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

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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

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