Rigorous Evaluation of Crosstalk Requirements for Large Optical Space Switches Based on Directional Couplers

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1 Rigorous Evaluation of Crosstalk Reuirements for Large Optical Space Switches Based on Directional Couplers Luís G. C. Cancela and João J. O. Pires Dept. of Electrical and Computer Engineering, nstituto Superior Técnico, 9- Lisboa, Portugal Tel: --887, Fax: --887, Abstract The impact of homodyne and heterodyne crosstalk on transmission performance of optical space switches based on a Benes network is studied. The system power penalty is evaluated as a function of the switch size using a rigorous approach based on the Gaussian uadrature rules method. These results are used as a reference to assess the accuracy of other relevant methods, such as saddlepoint, modified Chernoff bound, and Gaussian.. NTRODUCTON The increasing demand for bandwidth, mainly due to the tremendous growth in nternet traffic, in parallel with the maturation of WD technology have placed a great emphasis on the development of all-optical networks [. n these networks, it is expected that some functions such as routing, reconfiguration, and protection can be realised in optical domain. Optical switching is an area of crucial importance in realising these functions optically. A number of technologies such as mechanical, thermooptic, directional coupler, semiconductor optical amplifier gates, and ES (micro-electronic-mechanical systems) have been proposed to implement optical switches [. Using basic building blocks (x switching elements) it is expected that larger space switch matrices (up to hundreds of input/output fibers) can be built, at least with some of those technologies. The ultimate size limitation comes from the fact that optical switches are analogue devices and impairments such as losses and crosstalk accumulates as optical signals are routed through these devices. Crosstalk is mainly due to imperfect isolation of the switching elements. This imperfection originates leakage signals, whose wavelengths can be identical or different from the desired signal resulting in homodyne or heterodyne crosstalk, respectively. Contrary to signal attenuation, crosstalk can not be resolved by optical amplification, and in this way becomes a major problem to face in designing large space switch matrices. t is, therefore, reuired an accurate modelling of the impact of crosstalk on signal uality, as well as a detailed analysis of the scaling of component crosstalk with switch size. The impact of crosstalk on system transmission performance has been studied in the context of optical networks using different approaches. These include, namely, the Gaussian [, the modified Chernoff bound (CB) [, and the saddlepoint (SP) [. The purpose of this paper is to analyse the constraints imposed by crosstalk on the design of optical switches using a Gaussian uadrature rules (GQR) method, since this techniue allows the evaluation of the average bit error rate (BER) with any desired accuracy [. t is considered a space switch matrix with an architecture based on a Benes network and implemented using directional couplers. A detailed analysis of the signal and crosstalk propagation through the switch, and a comparison between the and other published methods are also provided.. ODEL OF THE SGNAL AND CROSSTALK The Benes network has been considered by many authors [7 for designing optical space switches based on guided-wave directional couplers. This network despite being rearrangeable non-blocking offers the advantage of reuiring fewer switching elements than other structures [8. The Benes network reuires = log N stages (N is the number of inputs/outputs) with N switching elements in each stage [8. An example of an 8x8 Benes optical switch is given in Figure, in which all the switching elements are in bar state, and the desired signal (black arrows) is routed from the first input to the first output. N P U T S 7 8 x switching element Fig. Structure of a 8x8 Benes optical switch. t is assumed that each switching element is constructed using a x directional coupler, characterised by the switching element crosstalk X c. This crosstalk is due to undesirable coupling between the two waveguides of the coupler and gives rise to multiple leakage paths between any input and any output in a large optical space switch. t can be shown that the electrical field at the selected output for a NxN Benes switch can be given by m m m () = α s() + ( c ) ( c ) km() Et E t j X X E, t () m= Signal path Crosstalk path k = O U T P U T 7 S 8 with α = ( X c ), Es ( t) the electrical field entering the switch at the selected input, and Ekm, ( t) the interfering electrical fields arising from the others N optical channels (inputs of the switch), as well as from leakage from

2 the selected input. The interfering fields can be categorised as primary ( m = ), or higher order ( m > ). For the case of the 8x8 Benes switch there are five primary interfering fields, which are represented by grey arrows in Figure. The first term of the right hand side of () represents the field of the selected data signal and the second one the crosstalk terms due to leakage paths inside the switch. Euation () assumes that all the switch inputs are active and the state of polarisation for the signal and crosstalk terms is identical (i.e. worst case situation). For an intensity modulated single-freuency laser diode source, Es ( t) and Ekm, ( t) can be given by [ ( ) exp ( ) E () t = Pd () t exp j ω t + φ () t () s s s s [ E () t = Pd () t j ω t + φ () t () km, km, km, km, respectively, P is the optical power, ω is the optical freuency, φ( t ) is the laser phase noise, and dt ( ) is the binary data seuence taking on values of and, with eual probability, in a bit period T b. The subscript s of these parameters states for the selected signal, while the subscript k,m corresponds to the crosstalk terms. Assuming that the switching element crosstalk is small enough, so that only primary interferes are relevant, the current resulting from the photodetection of the lightwave given by (), can be approximated as it () = sα ds() t + ds() tε rs, k() t dk() t () k = ε = Xc ( Xc) () and r () t = sin ω t + φ () t. () ( ) sk, sk, sk, For simplicity of notation, in the above euations the primary interferes have been characterised by the subscript k instead of k,. n () s = R λ P is the average photocurrent with R λ the photodiode responsivity, ωsk, = ωs ωk is the freuency spacing between the selected signal and interfering signals and φsk, () t = φs() t φk() t is the laser phase noise difference between the same signals. The first term on the right hand side of () is the selected data signal and the second one is the signal-crosstalk beat noise. This noise results from phase-to-intensity noise conversion and is an incoherent process, since the different interfering signals come from independent laser sources. Therefore, the laser phase noise difference, φ sk, ( t), may be treated as a uniformly distributed random process [9. The photocurrent described by () is amplified and filtered by a baseband filter with impulse response hr (), t that is assumed to provide a % raised cosine eualization. The decision variable after sampling the filtered signal at instant t is given by V = i t hr t + Nth = Z+ Nth. (7) n (7) the asterisk denotes convolution, Z is a random variable that includes the influence of signal-crosstalk beat noise, and N th is a filtered version of the receiver circuit noise. Here, it is assumed that the shot noise due to the signal can be neglected and the random variable N th is Gaussian with zero mean and variance σ th. Assuming that dt ( ) is modelled as a rectangular pulse, as well as a perfect extinction ratio, the random variable Z, when d s =, can be written, using () and (7), as Tb Z = AS + ε Y() τ hr( t τ) dτ (8) Y () τ = r () τ = sin ω τ + φ () τ (9) [ sk sk s, k,, k = k = with the number of interfering terms in the logical state one and As = α s. On the other hand, when d s =, that variable reduces to Z =.. THE OENT GENERATNG FUNCTON OF THE BEAT NOSE A common procedure to evaluate the impact of crosstalk on transmission performance comprises the BER calculation by using the probability density function (PDF) of the decision variable. n the present case this procedure involves a number of difficulties. Actually, as can be concluded from (8) and (9) the random variable Z depends on the beat noise components r sk,, that have a sinusoidal dependence on the uniformly distributed random process φ sk, (), t on the number of interfering terms that are on the logical state one, as well as on the baseband filtering effects. n the presence of heterodyne crosstalk, the filter is also responsible for the partial or total elimination of the signalcrosstalk beat noise, depending on the ratio between the beat freuency ω sk, and the filter bandwidth. These problems make very difficult to calculate the PDF of Z and as a conseuence the PDF of the decision variable V. An alternative methodology can be applied to describe the statistics of the decision variable. This methodology involves the use of the moment generating function (GF) and will be applied in this paper. n order to simplify the calculation of the GF of the random variable Z it is assumed that the baseband filter bandwidth is wider than the signal-crosstalk beat noise spectrum [9, and that all the optical channels have the same freuency separation ω, i.e. ωsk, = ω, k. Under these assumptions it can be shown that the GF of Z is given by () ( ) ( ) ( ) Z s = exp sas s ε Asu s ε Asu () (.) represents the modified Bessel function of the first kind of order zero and

3 Tb u = cos ωτ hr t τ dτ () Tb u = sin ωτ hr t τ dτ. () V. BER EVALUATON ETHODS n this section, the is introduced in order to evaluate the BER in the presence of signal-crosstalk beat noise and Gaussian noise. A brief description of other published methods is also provided, so that the results obtained with different techniues can be compared. A. Gauss uadrature rules method As seen before, the decision variable V depends on the random variables Z and N th. The last one is modelled as a Gaussian variable, while the former one is characterised by taking on + possible values depending on the number of interfering terms that are in the logical state one. n this way, the average BER can be obtained by averaging the conditional BER for interferes over all possible values of Z giving + BER = ( ) erfc D + () = σ th Z D ( ) = + erfc pz ( ) dz. () σ th n the above euations () denotes the conditional BER when the data signal is in the logical state one, erfc( x ) is the complementary error function, pz ( ) is the PDF of Z, and D is the decision threshold level. The evaluation of () reuires the numerical computation of an integral that depends on the PDF of Z, which is not known explicitly. To overcome such difficulty one can apply the [. This method is based on the calculation of the first N Q + ( N Q is the number of terms used in the ) moments of Z, and permits to write () in the following form [: ( ) N Q x j D wjerfc j= σ th () w j and x j are the weights and abscissas, respectively, of the integration method. t should be noted that the integral in () can be evaluated with any desired accuracy by using an appropriate number of moments. B. odified Chernoff bound and saddlepoint The modified Chernoff bound and the saddlepoint have been used by different authors to analyse the impact of crosstalk on optical networks [, [, [. Both methods have in common the fact that rely on the GF to statistically characterise the signal-crosstalk beat noise and Gaussian noise. Therefore, to apply these methods it is reuired the GF of the random variable V. For d s = it can be shown that the GF of this random variable is given by = Z s th s () ( s ) V = Z ( s ) is given by () and th ( s ) is the GF of the random variable N th [. For d s =, the GF of V reduces to V ( s) = th( s), since it is assumed a perfect extinction ratio. To derive the CB, it is applied a procedure similar to that described in [ and [, giving BER CB = V ( ) ( ) s exp sd + πσ th s (7) V ( ) ( ) s exp s D s s > and s >. The parameters s and s are evaluated so that the right hand side of (7) is minimised. For the case of the SP, the average BER can be written as [ exp BER π [ Φ ( s) Φ ( s) [ i exp + Φ i ( s i ) ln V ( s i ) s i D lns i [ Φ ( s) Φ ( s ) (8) = and Φ i ( s i ) is the second derivative of Φ i ( s i ), with i =,. The computation of the parameters s and s follows from ( s ) ( s ) Φ =, with s > Φ =, with s <. (9) C. Gaussian and power penalty The Gaussian due to its simplicity has been used extensively in crosstalk studies, e.g. [. Using this Z can be treated as Gaussian and () reduces to ( ) A erfc S D ( σs, + σth) () σ s, denotes the signal-crosstalk beat noise power. The Gaussian despite overestimating the BER permits to analyse, easily, how the switch matrices, in the presence of homodyne crosstalk, scale with respect to the number of inputs/outputs N, by using the euation [ N( Xc δ) log, ( δ )( X ) δ c Q Xc () Q for a BER of 9 and δ is the power penalty due to the homodyne crosstalk.

4 V. NUERCAL RESULTS t is assumed that all the optical channels present at the input of the switch matrix operate at. Gbit/s, and the GQR method, that is utilised as a reference to judge the precision of other methods, uses N Q = to ensure an appropriate accuracy [. Furthermore, it is considered a fixed decision threshold set at midway between the "one" and "zero". Figure shows the power penalty at a BER of 9, due to homodyne crosstalk, for a x switch, considering the referred methods, as well as the Gaussian. As expected, the Gaussian is very inaccurate and the CB and the SP predict very precisely the power penalty. CB SP Gaussian Switching Element Crosstalk (db) Fig. Power penalty for a x switch. Figure shows the power penalty, for a x switch. n this case, the Gaussian becomes more accurate, (because as the number of crosstalk terms increase the distribution of the decision variable rapidly becomes Gaussian) and the CB becomes less tight for larger power penalties. This discrepancy can be explained by noticing that the CB is a more simplistic approach than the SP or the [. CB SP Gaussian Fig.. Same as Fig., but for a x switch. Figure shows the dependency of the power penalty, due to homodyne crosstalk, on the switch size. The curves represented in this figure are obtained with the. These results show that a switch matrix should be possible if the switching element crosstalk is kept less than - 8 db, assuming db power penalty at a BER of 9. For the same power penalty a x switch can tolerate a switching element crosstalk 8 db higher. x x 8x8 x x Fig.. Power penalty with the switch size as a parameter. Figure represents the switching element crosstalk as a function of the switch size, considering both the optimum and the middle decision threshold level, for a db power penalty. The results are obtained with the. From this figure it can be observed that the optimisation of the threshold level can relax the switching element crosstalk by at least db. n Figure are also represented the results obtained with euation (). This euation constitutes an expedite way to calculate the maximum switch size, and can be undoubtedly applicable to switches larger than x. For smaller switches it overestimates significantly the real value as can be concluded from Figure iddle decision threshold - Optimum decision threshold - Euation () - Gaussian - Switch size Fig.. Switching element crosstalk versus switch size for a db power penalty.

5 Finally, the influence of the channel freuency spacing in the behaviour of heterodyne crosstalk can be analysed using Figure. This figure shows the power penalty, at the filter output, as a function of channel freuency spacing, ω, for a x and for a x switch, considering a switching element crosstalk of - db. From this figure, it can be concluded that the and the SP give very similar results, while the CB overestimates the power penalty, in particular for the x switch. Also, it can be observed that when the channel freuency spacing increases the power penalty due to heterodyne crosstalk decreases. For example, when the channel freuency spacing increases from GHz to GHz, the power penalty decreases from. db to. db for the x switch, while for the x switch a. db reduction is obtained. n the limit, when all the signal-crosstalk beat noise falls completely outside the receiver bandwidth, the heterodyne crosstalk is completely filtered out x x SP CB... Channel freuency spacing (GHz) Fig. Power penalty versus channel freuency spacing, considering a switching element crosstalk of - db. V. CONCLUSONS This paper has analysed the limitations imposed by homodyne and heterodyne crosstalk on the design of optical space switches based on a Benes network. The power penalty evaluation has been based on the Gauss uadrature rules method, which has been used as a reference to judge the precision of other methods such as modified Chernoff bound, saddlepoint, and Gaussian. The results have shown that the saddlepoint is extremely accurate for all the situations considered in this paper, and that the modified Chernoff bound is also very tight for a number of cases of practical interest. However, when the switch size and the switching element crosstalk increases the modified Chernoff bound tends to slightly overestimate the power penalty. The Gaussian, as it is well known, overestimates the power penalty, specially for small switch sizes. Nevertheless, a simple euation, that allow to analyse the scaling of crosstalk with switch size, can be deduced from it and has been proven to be very accurate for switch sizes larger than x. When the switch input channels operate at different wavelengths the crosstalk reuirements can be significantly relaxed. Actually, when the channel freuency separation is larger than the receiver electrical bandwidth, the impact of heterodyne crosstalk becomes insignificant. V. REFERENCES [ O ahony et al., The Design of a European Optical Network, J. Lightwave Technol., vol., pp , ay 99. [ A. Pattavina et al., Techniues and Technologies towards All- Optical Switching, Optical Networks, vol., pp. 7-9, April. [ H. Takahashi et al., mpact of Crosstalk in an Arrayed- Waveguide ultiplexer on NxN Optical nterconnection, J. Lightwave Technol., vol., pp. 97-, June 99. [ L. oura et al., odelling of nterferometric Crosstalk in Optical Networks, in Globecom 9, 99, pp. -7. [. onroy, and E. Tangdiongga, Performance Evaluation of Optical Cross-Connects by Saddlepoint Approximation, J. Lightwave Technol., vol., pp. 7-, arch 998. [ J. O Reilly, and J.R.F. da Rocha, mproved Error Probability Evaluation ethods for Direct Detection Optical Communications Systems, EEE Trans. nformation Theory, vol. T-, pp , November 987. [7 G. Wilfong et al., WD Cross-Connect Architectures with Reduced Complexity, J. Lightwave Technol., vol. 7, pp. 7-7, October 999. [8 C. Lu, and R. Thompson, The Double-Layer Network Architecture for Photonic Switching, EEE Trans. Lightwave Technol., vol., pp.8-89, August 99. [9 J. Gimlett, and N. Cheung, Effects of Phase-to-ntensity Noise Conversion by ultiple Reflections on Gigabit-per-Second DFB Laser Transmission Systems, J. Lightwave Technol., vol. 7, pp , June 989. [ K. Schumacher, and J. O Reilly, Relationship Between the Saddlepoint Approximation and the odified Chernoff Bound, EEE Trans. Comm., vol. 8, pp. 7-7, arch 99.

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