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Co-propagation of pulses with steepening and phase modulation effects
Optics Communications85 ( 1991 ) 306-310 North-Holland
OPT IC S
COMMUNICATIONS
Co-propagation of pulses with steepening and phase modulation effects P.C. Subramaniam Optical WaveguideGroup. Department of Physics, Indian Institute of Teehnology, New Delhi-11016, India
Received 9 November 1990;revised manuscript received 13 May 1991
Weconsiderthe copragatlonof a high intensity pump and a low intensity signal in the presenceof dispersion, phase modulation and steepening effects. The possibilityof obtaining large compressionfor the signal pulse is investigated.
1. Introduction
Solitons are formed by the mutual interaction between the dispersive effects and the self phase modulation (SPM) the pulse encounters in the medium. For short pulses (pulse width < 1 ps) and for long fiber lengths [ 1-4 ], the nonlinear term due to SPM alone is not enough to predict the propagation of such pulses. A sech pulse, which is a bright soliton solution with only the SPM term present as the nonlinearity, undergoes steepening effect which results in an asymmetric output spectrum, when the pulse width is reduced to about a few ps. Self steepening results in shock formation in the absence of dispersive effects. In the presence of dispersion, this shock formation is avoided. Dispersion can balance the self steepening effect resulting in a stable soliton formation, provided the pulse has a proper envelope [5]. As the pulse steepens, the accompanying increase of spectral width makes dispersion all the more important resulting in a dispersive velocity spread which tends to dissipate the shock and balances the nonlinear velocity change which has resulted in the steepening of the pulse. With self steepening present, a bright soliton can exist even in the normal dispersion region (NDR) of the fiber in contrast to the Schroedinger soliton which exists only in the anomalous dispersion region (ADR). Cross phase modulation (XPM) occurs when at least one of the two co-propagating beams is intense, XPM is important, since it results in modulational instability in the ADR as well as the NDR [6], whereas, in the absence of the XPM, SPM results in modulational instability only in the ADR. XPM can result in wave breaking and pulse compression [7,8] and often effects and induced frequency shift in the co-propagating pulses [9 ]. It has been possible to generate short pulses from CW employing XPM [ 10,1 1 ]. The co-propagating of two short pulses necessitates the inclusion of a cross steepening term as is shown in ref. [ 12]. This comes about due to the fact that the existence of one wave affects the propagation of the other by steepening it. Here we consider the co-propagation of two short pulses under the combined influence of linear dispersion, phase modulation (SPM and XPM) and steepening (self and cross) effects. We show that if one of the pulses is intense enough to propagate as a soliton by itself and the other a weak signal, the signal pulse can get compressed by large factors. This is in contrast to the case in ref. [8] where expanded pulses accompanied compressed pulses.
306
0030-4018/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 85, number 4
OPTICS COMMUNICATIONS
15 September 1991
2. Analysis The co-propagation of two pulses at frequencies g2p (pump) and g2s (signal) is described by the set of coupled nonlinear equations [ 12 ]: 0Ap ..bfl(pl) 0Ap LR(2)02Ap=i.cp[Avl2Ap - 2 0 0z - ~ - + 2 "p Ot z ~pprp~ (]AplZAp), 0As
1• R ( 2 ) O2As = 2 i z s ] A p ] Z A s
0As
Oz +/~')o~-
+ 2"s
- - -4
at 2
~s
zs
0
(1) (2)
(]Apl2As)
~
'
where Ap and As are the envelopes of the pump and the signal respectively, fl~2) is the inverse group velocity of the pump (signal), and f l ~ ) is the group velocity dispersion at the pump (signal) frequency. Zp(s) is the nonlinear coefficient at the pump (signal) frequency determined by the Kerr nonlinearity. In writing the above set of equations, we have assumed in the equations derived in ref. [ 12 ], that the pump intensity is much larger than the signal intensity, i.e.; ]Ap[2>>IAsl 2. In deriving the above set of equations in ref. [ 12 ], it has been assumed that the medium responds instantaneously to any excitation so that the self-frequency shift due to Raman scattering is neglected. This assumption is valid as long as the pulse spectrum is small compared to the central frequency of the individual pulses, which holds good as long as the pulse width is larger than a few tens of femtoseconds. The pump pulse suffers only SPM (no XPM) and has the bright soliton solution under self steepening given by [5]
lAp]2= [A 2 / ( 2 _ 6 )
] [cosh2/t(t_ fl(p~) z - m z )
+ ( 0 - 1 ) / ( 2 - 0 ) ] -1 ,
(3)
where m correspond to a shift in the group velocity of the pump pulse due to a frequency shift because of self steepening. If self-steepening is absent, m = 0, 0= 1 and the solution reverts to the usual sech Schroedinger soliton. The parameters/t and 0 are defined through the relations 2 2 2m
0=-
"cp
+
[ zpA
2m -
"~
2rp
-
4rpm -
+
-
r2A z >0.
(5,6)
Substituting the soliton solution (eq. (3)) for the pump in the evolution equation for the signal (eq. 2) ) and going over to the retarded time given by T = t - (fl(v ~) + m ) z , the equation reads 0As Oz
0Ao +Aft ()) ---z°
IR(~)" 02As
OT + 2~'s
OT 2
=2irslAp[2As
4rs 0 E2s OT
(IAplZAs)
(7)
'
where Afl( l ) = fl~ ' ) _ fl~ l ) -- m. We assume a soliton solution for the signal pulse of the form exp[iAz(z, T)] .
As(z, T ) = A , ( T )
(8)
Substituting this assumed solution into eq. (7) for the signal and equating real and imaginary parts separately results in the set of equations: OA2 fls(2) F d2A, {0A2"~ 2] . 4zs 0.42 OA20z+ A f t ( ' ) - - ~ + ~ LA-t d T 2 - k - O - T ) j=ZZs IA. I : - ~ IApl2 0T '
m,(1) d/~l dT
~s(2)[2 d'~l 0A2 2
d ~ O~-T-=[=AI
02A2l _
4Ts( 0 2+ 20 ) OT2j--~ A,-~-TIApl IAp[ ~-~Al
(9)
•
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Multiplying eq. (10) by A~ and integrating over T gives O A ~ / ~ l ' = A f l ~ " / f l ~ 2~ + (4r~/Qd3~2')lAp 12 .
(11 )
This implies a shift in the central frequency of the signal given by A]/( l )/,g~2j and a chirping given by the second term in eq. ( 11 ). This analysis is valid as long as IAfl (l)/fl~2) I << £2s, the signal central frequency. From eq. (9), along with eq. ( 11 ), it is obvious, that
O.42/Oz=K,
a
constant,
{ 12)
Using these results in eq. (9), we get the equation for Al as d2A,/dT2=,4~tM+N[ch2/zT+(~-l)/(2-~)]
'-Q[ch2/zT+(c~-l)/(2-fi)]-2'~,
(13)
where M = ( - 2/fl~ 2~ ) ( K + Afl~"2/2fi~ 2) ) ,
N = [ r ~ A 2 / ( 2 - ~ ) ] (4/fl~ 2) - 8Afl~')/£2~fl~ 2)2 ),
Q = 12(rs/f2~fl~ 2) )2A4 , and ch stands for the hyperbolic cos function. Making the transformation [ch2/tT+ ( 6 - 1 ) / ( 2 - ~ ) ] = X reduces eq. ( 13 ) to an algebraic one: dM2 A2' [ ( 2 X + - 2 - ~ d ) 2 -N 1 ] + 2 ~ - ( 2dX - _ ~ - d )X ='y4~-'(
X
Q2) •
(14,
The solution of eq. (14) can be written down as a series, keeping in mind that the sought-for function goes to zero as T goes to oo: L'| 1 ( X ) ~ - a 0 + a L / X + a 2 / X 2
+ ....
(
15 )
It can be seen by direct substitution that ao is always zero. One could choose the parameters M, N and Q such that any of the coefficients a, (in general, any combination of a,s) could be kept non zero and the others zero. In general, for a, :;e0 for i= q, (q determines the degree of compression ), the material parameters (and hence the operating wavelengths) are determined through the relations: 4q~-=M/l~ 2,
4(q2+q/2)[~5/(cS-2)]=N/li
2,
q(q+l)[c52/(~5-2)2-1]+Q/,u2=O,
so that the solution ofeq. (14) with a~¢0 for i = q is given by A,(T)=au[coshZllT+
((~- 1 ) / ( 2 - ( 5 ) ] - ~
where aq is an arbitrary constant. From the expressions for M, N and Q, one can easily determine the conditions (i.e. the operating wavelengths) for generating an arbitrary combination of compressed pulses all with different amounts of compression.
3. Discussion of the results The degree of compression the signal pulse undergoes is determined by the value of q, which can be very large. In contrast to the results in ref. [8], here we can obtain compressed pulses without any accompanying broadened pulses, The maximum value of q is limited by the fact that the shift in central frequency of the signal pulse should be much smaller than the central frequency itself (so that the slowly varying envelope approximation which forms one of the important assumptions involved in the derivation of the governing pair of equations is not violated), and also by the fact that for both -Qp and ~s, the fiber remains single moded, and so infinite compression is not possible. Also, in the derivation of the governing set of nonlinear equations in 308
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15 September 1991
ref. [ 12 ], we assumed that the nonlinear response of the fiber is instantaneous. But as the pulse gets compressed by large factors to a few tens o f femtoseconds width, the response of the fiber can no more be considered to be instantaneous. One has to take into account the effect of nonlinear inertia, which leads to a self frequency shift [ 13 ] due to stimulated Raman scattering. Also the Taylor series expansion of the propagation constant was truncated after the second derivative term offl with respect to ~. This would no more hold good as the higher order dispersion terms gain in importance as the pulse spectrum becomes comparable to the central frequency. Any arbitrary signal pulse would travel down the fiber (along with the p u m p ) as a linear combination of compressed pulses for general combinations of material parameters. For a pulse which closely approximates one of the eigenfunctions of the system, the pulse would evolve into one after propagation through some characteristic distance determined by the parameter values involved. The phase modulation needed by the signal pulse to compensate for the dispersion it suffers is provided for by the pump, enabling the signal pulse to travel as a soliton even with an arbitrary low power. This is a consequence of the fact that the governing equation for the signal pulse is linear in its amplitude. In the absence of the pump, the signal pulse would gradually disperse away. This effect, as pointed out in ref. [8 ] could be used for high speed optical switching employing a high speed saturable absorber. Comparing these results with the results in ref. [ 8] indicates the reason for obtaining only compressed pulses with no accompanying broadened pulses in contrast to the case in ref. [ 8 ]. The cross steepening effect provides an additional nonlinear chirp to the signal pulse (over and above the chirp generated by X P M ) which would result in pulse compression if the pulse is passed through a suitable dispersive element, which is the fiber itself in this case. For different material parameter combinations, the chirp induced due to the cross steepening effect is different resulting in different compression factors. The notable feature of the compressed pulses is the fact that these pulses are all chirped (see eq. ( 1 1 ) ), so that these pulses could be compressed further by external means. The possibility of compression of the signal pulse even though the group velocities of the pump and the signal are not the same allows a larger range of operating wavelengths for the signal pulse. A difference in the group velocities does not necessarily imply a walk-off between the interacting pulses as is shown in ref. [ 14 ]. A shift in the central frequency of one or both the pulses - the shift being such that the two pulses now start travelling at the same group velocity - would offset the walk-off effect. This results from the fact that for small initial differences in group velocities of the individual pulses, the attractive force between the pulses is strong enough to facilitate "locking" together of the pulses. In conclusion, we show that it is possible to attain large compression ratios for a weak signal when it copropagates with a strong p u m p pulse in the presence of dispersion, SPM, XPM and cross steepening. A nonzero value of the group velocity difference is not a deterrant in forming a stable pair as the two pulses propagate down the fiber.
Acknowledgements The author is thankful to prof. A.K. Ghatak for the many discussions and for this constant encouragement and to the referee for useful suggestions.
References [ 1] UF. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. [ 2 ] N. Tzoar and M. Jain, Phys. Rev. A 23 ( 1981 ) 1266. [3] D. Anderson and M. Lisak, Optics Lett. 7 (1982) 394. 309
Volume 85, number 4
OPTICS COMMUNICATIONS
[4] H. Nakatsuka, D. Grischkowsky and A.C. Balant, Phys. Rev. Lett. 47 ( 1981 ) 910. [ 5 ] D. Anderson and M. Lisak, Phys. Rev. A 27 ( 1983 ) 1393. [6] G.P. Agrawal, Phys. Rev. Lett. 59 (1987) 880. [7 ] G.P. Agrawal, P.L. Baldeck and R.R. Alfano, Optics Lett. 14 ( 1988 ) 137. [8 ] P.C. Subramaniam and A.K. Ghatak, communicated. [ 9 ] P.L. Baldeck, R.R. Alfano and G.P. Agrawal, Appl. Phys. Lett. 52 ( 1988 ) 1939. [ 10] D. Schadt and B. Jaskorzynska, Electron. Lett. 23 (1987) 1090. [ 11 ] V.L. da Silva and C.H. Brito Cruz, J. Opt. Soc. Am. B. 7 (1990) 7 (1990) 750. [ 12] P.C. Subramaniam, communicated. [ 13 ] Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23 ( 1987 ) 510. [ 14] P.C. Subramaniam, Optics Lett., to be published.
310
15 September 1991
OPT IC S
COMMUNICATIONS
Co-propagation of pulses with steepening and phase modulation effects P.C. Subramaniam Optical WaveguideGroup. Department of Physics, Indian Institute of Teehnology, New Delhi-11016, India
Received 9 November 1990;revised manuscript received 13 May 1991
Weconsiderthe copragatlonof a high intensity pump and a low intensity signal in the presenceof dispersion, phase modulation and steepening effects. The possibilityof obtaining large compressionfor the signal pulse is investigated.
1. Introduction
Solitons are formed by the mutual interaction between the dispersive effects and the self phase modulation (SPM) the pulse encounters in the medium. For short pulses (pulse width < 1 ps) and for long fiber lengths [ 1-4 ], the nonlinear term due to SPM alone is not enough to predict the propagation of such pulses. A sech pulse, which is a bright soliton solution with only the SPM term present as the nonlinearity, undergoes steepening effect which results in an asymmetric output spectrum, when the pulse width is reduced to about a few ps. Self steepening results in shock formation in the absence of dispersive effects. In the presence of dispersion, this shock formation is avoided. Dispersion can balance the self steepening effect resulting in a stable soliton formation, provided the pulse has a proper envelope [5]. As the pulse steepens, the accompanying increase of spectral width makes dispersion all the more important resulting in a dispersive velocity spread which tends to dissipate the shock and balances the nonlinear velocity change which has resulted in the steepening of the pulse. With self steepening present, a bright soliton can exist even in the normal dispersion region (NDR) of the fiber in contrast to the Schroedinger soliton which exists only in the anomalous dispersion region (ADR). Cross phase modulation (XPM) occurs when at least one of the two co-propagating beams is intense, XPM is important, since it results in modulational instability in the ADR as well as the NDR [6], whereas, in the absence of the XPM, SPM results in modulational instability only in the ADR. XPM can result in wave breaking and pulse compression [7,8] and often effects and induced frequency shift in the co-propagating pulses [9 ]. It has been possible to generate short pulses from CW employing XPM [ 10,1 1 ]. The co-propagating of two short pulses necessitates the inclusion of a cross steepening term as is shown in ref. [ 12]. This comes about due to the fact that the existence of one wave affects the propagation of the other by steepening it. Here we consider the co-propagation of two short pulses under the combined influence of linear dispersion, phase modulation (SPM and XPM) and steepening (self and cross) effects. We show that if one of the pulses is intense enough to propagate as a soliton by itself and the other a weak signal, the signal pulse can get compressed by large factors. This is in contrast to the case in ref. [8] where expanded pulses accompanied compressed pulses.
306
0030-4018/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 85, number 4
OPTICS COMMUNICATIONS
15 September 1991
2. Analysis The co-propagation of two pulses at frequencies g2p (pump) and g2s (signal) is described by the set of coupled nonlinear equations [ 12 ]: 0Ap ..bfl(pl) 0Ap LR(2)02Ap=i.cp[Avl2Ap - 2 0 0z - ~ - + 2 "p Ot z ~pprp~ (]AplZAp), 0As
1• R ( 2 ) O2As = 2 i z s ] A p ] Z A s
0As
Oz +/~')o~-
+ 2"s
- - -4
at 2
~s
zs
0
(1) (2)
(]Apl2As)
~
'
where Ap and As are the envelopes of the pump and the signal respectively, fl~2) is the inverse group velocity of the pump (signal), and f l ~ ) is the group velocity dispersion at the pump (signal) frequency. Zp(s) is the nonlinear coefficient at the pump (signal) frequency determined by the Kerr nonlinearity. In writing the above set of equations, we have assumed in the equations derived in ref. [ 12 ], that the pump intensity is much larger than the signal intensity, i.e.; ]Ap[2>>IAsl 2. In deriving the above set of equations in ref. [ 12 ], it has been assumed that the medium responds instantaneously to any excitation so that the self-frequency shift due to Raman scattering is neglected. This assumption is valid as long as the pulse spectrum is small compared to the central frequency of the individual pulses, which holds good as long as the pulse width is larger than a few tens of femtoseconds. The pump pulse suffers only SPM (no XPM) and has the bright soliton solution under self steepening given by [5]
lAp]2= [A 2 / ( 2 _ 6 )
] [cosh2/t(t_ fl(p~) z - m z )
+ ( 0 - 1 ) / ( 2 - 0 ) ] -1 ,
(3)
where m correspond to a shift in the group velocity of the pump pulse due to a frequency shift because of self steepening. If self-steepening is absent, m = 0, 0= 1 and the solution reverts to the usual sech Schroedinger soliton. The parameters/t and 0 are defined through the relations 2 2 2m
0=-
"cp
+
[ zpA
2m -
"~
2rp
-
4rpm -
+
-
r2A z >0.
(5,6)
Substituting the soliton solution (eq. (3)) for the pump in the evolution equation for the signal (eq. 2) ) and going over to the retarded time given by T = t - (fl(v ~) + m ) z , the equation reads 0As Oz
0Ao +Aft ()) ---z°
IR(~)" 02As
OT + 2~'s
OT 2
=2irslAp[2As
4rs 0 E2s OT
(IAplZAs)
(7)
'
where Afl( l ) = fl~ ' ) _ fl~ l ) -- m. We assume a soliton solution for the signal pulse of the form exp[iAz(z, T)] .
As(z, T ) = A , ( T )
(8)
Substituting this assumed solution into eq. (7) for the signal and equating real and imaginary parts separately results in the set of equations: OA2 fls(2) F d2A, {0A2"~ 2] . 4zs 0.42 OA20z+ A f t ( ' ) - - ~ + ~ LA-t d T 2 - k - O - T ) j=ZZs IA. I : - ~ IApl2 0T '
m,(1) d/~l dT
~s(2)[2 d'~l 0A2 2
d ~ O~-T-=[=AI
02A2l _
4Ts( 0 2+ 20 ) OT2j--~ A,-~-TIApl IAp[ ~-~Al
(9)
•
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OPTICS COMMUNICATIONS
15 September 1991
Multiplying eq. (10) by A~ and integrating over T gives O A ~ / ~ l ' = A f l ~ " / f l ~ 2~ + (4r~/Qd3~2')lAp 12 .
(11 )
This implies a shift in the central frequency of the signal given by A]/( l )/,g~2j and a chirping given by the second term in eq. ( 11 ). This analysis is valid as long as IAfl (l)/fl~2) I << £2s, the signal central frequency. From eq. (9), along with eq. ( 11 ), it is obvious, that
O.42/Oz=K,
a
constant,
{ 12)
Using these results in eq. (9), we get the equation for Al as d2A,/dT2=,4~tM+N[ch2/zT+(~-l)/(2-~)]
'-Q[ch2/zT+(c~-l)/(2-fi)]-2'~,
(13)
where M = ( - 2/fl~ 2~ ) ( K + Afl~"2/2fi~ 2) ) ,
N = [ r ~ A 2 / ( 2 - ~ ) ] (4/fl~ 2) - 8Afl~')/£2~fl~ 2)2 ),
Q = 12(rs/f2~fl~ 2) )2A4 , and ch stands for the hyperbolic cos function. Making the transformation [ch2/tT+ ( 6 - 1 ) / ( 2 - ~ ) ] = X reduces eq. ( 13 ) to an algebraic one: dM2 A2' [ ( 2 X + - 2 - ~ d ) 2 -N 1 ] + 2 ~ - ( 2dX - _ ~ - d )X ='y4~-'(
X
Q2) •
(14,
The solution of eq. (14) can be written down as a series, keeping in mind that the sought-for function goes to zero as T goes to oo: L'| 1 ( X ) ~ - a 0 + a L / X + a 2 / X 2
+ ....
(
15 )
It can be seen by direct substitution that ao is always zero. One could choose the parameters M, N and Q such that any of the coefficients a, (in general, any combination of a,s) could be kept non zero and the others zero. In general, for a, :;e0 for i= q, (q determines the degree of compression ), the material parameters (and hence the operating wavelengths) are determined through the relations: 4q~-=M/l~ 2,
4(q2+q/2)[~5/(cS-2)]=N/li
2,
q(q+l)[c52/(~5-2)2-1]+Q/,u2=O,
so that the solution ofeq. (14) with a~¢0 for i = q is given by A,(T)=au[coshZllT+
((~- 1 ) / ( 2 - ( 5 ) ] - ~
where aq is an arbitrary constant. From the expressions for M, N and Q, one can easily determine the conditions (i.e. the operating wavelengths) for generating an arbitrary combination of compressed pulses all with different amounts of compression.
3. Discussion of the results The degree of compression the signal pulse undergoes is determined by the value of q, which can be very large. In contrast to the results in ref. [8], here we can obtain compressed pulses without any accompanying broadened pulses, The maximum value of q is limited by the fact that the shift in central frequency of the signal pulse should be much smaller than the central frequency itself (so that the slowly varying envelope approximation which forms one of the important assumptions involved in the derivation of the governing pair of equations is not violated), and also by the fact that for both -Qp and ~s, the fiber remains single moded, and so infinite compression is not possible. Also, in the derivation of the governing set of nonlinear equations in 308
Volume 85, number 4
OPTICS COMMUNICATIONS
15 September 1991
ref. [ 12 ], we assumed that the nonlinear response of the fiber is instantaneous. But as the pulse gets compressed by large factors to a few tens o f femtoseconds width, the response of the fiber can no more be considered to be instantaneous. One has to take into account the effect of nonlinear inertia, which leads to a self frequency shift [ 13 ] due to stimulated Raman scattering. Also the Taylor series expansion of the propagation constant was truncated after the second derivative term offl with respect to ~. This would no more hold good as the higher order dispersion terms gain in importance as the pulse spectrum becomes comparable to the central frequency. Any arbitrary signal pulse would travel down the fiber (along with the p u m p ) as a linear combination of compressed pulses for general combinations of material parameters. For a pulse which closely approximates one of the eigenfunctions of the system, the pulse would evolve into one after propagation through some characteristic distance determined by the parameter values involved. The phase modulation needed by the signal pulse to compensate for the dispersion it suffers is provided for by the pump, enabling the signal pulse to travel as a soliton even with an arbitrary low power. This is a consequence of the fact that the governing equation for the signal pulse is linear in its amplitude. In the absence of the pump, the signal pulse would gradually disperse away. This effect, as pointed out in ref. [8 ] could be used for high speed optical switching employing a high speed saturable absorber. Comparing these results with the results in ref. [ 8] indicates the reason for obtaining only compressed pulses with no accompanying broadened pulses in contrast to the case in ref. [ 8 ]. The cross steepening effect provides an additional nonlinear chirp to the signal pulse (over and above the chirp generated by X P M ) which would result in pulse compression if the pulse is passed through a suitable dispersive element, which is the fiber itself in this case. For different material parameter combinations, the chirp induced due to the cross steepening effect is different resulting in different compression factors. The notable feature of the compressed pulses is the fact that these pulses are all chirped (see eq. ( 1 1 ) ), so that these pulses could be compressed further by external means. The possibility of compression of the signal pulse even though the group velocities of the pump and the signal are not the same allows a larger range of operating wavelengths for the signal pulse. A difference in the group velocities does not necessarily imply a walk-off between the interacting pulses as is shown in ref. [ 14 ]. A shift in the central frequency of one or both the pulses - the shift being such that the two pulses now start travelling at the same group velocity - would offset the walk-off effect. This results from the fact that for small initial differences in group velocities of the individual pulses, the attractive force between the pulses is strong enough to facilitate "locking" together of the pulses. In conclusion, we show that it is possible to attain large compression ratios for a weak signal when it copropagates with a strong p u m p pulse in the presence of dispersion, SPM, XPM and cross steepening. A nonzero value of the group velocity difference is not a deterrant in forming a stable pair as the two pulses propagate down the fiber.
Acknowledgements The author is thankful to prof. A.K. Ghatak for the many discussions and for this constant encouragement and to the referee for useful suggestions.
References [ 1] UF. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. [ 2 ] N. Tzoar and M. Jain, Phys. Rev. A 23 ( 1981 ) 1266. [3] D. Anderson and M. Lisak, Optics Lett. 7 (1982) 394. 309
Volume 85, number 4
OPTICS COMMUNICATIONS
[4] H. Nakatsuka, D. Grischkowsky and A.C. Balant, Phys. Rev. Lett. 47 ( 1981 ) 910. [ 5 ] D. Anderson and M. Lisak, Phys. Rev. A 27 ( 1983 ) 1393. [6] G.P. Agrawal, Phys. Rev. Lett. 59 (1987) 880. [7 ] G.P. Agrawal, P.L. Baldeck and R.R. Alfano, Optics Lett. 14 ( 1988 ) 137. [8 ] P.C. Subramaniam and A.K. Ghatak, communicated. [ 9 ] P.L. Baldeck, R.R. Alfano and G.P. Agrawal, Appl. Phys. Lett. 52 ( 1988 ) 1939. [ 10] D. Schadt and B. Jaskorzynska, Electron. Lett. 23 (1987) 1090. [ 11 ] V.L. da Silva and C.H. Brito Cruz, J. Opt. Soc. Am. B. 7 (1990) 7 (1990) 750. [ 12] P.C. Subramaniam, communicated. [ 13 ] Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23 ( 1987 ) 510. [ 14] P.C. Subramaniam, Optics Lett., to be published.
310
15 September 1991