Spectral linewidth in microcavity surface-emitting 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 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 arerequired to 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 issueof the linewidth in microcavity lasershas the differential susceptibility,4measurements of which5 pro- receivedrelatively 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 asymmetryof the differential 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 hasbeen calculated 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 quantumconfinement of carriers ous emission rate per mode to the rate and photons is likely to serve as a test of its validity. per photon) and the linewidth enhancementfactor. From a Microcavity lasershave attracted considerable 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 examined for 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 emitting laser with a cavity width of 20 pm is also displayed. The required value of the reflectivity can be achieved by a x-t1-f ”[~~(Wh 6) variety of designs, such as Al(Ga)As/GaAs stacks or com- where P,,Jk) is the momentum matrix element between the pact dielectric multilayer mirrors. All quantum well lasers nth subband in the conduction band and the mth subband in have an active medium consisting of a single 50 A the valence band, f and f” the electron and hole distribution b$2Gao.8As quantum well with built-in compressive strain functions, respectively, p;- the density of states for polariza- on a GaAs substrate. tion 2, fiw the photon energy, k the electron wave vector in DISCUSSION the plane of the quantum well, Ee and Eh the electron and hole energies, respectively, in the indicated subbands, v the The derivation of Eq. (1) assumes that spontaneous refractive index in the active layer, m. the free-electron emission is irreversible owing to the coupling to the thermal mass, e the electron charge, and +i the reduced Planck con- reservoirs of the modes of the universe and of the crystal stant. The method and quantitative results for the calcula- lattice. The interaction of a two-level system at 0 K and of a tions of the photon density of states from solutions of the single-mode field is, in principle, reversible with the continu- wave equation in the surface-emitting microcavity have been ous energy exchangebetween the field and the system (Rabi presentedelsewhere.14115 In a brief summary, the wave equa- oscillations).‘* However, in practical microcavity lasers, the tion is separated in the transverse and longitudinal coordi- photon lifetime (a few ps) and the dipole lifetime (tenths of a nates, with the solutions for the former computed analyti- ps) are much smaller than the period of Rabi oscillations cally for a square cross section of the surface-emitting laser (tens of ps) resulting in a loss of coherence.The dipole life- cavity. All of the transversemodes are assumedto be discrete time cannot be greatly increasedby lowering the temperature and correspond to discrete eigenvalues owing to the total due to the strong carrier-carrier interaction. From Eq. (l),

J. Appl. Phys., Vol. 76, No. IO, 15 November 1994 I. Vurgaftman and J. Singh 5637 total, 0.5 fxrn A----total 20 pm losing______-mode, ------O.Spm ---. .lasing..mode,..~~.~.~......

lo9 i kity

ii lo*: 1.0 pm cavity

I 10+-L..,

0.0 2.0 4.0 6.0 8.0 PHOTON DENSITY (cm-‘) 0.0 100.0 200.0 300.0 400.0 CURRENT DENSITY (A/cm’) FIG. 2. The spectral linewidth vs the photon density in microcavity lasers with cavity widths of 0.5 and 1.0 pm, a bulk DFB laser with a 100 nm thick active layer, and a strained quantum well DFB laser. FIG. 1. The photon density vs the injected current density for the 0.5 and 20 J.L~ wide cavities in all modes and in the lasing mode only. order-of-magnitudereduction has been demonstratedin mul- the linewidth of a semiconductor laser is directly propor- tiquantum well DFB lasers.” By substituting typical values tional to the rate at which spontaneousemissions perturb the for the active areaand quantumefficiency, the linewidth cal- phaseof the optical field in the lasing mode. At higher field culated in Fig. 2 for DFB lasersis smaller by a factor of 2-4. intensities (higher photon densities), the dephasingeffect of The discrepancymay be attributed to [i) uncertainty in the p quantum noise becomes less important. Since the rate of factor, for which only an order-of-magnitude estimate is spontaneousemission into the lasing mode is greatly en- available, (ii) the neglect of the effects of gain saturation at hancedin a microcavity laser,it is natural to expect linewidth high photon densities, and (iii) the neglect of the additional broadeningby several orders of magnitude. For example, enhancementfactors derived from nonorthogonalityof trans- p=O.l, 0.05 for the microcavity laserswith a cavity of width verse and longitudinal cavity modes.” Note also that, on the of 0.5 and 1.0 m, respectively, comparedwith @=1O-5 in basis of this model, the linewidth is expectedto be approxi- the DFB laser. If the active regions in the DFB and micro- mately inversely proportional to the cavity length. The re- cavity lasers are identical, it might appearfrom ELq.(1) that sults of Fig. 2 also indicate almost an order-of-magnitude the linewidth is proportional to the spontaneousemission line narrowing in the strained quantum well laser as com- coupling factor. Nevertheless, numerical simulations show pared with the double heterostructurelaser. The linewidth that linewidth broadening is much slower than /? enhance- reduction is solely due to the smaller value of the linewidth ment. For p sufficientIy close to unity, the total spontaneous enhancementfactor. emission rate is modified along with its distribution between The results of Fig. 2 demonstratethat a spectral line- the cavity modes. Owing to the insufficient filling of the width of severalhundred MHz is to be expectedtheoretically wave-vector space by discrete transverse modes in the from surface-emittingmicrocavity lasers with strained quan- surface-emitting geometry, the total spontaneousemission tum well active regions.Another useful comparisonthat can rate may be reducedby a factor of 2-3, a phenomenonalso be made is in terms of the injected current density. The re- manifested in the suppression of the threshold current sults of theoretical calculations are summarized in Fig. 3. density.14A combination of a microscopic cavity and a Only the linewidth abovethreshold is shown since the use of strained active layer also allows a slight reduction in the the Schawlow-Townes formula below threshold requires threshold current density with the concomitant reduction in some care.‘l The relation betweenlaser threshold and line- the linewidth enhancementfactor. The spectral linewidth ver- width narrowing is discussedfurther below. The threshold sus the photon density is shown in Fig. 2 for two microcavity current density in bulk DFB lasers is close to 1000 A/cm2, lasers, a DFB laser with a 100 nm thick active region and a while it is only 160 A/cm’ in strained quantum well lasers. strained quantum well active layer. The majority of experi- The threshold current density of the microcavity lasers is mental measurementsof the linewidth to date have focused further reduced to 60-80 A/cm’, depending on the lateral on the bulk DFB laser. The reported linewidths are in the width. Therefore, as can be seenfrom Fig. 3, the linewidth lo-100 MHz range for output powers of several mW. An enhancementin microcavity lasers is further countervailed

5638 J. Appl. Phys., Vol. 76, No. IO, 1.5 November 1994 I. Vurgaftman and J. Singh intrinsic losses in the cavity.22In a macroscopiccavity laser, the threshold current obtainedby finding the current density intercept as discussedabove is close to the point at which the gain-loss balance becomesnearly perfect. However, this is no longer true in a thresholdlessmicroscopic cavity laser.A finite injection current is always necessaryto achieve trans- parency in the active medium. Sufficiently far below that point, linewidths of the order of those in a “cold” cavity are to be expected.

CONCLUSIONS

DFB bulk To summarize, in this paper we have presentedthe re- 4 L . I sults of our numerical calculations for the spectral linewidth in micrometer-sizedmicrocavity surface-emitting lasers not- ‘“i \B SQW / ing the reasonsfor the absenceof the “catastrophic” line- width broadeningexpected from spontaneousemission cou- pling factor considerations alone and given an intuitive I I IO5 I discussion of the various factors that must be taken into ac- 0.0 500.0 1000.0 1500: 0 count if an adequateunderstanding of the linewidth in mac- CURRENT DENSITY (A/cm’) roscopic and microscopic cavity lasers is desired.

FIG. 3. The spectral linewidth vs the photon density in microcavity lasers ACKNOWLEDGMENT with cavity widths of 0.5 and 1.0 w, a bulk DFB laser with a 100 nm thick active layer, and a strained quantum well DFB laser. This work was supportedby the Army ResearchOffice under Contract No. DAAL 03-92-G-0109.

‘M. W. Fleming and A. Mooradian, Appl. Phys. Lett. 38, 511 (1981); 40, by threshold reduction, i.e., the difference in the spectral 86.5 (1982). linewidth is reduced if the comparison is made at the same ‘Originally presented by A. L. Schawlow and C. H. Townes, Phys. Rev. current injection. The calculations made for the DFB laser 112, 1940 (1958); corrected by M. Lax, in Physics of Quantum Electron- assumethat the laser oscillates in a single mode asplong as it ics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1966), p. 735; H. Haug and H. Haken, Z. Phys. 204, 262 is biased above threshold. It must be recognizedthat not for (1967). all so-called single-mode lasers, is this assumption sound. ‘C H Henry, IEEE J. Quantum Electron. QE-18, 259 (1982); QE-19, On the other hand, properly designedmicrocavity lasers are 1391’11983). intrinsicaIly single mode both above and below the lasing 4K. Vahala, L. C Chiu, S. Margalit, and A. Yariv, Appl. Phys. L&t. 42, 631 (1983). threshold since only one resonantmode acceptsa significant ‘C. Harder, K. Vahala, and A. Yariv, Appl. Phys. Lett. 42, 328 (1983). portion of spontaneousemission. ’ Y. Arakawa, K. Vahala, and A. Yariv, Appl. Phys. L&t. 45, 950 (1984); Y. In a macroscopic cavity laser, the relation between the Arakawa and A. Yariv, IEEE J. Quantum Electron. QE-21, 1666 (1985). linewidth far below and far above lasing threshold is easily 7W. Elsisser and E. 0. Gijbel, IEEE J. Quantum Electron. QE-21, 687 (1985); K. Vahala and A. Yariv, Appl. Phys. Lett. 43, 140 (1983). understood.In the former case, the linewidth is of the order ‘W. Elsiisser, Appl. Phys. Lett. 44, 1126 (1984). of the “cold” cavity linewidth inversely related to the cavity ‘F. Herzel, K. Henneberger, and W. Vogel, IEEE J. Quantum Electron. quality factor. In the latter case, the linewidth is inversely QE-29, 2891 (1993). *‘G. P. Agrawal, IEEE J. Quantum Electron. QE-23, 860 (1987); J. Appl. proportional to the output optical power. In the vicinity of the Phys. 63, 1232 (1988). lasing threshold, more complex behavior has been observed “G. P. Agrawal, IEEE J. Quantum Electron. QE-26, 1901 (1990). and analyzed.‘l In microcavity lasers whose light-current “Y. Yamamoto, S. Machida, and G. Bjork, Opt. Quantum Electron. 24, characteristic is shown in Fig. 1, the lasing transition is less S215 (1992); H. Yokoyama and S. D. Brorson, J. Appl. Phys. 66, 4801 (1989). pronounced,but the general conclusions arrived at for mac- t3T Baba, T. Hamano, F. Koyama, and K. Iga, IEEE J. Quantum Electron. roscopic cavity lasers are still valid. In truly “thresholdless” QE-27, 1347 (1991); QE-28, 1310 (1992). microcavity lasers, for which p must exceed O.&l5 the cur- 141.Vurgaftman and J. Singh, Appl. Phys. I.&t. 64, 1472 (1994). vature in the light-current curve disappears,and the threshold “I. Vurgaftman and J. Singh (unpublished). I6 G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nos- current density defined as the intersectionof the light-current trand Reinhold, New York, 1993). curve with zero photon density is almost exactly zero. The t7J. P. Loehr and J. Singh, IEEE J. Quantum Electron. QE-27, 708 (1991). question arises: Can a linewidth significantly narrower than “M Sargent III, M. 0. Scully, and W. E. Lamb, Jr., Laser Physics the “cold” cavity linewidth be expectedin microcavity la- (Addison-Wesley, London, 1974). r9K Uomi, S. Sasaki, T. Tsuchiya, M. Okai, M. Aoki, and N. Chinone, sers for any value of injected current? The answer can be Electron. Lett. 26, 52 (1990). obtainedby recollecting that the Schawlow-Townes expres- “W. A. Hamel and J. P. Woerdman, Phys. Rev. A 40, 2785 (1989); Phys. sion accountsfor the linewidth due to quantumnoise arising Rev. I.&t. 64, 1506 (1990). from spontaneousemission processes.If the optical field is “R Hui, N. Caponio, S. Benedetto, and I. Montrosset, IEEE Photon. Tech- nol. Lett. 4, 841 (1992). treated classically, the laser emission line is a delta function “For a “philosophical” discussion of this point, see R. Lang, M. 0. Scully, if the gain of the active medium balancesthe reflection and and W. E. Lamb, Jr., Phys. Rev. A 7, 1788 (1973).

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