realistic model for the output beam profile of stripe and tapered superluminescent light-emitting...

Realistic model for the output beam profileof stripe and tapered superluminescentlight-emitting diodes

Frederica Causa and Jayanta Sarma

We present a new model to analyze the spatial characteristics of the output beam of conventional�straight-stripe� and tapered superluminescent light-emitting diodes. The device model includes bothspontaneous and stimulated emission processes as well as a nonuniform carrier density distribution tocorrectly represent current spreading and carrier diffusion effects. Near- and far-field intensity profilescomputed with this model are accurately verified over a wide range of injection currents by comparisonswith experimental results measured from in-house fabricated devices. © 2003 Optical Society ofAmerica

OCIS codes: 230.0250, 230.3670, 000.4430.

1. Introduction

High-power superluminescent light-emitting diodes�SLEDs� have become topical after recent advances invarious application areas including short-haul opticalcommunications,1–3 optical coherence tomography,4and fiber gyroscopes.5 To obtain high output powersfrom SLEDs it is necessary to maintain very low facetreflectivities to inhibit lasing. Facet reflectivity ismost commonly reduced by incorporating an un-pumped absorbing region at the rear of the device.6–10

However, the rear absorbing region does not presentthe ideal zero reflectivity, and beyond a level of �in-herent� optical pumping the feedback from the rearfacet is sufficient for optical gain in the electricallypumped region to induce lasing. A convenient expe-dient for obtaining a much reduced effective absorb-ing region and minimizing device costs is to use anetched V-groove as a deflector at the rear of the de-vice11; such a structure was demonstrated to producerecord high output powers.

In addition, for the design of high-power SLEDsthere is also the need to overcome the limitations ofoptical gain saturation and catastrophic optical dam-

The authors are with The Department of Electronic and Elec-trical Engineering, University of Bath, BA2 7AY, England, UK.e-mail for F. Causa, [email protected].

Received 27 January 2003; revised manuscript received 17 April2003.

0003-6935�03�214341-08$15.00�0© 2003 Optical Society of America

age that typically occur at high-power regimes.Such problems were most commonly resolved by us-ing broad-area devices but note that the largest out-put powers were achieved with tapered geometrySLEDs �TSLEDs�.1,11,12 In addition, an even mod-erate antireflection �AR� coating on the output �front�facet of the TSLED is effective in reducing the carrierdepletion at the rear of the device that otherwiseresults at larger injection currents because of thereflectivity at the output facet.11 Similar conditionshave been achieved by using angled-contact straight-stripe SLEDs13,14 and angled-contact TSLEDs�ATSLEDs�.2,15

We present details of a new model based on the rayanalysis that was successfully used to interpret theexperimental characteristics of the ATSLEDs dis-cussed in detail in Ref. 15. In this paper resultsfrom the model are validated by comparisons withexperimental near and far intensity characteristicsmeasured from inhouse-fabricated straight-stripeSLEDs �SSLEDs� and TSLEDs. The convenient fea-ture of the model presented in this paper is that onecan analyze any injection contact geometry to readilydetermine both near- and far-field intensity profiles.

The device model is discussed in Section 2 togetherwith details of device geometry and key parameters.Results of output power versus injected current �lightcurrent� characteristics measured from in-house fab-ricated SSLEDs and TSLEDs, presented in Section 3,provide an experimental estimate of some of the ma-terial�device parameters to be used in the model.Representative near- and far-field intensity curves

20 July 2003 � Vol. 42, No. 21 � APPLIED OPTICS 4341

computed by the model are compared in Section 4 withcorresponding experimental profiles measured on sev-eral in-house fabricated devices. Finally the conclu-sions are summarized in Section 5.

2. Device Model

The model presented here is based on the ray anal-ysis and has been developed to simulate various de-vice geometries including, most important, taperedinjection contacts and angled devices for which theinjection contact is at an angle with respect to thenormal to the facet.15 To conveniently represent thedevice of interest, the carrier density distributionN�x, z� used in this model is assumed to depend onboth the lateral x and the longitudinal z variables toallow for nonuniformities along both axes. In par-ticular, the lateral nonuniformity is essential to rep-resenting both current spreading and carrierdiffusion effects.

Both spontaneous and amplified spontaneous�stimulated� emission processes are included in themodel to estimate the near- and far-field intensityprofiles of SLEDs as a function of the device geometryand the injection current. But, to minimize compli-cated analysis without significant loss in the gener-ality of the results, the steady-state model presentedhere does not incorporate wavelength and polariza-tion dependencies.

The devices studied in this paper are edge emittingand have been fabricated from conventional GaAs�AlGaAs double-heterostructure �DH� semiconductorlaser material. The dielectric step across the DHmultilayer produces strong optical guiding along thevertical y direction, a feature accounted for in thepresent model through the vertical optical confine-ment factor � to obtain the expression for the modalgain:

G� x, z� � �Ag�N� x, z� � Ntr� , (1)

where Ag is the gain constant and Ntr is the trans-parency carrier density. Use of the optical confine-ment factor greatly simplifies the analysis so that anydevice of interest is represented in this model by atwo-dimensional geometry in the �x, z� plane thatessentially corresponds to the injection metal contact.

The fabricated devices typically have a shallow ribetched in the p2� cladding layer to reduce currentspreading �Fig. 1�. However, it has been verifiedthat the corresponding lateral effective refractive-index step is negligible so that there is no index guid-ing along the x direction.

A top-view schematic of a straight-stripe SLED ispresented in Fig. 2. The shaded rectangle repre-sents the current injection contact, and the overallrectangle ABCD is the computational window usedfor the model. The analysis progresses by computa-tions of the amplified spontaneous emission corre-sponding to each ray. Therefore the formulationbecomes simple and straightforward once the carrierdensity profile �which in effect defines the device ge-

ometry� and the computational window have beendefined.

The output facet of the device of interest is as-sumed to be on the AB side of the computationalwindow. The computed rays originate at any pointon the three sides, BC, CD, DA, of the computationalwindow, for example, T1 in Fig. 2. Only those rayswhose end points are in the AB segment contribute tothe output beam of the device. �For example, in Fig.2 the end point of the considered ray is denoted T2.�In the model discussed here the reflectivity of bothrear and front facets are considered to be zero. Thelength R of any ray depends on the coordinates ofboth the origin T1 and the end point T2 and can bedetermined by simple geometrical considerations, asfollows.

Let Wtot be the length of the AB segment, Wout theoutput facet width, L the length of the device �corre-sponding to the length of BC�, � the angle between theray and the AB side, and �xf , L� the coordinates of T2.An important point here is that the width Wtot of thecomputational window is generally larger than thedevice output facet width Wout to account for currentspreading and carrier diffusion effects.

With reference to Fig. 2 it is possible to identify thefollowing two particular cases for the position of theray origin T1:

�i� T1 � D corresponding to � �1�xf� arctan�L�xf�;

�ii� T1 � C corresponding to � �2�xf� �2 �arctan��Wtot � xf��L�.

Fig. 1. Schematic of a typical edge-emitting SLED.

Fig. 2. Top view schematic of the geometry utilized for modelingSLEDs.

4342 APPLIED OPTICS � Vol. 42, No. 21 � 20 July 2003

Note that the expressions for both angles abovedepend on the position xf of the end point T2 on thedevice facet.

The length R R�xf , �� of a given ray is thus givenby

R �xf

cos��for 0 � � � �1 , (2)

R �L

cos�

2� �� for �1 � � � �2 , (3)

R �Wtot � xf

cos� � ��for �2 � � � . (4)

The optical power associated with a given ray is cal-culated by summing the contributions of spontaneousand stimulated emissions along the ray. For thispurpose a new variable, denoted s, is defined thatrepresents the spatial variable along the ray.

The optical power P associated with any ray iscalculated by solving the following one-dimensionalequation16:

dPds

� G�N�s��P�s� � �N�s�

sp�, (5)

where G�N�s�� is the modal gain, N�s� is the carrierdensity expressed in terms of the new variable s, �is the fraction of spontaneous emission couplinginto the forward traveling photon flux, sp is thespontaneous recombination time constant, and �� c�nmat is the group velocity with nmat the �vertical�mode effective refractive index. The fraction � isapproximated here by an averaged constant �� v�h��4 where the vertical and horizontal cap-ture angles are, respectively, �v � 2 arctan�d�L�and �h � arctan�2Wout�ML�, with d the active layerthickness and M the number of discretization pointsused in the model. The material and device param-eters used in the simulations are documented by theresults in Sections 3 and 4.

A simple case to analyze is when, with reference toFig. 2, the gain is constant in the shaded area �andzero outside that area�, that is, N�s� N const andG�N�s�� G�N� G const. In this case Eq. �1� isamenable to a simple analytic solution16:

P� xf, L; �� exp�GR� � 1�Psp , (6)

where Psp � N�� sp�G�. The closed-form solution,Eq. �6�, is used in conjunction with the results dis-cussed in Section 3 to show that only by includingcurrent spreading and carrier diffusion the deviceoutput beam characteristics can be accurately repre-sented.

The carrier densities and gain distributions thatare of interest here are �laterally and, e.g., for tapereddevices, longitudinally� nonuniform, and therefore

Eq. �5� must be solved numerically by using, for ex-ample, a Runge–Kutta-type method.

In principle, the carrier density profile N�x, z� inthe active layer can be obtained by numerically solv-ing the nonlinear diffusion equation inclusive of cur-rent spreading all along the length of the device.17

However, the method above is too detailed for thispurpose especially in longitudinally nonuniform de-vices because storage of large arrays of data are nec-essary to represent the carrier density distribution inthe active layer.

An alternative method that greatly simplifies thenumerical procedure is to find a convenient analyticalrepresentation for the carrier density profile at alllongitudinal positions. In this model the followingsuper-Gaussian �SG� function18 has been used forthis purpose:

N� x, z� � Nmax� z�exp�� 2xWN� z��

q� z�� , (7)

where the maximum value Nmax�z�, the width WN�z�,and the exponent q�z� of the SG function are z depen-dent so that the carrier density profile can be accu-rately described at all longitudinal positions. Theflexibility of having z-dependent parameters is spe-cially important for taper geometry devices since thecarrier density profile may vary from Gaussianlikewhere the rib width is small to almost rectangularwhere the rib width is large compared with the car-rier diffusion length.

To further increase the accuracy of the model theSG parameters used in Eq. �7� are determined bymatching the SG representation of the carrier densityprofile to the spontaneous emission profile measuredfor material gain below transparency at the devicefacet,19 as discussed in Section 3.

For implementation of the numerical model, notethat Eq. �5� is effectively written in terms of the raycoordinate system �s, �� with � fixed for an individualray, whereas the device�material parameters aregenerally given in terms of the conventional Carte-sian coordinates, e.g., Eq. �1� and �7�. It is thereforenecessary for one to convert from one system of ref-erence to the other to properly solve Eq. �5�, by using

x � xF � �R � s�cos�� , (8a)

z � L � �R � s�sin�� . (8b)

The considerations above highlight the importance ofusing an analytic function to represent N�x, z�, Eq.�7�. This expedient in fact greatly simplifies the nu-merical solution of Eq. �5�, which requires the func-tion G�x, z� to be sampled at arbitrary points alongthe length of the ray.

The near intensity profile INF�xf�, that is, the powerdensity distribution across the facet, is obtained by


integrating the contributions of rays at all angles � ateach point �xf, L� on the facet. Thus

INF� xf� � ��a

�a

P� xf, L; ��d� . (9)

In Eq. �9� the limits of integration are determined bythe lens acceptance angle �a � arcsin�NA�nmat�,where NA is the numerical aperture of the collectinglens.

The far intensity profile IFF��, that is, the angulardistribution of the power density, is found by inte-grating the contributions of the rays at a particularangle from all the points on the output facet. Thus

IFF��

�

P� xf, L; ��dxf . (10)

3. Experimental Estimate of Model Parameters

For reliable and realistic modeling it is useful to es-timate experimentally as many device and materialparameters as possible. The results in this sectionare from devices fabricated from DH bulk GaAs semi-conductor material �Table 1� and have the followingdimensions: SSLEDs, Wout 50 �m, L 1 mm;TSLEDs, Wout 100 �m, L 1 mm. Therefore thetwo above categories of devices have the same injec-tion contact areas.

The transparency carrier density can be evaluatedby comparing the measured light output versus in-jection current �L–I� characteristics of SSLEDs andTSLEDs, as described in the following. Representa-tive experimental L–I curves for both SSLEDs andTSLEDs are presented in Fig. 3. At very low injec-tion currents, below transparency, the output power�spontaneous emission� comes essentially from theregion close to the facet; thus at these values of theinjection current the TSLED generally producesmore power than the SSLED �Fig. 3�a��. At trans-parency the two categories of devices emit the samepower since they have the same active volume. Thevalue of the current at which the two L–I curves crossshould therefore give quite an accurate experimentalcharacterization of the injection current required fortransparency Itr. For the material of interest herethe transparency current density is Jtr � 2.4 kAcm�2. At injection currents just above transparencythe devices begin to have optical gain and it is ob-served that the SSLED emits more optical powerthan the TSLED. A qualitative explanation for thisobservation can be provided by assuming that raysperpendicular to the facet more significantly contrib-ute to the overall output power at low gain. In thiscase the difference in output power between SSLEDsand TSLEDs derives from the fact that in the formerall such rays are identical while in the latter thedominant contribution comes from shorter rays. Athigher injection currents �Fig. 3�b�� it is observed thatthe TSLED produces the highest output power, sug-gesting that optical saturation effects begin to take

place in the SSLED and that therefore the taper ge-ometry yields a more efficient device design.

An essential part of the experimental activity in-volves detection of the carrier density profile at lowinjection current densities �below transparency� toestimate the SG parameters, Eq. �6�, to be used in themodel.

At low injection currents the �optical� near inten-sity profile is due to spontaneous photon generation.From the rate equations it can be seen that in thiscase the output photon density profile is proportional

Fig. 3. Output optical power versus input current for in-housefabricated devices: dashed curve, SSLED �W 50 �m, L 1000�m�; solid curve, TSLED �W 100 �m, L 1000 �m�; �a� for lowinjection currents �b� for a wide range of injection currents.


to the square of the carrier density profile through the�bimolecular� spontaneous recombination term.The near intensity profile can be detected experimen-tally with the apparatus shown schematically in Fig.4. The experimental profile is then matched withthe theoretical profile obtained from the solution ofthe carrier distribution model briefly described in thefollowing to estimate the SG parameters.

The carrier density distribution can be determinedby solving the diffusion equation. The particularfunction expansion method used to obtain the resultspresented in this paper was described in Ref. 17. Inthe method adopted here diffusion along the longitu-dinal axis is neglected, so that the z variable is nowparameterized, that is, D��2N� x, z�� z2 � 0 andtherefore D��2N� x, z�� x2 � D�d2N� x; z��dx2.At low injection current regimes the stimulated re-combination term becomes insignificant, and there-fore the equation describing the time-independentlateral carrier diffusion in the active layer takes theform20, 21

Dd2N� x; z�

dx2 �N� x; z�

�N��

Jy� x; z�

qd� 0 , (11)

where N�x; z� is the carrier density distribution to beevaluated at any given longitudinal position, D is theeffective diffusion coefficient �that approximates theambipolar diffusion�, �N� � �Br�n0 � N� x; z�� N2� x; z��1 is the recombination rate inclusive ofthe quadratic and cubic recombination terms, Br isthe bimolecular recombination constant, n0 is the

doping density, � is the Auger recombination coeffi-cient, q is the electronic charge, d is the thickness ofthe active layer, and Jy�x� is the top junction injectioncurrent density distribution �Fig. 5�. It is assumedthat Jy�x� is explicitly known from an approximateanalysis as in Ref. 22.

For structures that are unbounded along x theboundary conditions associated with Eq. �11� are

N� x; z� 3 0,dN� x; z�

dx3 0 for x 3 � . (12)

The near intensity profile measured below trans-parency is used to estimate the diffusion coefficient Dand the current density distribution Jy�x� for the de-vices of interest. A comparison of the experimentalnear intensity profile �below threshold� with the cor-responding theoretical result is shown in Fig. 6�a� foran inhouse-fabricated SSLED. By adjusting the in-put parameters to match the experimental curve, it ispossible to compute the carrier density profile corre-sponding to a given injection current.17

The next step is to approximate the carrier densityprofile derived from the diffusion equation model to aSG function that is then used in the SLED model.The best-fitting SG for the carrier density profile cal-culated from the experimental curve of Fig. 6�a� ispresented in Fig. 6�b� together with the correspond-ing carrier density profile computed by solving thediffusion equation. For the case shown in Fig. 6 thebest-fit diffusion coefficient is D 40 cm2 s�1.

4. Near and Far Intensity Profiles

Results from the SLED model described in Section 2are validated by the experimental near and far in-tensity profiles measured from several inhouse-fabricated SSLEDs and TSLEDs. The ray modelpresented in this paper is used to simultaneouslycalculate both the near and the far intensity profilesfor a given device. Hence, once the parameters arechosen to match for either the near or the far inten-sity of a device at a particular current, it is found thatthe near and far intensity curves subsequently cal-culated for that device are in good agreement withthe corresponding experimental profiles for a widerange of currents.

The comparison between measured and theoreticalnear intensity profiles of SSLEDs and TSLEDs fordifferent injection currents is shown in Figs. 7 and 8,respectively. The dotted curves in both figures rep-resent the closed-form solution calculated with Eq.�6�. Note that in neither case does the closed-formsolution match the corresponding experimental pro-file, demonstrating that it is essential to include cur-rent spreading and carrier diffusion effects tocorrectly represent the operation of these devices.On the other hand, in Figs. 7 and 8 the matchingcurves have been calculated including the SG carrierdensity distributions in the SLED model.

The adequacy of the model presented in this paperis substantiated by the interpretation of the far-fieldintensity profiles corresponding to the near intensity

Fig. 4. Experimental apparatus for detection of the near intensityprofile.

Fig. 5. Schematic illustrating current spreading in a typical DHsemiconductor material SLED.


profiles shown in Figs. 7 and 8. The comparisonbetween far intensity profiles obtained from themodel with those experimentally measured frominhouse-fabricated devices are presented in Fig. 9 forSSLEDs and in Fig. 10 for TSLEDs. It is useful todiscuss some of the features observed from the exper-imental far intensity profiles to underline the poten-

tial of the model. The closed-form solutioncorresponding to a uniform gain in the SSLED pro-duces a far-field intensity profile showing a sharptriangular shape on the beam on axis �dashed curvein Fig. 9�. However, the far intensity profiles mea-sured from SSLEDs present a more rounded top.

Fig. 6. �a� Near intensity profile for a SSLED at low injectioncurrent, I 60 mA: solid curve, experimental near intensityprofile; dashed curve, numerical solution of the diffusion equation;dotted curve, best-fitting SG representation. �b� Correspondingcarrier density profile: dashed curve, numerical solution of thediffusion equation; dotted curve, best-fitting SG representation.SSLED device geometry, Wout 50 �m, L 1000 �m; modelparameters, q�z� q 4, WN 55 �m, Wtot 136 �m, d 0.35�m, Ag 1.5 � 10�16 cm2, nmat 3.5.

Fig. 7. Near intensity profile of an SSLED at I 0.2 A and I 2A �shifted for clarity�: dots, experimental profiles; solid curve,solution from the model; dashed curve, analytic closed-form solu-tion. SSLED device geometry; Wout 50 �m, L 1000 �m.Model parameters: Ntr 1.5 � 1018 cm�3, q�z� q 6, WN 45�m, Wtot 136 �m, � 0.6, lens �NA� 0.54.

Fig. 8. Near intensity profile of a TSLED at I 0.5 A and I 1.5A �shifted for clarity�: dots, experimental profiles; solid curve,solution from the model; dashed curve, analytic closed-form solu-tion. TSLED device geometry: Wout 100 �m, L 1000 �m.Model parameters: Ntr 1.1 � 1018 cm�3, q�z 0� 2, q�z L� 8, WN 140 �m, Wtot 200 �m, � 0.7, lens �NA� 0.54.


This is explained with the present model by consid-ering a realistic, nonuniform carrier density profile inthe device. The correct far intensity profile calcu-lated by the realistic model satisfactorily reproducesthe measured profile, as shown by the continuouscurve in Fig. 9.

A similar argument can be used with reference toTSLEDs. In this case, however, note that theclosed-form solution is flat in the center over a range

of angles that is determined by the taper angle. Themore realistic profile �dashed curve� produced by themodel compares well with the experimental curve,showing a smoother profile.

5. Conclusions

We have presented a new ray-model analysis for thedetailed interpretation of the output-beam character-istics of straight-stripe and tapered superlumines-cent LEDs. The model is used to calculatesimultaneously both near- and far-field intensity pro-files for any given device. A super-Gaussian func-tion is used to conveniently represent the carrierdensity profile in the model to account for both cur-rent spreading and carrier diffusion effects. Thisexpedient analytical formulation makes the com-puter model fast and accurate, as demonstrated bythe excellent agreement between the computed andvarious experimental results measured from severalinhouse-fabricated devices.

The model parameters are chosen by matching ei-ther the near or the far intensity profile for a givendevice at a specified current; the subsequently calcu-lated intensity profiles for that device are found tomatch the corresponding experimental profiles for awide range of injection currents. In addition, andmost important, the flexibility of having z-dependentparameters permits one to also analyze taperedgeometry SLED devices where the carrier densityprofile varies for increasing rib widths from Gaussianlike to almost rectangular.

The extension of this procedure to include finitefacet reflectivity and optically induced carrier satu-ration �hole burning� is being pursued and will bereported.

The authors thank S. Yunus and L. D. Burrow forproviding the experimental results and J. S. Robertsof the III–V Semiconductor Central Facility, Univer-sity of Sheffield, UK, for providing the semiconductormaterial.

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realistic model for the output beam profile of stripe and tapered superluminescent light-emitting...

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