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Optimization of soliton transmissions in dispersion-managed fiber links
1 January 1998
Optics Communications 145 Ž1998. 48–52
Optimization of soliton transmissions in dispersion-managed fiber links M. Wald a , I.M. Uzunov a
a,1
, F. Lederer a , S. Wabnitz
b
Institut fur und Theoretische Optik, UniÕersitat ¨ Festkorpertheorie ¨ ¨ Jena, Max-Wien-Platz 1, D-07743 Jena, Germany b Laboratoire de Physique, UniÕersite´ de Bourgogne, AÕ. A. SaÕary 9, BP 400, 21011 Dijon, France Received 25 April 1997; accepted 23 June 1997
Abstract We propose a simple optimization criterion Žincluding the best launch point position in-between amplifiers. for the design of soliton transmission lines. The present approach is shown to minimize energy scattering from the solitons into the continuum. q 1998 Elsevier Science B.V. PACS: 42.65.–k; 42.81.Dp Keywords: Optical solitons; Dispersion management
Stable propagation of optical soliton pulses in periodically amplified telecommunication links requires the average Žanomalous. group-velocity dispersion ŽGVD. length to be a few times longer than the amplifier spacing Žf 30–50 km. w1–3x. In the absence of in-line timing or frequency controls such as synchronous modulation or filtering, the above constraint on the GVD length limits the bit-rate of either standard or dispersion-shifted fiber-based soliton transmissions to about 2.5 or 10 Gbitrs, respectively. Otherwise, if the GVD length is comparable or even shorther than the amplifier spacing, soliton reshaping occurs through the emission of radiative waves at those frequencies where the periodic amplification matches the linear and the soliton wave numbers w1,4x. To overcome the above bit-rate limitation, the fiber GVD in each amplifier span may be adapted to the decreasing soliton power w5,6x. A practical way to achieve this task is stepwise GVD profiling w7x. A complementary approach to control the fiber GVD, as discussed in several theoretical w8–10x and experimental works w11,12x, applies to solitons the technique of dispersion compensation. This method, which was introduced in linear Žnon-return-to-zero, or NRZ. communications, involves a periodic sequence of fibers with alternating GVD signs. In contrast to the NRZ case, for soliton transmissions a small residual anomalous GVD is left uncompensated, to balance fiber nonlinearity. In single-channel systems, optimization of the GVD compensation method involves a minimization of the radiation which is scattered by the periodically amplified pulses 2 : first-order soliton perturbation theory Žincluding radiation w13x. was used to derive a simple formula for the relative values of dispersion in the two sections w14x. We are concerned here with the practical problem of the optimal design of a realistic transmission system, where one should also take into account: Ža. The variations of the nonlinear coefficient and loss Žas both quantities are considerably larger in a dispersion compensating fiber ŽDCF. than in a dispersion-shifted fiber.; Žb. the relative position of the two fibers in each span w10,15x; Žc. the choice of the launch point of the signal w16x Žthis is equivalent to pre-chirping the input pulses w17x.. The large number of parameters involved requires simple and possibly analytical optimization criteria w18x. This Letter
1 2
Permanent address: Institute of Electronics, Bulgarian Academy of Sciences, boul. Tsarigradsko shosse 72, Sofia 1784, Bulgaria. In the WDM case, the asymmetric collision-induced frequency shifts should also be minimized.
0030-4018r98r$17.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 3 6 3 - 5
M. Wald et al.r Optics Communications 145 (1998) 48–52
49
is aimed at deriving such a formula for the design of a soliton transmission line with minimum losses into the continuum. The approach presented here applies to a periodic link of N identical unit cells representing the amplifier spans. Within each unit cell, there may be an arbitrary variation of the GVD, losses and effective nonlinearity; moreover, there is a freedom to launch the input signal at any point within the cell. In the simplest, yet most practical case where the unit cell consists of two homogeneous fiber sections, our approach leads to simple analytical expressions. These predictions were double-checked by simulations, involving commercially available dispersion-shifted fibers ŽDSF. and DCFs. ŽNLS. Nonlinear pulse dynamics in the unit cells is described by the dimensionless perturbed nonlinear Schrodinger ¨ equation iq z y 12 D Ž z . qtt q R Ž z . < q < 2 q s yi G Ž z . q. 2
Ž1.
2<
The local GVD reads b Ž z . s E krEv Ž v s v 0 . , where k is the fiber mode propagation constant and v 0 is the carrier frequency. The length scale in Eq. Ž1. is set by the average GVD ² b : s w H0zab Ž z . d z xrz a. Moreover, z s ZrZ0 s <² b :< ZrT02 , t s ŽT y ZrÕg .rT0 , and Õg is the constant group velocity. Whereas DŽ z . s b Ž z .r<² b :< and RŽ z . s g Ž z .r²g : relate local and average values of dispersion and effective nonlinearity, respectively. G Ž z . is the loss coefficient, normalized with Z0 . The amplifier gain exactly compensates the fiber loss, i.e. G s expw2H0za G Ž z . d z x. If the average dispersion length exceeds a few times the amplifier spacing Ž ZarZ0 s z a < 1., and moreover < DŽ z .< z a , OŽ1., one may neglect pulse width variations due to the local GVD, and an average or guiding center soliton is allowed to propagate through the system w8x. The linear loss may be factored out by writing q Ž z,t . s aŽ z . uŽ z,t . with d aŽ z .rd z s yG Ž z . aŽ z .. aŽ z . s a 0 expwyH0z G Ž zX . d zX x. The amplitude enhancement factor a 0 is adjusted to the average soliton requirement ² Ra2 : s 1. The additional transformation w1,8x z
j Ž z. s
X
X 2
H0 R Ž z . aŽ z .
d zX
Ž2.
yields the basic equation iuj y 12 E Ž j . utt q < u < 2 u s 0,
Ž3.
where the effective dispersion parameter EŽ j . s
Dw z Ž j . x
Ž4.
R w z Ž j . x a2 w z Ž j . x
contains all the information on the periodic variation of GVD, loss Žor gain. and nonlinearity. Since ² EŽ j .: s w H0za EŽ j . d j xrz a s y1, the fluctuation z Ž j . s EŽ j . y ² EŽ j .: s EŽ j . q 1 is a source of deviations of the envelope uŽ z,t . from the average soliton shape. The overall distortion of a soliton signal up to some distance j within the unit cell may be measured in terms of the accumulated deviation f Ž j 0 , j . s Hjj00qj z Ž j X . d j X , where j 0 s H0z 0 RŽ zX . aŽ zX . 2 d zX , and z 0 is the coordinate of the launch point in soliton units; note that f Ž j 0 , ja . s 0. We are thus naturally led to adopt as optimization criterion the requirement of a zero average Žover each span. accumulated deviation, viz. F Ž j 0 . ' y² f Ž j 0 , j . : ´ 0.
Ž5.
This equation may be rewritten in its final form by undoing the transformation Ž2., F Ž z0 . s
H0
z0
D Ž z . q R Ž z . aŽ z .
2
d zq
1 za
H0
za
D Ž z . q R Ž z . aŽ z .
2
j Ž z. d z,
Ž 6.
where the second term is obtained after carrying out one integration by parts. For the periodic system considered here F Ž z 0 . exhibits always two zeros within a unit cell. At these zeros the deviation from the ideal averaged soliton is expected to be minimum. The above formula represents the main result of this Letter: it holds for any variation of the fiber parameters within the unit cell. Already at this stage, some interesting results may be read off by inspecting Eq. Ž6.. Clearly, whenever the launch point is set right behind the amplifier the first integral vanishes, and F Ž0. remains the only quantity to be canceled out. However, the adjustment of the launch point adds another degree of freedom for canceling F Ž z 0 .. It is interesting to note that the integrands in Eq. Ž6. are equal to zero for an ideal continuous dispersion profiling Ži.e., where the ratio of dispersion and nonlinearity exactly follows the peak power.. Moreover, numerical simulations show that < F Ž z 0 .< is related to the level of generated radiation and F Ž z 0 . measures the chirp acquired by the input pulse at the end of the first cell. To give a specific example of application of Eq. Ž6., let us consider the simplest and, for practical applications, most interesting case of a two-step dispersion compensation scheme, where the unit cell is divided in two sections with constant
M. Wald et al.r Optics Communications 145 (1998) 48–52
50
GVD of opposite signs, and parameters z1,2 , D1,2 , R1,2 , G 1,2 , with z a s z 1 q z 2 . In order to meet the requirement of negligible pulse-width Žand amplitude. changes Žor pulse breathing w19x. owing to the relatively high Žwith respect to the average value. fiber GVD, the lengths Z1,2 of the two sections should not considerably exceed the local dispersion length, 1,2 say, Z loc ' Z0r< D1,2 <. As a rule of thumb, one may take z1,2 < D1,2 < F 2 w20x. Averaging over the amplifier span simplifies now to, e.g., ² b : s Ž b 1 z 1 q b 2 z 2 .rz a; the input power enhancement factor w1–3x to generate a unit-amplitude guidingcenter soliton reads as a20 s
2 za R 1rG 1 y R 2r Ž GG 2 . y Ž R 1rG 1 y R 2rG 2 . exp Ž y2 G 1 z 1 .
.
Ž7.
Moreover, the integration in Ž6. may be easily carried out: one obtains an explicit expression for the quantity to be cancelled out
°F Ž0. q D z q R a 1 y expŽy2 G z . , 1 0
F Ž z . s~
2 1 0
1 0
z 0 F z1 ,
2 G1
0
¢
F Ž z 1 . q D 2 Ž z 0 y z 1 . q R 2 a 20 exp Ž y2 G 1 z 1 .
1 y exp w y2 G 2 Ž z 0 y z 1 . x 2 G2
Ž8. ,
z 0 ) z1 ,
where F Ž0. s
za 2
y
R 1 a 02 2 G1
y
D2 2 G2
q
a02 2 za
D2 z2
ž
R2
G2
y
R1
G1
/
exp Ž y2 G 1 z1 . q R 1
ž
1 y exp Ž y2 G 1 z 1 . 2 G1
/ž
D2
G2
y
D1
G1
/
.
Ž9. The optimum system design corresponds to the straightforward task of finding the zeros of Ž8.. If the launch point is set to zero, the procedure simplifies to canceling Ž9.. Before proceeding to consider a specific example of application, it is noteworthy to point out that the general method outlined above reproduces, in some particular instances, earlier prescriptions for the optimization of soliton systems. In fact, in the case of lumped amplification without dispersion management, F Ž z 0 . equals function A1Ž z s z 0 . that was introduced in Ref. w1x Žsee Eq. Ž23. there. to measure pulse deviation from an average soliton: canceling F Ž z 0 . may be achieved by launch point adjustment w16x. A similar situation occurs for pure dispersion management with R 1 s R 2 s 1 and G Ž z . s 0 w8x. In both cases, the pulse distortion function was derived by means of the Lie transform method. A different approach for the minimization of radiation scattering in dispersion-managed fiber links may employ first-order soliton perturbation theory: Eq. Ž17. of Ref. w14x Žwhere no launch point optimization was considered and R 1 s R 2 s 1, G 1 s G 2 s G . coincides with the condition F Ž0. s 0. In what follows, we will demonstrate by simulations the effectiveness of the result Ž8., Ž9.. To this end, we considered a two-step GVD compensated fiber link: each cell consists of a concatenation of a DSF and a DCF. The zeros of Eq. Ž8. correspond to the best suppression of the generated radiation: to confirm this, we numerically solved Ž1. by means of a standard split-step algorithm. The pulse shapes at the end of each unit cell were used as initial ‘‘potential’’ in the direct Zakharov–Shabat scattering problem, whose solution yields the complex soliton eigenvalue, say, l. The relative non-soliton Žor radiation. energy content of the field is then given by the formula s s 1 y 2 ImŽ l. w14,15x. Let us consider at first the specific case of a 20 GBitrs soliton transmission Žwith a pulse width Tfwhm s 11 ps, and spacing DT s 50 ps.: the length DSF of the DSF Žwhich is nearly equal here to the amplifier spacing. is about 1.7 times larger than Z loc . The DSF fiber has the 2 DSF loss a DSF s 0.24 dBrkm, the GVD b DSF s y1.28 ps rkm Žso that Z loc s 30 km., and the length is Z DSF s 50 km; for the DCF, the loss is a DCF s 0.6 dBrkm, the GVD is b DCF s 20 ps 2rkm, and the length Z DCF s 2.53 km; the ratio of nonlinear coefficients is g DCFrg DSF s 1.6, and the amplifier spacing Za s 52.53 km, so that the average dispersion is ² b : s y0.26 ps 2rkm, which yields Z0 s 158 km, and z a s 0.332. Without dispersion compensation, the resonant periodic amplification leads to a substantial growth of radiation which, in turn, strongly interacts and distorts the soliton signal after a fairly short distance. This detrimental radiation may be strongly reduced by using a GVD map, with the proper choice of the launch point: in Fig. 1, we show the energy fraction of radiation s ' 1 y 2 ImŽ l. that is generated at the distance z s 35 z a Žor Z s 1800 km., versus the input coordinate z 0 . Here the DCFs are placed before the amplifiers: as can be seen, the radiation is minimized around the launch points z 01, z 02 which are chosen according to Eqs. Ž8., Ž9., viz. F Ž z 0 . s 0. On the other hand, Fig. 2 shows the dependence of s with distance, for different choices of z 0 . In Figs. 2a, 2b, the DCFs are placed right before or after the amplifiers, respectively. Whereas the dots indicate a launch point at the amplifiers Ž z 0 s 0., and the triangles or diamonds are obtained for the optimal choices of z 0 , again according to Eqs. Ž8., Ž9.. As can be seen, the radiation is strongly reduced when the proper launch points are selected, in particular for compensation in the high-power region. Fig. 2 also shows that, in contrast with the case of a launch
M. Wald et al.r Optics Communications 145 (1998) 48–52
51
Fig. 1. Radiation energy versus launch point position for DCFs before the amplifiers.
point at the amplifier Žwhere the DCF should be placed in the low-power region w15x., whenever the optimum launch point is chosen the level of radiation is independent of the relative position of the DCF in the unit cell. Indeed we observed that, for an optimized 20 GBitrs stream, soliton interactions are virtually unaffected by the presence of a small radiation level. For strong dispersion compensation, where z 1,2 < D1,2 < 4 1, the present optimization scheme is not sufficient to guarantee stable soliton transmission. For example, Fig. 3a shows the distortion of a 40 GBitrs Ži.e., Tfwhm s 5 ps, DT s 25 ps. stream DSF through a link with optimal z 0 s 0.192, a DSF dispersion distance Z loc s 6.3 km, the lengths Z DSF s 50 km, Z DCF s 2.9 2 km, Za s 52.9 km, and ² b : s y0.1 ps rkm, so that Z0 s 80 km. As pointed out in Ref. w22x, in such cases the dispersive pulse width variations within each cell do introduce an additional oscillation of the soliton amplitude, which ultimately hampers the long-scale balance between average GVD and nonlinearity. This balance may be restored by further enhancing the amplitude of the input signal with respect to the usual average fundamental soliton value of Eq. Ž7. w18x. In order to show the possibility of extending the present optimization scheme to the regime of strong dispersion management, we took advantage of the variational approach w19x for calculating the optimal input soliton amplitude. As usual, this involves considering a perturbed fundamental soliton with z-evolving time width, amplitude, phase and chirp as a trial function. By solving the resulting ordinary differential equations for the pulse parameters, the optimal launch points and input amplitude may be found by the shooting method: here one imposes the recovery of the initial amplitude along with the minimization of the chirp upon propagation through the unit cell. The resulting optimal launch point differs from the prediction of Eqs. Ž8., Ž9.; whereas the amplitude enhancement factor turns out to be equal to about one and a half the value of Eq. Ž7.. With these optimal data, we found that a 40 GBitrs signal can be distortion-free propagated over more than 2000 km Žsee Fig. 3b.. Note that, for the previously considered 20 GBitrs scenario, the amplitude enhancement is equal to just
Fig. 2. Energy of radiation scattered by a soliton versus distance z for different launch points. DCFs either Ža. before Žand z 01 s 0.102, z 02 s 0.328. or Žb. after Žand z 01 s 0.011 or z 02 s 0.135. the amplifiers.
M. Wald et al.r Optics Communications 145 (1998) 48–52
52
Fig. 3. Propagation of a 40 GBitrs Ž..1101... soliton stream Ža. without amplitude enhancement Ž z 0 s 0.192., Žb. with amplitude enhancement factor 1.45 Žand z 0 s 0.136..
1.03, which justifies a posteriori the use of Eq. Ž7.. Details on the use of the variational method for the optimization of dispersion-compensated 40 GBitrs single channel soliton transmissions will be described elsewhere. In conclusion, in this work we presented a simple analytical formula for the simultaneous optimization of the launch point and of the dispersion map in soliton transmission lines. We showed that the method permits a minimization of the generated radiation; several related optimization schemes appear as particular cases of the present general prescription. In the case of strong GVD management Žthis may also apply to upgrade standard fiber lines beyond 10 Gbitrs., a variational approach may be used to extend the optimization to solitons with properly enhanced power.
Acknowledgements M. Wald, I.M. Uzunov, and F. Lederer acknowledge a grant from the Deutsche Forschungsgemeinschaft ŽInnovationskolleg Ink 1rA1.. The work was carried out in the framework of the COST 241 initiative of the European Union.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x
A. Hasegawa, Y. Kodama, Optics Lett. 15 Ž1990. 1443. L.F. Molenauer, S.G. Evangelides, H.A. Haus, Lightwave Technol. 9 Ž1991. 194. K.J. Blow, N.J. Doran, IEEE Photon. Technol. Lett. 3 Ž1991. 369. J.P. Gordon, J. Opt. Soc. of Am. B 9 Ž1992. 91. K. Tajima, Optics Lett. 12 Ž1987. 55. H.H. Kuehl, J. Opt. Soc. Am. B 5 Ž1988. 709. W. Forysiak, F.M. Knox, N.J. Doran, J. Lightwave Technol. 12 Ž1994. 1330. A. Hasegawa, Y. Kodama, Optics Lett. 16 Ž1991. 1385. M. Nakazawa, H. Kubota, Electron. Lett. 31 Ž1995. 216. F.M. Knox, W. Forysiak, N.J. Doran, J. Lightwave Technol. 7 Ž1995. 1955. M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto, H. Taga, S. Akiba, Electron. Lett. 31 Ž1995. 2027. F.M. Knox, P. Harper, P.N. Kean, I. Bennion, N.J. Doran, 22-nd European Conference in Optical Communication ECOC’96, Oslo, paper WeC.3.2. Y. Kodama, S. Kumar, A. Hasegawa, in: Physics and Applications of Optical Solitons in Fibers ’95, Kluwer, 1996, pp. 15, 16. T. Georges, B. Charbonnier, Optics Lett. 21 Ž1996. 1232. S. Wabnitz, I. Uzunov, F. Lederer, IEEE Phot. Technol. Lett. 8 Ž1996. 1091. W. Forysiak, N.J. Doran, F.M. Knox, K.J. Blow, Optics Comm. 117 Ž1995. 65; 119 Ž1995. 697 ŽErratum.. T. Georges, B. Charbonnier, Photonics Techn. Lett. 9 Ž1997. 127. Optimization conditions based on a second-order Lie transform expansion have been recently derived in: Y. Kodama, A. Maruta, Optimal design of dispersion managements for a soliton-WDM system, submitted to Electronics Letters Ž1997.. I. Gabitov, E. Shapiro, S.K. Turitsyn, Optics Comm. 134 Ž1997. 317. M. Midrio, M. Romagnoli, S. Wabnitz, P. Franco, Optics Lett. 21 Ž1996. 1351. G. Boffetta, A.R. Osborne, J. Comp. Phys. 102 Ž1992. 252. N.J. Smith, N.J. Doran, F.M. Knox, W. Forysiak, Optics Lett. 21 Ž1996. 1981.
Optics Communications 145 Ž1998. 48–52
Optimization of soliton transmissions in dispersion-managed fiber links M. Wald a , I.M. Uzunov a
a,1
, F. Lederer a , S. Wabnitz
b
Institut fur und Theoretische Optik, UniÕersitat ¨ Festkorpertheorie ¨ ¨ Jena, Max-Wien-Platz 1, D-07743 Jena, Germany b Laboratoire de Physique, UniÕersite´ de Bourgogne, AÕ. A. SaÕary 9, BP 400, 21011 Dijon, France Received 25 April 1997; accepted 23 June 1997
Abstract We propose a simple optimization criterion Žincluding the best launch point position in-between amplifiers. for the design of soliton transmission lines. The present approach is shown to minimize energy scattering from the solitons into the continuum. q 1998 Elsevier Science B.V. PACS: 42.65.–k; 42.81.Dp Keywords: Optical solitons; Dispersion management
Stable propagation of optical soliton pulses in periodically amplified telecommunication links requires the average Žanomalous. group-velocity dispersion ŽGVD. length to be a few times longer than the amplifier spacing Žf 30–50 km. w1–3x. In the absence of in-line timing or frequency controls such as synchronous modulation or filtering, the above constraint on the GVD length limits the bit-rate of either standard or dispersion-shifted fiber-based soliton transmissions to about 2.5 or 10 Gbitrs, respectively. Otherwise, if the GVD length is comparable or even shorther than the amplifier spacing, soliton reshaping occurs through the emission of radiative waves at those frequencies where the periodic amplification matches the linear and the soliton wave numbers w1,4x. To overcome the above bit-rate limitation, the fiber GVD in each amplifier span may be adapted to the decreasing soliton power w5,6x. A practical way to achieve this task is stepwise GVD profiling w7x. A complementary approach to control the fiber GVD, as discussed in several theoretical w8–10x and experimental works w11,12x, applies to solitons the technique of dispersion compensation. This method, which was introduced in linear Žnon-return-to-zero, or NRZ. communications, involves a periodic sequence of fibers with alternating GVD signs. In contrast to the NRZ case, for soliton transmissions a small residual anomalous GVD is left uncompensated, to balance fiber nonlinearity. In single-channel systems, optimization of the GVD compensation method involves a minimization of the radiation which is scattered by the periodically amplified pulses 2 : first-order soliton perturbation theory Žincluding radiation w13x. was used to derive a simple formula for the relative values of dispersion in the two sections w14x. We are concerned here with the practical problem of the optimal design of a realistic transmission system, where one should also take into account: Ža. The variations of the nonlinear coefficient and loss Žas both quantities are considerably larger in a dispersion compensating fiber ŽDCF. than in a dispersion-shifted fiber.; Žb. the relative position of the two fibers in each span w10,15x; Žc. the choice of the launch point of the signal w16x Žthis is equivalent to pre-chirping the input pulses w17x.. The large number of parameters involved requires simple and possibly analytical optimization criteria w18x. This Letter
1 2
Permanent address: Institute of Electronics, Bulgarian Academy of Sciences, boul. Tsarigradsko shosse 72, Sofia 1784, Bulgaria. In the WDM case, the asymmetric collision-induced frequency shifts should also be minimized.
0030-4018r98r$17.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 3 6 3 - 5
M. Wald et al.r Optics Communications 145 (1998) 48–52
49
is aimed at deriving such a formula for the design of a soliton transmission line with minimum losses into the continuum. The approach presented here applies to a periodic link of N identical unit cells representing the amplifier spans. Within each unit cell, there may be an arbitrary variation of the GVD, losses and effective nonlinearity; moreover, there is a freedom to launch the input signal at any point within the cell. In the simplest, yet most practical case where the unit cell consists of two homogeneous fiber sections, our approach leads to simple analytical expressions. These predictions were double-checked by simulations, involving commercially available dispersion-shifted fibers ŽDSF. and DCFs. ŽNLS. Nonlinear pulse dynamics in the unit cells is described by the dimensionless perturbed nonlinear Schrodinger ¨ equation iq z y 12 D Ž z . qtt q R Ž z . < q < 2 q s yi G Ž z . q. 2
Ž1.
2<
The local GVD reads b Ž z . s E krEv Ž v s v 0 . , where k is the fiber mode propagation constant and v 0 is the carrier frequency. The length scale in Eq. Ž1. is set by the average GVD ² b : s w H0zab Ž z . d z xrz a. Moreover, z s ZrZ0 s <² b :< ZrT02 , t s ŽT y ZrÕg .rT0 , and Õg is the constant group velocity. Whereas DŽ z . s b Ž z .r<² b :< and RŽ z . s g Ž z .r²g : relate local and average values of dispersion and effective nonlinearity, respectively. G Ž z . is the loss coefficient, normalized with Z0 . The amplifier gain exactly compensates the fiber loss, i.e. G s expw2H0za G Ž z . d z x. If the average dispersion length exceeds a few times the amplifier spacing Ž ZarZ0 s z a < 1., and moreover < DŽ z .< z a , OŽ1., one may neglect pulse width variations due to the local GVD, and an average or guiding center soliton is allowed to propagate through the system w8x. The linear loss may be factored out by writing q Ž z,t . s aŽ z . uŽ z,t . with d aŽ z .rd z s yG Ž z . aŽ z .. aŽ z . s a 0 expwyH0z G Ž zX . d zX x. The amplitude enhancement factor a 0 is adjusted to the average soliton requirement ² Ra2 : s 1. The additional transformation w1,8x z
j Ž z. s
X
X 2
H0 R Ž z . aŽ z .
d zX
Ž2.
yields the basic equation iuj y 12 E Ž j . utt q < u < 2 u s 0,
Ž3.
where the effective dispersion parameter EŽ j . s
Dw z Ž j . x
Ž4.
R w z Ž j . x a2 w z Ž j . x
contains all the information on the periodic variation of GVD, loss Žor gain. and nonlinearity. Since ² EŽ j .: s w H0za EŽ j . d j xrz a s y1, the fluctuation z Ž j . s EŽ j . y ² EŽ j .: s EŽ j . q 1 is a source of deviations of the envelope uŽ z,t . from the average soliton shape. The overall distortion of a soliton signal up to some distance j within the unit cell may be measured in terms of the accumulated deviation f Ž j 0 , j . s Hjj00qj z Ž j X . d j X , where j 0 s H0z 0 RŽ zX . aŽ zX . 2 d zX , and z 0 is the coordinate of the launch point in soliton units; note that f Ž j 0 , ja . s 0. We are thus naturally led to adopt as optimization criterion the requirement of a zero average Žover each span. accumulated deviation, viz. F Ž j 0 . ' y² f Ž j 0 , j . : ´ 0.
Ž5.
This equation may be rewritten in its final form by undoing the transformation Ž2., F Ž z0 . s
H0
z0
D Ž z . q R Ž z . aŽ z .
2
d zq
1 za
H0
za
D Ž z . q R Ž z . aŽ z .
2
j Ž z. d z,
Ž 6.
where the second term is obtained after carrying out one integration by parts. For the periodic system considered here F Ž z 0 . exhibits always two zeros within a unit cell. At these zeros the deviation from the ideal averaged soliton is expected to be minimum. The above formula represents the main result of this Letter: it holds for any variation of the fiber parameters within the unit cell. Already at this stage, some interesting results may be read off by inspecting Eq. Ž6.. Clearly, whenever the launch point is set right behind the amplifier the first integral vanishes, and F Ž0. remains the only quantity to be canceled out. However, the adjustment of the launch point adds another degree of freedom for canceling F Ž z 0 .. It is interesting to note that the integrands in Eq. Ž6. are equal to zero for an ideal continuous dispersion profiling Ži.e., where the ratio of dispersion and nonlinearity exactly follows the peak power.. Moreover, numerical simulations show that < F Ž z 0 .< is related to the level of generated radiation and F Ž z 0 . measures the chirp acquired by the input pulse at the end of the first cell. To give a specific example of application of Eq. Ž6., let us consider the simplest and, for practical applications, most interesting case of a two-step dispersion compensation scheme, where the unit cell is divided in two sections with constant
M. Wald et al.r Optics Communications 145 (1998) 48–52
50
GVD of opposite signs, and parameters z1,2 , D1,2 , R1,2 , G 1,2 , with z a s z 1 q z 2 . In order to meet the requirement of negligible pulse-width Žand amplitude. changes Žor pulse breathing w19x. owing to the relatively high Žwith respect to the average value. fiber GVD, the lengths Z1,2 of the two sections should not considerably exceed the local dispersion length, 1,2 say, Z loc ' Z0r< D1,2 <. As a rule of thumb, one may take z1,2 < D1,2 < F 2 w20x. Averaging over the amplifier span simplifies now to, e.g., ² b : s Ž b 1 z 1 q b 2 z 2 .rz a; the input power enhancement factor w1–3x to generate a unit-amplitude guidingcenter soliton reads as a20 s
2 za R 1rG 1 y R 2r Ž GG 2 . y Ž R 1rG 1 y R 2rG 2 . exp Ž y2 G 1 z 1 .
.
Ž7.
Moreover, the integration in Ž6. may be easily carried out: one obtains an explicit expression for the quantity to be cancelled out
°F Ž0. q D z q R a 1 y expŽy2 G z . , 1 0
F Ž z . s~
2 1 0
1 0
z 0 F z1 ,
2 G1
0
¢
F Ž z 1 . q D 2 Ž z 0 y z 1 . q R 2 a 20 exp Ž y2 G 1 z 1 .
1 y exp w y2 G 2 Ž z 0 y z 1 . x 2 G2
Ž8. ,
z 0 ) z1 ,
where F Ž0. s
za 2
y
R 1 a 02 2 G1
y
D2 2 G2
q
a02 2 za
D2 z2
ž
R2
G2
y
R1
G1
/
exp Ž y2 G 1 z1 . q R 1
ž
1 y exp Ž y2 G 1 z 1 . 2 G1
/ž
D2
G2
y
D1
G1
/
.
Ž9. The optimum system design corresponds to the straightforward task of finding the zeros of Ž8.. If the launch point is set to zero, the procedure simplifies to canceling Ž9.. Before proceeding to consider a specific example of application, it is noteworthy to point out that the general method outlined above reproduces, in some particular instances, earlier prescriptions for the optimization of soliton systems. In fact, in the case of lumped amplification without dispersion management, F Ž z 0 . equals function A1Ž z s z 0 . that was introduced in Ref. w1x Žsee Eq. Ž23. there. to measure pulse deviation from an average soliton: canceling F Ž z 0 . may be achieved by launch point adjustment w16x. A similar situation occurs for pure dispersion management with R 1 s R 2 s 1 and G Ž z . s 0 w8x. In both cases, the pulse distortion function was derived by means of the Lie transform method. A different approach for the minimization of radiation scattering in dispersion-managed fiber links may employ first-order soliton perturbation theory: Eq. Ž17. of Ref. w14x Žwhere no launch point optimization was considered and R 1 s R 2 s 1, G 1 s G 2 s G . coincides with the condition F Ž0. s 0. In what follows, we will demonstrate by simulations the effectiveness of the result Ž8., Ž9.. To this end, we considered a two-step GVD compensated fiber link: each cell consists of a concatenation of a DSF and a DCF. The zeros of Eq. Ž8. correspond to the best suppression of the generated radiation: to confirm this, we numerically solved Ž1. by means of a standard split-step algorithm. The pulse shapes at the end of each unit cell were used as initial ‘‘potential’’ in the direct Zakharov–Shabat scattering problem, whose solution yields the complex soliton eigenvalue, say, l. The relative non-soliton Žor radiation. energy content of the field is then given by the formula s s 1 y 2 ImŽ l. w14,15x. Let us consider at first the specific case of a 20 GBitrs soliton transmission Žwith a pulse width Tfwhm s 11 ps, and spacing DT s 50 ps.: the length DSF of the DSF Žwhich is nearly equal here to the amplifier spacing. is about 1.7 times larger than Z loc . The DSF fiber has the 2 DSF loss a DSF s 0.24 dBrkm, the GVD b DSF s y1.28 ps rkm Žso that Z loc s 30 km., and the length is Z DSF s 50 km; for the DCF, the loss is a DCF s 0.6 dBrkm, the GVD is b DCF s 20 ps 2rkm, and the length Z DCF s 2.53 km; the ratio of nonlinear coefficients is g DCFrg DSF s 1.6, and the amplifier spacing Za s 52.53 km, so that the average dispersion is ² b : s y0.26 ps 2rkm, which yields Z0 s 158 km, and z a s 0.332. Without dispersion compensation, the resonant periodic amplification leads to a substantial growth of radiation which, in turn, strongly interacts and distorts the soliton signal after a fairly short distance. This detrimental radiation may be strongly reduced by using a GVD map, with the proper choice of the launch point: in Fig. 1, we show the energy fraction of radiation s ' 1 y 2 ImŽ l. that is generated at the distance z s 35 z a Žor Z s 1800 km., versus the input coordinate z 0 . Here the DCFs are placed before the amplifiers: as can be seen, the radiation is minimized around the launch points z 01, z 02 which are chosen according to Eqs. Ž8., Ž9., viz. F Ž z 0 . s 0. On the other hand, Fig. 2 shows the dependence of s with distance, for different choices of z 0 . In Figs. 2a, 2b, the DCFs are placed right before or after the amplifiers, respectively. Whereas the dots indicate a launch point at the amplifiers Ž z 0 s 0., and the triangles or diamonds are obtained for the optimal choices of z 0 , again according to Eqs. Ž8., Ž9.. As can be seen, the radiation is strongly reduced when the proper launch points are selected, in particular for compensation in the high-power region. Fig. 2 also shows that, in contrast with the case of a launch
M. Wald et al.r Optics Communications 145 (1998) 48–52
51
Fig. 1. Radiation energy versus launch point position for DCFs before the amplifiers.
point at the amplifier Žwhere the DCF should be placed in the low-power region w15x., whenever the optimum launch point is chosen the level of radiation is independent of the relative position of the DCF in the unit cell. Indeed we observed that, for an optimized 20 GBitrs stream, soliton interactions are virtually unaffected by the presence of a small radiation level. For strong dispersion compensation, where z 1,2 < D1,2 < 4 1, the present optimization scheme is not sufficient to guarantee stable soliton transmission. For example, Fig. 3a shows the distortion of a 40 GBitrs Ži.e., Tfwhm s 5 ps, DT s 25 ps. stream DSF through a link with optimal z 0 s 0.192, a DSF dispersion distance Z loc s 6.3 km, the lengths Z DSF s 50 km, Z DCF s 2.9 2 km, Za s 52.9 km, and ² b : s y0.1 ps rkm, so that Z0 s 80 km. As pointed out in Ref. w22x, in such cases the dispersive pulse width variations within each cell do introduce an additional oscillation of the soliton amplitude, which ultimately hampers the long-scale balance between average GVD and nonlinearity. This balance may be restored by further enhancing the amplitude of the input signal with respect to the usual average fundamental soliton value of Eq. Ž7. w18x. In order to show the possibility of extending the present optimization scheme to the regime of strong dispersion management, we took advantage of the variational approach w19x for calculating the optimal input soliton amplitude. As usual, this involves considering a perturbed fundamental soliton with z-evolving time width, amplitude, phase and chirp as a trial function. By solving the resulting ordinary differential equations for the pulse parameters, the optimal launch points and input amplitude may be found by the shooting method: here one imposes the recovery of the initial amplitude along with the minimization of the chirp upon propagation through the unit cell. The resulting optimal launch point differs from the prediction of Eqs. Ž8., Ž9.; whereas the amplitude enhancement factor turns out to be equal to about one and a half the value of Eq. Ž7.. With these optimal data, we found that a 40 GBitrs signal can be distortion-free propagated over more than 2000 km Žsee Fig. 3b.. Note that, for the previously considered 20 GBitrs scenario, the amplitude enhancement is equal to just
Fig. 2. Energy of radiation scattered by a soliton versus distance z for different launch points. DCFs either Ža. before Žand z 01 s 0.102, z 02 s 0.328. or Žb. after Žand z 01 s 0.011 or z 02 s 0.135. the amplifiers.
M. Wald et al.r Optics Communications 145 (1998) 48–52
52
Fig. 3. Propagation of a 40 GBitrs Ž..1101... soliton stream Ža. without amplitude enhancement Ž z 0 s 0.192., Žb. with amplitude enhancement factor 1.45 Žand z 0 s 0.136..
1.03, which justifies a posteriori the use of Eq. Ž7.. Details on the use of the variational method for the optimization of dispersion-compensated 40 GBitrs single channel soliton transmissions will be described elsewhere. In conclusion, in this work we presented a simple analytical formula for the simultaneous optimization of the launch point and of the dispersion map in soliton transmission lines. We showed that the method permits a minimization of the generated radiation; several related optimization schemes appear as particular cases of the present general prescription. In the case of strong GVD management Žthis may also apply to upgrade standard fiber lines beyond 10 Gbitrs., a variational approach may be used to extend the optimization to solitons with properly enhanced power.
Acknowledgements M. Wald, I.M. Uzunov, and F. Lederer acknowledge a grant from the Deutsche Forschungsgemeinschaft ŽInnovationskolleg Ink 1rA1.. The work was carried out in the framework of the COST 241 initiative of the European Union.
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