Equalizer tap length requirement for mode group delay-compensated fiber link with weakly random mode coupling

Transcription

1 Equalizer tap length requirement or mode group delay-compenated iber link with weakly random mode coupling eng Bai,2,* and Guiang Li,3,4 CREOL, The College o Optic & Photonic, Univerity o Central Florida 4 Central Florida Blvd. Orlando FL, , USA 2 Ininera Corp., 69 Wet Java Dr. Sunnyvale CA, 9489, USA 3 College o Preciion Intrument and Opto-Electronic Engineering, Tianjin Univerity, Tianjin, China 4 li@creol.uc.edu * bneng@creol.uc.edu Abtract: The equalizer tap length requirement i invetigated analytically and numerically or dierential modal group delay (DMGD) compenated iber link with weakly random mode coupling. Each pan o the DMGD compenated link comprie multiple pair o iber which have oppoite ign o DMGD. The reult reveal that under weak random mode coupling, the required tap length o the equalizer i proportional to modal group delay o a ingle DMGD compenated pair, intead o the total modal group delay (MGD) o the entire link. By uing mall DMGD compenation tep ize, the required tap length (RTL) can be potentially reduced by 2 order o magnitude. 24 Optical Society o America OCIS code: (6.423) Multiplexing; (6.233) Fiber optic communication; (6.236) Fiber optic link and ubytem. Reerence and link.. Bai and G. Li, Adaptive requency-domain equalization or mode-diviion multiplexed tranmiion, Photonic Technology Letter 24(2), (22). 2. K. Ho and J. M. Kahn, Statitic o group delay in multimode iber with trong mode coupling, J. Lightwave Technol. 29(2), (2). 3. F. Ferreira, D. Foneca, A. Lobato, B. Inan, and H. Silva, Reach improvement o mode diviion multiplexed ytem uing iber plice, Photonic Technology Letter 25(2), 9 94 (23). 4. M. Li, B. Hoover, S. Li, S. Bickham, S. Ten, E. Ip, Y. Huang, E. Mateo, Y. Shao, and T. Wang, Low delay and large eective area ew-mode iber or mode-diviion multiplexing, In Opto-Electronic and Communication Conerence (OECC), (22). 5. T. Mori, T. Sakamoto, M. Wada, T. Yamamoto, and F. Yamamoto, Low DMD Four Mode Tranmiion Fiber or Wide-band WDM-MIMO Sytem, in Optical Fiber Communication Conerence/ational Fiber Optic Engineer Conerence, (Optical Society o America, 23), paper OTh3K.. 6. T. Sakamoto, T. Mori, T. Yamamoto, and S. Tomita, Dierential Mode Delay Managed Tranmiion Line or WDM-MIMO Sytem Uing Multi-Step Index Fiber, J. Lightwave Technol. 3(7), (23). 7. S. Randel, R. Ry, A. Gnauck, M. Metre, C. Schmidt, R. Eiambre, P. Winzer, R. Delbue, P. Pupalaiki, A. Sureka, Y. Sun, X. Jiang, and R. Lingle, Mode-multiplexed 6 2-GBd QPSK tranmiion over 2-km DGD-compenated ew-mode iber, in Optical Fiber Communication Conerence, OSA Technical Diget (Optical Society o America, 22), paper PDP5C C. Antonelli, A. Mecozzi, M. Shtai, and P. J. Winzer, Random coupling between group o degenerate iber mode in mode multiplexed tranmiion, Opt. Expre 2(8), (23). 9. F. Yaman, E. Mateo, and T. Wang, Impact o Modal Crotalk and Multi-Path Intererence on Few-Mode Fiber Tranmiion, in Optical Fiber Communication Conerence, OSA Technical Diget (Optical Society o America, 22), paper OTuD.2.. F. Ferreira, D. Foneca, and H. Silva, Deign o ew-mode iber with arbitrary and lattened dierential mode delay, Photonic Technology Letter 25(5), (23).. L. Grüner-ielen, Y. Sun, J. W. icholon, D. Jakoben, K. G. Jeperen, R. Lingle, Jr., and B. Páldóttir, Few mode tranmiion iber with low DGD, low mode coupling, and low lo, J. Lightwave Technol. 3(23), (22). (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4247

2 . Introduction Mode-diviion multiplexed tranmiion uing ew-mode iber (FMF) ha been conidered a a promiing candidate to overcome the undamental capacity limit o the ingle-mode iber. However, due to mode coupling and dierential mode group delay (DMGD), complicated multiple-input- multiple-output (MIMO) equalizer i required in the coherent receiver to cancel multimode intererence. To reduce the complexity o MIMO equalizer, our main approache have been propoed. The irt approach ue computational eicient algorithm, uch a requency-domain equalization (FDE) to ave the number o multiplication per ymbol []. By uing FDE, the algorithmic complexity o the equalizer cale logarithmically with the mode group delay (MGD) intead o linearly uing time-domain equalization. However, or iber link whoe total MGD i large, FDE till require long memory length thereore high hardware complexity o FDE. The econd approach i to deign and abricate a iber with ultra-low DMGD. Although, a low DMGD iber with two mode ha been reported, or FMF upporting 4 mode or more, ultra-low DMGD i diicult to be achieved o ar. The third approach i to introduce trong coupling to reduce the channel impule repone pread (CIRS), or the equalizer tap length [2]. Due to the eective index dierence between dierent mode, FMF are naturally weakly coupled. To intentionally enhance mode coupling, artiicial perturbation have to be applied on the iber. O-center plicing wa propoed in [3] to induce mode coupling. However, the o-center plicing introduce mode-dependent lo which degrade the ytem perormance. The ourth approach i uing DMGD-compenated iber [4 7]. In a DMGD-compenated iber link, two type o iber are pliced together. They are: P-type whoe DMGD i poitive and -type whoe DMGD i negative. By center plicing multiple pair o thee two iber, the aggregate MGD can be achieved to be very low. So ar, DMGD-compenated iber which can guide 4 mode ha been demontrated [5]. I random mode coupling can be neglected, the CIRS o a ully DMGD compenated link virtually vanihe. To minimize random mode coupling, it i poible to deign a FMF with large eective reractive index dierence. However, with mode coupling, the CIRS could be much larger a wa hown in a reported experiment [7]. In thi paper, the CIRS and required equalizer tap length or DMGD-compenated link with weak mode coupling i invetigated analytically and numerically. Our reult how that by decreaing the DMGD compenation tep ize, the CIRS o a DMGD-compenated link can be igniicantly reduced even under moderate mode coupling. The remainder o the paper i organized a ollow. Section 2 decribe an analytical model o DMGD-compenated link under the weak mode-coupling aumption. Section 3 preent imulation reult o a long-ditance DMGD-compenated link and compare the required equalizer tap length between analytical and numerical reult. Section 4 provide a concluion. 2. Theory For a FMF which can guide mode, dierent mode propagate with dierent group velocitie reulting in dierent group delayτ i. The DMGD or the i th mode can be deined a the dierence between τ i and the average MGD o all the mode. Δ τ = τ τ. () i i j j= A DMGDC link comprie 2 type o FMF: P-type and -type. The DMGD o i poitive or P-type iber while negative or -type iber. In a ingle pan o a DMGDC link, two type o FMF are pliced alternatively a hown in Fig.. The adjacent two FMF ection orm a DMGD-compenation pair which i the building block o the DMGDC pan. (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4248

3 In the zero reidual DMGD cae, the accumulate DMGD or every mode group o thee two ection atiy the ollowing Eq. Δτ i and where repectively; equal to ( ) τ i ( ) i τi L Δ τ = Δ (2) Δ are the DMGD o the i th mode o the P-type and -type FMF, ( ) and L are the length o P-type and -type iber. I Δ τi = Δ τi, L which can be deined a compenation tep-ize. Although DMGD compenation or 4 mode group ha been demontrated, it may be hard to ully compenated DMGD or all the mode in many-mode iber in practice. A an approximation, analytical derivation and numerical analye in thi paper conider only zero reidual DMGD link. In addition, the delay pread due to degenerate mode i aumed to be neglected mall compared to CIRS. Thu we only invetigated delay pread between non-degenerate mode group. Comp. Step-ize P P P Fig.. A ingle pan o DMGDC link. 2. Impule repone o a DMGD uncompenated link A FMF link which ha D patial degree o reedom can be characterized uing a D D matrix H. Each element o the matrix ( h ij ) i an impule repone which lat a certain period o time. The input-output relationhip can be expreed a D j = ij i i= y h x (3) where xi and y j are input ignal in the i th mode and output ignal in the j th mode. The CIRS equal to the longet length o h ij.in the P type iber, the lowet mode i named the S mode and the atet mode i named the F mode. Due to oppoite ign o DMGD, S mode in the type iber i actually the atet mode and the F mode i the lowet one. For the cae o weakly coupled FMF, the CIRS correpond to delay pread ΔT o the FMF: ( τ τ ) Δ T = Δ Δ L (4) where Δ τ i the MGD o the S mode, Δ τ i the MGD o the F mode and L i the iber length o the link. The DMGD i denoted by Δ τ =Δτ Δ τ. Since group delay dierence between the S and F mode determine the MGD o the link, thee two mode are the mot critical pair that determine CIRS. The impule repone h decribe the linear coupling rom the S to the F mode along the link, which include direct coupling between them and indirect coupling via other mode. Due to the weak coupling aumption, indirect coupling i negligible compared to direct coupling. Another conequence o weak coupling i that all mode can be conidered a un-depleted and mode coupling can be decribed by irt order perturbation analyi a in [8].Thereore, the link can be impliied a a two-mode ytem which i illutrated in Fig. 2. (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4249

4 Fmode Smode Δτ Δτ z Fig. 2. A ingle pan o DMGD uncompenated iber. The two traight line ymbolize the F and the S mode. At location z, ignal in the S z z, L. The mode i coupled to that in the F mode with a coupling coeicient κ where [ ] tatitic o κ ( z) σ which equal to mode cattering coeicient (MSC) deined in [9]. The path delay ( 2 κ i aumed to obey normal ditribution with zero mean [8] and variance o o the coupling light via coupling location z can be calculated p T z =Δ τz+δ τ L (5) T z ) The impule repone unction h thereore can be obtained by integrating over all poible coupling path rom beginning to end a where β and L () = δ ( Δτ Δτ ) κ exp( Δ β + β ) h t t z L z j z j L dz t Δτ L Δβ Δτ β Δτβ = κ exp j t j L Δτ Δτ Δτ Δτ β are the propagation contant o the two mode, repectively; p (6) Δβ i the dierence between them. Becaue κ ( z) i nonzero or z L, the range o t can be calculated a Δτ L t Δ τl. Inide the integral, the expreion decribe a decompoed impule repone induced by a ingle coupling event at location z with coupling trength κ ( z). The exponential term denote an accumulated phae o the elected path. Since the equalizer need to cover the non-zero range o the impule repone, the required tap length (RTL) o DMGD uncompenated link, auming overampling rate o 2, can be expreed a ollowing where R i the ymbol rate. tap = 2Δ τ LR (7) 2.2 Impule Repone o a DMGD compenated pair Figure 3(a)-3(b) how a iber pan with one DMGD compenation pair. (a) P type ( Δτ ) ( ) type ( Δτ ) (b) P type ( Δτ ) ( ) type ( Δτ ) Fmode Smode F mode S mode z z Fig. 3. A ingle pan with one DMGD compenation pair with mode coupling in (a) P type ection (Cae I), (b) type ection (Cae II). Two cae are dicued in thi ection: I) mode coupling occur in the P ection; II) mode coupling occur in the ection. The impule repone or the entire link i thu given by: (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 425

5 ( () I ) ( II) () () h t = h t + h t (8) ( where I ) h () t and h II () t are the impule repone or cae I and cae II, repectively (not the impule repone o the repective ection). Following the imilar procedure a the ( derivation o Eq. (6), I ) h () t and h II () t can be expreed a the ollowing integral: L P () = δ ( Δτ ) κ exp( Δ β + ( β + β )) h t t z z j z j L L dz (9) ( I ) ( P ) ( P ) ( P ) ( P ) ( ) p + L ( II ) ( ) ( ) () = δ Δτ ( P ) κ h t t z L L z ( ) ( ) ( ) ( ) exp jδ β z+ j β Δ β Lp + β L dz () where Δ τ i overall DMGD or the P-type or the -type FMF, β ( ) and β are propagation contant or lowet and atet mode in the P-type or the -type FMF repectively, Δβ i the dierence between β and β. Subtituting Eq. (9-) into Eq. (8), we obtain h a: t t ( ) h () t = κ exp jδ β + j ( β + β P L ) Δτ Δτ Δτ () ( ) t ( ) t ( ) ( ) + κ exp ( ) + L + j β ( ) Δ + j ( ) ( β Δ β ) + jβ L Δτ Δτ Δτ P The non-zero temporal range o h can be ound to be t Δ τ L P. Similarly, h can be derived and it non-zero temporal range i Δτ t. Thereore, the overall temporal range o interaction between S and F mode i ( P ) Δτ, Δτ L P. The CIRS thu equal to 2Δ τ. L P 2.3 Impule Repone o DMGD compenated link A DMGD compenation link contain multiple pan and each pan comprie multiple compenation pair a hown in Fig.. Due to the weak coupling aumption, only irt-order coupling need to be account. Thereore, in any poible path rom mode S at the tranmitter to mode F at the receiver hould contain only one coupling event. Hence h can be calculated a a um o the intererence rom all the direct coupling path rom the S mode to the F mode, i.e., h can be expreed a the ummation o K component where K i the total number o compenation pair: ( k ) () () h t = h t. (2) K k = Each component repreent intererence rom all poible coupling path in one compenation pair. Except or the DMGD compenating pair where the coupling occur, the ignal tay in one mode when paing other pair. Thu, zero MDG i accumulated on other pair. Conequently, or the path with it coupling location in the k th pair, the path delay o the (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 425

6 coupling light via coupling location z = ( k )( + L) +Δ z ( k,2... K) the local poition Δ z ( <Δ z< L + L ) inide that pair. P = i determined by T p ( z) Δ τ Δ z < Δ z< Δ ( ) τ ( Δ z L ) <Δ z < + L = (3) Thereore, the CIRS o the DMGD compenated link i the ame a a ingle compenation pair. I an overampling rate 2 i aumed, the RTL or time domain equalization can be calculated a ollowing tap 4 τ R = Δ (4) According to Eq. (4), the RTL depend on MGD o a ection o P-type iber rather than the MGD o the whole link. It indicate that by uing mall compenation tep- ize, the RTL could be very low even with weak random mode coupling. Thi i a main reult o thi paper. 3. Simulation For implicity, P-type and -type FMF which can guide two mode are deigned and imulated. In P-type iber, the ater mode i labeled a mode. the lower mode i labeled a mode 2. Trench aited graded index proile i ued a in []. By adjuting the power indexα, DMGD can be tuned to be p/km or P-type FMF and p/km or - type acro the C-band. The index proile and the deign parameter are hown in Fig. 4. The eective index dierence between the two mode are and the chromatic diperion coeicient are 2 p/nm/km or both type o iber. a a 2 α Δ Δ 2 a = 8μm a = 9.875μm 2 a = 6.875μm 3 Δ =.45% Δ =.395% 2 a 3 Fig. 4. Trench aited graded index proile o P type ( α = 2.79 ) or type ( α = 2.96 ). Multi-ection ield propagation model wa ued to imulate two-mode tranmiion in FMF [9]. The ection length wa et to be 2m a ame a []. For other ection length uch a m or 5m, negligible dierence wa oberved. MSC wa et to be 35dB/km which i lightly higher than the iber with imilar index proile ued in [] ( 25dB crotalk or 3km iber, or 39.8dB/km). Loe were.2db/km or both mode. o crotalk wa aumed rom mode MUX/DEMUX or plicing. The imulation etup i imilar to []. In the tranmitter, two independent 28 Gbaud QPSK ignal tream were generated and multiplexed into two mode o FMF link compriing pan with a pan length o 28km. Each pan wa contructed by plicing multiple compenation iber pair. At the end o each pan, an ideal multimode EDFA wa ued to compenate loe or both mode. At the receiver, ignal in the two mode were detected ater an ideal mode de-multiplexer and two coherent receiver. Chromatic diperion wa compenated uing a tatic requency-domain equalizer beore adaptive equalization. Timedomain equalization (TDE) wa ued to equalize the ignal and etimate the impule repone o the link due to it reedom to have arbitrary tap length. (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4252

7 To veriy the analytical olution, impule repone were numerically analyzed or DMGD compenated and uncompenated link uing the leat mean quare (LMS) method. Some typical impule repone are hown in Fig. 5 For the DMGD uncompenated link, all pan were contructed uing the P-type iber with a DMGD o p/km. A hown in Fig. 5(a), impule repone h 2 i conined in a center rectangular region. The width o region i 762 tap which matche the etimated 768-tap width by uing Eq. (7). Figure 5(b)-5(c) how h 2 and h2 or the DMGD compenated link. The compenation tep-ize wa et to be 64km which wa hal the pan length. It i oberved that the length o h 2 and h2 i about 2 time horter than that or the uncompenated link which agree with theoretical analyi. Figure 5(d) illutrate the impule repone o h. The center peak correpond to the ignal rom the uncoupled path. The pedetal, which i 3 db down rom the center peak o h and 5 db down rom the magnitude o h2 and h 2 i due to multipath intererence which cover both the t > and the t < ide. In the model dicued in ection 2, the multipath intererence wa ignored due to the weak coupling aumption. Thi impliication in the analytical model can introduce error to the etimation o CIRS, particularly when MSC i high. Magnitude (db) (a) umber o Tap period Magnitude (db) (b) umber o Tap period -2 (c) -2 (d) Magnitude (db) umber o Tap period Magnitude (db) umber o Tap period Fig. 5. Magnitude o impule repone V. number o tap period or (a) h2 o 28km P- type iber link; (b) h 2, (c) h2 and (d) h o (64km(P) + 64km()) DMGDC iber link. (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4253

8 6 4 2 Q 2 (db) Lp = km 8 Lp = 2km Lp = 4km Lp = 8km 6 Lp = 6km Lp = 32km Lp = 64km Tap number Fig. 6. Q 2 (db) V. Tap number ued in LMS equalizer when MSC = 35dB/km or pan. To rigorouly analyze the RTL or the link, the receiving data are proceed by equalizer with variou tap length. The Q 2 actor a a unction o tap length i plotted in Fig. 6. A the tap length increae, more ditributed mode coupling or intererence are canceled leading to higher Q 2. When the tap length exceed the CIRS, Q 2 converge to the maximum value determined by the OSR at the receiver. Figure 6 alo how Q 2 curve or variou compenation tep ize. For each curve, the RTL can be deined a the minimum tap length o the equalizer to achieve a.db Q 2 penalty compared to maximum achievable Q 2. Thereore, RTL a a unction o compenation tep-ize can be plotted and i hown in Fig. 7 or dierent MSC. Required tap number Simulation(MSC=-25dB/km pan=) Simulation(MSC=-25dB/km pan=2) Simulation(MSC=-3dB/km pan=) Simulation(MSC=-3dB/km pan=2) Simulation(MSC=-35dB/km pan=) Simulation(MSC=-35dB/km pan=2) Analytical Compenation tep ize (km) Fig. 7. Required tap number V. compenation tep-ize or variou MSC. It i oberved that or the cae that MSC equal to 35dB/km, the numerical reult agree with analytical etimation well. According to Fig. 7, RTL can be reduced by decreaing compenation tep-ize. A imilar concluion i alo reached in [3]. Here, we derived and urther veriied the linear caling rule between RTL and the compenation tep-ize. When compenation tep-ize equal to km, the RTL i only 4 tap or pan which i /52 o (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4254

9 that or uncompenated link when MSC equal to 35dB/km. A MSC increae, the analytical model underetimate RTL. Due to tronger random coupling, indirect coupling cannot be neglected any more. For MSC equal to 3dB/km or 25dB/km, the numerical reult how that the RTL i till linearly proportional to the compenation tep ize while lope o the curve are larger when MSC i increaed. Reult or 2 pan are alo plot in Fig. 7. Due to higher cumulated mode coupling, RTL or 2 pan link i larger than the one or pan while the linear dependence on compenation tep ize till remain. It i worth to noting that by uing ever maller compenation tep-ize, CIRS o the entire link can be reduced to the level which i comparable to delay pread between degenerate mode. To achieve accurate etimation o CIRS in thi regime, the degenerate mode diperion mut be taken account. 4. Concluion The equalizer tap length requirement ha been invetigated analytically and numerically or DMGD compenated link with weak random mode coupling. Unlike the DMGD uncompenated counterpart, the RTL o DMGD compenated link depend on the MGD o a ingle P-type or -type iber ection. Although DMGD compenating iber cannot reduce CIRS to zero when mode coupling ha to be conidered in practice, by uing mall compenation tep-ize, the RTL o DMGD-compenated link can be order o magnitude horter than that o DMGD uncompenated link. Acknowledgement Thi reearch wa upported in part by the ational Baic Reearch Programme o China (973) Project #24CB34. (C) 24 OSA 24 February 24 Vol. 22, o. 4 DOI:.364/OE OPTICS EXPRESS 4255