Solitons in optical fibers
Szilvia Nagy
Department of Telecommunications
Széchenyi István University
Győr
Hungary
Outline – General Properties
Nonlinear effects in fibers
History of solitons
Korteweg—deVries equations
Envelop solitons
Solitons in optical fibers
Amplification of solitons – optical soliton
transmission systems
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Nonlinear effects in fibers
Brillouin scattering:
 acoustic vibrations caused by electromagnetic field
(e.g. the light itself, if P>3mW)
 acoustic waves generate refractive index
fluctuations
 scattering on the refraction index waves
 the frequency of the light is shifted slightly
direction dependently (~11 GHz backw.)
 longer pulses – stronger effect
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Nonlinear effects in fibers
Raman scattering:
 optical phonons (vibrations) caused by
electromagnetic field and the light can
exchange energy (similar to Brillouin but
not acoustical phonons)
Stimulated Raman and Brillouin scattering
can be used for amplification
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Nonlinear effects in fibers
(Pockels effect:
 refractive index change due to ecternal
electronic field
 Dn ~ |E| - a linear effect)
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Nonlinear effects in fibers
Kerr effect:
 the refractive index changes in response
to an electromagnetic field
2
 Dn = K l|E|
 light modulators up to 10 GHz
 can cause self-phase modulation, selfinduced phase and frequency shift, selffocusing, mode locking
 can produce solitons with the dispersion
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Nonlinear effects in fibers
Kerr effect:
 the polarization vector
3
3
3
j 1
j 1 k 1
3
3
3
Pi   0  ij1E j   0  ijk2 E j Ek   0  ijk1 E j Ek E  
j 1 k 1  1
Pockels

Kerr
if E=Ew cos(wt), the polarization in first
order is

1
3 
P   0    Ew
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 E coswt
w
7
Nonlinear effects in fibers
Kerr effect:

1
3 
P   0    Ew

the susceptibility
2
 E coswt
w
3 3  2
     Ew
4
1

the refractive index
n  n0 

3 3  2
 Ew  n0  n2I
8n0
n2 is mostly small, large intensity is needed
(silica: n2≈10−20m2/W, I ≈109W/cm2)
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Nonlinear effects in fibers
Gordon-Haus jitter:
 a timing jitter originating from fluctuations
of the center frequency of the (soliton)
pulse
 noise in fiber optic links caused by
periodically spaced amplifiers
 the amplifiers introduce quantum noise,
this shifts the center frequency of the
pulse
 the behavior of the center frequency
modeled as random walk
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Nonlinear effects in fibers
Gordon-Haus jitter:
 dominant in long-haul data transmission
3
 ~L ,
 can be suppressed by
regularly applied optical filters
amplifiers with limited gain bandwidth

can also take place in mode-locked
lasers
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History of solitons
John Scott Russel (1808-1882)
1834, Union Canal, Hermiston near
Edinbourgh, a boat was pulled
after the stop of the boat a
„wave of translation” arised
8-9miles/hour wave velocity
traveled 1-2 miles long
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History of solitons
J. S. Russel,
Report on Waves,
1844
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History of solitons
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Snibston Discovery Park
13
History of solitons
Scott Russel Aqueduct,
1995
Heriot-Watt University
Edinbourgh
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History of solitons
1870s J. Boussinesq, Rayleigh both deduced
the secret of Russel’s waves: the dispersion
and the nonlinearity cancels each other
1964 Zabusky and Kruskal solves the KdV
equation numerically, solitary wave
solutions: soliton
1960s: nonlinear wave propagation studied
with computers: many fields were found
where solitons appear
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History of solitons
1970s A. Hasegawa proposed solitons in
optical fibers
1980 Mollenauer demonstrated soliton
transmission in optical fiber (10 ps, 1.5 mm,
700 m fiber)
1988 Mollenauer and Smith sent soliton light
pulses in fiber for 6000 km without electronic
amplifier
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Korteweg—deVries equations
In 1895 Korteweg and
deVries modeled the
wave motion on the
surface of shallow water
by the equation
3
h
h  h
h  3 0
t
x x
where
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h
t
x
wave height
time in coordinates
space coordinate
moving
with
the wave
17
Korteweg—deVries equations
Derivation of the KdV equation
a wave h propagating in x direction can be
described in the coordinate system (x,t)
traveling with the wave as
h
0
t
Using the original (x,t) coordinates:
h
0
t
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x  x  vt ,
tt
h
h
v
0
t
x
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Korteweg—deVries equations
Stationary solution of the KdV equation
Dispersive and nonlinear effects can balance
to make a stationary solution
3
h
h  h
h  3 0
t
x x
vh  v0  const  h
w  kv  w0  const  k 2 w0
 w0  const  k 3 v0
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Korteweg—deVries equations
Stationary solution of the KdV equation
Dispersive and nonlinear effects can balance
to make a stationary solution
3
h
h  h
h  3 0
t
x x
ht , x   3h sech
2
h
x  ht
2
where h is the velocity of the solitary wave in
the (x,t) space
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Korteweg—deVries equations
Stationary solution of the KdV equation
ht , x   3h sech
h 1
ht,x
2
h
x  ht
2
h  10
ht,x
t
x
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t
x
21
Korteweg—deVries equations
The KdV equation and the inverse scattering
problems
the Schrödinger equation:
 2
 l  ux ,t   0
2
x
if „potential” u(x,t) satisfies a KdV equation,
 l is independent of time
 u(x,0) → 0 as |x|→ ∞
 the Schrödinger equation can be solved for
t=0 for a given initial u(x,0)
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Korteweg—deVries equations
The KdV equation and the inverse scattering
problems
 t=0 scattering data can be derived from
the t=0 solution
 the time evolution of  and thus the
scattering data is known


 3
u

 A 3  B   Cu
t
x
x
x
u(x,t) can be found for each (x,t) by
inverse scattering methods.
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Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
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soliton propagating and scattering
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Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
soliton1.mpeg
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soliton wave in the sea (Molokai)
25
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
soliton1.mpeg
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soliton wave in the sky
26
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
soliton1.mpeg
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two solitons 1D
27
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
soliton1.mpeg
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two solitons 2D
28
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
soliton1.mpeg
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crossing solitons
29
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
soliton1.mpeg
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crossing solitons
30
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
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airball soliton scattering
31
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
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airball soliton scattering – a pinch
32
Korteweg—deVries equations
Solutions of KdV equations with various
boundary conditions in various dimensions
higher
order
soliton
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Envelop solitons
Envelop of a wave
if the amplitude of a wave varies (slowly)
envelop
of the
wave
ht , x 
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complex
amplitude
34
Envelop solitons
If the wave can be described by

i k0 x w0t 
ˆ
Ex ,t   Re Ex ,t   e

ˆ x , t 
the wave equation for the envelop E
2
ˆ E
ˆ
E
ˆ k  2 E
ˆ
E
i

g 2 0
2
x 2 t

with
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2k
k  2
w
w  w0
reduction
factor, ~1/2
l
Dw0
D , 
and g  2n2 .
w
w0
l
35
Envelop solitons
Normalization
2
ˆ
ˆ
2ˆ
E
E
ˆ


E k  E
i

g 2 0
2
x 2 t

gl ˆ
q
E,

T
X
t
lk
,
x
l
2
q 1  2 q
i

 q q0
2
X 2 T
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Envelop solitons
Solving the non-linear Schrödinger equation
2
q 1  2 q
i

 q q0
2
X 2 T

test function qT , X   T , X eiT ,X 

the new equation
    
i
    0
X T  T 
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Envelop solitons

looking for solitary wave solution of the
new equation
    
i
    0
X T  T 

if
2
q   is a stationary solution

0
X


 CX 
T
it can be shown, that C is
independent of X
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Envelop solitons


the solutions
  0 sech 2 0 T
0 and 0 are
phase constants
0

2
which give
qT , X   h sechhT  X  0 e
h=1/2 :
amplitude and pulse
width
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2
2


h


i   T 
X  0 


2


:
transmission speed
39
Solitons in optical fibers
envelop equation of a light wave in a fiber
2
ˆ
ˆ
2ˆ
E
E
ˆ
E k  E
i

g 2 0
2
x 2 t

fiber loss rate per unit length: g
2
ˆ E
ˆ
E
ˆ k  2 E
ˆ
ˆ
E
igE
i

g 2  2
2
x 2 t


2k
with k  2
w
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w  w0
Dw0
2n2 
, 
, g
.
w0
l
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Solitons in optical fibers
Solitons can arise as solution of
2
ˆ
ˆ
2ˆ
E
E
ˆ
ˆ
E k  E
igE
i

g 2  2
2
x 2 t


if the real part of the nonlinear term is
dominant,
2
g
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ˆ E
ˆ
E

2
ˆ
gE
 2

2n2  n2
g

l
l
ˆ
n2 E
l
2
g
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Solitons in optical fibers
the condition for existence of a soliton:
ˆ
n2 E
l
2
g
example:
l ≈ 1500 nm
|Ê| ≈ 106 V/m
n2 ≈ 1.2×10−22 m2/V2
g < 2 ×10−4 m−1
1.7 dB/km
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Solitons in optical fibers
the normalized equation, with
2
q 1  2 q
i

 q q  iGq
2
X 2 T
gl
G 2

if G is small enough, perturbation techniques
can be used
qT , X   h X  sechhX T e iX   OG
h X   q0 e
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2 GX
2

q0
 X  
1 e4 GX
8G

43
Solitons in optical fibers
The solution of the normalized soliton
equation in fibers with loss predicts
 the amplitude h of the soliton decreases
as it propagates:
h X   q0 e2GX

the width  of the soliton increases
2



q0
 4 GX
 X  
1 e
8G
their product remains constant
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Solitons in optical fibers
Effects of the waveguide manifest as
2
 q
 1  2q
i
 Gq  
q q
2
 X
 2 T
  3q
2

 2
 i 1 3  2
q q  3 q q   0
T
T
 T


higher order
linear dispersion
nonlinear dispersion of
the Kerr coefficient
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
nonlinear dissipation
due to Raman
processes
(imaginary!!!)
45
Solitons in optical fibers
Necessary condition for existence of a
soliton
t0 P0  9.3 102 l3 / 2 D S





t0 :
P0:
l :
D:
S :
pulse length [ps]
required pulse power [W]
wavelength [mm]
dispersion [ps/(nm km)]
cross-sectional area [mm2]
e.g., S=60 mm2, l=1.5 mm, |D|=10 ps/(nm km)
t0=10 ps, P0=180 mW
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Solitons in optical fibers
Soliton generation needs
 low loss fiber (<1 dB/km)
 spectral width of the laser pulse be
narrower than the inverse of the pulse
length
 Mollenauer & al. 1980, AT&T Bell Lab.
700 m fiber, 10−6 cm2 cross section
7 ps pulse,
F2+ color center laser with Nd:YAG pump
1.2 W soliton threshold
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Amplification of solitons
For small loss the soliton propagates with the
product of its pulse length and height being
constant
reshaping is needed for long-distance
communication application
reshaping methods:
 induced Raman amplification – the loss
compensated along the fiber
 repeated Raman Amplifiers
 Er doped amplifiers
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Amplification of solitons
Experiment on the long distance
transmission of a soliton by repeated Raman
Amplification (Mollenauer & Smith, 1988)
3 dB coupler
41.7 km
pump in
1500 nm
signal
out
all fiber MZ
interferometer
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l dependent
coupler
signal in
1600 nm
filter, 9 ps diode,
spectrum
analyzer
49
Amplification of solitons
Erbium doped fiber amplifiers, periodically
placed in the transmission line
 distance of the amplifiers should be less
then the soliton dispersion length
 dispersion shifted fibers or filters for
reshaping
quantum noise arise
 spontaneous emission noise
 Gordon—House jitter
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Optical soliton transmission systems
The soliton based communication systems
mostly use on/off or DPSK keying
In soliton communication systems the timing
jitters which originate from frequency
fluctuation are held under control by narrow
band optical filters
 frequency guiding filter
 e.g., a shallow Fabry-Perot etalon filter
(in non-soliton systems, these guiding filters
destroy the signal, they are not used)
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Optical soliton transmission systems
It is possible to make the soliton “slide” in
frequency
 sliding frequency guiding filters
 each consecutive narrow-band filter has
slightly different center frequency
 center frequency sliding rate: f’= df/dz
 the solitons can follow the frequency shift
 the noise can not follow the frequency
sliding, it drops out
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Optical soliton transmission systems
Wavelength division multiplexing in soliton
communication systems
 solitons with different center frequency
propagate with different group velocity
 in collision of two solitons, they propagate
together for a while
 collision length:
Lcoll
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2t

DDl
53
Optical soliton transmission systems



during the collision both solitons shifts in
frequency (same magnitude, opposite
sign)
first part of the collision: the fast soliton’s
velocity increases, while the slow one
becomes slower
at the second part of the collision, the
opposite effect takes place,
symmetrically
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Optical soliton transmission systems


if during the collision the solitons reach an
amplifier or a reshaper, the symmetry
brakes
the result is non-zero residual frequency
shift can arise, unless
Lcoll  2Lamp
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Optical soliton transmission systems



if a collision of two solitons take place at
the input of the transmission
half collision
it can be avoided by staggering the pulse
positions of the WDM channels at the
input.
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J. C. Russel,
Report of the fourteenth meeting of the British Association for the
Advancement of Science, York, September 1844, p. 311
London, 1845.
Boussinesq
J. Math. Pures Appl., vol. 7, p. 55, 1972.
Lord Rayleigh
Philosophical Magazine, s5, vol. 1, p. 257, 1876,
Proc. London Math. Soc. s1, vol. 17, p. 4, 1885.
N.J. Zabusky, M.D. Kruskal,
Phys. Rev. Lett., vol. 15, p. 240, 1965.
A. Hasegawa, F.D. Tappert,
Appl. Phys. Lett., vol. 23, p. 142, 1973.
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L.F. Mollenauer, R.H. Stolen, J.P. Gorden,
Phys. Rev. Lett., vol. 45, p. 1095, 1980.
J.P. Gordon, H.A. Haus,
Opt. Lett., vol. 11, p. 665, 1986.
D.J. Korteweg, G, deVries,
Phil. Mag. Ser. 5, vol. 39, p. 422, 1895.
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J. Hecht,
Understanding fiber Optics (fifth edition),
Pearson Prentice Hall, Upper Saddle River, New Jersey, Columbus,
Ohio, 2006.
J. Gowar,
Optical Communication Systems (second edition)
Prentice-Hall of India, New Delhi, 2004.
A. Hasegawa,
Optical Solitons in Fibers
Springer-Verlag, Berlin, 1989.
Fiber Optic Handbook, Fiber, Devices, and Systems for Optical
Communications,
editor: M. Bass, (associate editor: E. W. Van Stryland)
McGraw-Hill, New York, 2002.
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J. Hietarinta, J. Ruokolainen,
Dromions – The Movie
http://users.utu.fi/hietarin/dromions/index.html.
E. Frenkel,
Five lectures on soliton equations
arXiv:q-alg/9712005v1 1997
Contribution to Survays in Differential Geometry, vol. 3,
International Press.
Encyclopedia of Laser Physics
http://www.rp-photonics.com/solitons.html,
http://www.rp-photonics.com/higher_order_solitons.html,
Light Bullet Home Page,
http://www.sfu.ca/~renns/lbullets.html,
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