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The dynamic characteristics, including the crosstalk and relaxation oscillation, of linear optical amplifiers (LOAs) are investigated by small-signal analysis under an averaging carrier density approximation and compared with the results of numerical

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636 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY 2005
Numerical and Theoretical Analysis of the Crosstalk in Linear Optical Ampliﬁers
Chao-Yuan Jin, Yong-Zhen Huang
, Senior Member, IEEE
, Li-Juan Yu, and Sheng-Ling Deng
Abstract—
The dynamic characteristics, including the crosstalkand relaxation oscillation, of linear optical ampliﬁers (LOAs) areinvestigated by small-signal analysis under an averaging carrierdensityapproximationandcomparedwiththeresultsofnumericalsimulation. Thegoodagreementbetweenthenumericalsimulationand the small-signal analysis indicated the averaging carrier den-sityisanappropriateapproximationforanalyzingLOAs.Theoret-ical analyzes also show that the dynamic properties of the verticallaser ﬁelds dominate the dynamic performance of LOAs. Based onthesmall-signalanalysis,aconciseequationforthecrosstalkunderhigh bit rate was derived, which can be applied to measure the dif-ferential gain of LOAs.
Index Terms—
Crosstalk, linear optical ampliﬁer (LOA), re-laxation oscillation, semiconductor optical ampliﬁer (SOA),small-signal analysis.
I. I
NTRODUCTION
T
HE linear optical ampliﬁer (LOA), which achievesgain clamping by integrating vertical-cavity lasers(VCLs) on the direction perpendicular to the cavity, was in-troduced recently [1]. It was demonstrated that LOAs wouldbe hopefully implemented into the wavelength-division mul-tiplexing (WDM) networks [2]–[4]. Great advantages, such
as low crosstalk, no switch transient [1], low noise ﬁgure andwide-band gain clamping [5], have been studied experimentallyand theoretically for LOAs.MathematicalmodelsarerequiredtohelpthedesignofLOAsand to predict their operational characteristics. We have devel-opeddetailedmodelsandalgorithmstosimulatethestate-steadypropertiesofLOAs[5]andgain-clampedsemiconductoropticalampliﬁers (GCSOAs) [6]. Dynamic simulations for the relax-ation oscillation of LOAs were introduced in [7]. However, thedynamic simulation of LOAs is still a time consuming task. Itwould be advantageous if a simple and convenient method wasdeveloped to theoretically predict the properties of LOAs.For semiconductor lasers, the small-signal analysis is a pow-erful method to predict the dynamic performance of the lasers.However, the signiﬁcant longitudinal carrier spatial inhomo-geneityaswellasgainsaturationpreventsthesmall-signalanal-ysis beingused for the analysisof traditional semiconductor op-tical ampliﬁers (SOAs). An approximate approach, which phe-nomenologically took into account the inhomogeneous gain,
Manuscript receivedSeptember28, 2004;revisedJanuary 4,2005.This work was supported in part by the National Nature Science Foundation of Chinaunder Grant 60225011, the Major State Basic Research Program under GrantG2000036606, and the project of “863” plan under Grant 2003AA311070.The authors are with the State Key Laboratory on Integrated Optoelectronics,Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,China (e-mail: yzhuang@red.semi.ac.cn).Digital Object Identiﬁer 10.1109/JQE.2005.845354
was developed for the small-signal analysis of SOAs [8]. Thelongitudinal carrier density of LOAs keeps nearly constant be-fore gain saturation [5], so we can expect that the averaging car-rier densities will be a reasonable approximation for dynamicanalysis of LOAs.In this paper, we propose a small-signal method using av-eraging carrier density approximation based on an averagingphoton density model [9] to quantitatively investigate the dy-namic characteristics of LOAs, and numerically simulate thedynamic performance of LOAs including the crosstalk and re-laxation oscillations. The good agreement between the numer-ical results and small-signal analysis demonstrates that aver-aging carrier densities is an appropriate approximation for thesmall-signalanalysisofLOAs.Thetheoreticalresultsalsoshowthat the dynamic characteristics of LOAs are greatly dominatedby the properties of vertical cavity lasers.II. D
YNAMIC
M
ODEL
In this section, we develop a detailed dynamic model forLOAs based on the steady-state model of LOAs [5] and the dy-namic model of SOAs [10]. As a matter of convenience, the rateequations are written down in the form of photon densities asfollows:(1)(2)(3)where isthephotonfrequency, isthefrequencyofthelaserﬁeld, aretheforwardandbackwardpropagatingphotonden-sities of the signal and spontaneous emission, is the photondensity of the lasing light in VCLs, and representthe spontaneous emission densities coupled into the ampliﬁedspontaneous emission (ASE) and the vertical laser ﬁeld, is thematerial gain coefﬁcient, is the volume of the active region,is the optical conﬁnement factor, is the optical conﬁnementfactoroftheverticalcavitylaser(VCL), isthegroupvelocity,
0018-9197/$20.00 © 2005 IEEE
JIN
et al.
: NUMERICAL AND THEORETICAL ANALYSIS OF THE CROSSTALK IN LOAs 637
TABLE IL
IST OF
P
ARAMETER
V
ALUES
Fig. 1. Schematic structure of a LOA longitudinally divided into
sectionsof equal length for numerical simulation.
is the absorption loss, is the threshold gain of the VCL.And the recombination rates is given by(4)where , , and are the coef
ﬁ
cients of defectrecombination, radiative recombination and Auger recombina-tion, respectively. The values of these parameters are listed inTable I.We can investigate the dynamic characteristic of the LOAsby solving the rate (1)
–
(3) in the time domain. However, as therate equations cannot be solved analytically, we should dividethe longitudinal direction of the LOA into a number of sec-tions for numerical solutions. Fig. 1 is a schematic structureof LOAs divided into sections with equal length and eachsection is treated as a small VCL, where and are thephoton densities of the signal and spontaneous emissions,and are the power re
ﬂ
ection coef
ﬁ
cients at the LOAs
’
inputand output facets, is the power re
ﬂ
ectivity of the VCL, andis the length of each VCL. is used in the numericalsimulation.We simply take the material gain spectrum of bulk materialsfor simulation [11], [12] as follows:
(5)where and are the Fermi
–
Dirac distributions in theconduction band and valence band, is the refractive indexof the active region, and are the effective mass of theelectron and the heavy hole, the radiative carrier recombinationlifetime is de
ﬁ
ned by ,and is the bandgapenergy with the following bandgap shrinkage [13]:(6)where is the bandgap shrinkage coef
ﬁ
cient, and is thebandgap energy without the injected carriers.The material gain can also be expressed as the stimulatedemission minus the stimulated absorption as follows:(7)where and are the rates per unit length of stimulatedemission and absorption. can be written as(8)The spontaneous term can be written as [5](9)where and are the width and thickness of the active region,respectively, and is the frequency interval of the ASE spec-trum. of the vertical cavity laser can take the sameform as(10)where is the longitudinal mode interval of the VCL.Takingintoaccountthenonlineargainduetothecarrierspec-tral-hole-burning, we simply rewrite the material gain as [14](11)where is the gain compression coef
ﬁ
cient and is the totalphoton density in the cavity of each VCL.III. S
MALL
-S
IGNAL
A
NALYSIS
In the steady-state simulation, the carrier density along thedevice
’
scavitykeepsa nearlyconstantvalue beforegainsatura-tion[5].Thus, indynamicsmall-signalanalysis, weassumethatcarrier density maintains the same value in the cavity directionapproximately. The approximation of averaging photon density
638 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY 2005
[9] is introduced, which is more suitable for LOAs than for con-ventionalSOAs.Ignoringthespatialvariationsofthecarrierandphoton densities, we can write (1) and (2) under averaging car-rier densities and photon densities approximations(12)(13)where istheASEpartinthecarrierrateequation, andare the material gain of signal and lasing light respectively.Ignoring the cavity re
ﬂ
ectivities, can be approximatelywrote as [9](14)where isthepeakgainofthegainspectrum, isthelengthof LOA, istheequivalentspontaneousemissionfactorforLOAs,and is the single-pass gain of peak gain wavelength(15)In the simulation, we choose the peak gain wavelengthm. Under small-signal approximation, the aver-aging signal photon density in the cavity can be written as [9](16)where presents the input signal density and is thesingle-pass gain of signal wavelength.ToobservethedynamicalresponseofLOAstoaperturbation,wetakethedifferentialof(12)and(13)andobtainthefollowingequations by ignoring the change of relative to carrierdensities(17)(18)where represents the differential of variables, and the super-script0representsthesteady-statetermoftheTaylorexpansion.and are the differential gain coef
ﬁ
cients of signal andlasing light, respectively, and the differential carrier lifetimecan be de
ﬁ
ned as(19)Combining (17) with (18) and eliminating the , we get asecond-order differential equation of (20)where(21)(22)Equation(20) is asecond-order differentialequationintheformof over damped vibration, where is the relaxation resonancefrequency and is the damping factor. If the signal photondensity be much smaller than lasing light photon density alongthe longitudinal direction, the relaxation resonance frequencyand the damping factor can be approximately written as(23)which are equal to the relaxation resonance frequency and thedamping factor of vertical cavity lasers [15], so the propertiesof the vertical laser
ﬁ
elds will dominate the LOAs
’
dynamiccharacteristics.Equations (17), (18), and (20) are main results for the LOAsunder the small-signal approximation. Solving these equationsunder different initial conditions, we can conveniently predictthe dynamic performance of LOAs.Taking into account the nonlinear gain in the small-signalanalysis, the correction of nonlinear gain will be determinedby practical value of . We simply choose the as the aver-agingvalue of wholeLOA cavity. The nonlinear gain correctionis about 0.999 as 10 m .IV. R
ELAXATION
O
SCILLATIONS
First, we investigate the relaxation oscillations in a systemwith two signal channels. The probe signal has a constant inputpower and the pump signal is a square-wave, which is switchedon at ns and off at ns. We try to describe themodulation process of the relaxation oscillation by the abovesmall-signal analysis. At the rising edge of the square-wavepump signal, the initial conditions are(24)whereweassumethat keepsthesamevalueaftertherisingedge of the input square-wave signal.Solving (20) under these initial conditions, we can get thefollowing results:(25)
JIN
et al.
: NUMERICAL AND THEORETICAL ANALYSIS OF THE CROSSTALK IN LOAs 639
Fig. 2. (a) Carrier density, (b) output power of the probesignal, and (c) photondensityofthelasinglightinthecavityasfunctionsoftimeataninjectioncurrentof 150 mA with a pump signal of
0
20 dBm. The pump signals are switched onat 0 ns andoff at2 ns. Thesolid lineis the resultsof numerical simulation,whilethe dashed line is the results of small-signal analysis.
where is the actual peak frequency of the resonance, whichis slightly less than the(26)Input into the (18), we can get(27)For the falling edge, the solutions will be the minus form of the(25) and (27). The expression of the probe signal can be derivedout by(28)Taking the pump signal input of 20 dBm and a constantinput probe signal of 40 dBm, we investigate the effect of thepumpsignalactingonthecarrierdensity,theoutputprobesignaland the lasing light by the small-signal analysis, and plot the re-sultsinFig.2asdashedlines.Inourmodel,wechoosethelasingwavelength at 1530 nm and the signal wavelength at 1540 nm.The device gain in this situation is about 23 dB. The modulationof the pump signal results in a modulation of device gain. How-ever,as we can seein Fig. 2, because thecarrier density changesin a very small range about 0.006 10 m , the device gainwill change very little. As a result, the probe signal output os-cillates in a small range of 0.3 dBm, and the mean value of theprobe signal keeps the same in the oscillations of the modula-tion. For the lasing light, because the lasing light will be greatlyaffected by the changes of the carrier density near threshold [5],the lasing light drops about 0.04 10 m while the carrierdensity oscillates in a verysmall rangeof 0.006 10 m andthe change of the average carrier density cannot be read out inthe
ﬁ
gure.Taking , we numerically solve rate (1)
–
(3) with thepumping square wave turns on and off at and ns, respec-tively. In Fig. 2, we also plot the results of numerical simulationas solid lines, where the carrier density and lasing photon den-sity of VCLs are averaging values over regions of theLOA. The results of small signal analysis are agreement verywell with the numerical simulations when we choose the equiv-alent spontaneous emission factor used in (14). Forthe lasing light, there is a small difference between the numer-ical and theoretical results, because little difference between thecarrier densities will affect the lasing light greatly. According to(21) and (22), is 3.2 10 s and is 3.8 10 s inthe relaxation oscillations. In our model, the small-signal anal-ysis for the relaxation oscillation is available as the input signalis less than about 16 dBm.For a large-signal injection with a square-wave of 10 dBmfrom to ns and a constant input probe signal of 40 dBm, we plot the numerically simulated responses andthe results of the small-signal model of LOA in Fig. 3 as thesolid lines and the dashed lines, respectively. In this case, thenumerical simulation shows that the average carrier densitydrops about 0.03 10 m due to the large-signal injection,which cannot be predicted by the small-signal model. And thelarge-signal injection results in the drop of the lasing light of VCLs about 2.5 10 m . As the large-signal turn off atns, the carrier densities in different regions increase andreach peak values at different times due to the different drop of carrier density at each region, and the photon density of lasinglight in different VCL reaches the
ﬁ
rst peak at different timeafter ns.V. C
ROSSTALK
To obtain the small-signal response of sinusoidal optical in- jection, we assume that solutions of all variables have the formof harmonic modulation(29)
640 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY 2005
Fig. 3. (a) Carrier density, (b) output power ofthe probesignal, and (c)photondensityofthelasinglightinthecavityasfunctionsoftimeataninjectioncurrentof 150 mA with a pump signal of
0
10 dBm. The pump signals are switched onat 0 ns andoffat 2 ns. Thesolid line is the resultsof numericalsimulation, whilethe dashed line is the results of small-signal analysis.
The crosstalk in this situation is de
ﬁ
ned as the normalizedchange of devices gain divided by the normalized change of thesinusoidal signal injectionCrosstalk (30)Input (29) into (20), we can get the solution(31)(32)Input (32) into (30), we can derive outCrosstalk (33)Equation (33) illuminates that the crosstalk of LOAs directlyproportionate to the pump signal power and the device gain.When is equal to , the crosstalk will reach a peak value.Fig. 4 compares the results of small-signal analysis with thenumerical simulation of the crosstalk. These results also agree
Fig. 4. Crosstalk as a function of the input pump signal frequency. The inputpumpsignal is
0
30dBm.Thesolid lines andhollow circlesare the small-signalanalysis results and the numerical results, respectively.
very well with the experiment results of intrinsic intermodu-lation distortion [16]. The results show that the averaging car-rier density is an appropriate approximation for the small-signalanalysis of LOAs. In Fig. 4 the crosstalk reaches a peak valuewhen the external modulation frequency is equal to the intrinsicresonance frequency . If we want to reduce the crosstalk athigh bit-rates, we should reduce . is 5.0 and 7.8 GHz atinjection current of 150 and 250 mA, respectively.The crosstalk at high bit-rates 10 GHz can be approxi-mately derived from (33)Crosstalk (34)Fordevicemeasurements,assumingthereis nomuchdifferenceforthematerialgainofthesignalandlasinglight,wecanrewrite(34) asCrosstalk (35)where presents the output signal density. So the differ-ential gain can be measured from the crosstalk in LOAs.Fig.5isthecomparisonofthesmall-signalanalysisfrom(34)and the numerical results of the crosstalk. The analytical resultsagree very well with the results obtained by the numerical sim-ulation in high bit-rates and the error is less than 1 dB. Basedon (34), we can clearly see that the crosstalk of LOAs is linearwith the input power of the pump signal, which is different withthe traditional SOAs [10].VI. C
ONCLUSION
We have developed detailed theoretical methods to investi-gatethedynamicpropertiesofLOAs.Itis demonstratedthatav-eraging carrier densities is an appropriate approximation for thesmall-signal analysis of LOAs. The large lasing power of ver-tical lasers dominated the dynamic properties of LOAs, whichis the main reason for the linear crosstalk. We also give some

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