IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 6, JUNE 2004 607

Dynamic Behavior of Fundamental-Mode StabilizedVCSELs Using a Shallow Surface ReliefJohan S. Gustavsson, Åsa Haglund, Jörgen Bengtsson, Peter Modh, and Anders Larsson

Abstract—An extensive theoretical study was performed onthe dynamic behavior of 850-nm-wavelength oxide-confinedfundamental-mode stabilized vertical-cavity surface-emittinglasers (VCSELs), using a shallow surface relief. The surface reliefis used to provide lower mirror loss for the fundamental mode,thus acting as a mode discriminator. In this way, single-modeoperation at high power levels can be obtained. We utilized acomprehensive model that includes the detailed epitaxial layerstructure and device geometry when calculating the opticalfields and that accurately accounts for the dynamic effects ofcarrier density and temperature on the modal distributions.Modulation response, eye diagrams, bit error rate (BER), andrelative intensity noise (RIN) were simulated and comparedto the performance of VCSELs without a mode discriminator,i.e., conventional multimode VCSELs. The fundamental-modestabilized VCSELs are associated with a higher out-coupling,which lowers the relaxation oscillation frequency and damping,and strong spatial hole burning, which induces a low-frequencyroll-off in the modulation response and contributes to the dampingof the relaxation oscillation at low bias. However, their dynamicsis fully competitive with conventional multimode VCSELs at both2.5 and 10 Gb/s although they exhibit a slightly higher eye closure.We only found a 0.5-dB power penalty in the BER. The RIN isenhanced, with a peak that is about 10–15 dB higher, caused bythe lower damping of the relaxation oscillation. It should be notedthat in the comparison we assume that all modes are equallycaptured from the multimode VCSEL. A mode-selective loss canseverely degrade its performance.

Index Terms—Dynamics, modeling, single mode, surface relief,vertical-cavity surface-emitting laser (VCSEL).

I. INTRODUCTION

THE vertical-cavity surface-emitting laser (VCSEL) hasestablished itself as a light source of choice for cost-sensi-

tive fiber-optic communication links and networks. While mostlinks and networks today use multimode VCSELs, more de-manding communication applications will require single-modeVCSELs with good dynamic performance. Examples includehigh-speed links where chromatic dispersion in the fiberis a limitation, long-wavelength VCSEL-based links wheresingle-mode fibers are used, and free space optical intercon-nects where the beam quality is of importance for efficiencyand crosstalk. Many of these future “single mode” applicationsfor VCSELs will also necessitate high output power. This has

Manuscript received October 20, 2003; revised February 25, 2004. Thiswork was supported by the Foundation for Strategic Research (SSF) throughthe Chalmers Center for High Speed Electronics and Photonics.

The authors are with the Photonics Laboratory, Department of Microtech-nology and Nanoscience, Chalmers University of Technology, SE-412 96 Göte-borg, Sweden (e-mail: johan.gustavsson@mc2.chalmers.se).

Digital Object Identifier 10.1109/JQE.2004.828273

been a challenge since the VCSEL is by nature multimode,because of its relatively large transverse dimensions.

Considerable effort has been invested to develop VCSELswith high single-mode output powers. Several techniques havebeen developed with the objective to change the transverseguiding and/or introduce mode-selective loss or gain. Twosuccessful methods include the monolithic coupled resonatorVCSEL using proton implantation [1] and the antiresonantreflecting optical waveguide (ARROW) VCSEL [2]. Theformer has achieved a single-mode output power of 6.0 mWand the latter 7.1 mW. The drawback with these devices is thattheir manufacture involves a more complex epitaxial growthand fabrication as compared to standard VCSELs, adding to themanufacturing costs. Addressing less complex solutions, themost common technique for achieving single-mode emission issimply to decrease the diameter of the index-guiding apertureof a standard oxide-confined VCSEL until higher order modesare no longer supported by the waveguide or prevented fromlasing due to high diffraction losses. A single-mode power of4.8 mW has been demonstrated using an oxide aperture 3.5 min diameter [3]. This method limits the output power due to asmall current confining region, which increases the differentialresistance and thereby the self-heating. The latter also resultsin a poor reliability. A further problem is the difficulty inreproducing a small oxide aperture and thereby reproducingthe performance. Another successful but less complex solutionis the surface relief technique [4], which allows for higheroutput power by a larger current aperture and thereby a lowerself-heating. This method only involves a slight modificationto standard VCSEL processing. By etching a shallow surfacerelief into the top mirror, lateral differences in the mirrorloss can be achieved, with minor change in the waveguideproperties. The mirror losses can then be tailored to give highermodal losses for the higher order transverse modes, whichprevent or delay their onset. Using an “inverted” surface relieftechnique, described in Section III, a single-mode power of6.5 mW has been achieved [5].

The static behavior of fundamental-mode stabilized VCSELswith a shallow surface relief has been studied extensively,e.g., [6]–[9]. In this paper, we report on a theoretical study ofimportant dynamic characteristics. These include modulationresponse, eye diagram, bit error rate (BER), and relative inten-sity noise (RIN). The fundamental-mode stabilized VCSELsare associated with a higher out-coupling. This results in alower photon density in the cavity, and the devices are thereforebelieved to have a lower speed. Further, the high degree of localstimulated recombination causes severe spatial hole burning(SHB), which will affect, for instance, the linearity of the

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device. The purpose of this study is to evaluate the influence ofthese effects on the dynamic performance of fundamental-modestabilized VCSELs, using a surface relief, compared to VC-SELs without a surface relief, i.e., conventional multimodedevices. The paper is organized as follows. In Section II, asummary is made of the quasi-three-dimensional (quasi-3-D)time-domain model that is applied in the simulations. Sec-tion III gives an introduction to the surface relief technique, andSection IV describes the geometry and epitaxial structure ofthe investigated VCSELs. The numerical results are presentedand discussed in Section V. Finally, a conclusion is given inSection VI.

II. MODEL

The original model is described in mathematical detail in[10]. Therefore, only a brief summary is made here, followedby a more thorough discussion on the extensions made in thiswork. The quasi-3-D dynamic model consists of a number ofinterdependent submodels that treat the optical fields, current,and heat transport in the device. By an iterative procedure thatis outlined in [11], a self-consistent solution is obtained for eachpoint in time and space.

The scalar optical model is based on an effective indexmethod developed by Hadley et al. [12]. The optical fieldsin weakly index-guiding VCSELs, such as oxide-confinedVCSELs, are characterized by near-paraxial propagation withpolarization states in the plane of the mirrors. Assuming thatthe structure only depends upon the longitudinal position in anumber of transverse regions, the electrical field is separatedinto a time-independent longitudinal and a nearly harmonictime-dependent transverse component. By further assumingthat the electro-magnetic field settles momentarily from aperturbation in the refractive index, which is generally a goodapproximation as long as the refractive index changes areon a time scale longer than the photon lifetime, the waveequation is transformed into eigenvalue equations in both thelongitudinal and transverse directions. The eigenvalue problemin the longitudinal direction is solved once for each transverseregion. The solution gives the longitudinal field distributionand an eigenvalue, the real and imaginary parts of which arerelated to the effective index and cavity loss, respectively.Regarding the effective index approximation, it is assumed thatthe longitudinal field distributions in the different transverseregions are all approximately the same. The longitudinal fielddistribution and the transverse distribution of effective indexand cavity loss are then input, together with the index pertur-bations from the other submodels, to the eigenvalue problemin the transverse direction, which is solved for each point intime. The solution gives the instantaneous transverse fielddistribution and an eigenvalue for each mode, whose real partis the deviation from the nominal angular oscillation frequencyand the imaginary part is the growth/decay rate of the fieldamplitude. The different modes of this weakly index-guidingstructure are designated as linearly polarized (LP) modes. Itshould be noted that the optical model does not account fordiffraction losses, which become significant when the apertureof an index-guided VCSEL is below 3–4 m.

In the current transport model, the drift transport of carriersfrom the contacts through the Bragg mirrors to the separate con-finement heterostructure (SCH), which contains the quantumwells (QWs), is approximated to be quasi-stationary, and mo-tivated by the inherent small dimensions. The current densitydistribution in the Bragg mirrors is obtained from the solution ofthe Laplace equation for the local electrostatic potential, wherehighly anisotropic electrical conductivities are applied due tothe large band discontinuities in the longitudinal direction. Thecarriers injected into the SCH will be diffusively transported inrandom directions, where some will be captured by the QWsbefore recombining. To account for this transport limiting ef-fect, two coupled one-dimensional rate equations are used to de-scribe the local carrier density in the SCH and in the QWs. Forthe particular VCSELs studied in this study, the carrier leakage(carriers escaping the SCH) is neglected because of the rela-tively large effective barrier height. The high density of carriersin the QWs changes the refractive index and provides gain tothe optical fields. The gain is modeled by an empirical formulathat accounts for dependencies on carrier density, temperature,and wavelength [13]. To include the nonlinear dependency ofthe gain on the photon density, due to spectral hole burning andhot carrier effects, a phenomenological gain suppression factoris used. It should be noted that the current transport model doesnot include the effects of depletion capacitance and parasitic ca-pacitances (e.g., bond pad and oxide layer). The former effectsare generally small when studying dynamics above the thresholdcurrent, and the latter effects can be made small by properly de-signing the VCSEL.

The heat transport model considers the local heat generationand dissipation in the VCSEL structure. There are a number ofprocesses that contribute to the self-heating of the device [14].The model accounts for Joule heating from the current transportin the resistive mirrors and nonradiative recombination of car-riers in the active region. The time-dependent temperature dis-tribution is obtained by solving the thermal conductivity equa-tion, where anisotropic thermal conductivities are used to de-scribe the mirror layers. When the VCSEL is current modu-lated at frequencies above 10 MHz, the temperature distribu-tion will be time-independent and based on the time-averagedvalue of the heat source fluctuations, due to the relatively largethermal time constant. Elevated temperatures increase the re-fractive index and shift the gain spectrum toward longer wave-lengths.

Noise is included in both the optical and current transportmodel by allowing for the deviation of the photon and carrierdensities from their deterministic values. This is done in the rateequations for the respective type of particles where a stochasticcontribution is added, which gives a statistically correct simu-lation of the fluctuations in particle density during the time stepused in the numerical algorithm [15].

A. Model Extensions

Two additions to the model have been made in this study.First, in the optical model, a more accurate expression for therate of spontaneously emitted photons that is coupled into mode

, , is applied in the rate equation for the modal inten-sities. Previously we simply assumed that a rather arbitrary, but

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small, fraction of the spontaneously emitted photons werecoupled into the mode. A more accurate expression is obtainedusing the fact that equals the stimulated recombination ratein mode , had there been only one photon in the mode.Thus,

(1)A detailed explanation of the symbols in the integral canbe found in [10]. In short, and are the refractive indexand material gain coefficient of the QWs, respectively,

assuming , is theweighted value of the relative permittivity in the longitudinaldirection for the different transverse regions , and is thepopulation-inversion factor. The remaining factors describe thenumber of photons in a volume element of the QWs. The pop-ulation-inversion factor is determined by the quasi-Fermi-levelseparation, ,however, in the simulations we utilize the approximation

[16] to reduce the computationaltime. and are the local carrier and transparency carrierdensity in the QWs, respectively, where the latter is defined by

. The spatial integral of the population-inversionfactor, material gain distribution, and the modal intensitydistribution results in a mode-dependent rate of spontaneouslyemitted photons that is coupled into a mode. It should be notedthat in the expression for we have assumed a Petermannfactor equal to one [17].

Second, in the thermal model, the heat generation caused byfree-carrier absorption is accounted for by including a third heatsource, , in the thermal conductivity equation. Simulationsshow that this heat source generally contributes more to the heatgeneration than does the nonradiative recombination, but lessthan does the Joule heating, for our VCSELs. The modal lossfrom free-carrier absorption is determined by the spatial overlapof the modal intensity distribution with the material absorptionin the doped mirror layers

(2)

where [10]. Bysimply identifying the local free-carrier absorption losses for theindividual modes from (2), one obtains

(3)

In the simulations, we use a significantly coarser discretizationin the longitudinal direction when solving the temperature dis-tribution as compared to solving the optical field to reduce the

Fig. 1. Calculated cold-cavity mirror loss and resonance wavelength asa function of etch depth for the VCSEL structure. The inset shows thelongitudinal standing wave pattern in the proximity of the surface.

computational time. Therefore, we approximate by calcu-lating the total modal losses from free-carrier absorption in thetop and bottom mirrors and distributing them with the envelopefunction of the intensity distribution in the longitudinal direc-tion. This is motivated by the relatively long diffusion lengthof the heat transport. We separate the absorption losses into topand bottom mirror absorption losses since the free-carrier ab-sorption is generally much higher in the p-doped mirror.

III. SURFACE RELIEF TECHNIQUE

The idea of the surface relief technique, as mentioned in theIntroduction, is to introduce mode-selective loss without dis-turbing the waveguide properties of the VCSEL. In order tofavor the fundamental mode, the technique is applied in twodifferent ways [8]. The first and most pursued way is to etch ashallow surface relief, in the form of a ring, in the top layer of aconventional VCSEL structure. The second way, what we referto as the “inverted” method, is to add an extra -thick layer ontop of a conventional VCSEL structure during epitaxial growth.The reflections from this topmost layer will be in anti-phasewith respect to the reflections further down in the mirror stack.A shallow surface relief, in the form of a disk, is then etched inthe center of the device. By choosing an appropriate etch depthfor the surface relief, a relatively high mirror loss contrast be-tween the etched and unetched areas can be accomplished, withlittle effect on the waveguide properties. This is illustrated inFig. 1, where the calculated mirror loss and resonance wave-length as a function of etch depth are shown for the investigatedVCSEL structure, in which the second “inverted” method is ap-plied. The inset demonstrates the effect of the anti-phase reflec-tions on the longitudinal standing wave pattern in the VCSEL.The “inverted” method has a fabricational advantage. It utilizesthe high thickness precision in the epitaxial growth, to reach anarrow local maximum in the mirror losses. This will then relax

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Fig. 2. Calculated cold-cavity loss and relative effective index as a functionof transverse position. The intensity distribution of the LP , LP , and LPmodes are also displayed. The cold cavity modal loss for the three respectivemodes are 0.405, 0.563, and 0.601 ps .

the required etch depth precision since a local minimum in themirror loss is much broader (see Fig. 1). As the etched surface isresponsible for most of the reflection, one might be concernedabout scattering losses resulting from the surface roughness. Inpractice, this should not be a problem since standard ion beametching can produce excellent surfaces with depth fluctuationsof less than 5 nm rms.

As mentioned, the “inverted” surface relief is used to intro-duce lower mirror loss in the central region. Fig. 2 shows thetransverse cavity loss distribution for four different transverseregions, for a VCSEL with a surface relief having a depth of57.5 nm and a diameter of 3.5 m. The oxide aperture, witha diameter of 6 m, induces a guiding effective index distri-bution, which is also included in the figure together with thetransverse distribution of the , , and modes. Allof the modes experience a higher loss in the periphery. How-ever, the higher order modes are discriminated because of theirlarger overlap with the high loss region. The large modal losseswill prevent the higher order modes from lasing or delay theironset to larger drive currents, which will allow for higher funda-mental-mode output powers. This is illustrated in Fig. 3, wherethe light–current characteristics for two VCSELs with surfacerelief with diameters of 4 and 11 m, respectively, are shown.The oxide aperture diameter is 6 m. The former case corre-sponds to a fundamental-mode stabilized VCSEL and the lattercase, with an effectively infinite surface relief diameter, corre-sponds to a conventional multimode VCSEL, where the modalloss for the different modes is lower and very similar to eachother. As can be seen, the fundamental-mode stabilized VCSELhas a higher threshold current and slope efficiency due to itshigher modal loss.

When designing a VCSEL with a fundamental-mode stabi-lizing surface relief, there exists an optimal combination of re-lief depth, relief diameter, and oxide aperture diameter, to pro-

Fig. 3. Simulated CW output power as a function of current for a VCSEL witha surface relief depth of 57.5 nm. The surface relief and oxide aperture diametersare (a) 4 and 6 �m and (b) 11 and 6 �m, respectively. The contributions fromindividual modes are also indicated.

duce the highest single-mode output power [7]. Too shallow arelief will provide insufficient difference in the modal loss, andtoo deep a relief will obstruct the current injection and alterthe waveguide properties. Moreover, a smaller relief diameterwill postpone the onset of higher order modes as they now havea large overlap with the unetched high loss region. However,this will also increase the threshold current of the modeand thereby reduce the maximum output power, which is often

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Fig. 4. Schematic cross-sectional view of the VCSEL geometry. A circularsurface relief is etched in the center of the device.

TABLE IVCSEL STRUCTURE

limited by the thermal roll-over. For larger relief diameters, thehigher order modes will reach threshold at lower currents, andthus limit the maximum single-mode output power. Finally, alarger oxide aperture will in general increase the difference inmodal loss by the decreased overlap between the mode distribu-tions. On the other hand, the larger current-confining apertureleads to a more nonuniform current injection, which favors theonset of higher order modes. Other effects that contribute to adecreased mode discrimination are SHB and thermal lensing,which become increasingly important as the drive current is in-creased.

IV. DEVICE STRUCTURE

A schematic cross-sectional view of the VCSEL geometryis shown in Fig. 4, and the detailed epitaxial structure, whichis designed for 850-nm emission, is listed in Table I. To en-able high output powers from VCSELs, it is important to op-timize the epitaxial structure. The top mirror should have a rel-atively low reflectivity to increase the out-coupling. However,too high an out-coupling will result in large threshold and drivecurrents and therefore excessive device heating which limits

TABLE IIDEVICE AND MATERIAL PARAMETERS

TABLE IIIOPTICAL GAIN PARAMETERS

the output power. The SCH contains three GaAs QWs, and thep-doped top and n-doped bottom Bragg mirrors are built up by22 and 34 mirror pairs, respectively, with graded interfaces. Tothe top mirror, a topmost -thick GaAs layer is added to ob-tain the above-mentioned anti-phase reflection of the unetchedsurface. A 30-nm-thick Al Ga As layer, positioned 30.5nm above the SCH, is selectively oxidized to form the oxideaperture, which provides both optical and current confinement.The oxide layer is purposely located in a node of the longitudinaloptical field. Further, a modulation doping scheme is applied tothe top mirror in order to decrease the differential resistance,while maintaining a low free-carrier absorption loss. A low dif-ferential resistance is desirable since it reduces self-heating andthereby delays the thermal roll-over, which allows for higheroutput powers. Finally, a circular surface relief of depth 57.5nm is etched in the center of the device.

V. SIMULATIONS

In the simulations of the VCSELs with a surface re-lief, we use the device and material parameters that arelisted in Table II and the material gain parameters inTable III. Details on these parameters and the model theydescribe are found in [10]. The free-carrier absorption coef-ficient for the doped mirror layers is crudely estimated by

cm , where cmand cm are the n- and p-doping concentrations, respec-tively [18].

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Fig. 5. Calculated maximum single-fundamental-mode output power as afunction of surface relief diameter for different oxide aperture diameters.Single-mode operation is defined by an SMSR of >20 dB. The surface reliefdepth is 57.5 nm.

A numerical study was first performed to optimize the sur-face relief and oxide aperture diameters. From simulated staticlight–current characteristics, we found that the VCSELs withan oxide aperture between 4–8 m produced the highest single-fundamental-mode output powers. Fig. 5 shows the maximalvalues when a side-mode suppression ratio (SMSR) of 20 dBis required. For each oxide aperture diameter, there is an op-timum surface relief diameter. Further, the optimal combinationis obtained by an oxide aperture and surface relief diameter of 5and 3 m, respectively, producing 6.1 mW. In the following, wewill study three fundamental-mode stabilized VCSELs havingan oxide aperture of 4, 6, and 8 m with an optimized surface re-lief diameter, referred to as single-mode VCSELs. We will alsostudy three VCSELs with the same oxide aperture diameters butwith surface relief diameters that do not introduce any mode dis-crimination, referred to as multimode VCSELs (because of thetop epitaxial layer of GaAs, also the multimode VCSELsmust have a surface relief to remove this layer, but the relief di-ameter is so large that it is effectively infinite). The importantdevice dimensions are listed in Table IV.

Thesimulatedlight–currentcharacteristicsforthesingle-modeand multimode VCSELs are shown in Fig. 6. The single-modeVCSELs are fundamental-mode stabilized over the entire op-erational range and produce powers above 5 mW at the thermalroll-over. The threshold current and initial slope efficiency forthe three single-mode and three multimode devices are 0.5, 1.1,2.5 and 0.2, 0.3, 0.5 mA and 0.78, 0.64, 0.55 and 0.55, 0.55, 0.55W/A, respectively. The higher modal loss for the single-modeVCSELs is responsible for the higher threshold currents andslope efficiencies. Also, the larger difference in the modal lossfor the single-mode VCSELs contributes to the larger variationin the threshold current and slope efficiency with oxide aperture

TABLE IVDEVICE GEOMETRY

diameter. A larger oxide aperture reduces the mode confinement,which increases the overlap between the mode and the highmirror loss region. The cold cavity losses for the three respectivesingle-mode VCSELs are 0.400, 0.405, and 0.479 ps . More-over, the output power of the single-mode devices exhibits amore nonlinear behavior with current and is aggravated for largeroxide aperture diameters. This is mainly caused by the enhancednonlinear effects of SHB and thermal lensing. The effect ofSHB is to depress the carrier density in the center of the activeregion by the intense stimulated recombination in this region.At low currents, the stimulated recombination rate is small andthe carrier distribution resembles the injected current densitydistribution, but with higher currents SHB becomes increasinglyeffective. As a consequence, the carrier density in the peripheryinstead increases strongly because the modal gain must still beequal to the modal loss even though the local gain in the center hasdecreased due to SHB. An increasing number of injected carrierswill thus be lost to the peripheral reservoir of carriers where theyare unlikely to cause stimulated recombination because of thelow photon density. This leads to a slight sublinear light–currentcharacteristic. Further, for lasers with large oxide diameters, theinjected current is larger at the periphery which enhances theSHB effect. At higher currents, thermal lensing adds to the effectof SHB. It is caused by the heating in the central part of the laser,leading to an increase of the refractive index in this region andtherefore a higher confinement of the mode to the center. Thiseffect is most outspoken for devices with a large oxide aperture.In this way, the optical mode abandons the peripheral region,where the losses are high, and therefore the modal loss is reduced,as well as the slope of the light–current characteristic as a smallerfraction of the generated photons are lost through the top mirror.As a rule, the multimode VCSELs normally reach higher outputpowers by the coexistence of several transverse modes, whichcan more effectively utilize the laterally distributed carriers in theactive region for stimulated emission. However, in some cases,the single-mode devices can produce a higher output power bytheir higher out-coupling. This is the case for the single-modeVCSEL with an oxide aperture of 4 m and has been shownexperimentally [9].

A. Modulation Response

The modulation response was computed from a small-signalanalysis. Fig. 7 shows the modulation response for thesingle-mode and multimode VCSELs at five different biascurrents, which correspond to a steady-state output power of0.25, 0.5, 1, 2.5, and 5 mW. The relaxation oscillation (RO)

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Fig. 6. Simulated output power as a function of current. (a) Fundamental-modestabilized VCSELs. (b) Multimode VCSELs.

frequency of the three single-mode devices is lower than that ofthe three respective multimode devices. This is mainly causedby the lower photon density in the cavity, compelled by thehigher modal loss. The RO frequency decreases with increasingoxide aperture diameter due to the reduced photon density andthe fact that the RO frequency depends on the square root ofthe photon density. The damping of the RO normally dependslinearly on the photon density. However, the single-modeVCSELs deviate from this dependency, especially at low outputpowers. This is most clearly seen in Fig. 7(a) for the VCSEL

Fig. 7. Calculated modulation response. The indices (a)–(e) correspond toa steady-state output power of 0.25, 0.5, 1, 2.5, and 5 mW, respectively. (a)Fundamental-mode stabilized VCSELs and (b) multimode VCSELs. The dottedline indicates the �3-dB limit for bandwidth determination.

with an oxide aperture of 8 m, where the RO peak decreasesat the lowest output power, indicating increased damping. Also,the single-mode VCSELs show a clear low-frequency roll-offin the modulation response. The increased damping and thelow-frequency roll-off are caused by the dynamic effects ofSHB. The influence of SHB on the modulation response ofsingle-mode VCSELs has been studied in [19] and [20]. Also,there SHB was found to induce a low-frequency roll-off andstrong damping of the RO. These effects were more pronounced

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in a smaller device, where the spatial distribution of the photonsvaries more abruptly.

To explain the influence of SHB, it is important to distinguishbetween the static and dynamic effects. At dc operation, the car-rier density decreases slightly in the central parts and increasesstrongly in the periphery as the current is increased. This is anobvious effect of the higher photon density which depresses thecarrier density in the center, whereas the higher carrier injectionincreases the carrier number in the periphery where stimulatedrecombination is negligible. Only at high currents does stimu-lated recombination also become significant in the periphery,which leads to a more uniform increase in the carrier density asthe current is further increased. This is also reflected in the dy-namic behavior at quasi-static modulation frequencies—smallcarrier density fluctuations in the center and large in the pe-riphery, especially at low and moderate bias. However, as willbe explained shortly, the long lifetime of the carriers in the pe-riphery leads to a dramatic reduction of the fluctuations in thisregion as the modulation frequency is increased. Above a fewgigahertz, the carrier density fluctuations are of roughly equalmagnitude irrespective of the lateral position. Therefore, the dy-namic effects of SHB are most pronounced at low frequencies.

With these observations, the SHB-induced low-frequencyroll-off can be rather simply understood. In the central regionof the VCSEL, the fluctuations in the carrier density are smalland their amplitude is independent of the modulation frequency.This is because the high photon density causes intense stim-ulated recombination which makes the carrier lifetime muchshorter than the modulation period. The small-signal carrierdensity is therefore determined by the instantaneous value of thesmall-signal injection current. The situation is quite different inthe peripheral region. Because of the low photon density, stimu-lated recombination is small and carrier lifetime longer than themodulation period. The small-signal carrier density is thereforesolely determined by the small-signal injection current whichadds carriers during the positive part of the modulation cycle andremoves an equal amount of carriers during the negative part. Thecarrier density is therefore not dependent on the instantaneousvalue of the injection current but by the time that has elapsed fromthe start of the last modulation cycle. Obviously, with increasingfrequency, the modulation period becomes shorter, which meansless time to add and remove carriers, and thus the fluctuationsin carrier density become smaller. Although the intensity is lowin this region of the VCSEL, the reduced fluctuations in carrierdensity do reduce the modal gain fluctuations somewhat. At thesemoderate frequencies, the photon number of the mode directlyfollows the fluctuations in the modal gain. Therefore, we observea slightly reduced modulation of the photon number with themodulation frequency, which explains the roll-off of the modula-tion response at low frequencies. At frequencies above the cut-offfrequency for SHB, which is typically 1–3 GHz, the fluctuationsin the peripheral region are down to the same low level as those inthe central region. Instead of SHB, the small-signal modulationis now increasingly influenced by RO effects.

The increased damping of the RO by SHB effects is under-stood as follows. Far below the RO frequency, the modulationis so slow that the negative feedback mechanism, the stimulatedrecombination, that determines the carrier density and photon

number have had time to settle at each point in time. Intuitivelythen, the RO frequency could be described as the modulationfrequency at which the period is still long enough for the photonnumber to rise significantly in the direct response to an increasein the carrier density, but short enough for any negative feed-back effects (the consecutive lowering of the carrier density fol-lowed by a lower photon number, resulting from the increasedstimulated recombination) not to occur within that period. Wemay thus say that the ROs occur at a modulation period roughlyequal to the feedback time. However, for a VCSEL strongly af-fected by SHB, the feedback time is not a single well-definedentity: since the feedback time is longer the lower the rate ofstimulated recombination, the feedback time in the periphery isconsiderably longer than in the center. Thus, the onset of ROs isnot sharply defined which leads to a less pronounced resonancepeak, i.e., an increased SHB-induced damping [19].

B. Eye Diagram

The digital modulation characteristics are commonly eval-uated by studying eye diagrams and BERs at various modu-lation conditions. In this study, we focus on the large-signalperformance at 2.5 and 10 Gb/s. The eye diagrams are gener-ated by first recording the response in the output power froma pseudorandom bit sequence modulation current withnonreturn-to-zero (NRZ) data format. Second, the response isdivided into 2-b intervals and overlayed on each other. In thesimulations, we used a time step of 0.4 and 0.1 ps for the 2.5-and 10-Gb/s modulation, respectively.

It is often desirable to reduce the OFF-state output power,for instance, to improve the extinction ratio. However, it isaccompanied with increased timing jitter and bit pattern effects,particularly at higher bit rates [21]. Fig. 8 shows simulated eyediagrams at 2.5 and 10 Gb/s for the single-mode and multimodeVCSELs. For each device, two eye diagrams are displayed,where the OFF and ON currents correspond to a steady-stateoutput power of 1 and 5 mW and 0.25 and 5 mW, respectively.Note that there is a discrepancy in the actual output power,especially in the ON state, due to a different self-heating. Sincethe output power exhibits intensity noise, the eye diagramshave a fuzzy appearance. At 2.5 Gb/s, the multimode VCSELsexhibit more well-defined ON and OFF states and less timingjitter. This is caused by the larger number of possible ways forthe laser to reach a certain state, by the coexistence of multipletransverse modes. The multimode devices have a higherdamping of the RO, which results in a shorter settling time.When the OFF-state output power is reduced from 1 to 0.25mW, the RO frequency and damping in the OFF state decrease.For the VCSELs with a larger oxide aperture diameter, theprevious bit pattern starts to influence the timing jitter. Thesettling time of the RO is then on the order of the bit period.Depending on when the laser is switched from the OFF to theON state, the initial conditions will be different and the laserresponds accordingly. This often results in a turn-on transitionin the form of multiple, almost parallel, discrete paths, eachcorresponding to a certain previous bit pattern. Two such pathscan be barely distinguished for the multimode VCSEL with anoxide aperture diameter of 8 m at 2.5 Gb/s [see the inset in the

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Fig. 8. Simulated eye diagrams. (a) 2.5-Gb/s fundamental-mode stabilized VCSELs. (b) 10-Gb/s fundamental-mode stabilized VCSELs. (c) 2.5-Gb/s multimodeVCSELs. (d) 10-Gb/s multimode VCSELs. For each VCSEL, two eye diagrams are displayed, where the diagram to the right corresponds to a lower OFF-stateoutput power.

bottom right eye diagram of Fig. 8(c)]. Naturally, at 10 Gb/s,the influence of the previous bit pattern on the timing jitter iseven stronger. The number of paths are increased and the ON

and OFF states are further broadened. The lower RO frequencyand larger timing jitter for the single-mode VCSELs result ina somewhat higher degree of eye closure. When the OFF-stateoutput power is reduced to 0.25 mW, the single-mode devicesshow only a small degradation of the timing jitter, while themultimode devices are more influenced. It should be noted thatthe multimode devices have a somewhat lower actual outputpower in the OFF state, especially the devices with a largeroxide aperture diameter, which increases the timing jitter.

In Fig. 9, the eye diagrams of the total output power togetherwith the individual eye diagrams of the two most dominantmodes are shown for the multimode VCSEL with a 6- m

oxide aperture at 2.5 and 10 Gb/s. The eye qualities of theindividual modes are appreciably deteriorated compared to theeye diagrams of the total output power. The degradation is aconsequence of anticorrelated power fluctuations of the indi-vidual modes due to the strong competition for carriers from acommon reservoir (mode-partition noise). The effect becomesparticularly pronounced for modes with a high degree of spatialoverlap due to increased carrier competition. This exampledemonstrates the importance of collecting the power from allexisting modes in optical links using multimode VCSELs.

C. BER

Studying the BER versus average received optical power is astandard procedure for evaluating the transmission performanceof an optical link. Here we study the BER for back-to-back

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

Fig. 9. Modally resolved eye diagrams for the multimode VCSEL with a 6-�m oxide aperture. The eye diagrams to the left and right correspond to 2.5 and 10Gb/s, respectively. The top eye diagrams show the total output power and the lower eye diagrams show the output power in each of the two most dominating modes.

transmission. The receiver consists of a photodetector, low-passfilter, and decision circuitry. In this example, we use receiverspecifications from a measurement setup that we will use infuture comparative experiments. The photodetector is assumedto have a responsivity of 0.4 A/W, output impedance of 50 ,and dominating thermal noise with a noise-equivalent power of50 pW . The low-pass filter is assumed to be a fifth-orderBessel filter with a 3-dB bandwidth of , whereis the bit rate. This corresponds approximately to the optimumbandwidth for the NRZ data format [22]. The fifth-order Besselfilter is described in the frequency domain by

(4)

where [23]. The constants are ,, , , , and. A semi-analytical approach described in [22] is

used to calculate the BER as follows:

(5)

where is the word length, is the low-pass filtered pho-tocurrent, and is the standard deviation of the superimposednoise current from the receiver. The BER depends on the deci-sion level and time of the decision circuitry, and , respec-tively. A minimum BER is obtained by optimizing and .

Fig. 10 shows the calculated minimum BER versus averagedreceived optical power at 2.5 and 10 Gb/s for the single-modeand multimode VCSELs. The BER is calculated for the outputpower responses generating the eye diagrams in Fig. 8. At 2.5Gb/s, the single-mode and multimode devices have similar re-ceiver sensitivities (received optical power at a BER of )that are better than 17 dBm for an OFF-state power of 1 mW.The small variation in receiver sensitivity between devices withdifferent oxide apertures is mainly due to the previously men-tioned discrepancy in actual output power in the ON and OFF

states, especially for the single-mode devices. This results in asmall variation in the extinction ratio between the devices. TheBER value cannot be attributed to a particular feature of the eyediagram. For instance, the receiver sensitivity is strongly influ-enced by both the extinction ratio and pulse overshoot. The latteris particular pronounced for single-mode VCSELs at 10 Gb/s,which can partly explain the larger variation in receiver sensi-tivity for these devices.

When the OFF-state power is reduced to 0.25 mW at 2.5Gb/s, the receiver sensitivity is improved with about 1 and 1.5dB for the single-mode and multimode VCSELs, respectively.This is caused by the increased extinction ratio. The morewell-defined ON and OFF states for the multimode VCSELsresult in a somewhat higher quality of the eye. Going to 10Gb/s, the receiver sensitivity degrades by about 3 to 3.5 dB forboth the single-mode and multimode VCSELs. The main partof this degradation (3 dB) is a result of the necessarily increasedbandwidth of the receiver, which increases the thermal receivernoise. The additional degradation ( 0.5 dB) is caused by theincreased eye closure. It should be noted that we have assumedthat there is no external optical feedback to the laser. Opticalfeedback from external reflections can dramatically degradethe system performance.

GUSTAVSSON et al.: DYNAMIC BEHAVIOR OF FUNDAMENTAL-MODE STABILIZED VCSELs USING A SHALLOW SURFACE RELIEF 617

Fig. 10. Calculated minimum BER as a function of average received optical power. (a) 2.5-Gb/s fundamental-mode stabilized VCSELs. (b) 10-Gb/sfundamental-mode stabilized VCSELs. (c) 2.5-Gb/s multimode VCSELs. (d) 10-Gb/s multimode VCSELs. The BER is plotted for two different OFF-state outputpowers. The horizontal line indicates the BER of 10 , the limit for receiver sensitivity determination.

D. RIN

The laser RIN is of great importance for optical links sinceit ultimately limits the signal-to-noise ratio (SNR) and the dy-namic range. The RIN spectrum was calculated by Fourier trans-forming the time fluctuating output power, which was simulatedusing a temporal resolution of 1 ps. By averaging over 50 tra-jectories, each of duration 40 ns, we obtain a spectral resolutionof 25 MHz.

Fig. 11 shows the RIN spectra for the single- and multimodeVCSELs at three different bias currents, which correspond toa steady-state output power of 0.5, 2.5, and 5 mW. The peakRIN occurs at the RO frequency and is strongly dependent onthe damping of the RO. For the single-mode devices, whichhave a lower damping, the peak RIN varies from 140 dB/Hz,for the device with an oxide aperture diameter of 4 m at anoutput power of 5 mW, to 105 dB/Hz for the device with anoxide aperture diameter of 8 m at an output power of 0.5 mW.For the corresponding multimode devices, which have a higherdamping, the peak RIN is 10–15 dB lower. Below the RO fre-quency, the RIN decreases and reaches a low-frequency RINthat is approximately 10–20 dB lower than the peak RIN, de-pending on the damping of the RO. In this frequency region,faint multiple peaks can be distinguished for the multimode VC-SELs, especially for the devices with a larger oxide aperturediameter. The peaks have been referred to as “mode partitionfrequencies,” which result from the carrier interchange betweenthe modes [24], [25]. Finally, the single-mode devices did notexhibit squeezing (noise levels lower than the lowest possiblehad the photon emission from the VCSEL been a pure Poissonprocess) in the low-frequency RIN, which is probably due to thehigh free-carrier absorption loss and strong SHB. The phenom-

enon has been observed in more efficient single-mode VCSELsat high output powers [26]–[28].

VI. CONCLUSION

We have performed a detailed numerical study on the dy-namic behavior of fundamental-mode stabilized VCSELs, usinga shallow surface relief. The surface relief spatially modulatesthe mirror loss, such that the higher order transverse modes ex-perience a significantly higher modal loss. However, the methodinevitably contributes to a higher modal loss also for the fun-damental mode. The higher out-coupling gives rise to a lowerphoton density in the cavity. It can be anticipated that this effect,together with the strong SHB, might impair the dynamic perfor-mance of these VCSELs. To investigate this numerically, a com-prehensive quasi-3-D model that includes the detailed epitaxiallayer structure and device geometry when calculating the opticalfields was used. The model accurately accounts for the dynamiceffects of carrier density and temperature on the modal distri-butions. Modulation response, eye diagram and BER, and RINof the fundamental-mode stabilized VCSELs were simulatedand compared to simulated characteristics of VCSELs withouta mode discriminator, i.e., conventional multimode VCSELs.

The fundamental-mode stabilized VCSELs exhibit a lowerRO frequency and damping, induced by the lower photondensity in the cavity. The strong SHB gives rise to a low-fre-quency roll-off in the modulation response and contributesto the damping of the RO, particularly at low output powers.Simulated eye diagrams show that the fundamental-mode sta-bilized VCSELs are able to operate at 10 Gb/s although the eyeclosure is somewhat higher than for the multimode VCSELs.

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

Fig. 11. Simulated RIN spectra. The indices (a)–(c) correspond toa steady-state output power of 0.5, 2.5, and 5 mW, respectively. (a)Fundamental-mode stabilized VCSELs. (b) Multimode VCSELs.

This is confirmed by calculated minimum BER, where onlya 0.5-dB power penalty is observed. The RIN is found to beenhanced for the fundamental-mode stabilized VCSELs. Thepeak RIN is about 10–15 dB higher, due to lower damping ofthe RO. In short, even at 10 Gb/s, the dynamic performance ofthe fundamental-mode stabilized VCSELs is competitive withthat of conventional multimode VCSELs. Moreover, in theanalysis, we have assumed that the output from all modes inthe multimode VCSELs is captured. In reality, mode-selective

losses frequently occur along an optical link, from the laserto the photodetector, which can dramatically degrade thedynamic performance.

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Johan S. Gustavsson received the M.Sc. degree in electrical engineering fromChalmers University of Technology, Göteborg, Sweden, in 1998, where he iscurrently working toward the Ph.D. degree.

He joined the Photonics Laboratory, Department of Microtechnology andNanoscience, Chalmers University of Technology. His research is focused oncharacterizing and modeling the dynamics of vertical-cavity surface-emittinglasers.

Åsa Haglund received the M.Sc. degree in physics from Göteborg University,Göteborg, Sweden, in 2000. She is currently working toward the Ph.D. degreeat Chalmers University of Technology, Gothenburg, Sweden.

She is currently with the Photonics Laboratory, Department of Microtech-nology and Nanoscience, Chalmers University of Technology, where her re-search is focused on design, fabrication and characterization of vertical-cavitysurfac-emitting lasers for high-speed optical interconnects.

Jörgen Bengtsson received the M.Sc. degree in engineering physics and thePh.D. degree in photonics from Chalmers University of Technology, Göteborg,Sweden, in 1993 ad 1997, respectively.

Presently, he is an Assistant Professor with the Photonics Laboratory,Chalmers University of Technology. A major research topic has been thedevelopment of novel passive optical components using the inherent multi-functional capabilities of diffractive optics. Earlier work focused on free-spaceapplications, but more recent work has also included diffractive integratedoptics. A current area of research is the modeling and design of devices withdiffractive structures integrated with, or being part of, active components suchas distributed Bragg reflector lasers and vertical-cavity surface-emitting lasersto create laser sources with new desired characteristics.

Peter Modh received the M.Sc. degree in electrical engineering and thePh.D. degree in photonics from Chalmers University of Technology, Göte-borg, Sweden, in 1997 and 2002, respectively. His Ph.D. work focused ongrating-coupled surface-emitting semiconductor lasers with emphasis onbroad-area spatially single-mode lasers integrated with advanced beam-shapingcouplers.

Presently, he is a Research Associate with the Photonics Laboratory,Chalmers University of Technology. His current research includes fabricationof vertical-cavity surface-emitting lasers for high-speed digital and analogapplications with focus on high-power single-mode devices.

Anders Larsson received the M.Sc. and Ph.D. degrees in electrical engineeringfrom Chalmers University of Technology, Göteborg, Sweden, in 1982 and 1987,respectively.

In 1991, he joined the faculty at Chalmers University of Technology where bebecame a Professor in 1994. From 1984 to 1985, he was with the Department ofApplied Physics, California Institute of Technology, Pasadena, and from 1988to 1991 he was with the Jet Propulsion Laboratory, Pasadena, CA. His back-ground is in the area of quantum-well materials and devices for optical commu-nication, optical information processing, and infrared detection. Currently, hisresearch is focused on surface-emitting lasers (vertical-cavity surface-emittinglasers and grating-coupled surface-emitting lasers) and new gain materials forsemiconductor lasers.