## Cavity *Q*, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum
dots

Optics Express, Vol. 14, Issue 3, pp. 1094-1105 (2006)

http://dx.doi.org/10.1364/OE.14.001094

Acrobat PDF (485 KB)

### Abstract

The quality factor (*Q*), mode volume (*V _{eff}
*), and room-temperature lasing threshold of microdisk cavities with embedded quantum dots (QDs) are investigated. Finite element method simulations of standing wave modes within the microdisk reveal that

*V*

_{eff}can be as small as 2(λ/

*n*)

^{3}while maintaining radiation-limited

*Q*s in excess of 10

^{5}. Microdisks with a 2

*μ*m diameter are fabricated in an AlGaAs material containing a single layer of InAs QDs with peak emission at λ = 1317 nm. For devices with

*V*

_{eff}~2 (λ/

*n*)

^{3},

*Q*s as high as 1.2× 10

^{5}are measured passively in the 1.4

*μ*m band, using an optical fiber taper waveguide. Optical pumping yields laser emission in the 1.3

*μ*m band, with room temperature, continuous-wave thresholds as low as 1

*μ*W of absorbed pump power. Out-coupling of the laser emission is also shown to be significantly enhanced through the use of optical fiber tapers, with a laser differential efficiency as high as ξ ~ 16% and out-coupling efficiency in excess of 28% measured after accounting for losses in the optical fiber system.

© 2006 Optical Society of America

## 1. Introduction

01. P. Michler, A. Kiraz, C. Becher, W. Schoenfeld, P. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A Quantum Dot Single-Photon Turnstile Device,” Science **290**, 2282–2285 (2000). [CrossRef] [PubMed]

02. C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered Single Photons from a Quantum Dot,” Phys. Rev. Lett. **86**, 1502–1505 (2001). [CrossRef] [PubMed]

03. E. Moreau, I. Robert, J. Gérard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. **79**, 2865–2867 (2001). [CrossRef]

04. J. Reithmaier, G. Sek, A. Loffer, C. Hoffman, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature **432**, 197–200 (2004). [CrossRef] [PubMed]

05. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, Q. Schenkin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature **432**, 200–203 (2004). [CrossRef] [PubMed]

06. E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. Gérard, and J. Bloch, “Exciton photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. **95** (2005). [CrossRef] [PubMed]

07. H. Cao, J. Xu, W. Xiang, Y. Ma, S.-H. Chang, S. Ho, and G. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. **76**, 3519–3521 (2000). [CrossRef]

08. T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Room temperature continuous wave lasing InAs quantum-dot microdisks with air cladding,” Opt. Express **13**, 1615–1620 (2005). [CrossRef] [PubMed]

*Q*), mode volume (

*V*

_{eff}), and the efficiency of light collection from the microcavity (η

*).*

_{o}*Q*and

*V*

_{eff}describe the decay rate (κ) and peak electric field strength within the cavity, respectively, which along with the oscillator strength and dephasing rate of the QD exciton determine if the coupled QD-photon system is in the regime of reversible energy exchange (strong coupling) or in a perturbative regime (weak coupling) characterized by a modification of the QD exciton radiative lifetime (the Purcell effect)[10

10. H. J. Kimble, “Strong Interactions of Single Atoms and Photons in Cavity QED,” Physica Scripta **T76**, 127–137 (1998). [CrossRef]

_{0}is of great importance for quantum networking[11

11. J. Cirac, P. Zoller, H. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. **78**, 3221–3224 (1997). [CrossRef]

12. E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature **409**, 46–52 (2001). [CrossRef] [PubMed]

13. A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A **69** (2004). [CrossRef]

*Q*whispering-gallery resonances were first studied in the context of semiconductor microlasers in the early 1990s[14

14. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. **60**, 289–291 (1992). [CrossRef]

01. P. Michler, A. Kiraz, C. Becher, W. Schoenfeld, P. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A Quantum Dot Single-Photon Turnstile Device,” Science **290**, 2282–2285 (2000). [CrossRef] [PubMed]

06. E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. Gérard, and J. Bloch, “Exciton photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. **95** (2005). [CrossRef] [PubMed]

07. H. Cao, J. Xu, W. Xiang, Y. Ma, S.-H. Chang, S. Ho, and G. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. **76**, 3519–3521 (2000). [CrossRef]

08. T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Room temperature continuous wave lasing InAs quantum-dot microdisks with air cladding,” Opt. Express **13**, 1615–1620 (2005). [CrossRef] [PubMed]

15. B. Gayral, J. M. Gerard, A. Lemaître, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. **75**, 1908–1910 (1999). [CrossRef]

07. H. Cao, J. Xu, W. Xiang, Y. Ma, S.-H. Chang, S. Ho, and G. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. **76**, 3519–3521 (2000). [CrossRef]

08. T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Room temperature continuous wave lasing InAs quantum-dot microdisks with air cladding,” Opt. Express **13**, 1615–1620 (2005). [CrossRef] [PubMed]

*Q*have resulted in RT operation in devices containing a single layer of QDs[16

16. K. Srinivasan, M. Borselli, T. Johnson, P. Barclay, O. Painter, A. Stintz, and S. Krishna, “Optical loss and lasing characteristics of high-quality-factor AlGaAs microdisk resonators with embedded quantum dots,” Appl. Phys. Lett. **86**, 151106 (2005). [CrossRef]

17. K. Srinivasan, A. Stintz, S. Krishna, and O. Painter, “Photoluminescence measurements of quantum-dot-containing semiconductor microdisk resonators using optical fiber taper waveguides,” Phys. Rev. B **72**, 205318 (2005). [CrossRef]

*Q*and η

_{0}are not only important for lasers, but for future experiments in cavity quantum electrodynamics (cQED).

*Q*and

*V*

_{eff}as a function of disk diameter. We relate these parameters to those used in cQED, and from this, determine that disks of 1.5 – 2

*μ*m in diameter are optimal for use in future experiments with InAs QDs. Section 3 briefly outlines the methods used to fabricate and test devices consisting of a 2

*μ*m diameter disk created in an Al-GaAs heterostructure with a single layer of self-assembled InAs QDs. In Section 4, we present experimental measurements of the fabricated devices. Through passive characterization, cavity

*Q*s as high as 1.2× 10

^{5}are demonstrated for devices with a predicted

*V*

_{eff}~ 2.2(λ/

*n*)

^{3}. In addition, photoluminescence measurements show that the devices operate as lasers with RT, continuous-wave thresholds of ~1

*μ*W of absorbed pump power. Finally, the optical fiber taper is used to increase the efficiency of out-coupling by nearly two orders of magnitude, so that an overall fiber-coupled laser differential efficiency of ξ, ~ 16% is achieved. We conclude by presenting some estimates of the number of QDs contributing to lasing and the spontaneous emission coupling factor (β) of the devices.

## 2. Simulations

*Q*and

*V*

_{eff}of the microdisk cavities, finite-element eigenfrequency simulations[18

18. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A **71**, 013817 (2005). [CrossRef]

20. L. Andreani, G. Panzarini, and J.-M. Gerard, “Strong-coupling regime for quantum boxes in pillar microcavi-ties:Theory,” Phys. Rev. B **60**, 13276–13279 (1999). [CrossRef]

**r**) is the dielectric constant, |

*E*(

**r**)| is the electric field strength, and

*V*is a quantization volume encompassing the resonator and with a boundary in the radiation zone of the cavity mode under study. The resonance wavelength λ

_{0}and radiation limited quality factor

*Q*

_{rad}are determined from the complex eigenvalue (wavenumber) of the resonant cavity mode,

*k*, obtained by the finite-element solver, with λ

_{0}= 2π/ℜℯ(

*k*) and

*Q*

_{rad}= ℜℯ(

*k*)/(2ℑm(

*k*)).

16. K. Srinivasan, M. Borselli, T. Johnson, P. Barclay, O. Painter, A. Stintz, and S. Krishna, “Optical loss and lasing characteristics of high-quality-factor AlGaAs microdisk resonators with embedded quantum dots,” Appl. Phys. Lett. **86**, 151106 (2005). [CrossRef]

_{p,m}to label whispering-gallery-modes (WGMs) with electric field polarization dominantly in the plane of the microdisk, radial order

*p*, and azimuthal mode number

*m*. The refractive index of the microdisk waveguide is taken as

*n*= 3.36 in the simulations, corresponding to the average of the refractive indices of the GaAs and AlGaAs layers at λ = 1200 nm.

*standing wave*modes that are superpositions of the standard clockwise (CW) and counterclockwise (CCW)

*traveling*wave modes typically studied in microdisks. These standing wave modes form when surface scattering couples and splits the initially degenerate CW and CCW traveling wave modes[21

21. D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. **20**, 1835–1837 (1995). [CrossRef] [PubMed]

22. T. Kippenberg, S. Spillane, and K. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. **27**, 1669–1671 (2002). [CrossRef]

23. M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530 (2005). [CrossRef] [PubMed]

16. K. Srinivasan, M. Borselli, T. Johnson, P. Barclay, O. Painter, A. Stintz, and S. Krishna, “Optical loss and lasing characteristics of high-quality-factor AlGaAs microdisk resonators with embedded quantum dots,” Appl. Phys. Lett. **86**, 151106 (2005). [CrossRef]

17. K. Srinivasan, A. Stintz, S. Krishna, and O. Painter, “Photoluminescence measurements of quantum-dot-containing semiconductor microdisk resonators using optical fiber taper waveguides,” Phys. Rev. B **72**, 205318 (2005). [CrossRef]

23. M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530 (2005). [CrossRef] [PubMed]

*g*~ 1√

*V*

_{eff}. A QD positioned at an anti-node of the standing wave will have an exciton-photon coupling rate which is √2 times larger than for the traveling wave mode.

*V*

_{eff}for these standing wave modes can be as small as 2(λ/

*n*)

^{3}while maintaining

*Q*

_{rad}> 10

^{5}. Indeed, for microdisk average diameters

*D*> 2

*μ*m[24], radiation losses are not expected to be the dominant loss mechanism as

*Q*

_{rad}quickly exceeds 10

^{7}, and other sources of field decay such as material absorption or surface scattering are likely to dominate . To translate these results into the standard parameters studied in cQED, we calculate the cavity decay rate κ/2π = ω/(4π

*Q*) (assuming

*Q*=

*Q*

_{rad}) and the coherent coupling rate

*g*between the cavity mode and a single QD exciton. In this calculation, a spontaneous emission lifetime τ

*= 1 ns is assumed for the QD exciton, and*

_{sp}*g*=

**d**∙

**E**/

*h*̄ is the vacuum coherent coupling rate between cavity mode and QD exciton, given by[10

10. H. J. Kimble, “Strong Interactions of Single Atoms and Photons in Cavity QED,” Physica Scripta **T76**, 127–137 (1998). [CrossRef]

20. L. Andreani, G. Panzarini, and J.-M. Gerard, “Strong-coupling regime for quantum boxes in pillar microcavi-ties:Theory,” Phys. Rev. B **60**, 13276–13279 (1999). [CrossRef]

*c*is the speed of light and

*n*is the refractive index at the location of the QD. This formula assumes that the QD is optimally positioned within the cavity field, so that the calculated

*g*is the maximum possible coupling rate. The resulting values for

*g*and κ are displayed in Fig. 2(b), and show that

*g*/2π can exceed κ/2π by over an order of magnitude for a range of disk diameters. In addition, for all but the smallest-sized microdisks, κ/2π < 1 GHz. A decay rate of 1 GHz is chosen as a benchmark value as it corresponds to a linewidth of a few

*μ*eV at these wavelengths, on par with the narrowest self-assembled InAs QD exciton linewidths that have been measured at cryogenic temperatures[25]. Indeed, because dissipation in a strongly-coupled QD-photon system can either be due to cavity decay or quantum dot dephasing, in Fig. 3 we examine the ratio of

*g*to the maximum decay rate in the system assuming a fixed QD dephasing rate γ/2π=1 GHz[26]. This ratio is roughly representative of the number of coherent exchanges of energy (Rabi oscillations) that can take place between QD and photon. We see that it peaks at a value of about 18 for a disk diameter

*D*~ 1.5

*μ*m. For diameters smaller than this, loss is dominated by cavity decay due to radiation (so that

*g*/

*max*(γ,κ) =

*g*/κ), while for larger diameters, the dominant loss mechanism is due to dephasing of the QD (

*g*/

*max*(γκ) =

*g*/γ).

*Q*

_{rad}and approximately linear dependence of

*V*

_{eff}on microdisk diameter,

*Q*

_{rad}/

*V*

_{eff}rapidly rises above 10

^{7}for microdisks of diameter only

*D*= 2.5

*μ*m. These values of

*Q*

_{rad}and

*V*

_{eff}are comparable to those found in recent high-

*Q*photonic crystal microcavity designs[27

27. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express **10**, 670–684 (2002). [PubMed]

28. H.-Y Ryu, M. Notomi, and Y.-H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. **83**, 4294–4296 (2003). [CrossRef]

29. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials **4**, 207–210 (2005). [CrossRef]

31. Z. Zhang and M. Qiu, “Small-volume waveguide-section high *Q* microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

*Q*planar photonic crystal micro-cavities, in which one may trade-off a linear increase in

*V*

_{eff}for an exponential increase in

*Q*, has recently been described by Englund, et al., in Ref. [32

32. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

*g*/max(γ,κ) with γ/2π = 1 GHz as our metric, and as such focus on 1.5-2

*μ*m diameter microdisks.

## 3. Growth, fabrication, and test set-up

17. K. Srinivasan, A. Stintz, S. Krishna, and O. Painter, “Photoluminescence measurements of quantum-dot-containing semiconductor microdisk resonators using optical fiber taper waveguides,” Phys. Rev. B **72**, 205318 (2005). [CrossRef]

*μ*m. It is mounted onto an acrylic mount that is attached to a motorized Z-axis stage (50 nm encoded resolution), so that the fiber taper can be precisely aligned to the microdisk, which is in turn mounted on a motorized XY stage. When doing passive measurements of cavity

*Q*, the taper input is connected to a scanning tunable laser (5 MHz linewidth) with a tuning range between 1420–1480 nm, and the taper output is connected to a photodetector to monitor the transmitted power. Alternately, when collecting emission from the microdisk through the fiber taper, the taper input is left unconnected and the output is sent into the OSA.

## 4. Experimental results

*Q*of the microdisks. Based on the simulations presented in Section 2, we have focused on 2

*μ*m diameter microdisks. Due to the small diameter of these microdisks, the finite-element-calculated free-spectral range of resonant modes is relatively large, with resonances occurring at 1265, 1346, and 1438 nm for the TE

_{p=1}WGMs with azimuthal mode numbers

*m*= 11,10, and 9, respectively. The simulations presented in Section 2 were done for the TE

_{1,11}mode in the λ = 1200 nm band due to the applicability of that wavelength region for future low temperature cQED experiments. However, for the current room-temperature measurements, the absorption due to the QD layer at those wavelengths is significant, so we probe the devices within the λ = 1400 nm band (~100 nm red-detuned from the peak ground-state manifold QD emission). At these longer wavelengths the radiation-limited

*Q*

_{rad}for a given disk diameter will be smaller than its value in the shorter λ = 1200 nm band. Table 1 summarizes the properties of the TE

_{p=1}WGMs within the 1200–1400 wavelength band for a

*D*= 2

*μ*m microdisk with shape as shown in Fig. 1.

21. D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. **20**, 1835–1837 (1995). [CrossRef] [PubMed]

*Q*factors of 1.2 × 10

^{5}, and in general

*Q*s of 0.9–1.3 × 10

^{5}have been measured for these 2

*μ*m diameter microdisks. The

*Q*s of these modes are approaching the radiation-limited value of 3.7× 10

^{5}, and are some of the highest measured values for near-IR wavelength-scale microcavities in AlGaAs[5

05. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, Q. Schenkin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature **432**, 200–203 (2004). [CrossRef] [PubMed]

15. B. Gayral, J. M. Gerard, A. Lemaître, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. **75**, 1908–1910 (1999). [CrossRef]

**86**, 151106 (2005). [CrossRef]

34. A. Loffler, J. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. **86**, 111105 (2005). [CrossRef]

*g*for an optimally placed QD. In addition, these

*Q*s, if replicated within the QD emission band at λ = 1300 nm, are high enough to ensure that room-temperature lasing should be achievable from the single layer of QDs in these devices[33

33. A. Stintz, G. Liu, H. Li, L. Lester, and K. Malloy, “Low-Threshold Current Density 1.3-*μ*m InAs Quantum-Dot Lasers with the Dots-in-a-Well (DWELL) structure,” IEEE Photonics Technol. Lett. **12**, 591–593 (2000). [CrossRef]

*Q*

_{rad}of 1.9 ×10

^{6}, although surface scattering may also slightly increase due to its approximate cubic dependence on frequency[23

23. M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530 (2005). [CrossRef] [PubMed]

*μ*m

^{2}, is sufficiently large to pump the entirety of the disk area. A light-in versus light-out (L-L) curve for one of the

*D*~ 2

*μ*m microdisks with a resonant emission peak at λ ~ 1345 nm is shown in Fig. 4(b), and displays a lasing threshold kink at approximately 1.0

*μ*W of absorbed pump power. The laser mode wavelength corresponds well with the TE

_{p=1 m=10}mode from finite-element simulations (see Table 1). The absorbed pump power is estimated to be 11% of the incident pump power on the microdisk, and was determined by first taking the fraction of the pump spot that actually impinges on the disk, and then assuming an absorption coefficient of 10

^{4}cm

^{-1}for the GaAs layers and quantum well layer and accounting for reflections at the GaAs/air interfaces[17

**72**, 205318 (2005). [CrossRef]

**13**, 1615–1620 (2005). [CrossRef] [PubMed]

_{p=1 ,m=10}WGM emission peak at λ = 1330 nm and has a threshold absorbed pump power of 1.1

*μ*W. As demonstrated in Ref. [17

**72**, 205318 (2005). [CrossRef]

*Q*can efficiently out-couple light from the lasing mode. We do this by maintaining the free-space pumping used above while contacting a fiber taper to the side of the microdisk as shown in the inset of Fig. 5(b). From the corresponding L-L curve (Fig. 5(b)) we see that the laser threshold under fiber taper loading has increased from 1.1

*μ*W to 1.6

*μ*W, but in addition the differential laser efficiency ξ, is now 4% compared to 0.1% when employing free-space collection (Fig. 5(a)–(b)). Furthermore, because the microdisk modes are standing waves they radiate into both the forwards and backwards channels of the fiber. With collection from both the forward and backward channels the differential efficiency was measured to be twice that of the single forward channel. Collecting from both channels and adjusting for all fiber losses in the system (roughly 50% due to fiber splices and taper loss in this case; tapers with losses < 10% are routinely fabricated in our lab), the total differential laser efficiency with fiber taper collection is 16%. Due to the difference in photon energy of the pump laser and microdisk emission, this laser differential efficiency corresponds to a conversion efficiency of 28% from pump photons to fiber-collected microdisk laser photons. 28% is thus a

*lower*bound on the fiber-taper collection efficiency and/or quantum efficiency of the QD active region.

_{p=1 m=10}laser mode when using the fiber taper to out-couple the laser light, we also see in the below-threshold spectrum of Fig. 5(b) that two additional resonances appear at λ = 1310 nm and λ = 1306 nm. The long wavelength mode is identified as TM

_{p=1,m=8}and the short wavelength mode as TE

_{p=2,m=7}from finite-element simulations. These modes are not discernible in the free-space collected spectrum due to their low radiation-limited

*Q*factors (800 and 5000 for the TE

_{2,7}and TM

_{1,8}, respectively), but show up in the taper coupled spectrum due to their alignment with the QD ground-state exciton emission peak and the heightened sensitivity of the taper coupling method. The single-mode lasing and limited number of WGM resonances (6 when including the degeneracy of the WGMs) in the emission spectrum in these

*D*= 2

*μ*m microdisks is a result of the large 80–100 nm free-spectral-range of modes in the 1300–1500 wavelength band. As a result, one would expect the spontaneous emission factor (β) of these microdisk lasers to be relatively high.

*weak lower bound*β′ for the (β-factor directly from the L-L curve. From Fig. 6 we estimate β ≥ β′ ~ 3%.

*N*

^{2}and surface recombination with a

*N*

^{1.22}carrier dependence (the ratio of the power law dependences is set equal to the measured sub-threshold slope of m=1.67) is fit to the data and shown as a solid curve in Fig. 6. In this model the measured fiber taper collection efficiency was used, along with the previously measured and estimated QD density, maximum gain, and quantum efficiency from stripe lasers[33

33. A. Stintz, G. Liu, H. Li, L. Lester, and K. Malloy, “Low-Threshold Current Density 1.3-*μ*m InAs Quantum-Dot Lasers with the Dots-in-a-Well (DWELL) structure,” IEEE Photonics Technol. Lett. **12**, 591–593 (2000). [CrossRef]

*Q*WGM resonances within the QD ground-state manifold emission band[38

38.
This estimate was based upon considering Purcell enhancement at RT for QDs spatially and spectrally aligned with the WGMs (*F _{P}* ~ 6), and suppression of spontaneous emission for QDs spatially and spectrally misaligned from the WGMs (

*F*~ 0.4). This simple estimate is consistent with accurate finite-difference time-domain calculations of similar sized microdisks[39].

_{P}*= 1 ns. The data was fit by varying*

_{sp}*only*the effective surface recombination velocity. As seen in Fig. 6, the fit is quite good over the entire sub-threshold and threshold regions of the laser data. The inferred surface recombination velocity from the fit is

*v*~ 75 cm/s, extremely slow for the AlGaAs material system[40] but perhaps indicative of the fast capture rate of carriers, and consequent localization, into QDs[41

_{s}41. T. Sosnowski, T. Norris, H. Jiang, J. Singh, K. Kamath, and P. Bhattacharya, “Rapid carrier relaxation in In0.4Ga_{0.60}As/GaAs quantum dots characterized by differential transmission spectroscopy,” Phys. Rev. B **57**, R9423–R9426 (1998). [CrossRef]

42. D. Yarotski, R. Averitt, N. Negre, S. Crooker, A. Taylor, G. Donati, A. Stintz, L. Lester, and K. Malloy, “Ultrafast carrier-relaxation dynamics in self-assembled InAs/GaAs quantum dots,” J. Opt. Soc. Am. B **19**, 1480–1484 (2002). [CrossRef]

*D*= 2

*μ*m microdisks, even with this low velocity the model predicts that laser threshold pump power is dominated by surface recombination with an effective lifetime τ

*~ 300 ps. Such a surface recombination lifetime has also been estimated by Baba, et al., in their recent work on QD-microdisk lasers[43*

_{s}43. T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Lasing characteristics of InAs quantum-dot microdisk from 3K to room temperature,” Appl. Phys. Lett. **85**, 1326–1328 (2004). [CrossRef]

*μ*m

^{2}, and the predicted QD density for this sample is 300

*μ*m

^{-2}, so that ~ 300 QDs are spatially aligned with the cavity mode. Assuming a RT homogeneous linewidth on the order of a few meV[25], compared to a measured inhomogeneous Gaussian broadening of 35 meV, and considering the location of the lasing mode in the tail of the Gaussian distribution, we estimate < 10% of these dots are spectrally aligned with the cavity mode. By this estimate, on the order of 25 QDs contribute to lasing.

## 5. Conclusions

*μ*m diameter quantum-dot-containing microdisks that have a quality factor

*Q*in excess of 10

^{5}for a predicted mode volume

*V*

_{eff}as small as ~ 2.2(λ/

*n*)

^{3}. Such devices are predicted to be suitable for future experiments in single quantum dot, single photon experiments in cavity QED, where these

*Q*and

*V*

_{eff}values can enable strong coupling at GHz-scale speeds. An initial application of this work is in continuous wave, optically pumped microcavity lasers. Here, the high

*Q*ensures that lasing can be achieved with the modest gain provided by a single layer of quantum dots, and combined with the ultra-small

*V*

_{eff}, results in thresholds as low as 1.0

*μ*W of absorbed power. In addition, the fiber taper coupling is shown to be an efficient method to collect the laser emission with a measured 28% lower bound on out-coupling efficiency.

## References and links

01. | P. Michler, A. Kiraz, C. Becher, W. Schoenfeld, P. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A Quantum Dot Single-Photon Turnstile Device,” Science |

02. | C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered Single Photons from a Quantum Dot,” Phys. Rev. Lett. |

03. | E. Moreau, I. Robert, J. Gérard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. |

04. | J. Reithmaier, G. Sek, A. Loffer, C. Hoffman, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature |

05. | T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, Q. Schenkin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature |

06. | E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. Gérard, and J. Bloch, “Exciton photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. |

07. | H. Cao, J. Xu, W. Xiang, Y. Ma, S.-H. Chang, S. Ho, and G. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. |

08. | T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Room temperature continuous wave lasing InAs quantum-dot microdisks with air cladding,” Opt. Express |

09. | T. Yang, O. Schekin, J. O’Brien, and D. Deppe, “Room temperature, continuous-wave lasing near 1300 nm in microdisks with quantum dot active regions,” IEE Elec. Lett. |

10. | H. J. Kimble, “Strong Interactions of Single Atoms and Photons in Cavity QED,” Physica Scripta |

11. | J. Cirac, P. Zoller, H. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. |

12. | E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature |

13. | A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A |

14. | S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. |

15. | B. Gayral, J. M. Gerard, A. Lemaître, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. |

16. | K. Srinivasan, M. Borselli, T. Johnson, P. Barclay, O. Painter, A. Stintz, and S. Krishna, “Optical loss and lasing characteristics of high-quality-factor AlGaAs microdisk resonators with embedded quantum dots,” Appl. Phys. Lett. |

17. | K. Srinivasan, A. Stintz, S. Krishna, and O. Painter, “Photoluminescence measurements of quantum-dot-containing semiconductor microdisk resonators using optical fiber taper waveguides,” Phys. Rev. B |

18. | S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A |

19. | M. Borselli, T. Johnson, and O. Painter, “Measuring the role of surface chemistry in silicon microphotonics,” submitted for publication (2005) |

20. | L. Andreani, G. Panzarini, and J.-M. Gerard, “Strong-coupling regime for quantum boxes in pillar microcavi-ties:Theory,” Phys. Rev. B |

21. | D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. |

22. | T. Kippenberg, S. Spillane, and K. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. |

23. | M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express |

24. | The average diameter is taken at the center of the slab, or equivalently, is the average of the top and bottom diameters. |

25. | M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In |

26. |
Note that γ = γ |

27. | K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express |

28. | H.-Y Ryu, M. Notomi, and Y.-H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. |

29. | B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials |

30. | E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Photonic crystal nanocavity formed by local width modulation of line-defect with |

31. | Z. Zhang and M. Qiu, “Small-volume waveguide-section high |

32. | D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express |

33. | A. Stintz, G. Liu, H. Li, L. Lester, and K. Malloy, “Low-Threshold Current Density 1.3- |

34. | A. Loffler, J. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. |

35. | As has been discussed recently in Ref [36] this may not be an accurate model for QD state-filling, but for our simple analysis here it will suffice. |

36. | H. Pask, H. Summer, and P. Blood, “Localized Recombination and Gain in Quantum Dots,” In |

37. | G. P. Agrawal and N. K. Dutta, |

38. |
This estimate was based upon considering Purcell enhancement at RT for QDs spatially and spectrally aligned with the WGMs ( F ~ 0.4). This simple estimate is consistent with accurate finite-difference time-domain calculations of similar sized microdisks[39].
_{P} |

39. | J. Vučković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “FDTD Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quantum Electron. |

40. | L. A. Coldren and S. W. Corzine, |

41. | T. Sosnowski, T. Norris, H. Jiang, J. Singh, K. Kamath, and P. Bhattacharya, “Rapid carrier relaxation in In0.4Ga |

42. | D. Yarotski, R. Averitt, N. Negre, S. Crooker, A. Taylor, G. Donati, A. Stintz, L. Lester, and K. Malloy, “Ultrafast carrier-relaxation dynamics in self-assembled InAs/GaAs quantum dots,” J. Opt. Soc. Am. B |

43. | T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Lasing characteristics of InAs quantum-dot microdisk from 3K to room temperature,” Appl. Phys. Lett. |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(230.5750) Optical devices : Resonators

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: November 17, 2005

Revised Manuscript: January 13, 2006

Manuscript Accepted: January 24, 2006

Published: February 6, 2006

**Citation**

Kartik Srinivasan, Matthew Borselli, Oskar Painter, Andreas Stintz, and Sanjay Krishna, "Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots," Opt. Express **14**, 1094-1105 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1094

Sort: Journal | Reset

### References

- P. Michler, A. Kiraz, C. Becher, W. Schoenfeld, P. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A Quantum Dot Single-Photon Turnstile Device,” Science 290, 2282–2285 (2000). [CrossRef] [PubMed]
- C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered Single Photons from a Quantum Dot,” Phys. Rev. Lett. 86, 1502–1505 (2001). [CrossRef] [PubMed]
- E. Moreau, I. Robert, J. Gérard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. 79, 2865–2867 (2001). [CrossRef]
- J. Reithmaier, G. Sek, A. Loffer, C. Hoffman, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004). [CrossRef] [PubMed]
- T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, Q. Schenkin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004). [CrossRef] [PubMed]
- E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. Gérard, and J. Bloch, “Exciton photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. 95 (2005). [CrossRef] [PubMed]
- H. Cao, J. Xu, W. Xiang, Y. Ma, S.-H. Chang, S. Ho, and G. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000). [CrossRef]
- T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Room temperature continuous wave lasing InAs quantum-dot microdisks with air cladding,” Opt. Express 13, 1615–1620 (2005). [CrossRef] [PubMed]
- T. Yang, O. Schekin, J. O’Brien, and D. Deppe, “Room temperature, continuous-wave lasing near 1300 nm in microdisks with quantum dot active regions,” IEE Elec. Lett. 39 (2003).
- H. J. Kimble, “Strong Interactions of Single Atoms and Photons in Cavity QED,” Physica Scripta T76, 127–137 (1998). [CrossRef]
- J. Cirac, P. Zoller, H. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]
- E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]
- A. Kiraz, M. Atature, and A. Imamoglu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69 (2004). [CrossRef]
- S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. 60, 289–291 (1992). [CrossRef]
- B. Gayral, J. M. Gérard, A. Lemaître, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999). [CrossRef]
- K. Srinivasan, M. Borselli, T. Johnson, P. Barclay, O. Painter, A. Stintz, and S. Krishna, “Optical loss and lasing characteristics of high-quality-factor AlGaAs microdisk resonators with embedded quantum dots,” Appl. Phys. Lett. 86, 151106 (2005). [CrossRef]
- K. Srinivasan, A. Stintz, S. Krishna, and O. Painter, “Photoluminescence measurements of quantum-dot-containing semiconductor microdisk resonators using optical fiber taper waveguides,” Phys. Rev. B 72, 205318 (2005). [CrossRef]
- S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]
- M. Borselli, T. Johnson, and O. Painter, “Measuring the role of surface chemistry in silicon microphotonics,” submitted for publication (2005)
- L. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B 60, 13276–13279 (1999). [CrossRef]
- D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. 20, 1835–1837 (1995). [CrossRef] [PubMed]
- T. Kippenberg, S. Spillane, and K. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002). [CrossRef]
- M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005). [CrossRef] [PubMed]
- The average diameter is taken at the center of the slab, or equivalently, is the average of the top and bottom diameters.
- M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0:60Ga0:40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308(R) (2002).
- Note that γ≡γ is in general greater than half the total excitonic decay rate (γ√2) radiative decay rate (1/2τsp) for QD excitons, due to near-elastic scattering or dephasing events with, for example, acoustic phonons of the lattice.
- K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [PubMed]
- H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]
- B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials 4, 207–210 (2005). [CrossRef]
- E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Photonic crystal nanocavity formed by local width modulation of line-defect with Q of one million,” In LEOS 2005, Post-Deadline Session PD 1.1, (IEEE Lasers and Electro-Optics Society, 2005).
- Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004). [CrossRef] [PubMed]
- D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005). [CrossRef] [PubMed]
- A. Stintz, G. Liu, H. Li, L. Lester, and K. Malloy, “Low-Threshold Current Density 1.3-µm InAs Quantum-Dot Lasers with the Dots-in-a-Well (DWELL) structure,” IEEE Photonics Technol. Lett. 12, 591–593 (2000). [CrossRef]
- A. Loffler, J. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. 86, 111105 (2005). [CrossRef]
- As has been discussed recently in Ref. [36] this may not be an accurate model for QD state-filling, but for our simple analysis here it will suffice.
- H. Pask, H. Summer, and P. Blood, “Localized Recombination and Gain in Quantum Dots,” In Tech. Dig. Conf. on Lasers and Electro-Optics, CThH3, (Optical Society of America, Baltimore, MD, 2005).
- G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, New York, NY, 1993).
- This estimate was based upon considering Purcell enhancement at RT for QDs spatially and spectrally aligned with the WGMs (FP ~ 6), and suppression of spontaneous emission for QDs spatially and spectrally misaligned from the WGMs (FP ~ 0:4). This simple estimate is consistent with accurate finite-difference time-domain calculations of similar sized microdisks[39].
- J. Vučković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “FDTD Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quantum Electron. 35, 1168–1175 (1999). [CrossRef]
- L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, Inc., New York, NY, 1995).
- T. Sosnowski, T. Norris, H. Jiang, J. Singh, K. Kamath, and P. Bhattacharya, “Rapid carrier relaxation in In0:4Ga0:60As/GaAs quantum dots characterized by differential transmission spectroscopy,” Phys. Rev. B 57, R9423–R9426 (1998). [CrossRef]
- D. Yarotski, R. Averitt, N. Negre, S. Crooker, A. Taylor, G. Donati, A. Stintz, L. Lester, and K. Malloy, “Ultrafast carrier-relaxation dynamics in self-assembled InAs/GaAs quantum dots,” J. Opt. Soc. Am. B 19, 1480–1484 (2002). [CrossRef]
- T. Ide, T. Baba, J. Tatebayashi, S. Iwamoto, T. Nakaoka, and Y. Arakawa, “Lasing characteristics of InAs quantum-dot microdisk from 3K to room temperature,” Appl. Phys. Lett. 85, 1326–1328 (2004). [CrossRef]

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