Optical Communications

136 Pages·2001·8.034 MB·English

by  Freude W.

Optical Communications
Wolfgang Freude*
High-Frequency and Quantum Electronics Laboratory
(Institut fur Hochfrequenztechnik und Quantenelektronik)
University of Karlsruhe
International Department
SS 2000
24th April 2001
*Not to be published. All rights reserved. No part of this compuscript may be reproduced or utilized
in any form or by any means without permission in writing from the author.
Contents
1 Introduction 1
2 Light Waveguides 3
2.1 Fundamentals of wave propagation 3
2.1.1 Medium properties 3
2.1.2 Wave equation 5
2.1.3 Homogeneous medium 6
2.1.4 Properties of silica glass 10
2.1.5 Plane boundary 14
2.2 Principles of waveguiding 16
2.3 Slab waveguide 17
2.3.1 Eigenvalues in pictures 18
2.3.2 Eigenvalue equation 19
2.3.3 Vector solution 21
2.3.4 Scalar solution 22
2.3.5 Group delay dispersion 24
2.4 Strip waveguide 25
2.5 Fibre waveguide 25
2.5.1 Modal fields in a fibre 26
2.5.2 Weakly guiding fibre in scalar approximation 28
2.5.3 Step-index fibre 30
2.5.4 Parabolic-index fibre 35
2.5.5 Orthogonality and coupling efficiency 37
2.6 Intensity modulation 37
2.6.1 Gaussian impulse 38
2.6.2 Light source 38
2.6.3 Impulse response and transfer function 39
2.6.4 Noiseless light source 41
2.6.5 Noisy light source 47
2.6.6 Sinusoidal modulation 49
2.6.7 Multimode waveguide 50
3 Ligtht sources 53
3.1 Number of modes 54
3.2 Luminescence and laser radiation 55
3.2.1 Lifetime and linewidth 57
3.2.2 Laser action 57
3.2.3 Modulation 58
3.2.4 Amplification 58
3.2.5 Coupling efficiency 58
3.2.6 Noise 58
3.2.7 Laser active materials 58
3.3 Semiconductor light source 60
3.3.1 Energy bands and density of states 60
3.3.2 Filling of electronic states 63
3.3.3 Impurities and doping 64
3.3.4 Compound semiconductors and heterojunctions 67
3.4 Emission and absorption of light 72
3.4.1 Induced and spontaneous transitions 72
3.4.2 Induced and spontaneous transitions in a semiconductor 74
3.4.3 Radiative and nonradiative transitions 79
3.5 Light-emitting diode 82
3.5.1 Output power and modulation properties 82
3.5.2 Device structures 83
3.5.3 LED spectrum 85
3.6 Laser diode 86
3.6.1 Basic equations 86
3.6.2 Amplitude-phase coupling 89
3.6.3 Rate equations 91
3.6.4 Characteristic curves and efficiencies 93
3.6.5 Threshold current 95
3.6.6 Device structures 96
3.6.7 Spectrum 97
3.6.8 Small-signal intensity modulation 97
3.6.9 Large-signal intensity modulation 100
4 Pin-photodiodes 103
4.1 Basic equations 103
4.1.1 Short-circuit photocurrent 104
4.1.2 Equivalent electrical circuit 107
4.2 Materials 107
4.3 Time and frequency response 108
4.4 Cutoff frequency, quantum efficiency and responsivity 112
4.5 Device structures 114
5 Amplifiers 116
6 Noise 118
6.1 Noise mechanisms 118
6.2 Photocurrent noise 118
6.3 Thermal noise 119
6.4 Noise figure 120
7 Receivers and systems 121
7.1 Pin-photodiode receiver limits 121
7.2 Detection errors 122
7.2.1 Optimum decision 124
7.2.2 Signal-to-noise ratio 125
7.2.3 Power penalty 125
7.2.4 Quantum limit 127
7.3 System design 128
7.3.1 Loss-limited systems 129
7.3.2 Dispersion-limited systems 129
7.4 System architectures 130
Preface
Lightwave technology developed over the last 25 years has greatly influenced our needs for com-
munication. Most spectacular is the explosion in internet traffic. Resources made accessible in
the World Wide Web (WWW) have changed our attitude towards information acquisition, which
is more and more being regarded as a an everyday's necessity, and even as a natural right for
everybody. The interpretation of WWW as „World Wide Wait" reflects the impatience of users,
demanding — ahead of the actual technical implementation — more and more bandwidth at less
and less cost. Today's undersea and underground optical cables provide the large-capacity links
nobody could have dreamt of five years back, carrying more than 90 % of the communication
traffic. So lightwave technology has indeed greatly changed our lives.
Because of its practical importance, and because of its extreme broad disciplines, optical com-
munication is a worthwhile study subject. The present course tries to give a timely overview,
exploiting results from electromagnetic field theory (for describing wave propagation in light
waveguides), solid-state physics (for understanding laser diodes, LED, and photodiodes), and
communication theory (receivers, limitations by noise, system and networks), to mention just a
few.
Obviously, this one-semester course has to concentrate on the basic concepts. Some minimal
background is required: Calculus, differential equations, time-invariant linear systems, Fourier
transforms, p-n junction physics, and traveling waves. For further reading, the following list
provides some material. For more specialized topics a few publications were cited in the text.
Texbooks: ADAMS, M. J.: An introduction to optical waveguides. Chichester: John Wiley &
Sons 1981 — ARNAUD, J. A.: Beam and fiber optics. New York: Academic Press 1976 — EBE-
LING, K. J.: Integrierte Optoelektronik, 2. Ed. Berlin: Springer-Verlag 1992. In German. English
translation also available. — GRAU, G.; FREUDE, W.: Optische Nachrichtentechnik, 3. Ed.
Berlin: Springer-Verlag 1991. In German. Since 1997 out of print. Corrected reprint from Uni-
versity Karlsruhe 1999, available via W. F. ([email protected]). The present
course is based mainly on this book and on material found in: AGRAWAL, G. P.: Fiber-
optic communication systems. Chichester: John Wiley & Sons 1997 — GHATAK, A.; THYA-
GARAJAN, K.: Introduction to fiber optics. Cambridge: University Press 1998 — Liu, M. M.-
K.: Principles and applications of optical communications. Chicago: McGraw-Hill 1996
— SINGH, J.: Physics of semiconductors and their heterostructures. New York: McGraw-
Hill 1993. — SNYDER, A. W.; LOVE, J. D.: Optical waveguide theory. London: Chapman and
Hall 1983 — SZE, S. M.: Physics of semiconductor devices. New York: John Wiley & Sons
1985 — UNGER, H.-G.: Planar optical waveguides and fibres. Oxford: Clarendon Press 1977 —
UNGER, H.-G.: Optische Nachrichtentechnik, Part I and II, 2. Ed. Heidelberg: Dr. Alfred Huthig
1990 and 1992. In German.
This course is still under development, so the compuscript provides only partially fully formu-
lated text. As for the rest, an outline of the main points is given. Many figures are taken from
„Optische Nachrichtentechnik" (see above) carrying German lettering and decimal commas. To
avoid a complete redesign, the appropriate translations are given in the figure captions.
In parallel, there are courses in Optical Communications (Optische Nachrichtentechnik, ONT)
held in German, which cover the material in more detail: Waveguides and Transmitters (ONT-1),
Receivers, Systems and Measurements (ONT-2), Selected Components and Technologies (ONT-
3), and Nonlinear Optical Pulse Generation and Transmission (ONT-4). Courses ONT-1 and
ONT-2 may be attended in arbitrary sequence. In each of these courses, an introductory chapter
summarizes the respective prerequisites.
Chapter 1
Introduction
An optical or lightwave communication system uses lightwaves in a vacuum wavelength range
0.6/um... 1.2/tin < A < 1.6/ffli corresponding to carrier frequencies / = c/A (vacuum speed of
light c) of 500 THz ... 250 THz > / > 190 THz. A communication system is referred to as a point-
to-point transmission link. When many transmission links are interconnected with multiplexing
or switching functions, they are called a communication network. The principle of an optical
transmission link is shown in Fig. 1.1. A light-emitting semiconductor device (/aser diode = LD
Laserdiode, Photodiode
Lumineszenzdiode
Fig. 1.1. Optical point-to-point transmission link with direct (incoherent) detection. Strom = current, Lumines-
zenzdiode = LED, Licht = light, Glasfaser = glass fibre
or Hght-emitting diode = LED) is excited by an electric current, thereby converting the electrical
signal (information) to light. The signal is transmitted as an analog or digital modulation of the
light power P(t), the classical power resulting from an average over a few optical cycles. The
light is transported through a dielectric light waveguide (LWG), consisting of a cladding and a
core, which confines the light. For long-distance communication, optical quartz glass fibres are
used. At the receiver end, a photodetector (PD) reconverts the light to an electric photocurrent
i(t) ~ P(t) in proportion to the light power. The very straightforward type of reception in Fig. 1.1
is called „direct" or incoherent, as opposed to coherent receivers using a heterodyne or homodyne
technique with a local laser oscillator (LO).
Obviously, optical communication systems can replace conventional electrical systems only, if
there is some advantage to be gained, which justifies the additional expenses of a twofold conversion
current-light and light-current. Some important advantages of optical signal transport are:
• Large transmission capacity because of the large fibre bandwidth in the order of (250 —
190) THz = 60 THz
• Low fibre loss, about 2.2,0.35, 0.15 dB / km at A = 0.85,1.3,1.55 /mi, i. e., down to 3 dB loss
for a fibre length of L = 20 km corresponding to a power attenuation by a factor of only 2
• Immunity to interference because of the high carrier frequency, and because of the strong
confinement of the light inside the fibre
In the following, the most basic point-to-point optical communication blocks are discussed: optical
transmitters, channels, amplifiers and receivers. Simple system properties and some more advanced
components are treated shortly in Chapter 6 and Chapter 7.
Optical waveguiding is important for transmitters, receivers, and, naturally, for the channel
itself. Because the mathematics of dielectric fibre waveguides is more involved, we start in Chap-
ter 2 with an important, yet simply to be described device, namely the slab waveguide, which also
represents a good model for the waveguiding portions of optical sources and detectors. Having
understood the concept of waveguiding, we discuss wave propagation in optical fibres.
The main component of an optical transmitter is its light source for electro-optic (EO) con-
version. In Chapter 3, the principles of laser diodes and LED are explained.
Optical amplifiers (OA) overcome the power loss in very long communication links. They
have bandwidths in the order of Af = 5...10THz centred at A = 1.3/im and A = 1.55/яп,
and remove the speed bottleneck from electronics by optics implementation. Chapter 4 explicates
the properties of pin-photodiodes. Two primary types of О A are discussed shortly in Chapter 5,
semiconductor ^aser amplifiers (SLA), and doped/ibre amplifiers (DFA). Among all DFA, Er3+-
doped /ibre amplifiers (EDFA) that amplify light around A = 1.55 /um are the most mature. In
fact, it was only after the invention of the EDFA in 1987 that optical communication became so
powerful as it is today.
Chapter 6 reviews some important noise mechanisms. Finally, Chapter 7 discusses system
aspects.
Chapter 2
Light Waveguides
2.1 Fundamentals of wave propagation
Electromagnetic waves (magnetic, electric field vector H, E, electric displacement D, polarization
P, magnetic induction B) as solutions of Maxwell's equations as functions of time in space. No
currents, no electric space charges. Medium at frequencies of interest is isotropic, linear and non-
magnetic (medium properties given by scalar, amplitude-independent quantities, relative magnetic
permeability \J, = 1). Dielectric constant and magnetic permeability as well as velocity of light,
T
wavelength for frequency /, angular frequency ш = 2тг/ in vacuum are CQ, Цо, с = l/^eo/uo,
A = c/f. Wave impedance of vacuum is ZQ = ^//.to/eo- With this notation, Maxwell's equations
are:
curl Я = —, curl£; = --г-,
dt dt
divD = 0, divB = О, (2Л)
D = eE + P, В = цоН.
0
All vector quantities are functions of time t and position vector r. Time-frequency Fourier trans-
form relation (FT = #(/) = F{V(t)}) and inverse FT (IFT = Ф(1) = ?~1 {£(/)}):
+ 00 f + 00
^(/)e+j277/td/, Ф(/)= ^(t)e-j27r/*dt. (2.2)
/ -oo J — oo
Functions often discriminated only by argument: <?(t) ф \F(f = t), \F(f) = *P(f)-
2.1.1 Medium properties
Medium made of positive and negative charges (e.g., protons and electrons). When an electric
field is present, it separates the charges of opposite polarity (periodically in the case of a time
periodic field). This charge separation results in an additional electric field, called (induced)
polarization. Polarization vector P in Eq. (2.1) follows E in real media with time-delay (causal
influence function u(t) with u(t < 0) = 0 describes relaxation processes of free and bound charges,
„memory" of real medium), therefore:
P(t,f) = eof™u(T,f)E(t-T,?)dT,
P(f) = *ox(f)E(f), x(f) = С «(*) e-J2^4 dt, (2.3)
X(f) =x(/)+JXi(/) = e(/)-l-Jeri(/), *(/) =**(-/)•
r
Proportionality constant between spectra E(f) and P(f) called electric susceptibility % (real
part x(/), imaginary part Xi(/)); defines complex relative dielectric constant c (real part e(/),
r r
imaginary part — e«(/))- A complex refractive index ra (real part ra(/), imaginary part —rij(/)) is
r
defined from the relation e = n2:
r
ra = ra-jraj, e = e -jc ,
r r ri
c = n2—n?, ci = 2nrii,
r r
n2 = |e (l + V1 + <&/e?) , "i = e /(2n), (2.4)
r ri
(for |e < e) raj и е /(2^/ё7),
r н
1^?2 (for |e » e) га» и sgn(e )v/eri|/2-
ri r ri
Definition of analytic (or regular or holomorphic) functions:
Cauchy's principle value integral (P means valor principalis, Latin for principle value):
+ 00 / X—£ + 00 \
0
P /^-d*=lim( / -^-d* / ^Lj (2.5)
J X - XQ e^O I j z - Жо + J X - X I
— oo \—oo xo+e 0 /
Given a causal time function «?(i < 0) = 0. Then, the Fourier spectrum of \P(t) is a
complex analytic function #(/) with real part !?(/) and imaginary part #»(/),
1 /-+00 if-ff) „ l /-+00 ip.(f)
*M = -P ^f^' *(/<>) = —P/ J^d/. (2.6)
7Г J.oo / - /О 7Г J_ } - /o
OQ
Real and imaginary parts are interconnected (because of causality) by the so-called
Hilbert transform and its inverse, Ф = Пр{Ф} and Ф = Пр1^} in Eq. (2.6).
Because of Eq. (2.6), %(/) is an analytic spectral function (FT of causal function u(t)), and the
so-called Kramers-Kronig relation follows:
1 Г+°° (f'} - 1
-c (f) = n {c(f)-l} = -P f _ d/' (2.7)
ri F r ^ J—CO fJ l f J
For x(f) = £r(/) - 1 = const, we have from Eq. (2.7) e (f) = 0 and therefore u(t) = (c - 1) S(t),
ri r
i.e., (5-memory = no memory ~~* no medium ~^ no polarization ~^ from Eq. (2.6), (2.7) e =
r
1, because ej = X« = 0. Real passive media always with memory, with Eq. (2.3) frequency
r
dependence of %, Xi follows. Possible: in certain frequency range is % constant (or approximately
constant), Xi = 0 (lossless). For this /-range the medium may be described by a real, constant (и
constant) refractive index. This is assumed in the following. Ansatz for D(t):
D = c eE, n = JT . (2.8)
0 r T
Model for real, lossy medium: E forces atoms of localized molecules as well as electrons bound
to nuclei to vibrate, also free electrons are forced to vibrate. Bound charges accumulate energy near
their resonance frequencies. Bound and free charges relax by interaction with the medium (heating,
absorption). Vibrating charge dipoles radiate like Hertz dipoles ~» phase-shifted, attenuated
secondary field described by polarization vector P. Number of free carriers f: refractive index ra \.
(n к д/ё^ for ej| 'C e). Number of bound charges t: refractive index n deviates stronger from
r r
ra = 1. Typical frequency dependence of n and raj depicted in Fig. 2.1.
Only free carriers, Fig. 2.la: n and raj have a pole at / = 0. When / ~\: minimum n (for
relaxation time TO —>• oo an refractive index n = 0 becomes possible), then for / —>• oo n —>• 1
(carrier mass because of inertia not moved by field, absorption disappears). п(ш) = О defines so-
р
called plasma frequency UJ ~^ n2 и 1 — (ш/ш)2. For ш < UJ large rij, large absorption; for ш > OJ
P р P P
small raj, small absorption (thin metal foils reflect infrared (IR) light, but absorb visible radiation
4
©
t1
0,2 <y, 0,3 «><
Fig. 2.1. Real part n und negative imaginary part (rtj) of complex refractive index n = n — jn;: Frequency
dependence (a) only free carriers (b) two collectives of bound charges with high mass (ions, low angular
resonance frequency wi) and low mass (electrons, high angular resonance frequency 0^2)
(VR) ~-* heat trap; metals transparent at X-rays; space communication through ionosphere with
high-frequency signals, along ionosphere with short waves; simple sun glasses with metal coating
may be transparent for ultraviolet light (UV), eye hazard!).
Only two resonant charge collectives regarded, Fig. 2.1b: Low-loss frequency ranges with nor-
mal dispersion An/ Аш > 0, dn/ dA < 0 (Latin norma, carpenter's square, i.e., an L-shaped or
T-shaped instrument used for obtaining or testing right angles, conforming to a standard). Absorp-
tion frequency ranges near the resonances uj\^ with anomalous dispersion An/ AUJ < 0, An/ A\ > 0
(Greek av-uj/ja\o<;, un-even, deviant, different),
dn
normal dispersion
?аи}>° dA <°
(2.9)
dn
anomalous dispersion
?аи}<° dA >0
In the region of normal dispersion, the deviation angle of a dispersing prism decreases continuously
with A, while near the infrared (wi) and ultraviolet (u>z) absorption lines in the region of anomalous
dispersion unusual dependencies of the deviation angle are to be observed. If / f> then n f
starting from n2(0) > 1 to maximum before resonance, after resonance minimum reached, moves
for another maximum. Tendency: / t ~^ n J,. After last minimum n —> 1 from below.
2.1.2 Wave equation
Reshaping Eqs. (2.1), (2.8) ~^ exact vector wave equations for real instantaneous quantities E, H,
2
i2d2E
V2E + gradf(gradlnn2) • E] = \
\ / cc2z dt2 '
(2.10)
82H
V2H + (gradlnn2) x curl Я = —
c2 dt2 '
Differential operator V2 operates on vector E, not identical with Laplace operator Д as applied
to scalar function f,
V2E = grad div Ё - curl curl Ё,
АФ = div grad Ф.
Vector components subscripted with corresponding coordinates. In Cartesian coordinates x,y,z:
unit vectors e ,e ,e. Only in Cartesian coordinates:
x y z
V2E = grad div Ё - curl curl Ё =
= e div grad E + e div grad E + e div grad E = (2.11)
x x y y z z
= eV2E + eyV2E + eV2E = e AE + eAE + eAE.
x x y z z x x y y z z
V2 in cylindrical and spherical coordinates given in tables.
2.1.3 Homogeneous medium
Differential equations (DE) for components of E and H are coupled for inhomogeneous media.
Lossless homogeneous medium assumed (n = const) ~^ simplification: grad Inn2 = 0 in Eq. (2.10).
Only in Cartesian coordinates: 2x3 decoupled wave equations for three Cartesian components
each for both field vectors E and H; in cylindrical coordinates r,ip,z: only the DE for the z-
components becomes decoupled from the other DE. Scalar wave equations with identical structure
for these decoupled components result,
&(t,x,y,z) = Eg(t,x,y,z),H(t,x,y,z), q = x,y,z,
q
&(t,r,<p,z) = E(t,r,<p,z),H(t,r,p,z), ^ ^
z z 2Л2
n2 <92
V2*^) = -^*(*,0-
Other field components via Maxwell's equations derived from ^-components. Further interdepen-
dencies of components by initial (1C) and boundary conditions (BC). For the following: Cartesian
coordinates x,y,z assumed.
Monochromatic waves
Monochromatic waves (A,tp,ipi real):
f(t,f) = A(f) e^-^'^ = ejaIte-j<?(F), (p(f) = (p(r )+j ^(f) (2.13)
A(r) and <p(r): Amplitude and phase of wave. Surfaces A(r) = const (or <pi(r) = const): am-
plitude surfaces. Surfaces f(f) = const: phase surfaces. Amplitude decreases fastest in direction
of amplitude vector a = — grad </?,, phase increases fastest in direction of phase vector b = grad if.
Propagation vector <p (real a, b):
grad</3 = gradc/?+j gradc/^ = 6 —j a (2-14)
Propagation direction of wave: not direction of propagation vector grad</3, but direction of phase
vector b normal to surfaces of constant phase (unit vector еъ = b/\b\ = grad</?/| grad</?|). Velocity
of phase surface (phase front) in direction of propagation defines phase velocity v of wave. From
ш dt — gradip • dr = Q, df = ej, ds follows
(grady). (grad ) | gradM' ^ ds и_ ^
ud t = y dg= ц = = ш ( }
jrad^l | gradip\ dt \&cadip\ \b\
Waves classified according to shape of phase surfaces. If phase surfaces are plane ~^ plane waves.
Homogeneous plane waves, if phase and amplitude surfaces coincide.
Plane waves Solution of wave equation (2.12) in homogeneous medium with separation ansatz:
<P(t,f) =exp(jwt)exp[-jfc-r] =exp(jwt)exp[-i(kx + ky + kz)]
x y z
Valid for:
(-%-kl-k*)V(t,f) = -n2(-)V(t,r),
z
4c/ (2.16)
fc2 2 2 =n2(^) =U2kl
+fc +fc
Separation condition Eq. (2.16) leads to separation constant ko, thus:
<p(r) = k-r, k2 = k-k = n'2kl (real!), k = -. (2.17)
0 с