accurate modeling of optical multiplexer/demultiplexer...

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 6, JUNE 2002 921

Accurate Modeling of OpticalMultiplexer/Demultiplexer Concatenation

in Transparent Multiwavelength Optical NetworksI. Roudas, Member, IEEE, N. Antoniades, Member, IEEE, T. Otani, T. E. Stern, Life Fellow, IEEE,

R. E. Wagner, Member, IEEE, and D. Q. Chowdhury, Member, IEEE

Abstract—This paper presents an accurate theoretical modelfor the study of concatenation of optical multiplexers/demulti-plexers (MUXs/DMUXs) in transparent multiwavelength opticalnetworks. The model is based on a semianalytical technique forthe evaluation of the error probability of the network topology.The error-probability evaluation takes into account arbitrarypulse shapes, arbitrary optical MUX/DMUX, and electroniclow-pass filter transfer functions, and non-Gaussian photocurrentstatistics at the output of the direct-detection receiver. To illustratethe model, the cascadability of arrayed waveguide grating (AWG)routers in a transparent network element chain is studied. Theperformance of the actual network is compared to the perfor-mance of a reference network with ideal optical MUXs/DMUXs.The optical power penalty at an error probability of 10 9 iscalculated as a function of the number of cascaded AWG routers,the bandwidth of AWG routers, and the laser carrier frequencyoffset from the channel’s nominal frequency.

Index Terms—Error analysis, optical filters, optical receivers.

I. INTRODUCTION

I N TRANSPARENT multiwavelength optical networks,each lightwave signal may be optically multiplexed/de-

multiplexed several times during propagation from its sourceto its destination [1]. Optical multiplexers/demultiplexers(MUXs/DMUXs) exhibit nonideal amplitude and phasetransfer functions within the optical signal band. That is, theiramplitude transfer functions might present passband curvature,tilt, and ripple. In addition, their phase transfer functionsmight not vary linearly with frequency. These impairments areenhanced when a large number of these devices are cascadedtogether. Consequently, optical MUX/DMUX concatenation

Manuscript received May 4, 2001; revised February 21, 2002. This work wassupported by the Defense Advanced Research Projects Agency under MDA972-95-3-0027 and performed as a part of the MONET consortium.

I. Roudas was with the Telcordia Technologies, Red Bank, NJ 07701 USA.He is now with Corning Inc., Somerset, NJ 08873 USA.

N. Antoniades was with the with the Department of Electrical Engineering,Columbia University, New York, NY 10027 USA. He is now with Corning Inc.,Somerset, NJ 08873 USA.

T. Otani was with the Department of Electrical Engineering, Columbia Uni-versity, New York, NY 10027 USA. He is now with the KDD Research andDevelopment Laboratories, Saitama 356-8502, Japan.

T. E. Stern is with the Department of Electrical Engineering, Columbia Uni-versity, New York, NY 10027 USA.

R. E. Wagner was with the Telcordia Technologies, Red Bank, NJ 07701USA. He is now with Corning Inc., Corning, NY 14831 USA.

D. Q. Chowdhury is with Corning Inc., Corning, NY 14831 USA.Publisher Item Identifier S 0733-8724(02)05399-9.

causes signal attenuation and distortion, i.e., intersymbolinterference (ISI), and eventually limits the maximum numberof optical network elements that can be cascaded.

In the past, the cascadability of different types of opticalMUXs/DMUXs was studied both theoretically [2]–[18] andexperimentally [11], [18]–[24] for several bit rates, channelspacings, modulation formats, and other system considera-tions. Most attention was concentrated on arrayed waveguidegrating (AWG) routers [6], [11], [13], [17], [19]–[22], [24],MUXs/DMUXs composed of multilayer interference (MI)filters [4], [5], [8], [10], [14], [17], [18], and MUXs/DMUXscomposed of fiber Bragg grating filters [14], [15], [23]. Generalstudies, not bound to a specific optical MUX/DMUX type, butbased on arbitrary transfer functions, also exist, e.g., [7] and[16].

Previous theoretical studies contain approximations con-cerning the network topology and the network performanceevaluation. More specifically, [4]–[6], [9], [10], and [12]–[18]study isolated chains of optical MUXs/DMUXs instead of re-alistic network topologies. In the latter case, the signal spectralclipping due to the concatenation of optical MUXs/DMUXsintroduces an excess loss which changes the operating pointof optical servo-controlled attenuators and erbium-dopedfiber amplifiers (EDFAs) in the network elements. There-fore, it is necessary to study the interaction between opticalMUXs/DMUXs, optical servo-controlled attenuators andEDFAs, and its impact on the power levels of the opticalsignal and amplified spontaneous emission (ASE) noise in thenetwork.

In addition, little effort was devoted to evaluate accurately thenetwork performance degradation due to the concatenation ofoptical MUXs/DMUXs. References [2], [4], [5], [7], [9], [10],[12]–[14], [16], and [17] compute the eye pattern at the outputof the direct-detection receiver by noiseless simulation and usethe eye opening as a qualitative criterion of the network per-formance. Other authors [3], [6], [8], [11], [18] include ASEnoise in their calculation with various degrees of accuracy, butassume Gaussian receiver noise statistics at the output of the di-rect-detection receiver. This assumption is strictly valid only forthermal-noise-limited receivers, but not for the most commoncase of optically preamplified direct-detection receivers.

This paper presents an accurate theoretical model, free ofboth aforementioned limitations, for the study of the concatena-tion of optical MUXs/DMUXs in transparent multiwavelengthoptical networks. The key features of the proposed model are

0733-8724/02$17.00 © 2002 IEEE

922 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 6, JUNE 2002

the following: 1) it is based on a transparent chain of wave-length add-drop multiplexers (WADMs) or wavelength selec-tive cross-connects (WSXCs) and 2) it uses the error probabilityas a criterion for the cascadability of optical MUXs/DMUXsin the network. The error-probability evaluation takes into ac-count arbitrary pulse shapes, arbitrary optical MUXs/DMUXsand electronic low-pass filter (LPF) transfer functions, and non-Gaussian photocurrent statistics at the output of the direct-de-tection receiver. For the error-probability evaluation, the modelcombines a semianalytic method [25] with an accurate statis-tical analysis of square-law detection [26]–[41]. The compu-tation involves several steps, including expansion of the ASEnoise impingent upon the photodiode in Karhunen–Loève series[42]–[44], numerical solution of the associated homogeneousFredholm equation of the second kind [44]–[46], diagonaliza-tion of a quadratic form [37], [45], and asymptotic evaluationof the probability density function (pdf) at the output of the di-rect-detection receiver from the characteristic function using themethod of steepest descent [26], [38], [41], [46].

To illustrate the model, the concatenation of AWG routers ina transparent network element chain is studied. The error proba-bility is calculated as a function of the number of AWG routers,the optical bandwidth of AWG routers, and the laser carrier fre-quency offset from the channel’s nominal frequency. These re-sults are compared to the performance of a reference networkcomposed of optical MUXs/DMUXs with rectangular ampli-tude response and zero phase response.

The following conclusions are drawn.

1) It is possible to concatenate an arbitrary number of con-ventional AWG routers, despite the inherent passbandcurvature of these devices, by increasing the equivalentnoise bandwidth of individual AWG routers for a givenbit rate at the expense of channel spacing.

2) In the case that the equivalent noise bandwidth of the indi-vidual AWG routers is much larger than the bit rate, whenthe number of cascaded AWG routers increases, initiallythe network performance is improved because the opticalbandwidth narrowing due to AWG router concatenationreduces the power of the ASE noise at the direct-detec-tion receiver without essentially distorting the signal. Asthe number of cascaded AWG routers continues to in-crease, the effective equivalent noise bandwidth of theAWG router cascade becomes comparable to the bit rateand the network performance slowly degrades due to theincrease of ISI.

3) Using the current model, it is possible in principle to max-imize the network performance for a given number ofconcatenated AWG routers by jointly optimizing the op-tical bandwidth of the individual AWG routers and theelectronic bandwidth of the receiver.

4) Finally, for narrow optical MUX/DMUX bandwidths,the network performance can be maximized by mis-aligning the laser carrier frequency from the opticalMUXs/DMUXs center frequency.

The rest of the paper is organized as follows. Section IIpresents the theoretical model for the study of concatenation ofoptical MUXs/DMUXs in transparent multiwavelength optical

networks. Section III presents a study case for the concatenationof AWG routers in a chain of transparent network elements.The details of the calculations are given in Appendixes A–C.

II. THEORETICAL MODEL

This section is divided into five parts. The first topic that istreated is the representation of an optical network topology bytwo separate equivalent channels for the signal and the ASEnoise, respectively. For illustrative purposes and without loss ofgenerality, a chain of optical network elements is studied, sinceit is the elementary building block of more complex networktopologies. In addition, a reference network is defined and isused for comparison with the actual network.

The second topic of this section is the mathematical descrip-tion for the signal and ASE noise propagation through theirequivalent channels and the derivation of an analytical expres-sion for the photocurrent at the output of the direct-detectionreceiver. This treatment highlights the underlying mechanismsfor performance degradation and provides a rigorous definitionfor the excess loss resulting from the optical MUXs/DMUXs.In addition, we introduce the notion of the excess noise factor,originating from a decrease in the insertion loss of servo-con-trolled optical attenuators.

The third topic of this section is the evaluation of the statis-tics of the photocurrent at the output of the direct-detection re-ceiver. The fourth topic of this section is the description of thesemianalytical method for the evaluation of the average errorprobability. The fifth topic of this section is the derivation of ananalytical expression for the error probability of the referencenetwork. We begin our discussion with a description of the net-work topology.

A. Network Representation

The network topology is shown in Fig. 1(a). It consists of achain of equidistant network elements separated byfiber spans of loss . All fibers carry wavelength division mul-tiplexed (WDM) optical signals. One optical signal, at nominalcarrier frequency , is added at the first network element andpropagates through the entire chain (dotted line).

An exhaustive study of network element architectures is be-yond the scope of this paper. Fig. 1(b) shows an example ar-chitecture of a reconfigurable WADM proposed in the multi-wavelength optical networking (MONET) project [47]. It con-sists of two EDFAs, an optical MUX/DMUX pair, an opticalswitch fabric for signal adding/dropping, and servo-controlledattenuators for power equalization. An example architecture ofan eight wavelength 4 4 WSXC can be found in [48]. It canbe shown that in the WSXC the optical signal passes from thesame number of optical components as shown in Fig. 1(b).

The optical transmitter and receiver, notionally, are not partsof the optical transport layer [1], so they are not included inFig. 1(b). Fig. 1(c) shows the structure of an optically pream-plified direct-detection receiver. The EDFA at the input of the

th network element is playing the role of the receiver op-tical preamplifier and is omitted in Fig. 1(c). The remainder ofthe receiver consists of a polarizer (optional), a tunable optical

ROUDASet al.: ACCURATE MODELING OF OPTICAL MUX/DMUX CONCATENATION 923

(a)

(b) (c)

Fig. 1. (a) Chain network topology. Dotted line indicates the optical path of the signal under study (NE: network element;� : signal’s nominal carrier wavelength).(b) Architecture of a WADM (EDFA: Erbium-doped fiber amplifier; DMUX: demultiplexer; MUX: multiplexer; VOA: servo-controlled attenuator). (c) Blockdiagram of an optically preamplified direct-detection receiver (Pol: Polarizer; BPF: tunable optical BPF with low-pass equivalent transfer function denoted byH (f); LPF: electronic low-pass filter with transfer function denoted byH (f); T : bit period;� propagation delay;� : sampling instant).

bandpass filter (BPF) with transfer function denoted by ,a photodiode, an electronic LPF with transfer function denotedby , a sampler, and a decision device, whose threshold isautomatically set to minimize the error probability. The signalis sampled at times , where , isthe bit period, is the propagation group delay through the net-work, and is the specific sampling position withinthe bit that minimizes the error probability.

For simplicity, we consider the case of single channel trans-mission. It is assumed that signal attenuation induced by the op-tical fibers is compensated fully by the EDFAs at the input ofthe network elements. The generalization to the case when thesignal attenuation induced by the optical fibers is not compen-sated fully by the EDFAs at the input of the network elements(e.g., in the case of WDM transmission and EDFA gain ripple) isstraightforward. Signal attenuation within the network elementsarises from the insertion loss of the optical components and theband-limiting operation of the optical MUXs/DMUXs. To iso-late the impact of the filtering by the optical MUXs/DMUXs,we assume that the network elements are designed so that theEDFAs at the output of each network element compensate fullythe total intranode insertion loss, if the optical MUXs/DMUXsare replaced by optical attenuators with the same insertion lossas the optical MUXs/DMUXs exhibit at their center frequency.Any additional signal attenuation caused by signal filtering bythe optical MUXs/DMUXs is defined as excess loss [4]. The ex-cess loss is compensated by adjusting the insertion loss of theservo-controlled attenuator following the MUX/DMUX pair inorder to keep the average signal power at the input of the boosterEDFA constant. Without loss of generality, it is assumed thatthe excess loss per node is within the operating range of theservo-controlled attenuators.

The above assumptions enable the reduction of the networktopology to the simplified block diagram shown in Fig. 2(a).All intranode optical components and fibers from the initial

Fig. 1(a) and (b) is eliminated.1 Optical MUXs/DMUXs are rep-resented by filters with transfer functions normalized to unityand denoted by , . The receiver tunableoptical BPF with transfer function is also included. Theservo-controlled attenuators are represented by gain elementswith gain , , equal to the excess loss causedby the optical MUX/DMUX pair of the th node. The EDFAsare represented by independent white Gaussian noise sources

, . The electric fields at the output of thetransmitter and the input of the receiver are denoted byand , respectively.

Inspection of the simplified block diagram shown in Fig. 2(a)reveals that the signal passes through MUXs/DMUXs and

servo-controlled attenuators, which compensate for the ex-cess loss caused by the first MUXs/DMUXs, but notfor the excess loss resulting from the th DMUX at the dropsite. Consequently, the excess loss cannot be fully compensatedas assumed by previous authors [4], [5].

The ASE noise generated by the EDFAs is amplified bythe effective gain provided by the last servo-controlledattenuators and is filtered by the last MUXs/DMUXs.The effective gain provided by the servo-controlled attenuatorsin the ASE noise path causes an additional ASE noise amplifi-cation. This effect is described by an excess noise factor, whichis introduced for the first time here and will be defined below.

Fig. 2(b) shows the final block diagram of the networktopology and direct-detection receiver. The input signaland equivalent input ASE noise are propagating throughdifferent channels which are represented by the transferfunction and the equivalent transmittance ,respectively.

1This paper focuses exclusively on the impact of optical MUX/DMUXconcatenation on the network performance. The interplay between bandwidthnarrowing due to optical MUX/DMUX concatenation and fiber effects, whichcause spectral broadening (e.g., self-phase modulation), is not considered hereand will be addressed in future work.


(a)

(b)

Fig. 2. (a) Simplified block diagram of the network topology shown in Fig. 1(a) (E (t): input signal;g : ith attenuator gain;n (t): ith EDFA ASE noise). (b)Final block diagram of the network topology shown in Fig. 1(a) (H (f) = transfer function of the signal channel;T (f) = equivalent transmittance of the ASEnoise channel;n (t) = equivalent input ASE noise).

The equivalent input ASE noise is defined here as afictitious white Gaussian noise source that, if placed at the inputof the network, will produce the same noise at the receiver inputas the one provided by all the optical amplifiers in the network.The equivalent transmittance can be expressed as a func-tion of the last optical MUXs/DMUXs transmittancesand the gains of the last servo-controlled attenuators.

We will compare the performance of the actual network el-ement chain with the performance of a reference network ele-ment chain, which is defined for the first time here as follows.All optical MUXs/DMUXs are replaced with perfectly aligned,ideal optical MUXs/DMUXs exhibiting rectangular amplitudetransfer function, and linear or zero phase transfer function. Theamplitude transfer functions have the same insertion loss at thecenter frequency and the same equivalent noise bandwidthas the individual actual optical MUXs/DMUXs. In addition, itis assumed that the ideal optical MUXs/DMUXs do not intro-duce any ISI. Finally, it is assumed that there is a polarizer infront of the tunable optical BPF in the optically preamplifiedreceiver aligned along the received signal polarization and thatthe receiver electronic LPF is an integrate and dump filter withimpulse response duration equal to the bit period.

It is well known [49] that an optimum direct-detection re-ceiver is composed of a matched optical BPF and a wide-bandelectronic LPF. The reason for the adoption of a suboptimumreference network as a fair standard for comparison is thatcommercially available optical MUXs/DMUXs are designedin order to obtain maximally flat passband and minimumchromatic dispersion rather than matched filter characteristics.Nevertheless, the choice of reference network does not affectthe essence of the results in Section III, but only the relativepower penalties in Figs. 9–11.

B. Signal and Noise Representation

For mathematical convenience, a low-pass equivalent repre-sentation [49] of the optical signals and components is used inthe following. To distinguish vectors from scalars, we identifyvector quantities with boldface type. Matrices are also denotedby boldface type. The distinction should be clear from the con-text.

The electric field at the output of the transmitter can bewritten as , where is the average signalpower, is the normalized Jones vector for the transmitted stateof polarization, and is the normalized complex envelopethat can be written as

(1)

where is the amplitude of the normalized complex envelope(taking values in the range ), is the offset of the

transmitter’s actual carrier frequency from the channel’s nom-inal frequency , is the deviation of the transmitter’s in-stantaneous frequency from the transmitter’s actual carrier fre-quency (i.e., chirp), and is the transmitter’s phasenoise.

The received electric field vector can be written as, where is the normalized Jones vector for the

received state of polarization, and is the normalized com-plex envelope of the received electric field that can be writtenas a function of the normalized complex envelope of the trans-mitted electric field

(2)

where is the signal effective gain provided by the reduc-tion of the insertion loss of the servo-controlled attenua-


tors, specifically , denotes convolution, andis the impulse response of the cascade of optical

MUXs/DMUXs and the receiver’s tunable optical BPF

(3)

where the operator denotes the inverse Fourier transform.The th attenuator gain is equal to

(4)

where and denote the total average input/outputpowers to/from the optical MUX/DMUX pair of theth node,respectively.

The received ASE noise is unpolarized and can be analyzedinto two orthogonal states of polarizationand , parallel andperpendicular to the received signal’s electric field, respectively

(5)

where and are independent identically distributed (i.i.d.)complex bandpass Gaussian processes with two-sided powerspectral density (psd)

(6)

where is the two-sided equivalent input ASE noise psd foreach polarization and is the transmittance of the equiva-lent channel seen by the equivalent input ASE noise. In the gen-eral case, the two EDFAs within each network element have dif-ferent gains and noise figures. Assuming, without loss of gener-ality, that all EDFAs are identical, that the spontaneous emissionfactor and the EDFA gain are independent of frequencywithin the signal band, and that , can be writtenas

(7)

where is the Planck’s constant.In (6), is calculated through the relationship

(8)

where is an auxiliary variable defined by the recursion

(9)

with initial condition . The subscript denotes thenumber of fiber spans.

Inspection of (8) reveals that has gain due to the actionof servo-controlled attenuators. We can rewrite in theform , where is normalized tounity and is defined as the excess noise factor

(10)

The variance of the orthogonally polarized components,of the ASE noise at the output of the chain of network ele-

ments can be calculated by integration of (6)

(11)

where is the equivalent noise bandwidth of the chain ofnetwork elements defined as

(12)

The total received electric field vector impingent uponthe photodetector can be written as

(13)where and are i.i.d. complex bandpass Gaussian pro-cesses with zero mean and unit variance and the multiplicationcoefficient is written as

(14)

where denotes the average signal energy anddenotes the normalized equivalent noise

bandwidth. The quantity represents the received op-tical signal-to-noise ratio (OSNR) measured in a resolutionbandwidth equal to the bit rate.

Relationship (13) can be rewritten in terms of the quadraturecomponents of the signal and ASE noise

(15)

where , and are the real parts of the signaland ASE noise in the and polarizations, respectively, and

, and are the imaginary parts of the signaland ASE noise in the and polarizations, respectively.

The photodiode is modeled as a square law detector. The op-tical power measured by the photodetector is defined as

(16)

In the presence of a polarizer in front of the tunable opticalBPF in the optically preamplified receiver aligned along the

polarization, the last two terms and on theright-hand side of (16) are omitted.

The final expression for the output photocurrent is

- –

(17)

where is the responsivity of the photodiode, is the termassociated with the direct-detection of the signal,- de-


notes the signal-ASE noise beating, and – denotesthe ASE–ASE noise beating in dimensionless form

-(18)

–(19)

In (17), and are i.i.d. real Gaussian processes withzero mean and unit variance that represent the shot and thermalnoise, respectively.

In reality, shot noise follows Poisson statistics, e.g., [31], [33],[35], [36], [38], and [39]. Here, the shot-noise varianceiscalculated approximately assuming that the level of the receivedsignal is constant within the bit period with amplitudeequal to . If we neglect the dark current, the shot noisevariance can be written as [50]

(20)

where is the electron charge, is an auxiliary variable thattakes the values in the presence or absence of polarizer,respectively, and is the equivalent noise bandwidth of theelectronic filter defined as [51]

(21)

The above expressions can be easily modified for avalanchephotodiodes [50].

The thermal noise is a zero mean Gaussian noise with vari-ance [50]

(22)

where is the Boltzman’s constant, is the absolute temper-ature, is the electronic preamplifier noise figure, and isthe load resistor.

C. Photocurrent Statistics

At the sampling instant , the value of thephotocurrent at the output of the sampler is . Forthe evaluation of the error probability, it is necessary to calculatethe statistics of the random variable (RV). This can be donethrough the characteristic function , where

denotes the expected value.The analytic evaluation of the characteristic function

can be done using the same formulation as in [26]–[41]. Ourderivation is based on a generalization of the analysis by [37].Our modifications of [37] allow to take into account arbitraryoptical MUXs/DMUXs transfer functions, the polarizer at thereceiver, and the shot noise (assuming Gaussian pdf), which arenot included in [37].

As shown in the Appendix A, the characteristic functionof is given by

(23)

where the coefficients , , , and , are de-fined in Appendix A.

From the characteristic function, it is possible to evaluate thepdf of the photocurrent by taking the Fourier transformof , i.e., . In the general case, aclosed-form analytic solution for does not exist. Theasymptotic value of can be found using the steepestdescent method [46]

(24)

where the prime denotes differentiation with respect to, isthe root of the first derivative and is an auxiliaryphase function defined as

(25)

It is also commonplace to approximate by aGaussian pdf with mean and standard deviation . Themean and standard deviation can be evaluated fromthe characteristic function using the relationship [49]

(26)

where are the noncentral moments and ,.

D. Error-Probability Evaluation

Finally, we use a semianalytic method [25] for the accurateevaluation of the error probability of the network. As its nameindicates, the method consists of a combination of simulationand analysis. Noiseless simulation is used to evaluate the ac-cumulation of ISI during signal propagation. Given the signalwaveform at the receiver, the noise cdf and the error probabilityare calculated analytically.

The semianalytic method assumes that the ISI is caused pri-marily by a limited number of bits. Here, it is assumed thatthe transmission channel is causal, so each bit is affected by theprevious bits. For example, consider that is the bit thatwe want to detect and is a sequenceof the last received bits. Then, the mean error probabilitycan be calculated by averaging over all thepossible combi-nations of bits

(27)

where is the probability of occurrence of the combi-nation of bits and is the conditional probabilityto receive the complementary symbol, given that the partic-ular combination of bits was sent.


The conditional error probability is defined asfollows:

(28)

where is the decision threshold. The optimum value of thedecision threshold , which minimizes the mean error proba-bility , is the root of the first derivative of (27) with respect to

. The integrals (28) must be evaluated numerically.An alternative expression for the conditional error

probability can be found by substitution ofin (28)

(29)

The asymptotic value of the integrals in (29) can be foundusing, again, the steepest descent method [26], [38], [41], [46].

If we approximate by a Gaussian pdf with meanand standard deviation , substitution in (28) gives

erfc

erfc(30)

where erfc is the complementary error function [52] definedas erfc .

E. Error Probability for the Reference Network

For the reference network, the mean error probability is ap-proximately given by [31]

(31)

where is an integer number, approximately equal to the bitperiod-optical bandwidth product, i.e., , de-notes the least integer not smaller than, is the normalizedoptimum threshold defined in the interval , andis the generalized Marcum function [49] defined by

(32)

where denotes the modified Bessel function of the firstkind of order .

For , (31) becomes identical to the error probabilityof a matched filter direct-detection receiver.

III. STUDY CASE—CONCATENATION OFAWG ROUTERS

AWG routers are a promising technology for transparentdense wavelength-division multiplexing networks due to theirability to multiplex/demultiplex a large number of channels,their negligible chromatic dispersion, their compactness (i.e.,

integrated form) and their low cost [53]. Disadvantages ofconventional AWG routers are the relatively high insertion lossand the passband curvature [54]. Several techniques to increasethe passband flatness are proposed in the literature. However,there is a tradeoff between passband flatness and insertion loss[54].

This section presents an accurate study of the concatenationof conventional AWG routers (i.e., with round passband top) in atransparent multiwavelength optical network chain like the onepresented in Fig. 1(a). This subject was partly studied previouslyby [6] and [11] for 10-Gb/s nonreturn-to-zero (NRZ) and by[17] for 40 Gb/s return-to-zero (RZ) transmission. The mainpurpose of this study case is to illustrate the theoretical modelof Section II.

This section is divided into two parts. In the first part, we ex-amine whether a Gaussian function can describe well the top ofthe transmittance of commercially available conventional AWGrouters. In addition, we investigate the importance of randomcentral frequency offsets from the channels’ nominal frequen-cies and the variations of the 3-dB filter bandwidth from channelto channel and from device to device.

The second part is devoted to the evaluation of the error prob-ability of a chain network with AWG routers as a function of thenumber of fiber spans, the equivalent noise bandwidth of AWGrouters, and the laser carrier frequency offset from the channel’snominal frequency.

A. Transmittance of AWG Routers

Fig. 3 shows the measured transmittance from an input port toall output ports of a commercially available 88 conventionalAWG router, with channels spaced 100-GHz apart. Dotted linesindicate the periodic transmittance of one of the channels. It isobserved that the transmittance for each channel presents roundtop in the passband and side lobes in the stopband. In addition, itis observed that there is a systematic insertion loss difference be-tween channels (loss imbalance [55]), i.e., edge channels withinthe free spectral range of the device exhibit more loss than themiddle channel.

In equivalent low-pass formulation, the magnitudeof the transfer function of each channel can

be approximated at the center of the passband by a Gaussianfunction (see, e.g., [56])

(33)

where is the insertion loss, is the central frequency offsetfrom the channel’s nominal frequency, and is the cutofffrequency, which is related to the 3-dB bandwidth (full-width athalf-maximum) of the transmittance through

dB (34)

More relationships about the properties of this type of AWGrouter are given in Appendix B.

Obviously, the Gaussian approximation (33) does not de-scribe the side lobes in the stopband (see Fig. 3). However,this discrepancy becomes less important as the number ofconcatenated optical MUXs/DMUXs increases. For example,accurate description of the transmittance of an individual


Fig. 3. Transmittance of a commercially available 8� 8 AWG router with channels spaced 100-GHz apart from an input port to all eight output ports. Dottedlines indicate the periodic transmittance of one of the channels.

Fig. 4. Measurements of the normalized transmittance of a sample AWG router channel from Fig. 3 (points) and their fitting by the square of (33) (solid line).

optical MUX or DMUX within only 10 dB from the top of thepassband is sufficient for the description of the overall transferfunction of a chain of ten optical MUXs/DMUXs within 100dB from the top of the passband.

In addition, in (33), it is assumed that the optical AWG routersare not dispersive devices within the signal band and the phaseis set equal to zero. In practice, small variations of refractiveindixes, width, and thickness from waveguide to waveguide in-duce amplitude and phase errors in the transfer function. Never-theless, these errors are assumed small and the behavior of thedevice is dominated by the curvature of the passband [21], [57],[58].

Fig. 4 shows measurements of the normalized transmittanceof one AWG router channel from Fig. 3 (points) and their fit-

ting by the square of (33) (solid line). Measurements within 10dB from the top of the passband are used for the fitting; the fitis excellent. We conclude that the Gaussian approximation ac-curately describes the behavior of the transmittance of conven-tional AWG routers in the center of the passband and its vicinity.

Fitting of the normalized transmittance measurements isused next for the extraction of the center frequency and the3-dB bandwidth and the evaluation of their statistics.2 For thispurpose, measurements of the transmittance of two commer-

2It is worth noting that there is no universally accepted method for themeasurement of the center frequency and the 3-dB bandwidth of opticalMUXs/DMUXs. For example, commercially available devices are oftencharacterized by measuring the 3-dB points from the top of the passband anddefining their average as the center frequency.


Fig. 5. Histogram of AWG router central frequency offsets from the channels’ nominal frequencies (based on measurements of 96 transmittances).

Fig. 6. Histogram of 3-dB bandwidths of AWG router channels (based on measurements of 96 transmittances).

cially available 8 8 conventional AWG routers, with channelsspaced 100-GHz apart, are fitted by the square of (33). Due tothe lack of a large sample of devices, we made the (arbitrary)assumption that the transfer functions of different channelsand different input/output ports within the same device areindependent. This assumption is not strictly correct so theextracted center frequencies and 3-dB bandwidths are partlycorrelated. In addition, as a result of our assumption, we makeno distinction between random and systematic error. Conse-quently, the following statistics provide only a rough estimation

of the importance of random variations in the transmittance ofAWG routers.

Histograms for the central frequency offsets from the chan-nels’ nominal frequencies and the 3-dB bandwidths for twocommercially available 8 8 conventional AWG routers, withchannels spaced 100-GHz apart, are presented in Figs. 5 and 6,respectively. Fig. 5 shows that central frequency offsets from thechannels’ nominal frequencies are concentrated around zero in arange ( 8 GHz, 8 GHz), with standard deviation of 2.7 GHz,which is about 7% of the mean 3-dB filter bandwidth. Fig. 6


Fig. 7. Signal transfer functionH (f) (solid lines) and normalizedtransmittanceT (f) (dotted lines) of the chain of network elements seen bythe equivalent input ASE noise for one, five, ten, and 15 fiber spans.

shows that the mean 3-dB bandwidth is 38 GHz, with a stan-dard deviation of 1.6 GHz, which is 4% of the mean 3-dB filterbandwidth.

Based on the above, we conclude that the random central fre-quency offsets from the channels’ nominal frequencies and thevariations of the 3-dB filter bandwidth of commercially avail-able AWG routers are small and, in a first approximation, canbe neglected. In the following, the transfer functions of concate-nated AWG routers will be considered identical and perfectlyaligned. The impact of random variations on the system’s per-formance will be addressed in a future paper.

B. AWG Router Cascadability

The AWG router cascadability depends on the selectionof network design parameters, e.g., transmitted optical pulseshape, extinction ratio, chirp, fiber chromatic dispersion andnonlinearities, receiver type, and so forth. An exhaustivestudy of the influence of all possible combinations of theaforementioned parameters on the network performance isbeyond the scope of this example. To isolate the impact of theAWG router filtering from other transmission effects, the fol-lowing network parameter set is used in this study: ideal NRZpulses with infinite extinction ratio and zero chirp, absence ofchromatic dispersion and nonlinearities, ASE-noise-limiteddirect-detection receiver, negligible shot and thermal noises,and a fourth-order Bessel electronic LPF with cutoff frequency

(i.e., equivalent noise bandwidth ).Appendix C presents the details of the implementation of the

semianalytic method for the error-probability evaluation in thisspecific case study.

Fig. 7 shows the signal transfer function (solid lines)and the normalized transmittance seen by the equiv-alent input ASE noise (dotted lines) for one, five, ten, and 15fiber spans. Since all curves present symmetry around the origin,only the positive frequency semiaxis is displayed. It is observedfrom (6) that depends on the effective attenuator gains

, . Here, we assumed that all attenuator gains,are unity (case of wide AWG routers compared to

the signal spectral occupancy). It is observed that andcoincide in the proximity of the origin for any number

(a)

(b)

Fig. 8. Mean error probability�P as a function of the received OSNR�E =Nmeasured in a resolution bandwidth equal to the bit rate for one (dotted line),three (dashed-dot line), and 12 (dashed line) fiber spans. (a) In the presence ofpolarizer at the receiver. (b) In the absence of polarizer at the receiver (condition:B = 4R ).

of fiber spans. Away from the origin, they are very different forand their discrepancy increases as a function of the

number of spans. At the limit , .The behavior of implies that the filtering of the ASEnoise by the optical MUX/DMUX chain is less severe than thefiltering of the signal due to the distributed generation of theASE noise in the network. We conclude that, for a large numberof fiber spans, it is beneficial to use a narrow-band optical BPFat the optically preamplified direct-detection receiver for rejec-tion of the long ASE noise tails.

Fig. 8(a) and (b) shows the mean error probabilityas afunction of the received OSNR measured in a resolu-tion bandwidth equal to the bit rate for one (dotted line), three(dashed-dot line), and 12 (dashed line) fiber spans, in the pres-ence and in the absence of polarizer at the receiver respectively.The equivalent noise bandwidth of individual AWG routers inthese graphs is assumed . (For example, in the case ofthe AWG routers shown in Fig. 3, which exhibit a mean equiv-alent noise bandwidth equal to 40.5 GHz, the above assump-tion implies a bit rate Gb/s.) For a comparison, theerror probability for the reference network (31) is also shown(solid line). In Fig. 8(a), for one fiber span, 1.4 dB of addi-tional power compared to the reference network are requiredto achieve an error probability of 10. When the number ofspans increases, initially the sensitivity is improved because


Fig. 9. Optical power penalty compared to the reference network, at an errorprobability of 10 , as a function of the number of fiber spans, for threeindividual AWG router equivalent noise bandwidths, i.e.,B = 4R , 6R ,and8R , in the presence (solid lines) or absence (dotted lines) of polarizer,respectively.

the optical bandwidth narrowing reduces the power of the ASEnoise without essentially distorting the signal. A maximum sen-sitivity is achieved after three fiber spans, where there is only0.61 dB power penalty compared to the reference network. Asthe number of spans continues to increase, the power penaltyslowly increases due to the increase of ISI. It is observed thatthe slope of the error-probability curves changes as a functionof the number of fiber spans. This is due to the fact that AWGfiltering changes the properties of the ASE noise at the input ofthe direct-detection receiver. Similar conclusions can be drawnfrom Fig. 8(b). Notice that, in Fig. 8(a), at high error probabili-ties (i.e., low OSNRs), the actual network for three fiber spansperforms slightly better than the reference network. This ap-parent paradox is due to the fact that the reference network issuboptimum. A matched filter direct-detection receiver [not de-picted in Fig. 8(a) and (b) to avoid clutter] presents superior per-formance than both the actual and the reference network at allerror probabilities. For a matched filter direct-detection receiver,the OSNR measured in a resolution bandwidth equal to the bitrate required to achieve an error probability of 10is approx-imately 38.5 (15.85 dB).

Fig. 9 shows the optical power penalty compared to the refer-ence network at an error probability of 10as a function of thenumber of fiber spans for three individual AWG router equiv-alent noise bandwidths, i.e., , , and , in thepresence (solid lines) or absence (dotted lines) of polarizer atthe receiver. In the case, in the presence of polar-izer, a minimum power penalty 0.61 dB is achieved after threefiber spans. In the absence of polarizer, a broad maximum sen-sitivity is achieved after four fiber spans, where there is only0.87-dB power penalty compared to the reference network. Inthe case, in the presence of polarizer, a broad max-imum sensitivity is achieved after 12 fiber spans, where thereis only 0.16-dB power penalty compared to the reference net-work. In the absence of polarizer, a broad maximum sensitivityis achieved after fifteen fiber spans, where there is only 0.41-dBpower penalty compared to the reference network. The resultsfor the case fall in between the and

cases. These results indicate that it is possible to

Fig. 10. Optical power penalty compared to the reference network, at an errorprobability of 10 , as a function of the equivalent noise bandwidth of theindividual AWG routers for five (solid line), ten (dotted line), and 15 (dashed-dotline) fiber spans in the presence of polarizer at the receiver.

concatenate an arbitrary number of conventional AWG routers,despite the inherent passband curvature of these devices, byincreasing the equivalent noise bandwidth of individual AWGrouters for fixed bit rate at the expense of channel spacing. Inaddition, it is observed that an ideal polarizer always improvesthe performance of the optically preamplified receiver, as ex-pected, due to the elimination of the ASE–ASE noise beatingterms and on the right-hand side of (16). The roleof the polarizer is more important for larger individual AWGrouter equivalent noise bandwidths and smaller number of fiberspans. However, in all cases examined in the present paper, theperformance improvement due to the polarizer is marginal (lessthan 0.5 dB) and does not justify the implementation cost of thisdevice.

Using the current model, it is possible, in principle, tomaximize the network performance for a given number ofconcatenated AWG routers by jointly optimizing the opticalbandwidth of the individual AWG routers and the electronicLPF bandwidth of the receiver. However, such optimizationis impractical because in transparent optical networks, eachoptical signal passes through a different number of concate-nated AWG routers. In addition, in reconfigurable opticalnetworks, each optical signal may be rerouted through severaldifferent paths during the network’s lifetime, depending ontraffic demands, equipment failures, and so forth. Nevertheless,it is instructive to examine the impact of the equivalent noisebandwidth of individual AWG routers on the performance ofthe network.

Fig. 10 shows the optical power penalty at an error proba-bility of 10 (compared to the corresponding reference net-work at the optimum operating point) as a function of the equiv-alent noise bandwidth of individual AWG routers for constantelectronic LPF equivalent noise bandwidth . We considerthree different paths in the network, where the signal passesthrough five (solid line), ten (dotted line), and 15 (dashed-dotline) fiber spans in the presence of polarizer at the receiver. Forfive fiber spans, a minimum sensitivity is achieved at ,where there is only 0.47-dB power penalty compared to the cor-responding reference network, which requires an OSNR of 16.6dB at an error probability of 10 . In the case of ten fiber spans,


Fig. 11. Optical power penalty compared to the reference network, at an errorprobability of 10 , as a function of the laser carrier frequency misalignmentfrom the channel’s nominal frequency, when the signal passes through five (solidline), ten (dotted line), and 15 (dashed-dot line) fiber spans in the presence ofpolarizer at the receiver (condition:B = 4R ).

a minimum sensitivity is achieved at , where thereis only 0.25-dB power penalty compared to the correspondingreference network, which requires an OSNR of 16.9 dB at anerror probability of 10 . In the case of 15 fiber spans, a min-imum sensitivity is achieved at , where there is only0.06-dB power penalty compared to the corresponding refer-ence network, which requires an OSNR of 17 dB at an errorprobability of 10 . The sensitivity degrades sharply for nar-rower equivalent noise bandwidths. We conclude that a possiblecompromise is to choose the AWG router equivalent noise band-width equal to , which guarantees an optical powerpenalty less than 0.5 dB in all cases.

Fig. 11 shows the optical power penalty compared to the ref-erence network, at an error probability of 10, as a functionof the laser carrier frequency offset from the channel’s nominalfrequency when the signal passes through five (solid line), ten(dotted line), and 15 (dashed-dot line) fiber spans in the pres-ence of polarizer at the receiver. Due to the symmetry in the op-tical signal spectrum and the AWG transfer function, the powerpenalty is the same for laser misalignments of the same mag-nitude, but different signs. Therefore, only the optical powerpenalty for positive misalignments is displayed. In the case offive fiber spans, the model predicts a monotonic increase in theoptical power penalty as the laser carrier frequency offset in-creases. However, in the case of ten and 15 fiber spans, themodel predicts a minimum optical power penalty of 1.34 and1.97 dB, respectively, compared to the reference network for alaser carrier frequency offset and , respectively,from the channel’s nominal frequency. This surprising result,which occurs when the effective equivalent bandwidth of op-tical MUXs/DMUXs cascade becomes comparable to the bitrate, was first observed theoretically and experimentally by [6]and was attributed to the filtering of the vestigial side band.Preliminary simulations performed by the authors indicate thatthis effect is universal and can be observed for other modu-lation formats and optical MUXs/DMUXs types as well, e.g.,Gaussian RZ pulses and MUXs/DMUXs composed of MI fil-ters. Recently, this effect was proposed as a means to achieve

ultrahigh-capacity WDM transmission [59] and was used to ob-tain record spectral efficiency 1.28 b/s/Hz in a 10.2-Tb/s trans-mission experiment [60].

IV. SUMMARY

This paper presents a general theoretical model for the studyof concatenation of optical MUXs/DMUXs in transparent mul-tiwavelength optical networks. The model is based on a semian-alytical technique [25] for the evaluation of the error probability.Approximate calculations of the error probability with semian-alytical techniques, as a criterion for the cascadability of opticalMUXs/DMUXs, were used in the past, e.g., [3] and [6]. How-ever, the present technique offers superior accuracy, taking intoaccount arbitrary pulse shapes, arbitrary optical MUX/DMUXand electronic LPF transfer functions, and non-Gaussian pho-tocurrent statistics at the output of the direct-detection receiverassuming ideal square-law detection [26]–[41]. The computa-tion of the error probability involves several steps, includingexpansion of the ASE noise impingent upon the photodiode inKarhunen–Loève series [42]–[44], numerical solution of the as-sociated homogeneous Fredholm equation of the second kind[44]–[46], diagonalization of a quadratic form [37], [45], andasymptotic evaluation of the pdf at the output of the direct-detec-tion receiver from the characteristic function using the methodof steepest descent [26], [41], [46].

The model helps to identify the underlying mechanisms forperformance degradation and to provide a rigorous definitionfor the excess loss due to spectral clipping resulting from opticalMUX/DMUX concatenation. In addition, for the first time, thenotions of effective gain and excess noise factor due to servo-controlled attenuators are introduced. Finally, a reference net-work with ideal optical MUXs/DMUXs is defined for compar-ison.

The model can be used to derive specifications for arbitraryoptical MUXs/DMUXs in order to achieve a prescribed powerpenalty in conjunction with different modulation formats. Toillustrate the model, the concatenation of conventional AWGrouters in a transparent multiwavelength optical network chainis studied. First, measurements of the transmittance of com-mercially available AWG routers are fitted in order to extractthe values of the channels’ central frequencies and 3-dB band-widths and evaluate their statistics. Statistical results indicatethat first-order random variations in the transfer functions ofAWG routers from device-to-device can be neglected. Then,the error probability is evaluated as a function of the numberof AWG routers, the bandwidth of AWG routers, and the lasercarrier frequency offset from the channel’s nominal frequency.

APPENDIX A

DERIVATION OF THE CHARACTERISTIC FUNCTION OF THE

OUTPUT PHOTOCURRENT

The analytical evaluation of the characteristic function of thefiltered noise at the output of a square-law detector was per-formed initially in the context of radio communications (e.g.,see [26] and the references therein) and later in the context of


optically preamplified direct-detection receivers, e.g., [27]–[41]for different types of optical and electronic LPFs.

Here, we adapt the formalism of [37] to the study of opticalMUX/DMUX concatenation. For the sake of completeness, thederivation of the expression (23) for the characteristic function

is described briefly. The interested reader can find moredetails in [37]. Wherever possible, the same notation as in [37]is used. Our modifications of [37] consist of the inclusion of ar-bitrary optical MUXs/DMUXs transfer functions, the polarizerat the receiver, and the shot noise variance, borrowing elementsfrom [27]–[41].

The derivation is divided into the following parts. In thefirst part, the convolutions in (18) and (19) describing thelow-pass filtering of the signal–ASE and ASE–ASE noisebeatings are transformed into sums of i.i.d. Gaussian RVsusing a Karhunen–Loève expansion of the equivalent inputASE noise. In the second part, the characteristic function ofthe signal–ASE and ASE–ASE noise beatings is calculated.Finally, the shot and thermal contribution are added to thecharacteristic function.

As a starting point, we rewrite the photocurrent given in (17)at the output of the sampler

- –

(35)

where

-(36)

–(37)

For the evaluation of the statistics of (37), the equivalent inputASE noise quadrature components ,are represented in the time interval , where is theduration of the impulse response of the electronic LPF, by aKarhunen–Loève expansion [42], [43]

(38)

where the coefficients are independent Gaussian RVs withzero mean and variance [42]–[44]. The orthonormal func-tions and the variances of are the eigenfunctionsand the eigenvalues, respectively, of the homogeneous Fred-holm equation of the second kind [45]

(39)

where is the autocorrelation function of and canbe evaluated by the real part of the inverse Fourier transform of

, i.e., .It is worth noting that the eigenfunctions and the eigenvalues

for the real and imaginary parts of theand polarizations arealways degenerate, i.e., ,

. However, if does not present Hermitian symmetryaround the origin, e.g., due to optical MUX/DMUX centerfrequency offsets or systematic passband tilts, then the real andimaginary parts of the ASE noise along theand polariza-tions and the corresponding coefficients ,are cross correlated [49, pp. 153–157]. In the following, it isassumed that presents Hermitian symmetry, as isthe case for perfectly aligned AWG routers with symmetrictransfer functions, so this cross correlation is negligible, i.e.,

, where denotes imaginary part.Equation (39) can be solved analytically only in certain spe-

cial cases of [44]. In the case of arbitrary opticalMUX/DMUX transfer functions, (39) must be resolved numeri-cally [46]. The method consists in using a quadrature rule for theintegral in the left-hand side of (39) and in discretizing the timeat the same nodes in order to create a matrix eigenvalue equa-tion. For the quadrature, we use the trapezoidal rule because itallows for equally spaced nodesand is, therefore, well suitedfor the evaluation of integrals (44), involving the signalprovided from digital simulation.

Substituting the Karhunen–Loève expansion of into(37) yields

– (40)

where we defined .Relation (40) is a quadratic form that can be rewritten in ma-

trix formulation [37]

– (41)

where denotes transpose, is a diagonal matrixwith diagonal elements equal to the standard deviations of,and is a column vector with elements , where

are new RVs following a normal distribution.The quadratic form (41) can be diagonalized using a unitary

transformation , where is a transformation ma-trix whose columns are the eigenvectors of and is anew column vector with elements , where arenew RVs following a normal distribution. It can be shown thatthe new RVs are uncorrelated and, thus, independent [61].Finally, substituting in (41) yields

–

(42)

where are the eigenvalues of .Similarly, - can be written in series in terms of the

new i.i.d. RVs

-

(43)where

(44)

Obviously, in (43) are multiplication coefficients that de-pend on the unitary transformation and are a function of theprojection of the received signal on the orthonormal functions


. Notice that depend also on the sampling instant,but this dependence is implied in order to simplify the notation.

From (42) and (43), the characteristic function of the- and – can be evaluated as

(45)

where takes the values in the presence or absence ofpolarizer, respectively.

The characteristic functions for the shot and thermalnoises can be readily evaluated [49]

(46)

Finally, the conditional characteristic function ofis given by

(47)

APPENDIX B

ANALYTICAL EXPRESSIONS FORAWG ROUTERPARAMETERS

The equivalent noise bandwidth of a BPF is defined as

(48)

where denotes the equivalent low-pass transmittance ofthe BPF.

If we neglect the periodicity of the AWG router transmittance,we can evaluate analytically the equivalent noise bandwidth ofone AWG router by substitution of (33) in (48) and use of therelationship in [62]

(49)

Notice that, for one AWG router, the equivalent noise band-width is slightly larger than the 3-dB bandwidth, which isequal to dB .

The impulse response of concatenated AWG routers isgiven by

(50)

where is the transfer function of the cascade ofopticalMUXs/DMUXs.

By substitution of (33) in (50) and use of the relationshipin [62], we find

(51)

where .

For the simulation described in Appendix C, the duration ofthe impulse response (51) is arbitrarily defined as eight timesthe root-mean-square (rms) width [63]

(52)

where is defined as

(53)

It is straightforward to show that and .

APPENDIX C

IMPLEMENTATION OF THESEMI-ANALYTIC METHOD

The algorithm for the semianalytic evaluation of the errorprobability involves several computational steps.

1) Evaluation of the signal waveform at the output of thereceiver by noiseless simulation of the the block diagramof Fig. 2(b).

2) Analytical evaluation of the ASE noise psd at the photo-diode.

3) Analytical evaluation of the characteristic function of thenoise at the output of the receiver for each bit, assumingideal square-law detection and using an expansionof the ASE noise impingent upon the photodiode inKarhunen–Loève series [42]–[44].

4) Asymptotic evaluation of the pdf of the photocurrentat the output of the receiver for each bit from the cor-responding characteristic function using the method ofsteepest descent [26], [38], [41], [46].

5) Evaluation of the conditional error probability for each bitby numerical integration of the pdf tail.

6) Evaluation of the mean error probability by averagingover the conditional error probabilities for all bits.

7) Finally, numerical minimization of the mean error proba-bility by optimization of the decision threshold.

In the simulation, a pseudorandom sequence with period upto 2 is used to modulate the optical signal. This is a de Bruijnsequence [64], i.e., a maximal-length pseudorandom sequencewith an additional zero added after the longest string of zeros.Successive iterations with different sequence lengths show thatlonger sequences increase computing time and provide a neg-ligible increase on accuracy. The optical waveform is assumedNRZ with perfect rise and fall times, zero chirp, and infiniteextinction ratio. AWG routers are represented by finite impulseresponse (FIR) filters. The number of coefficients of each FIRfilter is calculated so that the duration of the impulse responseis equal to eight times the rms width (see Appendix B). Forthe evaluation of the effective gain provided by each servo-con-trolled attenuator, relationship (4) is used. For simplicity, onlythe signal is considered in the calculation of the average power,since it is assumed it is much higher than the ASE noise powerfor normal network operating conditions. This will introduce asmall error for high error probabilities, where the ASE noise ismore important.

Commercially available optically preamplified receivers usu-ally have a narrow tunable optical BPF (e.g., fiber Fabry–Perot


with optical bandwidth of the order of [65]) with an auto-tracking circuit that allows the filter to align its transmissionpeak with the signal carrier frequency. For simplicity, in thefollowing simulations, it is assumed that the equivalent noisebandwidth of the receiver optical BPF is much wider than theequivalent noise bandwidth of each optical MUX/DMUX. Itssole purpose is to eliminate the periodicity of the ASE spectruminherent in AWG routers and does not cause additional insertionloss or signal filtering. A fourth-order Bessel LPF with cutofffrequency is used at the receiver.

The optimum sampling instant must be found through anumerical minimization of the error probability. Here, for sim-plicity, the noiseless eye opening at the output of the receiver isused to determine the optimum sampling instant. The evalu-ation is performed in two steps. First, the propagation delayiscalculated by correlation of input and output sequences;. Then,

is calculated in order to maximize the eye opening. A jitterwindow around , due to the clock recovery circuit, can be as-sumed.

The evaluation of the equivalent transmittance is done analyt-ically using the recursive relationship (8). The autocorrelation ofthe ASE noise is evaluated using the inverse fast Fourier trans-form (FFT) with 1024 points.

For the evaluation of the characteristic function from (23),the coefficients , , , and , are needed. Asshown in Appendix A, this requires the numerical solution of thehomogeneous Fredholm equation of the second kind (39) [46].For the quadrature, we use the trapezoidal rule with 64–512nodes. The thermal and shot noise variances are set to zero.

The evaluation of pdf’s from the characteristic function isdone using the method of steepest descent. The pdf’s computedby the steepest descent method are in excellent agreement withthe pdf’s given by the FFT of the characteristic function.

The optimum threshold occurs at the intersection of the sumof pdf’s of the ZEROs and the sum of pdf’s of ONEs of the deBruijn sequence. The optimum threshold is evaluated numeri-cally.

The evaluation of conditional error probabilities directly fromthe characteristic function using (29) is oscillatory, unless a suc-cessful guess of the vicinity of the root is found. Therefore,numerical integration of the pdf tail is preferable.

ACKNOWLEDGMENT

The authors would like to thank Dr. L. D. Garrett for pro-viding the measurements of Fig. 3, Drs. S. F. Habiby, N. N.Khrais, A. F. Elrefaie, J. B. D. Soole, J. C. Cartledge, S. P.Burtsev, M. Lee, I. Tomkos, and A. Boscovic for helpful dis-cussions, and Ms. L. M. Szemiot for corrections in the syntaxof the manuscript.

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accurate modeling of optical multiplexer/demultiplexer...

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