Scalability Limitations Of Optical Access And Metro .
Invited PaperScalability limitations of optical access and metro networks due to thepolarization-dependent gain of semiconductor optical amplifiersI. Roudas*a and N. AntoniadesbDepartment of Electrical and Computer Engineering, University of Patras, Rio 26500, Greece.bDepartment of Engineering Science and Physics, City University of New York, College of StatenIsland, Staten Island, NY 10314, USA.aABSTRACTThis article presents, for the first time, the derivation of approximate analytical formulae for the probability densityfunction and the cumulative density function of the optical signal-to-noise ratio variation in optical local andmetropolitan area networks due to the weakly polarization-dependent gain of cascaded semiconductor opticalamplifiers. The cumulative density function is used to calculate the outage probability and derive specifications for themaximum allowable value of polarization-dependent gain per semiconductor optical amplifier in order to achieve agiven network size.Keywords: Optical communications, optical signal-to-noise ratio (OSNR), polarization-dependent gain (PDG),semiconductor optical amplifiers (SOA), optical metropolitan area networks (MAN).1. INTRODUCTIONSemiconductor optical amplifiers (SOAs) are increasingly considered for multi-channel amplification in optical localarea networks (LANs) and metropolitan area networks (MANs) due to their low cost, small size, wide bandwidth, andcapability of operation in several wavelength bands [1]. However, SOAs can cause severe distortions of wavelengthdivision multiplexed (WDM) optical signals, i.e., due to their chirp, self-gain modulation, cross-gain modulation, selfphase modulation, cross-phase modulation and four wave mixing, resulting in a decrease of the optical networkperformance [1]. Thanks to the small size of optical LANs and MANs, the aforementioned transmission impairmentscan be tolerated or mitigated to a certain extent with appropriate design and biasing of the SOAs. Then, the dominantresidual signal degradation induced by the SOAs is related to their polarization-dependent gain (PDG).PDG leads to a variation of the optical signal-to-noise ratio (OSNR) at the output of a chain of semiconductor opticalamplifiers. This, in turn, can cause significant fluctuations of the bit error rate and outages in the performance of opticalcommunications systems and networks [2]. This effect becomes more pronounced as the number of cascaded SOAsincreases and sets an upper limit to the scalability of the optical communications systems and networks.The impact of PDG and polarization-dependent loss (PDL), in the presence or absence of polarization mode dispersion(PMD), was extensively studied, both theoretically and experimentally, in long-haul and ultra-long-haul opticalcommunications systems, e.g., [3]-[26]. In contrast, there is no analogous study for the case of optical LANs and MANsbecause polarization effects are usually considered negligible. However, if these networks contain even a small numberof SOAs concatenated in series, the impact of PDG can become significant, since commercially available SOAs forthese applications exhibit PDG typically of 0.5-1.5 dB [27].A recent comprehensive study [21] derived analytical expressions for the outage probability due to polarizationdependent loss (PDL) in long-haul terrestrial optical communications systems, where the number of cascaded PDLelements is large and the PDL can be considered to be continuously distributed along the system.This paper extends the model of [21] to the case of optical LANs and MANs with a small number of SOAs exhibitingweakly PDG. The formalism of [21] is altered to consider lumped PDG elements along the system and is simplified byusing the approximate PDL vector concatenation rule of [19], [20] for the case of weakly PDG. It is shown that theOSNR variation after N SOAs, in the case of weakly PDG, is approximately equal to the sum of N independent,*roudas@ece.upatras.gr; phone (30) 2610-996-484; fax (30) 2610-997-342.Optical Transmission Systems and Equipment for Networking V, edited by Benjamin B. Dingel,Ken-ichi Sato, Werner Weiershausen, Achyut K. Dutta, Proc. of SPIE Vol. 6388, 63880J,(2006) · 0277-786X/06/ 15 · doi: 10.1117/12.684893Proc. of SPIE Vol. 6388 63880J-1Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
uniformly distributed random variables. Therefore, it is straightforward to derive analytical formulae for the probabilitydensity function (pdf) and the cumulative density function (cdf) of the ONSR variation. The latter is used to calculatethe outage probability and derive specifications for the maximum allowable PDG per SOA as a function of the numberof SOAs which can be cascaded.The remainder of the paper is organized as follows: in Section 2, the optical communications system is described andthe equivalent system model is outlined and formulated. Section 3 presents analytical and simulation results thatdemonstrate the validity of the model and a plot of the maximum allowable PDG per SOA, as a function of the numberof SOAs which can be cascaded.2. THEORETICAL MODEL2.1. System descriptionThe system topology under study consists of a chain of N stages (Fig. 1 (a)). Each stage is composed of an SOA withaverage gain g 0,i , followed by passive optical components and short spans of optical fibers with total insertion lossli ( 0 li 1) , such that g0,i li 1 , i 1,K , N . In addition, SOAs exhibit weakly PDG. This is due to thedifference in the confinement factors of the transverse electric (TE) and transverse magnetic (TM) modes, since theactive region is not rotationally symmetric [28].In the subsequent analysis, a number of simplifying assumptions, which are frequently satisfactory in practice, areadopted: (i) SOAs are operating at the linear (i.e., unsaturated) regime. (ii) All transmission impairments other thancomponent insertion loss/gain and SOA PDG are neglected due to the small size of the network. (iii) Due to thebirefringence of interconnecting fibers, the signal state of polarization (SOP) is fully randomized between consecutiveSOAs. However, fiber PMD is not taken into account. (iv) There is no dynamic gain equalization or PDG equalizationin order to minimize cost.Conforming to the above assumptions, SOAs can be considered as equivalent partial polarizers with gain instead ofinsertion loss. Therefore, the models of [19]-[21], initially intended for the description of PDL, can be readily adaptedto the problem under study.The system topology can be reduced to the one shown in the simplified block diagram of Fig. 1 (b). The SOAs, thepassive optical components and the optical fibers are eliminated since the optical attenuation induced by the passiveoptical components and the optical fibers is compensated fully by the average gain of the SOAs. SOAs are representedby independent equivalent noise sources with total power Pn at the input of each SOA followed by a PDG element.For the derivation of OSNR statistics, two cases can be distinguished, according to whether the OSNR at the outputof the SOA chain is calculated taking into account only the ASE noise parallel to the received signal SOP or both ASEnoise components. The former definition of OSNR is better correlated to the error probability in optically preamplifieddirect-detection receivers when the ASE-ASE noise beating is negligible [18], [21]. However, in the case of CWDMoptical LANs and MANs, employing optical multiplexers-demultiplexers with 13 nm bandwidth [29], the contributionof the ASE noise orthogonal to the received signal SOP in the error probability might be significant. Therefore, wefocus here on the OSNR statistics when the total ASE noise is taken into account.2.2. Definitions and notationsi -th SOA in Stokes space are denoted by the unit Stokes vectors pˆ i , i 1,K , N . Thegains associated with these eigenaxes are Gmax,i , Gmin,i , respectively. Both the eigenaxes and the gains are assumedThe PDG eigenaxes of theindependent of frequency. The SOA PDG is defined in dB units asρi ρi ,dB 10 log ρiwhere [2]Gmax,iGmin,iProc. of SPIE Vol. 6388 63880J-2Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: )
Nth stage1st stageE(i)E (1) SOA PC 0FSOA PC 0F(a)E(i)E0 (1) n0 (1)(b)Fig. 1. (a) System topology; (b) Equivalent system model. (Symbols: SOA: semiconductor optical amplifier,PC: passive optical components, OF: optical fiber span, Ein (t ) : input signal electric field, Eout (t ) :output signal electric field, n eq ,i (t ) : equivalent input ASE noise electric field, n out (t ) : output ASEnoise electric field).Following [21], it is possible to define the following auxiliary quantities: Average gain:g 0,i PDG coefficient:Γi PDG vector in Stokes spaceGmax,i Gmin,i2Gmax,i Gmin,iGmax,i Gmin,irΓi Γi pˆ iProc. of SPIE Vol. 6388 63880J-3Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: )(3)(4)
2.3. Mathematical formulationIn the following, we will use the first-order Taylor expansion of the transmittance of N concatenated stages in terms ofΓi (cf. expressions (6), (10), (11) in [19])rTtot ( N ) 1 Γtot ( N ) sˆ O ( Γi 2 )(5)where ŝ is the launched signal SOP at the input of the chain and we defined the total PDG vector of the SOA chain asN rrΓtot ( N ) Γi(6)i 1The power of the optical signal at the output of the chain is (cf. expression (1) of [21] with slight changes in notation)rPs ,out ( N ) Ttot ( N ) Ps Ps 1 Γtot ( N ) sˆ where(7)Ps is the launched optical signal power.The power of the total ASE noise at the output of the SOA chain is given by (cf. expression (12) of [21] with slightchanges in notation)Pn ,out ,tot ( N ) NPnwhere(8)Pn denotes the total equivalent ASE noise power at the input of the individual SOAs in both polarizations.The OSNR after N stages taking into account both ASE noise components is given byR(N ) wherePs ,out ( N )Pn ,out ,totr RN 1 Γtot ( N ) sˆ (N)RN is the OSNR after N stages in the absence of PDGPRN sNPn(9)(10)2.4. OSNR statisticsIt is convenient to introduce the relative OSNR variation y defined asy R( N ) RNRN(11)pˆ i with random direction in the Stokes space on afixed unit vector ŝ is uniformly distributed over the interval [0,1] [30], [31]. Then, the relative OSNR variation y canrbe expressed as a weighted sum of N independent, uniformly distributed random variables Γ i sˆ , each taking values inIt can be shown that the length of the projection of a unit vector[]the interval Γ i , Γ i .Proc. of SPIE Vol. 6388 63880J-4Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
We distinguish the following partial cases:A. Chain of identical SOAsΓ i Γ , the pdf of the relative OSNR variation at theIn the case of identical SOAs with PDG coefficientsoutput of the N -th stage can be expressed in a variety of closed forms, e.g., [30]py ( y) where the “plus” functionN1( 2Γ ) (N ( 1)N 1 !)k 0kN 1 N y ( N 2k ) Γ k (12)x n vanishes for x 0 and equals x n for x 0 [30].The cdf of the relative ONSR variation in the case of identical SOAs with PDG coefficientsFy ( y ) 1( 2Γ )NN ( 1)N!kk 0Γi Γ is [30]N N y ( N 2k ) Γ k (13)B. Chain of non-identical SOAsIn the case of non-identical SOAs with PDG coefficientsΓi , the relative ONSR variation pdf cannot becalculated in closed form. However, it might be approximated in its central region by a Gaussian, in the case of alarge number of concatenated SOAs, due to the central limit theorem [32]. It is straightforward to show that therelative ONSR variation has zero mean and variance given by1 N 2 Γi3 i 1σ y2 (14)The cdf of the relative ONSR variation in the case of non-identical SOAs with PDG coefficientsΓi can becalculated analytically byN 1 Fy ( y ) Fu y Γi i 1 2 (15)where Fu ( u ) is the auxiliary cdf (cf. expression (26.57b) in [33])Fu ( u ) 1NN ! Γi sgn ( v )( u Γv )v CN (16)i 1In (16), Γ ( Γ1 ,K , Γ N ) , C is the hypercube{x RN;0 xi 1 for i 1,K , N } , the summation isnover the 2 vertices v of C , and we defined the hypervector sign function assgn ( v ) ( 1)m(17)Nm ν ii 1Proc. of SPIE Vol. 6388 63880J-5Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: 8)
Similar to [21], the outage probability can be approximately defined as the probability that the normalized OSNRR ( N ) / RN falls below a threshold χ (referred to as OSNR margin) and is given byPoutage Fy ( χ 1)(19)It must be stressed that the above definition of the outage probability is unconventional [2]. Formally, one needs to firstcalculate the error probability, e.g., generalizing the formalism of [22], [23]. However, this calculation is outside of thescope of the current paper and will be part of future work.The validity of the theoretical expressions is confirmed by comparison with the simulation results of [26]. To acceleratethe simulation, the random orientation of the PDG vector is chosen as follows: a constellation of n approximatelyequidistant points on the surface of the Poincaré sphere is a priori selected so the Poincaré sphere is sufficiently coveredeven with a small number of points n . The optimal configuration of n points on the surface of a sphere can becomputed using various optimization algorithms (see e.g., [34] and the references therein). If n is the number ofrealizations of the PDG vector of each SOA and N is the number of spans, there are nNpossible combinations oforientations of the PDG vectors of SOAs of consecutive stages, i.e., the simulation is repeated nNtimes.3. RESULTS AND DISCUSSIONFig. 2-3 show semi-logarithmic plots of the relative OSNR variation pdf, as given by (12), at the output of the SOAchain, for N 2 5 cascaded stages (curves). The validity of the theoretical expression (12) is checked bycomparison with Monte Carlo simulation for SOA PDG equal to ρi 0.5 dB and ρi 1 dB (points) (see [26] fordetails). It is observed that, for small values of N Γ , there is excellent agreement between the analytical and numericalresults. For larger values of N Γ , the left tail of the numerical pdf decreases more rapidly than the right tail and thenumerical results deviate from the theoretical prediction. This discrepancy is due to the fact that the actual relative{} {}NNOSNR variation lies in the interval (1 Γ ) 1 / Γ, (1 Γ ) 1 / Γ , whereas (12) takes values in the interval [ N Γ, N Γ ] . This indicates that (13), (19) yield slightly pessimistic results for the calculation of outage probability.Fig. 4 shows the maximum allowable PDG per SOA, calculated using (13), (19), as a function of the number of stagestraversed in order to achieve an outage probability of 1/17,520, which corresponds to an outage time of 30 min per year[2]. The OSNR margin χ allocated for PDG is assumed 1 dB (solid curve) and 2 dB (dash-dotted curve). Forexample, if the OSNR margin allocated for PDG is 1 dB and the maximum allowable PDG per SOA is 0.5 dB, up tofour SOAs can be cascaded in series. If the OSNR margin is increased to 2 dB, then the maximum allowable PDG perSOA, for a network containing four SOAs in series, is 0.89 dB.4. SUMMARYThis article presents, for the first time, the derivation of approximate analytical formulae for the pdf and cdf of therelative OSNR variation in optical access and metro networks comprising N cascaded SOAs with small PDGcoefficients. The cdf is used to calculate the outage probability and derive specifications for the maximum allowablePDG per SOA. For large values of N Γ i , the aforementioned expression for the outage probability becomespessimistic, so the specifications can be applied a fortiori in all cases.Proc. of SPIE Vol. 6388 63880J-6Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
0—0.50)—3 —2 —1 0yir(a)123(b)0)00)0—4—4—5—4—220—6 —4 —2 04426yiryir(c)(d)Fig. 2. Analytical pdf’s of the relative OSNR variation for (a) N 2 , (b) N 3 , (c) N 4 , (d) N 5 SOAstages plotted in logarithmic scale using (12) and verification by Monte Carlo simulation. (Symbols:curves: analysis, points: simulation). (Conditions: SOA PDG ρi 0.5 dB , number of simulationruns: 122 N ).0—0.5—2.5—3—2—1021L 1—3 —2 —1 —5—5—4—2024yir(c)Fig. 3. Same as Fig. 2 for SOA PDG ρi 1 dB .—6 —4 —2 02yir(d)Proc. of SPIE Vol. 6388 63880J-7Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx46
21.5 IEE\ x 2 dB\ \ ,.o: t:H0Number of SOAsFig. 4. Maximum allowable PDG per SOA as a function of the number of SOA stages for OSNR marginχ 1 dB (solid curve) and χ 2 dB (dash-dotted curve).5. ACKNOWLEDGMENTThe authors wish to thank Drs. K. C. Reichmann, P. P. Iannone, N. J. Frigo, and A. M. Levine of AT&T LabsResearch, and Drs. B. R. Hemenway and M. Sauer of Corning Inc., for stimulating discussions.Proc. of SPIE Vol. 6388 63880J-8Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/24/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
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