Post on 03-Feb-2018
CHAPTER 1
SOLITONS IN OPTICAL FIBERCOMMUNICATIONS
1.1 INTRODUCTION
In recent times, many countries have moved from the postindustrial era
to the information era. Incredible as this would have seemed just a few
decades ago, these countries now produce more information than they do
tangible products, relegating manufacturing to a secondary role in their
economies. The more information we produce, the greater the need for its
delivery because, obviously, information works only when it is delivered to
the right place, at the right time, in the right form. And it is the business of the
telecommunications industry to do just that [1]. Hence, it will not be
incorrect to say that the communication technology is one of the current areas
of interest. Usage of light for communication is not new and has been in
practice for several hundreds of years, although in a crude form. The advent
of fiber optics has undoubtedly revolutionized telecommunication systems
around the world, enabling an unprecedented amount of information
exchange, all at almost an amazing speed of light. However, we are just at the
beginning of what will likely to be known as photonics century. Just as
electronics dramatically improved the quality of life in the last century,
photonics promises to do the same in the 2 1st century.
2
Nowadays, in addition to the telephone, people use the mobile phone
and the internet for their everyday activities such as e-shopping, e-business
transaction, playing games and downloading required materials (music and
scientific articles). They also communicate by e-mail and voice chatting with
other people anywhere on the globe. These services ultimately demand high
bandwidth information transmission networks. Undoubtedly, optical fiber
communication (OFC) system is the only answer to cope with such a
phenomenal growth in the bandwidth requirement.
Despite being a wired communication, an OFC system offers several
advantages compared to copper cable communications or co-axial or even
satellite communications. Of all the advantages, the most important is the
extremely higher bandwidth (of the order of T b/s) that can be possible only
with OFC system. Another important advantage with optical communication
is the speed with which data can be transmitted. As we know, light is so fast
that it takes less than two seconds to travel from earth to moon. So, light as
information through fibers can also ensure relatively faster information
transfer when compared to copper or co-axial communications. However the
strength of a signal traveling through an optical fiber weakens with distance
due to attenuation, dispersion and nonlinear effects. Nowadays, attenuation is
being tackled by optical amplifiers such as Erbium Doped Fiber Amplifier
(EDFA) and Raman Fiber Amplifier (RFA). Dispersion turns out to be one of
3
the major problems in optical communications. Since, only the Single Mode
Fibers (SMF) are being extensively used for long-haul communications, the
type of dispersion to be dealt with is chromatic dispersion. Over the time,
dispersion can lead to a phenomenon known as Inter-symbol Interference
(ISI), wherein the pulses broaden to a point that they interfere with one
another. Therefore, dispersion ultimately limits the bandwidth of the fiber,
reducing the amount of information it can reliably carry. Though there are
several novel fibers designed to tackle the problem of dispersion, none of the
fibers can completely eliminate the dispersion experienced by the signal. The
use of high intense laser sources in optical communications has resulted in
many nonlinear effects, which are usually detrimental. A clever configuration
of chromatic dispersion in the anomalous regime with a nonlinear effect
called Self Phase Modulation (SPM) has led to the realization of so-called
optical soliton [2]. Not only do solitons not disperse but an encounter with a
perturbation will usually leave the soliton unaltered. One of the keys to the
success of the ensuing photonics revolution will be the use of optical solitons
inOFC.
Alexander Graham Bell may have secured his place in history but
another Scott known for his ship hull designs, is an unexpected addition to the
communications Hall of Fame. In 1834 a Scottish engineer named Scott
Russell observed a boat being drawn rapidly along Edinburgh's Union Canal
4
by a pair of horses. When the boat stopped, he noticed that the bow wave
continued forward at great velocity, assuming the form of a large solitary
elevation, which continued its course along the channel apparently without
change of form or diminution of speed. He followed such an undiminishing
wave on a horse back for several miles until he lost sight of it in the windings
of river. He called those waves as "solitary waves". Then followed several
equations, one in 1872 namely Boussinesq's equation and another in 1895
namely Korteweg-deVries (KdV) which proved theoretically the existence of
solitary waves.
Later, in 1965, Martin Kruskal and Norman Zabusky studied the KdV
equation numerically and revealed the nature of these solitary waves that they
could reemerge without change in shape and velocity even after the collision
among themselves. They named these waves as solitons to sound like protons,
electrons, photons in order to impress on their particle-like nature. It was
Hasegawa and Tappert [3, 4], who predicted the existence of optical solitons
in 1973 and the same was confirmed experimentally by Mollenauer and his
group in 1980 [5]. For communication applications, optical soliton is
modified so that it is more immune to external perturbations by applying
proper variation of the fiber dispersion profile. Solitons that are created in
such fibers are often called Dispersion Managed Solitons (DMS). We have
experimental results of one, 40 G b/s single channel DMS transmission over
5
10,000 km [6] and another, 1.4 T b/s single channel DMS transmission over
6,000 km [7].
The qualities of the 'soliton wave' that it does not break up, spread out
or lose strength over distance - make it ideal for fiber optic communications
networks. In these networks, billions of solitons per second carry information
down the fiber circuits for telephones, computers, cable television etc.
Now, it has been proved beyond doubt that solitons do exist in many
areas of science namely, particle physics, molecular biology, quantum
mechanics, geology, meteorology, oceanography, astrophysics, optics and
cosmology. But solitons that exist in optics (so called "optical solitons") have
been drawing a greater attention among the scientific community for the
reason that these solitons seem to be right candidates for transferring
information (audio or video or data) across the world through optical fibers.
An optical soliton is basically a short, bell shaped laser pulse, which
has the ability to travel down the fiber several thousands of km without
dispersion when the loss in the system is taken care of. These solitons are
realized only in nonlinear regime.
In optics, the terms linear and nonlinear mean "power-independent"
DI
and 'power-dependent' phenomena, respectively. In optical fibers, a
nonlinear effect would mean that the refractive index of the fiber not only
depends on the frequency of the signal but also on the intensity of the light.
Under low intensity limit, the refractive index of the fiber will depend only on
the frequency of light. But, since in long-haul optical fiber communications,
only laser sources are used, the resultant intensity of light is large enough to
induce nonlinear phenomena in optical fibers. As already mentioned, attempts
to turn a detrimental nonlinear effect called SPM for some useful end have
brought in the concept of optical solitons.
1.2 OPTICAL FIBER COMMUNICATIONS
Undoubtedly, fiber optic technology has pervaded into every walk of
our life. For instance, each time we pick up our telephone, turn on our
television, transmit documents over a fax, give a cashier our credit card, use a
bank ATM, or surf the World Wide Web (WWW), we are undoubtedly using
fiber optic communications technology [8 - 10]. The importance of fiber optic
communications technology has been growing at a faster rate since the day of
realization of low loss fibers in 1970 and today optical fibers handle about
80% of current telecommunications traffic.
Optical fibers have many distinct advantages over their metallic
counterparts viz., copper cables (both twisted pair and coaxial). Optical fibers
7
are compact, lightweight and have the ability to transfer huge amounts of
information. The optical fibers are 200 times lighter, occupy 150 times lesser
volume and have bandwidth 10,000 times greater than that of coaxial cables.
With these numbers in mind, the overwhelming advantages of optical fiber
technology become clearly apparent. Apart from the economic advantages,
optical fibers also exhibit technological superiority. Optical fibers are made
up of silica-based glass or plastic, which are insulators and therefore have no
currents flowing in them. As a result, fibers are immune to electromagnetic
interference. In addition, fiber systems cannot be tapped into without being
detected. This degree of security makes fiber optic communications a choice
of preference especially in defense services. Another advantage of optical
fibers is that they do not corrode.
As we live in the information era, the amount of data produced keeps
on doubling every year. Hence every communication system looks for all
possible ways to increase its information carrying capacity, otherwise called
bandwidth. As a general rule, the information carrying capacity of any
communication system is roughly about 10% of the frequency with which it
operates. As light falls in the highest frequency range compared to the rest of
the carrier frequencies of other communication system, the bandwidth of an
optical fiber communication system can be as high as 50 Th/s. There is no
indication that any other communication systems, including satellite
8
communication, can ever achieve bandwidth as high as that of optical fiber
communication. As a result, the technology of fiber optic communications has
been evolving at a faster rate since the commercial deployment of optical
fibers.
There are, however, a few challenges that optical fibers have to deal
with compared to copper cables. Working with fibers requires a great deal of
skill and costly equipment. Though we have several above mentioned
advantages, in the case of long distance communication, the information
carrying capacity is quite severely affected due to attenuation, dispersion,
nonlinearity and amplifier induced noise. The attenuation, nowadays, is being
taken care by EDFA. When an optical pulse propagates through a fiber its
pulse width increases due to dispersion. Because of this, we will not be able to
distinguish between pulses and hence leading to Bit Error Rate (BER). So, the
dispersion is considered to be the most threatening aspect of OFC system. In
order to fight out the problem of dispersion, in recent years, there have been
many dramatic improvements with regard to the design of fiber such as
Dispersion Shifted Fiber (DSF), Non-Zero Dispersion-Shifted Fiber (NZ-
DSF), Dispersion-Flattened Fiber (DFF), Dispersion Compensating Fiber
(DCF), etc. But none of the fibers mentioned hitherto could completely
eliminate the dispersion. Moreover, Single Mode Fibers (SMF) which
constitute more than 80% of the fibers laid all over the world, suffer from so
called Polarization-Mode Dispersion (PMD). Thanks to the discovery of
optical solitons, the future fiber links are expected to experience
dispersionless pulse propagation.
1.2.1 THE OPTICAL FIBER
An optical fiber is basically a thin, transparent flexible strand that can
carry light within it by means of total internal reflection. The simplest optical
fiber is a cylindrical structure consisting of a central core of doped silica
(Si02) surrounded by a concentric cladding of pure silica. Such a fiber is
referred to as a step index fiber. The refractive index of the core (n i) is
slightly greater than that of the cladding (n2) and optical pulses get transmitted
through fibers by means of total internal reflection, shown in Fig.(1.1). All
those light rays that are incident at the core-cladding interface at an angle
greater than the critical angle given by
Oc = sin'(n2 /n1 ). (1.1)
will get totally internally reflected and will be guided within the core. If the
corresponding maximum angle of incidence at the entrance face of the fiber is
Oa, then the numerical aperture (NA) of the fiber is defined as
NA = sin 0,,= jn - n. (1.2)
Here, °a is referred to as 'acceptance angle'. Telecommunication optical
fibers have typically NA = 0.2 which corresponds to a maximum angle of
acceptance of about 11.5°.
Anglegreater than
10
Figure 1.1 Light guidance takes place through the phenomenon of
total internal reflection
Optical fibers can be broadly classified into two types namely
multimode and SMF. Multimode fibers are characterized by core diameters of
50 jnn and cladding diameters of 125 ,um while SMF have typically core
diameters of 8 to 12 pm and cladding diameters of 125 1um. There are two
main types of multimode fibers namely step index and graded index fibers.
Step index fibers are characterized by a homogeneous core of constant
refractive index while graded index fibers have an inhomogeneous core in
which the refractive index decreases in an almost parabolic fashion from the
center of the core to the core-cladding interface.
1.2.2 OPTICAL FIBER COMMUNICATION SYSTEM
In a typical OFC system, the information to be sent is first coded
into a binary sequence of electrical pulses which then are used to modulate a
laser beam to produce a sequence of ones and zeroes, represented by the
11
presence and absence of light respectively. The rate of information transfer is
expressed as the bit rate, which is nothing but the number of bits being sent
per second. Unfortunately, optical fiber transmission systems are subject to
three main effects that individually as well as collectively adversely affect
how much data - and how far the data - can be sent down the fiber. Power
loss, dispersion and nonlinearity all constrain both single-channel and
multiple wavelength transmissions. Let us discuss them one by one:
• Attenuation: The power carried by a light pulse propagating through the
fiber continuously decreases as it propagates along the fiber. The main
mechanisms responsible for this include Rayleigh scattering, absorption
by impurities, mainly water, waveguide imperfections such as bends, etc.
and intrinsic infrared and ultraviolet absorption. Hence, a fiber-optic cable
gradually reduces the power of the light traveling down it, typically at a
rate of 0.16 dB/km.
• Pulse dispersion: The speed that the light travels down the fiber depends
oi the wavelength. Chromatic dispersion is an important type of
dispersion that leads to broadening and overlapping of the data bits in a
signal as it propagates along the fiber. Dispersion is a linear effect. The
sign of the dispersion refers to whether the velocity increases (negative) or
decreases (positive) with wavelength.
• Nonlinear effects: The greater the intensity of light in the core, the higher
is the value of refractive index of the core. In other words, the refractive
index increases by an amount that is in proportion to the intensity of the
13
possibility of sending multiple optical signals at different wavelengths
through a single fiber, which is nothing but a technology called Wavelength
Division Multiplexing (WDM).
Thanks to the discovery of optical amplifiers, the technology of
boosting the optical signal has become relatively very simple. Until recently,
electronic repeaters were used for amplifying the signals as and when the
signal went weak. The most popular optical amplifier as on today is EDFA.
1.2.4 PULSE DISPERSION
The pulse dispersion is one of the troublesome linear effects in OFC. It
ultimately leads to widening of the pulse in the time domain. There are two
types of dispersions, namely, intermodal and intramodal dispersions.
The very first fiber, which was employed for communication, was
referred to as a step index multimode fiber. But it suffered from intermodal
dispersion whose details are as follows: When an optical pulse is launched
into the fiber, the power contained in the pulse is distributed into various
modes within the fiber. Each of these modes travels with constant speed but
takes different routes within the fiber. As a result, they arrive at the end of the
fiber with different timings. In short, the pulse widening, caused by the mode
structure of a light beam inside the fiber, is called intermodal dispersion.
14
Ultimately, intermodal dispersion became the bottleneck for the bit rate of the
fiber systems. Hence, a better fiber to cope with the problem of intermodal
dispersion was thought of and subsequently realized too. Such a fiber is called
a graded index fiber. In this fiber, the refractive index is maximum for the
core at the centre and decreases in a parabolic fashion until it meets the core-
cladding boundary. This profile ensures that the mode propagating along the
centerline of the fiber - the shortest distance - travels at the lowest speed
because it meets the highest refractive index. On the other hand, the mode
traveling closer to the fiber cladding - the longer distance - propagates at the
higher speed because it meets a lower refractive index. Hence the fractions of
an input pulse, delivered by the different modes, arrive at the receiver end
more or less simultaneously. Therefore, intermodal dispersion was
considerably reduced in this fiber and the bit rate was also appreciably
increased. Though this profile ultimately decreased the intermodal dispersion
to a larger extent, it could not completely eliminate the intermodal dispersion.
The best solution to handle the problem of intermodal dispersion came with
the realization of SMF, which constitute the major links throughout the world,
especially for long haul communications. Since this fiber sustains only one
mode of propagation, the intermodal dispersion is completely absent in this
case. However, SMF do suffer from the so called intramodal dispersion
(chromatic dispersion), which is discussed below.
n
15
Since majority of the fiber links all over the world use SMF, the most
worrisome aspect of optical communications is none other than chromatic
dispersion. Because of the fact that small thermal fluctuations and quantum
uncertainties prevent any light source from being truly monochromatic, even
the best available laser source does possess a finite spectral width. Hence,
different spectral components of the pulse experience different amount of
indices and hence travel through the fiber with different group velocities,
resulting in pulse widening. This phenomenon is called chromatic dispersion
or intramodal dispersion. It is also called Group Velocity Dispersion (GVD),
since the dispersion is a result of the group velocity being a function of the
wavelength. The two main causes of intramodal dispersion are as follows:
• Material dispersion, which arises from the variation of the refractive index
of the core material as a function of wavelength. This causes a wavelength
dependence of the group velocity of any given mode; that is, pulse
spreading occurs even when different wavelengths follow the same path
within the fiber.
• Waveguide dispersion, which occurs because a SMF confines only about
80% of the optical power to the core. Dispersion thus arises, since the
remaining 20% of the light propagating in the cladding travels faster than
the light confined to the core. Since this dispersion arises on account of the
refractive index profile of the optical waveguide, it is being referred to as
waveguide dispersion.
16
In order to understand pulse dispersion in a SMF, we consider a
Gaussian input pulse described by
'P(z = O,t) = c e_12ei0)0t. (1.4)
where 'r 0 is the input pulse width, coo is the central frequency of the light wave
and C is a constant. The frequency spectrum of such a pulse can be obtained
by taking a Fourier transform of Eqn.(1.4),
2/
Cr0 e-'02 (,O_WO)2 /4(1.5)
If fl(o) represents the frequency dependent propagation constant of the
mode, then each frequency component of the incident pulse suffers a phase
shift of fl(w)z after propagating through a distance 'z' in the fiber. Thus the
output pulse can be written as
W(z,t) = fA(w)eit_t1dw. (1.6)
Since the frequency spectrum given by Eqn.(1.5) is usually very
sharply peaked, we make a Taylor series expansion of 8(w) around w0:
d/3 ldfl(w)=fl(w)+—J
(CO -w)+ (ww)2+ (1.7)dwl do)
Substituting the expansion given by Eqn.(1.7) in Eqn.(1.6) and
integrating, we obtain the following expression for the output pulse
[11, 12].
/
I zl___ )
It
çv(z,t)
--Vg
I
(1 + a2 )4
r2 (z)= exp-
17
exp [i( .I(z,t) - (1.8)
where (z, t) = coot + K1t-
tan-](a),Vg ) 2
(1.9)
C 2azK = (1+a2)rT;
0 TO
d2181i- 2 (z) = r 2(1+ a); a = —I
do) 2CD0
1 = d/3Vg dw%
= ---D2,-c
(1.10)
We notice from Eqn.(1.8) that as the pulse propagates, it gets
broadened in time; the pulse width at any value of z is given by 'r(z). We also
notice that the phase of the pulse is no more proportional to time t but varies
quadratically with time t. This implies that the instantaneous frequency of the
pulse varies with time and such a pulse is referred to as a chirped pulse.
Fig.(1 .2) shows the chirping in the normal and anomalous dispersion regions
of propagation. The temporal broadening and chirping of the pulse are
determined by the value of the dispersion coefficient D (usually measured in
units of ps/km-nm, i.e., the dispersion suffered in picoseconds per kilometer
of propagation length per nanometer of spectral width of the source), which in
turn depends on the variation of 8 with frequency. The dependence of 8 on
frequency or wavelength can be due to material and wave guide dispersions.
The algebraic sum of material dispersion and waveguide dispersion gives the
18
total dispersion. As already mentioned, though there are several novel fibers
available to handle the problem of dispersion, none can ensure dispersionless
pulse propagation.
Unchirped input pulse Chirped and broadenedOut put pulse
Anamalous dispersion region ( D > 0)
Normal dispersion region ( D < 0)
Figure 1.2 Typical chirping caused in the anomalous and normal dispersionregions of an optical fiber
Apart from the chromatic dispersion, there is another dispersion called
PMD that arises only with SMF. Even though we call the fiber 'single mode',
it actually carries two modes under one name. These modes are linear-
19
polarized waves that propagate within a fiber in two orthogonal planes.
Ideally, each of the modes carries half of the total light power. If the fiber has
ideal symmetric cross-sectional properties both the modes propagate at the
same velocity and arrive at the fiber end simultaneously.
Thus, signal travel along the fiber remains undisturbed and the
presence of the polarized modes goes unnoticed. But there is some asymmetry
in every fabricated fiber, but the most likely times for serious asymmetry to
occur are during the fiber-cabling and splicing processes. Under this
condition, both the modes do not travel with same velocity and hence come at
the end of the fiber with different timings. In a nutshell, the pulse spreading
caused by a change of fiber polarization properties is called PMD. As already
mentioned, PMD in optical fiber arises from the modal birefringence caused
by geometrical core deformation and external stresses. It is known to be a
dynamic problem that changes with time, owing to different environmental
factors such as temperature and stress. PMD is a complex phenomenon, but
fortunately its impairments become significant only in high-bit-rate signals at
10 G b/s and beyond and in relatively long-haul transport. Its complexity is
further compounded by frequency-dependent higher-order contributions [13].
Though the dispersion due to PMD is less compared with chromatic
dispersion, unfortunately former is a random process. This is why there is no
20
real means for its compensation like the case for chromatic dispersion. Hence
PMD forms another worrying factor in the long-haul communication links.
1.2.5 NONLINEARITY IN OPTICS
Physics would be dull and life most unfulfilling if all physical
phenomena around us were linear. Fortunately, we are living in a nonlinear
world. While linearization beautifies physics, nonlinearity provides
excitement in physics [14]. This sub-section is devoted to discuss the study of
nonlinear electromagnetic phenomena in the optical region, which normally
arise while employing high intense laser sources.
Nowadays, nonlinear optical effects are unavoidable and they are
becoming increasingly important as the optical power density available from
lasers has increased tremendously in recent years, from 1012 to 1018 W/cm2
[15]. Such a high power optical beam propagating through optical fibers
induces many nonlinear effects which are usually detrimental but
unfortunately, unavoidable. They arise from the interaction of the external
electric field 'E' with the molecular dipole moment, which rotates those
dipoles and creates a polarization field 'P'. Now we discuss the physics of
nonlinear optics. When a beam of light is launched into a material, it causes
the charges of the atoms to oscillate. The polarization field is linearly
dependent on the magnitude of the external field as long as the magnitude of
PA
the field is small and the corresponding constant of proportionality is called
the electric susceptibility ' X' .This is the regime of linear optics. P = co E.
Thus, in the regime of the linear optics, as long as the intra-atomic electric
field strength is greater than the field strength of input light, the amount of
charge displacement is proportional to the instantaneous magnitude of the
electric field. The charges oscillate at the same frequency as the frequency of
the incident light and they either radiate light at that frequency or the energy
is transferred into non-radiative modes that result in material heating or other
energy transfer mechanisms. The light is effectively bound to the material; the
light excites charges that re-radiate light, which, in turn, excites charges, and
so on. As a result, the light travels through the material at a lower speed than
it does in the vacuum.
However, when the intra-atomic field strength is less than the field
strength of input light, the situation is drastically different as the external field
is increased. At this stage, the linearity eventually breaks down, as the
displacement of a charge from its equilibrium value is a nonlinear function of
the electric field. For the small forces, the displacement of the charge is small
and is approximated by a harmonic potential and a linear force. When the
displacement from equilibrium is large, the harmonic approximation breaks
down and the force is no longer a linear function of the displacement, i.e.,
response of the material is nonlinear [16] due to the anharmonic motion of
22
bound electrons. When an electric field is applied to a bulk material, a dipole
moment is also induced. The polarization, P, defined as the induced dipole
moment per unit volume, is a power series in the applied electric field E.
Thus, in the regime of nonlinear optics, higher order terms are needed to
describe the polarization field. By expanding in a Taylor series one
obtains:
P = e0 E + evoX E2 + E3 +
where F0 is the vacuum permittivity and and X (3) are the nonlinear second
and third order electric susceptibilities respectively.
The nonlinearity reaches a maximum just prior to ionization of the
molecule, when the external electric field equals the internal coulomb field of
the molecule - typically around 10 9 V/cm. Once a free electron is created, the
nonlinear effects are reduced until the electric field is increased to the extent
that the liberated electron gains sufficient energy to create secondary
ionization [17]. This gives rise to a wide range of new nonlinear effects and is
known as the 'strong field regime' in contrast to the 'perturbative regime'
below the ionization threshold. In the perturbative regime, much of the
interest lies in effects generated through ,%.2) and ,%3) whereas in the strong
field regime many higher orders of harmonics can be created [18].
23
The dielectric constant, c, is calculated using the relation
s = 1+ The dielectric constant combines the effects of the external field
together with the polarization field, and can be used to determine the
refractive index, n, of the material (n2 = deo). When the nonlinear terms are
included, they have important consequences for the propagation of light since
the intensity of light is dependent on the amplitude of the electric field. Thus
the velocity becomes intensity dependent and gives rise to new phenomena
that are discussed in what follows.
The second-order susceptibility (2) is the source of the second-order
nonlinearities, such as the second harmonic generation (SHG) i.e., frequency
doubling, up- and down-conversion of wavelengths, parametric amplification
and the Linear Electro-Optic effect (LEO). The third-order susceptibility
(,3)), in turn, is the source of third order effects, such as Third Harmonic
Generation (THG), electrochromism (EC), and Kerr effects that include
Stimulated Raman scattering (SRS), Stimulated Brillouin scattering (SBS),
SPM, XPM and FWM.
1.2.6 NONLINEAR EFFECTS IN FIBERS
As discussed, high intensity pulses propagating through fibers induce
many nonlinear effects. The fundamental nonlinear effect that arises in optical
fibers is due to the term (3) . The contribution due to is zero in the case of
24
fiber since Si02 molecules with which fibers are made are essentially Centro-
symmetric molecules. For instance, let us consider a light beam having a
power of 100mW propagating through an optical fiber having an effective
mode area of 50 n2 . The corresponding optical intensity is 2 x i09 W1The At
such high intensities, the nonlinear effects in optical fibers start influencing
the propagation of the light beam and can significantly affect the capacity of a
WDM OFC system [19]. The most important nonlinear effects that affect
OFC systems include SPM, XPM and FWM.
Besides the above mentioned nonlinear effects, there are two more
nonlinear phenomena namely SRS and SBS. Though several nonlinear effects
arise in optical fibers, in this chapter, we restrict to the discussion of only the
SPM, which is an important effect that helps generate optical solitons.
1.2.7 SELF PHASE MODULATION (SPM)
Since the lowest order nonlinearity present in an optical fiber is the
third order nonlinearity, the polarization produced consists of a linear and a
nonlinear term as follows:
P = E +80 ' (3)E3 . (1.11)
where x and X'3 represent the linear and third order susceptibility of the
medium (silica) and E represents the electric field of the propagating light
wave. If we assume the incident electric field to be given by
n=no +n2 I, (1.16)
(3)
where n2 - -- 4 ce0n
(1.17)
25
E=E0 Cos (tt—/3z). (1.12)
where fi is the propagation constant, then substituting in Eqn.( 1.11), we obtain
the following expression for the induced polarization at frequency co:
P=e0
x+x (3)EJEo cocost — flz). (1.13)
Now, the intensity of the propagating light wave is given by
I = c 60 no E. (1.14)
Substituting Eqn.(1.14) in Eqn.( 1.13) and using the fact that polarization and
the refractive index are related through the relation,
P=6 (n2 —1)E. (1.15)
we obtain the following expression for the refractive index of the medium in
the presence of nonlinearity:
Here we have assumed the second term in Eqn.(1.16) to be very small
in comparison to n0. Eqn.(1.16) gives the expression for the intensity
dependent refractive index of the medium due to the third order nonlinearity.
It is this intensity dependent refractive index that gives rise to SPM.
In the case of an optical fiber, the light wave propagates in the form of
a mode having a specific field distribution in the transverse plane of the fiber.
PRI
For example, the fundamental mode is approximately Gaussian in the
transverse distribution. Thus in optical fibers, it is more convenient to express
the propagation in terms of modal power rather than intensity, which is
dependent on the transverse coordinate. If Aeff is the effective cross sectional
area of the mode, then I = P/Aeff, where P is the power carried by the optical
beam.
If a represents the attenuation coefficient of the optical fiber, then the
power propagating through the fiber decreases exponentially as P(z) = P0 e
where P0 is the input power. In such a case, the phase shift suffered by an
optical beam in propagating through a length L of the optical fiber is given by
= SflNL dz= fl L + yPoL ff, (1.18)
- ()where Leff
ea L
- . (1.19)a
is called the effective length of the fiber. If aL >>1 then Leff 1/a and if
aL << 1 then Leff - L. The effective length gives the length of the optical fiber
wherein most of the nonlinear phase shift is accumulated. For SMF operating
at 1550nm, a 0.25 dB/km (= 5.8 x 10 m') and thus L ff - L for L << 17 km
and Leff 17 km for L>> 17 km. Since the propagation constant, /JNL, of the
mode depends on the power carried by the mode, the phase cJ of the emergent
wave depends on its power and hence this is referred to as SPM. Let us
27
consider a Gaussian input pulse at a center frequency of 'con' with an electric
field given by
E = E0 e t e'°"0 (1.20)
entering into an optical fiber of length L. In the presence of only SPM (i.e., no
dispersion), the output electric field distribution would be
E = Eoe_122e_12
(1.21)
For a pulse, P0 is a function of time and thus the phase of the output
pulse is no more a linear function of time. Thus the output pulse is chirped
and the instantaneous frequency of the output pulse is given by
d (coot _YPOLeff) coo _7Leff dP0_j_. (1.22)
Figure.( 1.3) shows the temporal variation of P0(t) and —dPçIdt for a
Gaussian pulse. The leading edge of the pulse corresponds to the left of the
peak of the pulse while the trailing edge corresponds to the right of the peak.
Thus in the presence of SPM, the leading edge gets downshifted in frequency
while the trailing edge gets up-shifted in frequency. The frequency at the
center of the pulse remains unchanged from W.
L
0.6
CA
0.4
0.2
U-4 -Z 0 2 4
-1 -2 0 2 4
0.
OA
0.2
0
-c .4
t
28
Figure 1.3Temporal variation of P0(t) and —dP0/dt for a Gaussian pulse.
Figure 1.4 Chirping due to self phase modulation.
29
Figure.(1.4) shows an input unchirped and the output chirped pulse
generated due to SPM. It should be noted that the SPM only broadens the
pulse in the frequency domain, not in the time domain. The chirping due to
nonlinearity without any corresponding increase in pulse width leads to
increased spectral broadening of the pulse. This spectral broadening coupled
with the dispersion in the fiber leads to modified dispersive propagation of the
pulse.
1.3 FORMATION OF SOLITONS IN FIBERS
In order to increase the amount of information carrying capacity of
optical fiber communication system, it is necessary to reduce the pulse width
as short as possible. As an optical pulse travels down a fiber, the longer
wavelength components of the light pulse tend to fall behind since the
wavelength range is in JR region. Thus, extending the trailing edge increases
the width of the pulse, so that the frequency at the leading edge is higher than
at the trailing edge. This is called optical (anomalous) dispersion and causes
conventional optical pulses to broaden. The effect of this is to limit the data
rates that can be achieved on monomode fibers. If too many short pulses are
injected into a fiber, they will overlap after propagating over some distance
and this is known as ISI [20]. As a result of which it is almost impossible to
distinguish between pulses - and the information will be lost or cross-talk will
take place in the case of co-propagating signals [11].
30
Such limitations can be overcome, if one manipulates the effects of
nonlinearity, occurring due to the intensity dependence of the refractive index.
When the intensity of the pulse is strong enough, the width of the pulse is
shortened, and the pulse becomes compressed, thereby making a counter-
effect to the broadening effect of dispersion. The result is a pulse that can
keep its shape for a long propagation distance. These steady pulses are called
optical solitons. The physical explanation of formation of such solitons in a
fiber is discussed as follows [2, 21].
When an optical pulse is transmitted in a fiber, it suffers from pulse
broadening due to dispersion. The optical pulse has a spectrum of Fourier
frequency components. As the index of refraction of any optical medium is a
function of frequency, various Fourier components of the pulse will
experience different indices of refraction in a dielectric medium like silica
fiber. As the refractive index is a measure of the velocity of the pulse
propagating in the dielectric medium, different Fourier components travel
with different velocities called group velocity. Because of this, the optical
pulse will spread in the time domain during the course of propagation. This is
called GVD or chromatic dispersion as shown in Fig.(1.5). This pulse
broadening is a major problem in fiber optic communication systems. A
broadened pulse has much lower peak intensity than the initial pulse launched
into the fiber, making it more difficult to detect. In the worst case, the
C,
E
e
31
broadening of two neighboring pulses may cause them to overlap, leading to
errors at the receiving end of the system.
Figure 1.5 Pulse broadening due to chromatic dispersion
As discussed, a material's refractive index is not only dependent on the
frequency of the light but also on the intensity of the light. This is due to the
fact that the induced electron cloud polarization in a material is not actually a
linear function of the light intensity. The degree of polarization increases
nonlinearly with light intensity so that the material exerts greater slowing
forces on more intense light. The result is that the refractive index of a
material increases with the increasing light intensity. Phenomenological
consequences of this intensity dependence of refractive index in fiber optics
are known as fiber nonlinearities.
There exist many different types of fiber nonlinearities as we
discussed. But, the one of most concern to soliton theory is SPM. With SPM,
32
the optical pulse exhibits a phase shift induced by the intensity-dependent
refractive index. The most intense regions of the pulse are slowed down the
most, so they exhibit the greatest phase shift. Since a phase shift changes the
distances between the peaks of an oscillating function, it also changes the
oscillation frequency along the horizontal axis. That is in any pulse
propagation there will be a generation of phase shift between different
frequency components. This phase shift depends on the refractive index of the
medium. As the refractive index of the medium depends on the intensity of
the pulse, which is a time varying quantity, the induced phase shift will also
vary with time. This can be considered as a generation of newer frequency
components in the front and back end of the optical pulse called chirping.
Thus the phase modulation to the pulse is due to its intensity itself, this effect
is called SPM. This can be considered as a spread in frequency domain. SPM
leads to chirping with lower frequencies on the leading side and higher
frequencies on the trailing side of the pulse as shown in Fig.(1.6). Like
dispersion, SPM may lead to errors at the receiving end of a fiber optic
communication system. This is particularly true for WDM system, where the
frequencies of individual signals need to stay within strict upper and lower
bounds to avoid encroaching on the other signals [11]. SPM leads to lower
frequencies at the leading side of the pulse and higher frequencies at the
trailing side of the pulse. Anomalous dispersion causes lower frequencies to
travel slower than higher frequencies.
33
Figure 1.6 Chirping of pulse due to self phase modulation
Therefore, anomalous dispersion causes the leading side of the pulse to
travel slower than the trailing side, effectively compresses the pulse and
undoing the frequency chirp induced by SPM. If the properties of the pulse
are just right when the instantaneous effects of SPM and anomalous
dispersion cancel each other out completely. Then the pulse remains
unchirped and retains its initial width along the entire length of the fiber and it
is clearly shown Fig.(1.7). In other words, a soliton is said to have been
formed. Fig.(1.8) shows an optical soliton with a hyperbolic-secant envelope
The credit of discovering, such optical solitons as a communicating medium,
goes to Hasegawa and Tappert [3,4]. The soliton pulse is a bell shaped
pulse. These solitons have other interesting properties. As described
earlier, they have a particle-like nature. Another prominent property
of solitons is that they have stable propagation characteristics. They are very
robust to perturbations in the transmission path and the perturbed
E
34
Figure 1.7Soliton pulse neither broadens nor in its spectrum.
Figure 1.8An optical soliton with a hyperbolic-secant envelope.
pulses will eventually evolve into stable solitons. The propagation of such
optical soliton in a fiber medium is governed by NLS. In the next section, we
derive the NLS, a master equation for information transfer in optical fiber.
1.3.1 SOLITON BASED OPTICAL FIBER COMMUNICATIONS
As mentioned already, to cope with the problem of linear dispersion
effect, fortunately there is a nonlinear, counter effect, which shortens the
35
width of the pulse. This effect is called SPM. It is well known that
nonlinearity in optics comes into play only when using high intense light
sources. Hence obviously, the study of nonlinear optics gained momentum
only after the invention of lasers in 1960. The nonlinearity in optics gives rise
to many new phenomena such as SHG, SPM, etc. SPM is the phenomenon
wherein the phase of the pulse gets modulated by its own intensity. This leads
to frequency chirping, ultimately expands the pulse in the frequency domain.
In a fiber, a clever configuration of both the linear (dispersion) and the
nonlinear effect (SPM) lead to the generation of a pulse that can maintain its
width and shape over a long propagation distance-provided the loss in the
system is taken care of The steadiness of these pulses is called optical fiber
solitons. Due to their short pulse duration and high stability, solitons could
form the backbone of the high speed communications of tomorrow's
information super highway.
1.4 THE NONLINEAR SCHRODINGER EQUATION: A MASTEREQUATION FOR INFORMATION TRANSFER IN OPTICALFIBERS
We now proceed to derive the equation that describes the evolution of
E' along the direction 'z' of the propagation of information. The most
convenient way to derive the envelop equation is to Taylor-expand the wave
36
number k(co,1E1 2 ) around the carrier frequency 'co.' and the electric field
intensity El [21],
kko=k(ooXowo)+°° - 2 (w a) + 21E12.2alEl
(1.23)
and to replace k - k0 with the operator i - and co - w0 with - i -, and toat
operate on the electric field envelope, q(z,t). The resulting equation reads
(1.24)
•I'8E +k
•aEk"a akiI — —j +az at ) 2 at alEl2
El 2 = 0. (1.24)
The refractive index n(k,w,1E12) for a plane electromagnetic wave in Kerr
media is given by
ck n2JEl2= n0 (CO) +CO 2
Thus k, k, in Eqn.(1.24) are given approximately by3IEI
khb0(00 k' - ——n2C ' C aU) aq 2 2c
(1.25)
(1.26)
We note that to obtain k' in this expression, we should go back to
k - = wn0(a))wn2JE
C c c 2(1.27)
37
and take the second derivative of 'k' with respect 'co '. It is often convenient
to study the evolution of 'E' in the co-ordinate moving at the group
velocity = t - k z. Then the envelope equation becomes,
.E k' 2E (OOn2IEI2E0
Z 2&2 2c(1.28)
Here, 'E' is the slowly varying amplitude of the pulse envelope and the
subscripts of 'z' and 't' denote partial differentiations of space and time
coordinates respectively. The above equation is the master equation that
describes information transfer in a fiber with group dispersion and
nonlinearity, first derived by Hasegawa and Tappert in 1973 [3, 4]. For a light
wave envelop in a fiber, the coefficients of this equation depends on the fiber
geometry and modal structure of the guided light wave. This equation is
found to have many applications not only in optics but also in field theories
and spin systems as well. In the above equation, the second term refers to
dispersion - a linear effect and the third term refers to Kerr effect - a
nonlinear effect. It plays the role of the attractive potential of the Schrodinger
equation, which leads to self-trapping of the pulses. In the anomalous
dispersion regime, the solitary wave solutions are commonly known as bright
solitons having sech profile. In the case of normal dispersion regime, the
solutions are called dark solitons. They appear as dips on a bright white
background and have a tanh profile. After the theoretical prediction of optical
solitons, it took seven years for the first experimental demonstration of
solitons by Mollenauer et al [5]. This was due to the lack of availability of
38
suitable sources. The invention of suitable lasers solved it later. After the
experimental confirmation of solitons, researchers started looking for
nonlinear systems which can allow solitons to propagate through it. Their
propagation is governed mainly by Nonlinear Partial Differential (NPDE)
Equations.
1.5 APPLICATIONS OF SOLITONS
Having discussed the formation of optical solitons and the NLSE, the
master equation governing the information transfer in fiber, we proceed to
discuss the applications of optical solitons. For brevity, we mention some of
the important applications.
The effects due to nonlinearity and dispersion are destructive in OFC
but useful in Optical Soliton Fiber Communication (OSFC) systems. The
soliton type pulses are highly stable. Their transmission rate is more than 100
times better than that in the best linear system. They are not affected by the
imperfections in the fiber geometry or structure. Soliton can be propagated
without any distortion if the nonlinear characteristics like amplitude, intensity
of the pulse-depending on velocity and the dispersion characteristics like
frequency-depending on velocity of the media, are balanced.
Soliton can also be multiplexed at several wavelengths without
interaction between the channels, though they usually suffered in Non Return
to Zero (NRZ) systems. Nowadays, most of the communication systems use
RZ format, for example Transoceanic Transmission (TOT) where the
transmission rate is 10 G b/s per channel, transmits the information transfer
in dispersion managed fibers. This format is the only stable form for pulse
propagation through the fiber in the presence of fiber nonlinearity and
dispersion in all optical transmission lines with minimum loss. In dispersion
managed fibers, a large pulse width is allowed, pulse height is reduced and
nonlinear interactions between adjacent pulses as well as among different
wavelength channels are reduced [22, 23]. Not only in he field of
communication, solitons also find application in the construction of optical
switches [24], soliton laser [25], pulse compression [26] and the like.
1.5.1 ADVANTAGES OF SOLITON BASED COMMUNICATIONSYSTEMS
In the previous section, some of the major applications of optical
solitons are mentioned. In this section, the advantages of soliton based
communication systems which are expected to be the preferred choice for
future communication systems, are presented [2, 21, 22, 23].
. Solitons are unaffected by an effect called PMD due to the imperfection in
the circular symmetry fiber which leads to a small and variable difference
40
between the propagation constants of orthogonal polarized modes. This
dispersion becomes a major problem over long distances and at high data
rates.
• Solitons are well matched with all optical processing techniques. Our long
term goal is to create networks in which all of the key high-speed
functions, including routing, demultiplexing and switching are performed
in the optical domain. So the signals need not be converted into an
electrical form on the way. Most of the devices and techniques designed
for these tasks work only with well-separated optical pulses, which are
particularly effective with solitons.
• If the solitons are controlled properly they can be more robust than NRZ
pulses. Schemes have been devised that can not only provide control over
the temporal positions of the solitons, but also remove noise added by
amplifiers. Such schemes would allow the separations between amplifiers
to be many times greater than in the schemes that are used with NRZ
pulses.
The particle nature of solitons can be employed for sliding-frequency
guiding optical filters [27] along the link. With these centered at slightly
reducing wavelengths along the path, the soliton is capable of following
this change without any degradation.
• The use of in-line saturable absorbers, which work in the time domain to
suppress noise.
41
The particle feature of solitons is also very useful to perform various all
optical functions such as switching.
. Yet another and very important particle feature is the fact that solitons
tend to stay together in presence of a walk-off between different
polarization components - so called PMD [28]. The soliton PMD
robustness may be a key to success when upgrading existing fiber links to
high speed.
Solitons would replace the traditional NRZ with RZ modulations,
which are used in almost all commercial terrestrial WDM systems. Typically
the design of a conventional WDM system involves an effort to increase the
power as much as possible to counteract attenuation and noise without
introducing too much nonlinearity. Thus NRZ and RZ systems are often
called linear system. Recent advancements in soliton communication with
3.2T b/s have been demonstrated
42
1.6 REFERENCE
1. Mynbaev.D.K. and Schiener.L.L. (2001), 'Fiber communicationstechnology', Pearson Education Asia.
2. Agrawal.G.P. (2001a), 'Applications of Nonlinear Fiber Optics', SecondEdition, Academic Press, New York.
3. Hasegawa.A. and Tappert.F. (1973a), 'Transmission of stationarynonlinear optical pulses in dispersive dielectric fibers. I. Anomalousdispersion', Appl. Phys. Lett., Vol. 23, pp. 142 - 144.
4. Hasegawa.A. and Tappert.F. (1973b), 'Transmission of stationarynonlinear optical pulses in dispersive dielectric fibers. II. Normaldispersion', Appi. Phys. Lett., Vol. 23, pp. 171 - 172.
5. Mollenaur.L.F, Stolen.R.H. and Gordon.G.P. (1980), 'Experimentalobservation of picosecond pulse narrowing and soliton in optical fibers',Phys. Rev. Lett., Vol. 45(13), pp. 1095 - 1098.
6. Morita.I, Tanaka.K, Edagawa. and Suzuki.M (1998), '1998 Europeanconference on optical communication', Vol. 3, pp. 47 - 52, Madrid Spain.
7. Sugahara.H, Fukuchi.K, Tanaka. A, Inada.Y. and Ono.T (2002), 'Opticalfiber conference post deadline paper', FC 6-1, Anaheim CA.
8. Iannone.E, Matera.F. and Settembre.M. (1998), 'Nonlinear opticalcommunication networks', John Wiley, New York.
9. Kazovsky.L, Benedetto.S. and Willner.A. (1996), 'Optical fibercommunication systems', Artech House, Boston.
10.Ramaswamy.R. and Sivarajan.K.N. (2001), 'Optical networks:A practical perspective', Morgan Kaufmann Harcourt Asia, Singapore.
11.Ghatak.A. and Thyagarajan.K. (1998), 'Introduction to Fiber Optics',Cambridge University Press.
12.Thyagarajan.K. (2003), 'Linear and nonlinear propagation effects inoptical fibers', in Recent advances in optical solitons: Theory andExperiments', Vol. 613, pp. 34 - 70, edited by, Porsezian K. andKuriakose V. C., Springer-verlag, Lecture Notes in Physics.
13.Yasin Akhtar Raja.M. and Sameer K.Arabasi. (2003), 'Design andsimulations of a dynamic polarization-mode dispersion compensator forlong-haul optical networks', Opt. Exp., Vol. 11, pp. 1166 - 1174.
43
14.Shen.Y.R. (2003), 'The Principles of nonlinear optics', Wiley, New York.
15.Brabec.T. and Krausz F. (2000), 'Intense few-cycle laser fields: Frontiersof nonlinear optics', Rev. Mod. Phys., Vol. 72, pp. 545 - 591.
16.Bloembergen.N. (1965), 'Nonlinear optics', Benjamin, New York.
17.Corkum.P.B. (1994), 'Plasma perspective on strong field multiphotonionization', Phys. Rev. Lett., Vol. 71, pp. 1994 - 1997.
18.Chang.Z, Rundquist.A, Wang.H, Murnane.M.M. and Kapteyn.H.C.(1997), 'Generation of coherent soft X rays at 23 nm using highharmonics', Phys. Rev. Lett., Vol. 79, pp. 2967 - 2970.
19.Chraplyvy.A.R. (1990), 'Limitations on lightwave communicationsimposed by optical fiber nonlinearities', J. Lightwave Tech., Vol. 8, pp.1548 - 1557.
20. Senior.J.M. (1999), 'Optical fiber communications: Principles andpractice', Prentice-Hall of India, Pvt. Ltd., New Delhi.
21.Hasegawa.A. and Matsumoto.M. (2002), 'Optical solitons in fibers',Springer-verlag.
22.Hasegawa.A. and Kodama.Y. (1995), 'Solitons in opticalcommunications', Oxford University Press, New York.
23.Boyd.RW. (2003), 'Nonlinear optics', Academic Press, UK.
24.Enns.R.H, Edmundson.D.E. and Rangnekar.S.S. (1987), 'Bistable solitonsand optical switching', IEEE J. Quant. Electron., Vol. 23, pp. 1199 -1204.
25.Mollenauer.L F. and Stolen.R.H. (1984), 'The soliton laser', Opt. Lett.,Vol. 9, pp. 13 - 15.
26. Sarukura.N, Ishida.Y. and Nakano.H. (1991), 'Generation of 50 fseçpulses from a pulse-compressed, cw passively mode-locked Ti: sapphirelaser', Opt. Lett., Vol. 16, pp. 153 - 155.
27.Mollenauer L. F, Lichtman E, Neubelt M. J. and Harvey G.T. (1993),'Demonstration, using sliding-frequency guiding filter, of error-freesoliton transmission over more than 20 Mm at 10 Gb/s, single channel,and over more than 13 Mm at 20 Gb/s, in a two-channel WDM', Electron.Lett., Vol. 29, pp. 910 - 912.
44
28.Mollenauer.L.F, Smith K, Gordon.J.P. and Menyuk.C.R. (1989),'Resistance of solitons to the effects of polarization mode dispersion inoptical fibers', Opt. Left., Vol. 14, pp. 1219 - 1221.