Micwave and Optical Technlgy Letters, Vl. 1, N. 3, 1999. pp. 05-08 Mdelling f NOLM Demultiplexers Emplying Optical Slitn Cntrl Pulse Z. Ghassemly, C. Y. Cheung & A. K. Ray Electrnics Research Grup, Schl f Engineering, Sheffield Hallam university, Pnd St., Sheffield, S1 1WB, UK E-mail: z.f.ghassemly@shu.ac.uk ABSTRACT: An ptical slitn pulse may be used as a cntrl signal in nn-linear ptical lp mirrr demultiplexers in rder t reduce timing-jitter nise and crss-talk. A mathematical mdel fr calculating the width f the slitn switching windw and ptimizing critical parameters such as walk-ff time and pulse width has been presented, and its accuracy is verified by slving the nnlinear Schrdinger equatin. Key wrd: Semicnductr laser amplifier, ptical switching, slitn, ptical time divisin demultiplexer, ptical lp mirrrs. 1- INTRODUCTION The nn-linear ptical lp mirrr (NOLM) demultiplexer is a prmising cnfiguratin fr achieving all-ptical time divisin demultiplexing because f its high perating speed [1]. Channel demultiplexing is realised by the phase difference beeen the clckwise (CW) and cunter-clckwise (CCW) signal pulses prpagating within the fibre lp, see Figure 1. In ultra-high speed systems, there are majr prblems: 1) difficulty in achieving cmplete switching f the signal pulses; and ) timing jitter beeen the cprpagating cntrl and signal pulses. The latter is reduced t a certain extent by intrductin f walk-ff time T w beeen the cntrl and signal pulses. This results in a square switching windw shape, with a width f T w L, where L is the fibre lp length, rather than bell shape, thus enabling imprved switching and cnsequently better tlerance t timing jitter []. A 100% switching is difficult t achieve because f the cntrl pulse experiencing a slitn cmpressin effect due t interactin beeen the fibre dispersin and self phase mdulatin (SPM) [4]. This will result in an increased cntrl signal peak pwer which in turn leads t an asymmetrical switching windw with an increasing phase shift twards the end f the switching prfile. Maximum switching may be achieved by emplying ptical slitn as a signal r cntrl pulse, since the fundamental prperties f the slitn are unifrm phase ver the entire pulse and cnstant pulse shape ver the entire prpagatin length. Here, we investigate the later ptin, and shw that the prblem f pulse defrmatin during prpagatin within the lp is slved, thus resulting in a much imprved switching windw with high transmittance and reduced timing jitter effects. - THEORY T study the utput transmittance f the NOLM demultiplexer it is best t cnsider the cmbined impact f the fllwing effects: 1) crss phase mdulatin (XPM), ) prpagatin f ptical slitn pulse within the fibre lp; and 3i) walk-ff time beeen the cntrl and signal pulses. The switching windw is created by the phase difference beeen the CW and CCW signals. The phase f the signal pulse cprpagating with the cntrl pulse is substantially changed by the XPM effect. With the inclusin f walk-ff time, the phase change can be represented by L φ = γ PT ( Tw x ) dx (1) 0 where γ is the nn-linear cefficient, P is the ptical pwer prfile, and T w = the walk-ff time per unit length. The cntrl pulse has t meet the fllwing slitn cnditins in rder t prpagate undisrtedly alng a lssless fibre [6]:
1- the prpagatin equatin f a fundamental slitn wave u( ξ, τ) = sec h( τ)exp( iξ/ ) - the peak pwer P = β / γt where u is the nrmalised amplitude, τ = T/T and ξ = z/l D, z is the travelling distance, L D is the dispersin length = T / β, T FWHM 1763. is the pulse width f the ptical slitn cntrl pulse, and β is the first rder dispersin cefficient. The cntrl pulse walks thrugh the signal pulse due t difference in the grup velcity, and with the phase change f the signal depending n the value f the ttal walk-ff time, the time varying ptical pwer prfile in Eq. (1) can be replaced by the pulse average pwer ver the duratin f ttal walk-ff time. The maximum average pwer f the cntrl pulse, resulting in XPM, is given as P ave(max) = T PT ( ) dt T / / T () Fr slitn pulse, P(T) = P. u, therefre the peak phase change can be shwn t be T / T φ p = [ γlp T sec h ( T / T ) d( T / T )]/ T (3) T / T where T = T w L is the ttal walk-ff time. Substituting fr P and replacing φ p by π fr maximum transmittance and slving fr sech (x), we btain an expressin fr lss less fibre lp as ( T / T ) T T π = L β tanh (4) 4 The utput transmittance wuld be equal t 1 if the cnditin set by Eq.(4) is satisfied. In Eq. (4) it is assumed that fibre lp is lss free. Hwever, in NOLM demultiplexers where a lng length f fibre is used, it is necessary t include fibre lss α in the analysis. Assuming that the rate f decrease f pwer is s small, the increase in chrmatic dispersin can be neglected, and therefre the peak pwer can be expressed as P 1 L = L P ( 0)exp( α z ) dz (5) 0 Slving Eq. (5) and substituting it int Eq. (3), we can btain a mdified expressin fr maximum transmittance as [ 1 exp ( L )] tanh ( T / T ) T T πα = 4 β α (6) The utput transmittance wuld be equal t 1 if the cnditin set by Eq. (4) is satisfied. In NOLM emplying a Gaussian pulse, the switching windw prfile is a bell shape with a nn-flat tp. Therefre, relative timing jitter beeen the cntrl and signal pulses and cntrl pulse defrmatin within the lp will induce intensity fluctuatin f the demultiplexed utput signal, thus resulting in switching pwer penalty [5], whereas with an ptical slitn pulse, the shape f the switching windw becmes symmetrical with a flat-tp transmittance prfile, thus resulting in reduced timing jitter nise
and inter-channel crss-talk. T achieve the frmer, the variatin f transmittance at the tp f the switching windw has t be kept within the perturbatin-tlerance factr F p ; see Figure. The shaded areas represent the prtin f the cntrl pulse prfile walking thrugh the signal pulse alng the whle prpagating distance. At time t 1, the transmittance level is belw the threshld level. Frm t 1 t t 3, it remains abve the threshld level and reaches peak value at time t. After t 3, the transmittance level begin t drp. Therefre, the width f the transmissin windw is equal t t 3 - t 1. By cmparing the walk-ff regin f the pulse prfile at t 1, t, and t 3, the walk-ff regin f the pulse prfile at t 1 is just left shifted with t = (T /) -T 1 and t 3 is just right shifted with t = T -(T /) frm the symmetric walk-ff regin f the pulse prfile at t, respectively. T -(T /) = (T /) -T 1 because f the symmetrical shape f a slitn pulse. The transmissin windw s perturbatin-tlerance width is defined as the perid f time at which the nrmalised transmittance level remains abve the threshld value 1 - F p, which is given as W = ) ( T / T1 ) = ( T T / ) = ( T / Ti (7) where T 1 and T are referred t in Figure. Fr F p = 0.5, W = FWHM width. A mdel was develped fr calculating the width f a slitn switching windw assuming that α = 0. Fr a given set f system parameters, F p can be related t T 1 and T by the fllwing equatin: 1 ( 4cs ( F )/ ) tanh( T / T ) = tanh( T / T ) tanh[( T T ) / T ] p π (8) i Equatin (8) yields slutins fr T i : ne is T 1, and the ther is fr T. W can then be calculated by substituting either f the slutins fr T i in Eq. (8) int Eq. (7). The expressin fr W hlds true n matter which ne f the slutins f T i is used in the substitutin. i 3- SIMULATION RESULTS AND DISCUSSION Equatin (4) was slved numerically by the Newtn-Raphsn methd (NRM) [3], and the slutin was checked by applying the crrespnding values f T and T t Eq. (4) fr verificatin. Fr L = 3 km, the results fr the slitn pulse width versus the ttal walk-ff time fr different values f β and α fr a slitn switching windw at maximum transmittance are shwn in Figure 3. The slitn pulse width decreases with the increase f the ttal walk-ff time. This is due t the decrease in the interactive time fr XPM beeen signal pulses and the central peak f cntrl pulses. The reduced pulse width cmpresses mre ptical pwer arund the central peak f the cntrl pulses, and as a result, the phase change φ is increased. The increased phase change due t reduced values f the slitn pulse width is exactly equal t the decreased phase change due t the larger value f the ttal walk-ff time, thus maintaining maximum transmittance. Figure 3 als shws the affects f the first rder dispersin cefficient β n the slitn pulse width and the ttal walk-ff time. Fr a fixed value f ttal walk-ff time, a smaller dispersin cefficient results in a reduced slitn pulse width. This is because the peak pwer f a ptical slitn pulse is directly prprtinal t the magnitude f the first rder dispersin cefficient β and inversely prprtinal t the square f the slitn pulse width. A reductin in peak pwer, due t lwer values f β, is partially cmpensated by the decrease f the slitn pulse width. Fr lssy fibre Eq. (6) is slved numerically and the results fr a range f α are als shwn in Figure 3. With the inclusin f α, the peak pwer f a pulse is gradually reduced as the pulse prpagates alng the fibre lp and, as a result, the phase change due t XPM is als reduced. Fr α = 0. db/km, a narrwer pulse width may be used t cmpensate fr the lss f phase due t α. Equatin (8) was slved numerically by a cmbinatin f the functin iteratin and NRM and the results fr T i were used t calculate the transmissin windw width. Values f the slitn pulse width and the ttal walk-ff time were btained frm Eq. (4), and the results fr the transmissin windw width fr different values f β are illustrated in Figure 4, shwing a threshld level abve which the slpe increases significantly. The threshld level dependence n the β can be used as an imprtant parameter t predict the width f a slitn switching windw. T create a wider switching windw, if the ttal walk-ff time is greater than the threshld level, then a smaller value f β can be used, and if the ttal walk-ff time is smaller than the threshld level, then a larger value f β may be used. By
increasing the width f the switching windw, the timing jitter nise can be reduced, but at the cst f increased inter-channel crss-talk by allwing sme prtin f the adjacent pulses t be demultiplexed t the utput prt, which is the subject f further study. Finally, the mathematical mdel develped is verified by slving the nn-linear Schrdinger equatin with beam prpagatin methd (BPM) [], and the results are shwn in Table 1. An interactive fibre length f 3 km and first-rder dispersin parameters f 5 ps /km are used fr calculatin, and the pulse width fr lss-free and lssy fibre are btained frm Eqs. (4) and (6), respectively with T = 10 ps. The results btained agree very clsely, thus verifying the accuracy f the mdel develped. Hwever, the deviatin f the windw width increases as the fibre lss per kilmetre increases. 4- CONCLUSIONS A mathematical mdel fr NOLM emplying a slitn cntrl pulse has been presented, where the prblem f pulse defrmatin during prpagatin within the lp is vercme. It has been shwn that the mdel can be used t ptimise critical parameters, such as walk-ff time and pulse width, fr maximising utput transmittance with much reduced timing jitter effects. The accuracy f the mathematical mdel develped is verified by slving the nn-linear Schrdinger equatin. Acknwledgement One f the authrs Mr C. Y. Cheung is financially supprted by Overseas Research Students Award. REFERENCES 1) M. Jinn, and T. Matsumt, Nnlinear Sagnac Interfermeter switch and its applicatins, IEEE J. f Quantum electrnics, 8 (199), 875-88. ) K. Uchiyama, et-al., Effects f cntrl-signal pulse walk-ff n BER perfrmance f nnlinear ptical lp mirrr demultiplexer, Electrn. Lett 9 (1993), 1870-1871. 3) P. R. Turner, Numerical analysis, Macmillan, New Yrk, 1994. 4) L. P. Barry, et-al., Effect f cntrl pulse defrmatin n the switching characteristics f a NOLM, Prcs. f the 1st Australian Cnf. n Optical Fibre Technlgy, Dec. 1996, pp. 197-00. 5) K. Uchiyama, et-al., Signal-t-nise rati analysis f 100 Gb/s demultiplexing using nnlinear ptical lp mirrr, IEEE J., Lighave Technl 15 (1997), 194-01. 6) G. P. Agrawel, Nnlinear fiber ptics, Academic Press, New Yrk, 1989. 1999 Jhn Wiley & Sns, Inc. CCC 0895-477/99
Cntrl cupler Cntrl Data in (signal CW Lng Fibre Lp CCW I/O Cupler Data ut Figure 1 A typical cnfiguratin f NOLM demultiplexer Transmissin windw At time t 1 At time t 3 F p T W T W T 1 T 1 -T W T -T W T At time t -T W / T W T W / Figure Typical slitn transmissin windw: the shaded areas representing the prtin f the cntrl pulse verlapping with the signal pulse thrughut the cmplete prpagatin
β = -0 ps /km β =-10 ps /km β = -5 ps /km Fibre Lss α 0 db/km 0. db/km 0.5 db/km 0.8 db/km Figure 3 Slitn pulse width versus ttal walk-ff time fr different values f β and α β (ps /km) -5-10 -15-0 Figure 4 Perturbatin-tlerance width versus ttal walk-ff time fr different values f β
TABLE 1 Results Using BPM fr L = 3 km and β = -5 ps /km Fibre lss (db/km) Parameter BPM Mathematic al Mdel 0 Transmittance 0.9998 1.0000 Windw width (F p = 5%) ps 6.73 6.75 0. Transmittance 1.0000 1.0000 Windw width (F p = 5%) ps 6.66 6.9 0.5 Transmittance 1.0000 1.0000 Windw width (F p = 5%) ps 6.3 7.17