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Nonlinear pulse propagation in a monomode composite glass optical fibre
Volume 84, number 5,6
OPTICS COMMUNICATIONS
1 August 1991
Nonlinear pulse propagation in a monomode composite glass optical fibre Ajit Kumar Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 1I0016, India
Received 8 November 1990; revised manuscript received 4 March 1991
Starting from the nonlinear wave equation the basic dynamical equation for pulse propagation in a monomode compositeglass optical fibre with saturating nonlinearity is derived. The qualitative analysis based on the integrals of motion of this equation shows that this new equation also should support soliton propagation.
During the last decade nonlinear pulse propagation [ 1-14 ] in a monomode optical fibre has attracted much attention due to its many applications in high-bitrate communication systems. The main activity has been centered around the pioneering work of Hasegawa and Tappert [ 1 ] who showed that it was possible to have stable soliton pulse propagation in a monomode optical fibre with Kerr nonlinearity. The existence of such a soliton was experimentally established by MoUenauer et al. [ 7 ] in 1980. Recently there has been a great deal of interest in composite material (especially semiconductor-doped glass) optical fibres in connection with the possibility of the construction of switches and other bistable devices [ 15-18 ]. As it is well known [ 19 ], due to the fact that nonlinear phase shift usually follows intensity envelope, only partial switching of the pulse is possible in optical fibres. However, if one used solitons one could achieve switching of the entire pulse since in the anomalous dispersion region N = 1 solitons show a uniform phase shift over the entire waveform [ 19 ]. Hence it is desirable to study pulse propagation in composite glass fibres. Furthermore it is known [20] that in semiconductor-doped materials higher than third-order nonlinear effects become involved through saturation in the absorption. Associated with this nonlinear absorption there is a change in the refractive index which, when calculated through Kramers-Kronig transformation, shows saturation [21 ]. Apart from that, it has recently been shown [22 ] that the effective nonlinearity in a composite material must saturate. This suggests that for the description of nonlinear processes in a composite material fibre the nonlinear Schr'6dinger equation is a rather poor approximation. Keeping this point in mind, in the present work we derive the basic nonlinear dynamical equation describing pulse propagation in composite glass fibre by taking into account the saturating nonlinearity and using the slowly varying envelope approximation (SVEA). A qualitative analysis of this equation based on the integrals of motion shows that this new equation also should support soliton propagation. Consider a monomode isotropic fibre with a circular cross section. Let ro be the radius of its core and x the axis along the fibre. In the case of saturating nonlinearity the wave equation in the core r e # o n of the fibre may be written as
V2~-
1 02D L C2
0t 2
l - - C2
0 2 D NL 0t 2
'
(1)
where c is the speed of light and D L and D NL are the linear and nonlinear part of the electric induction vector D, respectively, 346
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oo
DL(t)=
J e(t')E(t-t') dt' ,
(2)
o
DNL(t) = ~s[E--E exp( - z [ E [ 2) ] .
(3)
Here, ~ stands for the linear dielectric constant and Z= e:/es. ~2 is the usual Kerr coefficient for the dielectric constant and es is a constant which makes xIEI 2 dimensionless. Note that the dielectric functions [23,24] eNL( IE[ 2) = a l E I / ( 1 +blEI 2) ,
(3a)
(a/b)
(3b)
6NL( IEI 2) =
[1 - e x p ( - h i E [ z) ] ,
where a and b are appropriate constants, are the most frequently used forms of nonlinear dielectric functions involving saturation. Both of them are Kerr-like for small field intensities and reveal a common saturation. Our expression for D NL in eq. (3) results from the choice of ~NLin the form (3b). This is because we consider our solitary wave to be supported entirely by the HEI~ mode (see below) and here it should be confined entirely in the core region. For such a situation, the analysis of Langbein et al. [ 23 ] shows that the choice of ENL( IEI 2) in the form (3b) should be better. Now, for simplicity, we assume that the solitary wave is entirely supported by the HE11 mode of the fibre far from cutoff. Under these conditions the electric field is confined entirely in the core region and its major component is the transverse one, which is linearly polarized. Hence, keeping in mind the SVEA, we look for the solution of eq. ( 1 ) in the form [ 13 ]
E( t, x, r) =eR(r)A( t, x)
exp[
-i(oJt-flx)
],
(4)
where e is the unit vector in the direction of polarization, fl is the propagation constant, R(r) is the mode function describing the transverse distribution of the electric field in the mode and A (t, x) is the slowly varying complex envelope amplitude. Here r represents the transverse coordinate r=~yy+Gz, iy and G being the unit vectors along the y and z axis, respectively. From eqs. ( I ) - ( 4 ) , assuming the temporal dispersion of the dielectric permittivity to be small, and using the procedure of section 2.1 of ref. [ 4 ] we arrive at the following nonlinear differential equation for A (t, x):
R[i(A~+ Vg 1 A, )t -
~
Att-
t ~ko,oA, + ~ Axx _--ik~° 2kvgAttt- i ~
Ant ]
~s = - ~-~c2 {R(w2A+ 2koA,) [1-exp(-zIRI21AI2) ] + 2iogzIRI2RA( IAI2),exp(-zIRI21AI2) } ,
(5)
where vg is the group velocity and k is the wave number. From here onwards a suffix stands for the partial derivative with respect to it unless stated otherwise. Note that in deriving eq. (5) we have made use of the following: (a) for the HE~t mode fl=cox/Q/c=k and (b) following Hasegawa and Tappert [1] we have assumed that the mode function R is given by the eigenmode of the linear fibre. Now using SVEA and the fact that in a monomode optical fibre the difference between the phase and group velocities is negligible [25] we can make the simplification [5]
O9~s
(.t)~s
- iR 2--k-~c2A, [ 1 - e x p ( - z [ R [ Z l A 12) ] - i 2--k~c2ZI R 12RA(IA 12), exp( -zIRI:[A 12) .
(6)
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Eqs. (5) and (6) yield 1 l 1 R[i(Ax+ -~sA,)-~ko~o~A,,-giko,.~,At,, ]
=-Es R -~-~c2At+~-~sA
[1-exp(-zIRI2IAI2)]+-~-~c2XIRI2RA(IAI2),exp(-xIRI2IAI2)
.
(7)
Let us now average this equation over the cross section of the fibre. In doing so I assume R(r).~ exp ( - r 2 / r 2) since for silica glass fibres the difference An in the refractive indices of the core and the cladding is usually very small: An << nc; nc being the refractive index of the core. As a result we get 1 1 1 i(Ax+-~.At)-~ko,o~A.-gik~o,o~Am= io)~-/cAz' + -"L~[ / 2---~' ito ZA ( A I2),(~21-AI 1 4 +2--~5c2
exp(-zIAIZ)x[AI 2
_ _1 c°2 A) (1 _ zIA' 2___~5c2
exp(-zlAI
2 + :" S
2))
exp(-xIAI2))]x21AI4 ]j.
(8)
Now we introduce the following dimensionless variables:
q=A/Ao,
Ao=Z -1/2,
~=(kn2/Zno)x,
r=[knz/no)C(-k~oo,)]l/2(t-x/vg),
(9)
where Ao is the maximum field, no =x/~ and n2 is the Kerr coefficient for the refractive index. If we take into account that Es=~(-IE2 and e2 = 2non2, then eq. (18) can be written in the dimensionless form
iq¢+½q~+q[1-1/lql2+exp( - IqlZ)/lqlZ]=-i6q~+iaq~[1-1/IqlZ+exp( - Iql2)llql 2] - i a q ( Iq12)~[ 1 / I q l 4 - e x p ( - Iql 2) ( 11 Iql2+ l / I q l 4) ] ,
(10)
where
a=(llv,)[n2/noz(_ko~o~)]l/2,
g= -g[ko,~o/(-k~oo~) 1 ] [kn2/no)¢(-k~o)
]1/2
( 11 )
Note that in the absence of the perturbation terms on the right-hand side eq. (10) will read iq¢+½q**+q[1-1/Iq12+exp(-
Iql2) / ]ql 2]=0.
(12)
This is our basic nonlinear dynamical equation for pulse propagation in composite glass fibres. Firstly, we would like to note here that a strict mathematical proof the existence of a soliton solution to eq. (12) is difficult unless one succeeds in solving it by the inverse scattering method (ISM). But physically, as discussed below, the conservation of 11 and 12 should prevent the pulse from decaying into plane waves and lead to pulse propagation with oscillating pulse width. If so, then the physical interpretation of ISM tells us that the pulse should give rise to a soliton or to many solitons depending on the input pulse energy. To gain insight whether eq. (12) can support soliton pulse propagation or not let us now proceed in the following way: For a localized pulse like a bright soliton, eq. (12) has the integrals of motion [5,26] dll/d~=O,
IqlZdz,
Ii = --oo
dlz/d~=0,
/2= i
[½[q~[2-F( Iq[Z) ] dz ,
(13)
--oo
with F ( l q l 2 ) = Iql2-Ein(lql 2) 348
(14)
Volume 84, number 5,6
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where Ein(x) = E l (x) + I n x + 7 . Here E~ (x) is the well known integral exponential function and XEuler's constant. It follows from this that during the propagation of the pulse the total intensity as well as the combined effect of dispersion and nonlinearity are conserved. For linear fibres F = 0 and I2> 0. In this case, for a given form of q, using dI2/d~=0, one recovers the usual pulse broadening. For nonlinear fibres F # 0 and I2 can he positive as well as negative and since/2 is conserved during propagation the pulse width will oscillate around a certain average value due to the oscillation in the relative strength of dispersion and nonlinearity (as we shall see below this is confirmed by our numerical results). To study this let us take a gaussian pulse A (~=0, z) =Ao exp( - z212a 2)
( 15 )
with initially plane wavefront launched at the input end of the fibre (~--0). Here Ao and ao are the initial amplitude and width respectively. In what follows we shall study the evolution of this gaussian pulse through the fibre described by eq. (12) using the invariants Ii and I2. In doing so we shall adhere to the usual adiabatic approximation which assumes the pulse to preserve its shape while its amplitude and width might vary with distance during propagation [ 27 ]. It is a kind of stability analysis in the adiabatic approximation. Hence, we take the current pulse in the form q(~, r) =/~(~) exp[ -z2/2a2(~) + ½ib(~)z 2 ] ,
(16)
where b (~) = - ( 1/a) da/d~. In fact there is no restriction on the choice of the test function. One can choose any suitable form of the input pulse. Our choice of the gaussian pulse is related to the fact that it has been used frequently for such analyses [ 8,11,27 ] and to a good degree of accuracy approximates a natural laser pulse. Let us now recall that 0 < I ql 2< 1 because q=A/Ao where Ao is the maximum field amplitude. Hence the approximation holds [ 15 ]
Ein( Iql 2) =el Iql2+c21q14+c3 Iql6+c4 Iql 8+c5 Iql ~°+o(~( I q i 2 ) ) ,
(17)
with Cl =0.9999, c 2 = - 0 . 2 4 9 9 1 , c3=0.05519, c 4 = - 0 . 0 0 9 7 6 , c5=0.00107 and ~( Iql 2) < 2 × 10 -7. Now after simple algebra we get from eqs. ( 13)- (17) the differential equation for the pulsewidth, dd2a ~ 2 = - - 0 / l l l a -1Z + ( 0 / 2 I I 2 + l ) ~ 1 _a313 -1_- +a414 - - 4~ ,
(18)
where I~ = IAI 2 is the integral of the energy of the pulse, 0/i = 0.0884, 0/2 = 0.0318, 0/3 = 0.00732, and 0/4 = 0.0009. We have integrated eq. ( 18 ) by the second-order Runge-Kutta method for different values of the parameter
t
2.3
.-r-
S .~
t
:
2.0
I
~ '
\
/ ,
c2
•
Q=
N
"-',.
1.5
,it ',z-
.,m<, }-.I L
,,.,,. 0 Z
1.C 80
100 112 PROPAGATION DISTANCE (DIMENSIONLESS)
128
136
Fig. 1. Normalized pulsewidth variation with dimensionless propagation distance after the pulse has covered 80 units of distance. The curves Cl, C2, C3 and C4 correspond to values Ofll equal to l, 1.5, 2 and 2.5, respectively. 349
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I~ c o r r e s p o n d i n g to different v a l u e s o f the i n p u t pulse energy a n d the initial pulse w i d t h a o = I. T h e results are d e p i c t e d in fig. 1. It is clear f r o m fig. 1 that the p u l s e w i d t h oscillates a r o u n d its i n i t i a l value r e g a i n i n g it periodically. T h e spread a r o u n d the i n i t i a l p u l s e w i d t h decreases as the i n p u t pulse energy is increased. H e n c e a s y m p t o t i c a l l y , if o n e chooses the a p p r o p r i a t e i n p u t pulse p a r a m e t e r s , soliton p r o p a g a t i o n will result. Conclusion: I n the p r e s e n t work, s t a r t i n g f r o m the n o n l i n e a r wave e q u a t i o n o f classical e l e c t r o d y n a m i c s , a rigorous d e r i v a t i o n o f the basic n o n l i n e a r e v o l u t i o n e q u a t i o n for pulse p r o p a g a t i o n i n a m o n o m o d e c o m p o s i t e glass optical fibre w i t h s a t u r a t i n g n o n l i n e a r i t y is presented. It is s h o w n that soliton p r o p a g a t i o n t h r o u g h a fibre d e s c r i b e d b y this n e w e q u a t i o n c o u l d be possible. I express m a y sincere t h a n k s to the referees for c o n s t r u c t i v e critical c o m m e n t s which h e l p e d i n i m p r o v i n g the clarity o f the m a n u s c r i p t .
References [ 1 ] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23 (1973) 142, 171. [ 2 ] A. Hasegawa and Y. Kodama, Proc. IEEE 69 ( 1981 ) 1145. [ 3 ] Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23 ( 1987 ) 510. [4]A. Kumar, Phys. Rep. 187 (1990) 63. [5] A. Kumar and M.S. Sodha, Elect. Lett. 23 (1987) 275. [6] L.F. Mollenauer, J.P. Gordon and M.N. Islam, IEEE J. Quantum Electron. QE-22 (1986) 157. [7] L.F. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Len. 45 (1980) 1095. [ 8 ] A. Kumar, S.N. Sarkar and A.K. Ghatak, Optics Len. 11 (1986) 321. [ 9 ] A. Hasegawa, Optical Solitons in Fibers (Springer, Berlin, 1989 ). [ 10 ] K.J. Blow and N.J. Doran, IEEE Proc. J. 134 ( 1987 ) 137. [ 11 ] D. Anderson, M. Lisak and T. Reichel, Phys. Rev. A 38 ( 1988 ) 1618. [ 12] N.J. Doran and K.J. Blow, IEEE J. Quantum Electron. QE-19 (1983) 1183. [ 13] D.N. Christidoulides and R.I. Joseph, Optics Lett. 9 (1984) 229, 408. [ 14] D. Krokel, N.J. Halas, G. Giuliani and D. Grisehkowsky, Phys. Rev. Lett. 60 (1988) 29. [ 15 ] A. Abramowitz and I. Stegun, Handbook of Mathematical Physics (Dover, New York, 1972 ) ch. 5, 5.1.53. [ 16] D. Anderson, M. Lisak andT. Reichel, J. Opt. Soc. Am. B (1989) 207. [ 17] D. Cotter, Electron. Lett. 22 (1986) 693. [ 18 ] B.J. Ainslie, H.P. Girdelstone and D. Cotter, Electron. Lett. 23 (1987) 405. [ 19 ] M.N. Islam, E.R. Sunderman, R.H. Stolen. W. Pleibel and J.R. Simpson, Optics Lett. 14 ( 1989 ) 811. [20] C. Flitzanis, F. Hache, M.C. Klein and D. Richard, in: Optics in Complex Systems, SPIE, Vol. 1319 (1990) pp. 75-84. [21 ] M.A. Fisher, D.A.H. Mace, J.N, Tothill and K.J. Monserrat, in: Optics in Complex Systems, SPIE, Vol. 1319 (1990) pp. 106-109. [22] N.C. Kothari, Phys. Rev. A 41 (1990) 4138. [23] U. Langbein, F. Lederer, T. Peschell and H.E. Ponath, Optics Lett. l0 (1985) 571. [ 24 ] V.E. Woods, E.D. Evans and R.P. Kenan, Optics Commun. 69 (1988) 156. [25 ] A.H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983). [ 26 ] A. Kumar, Invariant of motion method for nonlinear pulse propagation in optical fibres, Phys. Rev. A, to be published. [27] D. Anderson, Phys. Rev. 27 (1983) 3135.
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1 August 1991
Nonlinear pulse propagation in a monomode composite glass optical fibre Ajit Kumar Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 1I0016, India
Received 8 November 1990; revised manuscript received 4 March 1991
Starting from the nonlinear wave equation the basic dynamical equation for pulse propagation in a monomode compositeglass optical fibre with saturating nonlinearity is derived. The qualitative analysis based on the integrals of motion of this equation shows that this new equation also should support soliton propagation.
During the last decade nonlinear pulse propagation [ 1-14 ] in a monomode optical fibre has attracted much attention due to its many applications in high-bitrate communication systems. The main activity has been centered around the pioneering work of Hasegawa and Tappert [ 1 ] who showed that it was possible to have stable soliton pulse propagation in a monomode optical fibre with Kerr nonlinearity. The existence of such a soliton was experimentally established by MoUenauer et al. [ 7 ] in 1980. Recently there has been a great deal of interest in composite material (especially semiconductor-doped glass) optical fibres in connection with the possibility of the construction of switches and other bistable devices [ 15-18 ]. As it is well known [ 19 ], due to the fact that nonlinear phase shift usually follows intensity envelope, only partial switching of the pulse is possible in optical fibres. However, if one used solitons one could achieve switching of the entire pulse since in the anomalous dispersion region N = 1 solitons show a uniform phase shift over the entire waveform [ 19 ]. Hence it is desirable to study pulse propagation in composite glass fibres. Furthermore it is known [20] that in semiconductor-doped materials higher than third-order nonlinear effects become involved through saturation in the absorption. Associated with this nonlinear absorption there is a change in the refractive index which, when calculated through Kramers-Kronig transformation, shows saturation [21 ]. Apart from that, it has recently been shown [22 ] that the effective nonlinearity in a composite material must saturate. This suggests that for the description of nonlinear processes in a composite material fibre the nonlinear Schr'6dinger equation is a rather poor approximation. Keeping this point in mind, in the present work we derive the basic nonlinear dynamical equation describing pulse propagation in composite glass fibre by taking into account the saturating nonlinearity and using the slowly varying envelope approximation (SVEA). A qualitative analysis of this equation based on the integrals of motion shows that this new equation also should support soliton propagation. Consider a monomode isotropic fibre with a circular cross section. Let ro be the radius of its core and x the axis along the fibre. In the case of saturating nonlinearity the wave equation in the core r e # o n of the fibre may be written as
V2~-
1 02D L C2
0t 2
l - - C2
0 2 D NL 0t 2
'
(1)
where c is the speed of light and D L and D NL are the linear and nonlinear part of the electric induction vector D, respectively, 346
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V o l u m e 84, n u m b e r 5,6
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1 August 1991
oo
DL(t)=
J e(t')E(t-t') dt' ,
(2)
o
DNL(t) = ~s[E--E exp( - z [ E [ 2) ] .
(3)
Here, ~ stands for the linear dielectric constant and Z= e:/es. ~2 is the usual Kerr coefficient for the dielectric constant and es is a constant which makes xIEI 2 dimensionless. Note that the dielectric functions [23,24] eNL( IE[ 2) = a l E I / ( 1 +blEI 2) ,
(3a)
(a/b)
(3b)
6NL( IEI 2) =
[1 - e x p ( - h i E [ z) ] ,
where a and b are appropriate constants, are the most frequently used forms of nonlinear dielectric functions involving saturation. Both of them are Kerr-like for small field intensities and reveal a common saturation. Our expression for D NL in eq. (3) results from the choice of ~NLin the form (3b). This is because we consider our solitary wave to be supported entirely by the HEI~ mode (see below) and here it should be confined entirely in the core region. For such a situation, the analysis of Langbein et al. [ 23 ] shows that the choice of ENL( IEI 2) in the form (3b) should be better. Now, for simplicity, we assume that the solitary wave is entirely supported by the HE11 mode of the fibre far from cutoff. Under these conditions the electric field is confined entirely in the core region and its major component is the transverse one, which is linearly polarized. Hence, keeping in mind the SVEA, we look for the solution of eq. ( 1 ) in the form [ 13 ]
E( t, x, r) =eR(r)A( t, x)
exp[
-i(oJt-flx)
],
(4)
where e is the unit vector in the direction of polarization, fl is the propagation constant, R(r) is the mode function describing the transverse distribution of the electric field in the mode and A (t, x) is the slowly varying complex envelope amplitude. Here r represents the transverse coordinate r=~yy+Gz, iy and G being the unit vectors along the y and z axis, respectively. From eqs. ( I ) - ( 4 ) , assuming the temporal dispersion of the dielectric permittivity to be small, and using the procedure of section 2.1 of ref. [ 4 ] we arrive at the following nonlinear differential equation for A (t, x):
R[i(A~+ Vg 1 A, )t -
~
Att-
t ~ko,oA, + ~ Axx _--ik~° 2kvgAttt- i ~
Ant ]
~s = - ~-~c2 {R(w2A+ 2koA,) [1-exp(-zIRI21AI2) ] + 2iogzIRI2RA( IAI2),exp(-zIRI21AI2) } ,
(5)
where vg is the group velocity and k is the wave number. From here onwards a suffix stands for the partial derivative with respect to it unless stated otherwise. Note that in deriving eq. (5) we have made use of the following: (a) for the HE~t mode fl=cox/Q/c=k and (b) following Hasegawa and Tappert [1] we have assumed that the mode function R is given by the eigenmode of the linear fibre. Now using SVEA and the fact that in a monomode optical fibre the difference between the phase and group velocities is negligible [25] we can make the simplification [5]
O9~s
(.t)~s
- iR 2--k-~c2A, [ 1 - e x p ( - z [ R [ Z l A 12) ] - i 2--k~c2ZI R 12RA(IA 12), exp( -zIRI:[A 12) .
(6)
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Eqs. (5) and (6) yield 1 l 1 R[i(Ax+ -~sA,)-~ko~o~A,,-giko,.~,At,, ]
=-Es R -~-~c2At+~-~sA
[1-exp(-zIRI2IAI2)]+-~-~c2XIRI2RA(IAI2),exp(-xIRI2IAI2)
.
(7)
Let us now average this equation over the cross section of the fibre. In doing so I assume R(r).~ exp ( - r 2 / r 2) since for silica glass fibres the difference An in the refractive indices of the core and the cladding is usually very small: An << nc; nc being the refractive index of the core. As a result we get 1 1 1 i(Ax+-~.At)-~ko,o~A.-gik~o,o~Am= io)~-/cAz' + -"L~[ / 2---~' ito ZA ( A I2),(~21-AI 1 4 +2--~5c2
exp(-zIAIZ)x[AI 2
_ _1 c°2 A) (1 _ zIA' 2___~5c2
exp(-zlAI
2 + :" S
2))
exp(-xIAI2))]x21AI4 ]j.
(8)
Now we introduce the following dimensionless variables:
q=A/Ao,
Ao=Z -1/2,
~=(kn2/Zno)x,
r=[knz/no)C(-k~oo,)]l/2(t-x/vg),
(9)
where Ao is the maximum field, no =x/~ and n2 is the Kerr coefficient for the refractive index. If we take into account that Es=~(-IE2 and e2 = 2non2, then eq. (18) can be written in the dimensionless form
iq¢+½q~+q[1-1/lql2+exp( - IqlZ)/lqlZ]=-i6q~+iaq~[1-1/IqlZ+exp( - Iql2)llql 2] - i a q ( Iq12)~[ 1 / I q l 4 - e x p ( - Iql 2) ( 11 Iql2+ l / I q l 4) ] ,
(10)
where
a=(llv,)[n2/noz(_ko~o~)]l/2,
g= -g[ko,~o/(-k~oo~) 1 ] [kn2/no)¢(-k~o)
]1/2
( 11 )
Note that in the absence of the perturbation terms on the right-hand side eq. (10) will read iq¢+½q**+q[1-1/Iq12+exp(-
Iql2) / ]ql 2]=0.
(12)
This is our basic nonlinear dynamical equation for pulse propagation in composite glass fibres. Firstly, we would like to note here that a strict mathematical proof the existence of a soliton solution to eq. (12) is difficult unless one succeeds in solving it by the inverse scattering method (ISM). But physically, as discussed below, the conservation of 11 and 12 should prevent the pulse from decaying into plane waves and lead to pulse propagation with oscillating pulse width. If so, then the physical interpretation of ISM tells us that the pulse should give rise to a soliton or to many solitons depending on the input pulse energy. To gain insight whether eq. (12) can support soliton pulse propagation or not let us now proceed in the following way: For a localized pulse like a bright soliton, eq. (12) has the integrals of motion [5,26] dll/d~=O,
IqlZdz,
Ii = --oo
dlz/d~=0,
/2= i
[½[q~[2-F( Iq[Z) ] dz ,
(13)
--oo
with F ( l q l 2 ) = Iql2-Ein(lql 2) 348
(14)
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where Ein(x) = E l (x) + I n x + 7 . Here E~ (x) is the well known integral exponential function and XEuler's constant. It follows from this that during the propagation of the pulse the total intensity as well as the combined effect of dispersion and nonlinearity are conserved. For linear fibres F = 0 and I2> 0. In this case, for a given form of q, using dI2/d~=0, one recovers the usual pulse broadening. For nonlinear fibres F # 0 and I2 can he positive as well as negative and since/2 is conserved during propagation the pulse width will oscillate around a certain average value due to the oscillation in the relative strength of dispersion and nonlinearity (as we shall see below this is confirmed by our numerical results). To study this let us take a gaussian pulse A (~=0, z) =Ao exp( - z212a 2)
( 15 )
with initially plane wavefront launched at the input end of the fibre (~--0). Here Ao and ao are the initial amplitude and width respectively. In what follows we shall study the evolution of this gaussian pulse through the fibre described by eq. (12) using the invariants Ii and I2. In doing so we shall adhere to the usual adiabatic approximation which assumes the pulse to preserve its shape while its amplitude and width might vary with distance during propagation [ 27 ]. It is a kind of stability analysis in the adiabatic approximation. Hence, we take the current pulse in the form q(~, r) =/~(~) exp[ -z2/2a2(~) + ½ib(~)z 2 ] ,
(16)
where b (~) = - ( 1/a) da/d~. In fact there is no restriction on the choice of the test function. One can choose any suitable form of the input pulse. Our choice of the gaussian pulse is related to the fact that it has been used frequently for such analyses [ 8,11,27 ] and to a good degree of accuracy approximates a natural laser pulse. Let us now recall that 0 < I ql 2< 1 because q=A/Ao where Ao is the maximum field amplitude. Hence the approximation holds [ 15 ]
Ein( Iql 2) =el Iql2+c21q14+c3 Iql6+c4 Iql 8+c5 Iql ~°+o(~( I q i 2 ) ) ,
(17)
with Cl =0.9999, c 2 = - 0 . 2 4 9 9 1 , c3=0.05519, c 4 = - 0 . 0 0 9 7 6 , c5=0.00107 and ~( Iql 2) < 2 × 10 -7. Now after simple algebra we get from eqs. ( 13)- (17) the differential equation for the pulsewidth, dd2a ~ 2 = - - 0 / l l l a -1Z + ( 0 / 2 I I 2 + l ) ~ 1 _a313 -1_- +a414 - - 4~ ,
(18)
where I~ = IAI 2 is the integral of the energy of the pulse, 0/i = 0.0884, 0/2 = 0.0318, 0/3 = 0.00732, and 0/4 = 0.0009. We have integrated eq. ( 18 ) by the second-order Runge-Kutta method for different values of the parameter
t
2.3
.-r-
S .~
t
:
2.0
I
~ '
\
/ ,
c2
•
Q=
N
"-',.
1.5
,it ',z-
.,m<, }-.I L
,,.,,. 0 Z
1.C 80
100 112 PROPAGATION DISTANCE (DIMENSIONLESS)
128
136
Fig. 1. Normalized pulsewidth variation with dimensionless propagation distance after the pulse has covered 80 units of distance. The curves Cl, C2, C3 and C4 correspond to values Ofll equal to l, 1.5, 2 and 2.5, respectively. 349
Volume 84, number 5,6
OPTICS COMMUNICATIONS
1 August 1991
I~ c o r r e s p o n d i n g to different v a l u e s o f the i n p u t pulse energy a n d the initial pulse w i d t h a o = I. T h e results are d e p i c t e d in fig. 1. It is clear f r o m fig. 1 that the p u l s e w i d t h oscillates a r o u n d its i n i t i a l value r e g a i n i n g it periodically. T h e spread a r o u n d the i n i t i a l p u l s e w i d t h decreases as the i n p u t pulse energy is increased. H e n c e a s y m p t o t i c a l l y , if o n e chooses the a p p r o p r i a t e i n p u t pulse p a r a m e t e r s , soliton p r o p a g a t i o n will result. Conclusion: I n the p r e s e n t work, s t a r t i n g f r o m the n o n l i n e a r wave e q u a t i o n o f classical e l e c t r o d y n a m i c s , a rigorous d e r i v a t i o n o f the basic n o n l i n e a r e v o l u t i o n e q u a t i o n for pulse p r o p a g a t i o n i n a m o n o m o d e c o m p o s i t e glass optical fibre w i t h s a t u r a t i n g n o n l i n e a r i t y is presented. It is s h o w n that soliton p r o p a g a t i o n t h r o u g h a fibre d e s c r i b e d b y this n e w e q u a t i o n c o u l d be possible. I express m a y sincere t h a n k s to the referees for c o n s t r u c t i v e critical c o m m e n t s which h e l p e d i n i m p r o v i n g the clarity o f the m a n u s c r i p t .
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