OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 25 — Dec. 7, 2009
  • pp: 23323–23331
« Show journal navigation

Subwavelength silicon microcavities

Jeffrey Shainline, Stuart Elston, Zhijun Liu, Gustavo Fernandes, Rashid Zia, and Jimmy Xu  »View Author Affiliations


Optics Express, Vol. 17, Issue 25, pp. 23323-23331 (2009)
http://dx.doi.org/10.1364/OE.17.023323


View Full Text Article

Acrobat PDF (639 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

In this work we studied silicon microdisks of diameters ranging from smaller to larger than the wavelength corresponding to resonances of azimuthal mode number (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

Resonances in microdisks can be classified according to three mode numbers corresponding to quantization in the vertical, radial and azimuthal directions [13

13. K. Zhang, and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 1998).

]. To completely specify the mode one must also specify the polarization as TE-like (Hz is the dominant field component) or TM-like (Ez is the dominant field component). The highest-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]

]. Therefore, we consider only modes of fundamental order in the vertical and radial direction and classify modes by m. Further, the partial differential equation describing the system is separable if one assumes exp(imf) 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

15. Amnon Yariv, Quantum Electronics (John Wiley and Sons, 1989), Chap. 22.

]. 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 cosβz for even modes), thenβ=k0(n2n¯2)1/2, where 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 n¯is the effective index of refraction which enters the radial equation. Satisfying the boundary conditions at the top or bottom of the disk leads to the transcendental equation
n2n¯2tan(k0d2n2n¯2)=ξn¯21
(1)
where d is the thickness of the silicon disk andξ=n2for TM modes and unity for TE modes. For a silicon disk of thickness d = 250nm operating at λ = 1550nm, n¯ = 2.015 (2.914) for TM (TE) modes. The objective of this study was to explore the regime where the size of the disk is smaller than the free space wavelength. To this end, the substantially larger effective index for TE modes is an indispensable attribute. For this reason we consider only TE modes in this study. For sufficiently thin disks (h300nm) at λ1550nm, TE modes have significantly larger radiation-limited quality factors, have larger m for a given disk radius, and therefore all modes populating thin subwavelength disks are TE-polarized.

Having obtained the effective index of refraction through consideration of the z equation alone, the radial equation reads
1rddrrddrR(r)+Veff(r)R(r)=k02R(r),
(2)
where

Veff(r)=k02[1n¯2(r)]+m2r2.
(3)

The solution to Eq. (2) inside the disk is a Bessel function, Jm(Tr),where T=n¯k0, k2=T2+β2and k=nk0. The effective index of refraction, which enters Eq. (2) through Eq. (3), is a function of 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, k02plays the role of the energy eigenvalue of the quantum eigenstate. The limit of the analogy is that k02 also enters into the effective potential. Special consideration of the curvilinear kinetic energy term must be given if one is to justify the definition of the effective potential as in Eq. (3). Discussion of this point is held in Ref [16

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

]. Further consideration of the effective potential picture as applied to microdisks is given in Section 5.

3. Fabrication of subwavelength silicon microdisks

SEM images of two completed disks are shown in Fig. 1
Fig. 1 Fabricated structures and calculated field profiles of microdisks for m = 7 and m = 4 modes. a) Disk of 1.89μm diameter. b) Re(Hz) field profile of m = 7 mode in the y-z plane calculated with the FDFD method. Q = 18,200. c) Far field (log10(|Re(Hz)|)) for the same disk. d) Re(Hz) in the x-y plane calculated analytically. e) Disk of 1.35μm diameter. f-h) Calculations for the 1.35μm disk corresponding to those in b-d.
along with the calculated near- and far-field profiles of their resonant modes. Contrasting Fig. 1(b) and Fig. 1(f) one sees the near-field amplitude extending further radially and axially in the subwavelength disk. Contrasting Fig. 1(c) and Fig. 1(g) one observes the increased far-field radiation emanating from the subwavelength cavity.

4. Experimental details and results

To experimentally characterize the disks we utilize the technique of tapered fiber spectroscopy [8

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

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]

]. A tunable laser sweeps the wavelength range from 1.52μm to 1.62μm. The transmission through a tapered fiber coupled to a disk is monitored with an optical spectrum analyzer. The transmission dips are fit to a Lorentzian function and the Q factor is extracted from the width and center wavelength.

In Fig. 3
Fig. 3 Microdisk Q versus m. a) Experimental data, theoretical values obtained with FDFD and a fit to an exponential function are shown. The linear relationship between the disk radius, r 0, and m is shown in the inset. b) The effective potential and tunneling parameters.
we present a comparison of the experimentally-obtained 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

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

,22

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]

].

υeff=ϵ(r)|E(r)|2d3rmax[ϵ(r)|E(r)|2].
(4)

To calculate the volume integral in the numerator of Eq. (4), exp(imf) 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

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]

] wherein standing waves with sin(mf) 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

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.

] 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.

In using evanescent coupling in transmission measurements to determine the radiation-limited 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 as
Qmeas1=Qrad1+Qc1+Qp1,
(5)
where 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). Qc 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 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]

], increasing the gap resulted in a decrease in the depth of the transmission dip and increase in the measured 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.

It is important to note that contributions to error in the tapered fiber measurements are numerous. They include coupling between the fiber and disk, noise due to vibrations of the fiber, substrate and disk coupling and perturbation due to the pedestal. Additionally, uncertainty in the measurement of disk diameters with SEM affects comparison between experiment and theory. Because the contributions to error are so varied and each is so difficult to quantify, we do not venture to guess at the uncertaintly in the data in Table 1. We note that measured resonant wavelengths are within 1-3% of predicted values, measured 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.

The effect of the substrate is more pronounced if one considers non-undercut resonators in this low-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 λsim = 1.630μm. For the m = 7 disk with no undercut Q sim = 5265 and λsim = 1.555μm.

5. Analysis

Analysis of our data reveals that 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(χm), 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]

].

The exponential dependence of Q on m can be understood in the framework of the effective potential [16

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

,18

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

], which is given by Eq. (3) and is plotted in Fig. 3(b). This quantity enters the scalar radial equation for the dominant field component if TE or TM polarization is assumed. As discussed in Sec. 2, 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/r0 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 dT = m/k-r 0. Using the linear relationship between m and r 0 (r 0 = sm + b) the tunneling thickness is dT = (1/k-s)m-b. Thus,

1Q(m)T(m)exp[κdT(m)]exp(κm),
(6)

Where 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

In conclusion, we have investigated the resonant modes of silicon microdisks that are smaller in every dimension than the free-space wavelength of light being stored in the cavities. We have presented a systematic characterization of the 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

We are grateful to Dr. Gernot Pomrenke and the grant support of AFOSR (FA9550-07-1-0286) and to the WCU program at SNU, Korea.

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. 60(3), 289–291 ( 1992). [CrossRef]

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. 28(11), 1010–1012 ( 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]

5.

Q. Song, H. Cao, S. T. Ho, and G. S. Solomon, “Near-IR subwavelength microdisk lasers,” Appl. Phys. Lett. 94(6), 061109 ( 2009). [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]

13.

K. Zhang, and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 1998).

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]

15.

Amnon Yariv, Quantum Electronics (John Wiley and Sons, 1989), Chap. 22.

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. 34(10), 1933–1937 ( 1998). [CrossRef]

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]

21.

H. Benisty, J.-M. Gerard, R. Houdre, J. Rarity, and C. Weisbuch, eds., Confined Photon Systems (Springer, New York, 1998).

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]

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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]
  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. 28(11), 1010–1012 (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]
  5. Q. Song, H. Cao, S. T. Ho, and G. S. Solomon, “Near-IR subwavelength microdisk lasers,” Appl. Phys. Lett. 94(6), 061109 (2009). [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]
  13. K. Zhang, and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 1998).
  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]
  15. Amnon Yariv, Quantum Electronics (John Wiley and Sons, 1989), Chap. 22.
  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. 34(10), 1933–1937 (1998). [CrossRef]
  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]
  21. H. Benisty, J.-M. Gerard, R. Houdre, J. Rarity, and C. Weisbuch, eds., Confined Photon Systems (Springer, New York, 1998).
  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]
  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]
  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.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited