chapter 5 solitons in nonlinear directional...
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126
CHAPTER 5
SOLITONS IN NONLINEAR DIRECTIONALCOUPLER
5.1 INTRODUCTION
Motivation of this chapter is to investigate the behavior of solitons in a
directional coupler both because of its intriguing character as a physical and
mathematical phenomenon and from its possible applications. One of these is
an all optical switching which can be used as a routing device in many
applications, for example, in data transport systems. In one possible
realization of all optical switching, the signal for switching comes with the
pulse itself; its energy determines whether the pulse at the end of the coupler
arrives at the original or at the other channel. This chapter deals with the
soliton dynamics related to such a device. The aim of this chapter is primarily
to understand this switching phenomenon in a directional coupler by
investigating the geometry of solutions in the infinite dimensional function
space in which the system evolves. This is in analogy with the so-called phase
portrait analysis common in simple finite dimensional systems. This is done
by mathematical analysis known as Stokes polarization parameters.
Nowadays the nonlinear couplers are the backbone of all optical
processing. The nonlinear coupler has certainly been the most frequently
studied device, since it was proposed [1, 2]. The most frequently intense
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theoretical and experimental research activities are going on all around the
world. Use of soliton like pulse with couplers dramatized the field of
nonlinear optics. Most directional couplers are twin core couplers with a
permanent coupling between the cores. The coupling of energy from one
guide to the other occurs because of the overlap of the evanescent fields
between cores. Considering the linear coupling between the fields, and
assuming that the two interacting waveguides are equivalent, it can be shown
that pulse propagation in this type of device is described by two coupled
NLSEs [2, 3]. The physics of nonlinear couplers with continuous wave inputs
has been presented by Snyder et al., [4]. They establish the basic principles of
couplers operation and describe the main features of switching. Optical
nonlinear couplers are very useful devices which allow fast switching and
signal coupling in optical communication links. The nonlinear coupler
response to solitons has been described by Pare and Florjanczyk [5].
Nonlinear couplers also have applications as intensity dependent switches,
and they can also be used to multiplex and demultiplex the pulses. Optical
couplers are made as planar devices using semiconductor material [61 or dual
core single mode fibers [7]. Soliton propagation in fiber waveguides
supporting two coupled modes has been studied theoretically by several
authors [1-8] in the past years. The fast optical computers can be designed
using soliton switching and logic components [8, 9], perhaps involving
couplers. From theoretical aspects, an important point is the stationary pulses
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(soliton states), and their stability. The stability of the soliton states
determines the device properties for long term propagation and the behavior
of pulses in the coupler. Akhmediev and Ankiewicz [10] discovered different
families of soliton states in nonlinear fiber couplers. These soliton states are
pairs of unequal pulses which can propagate in a directional coupler without
changing their shapes. Soto-Crespo and Akhmediev [11] made a detailed
study of the stability of soliton states in nonlinear fiber couplers for lower
order soliton states. Ankiewicz et al., [12] also analyzed the possible states
formed by pairs of dark solitons propagating in dual core nonlinear optical
fiber couplers. These dark solitons propagating in two waveguide cores can
form coupled stationary states in a manner some what analogous to the
coupled states possible for bright solitons.
Port 3Port I
^^ r 16,Core 1
Core 2p ott PnA4
L6
(a)
Figure 5.1(a) The Nonlinear directional coupler (NLDC)
5.1 (b) Cross section of NLDC
5.1.1 FIBER COUPLERS
Fiber couplers are devices that consist of two parallel optical fibers
brought very closely together, as in Fig.(5.1). If 's' is the center-to-center
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separation between the two cores and 'q' the core radius, then the number
's/q' usually varies[13] between the values 2 and 4, with s/q = 2
corresponding to touching fibers. Figure (5.1) reveals that fiber couplers are
four-port devices: they have two input and two output ports. In the framework
of the light wave technology, their operation is based on splitting coherently
an optical field incident on one of the input ports and directing the two parts
to the output ports.
5.1.2 DIRECTIONAL COUPLERS
Since the output is directed in one of the two different directions, such
devices are known as directional fiber couplers or simply directional couplers.
Their full name 'nonlinear directional couplers' (NLDC) is due to the fact that
they exhibit nonlinear phenomena of Kerr type. These structures have been
studied extensively for their potential applications in all optical switching [2,
3, 5, 7, 14-22] in the context of two general theories: the normal mode theory
and the coupled mode theory. On the one hand, the normal mode theory
considers the coupler (that is made of two SMF) as a bimodal waveguide,
which supports two normal modes: the even mode with a symmetric field
distribution and the odd mode with an anti-symmetric one. In other words, the
coupler is considered as a single element device which supports two modes
(widely known as super modes [23]). Transfer of optical power between the
two cores is then described by the beating between these two super modes
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[24-26]. The coupled mode theory on the other hand considers the coupler as
a double element device with each one of its elements supporting one mode.
The two elements are the two single mode optical fibers and the mode that
supports each element is the actual propagating field distribution, in contrast
with the super modes whose linear combination represents the actual field
distributions and not the super modes themselves [23, 24, 27]. Here, the
transfer of optical power between the two cores is explained as evanescent
field-coupling between the modes of the individual cores of the coupler. The
mechanism is characterized by a parameter known as the coupling coefficient.
In general, the coupling coefficient is wavelength dependent (dispersive).
As far as the Inter Modal Dispersion (JMD) is concerned, it is a
phenomenon that arises from the coupling of the propagating fields inside the
two cores. According to the normal mode theory, on the one hand, the IMD
arises from the group delay difference between the two super modes of the
composite fiber. On the other hand, according to the coupled mode theory, the
IMD has to do with the frequency dependency of the coupling coefficient.
Notice that in [2, 3, 5, 7, 14-22] where no IMD is taken into account, the
coupling coefficient K is considered as a constant. This apparent discrepancy
between the two approaches is nothing but a matter of mathematical
perspective. In fact, Chiang [13, 24] showed that the group delay difference &
per unit length of the fiber is analogous to the first derivative of the coupling
131
coefficient with respect to the free space wave number or, in other words, that
the IMD in the normal mode theory is in fact equivalent to the coupling
coefficient dispersion of the coupled mode theory. It should be noted here that
the term IMD emanates from the normal mode theory as it literally means
'delay between the two supermodes', but is used properly as a term in the
coupled-mode theory, indicating its strong relation with the frequency
dependence of the coupling coefficient. Thus, both theories are unified in the
framework of this terminology. As far as previous works are concerned, the
coupling coefficient in all approaches [2, 3, 5, 7, 14 - 22] is considered as a
constant with respect to frequency: it is a parameter which depends on the
geometry of the waveguides of the coupler and the proximity of the two fibers
but is independent of the bandwidth coefficients of the pulse that is being
transmitted in the device. In order to take into account the frequency
dependence of the coupling coefficient a generalization of the model
equations is required. In References [24, 27], this generalization is made in
terms of the coupled mode theory. In fact the equation terms that describe the
IMD have been added after proper normalizations. The walk-off phenomenon
that is caused by IMD is described and the critical distance for the pulse
breakup is estimated. In these two works the possible dominance of IMD over
group velocity dispersion is highlighted. Inter-modal dispersion is also
experimentally demonstrated in [28] and numerical simulations have also
been presented in [25, 26, 29]. In these works, the evolution of the pulses in
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the coupler is observed under the influence of IMD. In References [25, 29],
the relation of the pulse width and the IMD strength is investigated and its
significant role in the model is explained. An analytic procedure of deriving
the generalized equations is presented in [30] where the drift phenomenon on
the time axis is described again via numerical simulation. In order to obtain
conclusions that can be useful for ultra-fast switching, optical pulses are
launched only in one of the input ports of the coupler. Hence the study
embraces the ways this optical power switches between the cores and how the
coupler directs it to the output ports. A coupler with specific (normalized)
constant parameters is considered and only the IMD coefficient is left
unbound to vary.
5.2 THEORETICAL MODEL
In search of an appropriate mathematical model to represent the
propagation of optical pulses inside the coupler, one can begin either from the
normal-mode theory or from the coupled mode theory [24, 31]. The normal
mode theory has the advantage of being more general as a method, since it
can be applied even to strongly coupled devices [24], to which the coupled
mode equations fail to apply. Nevertheless, when weakly coupled devices are
considered the coupled mode theory provides us the same results, although
making use of different equations, the coupled-mode equations. In this
chapter, guiding fibers leading to evanescent coupling have been considered,
133
where it is assumed that the nonlinear change in the refractive index does not
modify the cross sectional distributions of the mode fields. This feature has
already been discussed in section 5.1, when couplers with 2 < 4 wereq
referred (slq <2 would mean strong coupling). In fact, conventional single
mode fibers with slq > 2 are subject to evanescent field coupling, since the
HE,, mode that they support decays exponentially outside the fiber. As a
result, the optical field of the one fiber that enters the neighbor fiber is
extremely weak when the relation s/q> 2 is satisfied. Hence, both approaches
(normal mode, coupled mode) are valid and available for the evanescent field
directional coupler. Although the choice of the method is trivial at this point
(since both methods obtain the same results) the coupled mode theory is
chosen in this work, for it gives a better physical insight into the nature of
propagation. This is due to the fact that it represents the actual modes that
travel separately inside each of the two single mode fiber that consists the
coupler. A quite general model that describes pulse propagation inside an
evanescent field coupler is that of the coupled nonlinear Schrodinger
equations (CNLSE) and can be written as
A1 +ifl11 L_'+(CI A1J2+C2A22)A1
3A 2I +K01 A2 +iK11 ------- 2 0,at 2 at
134
aA aA2/3a2A2 +(c2, A,J 2 +C22 A2 1 2 )A2i—+fi02 A2 +11312
az at 2 ôt
aA,Ku a2A_ '=0. (5.1)+ 1CO2 Al + I K,2 at at
where 13o, PI., 132n and i, Kin, K2n, are the coefficients of the respective
Taylor expansions: )6,(o)) = )60, + (w - w,, ),8 + - (w - )2 fi2 +
n=1,2 with 6, =['V3J() = K0,, + (w - K1 +
1- (CO- 2 K2,, +
( 11K^n=l2wlthK --
dic
and Co0 is the carrier frequency. The coefficients fi .802 and /32n are
respectively, the wave number calculated at co = , the inverse group velocity
and group velocity dispersion, while iç, , Kin and K2n are, respectively, the
linear coupling coefficient, the IMD and a higher order term not specifically
named in the scientific literature. The subscript n refers to the pulse A n that
propagates inside the core n. More over C 11 , C 12 , C21 and C22 are the Kerr
coefficients that usually acquire the following values [32]. C 11 = C22 = C1
and C 12 = C21 = C2, where C 1 and C2 account for self-phase modulation and
cross-phase modulation respectively. The terms containing C 12 and C21 are
omitted, since these coefficients are associated with an overlap integral
leading to extremely weak XPM. Adapting transformation
A — U ExPL 1801
-2 )'
n=1,2,...
T=t_[
+1812))
2
The coupled mode Eqn.(5. 1) become
iL+U j 2LTl +(csPMuj2 +CxPMU2j)Ul
ôz 2 2T 28T2
5U2 !aLU2 0+K K01U2+i' 1 8T 2 ¶2
• au2 . öfl, au2 18a2u2 +CXpM U1 12+csPMlu2j2)u2
az2 2 - 2 ¶2
(5.2)au, K2 8U1
=0.+K02 U1 +iK12------- ¶2
where 8180 = /] - 1802 and 88 = 181I 1812.
Thus it is evident that 15,8, and 8/3k represent phase velocity mismatch and
group velocity mismatch respectively between the two propagating pulses.
When the two pulses coincide in wavelength these two terms vanish from the
equations, as long as the two fibers have the same material and geometrical
characteristics. By applying the following transformation to Eqn.(5 .2)
T z U.U
I;) ZO 'Uo
the coupled mode equations in normalized form are obtained. The quantities
T0, Z0, U0 are the reference time, distance and amplitude respectively that are
135
chosen at will. Usually To coincides with the initial temporal width of the
136
pulse, U0 with its initial amplitude and Z 0 with the dispersion length. The
Eqn.(5.2) becomes
jL+A/3u 2 +NLx,Mu2 2 )u
&z 22
+ 'COI U2 +i 'CI +-•=O,
&r 2 ôr
2 2"IÔU2AflU +NLpMu2 ) u2&r 2 al-2
(5.3)au, K22ÔU20
+K02U1az-2
The coupler in hand is symmetrical with identical cores through which pulses
propagate operating at the same wavelength. Therefore, the two pulses will
not suffer from phase velocity or group velocity mismatch. Taking into
consideration that, due to core separation, XPM is extremely weak [23, 25]
the NLXPM coefficient can be set to zero. One last adjustment concerns the
coefficients 1(21, K22 which are usually very small compared to the rest of the
coefficients [24, 27]. Under all the assumptions, the system of coupled
Eqn.(5.3) reduces to the following
U 1 +K0 U 2 +iK1O,--2z- ar
u2+K0u,au
+iK—'-2&r 25z
O. (5.4)-
fl2 zo U2Z
;NL = C(, = K (, Z() and K1 = K 1with D=– SPM 0 0
T2,M; K
T
137
T2Z0 =---- and U -
1)62 1 thenforfl 2! - T 2 CSPM
Further more, if Z0, U0 are chosen as
the case of anomalous dispersion (fl2 <0) the following results are obtained:
KI 2 KTD = 1, NLSPM = 1, K< =
0 and K - -'---.
1)621 11821
And Eqn.(5.4) takes the following form,
i ( U—' +2
u1+ic0u2=0,
a 'oJ 28r2
1 ÔU 2u2+1c0u1=0.
( a^ 'az) 2a
Having discussed the required theoretical model for the pulse propagation in
the nonlinear directional coupler, in what follows, the existence of symmetric
and anti-symmetric soliton states will be investigated. In addition, by
considering the cross-phase modulation effect in the coupler, the switching
mechanism using the well known polarization Stokes polarization parameters
will also be discussed in the forthcoming section.
5.2.1 SYMMETRIC AND ANTI-SYMMETRIC SOLITON STATES
In this section, it is intended to investigate the symmetric and anti-
symmetric states of the solitons which will be used to explain the switching
mechanism in the directional coupler. Here, the pulse propagation under the
influence of non-Kerr nonlinearity called quintic nonlinearity is considered
(5.5)
whose importance in the optical fiber communication has already been
138
discussed in the first chapter. Soliton like pulse propagation in dual core fiber
is expressed as two linearly CNLSE by including the effects of second order
dispersion, cubic and quintic self- and XPM effects. In the absence of quintic
nonlinear effect, the results are well known [11]. Linear coupling occurring
between the cores is described by a normalized coupling coefficient K. In the
normal dispersion case, the complex amplitudes U and V in the cores are
described by
.au ia2ul— +- --+aU2U+J3JUI4U+K V = 0,az 2at2
.av ia2vi—+-----+aV2V+/3JV4V+KU_–O. (5.6)az 2at2
Here U(z,t) and V(z,t) are the electrical field envelope, K is the
normalized coupling coefficient between the two cores. Z is the normalized
longitudinal coordinate; t is the normalized retard time. To get the solution of
the Eqn.(5.6), the form is chosen to be
U(z,q,t) = u(z,t) e' ; V(z,q,t) = v(z,t) e" (5.7)
The equations for pulses in a coupler can be obtained from the equations for
birefringent fiber using simple transformation of Eqn.(5 .6), by separating the
fast oscillatory part ejz from the envelope functions. It is possible to represent
the solution for Eqn.(5.6) in the form of Eqn.(5.7). Here 'q' is the signal wave
number shift, and 'u' and 'v' are the new envelopes functions. The equation is
modified as
a2u---=2(q—K)u-2a u 3 —2flu5. (5.10)
139
au 12uu +— ---+a uj2u +/3Iu4u + K v = 0,
8z 2at2
3v la2vi.--q v+----+alv2v+/3v4v+K u = 0. (5.8)
az 28t2
The above equations have stationary solutions UO(t) and Vo(t). These solutions
can be found by solving Eqn.(5.8), excluding the term involving the
derivatives with respect to z. The Eqn.(5.8) is symmetric when u = v; and in
this case it is rewritten as,
1 ô2u-----(q—K)u+au3+/3u =0. (5.9)
After a small algebra, the above equation is becomes
(dU)2 =-2(K—q)u 2 —au 4 _flu6 +C. (5.11)di'
where, C is the integration constant. In the following section, the existence of
different states of the soliton pulse in the nonlinear directional coupler is
investigated.
5.2.2 BRIGHT SOLITON FOR SYMMETRIC AND ANTI-SYMMETRIC STATES OF NONLINEAR COUPLERS
In this section, the existence of the symmetric and anti-symmetric states of
the soliton pulse in the NLDC under the influence of CQ nonlinearity is
discussed. When the constant of integration C = 0 the Eqn.(5.11) is modified
as
1
0.8
.. 0.6
0.4
0.2
0
140
dt = I2(q_K)u2_au4_/3u6. (5.12)
On solving the above equation for symmetric case when u = v
a(-i)
[ U" (t) = V0 (t) = + ,il
a+ cosh[8(q - K)
t]
2(q—K) 2(q—K))2 3(q—K)
(5.13)
On solving the Eqn.(5. 12) for anti-asymmetric case when u -v
[a+
a 1 [8(q + K) ]]
I
+ cosh
(-i)
u0 (t) = -v0 (t) = , I I I
2(q + K) \l 2(q + K)) 3(q + K)
(5.14)
-4 -2 0 2 4Time
Figure 5.2Bright soliton plot of Eqns(5. 13 and 5.14) for the values
= 0.5678 1, a = 0.82332, K = 0.678, q = 0.8
For the above two cases the solutions are obtained when the integration
constant is assumed to be zero. The bright soliton plot for both symmetric and
141
anti-symmetric cases is represented in Fig.(5.2). In the absence of the physical
parameters like linear coupling coefficient K and quintic nonlinearity /3,
Eqns.(5.13) and (5.14) degenerate into a soliton of a single NLSE.
5.2.3 DARK SOLITON FOR SYMMETRIC AND ANTI-SYMMETRICSTATES OF NON LINEAR COUPLERS
Now, one more soliton called dark soliton is discussed. To derive dark
soliton solution analytically for Eqn.(5.6), the Eqn.(5.1 1) is modified with a
small transformation u = 1/
(dRy (5.15)( dt) R ) R2 3R3 )
After a small manipulation of the above equation,
dR-- J- 4C h1+R+2( K -R3. (5.16)dt 3C) C) c)
R'
(dRj4CJ(RXRXR)dt
(5.17)
(^''\where e1e2e3 =-j -e1 e2 -ee3 -e2e3 =( a );
e1 +e2 +e3 = 2(K-q).
The solution oscillates between e 2 and e 3 where e 1 < e2 e3 are the real
roots of e.
e3 __________
dR= J.J-4Cdt.
l2 fCJ?_e1 XR_e2 XR_e3) e
(5.18)
Then the solution of the above Eqn.(5.18) is written as in Jacobi elliptical
function.
2lul 0.0.0.
142
R = e 3 +(e2 - e3)sn2 [jC(e i —e3)t,ni], (5.19)
2—k 22k2-1 k2+I 2 e3 _e2where e1 = ;e2 = ;e3 = - ; m = k =3 3 3 e3—e1
where in the modulation parameter. Equation (5.19) represents the dark
soliton ease for symmetric state when u = v; on the other hand for anti-
symmetric ease u = -v, the change in condition is
+e2 +e 3 = - 2(K+q)
C
Figure 5.3Dark soliton plot of Eqn.(5.19)
for the values e 1 = 0.3, e = 0.3, e 3 = 0.7, v = 0.3
and the solution is same as in the case of Eqn.(5.19). The dark soliton plot is
represented in Fig.(5.3) for the Eqn.(5.19). So far, the existence of symmetric,
anti-symmetric and asymmetric states of the bright and dark solitons has been
discussed. As pointed out in the introduction, in the following section, the
mechanism of switching is explained by considering an important nonlinear
143
effect called cross-phase modulation which actually influences the switching
dynamics in the nonlinear directional coupler.
5.3 SOLITON IN COUPLERS: QUINTIC SELF- AND CROSS-PHASE MODULATION EFFECTS
In order to explain the switching mechanism based on the self- and cross-
phase modulation effects, the following theoretical model has been
implemented under the influence of non-Ken nonlinearity. The propagation
of two coherent waves in a nonlinear dual core fiber can be described in terms
of two linearly coupled NLSEs. The coupled NLSEs are,
.au 1 82U +2V2)2U+KV=0,
ÔZ 28t2
• 8V 1 av (lV12 +2IU2)V+/3V2 +2U2)2V+KU=0. (5.20)3Z 2at2
Here U and V are the envelope functions and K is the normalized coupling
coefficient between the cores. The solution is of the form of
U(z,q,t) = u(z,t)e" ; V(z,q,t) = v(z,t)e'' Z (5.21)
where u and v are the real functions of t and q is the real parameter of the
solution. Then Eqn.(5.20) is modified with the help of Eqn.(5.21) and
rewritten as,
iu- —qu+Kv+^—'t +a (JU12
+2 IV1 2) U +,8 (JUll
+41vl' +4 I U 12 IV12) U = 0,
iv + ,fl
(lV11 + 4 Jul ' + 4 IU 121VI2) V
= O. (5.22)
144
From the above Eqn.(5.22), one can get u and v, and they are given below
with the derivative of z.
4 2u- =i qu+Kv+^—" +a
(JU12 +21VI2)U +,fl (U14
+4uIv)uJ
v =i
v + Ku +qV
-+a(lV 1 2
+2 u12)v+flvj+4Iu4 +4Iu 2 v 2 ) v (5.23)-
2
A convenient way to solve the Eqn.(5.23) is to use stokes parameters[33]
S" = Jul' +Jv12,
S i =k 2 —v2,(5.24)
S 2 =UV+UV,
S 3 =i(uv _u*v).
All these four parameters are real functions of z and t. They vary along the
fiber as well across the pulses. Hence we call it as differential stokes
parameters. Using stokes parameters Eqn.(5 .23) can be written in the form
dz fs,,dt=O,
- Js1 dt = —2K fs3dt,
(5.25)
fs2 dt =-3/3Js0s1s3dt_afs1s3dt,
doo10 10
_fs3 dt = 2KJs 1 dt + 3,8Js(,s l s2 dt + ajs1s2dt.
The above equations are integro differential equation and they become,
dz =o. (5.26)
dz = —2K S3 g. (5.27)
145
(5.28)= –3/1 SQ S1 S3 g –aS1 S3 g.dz
dS i=2KS1 g+3flS(, S 1 S29+aS1 S29-dz
Using Eqns.(5.27 and 5.28), we get
S1(3flS0+a)---'--2K--=O.dz dz
Integrating the above equation we get,
(3/3s0 +a i--2KS2 +C=O,
(5.29)
(5.30)
(5.31)
where C is the constant of integration. From the above equation we get,
IS2 =(3,8S0+a) S 2
C—+--.
4K 2K
Differentiating Eqn.(5 .27), we get
d2SL= 2K!g.
dz 2 dz
(5.32)
(5.33)
By using the Eqn.(5.29) and Eqn.(5.32), Eqn.(5.33) is modified as
= IF S - AS,dz2
where
F=-4K 2 9 2 -3/3g 2 CS0 –g 2 Ca and
A=9g 2 /3 2 S +6ag2 ,8 S^ +g2a2.
The above equation (5.34) is written as
= ± IF S12 - +28.dz V 2
(5.34)
(5.35)
1.75
1.5
1.25
1-I
In0.75
0.5
0.25
0
-4 -2 0 2 4Time t
146
5.3.1. BRIGHT SOLITON SOLUTION
This section is devoted to investigate the bright soliton in the NLDC in
the presence of coupling parameter. On solving the Eqn.(5.35), when the
integration constant ö is assumed to be zero, the solution is
F2
2— K 2 g 2 - 3/3 g 2 C S 1 - g 2 C a sech[ z], (5.36)S =fl2s2+6ag2fls+g2a2)
where 93=..j(_4K2g2_3flg2CS0_g2Ca).
The above solution is in the form of bright soliton, for which soliton plot is
represented in Fig.(5.4).
Figure 5.4Bright soliton 2D plot of Eqn(5.36) for the values
C = 0.984, K = 0. 11, 8 = 0.7, a = 0.1 S. = 1.02 and g = - 1.1
5.3.2 DARK SOLITON SOLUTION
So far, generation of dark soliton has been discussed for both ideal and
real world fiber systems (Nonuniform fibers). This section is devoted to
07
0.6
0-s
0.4-I
(1_i
03
0.2
0.1
0
147
investigate the dark soliton in the NLDC in the presence of coupling
parameter. The Eqn.(5.35) is modified as
dS1I_2FS12 +_
dz F2 A A
When 6 = we get,
L(IFiS2
dz - 2 4
On solving the above equation the solution for the dark soliton is
(5.37)
(5.38)
S, = etath[F—j2 +3/3 g 2 CS0 +g 2 C a)
(5.39)z],
2
_4K2g2_3fig20_g2Ca\where O9g2fl2s26g2fis+g22J
-4 -2 U 2 4Time t
Figure 5.5Dark soliton 2D plot of Eqn(5.39) for the values
C = 0.99, K = 0. 11, /3 = 0.78, a = 0. 13, S 0 = 1.1 and g = 0.987
-x -L 2
A0
A
V(A0)
(a)
(b)
V(A0) V(A0)
(c)
(d)
Figure 5.6.
148
The Eqn.(5.39) shows the dark soliton in the NLDC in the presence of
XPM effect under the influence of quintic nonlinearity for which soliton plot
is presented in Fig.(5.5).
Transformation of potential wells (21) view) from symmetry to asymmetry as
the nonlinearity and detuning parameters are increased
149
A
A
(a) (b)
(c) (ii)
Figure 5.7
Phase-plane diagrams for the potential well
plots for the above mentioned values
5.4 SWITCHING MECHANISM BASED ON ANHARMONICOSCILLATOR EQUATION
Now, the switching mechanism is discussed based on the anharmonic
oscillator Eqn.(5.34) which contains all the information about the CNLS
system under consideration. From Figs.(5.6) and (5.7), it is evident that the
qualitative aspect of the potential well drastically changes as we increase the
nonlinearity parameter and thereby it also increases the coupling between the
two modes. In this connection, the intensions to analyze how the energy is
shared and transferred between the co-propagating modes in terms of
150
potential well. To investigate the impact of nonlinearity in NLDC through the
potential well plots, the linear case is discussed first. Under this condition, the
system obeys the harmonic oscillator equation. Now, by applying the stability
condition obtained from LSA to Eqn.(5.20) it is found that the system
possesses a single well potential. This is clearly evident from Figs.(5.6a) and
(5.7a) where the energy transfer is periodic and complete. From the shape of
the potential, a stable behavior is expected.
Further, at low intensities of the incident light, the particle is initially at
rest at the bottom of the quasi-harmonic potential well. When a small amount
of nonlinearity is introduced into the system, it is found that the system
possesses the double well potential as shown in Figs. (5.6b) and (5.7b). If the
nonlinearity is increased the anharmonicity becomes dominant and it plays an
indispensable role in the potential well plots. Hence the light i.e., photons
migrate from one mode to the other and causes the switching mechanism in
NLDC. Consequently, under this condition, the energy is shared between the
co-propagating modes. This kind of energy transfer is explained in Figs.
(5.6c) and (5.7c). This type of energy transfer is mainly because of the
anharmonicity, which breaks the symmetry property of the potential well.
Moreover, the migration of photons from one potential well at A 0 = +1 to the
other at A0 = -1 also confirms the transfer of photons from one mode to the
151
other in the NLDC. At this juncture; it is pointed out that Anderson et al., [34]
also reported this type of stability and instability in the NLDC.
For further increase in nonlinearity and linear coupling parameters,
there will be more unequal sharing of energy between the co-propagating
modes. This is shown in Fig. (5.6d) here the energy is unequally transferred
from one potential well to the other. This means that more photons migrate
from one potential side to the other. Here the photons take longer time to
reach the position at A0 = -1, which means that the period is increased. From
Fig.(5.7d) it is to be noted that the photons available in a well where the
potential function has minimum at A 0 = 4 are more stable when compared to
the other well. So the migration of photons from one potential to the other
makes more asymmetry in the potential well diagram. From this process,
symmetry breaking instability can easily be anticipated. This kind of unequal
sharing of energy between the modes can be thought of as switching
mechanism. This finds application in nonlinear grating structures as optical
switches as pointed out by Winful et al., [35]. In addition to the above
potential well discussion, the phase-plane diagrams are also plotted and they
provide clear information about the energy transfer between the co-
propagating modes in NLDC.
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5.5 RESULTS AND DISSCUSSION
In this chapter, two methods have been used to investigate the pulse
propagation in the NLDC for the co-propagating modes. In the first step of
the analysis, explicit soliton solutions for the symmetric and anti-symmetric
cases have been found. In addition, asymmetric soliton state has also been
discussed. Besides, another approach was used by using the well known
Stokes polarization parameters so as to describe the switching mechanism.
For this purpose, the governing system of equations has been reduced to an
anharmonic oscillator equation by using the Stokes parameters. Further, the
energy transfer/sharing between the co-propagating modes in NLDC in terms
of potential well plots and phase-plane diagrams are explained. From this
process, the existence of symmetry breaking instability of solitons state in
nonlinear directional coupler is predicted. This approach can also be applied
to other interesting physical systems like quadratic nonlinearity.
153
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