GENERAL ARTICLE Elements of Optical Solitons: An...

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643 RESONANCE July 2010 GENERAL ARTICLE Elements of Optical Solitons: An Overview V C Kuriakose and K Porsezian K. V. Rashmi is currently an M. E. student at the Indian Institute of Science. She received her B. Tech. degree from the National Institute of Technology Karnataka, Surathkal. Her current research interests are in coding theory, error-correction in networks and wireless communication.“ Keywords Optical solitons, nonlinear opti- cal fibers. (left) V C Kuriakose retired as professor from the Department of Physics, Cochin University of Science and Technology, Kochi and currently is CSIR Emeritus Scientist. His areas of research interest include nonlinear dynamics, gravitation and cosmology. (right) K Porsezian is currently working as Professor in the Depart- ment of Physics, Pondicherry University (a Central University), Puducherry. His areas of research interest include solitons and modulational instability in nonlinear fiber optics and Bose–Einstein condensation. Existence of solitons in nonlinear optical ¯bres was ¯rst predicted by Hasegawa and T appert in 1973 and its experimental veri¯cation came in 1980. Ever since, both experimental and the- oretical physicists have shown keen interest in this topic due to its versatile applications. Op- tical solitons are formed due to the balance be- t ween the group velocity dispersion and the self phase modulation in the anomalous dispersion regime, and the governing wa ve equation is the Nonlinear SchrÄ odinger (NLS) equation. Optical solitons can exist in v arious systems like photonic crystal ¯bres, bulk materials like photorefractive materials, photopolymers, etc. What follows is an introduction to optical soiltons and their ap- plications. 1. Introduction The word soliton was born in science vocabulary in 1964 mainly through the work of M Krusk al and N Zabusky and it refers to highly stable localized solutions of cer- tain nonlinear partial di®eren tial equations describing physical phenomena. The soliton concept has deeply penetrated into almost all branches of science wherever scienti¯c phenomena are described by nonlinear partial di®erential equations. This concept was ¯rst discov- ered in hydrodynamics in the 19th century and entered into other branches in physics. Scottish Nav al Architect John Scott Russell is the father of this novel idea. In 1834 when the boat in which he was traveling across the Union Canal connecting Glasgow and Edinburgh was suddenly stopped, he observed a lump of water leaving the barge of the boat and moving without changing its

Transcript of GENERAL ARTICLE Elements of Optical Solitons: An...

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643RESONANCE July 2010

GENERAL ARTICLE

Elements of Optical Solitons: An Overview

V C Kuriakose and K Porsezian

K. V. Rashmi is currently an M. E. student at the Indian Institute of Science. She received her B. Tech.

degree from the National Institute of Technology Karnataka, Surathkal. Her current research

interests are in coding theory, error-correction in networks and wireless communication.“

Keywords

Optical solitons, nonlinear opti-

cal fibers.

(left) V C Kuriakose

retired as professor from

the Department of Physics,

Cochin University of

Science and Technology,

Kochi and currently is

CSIR Emeritus Scientist.

His areas of research

interest include nonlinear

dynamics, gravitation and

cosmology.

(right) K Porsezian is

currently working as

Professor in the Depart-

ment of Physics,

Pondicherry University

(a Central University),

Puducherry. His areas of

research interest include

solitons and modulational

instability in nonlinear

fiber optics and

Bose–Einstein

condensation.

Existence of solitons in nonlinear optical ¯breswas ¯rst predicted by Hasegawa and Tappert in1973 and its experimental veri¯cation came in1980. Ever since, both experimental and the-oretical physicists have shown keen interest inthis topic due to its versatile applications. Op-tical solitons are formed due to the balance be-tween the group velocity dispersion and the selfphase modulation in the anomalous dispersionregime, and the governing wave equation is theNonlinear SchrÄodinger (NLS) equation. Opticalsolitons can exist in various systems like photoniccrystal ¯bres, bulk materials like photorefractivematerials, photopolymers, etc. What follows isan introduction to optical soiltons and their ap-plications.

1. Introduction

The word soliton was born in science vocabulary in 1964mainly through the work of M Kruskal and N Zabuskyand it refers to highly stable localized solutions of cer-tain nonlinear partial di®erential equations describingphysical phenomena. The soliton concept has deeplypenetrated into almost all branches of science whereverscienti¯c phenomena are described by nonlinear partialdi®erential equations. This concept was ¯rst discov-ered in hydrodynamics in the 19th century and enteredinto other branches in physics. Scottish Naval ArchitectJohn Scott Russell is the father of this novel idea. In1834 when the boat in which he was traveling across theUnion Canal connecting Glasgow and Edinburgh wassuddenly stopped, he observed a lump of water leavingthe barge of the boat and moving without changing its

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GENERAL ARTICLE

Figure 1. Solitary wave.

Inte

nsi

ty

Time t

Soliton in optics refers

to a situation where

light beam or pulse

propagates through

a nonlinear optical

medium without any

change in its shape

andvelocity.

shape and speed. He could see this lump moving with-out any change over a distance of 2 miles. He called sucha wave as solitary wave (Figure 1). Later on, this excit-ing phenomenon was observed in di®erent branches ofscience through both analytical and experimental meth-ods. Soliton in optics refers to a situation where lightbeam or pulse propagates through a nonlinear opticalmedium without any change in its shape and velocity.In this article, we will see that this happens because thetendency for the pulse to spread, caused by dispersionin the medium, is compensated by the nonlinear depen-dence of the refractive index on the amplitude, whichcauses the pulse to compress. Solitons in nonlinear op-tical ¯bers have important applications in communica-tion, optical computing, optical switching, etc.

2. Group Velocity Dispersion

Optical ¯ber communication has great advantages overthe conventional cable communication system. The mostimportant advantage is the availability of enormous com-munication bandwidth which is a measure of the infor-mation carrying capacity. Some of the other advantagesare: i) less weight, ii) no hazards of short circuits, iii)low cost and iv) immunity to adverse temperature andmoisture conditions. But there are limitations to the dis-tance over which the information can be conveyed with-out any distortion. The inevitable distortion is due tothe optical phenomenon called dispersion which causesan optical signal to get dispersed as it passes throughthe ¯ber. This actually limits the transmission capacityof the system. There are mainly two types of dispersion,one due to intramodal dispersion and the other due tointermodal delay e®ects. These dispersions cause thespreading of the pulse passing through a ¯ber and thee®ects can be quanti¯ed by considering group velocitiesof the guided modes. Group velocity gives the speedat which energy in a particular mode travels along the¯ber. The intramodal dispersion can be due to material

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GENERAL ARTICLE

645RESONANCE July 2010

An immediate

effect of dispersion

is that as the light

pulse travels

through the fiber,

its shape gets

broadened.

and waveguide dispersions.

Material dispersion, also known as chromatic disper-sion, arises due to the variation of refractive index ofthe core of the ¯ber as a function of wavelength. How-ever, waveguide dispersion which occurs in single mode¯bers is due to the fact that only 80% of the opticalpower is con¯ned to the core while 20% of the opticalpower is con¯ned to the cladding, resulting in distortionas the velocities of the pulse in the cladding and thatin the core have di®erent values. Distortion due to in-tramodal delay occurs in multimode ¯bers as each modehas di®erent values for the group velocities. The full ef-fects of these three distortions are seldom observed inpractice because of other factors such as nonideal indexpro¯les, optical power-launching conditions, nonuniformmode attenuation, etc. An immediate e®ect of disper-sion is that as the light pulse travels through the ¯ber,its shape gets broadened. The group velocity, the veloc-ity at which the energy in a pulse travels in a ¯ber, isgiven by

Vg =d!

dk; (1)

which can also be called as the velocity of the envelopewave. In a nondispersive medium, the component wavesand their superposition (the envelope) will travel withthe same velocity which implies that the shape of thepulse (envelope) remains unchanged as the pulse prop-agates through the medium. But this is a rare situa-tion and in an optical medium the velocity depends onthe refractive index which in turn depends on the fre-quency (wavelength) of the wave. The velocity (phasevelocity) of a wave is !=k = v = c=n; where c is thevelocity of the wave in vacuum and n is the refractiveindex of the medium. As the signal propagates alongthe ¯ber, each spectral component can be assumed totravel independently and to undergo a time delay perunit wavelength. We de¯ne the propagation time over

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In the normal

dispersion regime

the high frequency

components (blue

shifted) of an optical

pulse travel slower

than the low

frequency

components (red

shifted). The

opposite occurs in

the anomalous

regime.

a distance L for a given group velocity Vg as ¿ = L=Vg:If the spectral width of the pulse is not wide, it may beapproximated that the delay di®erence per unit wave-length along the propagation path is given by d¿=d¸: Ifthe wavelengths of spectral components are spread overa wavelength range ±¸, the total delay di®erence ±¿ overa distance L is given by

±¿ =d¿

d¸±¸ = ¡

L

d2n

d¸2±¸: (2)

Thus the spread in arrival times depends on (d2n=d¸2):This means that the pulse gets spread out as it movesalong the ¯ber, because di®erent component waves whichconstitute the pulse have di®erent phase velocities. Thisphenomenon is known as Group Velocity Dispersion(GVD). We introduce a quantity called mode propaga-tion constant ¯ = n(!)!=c and give a Taylor series ex-pansion:

¯(!) = ¯0 + ¯1(! ¡ !0) +1

2¯2(! ¡ !0)

2 + :::; (3)

where ¯m = (dm¯=d!m);m = 1; 2; 3;... . A little bit ofalgebra will show that

¯2 =d¯1

d!=

¸3

2¼c20

d2n

d¸2: (4)

Here ¯2 is known as GVD parameter. Another quantityknown as total dispersion parameter widely used in theliterature is de¯ned as,

D¸ =d¯1

d¸= ¡

2¼c

¸2¯2: (5)

From Figure 2, it can be seen that D¸ increases as ¸increases and vanishes at a wavelength around 1310 nm.This wavelength is called zero dispersion wavelength ¸D.For ¸ < ¸D; ¯2 > 0, and the ¯ber is said to exhibit nor-mal dispersion, while for ¸ > ¸D; ¯2 < 0, and the ¯ber

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Figure 2. Wavelength de-

pendence of the dispersion

parameter D for standard

fiber.

is said to exhibit anomalous dispersion. In the normaldispersion regime the high frequency components (blueshifted) of an optical pulse travel slower than the low fre-quency components (red shifted). The opposite occursin the anomalous regime. The anomalous regime playsan important role in nonlinear ¯ber optics, in particu-lar, for the formation of highly stable and distortionlesssolitons.

3. Fiber Nonlinearities

The other factor which puts a limit on the performanceof optical ¯bers regarding the length over which an opti-cal pulse could be transmitted, arises from the nonlinearprocesses taking place in the ¯ber. The major nonlinearprocesses to be considered are: i) Self Phase Modulation(SPM), ii) Stimulated Raman scattering (SRS) and iii)Stimulated Brillouin Scattering (SBS). SPM gives riseto spectral compression, an e®ect opposite to that dueto GVD and limits the maximum achievable informa-tion transfer rate. SRS limits the maximum operationalpower levels or gives rise to information loss and intro-duces noise into the system. In SBS, acoustic phononsrather than high frequency optical phonons give rise tofrequency-shifted scattering radiation.

Many of these nonlinear processes which act as limitingfactors in communication systems can be used for otherpotential applications. SPM can be used for pulse com-pression while SRS provides sources of new and tunableradiations. Now, let us brie°y discuss the above e®ects.

Many of these

nonlinear processes

which act as limiting

factors in

communication

systems can be

used for other

potential applications.

SPM can be used

for pulse

compression while

SRS provides

sources of new and

tunable radiations.

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The polarizability of

a dielectric medium in

a normal situation is

proportional to the

electric field

strength, but if the

strength of the light

beam is sufficiently

high, which is

possible if we use a

laser beam, it is

found that the

above relationship is

no longer valid and

the polarizability of

the medium depends

on higher powers of

the electric field

strength and the

medium is said to

be nonlinear.

The lowest order

nonlinear effects in

optical fibers

originate from the

third order

susceptibility.

The invention of lasers in the 1960s led to a new branchin optics called nonlinear optics. The polarizability of adielectric medium in a normal situation is proportionalto the electric ¯eld strength, but if the strength of thelight beam is su±ciently high, which is possible if weuse a laser beam, it is found that the above relationshipis no longer valid and the polarizability of the mediumdepends on higher powers of the electric ¯eld strengthand the medium is said to be nonlinear. Thus, we canwrite the polarizability of a nonlinear medium as:

P = ²0(Â(1)E + Â(2)E2 + Â(3)E3 + ::::::::); (6)

where ²0 is the vacuum permittivity and Â(j)(j = 1; 2; 3;: : : ) is the jth order susceptibility tensor. The linearsusceptibility Â(1) contains the dominant contributionto P . Its e®ects are included through the refractive in-dex n. The second order susceptibility Â(2) is responsiblefor such nonlinear e®ects as second harmonic generationand sum and di®erence frequency generation. However,it is nonzero only for media which lack an inversion sym-metry at the molecular level. As SiO2 is a symmetricmolecule, Â(2) or in general, even powered terms in E in(6) vanish for silica glasses. As a result, optical ¯bersdo not normally exhibit second order nonlinear e®ects.The lowest order nonlinear e®ects in optical ¯bers orig-inate from the third order susceptibility Â(3), which isresponsible for the phenomena such as third harmonicgeneration, four-wave mixing and nonlinear refraction.The refractive index and the susceptibility for a nonlin-ear medium are related through the relation

n = (1 + Â(1) + Â(2)E + Â(3)E2 + ::::::::)12 : (7)

For an optical ¯ber, this relation takes the form

n = (1 + Â(1) + Â(3)E2)12 = n0 +

1

2Â(3)E2 = n0 + n2I:

(8)

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The refractive

index of the

medium depends

on the intensity of

light propagating

through it.

The most interesting

application of

intensity dependent

refractive index is

SPM which refers to

a situation where an

optical pulse passing

through an optical

fiber experiences an

additional phase shift.

This equation says that the refractive index of themedium depends on the intensity of light propagatingthrough it. Light is in°uencing its own velocity as ittravels { light can interact with light in a nonlinearmedium. For fused silica, n0 » 1:46 and n2 » 3:2 £10¡20 m2W¡1. If we consider the propagation of a modecarrying 100 mW of power in a single mode ¯ber witha core diameter of 5¹m, the change in refractive indexdue to nonlinear e®ects is ¢n = n2I = 1:6£10¡10. Thisshows that we require pico or femto second lasers toexcite nonlinearities in optical ¯bers. The intensity de-pendence of refractive index has a number of interestingapplications like self phase modulation (SPM) and crossphase modulation (XPM).

Self Phase Modulation: The most interesting appli-cation of intensity dependent refractive index is SPMwhich refers to a situation where an optical pulse pass-ing through an optical ¯ber experiences an additionalphase shift. Its magnitude can be obtained by notingthat when the optical pulse travels over a distance Lthrough an optical ¯ber, its shift in the phase is givenby

' =2¼

¸nL =

¸(n0 + n2I )L: (9)

The intensity dependent nonlinear shift is (2¼=¸) n2ILand is due to SPM. I is time dependent and hence 'is also time dependent. Therefore, an additional fre-quency term ! = d'=dt is introduced into the dynam-ics of the optical pulse resulting in a change in the fre-quency spectrum and this change in frequency is knownas SPM. This will result in the shape of the pulse, pulsebroadening or pulse compression can be achieved un-der appropriate conditions. If the original frequency ofthe pulse is !0, as a result of SPM, the instantaneousfrequency of the pulse becomes !0 = !0 + (d'=dt) =!0¡(2¼=¸) zn2 (dI=dt); where we have assumed that thepulse is propagating in the positive z-direction through a

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GENERAL ARTICLE

Figure 3.(left) Self-phase

modulation.

Figure 4. (right) Linear and

nonlinear effects on

Gaussian pulses.

The optical pulse

can be sent through

a nonlinear optical

medium without any

distortion; the

optical pulse is

then said to be an

optical soliton.

distance z. At the leading edge of the pulse (dI=dt) > 0while at the trailing edge of the pulse (dI=dt) < 0.Hence the pulse is said to be chirped, i.e., the frequencyvaries across the pulse (Figure 3).

We have now two independent phenomena occurring innonlinear optical ¯bers and these two phenomena inde-pendently distort the shape of the optical pulse as itpasses through the ¯ber. But it is found that un-der appropriate conditions these two phenomena can bemade competing such that the e®ect due to one can becanceled by the e®ect due to the other. This will re-sult in a situation where the optical pulse can be sentthrough a nonlinear optical medium without any dis-tortion; the optical pulse is then said to be an opticalsoliton (Figure 4).

4. Wave Propagation Through an Optical Fiber

The wave equation that can describe propagation of op-tical pulses along an optical ¯ber can be derived start-ing from Maxwell's equations. The study of nonlineare®ects in optical ¯bers involves the use of short pulseswith widths ranging from » 10 ns to 10 fs. When suchpulses propagate through the ¯ber both dispersive and

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The NLS equation

is a fundamental

equation in soliton

theory and appears

in many branches of

science.

nonlinear e®ects in°uence the shape and spectrum. IfA(z; t) is the slowly-varying amplitude of the pulse en-velope, then it can be seen that the di®erential equationwhich describes the wave propagation through the opti-cal ¯ber can be cast in the form:

i@A

@z¡¯2

2

@2A

@T 2+ ° jAj2A = 0: (10)

This equation represents an optical pulse passing in thez-direction through an optical ¯ber exhibiting GVD andSPM, while other e®ects like absorption, higher ordernonlinearities, higher order dispersion, SRS, SBS havenot been taken into consideration. If these e®ects weretaken into account, the equation would not have been assimple as given by (10). Equation (10) is referred to asNonlinear SchrÄodinger (NLS) equation because it resem-bles the SchrÄodinger equation with a nonlinear potentialterm (variable z playing the role of time and vice-versa).In (10) the second term represents dispersion while thethird term accounts for the nonlinearity of the ¯ber. ¯2

is the GVD parameter while the parameter ° is a mea-sure of the third order nonlinearity of the medium. TheNLS equation is a fundamental equation in soliton the-ory and appears in many branches of science. There ex-ist di®erent analytical methods to solve nonlinear partialdi®erential equations. Inverse scattering method is oneamong them. Only certain nonlinear partial di®erentialequations can be solved by inverse scattering methodand NLS equation belongs to this special category. Za-kharov and Shabat in 1971 solved NLS equation usinginverse scattering method. Introducing dimensionlessvariables, U = (A=

pP0); » = (z=LD); ¿ = (T=T0), (10)

can be written as,

i@U

@»= ¯2

1

2

@2U

@¿ 2¡N2 jU j2 U; (11)

where P0 is the peak power, T0 is the width of theincident pulse and the parameter N is introduced as

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GENERAL ARTICLE

Figure 5. Bright soliton.

If a hyperbolic-secant

pulse whose width T0

and the peak power

P0

is launched into

an ideal lossless

optical fiber, the

pulse will propagate

undistorted without

any change in shape

for arbitrarily long

distances.

N2 = (LD=LNL) = (°P0T0=j¯2j). For anomalous disper-sion regime ¯2 < 0, while for normal dispersion regime¯2 > 0. The dispersion length LD and the nonlinearlength LNL are de¯ned as LD = (T 2

0 =j¯2j); and LNL =(1=°P0):

For the anomalous dispersion regime, (11) now takes theform, putting u = NU ,

i@u

@»+

1

2

@2u

@¿2+ juj2 u = 0: (12)

The solution of this equation is given by u(»; ¿) = sech(¿ )exp(i»=2), it is called fundamental soliton solution of(12) and has a bell-shaped form (Figure 5). These typesof solitons are called bright solitons. In the context ofoptical ¯bers, this solution indicates that if a hyperbolic-secant pulse whose width T0 and the peak power P0 islaunched into an ideal lossless optical ¯ber, the pulse willpropagate undistorted without any change in shape forarbitrarily long distances. It is this fundamental prop-erty of solitons that makes them attractive for opticalcommunication systems. This remarkable idea was ¯rstproposed in 1973 by Hasegawa and Tappert. However,the non-availability of suitable picosecond lasers andlow-loss ¯ber delayed experimental con¯rmation of thisresult. The peak power to support the fundamental soli-ton inside an optical ¯ber is given by P0 = (j¯2j=°T

20 ) ¼

(3:11 j¯2j=°T2FWHM); where the FWHM of the soliton is:

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GENERAL ARTICLE

Figure 6. Dark soliton.

Mollenauer and his

group in 1980

first observed

solitons in optical

fibers.

TFWHM ¼ 1:76T0: Using typical parameter values: ¯2 =¡1ps2/km, ° = 3 W¡1/km for dispersion-shifted ¯bersnear the 1:55¹m wavelength, P0 is » 1W for T0 = 1 psbut reduces to 10 mW when T0 = 10 ps. This showsthat optical solitons can form in optical ¯bers at powerlevels available from semiconductor lasers.

Mollenauer and his group in 1980 ¯rst observed solitonsin optical ¯bers. They used a mode-locked color-centrelaser capable of emitting short pulses (TFWHM » 7 ps)near 1:55 ¹m. The pulses were propagated through anoptical ¯ber of length 700 m, the core diameter of the¯ber being 9:3 ¹m. If we consider high dispersion, higherorder nonlinearities, SRS, SBS, ¯ber loss, etc., the waveequation takes a complicated form and to analyze theresulting equation through inverse scattering techniquesmay not be su±cient and we have to use other analyticalmethods.

If we consider normal dispersion regime instead of anom-alous regime, in place of (12) we have,

i@u

@»¡

1

2

@2u

@¿ 2+ juj2u = 0: (13)

The solution of this equation is given by u(»; ¿) = tanh(¿ )exp(i»). The intensity pro¯les associated with such so-lutions show a dip in a uniform background and thusthe solutions are called dark solitons (Figure 6).

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In 1988, Mollenauer

and his group

succeeded in

propagating a

soliton over 6000

km without any

repeaters.

The experimental veri¯cation of soliton phenomena inoptical ¯bers by Mollenauer et al in 1980 and the ex-perimental demonstration of soliton reshaping by op-tical ampli¯cation and the long distance transmissionby means of Raman ampli¯cation in 1985 by the Mol-lenauer group have led people to believe that solitonscould be used for ultra high speed transmission pur-poses. In 1988, Mollenauer and his group succeededin propagating a soliton over 6000 km without any re-peaters. Later on, solitons have been propagated with-out deterioration over many thousand of kilometers.

5. Other Types of Optical Solitons

Dispersion-Managed Solitons: As we have discussedearlier, Group Velocity Dispersion (GVD) in a nonlin-ear optical ¯ber plays a very crucial role in the shapingof the pulse propagating through the ¯ber. When theNLSE and its solutions are studied, it is assumed thatthe GVD parameter is constant throughout the lengthof the ¯ber, but it is not so. To meet this challenge,a technique called dispersion management is often usedin the design of modern ¯ber optic communication sys-tems. This technique consists of connecting ¯bers hav-ing various values of dispersion parameters such thatthe average value of GVD in each period is quite low.The ensuing nonlinear stationary pulse generated by thebalance between e®ective dispersion and the averagednonlinearity is termed as dispersion-managed soliton.In addition to this, suitable theoretical equations havebeen developed to study the dynamics of optical solitonspropagating through dense wavelength-division multi-plexed (WDM) systems with dispersion-management,damping and ampli¯cation. Recent studies provide anin-depth understanding of the soliton propagation invarious types of nonlinear optical ¯bers such as polar-ization preserving ¯bers, birefringent ¯bers, dispersion-°attened ¯bers, etc. Very recently, experimental obser-vation of the soliton dispersion management in a ¯ber

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Figure 7. Schematic dia-

gram of photonic crystal

fiber.

The arrangement of

air holes in the

cladding region

gained more

importance due to

optical properties of

PCFs such as highly

nonlinear, highly

birefringent, large

mode field area, high

numerical aperture,

ultra flattened

dispersion,

adjustable zero

dispersion, etc.

with sine-wave variation of the core diameter along thelongitudinal direction of propagation of the dispersionoscillating ¯ber, has been reported. Research studieson soliton dispersion management have been initiatedby Hasegawa and his group and ever since various re-search groups have pursued studies showing rich varietyof properties.

Solitons in Photonic Crystal Fiber: In recent years,photonic crystal ¯bers (PCFs) (Figure 7) have received agreat deal of scienti¯c attention because of their numer-ous and invaluable nonlinear applications in sensor andcommunication ¯elds. The arrangement of air holes inthe cladding region gained more importance due to op-tical properties of PCFs such as highly nonlinear, highlybirefringent, large mode ¯eld area, high numerical aper-ture, ultra °attened dispersion, adjustable zero disper-sion, etc. The microstructure of the cladding can bevaried in order to tailor the e®ective refractive indexand so the propagation properties of PCF. In PCF, therefractive index di®erence between core and cladding ismuch higher than in conventional ¯bers, which can beobtained by changing the size of the air hole. Thissigni¯cant variation of air hole diameter, pitch anddesign of the PCF have several applications like wave-length conversion using four wave mixing, optimizationof the pump spectra to achieve °at-Raman gain, min-imization of noise ¯gure of the PCF ampli¯er, Ramanlasing characteristics, narrow or broad band pass ¯lters,etc. The recent widespread research on PCF was mainlymotivated by their nonlinear optical applications suchas supercontinuum generation, pulse compression, op-tical switching, ¯ber laser, parametric ampli¯er, soliton

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Figure 8. Soliton pulse

propagation through pho-

tonic crystal fiber for differ-

ent design parameters.

Much of the recent

work has been

motivatedby

experiments in PCFs,

where the

combination of

engineered

dispersion-induced

frequency chirp and

elevatednonlinearity-

induced frequency

has led to the soliton

pulse propagation.

propagation, modulational instability, etc. Among theseapplications, soliton propagation in PCF with di®erentstructure is attracting many researchers, and a greatdeal of numerical simulation work has already been donein this emerging technological area. Much of the re-cent work has been motivated by experiments in PCFs,where the combination of engineered dispersion-inducedfrequency chirp and elevated nonlinearity-induced fre-quency has led to the soliton pulse propagation, thathave found many important applications. For instance,the soliton with variation of both dispersion and non-linear parameters has been investigated by employingthe exact solution of nonlinear SchrÄodinger equation.Researchers have studied elaborately the nontopologi-cal soliton wave solution through gas-¯lled PCF withnewly designed structures. Recent literature also pro-vide supercontinuum generation and pulse compressionup to 6 fs using optical soliton in PCF. Besides, Raman-induced soliton pulse propagation has been predictedin PCF combining with higher order nonlinear e®ects.On employing femtosecond solitons in PCF, energy ex-change between the solitons and polarization instabilityhave been demonstrated. In addition to this, pioneeringexperiments on bound soliton pairs in a highly nonlin-ear PCF have been successfully conducted. Apart fromthe soliton in PCF, the stability of the pulse due to thein°uence of frequency shift has been analyzed in detail(Figure 8).

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An important

application of

nonlinear fiber

optics consists of

compressing

optical pulses.

6. Applications of Optical Solitons

Soliton Ampli¯cation: In real situations, one has toconsider the loss of energy as the pulse propa-gates through a ¯ber due to the absorption of energyby the ¯ber and this e®ect is usually termed as ¯berloss. Fiber losses lead to broadening of solitons andthis is a drawback for many practical applications, es-pecially optical communications. This loss in energycan be restored by ampli¯cation of the optical pulsesand usually two schemes are employed for soliton am-pli¯cation. These are known as lumped and distributedampli¯cation schemes. In the lumped scheme, an opti-cal ampli¯er boosts the soliton energy to its input levelafter the soliton has been propagated through a certaindistance. Two methods, stimulated Raman scattering orerbium-doped ¯bers are used in the distributed ampli¯-cation scheme. Both methods require periodic pumpingalong the ¯ber length. Since erbium-doped ¯bers be-came available commercially after 1990, they have beenused widely for loss compensation.

Pulse Compression: An important application of non-linear ¯ber optics consists of compressing optical pulses.Pulses shorter than 5 fs have been produced by using dis-persive and nonlinear e®ects occurring in optical ¯bers.There are two types of pulse compressors based on non-linear ¯ber optics: grating-¯ber and soliton-e®ect com-pressors. A soliton-e®ect pulse compressor consists ofonly a piece of ¯ber whose wavelength is suitably cho-sen. Pulse compression takes place as a result of an in-terplay between GVD and SPM. This method has beenused since 1983. Dispersion-decreasing ¯bers (DDFs)are found to be useful for achieving pulse compression.Using DDF pulse compression mechanism, it is possibleto generate a train of ultrashort pulses.

Soliton Bit Rate: Solitons, if used with suitable care,would replace the traditional non-return to zero (NRZ)

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For the past two

decades or so, many

leading research

groups in USA,

Japan, France,

Germany, UK and

Sweden have been

working on soliton

technologies for high-

bit-rate, long haul

communications.

and return to zero (RZ) modulations, which are widelyused in almost all commercial terrestrial wavelength di-vision multiplier (WDM) systems. NRZ is a binary mod-ulation with square pulses in which the signal is o® fora 0 bit and on for a 1 bit. RZ is the same, except thatone pulse is shorter than the bit time. Although Kerrnonlinearities (SPM, Cross Phase Modulation (XPM)and Four-Wave Mixing (FWM)) exist in NRZ and RZsystems, they are undesirable e®ects limiting the perfor-mance and distorting signals at high speeds. Typicallythe design of a conventional WDM system involves in-creasing the power as much as possible (to counteract at-tenuation and noise) without introducing too much non-linearity. Thus, NRZ and RZ systems are often calledlinear systems. The main di®erence between encodingdigital signals with solitons and with NRZ is that in thelatter case if two 1s are close together, the signal in-tensity does not drop back to 0 between the individualbits as it does with solitons. For the past two decadesor so, many leading research groups in USA, Japan,France, Germany, UK and Sweden have been workingon soliton technologies for high-bit-rate, long haul com-munications, even while a heated debate took place overwhether solitons were appropriate or even feasible forthese networks. For this purpose, many billions of dol-lars are being invested and active research is in progress,mainly on solitons. Many argued that the conventionalNRZ or RZ modulation formats were a better choicethan the soliton-based technology.

In recent years, after the invention of dispersion-managedsolitons, the debate changed as theoretical understand-ing improved, network distances increased, bit ratesclimbed and certain technologies became commerciallypractical. NRZ experiments worked very well becauseof dispersion management. In this way, it is possible tohave both high local and low average dispersion in thesystem. The high local dispersion tends to reduce an

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Soliton light can be

used to transmit

data at rates in

excess of 50 Gbit/s

over distances

greater than

19,000 km of

Dispersion-Shifted

Fiber, requiring no

repeaters and with

no errors.

The existence of

an ideal soliton in a

fiber demands that

soliton pulses

should be well

separated from

each other.

e®ect known as four-wave mixing (which tends to dis-tort signals and produce intersymbol interference) anddisrupts the phase matching of the di®erent optical fre-quencies making up a signal, thus reducing the interac-tions among them. Moreover, the low average disper-sion reduces its net cumulative e®ects over long spansof optical ¯ber. Because of this, solitons have graduallyemerged as a viable candidate for the next generation ofterrestrial transport systems. Soliton light can be usedto transmit data at rates in excess of 50 Gbit/s overdistances greater than 19,000 km of Dispersion-ShiftedFiber, requiring no repeaters and with no errors. In-terestingly however, a form of RZ modulation calledchirped RZ (CRZ) has emerged as the chosen candi-date for the next generation undersea systems. Solitonsand the CRZ modulation format, like other `hard op-tic' technologies, are just one step forward in creatingthe next generation of optical networks. Research sci-entists are actually working on other technologies thatimprove components in optical networks. And at thesame time, advances in optical networking software, or`soft optics' continues to be the umbrella that harnessesand manages the light created by hard optics and willbe used to build future all-optical networks. Systemsemploying solitons have also been able to exploit WDMto increase the total bit rate, but not yet to the extentobtained with NRZ. Mollenauer's team has transmittedeight separate 10Gbit/s channels using solitons, for atotal of 80 Gbit/s, and Nakazawa of NTT Laboratorieshas achieved ¯ve separate 20-Gbit/s channels, for a to-tal capacity of 100 Gbit/s and, very recently, becauseof the constant e®orts, several groups have enormouslyincreased the number of channels and bit rates.

Timing Jitter: The existence of an ideal soliton in a¯ber demands that soliton pulses should be well sep-arated from each other. Thus if each soliton pulse isdesignated to carry one digit of information, in order

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GENERAL ARTICLE

In the Tokyo

Metropolitan Network

the rate of

information

transmission was as

high as 40 Gbit/s

while operating on

the soliton mode,

whereas while

operating on an

ordinary linear mode

using the same

network, the

information rate was

only about 2.4 Gbit/s.

The major detrimental

factor to the use of all

optical soliton

communication lines is

the occurrence of

soliton timing jitter

which is due to

amplifier noise or due

to interactions with

neighboring solitons.

to identify each information bit, it should be separatedsu±ciently from adjacent digits. This is possible onlywhen the soliton pulse width becomes much shorter thanthe bit rates, thereby demanding a much larger band-width when compared with a linear pulse having thesame bit rate. As a testimony to the experimental real-ization on information transmission by optical solitons,the Japanese ¯rm NTT has implemented informationtransfer through all optical soliton communication linesin the Tokyo Metropolitan Network and observed thatthe rate of information transmission through this net-work was as high as 40 Gbit/s while operating on thesoliton mode, whereas while operating on an ordinarylinear mode using the same network, the informationrate was only about 2.4 Gbit/s. In addition to this,in laboratory conditions, the same Japanese ¯rm wasable to increase the transmission bit rate to terabits persecond. The major detrimental factor to the use of alloptical soliton communication lines is the occurrence ofsoliton timing jitter which is due to ampli¯er noise ordue to interactions with neighboring solitons and thusis responsible for bit rate error. Soliton jitter can bee®ectively controlled by exploiting the robust nature ofsolitons and by reducing the average dispersion close tozero by means of dispersion compensation.

Soliton Photonic Switches: As with optical regen-eration, the need to convert optical data back to baseband electrical signals is tremendously cumbersome andslow. New techniques hold out the hope that it willsoon be possible to execute logic circuits at the opticallevel in the form of switches or binary logic gates suchas NOR and NAND gates. Although this technologystill lies in the realm of the laboratory, a most interest-ing application of solitons can be seen in the SOLITONNOR GATE. It does not take too much imaginationto see what could be achieved if this approach to op-tical logic circuits can be commercialized in the near

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GENERAL ARTICLE

Not only does this

fiber enable the

laser generation of

high-power

solitons and

amplify low-power

optical data

streams, but it can

act as a logic gate

as well!

future. So, how does this optical logic work? Again, itis based on erbium-doped ¯ber. Not only does this ¯berenable the laser generation of high-power solitons andamplify low-power optical data streams, but it can actas a logic gate as well! It is hard to believe that all thiscan be achieved from a piece of passive glass! Erbium¯ber is an attractive medium to make all-optical logicgates, because such gates have an almost instantaneousresponse. By using long lengths of inexpensive ¯ber, lowswitching energies can be achieved. The technology iscalled soliton-dragging logic gates and can satisfy all theneeds for logic gates or switches in an optical computeror digital communications data switch. It is possible tocascade gates, connect the output of one gate to inputsof many others and vice versa. It is also possible toperform the complete range of Boolean logic operationsincluding addition and subtraction.

In recent years, an even newer type of optical soliton hasbeen discovered. These solitons are created with a verystrong nonlinear e®ect found in some crystals in whichtwo light ¯elds can `shake hands' and cooperate by trav-eling together without dispersion. These are much morestable than previously known solitons, because they areadapted to travel in many other types of environmentrather than just inside optical ¯bers. For example, theycan travel stably in the surface region of an optical in-tegrated circuit. Because of the very strong nonlineare®ect (which is several thousand times larger than theprevious nonlinear e®ect), optical solitons of this typecan also be produced with extremely low laser powers.Possible applications include optical information stor-age of enormous quantities of data, all-optical switches,much faster optical devices than any known electronicdevices, etc.

7. Spatial Solitons

When optical beams travel, say along z-direction, through

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GENERAL ARTICLE

A laser beam when

focused onto the

edge of a

photosensitive

material can write its

own waveguide and

then gets guided by

this waveguide.

When illuminated, a

space-charge field is

formed in the

photorefractive

material which

induces nonlinear

changes in the

refractive index of

the material by the

electro-optic

(Pockels) effect. This

change in refractive

index can counter

the effect of beam

diffraction and form

a PR soliton.

a bulk homogeneous optical medium, there exists everypossibility that the beam di®racts along x and y direc-tions. However, if this di®raction can be compensatedby an increase of the refractive index of the material inthe transverse directions occupied by the beam, whichis possible if the medium is nonlinear, the medium be-comes an optical wave guide con¯ning the wave in thehigh index region. There has been tremendous growth inthe ¯eld of optical spatial solitons since the ¯rst observa-tion of self-trapping of light. A laser beam when focusedonto the edge of a photosensitive material can write itsown waveguide and then gets guided by this waveguide.In this process, the beam of light which initially dif-fracts, changes the refractive index of the medium. Astime passes on, the beam dynamically creates a chan-nel that counteracts di®raction and guides the beamthrough the material. When an optical beam propagatesin a suitable nonlinear medium, solitons can be formedand can be propagated without any di®raction e®ect.Spatial solitons with various dimensionality have beenobserved in various nonlinear media. Below, we discussspatial solitons formed in photorefractive materials andphotopolymers because of their practical importance.

Photorefractive Solitons: The study of spatial soli-tons is considered to be important because of possibleapplications in optical switching and routing. Segev et alproposed a new kind of spatial soliton, the Photorefrac-tive Soliton (PR) in the year 1992. When illuminated, aspace-charge ¯eld is formed in the photorefractive mate-rial which induces nonlinear changes in the refractive in-dex of the material by the electro-optic (Pockels) e®ect.This change in refractive index can counter the e®ectof beam di®raction and form a PR soliton. The lightbeam e®ectively traps itself in a self-written waveguide.As compared to the Kerr-type solitons, these solitonsexist in two dimensions and can be generated at lowpower levels of the order of several microwatts. The PR

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GENERAL ARTICLE

The PR soliton has

been investigated

extensively by

various groups as

it has potential

applications in all-

optical switching,

beam steering,

optical

interconnects, etc.

soliton has been investigated extensively by variousgroups as it has potential applications in all-opticalswitching, beam steering, optical interconnects, etc. Atpresent, three di®erent kinds of PR solitons have beenproposed: quasi-steady state solitons, screening solitonsand screening photovoltaic solitons. The screening PRsolitons are one of the most extensively studied solitons.They are possible in steady state when an external biasvoltage is applied to a non-photovoltaic PR crystal. This¯eld is partially screened by space charges induced bythe soliton beam. The combined e®ect of the balancebetween the beam di®raction and the PR focusing ef-fect results in the formation of a screening soliton. Anew kind of PR soliton, the photorefractive polymericsoliton, has been proposed and observed in 1991 in aphotorefractive polymer made from a mixture of two di-cyanomethylenedihydrofuran (DCDHF) chromospheres.Since then it has attracted much research interest owingto the possibility of using them as highly e±cient activeoptical elements for data transmission and controllingcoherent radiation in various electro-optical and opticalcommunication devices when compared to PR crystals.

Solitons in Self-Writing Waveguides: Light inducedor self-written waveguide formation is a recognized tech-nology by which we can form an optical waveguide as aresult of the self-trapping action of a laser beam passedthrough a converging lens or a single mode ¯ber. Self-writing is a relatively new and emerging area of researchin optics and the ¯rst experiment was demonstrated byFrisken in 1993. Since then the phenomenon of self-writing has been reported in a number of photosensitiveoptical materials including UV-cured epoxy, germano-silicate glass, planar chalcogenide glass, etc. The physicsof self-writing in all cases is very similar to that of spa-tial solitons discussed above, which occur due to bal-ance between linear di®raction e®ect and nonlinear self-focusing e®ect. These waveguides o®er a number of

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GENERAL ARTICLE

The self-written

waveguides so

formed are of

particular interest as

these waveguides

can be formed at low

power levels and the

material response is

wavelength

sensitive; therefore a

weak beam can

guide an intense

beam at a less

photosensitive

wavelength.

advantages in comparison with other methods of fab-ricating waveguides such as epitaxial growth, di®usionmethods and direct writing.

The self-written waveguides so formed are of particularinterest as these waveguides can be formed at low powerlevels and the material response is wavelength sensi-tive; therefore a weak beam can guide an intense beamat a less photosensitive wavelength. These waveguidesevolve dynamically, and they experience minimal radi-ation losses due to the absence of sharp bends. Manyprocessing steps are needed to form buried waveguides.But, if the bu®er layer is transparent at the writingwavelength, self-writing can be used to form these buriedwaveguides. Another application of self-written wave-guides is in telecommunication. The major obstacles inthe widespread use of single mode ¯bers in the telecom-munication industry are problems associated with e®ec-tive low cost coupling of light into single mode optical¯bers and the incorporation of bulk devices into optical¯bers. A self-written waveguide structure induced bylaser-light irradiation is considered as a candidate forconvenient coupling technique between the optical ¯berand waveguide .

Polymer optical waveguides have attracted considerableattention for their possible application as optical compo-nents in future optical communication systems; becausefabricating waveguides from polymers is much easierthan fabricating them from inorganic materials. In re-cent years, experiments with photopolymerizable mate-rials have produced promising results. Self-trapping andself-focusing phenomena in photopolymerizable materi-als (a liquid diacrylate photopolymer) have been demon-strated both experimentally and theoretically. Thesephenomena have been observed after an initial di®rac-tion period lasting approximately 20 s. A directionalcoupler using a three-dimensional waveguide structurehas been fabricated. Recently, successful fabrication of

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The change in

refractive index

occurs both due to

photo bleaching of

methyleneblue

and the photo

polymerization

effect.

arti¯cial ommatidia (imaging unit of insect's compoundeyes) by use of self-writing and polymer integrated op-tics has been reported. These biomimetic structureswere obtained by con¯guring micro lenses to play dualroles for self-writing of waveguides (during the fabrica-tion) and collection of light (during the operation).

When the photopolymer is illuminated with light of ap-propriate wavelength (632.8 nm), the polymer chainsbegin to join. The length of these chains determinesthe density of these polymers. As a result, the refrac-tive index of the exposed part of the material changes.The change in refractive index occurs both due to photobleaching of methylene blue and the photo polymer-ization e®ect. Photo bleaching is a term that appliesto techniques of exposing dye-doped materials to lightwhose wavelength lies within the spectral absorptionbands of the composite dye-polymer system. The changein refractive index so produced is much larger than thatof traditional nonlinear optical phenomena such as Kerror photorefractive e®ects. However, the index changeupon illumination is not instantaneous as compared tothese two e®ects. In recent years, many types of pho-topolymerizable systems have been developed as holo-graphic recording media. These materials have charac-teristics such as good spectral sensitivity, high resolu-tion, high di®raction e±ciency, high signal to noise ra-tio, temporal stability and processing in real time, whichmake them suitable for recording holograms. Because ofthese properties, photopolymer materials are useful inapplications such as optical memories, holographic dis-plays, holographic optical elements, optical computingand holographic interferometry. The most widely usedphotopolymer recording medium consists of acryl amideand poly (vinyl alcohol). Very recently (C P Jisha etal, Applied Optics, Vol. 47, pp6502-07, 2009), a self-written waveguide inside a bulk methylene blue sensi-tized poly/vinyl alcohol)/acrylamide photopolymer ma-terial has been observed (Figure 9).

Figure 9. Propagation of

the beam through the me-

dium with no diffraction.

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Suggested Reading

[1] A Hasegawa, Optical Solitons in Fibers, Springer-Verlag, 1989.

[2] G P Agrawal, Nonlinear Fiber Optics, Academic Press, 2001.

[3] G P Agrawal, Applications of Nonlinear Fiber Optics, Academic Press,

2001.

[4] Y S Kivshar and G P Agrawal, Optical Solitons: From Fibers to Photonic

Crystals, Academic Press, 2003.

[5] K Porsezian and V C Kuriakose (Eds) Optical Solitons: Theoretical and

Experimental Challenges, Springer-Verlag, 2003.

[6] L Solymar et al, The Physics and Applications of Photorefractive Mate-

rials, Oxford University Press, 1996.

Address for Correspondence

V C Kuriakose

Department of Physics

Cochin University of Science

and Technology

Kochi 682 022, India.

Email:

[email protected]

K Porsezian

Department of Physics

Pondicherry University

Pondicherry 605 014, India.

Email:

[email protected]

9. Conclusion

The use of solitons (both temporal and spatial) as infor-mation carriers in optical systems is still being heavilyresearched. As the pulses broaden, neighboring solitonswill overlap and this overlap is not fully understood.Systems, which allow the propagation of envelope soli-tons, operate in the region of negative GVD. However, ithas been recently found that dark solitons can propagatethrough optical ¯bers. The characteristics of these ¯bersare currently being investigated. Some of the topics likehigher order nonlinear and dispersive e®ects, cross phasemodulation, SRS and SBS, polarization e®ects, birefrin-gence have not been discussed in this article. Interestedreaders can look into the references cited below for moreinformation on optical solitons. This article discussessome of the directions that theoretical and experimen-tal investigations have taken over the last few decadesin this rapidly growing ¯eld. It is clear that solitonswill play a major role in the next generation of opticalcommunication systems.