Rate-equation model for coupled-cavity surface-emitting lasers

1646 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 12, DECEMBER 2004

Rate-Equation Model for Coupled-CavitySurface-Emitting Lasers

Vlad Badilita, Jean-Francois Carlin, Marc Ilegems, Member, IEEE, and Krassimir Panajotov

Abstract—We present a detailed theoretical study of a vertical-cavity surface-emitting laser (VCSEL) with two optically coupled,active cavities. The study is based on a rate-equation model writtenfor carriers and photons under steady-state conditions. The modelallows one to determine all the relevant parameters—carrier den-sities, gains, and output powers—starting from two input parame-ters: the injection currents in each cavity. The system of equationsis solved for different operating regimes of the device and the re-sults provided by the model are shown to be in very good qualitativeand quantitative agreement with the experimental data.

Index Terms—Multisection devices, rate equations, surface-emitting lasers.

I. INTRODUCTION

VERTICAL-CAVITY surface-emitting lasers (VCSELs)are a new generation of semiconductor lasers that differ

considerably from the conventional edge-emitting lasers [1].The very short cavity length and near-circular geometry leadsto single-mode emission, extremely low threshold currents,and high-speed operation [2]. With the maturing of the VCSELtechnology it became possible to manufacture more complexstructures, consisting of more than one cavity [3]–[5]. Thevertical design of the VCSEL structures facilitates the manu-facturing of such coupled cavity devices within a monolithicallygrown wafer.

Coupled semiconductor microcavities consisting of twoVCSEL-type cavities sharing a common mirror, were first inves-tigated by Stanley et al. in 1994 [3]. Reflectance spectraof the twoidentical -sized cavities each containing three InGaAs quantumwells (QWs) showed a pronounced anticrossing effect due to theoptical coupling between the two cavities. The separation of thecoupled modes depends on the coupling strength between thetwo cavities, i.e., the transmission of the common distributedBragg reflector (DBR). Light emission in coupled-cavity VC-SELs (CC-VCSELs) was first demonstrated by Pellandini etal. [4] with optically pumped devices. Depending on the twocavities, the CC-VCSELs can operate on either of the two longi-tudinal modes supported by the structure, or on both wavelengths

Manuscript received June 10, 2004; revised September 13, 2004. This workwas supported by the European Commission through the VISTA project and inpart by the Swiss National Science Foundation. The work of K. Panajotov wassupported in part by the IAP “Photon” network of the Belgian government andin part by FWO Flanders, GOA, and OZR of the Vrije Universiteit Brussels(VUB).

V. Badilita, J.-F. Carlin, and M. Ilegems are with the Institute for QuantumElectronics and Photonics, Swiss Federal Institute of Technology, LausanneCH-1015, Switzerland (e-mail: [email protected]).

K. Panajotov is with the Department of Applied Physics and Photonics, VrijeUniversiteit Brussel, Brussels B-1050, Belgium, and also with the Institute ofSolid State Physics, Sofia1784, Bulgaria (e-mail: [email protected]).

Digital Object Identifier 10.1109/JQE.2004.837787

simultaneously [6], [7]. The dynamics of the dual wavelengthoperation under optical excitation has been investigated in [5].The QWs in this case were slightly detuned while the cavitieswere of the same lengths. Lasing in pulses as short as 4.8 ps wasachieved with a well pronounced asymmetry in the time-decayfor the short- and long-wavelength modes.

Dual wavelength operation of CC-VCSELs with electricalpumping was first demonstrated by Carlin et al. [6]. The firstelectrically pumped CC-VCSELs emitting simultaneously ontwo different wavelengths and operating in continuous modeat room temperature were demonstrated by the same group(Brunner et al.) in 2000 [7]. CC-VCSELs consisting of an activecavity containing InGaAs QWs coupled to a passive GaAs cavitywere realized by Fisher et al. [8]. Modulation of the laser inten-sity by both forward and reverse biasing of the passive cavitywas demonstrated and attributed to carrier induced changes inthe refractive index and to field dependent cavity absorption inforward and reverse bias, respectively [8]. These CC-VCSELsstructure have also been shown to operate in a -switch modewith a pulse width as small as 150 ps [9]. In contrast to conven-tional single-cavity VCSELs, the -switching is achieved bymodulation of the reverse bias applied to the passive cavity, i.e.,by modulation of the optical losses rather than the optical gain[9]. Subsequently, CC-VCSELs with identical GaAs QWs inthe two cavities were shown to exhibit bistable behavior in thelight-versus-current (L–I) characteristics even when operatingin the multiple transverse mode regime [10]. Such bistabilityis only observed when operating in a continuous-wave (CW)mode, suggesting a strong influence of current induced heating inthe CC-VCSELs. Cavity detuning induced by such heating andthe shift of the absorption resonance in the passive cavity weresuggested as a possible mechanism for the observed abruptswitching-off of the lasing action [10].

The threshold characteristics of CC-VCSELs were investi-gated in [11]. Lasing into two longitudinal modes and multipletransverse modes was reported and the effect of current injec-tion in one cavity on the threshold current in the other cavity wasdemonstrated [11]. Single fundamental mode operation with 6.1mW output power was achieved for these structures by reducingthe aperture size [12]. In both cases, the two cavities are of thesame length each containing five GaAs QWs. The electrical andoptical confinement consists in an oxide layer for one cavity andproton implanted region for the other. When the current confine-ment apertures are of the same size strong mode competitionis observed, resulting in mode hopping between the short- andlong-wavelength modes [12].

The threshold characteristics of an asymmetric CC-VCSELstructure consisting of the two active cavities of slightly dif-ferent lengths were previously reported by our group [13].

0018-9197/04$20.00 © 2004 IEEE

BADILITA et al.: RATE-EQUATION MODEL FOR COUPLED-CAVITY SURFACE-EMITTING LASERS 1647

Fig. 1. Schematic of the coupled-cavities VCSEL.

Single wavelength threshold curves and the double-wavelengththreshold point were identified. Simultaneous lasing on twomodes was demonstrated.

It is well known that the polarization behavior is an importantissue in VCSELs research. The cylindrical symmetry makes itdifficult to be predicted. The control of polarization-switchingwas reported in a CC-VCSEL through the reverse bias appliedto one cavity while direct biasing the other [14]. The lengths ofthe cavities are slightly detuned (5%) and both contain InGaAsQWs as active-absorption media.

In this paper, we present a comprehensive study of theCC-VCSEL structure. We build up a rate-equation modelthat describes the behavior of the CC-VCSEL both at andabove threshold, under steady-state conditions. The work isstructured as follows. Section II presents the structure and theworking principles of the CC-VCSEL. In Section III, we buildthe rate-equation model under steady-state conditions and inSection IV, we study the system of equations for differentworking regimes of the device. The comparison between theexperimental and theoretically calculated data is presented inSection V, and conclusions are given in Section VI.

II. DEVICE STRUCTURE AND OPERATING PRINCIPLES

The device, identical with the one studied in [15], comprisestwo optical cavities, each of them containing InGaAs QWs asactive-absorptive media. The cavities share a common, semi-transparent mirror that determines the mutual coupling. Thestructure is completed with a top and a bottom Bragg mirror todelimitate the cavities.

Fig. 1 shows a schematic of the coupled-cavities VCSEL.This is a double-mesa structure with three electrical contactsto independently pump the two cavities. The layers are grownby metal–organic vapor-phase epitaxy (MOVPE) on n-typeGaAs substrates. The Bragg reflectors (940 nm—resonantwavelength) are made up of GaAs–Al Ga As layers,where compositional grading has been used to achieve lowresistance. Bottom and top DBRs are n-type, with Si used fordoping, and they have 14 and 26.5 periods, respectively. Thedevice is designed to be bottom-emitting because the reflec-tivity of the top mirror, together with a Au cap layer, is 99.8%versus the reflectivity of the bottom mirror which is 99.6%. Themiddle Bragg reflector, which controls the coupling between

the cavities, has only 8.5 periods and a reflectivity of 67%. Thismirror is p-type with C used for doping. The first and last lowindex layers, adjacent to the cavities in the middle DBR, consistof pure AlAs to provide optical and electrical confinement as aresult of subsequent wet oxidation.

Both cavities are optically active having identical 8-nm-thickstrained InGaAs QWs as gain/absorption media, embedded in10-nm Al Ga As barriers. The top cavity containsQWs, the bottom cavity contains QWs. Such a structure,if symmetrical, i.e., with cavities of the same length, will supporttwo identical modes that are equally distributed in the cavities.Because of the large field overlap, the gain/loss in the cavitieswill equally affect both modes making simultaneous lasing diffi-cult. If a small detuning is introduced between the lengths of thecavities, the overlap is reduced, each mode being predominantlyconfined in the corresponding cavity. The present structure isdesigned with a 4% detuning between the cavities. The top(bottom) cavity will be also referred to as the short (long) cavity.The gain/absorption media from one cavity will predominantlyaffect the corresponding mode, significantly decreasing the com-petition and allowing for simultaneous lasing in the two modes.As shown in [16], the optical field intensity of the short/longwavelength mode (from now on referred to as ) is moreconfined in the short/long cavity. The interplay between thegain/absorption in the cavities determines the working regimesof the device as will be presented in the next section.

This specific structure studied supports two longitudinalmodes: the short-wavelength mode located at nmand the long-wavelength mode located at nm.

III. RATE-EQUATION MODEL

A. Rate Equations—General Form

The operation of the coupled-cavity device is describedstarting from simple, intuitive arguments, developing the rateequations for both the number of carriers and the number ofphotons for the two optical modes. The reservoir model[17] used to develop the rate equations is shown below (Fig. 2).From now on, we will use the subscript index , 2 to denotethe cavity (1 = top, shorter cavity; 2 = bottom, longer cavity),and the superscript index “ ” or “ ” to denote the short orthe long mode, respectively. We recall that the short mode ispredominantly confined in the short/top cavity andthe long mode is predominantly confined in the long/bottom

cavity. The number of carriers injected into eachcavity of the laser is but only a fraction of reachesthe corresponding QW active region , where designatesthe carrier injection efficiency of the laser. The carriers thatsucceed to get in the reservoir may experience: 1) nonradiativerecombination at the rate ; 2) spontaneous recombinationat the rate of which a certain fraction, given by thespontaneous emission factor, recombine in the mode of interest

; 3) stimulated recombination at the rate ; and4) stimulated absorption at the rate .

In the following, the spontaneous recombination into thecavity modes will be neglected. The recombination rates will beexpressed as usual per unit volume, and therefore are multipliedby the volume of the active region in Fig. 2.


Fig. 2. Reservoir model of the coupled-cavities VCSEL.

In the two photon-reservoirs, the number of photons inmodes “ ” and “ ” varies as follows: stimulated and spon-taneous emission provide the emission of photons at the rate:

while the absorption in the active regiondepletes photons at the rate . All the other photons eitherleave the cavity through the mirrors or are internally absorbed.The photon population decays at the rate , where

is the number of photons in cavity in the mode “ ” or“ ” and is the photon lifetime in the mode “ ” or “ ”.Following the usual procedure we obtain the carrier and photonnumber rate equations, setting the time variation rate of thenumber of carriers and photons equal to the sum of rates into,minus the sum of rates out of the respective reservoirs.

The rate equations for the carriers are written for each cavitywhile those for the photons are written for each mode ( —shortand —long)

(1)

valid for 1, 2.The rate equations for the photons in the short, respectively

long-wavelength modes are given by

(2)

(3)

where the total number of photons in the mode “ ” or “ ”has been introduced.

Before proceeding further, let us develop some preliminaryconsiderations: the variation of the carrier density in cavitydue to stimulated emission and stimulated absorption is givenby [17]

(4)

valid for , 2, where represents the gain coefficientsfor the modes “ ” or “ ” in the cavity , the group ve-locity corresponding to the photons in each mode, and thevolume of cavity . The factor 2 accounts for the fact that onephoton crosses the active region two times per round trip. Theother parameter— —stands for the standing wave enhance-ment factor that depends on the placement of the QWs with re-spect to the field distribution; the value of varies between0 if the QWs are placed at a field node, and 2 if the QWs arelocated at a field maximum. The standing wave enhancementfactors are design parameters calculated using a transmissionmatrix program.

B. Steady-State Conditions

We will continue to study the rate equations under steady-state conditions, separately for the carriers and photons, settingthe time derivatives in (1)–(3) to zero. We start with (1) understeady-state conditions

(5)

valid for 1, 2.The fraction of the number of photons in the mode “ ” or

“ ” existing in the cavity is given by the confinement factor


Fig. 3. Schematic of the field distribution for the “S” and “L” modes insidethe CC-VCSEL [7].

introduced by the lumped mirror model [7], [16] or calcu-lated by the transfer-matrix method. These coefficients satisfy anormalization condition for each mode, respectively

(6)

and by design

(7)

for .In order to visualize better this confinement coefficient, we

show in Fig. 3 a schematic of the field distribution of the twomodes inside the CC-VCSEL, reproduced from [7].

So the confinement of the short mode in the cavity 1 (top)and of the long mode in the cavity 2 (bottom) are described bya single independent confinement coefficient. We introduce thefour coefficients explicitly in (5)

(8)

valid for 1, 2.The spontaneous emission rate together with the nonradiative

recombination rate in cavity are proportional to the total carrierdensity in cavity , , the proportionality factor being thecarrier lifetime assumed to be the same in the two cavities

(9)

valid for 1, 2.The two rate equations for carriers become

(10)for 1, 2.

We explicitly write the volume of the active region, where is the section of the device and is the width of

the active region; , with —the number of QWs in thecavity and —the width of one QW. The cross section area ofthe device is assumed constant, i.e., . The volumeof the optical mode in cavity is given by , butthe effective cavity lengths may be considered

equal for the two cavities taken into account the relatively smalldetuning—typically a few percent—as well as the penetrationdepths in the mirrors. Equation (10) becomes

(11)

valid for 1, 2.The rate equations for photons in the mode “ ” and “ ,” re-

spectively (2) and (3), are rewritten using (4) and neglecting thecontribution of the spontaneous emission in the mode of interest(the terms weighted by )

(12)

with an identical equation for the long-wavelength mode.Introducing again explicitly the volume of the active regionand the volume of the optical mode we get

(13)The rate equations for carriers and photons are rewritten in

the compact form below

(14)

(15)

and

(16)

(17)

IV. SOLUTIONS FOR DIFFERENT OPERATING REGIMES

For numerical calculations we will write the system (14)–(17)in a simpler way as follows:

(18)

(19)

(20)

(21)

The constants introduced in the equations above contain thedesign parameters of the device and are given below

(22)

(23)


We express the gain versus carrier density variation by a loga-rithmic dependence with three fitting parameters [17] which canbe resumed in the following equations:

(24)

(25)

(26)

(27)

where designates the carrier densities in thequantum wells. The fitting parameter represents the trans-parency carrier density at which the corresponding gain is zero.The fitting parameter determines the quantum well absorp-tion at the corresponding wavelength, i.e.,

(28)

Equations (18)–(21) form a system of four equations havingfour unknowns: the two photon numbers , the two carriernumbers , and two input parameters—the two bias currents

. In fact, one can set any other two parameters as “input param-eters” and solve the system with respect to the remaining 4 un-knowns. For instance, for any pair of injection currentsone can determine the corresponding number of photons, i.e.,the output powers for each mode, and the number of carrierswhich give also the gain coefficients. We refer to this system asa starting point for the subsequent discussion regarding differentoperating regimes.

In this section, we separately consider the following cases:

1) the “double-threshold” point which is the point whereboth modes start lasing simultaneously;

2) one mode lasing threshold while the other mode remainsbelow threshold;

3) one mode lasing threshold provided the other mode is wellabove threshold;

4) one mode lasing well above threshold while the othermode remains below threshold;

5) both modes well above threshold;

For each situation, the dependency between the two injectioncurrents to fulfill the corresponding lasing conditions is found.

A. “Double-Threshold” Point

An important point in the description of a CC-VCSEL is the“double-threshold point”, the point where the two modes startlasing simultaneously. At this point the numbers of photons foreach mode are zero, , so (18) and (19) can bewritten

(29)

(30)

Equations (20) and (21) must be satisfied whatever the valuesof and , whenever the corresponding mode is lasing

(31)

(32)

where the gain-carrier density dependences have alreadybeen inserted. Equations (31) and (32) determine the double-threshold carrier numbers , and, consequently, thegains corresponding to the double-threshold , . To-gether with (29) and (30), they determine the minimum biascurrent values needed to achieve simultaneous dual-wavelengthlasing . This is the point labeled by “A” in Fig. 4,which summarizes the theoretical results of the model. Fig. 5presents the experimental results for comparison.

B. Borders of the “One-Mode Lasing” Regions

In order to find the relation for the long-wave-length mode threshold with the short-wavelength moderemaining below threshold, we solve the system (18)–(21) for

. The difference with respect to the previouscase is that one has to take into account only the thresholdequation for the long-wavelength mode, (21), whether thethreshold (20) for the short-wavelength mode is not valid sincethis mode is under threshold. In this way, one obtains a setof equations specific to this particular case which yields arelationship between the two bias currents. As everything stemsfrom the initial system (18)–(21), we do not rewrite it for eachparticular case.

The procedure to find the relation for the short-wavelength mode threshold while the long-wavelength mode isbelow threshold is similar. The theoretical curves correspondingto this case are labeled by “B” in Fig. 4.

C. Borders of the “Both Modes Lasing” Region

In order to find the relation for the threshold ofthe long-wavelength mode while the short-wavelength mode islasing, we solve (18) and (19) from the above system for

and . Both threshold (20) and (21) are valid since onemode is above threshold (short-wavelength mode) and the otheris exactly at threshold (long-wavelength mode).

One can notice that the threshold equations are identical with(31) and (32), used to characterize the double-threshold pointof the device. This means that the number of carriers and thegain values for the whole dual-wavelength region—where boththreshold equations hold simultaneously—will be equal to thethreshold number of carriers and the corresponding gains. Anal-ogous with a single-cavity laser, the gain and the number of car-riers in each cavity are pinned in the case of simultaneous lasingon both modes. With the gain and number of carriers pinned attheir threshold value, we get a linear dependence .The analytical dependence between the two bias currents for


Fig. 4. Threshold for each mode as a function of the bias currents (calculated).

the threshold of the “ ” mode while the “ ” mode is abovethreshold is given below

(33)

The procedure is similar to find the threshold of theshort-wavelength mode, while the long-wavelength mode islasing, yielding also a linear dependence between the two biascurrents

(34)

Equations (33) and (34) delimitate in the coordinatesthe region where both modes are lasing at the same time, i.e.,the dual-wavelength working region and they are labeled “C” inFig. 4. We will see in Fig. 6, presented in the next section, theresults obtained for a different parameterization, a linear depen-dence of the gain versus carrier density.

One may also be interested to find the dependence betweenthe two bias currents for a certain, constant output power emittedin either mode.

D. Inside the “One-Mode Lasing” Regions

In order to find the relation for a constant(nonzero) output power in the short-wavelength mode while thelong-wavelength mode is below threshold, we solve the system(18)–(21) for and . Only the threshold (20)is valid since the long-wavelength mode is under threshold.

The equation for carriers in the first active region is nonlinearand has to be solved for the carrier density as a function of theinput current. This can be done both numerically and in closed-form using the “Lamberts W” function [18], [19]. The thresholdequation gives the dependency between the two carrier numbers

. In this manner, the equation for carriers in thesecond active region yields the dependency.

A similar procedure has been carried out to find thefor a constant (nonzero) output power in the

Fig. 5. Threshold for each mode as a function of the bias currents(experimental).

Fig. 6. Threshold for each mode as a function of the bias currents—calculatedwith a linear gain versus current density parameterization.

long-wavelength mode while the short-wavelength mode isbelow threshold. These theoretical curves are labeled with “D”in Fig. 7.

A special case is realized when the top cavity lases and thebottom cavity is biased such that all the carriers are extractedfrom the corresponding QWs (typically bottom cavity under re-verse bias). In this situation, the meaning of is not of an injec-tion current any longer, but that of a photocurrent which is ex-tracted from the system, therefore it will be negative. In order totreat this specific case, one has to add to the previous mentionedconditions ( and ) the fact that all the carriersare extracted from the bottom active region, i.e., . Thesolution of this particular case is a single point in thediagram which gives the maximum photocurrent (i.e., all thecarriers are extracted) detected in the bottom cavity for a cer-tain output power in the lasing mode due to the direct bias ofthe top cavity. For different values of the input current in the top


Fig. 7. Quasi-3-D phase-diagramspectrally resolved curves at constant output power as a function of the bias currents (calculated).

cavity, the dependence (labeled with “D1” in Fig. 7)is given by

(35)

E. Inside the “Both Modes Lasing” Region

The last investigation concerns the dependencyfor a constant nonzero output power in the long-wavelengthmode while the short-wavelength mode is also above threshold,i.e., the output power for the “ ” mode is also nonzero. Thesystem (18)–(21) is used in its most general form. As thethreshold equations are identical with those for the previouslyinvestigated “double-threshold point”, the carrier densities andgain values governing this operating regime will be those cor-responding to the threshold carrier densities and the thresholdgain. The analytical dependence between the two bias currentswill be also linear. The straight lines describing thedependency for constant output power in the “ ” mode areparallel with those given by (33), the threshold for the “ ”mode while the “ ” mode is lasing

(36)

Analogously, we obtain the dependency for the case when aconstant output power is set for the “ ” mode, while the “ ”mode is also above threshold

(37)

Fig. 8. Quasi-3-D phase-diagram spectrally resolved curves at constant outputpower as a function of the bias currents (experimental).

One can notice that inside the dual-wavelength operation re-gion, the constant-output power curves in the diagramare straight lines parallel to the corresponding threshold linesthat delimitate this working regime labeled with “E” in Fig. 7.

Table I summarizes the operating regimes investigated above,indicating the labels that identify them on the graphic in Fig. 4and Fig. 7 as well as the conditions to be imposed to the systemof (18)–(21) for each case.

V. EXPERIMENT AND THEORY—COMPARISON

As explained in the description of the device (Section II),the double-mesa structure of the CC-VCSEL allows indepen-


TABLE IOPERATING REGIMES OF THE CC-VCSEL

dent electrical pumping of the two cavities. The experimentalresults presented here are obtained under pulsed operation forboth cavities with frequency 1 kHz and duty cycle 1%, whichcorresponds to a period of 1 ms and a pulsewidth of 10 s. Aswe demonstrated in the previous section using the rate equa-tions, the threshold for each mode is a function of both biascurrents. The experimental threshold curves are presented inFig. 5 in the form of an diagram. One can clearly no-tice the four operating regimes of the device: the “no-lasing”region, the single-mode lasing region at either wavelength orthe dual-wavelength lasing region. For comparison, we presentin Fig. 4 the calculated threshold curves that show a very sim-ilar behavior as the experimental ones. The following param-eters have been used in order to plot the threshold curves inFig. 7: ; ; ;

; mm; ; ; m ;ns, ps, ps, cm ;

cm ; cm ; cm ;cm ; cm .

Both experimental and theoretical results in Figs. 4 and 5 con-firm the results reported previously [13] on a similar structure.The threshold being a function of both currents, one can see thata decrease of the bias current in one cavity, increases the currentin the other cavity necessary to reach threshold. The thresholdfor the short-wavelength mode can be reached even for a zerocurrent in the bottom cavity, i.e., with the bottom cavity actingas a passive absorber.

We have to mention here that the specific shape of thediagram presented in Fig. 4, showing the different operatingregimes of the CC-VCSEL, would depend on the gain-carrierdensity parameterization chosen. For example, we show inFig. 6 the same diagram but for a linear dependence of the gainwith the carrier density, i.e., the (24)–(27) become

(38)

(39)

(40)

(41)

where is the differential gain coefficient.As one can see comparing Figs. 4 and 6, the general behavior

with the different regimes of operation remains essentially thesame, characteristic border lines and the double threshold pointdo not change. Moreover there is no significant modification inthe region where the device emits two wavelengths simultane-ously, i.e., the region labeled “E” and bordered by the curves“C”. This region is essentially described by the pinned gainvalues. The four values for the gain, as well as the correspondingcarrier densities are constant in this operation regime, yieldinga linear dependence between the two injection currents, withthe slopes determined by the parameters of the structure and bythe pinned gain values and independent on the parameterizationchosen.

However, the single-mode lasing regions of the device, i.e.,the regions labeled “B” and “D” are more influenced as onlyone threshold equation holds therein (only for the lasing mode).As can be seen from Fig. 6, for a linear dependence of the gainon the carrier density the threshold curves for single mode op-eration become straight lines.

Fig. 8 presents a quasi-3-D diagram (experimental data)where one can see the dependency between the bias currents fordifferent constant values of the output power for each mode. Aspredicted by the rate-equation model in Section IV, the curvescorresponding to different constant output powers for the “ ”mode inside the dual-wavelength operating region are parallelto the threshold curve for the “ ” mode that delimitates thisregion. The curves for the “ ” mode show the same behaviorin agreement with the theory. For comparison Fig. 7 shows thecalculated curves that correspond to three different numbersof photons for each mode respectively, which means actuallythree different output powers. The two families of curves inFig. 7 are calculated according to the model presented in theprevious section.

In order to completely characterize the CC-VCSEL device,one can perform a scan over one of the bias currents (for in-stance ) for a constant value of the other current fordifferent parameters of interest such as the spectrally resolvedoutput power (Fig. 9) or the carrier density in each active region


Fig. 9. Calculated L–I curves for a constant value of the current in the bottomcavity I = 1:5 mA.

Fig. 10. Experimental L–I curves for a constant value of the current in thebottom cavity I = 0:4 mA.

(Fig. 11). Excellent agreement is obtained between the experi-mental L–I curves (Fig. 10) and the calculated ones. Scanningalong the specified value for mA, the threshold forthe long-wavelength mode is reached at approx. 0.4 mA. Theshort-wavelength mode starts lasing at around 0.75 mA, the de-vice entering the dual-wavelength lasing regime. The compe-tition between the two modes determines the switch-off of thelong-wavelength mode at mA. A discontinuity in theslope of the short-wavelength mode is noticed simultaneouslywith the switch-off of the “ ” mode. This is because of the stopof the optical pumping due to the “ ” mode.

Fig. 11 shows the variation of the carrier densities in the activeregions of the two cavities as a function of the current in the topcavity for a constant value of the current in the bottom cavity,

mA. According to (31), the number of carriers in thetop cavity varies linearly with the current starting from zero upto the double-threshold point. This dependency remains valid inthe one-mode lasing regime as well—(35). As soon as the de-vice enters the dual-wavelength lasing regime at mA,the carrier densities in both cavities are pinned to their double-threshold point values, as both threshold equations are satisfiedsimultaneously. The value is constant until the long-wavelengthmode is quenched. After that, the carrier density in active re-gion “1” continues to increase linearly with the top current, .The number of carriers in the bottom cavity (maintained at abias current mA) starts from a constant, nonzero valuegiven by the constant current. When the mode predominantly

Fig. 11. Carrier density evolution versus the bias current of the top cavity fora fixed current in the bottom cavity.

Fig. 12. Light output power versus pulsed current intensity for a singleactive-cavity CC-VCSEL (experimental) [6].

located in the bottom cavity—the “ ” mode—starts lasing, thenumber of carriers in the bottom cavity decreases until the valuecorresponding to the double threshold is reached and then re-mains constant during the dual-wavelength lasing. The numberof carriers in the bottom cavity continues to decrease when isfurther increased beyond the switch-off of the long-wavelengthmode.

This rate-equation model can be used to simulate differentCC-VCSEL configurations with only the top (short) cavityelectrically pumped [6]. Lasing starts first on the short-wave-length mode. The top cavity then optically pumps the QWsin the bottom cavity. This provides additional gain for thelong-wavelength mode resulting in subsequent lasing emissionon the long-wavelength mode. Figs. 12 and 13 demonstratethe dual-wavelength emission from a two-terminal CC-VCSELdevice, i.e., with only one active cavity. This result theoreticallyconfirms the previously reported experimental data in [6] andFig. 12.

This case is treated starting from the usual system of(18)–(21), imposing as well as the previously discussedconditions for different operating regimes.

One can notice in Fig. 13 a region of bistability close to thethreshold current corresponding to the short-wavelength mode.


Fig. 13. Calculated light output power versus current intensity for a singleactive-cavity CC-VCSEL.

While the device is still below threshold, the absorption in thepassive cavity is very high (28). As soon as the threshold isreached, the passive cavity is photo-pumped by the radiation onthe short-wavelength mode, thus approaching the transparency.Therefore a current which is lower than the initial threshold isenough to maintain the laser emission. This result is also similarto what has been experimentally observed in [9] by Fisher etal.—a monolithic coupled-resonator vertical-cavity laser whichexhibits bistable behavior in the light output versus injectioncurrent. The bistability regions reported in [9] are quite large andare positioned at the peak of the lasing output power. Accordingto our model the width of this bistable region is very sensitive todifferent design parameters. Smaller detuning between the twocavities is expected to increase the bistability as the optical fieldof the “ ” mode penetrates more into the passive cavity.

VI. CONCLUSION

In this work, we report a detailed theoretical study of theCC-VCSEL as well as a comparison of the calculations in goodagreement with experimental data. We start from the intuitivemodel of rate equations both for photons and for carriers writtenfor the two cavities under steady-state conditions. Using a log-arithmic gain versus carrier density dependence, the system ofequations obtained is solved imposing different conditions thatcorrespond to different working regimes of the CC-VCSEL. Thetheoretical results are in good agreement with the experimentaldata. We demonstrate that our model is a good instrument to in-vestigate different CC-VCSEL configurations. Such a structurewith only one electrically pumped cavity is also studied usingthe present rate-equation model and the results confirm the ex-periment reported before.

In conclusion, this model represents an important toolbox inthe study of CC-VCSELs under steady-state conditions. We ex-pect to further increase the functionality of the model intro-ducing the time dependence, thus making it applicable for dy-namic analysis of such structures.

REFERENCES

[1] K. Iga, “Surface-emitting laser—its birth and generation of new opto-electronic field,” IEEE J. Select. Topics Electron., vol. 6, pp. 1201–1215,Mar.-Apr. 2000.

[2] G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott,and S. W. Corzine, “Comprehensive modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron., vol. 32, pp. 607–616, Apr.1996.

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[5] P. Michler, H. Hilpert, and G. Reiner, “Dynamics of dual-wavelengthemission from coupled semiconductor microcavity laser,” Appl. Phys.Lett., vol. 70, pp. 2073–2075, 1997.

[6] J.-F. Carlin, R. P. Stanley, P. Pellandini, U. Oestertle, and M. Ilegems,“The dual wavelength bi-vertical cavity surface-emitting laser,” Appl.Phys. Lett., vol. 75, pp. 908–910, 2000.

[7] M. Brunner, K. Gulden, R. Hovel, M. Moser, J.-F. Carlin, R. P. Stanley,and M. Ilegems, “Continuous-wave dual-wavelength lasing in a two-section vertical-cavity laser,” IEEE Photon. Technol. Lett., vol. 12, pp.1316–1318, Oct. 2000.

[8] A. J. Fisher, K. D. Choquette, W. W. Chow, H. Q. Hou, and K. M. Geib,“Coupled resonator vertical-cavity diode,” Appl. Phys. Lett., vol. 75, pp.3020–3022, 1999.

[9] A. J. Fisher, W. W. Chow, K. D. Choquette, A. A. Allerman, and K.M. Geib, “Qswitched operation of a coupled-resonator vertical-cavitydiode,” Appl. Phys. Lett., vol. 76, pp. 1975–1977, 1999.

[10] A. J. Fisher, K. D. Choquette, W. W. Chow, A. A. Allerman, and K. M.Geib, “Bistable output from a coupled-resonator vertical-cavity diode,”Appl. Phys. Lett., vol. 77, pp. 3319–3321, 2000.

[11] D. M. Grasso and K. D. Choquette, “Threshold and modal character-istics of composite-resonator vertical-cavity lasers,” IEEE J. QuantumElectron., vol. 39, pp. 1526–1530, Dec. 2003.

[12] A. J. Fischer, K. D. Choquette, W. W. Chow, A. A. Allerman, D. K.Serkland, and K. M. Geib, “High single-mode power observed from acoupled-resonator vertical-cavity laser diode,” Appl. Phys. Lett., vol. 79,pp. 4079–4081, 2001.

[13] V. Badilita, J.-F. Carlin, M. Brunner, and M. Ilegems, “Light-currentcharacterization of dual-wavelength VCSELs,” Proc. SPIE, vol. 4649,pp. 87–95, 2002.

[14] V. Badilita, J.-F. Carlin, M. Ilegems, M. Brunner, G. Vershaffelt, and K.Panajotov, “Control of polarization switching in vertical coupled-cavi-ties surface-emitting lasers,” IEEE Photon. Technol. Lett., vol. 16, pp.365–367, Feb. 2004.

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[16] V. Badilita, J.-F. Carlin, and M. Ilegems, “Lumped mirror model forCC-VCSELs,” in preparation.

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Vlad Badilita received the B.S. and M.S. degrees in solid-state physics fromthe University of Bucharest, Bucharest, Romania, in 1997 and 1999, respec-tively. In 2000, he joined the Swiss Federal Institute of Technology, Lausanne,Switzerland, to pursue the Ph.D. degree, working on coupled-cavity VCSELs.

From 1997 to 2000, he was with National Institute for R&D in Microtech-nologies, Bucharest, working on micromachined circuits for microwave andmillimeter-wave applications.


Jean-Francois Carlin received the M.S. degree from the Ecole Centrale deLyon, Lyon, France, in 1986 and the Ph.D. degree in 1993 from the Swiss Fed-eral Institute of Technology, Lausanne, Switzerland, with a dissertation entitled“Chemical Beam Epitaxy of Ga–In–As–P Quaternary Compounds for Opto-electronic Devices.”

Since then, he has been working on the growth and characterization of long-wavelength vertical-cavity lasers and microcavity light-emitting diodes, and hasled the development of the first dual-wavelength coupled-cavity surface-emit-ting laser operating continuously at room temperature. He is presently leadingthe research effort on the growth of wide bandgap nitride semiconductors.

Marc Ilegems received the degrees in electrical engineering and telecommuni-cations from the University of Brussels, Brussels, Belguim, and the Ph.D. degreein electrical engineering from Stanford University, Stanford, CA.

From 1969 to 1977, he was a Member of the Technical Staff at Solid-StateElectronics Research Laboratory, Bell Laboratories, Murray Hill, NJ. He joinedthe Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1977 asProfessor of electrical engineering, and subsequently as Director of its Insti-tute of Micro- and Opto-Electronics (1988–1998) and as Head of the PhysicsDepartment (1998–2000). His research group has been involved in several EUresearch programs focused on transport and optical properties of semiconductormaterials, physics of microcavities and quantum-well/quantum-dot structures,electronic and photonic devices (light-emitting diodes and vertical-cavity lasers,two-dimensional photonic crystal structures for integrated optics), and widebandgap semiconductor nitrides.

Krassimir Panajotov was born in Targoviscte, Bulgaria, on July 11, 1957.He recevied the M.S., Ph.D., and D.Sc. degrees in physics from Sofia Univer-sity, Sofia, Bulgaria, in 1982, 1988, and 2002, respectively. His thesis and laterwork are in the field of nonlinear optics and optical bistability. Among them arethe demonstration of hybrid and all-optical polarization bistability and trista-bility. The D.Sc. dissertation is about polarization of light as a mechanism forswitching and control in photonics.

Since 1982, he has been appointed at the Institute of Solid State Physics,Bulgarian Academy of Sciences, Sofia. During 1997–1998, he was a Guest Re-searcher at the Vrije Universiteit Brussels, Brussels, Belguim, working on thetopic of polarization switching in vertical-cavity surface-emitting lasers. During2001–2004, he again joined VUB as a guest professor. His research interests in-clude thin film and fiber optics, nonlinear optics, VCSELs, and nonlinear laserdynamics in which fields he has coauthored more than 140 publications in in-ternationally reviewed journals and conference proceedings.

Dr. Panajotov has served as referee for Physical Review, Journal of AppliedPhysics, Optics Letters, IEEE JOURNAL OF QUANTUM ELECTRONICS, and theJournal of the Optical Society of America B, Optical Physics.

Rate-equation model for coupled-cavity surface-emitting lasers

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