Accurate noise characterization of wavelength converters based on XGM in SOAs

182 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 1, JANUARY 2003

Accurate Noise Characterization of WavelengthConverters Based on XGM in SOAsManuel Muñoz de la Corte and Jaafar M. H. Elmirghani, Senior Member, IEEE

Abstract—Wavelength conversion (WCR) has recently emergedas an important technique that can be used to manage the spectac-ular increase in traffic in dense wavelength division multiplexing(DWDM) networks. WCR is extremely useful to solve contention,reduce wavelength blocking and a wide range of WCR methodshave been reported. The optimum placement of these deviceswithin a network remains an unexplored area and accurate modelsto measure the performance of the system are becoming vital.Therefore, this paper presents an original noise characterization ofa class of WCRs based on cross-gain modulation (XGM) in semi-conductor optical amplifiers (SOAs) when the components fromspontaneous emission of the SOAs at the receiver are dominantover thermal and shot noise terms. A new and simple expressionfor the error probability � � is presented offering considerableadditional accuracy in sensitivity assessment compared with theGaussian approach when ASE noise is dominant. Simulationcomparisons are presented in different scenarios for a standardsystem working at 2.5 Gb/s in a metropolitan area network takinginto account the key parameters when such systems are designed.

Index Terms—Performance predictions, probability of error( ), semiconductor optical amplifier (SOA), wavelength conver-sion (WCR), wavelength division multiplexing (WDM).

I. INTRODUCTION

WAVELENGTH conversion (WCR) has recently emergedas an important technique that can be used to manage

the spectacular increase in traffic in WDM networks. WCRis extremely useful in resolving contentions and in reducingwavelength blocking. A wide range of WCR methods havebeen reported [1]–[3]. One of the most widely used methods(due its simplicity) exploits XGM in SOAs and commonly usesthe configuration shown in Fig. 1. This scheme is not free fromlimitations and the resultant WCR can introduce severe signaldegradation. The most important parameters that can be used tomeasure this degradation, when the SOA is operated far fromits bandwidth limit, are the extinction ratio and the amplifiedspontaneous emission (ASE) introduced by the semiconductoroptical amplifier (SOA). The optimum placement of thesedevices within a network remains an unexplored area andaccurate models that can measure the performance of thesystem are becoming vital.

Therefore, this paper presents an original expression for thesystem probability of error that takes into account these twomain impairments and offers considerable additional accuracy

Manuscript received August 20, 2002.The authors are with the Department of Electrical and Electronic En-

gineering, University of Wales Swansea, Swansea SA2 8PP U.K. (e-mail:[email protected]).

Digital Object Identifier 10.1109/JLT.2003.808618

Fig. 1. WCR based on XGM in SOA.

in sensitivity assessment if ASE noise components are assumeddominant at the receiver.

Following the introduction, this paper is organized as follows.In Section II, the system model is presented; including the gainmodel for the SOA, the components of noise in the electricaldomain as well as the different parameters used in the study. InSection III, a brief review of the existing Gaussian expression isoutlined as well as the complete derivation of two new expres-sions using a Chi-squared distribution. In Section IV, the scopeof validity of each approach is studied. In Section V, results arepresented that assess the improvement offered by the new statis-tical model and finally Section VI summarizes the main findingsof the paper.

II. SYSTEM MODEL

The WCR model is shown in Fig. 1. Two optical signal are in-jected at the input of the SOA combined by an optical coupler.The pump signal (information carrying signal) modulates theSOA gain and is described using an optical power cen-tered at and the extinction ratio . This signal is modulatedfollowing a binary ASK method and is the signal whose wave-length is to be changed. The signal at the output of the probelaser (probe signal), is a continuous-wave (CW) signal, and isdescribed by a power centered at the target wavelength

. As can be observed in Fig. 1, the SOA is described by aspontaneous emission factor , an effective gain (whenthe device is operated far from the bandwidth limit), and anSOA saturation power . An optical tunable band pass filter(OTBPF) at the SOA output is used to select the signal cen-tered at the target wavelength. This filter is tuned in harmonywith the probe laser to select . The optical signal is receivedafter a link loss measured in decibels. This loss can be mea-sured in kilometers assuming a certain fixed loss per kilometer,where a typical combined value for this is 0.2–0.4 dB/km. Thisloss could be caused by an optical fiber or by the fiber loss aswell as the loss due to passive devices such as filters, circulators,and couplers. The optical receiver is described by an electricalbandwidth that is a function of the system operating bit rate.The receiver is also described using a minimum optical power

0733-8724/03$17.00 © 2003 IEEE

DE LA CORTE AND ELMIRGHANI: ACCURATE NOISE CHARACTERIZATION OF WAVELENGTH CONVERTERS 183

at the input required to maintain the probability of errorat where is the receiver sensitivity. This studywill determine as a function of the different parametersshown in Fig. 1, such as the pump and probe signal powers,SOA parameters and losses in the fiber . The improvement inperformance predictions will be quantified using originalstatistical expressions for the error probability. We will checkthat the new expressions offer a high level of accuracy com-pared to the Gaussian expression that is commonly used. TheGaussian expression does not offer reliable predictions whenused in applications where WCR based on XGM in SOAs arepresent. Additionally we will assume that the system is operatedat 2.5 Gb/s per channel in a metropolitan area network (MAN)and therefore it is working far from the bandwidth limit of theSOA. This assumption allows us to ignore the second-order ef-fects (for example carrier dynamics) related to this limitationthat takes special importance in national and international widenetworks where very high bit rates are compulsory.

A. Gain Model

As can be seen in Fig. 1, the SOA is described by , that isassumed to be constant with the input wavelength and equal to27 dB [4], and by that takes a typical value of 0.1 mW.depends on the amplifier parameters such as the active cavitydimensions or the polarization injected [5].

The gain saturation in SOAs follows [5]:

(1)

where the pump power will be assumed to be dominant overthe probe power. Therefore the gain saturation process in theSOA will be directed by the variations in the information car-rying signal (pump signal). For instance, Fig. 2 shows the gainsaturation process in a SOA when typical input powers in therange from 25 dBm to 1.7 dBm are injected. At this point it isimportant to observe the following aspects. First, the strong de-pendence of the gain on the input power is manifested and thismay be an inconvenience in other type of applications (in-lineamplifiers, preamplifier, etc.) but not in XGM with SOAs wherethis modulation is used to achieve wavelength conversion. Sec-ondly gains around are obtained for small powers at the inputand when the input power is increased up to values in the orderof dBm the gain falls rapidly. In this way, the gainin the SOA is modulated by the total input power.

B. Noise Components

The ASE noise power at the output of a SOA may be writtenas

(2)

where is the Plank constant, is the probe signal frequency,is the gain given by the (1) and is the OTBPF optical band-width. The beat components obtained in the electrical domainare and , thus the following noise componentsare present at the receiver:

(3)

Fig. 2. Gain saturation in a SOA.

where is the shot noise term that will be discussed laterand is the thermal noise term. The thermal noise in thiskind of systems determines the sensitivity of the optoelectronicreceiver since it is dominant when the input power (on the de-tector’s face) is very small [6]. The following expression estab-lishes the relation between sensitivity, bit rate, and thermal noisepower:

with

(4)

where is the responsivity of the optical receiver and Gaussianstatistics are assumed for the thermal noise (hence, the ratio sixat ). Equation (4) represents a simple method forquantifying the thermal noise using which can easily bemeasured in most cases. is the shot noise term and is splitinto two different terms. One due to signal power and the otherdue the ASE power at the output of the SOA. is the spon-taneous–spontaneous beat noise and is the signal-spon-taneous beat noise. These two terms are dominant when eitherthe signal at the input of the receiver or the spontaneous emis-sion factor take high values [6]. Fig. 3, depicts the receiver sen-sitivity (assuming Gaussian statistics [4], [6]) versus the SOAgain for input and output coupling efficiencies of 0.24 and 0.31,respectively. Here, it is clearly seen that for low values of thegain (10 to 15 dB) an increment in the gain leads to a decre-ment in and therefore an improvement in the system per-formance. This can be explained by realizing that at low gainsthe ASE is low, thermal noise dominates and an increase in theSOA gain leads to an improvement in the signal power and areduction in the probability of error. On the other hand, whenthe SOA is operated at large gains the ASE components be-come dominant and no improvement in performance is obtainedwhen the gain is increased as shown by the saturation of thecurves. In this paper, we will assume high signal powers and ahigh spontaneous emission factor. As a consequence, we will beworking in the right-hand side of Fig. 3. Here, the ASE compo-nents remain dominant over thermal and shot noise components.This assumption leads to a very accurate and simple expressionfor the probability of error and is useful in most systems usingWCR based on XGM with SOAs. Additionally, Fig. 3 showsthe degradation in performance when is increased. Fig. 3


Fig. 3. Sensitivity at the receiver versus the gain in the SOA for different valuesof � and � � �� dBm, � � �� m, � � ��, � � ��, and � �

�� GHz.

was obtained using the following noise expressions representingnoise components in the electrical domain [4], [6]:

(5)

(6)

(7)

where negligible losses at the input and output couplings havebeen assumed as well as in the optical coupler. This does not leadto any limitation in the study since effective input and outputpowers and gain in the SOA can be assumed. The expressions(5)–(7) are given in which is equally useful, as will be seen,when the signal to noise ratio is evaluated. The optical filterbandwidth will be fixed at five times . This value is physicallyrealizable and will reduce the noise at the receiver (electrical do-main). Optical tunable filters have improved in design and morerecently structures have been proposed with very good charac-teristics. For example, tunable optical filters have been studiedwhere the high selectivity of fiber Fabry-Perot filters and thefiber Bragg grating bandpass nature are combined to yield band-widths down to 170 MHz (centered at 1550 nm), extinction ra-tios of 20 to 30 dB and a good tuning range [7]. A detailed studyof the impact of the profile of such filters on the performanceof the wavelength converter is beyond the scope of the currentwork. will be fixed at 0.7 times the bit rate of each channelin the WDM network (2.5 Gb/s) where binary ASK modulationis used. This relation between electrical bandwidth and bit rategives a good balance between noise and ISI [8]. is the gainexperienced by the probe signal (CW) when it goes through theSOA and is given by (1). It is important to emphasize that the

probe signal will be much weaker than the pump signal so thatthe gain modulation can be assumed to be exclusively caused bythe pump. We should observe that the gain saturation process isdue to the decrement in the carrier concentration when a largepower such as is injected at the input. When this signalis logic zero a high concentration of carriers is present in the ac-tive cavity and therefore a large gain is experienced by the probesignal. On the other hand, when the pump power at the input ishigh, very few free carriers are left to amplify the probe signal.This gain modulation is mapped to the probe and is manifestedat the WCR output as a new signal at the target wavelength. Thissignal will have the same digital information as the pump signalbut will suffer the inconvenience of being inverted. The squaredgenerated currents at the receiver will be

" " (8)

" " (9)

where is the electron charge and unity quantum efficiency isassumed. `` '' and `` '' are the gainswhen the zero and one signal levels are injected at the input,respectively. A distinct WCR design requirement is that of ahigh output extinction ratio. We will set the minimum value ofthe input extinction ratio to around 8 to 10 dB. The extinctionratio at the output of the WCR can be expressed as follows:

" "" "

(10)

This then specifies the range of gains desired. The signal tonoise ratio (SNR) at the output of the WCR can be written as

dB (11)

The SNR will increase if the difference between the currentsgenerated by logic one and zero cases become larger or alter-natively when the total noise at the receiver (electrical domain)becomes smaller. It is important to mention that the extinctionratio degradation and ASE generated in the SOAs are the mostserious drawbacks in these systems provided the WCRs are op-erated far from the SOAs bandwidth limit.

III. STATISTICAL DESCRIPTION

A. Statistical Model

The optical signal in the bit interval can be written asfollows:

(12)

(13)

where is the carrier frequency at the output of the SOA (targetfrequency) and where the one and zero cases have been dis-tinguished. It is possible to set to zero which is logical inother kinds of systems. However, when WCR based on XGMare present in the system the zero value cannot be neglected


since low extinction ratios at the output of the WCR are typ-ical (e.g., 8–10 dB). The electric field components at the outputof the SOA can be written using its Fourier series expansion1

(14)

The real and imaginary part of each complex coefficient in(14) are assumed to be independent Gaussian distributed vari-ables with zero mean and a variance of (that is obviously re-lated to the total noise in the electrical domain ). After theoptical tunable bandpass filter (OTBPF), with bandwidth ,only components are detected at the receiver where canbe expressed as follows:

assuming and (15)

where is the bit period. This choice for the spectral separationbetween components in the Fourier series of is

not arbitrary and will lead to orthogonal expansion componentsand hence simplified results once the electric field is squaredand averaged. After the filtering by the OTBPF the electric fieldgenerated by the ASE can be expressed [9] as

(16)

The diode at the receiver follows a quadratic rule in pro-cessing the electric field components. Taking into account allsources of electric field (signal and ASE), the generated currentto be detected can be written as follows:

with

for

for

odd (17)

where one and zero logic have been distinguished and an oddvalue for has been assumed which does not limit the scopeof the study because it can be used even with noninteger valuesgiving excellent results [10]. In the detection process the gen-erated current is averaged over the time duration of one bit andthis value is used to decide whether a one or zero logic is presentat the receiver

(18)

where because of the orthogonality of the functions inthe time interval only the diagonal terms in the sum

1We will consider only the case when ASE dominates. In Section V, the sta-tistical derivation for the exact expression that takes into account all the noisesources is outlined.

(17) will remain after the integration. This leads to the followingexpression:

where

(19)

The coefficients in (19) are assumed to be independentGaussian variables. As a consequence of the electric fieldgenerated by the information pulse, one of these variableswill have nonzero mean as can be deducted easily from (17).Setting the constant , which will not restrict our study,the generated current in the detector will be given by the sumof squared Gaussian variables. of them withzero mean, one with nonzero mean (due to the signal) and allof them with variance . This sum leads to a Chi-squareddistribution with degrees of freedom whose characteristicfunction may be written as follows:

(20)

where is the mean of the th component, with only theth different to zero. The density function of the Chi-squared

can be derived using the Inverse Fourier transform of (20)

(21)

which has the following close solution:

where

(22)

and where is the squared current generated by the signal atthe receiver (8) and (9) and are the modified Besselfunctions of order .

Now (22) should be integrated to obtain the system asfollows:

(23)

where is the detection threshold (amps). However, there is noclosed solution for (23) and approximation methods should beused (this paper is geared in this way) to derive them.

B. Gaussian Approximation

This approach makes use of the Central Limit Theorem whichstates that the probability density function of the sum of in-dependent variables with means and variances , can be


considered when is large enough to follow a Gaussian distri-bution. The mean and the variance of the Chi-squared distribu-tion can be obtained using the characteristic function. Withoutgoing into details about the calculation we obtain

(24)

where the one and zero logic cases have been distinguished sincedifferent signal and noise powers apply [as can be checked in(5)–(7)].

Therefore, when the following Gaussian distribu-tion is obtained:

(25)

which can be analytically evaluated, giving the following result:

(26)

where is a form of the complementary error functionthat can be written as

(27)

where the term on the right can be used when .Equation (26) is commonly approximated equating the

threshold to a value that equates its two terms leading to asimplified expression (and a threshold near optimum) given by

with

(28)

where the argument of the complementary error function is, shown in logarithmic scale in (11) [6]. Hence, the

positive effect over the system when the signal to noiseratio is increased is clearly shown since strictlydecreases with increase in its argument. The numerical valuefor the variance of the Gaussian expression is usually estimatedusing the total noise in the electrical domain [6]. This leads toaccurate predictions and has been experimentally confirmed[10]. The Gaussian expression remains valid even when ASEnoise is dominant if a large optical bandwidth is used (in theorder of 100 times or more the electrical bandwidth). This isbecause a large number of noise components will pass throughthe wideband OTBPF and reach the receiver. Therefore theCentral Limit Theorem can be invoked. However it is currentlytechnologically possible to manufacture OTBPF with narrowoptical bandwidths; in this paper an optical bandwidth equalto five times the electrical bandwidth is assumed leading to avalue of . These obviously are not the best conditionsto invoke the theorem. Another reason for this loss of accuracyof the Gaussian expression is associated with the negligiblemagnitude of the Gaussian noise sources like thermal and shot

noise (in comparison to the other sources) when SOAs areoperated in the network. However when the losses between theWCR and the receiver are large enough, the Gaussian sourcesof noise become dominant (because of the attenuation of theASE components introduced by the SOA) which makes thisexpression the most accurate. A complete expression that takesinto account both sources of noise (ASE and post-detectionGaussian sources) is presented in Section V. However, this iscomplex and its use is only advised if neither the Gaussianelectrical noise nor the ASE dominates. It is the system designerwho is to decide which model (Chi-squared if ASE is dominantand Gaussian if thermal and shot noises are dominant) is moreappropriate.

C. Saddle-Point Approximation

An accurate and tight approximation of (21) can be obtainedusing the Saddle-point approximation (also known as steepestdescent method). This is achieved by carrying out the followingsimplification in the denominator of the integrand:

(29)

where and are imposed as conditions.These conditions are not strict since the resultant expressionprovides excellent results even if they are not fulfilled [9].

The Saddle point approximation then leads to the followingexpression for the probability density function:

(30)

where obviously the one and zero cases must be distinguishedby using the correct signal and noise powers in each case.

To make the different approximations comparable, a varianceequal to should be imposed in (30), so that the followingexpression for is determined:

(31)

(32)

where once again the zero and one cases have been distin-guished. We should emphasize that (31) and (32) representthe variance of the real and imaginary parts in the coefficientspresented in (17) and not those applied in (26) and (28).

The Gaussian and Saddle-point probability density functionsare shown in Fig. 4 for a typical set of values. It can be seenthat the Gaussian expression becomes more inaccurate (ASEdominant case) when points away from the distribution max-imums ( and ) are considered. Another impor-tant detail is the threshold position at the receiver. This will beplaced in a certain optimal point that gives the minimum valuefor the system . This point corresponds to the intersection ofthe zero and one probability density functions which is a pointbetween about 0.3 and 0.7 times . It is shown in Fig. 4that using the Gaussian expression, the threshold would be set


Fig. 4. Gaussian and Chi-squared probability density functions for � �

� dBm, � � �� dBm, � � �, � � �� dB and � � ��.

at whereas with the more accurate one it wouldbe at . As a general rule the Gaussian expres-sion will find the optimum threshold position below comparedto more accurate expressions. To derive one expression for thesystem , expression (30) has to be integrated. This operationis simplified using the following change of variables:

(33)

which leads to the following integral expression:

(34)

that has to be integrated in the logic zero and one cases withinthe following limits:

(35)

Expanding the denominator of the integrand in (34) in a first-order Taylor series around the lower limit (higher contribu-tion to the integral) and letting the upper one go to infinity, aclose solution for (34) is found. This is an original and accuratecontribution which is useful in quantifying the performance ofWCRs based on XGM in SOAs as used in WDM networks

(36)

(37)

where

(38)

then the system may be written as follows:

(39)

As will be shown in Sections III-D and E this expression of-fers excellent results when the ASE noise dominates over theother sources. This is a common case when SOAs are present inthe system.

D. Chernoff Bound

The Chernoff bound is presented as an alternative to derive. An upper and extremely tight bound can be obtained

with this method when a density function tail is integrated fromto infinity [11]. The resultant expression, as will be seen when

the results are presented, will offer higher (pessimistic) valuesthan (37). The Chernoff bound states

with (40)

where is a parameter to be optimized in order to get thetightest bound. This can be achieved by carrying out thefollowing operation:

(41)

where is the moment generating function (MGF) of aChi-squared statistical distribution. The MGF is closely relatedwith the characteristic function (20) [11]. In this case, the MGFis easily obtained from (20) through the change of variablesto

(42)

Substituting (42) in (41), the following value for is obtained:

(43)

Then the new expression for the probability of detecting alogic one when a zero is being transmitted may be written as

(44)

where (43) has been used. Combining (44) and (36) a new andextremely accurate expression for the system probability oferror is obtained.


Fig. 5. � versus normalized threshold for � � � dBm, � � �� dBm, � � � and � � ��.

E. Detection Threshold

In Fig. 5, three sets of curves can be observed. In eachone has been represented using the three available closedexpressions for (Gaussian, Saddle point, and Chernoff)versus the normalized threshold for three values of the lossbetween the WCR and the detector. When the normalizedthreshold moves from zero to one the threshold at the receivermoves from to . The first observation to be madein Fig. 5, is that the probability of error in the system increaseswith increase in the losses. Also, when this happens, theoptimum threshold at the receiver become higher. It is clearlyshown that for dB and the threshold isplaced around halfway between and almostdouble than obtained when dB. This shift is explained(Fig. 4) by the variation in the level of the electrical currentgenerated when a zero or one level is transmitted. Althoughthe extinction ratio remains (as can been checked using(8) and (9)) the difference between themis reduced when is increased. Another reason is that thenoise terms for both cases (logic one and zero) will becomecloser when the losses are increased since there will be weakcontribution from the signal to the noise at the receiver. Fur-thermore it is confirmed that the Gaussian approximation offerslower threshold values than those that are assumed by usinga Chi-squared distribution. The expression (38) (referencedin the rest of the paper as Saddle point approximation) offersslightly lower values for the threshold than the approximationthat also uses the Chernoff bound (44) (referenced in the rest ofthe paper as Chernoff approximation) as may be deduced fromthe preceding discussion. The position of the threshold usingthe Gaussian expression when the ASE noise componentsare dominant over other Gaussian sources could lead to adeficient system performance even when the former maypredict the opposite (proper functioning). This can be easily

checked by observing the set of curves for dB. Usingthe Gaussian expression, the normalized threshold is set at

; hence . However using a more accurateexpression like the Chernoff approximation it can be seen thatat this threshold value the system actually works at ,five orders of magnitude above the Gaussian prediction and atthe limit of the system specifications. Therefore as a summaryof Fig. 5, we note the high sensitivity and dependence of thesystem on the position of the threshold at the receiver.

IV. RESULTS AND DISCUSSION

A. System Description

In this section, we will present a variety of results to show theinterdependence between the most important parameters of thesystem giving special attention to the pump and probe powers,extinction ratio and the spontaneous emission factor. Addition-ally, a comparison between the expressions for obtained ear-lier will be given. Those are, Gaussian approximation with the-oretical threshold (28), Gaussian approximation with variablethreshold (26), Saddle point plus Taylor approximation (refer-enced in the rest of the document as Saddle point approxima-tion) (38), and Saddle point plus Taylor plus Chernoff bound(referenced in the rest of the paper as Chernoff approximation)(44). A constant SOA gain with the input wavelength will beassumed. In practice, however, the SOA gain shows spectral de-pendence. This means that the extinction ratio degradation ob-served is wavelength dependent and hence the performance ofthe wavelength converter in terms of error rate depends on thewavelength of operation. This aspect is not considered in thecurrent work. The inputs will be centered at nm and

nm for the pump and probe signals [4]. It will alsobe assumed that our system is operated in a MAN working at2.5 Gb/s per channel and where which lets us ignore


TABLE IVALUES FOR THE PARAMETERS IN THE SIMULATION

the limitations imposed by the limited SOA bandwidth. There-fore in our system the signal will be degraded mainly in twoways, extinction ratio and added ASE noise. In this section wewill also assume large values for in the order of ten andsmall extinction ratios in the order of five or ten. In such sys-tems, the maximum number of conversions will be determinedby the extinction ratio degradation and added ASE noise in eachstage of conversion. These will affect the SNR at the receiverand therefore the system . Table I, summarizes the values usedfor the simulation. The rest of the parameters are functions ofthose presented in Table I.

B. Results and Comparison

In Fig. 6, the dependence of the bit error probability on thespontaneous emission factor can be seen for two extreme valuesof the extinction ratio. It is observed how increases when thespontaneous emission increases. An improvement of ten ordersof magnitude in is observed when is varied from five to tenregardless the value of . We see that when the admis-sible limit for is two if the Chernoff approximation is usedand ten if the Saddle point is used. Furthermore whenit will be highly unlikely that the system will reach the limit10 even if increases. Therefore, the system in this casewill be robust with respect to the noise. The Gaussian expres-sions (referenced in the figures as Gaussian and Gaussian with

theoretical threshold) stay all the way under the other approx-imations and this distance is larger when lower values forare reached. Then, a difference smaller than two orders of mag-nitude is found between the Saddle and Chernoff expressions.This difference remains constant when is increased for bothvalues of . On the other hand, the Gaussian approximationsseparate from the other approaches by two orders of magnitudewhen to more than five when . As a summary,we can realize the high sensitivity the system performance haswith respect to a variation in the extinction ratio and ASE. InFig. 7, the bit error probability is shown versus the extinctionratio for two extreme values of . It can be observed thatdecreases fast when is increased for both values of . When

varies from five to nine with a value of decreasesaround eight orders of magnitude if the Chi-squared model isused (Saddle and Chernoff approximations). It is seen how thedistance between the Gaussian approximations increases whenhigher values of are reached. A subtle effect, hidden in Fig. 6,is clearly shown in this one, the convergence between the twosets of curves ( and ) when reaches very low values.This occurs because when is too low it becomes the main causeof system errors independently of the particular value of .For instance, when the distance between the two sets ofcurves is around four orders of magnitude while it is less thantwo orders when . Fig. 6, in fact will help in setting themaximum allowed degradation in the extinction ratio. The min-imum extinction ratio for is around 5 for the new ex-pressions and around 4.5 for the Gaussian. When , theminimum extinction ratio is around 5.5 for the new expressionsand around 5 for the Gaussian expressions. It is also shown thatthe Chernoff approximation is always above (in ) the otherapproaches, being one order of magnitude over the Saddle pointapproximation but yielding comparable results. This graphwill also be quite useful to study aspects related to cascadabilityin such systems.

In Fig. 8, the dependence of on the pump signal powercan be seen for two extreme values of the spontaneous emissionfactor. The existence of an optimum pump power that is higherwhen increases is shown. This optimum power is around

5 dBm and 3 dBm for and , respectively. Anexplanation for the existence of the optimum power can beoffered by observing the low value used for . If the peakpower of the pump signal is high then the power for thezero logic is also high. That high logic zero power saturatesthe amplifier even when there is a zero at the input (pumpsignal). Therefore the extinction ratio at the output of theSOA is degraded and consequently the SNR and system .This excessive saturation of the SOA is even more evidentwhen reaches 0 dBm, where the two sets of curvesconverges at the limit of the specifications due the inefficientmodulation of the probe signal by the WCR. Here again theChernoff bound stays high all the way offering the highestvalues for the bit error probability and is one order of magnitudefrom the Saddle approximation. A good agreement betweenthe four approximations is found when assumes lowvalues, while they separate when rises. Linked to thepreceding idea, the Gaussian approximations separate fromthe others when the bit error probability reaches lower values.


Fig. 6. � versus � for � � � dBm, � � �� dBm, and � � �� dB.

Fig. 7. � versus � for � � � dBm, � � �� dBm, and � � �� dB.

Fig. 8, may be used mainly to answer two questions. First, tofind out the minimum pump signal power needed to maintainthe specification in terms of the error probability. Second, ifthe possibility of amplification of the incoming signal exists,or if the losses before WCR can be controlled (e.g., throughcareful placement of WCR), the pump signal power can beadjusted to reach the optimal system performance. As can bededuced from the discussion in this graph WCRs based onXGM in SOAs cannot be placed arbitrarily within the opticalnetwork. Depending on the pump signal power different systemperformance is obtained and if this power is too high or weakthe system may be driven out of the specification. Fig. 8, showsthat there is an error of around 1 dB in sensitivity predictionbetween the Gaussian and Saddle approximation and 1.5 dBbetween the Gaussian and the Chernoff approximations. Itshows a discrepancy between the two expressions based onthe Chi-squared model of about 0.5 dB.

In Fig. 9, the probability of error versus the pump signalpower is represented for three values of the extinction ratio. As

can be deduced from the discussion above, an improvement inthe extinction ratio shifts to the right the optimal pump power tobe injected. This is accompanied by a movement of the curvesto lower values of the probability of error as a consequence oflarger values of and hence larger SNRs. For the lower set ofcurves the system will decrease with until anoptimal value around dBm and the system remainswithin the specifications even for dBm (wherestays around 25 orders of magnitude under the limit). In thiscase, it will be more difficult for the pump signal to saturatethe device when logic zero is transmitted. Unfortunately, it isnot always possible to reach such high values of in this typeof systems and the designer will have to set the parameters inorder to have the most robust working point. Now again it is ob-served that the Gaussian approximations deviates considerablyfrom the results that follow the Chi-squared model when lowervalues for are reached. A final observation to be made inFig. 9, is the difference in sensitivity predictions obtained witheach expression. The Gaussian expression leads to sensitivity


Fig. 8. � versus � for � � �� dBm, � � �, and � � �� dB.

Fig. 9. � versus � for � � �� dBm, � � �, and � � �� dB.

values different by 1 dB from the Saddle point approximation,and 1.25 dB compared with the Chernoff approximation. TheChernoff approximation remains all the time one order of mag-nitude over the Saddle point approximation offering differencein the predictions of about 0.25 dB.

In Fig. 10, the system probability of error is shown versus thelosses between the WCR and the optical receiver with the pumpsignal power as a parameter. It can be observed once more thatthe best performance is obtained with the intermediate value forthe pump signal power dBm which is also deducedin the results above. A steep increase in occurs as a result ofthe increase in the losses. This effect is more pronounced for thebest cases ( and 0 dBm) where the system goes outof the specifications by a few dB showing in this way a high sen-

sitivity to the losses. For instance for dBm, whenmoves from 21 to 24 dB an increment of around 20 orders

of magnitude is found for all of the approximations. Therefore,it is clear that the system is more sensitive to the losses whenthe initial conditions are better (lower probability of error). Thecurves tend to converge in the worst case dBmwhen increases, since the losses are then the dominant causeof error in the system. This kind of behavior has already beenstudied above. In the case dBm the Gaussian ap-proximations are one order of magnitude under those given bythe Saddle point expression offering sensitivities 1 dB lowerthan the Gaussian. The difference is slightly higher using theChernoff approximation which gives a value 1.25 dB lower forthe sensitivity. When dBm, the values offered for


Fig. 10. � versus � for � � �� dBm, and � � �.

the Gaussian approximation for the sensitivity are under thoseobtained by the Saddle point and Chernoff expressions; around0.25 dB, and 0.5 dB, respectively. It is worth noting that for

dBm the Gaussian and Saddle point approxi-mations offer the same values for sensitivity while the Chernoffexpression deviates by around 0.25 dB.

In Fig. 11, the system probability of error is presented versusthe pump signal power for three typical values of the probe signalpower. Observing the curves presented, the strong dependenceof the system performance on is clear. For instancefor dBm and dBm estimationsaround are found while is predictedwhen dBm. Therefore, only 5 dB additionalprobe signal power leads to an improvement of around 40orders of magnitude in the probability of error. Furthermore inthis case, as in Fig. 9 a shift in the optimal value of isobserved when the pump signal power decreases, although lesspronounced now; there is a difference of around 1 dB betweenoptimal values for dBm and 15 dBm.This shift can be explained by the higher level the signal has nowwith respect to the noise which allows lower extinction ratios.The convergence and even cross between the Saddle point andGaussian approximations should be noted on the left hand sidein Fig. 11. In the right-hand side the Gaussian approximationsseparate from the Saddle point by more than three orders ofmagnitude and three and a half from the Chernoff approximation(when dBm). Furthermore, when the system

is at the limit of the specifications dBm , itis observed that the Chernoff approximation offers values ofsensitivity around 0.5 dB and 1 dB over the Saddle pointand Gaussian approximation, respectively. Fig. 11, offers amethod to adjust the power at the input of the probe laseras a function of the incoming signal power (pump power)injected in the SOA for a determined value of noise, lossesand extinction ratio. Alternatively Fig. 11, may be used to findthe necessary pump power at the input for a fixed value ofthe probe power. To do so the system designer could draw ahigher number of curves for different values of , and thenaccurately determine the system power requirements. Whathas been observed in the paragraph above can be extendedto any of the figures presented in this section which makesthis study a powerful method to design systems that utilizeWCRs based on XGM in SOAs.

In Fig. 12, the probability of error is represented versus theextinction ratio at the input of the WCR for typical values of theprobe signal power. The improvement grows when increasesand this is more marked when the probe signal power is higher.For instance if dBm and increases from 6 to 8,

will decrease around 20 orders of magnitude which clearlyshows high sensitivity in our system on this parameter. There-fore once more we derive the same conclusion; WCRs basedon XGM in SOAs cannot be placed arbitrarily within an opticalWDM network. The Saddle point and Chernoff approximationsoffer values for higher than the Gaussian for the same system


Fig. 11. � versus � for � � �, � � ��, and � � �� dB.

specifications since the curves for the first are always over thoseobtained with the Gaussian approximations. Convergence is ob-served between the three sets of curves when takes lowervalues and starts being the main cause of error in the system.For the lowest values of , the Gaussian approximations sep-arate from those that follow the Chi-squared model. This dif-ference between predictions is around five orders of magnitudewhen and dBm and around twenty when

dBm. However, the Saddle point and Chernoffapproximations present good consistency with the Chernoff al-ways predicting higher probabilities of error. Therefore, Fig. 12,may be used to find the maximum degradation (as far as extinc-tion ratio is concerned) that a travelling optical signal can sufferwhen it goes through a determined number of conversion stages.In systems that offer low values of ASE, the maximum numberof stages is determined by studying the extinction ratio degrada-tion in each conversion. This problem is called cascadability andfurther work is needed in the type of systems we are considering.

In Fig. 13, the system probability of error is representedversus the spontaneous emission factor for a set of valuesfor the probe signal power. In analogy to Fig. 12, a greatdependence of the system performance on the values of thisparameter is observed that is even higher when growsor alternatively when we have better initial conditions (lower

). It is noticed how a higher injection of power from theprobe laser is extremely effective in compensating the effectof the noise on the system. When the system is at the limit of

the specifications dBm and if the probe poweris increased to dBm we obtain a value for thesystem error probability of or less. Obviously,this is a consequence of the improvement experienced in theSNR. There is consistency between the Saddle point and Cher-noff approximations, however, the Gaussian approximationsseparates when takes the lowest values. This differencereaches its maximum value for and is around fiveorders of magnitude when dBm and aroundthirty when dBm. Once more we can studythe cascadability of these devices in a WDM network. If highvalues of (achieved by spectral optimization [5]) are assumed,Fig. 13 can be used to find the maximum noise allowed in thesystem and hence the maximum number of stages that can beplaced in an optical link. Furthermore by combining Figs. 12and 13 a complete method to study the cascadability can beformulated taking into account the two main undesired effectswhen the SOA is operated far from its bandwidth limit. It isalso worth observing that low values of the error probabilityhave been shown in Figs. 12 and 13 to highlight the fast andsignificant change is error rate as the extinction ratio, orthe probe power change in this class of wavelength converters.

V. RANGE OF APPLICABILITY

It is of interest to establish the conditions under which eachof the approximations discussed can and should be applied.


Fig. 12. � versus � for � � � dBm, � � �, � � �, and � � �� dB.

Fig. 13. � versus � for � � � dBm, � � ��, and � � �� dB.

The exact expression for the characteristic function of thenoise at the receiver can be found in [12]. This expressiontakes into account both kinds of noise sources: ASE noise andGaussian (thermal and shot noise) and assumes an arbitrarypulse shape (in this paper we have assumed a standard NRZcodification). A closed expression for this distribution can bederived using the Saddle point approximation (steepest descentmethod) yielding the following expression:

(45)

where

(46)

and

(47)


Fig. 14. Complete, Chi-squared, Gaussian probability density functions for � � �� dB �� .

The value for is obtained as the result of the followingequation:

(48)

The variance of the exact distribution (before applying theSaddle point approximation) can be written as [12]

(49)

with

(50)

If we equate the variance in (49) to the total noise in the electricaldomain a new expression for the variance may be presented as

(51)

The expression in (45) is too complex to be used to find theerror probability in the system and requires numeric methodsto be integrated. However, we can study it in three differentsituations: when ASE noise is dominant ,Gaussian noise is dominant and balancedcase . These three cases are presented inFigs. 14–16 using the Gaussian (25), Chi-squared (30), and

the complete (45) expressions. In Fig. 14 the accuracy ofthe Chi-squared model is clearly shown when ASE noise isdominant. In Fig. 16 it is shown how the Gaussian expressionoffer the best matching with the complete expression whenGaussian noise is dominant. In Fig. 15, a balanced case canbe seen. Furthermore as the results based on the Chi-squaredmodel offer larger values (pessimistic) for the error probability,it is recommended to use this model in the WCRs case since itwill ensure an error free functioning in our system.

VI. CONCLUSION

A more accurate statistical description of the noise asso-ciated with WCRs based on XGM in SOAs was presented.The analysis was used to demonstrate the effect of the varioussystem parameters on the probability of error. In particular,consideration was given to the effect of the pump power,probe power, extinction ratio, ASE noise, the propagation lossbetween WCR and receiver and the nonlinear SOA gain. Theresults have demonstrated that the Gaussian approximation,widely used, is optimistic and its use should be restricted tosystems in which Gaussian sources of noise (thermal and shot)are dominant. A more exact treatment using Chi-squared ap-proach was given valid for system where ASE noise is presentand that ensures safe and reliable predictions of performance.The current analysis allows further questions to be asked andperhaps answered relating to the placement of WCRs in WDMnetworks and the impact of other active components on WCRs.




REFERENCES

[1] J. M. H. Elmirghani and H. T. Mouftah, “Technologies and architecturesfor scalable dynamic dense WDM networks,” IEEE Commun. Mag., vol.38, pp. 58–66, Feb. 2000.

[2] , “All-optical wavelength conversion: Technologies and applica-tions in DWDM networks,” IEEE Commun. Mag., vol. 38, pp. 86–92,Mar. 2000.

[3] S. J. B. Yoo, “Wavelength conversion technologies for WDM networkapplications,” J. Lightwave Technol., vol. 14, pp. 955–965, June 1996.

[4] S. I. Pegg, M. J. Fice, M. J. Adams, and A. Hadjifotiou, “Noise inwavelength conversion by cross-gain modulation in a semiconductoroptical amplifier,” IEEE Photon. Technol. Lett., vol. 11, pp. 724–726,June 1999.

[5] E. Willner and W. Shieh, “Optimal spectral and power parameters forall-optical wavelength shifting: Single stage, fanout, and cascadability,”J. Lightwave Technol., vol. 13, pp. 771–781, May 1995.


[6] N. A. Olsson, “Lightwave systems with optical amplifiers,” J. LightwaveTechnol., vol. 7, pp. 1071–1082, July 1989.

[7] B. Ortega, J. Company, and J. L. Cruz, “Wavelength division multi-plexing all-fiber hybrid devices based on Fabry-Perot’s and gratings,”J. Lightwave Technol., vol. 17, pp. 1241–1247, July 1999.

[8] M. Shamoon, J. M. H. Emirghani, and R. A. Cryan, “Erbium dopedfiber amplifier systems with fiber Bragg grating optical filters,” J. Opt.Commun., vol. 20, pp. 188–193, Oct. 1999.

[9] D. Marcuse, “Derivation of analytical expressions for the bit-errorprobability in lightwave systems with optical amplifiers,” J. LightwaveTechnol., vol. 8, pp. 1816–1823, Dec. 1990.

[10] N. S. Bergano, F. W. Kerfoot, and C. R. Davidson, “Margin measure-ments in optical amplifier systems,” IEEE Photonic Technol. Lett., vol.5, pp. 304–306, Mar. 1993.

[11] J. G. Proakis, Digital Communications, 3rd ed, ser. Elect. Eng. Se-ries. New York: McGraw-Hill, 1995.

[12] D. Marcuse, “Calculation of bit-error probability for a lightwave systemwith optical amplifiers and post-detection Gaussian noise,” J. LightwaveTechnol., vol. 9, pp. 505–513, Apr. 1991.

Manuel Muñoz de la Corte was born in Cadiz, Spain, in 1977. He received theB.S. and M.S. degrees in telecommunications engineering from the Universityof Seville, Seville, Spain.

During 2000–2001, he was a Visiting Research Student in the Communica-tions System Laboratory, University of Wales Swansea, Swansea, U.K.

Jaafar M. H. Elmirghani (M’92–SM’99) workedin 1991 as a Research Engineer with the HighSpeed Optical Fiber Communications Group at GECHirst Research Center, U.K. Between January andSeptember 1994, he was a Postdoctoral ResearchFellow with the Manchester Metropolitan University,U.K. Subsequently, he joined the University ofNorthumbria, U.K., as a Senior Research Fellow. InFebruary 1995, he was appointed Senior Lecturer,and in February 1998, a Principal Lecturer and BTReader in Telecommunication Systems, all with the

University of Northumbria. He spent January to September 1999 as a RoyalSociety Sponsored Visiting Professor at Queen’s University, Kingston, ON,Canada. In 2000, he joined the University of Wales Swansea, Swansea, U.K.,as Professor and Chair of Optical Communications. He currently serves astechnical editor for the Journal of Optical Communications. He has publishedover 150 technical papers, co-edited Photonic Switching Technology—Systemsand Networks (New York: IEEE Press, 1998) and has research interests inoptical communication systems, signal processing, and communication theory.

During 1996 and 1997, he was secretary for the IEEE COMSOC SignalProcessing and Communications Electronics (SPCE) Technical Committee,then Vice Chairman, and, since 2000, has served as Chairman for the samecommittee. He is also Secretary of IEEE COMSOC Transmission, Access,and Optical Systems Technical Committee. He was an Editor of the IEEECommunications Magazine and has been on the technical program committeeof several IEEE ICC/GLOBECOM conferences. In particular, he acted asChairman for the IEEE GLOBECOM’99 Advanced Signal Processing forCommunication Symposium and will chair the same symposium in ICC’03 andICC’04. He was Chairman of IEEE GLOBECOM’00 and GLOBECOM’01Future Photonic Network Technologies, Architectures and Protocols Sym-posium. He is a member of IEEE Comsoc Technical Affairs Council and amember of IEEE Comsoc Nominations and Elections committee. In 1995,he was Co-chair for the SPIE Wireless Data Transmission Conference. Hewas a member of the technical program committee for SPIE Optical WirelessCommunication in 1997–1999. He is a Member of the Institution of ElectricalEngineers (lEE) and currently serves as Vice Chairman to the IEEE UK&RICommunications Chapter.

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Accurate noise characterization of wavelength converters based on XGM in SOAs

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