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

127

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

128

(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

129

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

130

[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

132

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.

152

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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