Dispersion Compensation and Soliton Transmission in Optical Fibers André Toscano Estriga Chibeles Dissertation submitted for obtaining the degree of Master in Electrical and Computer Engineering Jury President: Prof. Doutor José Bioucas Dias Supervisor: Prof. Doutor António Luís Campos da Silva Topa Co-Supervisor: Prof. Doutor Carlos Manuel dos Reis Paiva Members: Profª Doutora Maria Hermínia da Costa Marçal Abril 2011
89
Embed
Dispersion Compensation and Soliton Transmission in Optical ......Dispersion Compensation and Soliton Transmission in Optical Fibers André Toscano Estriga Chibeles Dissertation submitted
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Dispersion Compensation and Soliton Transmission in Optical
Fibers
André Toscano Estriga Chibeles
Dissertation submitted for obtaining the degree of Master in Electrical and Computer Engineering
Jury
President: Prof. Doutor José Bioucas Dias
Supervisor: Prof. Doutor António Luís Campos da Silva Topa
Co-Supervisor: Prof. Doutor Carlos Manuel dos Reis Paiva
Members: Profª Doutora Maria Hermínia da Costa Marçal
Abril 2011
ii
iii
Acknowledgements First I would like to thank to my supervisors Prof. Carlos Paiva and Prof. António Topa, to help
me in the process of completing my master, by giving me knowledge and guidance while I was making
this dissertation.
I would like to thank to all my friends, but specially
To my IST friends:
Nuno Couto, Rui Trindade, Filipa Henriques, André Neves, Gonçalo Carmo, João Cabrita, André
Esteves, Rafael Ferreira, Luís Pragosa, Francisco Pinto and Vera Silva.
To my friends from Évora:
Luís Almeida, Duarte Abêbora, Carlos Freixa, Tiago Toscano, Filipa Ribeiro, Daniel Engeitado,
João Rosa, Joaquim Faneca, Carlos Rosa, Tânia Pegacho, Luís Pegacho, João Roque, Luís Roque and
Filipe Louro.
To my 4th floor friends:
José Alves, Bruno Baleizão, João Vicente and António Eira.
To my friends of PIO XII university college:
André Patrão, Pedro Barata, João Caldinhas, João Fialho, Luis Rodrigues and Miguel Duarte.
And last but not least, to my family:
To my mum and dad, to my aunt Crisália, to my aunt Maria Antónia, to my uncle Zé, to my
cousin Manel and to my both grandmas, and my grandpa.
Wherever you are you will always be in my heart.
iv
Abstract In this dissertation, the propagation of pulses in a linear and non-linear regime is studied. The
topics of the temporal dispersion and laser chirp are taken into account for both the linear and non-linear
regimes. The pulse dispersion causes that the communication systems are not perfect.
To study the pulse evolution, the pulse propagation equations have to be determined, one equation
for the linear regime and another equation for the non-linear regime. The propagation equation for the
non-linear regime is the non linear Schrodinger equation (NLS). Only with this equation, the non-linear
effects can be taken into account.
In the linear regime, the techniques for dispersion compensation are addressed, the use of
dispersion compensating fibers (DCF). In this dissertation is mentioned other ways to compensate the
dispersion, they are not tested though.
Keywords:
Propagation, linear regime, non-linear regime, temporal dispersion, chirp, non-linear Schrodinger
Resumo Nesta dissertação, foi estudada a propagação de impulsos em regime linear e não linear. Quer a
dispersão temporal quer o “chirp”, são tidos em conta no regime linear e no regime não linear. A
dispersão dos impulsos faz com que os sistemas de comunicação não sejam perfeitos.
Para se estudar a propagação de impulsos, que a equação de propagação. Uma equação tem de ser
determinada para o meio linear, outra equação tem de ser determinada para o meio não linear.
Em regime linear analisa-se as técnicas de compensação da dispersão, nomeadamente, o uso de
fibras compensadoras de dispersão.
A equação de propagação que é usada para o meio não linear é a equação não linear de
Schrodinger. Com esta equação, os efeitos não lineares podem ser considerados.
Palavras chave:
Propagação de impulsos, meio linear, meio não linear, dispersão temporal, chirp, equação não
linear de Schrodinger, fibras compensadoras de dispersão, solitões.
vi
Table of contents Acknowledgements ......................................................................................................................... iii
Abstract ............................................................................................................................................ iv
Resumo ............................................................................................................................................. v
List of figures ................................................................................................................................. viii
List of tables ..................................................................................................................................... x
List ................................................................................................................................................... xi
List of acronyms ............................................................................................................................ xiii
Annex A .......................................................................................................................................... 63
A1. Modal equation of the hybrid modes ................................................................................... 63
Annex B .......................................................................................................................................... 71
B1. Numerical simulation of linear pulse propagation ............................................................... 71
Annex C .......................................................................................................................................... 74
C1. Numerical simulation of the NLS function: Split step Fourier method ............................... 74
viii
List of figures Figure 1.1: Increase in the bit-rate distance product during the period 1850 and 2000 [2]. ............. 2
Figure 1.2: Increase of the BL product over the period of 1975 and 2000. ..................................... 3
Figure 1.3: International undersea network of fiber-optic communication system [2]. ................... 4
Figure 1.4: Temporal dispersion that a pulse suffers after is propagated through a fiber. ............... 5
Figure 1.5: Train of pulses at the fiber input. ................................................................................... 5
Figure 1.6: Train of pulses at the receiver. ....................................................................................... 6
Figure 1.7: ISI of a train Train of pulses at the receiver input. ......................................................... 6
Figure 2.1: Waveguide structure of optical fiber. ............................................................................. 9
Figure 2.2: Normalized propagation constant as a function of the normalized frequency for the
fundamental mode LP01 for several values of ∆ . ........................................................................................ 15
Figure 2.3: Normalized propagation constant as a function of the normalized frequency, for the
first six LP modes. ....................................................................................................................................... 16
Figure 3.1: Absolute value of the pulse. The pulse at the input of the fiber is represented by a
dashed line. The impulse at the output of the fiber is represented by a full line. ........................................ 33
Figure 3.2: Absolute value of the pulse along the fiber, observed by two different angles. ........... 34
Figure 3.3: Pulse broadening along a fiber section, for different values of C. ............................... 35
Figure 3.4: Pulse amplitude at the input and output of the fiber for C=−2. ................................... 35
Figure 3.5: Evolution of the absolute value of the super Gaussian pulse for C=−2. ..................... 36
Figure 3.6: Pulse amplitude value at the input and output of the fiber, for C=0. ........................... 36
Figure 3.7: Evolution of the absolute value of the super Gaussian pulse, for C=0........................ 36
Figure 3.8: Pulse value at the input and output of the fiber for C=2. ............................................. 37
Figure 3.9: Evolution of the absolute value of the super Gaussian pulse for C=2......................... 37
Figure 3.10: Pulse amplitude for the entrance pulse and exit pulse, when the broadening is
List ∆ Fiber contrast a Radius of the core h Refractive index of the core α Refractive index of the cladding β Propagation constant
0k Propagation constant in vacuum u Normalized refractive index of the core w Normalized refractive index of the cladding n Modal refractive index v Normalized frequency
( )mJ u Bessel function of the first kind
( )mK u Bessel function of the second kind
modesN Number of modes
( ), , ,E x y z t Electric field
0E Amplitude of the electric field
( ),F x y Spatial distribution
( )0,B t Longitudinal variation
( )0,A t Pulse amplitude
Lβ Linear part of the propagation constant
NLβ Non-linear part of the propagation constant α Attenuation constant
gv Group velocity
DL Dispersion length ,τ ζ Normalized variables
2β Second-order dispersion
0τ Pulse initial width
3β Third-order dispersion 2σ Root-mean square t First order moment 2t Second order moment
gτ Group delay
NLφ Non-linear phase
inP Input power
0µ Magnetic permeability
0ε Electric permittivity
*E Fictional electric field 2
F Moment of the modal function
effA Effective area of the core
NLL Non-linear length
0q Initial separation
xii
r Relative amplitude
*y Admittance
xiii
List of acronyms EDFA Erbium-doped fiber amplifier ISI Inter-symbolic interference FSK Frequency shift keying DCF Dispersion compensating fiber GVD Group velocity dispersion RMS Root-mean square SPM Self-phase modulation QoS Quality of service XPM Cross-phase modulation FWM Four-wave mixing
xiv
1
1. Introduction
In this section, an historical overview, the motivation and objectives, and, the structure of the
work, as well as their contributions are presented
1.1. Historical overview
1.1.1. Optical fibers
Since the beginning of human kind, there was always a need to communicate over long distances.
Nowadays, the primary ways of communication are cell-phones, Internet and television. All this forms of
communication are supported by a network infrastructure.
A communication process consists in three main parts. The transmitter, from where the data is
generated, the communication link, responsible for transmitting the data over short, medium or long haul
distance, and the receiver, where the data is received. This work is specially focused in the communication
link.
Modern communication links of the global network are mostly optical fibers. Only the access
network is not yet completely implemented with optical fibers, but in future, the telecommunication
network will be completely composed by optical fibers.
Before the appereance of fiber optics, the communications were performed through coaxial
cables. With the use of coaxial cables in place of wires pairs, system capacity was increased considerably
[2]. In 1940, the first system using coaxial cable was implemented in the a 3 MHz band. The system was
capable of transmitting 300 voice channels and a single television channel [2]. The bandwidth of such
systems was considerably affected by the loss dependence on the frequency, which increased rapidly with
frequencies beyond 10 MHz [2].
With such a limitation in coaxial cable systen, a microwave communication system was
developed [2]. Then, both coaxial and microwave systems evolved considerably and both operated in bit
rates of the order of 100 Mb/s. Both systems had limitations in the spacing between repeaters, so that they
became very expensive.
The product BL , where B is the bit rate and L is the repeater spacing, is a system merit figure.
Figure 1.1 shows that the BL product has increased through technological advances during the last
century and a half.
2
Figure 1.1: Increase in the bit-rate distance product during the period 1850 and 2000 [2].
Such an increase in the BL product, in the second half of the twentieth century was only possible,
as optical waves were used as the carrier [2]. In May of 1960, Theodore Maiman performed the first
demonstration of a working laser [1]. The propagation medium was still needed to be invented. It was only
suggested in 1966 that optical fibers were capable of guiding the light in a manner similar to the guiding
of electrons in copper wires [2].
The first semiconductor laser was introduced by four independent groups, between September and
October of 1962 [1]. But the first lasers only worked with a cooling system of nitrogen. Only in 1970, the
first semiconductor lasers operating at room temperature made their appearance [2].
1.1.2. Evolution of lightwave systems
The research on fiber optics started near 1975. Between 1975 and 2000 an enormous progress has
happened. The communication systems evolution can be grouped into four categories, as can be seen from
Figure 1.2 which shows the BL product over the time period of 1975 and 2000.
3
Figure 1.2: Increase of the BL product over the period of 1975 and 2000.
Figure 1.2 shows a straight line corresponding to a doubling of the BL product every year and the
four generations so far developed [2]. The first generation of lightwave systems operated in the first
window (0.8 µm). Those systems became available for commercialization in 1980. The bit rate was
around 45 Mb/s and allowed a space between repeaters of up to 10 km [1]. The second generation
operated in the second window (1.3 µm) [1]. The attenuation for this generation was 1 dB/km and
dispersion was minimum [1]. The bit rate was 1.7 Gb/s and the spacing between repeaters was 50 km [1].
The second generation was first implemented at a bit rate of 100 Mb/s in the beginning of the 1980s, only
later, in 1987 the bit rate of 1.7 Gb/s was achieved [2].
Since 1979, it was known that optical fibers had a minimum loss around the 1.55 µm wavelength,
where the attenuation was 0.2 dB/km [1]. However, the dispersion in the third-order generation system
was higher [2]. In 1990, a combination of dispersion-shifted fibers and monomodals semiconductor lasers
led to the implementation of the third-order communication systems. These systems have bit rates of 10
Gb/s and the spacing between amplifiers is of 100 km [2].
The main problem of the third generation systems is that they require the use of electronic
repeaters, known as regenerators [1]. When several wavelengths were transmitted through a fiber, the
regeneration had to be performed by several regenerators, one for each wavelength. From an economical
point of view this was unconceivable. To resolve the problem of the amplification of several channels
4
using only one device, the first erbium-doped fiber amplifiers (EDFAs) was developed in 1986. EDFAs
operate in the third window and use semiconductor lasers for their pumping.
The fourth communication generation was the first all-optical generation, where the use of optical
amplifiers allowed the amplification of several wavelengths and with that, the wavelength division
multiplexing (WDM) was implemented [1]. With WDM the bit rates were increased to higher values.
In order to create a world-wide network, submarine links had to be launched; Figure 1.3 shows the
international submarine network [2].
Figure 1.3: International undersea network of fiber-optic communication system [2].
The first transatlantic submarine cable was deployed in 1956 [1], and it was called TAT-1. Later,
in 1988, the first submarine cable with optical fiber was deployed; the system was called TAT-8 and had a
monomodal fiber [1]. The TAT-8 belongs to the second generation communication system.
The submarine cables TAT-9 and TAT-10/11 were deployed in 1992, and belong to the third
generation communication systems. In 1996, the submarine cables of the fourth generation began to work;
they were the TPC-5 and TAT-12/13. They used EDFAs in the repeaters. The bit rates achieved were 5.30
Gb/s. The TPC-6 was installed in 2000 with a bit rate of 100 Gb/s.
1.1.3. Evolution of the Non-linear communication systems
The availability of low loss silica fibers led to the study of non-linear fiber optics. In 1972, the
stimulated Raman scattering and the Brillouin scattering were studied. The idea that optical fibers can
support soliton like pulses as a result of interplay between the dispersive and non-linear effects appeared
in 1973. In 1980 solitons were experimentally observed. The advances in generation and control of ultra-
5
short pulses was only possible due to the discover of solitons. The field of nonlinear fiber optics has
grown considerably since 1990s and is expected to continue during the twenty-first century [3].
1.2. Motivation and objectives
1.2.1. Dispersion limitation in optical communication systems
The main objective of this work is to study the pulse propagation in linear and non-linear regimes
for several types of pulses. In addition this work will address the dispersion effect. When the pulse
propagation is performed in a linear regime, a technique of dispersion compensation will be applied. In the
non-linear regime, soliton propagation will be studied.
When a pulse is propagating in an optical fiber, it can suffer a dispersion effect. This dispersion is
usually in the time domain. Figure 1.4 shows the dispersion suffered by a pulse can suffer after being
propagated through a fiber.
Figure 1.4: Temporal dispersion that a pulse suffers after is propagated through a fiber.
In a real situation, instead of a single pulse, a train of pulses is transmitted. Figure 1.5 shows a
train of pulses at the communication link input.
Figure 1.5: Train of pulses at the fiber input.
When the train of pulses of Figure 1.5 arrives at the receiver, the pulses shapes are not the same.
Figure 1.6 shows a possible shape of the pulses at the receiver input.
6
Figure 1.6: Train of pulses at the receiver.
In case of strong dispersion the pulse invades the bit slot of another pulse. This effect is called
inter-symbolic interference (ISI).When a pulse invades the bit slot of another pulse, the receiver may not
be able to distinguish whether the pulse corresponds to a bit “1” or “0”. Figure 1.7 shows is an example of
inter-symbolic interference in the propagation of a train of pulses situation.
Figure 1.7: ISI of a train Train of pulses at the receiver input.
Reducing the bit rate, can mitigate the effect of the IIE. So, it is very important to study the impact
of dispersion over a train of pulses and over single pulse, and limitations to the bit rate.
The pulse can be influenced by dispersion not only in the time domain, but also in the spectral
domain. When direct modulation is performed the pulse suffers a chirp effect, which consists in a
broadening of the spectrum of the pulse. The effect of the chirp in the pulse propagation will be study in
this dissertation.
1.2.2. Dispersion compensation schemes
Several techniques can be used to compensate the time dispersion and the chirp effect. To
compensate the chirp, a precompensation scheme, is generally used where the input pulse is changed at
the transmitter. There are several precompensation schemes, such as:
• Prechirp technique;
• Novel coding technique;
• Nonlinear prechirp technique.
7
The prechirp technique consists in inducing a certain amount of dispersion such that, once
conjugated with chirp, it will narrow the pulse spectrum. Then, when the pulse arrives at the receiver has
its original shape. The novel coding technique consists in the use of another propagation format, called the
frequency shift keying (FSK). The nonlinear prechirp technique amplifies the transmitter output using a
semiconductor (SOA) operating in the gain saturation regime. In practice, the nonlinear prechirp technique
induces a compression in the pulse.
The postcompensation technique consists in compensating the dispersion at the receiver, just
before the signal reaches the detector. This operation is preformed electronically, since the optical signal is
converted to its electrical form, and then, it is equalized.
The technique, considered in this dissertation is the use of dispersion compensating fibers (DCF).
The DCF fiber is between the propagation fiber and the receiver. The DCF is dimensioned to have a group
velocity dispersion (GVD) opposite to the existing of the fiber, so that the dispersion may be
compensated.
The dispersion compensating fibers introduce attenuation in a practical perspective. To counter
the imposed attenuation optical filters were designed to compensate the dispersion imposed by the optical
fibers. Since the (GVD) affects the optical signal through the spectral phase, it is evident that an optical
filter, whose transfer function cancels this phase, will restore the signal. Unfortunately, no optical filter
has a transfer function suitable for compensating the GVD exactly. Nevertheless, several optical filters
have provided partial GVD compensation by mimicking the ideal transfer function [4].
1.3. Structure This work is divided into 5 chapters. In the first chapter one, a historical overview of the optical
communication systems is presented. Then, the motivation and objectives of this work are presented.
Finally, the structure and the main contributions of this dissertation are presented.
The second chapter addresses the modal theory of optical fibers. The propagation equations are
obtained. Also in chapter 2, we address the number of modes propagating in an optical fiber, and how the
dimensions of the fiber influence the number of modes that are propagated. The hybrid nature of the
modes is also studied in this chapter.
In the third chapter the propagation equation for a linear regime is derived, the pulse broadening is
determined. It also presents several simulations for the propagation of different types of pulses. The pulses
that are used are the “bell-shaped” pulses and the super Gaussian pulses. The dispersion suffered by the
pulses is also studied. The GVD ( 2β ), the third order dispersion ( 3β ) and the chirp are also studied in
this chapter. Further in this chapter, the second order dispersion is compensated using a DCF.
8
In chapter four, the propagation of pulses is performed in a non-linear regime. The solitons are
studied as well as their interaction.
In chapter five, the main conclusions of this work are drawn.
1.4. Main contributions Efficient pulse propagation in optical fibers is, nowadays, very important, because most of the
telecommunications systems use them. With this work, where pulse propagation is studied in linear and
non-linear regimes, we hope to help to further increase the understanding of the effects that pulses can
suffer along their propagation through a fiber. Another important contribution is the systematic analysis of
the techniques used to compensate the dispersion that the pulses suffer either in the linear or non-linear
regime.
9
2. Optical fibers
When communications are performed using optical fibers, one or more modes can be propagated.
In the fiber the number of modes that can be propagated depends on the design of the optical fiber. In this
section, the parameters that influence the mode propagation and the number of modes are discussed.
2.1. Basic structure of an optical fiber The basic structure of an optical fiber is presented in Figure 2.1.
Figure 2.1: Waveguide structure of optical fiber.
Figure 2.1 shows that the optical fiber is composed by two main regions, the core region and the
cladding region. The core region is where the light is mostly propagated. The role of the cladding region is
to confine the light in the core region. The geometry of an optical fiber is cylindrical and, in most cases,
both the core and the cladding consist of silica. In order to maintain the light inside the core of the optical
fiber, the refractive index of the cladding has to be slightly lower than the refractive index of the core.
The dielectric contrast is a parameter that measures the difference between the refractive index of
the core and the cladding, and is expressed by [1]
2 21 2
212
n n
n
−∆ = (2.1)
10
where 1n is the refractive index of the core and 2n is the refractive index of the cladding.
For a step-index fiber, and using cylindrical coordinates are used (r, φ , z), the refractive index
varies along the coordinate r according to
( ) 1
2
,.
,
n r an r
n r a
≤=
> (2.2)
Then, a different transverse wave have to be considered for the core and for the cladding. They are respectively [1]
2 2 2 21 0h n k β= − (2.3)
2 2 2 22 0n kα β= − (2.4)
where β is the longitudinal wave number, and 0k is the vacuum wave number, given by
0
2.k
c
ω πλ
= = (2.5)
These two transverse wavenumbers are normalized in the next section.
2.2. Normalized frequency and wavenumber In this section, the normalization of the frequencies and of the wavenumber is presented.
Equations (2.3) and (2.4) can be normalized in order to obtain a simpler modal equation to be presented
further. The normalization of the core and the cladding wavenumber is respectively
u ha= (2.6)
w aα= (2.7)
where a is the radius of the core of the fiber.
To expand the equations (2.6) and (2.7) the longitudinal wavenumber β is defined as [1]
0nkβ = (2.8)
where n is the modal refractive index, given by
( )2 2 2 22 1 2n n b n n= + − (2.9)
where b is the normalized modal refractive index, to be defined ahead.
11
To define the normalized transverse wavenumber in the core and in the cladding in terms of the
modal refractive index, the longitudinal wavenumber β is replaced in equations (2.6) and (2.7). Then, the
parameters u and w yield
( )( )22 2 21 0u n n k a= − (2.10)
( )( )22 2 22 0 .w n n k a= − (2.11)
The parameters h and α are also called the transversal propagation constants in the core and the
cladding, respectively. Using the normalization of those parameters and summing equations (2.10) and
(2.11) a new normalized parameter is defined, which is called, normalized frequency, and is given by [1]
2 2 2v u w= + (2.12)
The parameter v can be expressed in terms of 1n , 2n , 0k and a according to
2 21 2 0 .v n n k a= − (2.13)
The equation (2.13) can also be expressed in terms of ∆ , as
1 0 2 .v n k a= ∆ (2.14)
The longitudinal wavenumber can be written in terms of v in the form
2
21.
2
vu
aβ = −
∆ (2.15)
After obtaining the previous parameters, the normalized modal refractive index is defined by [1]
2 22 2
22 2 2 2
1 2
1 .n nu w
bv v n n
−= − = =
− (2.16)
Due to the fact that 2 0 1 0n k n kβ≤ < and 2 1n n n≤ < , then the normalized modal refractive index can only vary
between 0 1b≤ < .
12
2.3. Hybrid modes modal equation In an optical fiber, the surface guided waves are generally hybrid modes. In hybrid modes the
axial electromagnetic field components zE and
zH are both different from zero [6]. It can be shown [1]
that the modal equation for the hybrid modes is given by [1]
( ) ( )42
22
1 2 ,m m
u vR u S u m
v uw
= − ∆
(2.17)
where the parameters ( )mR u and ( )mS u are defined, respectively as
( ) ( )( )
( )( )
m m
m
m m
J u K wR u
uJ u wK w
′ ′= + (2.18)
( ) ( )( )
( ) ( )( )
1 2 ,m m
m
m m
J u K wS u
uJ u wK w
′ ′= + − ∆ (2.19)
where ( )mJ u is the Bessel function of the first kind and ( )mJ u′ its derivative, ( )mK w is the Bessel function of
second kind and ( )mK w′ its derivative. The derivatives ( )mJ u′ and ( )mK w′ are, respectively, given by
( ) ( ) ( )1 1
1
2m m mJ u J u J u− +′ = − (2.20)
( ) ( ) ( )1 1
1.
2m m mK w K w K w− +′ = − + (2.21)
Derivation of equation (2.17) is presented in Annex 1.
The hybrid modes are classified into two categories: the HEmn modes and de EHmn modes. The m
parameter is the variation of the azimuthal coordinate, and the parameter n is the variation of the radial
coordinate.
While the general modal equation for the hybrid modes was derived in this section. In the next
section, a particular approximate modal equation will be presented.
13
2.4. Low contrast fibers When a fiber has a low dielectric contrast, i.e., 1∆ , the low contrast fiber linearly polarized
(LP) mode come from the approximation with 1
0
1n
n≅ . This means that the light confinement is not tight to
the core. These types of fibers are called, weakly-guided fiber. In this case the expression of ∆ can be
simplified to [2]
1 2
1
.n n
n
−∆ ≈ (2.22)
With low contrast fibers, the Gloge approximation for the modal equation can be used. The Gloge
approximation permits the use of the condition
( ) ( ).m mR u S u= (2.23)
Then, the equation (2.18) becomes
( )2
2 2m
mvR u
u w= ± (2.24)
where, when the ( + ) signal is used, only the EHmn are propagated. When the (− ) signal is used only the HEmn are
propagated.
When the EHmn modes are propagated the modal equation yields [1]
( )( )
( )( )
1 1 0m m
m m
J u K w
uJ u wK w
+ ++ = (2.25)
where the parameters ( )1mJ u+ and 1mK + are obtained using the following two equations [1]
( ) ( ) ( )1m m m
mJ u J u J u
u+′ = − + (2.26)
( ) ( ) ( )1 .m m m
mK w K w K w
w+′ = − + (2.27)
When the HEmn modes are propagated the modal equation yields
( )( )
( )( )
1 1 0m m
m m
J u K w
uJ u wK w
− −− = (2.28)
14
where the parameters 1mJ − and 1mK − are obtained through [1]
( ) ( ) ( )1m m m
mJ u J u J u
u−′ = − (2.29)
( ) ( ) ( )1m m m
mK w K w K w
w−′ = − − (2.30)
respectively.
When the objective is to propagate the modes TE0n and TM0n, the modal equation of both modes
is the same. Then, for 0m = the modal equation comes [1]
( )( )
( )( )
1 1
0 0
0.J u K w
uJ u wK w+ = (2.31)
There are two conditions that are obtained when these two modes are used, they are [1]
( ) ( ) ( )1m
m mJ u J u− = − (2.32)
( ) ( ).m mK w K w− = (2.33)
The HE0n modes are equivalent to the TE0n modes. The EH0n modes are equivalent to the TM0n
modes.
When the fibers are of low contrast the modes are linearly polarized, and are called LPpn modes.
When the propagated mode is the EHmn mode the correspondent LPpn mode has a 1p m= + . When the
propagated mode is the HEmn mode, the correspondent LPpn mode has a 1p m= − . The modal equation
that corresponds to the previous can be
( )( )
( )( )
1 1 0p p
p p
J u K wu w
J u K w
− −+ = (2.34)
or can be
( )( )
( )( )
1 1 0.p p
p p
J u K wu w
J u K w
+ +− = (2.35)
The equations (2.34) and (2.35) are equivalent [1].
After obtaining the modal equation for the LP modes, it can be simplified when the propagated
mode is the fundamental mode LP01 which is going to be done in the next section.
15
2.5. LP modes The study of the fundamental mode is very important, when single-mode fibers are used. The
SMF only support the fundamental mode HE11 or LP01. The fiber is designed such that all higher-order
modes are cut off the operating wavelength [2]. The modal equation of the fundamental mode, whether it
is equation (2.34) or equation (2.35), yields [1]
( ) ( ) ( ) ( )1 0 0 1 .uJ u K w wJ u K w= (2.36)
In order to obtain a ( )b v function, the parameters w and u are written in terms of b and v , and it is given values
to v . Then, the value of b is found in a way that the equation (2.36) has a solution. Figure 2.2 shows the ( )b v
function for the fundamental mode for various values of ∆ .
Figure 2.2: Normalized propagation constant as a function of the normalized frequency for the fundamental mode LP01 for several
values of ∆ .
Figure 2.2 shows that when the contrast increases, the required v increases so that the propagation
is possible. Then, for weakly guiding fibers with lower contrast the cut off v is lower. The propagation
coefficient increases as long as the normalized frequency increases.
Figure 2.3 shows the first six LP modes of an optical fiber when the solved equation is (2.34) or
(2.35).
16
Figure 2.3: Normalized propagation constant as a function of the normalized frequency, for the first six LP modes.
Figure 2.3 shows that a fiber with a large v supports many modes. An estimate to determine the
number of modes for a multimode fiber is to perform [2]
2
modes 2
vN = (2.37)
where modesN is the number of modes. Below a certain value of v , only the LP01 mode is propagated. The cut off
frequency 2.4048cv = is the value of v that turns the propagation of the LP01 mode possible [1]. The expression
that determines the maximum core radius is [1]
max
1
.2 2
cva an
λ
π≤ =
∆ (2.38)
The determination of the radius of the core influences the number of modes that can be
propagated. The fewer modes, the lower the core radius will be. Then, a monomodal fiber will have an
effective area lower than the effective area of the multimode fiber. Thus, the non-linear effects will be
more accentuated in the monomodal fiber than in the multimodal fiber.
2.6. Conclusions From this chapter the main conclusions that can be taken are:
• The number of modes that can be propagated in a fiber depends on the radius of the core
17
of the fiber;
• The core radius of the fiber determines whether the fiber is monomodal or multimodal;
• The cut-off frequency of a multimodal fiber is 2.4048cv = ;
• For low contrast fibers, when the contrast is lower the necessary v that is required, so that
the propagation is possible, is lower.
18
19
3. Pulse propagation in the linear regime
Even when the propagation of pulses is performed in a single-mode fiber, there is still a source of
dispersion that may influence the quality, speed, and bit rate of the communication. This type of fiber also
exhibits group velocity dispersion (GVD), and also may have higher order dispersion. The transmitted
pulses experience a time broadening due to the GVD. This pulse broadening may result in an inter-
symbolic interference (ISI) [1].
3.1. Propagation Equation in the linear regime In order to determine the shape of a pulse at the output of a communication link, the pulse
propagation equation has to be derived. Assuming that the electric field, at the input of the fiber, or 0z = ,
is linearly polarized in the direction of x, the electric field is given by
( ) ( )ˆ, ,0, , ,0,E x y y xE x y t= (3.1)
This expression be written as
( ) ( )0, ,0, , (0, )E x y t E F x y B t= ⋅ ⋅ (3.2)
where 0E is the amplitude of the electric field, ( ), F x y is the transversal distribution of the field in the
fundamental fiber mode and ( )0,B t is the time distribution of the field at 0z = . The approximation to the LP01
mode is reasonable because it is assumed that the fibers have low contrast, i.e. 1∆ << . The term ( )0,B t in equation
(3.2) defined at the entrance of the fiber, can be written as
( ) ( ) ( )00, 0, expB t A t i tω= ⋅ − (3.3)
where ( )0,A t is the pulse envelope at entrance of the fiber and 0ω is the angular frequency of the carrier.
The first point, it is important to determine the relation between ( )0,B t and ( )0,A t , using the
Fourier transform. In this thesis we use the following definitions of the Fourier transform and its inverse
Fourier transform are, respectively
( ) ( ) ( ), , expX z X z t i t dtω ω+∞
−∞= ⋅∫ (3.4)
20
( ) ( ) ( )1, , exp .
2X z t X z i t dtω ω
π
+∞
−∞= ⋅ −∫ (3.5)
Replacing ( ),X z t in equation (3.4), by the Fourier transform of ( ), ,0,E x y t yields
( ) ( ) ( ) ( ) ( )( )
0 0
0,
, ,0, , exp 0, exp .
A
E x y E F x y i t A t i t dt
ω
ω ω ω= ⋅ − ∫
(3.6)
Assuming a linear and time-invariant system and applying the properties of the Fourier transform, namely, the
frequency shift property, the equation (3.6) comes
( ) ( ) ( )0 0, ,0, , 0,E x y E F x y Aω ω ω= − (3.7)
which shows that
( ) ( )00, 0, .B Aω ω ω= − (3.8)
The spectral component ( ),B z ω propagates along the fiber with an propagation constant ( )pβ ω .
This propagation constant has several components [7], according to
( ) ( ) ( ) ( )2p L NL i
α ωβ ω β ω β ω= + + (3.9)
where ( )Lβ ω is the linear part of the propagation constant, ( )NLβ ω is the non-linear part of the propagation
constant and ( )α ω is the fiber loss parameter. In linear regime, the term ( )NLβ ω is zero. Assuming no losses, then
( )α ω is also zero. Then, equation (3.9) yields
( ) ( ).p Lβ ω β ω= (3.10)
To determine ( ),B z t , the inverse Fourier transform of ( ),B z ω needs to be performed. Replacing equation (3.10)
and, then, ( ),B z t yields
( ) ( ) ( )( )( )0
1, 0, exp .
2 LB z t A i z t dω ω β ω ω ωπ
+∞
−∞= − −∫ (3.11)
A simple way to show equation (3.11) is to make a variable change in which ω is replaced by Ω ,
such that
21
0
0
1.d
d
ω ω
ω ω
ω
Ω = −
=Ω+
=Ω
(3.12)
Then, by replacing (3.12), in (3.11), it comes
( ) ( ) ( ) ( )( )0 0
1, exp 0, exp .
2 LB z t i t A i z t dω β ωπ
+∞
−∞ = − Ω Ω + −Ω Ω ∫ (3.13)
To simplify the integral of equation (3.13), the term ( )0Lβ ω +Ω is expanded in a Taylor’s series. The result of the
expansion is
( ) ( )0 0Lβ ω β+Ω = +℘ Ω (3.14)
where ( )0 0β β ω= and
( )1 !
mm
m m
β∞
=
℘ Ω = Ω∑ (3.15)
where
0
.m
m m
d
dω ω
ββ
ω=
= (3.16)
With the expansion of ( )0Lβ ω +Ω , equation (3.13) results in
( ) ( ) ( )0 0, , expB z t A z t i z tβ ω= − (3.17)
where ( ),A z t is defined by
( ) ( ) ( )( )1, 0, exp .
2A z t A i z t d
π
+∞
−∞ = Ω ℘ Ω −Ω Ω ∫ (3.18)
In order to solve equation (3.18) the coefficients in equation (3.16) ( 1,2,m = …) have to be
determined. The first one, 1β , is physically related with the inverse of the group velocity gv as
0
1
1
g
d
d vω ω
ββ
ω =
= = (3.19)
22
Coefficients 2β and 3β are known as the second-order and third order dispersion terms and are related with gv by
0
2
2 2 2
1 g
g
vd
d vω ω
ββ
ω ω=
∂= = −
∂ (3.20)
0
32
3 3
d
dω ω
β ββ
ω β=
∂= =
∂ (3.21)
These parameters are responsible for the pulse broadening in optical fibers.
After computing the coefficients mβ , it is useful to determine ( ),A z t in terms of ( )0,A t . To do
that, it is useful to define
( ) ( ) ( ) [ ]1, 0, exp exp
2m
mA z t A i z i t dπ
+∞
−∞= Ω Ω ℘ Ω − Ω Ω ∫ (3.22)
Therefore from equation (3.18), results the general equation
( )1
, .!m
m
m
Ai A z t
z m
β∞
=
∂=
∂ ∑ (3.23)
Introducing now the fiber loss factor, equation (3.23) yields
( ) ( )1
, ,! 2m
m
m
Ai A z t A z t
z m
β α∞
=
∂= −
∂ ∑ (3.24)
where α is the attenuation constant.
After obtaining the derivative of ( ),A z t in order to z , it is necessary to derive the expression of
( ),mA z t as a function of
dAdtso that equation (3.24) can be completed. The first step is to derive
equation (3.22) in order to time for 1,2,3m = and 4 . Then the first four terms come
( )1 ,dA
iA z tdt
= − (3.25)
( )2
22,
d AiA z t
dt= (3.26)
( )3
33,
d AiA z t
dt= − (3.27)
23
( )4
44, .
d AiA z t
dt= (3.28)
Then, in a general way,
( )2 ,m
m
mm
d Ai A z t
dt
−= − (3.29)
in which ( ),mA z t is
( ) 2, .m
m
m m
d AA z t i
dt
−= − (3.30)
Due to the fact that equation (3.24) depends on ( ),mA z t , one has
( )1
1
, .! 2
mmm
mm
A d Ai i A z t
z m dt
β α∞−
=
∂= − −
∂ ∑ (3.31)
Then, by simplifying equation (3.31), it comes
( )1
1
, 0! 2
mmm
mm
A d Ai A z t
z m dt
β α∞−
=
∂+ + =
∂ ∑ (3.32)
If the fourth order and greater ( 4m ≥ ) propagation terms are ignored and the attenuation constant is neglected,
equation (3.32) comes
2 3
1 2 32 3
1 10.
2 6
A A d A d Ai
z t dt dtβ β β
∂ ∂+ + − =
∂ ∂ (3.33)
Equation (3.33) can be simplified, in order to easier derive its solution. To do so, a couple of
normalized variables is defined as [1]
0
12
0
D
D
z
LL
t z
ζτ
ββτ
τ
== →
− =
(3.34)
where DL is the dispersion length, τ0 is a measure of pulse width and 2β is the absolute value of the second-order
dispersion. In a first approach the terms dA
dzand
dA
dthave to be expressed in terms of τ and ζ , respectively
according to
24
1
1
1
1
D
D
A A A
z z z
z L
z
A A
z L z
ζ τζ τζ
βττβ τ
ζ τ
∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂∂
= ∂
∂ = − ∂ ∂ ∂ ∂ = − ∂ ∂ ∂
(3.35)
0
0
0
.1
1
A A A
t t t
t
t
A A
t
ζ τζ τζ
ττ
τ τ
∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂∂
= ∂ ∂ =
∂ ∂ ∂ = ∂ ∂
(3.36)
Then, equation (3.33) comes
2 3
322 2 2 30 0
1 10
2 6D D
A A Ai L L
A
ββζ τ τ τ
∂ ∂ ∂+ − =
∂ ∂ ∂ (3.37)
where the following definitions have been used
( )222
0
sgnDLββ
τ= (3.38)
330
1
6DLk
βτ
= (3.39)
By replacing equations (3.38) and (3.39) into (3.37), on has
( )2 3
2 2 3
1sgn 02
A A Ai kβ
ζ τ τ∂ ∂ ∂
+ − =∂ ∂ ∂
(3.40)
Where coefficient κ can be written as
330 2
1.
6k
βτ β
= (3.41)
The also called higher order dispersion coefficient.
25
After obtaining the basic propagation equation, the pulse shape can be determined at any distance
in the fiber, using simple computer simulation. Before the pulse propagation is performed, it is useful to
determine an expression that computes the broadening of the pulse along the fiber.
3.2. Analytical approach for pulse broadening The broadening of a pulse can induce ISI that limits the bit rate of a communication link. This
effect is more accentuated as long as the fiber length increases. Then, it is useful to determine an
analytical expression to measure the pulse in terms of the so called root-mean-square (RMS) width of the
pulse.
3.2.1. Second and third-order moments
In order to determine an expression for the width of a pulse, pulses of arbitrary shape have to be
take into account [7], because most pulses are not Gaussian and the dispersion coefficient 3β also affects
Gaussian pulses. Then, using the definition of root-mean square (RMS) value, the width of a pulse is
given by
22 2t tσ = − (3.42)
where t is the first order moment and 2t is the second order moment. The moments can be obtained by using a
general expression that is given by
( )
( )
2
2
,.
,
m
mt A z t dt
tA z t dt
+∞
−∞+∞
−∞
= ∫∫
(3.43)
In order to determine the pulse width, the non-linear effects have to be negligible. This is based on
the observation that the pulse spectrum does not change in a linear dispersive regime, irrespective of what
happens to the pulse shape [7].
The objective of this section is to determine t and 2t . In order to do that we will use the
following relation
( ) ( )22 1
, , 12
A z t dt A z dπ
+∞ +∞
−∞ −∞= Ω Ω =∫ ∫ (3.44)
26
Applying definition (3.43) and the relationship (3.44), t comes
( )2
, .t t A z t dt+∞
−∞= ∫ (3.45)
which is equivalent to write
( ) ( )*, , .t tA z t A z t dt+∞
−∞= ∫ (3.46)
By applying the Fourier transform to ( )* ,A z t , equation (3.46) can be written as
( ) ( ) ( )*1, , exp ,
2t tA z t A z i t d dt
π
+∞ +∞
−∞ −∞
= Ω Ω Ω ∫ ∫ (3.47)
And, changing the order of the integration, equation (3.47) finally yields
( ) ( ) ( )( )*1, , exp .
2t A z tA z t i t dt d
π
+∞ +∞
−∞ −∞= Ω Ω Ω∫ ∫ (3.48)
Because the system is assumed to be linear and time invariant, the derivative property can be
applied as
( ) ( ).
dX jtx t i
d
ωω
↔ (3.49)
Then, equation (3.48) becomes
( ) ( )*1, ,
2t i A z A z d
π
+∞
Ω−∞= − Ω Ω Ω∫ (3.50)
where
( ) ( ),, .
dA zA z
dΩ
ΩΩ =
Ω
(3.51)
After obtaining the first order moment, the second order moment can be easily determined.
Applying the definition of (3.43) and relation (3.44), 2t comes
( )22 2 ,t t A z t dt
+∞
−∞= ∫ (3.52)
Separating the terms inside the integral
27
( ) ( )2 , , .t tA z t tA z t dt+∞
−∞= ∫ (3.53)
After some algebra manipulation and using the Fourier transform, and its properties, equation (3.53) finally comes
( )22 , .t A z d
+∞
Ω−∞= Ω Ω∫ (3.54)
Expression for the first and second order moments were obtained in this section. In the next
sections the RMS width will be determined using these definitions and other parameters, such as, the
group delay.
3.2.2. RMS broadening of a function of the group delay
Although in the current approach the propagation constant does not include non-linear effects, it
includes dispersive effects of all orders. Considering the pulse at the input of the fiber
( ) ( ) ( )( )0, expA S iθΩ = Ω Ω (3.55)
where the spectral phase ( )θ Ω plays an important role as it is related to the frequency chirp of the pulse. In order to
take into consideration the chirp effects, the parameter ( )θ Ω and the group delay gτ have to be included in the
RMS expression of the broadening. To do that, the definitions derived in the previous sections will be used.
According to the definition of the group delay, one has
1L
g
g
d LL L
d v
βτ β
ω= = = (3.56)
where L is the fiber length and vg is the group velocity. Let’s define the derivative of ( )θ ω as
.d
d
θθΩ =
Ω (3.57)
To use the definitions (3.50) and (3.54), function ( ),A zΩ Ω has to be determined in terms of
( )S Ω and ( )θ Ω . Hence, using definition (3.51), ( ),A zΩ Ω comes
( ) ( ) ( )( ) ( ) ( )( )0 0, 0, exp 0, expd
A z A i z A iz i zd
ββ β β βΩ Ω ΩΩ = Ω − + Ω −
Ω (3.58)
where ( )0,AΩ Ω is
28
( ) ( )( ) ( ) ( )( )0, exp exp .A S i iS iθ θ θΩ Ω ΩΩ = Ω + Ω Ω (3.59)
Now, substituting equations (3.55), (3.58) and (3.59) into the definitions (3.50) and (3.54), the
moments 2t and t yield, in terms of ( )S Ω and ( )θ Ω , respectively