690 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL....

690 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 6, JUNE 2004

On the Noise Properties of Linear and NonlinearQuantum-Dot Semiconductor Optical Amplifiers:

The Impact of Inhomogeneously BroadenedGain and Fast Carrier Dynamics

Alberto Bilenca, Student Member, IEEE, and Gadi Eisenstein, Fellow, IEEE

Abstract—We present a detailed analytical model describing thenoise properties of quantum-dot (QD) optical amplifiers operatingin the linear and saturated regimes. We describe the dependenceof the optical noise on the main physical parameters characterizingthe QD gain medium as well as on operating conditions. The opticalnoise at the amplifier output shows a broad-band spectrum with anincoherent spectral hole due to the gain inhomogeneity. A coherentspectral dip stemming from noise–signal nonlinear interactions issuperimposed on that broad-band spectrum. The broad-band in-coherent component is also calculated using an approximate modelwhich makes use of an equivalent inhomogeneous population in-version factor. The validity of the approximation is examined indetail. We also calculate the electrical relative intensity noise andobserve a spectral hole corresponding to the spectral shape of theoptical noise. The most important characteristics of the optical andelectrical noise spectra are determined by the degree of inhomo-geneous broadening and by the fast carrier dynamics of QD am-plifiers. The fast dynamics causes a very wide noise spectral holewhich has important potential consequences for detection of fastdata and for all optical signal processing.

Index Terms—Noise, quantum dots (QDs), semiconductor op-tical amplifier (SOA).

I. INTRODUCTION

I NHOMOGENEOUS broadening of semiconductor opticalgain material was commonly encountered in the early days

of diode laser research [1] where it was a result of the poor epi-taxial growth capabilities. In fact, the successful developmentof defect-free, homogeneously broadened gain material markedthe dawn of high-quality diode lasers and optical amplifiers, em-ploying first bulk and later quantum-well (QW) structures.

The emergence of quantum-dot (QD) lasers [2], [3] andoptical amplifiers raised the issue of inhomogeneously broad-ened gain media once more [4]–[7]. Semiconductor QDs areusually self-assembled using the Stranski–Krastanov technique[3] which typically yields some fluctuations in dot dimensions,mainly their height, as well as some irregularities in theirspatial arrangement within the overall semiconductor structure.The gain spectra associated with the various dot populationsdiffer from each other, and the overall gain spectrum of a QD

Manuscript received December 22, 2003; revised March 5, 2004. This workwas supported in part by the BigBand project within the fifth framework of theEuropean Union and by the Israeli Ministry of Science.

The authors are with the Electrical Engineering Department, Technion IsraelInstitute of Technology, Haifa 32000 Israel (e-mail: [email protected]).

Digital Object Identifier 10.1109/JQE.2004.828260

ensemble is therefore inhomogeneously broadened and exhibitsa bandwidth which is much wider than that of QW structures[5]. The gain spectra associated with the various dot groupsare partially coupled since all dots are fed through the wettinglayer (WL) which serves as a common reservoir. Spectraloverlap due to homogeneous broadening adds to the coupling.A second unique property of QD ensembles is an inherently fastcarrier relaxation dynamics. Pump probe and four-wave mixing(FWM) experiments in QD amplifiers revealed a completesaturation recovery time of 1–3 ps [8], [9], vastly different fromwhat is common in homogeneous amplifiers [10].

The impact of inhomogeneously broadened gain and of fastcarrier dynamics has been modeled and investigated experimen-tally in both lasers [11]–[13] and optical amplifiers [14]–[16].However, there is one related issue which to date has not beensufficiently dealt with: the noise properties of QD optical am-plifiers. The noise of any optical amplifier is intimately relatedto the carrier distribution in energy, the gain spectrum, and thesaturation properties—all of which are strongly affected by gaininhomogeneity and gain dynamics. Moreover, in saturation, thenoise is determined by the deterioration of the population inver-sion [17], [18] and nonlinear interactions between the saturatingsignal and the noise which are mediated by carrier pulsation andform a coherent spectral hole whose width and depth are de-termined by the medium response time and the injected powerlevel [17], [19], [20]. Some linear noise properties of a uniformQD ensemble have been described in [21], but the important im-pact of gain inhomogeneity and the fast carrier dynamics has notbeen dealt with.

This paper describes a detailed semiclassical analysis of noiseproperties in an inhomogeneously broadened QD optical ampli-fier. We present an analytical noise formalism which addressesboth the optical and the electrical regimes for linear as well assaturated gain conditions. To simplify the analysis, we do not in-clude carrier noise. Also, since the model is semiclassical, contri-butions to the relative intensity noise (RIN) due to vacuum noise[20], [22] are ignored, so the appropriate RIN interpretation is aspresented in [23]. The noise we consider copropagates with theinjected signal and effects due to counterpropagating noise areneglected as are nonlinearities due to spatial hole burning. Thecalculated output optical noise spectra are consistent with severalpublished experimental results such as [4, Fig. 7] and [6, Fig. 4].We examine the roles played by thepower and spectral location of

0018-9197/04$20.00 © 2004 IEEE

BILENCA AND EISENSTEIN: ON THE NOISE PROPERTIES OF LINEAR AND NONLINEAR QD SOAs 691

Fig. 1. Schematic description of dynamics in the QD ensemble.

the saturating signal, the linewidth enhancement factor, the widthof the homogeneously broadened gain, and the degree of spectraloverlap between adjacent QD groups. We present an exact modelas well as some approximations based on an equivalent inhomo-geneous population inversion factor, , and gain. relates thegainandthebroad-band(incoherent)noisespectraldensity,muchas inconventionalhomogeneouslybroadenedamplifiers [24].Wecalculate the dependence of on the degree of gain inhomo-geneity, saturation level, and wavelength. Also, we study the im-pact of the WL density of states and the dot density on gain and theincoherent noise suppression on as well as on the populationpulsation process and the resultant coherent noise spectral hole.

II. CARRIER DYNAMICS

Fig. 1 describes schematically the energy band diagram ofthe conduction band in a QD gain medium with the WL ex-hibiting a single energy state. Shown are the processes of carriertransfer between the WL and each QD ground state (GS): cap-ture and re-emission as well as the spontaneous and stimulatedemission. For simplicity, we assume that the different QDscomprise a single, homogeneously broadened electron and holeGS. The partial coupling of spectrally adjacent gain regionsis described schematically on the left-hand side of Fig. 1. Theinhomogeneous broadening is considered by dividing the QDs’ensemble into a number of spectrally overlapping two-levelsystems, each having a different resonant wavelength of the GSinterband transition. The two-level systems are labeled by theindex , which denotes the th group of dots having resonantwavelength . The dynamics of the carriers are treated inthe framework of semiclassical density-matrix equations [25]using the rotating-wave approximation. The relaxation-cap-ture-re-emission processes are treated phenomenologically intheir corresponding rate–time approximations. In accordancewith reported models [2], [3], [12], [14], [15], we assume thatcharge neutrality always holds in each QD so that electrons andholes are not treated separately and we employ carrier density

rate equations. Finally, the amplifier is assumed to have ideallynonreflective facets.

Denoting the volumetric carrier density of the wetting layer,by , and that of the GS in the th dot group by( where is the total number of QD groups), theset of rate equations is

(1)

(2)

The first term on the right-hand side (r.h.s.) of (1) describes di-rect current injection from the contacts into the WL withbeing the injection efficiency. is the current density, is theelectron charge, and is the WL thickness. The second term rep-resents carrier capture from the WL into the various QD groupswith an average rate given by

(3)

where is the fraction of the th QD group type within an en-semble of different dot size populations, assumed to be Gaussiandistributed with a variance of is the capture ratefrom WL to the th QD group while

(4)

is the occupation probability of the ground state of the th QDgroup, where is the degeneracy of the GS level without spinand is the volumetric dot density. The ratio rep-resents the dot-to-wetting-layer volume ratio and is included inorder to ensure carrier conservation. Finally, denotes thecapture time into unoccupied dots. The third term on the r.h.sof (1) is the total carrier re-emission from the QDs into theWL. The re-emission rate, from the th QD group into the WL,

, is

(5)

where is the volumetric WL density of states (not includingspin degeneracy) and is the re-emission time when theWL occupation is zero. The relationship between and ,calculated under the condition of detailed balance is

(6)

where and are the transition angular frequencies ofthe WL and the GS of the th QD group, respectively. The last(fourth) term on the r.h.s of (1) describes spontaneous recombi-nation from the WL with a rate of .

Equation (2) describes GS carrier dynamics in the th QDgroup. The first and second terms on the r.h.s. of (2) describe re-spectively, capture and re-emission of carriers between the WL


and the GS of the th QD group. The third and fourth terms de-scribe respectively spontaneous relaxation and stimulated emis-sion due to the presence of an electric field in the GS of theth QD group. is defined such that has units of W/m .is the confinement factor, is the dephasing rate, is the

carrier angular frequency, and denotes the peak value ofthe GS gain coefficient in the th QD group as follows:

(7)

where represents the differential GS gain of the th QDgroup which is related to the dipole moment of the optical tran-sition, , by

(8)

where is the group refractive index. is a Heaviside stepfunction of unit amplitude, is the complex conjugate of ,and represents temporal convolution. Notice that the adia-batic approximation is only applied to changes of the occupa-tion probabilities while the spectral dependence of the gainis considered in full.

Equation (2) can be rearranged, and in accordance with [15]it becomes

(9a)

satisfies the relationship

(9b)

For , equals the steady-state value of . repre-sents the effective time response of gain saturation, given by

(10)

Extracting from (7) and substituting into (9a) leads to

(11)

Thus, the saturated gain coefficient at due to the contributionsof all QD groups in the presence of a strong injected signal,at , is

(12)

where

(13a)

(13b)

where is the homogeneously broadened gain bandwidth(FWHM) and is the gain saturation effective time response

Fig. 2. Calculated static characteristics. (a) Linear gain spectra. (b) Spectrallyresolved gain saturation.

calculated under steady-state conditions. Also, is com-puted by substituting (9b) into (7) and using the steady-state so-lutions of and .

Fig. 2 describes an exemplary calculated gain spectrum andwavelength-dependent gain saturation characteristics. Thisand all other calculations presented in this paper use typicalphysical parameters extracted from experiments and describedthroughout the literature. They are listed in Table I. The gainvalues calculated throughout this paper correspond to net valuesso we do not relate separately to optical losses. We defineas the input power for which the overall gain is compressed by3 dB. is used to normalize the input power throughout thispaper.

III. ELECTRIC FIELD PROPAGATION EQUATION

The slowly varying, complex envelope of the electric fieldof the optical signal has the form

c.c. (14)


TABLE IPHYSICAL PARAMETER VALUES USED FOR THE CALCULATIONS THROUGHOUT THE PAPER

Neglecting material dispersion and the term ,satisfies the following propagation equation in the shifted timecoordinate system:

(15)where is the internal waveguide loss, describesthe noise accompanying the propagation and amplificationprocesses, and represents the spectral dependence ofthe medium susceptibility with the Fourier transform ,which is given by

(16)is the linewidth enhancement factor which equals

(17)

The first term on the r.h.s. of (15) is closely related to the secondterm on the r.h.s. of (11), which involves the polarization matrixelement in the density-matrix formalism. The noise termin (15) is multiplicative [26] in its nature (i.e., it is coupled tothe state of the system) and contains the contributions of sponta-neous emission at the angular frequency from all QD groups.

is therefore

(18)where is the effective waveguide cross-section area andis the population inversion factor of the th QD group satisfyingthe relationship

(19)

where is calculated using (4). Also, andrepresent a set of independent identically

distributed, in-phase, and quadrature Gaussian noise processeswith the following statistical properties:

(20a)

(20b)

(20c)

denotes an ensemble average operator, is theFourier transform of whereis a real process, and is the Dirac delta function.

IV. SMALL-SIGNAL NOISE ANALYSIS

In this section, we derive the optical noise spectrum at theoutput of a QD amplifier due to the injection of either a singleCW signal or a modulated signal with a bit rate much lowerthan . The basic assumption governing the derivation ofthe output noise spectrum is that the noise has but a minor effecton the total output electric field and thus can be treated as a smallperturbation on the propagating electric field and the evolvinggain. Also, we assume that the WL is in steady state, i.e., whilestimulated emission affects the occupation level in the WL dueto the detailed balance between WL and ground state, the noiseaccompanying the saturating signal is sufficiently small so thatperturbations due to the nonlinear noise–signal interactions arenot sensed by the large WL carrier density.

An important mathematical issue that needs to be addressedis the interpretation of (15) which is a stochastic partial dif-ferential equation (SPDE) with a spatial white multiplicativenoise term. Based on physical arguments that consider Gaussianwhite noise as the limit of a realistic noise when its correlationlength decreases to zero, the noise term has to be interpretedin the Stratonovich sense, rather than the Ito sense [26]. In theStratonovich sense, the multiplicative noise term appearing in(15) has a nonzero mean value, namely, there exists some sys-tematic contribution, usually referred to as noise-induced driftor spurious noise term [26]. Using stochastic calculus, we canrewrite (15) and translate it into an Ito sense SPDE. An equa-tion similar to (15) is obtained but with a zero-mean noise termand an additional deterministic term, which compensates forthe nonzero mean of the multiplicative noise. This deterministicterm is proportional to the derivative of the gain with respect to


the electric field. It can be shown that, for the problem we areexamining, this term is negligible since the saturated amplifieracts as a hard-limiter which vastly reduces the gain variations.Also, this term reflects a second-order noise process and ouranalysis considers only first-order noise phenomena. We con-clude therefore that, in both the linear and saturated regimes, thesmall noise accompanying the amplification process acts effec-tively as an additive noise, and therefore the Ito and Stratonovichinterpretations are identical.

The starting point of the small-signal analysis is theexpansion of the solutions of (11) and (15), namely,

andwhere and are

the stationary solutions of the electric field and the th GSgain coefficient, respectively, and and representsthe electric field and the th GS gain coefficient fluctuations,respectively. Notice that we omit the bar over and inorder to simplify the notation. The stationary solution of theelectric field is found as the solution of (15) with and bysubstituting . This leads to

(21)

where is the initial condition of the electricfield at the amplifier input and is calculated using(16) and (17). , the stationary solution of the thGS gain coefficient, is given by (13) and is calculated bysetting the time derivative in (11) to zero and substituting

and . Next, we obtain the equationsfor and by substitutingand into (11) and (15),collecting first-order terms in and performing a Fourier

transform. We arrive at the frequency domain equations givenat the bottom of the page. The additive noise termappearing in (23) is the Fourier transform of , given by(18), with and replaced by the stationary solutionsof the population inversion factor and the gain (i.e., and

, respectively). Notice that appearing in (22)satisfies the complex conjugate of (23) with replaced by .Substituting (22) into the equation for andleads to the following set of equations:

(24)

where and are given in (25), shown at thebottom of the page, and

(26a)

(26b)

Equation (23) is essentially an inhomogeneous SPDE de-scribing the effect of population pulsation dynamics due tothe electric field of the noise. [(25a)] describes thesusceptibility at the frequency of the noise componentand its change induced by the injected signal, .

in (25b) represents the induced susceptibility atthe frequency of the noise component as a resultof the wave mixing between and .Equivalently, the contribution of the wave mixing between

and to the noise component isgoverned by [(26a)] whereas [(26b)] standsfor the susceptibility of and its induced changesdue to . The noise component at frequency at theamplifier output can be described as

(27)

(22)

(23)

(25a)

(25b)


where

(28a)

(28b)

(28c)

(28d)

with

(28e)

Essentially, is the homogeneous solution of the equa-tion set (24) and the coefficients and are calculated usingthe variation parameter technique [27]. Equation (27) states thatan output noise at the frequency comprises two components.The first one, , is the accumulated contribution of the sat-urated susceptibility and its change due to the injected signal

. The second, , is the generated noise field at frequencydue to the FWM process between and . We

invoke an approximation that states that FWM processes in-volving and any two conjugate symmetric noise bands

and are decoupled, thus, for example, thenoise field amplitudes at participating in the FWM processdo not change due to the FWM process between and

. We note that this (first-order) approximation is usedonly for calculating the spectral dip originating from the FWMprocess. The calculation of the incoherent broad-band noise isof course exact.

On the basis of (27) and (28), the power spectral density(PSD) of the amplified spontaneous emission (ASE) at the am-plifier output is calculated to be

(29)

where

(30a)

(30b)

with

(30c)At this stage, it is convenient to introduce an inhomogeneousinversion population factor describing the degree of popu-lation inversion at frequency due to all QD groups. This inho-mogeneous inversion population factor defined as

(31)

is then substituted into (30c) to obtain an expression for the op-tical noise spectrum per unit length having the familiar form

(32)

where is calculated using (12) and (13). Equation (29)includes the conventional incoherent contributions to the opticalnoise spectrum as well as a coherent contribution which is me-diated by population pulsation dynamics. The coherent contri-bution is governed by the second term of the r.h.s. of in (25a)and by in (25b), which are very small for frequencies muchhigher then . The broad-band optical noise at the ampli-fier output is therefore obtained by setting them to zero and then(29) becomes

(33)

The difference between linear and nonlinear amplifiers inwhat concerns the broad-band optical noise is only due to thespatial dependence of the gain and the population inversionfactor. The form of (33) is different from the common ex-pression for the noise in a linear inhomogenously broadenedamplifier whereand are the unsaturated inhomogeneouspopulation inversion factor and the small-signal gain, respec-tively, and can be calculated using the unsaturated values of

and in (31), (12), and (13). Essentially, we canapproximate the broad-band optical noise very accurately tohave the familiar form

(34)

where and we have introduced the av-erage longitudinal values of the gain coefficient factor and theinhomogeneous population inversion factor

(35a)

(35b)


It is also important to derive the RIN spectrum at the output ofthe amplifier, which is defined as

(36)

where

(37)

and

(38)

Substituting (38) into (37) and calculating its spectrum leads to

(39)

where is calculated using (29) and (30) andequals

(40)

with and computed by (30a) and (30b), respectively, andand given by (28a) and (28b), respectively.

V. RESULTS AND DISCUSSION

The small-signal noise analysis described in Section IV isused to calculate various noise properties under a variety of op-erating conditions. The optical noise spectrum at the amplifieroutput comprises broad-band ASE which in saturation containsan added spectral hole due to a nonlinear noise–signal inter-action [17]. The details of the entire optical noise spectrum,and of the corresponding RIN, depend on the exact gain ho-mogeneity and on the carrier dynamics. The various results de-scribed herein were calculated for the set of physical parameterslisted in Table I.

A. ASE and RIN Spectra

Fig. 3(a) shows calculated ASE noise spectra for (pre-dicted for an ideal QD) and for different input powers normal-ized to . Note that this value represents at the resonancewavelength of a specific QD group. The model accounts for con-tributions to the total at that wavelength from all other QDgroups. The broadband ASE shifts to long wavelengths as theinput power increases. This is due to the state filling effect stem-ming from the thermal coupling between the GS levels and theWL. This effect, also common in bulk and QW amplifiers [24],tends to homogenize the QD gain. Near the input wavelength,we observe a spectral hole which has two parts. The first re-sults from saturation of the inhomogeneously broadened gain.Its width depends on the degree of overlap between adjacentQD groups and on the input power. The narrower (and deeper)portion is due to a distributed nonlinear interaction between thesaturating signal propagating down the amplifier axis and theamplifier noise [19], [20] and depends on the efficiencies of thenonlinear processes and hence it is determined by the effective

Fig. 3. Power-dependent noise spectra for � = 0. (a) ASE spectra. (b) RINspectra: left—broad-band view; right—spectral hole center normalized to theRIN which considers only the incoherent noise.

response time of gain saturation which increases with the in-jected signal owing to stimulated emission.

Fig. 3(b) describes calculated RIN corresponding to the ASEnoise spectra of Fig. 3(a). The left-hand side of Fig. 3(b) showsa broad-band view which exhibits a spectral hole with two dis-tinct slopes. The part with the shallow slope is due to the generalASE reduction. The hole is more pronounced for larger powerssince stimulated emission exhausts the WL so that carrier re-plenishment in a particular saturated QD group originates fromother, spectrally neighboring, QD groups, typical of an inho-mogeneously broadened gain medium. The narrower part of thespectral hole signifies the nonlinear interaction between the sat-urating signal and the amplifier noise. The right-hand side ofFig. 3(b) shows a normalized zoom of the hole center. The RINis normalized with respect to the calculated RIN which con-siders only the incoherent noise contribution. This normaliza-tion enables to compare the coherent noise contribution at dif-ferent pump input powers. The observed hole is very wide: hun-dreds of gigahertz, due to the fast gain saturation dynamics, in


Fig. 4. Power-dependent noise spectra for � = 1:5. (a) ASE spectra. (b) RINspectra. Left: broad-band view; right: spectral hole center normalized to the RINwhich considers only the incoherent noise.

contrast to bulk and QW amplifiers where, due to the slow re-combination time constant, it extends over a bandwidth of onlya few gigahertz [17]. This wide-band dip in the RIN spectrum ofthe QD amplifier has numerous implications in detection of fastdata as well as in applications of fast optical signal processing.

Practical QDs have low but nonzero linewidth enhancementfactors [2]. In Fig. 4, we show calculated optical noise and RINspectra for an amplifier with . The main differencebetween the spectra in Fig. 4(a) and Fig. 3(a) is the spectralasymmetry near the input wavelength. This asymmetry is re-lated to the well-known Bogatov effect [28] and originates fromthe index grating, generated in the nonlinear noise–signal inter-action (for ), which dominates over the correspondinggain index grating. The RIN spectra are described in Fig. 4(b).The effect of a nonzero -parameter is quantitative in nature asit adds FM-to-AM noise conversion.

The optical noise and RIN spectra depend also on the inputwavelength. This dependence is described in Figs. 5 and 6. Forboth short and long pump wavelengths, the gain peak shifts to-

Fig. 5. Power-dependent ASE noise spectra for � = 1:5 and ashort-wavelength saturating signal.

Fig. 6. Power-dependent ASE noise spectra for � = 1:5 and along-wavelength saturating signal.

ward longer wavelengths as power increases due to state filling.For short wavelengths (Fig. 5), the spectral hole is small due tothe reduced FWM efficiency and the strong coupling to the WL.For long wavelengths (Fig. 6), the spectral hole is also smallerthan the one at the gain peak (Fig. 4) because of the larger satura-tion power. Calculation of the corresponding RIN spectra yieldsspectral holes similar to the one shown in Fig. 4(b) except thatthe depth is reduced for both short and long pump wavelengths.

B. Approximate Broad-Band (Incoherent) Noise Model

The broad-band incoherent noise has been approximated in(34) in terms of average gain coefficient [(35a)] and inhomo-geneous inversion population factor [(35b)] obtained by spatialaveraging along the amplifier axis. The validity of the approx-imation is examined in Fig. 7(a) which contains three curves.The complete exact ASE spectrum is shown in a solid line, thespectrum obtained from the same exact model but omitting the


Fig. 7. Comparison between exact and approximate models. (a) ASE spectra. Solid line: complete exact model; dashed line: exact model with nonlinearnoise–signal interaction omitted; dashed–dotted line: approximate model. (b) Spectral and power dependence of the ratio between approximate and exact ASEnoise. I: spectral overlap of 75% and P = P . II: spectral overlap of 88% and P = 10P .

nonlinear part is shown in a dashed line, and the spectrum cal-culated by the approximate model (which addresses only thebroad-band noise) is shown as a dash–dotted line. Case (I) is foran overlap of 75% and . The three spectra overlapeverywhere but near the pump wavelength where only the fullsolution shows the spectral hole. The approximate model devi-ates somewhat at wavelengths slightly longer than the pump butthis deviation is minor. Case (II) is calculated for an overlap of88% and . Here the two exact solutions overlap(obviously) everywhere but near the pump wavelength but theapproximate solution deviates measurably except at long wave-lengths where it still is valid. Fig. 7(b) described the ratio be-tween the approximate and exact incoherent noise power spectraas a function of input power and wavelength for two degrees ofoverlaps. Since the comparison relates only to the broad-bandnoise, the spectral regime near the pump wavelength is not con-sidered and appears in Fig. 7(b) as a plateau. We note that, forpractical input power levels, the approximated solution errs by amaximum of 15% for the smaller spectral overlap and 18% forthe larger one.

The equivalent inhomogeneous population inversion factorcan be further approximated following the simple common

interpretation and definition [24]:where and are the carrier density in the upper andlower levels of a two-level system, respectively. For theinhomogeneously broadened medium, the approximated equiv-alent inhomogeneous population inversion factor becomes

with calculated using (35a). This approximation adds anintuitive understanding but leads to larger deviations from theexact calculation, especially at short wavelengths.

C. Dependence on Spectral Overlap

An important characteristic of the inhomogeneously broad-ened QD amplifier is the spectral overlap of the various dotsize populations. A large spectral overlap between energeticallyclose QD groups represents a more homogeneous medium. Theoverlap is controlled in the calculations by changing the ho-mogeneously broadened width while maintaining the originallycalculated small-signal gain and ASE characteristics.

Fig. 8(a) and (b) shows, respectively, the ASE noise and RINspectra for three levels of spectral overlap calculated for a pumpsignal having an input power of and injected at the gainpeak wavelength. The difference between the ASE noise curves


Fig. 8. Noise spectra dependence on spectral overlap for � = 1:5 and P =

P . (a) ASE noise spectra. (b) RIN spectra normalized to the RIN value atf =1.25 THz.

is clear. The spectral hole is more pronounced for low overlapvalues owing to the increased decoupling between energeticallyclose QD groups. Also, the inset shows that the hole deepens forlow overlap values since the efficiency of the population pul-sation dynamics increases due to the improved discreetness ofthe QD states. In the corresponding RIN spectra [Fig. 8(b)], theoverlap affects both the width and the depth of the RIN dip. Inorder to highlight the difference between the curves, we nor-malized the RIN spectra to the RIN value calculated at 1.25THz. For a small spectral overlap, the hole is deeper (owing tothe enhanced inhomogeneity). At the same time, the superim-posed dip due to population pulsations is wider and deeper dueto the higher nonlinear interaction efficiency.

Additional aspects of the spectral overlap are described inFig. 9. Fig. 9(a) compares gain and incoherent noise powersaturation. As in any semiconductor amplifier, incoherent noisecompression (i.e., the ratio between the saturated and unsatu-rated broad-band (incoherent) noise power spectral densities,calculated throughout this paper at a specific wavelength),

Fig. 9. Gain and noise saturation dependence on spectral overlap. (a) �G(dashed line) and �P (solid line) calculated at wavelengths shorter andlonger than the pump wavelength for two overlap values. (b) Equivalentinhomogeneous population inversion factor calculated at wavelengths shorterand longer than the pump wavelength for two overlap values.

, is smaller than gain compression , since saturationcauses a deterioration of the population inversion factor. Thedifference between and is shown to depend on thespectral overlap and on the wavelength at which the comparisonis made, relative to the saturating pump wavelength. For asmall overlap (51%) and at wavelengths longer than the pumpwavelength (where the saturation power is high),and are essentially indistinguishable, but at wavelengthsshorter than the pump we note a small difference due tostronger coupling to the WL. For the large overlap (88%) case,both gain and noise saturate faster and the differences between

and are larger in both wavelength regimes, asexpected in a more homogeneous gain medium. The sameinformation stated in terms of dependence on wavelengthand overlap is shown Fig. 9(b). A large spectral overlap and ashort wavelength enhance the increase of with input power.

We further studied the equivalent population inversion factorspectrum expressed by (35b) as a function of input pump powerfor two degrees of overlap and three pump wavelengths asseen in Fig. 10. A small spectral overlap renders the materialmore inhomogeneously broadened and hence the dependenceof gain saturation on power reduces as is the increase in thebroad-band noise. Indeed, we note that is smaller for thelow 51% overlap. For either overlap, is always maximumat short wavelengths and increases with the degree of saturationand hence is largest for saturation at the gain peak. Sincerepresents only the broad-band ASE noise, it is not calculatednear the pump wavelength where the coherent spectral holedominates. Each of the six curves in Fig. 10 therefore contain aplateau near the pump wavelength where is not applicable.

D. Effect of Wetting Layer and Dot Density of States

The noise and gain saturation properties are affected of courseby the WL and the dot density. The WL is modeled by a singleenergy state having a volumetric density of states that can be


Fig. 10. Power and spectral dependence of the equivalent inhomogeneous population inversion factor for different saturating signal wavelengths, calculated fortwo spectral overlaps.

Fig. 11. Noise spectra dependence on the WL density of states. (a) Normalizedoptical PSD. (b) Normalized RIN.

varied. A large density of states (compared to the dot density)acts as an infinite reservoir while a smaller one can be the sourceof a carrier injection bottleneck affecting dynamics and noise.

Fig. 11 compares normalized optical noise and normalizedRIN for the WL density of states value used in all previous fig-ures with the cases of and . Note that thenormalization is with respect to the incoherent optical noise andRIN spectra. To make the calculation valid, we adjusted the biascurrent so as to preserve the occupation probabilities of the QDs

Fig. 12. Gain and noise saturation dependence on the WL density of states. (a)�P : solid line;�G: dashed line. (b) Equivalent inhomogeneous populationinversion factor.

under the condition of no input signal. Also, the input pump hada power of and was located at the peak wavelength of thesmall-signal gain spectrum. It is very clear that a reduced wet-ting layer density results in a narrower spectral hole consistentwith an injection bottleneck or a longer effective gain saturationtime constant which in turn means a reduced bandwidth. Fig. 12shows gain and noise compression as well as the equivalent pop-ulation inversion factor (sampled at a wavelength shorter than


Fig. 13. Noise spectra dependence on dots density. (a) Normalized opticalPSD. (b) Normalized RIN.

Fig. 14. Gain and noise saturation dependence on dots density. (a) �P :solid line; nd �G: dashed line. (b) Equivalent inhomogeneous populationinversion factor.

the pump wavelength) versus input power. A reduced WL den-sity of state causes deeper gain and optical noise saturation witha larger difference between and and an increased

, once more because of the carrier injection bottleneck.Finally, we investigate the issue of dot density. Fig. 13 de-

scribes the normalized optical noise and RIN spectra for the dotdensity used previously as well as for a doubled density. Inthis case, we lowered the bias current in order to obtain a similarsmall-signal gain but the small-signal ASE spectrum was obvi-ously changed. The saturating signal was located again at thepeak gain wavelength and its power was . It is more difficultto populate a larger dot density, which results in a different typeof carrier injection bottleneck and a reduced bandwidth for thespectral hole. Fig. 14 shows noise and gain compression as wellas the inversion factor as a function of input power for the two

dot densities. Once again we note that an increased dot density issynonymous to an injection bottleneck which causes larger gainand optical noise saturation, a larger difference between themand an increased .

Although reducing the WL density and increasing the dotdensity have the same effect on the ratio, their im-pact on the bandwidth of the population pulsation process in thecases we analyzed is different. Doubling the dot density whilekeeping the small-signal gain constant results in a more severebottleneck which has a larger effect on the bandwidth com-pared to reducing WL density (and also holding the small-signalgain constant). Also, the difference between and ishigher here for the increased dot density.

VI. CONCLUSION

This paper described a detailed analytical model for the linearand nonlinear noise properties of a QD optical amplifier. Thenoise properties are governed by two important characteristicsof QD gain media: the inhomogeneous gain broadening and thefast carrier dynamics. Gain and noise are modeled using a set ofcoupled rate equations for the carrier densities in the QDs andthe WL acting as a carrier reservoir and a wave equation de-scribing the evolution of the optical electric field and the noiseaccompanying its propagation and amplification. We presentedan analytical model which describes the incoherent broad-bandnoise as well as a noise spectral dip resulting from coherent non-linear interactions between the saturating signal and the noise.The optical noise spectra we calculate are consistent with pub-lished experimental results [4], [6]. The broad-band noise is alsoapproximated in terms of an equivalent inhomogeneous popula-tion inversion coefficient. The validity of the approximation isexamined under several conditions. The electrical RIN spectracorresponding to the optical noise spectra at the amplifier outputare also calculated. We examine the noise dependence on phys-ical parameters such as homogeneous broadening (which deter-mines the spectral overlap between gain spectra of different dotpopulations), WL density of states and the QD density. Also,we describe the noise spectra dependence on operating condi-tions concentrating on the saturating signal power and wave-length. The fast carrier dynamics is responsible for a very widecoherent noise spectral dip which may have important conse-quences for applications related to detection of fast data and alloptical signal processing.

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Alberto Bilenca (S’00) received the B.Sc. and M.Sc.degrees (from the Technion-Israel Institute of Tech-nology, Haifa, in 1996 and 2001, respectively, wherehe is currently working toward Ph.D. degree in thefield of nonlinear optoelectronic devices specializingin dynamical properties and noise.

Gadi Eisenstein (S’80–M’80–SM’90–F’99) re-ceived the B.Sc. degree from the University of SantaClara, Santa Clara, CA, in 1975 and the M.Sc. andPh.D. degrees from the University of Minnesota,Minneapolis, in 1978 and 1980, respectively.

In 1980, he joined AT&T Bell Laboratorieswhere he was a member of the Technical Staffin the Photonic Circuits Research Department.His research there was in the fields of diode laserdynamics, high-speed optoelectronic devices, opticalamplification, optical communication systems, and

thin-film technology. In 1989, he joined the faculty of the Technion, IsraelInstitute of Technology, Haifa, where he holds the Dianne and Mark SeidenChair of Electro Optics in Electrical Engineering and serves as the head of theBarbara and Norman Seiden Advanced Optoelectronics Center. His currentactivities are in the fields of quantum-dot lasers and amplifiers, nonlinearoptical amplifiers, compact short-pulse generators, bipolar heterojunctionphototransistors, wide-band fiber amplifiers, and broad-band fiber opticssystems. He published over 250 journal and conference papers, he lecturesregulary in all major fiber optics and diode laser conferences, and he serves onnumerous technical program committees.

Prof. Eisenstein is an Associate Editor of the IEEE JOURNAL OF QUANTUM

ELECTRONICS.

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