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Dark soliton trains formed by visible pulse collisions in optical fibers
Volume 82, number 1,2
OPTICS COMMUNICATIONS
1 April 1991
Dark soliton trains formed by visible pulse collisions in optical fibers J o s h u a E. R o t h e n b e r g IBM Watson Research Center, P.O. Box 218, YorktownHeights, NY 10598, USA Received 26 October 1990; revised manuscript received 17 December 1990
It is shown that, through the nonlinear collision of two visible pulses copropagatingin an optical fiber, a high-frequencytrain of dark pulses form, which are well described by the analytic dark soliton solution. Numerical solutions of the nonlinear Schroedinger equation show that the solitons form without secondarypulses and their pulsewidths adiabaticallybroaden in concert with the slowlydecreasingintensity of the backgroundpulse upon which they are situated. For equal input pulse intensities it is shown that the dark pulses asymptoticallyapproach the fundamental dark soliton. Optical solitons in fibers have received increasing attention recently, both because of their scientific and practical importance. Optical solitons are pulses which propagate with unaltered intensity owing to the balance between nonlinear self phase modulation and linear group velocity dispersion [ 1-4]. For anomalous dispersion (2~>1.3 ~tm), the simplest "bright" soliton is a pulse with intensity ~sech2t. Dark solitons, which occur for normal dispersion, are characterized by an intensity dip ( I = 1-~/2sech2t, where ~/2 is the contrast of the dip) on a continuous background which propagates unaltered. Although a true continuous background wave may be impractical, it has been shown that dark pulses on a background pulse of finite width can propagate adiabatically as solitons [5]. Dark solitons have been observed in optical fibers by a few different experimental techniques [ 6-8 ], however, the generation of the dark pulses required considerable experimental effort. The repetitive generation of solitons (soliton trains) was first considered in the generation of bright solitons (for anomalous dispersion) from modulational instability by Hasegawa [9], who found that sinusoidal intensity modulation evolved into self compressed pulses with some residual interpulse energy. More recently it was shown that continuous sinusoidal intensity modulation evolves into a train of bright (dark) solitons for anomalous (normal) dispersion only if one adds linear ampli-
fication to the propagation [ 10 ]. In a related experiment it was demonstrated that the nonlinear copropagation of two visible pulses in an optical fiber leads to rapid intensity modulation on the finite width background as the pulses overlap [ 11 ]. In this paper it is shown that trains of dark solitons form naturally in the nonlinear copropagation and collision of two pulses in an optical fiber. For pulses of equal input intensity, the solitons are found to asymptotically approach the theoretical shape of the fundamental (full contrast) dark soliton. The soliton widths vary adiabatically owing to the broadening, and hence diminished power, of the finite width background pulse upon which the solitons are situated. The propagation of the field, ¢(z, t) = Re{E(z, t) e x p [ - i ( c o o t - n o k o z ) ] } , in an optical fiber is accurately described by the nonlinear Schroedinger equation (NLSE) OE/Oz+ i( k<2) / 2 )O2E/Ot 2
= in2 ko [El 2 E - i T I E I 2E,
( 1)
where k<2>=82k/Sto 2 is positive for normal group velocity dispersion, and a retarded time frame travelling at v~=aog/Ok is assumed. For the numerical simulations presented here, parameters typical for a wavelength of 600 nm are assumed: k <2)=0.065 ps2/ m and 7=0.020 m -~ W -~ (with IEI 2 specified by
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
107
Volume 82, number 1,2
OPTICS C O M M U N I C A T I O N S
power in Watt). Now consider the following input field
This field is the in-phase sum of two equal sech 2 intensity pulses, and is easily formed by a Michelson interferometer [ 11 ]. The numerical integration of eq. (1) with this input is shown for various fiber lengths in fig. 1, where the fwhm of the input pulses is 2 ps (Zo= 1.13 ps), At=8 ps, Po= 500 W, and the intensity scale has been normalized at the fiber input. The results for small propagation distances ( ~<10 m) are in agreement with earlier experimental results [ 11 ], where each pulse broadens and acquires a linear frequency sweep owing to self phase modulation and dispersion, and, once overlap occurs, one observes rapid modulation owing to the interference between the large frequency shifts of the leading and trailing edges of the colliding pulses. With further propagation (note the time scale changes in fig. 1 ) the overlapping region broadens and the modulation reshapes into an expanding train of increasingly isoI
.43
.74
iNPUTfO 2.5 m 5m 20
.58 ~.14 Z LI.I
.05
.012
0 O.Sx TIME (psec, scale varies as :r)
1600 1.0x
Fig. 1. Calculation showing the copropagation in an optical fiber of two 500 W input pulses at the indicated distances. The intensity scale is normalized to unity at the input, and the peak intensity at each distance is indicated. The time scale is increasingly expanded as indicated.
108
((l-v~) ':~
+_iqtanh[(t-toT- V-~z)/zo]} exp(iyPoz), 4O0
--.5x
lated and narrow dark pulses. Although it will be shown that the dark pulses are adiabatically broadening (~~/:z) as they propagate, the expansion of the dark pulse separation is faster ( ~ z ) , and thus fig. 1 gives the relative appearance that the dark pulses are sharpening. The rapid spreading of the dark pulses is a result of the dispersion of the underlying background pulse. Note also the lack of modulation between the dark pulses, i.e. there is little apparent energy shed by the dark pulses as they reshape. Similar results are obtained for other values of Zo and Po, which determine the bandwidth generated by each input pulse, and At, which determines the beat frequency, and hence the dark pulse spacing. One might expect the dark pulses to be equally spaced since one expects regular interference between pulses which both have linear frequency sweeps [ 11 ]. However, it appears that the nonlinear interaction between the two pulses at small distances produces a slight nonuniformity in the dark pulse distribution. This is borne out by calculations using the input of eq. (2) but with a larger delay between the pulses. In this case one finds that a more dense and more regularly spaced dark pulse train forms. For different relative phases between the input pulses the same general results are obtained, however the timing of the dark pulses is modified. By varying the relative phase of the input pulses from 0 to 2n the dark pulse train shifts through one period, thereby enabling completely adjustable time shifting of the train. For a ~t phase difference (anti-symmetric case) the dark pulse distribution is nearly identical to that shown in fig. 1 except there is an additional dark pulse at t = 0. Dark solitons for normal dispersion have the form
[21 e,,a,,, =,,/~o
.024
I April 1991
(3)
where
z~ =k(2)/~ZPo, V - I = (k(Z)/rlZo) ( 1 -r/2) l/z,
(4) (5)
and 0 < q < 1. Thus the dark soliton intensity is IE~rk 12=Po {1 -r/2sech2 [ (t-to-T- V-lz)/zo]} •
(6)
Volume 82, number 1,2
OPTICS COMMUNICATIONS
The fwhm of the dark soliton is then 1.763Zo, and the parameter ~/determines its contrast and velocity. For ~1=1, E~rk~tanh(t-to)/Zo, so that the contrast is 100% (the intensity dips completely to 0) and the phase of this pulse has a discontinuous j u m p of n at t = to. The pulse for r/= 1 has been termed the fundamental or "black" soliton [5,8 ]. For ~/< 1, the intensity does not dip completely to zero (the contrast is given by r/2), and the phase jump is reduced and more gradual. Fig. 2 shows a comparison of the intensity and phase of the dark pulses found in the numerical solutions of fig. 1 (solid curves) with the analytic dark soliton form of eqs. ( 3 ) - ( 6 ) (dots), where the dark pulse in fig. 1 closest to t = 0 at a few propagation distances is examined. For this comparison, the average power in the vicinity of each dark pulse from fig. 1 is used forPo in eqs. (3), (4), and to is taken from the location of the dark pulse minimum. Then r/is found from the contrast of the dark
~
0.5.
0
-
.
-1
5
i
~
. . . . . . .
I
---
I
,
,
.
.
0
I
I
[
I
.
.
.
I
.
I
I
-.5 -4
0 4 TIME (psec)
-4
0 TIME
(psec)
4
Fig. 2. Solid curves showthe intensity and phase of the dark pulse nearest to t=0 in fig. 1. This is compared with the analytic dark soliton solution of eqs. (3), (4) (dots) for the distances indicated. Note that the time scale increases by a factor of 2 at successive distances. The background intensity level at each distance is indicated and corresponds to (a) 25, (b) 5.4, and (c) 1.3 W. Contrast parameter r/2is (a) 0.9892, (b) 0.9965, and (c) 0.9992, and the dark pulse fwhm is (a) 0.65, (b) 1.38, and (c) 2.85 ps.
1 April 1991
pulse intensity minimum calculated in fig. I. Note that the pulse width Zo is then determined from eq. (4) and is not adjusted. In the phase comparison the background linear phase shift (corresponding to a frequency shift) has been subtracted. The intensity fit is nearly exact for the distances examined. The remaining small discrepancy is a result of the sloping intensity of the expanding background pulse. Note that since the background pulse is spreading approximately linearly, the background power for a given soliton is decreasing like z - l , and thus, from eq. (4), one sees that the soliton width Zo~ x/~. The pulse widths found in fig. 2 agree with this approximation to good accuracy. The phase fit (with the same parameters as the intensity fit) is also nearly exact over the central part of the pulse. However, away from the pulse center the deviation in the phase is significant and is understood in terms of the quadratic phase variation of the background pulse owing to its linear frequency chirp. These results are similar with the numerical analyses of Tomlinson et al. [ 5 ], where it was also found that dark pulses propagated adiabatically as solitons in spite of the decreasing intensity and linear frequency chirp of a finite width background pulse. The parameter ~/found in fig. 2 is nearly unity in all cases and asymptotically approaches 1 for larger z. As is apparent from fig. 2, the phase fit is sensitive to extremely small changes in ~/as r/--,1, so it is clear that these dark pulses are approaching the fundamental black soliton as they propagate. Note that according to eq. (5), as r/--, 1, the relative soliton velocity V - l ~ 0 (i.e. it becomes temporally stationary in the propagating time frame of eq. ( 1 ) ) . However, from fig. 1 it is apparent that these solitons are not stationary. This is a result of the frequency chirp of the background pulse, which shifts the central frequency of each soliton and thus changes its relative group velocity. Eqs. ( 3 ) - (6) can likewise be generalized [ 4 ] for a frequency shift Aog, which then shifts the soliton velocity V - I by k(2)Aog. The fact that fundamental dark solitons form throughout the background pulse in fig. 1 is somewhat the result of the symmetry of the input field (eq. (2) ). To explore this feature, the input field of eq. (2) is modified by reducing the intensity of the trailing sech 2 pulse by a factor of 2. The results of the numerical propagation of this asymmetric field are 109
Volume 82, n u m b e r 1,2
OPTICS C O M M U N I C A T I O N S
shown in fig. 3. One sees rather similar results to that found in fig. 1, with the exception that the dark pulses have reduced contrast. In spite of this, these pulses still maintain nearly exact agreement with the analytic dark soliton form of eqs. ( 3 ) - ( 6 ), with similar small deviations owing to the non-uniform and frequency-swept background. It is interesting to point out that, similar to what one would expect in simple linear interference, for unbalanced input pulse intensities the interference contrast and, hence, the dark pulse depth are diminished. The effects of linear loss in eq. ( 1 ) on the soliton propagation have also been investigated numerically. For sufficiently small loss the width of the solitons adiabatically adjust to the reduced intensity owing to both the loss and to the spreading of the background pulse. However, for loss which is significant on the scale of the characteristic soliton distance (Zo ~ r2/k (2)), the solitons cannot "keep up" with the loss and begin to noticeably distort and radiate energy. For a given loss, after sufficient propagation distance, this regime always occurs since Zo is monotonically increasing (as the solitons broaden). For the example given in fig. 1, at a dis-
I :
~
.2
x 10
16 m
.09 z '"
40
_
.01 .0024
- i
~320 x
-.5x TIME ( p s e c ,
0 0.Sx scale varies as x)
8O
1 1250 1.0x
Fig. 3. Calculation showing the copropagation of two input pulses of peak powers 500 and 250 W at the indicated distances. The intensity and time scales are expanded as indicated.
110
1 April 1991
tance of I kin, significant distortion appears for loss ~ 5 - 1 0 dB/km. To insure distortion free soliton propagation for much longer distances it appears necessary to introduce gain into the fiber to balance the loss [ 12 ]. It has been noted that dark solitons can be produced, without threshold, from an arbitrarily modulated continuous wave [ 13 ]. Also it has been shown that dark solitons form on a sinusoidally modulated continuous wave in the presence of gain [ 10]. However, the generation of a train of dark solitons on a finite width background pulse is not generally achieved with an arbitrarily modulated input. For example consider the modulated input pulse
Ein =X/~o sin ( 2nt / Tl ) sech ( t / T2 ) ,
(7)
where T2 >> T~. The sinusoidal modulation supplies a phase change of n for each dark pulse, as is required for the dark soliton. If one takes the power Po to correspond to the soliton power for a dark pulse of width given by the modulation, then one would expect the input to naturally evolve into a train of dark solitons. However, the numerical solution for the test case 7"2=20 ps, T I = I ps, and Po=40 W, shows that, although the modulation is 100% initially, no solitons form and the contrast of the modulation decreases continuously until disappearing completely. This is seen to be a result of the positive and negative frequency components which comprise eq. (7) separating owing to dispersion. On the other hand, frequency chirped pulses (which form from the input of eq. (2) and which generally form from the linear dispersion of an arbitrary pulse after sufficient propagation) have the interesting property that superimposed intensity modulation of bandwidth less than the extent of the chirp does not generally disperse away from the background and will thus naturally evolve into dark solitary components On the finite width background. The experimental observation of these soliton trains would be most likely accomplished by a direct intensity measurement, since dark s01itons on a long background are difficult to observe via autocorrelation. A cross correlation intensity measurement averaged over many input pulses might require rather stringent requirements on the input amplitude and shape stability, since variation of the input pulse parameters would likely lead to time shifts of the dark
Volume 82, number 1,2
OPTICS COMMUNICATIONS
soliton train from pulse to pulse a n d thereby obscure the observation. Therefore, the intensity measurement would preferably be performed on a single pulse basis (e.g. a streak c a m e r a ) . F r o m the calculations presented here, a realistic resolution o f ~ 1 ps appears to be adequate for such a measurement. In conclusion, it has been shown that a straightforward m e t h o d can generate trains o f high contrast dark pulses with little a p p a r e n t interpulse energy. Although these dark pulses are situated on a finite-width frequency-chirped b a c k g r o u n d pulse o f d i m i n i s h i n g intensity, they are quite accurately described by high contrast d a r k solitons whose pulse width increases adiabatically as the b a c k g r o u n d intensity decreases. The t e m p o r a l position o f the generated solitons is d e p e n d e n t on the relative phase a n d shape o f the input pulses. It m a y therefore be possible to use input m o d u l a t i o n to m a n i p u l a t e the relative timing o f the solitons within the train, thereby creating the possibility o f a time shift digital signal.
1 April 1991
References [ 1] V.E. Zakharov and A.B. Shabat, Sov. Phys. JETP 34 (1972) 62 [Zh. Eksp. Teor. Fiz. 61 ( 1971 ) 118 ]. [2] V.E. Zakharov antl A.B. Shabat, Sov. Phys. JETP 37 (1973) 823 [Zh. Eksp. Teor. Fiz. 64 (1973) 1627]. [3] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23 (1973) 142. [4 ] A. Hasegawa and F. Tappert, Appl. Phys. Len. 23 (1973 ) 171. [5] W.J.Tomlinson, R.J. Hawkins, A.M. Weiner, J.P. Heritage and R.N. Thurston, J. Opt. Soc. Am. B 6 (1989) 329. [6] P. Emplit, J.P. Hamaide, F. Reynaud, C. Frochly and A. Barthelemy, Optics Comm. 62 (1987) 374. [ 7 ] D. Krokel, N.J. Halas, G. Guiliani and D. Grischkowsky, Phys. Rev. Lett. 60 (1988) 29. [8] A.M. Weiner, J.P. Heritage, R.J. Hawkins, R.N. Thurston, E.M. Kirschner, D.E. Leaird and W.J. Tomlinson, Phys. Rev. Lett. 61 (1988) 2445. [9] A. Hasegawa, Optics Lett. 9 (1984) 288. [ 10] E.M. Dianov, P.V. Mamyshev, A.M. Prokhorov and S.V. Chernikov, Optics Len. 14 (1989) 1008. [ 11 ] J.E. Rothenberg, Optics Lett. 15 (1990) 443. [ t2] A. Hasegawa and Y. Kodama, Optics Lett. 7 (1984) 285, 393. [ 13] S.A. Gredeskul and Y.S. Kivshar, Optics Lett. 14 (1989) 1281.
111
OPTICS COMMUNICATIONS
1 April 1991
Dark soliton trains formed by visible pulse collisions in optical fibers J o s h u a E. R o t h e n b e r g IBM Watson Research Center, P.O. Box 218, YorktownHeights, NY 10598, USA Received 26 October 1990; revised manuscript received 17 December 1990
It is shown that, through the nonlinear collision of two visible pulses copropagatingin an optical fiber, a high-frequencytrain of dark pulses form, which are well described by the analytic dark soliton solution. Numerical solutions of the nonlinear Schroedinger equation show that the solitons form without secondarypulses and their pulsewidths adiabaticallybroaden in concert with the slowlydecreasingintensity of the backgroundpulse upon which they are situated. For equal input pulse intensities it is shown that the dark pulses asymptoticallyapproach the fundamental dark soliton. Optical solitons in fibers have received increasing attention recently, both because of their scientific and practical importance. Optical solitons are pulses which propagate with unaltered intensity owing to the balance between nonlinear self phase modulation and linear group velocity dispersion [ 1-4]. For anomalous dispersion (2~>1.3 ~tm), the simplest "bright" soliton is a pulse with intensity ~sech2t. Dark solitons, which occur for normal dispersion, are characterized by an intensity dip ( I = 1-~/2sech2t, where ~/2 is the contrast of the dip) on a continuous background which propagates unaltered. Although a true continuous background wave may be impractical, it has been shown that dark pulses on a background pulse of finite width can propagate adiabatically as solitons [5]. Dark solitons have been observed in optical fibers by a few different experimental techniques [ 6-8 ], however, the generation of the dark pulses required considerable experimental effort. The repetitive generation of solitons (soliton trains) was first considered in the generation of bright solitons (for anomalous dispersion) from modulational instability by Hasegawa [9], who found that sinusoidal intensity modulation evolved into self compressed pulses with some residual interpulse energy. More recently it was shown that continuous sinusoidal intensity modulation evolves into a train of bright (dark) solitons for anomalous (normal) dispersion only if one adds linear ampli-
fication to the propagation [ 10 ]. In a related experiment it was demonstrated that the nonlinear copropagation of two visible pulses in an optical fiber leads to rapid intensity modulation on the finite width background as the pulses overlap [ 11 ]. In this paper it is shown that trains of dark solitons form naturally in the nonlinear copropagation and collision of two pulses in an optical fiber. For pulses of equal input intensity, the solitons are found to asymptotically approach the theoretical shape of the fundamental (full contrast) dark soliton. The soliton widths vary adiabatically owing to the broadening, and hence diminished power, of the finite width background pulse upon which the solitons are situated. The propagation of the field, ¢(z, t) = Re{E(z, t) e x p [ - i ( c o o t - n o k o z ) ] } , in an optical fiber is accurately described by the nonlinear Schroedinger equation (NLSE) OE/Oz+ i( k<2) / 2 )O2E/Ot 2
= in2 ko [El 2 E - i T I E I 2E,
( 1)
where k<2>=82k/Sto 2 is positive for normal group velocity dispersion, and a retarded time frame travelling at v~=aog/Ok is assumed. For the numerical simulations presented here, parameters typical for a wavelength of 600 nm are assumed: k <2)=0.065 ps2/ m and 7=0.020 m -~ W -~ (with IEI 2 specified by
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
107
Volume 82, number 1,2
OPTICS C O M M U N I C A T I O N S
power in Watt). Now consider the following input field
This field is the in-phase sum of two equal sech 2 intensity pulses, and is easily formed by a Michelson interferometer [ 11 ]. The numerical integration of eq. (1) with this input is shown for various fiber lengths in fig. 1, where the fwhm of the input pulses is 2 ps (Zo= 1.13 ps), At=8 ps, Po= 500 W, and the intensity scale has been normalized at the fiber input. The results for small propagation distances ( ~<10 m) are in agreement with earlier experimental results [ 11 ], where each pulse broadens and acquires a linear frequency sweep owing to self phase modulation and dispersion, and, once overlap occurs, one observes rapid modulation owing to the interference between the large frequency shifts of the leading and trailing edges of the colliding pulses. With further propagation (note the time scale changes in fig. 1 ) the overlapping region broadens and the modulation reshapes into an expanding train of increasingly isoI
.43
.74
iNPUTfO 2.5 m 5m 20
.58 ~.14 Z LI.I
.05
.012
0 O.Sx TIME (psec, scale varies as :r)
1600 1.0x
Fig. 1. Calculation showing the copropagation in an optical fiber of two 500 W input pulses at the indicated distances. The intensity scale is normalized to unity at the input, and the peak intensity at each distance is indicated. The time scale is increasingly expanded as indicated.
108
((l-v~) ':~
+_iqtanh[(t-toT- V-~z)/zo]} exp(iyPoz), 4O0
--.5x
lated and narrow dark pulses. Although it will be shown that the dark pulses are adiabatically broadening (~~/:z) as they propagate, the expansion of the dark pulse separation is faster ( ~ z ) , and thus fig. 1 gives the relative appearance that the dark pulses are sharpening. The rapid spreading of the dark pulses is a result of the dispersion of the underlying background pulse. Note also the lack of modulation between the dark pulses, i.e. there is little apparent energy shed by the dark pulses as they reshape. Similar results are obtained for other values of Zo and Po, which determine the bandwidth generated by each input pulse, and At, which determines the beat frequency, and hence the dark pulse spacing. One might expect the dark pulses to be equally spaced since one expects regular interference between pulses which both have linear frequency sweeps [ 11 ]. However, it appears that the nonlinear interaction between the two pulses at small distances produces a slight nonuniformity in the dark pulse distribution. This is borne out by calculations using the input of eq. (2) but with a larger delay between the pulses. In this case one finds that a more dense and more regularly spaced dark pulse train forms. For different relative phases between the input pulses the same general results are obtained, however the timing of the dark pulses is modified. By varying the relative phase of the input pulses from 0 to 2n the dark pulse train shifts through one period, thereby enabling completely adjustable time shifting of the train. For a ~t phase difference (anti-symmetric case) the dark pulse distribution is nearly identical to that shown in fig. 1 except there is an additional dark pulse at t = 0. Dark solitons for normal dispersion have the form
[21 e,,a,,, =,,/~o
.024
I April 1991
(3)
where
z~ =k(2)/~ZPo, V - I = (k(Z)/rlZo) ( 1 -r/2) l/z,
(4) (5)
and 0 < q < 1. Thus the dark soliton intensity is IE~rk 12=Po {1 -r/2sech2 [ (t-to-T- V-lz)/zo]} •
(6)
Volume 82, number 1,2
OPTICS COMMUNICATIONS
The fwhm of the dark soliton is then 1.763Zo, and the parameter ~/determines its contrast and velocity. For ~1=1, E~rk~tanh(t-to)/Zo, so that the contrast is 100% (the intensity dips completely to 0) and the phase of this pulse has a discontinuous j u m p of n at t = to. The pulse for r/= 1 has been termed the fundamental or "black" soliton [5,8 ]. For ~/< 1, the intensity does not dip completely to zero (the contrast is given by r/2), and the phase jump is reduced and more gradual. Fig. 2 shows a comparison of the intensity and phase of the dark pulses found in the numerical solutions of fig. 1 (solid curves) with the analytic dark soliton form of eqs. ( 3 ) - ( 6 ) (dots), where the dark pulse in fig. 1 closest to t = 0 at a few propagation distances is examined. For this comparison, the average power in the vicinity of each dark pulse from fig. 1 is used forPo in eqs. (3), (4), and to is taken from the location of the dark pulse minimum. Then r/is found from the contrast of the dark
~
0.5.
0
-
.
-1
5
i
~
. . . . . . .
I
---
I
,
,
.
.
0
I
I
[
I
.
.
.
I
.
I
I
-.5 -4
0 4 TIME (psec)
-4
0 TIME
(psec)
4
Fig. 2. Solid curves showthe intensity and phase of the dark pulse nearest to t=0 in fig. 1. This is compared with the analytic dark soliton solution of eqs. (3), (4) (dots) for the distances indicated. Note that the time scale increases by a factor of 2 at successive distances. The background intensity level at each distance is indicated and corresponds to (a) 25, (b) 5.4, and (c) 1.3 W. Contrast parameter r/2is (a) 0.9892, (b) 0.9965, and (c) 0.9992, and the dark pulse fwhm is (a) 0.65, (b) 1.38, and (c) 2.85 ps.
1 April 1991
pulse intensity minimum calculated in fig. I. Note that the pulse width Zo is then determined from eq. (4) and is not adjusted. In the phase comparison the background linear phase shift (corresponding to a frequency shift) has been subtracted. The intensity fit is nearly exact for the distances examined. The remaining small discrepancy is a result of the sloping intensity of the expanding background pulse. Note that since the background pulse is spreading approximately linearly, the background power for a given soliton is decreasing like z - l , and thus, from eq. (4), one sees that the soliton width Zo~ x/~. The pulse widths found in fig. 2 agree with this approximation to good accuracy. The phase fit (with the same parameters as the intensity fit) is also nearly exact over the central part of the pulse. However, away from the pulse center the deviation in the phase is significant and is understood in terms of the quadratic phase variation of the background pulse owing to its linear frequency chirp. These results are similar with the numerical analyses of Tomlinson et al. [ 5 ], where it was also found that dark pulses propagated adiabatically as solitons in spite of the decreasing intensity and linear frequency chirp of a finite width background pulse. The parameter ~/found in fig. 2 is nearly unity in all cases and asymptotically approaches 1 for larger z. As is apparent from fig. 2, the phase fit is sensitive to extremely small changes in ~/as r/--,1, so it is clear that these dark pulses are approaching the fundamental black soliton as they propagate. Note that according to eq. (5), as r/--, 1, the relative soliton velocity V - l ~ 0 (i.e. it becomes temporally stationary in the propagating time frame of eq. ( 1 ) ) . However, from fig. 1 it is apparent that these solitons are not stationary. This is a result of the frequency chirp of the background pulse, which shifts the central frequency of each soliton and thus changes its relative group velocity. Eqs. ( 3 ) - (6) can likewise be generalized [ 4 ] for a frequency shift Aog, which then shifts the soliton velocity V - I by k(2)Aog. The fact that fundamental dark solitons form throughout the background pulse in fig. 1 is somewhat the result of the symmetry of the input field (eq. (2) ). To explore this feature, the input field of eq. (2) is modified by reducing the intensity of the trailing sech 2 pulse by a factor of 2. The results of the numerical propagation of this asymmetric field are 109
Volume 82, n u m b e r 1,2
OPTICS C O M M U N I C A T I O N S
shown in fig. 3. One sees rather similar results to that found in fig. 1, with the exception that the dark pulses have reduced contrast. In spite of this, these pulses still maintain nearly exact agreement with the analytic dark soliton form of eqs. ( 3 ) - ( 6 ), with similar small deviations owing to the non-uniform and frequency-swept background. It is interesting to point out that, similar to what one would expect in simple linear interference, for unbalanced input pulse intensities the interference contrast and, hence, the dark pulse depth are diminished. The effects of linear loss in eq. ( 1 ) on the soliton propagation have also been investigated numerically. For sufficiently small loss the width of the solitons adiabatically adjust to the reduced intensity owing to both the loss and to the spreading of the background pulse. However, for loss which is significant on the scale of the characteristic soliton distance (Zo ~ r2/k (2)), the solitons cannot "keep up" with the loss and begin to noticeably distort and radiate energy. For a given loss, after sufficient propagation distance, this regime always occurs since Zo is monotonically increasing (as the solitons broaden). For the example given in fig. 1, at a dis-
I :
~
.2
x 10
16 m
.09 z '"
40
_
.01 .0024
- i
~320 x
-.5x TIME ( p s e c ,
0 0.Sx scale varies as x)
8O
1 1250 1.0x
Fig. 3. Calculation showing the copropagation of two input pulses of peak powers 500 and 250 W at the indicated distances. The intensity and time scales are expanded as indicated.
110
1 April 1991
tance of I kin, significant distortion appears for loss ~ 5 - 1 0 dB/km. To insure distortion free soliton propagation for much longer distances it appears necessary to introduce gain into the fiber to balance the loss [ 12 ]. It has been noted that dark solitons can be produced, without threshold, from an arbitrarily modulated continuous wave [ 13 ]. Also it has been shown that dark solitons form on a sinusoidally modulated continuous wave in the presence of gain [ 10]. However, the generation of a train of dark solitons on a finite width background pulse is not generally achieved with an arbitrarily modulated input. For example consider the modulated input pulse
Ein =X/~o sin ( 2nt / Tl ) sech ( t / T2 ) ,
(7)
where T2 >> T~. The sinusoidal modulation supplies a phase change of n for each dark pulse, as is required for the dark soliton. If one takes the power Po to correspond to the soliton power for a dark pulse of width given by the modulation, then one would expect the input to naturally evolve into a train of dark solitons. However, the numerical solution for the test case 7"2=20 ps, T I = I ps, and Po=40 W, shows that, although the modulation is 100% initially, no solitons form and the contrast of the modulation decreases continuously until disappearing completely. This is seen to be a result of the positive and negative frequency components which comprise eq. (7) separating owing to dispersion. On the other hand, frequency chirped pulses (which form from the input of eq. (2) and which generally form from the linear dispersion of an arbitrary pulse after sufficient propagation) have the interesting property that superimposed intensity modulation of bandwidth less than the extent of the chirp does not generally disperse away from the background and will thus naturally evolve into dark solitary components On the finite width background. The experimental observation of these soliton trains would be most likely accomplished by a direct intensity measurement, since dark s01itons on a long background are difficult to observe via autocorrelation. A cross correlation intensity measurement averaged over many input pulses might require rather stringent requirements on the input amplitude and shape stability, since variation of the input pulse parameters would likely lead to time shifts of the dark
Volume 82, number 1,2
OPTICS COMMUNICATIONS
soliton train from pulse to pulse a n d thereby obscure the observation. Therefore, the intensity measurement would preferably be performed on a single pulse basis (e.g. a streak c a m e r a ) . F r o m the calculations presented here, a realistic resolution o f ~ 1 ps appears to be adequate for such a measurement. In conclusion, it has been shown that a straightforward m e t h o d can generate trains o f high contrast dark pulses with little a p p a r e n t interpulse energy. Although these dark pulses are situated on a finite-width frequency-chirped b a c k g r o u n d pulse o f d i m i n i s h i n g intensity, they are quite accurately described by high contrast d a r k solitons whose pulse width increases adiabatically as the b a c k g r o u n d intensity decreases. The t e m p o r a l position o f the generated solitons is d e p e n d e n t on the relative phase a n d shape o f the input pulses. It m a y therefore be possible to use input m o d u l a t i o n to m a n i p u l a t e the relative timing o f the solitons within the train, thereby creating the possibility o f a time shift digital signal.
1 April 1991
References [ 1] V.E. Zakharov and A.B. Shabat, Sov. Phys. JETP 34 (1972) 62 [Zh. Eksp. Teor. Fiz. 61 ( 1971 ) 118 ]. [2] V.E. Zakharov antl A.B. Shabat, Sov. Phys. JETP 37 (1973) 823 [Zh. Eksp. Teor. Fiz. 64 (1973) 1627]. [3] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23 (1973) 142. [4 ] A. Hasegawa and F. Tappert, Appl. Phys. Len. 23 (1973 ) 171. [5] W.J.Tomlinson, R.J. Hawkins, A.M. Weiner, J.P. Heritage and R.N. Thurston, J. Opt. Soc. Am. B 6 (1989) 329. [6] P. Emplit, J.P. Hamaide, F. Reynaud, C. Frochly and A. Barthelemy, Optics Comm. 62 (1987) 374. [ 7 ] D. Krokel, N.J. Halas, G. Guiliani and D. Grischkowsky, Phys. Rev. Lett. 60 (1988) 29. [8] A.M. Weiner, J.P. Heritage, R.J. Hawkins, R.N. Thurston, E.M. Kirschner, D.E. Leaird and W.J. Tomlinson, Phys. Rev. Lett. 61 (1988) 2445. [9] A. Hasegawa, Optics Lett. 9 (1984) 288. [ 10] E.M. Dianov, P.V. Mamyshev, A.M. Prokhorov and S.V. Chernikov, Optics Len. 14 (1989) 1008. [ 11 ] J.E. Rothenberg, Optics Lett. 15 (1990) 443. [ t2] A. Hasegawa and Y. Kodama, Optics Lett. 7 (1984) 285, 393. [ 13] S.A. Gredeskul and Y.S. Kivshar, Optics Lett. 14 (1989) 1281.
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