Influence of radiation on soliton dynamics in nonlinear fibre couplers

15 August 1994

OPTICS COMMUNICATIC~IS ELSEVIER

Optics Communications 110 (1994) 287-292

Influence of radiation on soliton dynamics in nonlinear fibre couplers A.V. B u r y a k , N . N . A k h m e d i e v OpticalSciences Centre, Institute ofAdvancedStudies, Australian National University, Canberra,A.C.T. 2601, Australia Received22 March 1994

Abstract

The dynamics of pulse-likesolutions in nonlinear fiber couplers is studied analyticallyand numerically, taking radiation effects into account. The approach is based on generalisedcontinuity equations and the "approximation of averageprofile". Transitions between soliton solutions of nonlinear couplers predicted by this approach are in good agreement with numerical simulations.

1. Introduction

Pulse propagation in nonlinear couplers, and in two-mode fiber devices in general, is now an area of extensive study. For the case of nonlinear couplers, we can point to many numerical [1,2,4], analytical [ 3,5-8 ] and experimental works [ 9, t 0 ]. The practical reason for the current increase of interest in these kind of devices is the fact that they can potentially operate at speeds much higher than those possible with electronic or optoelectronic switches. In spite of a large number of works dealing with nonlinear couplers, until now we have no suitable dynamic picture of soliton pulse propagation which self-consistently takes into account the emission of radiation. However, it has been shown that radiation is an essential part of pulse evolution in nonlinear fiber couplers [ 8 ], Although the amount of radiation is small, it considerably influences the dynamics of soliton-like pulses. Radiation is one of the reasons for the instability of stationary soliton states in fibers [ 12 ]. It causes transitions from unstable stationary states to stable ones. In this paper we present an analytical approach which inherently includes the influence of radiation on the

dynamics of soliton pulses, clarifies the physical processes in nonlinear couplers and gives qualitative answers to many important questions.

2. Statement of the problem

Pulse propagation in a dual-core fiber coupler, including the effects of dispersion, to second order, and self-phase modulation, can be described in terms of two linearly-coupled nonlinear Schr~Sdinger equations [ 1 ]" iY~+½Y~+IYI2y+MZ=O,

iZ¢+ ½Z,T+ I Z I 2 Z + M Y = O ,

(1)

where Y(~, z) and Z(~, z) are the electric field envelopes, M is the normalised coupling coefficient between the two cores, ~ is the normalised longitudinal coordinate, z is the normalised retarded time, and the equations are written assuming anomalous group velocity dispersion regime. Using rescaling ~ = x / M , z= t / M 1/2, Y= UM 1/2,

0030-4018/94/$07.00 © 1994ElsevierScienceB.V. All rights reserved SSDI O030-4018 (94) 00242-M

288

A.V. Buryak, N.N. Akhmediev I Optics Communications 110 (1994) 287-292

Z= VM ~/2one can transform the system ( 1 ) into the

20

simpler form: 18-

iUx + ½u,, + l UI2U+ v=0,

ANTISYMMETRIC/

16-

iVx+½Vtt'~- l Vl2Vdr U=O .

(2)

Below we will analyse the system (2), noting that one can easily get all results for the system ( 1 ) from the results for the system (2) using the inverse scale transformations. The stationary soliton solutions of Eqs. (2) and their stability properties have been thoroughly investigated by Akhmediev et al. in Refs. [ 1 I, 12 ]. The system (2) has two invariants,

14. 12. a

10.

/

8-

6:

-/

2 2•.

Q= j (tel2q-lVlZ)dt

(3)

o

.

./

ASYMMETRIC (A-type)

°°°*

-

-2

--oo

~"-

2

4

6

8

10

q

(the energy) and

Fig. l. The energy-dispersion diagram for the three families of soliton states in the nonlinear coupler. The thin dashed lines show transitions between different branches obtained using the analytical approach of this paper. The thin solid lines show the results of direct numerical simulations of Eqs. (2).

+oo

H= ~ [½(Igtlz+lgtl2)-½(IUl4+lrl 4) --oo

- ( U V * + U*V) ] dt

(4)

(the Hamiltonian). We consider three lowest order one-parameter families of the stationary soliton solutions of the system (2) [ 11 ]. (The only parameter of these families is the nonlinear spatial frequency shift q. ) We present them on the energy-dispersion diagram of Fig. 1. They are the symmetric solution U=V= 2x~-l)sech(~l)t)e

iqx,

-5-

- 15 L

'..

-r- -20-

./

the antisymmetric solution

/

, ~SYM IETRIJF (A-J~ e ) ' ~ ~

/////

U= - V=x/~(q+ 1 ) sech(x/2 (q+ 1 )t) e iqx and the asymmetric solution (the solution of A-type) with I el # I VI. The asymmetric solution cannot be expressed in terms of known analytical functions. It is possible to represent these stationary soliton states on the plane ofinvariants of the system (2) i.e. on the (Q, H)-plane. This diagram has the form shown by bold curves in Fig. 2. The curves for symmetric and antisymmetric solutions on (Q, H) diagram are, respectively,

H=-Q3/96-Q,

(5)

H= -Q3/96+Q.

(6)

~,~

(

-35-

.,o./////////A .,, / . . / / / / K///,/. 0

1

2

3

4

6

7

8

9

10

Q Fig. 2. Hamiltonian versus energy for the three families ofsoliton states in the nonlinear coupler. The thin dashed curves show transitions between different branches obtained using the analytical approach of this paper. The thin solid lines show the results of direct numerical simulations of Eqs. (2). Soliton-like solutions do not exist in the hatched area of Q - H plane.

A. V. Buryak, N.N. Akhmediev /Optics Communications 110 (1994)287-292 In the case of the asymmetric A-type solutions we have to use numerical methods (the same as in Ref. [ 11 ] ) to find the corresponding curve. However, in the limit of Q>> 1 the asymptotic formula is H = - Q 3 / 2 4 (dashed bold curve in Fig. 2). Bifurcation of the asymmetric solutions from the symmOtric ones takes place at Q = 8/x/~ (Point M in Fig. 1 and in Fig. 2).

3. Generalised continuity equations System (2) is non-integrable in general. It is known that any nonstationary soliton-like pulse propagation is accompanied by radiation [ 8 ]. The stationary soliton states can propagate without radiation. However, if a stationary soliton state is unstable when a small perturbation is applied, it will evolve into another stable soliton state, while emitting small-amplitude quasi-linear waves. Hence, in practice, arbitrary initial condition in the form of soliton-like pulses will separate into a leading pulse and dispersed radiation. Due to this radiation, Q~o~and H, ol values of the leading pulse change with propagation. (H~ob and Q,ol are defined as in Eqs. (3), (4) but within a finite interval of integration where the leading pulse is located (see Fig. 3). ) To investigate the dynamics of solitonlike pulses we will calculate these changes. The over-

289

all values of Q and H are conserved, so that the rates of change of Q,o~ and H,o~ are related to the energy and Hamiltonian flows at the boundaries of the leading pulse. (Fig. 3 illustrates this idea: we divide the taxis into three parts, where the central part contains the leading part and two other parts contain the radiation. The energy and Hamiltonian flows are calculated at the points t = - a and t = at.) To find the explicit relation we obtain two differential conservation laws derived from the initial Eqs. (2), 0 0. ~xpQ=-- ~JQ,

0 0. ~ x p H = - ~tJH,

(7)

where pQ= I UI 2 + l V I 2 andpH=½(IUtl2+l vt 12) ½( I UI 4 + I vI 4) _ ( UV* + U* v) are the energy and Hamiltonian densities, respectively, and j Q = - (i/

2 )( UtU*-U~ U+ V t V * - V ~ ) and j x = - ½ ( U t U ~ + U~ Ux + lit V* + V~ Vx) are the energy and Hamiltonian flows. Therefore, Eqs. (7) are generalised onedimensional continuity equations. Integration of Eqs. (7) over the interval ( - at, at) gives 0 Ox Q~ot= - [JQ(at ) --jQ( --at ) ] ,

0

~xHsol = -- [JH(at) --J/~( --at)] •

(8)

Eqs. (8) determine the rate of change of H~o~, and Q~o~in terms of the energy and Hamiltonian flows outside the leading pulse. These flows are determined by radiation from the leading pulse. The radiation can be considered as a Fourier series of the small amplitude linear waves. The linearized Eqs. (2) give two possible types of linear waves (symmetric and antisymmetric) for each frequency to. of radiation:

> =)

u=a. e x p ( i t o . x + i t 2 . t ) , u = + a. exp (ion. x + i~2. t ) .

( 9a )

The dispersion relations for these waves have the form

0.0 RADIATION . . . .

i

. . . .

i

i LEADING PULSE . . . .

i

. . . .

i

. . . . . . . . .

RADIATION ~

. . . .

i

. . . .

t Fig. 3. A component of a solution of Eqs. (2) consisting of soliton-like pulse and radiation. The points t = - a and t = ot separate the main pulse from the small-amplitude radiation.

-2to, +2=t22 .

(9b)

Now substituting Eqs. (9) into Eqs. (8) one finds

OQ~d Ox

OHrad =-

y' s . , tl

Ox

-

y~ II

oJ.s.,

(10)

290

A. V. Buryak, N.N.Akhmediev /Optics Communications 1I0 (1994) 287-292

where S, is the total energy flow for the Fourier component at the frequency to = 09,. The values of 09, are defined by oscillation frequencies of the leading pulse. Now we can simplify the problem, assuming that: (i) The main contribution to the radiation comes from the leading pulse amplitude oscillations at the fundamental (lowest) frequency too- Contributions from all higher harmonics of 090 are small. (ii) The radiation waves have the same symmetry (i.e. antisymmetric) as the amplitude oscillations of the two leading pulse components. These assumptions (which are in agreement with direct numerical simulations) allow us to take into account only the So-term in each of Eqs. (10) and obtain the result:

OHsol - - too(Hsol, Qsol) OQ,ol

s3(x, t) = - i ( U * V - UV*).

These are differential Stokes parameters, depending on both variables x and t. Eqs. (2) can be written in terms of these parameters in the following form,

0 f So(X,t) d t = 0 Ox ' +~

-t-oo

of

Ox

f s3(x,t) dt,

s~(x,t) d t = - 2

-oo

--oo

woo

Ox

-t-~

s2(x,t) dt=

Sl(X,t) s 3 ( x , t ) d t ,

--oo

-co

+co

°f

0--£ ( 11 )

(13)

+oo

s3 (x, t ) dt = 2

- ~

I Sl (x, t ) dt

--oo +or)

A similar expression was obtained in Ref. [ 13 ] for the case of the slightly perturbed single non,linear SchriSdinger equation. In Eqs. ( 1 1 ) we have shown explicitly the dependence too(H~o,, Q~ol), since the frequency of the oscillations of the leading pulse depends on its energy and Hamiltonian. Note that 090 is always negative, due to dispersion relation (9b) "for the antisymmetric radiative waves. Thus we can write Eqs. ( 1 1 ) in the form:

OH~ol - I too(H~o,, O~o,) I • OQ~ol

(12)

Expression (12) is a dynamical equation, based on two main conservation laws of initial system (2). This equation assumes adiabaticity ((OQ/Ox) Tx<< Q and (OH/Ox) Tx << H where Tx = 2n/to) and its accuracy increases as the radiation emission rate decreases.

4. Approximation of average profile Now we concentrate on the calculation of too(Hso~, Q~ol) dependence for a soliton-like pulse. (Below we will refer only to Hso~,and Q~o~and will omit subscript names. ) A convenient way to solve this problem is to use Stokes parameters, which are defined by

So(X,t)=lUl2+lVI 2, s2(x, t) = U'V+ UV* ,

S l ( X , t ) = l U l 2 - l V [ 2,

-

f sl(x,t) s2(x,t)dt.

(14)

This system is equivalent to system (2). It has been shown, for the similar problem of soliton propagation in birefringent optical fibers [ 14 ], that Stokes parameters can be applied to a soliton as a whole. This is a consequence of the robustness of solitons in Hamiltonian systems [ 15 ]. We assume that solitons in a dual-core fiber have the same property, and seek the solution of the system (2) in the form

U(x,t)=A(x)f(t),

V(x,t)=B(x)f(t),

(15)

•w h e r e f ( t ) , which is the same in both channels is the envelope function of the soliton-like pulse and A and B are the pulse amplitudes. Direct numerical simulations show that envelope functions of the leading pulse components oscillate slightly around some average shape f ( t ) . These oscillations are small and can be neglected in the first approximation. Hence we call Eqs. (15) the "approximation of average profile". Using Eqs. ( 13 ) - (15) and carrying out some algebra, one can get the system

d

~x So(x) = 0 ,

d

S, (x) = - 2S3(x),

-~ S2(x) =gS, (x) S3(x),

A.K Buryak,N.N.Akhmediev/ OpticsCommunications110(1994)287-292 d s3(x) =2S, (x)-gSl (x) S2(x),

(16)

where

291

where K is the complete elliptic integral of the first kind, C= ( 1 -gW/2) and D = ( 1 -gW+g2S~o/4) 1/2 The fundamental frequency of radiation is given by tOo= 2n/Tx.

So(X)=IAlU+IBI u, S~(x)=IAIE-IBI u, S~(x)=A*B+AB*, S3(x)=-i(A*B-AB*), (17) and +oo

g=(f

+oo

f 2dr)

f4d/)(~ -oo

--1

.

-oo

There are two integrals of Eqs. ( 16): So(x)= IAI2+ IBI2=S~+S~+S~

(18)

and

The integral (18) is a consequence of energy conservation (3), while Eq. (19) is a consequence of the conservation of the Hamiltonian (4). Solutions of Eqs. (16) can be expressed in terms of elliptic Jacobi functions. All stationary points of the system (16) correspond to stationary solutions of the initial Eqs. (2). At small values of the parameter g (which is proportional to the leading pulse energy Q), there are only two stationary points (which are elliptic fixed points) $5~= (S~=0, $2=So, $3=0) and S~,tisym = (0, -So, 0), corresponding to symmetric and antisymmetric solutions respectively. At higher values ofg (higher leading pulse energy), there are two additional elliptic fixed points corresponding to asymmetric A-type states: Suym= ( +-x/S2 - 4 / g 2, 2/g, 0). The point $5~= (0, So, 0) in this instance turns into a fixed point of the saddle type. • The period of the oscillations of the solutions of the system (16) is given in terms of complete elliptic integrals. For - So < W< So, ~,N/--~--- ] ,

(20)

for W> So, Tx=~-~

Now we are in a position to find the general solution for transitions between different branches. The last step is to express W, So and g in terms of Q and H in tOo(IV, So, g). In this paper we are interested in a qualitative physical picture, rather than in an accurate solution of this problem. Thus we roughly approximate the average profile of the soliton-like pulse with the function

f(t) =$/cosh(2t). (19)

Wm~gS2 +S2 .

Tx=~r~

5. Dynamics of transitions

~

~-S--~,] '

(21)

(22)

This approximation becomes exact for the symmetric and antisymmetric solutions if we choose

So= IAI2+ IBI2=2. Using the expressions for the energy and Hamiltonian invariants, the ansatz (22) and the positions of fixed elliptic points, it is easy to construct the curves which correspond to the three families of approximate solutions on the (Q, H) plane. The curves for symmetric and antisymmetric solutions are the same as those defined by Eqs. (5) and (6). The curve for the asymmetric solutions in the approximation (22) can be obtained analytically, and is given by H= -Q3/32-12/Q (the curve Aavvrat Fig. 2). This agrees only qualitatively with the curve for exact asymmetric solution. The curve Aa~,r bifurcates from the curve of symmetric solutions at Q= 2x/~ (point M~ppr at Fig. 2). The hatched area in Fig. 2 (below the curve for exact asymmetric solutions and the curve for symmetric solutions above the point M) is the area of Q-H parameters where soliton-like pulses do not exist. (Any pulse with such initial values of Q and H will completely transform into dispersed radiation.) This allows us to suppose that the upper border line of this area corresponds to stable stationary soliton solutions. In fact this is correct, as numerical results on stability show [ 12 ]. Having an explicit form for the average profile of the soliton-like pulse (22) it is easy to find g(H, Q) and W(H, Q) dependencies analytically:

292

A. V. Butyak, N.N. Akhmediev /Optics Communications I10 (1994)287-292

g= Q2/24 and W = - Q 2 / 4 8 - 2 H / Q .

(23)

(One can see that the Hamiltonian H at fixed value of Q depends linearly on the evolution parameter/41. ) Thus we have solved the problem and found the righthand-side parts o f Eq. (12): d H / d Q = [trio(H, Q ) I : Eq. (12) gives the direction o f evolution (the slope o f the trajectory) at any point on the ( H - Q ) diagram (see Fig. 2). Note, that H can only decrease as Q decreases, so that the direction o f the evolution can only be to the left and down in the diagram o f Fig. 2. Examples o f dynamic transitions on the H - Q plane found by numerical integration of Eq. (12) are shown by the thin dashed lines in Fig. 2. These trajectories can be transferred into conventional energy-dispersion diagram o f Fig. 1 where they are also shown by dashed lines. We also present the final points of the exact transition trajectories (with the same starting points Qo, Ho as approximate ones) obtained by direct numerical simulations o f Eqs. (2) (standard Fourier split-step method, solid lines at Fig. 1 and Fig. 2). Qualitatively the behaviour o f exact transitions and approximate ones is the same. Their quantitative difference for the case o f transitions from symmetric to A-type asymmetric states is due to the fact that, in the approximation (22), the branch for Atype soliton states is considerably shifted relative to exact curve. We should note that in this paper we have considered a limited class o f transitions between soliton states. The "approximation o f average profile" works only when the perturbations o f the stationary solutions are even functions o f the time variable t. As a consequence, we cannot describe the transformations o f antisymmetric soliton solutions into pairs o f A-type soliton solutions which take place at sufficiently high values o f Q [ 8 ]. On the other hand, the transformations o f antisymmetric soliton solutions into symmetric soliton states at lower values o f Q are described with high accuracy (see Fig. 1 and Fig. 2 ). The symmetric solutions do not exhibit any growth o f even perturbations [ 12 ], and for transitions from them our approach is always correct.

6. Conclusions In conclusion, we considered soliton dynamics in nonlinear fiber couplers self-consistently taking ra-

diation effects into account. Our approach is based on generalised continuity equations and the "approximation o f average profile". It can also be used (with minor changes) for the description o f the soliton dynamics in a birefringent fibre or in any other two-mode fiber device.

Acknowledgements The authors are grateful to Prof. A.W. Snyder for fruitful discussions and to Dr. A. Ankiewicz for a critical reading of our manuscript. This work is supported by the Australian Photonics Cooperative Research Centre ( A P C R C ) .

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