Analytical Assessment of the Q-factor due to Cross-Phase Modulation (XPM) in Multispan WDM Transmission Systems

Transcription

1 Analytcal Assessment of the Q-factor due to Cross-Phase Modulaton (XPM) n Multspan WDM Transmsson Systems Stephan Pachncke 1*, Stefan Spälter, Jörg Rechert 3, and Edgar Voges 1 1 Unversty of Dortmund, Hgh Frequency Insttute, D-441 Dortmund, Germany Semens AG, ICN, Carrer Products, Optcal Solutons, D Munch, Germany 3 Semens AG, Corporate Technology, D Munch, Germany ABSTRACT In ths paper cross-phase modulaton (XPM) and ts mpact on multspan NRZ-modulated wavelength dvson multplexng (WDM) systems s examned. An analytcal model s presented to assess the sgnal degradaton. The mparments due to XPM are related to a Q-factor. The presented formulas are examned for dfferent dsperson compensaton schemes and also mxed-fber systems. The analytcal models are verfed by system smulatons employng the splt-step Fourer method. To determne the mparments due to XPM the pump-probe model s used, whch assesses the dstorton of an unmodulated probe channel. It s shown that ths model s applcable to NRZ systems. Exstng analytcal models are extended for certan condtons, e.g. gven dsperson compensaton schemes such as the dstrbuted under-compensaton scheme (DUCS) or the full-nlne and optmzed post compensaton scheme (FOCS). Furthermore, a formula wll be suggested to estmate the degradaton of performance, f further WDM channels are added to an exstng system. Keywords: cross-phase modulaton, nonlnear fber optcs, optcal crosstalk, optcal fber communcatons, wavelength dvson multplexng. 1. INTRODUCTION Cross-phase modulaton s one of the domnant degradaton effects n 10 Gbt/s NRZ WDM systems. Dense channel spacng as well as hgh sgnal powers lead to consderable sgnal degradatons. XPM s one of the man obstacles to achevng hgh transparent transmsson dstances. XPM causes a modulaton of the optcal phase of a gven channel through the refractve ndex varaton of the fber when other channels are ntensty modulated [1]. The modulaton of the optcal phase s converted nto ntensty dstorton of the sgnal by dsperson leadng to a nose-lke dstorton []. In ths paper the sgnal dstortons due to XPM have been solated from the other nonlnear effects, and a Q-factor due to the XPM degradaton s calculated. Several smplfed analytcal models are presented n ths paper and the boundary condtons for the dfferent models are outlned. Dfferent dsperson compensaton schemes are covered and also mxed fber systems have been analyzed.. ANALYTICAL MODELS FOR THE ASSESSMENT OF XPM In the past varous analytcal models for the assessment of the XPM degradaton have been presented [3-5]. These models have n common that they employ the pump-probe technque. Thereby the nfluence of the modulated pump channels on the unmodulated probe channel s assessed. From the nonlnear Schrödnger equaton t can be derved that the phase of the probe channel changes nfntesmally. dϕ ( z, ω ) = γ P ( 0, ω ) exp ( α + jωd λ ) ( z)dz (1) Probe Pump Channel * stephan.pachncke@un-dortmund.de; phone ; fax ; Unversty of Dortmund, Fredrch-Wöhler-Weg 4, D-441 Dortmund, Germany

2 Ths nfntesmal phase shft s converted nto ntensty modulaton by dsperson. In the followng, dfferent analytcal models for the assessment of the sgnal degradaton due to XPM are presented..1. The Cartaxo model The Cartaxo model [3] s a very versatle and flexble model. It assumes two copolarzed optcal waves, whch propagate n a sngle mode fber (SMF). At the nput of the fber, channel 1 s unmodulated (cw) and the optcal power of the second channel s snusodally modulated wth an angular frequency ω. Wth the help of the Wangmodel [], the change of the pump envelope due to dsperson (group-velocty dsperson, GVD) s taken nto account. A small sgnal approxmaton along the fber s assumed. The change of the ampltude due to dsperson s descrbed by the followng formula []: P Output ω D λ 4πc () t = cos P () t The phase modulaton (PM) of the probe channel at the poston z s converted nto ntensty modulaton (IM) due to the GVD. Because the PM s nfntesmal, a small sgnal approxmaton for the PM-IM converson can be appled, as t was outlned n []. Input () λ dp ( z, ω ) = P ( z) sn ω D( L z) dϕ( z, ω ) (3) Probe Pump 4πc It has been assumed that the sgnal propagates n the lnear regme from the poston z to the end of the fber secton, and only GVD affects the sgnal. The accumulated IM at the output of the fber segment can be calculated by the summaton of the dfferent contrbutons of the XPM nduced phase shfts at the poston z wth z = 0..L. Ths s equvalent to an ntegraton of eq. (3) from 0 to L. If the resultng equaton s subsequently dvded by the mean pump power, a transfer functon H(ω) can be defned. Fg. 1 shows that ths transfer functon has roughly hgh pass characterstcs. Fgure 1: Transfer functon for a 10 span NRZ 10 Gb/s transmsson system wth 0 dbm channel nput power nto the NZDSF and dbm nput power nto the DCF, 9 channels, D = 4.14 ps/(nm km), 50 GHz channel spacng, S = 0.08 ps/(nm² km), α = 0.3 db/km, γ = /(W km) wth DUCS (-50 ps/nm/span), a pre-compensaton of -170 ps/nm and 50 ps/nm resdual dsperson at the end of the system.

3 The equaton for -channel systems can be extended to an arbtrary number of channels. If there are more than one pump channels, the phase of the probe channel s modulated by each pump channel ndvdually. The phase shfts nduced by the dfferent pump channels are ndependent from each other. The aggregate transfer functon can be nterpreted as a superposton of the ndvdual transfer functons. Furthermore, t s possble to extend the transfer functon to an N-segment transmsson system. Because the system parameters (e.g. fber length, attenuaton, dsperson coeffcents, nonlnearty coeffcent, gan, etc.) may not be dentcal for each span, the analytcal equatons must be able to support arbtrary parameters for each span. The most mportant change for a multspan system s, though, that the nduced PM s now converted nto IM from the poston z to the total system length. The DCFs are consdered wth a negatve sgn accordngly. Because the equaton gven n [3] s very complex, t s not prnted at ths pont. A smplfed verson of that equaton s presented later n ths text (compare eq. (8)). It needs to be stressed that the Cartaxo model offers very hgh flexblty. It consders the change of the pump channel envelope due to GVD, as well as the dsperson slope. Furthermore, t s possble to nclude a pre- DCF as well as varyng dsperson compensaton schemes, whch may change on a span-by-span bass. Also systems wth mxed fbers (e.g. SSMF and NZDSF) may be modeled. Ths flexblty on the other hand leads to a very hgh computatonal effort. The computatonal tme for a 30 span system can be as hgh as several mnutes... The Bellott Model The Bellott model s a smplfed Cartaxo model, although both models have been developed ndependently from each other. The Bellott model s as the Cartaxo model based on the Wang model, whch enables to calculate an ampltude fluctuaton from the phase shft. In contrast to the Cartaxo model, though, the Bellott model neglects the dsperson slope. It assumes that the dsperson constant D s constant for all channels. Also the change of the envelope of the pump channels due to GVD s neglected. The Bellott model s used n recent publcatons [6][7]..3. The Shtaf Model In contrast to the prevous two models, the Shtaf model [5] s not based on the Wang model []. Thus, t does not use the smplfed model to calculate an ampltude fluctuaton from the phase shft. The Shtaf model does not consder a modulated pump channel, but t assumes only a sngle edge for the pump. As a result, the Shtaf model obtans a functon, whch descrbes the ntensty varaton of the probe channel..4. Comparson of the dfferent analytcal models The three models presented before are all based on pump-probe experments. Ths has the advantage that the sgnal degradaton due to XPM can be separated from other nonlnear effects lke self-phase modulaton (SPM), because the probe channel s cw. Furthermore the pump-probe model allows to fnd relatvely smple analytcal expressons. Essental s also the smplfed converson of PM to IM based on the Wang model. The Wang model offers the great advantage that not a dfferental operator of second order needs to be solved, but only smple sne and cosne terms need to be consdered. Due to these consderatons - n the followng - a combnaton of the Cartaxo model [3] and the Bellott model [4] s presented, whch has partly been publshed n [8].

4 3. SIMPLIFIED ANALYTICAL MODELS 3.1. Smulaton setup To verfy the results obtaned from the analytcal models, the followng smulaton system setup has been used (Fg. ). For the smulatons the program PHOTOSS [9] has been used. The smulatons used the separated channels model [10]. Ths model allows to swtch off all other nonlnear effects except for the XPM effect. In all smulatons 9 NRZ modulated channels were launched at the begnnng of the fber. At the end of the transmsson system, the optcal eye of the center channel has been analyzed. The center channel s the worst affected channel and thus defnes the worst-case scenaro. Fgure : Smulaton system setup. M-WDM channels and N-spans wth an optonal Pre-DCF. Only the center channel (193.1 THz) has been taken nto account for the assessment of the Q-factor. The consdered transmsson system conssts of M channels wth a channel spacng of 50 GHz or 100 GHz. 10 Gb/s NRZ sgnals wth a cos²-shape and a roll-off factor of R = 0.5 are transmtted. The length of the pseudo-random bt sequence (PRBS) was chosen to be 104 bts. The varance of the marks was analyzed at the end of the transmsson dstance. The optcal demux flter was chosen to be a frst-order Gaussan flter wth an FWHM of 1.5 GHz. 3.. Calculaton of the Q-factor As explaned before t, s possble to analytcally obtan a transfer functon H(ω) due to XPM. Because XPM s a nose-lke process t s convenent to descrbe the degradatons due to XPM wth the varance of the nose process. The varance can be calculated by multplyng the transfer functon wth the power spectral densty of the NRZ sgnal and applyng an nverse Fourer transformaton afterwards. The varance s calculated n the center of the pulse (t = 0). The power spectral densty of an NRZ sgnal s gven by [11]: P max cos( R Tω / ) sn( Tω / ) PSD NRZ ( ω) = + δ ( ω) (4) 4 1 ( R Tω / π ) Tω / P max s the sgnal peak power at the nput of the fber, R the roll-off factor of the pulse shape and T the bt duraton. Wth the help of eq. (4), the XPM-nduced varance can be calculated by: σ 1 = dω (5), N + P(0) H ( ω) H ( ω) PSD ( ω) XPM XPM, j opt. flter j j = 1, j π P(0) s the average channel power, H XPM,j (ω) the transfer functon due to XPM (compare eq. (8)) and H opt,flter (ω), the transfer functon of the optcal demultplex flter.

5 Wth the help of the varance afterwards a Q-factor due to XPM can be defned: 1 0 Q XPM = (6) σ +σ 1 0 In ths formula the nfluence of the XPM effect on the spaces can be neglected, because deally the extncton s qute hgh and thus the power of 0 s very low and the varance due to XPM on the spaces s also very low (compare Fg. 3). Fgure 3: Eye dagram and hstogram of the center channel for an 8-span SSMF system wth zero resdual dsperson at the end of the system. The XPM effect can be seen as an OSNR penalty. In the followng, a mnmal OSNR of 15 db and an addtonal OSNR penalty of db have been assumed. Wth the help of the followng formula from these fgures a mnmal tolerable Q-factor can be calculated: f opt Q = OSNR (7) f el Where f el (here: f el =14 GHz) and f opt (here: f opt = 1.5 GHz) are the two-sded bandwdths of the optcal and electrcal flters. Ths leads to a mnmal allowed Q-factor of approx Arbtrary dsperson compensaton schemes In the smplfed model for the calculaton of the transfer functon, whch s presented here, some assumptons were taken. The smplfed model neglects the dsperson slope (assumng D probe D pump ). It also assumes the fber attenuaton fber length product to be very much larger than 1. Ths leads to the followng formula: H N l 1 l 1 net ( l) = ( ) 1 ( l) sn( C ) ( ω ) g γ exp ω D λ + kl exp( L ) gk [ a ( C D ) b )] (8) l= 1 n= 1 n= 1 k XPM, k α ( ) k a + a k (b ) In ths formula stands for the ndex of the probe channel and k for the ndex of the pump channel. N s the total number of spans, g net the net gan from the transmtter to the recever, γ (l) the nonlnearty constant of the l th fber, D

6 the dsperson constant of the l th fber, λ k the channel spacng between channels and k, L the length of the L th fber, α the attenuaton constant, g k the gan n front of the l th transmsson fber and ω λ C = D R 4πc (9) ω λ ( l ) ( l 1) D = D I 4πc (10) a = α ωd λ (11) b k k = ω Dλ /(4πc) (1) Wth the wavelength of the th channel λ, the speed of lght n vacuum c, the resdual dsperson at the end of the system D R and the nlne dsperson n front of the l th segment D (l-1) I. Eq. 8 has the great advantage that transmsson systems wth arbtrary dsperson compensaton schemes, whch can also vary on a span-by-span bass, can be calculated. Furthermore mxed fber systems can be assessed. The nonlnearty on the DCF has been neglected n all formulas. That s why N denotes the number of spans and not the number of segments as orgnally n [3]. It wll be shown later that neglectng the nonlnearty on the DCF does not cause a hgher error. Furthermore, smplfcatons for the sne and cosne terms have been employed f the arguments of these were small. In the followng dagram the results of the analytcal calculatons are shown. Also the values obtaned from the numercal smulatons usng the splt-step Fourer (SSF) method have been plotted. Fgure 4: Comparson of the Q-factors for an 8-span NRZ 10 Gb/s transmsson system, 50 GHz channel spacng 9 channels, DUCS (-50 ps/nm/span), Pre-DCF of -170 ps/nm. NZDSF: D = 4.64 ps/(nm km), S = ps/(nm² km), α = 0.3 db/km, γ = /(W km), P ch = 0 dbm -.4 dbm channel nput power nto the NZDSF-DCF SSMF: D = 16.7 ps/(nm km), S = ps/(nm² km), α = 0.3 db/km, γ = /(W km), P ch = 0 dbm dbm channel nput power nto the SSMF-DCF As t can be seen from Fg. 4, the analytcal model s n good agreement wth the values obtaned from the SSF smulaton. The Q-factor has been plotted versus the DCF length at the end of the system. Fg. 4 also justfes the

7 smplfcatons, whch were taken. Especally mportant s the fact that the maxma of the smulaton curve and the analytcal curve fall together. Ths s the pont of optmum performance, whch s lkely to be used n real systems. In the mxed fber system, alternatng spans of SSMF and NZDSF were cascaded. It can be seen from Fg. 4 that mxed fber systems have a clear performance advantage (approx. 30%) compared to NZDSF systems. For systems wth a hgher number of spans (e.g. 4 spans), the Q-factor curves wll be flatter and the maxmum of the NZDSF curve wll le n the range of 13. Stll for the hgher number of spans the mxed fber system wll have a consderable performance advantage Dstrbuted undercompensaton scheme (DUCS) and dstrbuted overcompensaton scheme (DOCS) The model descrbed above can further be smplfed, f a dstrbuted undercompensaton scheme (DUCS) or a dstrbuted overcompensaton scheme (DOCS) s employed and the resdual dsperson at the end of the system s zero. In ths case the factor C becomes zero. It may also be assumed that the nput power for all spans s more or less constant and compensates for the occurrng fber attenuaton. In ths case the product exp(-αl)g K n eq. (8) can be set to one. In the followng equaton t has also been assumed that the undercompensaton (or overcompensaton) s constant per span. Wth these assumptons the transmsson functon can be smplfed: H XPM, k N ( l ) ( ) ( ) net a ( D ) b k ( ω ) = g γ exp ω λ D ( l 1) (13) k I 1 l = 1 a + (b ) k The uncompensated resdual dsperson after the frst span has been denoted n that equaton wth D I Full-nlne and optmzed post compensaton scheme (FOCS) In the case of full-nlne and optmzed post compensaton (FOCS) the analytcal equaton can be smplfed even further. It s possble to leave away all terms havng to do wth the resdual nlne dsperson (D (l) = 0). Ths also requres that no Pre-DCF s used. Furthermore, t s necessary to add the complete sne and cosne terms agan, because the small sgnal approxmatons are not vald anymore. Ths leads to the followng equaton: H net ak sn C b cos C cos C ω ) = g γ N + (14) a + 4b a k k ( k Fg. 5 shows that the analytcal equaton s stll n good agreement wth the values obtaned from the SSF-smulaton. The hgher dfferences to the smulaton (compare to Fg. 4) stem from the fact that the analytcal formula has been smplfed further.

8 Fgure 5: Comparson of the Q-factors due to XPM for a 14-span NRZ 10 Gb/s transmsson system wth 1.07 dbm channel nput power nto the NZDSF and 5.77 dbm nput power nto the DCF, 9 channels, D = 4 ps/(nm km), 50 GHz channel spacng, S = 0.08 ps/(nm² km), α = 0.5 db/km, γ =.056 1/(W km) wth FOCS. If the resdual dsperson at the end of the transmsson system s assumed to be very low, the transfer functon can be smplfed beyond that. All terms havng to do wth C can be set to zero. Ths leads to the followng equaton: H k net b ( ω ) = g γ N (15) a + 4b k Wth the help of ths equaton t s now possble to fnd a scalng law for the dependence of the degradaton on the number of WDM channels. The frequency dependent part of eq. 15 s domnated by the term a² k. That s why t can be assumed that also the Q-factor depends n some way from the term a² k. a k (compare eq. (11)) tself s defned by the fber attenuaton, the dsperson constant and the channel spacng. Heurstcally the followng formula could be derved: M W = ( a + 770) 1 exp M k ( ) a + 37 (16) k M s the number of pump channels and the ω used n a k s the angular frequency of the bt rate. The scalng law was computed for a 10 Gb/s squared-cosne nput sgnal wth a roll-off factor of 0.5. Wth the help of ths scalng law t s possble to calculate the Q-factor for a two channel system (one pump and one probe) and scale t to an arbtrary number of channels by smply dvng the two-channel Q-factor by the weghtng factor W M. Important to notce s the fact that the weghtng factor s ndependent from the total number of fber spans. Q XPM M Pumpchannels = Q XPM,1 pump, (17) W M The only assumpton s that the channels need to have equal channel spacng and that they are spaced symmetrcally around the probe channel. Theoretcally, though, the scalng law can be determned for an arbtrary poston of the probe channel. As you can see from Fg. 6, the heurstcally derved curve (eq. 17) s n good agreement wth the

9 curve, whch has been derved from SSF smulatons. The valdty of eq. 16 has been checked for dfferent fber types, dfferent channel spacngs and attenuaton coeffcents. Fgure 6: Dependence of the scalng law W M on the number of pump channels for a 10 Gb/s NRZ systems wth 14 spans, 1.07 dbm channel nput power nto the NZDSF and 5.77 dbm channel nput power nto the DCF, D = 4 ps/(nm km), 50 GHz channel spacng, α = 0.5 db/km, γ =.056 1/(W km). From Fg. 6 t can also be derved that t s suffcent to smulate a system wth approx. 8 pump channels, because for a hgher number of pump channels the Q-factor wll not degrade any further. Computatonal tme for a 8-span system Orgnal Cartaxo Orgnal Bellott Eq. (8) Eq. (15)+Eq. (17) 1 s 91 s 16 s s Table 1: Comparson of the computatonal tme needed for the dfferent analytcal models evaluatng a 8-span system usng MATLAB. The weghtng factor enables to decrease the computaton tme drastcally (compare Table 1), because only a sngle transfer functon has to be calculated for one pump channel and not an ndvdual transfer functon for each pump channel. 4. CONCLUSIONS In ths paper we have presented several models for the analyss of the degradaton effects due to XPM. The analytcal formulas employ the pump-probe technque. Varous smplfcatons for the analytcal formula have been ntroduced. Wth the help of these smplfcatons a major decrease n the computatonal tme has been reached. The analytcal results have been converted nto a Q-factor and compared to the results from SSF smulatons. Furthermore, a scalng law has been presented, whch descrbes the degradaton f further WDM channels are added. The equatons are sutable for offlne network plannng as well as constrant-based routng (CBR) n automatcally swtched optcal networks (ASON), because they are very fast. 5. ACKNOWLEDGMENTS The work was fnancally supported by Semens ICN, Munch.

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