Evolution of the Onset of Coherence in a Family of Photonic Crystal Nanolasers ͒ ͒ ͒ ͒ ͒ ͒ ͒ ͒ Y.-S

APPLIED PHYSICS LETTERS 91, 031108 ͑2007͒

Evolution of the onset of coherence in a family of photonic crystal nanolasers ͒ ͒ ͒ ͒ ͒ ͒ ͒ ͒ Y.-S. Choi,a ,b M. T. Rakher,c K. Hennessy,a S. Strauf,c A. Badolato,a P. M. Petroff,a ,d ͒ ͒ ͒ D. Bouwmeester,c and E. L. Hua ,d University of California, Santa Barbara, California 93106 ͑Received 25 April 2007; accepted 31 May 2007; published online 19 July 2007͒ The authors report on the systematic variation of the onset of lasing in high-␤ photonic crystal nanolasers. A series of nanocavities has been designed to systematically approach the high-␤ devices by controlling the number of modes in the s-shell spectrum of InAs quantum dots at 4 K. The lasing action is conﬁrmed by the observation of coherent-state transition to Poissonian photon statistics. The quantitative analysis reveals the high ␤ of 0.69, 0.44, and 0.19 for the nanocavities with one, two, and three modes, respectively. By mapping the observed lasing transitions to ␤ factors, the authors demonstrate the interplay of ␤ and lasing performance. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2751131͔

Recent advances in the ability to design and fabricate ergy, high-Q even ͑e#͒ modes, we specified a lattice con- active nanocavities with tailored modal characteristics1–3 stant of ϳ260 nm, a hole radius of ϳ65 nm, and a thickness have enabled the nanolasers with very high-␤ factors, mean- of ϳ126 nm. Microphotoluminescence ͑␮-PL͒ measure- ing virtually all spontaneous emission is directed into the ments were carried out at 4 K using a He cryostat and a lasing mode. Subsequently, substantial reduction in lasing monochromator under optical pumping with a 780 nm laser threshold1–6 and high-speed modulation7 have been demon- diode. The ␮-PL spectra in Fig. 1͑b͒ show that L3, L7, and strated, stimulating further research toward the highest- L11 cavities support one, two, and three nondegenerate performance nanolasers.8,9 However, the realization of high- modes polarized along the y direction, in accordance with ␤ nanolasers is complicated by the inherent difficulty in our cavity design. With a simple mode counting argument, characterizing their behavior. The characteristic nonlineari- this can stepwise change ␤ from approximately 1 to 0.5 to ͑ ͒ ties in the optical output power around the lasing threshold 0.33 and demonstrate systematic variation in the g 2 ͑0͒ become so subtle that conventional criteria used to determine transition near lasing thresholds. For the measurement of ͑ ͒ a lasing threshold are not well defined. Theoretical and ex- g 2 ͑␶͒, the ␮-PL was sent to two avalanche photodiodes ar- 15 perimental studies on second-order intensity correlation5,6,10 ranged in the Hanbury-Brown–Twiss configuration. or photon number fluctuation11,12 have demonstrated their We begin by describing measurements taken on a con- usefulness in confirming the high-␤ lasing actions in the ventional laser diode that calibrates our technique for a low- nanocavities5 and the microcavities.6,10–12 ␤ laser. This laser has a clearly defined threshold current of In this letter, we have investigated the evolution of the second-order intensity correlation function near the lasing thresholds of photonic crystal ͑PC͒ nanolasers with tailored modal characteristics. Our high-␤ PC nanolasers are com- posed of self-assembled InAs quantum dots ͑QDs͒, with a density of 5ϫ109 cm−2, grown by molecular beam epitaxy on a ͑100͒ GaAs substrate by the partially covered island technique.13 The QDs show sharp and distinct exciton and multiexciton spectra, with clearly delineated s-shell and p-shell regions at 4 K. The nanocavities were designed as triangular-lattice PC structures with 3, 7, and 11 missing holes in the ⌫-K direction, denoted as L3, L7, and L11 cavities, respectively. The cavity resonances are determined by the PC waveguide dispersion and the Fabry-Pérot condition.14 The finite-difference time-domain simulation shows that the cavity resonances exist on the dispersion curves of a single-line-defect PC waveguide, as shown in Fig. 1͑a͒. Thus, by changing the number of missing holes, the number of modes within the QD s-shell spectrum can be engineered. To match the s-shell spectrum with the low en-

FIG. 1. ͑Color online͒͑a͒ Dispersion curves of a PC waveguide and the ͒ a Department of Electrical and Computer Engineering. resonances of the L7 and L11 cavities; the shaded area is the leaky region. ͒ b Electronic mail: [email protected] A scanning electron microscopy inset shows the L7 cavity. ͑b͒ ␮-PL spectra ͒ c Physics Department. of the L3, L7, and L11 lasers far above threshold. Cavity modes are identi- d͒ Materials Department. ﬁed with the calculated Ey proﬁles.

0003-6951/2007/91͑3͒/031108/3/$23.0091, 031108-1 © 2007 American Institute of Physics Downloaded 05 Mar 2009 to 132.229.96.124. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp 031108-2 Choi et al. Appl. Phys. Lett. 91, 031108 ͑2007͒

FIG. 2. ͑Color online͒ Lasing characteristics of the laser diode. ͑a͒ The ͑ ͒ FIG. 3. ͑Color online͒ Lasing characteristics of the L3 laser. The g 2 ͑0͒ data output power is plotted as a function of the operating current. ͑b͒ The g͑2͒ and the output intensity are plotted as a function of incident power. The ϫ͑0͒ values as a function of threshold-normalized current for the shaded ͑ ͒ g 2 ͑␶͒ spectra taken at 0.6 and 6.2 ␮W are shown in the insets, where region of ͑a͒. The insets show the emission spectra at different currents. parallel lines indicate the standard deviation noise.

36.25 mA, as observed in the intensity versus current curve The observed evolution of the onset of coherence in the of Fig. 2͑a͒. Below threshold, the multi-quantum-well family of QD PC nanolasers can be analyzed by calculating ͑MQW͒ gain spectrum overlaps with about 100 cavity g͑2͒͑0͒ directly from the standard photon number probability ͟k=n ͓ ͑ −1 ͒␬͔ 11 modes, which are discriminated from the lasing mode by a distribution pn =p0 k=1Na / Nb +2TP R +k , where p0 is monochromator for tracking g͑2͒͑0͒ around the lasing thresh- the zero photon probability determined by normalization, n is ͑2͒ old. As shown in Fig. 2͑b͒, g ͑0͒ increases as the current is the number of photons in the cavity, and Na and Nb are the ϫ elevated to 0.986 Ith and rapidly converges to 1 as the cur- numbers of carriers in the upper and lower levels, respec- ͑⌳ rent is further increased to Ith. Therefore, a good correlation tively. TP, the effective pump rate, is defined as TP = 0 ␥ ͒−1 ⌳ can be found between the threshold current identified from + Ј where 0 is the external pump rate. R is defined by ⍀2 ⌳ Fig. 2͑a͒ and the narrow current region that the mode ap- R=4 TPT2, and Na and Nb can be given by Na =N 0TP and ͑2͒ ␥ ␬ proaches a coherent state with g ͑␶͒=1. Spectrally, the las- Nb =N ЈTP, respectively. is the cavity decay rate given by ing mode becomes dominant over the nonlasing modes, as ␬=2␲␯/Q, where Q is the cavity quality factor and ␯ is the shown in the insets of Fig. 2͑b͒. cavity frequency. In this calculation, the material-dependent In sharp contrast to the MQW laser diode, for a single laser parameters are determined experimentally. The cavity mode PC nanolaser based on a small number of QDs, the mode linewidth below threshold is used to find ␬. The value achievement of transparency and lasing threshold is expected of ␥Ј, i.e., the sum of all recombination rates that do not add to occur at very low pump powers. A conventional rate equa- a photon to the cavity, can be determined from the lifetime of tion analysis for a high-␤ ͑ϳ1͒ laser predicts linear intensity a QD uncoupled from the cavity mode, which is measured to versus pump characteristics without the standard “kink.”8,9 The actual intensity versus pump curve of the L3 structure shows a “soft-turn-on” behavior, distinctly different from that demonstrated by the laser diode. The initial sublinear increase of the output intensity is followed by an approximately linear region for incident powers between ϳ2 and ϳ20 ␮W, followed by the saturation region for powers greater than 20 ␮W, as shown in Fig. 3. In contrast to the ambiguous intensity versus pump characteristics, the g͑2͒͑0͒ data taken as a function of pump power clearly show the onset of lasing at this soft-turn-on region. The pronounced bunching behavior at low pump powers suggests the exis- tence of correlated stimulated emission events, i.e., amplified spontaneous emission. With further increase of pump powers, the L3 structure attains a coherent state, as indicated by the convergence to Poissonian statistics, i.e., g͑2͒͑␶͒=1, as shown in Fig. 3. The measurement of g͑2͒͑␶͒ in the L7 and L11 structures also demonstrated the onset of lasing while the output intensity showed similar soft-turn-on behaviors. FIG. 4. ͑Color online͒͑a͒ g͑2͒͑0͒ data of QD PC lasers are compared with Notably, the lasing transition region in the g͑2͒͑0͒ versus the theoretical curves obtained with ͓␤,␬,⍀,N͔=͓0.69±0.09,287, 40±10,͑3±1͒ϫ102͔ for L3, ͓␤,␬,⍀,N͔=͓0.44±0.10,432,27±7,͑9±3͒ pump data is broadest for the L3 structure, sharp for the ϫ102͔ for L7, and ͓␤,␬,⍀,N͔=͓0.19±0.08,396,13±2,͑3±0.7͒ϫ103͔ for L7 structure, and sharper for the L11 structure, as shown in L11, respectively. ͑b͒ g͑2͒͑0͒ data of the MQW laser are compared with the Fig. 4. theoretical curves obtained with ͓␤,␬,⍀,N͔=͓0.0001,200,0.7,5ϫ107͔. Downloaded 05 Mar 2009 to 132.229.96.124. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp 031108-3 Choi et al. Appl. Phys. Lett. 91, 031108 ͑2007͒

be 5 ns ͑␥Јϳ1.26 GHz͒. Direct optical measurement of the increase of the ␤, suggesting reduced photon number fluc- ␤ 6 characteristic dephasing time T2 depends on the exact state tuations in high- microlasers. The general trend found in of the QD and its interaction with the local environment, these controlled active nanocavities and the laser diode con- which is altered by the presence of the cavity mode.16 In- firms ultrahigh-␤ lasing and allows the analysis of effective deed, four-wave-mixing measurements of a QD ensemble QD-cavity coupling parameters. As the cavity decay rates are with no surrounding cavity have shown T2 values ranging larger than the effective coupling parameters, the observed from 0.33 ps to 1 ns depending on excitation.17 Our estimate lasing actions are in the weak-coupling regime. of T2 is obtained by measuring the coherence time using a In conclusion, we have demonstrated the systematic cor- Michelson interferometer and PL decay time T1 of the pho- relation between the coherent-state transition and the ␤ fac- tons emitted at the cavity wavelength at pump powers well tor in a family of QD PC nanolasers. The onset of coherence below the lasing threshold. In the bad cavity limit, where the is confirmed by the convergence to Poissonian statistics, i.e., ͑2͒ cavity linewidth is much greater than the QD linewidth, T2 g ͑0͒=1, with increasing pump powers. The quantitative can be determined from the coherence time and T1 using analysis reveals the high-␤ factors of 0.69, 0.44, and 0.19, ␶ 1/ c =1/2T1 +1/T2. We measured a coherence time of 4.3 ps generally in agreement with a simple mode counting argu- and a T1 of 280 ps, resulting in a T2 of 4.3 ps. Since the ment. This systematic approach of mapping the onset of co- coherence time matches the cavity linewidth, the cavity is herence to a series of controlled active nanophotonic devices most likely acting as a spectral filter and artificially increas- will be a critical measure of lasing performance of nanocavi- ing the coherence time, implying that we are in the good ties in general and, in particular, for devices reaching into the cavity limit. Therefore, we can take 4.3 ps only as an upper strong-coupling regime, where an emitter is coherently 20 bound for T2. As a lower bound, we note the literature value bound to a single mode. of 0.33 ps observed at high excitation.17 To analyze our data taken at a relatively low excitation, we choose T2 =1 ps and This work was funded by DARPA 972-01-1-0027, denote proper uncertainties for all variable fit parameters de- DMEA H94003-04-2-0403, and NSF NIRT 0304678. pending on T . The number of carriers N and the effective 2 1 coupling parameter between the cavity and carriers ⍀ are O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284, 1819 ͑1999͒. used as dependent fitting parameters. For the MQW laser 2 ͑ ͒ ͑2͒͑ ͒ M. Fujita and T. Baba, Appl. Phys. Lett. 80, 2051 2002 . diode, the g 0 data were analyzed with the typical values 3H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-G. Yang, J.-H. Baek, found in Ref. 11. S.-B. Kim, and Y.-H. Lee, Science 305,1444͑2004͒. The theoretical results are summarized in Fig. 4, along 4J. Hendrickson, B. C. Richards, J. 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Phys. 2, 484 ͑2006͒. is set by the amplitude fluctuations of the emitted light. Be- 8H. Yokoyama, Science 256,66͑1992͒. low threshold, this time scale is faster than our timing reso- 9G. Björk, A. Karlsson, and Y. Yamamoto, Phys. Rev. A 50, 1675 ͑1994͒. 10 lution and the expected bunching peak at ␶=0 is washed out. C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, Phys. Rev. A 75, 013803 ͑2007͒. The result is presented as a function of a normalized power 11 ͑ ͒ R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. M. Gibbs, and G. P/ Pth Ref. 19 and normalized current I/Ith. In the L3, L7, Khitrova, Phys. Rev. A 49, 4038 ͑1994͒. ͑2͒ and L11 lasers shown in Fig. 4͑a͒, the systematic g ͑0͒ 12P. R. Rice and H. J. Carmichael, Phys. Rev. A 50, 4318 ͑1994͒. variation near threshold can be compared with the theoretical 13J. M. García, T. Mankad, P. O. Holtz, P. J. Wellman, and P. M. Petroff, ͑ ͒ curves corresponding to ␤ factors of 0.69, 0.44, and 0.19, Appl. Phys. Lett. 72, 3172 1998 . 14S.-H. 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