Széchenyi István University Győr Hungary Solitons in optical fibers Szilvia Nagy Department of Telecommunications

Széchenyi István University

Győr

Hungary

Solitons in optical fibersSolitons in optical fibers

Szilvia NagySzilvia Nagy

Department of TelecommunicationDepartment of Telecommunicationss

22ESM Zilina 2008ESM Zilina 2008

Nonlinear effects in fibersNonlinear effects in fibers

History of solitonsHistory of solitons

Korteweg—deVries equationsKorteweg—deVries equations

Envelop solitonsEnvelop solitons


Amplification of solitons – optical soliton Amplification of solitons – optical soliton transmission systemstransmission systems

Outline – General PropertiesOutline – General Properties


Brillouin scattering:Brillouin scattering: acoustic vibrations caused by electro-acoustic vibrations caused by electro-

magnetic fieldmagnetic field(e.g. the light itself, if (e.g. the light itself, if PP>3mW)>3mW)

acoustic waves generate refractive index acoustic waves generate refractive index fluctuationsfluctuations

scattering on the refraction index wavesscattering on the refraction index waves the frequency of the light is shifted the frequency of the light is shifted

slightly slightly direction dependently direction dependently (~11 GHz(~11 GHz backw.backw.))

longer pulses – stronger effectlonger pulses – stronger effect



Raman scattering:Raman scattering: optical phonons (vibrations) caused by optical phonons (vibrations) caused by

electromagnetic field and the light can electromagnetic field and the light can exchange energy (similar to Brillouin but exchange energy (similar to Brillouin but not acoustical phonons)not acoustical phonons)

Stimulated Raman and Brillouin scattering Stimulated Raman and Brillouin scattering can be used for amplificationcan be used for amplification



((Pockels effect:Pockels effect: refractive index change due to ecternal refractive index change due to ecternal

electronic fieldelectronic field nn ~~ ||EE| - a | - a linear effectlinear effect))



Kerr effect:Kerr effect: the refractive index changes in response the refractive index changes in response

to an electromagnetic field to an electromagnetic field n n = = KK ||EE||22

light modulators up to 10 GHzlight modulators up to 10 GHz can cause self-phase modulation, self-can cause self-phase modulation, self-

induced phase and frequency shift, self-induced phase and frequency shift, self-focusing, mode lockingfocusing, mode locking

can produce solitons with the dispersioncan produce solitons with the dispersion



Kerr effect:Kerr effect: the polarization vectorthe polarization vector

if if EE==EE cos( cos(tt)), the polarization in first , the polarization in first order isorder is


3

1

3

1

3

1

10

3

1

3

1

20

3

1

10

j kkjijk

j kkjijk

jjiji EEEEEEP

PockelsKerr

t cos231

0 EEP


Kerr effect:Kerr effect:

the susceptibilitythe susceptibility

the refractive indexthe refractive index

nn22 is mostly small, large intensity is needed is mostly small, large intensity is needed (silica: (silica: nn22≈10≈10−20−20mm22/W, /W, II ≈10≈1099W/cmW/cm22))


231

43

E

t cos231

0 EEP

Innn

nn 20

23

00 8

3 E


Gordon-Haus jitter:Gordon-Haus jitter: a timing jitter originating from a timing jitter originating from

fluctuations of the center frequency of fluctuations of the center frequency of the (soliton) pulsethe (soliton) pulse

noise in fiber optic links caused by noise in fiber optic links caused by periodically spaced amplifiersperiodically spaced amplifiers

the amplifiers introduce quantum noise, the amplifiers introduce quantum noise, this shifts the center frequency of the this shifts the center frequency of the pulsepulse

the behavior of the center frequency the behavior of the center frequency modeled as random walkmodeled as random walk



Gordon-Haus jitter:Gordon-Haus jitter: dominant in long-haul data transmissiondominant in long-haul data transmission ~~LL33,, can be suppressed by can be suppressed by

regularly applied optical filtersregularly applied optical filters

amplifiers with limited gain bandwidthamplifiers with limited gain bandwidth can also take place in mode-locked can also take place in mode-locked

laserslasers




John Scott Russel (1808-1882)John Scott Russel (1808-1882)

1834, Union Canal, Hermiston near 1834, Union Canal, Hermiston near Edinbourgh, a boat was pulledEdinbourgh, a boat was pulled

after the stop of the boat a after the stop of the boat a „wave of translation” arised„wave of translation” arised

8-9miles/hour wave velocity 8-9miles/hour wave velocity

traveled 1-2 miles longtraveled 1-2 miles long



J. S. Russel, J. S. Russel, Report on Waves, Report on Waves, 18441844



Snibston Discovery ParkSnibston Discovery Park



Scott Russel Aqueduct,Scott Russel Aqueduct,19951995

Heriot-Watt University Heriot-Watt University EdinbourghEdinbourgh



1870s J. Boussinesq, Rayleigh both 1870s J. Boussinesq, Rayleigh both deduced the secret of Russel’s waves: the deduced the secret of Russel’s waves: the dispersion and the nonlinearity cancels dispersion and the nonlinearity cancels each othereach other

1964 Zabusky and Kruskal solves the KdV 1964 Zabusky and Kruskal solves the KdV equation numerically, solitary wave equation numerically, solitary wave solutions: solutions: solitonsoliton

1960s: nonlinear wave propagation 1960s: nonlinear wave propagation studied with computers: many fields were studied with computers: many fields were found where solitons appearfound where solitons appear



1970s A. Hasegawa proposed solitons in 1970s A. Hasegawa proposed solitons in optical fibersoptical fibers

1980 Mollenauer demonstrated soliton 1980 Mollenauer demonstrated soliton transmission in optical fiber (10 ps, 1.5 transmission in optical fiber (10 ps, 1.5 m, 700 m fiber)m, 700 m fiber)

1988 Mollenauer and Smith sent soliton 1988 Mollenauer and Smith sent soliton light pulses in fiber for 6000 km without light pulses in fiber for 6000 km without electronic amplifierelectronic amplifier


In 1895 Korteweg and In 1895 Korteweg and deVries modeled the deVries modeled the wave motion on the wave motion on the surface of shallow watersurface of shallow waterby the equationby the equation

where where hh wave heightwave height time in coordinates time in coordinates space coordinatespace coordinate

03

3

hhh

h

movingwith the wave




Derivation of the KdV equationDerivation of the KdV equation

a wave a wave hh propagating in propagating in xx direction can be direction can be described in the coordinate system (described in the coordinate system (,,) ) traveling with the wave astraveling with the wave as

Using the original (Using the original (xx,,tt) coordinates:) coordinates:

0

h

0

h0

xh

vth

t

,vx



Stationary solution of the KdV equationStationary solution of the KdV equation

Dispersive and nonlinear effects can Dispersive and nonlinear effects can balance to make a stationary solutionbalance to make a stationary solution

03

3

hhh

h

03

0

02

0

const

const

vk

kkv

hvhv const0




Dispersive and nonlinear effects can Dispersive and nonlinear effects can balance to make a stationary solutionbalance to make a stationary solution

where where is the velocity of the solitary wave is the velocity of the solitary wave in the (in the (,,) space) space

2

sech 3 2,h

03

3

hhh

h




2

sech 3 2,h

,h ,h

1 10



The KdV equation and the inverse scattering The KdV equation and the inverse scattering problemsproblems

the Schrödinger equation:the Schrödinger equation:

if „potential” if „potential” uu((xx,,tt)) satisfies a KdV equation, satisfies a KdV equation, is independent of timeis independent of time uu((xx,0) → 0,0) → 0 as as ||xx|→ ∞|→ ∞ the Schrödinger equation can be solved the Schrödinger equation can be solved

for for tt=0=0 for a given initial for a given initial uu((xx,0),0)

02

2

t,xu

x



The KdV equation and the inverse scattering The KdV equation and the inverse scattering problemsproblems tt=0=0 scattering data can be derived from scattering data can be derived from

the the tt=0=0 solution solution the time evolution of the time evolution of and thus the and thus the

scattering data is knownscattering data is known

uu((xx,,tt)) can be found for each can be found for each ((xx,,tt)) by by inverse scattering methods.inverse scattering methods.

xCu

xu

Bx

At

3

3



Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions

soliton propagating and scattering




soliton wave in the sea (Molokai)

soliton1.mpeg




soliton wave in the sky

soliton1.mpeg




two solitons 1D

soliton1.mpeg




two solitons 2D

soliton1.mpeg




crossing solitons

soliton1.mpeg




crossing solitons

soliton1.mpeg




airball soliton scattering




airball soliton scattering – a pinch




higher order soliton



Envelop of a waveEnvelop of a wave

if the amplitude of a wave varies (slowly)if the amplitude of a wave varies (slowly)

envelop of the wave

x,th

complex amplitude



If the wave can be described byIf the wave can be described by

the wave equation for the envelopthe wave equation for the envelop

withwith , , and and

txkiet,xEt,xE 00Re

t,xE

02 2

2

2

2

EE

gEkE

i

Dk

k0

2

2

.n

g

22

reduction factor, ~1/2

0

0



NormalizationNormalization

021 2

2

2

qqTq

Xq

i

02 2

2

2

2

EE

gEkE

i

X

,T

,Eq

k

g



Solving the non-linear Schrödinger Solving the non-linear Schrödinger equationequation

test functiontest function

the new equationthe new equation

021 2

2

2

qqTq

Xq

i

X,TieX,TX,Tq

0

TTXi



looking for solitary wave solution of the looking for solitary wave solution of the new equationnew equation

ifif is a stationary solutionis a stationary solution2

q

0X

0

TTXi

XCT

it can be shown, that C is independent of X


0

22

20sech

XTi

eXTX,Tq


the solutionsthe solutions

which givewhich give2

sech

0

02

0

T0 and 0 are

phase constants

=1/2 : amplitude and pulse

width

transmission

speed



envelop equation of a light wave in a fiberenvelop equation of a light wave in a fiber

fiber loss rate per unit length: fiber loss rate per unit length:

withwith

02 2

2

2

2

EE

gEkE

i

22

2

2

2

2

EiEE

gEkE

i

.n

g,,k

k

2

0

02

2 2

0



Solitons can arise as solution ofSolitons can arise as solution of

if the real part of the nonlinear term is if the real part of the nonlinear term is dominant,dominant,

2

2 En

22

2

2

2

2

EiEE

gEkE

i

222 nn

g

22

2

EEEg



the condition for existence of a soliton:the condition for existence of a soliton:

example: example:

≈ ≈ 1500 nm1500 nm

||ÊÊ| | ≈ 10≈ 1066 V/m V/m < 2 ×10< 2 ×10−4 −4 mm−1−1

nn22 ≈ 1.2×10≈ 1.2×10−22−22 m m22/V/V22

2

2 En

1.7 dB/km



the normalized equation, withthe normalized equation, with

if if is small enough, is small enough, perturbation perturbation techniquestechniques can be used can be used

qiqqTq

Xq

i

2

2

2

21

2

OeTXXX,Tq Xisech

XeqX 20 Xe

qX

4

20 1

8



The solution of the normalized soliton The solution of the normalized soliton equation in fibers with loss predictsequation in fibers with loss predicts the amplitude the amplitude of the soliton decreases of the soliton decreases

as it propagates: as it propagates:

the width the width of the soliton increases of the soliton increases

their product remains constanttheir product remains constant

XeqX 20

Xeq

X

42

0 18



Effects of the waveguide manifest asEffects of the waveguide manifest as

0

21

2

3

2

23

3

1

2

2

2

qT

qqqTT

qi

qqTq

qXq

i

higher order linear

dispersionnonlinear dispersion

of the Kerr coefficient

nonlinear dissipation due to Raman processes

(imaginary!!!)



Necessary condition for existence of a Necessary condition for existence of a solitonsoliton

0 0 : : pulse length [ps]pulse length [ps] PP00:: required pulse power [W]required pulse power [W] : : wavelength [wavelength [m]m] D D : : dispersion [ps/(nm km)]dispersion [ps/(nm km)] S S : : cross-sectional area [cross-sectional area [mm22]]

e.g., e.g., SS=60 =60 mm22, , =1.5 =1.5 m, m, ||DD||=10 ps/(nm km)=10 ps/(nm km)00=10 ps, =10 ps, PP00=180 mW=180 mW

SD.P / 23200 1039



Soliton generation needsSoliton generation needs low loss fiber (<1 dB/km)low loss fiber (<1 dB/km) spectral width of the laser pulse be spectral width of the laser pulse be

narrower than the inverse of the pulse narrower than the inverse of the pulse lengthlength

Mollenauer & al. 1980, AT&T Bell Lab.Mollenauer & al. 1980, AT&T Bell Lab.700 m fiber, 10700 m fiber, 10−6−6 cm cm22 cross section cross section

7 ps pulse,7 ps pulse,

FF2+2+ color center laser with Nd:YAG pump color center laser with Nd:YAG pump

1.2 W soliton threshold1.2 W soliton threshold


Amplification of solitonsAmplification of solitons

For small loss the soliton propagates with For small loss the soliton propagates with the product of its pulse length and height the product of its pulse length and height being constantbeing constant

reshaping is needed for long-distance reshaping is needed for long-distance communication applicationcommunication application

reshaping methods:reshaping methods: induced Raman amplification – the loss induced Raman amplification – the loss

compensated along the fibercompensated along the fiber repeated Raman Amplifiersrepeated Raman Amplifiers Er doped amplifiersEr doped amplifiers



Experiment on the long distance Experiment on the long distance transmission of a soliton by repeated transmission of a soliton by repeated Raman Amplification (Mollenauer & Smith, Raman Amplification (Mollenauer & Smith, 1988)1988)

41.7 km

3 dB coupler

dependent coupler

all fiber MZ interferomete

rsignal in

signalout

pump in

1600 nm

1500 nm

filter, 9 ps diode, spectrum analyzer



Erbium doped fiber amplifiers, periodically Erbium doped fiber amplifiers, periodically placed in the transmission lineplaced in the transmission line distance of the amplifiers should be less distance of the amplifiers should be less

then the soliton dispersion lengththen the soliton dispersion length dispersion shifted fibers or filters for dispersion shifted fibers or filters for

reshapingreshaping

quantum noise arisequantum noise arise spontaneous emission noisespontaneous emission noise Gordon—House jitterGordon—House jitter


Optical soliton transmission Optical soliton transmission systemssystems

The soliton based communication systems The soliton based communication systems mostly use on/offmostly use on/off or DPSK or DPSK keying keyingIn soliton communication systems the In soliton communication systems the timing jitters which originate from timing jitters which originate from frequency fluctuation afrequency fluctuation arre held under e held under control by narrow band optical filterscontrol by narrow band optical filters frequency guiding filterfrequency guiding filter e.g., a shallow Fabry-Perot etalon filtere.g., a shallow Fabry-Perot etalon filter(in non-soliton systems, these guiding (in non-soliton systems, these guiding

filters destroy the signal, they are not filters destroy the signal, they are not used)used)



It is possible to make the soliton “slide” in It is possible to make the soliton “slide” in frequencyfrequency sliding frequency guiding filterssliding frequency guiding filters each consecutive narrow-band filter has each consecutive narrow-band filter has

slightly different center frequencyslightly different center frequency center frequency sliding rate: center frequency sliding rate: ff’= ’= dfdf//dzdz the solitons can follow the frequency shiftthe solitons can follow the frequency shift the noise can not follow the frequency the noise can not follow the frequency

sliding, it drops outsliding, it drops out



Wavelength division multiplexing in soliton Wavelength division multiplexing in soliton communication systemscommunication systems solitons with different center frequency solitons with different center frequency

propagate with different group velocitypropagate with different group velocity in collision of two solitons, they in collision of two solitons, they

propagate together for a whilepropagate together for a while collision length:collision length:

D

L2

coll



during the collision both solitons shifts in during the collision both solitons shifts in frequency (same magnitude, opposite frequency (same magnitude, opposite sign)sign)

first part of the collision: the fast first part of the collision: the fast soliton’s velocity increases, while the soliton’s velocity increases, while the slow one becomes slowerslow one becomes slower

at the second part of the collision, the at the second part of the collision, the opposite effect takes place, opposite effect takes place, symmetricallysymmetrically



if during the collision the solitons reach if during the collision the solitons reach an amplifier or a reshaper, the an amplifier or a reshaper, the symmetry brakessymmetry brakes

the result is non-zero residual frequency the result is non-zero residual frequency shift can arise, unlessshift can arise, unless

ampcoll 2LL



if a collision of two solitons take place at if a collision of two solitons take place at the input of the transmissionthe input of the transmission

half collisionhalf collision it can be avoided by staggering the it can be avoided by staggering the

pulse positions of the WDM channels at pulse positions of the WDM channels at the input.the input.


J. C. Russel,J. C. Russel,Report of the fourteenth meeting of the British Association for Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844, p. 311the Advancement of Science, York, September 1844, p. 311London, 1845.London, 1845.

BoussinesqBoussinesqJ. Math. Pures Appl., vol. 7, p. 55, 1972.J. Math. Pures Appl., vol. 7, p. 55, 1972.

Lord RayleighLord RayleighPhilosophical Magazine, s5, vol. 1, p. 257, 1876,Philosophical Magazine, s5, vol. 1, p. 257, 1876,Proc. London Math. Soc. s1, vol. 17, p. 4, 1885.Proc. London Math. Soc. s1, vol. 17, p. 4, 1885.

N.J. Zabusky, M.D. Kruskal,N.J. Zabusky, M.D. Kruskal,Phys. Rev. Lett., vol. 15, p. 240, 1965.Phys. Rev. Lett., vol. 15, p. 240, 1965.

A. Hasegawa, F.D. Tappert,A. Hasegawa, F.D. Tappert,Appl. Phys. Lett., vol. 23, p. 142, 1973.Appl. Phys. Lett., vol. 23, p. 142, 1973.


L.F. Mollenauer, R.H. Stolen, J.P. Gorden,L.F. Mollenauer, R.H. Stolen, J.P. Gorden,Phys. Rev. Lett., vol. 45, p. 1095, 1980.Phys. Rev. Lett., vol. 45, p. 1095, 1980.

J.P. Gordon, H.A. Haus,J.P. Gordon, H.A. Haus,Opt. Lett., vol. 11, p. 665, 1986.Opt. Lett., vol. 11, p. 665, 1986.

D.J. Korteweg, G, deVries,D.J. Korteweg, G, deVries,Phil. Mag. Ser. 5, vol. 39, p. 422, 1895.Phil. Mag. Ser. 5, vol. 39, p. 422, 1895.


J. Hecht,J. Hecht,Understanding fiber Optics (fifth edition),Understanding fiber Optics (fifth edition),Pearson Prentice Hall, Upper Saddle River, New Jersey, Pearson Prentice Hall, Upper Saddle River, New Jersey, Columbus, Ohio, 2006.Columbus, Ohio, 2006.

J. Gowar,J. Gowar,Optical Communication Systems (second edition)Optical Communication Systems (second edition)Prentice-Hall of India, New Delhi, 2004.Prentice-Hall of India, New Delhi, 2004.

A. Hasegawa,A. Hasegawa,Optical Solitons in FibersOptical Solitons in FibersSpringer-Verlag, Berlin, 1989.Springer-Verlag, Berlin, 1989.

Fiber Optic Handbook, Fiber, Devices, and Systems for Optical Fiber Optic Handbook, Fiber, Devices, and Systems for Optical Communications,Communications,editor: M. Bass, (associate editor: E. W. Van Stryland)editor: M. Bass, (associate editor: E. W. Van Stryland)McGraw-Hill, New York, 2002.McGraw-Hill, New York, 2002.


J. Hietarinta, J. Ruokolainen,J. Hietarinta, J. Ruokolainen,Dromions – The MovieDromions – The Moviehttp://users.utu.fi/hietarin/dromions/index.html.http://users.utu.fi/hietarin/dromions/index.html.

E. Frenkel,E. Frenkel,Five lectures on soliton equationsFive lectures on soliton equationsarXiv:q-alg/9712005v1 1997arXiv:q-alg/9712005v1 1997Contribution to Survays in Differential Geometry, vol. 3, Contribution to Survays in Differential Geometry, vol. 3, International Press.International Press.

Encyclopedia of Laser PhysicsEncyclopedia of Laser Physicshttp://www.rp-photonics.com/solitons.html,http://www.rp-photonics.com/solitons.html,http://www.rp-photonics.com/higher_order_solitons.html,http://www.rp-photonics.com/higher_order_solitons.html,

Light Bullet Home Page,Light Bullet Home Page,http://www.sfu.ca/~renns/lbullets.html,http://www.sfu.ca/~renns/lbullets.html,

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Széchenyi István University Győr Hungary Solitons in optical fibers Szilvia Nagy Department of Telecommunications