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Transcript of Széchenyi István University Győr Hungary Solitons in optical fibers Szilvia Nagy Department of...
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
Solitons in optical fibersSolitons in optical fibers
Amplification of solitons – optical soliton Amplification of solitons – optical soliton transmission systemstransmission systems
Outline – General PropertiesOutline – General Properties
33ESM Zilina 2008ESM Zilina 2008
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
Nonlinear effects in fibersNonlinear effects in fibers
44ESM Zilina 2008ESM Zilina 2008
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
Nonlinear effects in fibersNonlinear effects in fibers
55ESM Zilina 2008ESM Zilina 2008
((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))
Nonlinear effects in fibersNonlinear effects in fibers
66ESM Zilina 2008ESM Zilina 2008
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
Nonlinear effects in fibersNonlinear effects in fibers
77ESM Zilina 2008ESM Zilina 2008
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
Nonlinear effects in fibersNonlinear effects in fibers
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
88ESM Zilina 2008ESM Zilina 2008
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))
Nonlinear effects in fibersNonlinear effects in fibers
231
43
E
t cos231
0 EEP
Innn
nn 20
23
00 8
3 E
99ESM Zilina 2008ESM Zilina 2008
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
Nonlinear effects in fibersNonlinear effects in fibers
1010ESM Zilina 2008ESM Zilina 2008
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
Nonlinear effects in fibersNonlinear effects in fibers
1111ESM Zilina 2008ESM Zilina 2008
History of solitonsHistory of solitons
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
1212ESM Zilina 2008ESM Zilina 2008
History of solitonsHistory of solitons
J. S. Russel, J. S. Russel, Report on Waves, Report on Waves, 18441844
1313ESM Zilina 2008ESM Zilina 2008
History of solitonsHistory of solitons
Snibston Discovery ParkSnibston Discovery Park
1414ESM Zilina 2008ESM Zilina 2008
History of solitonsHistory of solitons
Scott Russel Aqueduct,Scott Russel Aqueduct,19951995
Heriot-Watt University Heriot-Watt University EdinbourghEdinbourgh
1515ESM Zilina 2008ESM Zilina 2008
History of solitonsHistory of solitons
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
1616ESM Zilina 2008ESM Zilina 2008
History of solitonsHistory of solitons
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
1717ESM Zilina 2008ESM Zilina 2008
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
Korteweg—deVries equationsKorteweg—deVries equations
1818ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
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
1919ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
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
2020ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
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
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
2121ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Stationary solution of the KdV equationStationary solution of the KdV equation
2
sech 3 2,h
,h ,h
1 10
2222ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
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
2323ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
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
2424ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
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
2525ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
soliton wave in the sea (Molokai)
soliton1.mpeg
2626ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
soliton wave in the sky
soliton1.mpeg
2727ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
two solitons 1D
soliton1.mpeg
2828ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
two solitons 2D
soliton1.mpeg
2929ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
crossing solitons
soliton1.mpeg
3030ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
crossing solitons
soliton1.mpeg
3131ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
airball soliton scattering
3232ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
airball soliton scattering – a pinch
3333ESM Zilina 2008ESM Zilina 2008
Korteweg—deVries equationsKorteweg—deVries equations
Solutions of KdV equations with various Solutions of KdV equations with various boundary conditions in various dimensionsboundary conditions in various dimensions
higher order soliton
3434ESM Zilina 2008ESM Zilina 2008
Envelop solitonsEnvelop solitons
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
3535ESM Zilina 2008ESM Zilina 2008
Envelop solitonsEnvelop solitons
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
3636ESM Zilina 2008ESM Zilina 2008
Envelop solitonsEnvelop solitons
NormalizationNormalization
021 2
2
2
qqTq
Xq
i
02 2
2
2
2
EE
gEkE
i
X
,T
,Eq
k
g
3737ESM Zilina 2008ESM Zilina 2008
Envelop solitonsEnvelop solitons
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
3838ESM Zilina 2008ESM Zilina 2008
Envelop solitonsEnvelop solitons
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
3939ESM Zilina 2008ESM Zilina 2008
0
22
20sech
XTi
eXTX,Tq
Envelop solitonsEnvelop solitons
the solutionsthe solutions
which givewhich give2
sech
0
02
0
T0 and 0 are
phase constants
=1/2 : amplitude and pulse
width
transmission
speed
4040ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4141ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4242ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4343ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4444ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4545ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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!!!)
4646ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4747ESM Zilina 2008ESM Zilina 2008
Solitons in optical fibersSolitons in optical fibers
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
4848ESM Zilina 2008ESM Zilina 2008
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
4949ESM Zilina 2008ESM Zilina 2008
Amplification of solitonsAmplification of solitons
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
5050ESM Zilina 2008ESM Zilina 2008
Amplification of solitonsAmplification of solitons
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
5151ESM Zilina 2008ESM Zilina 2008
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)
5252ESM Zilina 2008ESM Zilina 2008
Optical soliton transmission Optical soliton transmission systemssystems
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
5353ESM Zilina 2008ESM Zilina 2008
Optical soliton transmission Optical soliton transmission systemssystems
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
5454ESM Zilina 2008ESM Zilina 2008
Optical soliton transmission Optical soliton transmission systemssystems
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
5555ESM Zilina 2008ESM Zilina 2008
Optical soliton transmission Optical soliton transmission systemssystems
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
5656ESM Zilina 2008ESM Zilina 2008
Optical soliton transmission Optical soliton transmission systemssystems
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.
5757ESM Zilina 2008ESM Zilina 2008
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.
5858ESM Zilina 2008ESM Zilina 2008
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.
5959ESM Zilina 2008ESM Zilina 2008
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.
6060ESM Zilina 2008ESM Zilina 2008
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,