Theoretical investigation of the role of optically induced carrier pulsations in wave mixing in semiconductor optical amplifiers

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168 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Kaunga-Nyirenda et al.

Theoretical investigation of the role of opticallyinduced carrier pulsations in wave mixing in

semiconductor optical amplifiers

Simeon N. Kaunga-Nyirenda,1 Michal P. Dlubek,2 Andrew J. Phillips,1,* Jun Jun Lim,1

Eric C. Larkins,1 and Slawomir Sujecki1

1Division of Electrical Systems and Optics, Faculty of Engineering, University Park, University of Nottingham,Nottingham NG7 2RD, UK

2M. P. Dlubek was with the School of Electrical and Electronic Engineering, University of Nottingham; now withOptical Networks, Ericsson, Coventry CV3 1JG, UK

*Corresponding author: [email protected]

Received September 4, 2009; accepted October 26, 2009;posted December 1, 2009 (Doc. ID 115862); published January 7, 2010

A theoretical investigation of the role of interband effects in wave mixing in semiconductor optical amplifiers(SOA) is reported. Carrier density modulation (CDM) caused by optical wave beating in SOAs is examinedalong with its dependence on different operating parameters. Unlike most wave mixing theories, in which theexistence and form of carrier pulsations are assumed a priori, we model the carrier dynamics and opticalpropagation in the time domain directly without invoking such an assumption. The dependence of CDM on thebias current, input power, and detuning between the pump and probe waves is investigated. Selected simula-tion results are verified experimentally. Good qualitative agreement is obtained between simulations and ex-periments for nearly degenerate wave mixing (restricted to 3 GHz by experimental limitations). © 2010 Op-tical Society of America

OCIS codes: 190.4223, 250.5980, 190.5970.

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. INTRODUCTIONemiconductor optical amplifiers (SOAs) are key elements

n modern optical communications for various applica-ions, ranging from linear amplification [1], wavelengthonversion [2], Boolean logic gates [3,4], phase conjuga-ion [5] and optical switching [6]. SOAs are particularlyttractive because of their large bandwidth, low cost, lowower consumption and ease of integration. Because ofhe superior linear amplification performance of erbium-oped fiber amplifiers (EDFAs), SOAs have mainly at-racted interest as photonic devices for nonlinear applica-ions. The strong nonlinear response of semiconductors6] compared with other materials (e.g., optical fiber)

akes the SOA well suited for optical signal processingunctions. The nonlinear optical response of the SOA,hich is a result of the interaction between strong optical

ignals and the gain medium, is responsible for self- andross-gain modulation (SGM/XGM), self- and cross-phaseodulation (SPM/XPM), and wave mixing (WM). While

ll these processes find applications in optical signal pro-essing, wave mixing has generated the most interest, be-ause of its data transparency (i.e., it preserves ampli-ude, phase, and frequency information) and bit-ratendependence [7].

In particular, WM in SOAs is expected to be importantor wavelength conversion, which increases network per-ormance by reducing the effects of lightpath blocking dueo the wavelength continuity constraint in both opticalacket and optical circuit switched wavelength-divisionultiplexed (WDM) networks [8,9]. The strength of the

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onlinearity and the intrinsic gain of the device makes ef-cient wavelength conversion possible using short inter-ction lengths, typically 1 mm. Hence, phase mismatch-ng is not the serious consideration it is for conversionsing fiber nonlinearities [10]. WM may also result inpectral inversion of the signal, which can be used for theompensation of chromatic dispersion induced by trans-ission through optical fiber, as demonstrated in [11].ther applications of WM in SOAs include all-optical de-ultiplexing of time-division multiplexed (TDM) data

treams, optical sampling, and understanding of basichenomena in inverted semiconductors.The numerous existing and potential applications of

OAs require that a clear understanding of their behaviore established. Numerical modeling provides a powerfulnd cost effective tool for performance simulation and de-ign optimization of devices operating under a wide rangef conditions. The knowledge obtained from studying theonlinear optical behaviour in SOAs can be essential forescribing and optimizing other enabling devices for newnd potential applications that rely on similar processes.n area of extreme interest is photonic integration, eithers photonic integrated circuits for performing complex op-ical signal processing to achieve all-optical functionality12] or integration with electronics as optical intercon-ects for high-speed cross-chip communications [13]. Tochieve high-density on-chip integration and integrationith electronic devices, the individual photonic compo-ents must be of the same order of size as the electronicomponents, typically one to two orders of magnitude

010 Optical Society of America

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Kaunga-Nyirenda et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 169

maller than conventional photonic devices. These inte-rated photonic devices will mainly be based on reso-antly enhanced nonlinear light–matter interactions.hotonic crystals are the most suitable candidates for thisole [12]. Becuase of the resonantly enhanced light–atter interactions and/or the small size of these devices,

he strength of the nonlinear optical effects can be veryigh even at low optical intensities. Hence, nonlinear ef-ects can become important at operating conditions thatould normally be considered linear in conventional pho-

onic devices. Although the study here is applied to a con-entional optical device (SOA), the general description ofhe optical nonlinearity presented can easily be applied tother devices.

In this paper, comprehensive modeling is used to obtainn understanding of WM phenomena in SOAs. We employuch an approach and compare the simulation resultsith experimental results from a commercial device. The

est of this paper is structured as follows. Section 2 pro-ides a general background to the WM theory, which dis-usses existing models and establishes the line of re-earch that forms the basis of our model. Sections 3 and 4riefly outline the model equations and the numerical ap-roach used to obtain their solution. In Section 5, impor-ant results from simulations and experiments are re-orted and discussed. Section 6 presents the conclusionserived from the work.

. BACKGROUNDhen two or more distinguishable optical waves of the

ame polarization state interact during their propagationhrough a nonlinear medium, new optical waves may beroduced as a result of WM. Most semiconductor crystalsre not centro-symmetric, which means that they exhibitoth second-order and third-order nonlinearities (withigher-order nonlinear contributions becoming increas-

ngly small). The new optical waves produced may be as aesult of direct four-wave interactions (intrinsic ��3� pro-ess) and/or effective four-wave interactions resultingrom two successive three-wave processes (cascaded�2� :��2� process) [14]. Experimental evidence suggestshat such cascaded second-order processes may indeed beresent in semiconductors [15]. The cascaded processesccur at different points in time and will be nonlocal if theuccessive processes take place in spatially separate re-ions. Such processes involve a real photon (an r.f. pho-on) generated by the first second-order process and ab-orbed by the second ��2� process. As discussed in [16] forhe case of carrier density modulation (CDM), the forma-ion of the dynamic gratings and the generation of theonverted signal (through scattering from these gratings)re entirely distinct processes. Externally, the overallcascaded ��2� :��2�) process appears as a third-order pro-ess. Further evidence that cascaded ��2� :��2� processeslay a role in CDM is the experimental detection of THzadiation in a four-wave mixing (FWM) experiment [17].he detected THz power was very low (sub-nW) since theeometry of standard SOA and semiconductor laser struc-ures provides wave guiding for near-infrared waves, butot THz waves, whose wavelength is generally in the tensf micrometers. Hence, the propagation of THz radiation,

nd consequently its effects away from the region of inter-ction, are neglected for the SOA dimensions considered.ince it is not possible to distinguish externally whetherhe experimentally observed new optical waves are due tontirely intrinsic ��3� processes or cascaded ��2� :��2� pro-esses or a combination of both, they can be described inhe same framework by emphasizing their effects. As aonsequence, we refer to all WM processes loosely asWM in the remainder of the paper.In non-degenerate FWM (NDFWM), optical beams at

ifferent frequencies are input into the SOA. A number ofhysical mechanisms have been proposed to mediate ND-WM in semiconductor media. These can generally belassified as either interband or intraband effects [18]. Inoth cases, beating between the different optical wavesxcites the material at the beat frequencies through theodulation of various parameters of the active medium

19]. This excitation creates dynamic index and gain grat-ngs. The dynamic gratings diffract a third signal, whichncidentally can also be one of the signals responsible forreating the beat frequency interaction, to generate newrequencies. The characteristics of the new waves gener-ted in the process largely depend on the physical fea-ure(s) of the active medium being modulated [20]. Whenhe frequency separation between the input waves is lesshan the reciprocal of the rate of carrier replenishmentnto the active region, the total carrier density respondso the beat frequency excitations, resulting in CDM. Atarger frequency separations, the response of the carrierensity decreases. At such fast time scales, dynamic grat-ngs also result from the modulation of the occupationalrobability within a band. These intraband dynamicffects—spectral-hole burning (SHB) and carrier heatingCH)—lead to a significant contribution to WM at detun-ngs in excess of the reciprocal of the carrier replenish-

ent rate. It is important to note that while CDM domi-ates at lower frequencies, it is still important at higherrequencies [21].

FWM studies in nonlinear media have largely focusedn observations in the optical domain [18,20], with em-hasis mainly on the issue of conversion efficiency [22]. Inost of the published models for FWM [18,20,23,24], theain task is to calculate the polarization of the gain me-

ium at each frequency that is induced by the interactionmong different frequency components of the optical sig-als and the active medium. The assumption in all theseodels is that the carrier density and the occupational

robabilities in the bands oscillate at the detuning fre-uency and its harmonics, thereby giving rise to the in-uced polarizations [18]. The idea of pulsations in the car-ier population arose from the similarities between thebserved data for relaxation oscillations in laser diodesnd FWM processes in semiconductor materials. Relax-tion oscillations are known to result from carrier pulsa-ions at the natural frequency of the material and the la-er resonator for a particular pumping current [25].

Typically, the solution to the carrier density rate equa-ion (and occupation probabilities) is expressed as a powereries involving the harmonics of the detuning frequency.n Agrawal’s seminal work, the carrier density rate equa-ion is linearized by truncating the power series at therst harmonic to make the solution tractable [5,18]. Lin-

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arizing the carrier density rate equation effectivelyeans a linear approximation for the nonlinear recombi-ation terms (caused by spontaneous emission �N2 anduger recombination �N3), which may not be sufficientlyccurate at high saturation levels [26] and cannot be ap-licable to devices in which these recombination mecha-isms are dominant, e.g., long-wavelength semiconductor

asers and amplifiers [27]. This also means that higher-rder mixing products cannot be accounted for by theheory. Higher harmonics may be neglected if relativelyow powers are considered. A common assumption to fur-her simplify the problem is that the probe power is muchmaller than the pump power, so that the gain experi-nced by the pump signal is unaffected by the presence ofhe probe signal (neglecting pump depletion). Pumpepletion and higher-order mixing products up to secondnd third harmonics of the detuning frequency were con-idered in [28,22], respectively. Other simplifying as-umptions include linear gain dependence on carrier den-ity and that all waves experience the same gain.urthermore, the number of mixing products is fixed ariori by truncating the power series solution of the car-ier density equation. Hereafter, this is referred to as thetruncated series assumption” (TSA). The common de-ominator for all the models described above is that theyre frequency-domain based and therefore essentiallyreat the steady-state behavior of the SOA.

Time-domain models for FWM in SOAs have also beenroposed, e.g., [29–31], but these still rely on the use ofhe TSA. Models based on the modified nonlinearchrödinger equation (MNLSE) have also been used toescribe FWM in SOAs [32]. These models are quite de-ailed and are able to model higher-order mixing productsithout invoking further assumptions. They also take

nto account different processes in the SOA, e.g., two-hoton absorption (TPA), self-phase modulation (SPM),arrier heating, and group velocity dispersion (GVD).owever, their inclusion of material dispersion in the

ime-domain simulations assumes that the gain is sym-etric about the peak gain wavelength. Realistic inclu-

ion of material dispersion is important especially if veryhort pulses or a large number of interacting signalshence covering a large bandwidth) are to be simulated inhe time domain, as is necessary if higher-order mixingroducts are to be accounted for properly.Different methods have been used to represent the

avelength-dependent gain in the time domain. The mostopular methods include: (i) finite impulse response (FIR)lter theory [33], (ii) a second-order Taylor series expan-ion about the gain peak wavelength with constant coef-cients [32], and (iii) a parabolic gain profile representedy a Lorentzian time-domain filter profile [34]. Theecond-order Taylor series expansion and the Lorentzianrofile are both inadequate, because of the asymmetricature of the material gain spectrum. Furthermore, it is

naccurate to use constant filter coefficients to representhe gain because of the spectral gain shift that is inevi-able whenever the carrier density in the cavity changes.

A general solution to the SOA FWM problem is there-ore necessary that relaxes some (and uses fewer) of thessumptions. We propose a time-domain model, in par-icular, to study the characteristics of CDM during FWM

n SOAs without using the TSA. The model incorporates aealistic dependence of the gain on the carrier density andolves the nonlinear carrier density rate equation to de-ermine the carrier density. CDM manifests itself natu-ally, if the beating between the optical waves is properlyaken into account in the description of the wave propa-ation, and also in their interaction with the gain mediumnd the carrier rate equation. Modulation of the carrierensity results in the modulation of the quasi-Fermi levelpacing and hence in a modulation of the voltage dropcross the active region of the laser or SOA [25]. Thisodulation must also show up in the stimulated emission

ate and hence the injection current. Thus, the spectrumf the injection current or voltage between the device ter-inals should reveal the presence of carrier pulsations at

he harmonics of the frequency determined by the detun-ng of the interacting waves [35]. The spectrum of the car-ier density should also show pulsations at the harmonicsf the detuning frequency. The CDM harmonics arise as aatural response of the nonlinear dependence of the car-ier recombination rate on the carrier density due to gainaturation, spontaneous emission, and Auger recombina-ion processes.

While the focus of most earlier work has been on theptical characteristics of FWM processes, we concentraten the characteristics of CDM. The main contribution ofhis work is the development of a full time-domain modelhat accurately calculates the carrier density and prop-rly incorporates the material dispersion. It is thereforeble to treat higher-order harmonics that will cover a sig-ificant bandwidth even in nearly-degenerate FWM con-gurations such that to neglect material dispersion or as-ume symmetric gain spectrum might be insufficient.his is particularly important if large input powers (typi-ally greater than 1 dBm) are used. Furthermore, carriereating effects can be included by incorporating the dy-amics of the total carrier energy (carrier temperatures)

n both the conduction and valence bands. The use ofime-dependent filter coefficients ensures that spectralhifts of the gain due to changes in carrier density are cor-ectly captured.

. THEORETICAL MODELhe theoretical model described below is applied to a mul-

iple quantum-well SOA (MQW-SOA). The device is fabri-ated into a buried heterostructure (BH). The propagationf the optical field in the active region of the SOA is de-cribed by the wave equation

� � � � E� +n2

c2

�2E�

�t2 = −1

�0c2

�2P�

�t2 , �1�

here n is the refractive index of the material (back-round index) in the absence of electrical pumping, c ishe speed of light in vacuum, �0 is the free-space permit-ivity, and P� �x ,y ,z , t� is the macroscopic polarization ofhe medium. It is assumed that the SOA structure is de-igned to support only the fundamental mode with aransverse distribution U�x ,y�, which is assumed to be in-ependent of wavelength and normalized such that��U�x ,y��2dxdy=1. The strong waveguiding of the BHs

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urther confines the optical field in the transverse direc-ions in addition to the optical confinement provided byhe separate confinement heterostructure (SCH) layersstrong index-guiding structures) [27]. The propagationedium is assumed to be of collinear geometry, i.e., a

traight waveguide. Thus, the fields can be reasonably as-umed to remain linearly polarized during propagationhrough the device. This implies that � ·E� =� ·P� =0, andhe left-hand side of Eq. (1) is ��E� =�� ·E� �−�2E� =�2E� . The optical propagation can be satisfactorily de-cribed by the scalar wave equation.

Considering the general case, the total optical fieldomprises both backward and forward propagating fields.sing the plane-wave expansion and dropping the vectorature, the total optical field in the active region can beritten as

E�x,y,z,t� = 12U�x,y��+�z,t�exp�i��t − kz��

+ �−�z,t�exp�i��t + kz�� + c.c�, �2�

here �+�z , t� and �−�z , t� are the forward and backwardropagating complex envelopes of the optical field, respec-ively, k=n� /c is the optical propagation constant in theongitudinal direction, and �=2�f, where f is a referencerequency for baseband transformation for all the signalsropagating through the active medium. The induced po-arization can be written in a similar form as the field

P�x,y,z,t� = 12U�x,y��P+�z,t�exp�i��t − kz��

+ P−�z,t�exp�i��t + kz�� + c.c.�, �3�

here P+�z , t� and P−�z , t� are the slowly varying forwardnd backward propagating complex polarizations. Substi-uting Eqs. (2) and (3) into the scalar wave equation (1)nd applying the slowly varying envelope approximationSVEA) leads to the propagation equations

n

c

��±

�t±

��±

�z= i

�xyk

2�0n2P±, �4�

hich are subject to the boundary conditions

�+�0,t� = R1�−�0,t� + 1 − R1�in1�0,t�, �5�

�−�L,t� = R2�+�L,t� + 1 − R2�in2�L,t�, �6�

here �in1�0, t� and �in2�L , t� are optical inputs injectedhrough the left and right facets. These facets have powereflectivities R1 and R2, respectively. The opticalaveguiding properties are represented by the confine-ent factor �xy, which is related to the waveguide trans-

erse mode distribution through �xy�−w/2

w/2 �−w/2w/2 �U�x ,y��2dydx /�−

�− �U�x ,y��2dydx [36], where

and d are the active region width and thickness, respec-ively.

The carrier dynamics are obtained by solving the ratequation [37]

�N�z,t�

�t=

J�z,t�

ed− �nr�N�N −

i�xy

4��+��P+�*

+ ��−��P−�* − c.c.�, �7�

here J is the total injected current density, e is the elec-ron charge, �nr represents nonradiative recombinationand generally depends on the carrier density), and ishe current injection efficiency. In Eq. (7), carrier diffusionffects have been neglected, assuming that the carrierensity is homogeneous �� ·N=0�. Thus, the total currentensity is due only to the external source from the elec-rodes. The current density from the external source is de-ermined as in [38]:

Jext�z,t� =Vbias − Vjcn�z,t�

rs, �8�

here rs is the series resistance of the p- and n-layers onither side of the active region and the p- and n-contactesistance. Vjcn is the voltage drop across the active layerp–n junction) and relates to the difference between thewo quasi-Fermi levels as

Vjcn�z,t� =EFn�z,t� − EFp�z,t�

e. �9�

Fn and EFp are the electron and hole quasi-Fermi levels,espectively. The quasi-Fermi levels are calculated withinhe gain model under the commonly used assumption ofharge neutrality. While charge neutrality is a valid as-umption in bulk material, it may not be a valid assump-ion for quantum wells—especially at high current pump-ng levels. A significant charge imbalance in the quantumell may strongly affect the optical properties of the sur-

ounding optical guiding layers (SCH layers) and the en-ire waveguide as a result of the carriers injected into theegion [39]. Nevertheless, charge neutrality has been as-umed to simplify the analysis and should be acceptableor moderate injection currents. EFn and EFp are esti-ated using the Joyce–Dixon approximation [40],

EFn,p = kTln� N

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2� N

Nc,v� +

1

24� N

Nc,v�2

− 0.0000347� N

Nc,v�3 . �10�

The induced macroscopic polarization (sum of micro-copic interband polarizations over the phase space) P± isfrequency-dependent quantity that is a function of the

otal optical field interacting with the medium via theime-varying carrier distributions. In the frequency do-ain, the polarization is related to the total electric field

n the cavity through

P±��,N,E� = �0�eff��,N,E�E±. �11a�

˜ �� and E�� are the Fourier transforms of P�t� and E�t�,espectively. The effective susceptibility takes into ac-ount all processes, both linear and nonlinear. No particu-ar functional dependence on the optical field is assumedn Eq. (11a). When the optical field intensity is sufficiently

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eak, the polarization can be expanded into a power se-ies of the optical field [14], and the effective susceptibil-ty can be written as

�eff = �n

��n+1�En:n = 0,1,2 . . . . �11b�

A general description of the optical nonlinearity is de-irable. The light–matter interactions should be properlyepresented in describing the polarization. Hence, the ef-ective susceptibility contains a total carrier-dependentet optical gain and refractive index change, both ofhich are also functions of the optical field:

�eff��,N,E� = −1

k�ignet��,N,E� − �1 − �xy

�xy� conf

+ k�n��,N,E�� , �12�

here conf represents the optical losses in the confine-ent regions.The nonlinear polarization contributions from pro-

esses occurring at much faster time scales than the casesf interest in this work, such as dynamic SHB and CH,ave been neglected. However, the effects of SHB on theain compression are considered phenomenologically.urthermore, optical nonlinearities arising from multi-hoton processes, e.g., two-photon absorption (TPA) haveeen neglected in our analysis. This is justifiable, as non-inearities arising from the processes considered in thisork are resonant with the bandgap energy, whereas TPA

s resonantly enhanced for photon energies of the order ofalf the bandgap energy. The changes in the induced po-

arization caused by the variations in refractive index andain act as sources for the new fields in the optical wavequation (4).

A model for the calculation of the material gain, spon-aneous emission, and refractive index spectra is requiredo complete the analysis. The band structure is calculatedsing a 4-band k·p-model [41,42]. The gain and sponta-eous emission rates are obtained for the material usingqs. (13a) and (13b), respectively:

g�� =1

��

ng�e2�

n2c�0m02 �

ic=iv�MT�2�red

mD�fn − fp�, �13a�

Rsp�� =1

��

�e2�

n2�0m02 �

ic=iv�Mavg�2�red

mDfn�1 − fp�,

�Mavg�2 =1

3 �all 3

polarizations

�MT�2. �13b�

Intraband scattering processes (carrier–carrier,arrier–phonon, etc.) lead to broadening of energy statesf the carriers. This results in broadening of the opticalpectra, as photons interact with carriers that do not haveell-defined energy states [41]. This energy broadening is

aken into account phenomenologically by directly broad-ning the optical spectra. This approach is inconsistentith the physics, since it broadens the optical spectra

ather than the energy of the electron states. However, its widely used [41] and has been adopted here, as it avoidsomplex and computationally intensive time-dependenterturbation theory for band structure calculations [43].he gain and spontaneous emission spectra are directlyroadened by convolving with a hyperbolic secant line-hape function,

L�Eeh − �� =�in

��sech�Eeh − ��

�/�in� , �14�

here �in is the intraband relaxation time mainly due tocattering processes. This method is found to eliminatehe unphysical non-zero gain/absorption tail at low en-rgy (which extends deep into the bandgap of the mate-ial), when the traditional method of directly broadeninghe gain using a Lorentzian lineshape function is em-loyed [43].In Eqs. (13a) and (13b), �red

mD is the m-dimensional re-uced density of states per unit volume per unit energym=2 for quantum wells and m=3 for bulk materials); fnnd fp are the Fermi factors for electrons and holes, re-pectively; m0 is the free electron mass; MT is the transi-ion matrix element; and ng is the group index of refrac-ion for the mode. The summation in Eqs. (13a) and (13b)s carried out over all the allowed subband transitionairs that have the same quantum number �ic= iv�. Theain is linked to the carrier density through the quasi-ermi levels. Gain compression due to SHB is includedhenomenologically by introducing a constant gain com-ression factor �shb:

g�� =g�N,T�

1 + �shbS��, �15�

here S is the density of photons of energy ��. In general,he contribution of SHB to the gain is spectrally depen-ent if dynamic effects are taken into account [44].The carrier-density-induced perturbations of the re-

ractive index spectra are calculated using the nonlinearramers–Kronig transformation [45],

�n��;�� = −c

2��

−

�g��;��

�� − �d��, �16�

here � is some form of external excitation, e.g., an elec-romagnetic field or temperature. This form is appropri-te for different nonlinear processes, e.g., self-action ef-ects (nonlinear refraction) and cascaded processes [46].

. NUMERICAL IMPLEMENTATIONhe wave equation (4) and the carrier rate equation (7)re coupled through the material equations (11a), (11b),12), (13a), (13b), and (14)–(16) and solved self-onsistently by dividing the active region into smallnough (shorter than the diffusion length) sections thatuantities can be considered constant within each section.he field propagation equations are marched in time us-

ng the Lax differencing scheme, which allows use of aarger time step than is allowed by the explicit differenc-ng scheme. The stability condition of the Lax differencingcheme is given by the Courant condition �c�t /n�z��1

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47]. The carrier density equation is integrated using aourth-order Runga–Kutta method. All carrier-dependentuantities are estimated at the midpoint of each section,hile the optical fields are evaluated at the boundaries of

he sections (Fig. 1). This is to ensure that the photonsraveling to the right and those traveling to the left in aarticular section interact with the same carrier-relateduantities [48]. The current density injected into the ac-ive region in Eq. (7) is obtained from Eq. (8).

To keep all the simulations in the time domain, therequency-dependent polarization (frequency-dependentain and refractive index changes) are represented in theime domain using a recursive digital filter [49],

y�n� = a0x�n� + a1x�n − 1� + a2x�n − 2� + . . . + b1y�n − 1�

+ b2y�n − 2� + . . . , �17�

here n is the time index, and x� � and y� � represent thenput and output, respectively. Digital filtering allows theomputation of long convolution integrals of two functionsn the time domain for discretely sampled signals.

The optimization of the filter coefficients involves theast Fourier transform. Hence, it can be argued that theverhead incurred in obtaining the filter coefficients at ev-ry spatial point and time step is much larger than Fou-ier transforming the field and multiplying by therequency-dependent quantities directly. The product ishen inverse Fourier transformed for use in Eq. (4). Thettractiveness of using the filter approach is that the filteroefficients can be obtained offline from the main simula-ion and stored in tables. For a given set of parameterse.g., carrier density and temperature), the gain and re-ractive index changes can be calculated and the filter co-fficients obtained for each case and stored in tables. Thelter coefficients can then be retrieved during simulationith negligible additional computational burden. This ap-roach is not possible with the direct energy–time trans-ormation approach, which must be done during the simu-ation.

The number of taps (previous inputs and outputs, theaximum of which determines the order of the filter) is

etermined by the accuracy required and the computa-ional complexity of the problem. A recursive filter with aesired frequency response was designed by starting with

Section

-1jSection

-2j

Node

-1j

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�+( )tn,zj-1

Fig. 1. Representation of longitudinal grid showing

generic set of recursion coefficients and using iterationo slowly mold them to fit the filter whose frequency re-ponse was sought. The filter coefficients were optimizedsing the least-squares minimization together with theteepest-descent iteration approach. This method is pow-rful without resorting to rigorous mathematics using-transforms [50]. A further advantage of this approach ishat it can be used to fit frequency responses that are inhe form of measured data as well as complicated re-ponses that cannot be represented in any analytical form50].

Figure 2 shows the fitted gain and refractive indexpectra for an 8-pole filter after applying the filter to anmpulse input and taking the Fourier transform of the re-ulting impulse response. A good fit is observed betweenhe filter response and the desired response. It was foundhat even a 4-pole filter gave a satisfactory fit for the gainpectrum. However, the fit for the index spectrum wasess satisfactory because the gain spectrum is smootherhan the index spectrum, which has a sharp turningoint. The number of poles for the filters for both spectraas to be the same [Eq. (18) below]. Therefore, the num-er of poles chosen was one that gave a satisfactory fit forhe index spectral profile, i.e., 8.

Once the recursive filter coefficients for the requiredrofile have been obtained, the filter Eq. (17) is applied tohe field equation at each time step as the field is up-ated. Discretizing the wave equation using the Laxcheme, the following advancement equation is obtainedusing ��±�j

n as shorthand for ��±�z , t�=�±�j�z ,n�t�)):

��±�jn+1 = �j+1

n ��±�j+1n + �j−1

n ��±�j−1n + �j

�g,ind�n �� r=0

Np−1

��iar�g��j�1/2

n+1/2

+ �kar�ind��j�1/2

n+1/2��in± �j

n−r + �r=1

Np−1

��ibr�g��j�1/2

n+1/2

+ �kbr�ind��j�1/2

n+1/2��out± �j

n−r� . �18�

he � are constants obtained from the discretization ofhe wave equation. The indices are used to indicate thepatial point and time step at which the constant multi-lies the field, while “g” and “ind” refer to constants relat-

Section

jSection

+1j

Node

+1j

N t( )n+½ ½,zj+

�-( )tn,zj+1

( )tn,zj�

+( )tn,zj+1

ns where fields and carrier densities are estimated.

Node

j

)j

�+

positio

its2Hnt

5TEaf=smhoatwwwtr

rmp

wctectsmioIOsdaw

asoFtrt

dAtTc

S

Ft8s


ng to the gain and refractive index change terms, respec-ively. Np is the number of poles (set to 8 in theimulations) for the filter. At every time step a total ofNp−1 storage points are required for the field profiles.owever, the computational overhead imposed by this isot excessive because of the linearity of the recursive fil-er equation.

. RESULTS AND DISCUSSIONo model FWM, the input to the SOA was of the formin�t�= ��p�t�+�s�t�exp�−j�t��exp�−j�reft�, where �=2�fd,nd fd is the detuning frequency between the pump laserrequency fp=�p /2� and the probe laser frequency fs�s /2�. The time dependence of the pump and probelowly varying fields is a general form to allow the use ofodulated input signals. In the simulations presentedere, however, CW input levels were used. Unless statedtherwise, the input power levels used were −0.8 dBmnd −0.7 dBm for the probe and pump signals, respec-ively, while the pumping current and detuning frequencyere set to 150 mA and 1.23 GHz, respectively. The pumpavelength was set to 1550 nm. The total simulation timeas 5 ns, which is enough to sample all detunings larger

han 0.1 GHz. It is important to note that since the car-ier density rate and optical wave equations are solved di-

ig. 2. Assessing recursive filters designed for (a) gain spec-rum, (b) refractive index change. The number of poles was set to. The fits were obtained by inputting a delta function to the de-igned filter.

ectly, there are no restrictions placed on the relativeagnitudes of the pump and probe waves. The CDM am-

litude is obtained as

�N�t� = N�t� − NSS, �19�

here NSS is the carrier density determined only by thearrier injection rate, non-radiative recombination, andhe amplified spontaneous emission in the absence of anyxternal optical injection. The spectrum of the injectionurrent/voltage is obtained by taking the discrete Fourierransform of the time-series obtained from Eq. (9). Thepectrum of the CDM amplitude is obtained in a similaranner from Eq. (19). Table 1 shows the parameters used

n all of the simulations. The experimental results werebtained using a nonlinear SOA made by the Centre forntegrated Photonics (CIP, Ipswitch, U.K.) (SOA-NL-EC-1550) with an active region consisting of InGaAs un-

trained quantum wells and InGaAs barriers [51]. As thisevice has a nonlinear geometry, an effective length withconfinement factor equal to that of the untapered regionas used [52].Figure 3 shows the simulated input field, output field,

nd carrier density spectra with a detuning of 2 GHz. Thepectra are obtained from the discrete Fourier transformsf the optical wave and the carrier density time series.ig. 3 shows a case where the reference frequency isaken as the average of the pump and probe frequencies,espectively, while the reference frequency coincides withhe frequency of the pump in the rest of the simulations.

Figure 3 shows that the carrier density pulsates at theetuning frequency between the pump and probe signals.similar CDM spectrum to Fig. 3(c) was obtained when

he carrier frequency coincided with the pump frequency.his confirms that the CDM spectra do not depend on thehoice of the carrier frequency used in applying the

Table 1. SOA Simulation Parameters

ymbol Description Value (Units)

L SOA length 1.1 mm [52]W SOA width 1.3 �m [52]D SOA thickness 0.1 �m [52]�z Longitudinal grid spacing 10 �m�xy Optical confinement factor 0.18 [52]R1 Input facet reflectivity 0.05% [51]R2 Output facet reflectivity 0.05% [51] int Internal loss 2.0 cm−1

a Linear gain coefficient 2.0�10−20 m2

N0 Carrier density at transparency 1.7521�1024 m−3

�s Effective carrier lifetime 0.7 nsLinear radiative recombination

coefficient1.0�107 s−1

Bimolecular recombinationcoefficient

5.6�10−16 m3 s−1

Auger recombination coefficient 3.0�10−41 m6 s−1

ng Group index of refraction for themode

3.6

n Index of refraction of active region 3.3 Current injection efficiency 0.7

�shb Gain compression factor 5.0�10−17 cm3

�in Intraband relaxation time 0.2 ps

SmtondtbrpscprsF

ttdtfrplfpf

fmamr

tqntppo

wcatFs

Fp

Ft−


VEA. Smaller secondary peaks are also observed at theultiples of the detuning frequency and are enhanced as

he input powers are increased [Fig. 3(d)]. These higher-rder components may be due to beating between theewly generated FWM products and the input beams orue to higher-order harmonics in the CDM due in turn tohe nonlinearity of the carrier rate equation caused byoth the carrier density-dependent lifetime and the satu-ation of the stimulated recombination rate. The centraleak in the carrier density spectrum corresponds to thetatic carrier density. It was confirmed that the CDM isaused by the interaction of the input waves, since theeaks at the detuning frequency disappeared in the car-ier density spectrum when one of the input signals waset to zero. The carrier population pulsations observed inig. 3 are consistent with the experimental results in [25].Since the effectiveness of interband effects depends on

he rate of carrier replenishment into the active region,he magnitude of CDM should decrease with increasingetuning frequency, as the carrier density cannot followhe fast oscillations of the stimulated emission rate. Thisact was tested by simulating the FWM in the SOA for aange of detunings frequencies up to 1 THz with all otherarameters the same. Figure. 4 shows a plot of the simu-ated carrier density modulation as a function of detuningrequency taken at the center of the device. The valueslotted are obtained at the first harmonic of the detuningrequency.

The CDM amplitude decreases sharply as the detuningrequency increases. The CDM amplitude at 1 THz is al-ost three orders of magnitude smaller than the value forfew GHz. The stimulated emission rate fluctuates at auch faster rate than the carriers injected into the active

egion. As a result, the carrier density cannot respond and

ig. 3. Simulated spectra (a) input optical field; (b) output optrobe power −0.8 dBm, Ibias=150 mA; (d) same as in (c) except h

he modulation efficiency decreases as the detuning fre-uency increases. Dynamic CH and SHB burning haveot been taken into account in the model. It is interestingo note the similarity in the dependence of the CDM am-litude on detuning frequency and the dependence of am-litude of the conjugate signal on the detuning frequencybserved in Fig. 5 (a) of [20].

Figure 5(a) shows the variation of the CDM amplitudeith bias current. As expected, the CDM amplitude in-

reases with bias current. However, both the simulatednd the experimental results show that the CDM ampli-ude saturates as the bias current is increased further.urthermore, a comparison of the experimental andimulated curves shows that the CDM approaches satu-

ld; (c) carrier density: detuning 2 GHz, pump power −0.7 dBm,ump power �+4.7 dBm�. Logarithmic scale used for clarity.

ig. 4. Normalized carrier density modulation as a function ofhe detuning frequency; pump power=−0.7 dBm, probe power.0.8 dBm, I =150 mA.

ical fieigher p

bias

rtmp

abotbs0ciwcerrsmTalns

tpd

wssgcto(tutCptrr

sdsardt

Fd−fib

FdP


ation at a faster rate in the experimental measurementshan is observed in simulations. The CDM from measure-ents and TSA shows the largest amplitude at 0 mA (ex-

eriment) and very close to 0 mA (TSA1).The qualitative behavior is similar in both experiment

nd simulation. The model predicts the behaviour of CDMetter than the TSAI in the gain region. At 0 mA, the ratef absorption is at its maximum and the net gain is equalo the absorption coefficient. The SOA behaves as an un-iased diode with an internal field and hence acts as aquare-law detector. As the bias current increases frommA toward transparency, the rate of absorption de-

reases while the rate of emission increases. This resultsn an overall decrease in the net absorption. This explainshy the CDM amplitude decreases as the current in-

reases from 0 mA toward transparency. Above transpar-ncy, the net gain increases. This leads to an increasedate of stimulated emission relative to absorption, whichesults in an increase in the CDM amplitude, exhibitingaturation at about 150 mA. This is consistent with theeasured gain-current dependence shown in Fig. 5(b).he absence of experimental data points between 0 andpproximately 50 mA in Fig. 5(a) can be attributed to theow level of CDM, such that its amplitude is below theoise level. Although there is significant CDM in the ab-orption regime, it is not useful for many applications as

ig. 5. (a) CDM amplitude as a function of bias current:etuning=1.23 GHz; pump power=−0.7 dBm; probe power=0.8 dBm; TSA1, truncated series assumption, truncated at therst harmonic. (b) Measured small signal gain dependence onias current for input power of −28 dBm.

he SOA is strongly absorbing in this regime, and both ex-eriment and simulation confirm that there are noetectable/measurable optical mixing products.The effectiveness of the CDM is expected to increase

ith input power, as the interaction between the differentignals should be enhanced. As the optical power inten-ity increases in the active region, carrier depletion andain saturation effects should prevent any further in-rease in the CDM amplitude. Figures 6(a) and 6(b) showhe simulated and measured CDM amplitude dependencen pump power, respectively, for two probe powers−0.8 dBm and +4.7 dBm). It is evident that CDM ampli-ude does increase with input power up to a point. The fig-res also reveal that the largest CDM is produced whenhe pump and probe have approximately equal power. TheDM amplitude is generally higher for the lower probeower �−0.8 dBm�. When one beam is much weaker thanhe other, the CDM amplitude decreases as the beating iseduced as the result of carrier depletion and gain satu-ation by the strong beam.

There is generally good qualitative agreement betweenimulation results and experimental measurements. Theiscrepancy between the simulation results and mea-ured data may, in part, be due to the limited knowledgebout the SOA used in the experiments. With additionalefinements to the model—e.g., the use of a two-imensional model to take into account the lateral varia-ion of power, carrier density, and current spreading and

ig. 6. Dependence of CDM amplitude on input pump power forifferent probe powers: detuning=1.230 GHz, Ibias=150 mA. (a)robe power=−0.8 dBm. (b) Probe power=+4.7 dBm.

ucaiocpftt

6Amfinfiterlpimbmvhohat

eamstodGp

ASsTiutiN

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2


se of a full bipolar electrical model that can account forharge imbalance in the quantum well—even closergreement should be achievable. Furthermore, in deriv-ng the wave equation (1) describing the propagation ofptical waves in SOA/lasers from Maxwell’s equations, aommon assumption is to neglect the current term in Am-ere’s law. While this assumption is plausible at opticalrequencies, it may contribute at THz frequencies, par-icularly for electrical measurements. These issues arehe focus of ongoing work.

. CONCLUSIONStime-domain model for a SOA has been presented. Theodel incorporates material dispersion using a recursivelter technique that correctly captures the asymmetricature of the gain coefficient. The use of time-dependentlter coefficients ensures appropriate tracking of the spec-ral shift of the gain caused by band-filling effects when-ver the carrier density changes. The treatment allowsepresentation of the light–matter interactions (e.g., non-inear index changes) without use of the power series ap-roximation (i.e., ��2�, ��3�, etc.), which has limited valid-ty. This general description of the optical nonlinearity

ay have interest beyond the current study. While it haseen developed for an SOA, the theory and numericalodelling employed can be useful for studying other de-

ices for potential new and exciting applications, such asigh-density photonics integration for achieving all-ptical functionality. In such devices, nonlinear optical be-avior is significant even at low optical power levels, andn accurate description is necessary to describe the opera-ion of these devices.

The model is used to study the behaviour of CDM gen-rated by wave mixing without an a priori assumptionbout the existence and form of carrier pulsations. Theodel confirms that the carrier density does indeed pul-

ate at the harmonics of the detuning frequency betweenhe input waves. An insight into the dependence of CDMn different parameters (input power level, bias current,etuning between the two input waves) is established.ood qualitative agreement between simulations and ex-eriments has been established.

CKNOWLEDGMENTS. Kaunga-Nyirenda thanks the Commonwealth Scholar-hip Commission (United Kingdom) for financial support.his work was also partly supported by the UK Engineer-

ng and Physical Sciences Research Council (EPRSC)nder grant GR/T09309/01. The authors are also gratefulo the Centre of Integrated Photonics (CIP) for usefulnformation and discussion about the SOA (SOA_L_OEC_1550).

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Documents

Theoretical investigation of the role of optically induced carrier pulsations in wave mixing in semiconductor optical amplifiers