Lecture I Introduction to Fiber Optic Communication …then proceed to review some elements of...

Lecture IIntroduction to Fiber Optic Communication

…then proceed to review some elements of signal analysis

…then review Electro-Magnetic Theory ,

… and guided wave optics

…in which we present a roadmap for the course …an overview of optical transmission,

ver. 3

046342

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


• 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 EXAMHomework 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: ”גלים ומערכות מפולגות“ 044148– Ideal (optional) prerequisites: Signals and Systems, Random

Signals, Analog and Digital Communication

מבוא לתקשורת בסיבים אופטיים grad/undergrad course #046342 - Spring 2010

mailto:[email protected]


• 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

Roadmap of 048951 (MAIN LECTURES) :

Overview of optical communication, review complex signals ,E-M, guided optics

Elements of Linear Propagation and Nonlinear OpticsNonlinear Schroedinger Equation Nonlinear impairmentsThe Photon Nature of Light + Direct Optical DetectionOptical Modulation Optical AmplificationPolarization Mode DispersionWavelength Division Multiplexed Direct Detection SystemsCoherent and Differentially Coherent Transmission

L1.

L2.

L3.

L4.

L5.

L6.

L7.

L8.

L9.

.L10.

.L11.

Moshe Nazarathy - All Rights Reserved

.L12.

Subject to Change

6

Acknowledgements:•P. Winzer, C. Chandrasekhar, ECOC’05,whose kind permission I was granted to incorporatetheir 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 communicationFrom meters to thousands of Kms, from Mb/s to multi Tb/s

9

History (+basic TIR principle)

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.

In the 19th century ….

10

Light is kept in the fiber core by TOTAL INTERNAL REFLECTION (TIR)

In the wave picture the fiber acts as a waveguide

0 2 2 0 1 1sin sink n k n Equal tangential k-components

1 2sin /TIR n n GeO 2 doped

SiO 2

…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 wavelength-division 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.


http://en.wikipedia.org/wiki/Charles_K._Kao

http://en.wikipedia.org/wiki/George_Hockham

http://en.wikipedia.org/wiki/Harlow

http://en.wikipedia.org/wiki/Corning_Glass_Works

http://en.wikipedia.org/wiki/Decibel

http://en.wikipedia.org/wiki/Kilometre

http://en.wikipedia.org/wiki/Laser_diode

http://en.wikipedia.org/wiki/Mbps

http://en.wikipedia.org/wiki/Km

http://en.wikipedia.org/wiki/Single-mode_optical_fiber

http://en.wikipedia.org/wiki/Gigabit

http://en.wikipedia.org/wiki/Transatlantic_telephone_cable

12

2006 2009 2012

100

10K

20K

תעבורהPB/Month

Source: CISCO Networking Report Q1/09

גידול בשנה60%

Why Optical Communication?

Current Economic Motivation

= 1.65 10.49

Initial motivation was telephony

Overview of photonic transmission Impairments - I

13

גם היא מעוותת את האותאי-ליניאריות –

ובמבנה מוליך הגל

נפיצה)) CHROMATIC DISPERSION (CD)

LOSS

NON-LINEARITYGets worse with TX power

POLARIZATION MODE DISPERSIONנפיצת אופני קיטוב

IMPAIRMENTS קילקולים NOISE SOURCESRX+OA dependentTX PowerOptical Signal/Noise

Pulse spreadingISIBER

14

Fiber loss spectrum

THIRD WINDOW (low Loss):Conventional, or C-band:~1525 nm - 1565 nm, Long, or L-band:1570 nm to 1610 nm..

SECOND WINDOW (low CD):1260 to 1360 nm(InGaAsP lasers, SMF)

FIRST WINDOW:800 to 900 nm (GaAs lasers, MMF)

16

Charles K. Kao working with fiber optics at the Standard Telecommunication Laboratories in England in the 1960s.

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.


self-study

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.


Fiber Loss spectrum – with individual loss mechanisms

Moshe Nazarathy Copyright 19[J. Buck, Fundamentals of Optical Fibers]

Fiber dispersion Pulse BroadeningInterSymbol Interference (ISI)degraded Bit Error Rate (BER)

FIBER WITH CHROMATIC DISPERSION (CD)

CDPulse Broadening (Animation)

“Different colors (wavelengths) travel at different velocities”1, 1, 0 1, 1, 1

THRESHOLD

1, 0, 01, 1, 0

Pulse BroadeningISIBER

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 suchtails 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

The proportionality constant called Dispersion Parameter, D, is plotted below:

DL

psec/( / k) m

nmgd L

Dd

Pulse spread [ps] is proportional to distance [Km] and to light bandwidth [nm]:

@1.3um, standard fiberis ideally CD-free(but is lossier)

@1.5um, standard fiberhas large CD(but has lowest loss)

21

>Lower disp

ersion

Hig

her

pow

er

The CD induced broadening is a larger percentageof 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]

980 or 1,480 nm

Optical gain in the 1,550 nm region.

24


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]

Self-study

Fiber links evolution

26

Overview of photonic transmission Impairments - II

27

28

Linear Impairments

MULTIPLEXING

In order to increase the capacity beyond the single

channel limit use Multiplexing in either time domain

(OTDM) or wavelength domain (WDM)

WDM became the most important development of the ‘90 ies

which enabled the Internet – in turn DWDM was enabled by

the emergence of broadband optical amplifiers in the late ’80 ies.

ריבוב

Wavelength Division Multiplexing (WDM)

WDM is the optical equivalent of FDM: Frequency Division Multiplexing (e.g. as used in Terrestrial Broadcast and Cable Television)

31

DWDM SystemsCurrent… …Soon

34

24 very wide (~350 GHz) WDM channels launched over the C- and L-bands


36/f c

Modern DWDM Core Network

37

Tx=TransmitterRx=Receiverpre= opt. preampNF= Noise FigureROADM= Reconfig.Opt. Add Drop Muxpost = opt. post-amp

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 backgroundThe 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) • FM• PM

RX(demod)

recovered“message”

Local oscillator “carrier”

TX(mod)

Channel“message”

“modulated carrier”

“carrier”

(audio)

(RF)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

cos( )t

AM Modulation (אפנון )

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

( )X t ( )Y t

0cos(2 )f t( )X t

0cos(2 )f t

( )Y t

Q: Why bother with trigo in highschool?

2cos ( ) ... 1 1cos(2 )

2 2

A: To understand AM radio@Technion, dummy!

cos cos ....

1 1... cos cos

2 2

AM modulation( )V t ( )R t

cos(2 )cA f tTransmitter

( ) 1 cos(2 )vV t m f t

( ) 1 cos(2 ) cos(2 )v cR t m f t A f t

cos(2 ) cos(2 )cos(2 )c v cA f t mA f t f t

cos(2 ) cos 22

cos 22

c c v

c v

mAA f t f f t

mAf f t

100cf KHz400vf Hz

AM modulation

( )V t ( )R t

cos(2 )cA f tTransmitter

( ) 1 cos(2 )vV t m f t

( ) cos(2 )

cos 22

cos 22

c

c v

c v

R t A f t

mAf f t

mAf f t

Freq.

Spectrumof ( )R t

2

mA

2

mAA

Spectrumof ( )V t

0 100f KHz400vf Hz

100.499.6100

[ ]KHz0.40.3

0.5

Single Tone amplitude modulated (AM) signal

Spectrum of AM signals

cf cf Wcf W

Complete AM radio link

( )V t

( )R t

cos(2 )cf tTransmitter

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

( )aR t ( )W t

cos(2 )cf t

Receiver

Low-PassFILTER

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

Complete AM DSB radio link( )V t

( )R t

cos(2 )cf t

Transmitter

( )aR t ( )W t

cos(2 )cf t

Receiver

( ) cos(2 ) ( )cV t f t R t

( ) ( ) cos(2 )cW t aR t f t

( ) cos(2 )cos(2 )c caV t f t f t

2( ) cos (2 )caV t f t 1 1

( ) cos(2 2 )2 2 caV t f t

( ) ( ) cos(2 2 )2 2 c

a aV t V t f t

Low-PassFILTER

( )2

aV t

-1 -0.5 0.5 1

-1

-0.5

0.5

1

1.5

2

2.5

3

2

Frequency Division Multiple Access (FDMA)

CH. 1TX

CH. 2TX

CH. NTX

...CH. 1

RX

CH. 2RX

CH. NRX

...

Upconverterx

cos 2f1t

Upconverterx

cos 2f2t

Upconverterx

cos 2fNt

Downconverterx

cos 2f1t

LPF

Downconverterx

cos 2f1t

LPF

Downconverterx

cos 2f1t

LPF

FDM MUX FDM DEMUX

...FDMA

CH

. 1

CH

. 2

CH

. N

analogmedium

Brief preview of the next generation: Coherent

Detection with Digital Signal Processing (DSP)

50

51

Differential Phase Shift Keying (DPSK)

A revolution akin to the transition from spark radioto (super)heterodyne radio [Armstrong, 1918]

COHERENTOPTICALTRANSMISSION

1 12 2cos cos cos cosr LO r LO r LOt t t t

LO laser

LO

mixer

r

LOIF IF

Tunable

rE

LOE

The photo-diodeacts as a mixer:

2

2

22

( )

2

tot

LO

LO

LO

r

r

r

i t E

E E

E E

E E

Mixing term

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

CmplxEnv.

Analyticsig.

Real sig.

Analyticsig.

Analyticsig.

Cmplx Env.

cc 0

carrierharmonic tone

( ) 2 ( ) ( )aY u Y

( ) 2 Re ( )ay t y t 2 Re ( ) cj ty t e

( ) ( ) cj tay t y t e

( ) ( ) cj tay t y t e

( ) ( ) ( )a c c aY Y v Shift v Y

( ) ( ) ( )a c cY Y v Shift v Y

( ) 2 ( ) ( )cY Shift v u Y

0j te

( )2u PHASESPLITTER

( )y t ( )y t(analytic) downconverter

( )ay t


Passband analog transmission and quadrature representations

0j te PHASESPLITTER

( )x t( )x t

(analytic) downconverter

( )ax t0j te

2 Re{}( )x t

(analytic) upconverter BP real sig.LP sig.(CE)

LP sig.(CE)

( )ax t

ReversibleTransformations

LPF

2 0cos 2π t

2 0- sin 2πν t

LPF

2 0cos 2πν t

2 0- sin 2π t

Ix t

Qx t

Ix t

Qx t

QUADRATURE SYNTHESIZER(MODULATOR)

QUADRATURE ANALYZER(DEMODULATOR)

Narrowbandreal signal

( )x t

002 2co sin 2( )= ( 2) )s - (I Qtx t x t x t t Quadrature representation of narrowband signal:

0 0cos 2 sin 2= Re ( ) ( ) +2 I Qj tx t x t t j 02Re ( )2 j tx t e

( )2u


2 0cos 2π t

2 0sin 2π t

CX t

SX t

QUADRATURE MOD

( )Cx t

( )Sx t

Quadrature and env/phase representations of narrowband signals

0

0 0

Re( )

Re

( )

( )

( )

( ) sicos( )n

2

2

QI

QI

j tx t j

j

x t

x t t j

x t

x t t

e

cmplx env.

analytic sig.

0 0( ) s( ) cos ( )( ) cos ( )in22 e cI Q nvxx t t x t tt ttx t

( )x t

( )x t

0j te

( )x t( )x t ( )ax t2 Re

0(( ))aj tx tx t e

Ix

Qx / 2envx

( )t

( )x t

Re

Im

( ) 2 ( )envx t x t

( ) ( )t x t

( )( ) j tex t

00( ) cos ( ) sin2 2I Qx t t x t t

0(Re ) j tx t e

) (( )QI jxx t t

0cos( ) ( )2 x t tt

0cos t

0sin tQ-comp.

I-comp.


Analog quadrature BP Link – complex representation

0j te

2 ( )u PHASESPLITTER

ˆ( )s t

0j te

2 Re{}( )s t

( )as tBPF

( )N t

LPF

optional

Assume an ideal BPF channeland no noise

00 0

( )s t

( )S

( )s t

00 0

PHASESPLITTER

output:

00

DEMOD

output:

( )s t

0

( )S analytic U/C analytic D/C

0 0

0 0

22

1

2

( )

( ) ( )

( ) ( )

j t j t

S

s t s tF

S S

e e

( )as t

0( ) ( )aS S

( )S

Self-study

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

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

( )

( )

( ) 0

( ) 0

t

t

E

E

E

H

H

H

In the absence of sources ( ):0 J

, [ , , ]t x y z

ff

t

( , , , )x y z t E Electric Field

( , , , )x y z t H Magnetic Field

permittivity permeability

ˆ

ˆ

ˆ

y z z y

z x x z

x y y x

v v

v v

v v

v x

y

z

x x y y z zv v v vDivergence:

Rotor (curl):


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).

( )

( )

( ) 0

( ) 0

t

t

E

E

E

H

H

H

Maxwell’s equations

B HD E

Note:


Wave equations in simple media( )t HE

{ ( )}t HE

Maxwell

2( ) E E

0 ( ) EMaxwell

0 E

2 2 2 2( ) ( )x y z E E E

[ , , ]x y zs s s s Gradient

[ , , ]x y z

Laplacian

Del

2 2 0t E E Wave equation

const

Maxwell

( )t E( ( ))t t E E

0

( )t Hconst; , commutet

Vector Identity2t E

E

2

1

v

self-study


Wave equations in simple media

2 22

10tv

E E Wave equation

2 22

10tv

H H

0( , , , ) ( / )x y z t g t z c E EParticular plane wave solutions (verify by substitution):

2

1

v

1v

0( , , , ) ( / )x y z t g t z c H H

(apply on 2nd Maxwell’s eq.)

2 2 2 2 2( )x y z x E E E2 2 2 0x tv E E 1-D wave equation

( )g arbitrary

v is the speed of lightin the medium.The speed of lightin vacuum is denoted c

self-study


Wave equations in simple media2 2 2 0tv E E Wave equation

0( , , , ) ( / )x y z t g t z vE E 2 2 2 2 2( )x y z z E E E

2 2 2 0z tc E EWave equation in 1-D:

Verify solution:

Laplacian:

2 2 2( / ) ( / )z tg t z c c g t z c Verify:

self-study


Wave equations in simple media

0( , ) ( / ) (0, / )z t g t z c t z c E E EInterpet solution:

2 2 2 0tv E E Wave equation

0z z

Travelling wave

0( , ) ( )sz t g z vtE E ( ) ( )s

zg z g

v

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

View all space at t (spatial snapshot at an instance t):

z-profile at t=0 is delayed by tz vt when observed at t

self-study


Wave equations - time-harmonic solutions

0 0( , ) ( / ) ( )sz t g t z v g z vt E E E ( ) cos( )g t t sinusoidal profiles

c

0( , ) cos ( ) )

zz t t

v

E E

0( , ) cos[ ]z t t z E E

wavenumber=phaseshiftper unit length

Spatial snapshot at an instance t:

Time evolution at location z: sinusoidal with frequency (time period ) 2 /T

sinusoidal with spatial frequency (spatial period ) 2 /

A distance zgenerates delayz/v, and phaseshift

/z v ( / )v z z

cos ( ) ) cos )t t Delaying a sinusoid by generates phaseshift

self-study


Wave equations - time-harmonic solutions

rad2

c m

0( , ) cos[ ]z t t z E E

Spatial (angular) frequencyWavenumberk-vectorPropagation constantPhase constant (phase/length)

Spatial snapshot at an instance t:

Time evolution at location z: sinusoidal with frequency (time period ) 2 /T

sinusoidal with spatial frequency (spatial period ) 2 /

cvT

f

rad

sec

22 f

T

vT

Phase velocity in material

Wavelength (spatial period)

Temporal (angular) frequency

1v

0 0

1c

Phase velocity in vacuum

2[ ] [ ]m rad 2z

self-study


Phase velocity, refractive index

0( , ) cos[ ]z t t z E E

vT

Phase velocity in vacuum

1v

0

0 0

1v

Phase velocity in material with : 0, v

Phase velocity in vacuum (with ): c0 0,

cn

vRefractive index:

(velocity slowdown factor)

0 0

00

1/

1/

20n

rRelative permittivity:

Phase velocity in material

1

r

susceptibility0 P E

self-study


Material dispersion relation

0( , ) cos[ ]z t t z E E c

nv

Refractive index:(velocity slowdown factor)

/n

v c n c

0 00

2k

c

Vacuum wavenumber

Wavenumber in the material

0

2n

v

0( ) nk nc

Material dispersion relation

0

0

self-study


Harmonic plane wave propagation

0( , ) cos[ ]z t t z E E Real optical (electric) field:

0( )( , ) 2 Re 2 Rej t z j t j zz t e e e E E E

Representation in terms of analytic signal and complex envelope:

analytic signal temporalcomplex-envelope

Spatio-temporalcomplex-envelope

( , )a z tE( , )z tE

0( )( , ) Re Rej t z j t j zz t e e e H H H

Similarly, for the magnetic field:

0je E E

real-valued

Note: linearly polarized, else phases of thecomponents would be different


Maxwell’s equations - time-harmonic (monochromatic) formulation

( )

( )

( ) 0

( ) 0

t

t

E

E

E

H

H

H

( , ) Re ( , , )

( , ) Re ( , , )

j t

j t

z t e x y z

z t e x y z

H H

E E

0

0

j

j

H

H

E

E

E

H


Wave equation - time-harmonic formulation

( , ) 2 Re ( , , )

( , ) 2 Re ( , , )

j t

j t

z t e x y z

z t e x y z

H H

E E

0

0

j

j

E

E

H

E

H

H

Simple media

2 22

10tv

E E

2 22

10tv

H H

22 2 2

2 2

1 1( )t j k

v v v

1 /k

v

2 2

2 2

0

0

k

k

E E

H H

Time-harmonic (Helmholtz)wave equations

2 2( ) k E E E

Alternative proof:

2( )

j

j j

E

E

H

E

wavenumber (in a medium):


Wave equation - time-harmonic solutions (I)2 2

2 2

0

0

k

k

E

H H

E

Time-harmonic wave equations

Example 1: Harmonic plane wave in a simple medium (along the z-axis):Assume uniform solution in the x-y plane:

( , , , ) Re ( )

( , , , ) Re ( )

j t

j t

x y z t e z

x y z t e z

H

E

H

E

2 2 2 2 2( )x y z z E E E

2 2

2 2

0

0

z

z

k

k

E

H H

E

0jkzeE E

2 2 2 2 2( )x y z z H H H

0jkzeH H

( ) ( )( , ) Re ( , ) Rej t kz j t kzz t e z t e H HE E


Modeling loss (I)Harmonic plane wave in a simple lossy medium

2 2

2 2

0

0

z

z

k

k

E

H H

E

0

jkzeE E

jkzeH H

( ) ( )0( , ) Re ( , ) Rej t kz j t kzz t e z t e H HE E

Loss is modeled by complexifying , ,n k

0re imk n k n n jn j

c c c

- the imaginary part is associated with the loss

0

re imn n jn

re imj


Modeling loss (II)Harmonic plane wave in a simple lossy medium

2 2

2 2

0

0

z

z

k

k

E

H H

E

jkzeE E

j

( )jkz j j z j z ze e e e E E E E

Right-propagating wave:

Attenuation

Attenuation is exponential in z hereWhen power is measured in dB, the power is linear decreasing in z

Phase-shift

2 2 2( ) ( )( ) (0) zP z z Pz e E E

[dB] 0[dB] [ / ] [ ]10log ( ) ( ) 2 dB km kmP z P z P z

Note: measure E in unitssuch that 2

( )P zE

Exercise: Relate [ / ],dB km

Guided wave optics


Wave equation - time-harmonic solutions (II)

2 2

2 2

0

0

k

k

E

H H

E

Time-harmonic (Helmholtz) wave equations

Example 2: Lossless waveguide / fiber (along the z-axis):Assume separable solution: general x-y pattern times z-harmonic variation

( , , , ) Re ( , )

( , , , ) Re ( , )

j z j t

j z j t

x y z t x y e e

x y z t x y e e

H H

E E

2 2 2 2 2 2 2( , ) ( ) ( , ) ( ( ) ) ( , )j z j z j zx y z Tx y e x y e j x y e E E E E

2 2 2

2 2 2

0

0

T

T

k

k

H H

E E E

H

2 2x y

t- and z-harmonic solutions

x

yz


tangential & normal are continuous at the interfaces

Guided wave solutions of the wave equation (I)

Lossless waveguide / fiber (along the z-axis):

( )

( )

( , , , ) Re ( , )

( , , , ) Re ( , )

j t z

j t z

x y z t x y e

x y z t x y e

H H

E E

2 2 2

2 2 2

( , ) ( ) ( , ) 0

( , ) ( ) ( , ) 0

T

T

x y k x y

x y k x y

E

H H

E

“Transverse” wave equations for the modes

“modes”

0kc

x

yz

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 ( , )k k x ypiecewise constant !

CORE

CLADDING

Boundary conditions stitch together the solutionsin the various piecewise constantregions (enforcing the same )

,HE


Guided wave solutions of the wave equation (II)

Lossless waveguide / fiber (along the z-axis):

( )

( )

( , , , ) Re ( , )

( , , , ) Re ( , )

j t z

j t z

x y z t x y e

x y z t x y e

H H

E E

2 2 2

2 2 2

( , ) ( , ) ( , )

( , ) ( , ) ( , )

T

T

k x y x y x y

k x y x y x y

H H

E E

“Transverse” wave equation expressed as an eigenvalue problem (analogous to Shroedinger’s eq.)

“modes”

x

yz

0

( , ) ( , )k k x y n x yc

piecewise constant !

CORE

CLADDING

( , )n x y

Typically, the eigenspectrum of allowed -s is discreteThe m-th eigen-solution (at a fixed ) is described by

( ,( )), ( ,,)m mm x y x y HE

In the core the transverse fields typically oscillate.In the cladding the fields decay away from the axis (evanscent)

coren

cladn

For consistent guidance

core cladn n


Guided wave solutions of the wave equation (III)

Example 2: Lossless waveguide / fiber (along the z-axis):

x

yz CORE

CLADDING

coren

cladn

( )

( )

( , , , ) Re ( , )

( , , , ) Re ( , )

j t z

j t z

x y z t x y e

x y z t x y e

H H

E E

“modes”

Ex Ey 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

Ez

MODES OF A MULTI-MODE FIBER(courtesy: Maxim Greenberg)

SINGLE-MODE fiber is used forlong-haul optical communication(sufficiently reducing the core diameter,just a single-mode is supported)

The eigenspectrum of -s meansthe various modes propagate with different speeds – modal dispersion:pulses get smeared, causing ISI degradation


Single-mode fiber

Mecozzi ECOC’05


Single-mode fiber (II)


Solving the wave equation forrotationally symmetric step-index

fibers

WHOLE SECTION IS SELF-STUDY(elements of it covered in TA class)

Wave propagation in optical fibers

0

0

0

0

BXE

t

DXH

t

D D E P

B B H

Maxwell equations in differential form

The polarization and electric field are linearly dependent

0( , ) ( , ') ( , ') 'P r t r t t E r t dt

2 2 2

02 2 2 2

1

B DX XE X

t t t

E PE P

t c t t

Fourier transformation of and E P

0 0( , ) ( , ) exp( ) , ( , ) ( , ) exp( )E r E r t i t dt P r P r t i t dt

22

0 02

2 2

2 2

( , ) ( , ) ( , )

( , ) 1 ( , ) ( , ) ( , )

X XE E r r E rc

E r r r E rc c

and substituting leads to

( , ) 1 ( , )r r Is defined as the complex frequency dependentDielectric constant

2( , ) ( / 2 )

1 Re Im

r n i c

nnc

Refractive index Absorption (loss)

n and frequency dependent.

The solution to the propagation problem is vastly simplified by introducingthe so called Gloge Approximation which assumes that

I or = n2

II) n is independent of r or n = 0

Using the identity

2 2( )

0

X XE E E E

DE

Leads to the wave equation

2 2 20 0 0( ) 0 / 2 /E n k E k c

In cylindrical coordinates, the equation for Ez (for example) is

2 2 22 2

02 2 2 2

1 10z z z z

zE E E E

n k Er rr r z

1

2

n r an

n 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

( , , ) ( ) ( ) ( )zE r z F r Z z

This leads to three regular differential equations

2

2

2

2

2 22 2 2

02 2

0 exp( )

0 exp( ) ( integer)

10

ZZ Z i z

z

m im m

F F mn k F

r rr r

Solution to the equation for F(r)

( ) ( ) ( )

( ) ( ) m m

m m

AJ r A Y r r aF r

CK r C I r r a

The field has to be finite at and zero for large 0r r The field Ez becomes then

( ) exp( )exp( ) ( )

( ) exp( )exp( ) m

zm

AJ r im i z r aE r

CK r im i z r a

2 2 2 2 2 2 2 21 0 2 0 0

0

2 n k n k k

Jm, Ym, Km, Im are Bessel functions and A,A’,C,C’ are constants

with

Similarly for Hz

( ) exp( )exp( ) ( )

( ) exp( )exp( ) m

zm

BJ r im i z r aH r

DK r im i z r a

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1J0(x)J1(x)J2(x)

x0 0.5 1 1.5 2 2.5 3

0

1

2

3

4

5

6

7

8

9

10K0(x)K1(x)K2(x)

x

The form of the Bessel functions is

The Maxwell equations are used to calculate the four other field components

02

02

202

202

z zr

z z

z zr

z z

i E HE

r r

i E HE

r r

i H EH n

r r

i H EH n

r r

There are now six equations describing all the fields in the core andin the cladding. There are four coefficients A B C D which need to becomputed. The coefficients are found using the boundary condition

need to be continuous at z zE E H H r a

The boundary conditions yield four equations which have to besatisfied simultaneously. The determinant of this set of equationsis set to zero and this leads to the important Eigenvalue Equation forthe propagation constant

2221

22

2 21 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1( )

[ ]

m m m m

m m m m

mm

J a K a J a n K a

J a K a J a K an

m JJ a

n ak a

The Eigenvalue equation is cumbersome relative to the case of a dielectricslab. Even for the dielectric slab, the solutions are not intuitive and have tobe 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 forthe 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

0 1 2 / n k n n n

Define modal index

A given mode with a given defines and this is the indexthat mode experiences. For example established the phase velocityof that mode.

When changes, say because the wavelength (and therefore k0) changesthe mode may reach cut off

n n

n

2 n n Cut off The mode is no more guided.

For a propagating mode, the field changes in the claddingAccording to

( ) exp( ) 12mK r r r

r

At cut off

22 0n n

and hence there is no exponential field reduction and no guiding.

At cut off 2 2

0 1 2( )k n n

Normalized Frequency V

Define 2 20 1 2 1

2( 2V k a n n an

V is proportional to or 1/or k0

Normalized propagation constant b

Define 0 2

1 2 1 2

/ k n nb

n n n n

B versus V

Given a frequency or wavelength, V is completely defines for a given fiberA large V number yield many modesA very approximate and crude rule of thumb states that the number ofmodes 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

0 0 0 0( ) ( ) ( ) ( ) 0J a K a J a K a

Cut off in the TM0n modes

2 22 0 0 2 0 0( ) ( ) ( ) ( ) 0n J a K a n J a K a

0 0

0

For 0 ( ) 0 ( ) 0

The properties of the Bessel function dictate that

the value for which ( ) 0 is 2.405

The condition for single mode operation is therefore

J a J V

V J V V

11

2.405

For 2.405, the only propagating mode is the TE

A standard single mode fiber designed for 1.3 m-1.6 m has

a cut off at 1.2 m.

V

V

31

2 1 2 2

2

For 1.45, =5 10 and 2.405, 3.2 m

Modal index ( ) (1 )

Approximate (empirical) expression for ( )

0.996( ) 1.428-

V

in the range 1.5 2.5, ( ) is accurate to within

n V a

n n b n n n b

b V

b V

V b V

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

xy

x

EnH

ziaKrK

ziaJrJEE

2/1

0

02

00

000 ))[exp((/)(

))[exp((/)([

ar

ar

Actually, there always exists another mode Ey and in theory the two modes have the same

Spot sizeThe 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

)exp(exp2

2zi

W

rAEx

W is the spot size

W and a are related by a formula

62/3 879.2619.165.0 VVa

w

Also, the spot size determines the confinement factor

2

2

2

0

2

0 2exp1

w

a

Edrr

Edrr

P

P

x

ax

total

core

Lecture I Introduction to Fiber Optic Communication …then proceed to review some elements of...

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Transcript of Lecture I Introduction to Fiber Optic Communication …then proceed to review some elements of...