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Pulse Propagation in Optical Fibers
Arleth Manuela Gonçalves
Departamento de Engenharia Electrotécnica e de Computadores
Instituto Superior Técnico
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
[email protected]
Abstrat — This paper addresses the pulse propagation through a fiber optic system, operating in the linear and
nonlinear regimes. After a brief introduction to optical fibers, we use the modal theory approach to understand the
operating principle for the pulses propagating in the fiber. Then, we try to understand the effect of dispersion, as the
unavoidable phenomenon to which pulses are subject, while propagating in the fiber, in the linear regime. Since
dispersion strongly limits the bandwidth of the transmitted signal and may cause interference between symbols, we
address several methods for reducing the effect of dispersion, which can be very efficient: the use of dispersion
compensating fibers and the propagation of solitons. In non-linear system, the non-linear Kerr effect explains the
appearance of self-phase modulation (SPM) which has a contrary action to the phenomenon of dispersion in the
anomalous dispersion region, allowing the appearance of solitons, which era pulses that conserve their shape along the
propagation. The propagation of pulses in nonlinear regime is governed by the nonlinear Shrondinger (NLS) equation for
which, their analytical solutions are calculated. An efficient method is used in the numerical simulations under the nonlinear regime: the split-step fourier Method (SSFM). Finally, a brief study of the fundamental soliton is presented.
Keywords — Fibre optics, dispersion, pulse propagation, solitons.
I. INTRODUCTION
T
he appearance of optical fibers drastically changed the
paradigm of data transmission, because of two main
advantages that make it particularly in relation to wires:
 The optical fibers are completely immune to
electromagnetic interference as they are made of
dielectric material capable of transmitting pulses of
light, which means that data is not corrupted during
transmission.
 Optical fibers do not conduct electric current, so there
will be no problems with electricity, as voltage
instability problems or issues with spokes.
The physical phenomenon known as total internal reflection
is the responsible for light pulses to travel along the fiber by
successive reflections [10].
Although since 1870 it was known that light could describe
a curved path within a material, only in 1952 the Indian
physicists Narinder Kapany could complete its experiments
leading to the invention the optical fibers. However, only with
the appearance of devices capable of converting electronic
pulses into pulses of light, it was possible to transmit
information through the fiber [9].
There are two types of fiber:
 Multimode fibers with core diameters greater than
the single mode, allowing light to traverse the fiber
through several paths, which are strongly affected
by the effect of dispersion and, because of this, are
most often used for data traffic within walking
distance.
 Single-mode fibers with core diameters of the order
of less than one strand of hair and let light travel in
the interior of fiber by a single path, and less
sensitive to the dispersion effect and are therefore
most often used for long distance communications
[6], [8].
With this work, we hope to have contributed to better
understanding of the effects to which light pulses propagate
along the fiber are subject, both in the linear and nonlinear
regimes.
II. MODAL THEORY
Because of the difference between the refractive indices of
the core and the cladding, it is possible to transmit of light
inside the fiber [3]. The core index of refraction of the core
must be always higher than the refractive index of the
cladding. The modal theory can explain the existence of an
angle of incidence that enables the phenomenon of total
internal reflection.
A. Fibers operated in the Linear Regime
Optical fibers operated in the linear regime (monomodal
regime), are the fibers that allow the use of only one light
signal through the fiber. They have a smaller radius and
greater bandwidth due to lower dispersion.
It is possible to define a maximum value for the angle of
incidence with respect to the fiber axis, called acceptance
angle maximum θ0max, imposing that only the rays entering the
fiber at an angle of less than θ0max are propagated along it [13].
Let n1 be the refractive index in the core and n2 be the
refractive index in the cladding. The parameter which
describes the ability to collect light in a fiber called a
numerical aperture (NA) which is related as θ0max follows
2
NA  n0 sin 0max  n12  n22  n1 2
there is a minimum frequency of operation (normalized cutoff
(2.1)
frequency
where n0 is the refractive index of the outer medium in which
the fiber is inserted, usually the air, where n0=1. Δ is the
dielectric contrast given by
n12  n22
(2.2)
2n12
For better understanding the wave propagation inside a step
index fiber with core radius a, one may consider the scheme
depicted in Fig. 2.1.

Fig 2. 1 – Rays propagation in fiber [10].
The longitudinal component of the electric field in
cylindrical coordinates has the following form
Ez (r, , z, t )  E0 F (r )exp(im )exp i(  z  t )
(2.3)
where exp[i(βz-ωt)] is the time and space dependence of
monochromatic light traveling along the fiber axis, E0 is the
amplitude of the field, m is the azimuthal variation index, F (r)
represents the radial modal eigenfunction, β is the longitudinal
propagation constant, which is related to the free-space
propagation constant as
k 2 ( r)  n2 ( r)k02   2
vc  2.4048 , which is
the cutoff value for the second propagating mode, indicating
whenever the fiber is operating below that value, the system
monomodal, or multimodal if above, i.e., for v  vc =>
singlemode fiber, for
v  vc => multimode fiber.
B. Propagation Modes in Fiber
Light, while propagating in the fiber, can be seen sa an
electromagnetic phenomenon, and the whole propagating
mechanism, which can be described by the electromagnetic
optical fields associated to it, is governed by Maxwell
equations [1].
Generally speaking, modes can be classified into TE
(Transverse Electric) modes, where there is no component of
electric field in the direction of propagation (Ez = 0), TM
(Transverse Magnetic) modes, where there is no magnetic
field component in the direction of propagation (Hz = 0), and
hybrid modes, for which there have both longitudinal field
components, i.e., they have electric field (Ez ≠ 0) and
magnetic field (Hz ≠ 0) components in the direction of
propagation. These latter modes can be classified into HEmn
and EHmn, where the parameter m refers to the azimuthal field
variation and the parameter n refers to the radial field variation
[10].
In general, the surface waves guided by an optical fiber are
hybrid modes, and their field components are governed by the
Bessel functions of first (Jm) and second (Km) species,
satisfying the following equation modal

u2   v 
Rm (u) Sm (u)  m2 1  2 2   
v   uv 

4
(2.8)
where
(2.5)
in
vacuum
is
Let k  q in the cladding and k  h in core. Since there
is a surface wave guided by the cladding sheath, suffering total
internal reflection at the core-cladding interface, under these
conditions, one must have q  i .
Introducing normalized (dimensionless) wavenumbers
u  ha and w  a , and a normalized frequency v , their
relation is given by
v 2  u2  w2
the fiber. A special value is given by
(2.4)
where
 n1 for r  a

n( r )  

n2 for r  a
and the propagation constant
k0   c  2  .
vc ) for which a certain mode starts to propagate in
(2.6)
Moreover, introducing the normalized modal refractive
index as
u 2 w2 (  k0 )2  n22
b  1 2  2 
(2.7)
v
v
n12  n22
Rm (u) 
Sm ( u ) 
J m' (u)
K ' ( w)
 m
uJ m (u) wKm ( w)
J m' (u)
K ' ( w)
 (1  2) m
uJ m (u)
wKm ( w)
(2.9a)
(2.9b)
The resolution of the Eq. (2.8), while being very complex,
turns it to be necessary to use numerical methods.
Only when the azimuthal index is zero (m = 0) it is possible
to propagate transversal TE0n and TM0n modes.
According to the Gloge approximation (i.e., for Δ << 1), in
which modes propagating in optical fibers are weakly guided,
one may take Rm(u)= Sm(u), which will cause the modal
Eq.(2.8) to be reduced to
mv 2
Rm (u)   2 2
(2.10)
uw
where the plus (+) sign corresponds to the modes EHmn and
the minus (-) sign corresponds to the modes HEmn.
Through the relationship between the Bessel functions and
their derivatives under the Gloge approximation, in which the
3
azimuthal index zero, the Eq (2.10) for both modes (EHmn
HEmn) reduces to
J 1 (u )
K ( w)
 1
0
(2.11)
uJ 0 (u) wK0 ( w)
inferring that the modes EH0n are actually TM0n and modes
HE0n are modes TE0n . Moreover for weakly guided fibers,
these modes are almost linearly polarized, and therefore will
be termed as LPpn.
One should stress that the only mode capable of propagating
in a single mode fiber system (the only mode whose cutoff
corresponds to vc=0), called the fundamental mode, is the
HE11 mode.
Mosr propagating modes are degenerated modes, i.e., they
have the same cut-off frequency but with different field
structure. In particular,
 HE1n modes give rise to modes LP0n
 Modes TE0n, and TM0n HE2n are degenerate and
give rise tomodes LP1n
 HEm+1,n modes and EHm-1,n with m ≥ 2 are also
giving rise to degenerate modes LPmn.
Therefore, one has: EHmn→LPpn (with p=m+1); HEmn→LPpn
(with p=m-1). Accordingly, one has the following modal for
LP modes:
J (u )
K ( w)
u p 1  w p 1
0
(2.12)
J p (u )
K p ( w)
For the fundamental mode (LP01), it reduces to
uJ1 (u) K0 ( w)  wJ 0 (u) K1 ( w)
(2.13)
Figure 2.2 shows the variation of the normalized modal
refractive index b as a function of normalized frequency v for
the first six modes propagating in the fiber.
Fig 2. 2 - First six LP mode fiber: diagrams b(v).
One can observe a general increase in the normalized
refractive index b modal with increasing normalized frequency
v, and the greater the normalized frequency v, more modes
propagating in the fiber.
It is also interesting to analyze the influence of the dielectric
contrast on the dispersion curves b(v). as in Figure 2.3
Fig 2. 3 - Influence of dielectric contrast on the dispersion curve b (v).
Noting that an increase in the dielectric contrast Δ causes an
increase in dispersion curves b(v), one may conclude that is
possible to reduce effects of dispersion by using fibers with a
smaller contrast.
III. PULSE PROPAGATION IN THE LINEAR REGIME
Any pulse, while propagating in an optical fiber, especially
in the linear regime, suffers the effect of time dispersion,
which causes its broadening and may create interference
between symbols, which can greatly limit the bandwidth of the
signal to be transmitted.
Let one consider an optical pulse with a finite spectral width
to be launched into the fiber. Each spectral component of the
pulse, while traveling along the fiber, has a different group
velocity which depends on its wavelength according to
1
2 c
vg 

(3.1)
 with


The dispersion due to the difference between the
propagation velocities of different spectral components is
called the Group Velocity Dispersion (GVD).
There are two main types of dispersion: the Intermodal
Dispersion (between the various modes of propagation) and
intramodal dispersion (under the same mode of propagation).
Intermodal dispersion is due to the various propagation
modes travelling in the fiber. Each mode has a different group
velocity for the same wavelength and the pulse width at the
output of the fiber will depend on the transmission times. The
time delay between the fastest (i.e., the fundamental mode)
and the slowest mode (mode of higher order, depending on
how many modes can propagate in this multimode fiber) is
responsible for the broadening of the pulse at the output of the
fiber.
In the case of the intermodal dispersion or, as commonly
referred to, the chromatic dispersion, the broadening of the
pulse occurs within the same mode, mainly because of the
dependence of the refractive index of the material with the
frequency.
However, the chromatic dispersion is a consequence of
combined effect of two factors: the material dispersion and the
waveguide dispersion [6], [11].
In the case of the material dispersion, the refractive index of
the constituent materials of the fiber has a non-linear variation
with the wavelength
dn
ng  n  
(3.2)
d
4
Waveguide dispersion follows from the fact that, for a given
mode, the energy distribution between the core and the
cladding is a function of wavelength. Generally, 80% of
optical power propagating in the fiber remains confined to the
core while the remaining 20% propagate along the cladding at
a speed greater than the core causing the broadening of the
pulse at the fiber output [6].
In the neighborhood of the zero dispersion wavelength
(ideal frequency band) the spectral components at different
wavelengths have almost the same propagation velocity, in
other words, propagation delay is quite constant at different
wavelengths.
In normal dispersion region, one has Dλ <0 => β2> 0. The
group velocity decreases with the frequency, i.e., in the red
spectral components travel faster than the blue ones.
Fig 3. 4 - Normal dispersion [5].
For a standard silicon optical fiber, the zero dispersion
wavelength occurs around 1300 nm. Around this value, there
is no pulse broadening. For this reason, the most current
optical communication systems have been developed to take
advantage is this characteristic [12].
Fig 3. 1 - Zero dispersion [5].
Discarding higher order dispersion effects, the parameter
which describes the fiber dispersion is given by
2 c
D   2  2
(3.3)

with the DGV coefficient β2 is given by
1 vg
2   2
(3.4)
vg (0 ) 
Let one take into consideration the electromagnetic
spectrum as depicted in Figure 3.2.
A. Pulse Propagation Equation in linear Regime
In this Section, the differential equation that governs the
propagation of pulses in the linear regime along a single mode
optical fiber with small contrast (Δ << 1), is derived ignoring
the effect of higher order dispersion.
Let A(0,t) be the pulse at the fiber input at z = 0. Assuming
that the pulse modulates a carrier, with an angular frequency
ω0, and that the electric field is linearly polarized along the x,
one may write
ˆ 0 F ( x, y) A(0, t )exp( i0t )
E( x, y,0, t )  xE
where F (x, y) is the modal function representing the
transversal variation of the fields of the LP01 mode.
With the help of the Fourier transform (numerically
computed through the FFT-Fast Fourier Transform and
Inverse Fast Fourier Transform-IFFT), the pulse envelope at a
generic point of the fiber is given by
Fig 3. 2 - Electromagnetic spectrum.
In the anomalous dispersion region, one has Dλ> 0 => β2
<0. The group velocity increases with the frequency, i.e., the
blue spectral components travel faster than those red (see
Figure 3.3.).
(3.5)
A( z, t ) 
1
2

 A( z, )exp(it )d

(3.6)
Thus the electric field in a generic point of the fiber is given
by
ˆ 0 F ( x, y ) A( z, t )exp i( 0 z  0t ) 
E ( x, y, z, t )  xE
(3.7)
where
0   (0 )
(3.7a)
In order to obtain A(z,t)) from A(0,t), we introduce the
following GVD coefficients
Fig 3. 3 - Anomalous dispersion [5].
m 
m 
 m
(3.8a)
5
where
1 
2 

1

 vg (0 )
 2  1
1 vg

 2
2


vg (0 ) 
(3.8b)
(3.8c)
For the numerical solution of Eq. (3.10) is useful to
introduce the normalized (dimensionless) space and time
variables
z
(3.14a)
 
LD

The frequency deviation over the carrier is given by
    0
(3.9)
As |Ω|<<ω0, it is reasonable to disregard the dispersion
coefficients above β2. Moreover, discarding the losses in the
fiber, the linear propagation equation, which allows
calculating A(z,t) from A(0,t), can be written by
t  1 z
0
(3.14b)
So equation (3.10) can be rewritten as
A( , ) i
 2 A( , )
 sgn(  2 )
0

2
 2
(3.15)
and the Fourier pair as
A( z, t )
A  2  A( z, t )
 1
i
0
z
t
2
t 2
2
(3.10)
A( , ) 

 A( , ) exp(i )d
(3.16a)

This equation can be solved using a simple algorithm which
allows the computation of pulse propagation along the fiber.
This algorithm is herein designated RIMF algorithm and is
applied in three steps:
First step: To compute
A( , ) 
1
2

 A( , ) exp(i )d
(3.16b)

where ψ is a normalized frequency given by

A(0, )  FFT  A(0, t ) 
 A(0, t ) exp(it )dt
   0  (  0 ) 0
(3.17)

Second step: Then compute
To apply the RIMF algorithm to Eq. (3.15), one has
A( z, )  A(0, )exp i() z 
First step: To calculate A(0, )  FFT  A(0, )
Third step: Finally
1
A( z, t )  IFFT  A( z, )  
2
with
()  1 
2
2
Second step: Then

 A( z, ) exp(it )d 

2
(3.11)
Because the pulses are usually narrowband |Ω|<<ω0, A(z,t) is
a slowly varying function in time and oscillates with exp(-iΩt).
In the absence of the dispersive effects, the pulse propagate
without distortion with a group delay
 g  1 z
 i

A( , )  A(0, ) exp   sgn( 2 ) 2 
2


Third step: Find A( , )  IFFT  A( , ) 
Figure 3.5 illustrates the effect of the GVD on a secanthyperbolic shaped pulse, i.e., one whose initial shape is
A0 ( )  A(0, )  sec h( )
(3.18)
(3.12)
Since it is reasonable to neglect the influence of the higher
order dispersion (β3=0), it is possible to isolate the effect of
the GDV in the pulse propagation.
Defining τ0 as the characteristic pulse time width, the
dispersion length is introduced as
LD 
 02
2
(3.13)
The dispersive effects in a optical link of length L are
negligible only if LD>L.
Fig 3. 5 - Comparison of the pulse input and output fiber.
6
A(0, )  FFT  A(0, t )
 A0 0

2 02 
2
exp  

1  iC
 2(1  iC ) 
(3.24)
The pulse spectral intensity will be
Fig 3. 6 - Evolution of the pulse.
2
B. Chirp
Since the optical signals emitted by a laser source suffer
from chirp, it is useful to introduce a chirp parameter C which
quantifies the variation in the carrier frequency
C  c
(3.19)
where βc is Henry factor, responsible for the enlargement of
the spectral line.
When considering the linear propagation of a Gaussian
pulse with the Chirp [3], the initial pulse can be written as
 1  iC  t 2 
A(0, t )  A0 exp  
  
2  0  


where A0 represents the pulse amplitude.
A(0, )  A02 02
 2 02 
exp  
2 
1 C2
 (1  C ) 
2
(3.25)
The spectral width Δω at 1/e maximum will be
 
1 C2
0
(3.26)
One may conclude that the larger the value of C, the larger
the pulse spectral width.
(3.20)
Fig 3. 8 - Evolution of Gaussian pulse with C=0.
Fig 3. 7 - Initial Gaussian pulse for C=0.
The frequency shift δω(z,t) caused by the existence of the
chirp is

 z t z
(3.21)
( z, t )  C  (1  C 2 ) 22  2 1
 0  1 ( z)

with
(3.22)
1 ( z )   0 ( z)
and η(z) is the pulse broadening factor
2

z   z 
 ( z )  1  sgn(  2 )C    
LD   LD 

Fig 3. 9 - Evolution of Gaussian pulse with C=2.
The pulse broadening factor η(z), introduced in Eq. (3.23),
also shows that a pulse may also suffer a time compression as
it propagates, as long as β2C<0 [3]. Figure 3.10 shows the
variation of the broadening factor with the travelled distance
for a Gaussian pulse in the anomalous dispersion region
(β2<0)
2
(3.23)
When applying the first step of the RIMF algorithm to the
initial impulse Gaussian, one has
Fig 3. 10 - Spatial evolution of spectral width pulse for different values of
C.
7
The pulse broadening due to the dispersion is sensitive to
the sharpness of the pulse. When considering pulses with
sharper steep edges, the broadening is generally greater, as is
the case Super-Gaussian pulse whose initial momentum is
given by
 1  iC  t 2 m 
A(0, t )  A0 exp  
  
2  0  


(3.27)
Fig 3. 14 - Evolution of Super-Gaussian pulse with C=2.
This expression is quite identical to the Gaussian pulse,
differing only in the value of the parameter m. For a Gaussian
pulse m is equal to 1. The edges of the pulses become
increasingly steep as m increases, as shown in Figure 3.11,
when compared with figure 3.7.
Fig 3. 11 - Initial Super-Gaussian pulse for C=0, to m=3.
The parameter m is related to the duration tr for which the
intensity of the pulse increases from 10% to 90% of its peak
value
tr 
0
m
(3.28)
Eq. (3.28) shows that a pulse with a smaller rise time
increases faster [2]. For this reason the super-Gaussian pulses
as well as expand faster than the Gaussian, also strongly
distort its original shape, as shown below, when considering
m=3.
Fig 3. 12 - Super-Gaussian comparison pulse input and output fiber.
Fig 3. 13 - Evolution of Super-Gaussian pulse with C=0.
C. Dispersion Compensation
A technique used to compensate for or minimize the
problem of dispersion is the use of dispersion compensating
fibers (DCF –Dispersion Compensation Fiber). This technique
is to combine optical fibers with different characteristics, such
that the average GDV of the link becomes quite small,
whereas the GDV of each section may be large [7]. In each
two consecutive sections, there are two types of fiber. A
section of greater length L1, operating in the anomalous
dispersion region (with β21<0), and a shorter section of length
L2, operating in the normal region (with β22>0). Nevertheless,
the two GDV coefficients are quite different (|β21|≠|β22|).
This technique takes advantage of the linear nature of the
system when considering the propagating optical pulse into
two sections of the fiber, whose impulse propagation equation
is given by
1
 i

A( L, ) 
A(0, ) exp    2 (  21L1   22 L2 )  i  d

2
 2

(3.29)
where L  L1  L2 .
The length of dispersion compensating fiber L2 is chosen so
that
21L1  22 L2  0  L2  
 21
L
22 1
(3.30)
Pick-up |β21|<<|β22| so that L2 is much lower than L1. When
using this technique it is common for L1 to be in the order of
100-1000 km, while L2 is in the order of 1 km. Provided that
they fulfill the condition β21L1+β22L2=0, A(L,τ)=A(0,τ) thus
recovering its original shape once every two consecutive
sections, although pulse width may change significantly in
each section. As an example of application, we consider
L1=20 km, β21= -1, L1=20 km e β21=1 and the pulse input of
the first fiber is given by
A0 ( )  A(0, )  sec h( )
(3.31)
Fig 3. 15 – Comparison input and output pulse in the 1st fiber.
8
The output pulse of the first fiber is the impulse input of the
second fiber (DCF).
So, taking into account the shape of the pulse to propagate
Fig 4. 1 - Power of a Gaussian pulse [7].
Fig 3. 16 - Comparison input and output pulse in the 2nd fiber.
IV. PULSE PROPAGATION IN NONLINEAR REGIME
Pulse propagation in nonlinear regime is affected by the
optical Kerr effect. The propagation of impulses in Nonlinear
Dispersive Regime (NLDR) is governed by the SPM and the
GDV simultaneously [4].
To disrupt the relative dielectric constant
(4.1)
 '   ( x, y )  
where
(4.2)
  2n( x, y )n
Whatever the process that led to that disturbance, the new
longitudinal propagation constant is given by
(4.3)
 '    
Introducing the parameter

wherein
n2' k0



n2' 2
ref2 
(4.4)
 is the effective area and ref is the effective radius.
The optical Kerr effect provides that for certain values of
n2'
(4.5)
   P( z, t )
P( z, t ) is the transported power, that is related to the power
of the internal fiber Pin (t ) and attenuation coefficient of the
fiber  as follows
(4.6)
P( z, t )  Pin (t )exp( z)
The nonlinear phase generated by Kerr effect will be
L
L
L
NL (t )   (  '   )dz    dz    P( z, t )dz
0
0
And because the effective length is
1
  1  exp(  L)
Comes to
 NL (t )
A. Non-Linear Equation Shrodinger
In non-linear regime, the equation of propagation of pulses
is governed by NLS equation, that neglecting effects of higher
order and losses is given by
u 1
 2u
2
(4.10)
i
 sgn(  2 ) 2  u u  0
 2

If incident peak power P0, observe the relationship
P0 

n2' 2
It has in front of the impulse dPin dt  0  (t )  0 ,
yielding a redshift. Similarly the tail of pulse
dPin dt  0  (t )  0 , causes a blueshift. Since this is an
effect contrary to what happens in GDV to the anomalous
dispersion, thus allows the propagation of pulses that retain
their shape along the propagation [4].

NL (t )   Pin (t )
(4.7)
0
(4.7)
(4.8)
Noting then that the nonlinear phase depends only on Pin(t),
hence the name of Self-Phase Modulation. The instantaneous
frequency deviation of the local (over carrier) caused by the
SPM pulse is
d
dP (t )
(t )   NL   in
(4.9)
dt
dt

2 n2' L
(4.11)
The propagation of pulses in a fiber of length L in simplistic
terms [4], is governed by the nonlinear regime for L>LNL and
the linear regime for L<LNL, for a grossly way, is possible to
distinguish four regimes of propagation in the fiber:
 Non-dispersive linear regime (NDLR), when L<LNL
and L<LD ( disregard GDV and APM effects)
 Dispersive linear regime (DLR), when L<LNL and
L>LD (disregard APM , only acts GDV)
 Non-dispersive nonlinear regime (NDNLR), when
L>LNL and L<LD (disregard GDV, only acts APM)
 Non-dispersive nonlinear regime (NDNLR), when
L>LNL and L>LD (acts GDV and APM
simultaneously, thus allowing the propagation of
solitons).
B. Analytical Solutions of NLS Equation
By limiting the analysis to just in case the solitons as they
occur (anomalous dispersion zone, where sgn(β2) = -1), comes
to NLS equation
u 1  2u
2
i

 u u0
(4.12)
2
 2 
Which has the analytical solution for solitary waves
 

u( , )   sec h   (  q0 ) exp i   2  0 
(4.13)

 2
Where

is the parameter which sets both the amplitude
and pulse width, q0 is pulse’s center in relation     0 and
0 is initial phase (in     0 ).
9
Treating a wave with localized surrounding ( u( , )
is
independent of  ), apart from that when  tends to  ,
u( , ) approaches 0.
When considering the loss (    LD  0 ) and 3rd
dispersion order (   3 6 2  0 ), variables
represents
Give name of fundamental soliton the canonical form of Eq.
(4.13) by making   1 and q0  0 to phase zero.
  
u( , )  sec h( ) exp  i 
 2
(4.14)
N
which
non-linearity and
D which represents the
dispersion, are given as a function of 

2
(4.18a)
N    i u
2
i
2
3
(4.18b)
D   sgn(  2 ) 2   3
2


What is not entirely true, but it is considered the variation of
u with negligible  .
When considering the incident pulse u0 ( )  u(  0, )
type, Eq.’s (4.17) solution will be
u( , )  exp  ( N  D ) u0 ( )
Making
u(  h, )  exp h( N  D ) u( , )
Fig 4. 2 - Evolution of fundamental soliton.
In case of periodic waves, must be taken into account that
using the inverse of the dispersion or IST (inverse scattering
transform), shows that any incident pulse shape
(4.15)
u0 ( )  u(  0, )  N sec h( )
Where N represents the soliton order. For soliton order N≥2,
unlike the fundamental soliton, pulses do not retain their shape
along the propagation, shows instead an evolution periodic
with period  0
  2 in real units represents
z0 

2
LD 
  02
2 2
(4.16)
(4.19)
(4.20)
Eq. (4.20) sets up an iterative scheme of longitudinal step
h , which enables the start (   0 ) at the end of the fiber
(    L  L LD ). Trying to divide the total space propagation
0     L in small sections elementary length h .
SSFM consists of two consecutive procedures, dividing the
Eq. (4.20) in
v( , )  exp(hN )u( , )
u(  h, )  exp(hD )v( , )
According to Eq. (4.23a),
(4.21a)
(4.21b)
v( , ) is given by
2
 h 
v( , )  exp     exp ih u( , )  u( , ) (4.22)


2


Using the Fourier transforms (1st step RIMF algorithm),
then defines
Fig 4. 3 - Evolution solution of the 3rd order.
v ( , )  FFT v( , ) 

 v( , ) exp(i )d
(4.23)

C. Numerical Simulation NLS Equation: Split-Step Fourier
Method
The disclosed method for solving nonlinear equations
propagating pulse is SSFM, which the optical apply the
numerical [4], is to separate the equation (4.10), non-linear of
dispersion.
Equation (4.10) can be written in a more compact form
u( , )
 ( D  N )u( , )  0

(4.17)
And these conditions
D becomes na algebraic operator
D , according to Eq. (4.18b) has been
i
sgn(  2 ) 2  i 3
(4.24)
2
By taking into account Eq. (4.21b) - 2nd step of the RIMF
algorithm
u(  h, )  exp(hD )v ( , ) 
D 
 h

exp i sgn(  2 ) 2  exp(ih 3 )v ( , )
 2

(4.25)
10
Each iteration to finally, apply 3rd step RIMF’s algorithm
u(  h, )  IFFT u(  h, ) 
1
2

 u(  h, ) exp(i )d
(4.26)

Thus SSMF summarizes the application of RIMF algorithm
in case of non-linear equations, using an iterative scheme
longitudinal h , showing such a method since it is very
effective to use longitudinal steps quite small, because the
smaller step size, greater iterations number and increasing the
number of iterations, the more efficient it becomes method.
Enhancing here that for simulation of all pulses in nonlinear regimes, used SSFM (as was case in Figures 21 and 22).
D. Characteristics of Fundamental Soliton


Area of soliton does not depend on any parameter
characteristic fiber [4]
(4.27)
A  2
Energy of the soliton is given by
2 2
(4.28)
Es 
 0
Shown to be inversely proportional to the temporal width
 Power spectral density is directly related to energy,
as follows

1
(4.29)
Es 
 S ( ) d 
2 
V. CONCLUSIONS
In this paper we have shown that an optical pulse
propagating along an optical fiber, suffers the effect of
dispersion which broads the pulse at the output of the fiber.
This is a limiting factor for the bandwidth of the transmitted
signal. In linear regime, the technique of using the dispersion
compensating fibers every two consecutive sections, shows to
be very efficient and to solve the effect of the dispersion.
In non-linear regime, pulses are subject to AMF (caused by
the optical Kerr effect). In the anomalous dispersion region,
this effect is contrary to the DGV, thereby allowing the
propagation of pulses very interesting because it does not
changes its shape over propagation in the fiber. These pulses
are called soliton. For the simulation pulse scheme is nonlinear regime, we have used the SSFM, which has proved to
be as efficient as higher the number of iterations used in the
method.
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[1]
[2]
[3]
[4]
[5]
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Paiva, C. (2010). Fotónica. Fibras Ópticas .
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