Abstract: In this research, we study a nonsimultaneous three-soliton collision in the presence of third-order dispersion in WDM systems. The interaction between solitons may be viewed as an inelastic collision in which energy is lost to continuous radiation owing to nonzero third-order dispersion. We develop a perturbation theory with two small parameters; the third order dispersion coefficient d3 andthe reciprocal of the interchannel frequency difference, 1/β. In the leading order the amplitude of the emitted radiation after each collision is proportional to d3/β2. In addition, the only other effects up to the combined third order of the perturbation theory are phase changes and position shifts of the solitons. It has been shown that after each collision the rate of emitted energy is the same.
INTRODUCTION
In recent studies, it has been shown that Wavelength-Division Multiplexing (WDM) can be a successful method to increase the transmission capacity of optical fiber systems (Evangelides and Gardon, 1996; Etrich et al., 2001). However, there are some limitations on the performance of these systems. One of the major ones is due to the nonlinear interchannel interaction of data signals from different channels (Agrawal, 2002). The increasing demand for the development of optical communication systems for high data rate transmission and high quality information leads to the increase in the number of channels used and a decrease in the width of the pulses launched. As a result, the importance of the interchannel collisions between optical pulses is expected to increase and an accurate description of the effects of these collisions is required.
In WDM method, solitons at different wavelengths are injected at the input of the fiber. Since, the spectral separation between these solitons (channels) is λ<=0.5 nm and they have different velocities in different channels, collisions among them inevitably take place (Agrawal, 2002). In here, full attention is given to the effect of third-order dispersion on the interchannel collisions between solitons, which is expected to be dominant (in comparison with other inelastic effects) near the zero-dispersion wavelength.
In an ideal fiber, interchannel collisions between solitons can be modeled
by using the nonlinear Schrödinger equation (NLSE) (Hasegawa and Matsumoto,
2002). In that case since the collisions are elastic no radiation is emitted.
Moreover, by using NLSE the amplitude, frequency and shape of the solitons do
not change. Using this model for an ideal collision only leaves us a phase shift
proportional to
Accurate analysis of the effects of perturbations on interchannel collisions is a very complicated and long-standing problem. The main technical issue in this case is how to develop a perturbation theory around the multisoliton solutions of the ideal NLSE. In spite of the existence of exact expressions for the multisoliton solutions of the ideal NLSE, direct perturbative analysis around the complex multisoliton solutions has not yet been successfully implemented.
Our aim in this work is to study the effects of the second collision the perturbative soliton from the origin channel (It means a soliton that has shifted its phase and position because of a collision with another soliton from the other channel and partially its energy is lost at continuous radiation) with stationary soliton from the other channel. Moreover, we calculate the dynamics and the total intensity of the continuous radiation emitted as a result of two collisions and also the change, induced by the collision, in the soliton parameters. That before is shown (Plege et al., 2003) interaction between stationary two-soliton in presence of third order dispersion lead to O(1/β) phase shift, O(1/β2) position shift. In addition, the amplitude of the emitted radiation is proportional to O(d3/β2). It should also be mentioned that the propagation of a single pulse in the presence of third-order dispersion was studied in detail by Kodama, (1985) and Horikis and Elgin (2001). It was found that even if the pulse launched into the fiber is not exactly of the stationary form it evolves into the stationary form after a transient (Elgin et al., 1995).
Notice that the major technical tool used in the analytical calculations is singular perturbation theory that is an appropriate extension of the technique developed (Kaup, 1990).
MATERIALS AND METHODS
Propagation of an electrical field wave packet ψ (t, z) through an optical fiber under the influence of third order dispersion is described by the following modification of the nonlinear Schrφdinger (Agrawal, 2001)):
(1) |
Where, z is the dimensionless position along the fiber
For d3≠0, Eq. 1 is not integrable. However, in many particle cases d3<<1, allowing a perturbative calculation about the integrable d3 = 0 limit. Fiber losses in Eq. 1 are neglected. In practice, this can be achieved by compensating for losses in a fiber span by means of distributed optical amplification, e.g., Raman amplification.
Let us assume that d3<<1 and derive perturbatively a z-independent (stationary) single-soliton solution of Eq. 1. When d3 = 0, the single-soliton solution of Eq. 1 in a β frequency channel is described by
(2) |
Fig. 1: | Propagation two solitons in channels 0 and β at a fiber |
Where, αβ, ηβ and yβ are the soliton phase, amplitude and position, respectively. Assuming that d3<<1, we will be looking for a stationary perturbative single-soliton solution of Eq. 1 in the form
(3) |
|
and
The term
Fig. 2: | Non simultaneous, Three-soliton collision: η2 = η1 = η0 = 1, β2 = 10, β1 = -10, β0 = 0, y2 = 0, y1 = 0, y0 = 10 |
Symbolically we assume the perturbative soliton as an ideal soliton Ψ0 in which there are some changes in the former one due to the first collision that can be used as initial conditions for the second collision. We are looking for a two-soliton solution of Eq. 1 in the form, Ψtwo = Ψ0+Ψβ+Φ where, ψ0 and ψβ are described by Eq. 3 with β = 0 and β, respectively and Φ΄ is a small correction due to collision. It is straightforward to check that the exact two-soliton solution of Eq. 1 at d3 = 0 acquires the form
(4) |
Where, Φ0 and Φβ are corrections of the leading order 1/β in the channels 0 and β, respectively. The terms Φ-β and Φ2β correspond to O (1/β2) corrections in channels-β and 2β, respectively; the two latter corrections are exponentially small outside the collision region. By analogy with the ideal d3 = 0 case, one substitutes a solution of the form Eq. 4 into Eq. 1 and calculates Φ΄0. Since Φ΄0 oscillates together with ψ0 and d3<<1, one neglects the exponentially small contributions from the terms rapidly oscillating with t and z. Then the equation describing Φ΄0 is
(5) |
Vicinity (in z) of the collision event is given by
Where
we use this region as
Even though the rigorous calculation of the effects of the collision in successive orders of the perturbation theory is quite complicated, the main result can be derived in a straightforward manner by use of just a few equations. The initial condition for Φ΄(0)0 before the second collision is equal to solution of after the first collision in the first order perturbation theory
(6) |
Where:
(7) |
In Eq. 7 and the following equations the superscript in stands for initial values of phase, position, etc., while the superscript out represents final values of the same parameters.
In the collision region Eq. 5 reduces to
(8) |
Integrating Eq. 8 over the collision region and using the initial condition 6 at
one arrives at
(9) |
And
(10) |
Comparing Eq. 9 and 6, we see that the only effect of the collision in the first order of the perturbation theory is a change of the soliton phase
(11) |
Notice that Eq. 11 is also equal to the result obtained from the first collision.
Calculation of higher order terms requires knowledge of the complete z dependence
of
(12) |
For second order perturbation theory similar to the first order perturbation
theory, the initial condition for the 0(1/β2) term
(13) |
Where:
(14) |
In the collision region the 0(1/β2) part of Eq.
5, one can show that the only change in the solitonss parameters comes
from the term
(15) |
Where:
(16) |
Integrating over the collision region and using the initial condition (13)
at
(17) |
Where:
(18) |
and
Because of,
Comparing Eq. 17 and 13, we see that the only effect of the collision in 1/β2 order is a position shift (time retardation) give by
(19) |
Also, we shown that after each collision the rate of position shift is the same.
Moreover, we see that the only effect of the collision in O(d3/β)
is a O(1/β) change of phase, on top of the O(d3) stationary
solution
Emission of radiation comes from the O(d3/β2) term. To analyze the O(d3/β2) correction, we first write it in the form
(20) |
Where
(21) |
The initial condition for is taken to be
(22) |
Where:
(23) |
i.e., the initial condition contains radiation.
The equation of source term for the emitted radiation Φ~´(1)R03 is
(24) |
Substituting Eq. 12 into Eq. 24 and integrating over the collision region, one obtains
(25) |
Where, the coefficient B is defined by
(26) |
and
(27) |
The functions
(28) |
Where, s = k/η0. After second collision, we find
(29) |
Dynamics of the coefficients ak(x) are given by
(30) |
Equations 29 and 30 describe the dynamics
of the term
The emitted radiation after each collision is of order d3/β2. The absolute value of this emission for four different values of z = z1, z = z1+1, z = z1+3 and z = z1+7 are shown in Fig. 3.
Since
(31) |
Fig. 3: | Absolute value of the radiation profile
function normalized to B after the second collision, i.e., |
Fig. 4: | Absolute value of the radiation profile
function normalized to B, i.e., |
So, according to the Fig. 4, for zz1+1, all z dependent contributions to ERO decay algebraically with z-z1. Thus, far away from the collision region the only nonvanishing contribution to ERO is
(32) |
Using Eq. 26, we can calculate the radiation energy emitted ERO by the reference channel soliton after each collision in far away from the collision region:
(33) |
RESULTS AND DISCUSSION
We finds that the only changes in the pulse parameters up to the third order of the theory are the O(1/β) phase shift, Δα0~ 4ηβ(1+3d3β)1/2 [(1+3d3β/2)|β|]-1 and the O (1/β2) position shift, Δy0 = - 4ηβ (1+3d3β)1/2 [(1+3d3β/2)2β|β|]-1.
Thus, the radiations propagate away from the soliton (in t) with velocity, is of the first order. The soliton retains its shape (such that, at each instant, the soliton is close to a stationary solution, width phase anaphase velocity) while evolving slowly. Note that the rate of emitted energy after each collision is always the same. Also, the rate of changes soliton parameters after each collision is always the same. Moreover, neglecting the decrease in the soliton amplitude, the total energy emitted by the reference channel soliton
as a result of many collisions with solitons from the β channel grows linearly
with the number of collisions. Taking η0 = 1 and requiring that
the widths of the colliding solitons are equal, we obtain ηβ
= (1+3d3β)1/2 with the assumption that (typical setup
for a short pulse optical fiber experiment) τ0 = 0.5 ps, β2
= -1 ps2 km-1, β3 = 0.1 ps3
km-1, d3 = 0.07, β = 10, p0 = 0.4 W, κ
= 10 W-1 km-1, Δv = 2.03x1012 Hz. Thus
for the parameters introduced, we calculated that the mean distance passed by
the soliton until it experiences 20000 collisions and loses about 10% of its
energy is approximately 2500 km (Plege et al., 2003). The soliton amplitude
and phase velocity do not acquire any change up to third order of dispersion
in WDM systems. The result for the soliton amplitude is consistent with the
conservation law for the total energy, which requires η = 1+O (d23/β4)
for both solitons. Results obtained from the second collision the perturbative
soliton from origin channel with stationary soliton from other channel at third
order perturbation collision is similar with ones after the first collision
the ideal soliton from origin channel with stationary soliton from other channel.
Moreover, after the second collision at this order of theory, we obtained also,
the O(1/β) phase shift on top of the O(d3) stationary solution