Transcript self-study - Technion moodle

046342
Lecture I
ver. 3
Introduction to Fiber Optic
Communication
in which we present a roadmap for the course
…an overview of optical transmission,
…then proceed to review some elements of signal analysis
…then review Electro-Magnetic Theory,
…and guided wave optics
…
L1 map
• Admin, Roadmap, References
• Overview of Optical Communication
• Elements of signal analysis – narrowband signal
representations
• Elements of electro-magnetic wave propagation
• Guided Optics:
• Solving the wave equation
for cylindrical step index fibers – self-study
Moshe Nazarathy Copyright
2
Admin, Roadmap
3
‫מבוא לתקשורת בסיבים אופטיים‬
grad/undergrad course #046342 - Spring 2010
• Lecturer: Prof. Moshe Nazarathy ‫משה נצרתי‬
•
•
•
•
•
Moshe’s reception hours: Tue 12:30-14:20
room 755
office phone ext. 3917
[email protected]
TA: Alex Tolmachev
Grading: HW+FINAL EXAM
Homework will be submitted by pairs of students.
- Homeworks will represent 15% of the final grade. During the semester 9
homeworks will be assigned and the homework grade will be based on the 6 best
grades. 2 out of the 9 homeworks will include computer exercises the submission
of which is mandatory.
• Prerequisite:
044148 “‫”גלים ומערכות מפולגות‬
– Ideal (optional) prerequisites: Signals and Systems, Random
Signals, Analog and Digital Communication
Moshe Nazarathy Copyright
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• References:
– Moshe’s class and lecture notes (.ppt) on the “Moodle”
website: http://moodle.technion.ac.il/
– L. Kazovsky, S. Benedetto, A. Willner, Optical Fiber
Communication Systems, Artech House, 1996.
– Gagliardi & Karp, Optical Communications, 2nd edition, Wiley,
1995 (also 1st edition OK).
– J. Buck, Fundamentals of Optical Fibers, Wiley, 2004.
– Okoshi&Kikuchi, Coherent Optical Fiber Communications, KTK,
1988.
– J.W. Goodman, Statistical Optics, Wiley, 1985.
– Papers from professional literature
Moshe Nazarathy Copyright
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Roadmap of 048951 (MAIN LECTURES) :
L1.
L2.
L3.
L4.
L5.
L6.
L7.
L8.
L9.
Subject to Change
Overview of optical communication, review complex signals ,E-M, guided optics
Elements of Linear Propagation and Nonlinear Optics
Nonlinear Schroedinger Equation
Nonlinear impairments
The Photon Nature of Light + Direct Optical Detection
Optical Modulation
Optical Amplification
Polarization Mode Dispersion
Wavelength Division Multiplexed Direct Detection Systems
Coherent and Differentially Coherent Transmission
.L10.
.L11.
.L12.
6
Moshe Nazarathy - All Rights Reserved
Acknowledgements:
•P. Winzer, C. Chandrasekhar, ECOC’05,
whose kind permission I was granted to incorporate
their notes into my current course.
•A. Mecozzi and M. Shtaif also granted me kind
permission to incorporate their notes from ECOC’05.
•Prof. G. Eisenstein the previous teacher of this course
for his course notes
We are going to greatly benefit from these great materials
8
Overview of optical communication
From meters to thousands of Kms, from Mb/s to multi Tb/s
9
History (+basic TIR principle)
In the 19th century ….
In 1870, John Tyndall, using a jet of water that flowed from one container
to another and a beam of light, demonstrated that light used internal
reflection to follow a specific path. As water poured out through the spout
of the first container, Tyndall directed a beam of sunlight at the path of
the water. The light, as seen by the audience, followed a zigzag path
inside the curved path of the water. This simple experiment, illustrated in
the figure below, marked the first research into the guided transmission
of light. Light is kept in the fiber core by TOTAL INTERNAL REFLECTION (TIR)
In the wave picture the fiber acts as a waveguide
SiO 2
k0n2 sin 2  k0n1 sin 1
Equal tangential k-components
GeO 2 doped
sin TIR  n1 / n2
10
…and the rest of the story [Wiki]
History
•
In 1966 Charles K. Kao and George Hockham proposed optical fibers at STC Laboratories (STL), Harlow, when they showed that the
losses of 1000 db/km in existing glass (compared to 5-10 db/km in coaxial cable) was due to contaminants, which could potentially be
removed.
•
Optical fiber was successfully developed in 1970 by Corning Glass Works, with attenuation low enough for communication purposes
(about 20dB/km), and at the same time GaAs semiconductor lasers were developed that were compact and therefore suitable for
transmitting light through fiber optic cables for long distances.
•
After a period of research starting from 1975, the first commercial fiber-optic communications system was developed, which operated at a
wavelength around 0.8 µm and used GaAs semiconductor lasers. This first-generation system operated at a bit rate of 45 Mbps with
repeater spacing of up to 10 km. Soon on 22 April, 1977, General Telephone and Electronics sent the first live telephone traffic through
fiber optics at a 6 Mbps throughput in Long Beach, California.
•
The second generation of fiber-optic communication was developed for commercial use in the early 1980s, operated at 1.3 µm, and used
InGaAsP semiconductor lasers. Although these systems were initially limited by dispersion, in 1981 the single-mode fiber was revealed to
greatly improve system performance. By 1987, these systems were operating at bit rates of up to 1.7 Gb/s with repeater spacing up to
50 km.
•
The first transatlantic telephone cable to use optical fiber was TAT-8, based on Desurvire optimized laser amplification technology. It went
into operation in 1988.
•
Third-generation fiber-optic systems operated at 1.55 µm and had losses of about 0.2 dB/km. They achieved this despite earlier difficulties
with pulse-spreading at that wavelength using conventional InGaAsP semiconductor lasers. Scientists overcame this difficulty by using
dispersion-shifted fibers designed to have minimal dispersion at 1.55 µm or by limiting the laser spectrum to a single longitudinal mode.
These developments eventually allowed third-generation systems to operate commercially at 2.5 Gbit/s with repeater spacing in excess of
100 km.
•
The fourth generation of fiber-optic communication systems used optical amplification to reduce the need for repeaters and wavelengthdivision multiplexing to increase data capacity. These two improvements caused a revolution that resulted in the doubling of system
capacity every 6 months starting in 1992 until a bit rate of 10 Tb/s was reached by 2001. Recently, bit-rates of up to 14 Tbit/s have been
reached over a single 160 km line using optical amplifiers.
•
The focus of development for the fifth generation of fiber-optic communications is <<here Wiki gets it wrong – deleted>>.
•
In the late 1990s through 2000, industry promoters, and research companies such predicted vast increases in demand for
communications bandwidth due to increased use of the Internet, and commercialization of various bandwidth-intensive consumer
services, such as video on demand. Internet protocol data traffic was increasing exponentially, at a faster rate than integrated circuit
complexity had increased under Moore's Law. From the bust of the dot-com bubble through 2006, however, the main trend in the industry
has been consolidation of firms and offshoring of manufacturing to reduce costs. Recently, companies such as Verizon and AT&T have
taken advantage of fiber-optic communications to deliver a variety of high-throughput data and broadband services to consumers' homes.
Moshe Nazarathy Copyright
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Why Optical Communication?
Current Economic Motivation
Initial motivation was telephony
‫תעבורה‬
PB/Month
20K
Source: CISCO Networking Report Q1/09
‫ גידול בשנה‬60%
1.65 = 10.49
10K
100
2006
2009
2012
12
Overview of photonic
transmission Impairments - I
13
‫קילקולים‬
IMPAIRMENTS
LOSS
NOISE SOURCES
RX+OA dependent
TX Power
Optical Signal/Noise
‫ובמבנה מוליך הגל‬
(‫(נפיצה‬
CHROMATIC DISPERSION (CD)
Pulse spreadingISIBER
‫נפיצת אופני קיטוב‬
‫היא מעוותת את האות‬
POLARIZATION MODE DISPERSION
14
NON-LINEARITY
‫ליניאריות – גם‬-‫ אי‬Gets worse with TX power
Fiber loss spectrum
FIRST WINDOW:
800 to 900 nm (GaAs lasers, MMF)
SECOND WINDOW (low CD):
1260 to 1360 nm
(InGaAsP lasers, SMF)
THIRD WINDOW (low Loss):
Conventional, or C-band:
~1525 nm - 1565 nm,
Long, or L-band:
1570 nm to 1610 nm..
Charles K. Kao working with fiber optics
at the Standard Telecommunication Laboratories in England in the 1960s.
16
self-study
Substantially reducing fiber loss is what
made fiber-optical communication possible
•
•
•
•
The masters of light: This year's (2009) Nobel Prize in Physics is awarded for two
scientific achievements that have helped to shape the foundations of today’s
networked societies. They have created many practical innovations for everyday life
and provided new tools for scientific exploration.
In 1966, Charles K. Kao made a discovery that led to a breakthrough in fiber optics.
He carefully calculated how to transmit light over long distances via optical glass
fibers. With a fiber of purest glass it would be possible to transmit light signals over
100 kilometers, compared to only 20 meters for the fibers available in the 1960s.
Kao's enthusiasm inspired other researchers to share his vision of the future potential
of fiber optics. The first ultrapure fiber was successfully fabricated just four years
later, in 1970.
Today optical fibers make up the circulatory system that nourishes our
communication society. These low-loss glass fibers facilitate global broadband
communication such as the Internet. Light flows in thin threads of glass, and it carries
almost all of the telephony and data traffic in each and every direction. Text, music,
images and video can be transferred around the globe in a split second.
If we were to unravel all of the glass fibers that wind around the globe, we would get
a single thread over one billion kilometers long – which is enough to encircle the
globe more than 25 000 times – and is increasing by thousands of kilometers every
hour.
Moshe Nazarathy Copyright
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Loss mechanisms
•
Intrinsic losses (fundamental to the glass material – wavelength and material
composition dependent):
– Si02 absorption resonances with tails extending in the communication wavelength range
with peaks located at UV (0.1um electronic origin) and mid IR (lattice vibrational
modes, at 7 and 11um)
– Rayleigh scattering (excitation and re-radiation of light by atomic dipoles of dimensions
much smaller than a wavelength) – only excited due to irregularities of the atomic structure
– rapid random variations in the density, hence in the refractive index of the glass and of
dopant materials introduced into the Si02 lattice structure
•
Extrinsic losses (impurities, structural imperfections):
– Metallic and Rare Earth Impurities (V, Cr, Ni, Mn, Cu, Fe – Er, Pr, Nd, Sm, Eu, Tb, Dy) –
must be kept to concentration levels to few parts per billion by the vapor-phase processing
technologies.
– OH (hydroxil group) losses – OH may enter in the fabrication process – OH resonates
between 2.7 and 3.0 um – the OH vibrational modes are slightly anharmonic, generating
intermodulation tones at 1.38 um (2nd harmonic) and 0.95 um (3rd harmonic). There is also
a sideband at 1.25 um from the coupling of the 2nd harmonic with the fundamental Si-O
vibrational resonance.
•
Bending losses:
– Macrobending - curved guiding fiber bent into a loop
– Microbending - when the fiber comes in contact with rough surface, or small random axial
deformations, fluctuations in the core radius) – all random along z.
Moshe Nazarathy Copyright
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Fiber Loss spectrum – with
individual loss mechanisms
[J. Buck, Fundamentals of Optical Fibers]
Moshe Nazarathy Copyright
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Fiber dispersion Pulse Broadening
InterSymbol Interference (ISI)degraded Bit Error Rate (BER)
CDPulse Broadening (Animation)
1, 1, 0
FIBER WITH CHROMATIC DISPERSION (CD)
1, 1, 1
“Different colors (wavelengths) travel at different velocities”
Pulse BroadeningISIBER
THRESHOLD
1, 0, 0
1, 1, 0
Note: The ISI effect is complex: Sometimes the tail (RX response to a single pulse),
may last even longer than a single pulse slot (bit duration), and several such
tails from a few past symbols may superpose in any given bit interval
(the details depend on the transmitted bit sequence, the basic transmitter pulse,
the fiber channel CD, and the RX electrical response)
20
Fiber dispersion spectrum
Pulse spread [ps] is proportional to distance [Km] and to light bandwidth [nm]:
  DL
The proportionality constant called Dispersion Parameter, D, is plotted below:
d ( g / L)  psec/km 
D
d   nm 
@1.3um, standard fiber
is ideally CD-free
(but is lossier)
@1.5um, standard fiber
has large CD
(but has lowest loss)
21
Higher power
>
The CD induced broadening
is a larger percentage
of the shorter pulses
at higher bitrate
22
Overview of Optical
Amplification and
Dense Wavelength Division
Multiplexing (DWDM)
23
ERBIUM DOPED FIBER AMPLIFIER (EDFA)
An optical amplifier is a device that amplifies an optical signal directly, without the
need to first convert it to an electrical signal. An optical amplifier may be thought of as
a laser without an optical cavity, or one in which feedback from the cavity is
suppressed. Stimulated emission in the amplifier's gain medium causes amplification
of incoming light. [WIKI]
Optical gain in the 1,550 nm region.
980 or 1,480 nm
24
Self-study
Doped fibre amplifiers (DFAs) are optical amplifiers that use a doped optical fibre as a gain medium to amplify an optical signal.
The signal to be amplified and a pump laser are multiplexed into the doped fibre, and the signal is amplified through interaction
with the doping ions. The most common example is the Erbium Doped Fiber Amplifier (EDFA), where the core of a silica fiber is
doped with trivalent Erbium ions and can be efficiently pumped with a laser at a wavelength of 980 nm or 1,480 nm, and exhibits
gain in the 1,550 nm region.
Amplification is achieved by stimulated emission of photons from dopant ions in the doped fibre. The pump laser excites ions into
a higher energy from where they can decay via stimulated emission of a photon at the signal wavelength back to a lower energy
level. The excited ions can also decay spontaneously (spontaneous emission) or even through nonradiative processes involving
interactions with phonons of the glass matrix.
The amplification window of an optical amplifier is the range of optical wavelengths for which the amplifier yields a usable gain.
The amplification window is determined by the spectroscopic properties of the dopant ions, the glass structure of the optical fibre,
and the wavelength and power of the pump laser. Although the electronic transitions of an isolated ion are very well defined,
broadening of the energy levels occurs when the ions are incorporated into the glass of the optical fibre and thus the amplification
window is also broadened. The broad gain-bandwidth of fibre amplifiers make them particularly useful in wavelength-division
multiplexed communications systems as a single amplifier can be utilized to amplify all signals being carried on a fiber and whose
wavelengths fall within the gain window.
Noise: The principal source of noise in DFAs is Amplified Spontaneous Emission (ASE), which has a spectrum approximately the
same as the gain spectrum of the amplifier. As well as decaying via stimulated emission, electrons in the upper energy level can
also decay by spontaneous emission, which occurs at random, depending upon the glass structure and inversion level. Photons
are emitted spontaneously in all directions, but a proportion of those will be emitted in a direction that falls within the numerical
aperture of the fibre and are thus captured and guided by the fibre. Those photons captured may then interact with other dopant
ions, and are thus amplified by stimulated emission.
Gain saturation: Gain is achieved in a DFA due to population inversion of the dopant ions. The inversion level of a DFA is set,
primarily, by the power of the pump wavelength and the power at the amplified wavelengths. As the signal power increases, or the
pump power decreases, the inversion level will reduce and thereby the gain of the amplifier will be reduced. This effect is known
as gain saturation - as the signal level increases, the amplifier saturates and cannot produce any more output power, and
therefore the gain reduces. Saturation is also commonly known as gain compression.
EDFAs have two commonly-used pumping bands - 980 nm and 1480 nm. The 980 nm band has a higher absorption cross-section
and is generally used where low-noise performance is required. The absorption band is relatively narrow and so wavelength
stabilised laser sources are typically needed. The 1480 nm band has a lower, but broader, absorption cross-section and is
generally used for higher power amplifiers. A combination of 980 nm and 1480 nm pumping is generally utilised in amplifiers.
[WIKI]
Moshe Nazarathy Copyright
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Fiber links evolution
26
Overview of photonic
transmission Impairments - II
27
Linear
Impairments
28
29
A
30
MULTIPLEXING
‫ריבוב‬
In order to increase the capacity beyond the single
channel limit use Multiplexing in either time domain
(OTDM) or wavelength domain (WDM)
Wavelength Division Multiplexing (WDM)
WDM is the optical equivalent of FDM: Frequency Division Multiplexing
(e.g. as used in Terrestrial Broadcast and Cable Television)
WDM became the most important development of the ‘90ies
which enabled the Internet – in turn DWDM was enabled by
the emergence of broadband optical amplifiers in the late ’80ies.
31
32
33
DWDM Systems
Current…
…Soon
34
24 very wide (~350 GHz) WDM channels
launched over the C- and L-bands
Moshe Nazarathy Copyright
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f c/
36
Modern DWDM Core Network
Tx=Transmitter
Rx=Receiver
pre= opt. preamp
NF= Noise Figure
ROADM= Reconfig.
Opt. Add Drop Mux
post = opt. post-amp
37
An elementary introduction to
AM radio
From Lecture I gave to 2nd semester students in the 1 point course
“Directions in Electrical Engineering”
Analog Communication background
The concept of modulation and demodulation
ESSENTIAL BACKGROUND FOR COHERENT DETECTION
“The
communication technology
of a sufficiently advanced
civilization is indistinguishable
from magic”
Modulation-Demodulation
• Modulation: (at the TX)
Alteration of one waveform (the “carrier”) according to the
characteristics of another waveform (the “message”)
The “modulated carrier” is an information-bearing waveform
best suited for transmission over the channel.
• Demodulation: (at the RX)
Extraction of the “message” out of the modulated carrier.
• Continuous-wave (CW) modulation systems:
– LINEAR MODULATION (direct freq. translation of message spectrum)
• Amplitude Modulation (AM) -alternatively called DSB-LC (Large Carrier)
• Double-Sideband (DSB) -alternatively called DSB-SC (Suppressed
Carrier)
• Single Sideband Modulation (SSB)
• Vestigial Sideband (VSB)
– NON-LINEAR MODULATION (Exponential)
recovered
“modulated carrier”
• FM
“message”
“message”
• PM
TX
RX
(audio)
Channel
(mod)
(demod)
(RF)
“carrier”
Local oscillator
“carrier”
40
Moshe Nazarathy Copyright
AM Modulation (‫) אפנון‬

cos(2 f 0t )
X (t )
X (t )
Y (t )
3
2.5
2
1.5
1
3
0.5
-1
cos(2 f0t )
2.5
-0.5
2
-0.5
1.5
-1
1
0.5
1
0.5
1
cos(t   )
0.5
-1
-0.5
-0.5
-1
Y (t )
3
2.5
2
1.5
1
0.5
-1
-0.5
0.5
-0.5
-1
1
Q: Why bother with trigo in highschool?
A: To understand AM radio
@Technion, dummy!
cos   cos     ....
... 
1
1
cos      cos    
2
2
cos ( )  ...
2
1 1
  cos(2 )
2 2
AM modulation
f v  400 Hz
R (t )

V (t )
A cos(2 f c t )
Transmitter
V (t )  1  m cos(2 f vt )
R(t )  1  m cos(2 fvt ) A cos(2 f ct )
 A cos(2 f ct )  mA cos(2 f vt ) cos(2 f ct ) 


 
 A cos(2 f c t ) 

mA
cos  2
2
mA
cos  2
2
 fc 
f v  t 
 
 fc 
f v  t  
f c  100 KHz
AM modulation R (t )
V (t )

A cos(2 f ct )
f v  400 Hz
f0  100 KHz
Transmitter
V (t )  1  m cos(2 f vt )
R (t )  A cos(2 f c t ) 
mA

cos  2  f c  f v  t  
2
mA

cos  2  f c  f v  t 
2
Spectrum
of V ( t )
A
mA
2
Spectrum
R (t )
of
mA
2
Freq.
0.3
0.4
0.5
100.4
99.6
100
[ KHz ]
Single Tone amplitude modulated (AM) signal
Spectrum of AM signals
fc  W
fc
fc  W
Complete AM radio link
R (t )

V (t )
3
2.5
2
cos(2 f ct )
3
2.5
2
1.5
1.5
1
0.5
-1
-0.5
0.5
1
-0.5
-1
1
0.5
-1
-0.5
0.5
1
-0.5
Transmitter
-1
 aR (t )

W (t )
cos(2 f ct )
Receiver
Low-Pass
FILTER
3
2.5
2
1.5
1
0.5
-1
-0.5
0.5
-0.5
-1
1
Complete AM DSB radio link
R (t )

V (t )
3
2.5
cos(2 f ct )
3
2.5
2
1.5
1
2
1.5
1
0.5
-1
-0.5
0.5
-0.5
-1
0.5
-1
-0.5
0.5
1
-0.5
Transmitter
-1
V (t ) cos(2 f ct )  R(t )
aR ( t )
Low-Pass
FILTER
W (t )

cos(2 f ct )
Receiver
W (t )  aR(t ) cos(2 f ct )
 aV (t ) cos(2 f ct ) cos(2 f ct )

 aV (t ) cos (2 f c t ) 
2
1
1

 aV (t )   cos(2  2 f c  t )  
2
2

2
a
a
 V (t )  V (t ) cos(2  2 f c  t )
2
2
a
V (t )
2
1
Frequency Division Multiple
Access (FDMA)
CH. 1
TX
CH. 2
TX
Upconverter
Downconverter
x
LPF
x
cos 2f1t
cos 2f1t
Upconverter
Downconverter
x
LPF
x
cos 2f2t
analog
medium
...

Upconverter
Downconverter
x
LPF
x
cos 2fNt
cos 2f1t
FDM DEMUX
CH. N
CH. 2
CH. 1
FDM MUX
FDMA
CH. 2
RX
cos 2f1t
...
CH. N
TX
CH. 1
RX
...
CH. N
RX
Brief preview of the next
generation: Coherent
Detection with Digital Signal
Processing (DSP)
50
The photo-diode
acts as a mixer:
i (t )  Etot
2
 Er  E LO
2
 Er  E LO
2 Er  E LO
2
2
Mixing term
COHERENT
OPTICAL
TRANSMISSION
Er
LO laser
ELO
A revolution akin to the transition from spark radio
to (super)heterodyne radio [Armstrong, 1918]
mixer
r
Tunable
LO
Differential Phase Shift Keying
(DPSK)
IF
IF
cos r t  cos LO t   12 cos r  LO  t   12 cos r  LO  t 
LO
51
52
Signal analysis background:
Representations of narrowband
signals and systems:
-Complex envelopes, analytic signals
Quadrature (I&Q) Components
53
Analytic signals, complex envelopes, in the time & freq. domains
Real sig.
Analytic
sig.
Analytic
sig.
y(t )  2 Re ya (t )  2 Re y (t )e j t
c
y ( t )  ya (t )e
ya ( t )  y ( t )e
Analytic
sig.
jc t
Cmplx carrier
Env.
harmonic
tone
Y ( )  Ya (  vc )  Shift  vc Ya ( )
Ya ( )  Y (  vc )  Shift vc Y ( )
Y ( )  Shift  vc  2u( )Y ( )
y (t )
(analytic) downconverter
Moshe Nazarathy Copyright
c
0
Ya ( )  2u( )Y ( )
Cmplx Env.
 jct
 c
2u( )
PHASE
SPLITTER
ya ( t )
e j0t
y (t )
54
Passband analog transmission and quadrature representations
xI  t 
2cos  2π


t
0 
2cos  2πν t 
0 


LPF
xI  t 
x(t )
xQ
t 
- 2sin  2π 0t 




Narrowband
real signal
QUADRATURE
SYNTHESIZER
(MODULATOR) Quadrature
x(t )=
- 2sin  2πν0t 

LPF
xQ
t 
QUADRATURE ANALYZER
(DEMODULATOR)
representation of narrowband signal:
2 xI (t )cos
 2 0t  -
2 xQ (t )
sin  2 0t 

= 2Re  xI (t )  jxQ (t )  cos  2 0t  +jsin  2 0t 
 2 Re x(t )e j 2 0t
LP sig.
(analytic) upconverter
(CE)
xa (t )
x(t )
e
j0t
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2 Re{}

LP sig.
BP real sig. (analytic) downconverter (CE)
x(t )
Reversible
Transformations
2u( )
a
x (t )
PHASE
SPLITTER
e j0t
x(t )
55
Quadrature and env/phase representations of narrowband signals
x (t )  xI (t ) 2 cos 0t  xQ (t ) 2 sin 0t  xenv (t ) cos ct   (t ) 
QUADRATURE
MOD
2cos  2π

xCX(tt )
t
0 

C


 2sin  2π
t
0 

 Re x(t )e j0t
Q-comp.
 sin 0t Im
x(t )
XS  t 
x S (t )
xQ
x(t )
xenv / 2
 (t )
xI
xa (t )  x(t )e j0t
x(t )
2 Re
e j0t
x(t )
xI (t )  jxQ (t )
analytic sig.
cmplx env. x ( t )
xenv (t )  2 x(t )
Re I-comp.
cos 0t
 (t )  x(t )
xa (t )
 x(t ) e j ( t )
x (t )  2 Re  xI (t )  jxQ (t )  e
j0t
 2 Re  xI (t )  jxQ (t )  (cos 0t  j sin 0t )
 xI (t ) 2 cos 0t  xQ (t ) 2 sin 0t  2 x(t ) cos 0t   (t ) 
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56
Analog quadrature BP Link – complex representation
s (t )
a
s (t )
s (t )
S ( )
e
2 Re{}

s (t )
BPF
j0t
N (t )
S ( )
analytic U/C
0
0
0
2u ( )
PHASE
SPLITTER
0
PHASE
SPLITTER
0
0
DEMOD
e
 j0t
Self-study
LPF
sˆ(t )
optional
analytic D/C
S ( ) 
Assume an ideal BPF channel
and no noise
output:
sa (t )
s (t )
0
 s(t )e j0t  s(t )e  j0t 
F 2

2


1

S (  0 )  S (  0 )
2
Sa ()  S (  0 )
S ( )
output:
0
0
EXERCISE: Show that the energy of pulses and the power of random waveforms
is preserved between the real and complex domains.Therefore, the energies and powers
are unafffected by up/down conversion
57
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Elements of
electro-magnetic wave propagation
(the least background for this course
-mainly brought here to establish notation)
WHOLE SECTION IS SELF-STUDY
(REVIEW OF KNOWN MATERIAL FROM PRIOR COURSES)
Maxwell’s equations
In the absence of sources (
J0
  E   t (  H )
  H   t ( E)
  ( E)  0
  ( H)  0
):
f
t f  ,
t
Divergence:
  v   x vx   y v y   z vz
Rotor (curl):
  v  xˆ   y vz   z v y 
E( x, y, z , t )  Electric Field
H( x, y, z, t )  Magnetic Field

 permittivity
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  [ x ,  y ,  z ]
  permeability
 yˆ   z vx   x vz 
 zˆ   x v y   y vx 
59
Simple media
(linear, homogeneous, isotropic)
Linear:  ,  independent of E, H
Homogeneous:  ( x, y, z, t )   (t )
 ( x, y, z, t )  (t )
Isotropic:  ,  are scalars, not tensors (matrices).
  E   t (  H )
Maxwell’s equations
  H   t ( E)
  ( E)  0
  ( H)  0
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Note:
D  E
B  H
60
self-study
Wave equations in simple media
 E  0
0    ( E)  E
 E  t (  H)
Maxwell
  const
Maxwell

  const; , t commute
   E    { t (  H)}  t ( H)
   t (   t ( E))
Vector Identity
2
2
(  E)   E     t E
 E
 t ( E)
Maxwell
0
  [ x ,  y ,  z ]
Del
s  [ x s ,  y s ,  z s ]
Gradient
 E  ()E  (     )E
2
1
  2
v
2
x
2
y
 E    E  0
2
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2
t
2
z
Laplacian
Wave equation
61
Wave equations in simple media
1 2
 E  2 t E  0
v
1 2
2
 H  2 t H  0
v
Wave equation
2
1
  2
v
(apply

v is the speed of light
in the medium.
The speed of light
in vacuum is denoted
v
c
1

on 2nd Maxwell’s eq.)
Particular plane wave solutions (verify by substitution):
E( x, y, z, t )  E0 g (t  z / c) g () arbitrary
H( x, y, z, t )  H0 g (t  z / c)
self-study
 E  (     )E   E
2
2
x
2
2
y
2
z
 Ev  E  0
2
x
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2
t
2
x
1-D wave equation
62
Wave equations in simple media
2
 Ev  E  0
2
2
t
Wave equation in 1-D:
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z / v)
 E  (     )E   E
2
Laplacian:
Verify:
Wave equation
E( x, y, z, t )  E0 g (t
Verify solution:
self-study
2
x
2
y
2
z
2
2
z
 Ec  E  0
2
z
2
2
t
 g (t  z / c)  c  g (t  z / c)
2
z
2
t
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Wave equations in simple media
2
 Ev  E  0
2
2
t
self-study
Wave equation
View time evolution at location z (connect scope to antenna at z):
t-waveform at z=0 is delayed by 
 z / c when received at z
Interpet solution: E( z, t )  E0 g (t
z / c)  E(0, t
z / c)
Travelling wave
z 0
z
View all space at t (spatial snapshot at an instance t):
z-profile at t=0 is delayed by zt
E( z, t )  E0 g s ( z vt )
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 vt
z
gs ( z)  g ( )
v
when observed at t
64
self-study
Wave equations - time-harmonic solutions
E( z, t )  E0 g (t z / v)  E0 g s ( z vt )
g (t )  cos(t   ) sinusoidal profiles

z

 c
E( z, t )  E0 cos  (t
)   )
v

 wavenumber
=phaseshift
E( z, t )  E0 cos[t  z   ]
Delaying a sinusoid by

generates phaseshift
cos (t  )   )  cos t     )
Time evolution at location z:
sinusoidal with frequency
(time period


   z / v
 ( / v ) z   z
T  2 /  )
Spatial snapshot at an instance t:
sinusoidal with spatial frequency  (spatial period
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per unit length
A distance z
generates delay
z/v, and phaseshift
  2 / 
)
65
self-study
Wave equations - time-harmonic solutions
E( z, t )  E0 cos[t  z   ]
Time evolution at location z:
sinusoidal with frequency
(time period

T  2 /  )
Spatial snapshot at an instance t:
sinusoidal with spatial frequency  (spatial period

2  rad 
 
c
  m 
Spatial (angular) frequency
Wavenumber
k-vector
Propagation constant
Phase constant (phase/length)
 
v 
 T
2
  2 f 
T
)
 rad 
 sec 
Temporal (angular) frequency
c
  vT 
f
v
Phase velocity in material
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  2 / 
1

Wavelength (spatial period)
 [m]  2 [rad ]
c
1
 z    2
0 0
Phase velocity in vacuum
66
self-study
Phase velocity, refractive index
E( z, t )  E0 cos[t  z   ]
 
v 
 T
1
Phase velocity
in material
v

 , 0 : v
Phase velocity in vacuum (with  ,  ): c
0
0
Phase velocity in material with
Refractive index:
(velocity slowdown factor)
c
n
v
v0 
1
0 0
Phase velocity
in vacuum
1/ 0  0



0
1/ 0 
 r
  n 0
2
Relative permittivity:
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 r  1  susceptibilityP   0  E
67
Material dispersion relation
self-study
E( z, t )  E0 cos[t  z   ]


c
0
Refractive index:


n
(velocity slowdown factor)
v 0 



2


 
n

  n 0
v c/n
c

v
Wavenumber in the material

 ( )  nk0    n
c
2 
0 
  k0  
Material dispersion relation
0 c
Vacuum wavenumber
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68
Harmonic plane wave propagation
Note: linearly polarized, else phases of the
components would be different
Real optical (electric) field:
E( z, t )  E0 cos[t  z   ]
Representation in terms of analytic signal and complex envelope:
E( z, t )  2 Re E0e
j (t  z  )
jt
 2 Re e Ee
analytic signal
Ea ( z , t )
j z
E( z, t ) temporal
complex-envelope
real-valued
E  E0e
j
Spatio-temporal
complex-envelope
Similarly, for the magnetic field:
H( z, t )  Re H0e
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j ( t  z  )
jt
 Re e He
j z
69
Maxwell’s equations time-harmonic (monochromatic) formulation
jt
E( z, t )  Re e E( x, y, z )
jt
H( z, t )  Re e H( x, y, z )
  E   t (  H )
  H   t ( E)
  ( E)  0
  ( H)  0
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  E   j H
  H  j E
 E  0
  H  0
70
Wave equation - time-harmonic formulation
E( z, t )  2 Re e jt E( x, y , z )
H( z, t )  2 Re e jt H( x, y , z )
1 2
 E  2 t E  0
v
1 2
2
 H  2 t H  0
v
2
2E  k2 E  0
 H  k H  0
2
2
Time-harmonic (Helmholtz)
wave equations
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  E   j H
  H  j E
E  0
Simple media
H  0
1 2
1
 
2
 2  t   2 ( j )     k 2  
v
v
v
2
wavenumber (in a medium):


k  
  
v 1 / 
Alternative proof:
    E   j  H
  j ( j E)   2  E
(  E)  2E  k2 E
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Wave equation - time-harmonic solutions (I)
2 E  k 2 E  0
Time-harmonic wave equations
2 H  k 2 H  0
Example 1: Harmonic plane wave in a simple medium (along the z-axis):
Assume uniform solution in the x-y plane:
E( x, y, z, t )  Re e jt E( z )
H( x, y, z, t )  Re e jt H( z )
 Ek E  0
2E  ( 2x   2y   2z )E   2z E
 Hk H 0
2H  ( 2x   2y   2z )H   2z H
2
z
2
z
2
2
E  E0e
jkz
E( z, t )  Re Ee j (t kz )
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H  H 0e
jkz
H( z, t )  Re He j (t kz )
72
Modeling loss (I)
Harmonic plane wave in a simple lossy medium
 Ek E  0
2
z
2
E  E0e
 Hk H 0
2
z
2
H  He
jkz
E( z, t )  Re E0e j (t kz )
Loss is modeled by complexifying
jkz
H( z, t )  Re He j (t kz )
 , n, k
- the imaginary part is associated with the loss
      j 
re
k  n k0  n
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
n 
 nre  jnim
0
im

c
 n
re

c
 jn
im

c
   j
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Modeling loss (II)
Note: measure E in units
2
such that
P  E( z )
Harmonic plane wave in a simple lossy medium
E  Ee
 Ek E  0
2
z
2
 Hk H 0
2
z
2
jkz
  j
Right-propagating wave:
E  Ee
 jkz
 Ee
Phase-shift
 j (   j ) z
 Ee
P( z )  E( z )  E( z )  P(0)e
2
2
 j  z  z
2 z
e
Attenuation
Attenuation is exponential in z here
When power is measured in dB, the power is linear decreasing in z
10log P( z )  P( z )[dB]  P0[dB]  2[ dB / km ] z[ km ]
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Exercise: Relate
[ dB / km ] , 
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Guided wave optics
Wave equation - time-harmonic solutions (II)
 Ek E  0
2
2
2 H  k 2 H  0
Time-harmonic (Helmholtz) wave equations
x
z
y
Example 2: Lossless waveguide / fiber (along the z-axis):
Assume separable solution: general x-y pattern times z-harmonic variation
t- and z-harmonic solutions
E( x, y, z, t )  Re E( x, y )e  j z e jt
H( x, y, z, t )  Re H( x, y )e
 j z
e
jt
 2x   2y
2E  2E( x, y)e j z  (2x   2y   2z )E( x, y)e j z  (T2  ( j )2 )E( x, y)e j z
T2 E   2E  k 2E  0
 H Hk H0
2
T
2
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2
76
Guided wave solutions of the wave equation (I)
Note: As assumed in our wave eq. derivation, we consider piecewise homogeneous media,
and further assume cylindrical geometry along z:
n ( x, y, z )  n ( x, y )
x
k  k ( x, y)
piecewise constant !
z
CORE
y
CLADDING
Lossless waveguide / fiber (along the z-axis):
“modes”
E( x, y, z, t )  Re E( x, y )e j (t  z )
H( x, y, z, t )  Re H( x, y )e j (t  z )
Boundary conditions
stitch together the solutions
in the various piecewise constant
regions (enforcing the same  )
k 

c
  0 
 E( x, y )  (k   )E( x, y )  0 E,H tangential & normal
are continuous
2
2
2
at the interfaces
T H( x, y )  (k   )H( x, y )  0
2
T
2
2
“Transverse” wave equations for the modes
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77
Guided wave solutions of the wave equation (II)

For consistent guidance
k  k ( x, y ) 
ncore  nclad
x
c0
n ( x, y )
n ( x, y) piecewise constant !
ncore
CLADDING nclad
z
CORE
y
Lossless waveguide / fiber (along the z-axis):
“modes”
E( x, y, z, t )  Re E( x, y )e j (t  z )
H( x, y, z, t )  Re H( x, y )e j (t  z )
Typically, the eigenspectrum of
allowed  -s is discrete
The m-th eigen-solution
(at a fixed ) is described by

 m ( ), Em ( x, y ), H m ( x, y )
T2  k2 ( x, y )  E( x, y )   2 E( x, y )
T2  k2 ( x, y )  H( x, y )   2 H( x, y )
In the core the
transverse fields
typically oscillate.
In the cladding the
fields decay away from
the axis (evanscent)
“Transverse” wave equation expressed as an eigenvalue problem
(analogous
to Shroedinger’s eq.)
Moshe
Nazarathy Copyright
78
Guided wave solutions of the wave equation (III)
x
z
y
Example 2: Lossless waveguide / fiber (along the z-axis):
Ex
“modes”
E( x, y, z, t )  Re E( x, y )e j (t  z )
H( x, y, z, t )  Re H( x, y )e
j ( t   z )
MODES OF A MULTI-MODE FIBER
(courtesy: Maxim Greenberg)
The eigenspectrum of  -s means
the various modes propagate
with different speeds – modal dispersion:
pulses get smeared, causing ISI degradation
SINGLE-MODE fiber is used for
long-haul optical communication
(sufficiently reducing the core diameter,
just a single-mode is supported)
Moshe Nazarathy Copyright
ncore
CLADDING nclad
CORE
Ey
Ez
I
m=2
p=2
m=2
p=1
m=0
p=2
m=0
p=1
m=2
p=2
m=2
p=1
m=4
p=2
m=4
p=1
79
Single-mode fiber
Mecozzi
ECOC’05
Moshe
Nazarathy
Copyright
80
Single-mode fiber (II)
Moshe Nazarathy Copyright
81
Solving the wave equation for
rotationally symmetric step-index
fibers
WHOLE SECTION IS SELF-STUDY
(elements of it covered in TA class)
Moshe Nazarathy Copyright
82
Wave propagation in optical fibers
Maxwell equations in differential form
B
t
D
XH 
t
 D0
D  0 E  P
 B 0
B  0 H
XE  
The polarization and electric field are linearly dependent

P ( r, t )   0   ( r, t  t ') E ( r, t ')dt '

B
  D 
X XE  X
  

t
t  t 
2
1 2 E
2 P
 2   0 E  P    2 2   2
t
c t
t
Fourier transformation of
E and P




E ( r, )   0  E ( r, t ) exp(it )dt, P( r, )   0  P( r, t ) exp(it )dt
and substituting leads to
X XE 
2
c
2
2
c2
E ( r,  )  0 2 0  ( r,  ) E ( r,  ) 
E ( r,  ) 1   ( r,  )    ( r,  )
2
c
2
E ( r,  )
 (r, )  1   (r, )
Is defined as the complex frequency dependent
Dielectric constant
 ( r,  )  ( n  i c / 2 ) 2
n  1  Re 
Refractive index


nc
Im 
Absorption (loss)
n and  frequency dependent.
The solution to the propagation problem is vastly simplified by introducing
the so called Gloge Approximation which assumes that
I
  0 or  = n2
II) n is independent of r or n = 0
Using the identity
X XE  ( E )  2 E  2 E
 E 
D

0
Leads to the wave equation
 E n
2
2
2
()k0 E
0
k0   / c  2 / 0
In cylindrical coordinates, the equation for Ez (for example) is
 2 Ez 1 Ez 1  2 Ez  2 Ez
2 2




n
k0 Ez  0
2
2
2
2
r r r 
r
z
n1 r  a
n
n2 r  a
There are similar equations for E Er Hz H Hr. Only two need to be solved
The wave equation is solved by separation of variables
Ez (r,  , z )  F (r )( ) Z ( z)
This leads to three regular differential equations
2Z
 Z  0
2
z
 2
 m  0
2

 Z  exp(i  z )
   exp(im ) (m integer)
 2 F 1 F  2 2
m2
2

  n k0    2
2
r r 
r
r

 F  0

Solution to the equation for F(r)
 AJ m ( r )  AYm ( r ) r  a
F (r )  
CK m ( r )  C I m ( r ) r  a
with
 2  n12 k02   2
 2   2  n22 k02
k0 
2
0
Jm, Ym, Km, Im are Bessel functions and A,A’,C,C’ are constants
The field has to be finite at r  0 and zero for large
r

The field Ez becomes then
 AJ m ( r ) exp(im ) exp(i  z ) r  a
Ez ( r )  
CKm ( r ) exp(im ) exp(i  z ) r  a
Similarly for Hz
 BJ m ( r ) exp(im ) exp(i  z ) r  a
H z (r)  
 DKm ( r ) exp(im ) exp(i  z ) r  a
The form of the Bessel functions is
10
1
J0(x)
J1(x)
J2(x)
K0(x)
K1(x)
K2(x)
9
8
7
0.5
6
5
4
0
3
2
1
-0.5
0
1
2
3
4
5
x
6
7
8
9
10
0
0
0.5
1
1.5
x
2
2.5
3
The Maxwell equations are used to calculate the four other field components
i  E z
 H z 
Er  2  
 0

r  
  r
i   E z
H z 
E  2 
 0

r 
  r 
i  H z
2  E z 
Hr  2  
  0n

r  
  r
i   H z
2 E z 
H  2 
  0n 

r 
  r 
There are now six equations describing all the fields in the core and
in the cladding. There are four coefficients A B C D which need to be
computed. The coefficients are found using the boundary condition
E z E H z H need to be continuous at r  a
The boundary conditions yield four equations which have to be
satisfied simultaneously. The determinant of this set of equations
is set to zero and this leads to the important Eigenvalue Equation for
the propagation constant 
 J m ( a )
Km ( a )   J m ( a ) n22 K m ( a ) 

 2



  J m ( a )  K m ( a )    J m ( a ) n1  K m ( a ) 
2
 m   1
1 

  2 2
 
 n1ak0   
2
J m
J m ( a ) 
[ a ]
The Eigenvalue equation is cumbersome relative to the case of a dielectric
slab. Even for the dielectric slab, the solutions are not intuitive and have to
be found numerically and some times graphically.
Given a fiber and an operating wavelength, n1 n2 a, k0 the Eigenvalue equation
can be solved (at least numerically) to yield the propagation constant  for
the specific mode solved for.
The solutions are periodic in m and are counted successively
so a mode is labeled mn n = 1, 2, 1 . . .
Each mn represents a field distribution described by the six field equations.
In general Ez and Hz are non zero, except for the case of m = 0.
The modes are labeled HEmn or EHmn and for m=0, TE0n or TM0n.
Some times the modes are labeled LPmn
Define modal index
n   / k0
n1  n  n2
A given mode with a given  defines n and this is the index
that mode experiences. For example n established the phase velocity
of that mode.
When n changes, say because the wavelength (and therefore k0) changes
the mode may reach cut off
n  n2
Cut off The mode is no more guided.
For a propagating mode, the field changes in the cladding
According to
Km ( r ) 

exp(  r )
2 r
 r  1
At cut off
n  n2
2 0
and hence there is no exponential field reduction and no guiding.
At cut off
  k0 (n12  n22 )
Normalized Frequency V
 2
 
Define V  k0a (n12  n22  

 an1 2

V is proportional to
 or 1/ or k0
Normalized propagation constant b
Define
b
 / k0
n1  n2

n  n2
n1  n2
B versus V
Given a frequency or wavelength, V is completely defines for a given fiber
A large V number yield many modes
A very approximate and crude rule of thumb states that the number of
modes is V2/2
For small V numbers the number of modes is small, V = 5 yields 7 modes
The most important case is that for which there is only one mode
Single Mode Conditions
A single mode, HE11 is obtained when all other, higher, modes are cut off.
Inserting m = 0 in the Eigenvalue equation
Cut off in the TE0n modes
 J 0 ( a ) K0 ( a )   J 0 ( a ) K0 ( a )  0
Cut off in the TM0n modes
 n22 J 0 ( a) K0 ( a)   n22 J 0 ( a)K0 ( a)  0
For   0 J 0 ( a )  0  J 0 (V )  0
The properties of the Bessel function dictate that
the V value for which J 0 (V )  0 is V  2.405
The condition for single mode operation is therefore
V  2.405
For V  2.405, the only propagating mode is the TE11
A standard single mode fiber designed for 1.3 m-1.6 m has
a cut off at 1.2  m.
For n1  1.45,  =5 103 and V  2.405, a  3.2  m
Modal index n  n2  b( n1  n2 )  n2 (1  b )
Approximate (empirical) expression for b(V )
2
0.996 

b(V )   1.428
V


in the range V  1.5  2.5, b(V ) is accurate to within 0.2%
Field distribution
The fundamental mode of the fiber is such that in general it
is linearly polarized along Ex or Ey.
For Ex
[ J 0 (r ) / J 0 (a)[exp( iz ) r  a
E x  E0 
 K 0 (r ) / K 0 (a)[exp( iz ) r  a
1/ 2
 0 
H y  n2  
 0 
Ex
Actually, there always exists another mode Ey and in theory the
two modes have the same
Spot size
The basic field distribution is a Bessel function for which is it
hard to develop a simple intuitive picture. The field distribution
can be approximated by a Gaussian so that
  r2 
 exp( iz )
E x  A exp 
W2 


W is the spot size
W and a are related by a formula
w
 0.65  1.619V 3 / 2  2.879V  6
a
Also, the spot size determines the confinement factor
a
 drr
2
Ex
 2a 2 
Pcore

 0
 1  exp  
 w2 
Ptotal 
2


 drr E x
0