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Spectral Linewidth in Microcavity Surface-Emitting Lasers

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Spectral Linewidth in Microcavity Surface-Emitting Lasers Spectral linewidth in microcavity surface-emitting lasers lgor Vurgaftman and Jasprit Singh Solid State Electronics Laboratory, Department of Electrical Engineering and Computer Science, The Unitiersity of Michigan, Ann Arbor; Michigan 48109-2122 (Received13 May 1994; acceptedfor publication 29 July 1994) The Schawlow-Townes expressionfor the laser linewidth predicts a substantial cw linewidth enhancementin microcavity lasers, in which a large fraction of spontaneousemission is directed into the lasing mode, in contrastwith conventionalsemiconductor lasers, in which the lasing mode acceptsonly a tiny fraction of spontaneouslyemitted photons.By performing a theoreticalanalysis of rigorous solutions of the wave equationin the surface-emittingcavity and of the band structure in the active region, it is shown that the increasein the linewidth is much slower than the increase in the spontaneous-emissionfactor p becauseof reductionsin the total spontaneousemission rate, the threshold carrier density, and the linewidth enhancementfactor and an increasedslope of the light-current characteristicobtained for the microcavity laser in steady state. Also much smaller driving currentsin a microcavity laser are requiredto achievethe samepower output comparedwith conventionalsemiconductor lasers, resulting in a linewidth of the order of severalhundred MHz at moderate driving currents for the former. We also discuss the various factors influencing the linewidth in microcavity and strainedquantum well lasersand the relation betweenlasing threshold and spectral linewidth in both macroscopicand microscopic cavity lasers. INTRODUCTION laser geometry.The efficiency of microcavity lasers is owed to the fact that a large fraction (>O.l) of spontaneouslyemit- The availability of single-mode semiconductor lasers ted photonsis directedinto the lasing mode comparedwith a made initial precise measurementsof the cw spectral width tiny fraction (-lo-‘) in conventional cavity laser diodes. of single-modeemission in laser diodes possible.’ Interest- When the fraction reachesunity the lasing threshold transi- ingly, the results indicated a factor-of-SOdiscrepancy with tion disappearsas the distinction between spontaneousand theoretical predictions based on the standard Schawlow- stimulatedemission becomes less relevant.The enhancement Townesexpression.2 In order to explain the experimentalre- of the spontaneousemission factor may be explainedby the sults, the theory of the linewidth in semiconductorlasers was formation of a resonancein the photon density of statesclose reexamined.3It was shown that additional linewidth broad- to the interbandtransitions in the active medium dictated by ening arisesfrom retardedamplitude fluctuations causedby the boundary conditions in the cavity. Although both static spontaneousemission induced refractive index variations. and dynamic characteristicsof microcavity surface-emitting Thesewere quantified in terms of the linewidth enhancement lasers with practically realizable dimensions have been factor CYequal to the ratio of the real and imaginary parts of studied,r5the issue of the linewidth in microcavity lasershas the differential susceptibility,4measurements of which5 pro- received relatively little attention. In this paper, a compara- duceda theoretically expectedvalue close to 5. Calculations tive study of microcavity surface-emitting lasers with of the linewidth enhancementfactor in quantumwell lasers strained quantumwell active regions and quantumwell and indicated a factor of 2-3 reduction6due to the decreased bulk DFB lasers is presentedusing a modification of the asymmetry of the differential gain aroundthe lasing energy. Schawlow-Townesformula with the linewidth enhancement Many refinementsof the original formalism have been pro- factor basedon the Langevm approach.r6 posed in order to account for the power-independent contribution7 and deviations from the inverse power proportionality.* Recently, the semiconductorlaser linewidth has beencalculated using a nonequilibrium Green’s-function FORMALISM formalism.’ The amountof linewidth broadeningin excessof The modified Schawlow-Townesformula as it is usually that predictedby the Schawlow-Townesexpression has been applied in theoretical fits of the linewidth measurementsis attributedto the effects of gain saturation,” currently thought couched in terms of the linewidth of the passive resonator to contribute only a small broadening.” Successfulapplica- and contains two parameterswhose precise values are not tion of the currently acceptedformalism to new types of directly measurable:the inversion factor (ratio of spontane- semiconductorlasers with quantum confinementof carriers ous emission rate per mode to the stimulated emission rate and photons is likely to serve as a test of its validity. per photon) and the linewidth enhancementfactor. From a Microcavity lasershave attractedconsiderable interest of theoretical perspective, the spontaneousemission rate per semiconductorlaser researchersof late that has been fueled mode can be readily calculated,and the presenceof the in- by the possibility of drastic thresholdreduction by meansof version factor is unnecessary.A modification of the linewidth modifying the optical-modedensity using a laser cavity with formula yields the following simple expression? all dimensionsof the order of the photon wavelength.r2The extent of spontaneousemission factor increaser3and thresh- --P&p old reduction14715has been examinedfor the surface-emitting 6Wsr4&cl+ a, 5636 J. Appl. Phys. 76 (IO), 15 November 1994 0021-6979/94/76(10)/5636/4/$6.00 Q 1994 American Institute of Physics where R,, is the total spontaneousemission rate, p the frac- internal reflection at the boundary between the semiconduc- tion of spontaneousemission coupled into the lasing mode, S tor and air.r3 The solutions in the longitudinal direction are the photon density in the lasing mode, and obtained via a transfer-matrix approach separatelyfor s- and p-polarized modes. Three classesof modes are recovered: (i) dXdn)ldn 0) resonant modes resulting from a quarter-wavelength phase a= dxr(n)ldn ’ slip in the center of the center of the cavity, (ii) leaky modes where the real and imaginary parts of the differential suscep- with wave vectors beyond the optical band gap induced by tibilities are functions of the carrier density and given by:4*6 the quasi-periodic mirror stacks, and (iii) horizontally propa- gating modes similar to the usual longitudinal modes in the dxrCn> d&E) WTZ edge-emitting laser cavity. Only the first class of modes is -= dE (3) dn I dn CE-EI)~-W/T~)~’ capable of lasing owing to the large outcoupling losses asso- ciated with the other two classes. The modes of the full dX&) = dE dgin,E) E-E, three-dimensional cavity are obtained as crosspoints of the (4) dn- I dn (E-E#+ (fil~,)~’ longitudinal and transversemodes, the derivation of which is outlined above. The leaky modes are essentially unconfined where dgldn is the differential gain, El the lasing energy, by the cavity and contribute a normalized 1D density of and T2 the collisional broadening time due to carrier-carrier and carrier-phonon interactions, taken to be 0.2 ps. If states for each transverse eigenmode. The resonant and propagating modes contribute a discrete mode each for each T a--@, Eq. (3) contains a delta function, and Eq. (4) is a representation of the fact that XR is the Kramers-Kronig transverse eigenvalue. A Lorentzian line shape is assumed (Hilbert) transform of xI . for discrete modes with the linewidth determined by the re- This formulation of the Schawlow-Townes expression flectivity associatedwith each mode class.‘4T’5The calcula- does not usually represent significant progress since one pa- tion of the band structure and optical transition-matrix ele- ments in strained quantum wells using the Kohn-Luttinger rameter nSp is replaced by another, p. The latter is difficult to compute exactly in a macroscopic cavity laser, primarily ow- Hamiltonian has also been discussed at length.17The spon- ing to its small magnitude. In strongly index-guided lasers, p taneous emission rate into a given mode can then be found has been estimatedI to equal 10e5. Nevertheless, p is ex- by restricting the integral of Eq. (5) over the appropriate pected to be approximately inversely proportional to the cav- wave-vector spread, inversely related to the cavity quality ity length since the number of allowed longitudinal modes is factor. The latter is determined from the reflectivities for the doubled when the cavity length is doubled. By an extension propagating and confined modes as discussedextensively in of the direct calculations of the electronic band structure in Ref. 13. The linewidth enhancementfactor can be readily the strained quantum well active region17and of the photon computed for a given carrier density in the quantum well density of states13-15in the surface-emitting microcavity, it is using Eqs. (2)-(4) and extending the numerical approach of possible, however, to determine p accurately.‘4*15The total Ref. 17. The laser rate equations16can be used to find the spontaneousemission rate is given by (in cgs units): output photon density versus the injected current density. The results for a microcavity surface-emitting laser with lat- 4rr2e2fi 1 eral widths of 0.5 and 1 ,um and a mirror reflectivity of 0.991 c j- dKc 12 -P:,n(k)12 for normal incidence~~areshown in Fig. 1. For comparison, ?ym;fio ii7 * m v,,i the light-current characteristic of a conventional surface- X~Xfiw) S[EW -E;(k) -~~lW-E;W)lI
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