Chirped nonlinear pulsepropagation in a dispersion-compensated system

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Chirped nonlinear pulsepropagation in a dispersion-compensated system
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  November 15, 1997 / Vol. 22, No. 22 / OPTICS LETTERS  1689 Chirped nonlinear pulse propagation in adispersion-compensated system Yuji Kodama, Shiva Kumar,* and Akihiro Maruta Department of Communications Engineering, Osaka University, Suita, Osaka 565, Japan Received June 30, 1997 We study nonlinear pulse propagation in an optical transmission system with dispersion compensation. A chirped nonlinear pulse can propagate in such a system, but eventually it decays into dispersive waves in a waysimilar to the tunneling effect in quantum mechanics. The pulse consists of a quadratic potential that is dueto chirp in addition to the usual self-trapping potential and is responsible for the power enhancement and thedecay.  󰂩  1997 Optical Society of America The dispersion-compensated soliton seems to be themosteffectivecandidate foruseinahigh-speedcommu-nication system. 1–5 Even though (quasi-)stable pulsepropagation in such a system with enhanced power hasbeenreported, 1 theunderlyingphysicalmechanismhasnot yet been understood. Here we analyze chirped-pulse propagation in a dispersion-compensated systemand show that the quadratic potential that is due tochirp,in additiontothe usualself-trappingpotential,isresponsible for the power enhancement. We also showthat the pulse eventually decays into dispersive wavesin a way similar to the tunneling effect. As a model of a dispersion-compensated system weconsider i  ≠ u ≠ Z  1  d  Z  2 ≠ 2 u ≠ T  2  1 j u j 2 u    0,  (1)where the dispersion function d  Z   is a two-step peri-odic function, withd  Z    ( d 1  0 , Z  2 nZ d  , Z 1 d 2  for  Z 1  , Z  2 nZ d  , Z d .(2)Here  Z  is the normalized distance defined by theaverage dispersion. In this Letter we assume that thelocal dispersions  d 1  and  d 2  are of order  $ 20 , withthe average dispersion   d     1  in the normalized unit,so  d 1    1  1 D D  Z d  2  Z 1  Z d  and  d 2    1  2 D DZ 1  Z d ,with D D   d 1  2 d 2 . Wealso assume that theperiod of dispersion is  Z d    0.1 . These are typical values usedin a practical system. 1 Because of the large values of dispersion, the so-called guiding center soliton 6 cannotexist. The main feature of the pulse in such a systemis a large deformation from sech shape of the solitonand the appearance of strong chirp owing to the large values of local dispersions. Because of the strong chirp, we first assume that the solution  u  of Eq. (1)takes the form u  Z , T      w  Z , T   exp ∑  i 2  C  Z  T  2 ∏ .  (3)Substituting Eq. (3) into Eq. (1), we have i µ ≠ w ≠ Z  1 dCT   ≠ w ≠ T  ∂ 1  d 2 ≠ 2 w ≠ T  2  1 j w j 2 w 2 1  2    C  1 dC 2  T  2 w   2 i 2  dCw .  (4)The term on the right-hand side of Eq. (4) implies thatthe amplitude of   w  oscillates because of the chirp thatresults from the dispersion compensation. The secondterm indicates the scale variation in time coordinates.Taking these effects into account, we introduce a newcoordinate t  and amplitude function a  Z  , so that t    p  Z  T  ,  w  Z , T      a  Z  v  Z , t  .  (5)Then we have i  ≠ v ≠ Z  1  dp 2 2 ≠ 2 v ≠t 2  1 a 2 j v j 2 v 2 k  Z  2  t 2 v    0,  (6)with the equations for  a  Z  , p  Z  , and C  Z  :  a  2 1  2 Cad ,  (7)  p   2 Cpd ,  (8) k  Z    C  1 C 2 dp 2  .  (9)First we note from Eqs. (7) and (8) that  a  Z    a p  p  Z   with constant  a . Then we look for a periodicsolution of Eqs. (8) and (9). Note here that  k  Z  in Eq. (9) can be chosen arbitrarily, and choosing a specific form of   k  Z   corresponds to a particularsolution of Eq. (6). We should like to mention that thefunction k  Z   canbealsodetermined bythe variationalmethod, 4 and we have  k    a 1 dp 2 2 a 2 p  for somepositive constants a 1  and a 2 . Assuming that C  Z   and p  Z   are periodic solutions,the average of the product k p  should be zero, i.e.,  k p    1 Z d Z   Z d 0 k  Z  p  Z  d Z    0,  (10)as can be shown from the equation  d  Cp 2 1  d Z    k p .Because  p  Z   is positive, Eq. (10) implies that  k  Z  oscillates its sign within a unit cell,  nZ d  ,  Z  ,   n  1 1  Z d . Let us then assume the following simplest formfor k  Z   as a two-step periodic function: k  Z    ( k 1  0 , Z  2 nZ d  , Z 1 k 2  Z 1  , Z  2 nZ d  , Z d .  (11)We determine the constants  k 1  and  k 2  by imposing acondition to close the orbit   C  Z  , p  Z   at  Z    nZ d . 0146-9592/97/221689-03$10.00/0  󰂩 1997 Optical Society of America  1690  OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997 With Eq. (11), we obtain the solution of Eqs. (8) and (9)as a curve in the p 2 C  plane: C 2   2 p 2  E  Z  2b  Z  ln  p  .  (12)Here b  and  E  are two-step periodic functions given by b  Z     k 1  d 1  and  E  Z     E 1  for  0  ,  Z  2  nZ d  ,  Z 1 ,and by  b  Z     k 2  d 2  and  E  Z     E 2  for Z 1  , Z  2 nZ d  , Z d .For the case  d 1  .  0 , one should choose  C  0   ,  0  forpulse compression, and the minimum pulse width isobtained at the distance of zero chirp [the midpoint of each fiber in the present (lossless) case]. This impliesthat  C 2  p 2 from Eq. (12) is a decreasing function andtherefore that  b  Z   .  0 . In the case of a linear pulsewe take d 1   2 d 2  and k 1   2 k 2  (i.e.,   d      k     0  and b    constant) for the existence of a periodic solution.However, in the case of a nonlinear pulse we may takenonzero average dispersion   d    1   to compensate forchirp generated by the nonlinearity in a similar way asinthesolitoncase. Thenwe canshowthatbychoosing the appropriate constants  k 1 , k 2  and  E 1 , E 2  we have aunique closed orbit in the p 2 C  plane. Figure 1 showsa closed orbit in the  p 2 C  plane with  d 1  .  0 ,  d 2  ,  0 ,and  Z 1    Z d  2 . For such a periodic orbit,  k 1  and  k 2 satisfy  0  ,  k 1  , 2 k 2  and therefore   k   ,  0 . As weshow below, the negativity of the average of   k  Z   isconsistent with an enhanced power of the stationarypulse solution compared with that of the soliton casethat has uniform dispersion with   d     1  and   k     0 .With the solution of Eqs. (8) and (9) we now solveEq. (6) to find the pulse shape. To do so we first notethatthevariation fromthe averagedequationofEq. (6)is small, unlike in the case of Eq. (1), where  D D  3  Z d is so large   5  that we cannot apply the guiding centertheory directly to Eq. (1). This is the main reasonthat we use transformation (3) to identify the core of the solution  u  of Eq. (1) and apply the guiding centertheory to Eq. (6) to construct the stationary solution of Eq. (1). The core then satisfies the averaged equationof Eq. (6): i  ≠ V  ≠ Z  1  D 0 2 ≠ 2 V  ≠t 2  1 A 0 j V  j 2 V   2  K  0 2  t 2 V     0,  (13)where  D 0     dp 2  ,  A 0    a 2  p  , and  K  0     k  . It isinteresting to note that D 0  may take a nonzero positive value even in the case   d   #  0 . A stationary solutionof Eq. (13) can be obtained in the form  V   t , Z     f   t  exp  i l 0 Z   with a real function  f   t   and a realconstant l 0 . The function  f   t   is a solution of  D 0 2 d 2  f d t 2  1 A 0  f  3 2  K  0 2  t 2  f    l 0  f   .  (14)In analogy with the Schr¨odinger equation in quantummechanics, Eq. (14) expresses a wave function of aparticle under the potential given by U   t    2 A 0  f  2  t  1  K  0  2  t 2 .  (15)With a pulselike solution of Eq. (14) the potentialconsists of a nonlinear self-trapping term (as in thecase of a soliton) and a quadratic nontrapping back-ground (with  K  0  , 0 ). Because of the nontrapping ef-fect one needs to have higher intensity of   f  2 than forthe soliton  K  0    0  . Thisquantum-mechanical poten-tial is equivalent to the (classical) chirp, and in factthe chirp is expressed by the derivative of the poten-tial,  Dv  t     d U   d t . It is then easy to see that thechirp generated by the quadratic potential is oppositethat by generated the nonlinearity, which implies anenhanced power of the pulse solution. Figure 2 plotsthe bound-state solution of Eq. (14) that corresponds tothe parameters used in Fig. 1; we found that the en-hanced peak power at  Z    nZ d  is 2.25, which is closeto the value 2.24 obtained by the numerical simulationof Eq. (1). The difference is due to the corrections onthe stationary solution  f  , and the core solution  V   canbe obtained by a Fourier-like expansion of the eigen-modes of Eq. (14). That is, we can cancel the tails inthe stationary solution by taking account of the higher-order eigenmodes. Then the solution  v  of Eq. (6) canbe constructed by use of the guiding center theory (Lietransform with averaging). Also note from Eq. (14) that, because  K  0  ,  0 , thetail of solution  f   has an oscillation instead of aGaussian-like decay, as shown in Fig. 2; as a result,the total energy of the stationary solution, R `2`  j V  j 2 d T  ,diverges. This result implies that an input pulsewith finite energy eventually decays to dispersive Fig. 1. Chirp  C  Z   versus  p  Z   with  D D    58 ,  k 1    47 , k 2   2 47.94 , and Z d    0.155 .Fig. 2. Bound-state solution (dotted curve) of Eq. (14),with  2 A 0  D 0    7.156 ,  K  0  D 0    2 0.858 , and  2 l 0  D 0   3.962 ; and the pulse shape (solid curve) at  Z  Z d    199.25 obtained by numerical simulation of Eq. (1).  November 15, 1997 / Vol. 22, No. 22 / OPTICS LETTERS  1691 Fig. 3. Energy of the pulse within the window  j T  j  ,  5 normalized by the input energy versus distance, with  Z d   0.155  and Z 1  Z d    0.5 .Fig. 4. Energy of the pulse renormalized by the energy at Z  Z d    100  for each  D D  versus distance. Parameters arethe same as for Fig. 3. wavesbecause of the tunnelingeffect. Figure 3 showsthe energy of the pulse within the window  j T  j  ,  5 obtained by direct numerical simulation of Eq. (1) withGaussian initial data. There are two steps in thedecay process: One step is due to a formation of thequasi-stationary state; the other step is the gradualdecay of that state because of the tunneling effectpredicted in this Letter. Figure 4 shows the energy of thepulserenormalizedbytheenergyat Z  Z d    100 foreach  D D , where the solution can be considered quasi-stationary. The decay that is due to the tunneling effect can be neglected in practical situations with D D    60 . But one can observe a gradual decay of theenergy for D D  $ 90 .In conclusion, we have analyzed the propagation of a chirped nonlinear pulse in a dispersion-compensatedsystem and shown that the core solution consists of a parabolic potential that is responsible for the powerenhancement and tunneling effects.*Present address, Institut f¨ur Festk¨orpertheorie undTheoretische Optik, Universit¨at Jena, Jena, Germany. References 1. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, andI. Bennion, Electron. Lett.  32,  54 (1996).2. M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto,H. Taga, and S. Akiba, Electron. Lett.  31,  2027 (1995).3. M. Nakazawa and H. Kubota, Electron. Lett.  31,  216(1995).4. I. Gabitov and S. K. Turitsyn, JETP Lett.  63,  863 (1996);Opt. Lett.  21,  327 (1996).5. T. Georges and B. Charbonnier, Opt. Lett.  21,  1232(1996).6. A. Hasegawa and Y. Kodama, Opt. Lett.  15,  1443 (1990).
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