## Subwavelength silicon microcavities

Optics Express, Vol. 17, Issue 25, pp. 23323-23331 (2009)

http://dx.doi.org/10.1364/OE.17.023323

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### Abstract

We present a study of the first silicon microdisk resonators which are smaller than the free-space resonant wavelength in all spatial dimensions. Spectral details of whispering gallery modes with azimuthal mode number *m* = 4-7 are measured in microdisks with diameters between 1.35 and 1.89μm and are studied at wavelengths from 1.52 to 1.62μm. For the structures considered here, *m* = 5 is the highest azimuthal mode order in a subwavelength cavity and has measured *Q* = 1250. These results agree well with theoretical calculations using a finite difference frequency domain method and fit an exponential scaling law relating *Q* to disk radius via *m*.

© 2009 OSA

## 1. Introduction

1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. **60**(3), 289–291 (
1992). [CrossRef]

3. A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometre radius disk laser,” Electron. Lett. **29**(18), 1666–1667 (
1993). [CrossRef]

4. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics **1**(10), 589–594 (
2007). [CrossRef]

6. C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. **44**(5), 435–447 (
2008). [CrossRef]

7. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express **16**(6), 4309–4315 (
2008). [CrossRef] [PubMed]

8. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high- Q photonic crystal microcavity,” Phys. Rev. B **70**(8), 081306 (
2004). [CrossRef]

9. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature **459**(7246), 550–555 (
2009). [CrossRef] [PubMed]

10. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling and optical loss in silicon microdisks,” Appl. Phys. Lett. **85**(17), 3693–3695 (
2004). [CrossRef]

11. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**(5), 1515–1530 (
2005). [CrossRef] [PubMed]

12. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express **14**(3), 1094–1105 (
2006). [CrossRef] [PubMed]

*m*) from 2 to 7 with resonant wavelengths from 1.52μm to 1.62μm. We have quantified the relationship between the radiation-limited quality factor (

*Q*) and

*m*using tapered fiber spectroscopy and compared our results to finite-difference frequency-domain (FDFD) simulations. The

*m*= 5 mode was the mode of highest

*m*in a subwavelength disk and was measured to have radiation-limited

*Q*= 1250, indicative of strong potential for room-temperature laser operation if fabricated from or coupled to a gain medium.

## 2. Theoretical considerations

*H*is the dominant field component) or TM-like (

_{z}*E*is the dominant field component). The highest-

_{z}*Q*resonances occur when the wave vector is predominantly in the azimuthal direction [14

14. J. E. Heebner, T. C. Bond, and J. S. Kallman, “Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators,” Opt. Express **15**(8), 4452–4473 (
2007). [CrossRef] [PubMed]

*m*. Further, the partial differential equation describing the system is separable if one assumes exp(

*im*f) angular dependence and in the approximation that modes have purely TE or TM character. The equation governing the

*z*-dependence is identical to that of a slab waveguide [15]. If one chooses –

*β*

^{2}as the separation constant for the equation governing

*z*-dependence (so the functional dependence on

*z*inside the disk is

*k*

_{0}is the free space spatial frequency,

*n*is the index of refraction of the disk (taken to be 3.48 throughout this study), and

*d*is the thickness of the silicon disk and

*d*= 250nm operating at λ = 1550nm,

*m*for a given disk radius, and therefore all modes populating thin subwavelength disks are TE-polarized.

*z*equation alone, the radial equation readswhere

*r*in that it changes abruptly from its value inside the disk [determined by solution of Eq. (1)] to unity beyond the disk. It is this abrupt change which produces a potential well in which the electromagnetic field is contained. In Eq. (2) we have written Bessel’s equation in a non-standard form to make an analogy to a one-dimensional Schrödinger equation. In this analogy,

18. N. C. Frateschi and A. F. J. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. **80**(2), 644 (
1996). [CrossRef]

19. P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. **12**(3), 487–494 (
1994). [CrossRef]

20. R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and bound modes of surface plasmon waveguides,” Phys. Rev. B **71**(16), 165431 (
2005). [CrossRef]

*im*f) dependence) to semi-analytically reduce the 3D resonator geometry to a transverse 2D eigenvalue problem. This full-vectorial method allowed for calculations of all electromagnetic field components as well as the effective mode volume [12

12. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express **14**(3), 1094–1105 (
2006). [CrossRef] [PubMed]

14. J. E. Heebner, T. C. Bond, and J. S. Kallman, “Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators,” Opt. Express **15**(8), 4452–4473 (
2007). [CrossRef] [PubMed]

## 3. Fabrication of subwavelength silicon microdisks

8. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high- Q photonic crystal microcavity,” Phys. Rev. B **70**(8), 081306 (
2004). [CrossRef]

11. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**(5), 1515–1530 (
2005). [CrossRef] [PubMed]

22. B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature **457**(7228), 455–458 (
2009). [CrossRef] [PubMed]

*Q*regime, as Rayleigh scattering from sidewall imperfections is usually the dominant loss mechanism in such disks [11

11. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**(5), 1515–1530 (
2005). [CrossRef] [PubMed]

23. A. C. F. Hoole, M. E. Welland, and A. N. Broers, “Negative PMMA as a high-resolution resist—the limits and possibilities,” Semicond. Sci. Technol. **12**(9), 1166–1170 (
1997). [CrossRef]

^{4}μC/cm

^{2}. At dosages of this magnitude, the polymer units of PMMA crosslink to form a material which is resistant to etchants. In our experiments, this technique has given smoother sidewalls than simply performing positive lithography without resist reflow. The electron exposure is followed by development in acetone and ICP RIE with SF

_{6}and C

_{4}F

_{8}. Photolithography and deep ICP RIE are then performed to etch through the 3μm SiO

_{2}insulator layer and through ~30μm of the underlying silicon substrate. This isolates the disks on a strip which is lifted above the rest of the substrate and thus makes it easier access the disks with the tapered fiber. An undercut is performed with buffered hydrofluoric acid.

## 4. Experimental details and results

8. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high- Q photonic crystal microcavity,” Phys. Rev. B **70**(8), 081306 (
2004). [CrossRef]

12. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express **14**(3), 1094–1105 (
2006). [CrossRef] [PubMed]

*Q*factor is extracted from the width and center wavelength.

*Q*on the order of 10

^{13}(the Rayleigh-scattering-limited

*Q*in this disk is ~10

^{5})—down to a subwavelength disk of approximately 1μm diameter with radiation-limited

*Q =*235. Next to each disk is a tapered fiber spectrum of the characteristic resonance of the disk. The SEM images depict the decrease in size over an order of magnitude while broadening of the resonances demonstrates the exponential decrease in

*Q*factor. In Figs. 2(f)-2(h) tapered fiber spectra of large and small disks are contrasted in more detail. Figure 2(f) shows a high-resolution scan of a 2nm wavelength window of a tapered fiber spectrum from a 10μm-diameter silicon disk. The doublet character of the high-

*Q*mode is apparent, as discussed in Refs. [10

10. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling and optical loss in silicon microdisks,” Appl. Phys. Lett. **85**(17), 3693–3695 (
2004). [CrossRef]

**14**(3), 1094–1105 (
2006). [CrossRef] [PubMed]

*Q*, fundamental-radial-order modes, lower-

*Q*, higher-radial-order modes are present. In Fig. 2(h) a tapered fiber spectrum of a 1.35μm disk is shown. The free spectral range is ~200nm, and only one resonance is present. If one wishes to use whispering-gallery-mode microcavities for devices such as filters and switches, the large free spectral range of wavelength-scale microdisks is an attractive feature. Another attribute is that higher-radial-order modes do not exist in this wavelength range for a disk of this size. The other features of the spectrum in Fig. 2(h) are due to coupling to the substrate. Because these small disks are only a few hundred nanometers from the substrate, coupling of the fiber to the substrate is more prominent than in larger disks.

*Q*factors with the theoretical radiation-limited

*Q*factors obtained with FDFD calculations. The experimental points are acquired using tapered fiber spectroscopy as described above, and the theoretical points are determined by using the values of disk diameter as measured with SEM in the FDFD mode solver. The thickness of each disk was 250nm. The data for each azimuthal mode order is from a disk of a different radius so the resonant wavelengths are nearly equal. The agreement between experiment and theory indicates that our silicon microdisks have radiation-limited

*Q*factors that are not significantly degraded by minor sidewall perturbations. It should be noted that the measured resonance at

*m*= 7 was a doublet mode. The splitting was only observed for the

*m*= 7 mode where the Lorentzian is narrow enough to differentiate the two peaks. We find that the

*m*= 5 mode is the subwavelength mode of highest

*m*with measured

*Q*= 1250. The diameter of the disk was 1.49μm and resonant wavelength was 1.543μm giving a ratio of d/

*λ*= 0.967. The

*m*= 4 mode was observed in a cavity of diameter 1.35μm at wavelength of

*λ*= 1.591μm giving d/

*λ*= 0.849. Due to the limited wavelength window accessible by our tunable laser, and the broad nature of the low-

*Q m*= 3 mode, it was not observed in our studies. The theoretical value of d/

*λ*= 0.683 for the

*m*= 3 mode. Below

*m*= 3 the resonances have

*Q*< 10. To create microdisks with well defined resonances for

*m*< 3 and d/λ <0 0.7, metallodielectric architectures, which trade ohmic loss for suppressed radiation, may offer advantages [4

4. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics **1**(10), 589–594 (
2007). [CrossRef]

6. C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. **44**(5), 435–447 (
2008). [CrossRef]

22. B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature **457**(7228), 455–458 (
2009). [CrossRef] [PubMed]

*im*f) angular dependence is assumed, as it was to arrive at Eq. (2). This is an important distinction when contrasted with the work of Ref. [12

**14**(3), 1094–1105 (
2006). [CrossRef] [PubMed]

*m*f) angular dependence are assumed. The mode volume calculated assuming standing waves is half that of the circulating waves. In light of the work in Ref. [24] there does not seem to be justification for assuming the circularly-propagating waves injected into the microcavity from the tapered fiber establish standing waves when scattered from sidewall imperfections. To calculate the values of the effective mode volume presented in Table 1, the integral in Eq. (4) was performed over the entire simulation space which extended 2μm above and below the disk and 6μm in the radial direction.

*Q*of a resonator it is important to consider other loss mechanisms contributing to the measured

*Q*. The relation between the various contributions to cavity loss can be expressed aswhere the three contributions to the measured quality factor are radiation, coupling to the tapered fiber waveguide and parasitic losses such as sidewall scattering. In the structures considered here,

*Q*

_{rad}< 20,000, and

*Q*

_{p}in larger disks fabricated with similar methods has been measured to be greater than 100,000. For this reason it is justified to neglect the contribution from

*Q*

_{p}in Eq. (5).

*Q*is a function of the gap between the resonator and the waveguide, and this fact can be exploited to determine the relative contributions to Eq. (5) from

_{c}*Q*

_{rad}and

*Q*

_{c}. In the experiments performed in this study, the tapered fiber position relative to the disk was controlled with 100nm resolution. As can be seen in Fig. 1, the near field of the modes decays over 1-2μm. To accurately determine

*Q*

_{rad}, transmission spectra were taken with the fiber very near the disk to determine resonance positions. The fiber was then incrementally moved away from the disk, and additional spectra were acquired as the gap between the fiber and disk was increased. As in Ref. [10

10. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling and optical loss in silicon microdisks,” Appl. Phys. Lett. **85**(17), 3693–3695 (
2004). [CrossRef]

*Q*. Convergence behavior was observed until the depth of the dip was at the level of the measurement noise; the reported values for

*Q*

_{meas}in Fig. 3 and Table 1 are those from the maximal fiber-disk gap which resulted in a value consistent with the convergence trend and with a deep enough dip to be fit to a Lorentzian. We did not directly measure the fiber-resonator gap because in measurements of such small disks so close to the substrate it is advantageous to hold the fiber above the plane of the disk, and the devices were only viewed directly from above.

*Q*factors are near 10-20% of predicted values, and this is true with or without consideration of the substrate and pedestal in the model. For the lower azimuthal mode numbers, the measured values are consistently higher than the theoretical values, indicating a tendency of the FDFD mode solver to underestimate the radiation-limited

*Q*.

*m*regime. The theoretical values of the

*m*= 4 device listed in Table 1 if the entire half space below the disk is filled with dielectric characterized by

*n*= 1.46 are

*Q*

_{sim}= 104 and λ

_{sim}= 1.630μm. For the

*m*= 7 disk with no undercut

*Q*

_{sim}= 5265 and λ

_{sim}= 1.555μm.

## 5. Analysis

*m*has a linear relationship with

*r*

_{0}over the range

*m*= 2-7 at a (nearly) constant wavelength.

*Q*has an exponential dependence on

*m*of the form

*Q*(

*m*) =

*A*exp(χ

*m*), where A = 0.374 and χ = 1.54 are parameters which have been determined with non-linear regression. As seen in Fig. 3, this expression fits our data quite well over the range of

*m*values considered here. The linear dependence of

*m*on

*r*

_{0}can be understood because the mode propagates around the circumference of the disk. The effective mode volume also has a linear dependence on

*m*. Thus, the Purcell enhancement decays exponentially with decreased

*m*, as does the finesse [14

14. J. E. Heebner, T. C. Bond, and J. S. Kallman, “Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators,” Opt. Express **15**(8), 4452–4473 (
2007). [CrossRef] [PubMed]

*Q*on

*m*can be understood in the framework of the effective potential [16,18

18. N. C. Frateschi and A. F. J. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. **80**(2), 644 (
1996). [CrossRef]

*V*

_{eff}changes discontinuously at the edge of the disk, creating a potential well in which the electromagnetic field predominantly resides. There are two features of a potential which affect the tunneling probability: the barrier height and barrier width. For a given wavelength, the height of the tunneling barrier scales as

*m*

^{2}/

*r*

_{0}^{2}. In a general one-dimensional tunneling scenario, the tunneling probability will have a power law dependence on barrier height. Therefore, the quadratic dependence of the barrier height on

*m*cannot explain the exponential dependence of

*Q*on

*m*. However, the tunneling probability through a one-dimensional barrier is exponentially dependent on the barrier width. As shown in Fig. 3, the radial distance at which the confined mode can emerge to free space is

*r = m/k*, and is determined by the condition

*V*

_{eff}=

*k*

^{2}. Thus, the tunneling barrier thickness is given by

*d*

_{T}= m/k-r_{0}. Using the linear relationship between

*m*and

*r*

_{0}(

*r*

_{0}=

*sm + b*) the tunneling thickness is

*d*Thus,

_{T}= (1/k-s)m-b.*T*is the tunneling probability and k and k’ are unknown constants which could be determined from a rigorous treatment. Thus, the exponential dependence of

*Q*on

*m*(and

*r*

_{0}) is consistent with the linear dependence of tunneling barrier width on

*m*.

## 6. Conclusions

*m*= 2-7 modes and have quantified the linear relationship between

*m*and

*r*

_{0}as well as the exponential dependence of

*Q*on

*m*. Our experimental data is in agreement with theoretical values for the radiation-limited

*Q*. We have demonstrated the

*m*= 5 mode is the highest-

*Q*mode in a subwavelength disk and was measured in our structures to have a

*Q*= 1250 at a wavelength of 1.543μm in a microdisk of diameter 1.490μm.

## Acknowledgements

## References and links

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

2. | A. F. J. Levi, R. E. Slusher, S. L. McCall, T. Tanbun-Ek, D. L. Coblentz, and S. J. Pearton, “Room temperature operation of microdisc lasers with submilliamp threshold current,” Electron. Lett. |

3. | A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometre radius disk laser,” Electron. Lett. |

4. | M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics |

5. | Q. Song, H. Cao, S. T. Ho, and G. S. Solomon, “Near-IR subwavelength microdisk lasers,” Appl. Phys. Lett. |

6. | C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. |

7. | Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express |

8. | K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high- Q photonic crystal microcavity,” Phys. Rev. B |

9. | M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature |

10. | M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling and optical loss in silicon microdisks,” Appl. Phys. Lett. |

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

12. | K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express |

13. | K. Zhang, and D. Li, |

14. | J. E. Heebner, T. C. Bond, and J. S. Kallman, “Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators,” Opt. Express |

15. | Amnon Yariv, |

16. | Jens Uwe Nöckel, “Resonances in nonintegrable open systems,” Ph.D. thesis (Yale University, 1997) pp 91–105. |

17. | R. P. Wang and M.-M. Dumitrescu, “Optical modes in semiconductor microdisk lasers,” IEEE J. Quantum Electron. |

18. | N. C. Frateschi and A. F. J. Levi, “The spectrum of microdisk lasers,” J. Appl. Phys. |

19. | P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. |

20. | R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and bound modes of surface plasmon waveguides,” Phys. Rev. B |

21. | H. Benisty, J.-M. Gerard, R. Houdre, J. Rarity, and C. Weisbuch, eds., |

22. | B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature |

23. | A. C. F. Hoole, M. E. Welland, and A. N. Broers, “Negative PMMA as a high-resolution resist—the limits and possibilities,” Semicond. Sci. Technol. |

24. | L. Deych, and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single scatterer: myths and reality.” arXiv:0812.4404v1 [physics.optics] 23 Dec 2008. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(130.3990) Integrated optics : Micro-optical devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 5, 2009

Revised Manuscript: November 6, 2009

Manuscript Accepted: November 25, 2009

Published: December 4, 2009

**Citation**

Jeffrey Shainline, Stuart Elston, Zhijun Liu, Gustavo Fernandes, Rashid Zia, and Jimmy Xu, "Subwavelength silicon microcavities," Opt. Express **17**, 23323-23331 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23323

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### References

- S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289–291 (1992). [CrossRef]
- A. F. J. Levi, R. E. Slusher, S. L. McCall, T. Tanbun-Ek, D. L. Coblentz, and S. J. Pearton, “Room temperature operation of microdisc lasers with submilliamp threshold current,” Electron. Lett. 28(11), 1010–1012 (1992). [CrossRef]
- A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometre radius disk laser,” Electron. Lett. 29(18), 1666–1667 (1993). [CrossRef]
- M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]
- Q. Song, H. Cao, S. T. Ho, and G. S. Solomon, “Near-IR subwavelength microdisk lasers,” Appl. Phys. Lett. 94(6), 061109 (2009). [CrossRef]
- C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]
- Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express 16(6), 4309–4315 (2008). [CrossRef] [PubMed]
- K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high- Q photonic crystal microcavity,” Phys. Rev. B 70(8), 081306 (2004). [CrossRef]
- M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009). [CrossRef] [PubMed]
- M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling and optical loss in silicon microdisks,” Appl. Phys. Lett. 85(17), 3693–3695 (2004). [CrossRef]
- M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005). [CrossRef] [PubMed]
- K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14(3), 1094–1105 (2006). [CrossRef] [PubMed]
- K. Zhang, and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 1998).
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