoded as 14 bit word

• Three merging bits (no information)

– Limit run length and distribute 1s and 0s

• Total of 17 channel bits per octet

236

Data Framing

• Needed for reliable self-clocking and synchronisation

– 24 bits pattern + 3 merging bits

– control & display octet (17 bits)

– 12 data octets (204 bits)

– 4 error correction octets (68 bits)

– 12 data octets (204 bits)

– 4 error correction octets (68 bits)

Total 588 channel bits

237

First Level Error Correction

• 14 to 8 demodulation

• Reed-Solomon decoder

• Interleave

• Second Reed-Solomon decoder

• Result is a high probability of correct data

– Up to 220 octets/second raw errors (0.12% per-

fectly corrected (almost always)

CD-ROM Error Correction

• Each 2048 octet block of data is protected by

– 32 bit parity code for error detection

– 2208 bit Reed-Solomon error correction code

• Result is a much lower probability of error

– 10−9 is achievable for a CD in good condition

• Mechanical requirements

– Manufacture

– Accuracy of recording and reading mechanism

– Physical condition and wear

238

DVD-Digital Versatile Disc

• Primary application is video

– 2 hours of broadcast quality video per layer

– Up to 9 hours of VHS quality video per layer

• Other applications

– Better-than-CD audio

– Data storage

∗ 4.38 Gigabytes (Single Sided, Single Layer) to

∗ 15.90 Gigabytes (Double Sided, Double Layer)

• Data rate of 11.1 Mbit/s

– Around 5 Mbit/s for broadcast quality video

Features of DVD-Video

• Broadcast quality 2 hours pwe layer per side

– 8 hours per DS/DL disc

– 30 hours of VHS quality on DS/DL

• Uses MPEG-2 video coding

– Widescreen or standard TV

• Up to 8 audio tracks

• Seamless branching of video

• Random access

239

DVD versus CD

• Increase in capacity due to

– Smaller pit length ×2.08

∗ 0.40 to 1.87 versus 0.83 to 3.05 micron

– Tighter trach pitch ×2.16

∗ 0.74 versus 1.6 micron

– Larger data area ×1.02

– Better channel modulation ×1.06

– Better error correction ×1.32

– Less sector overhead ×1.06

• Total increase ×7

240

Example of data asymmetry

• H.261/263

– Real-time video-conferencing

– Asymmetric processing, symmetric channel

– Very powerful data processing at encoder

– These days, can be handled in real time by a

modern PC

• MPEG-1

– CD-ROM based video storage & retrieval

– Asymmetric processing and channel

– 10 minutes of video may require hours of process-

ing on a PC

Conclusions

• Error detection and correction is needed beacuse stor-

age and transmission is not perfect

• CD, CD-ROM and DVD have different error correc-

tion requirements but operate on the same principles