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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 23 — Nov. 9, 2009
  • pp: 20998–21006
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Ultra-high quality-factor resonators with perfect azimuthal Modal-Symmetry

Nikolaj Moll, Thilo Stöferle, Sophie Schönenberger, and Rainer F. Mahrt  »View Author Affiliations


Optics Express, Vol. 17, Issue 23, pp. 20998-21006 (2009)
http://dx.doi.org/10.1364/OE.17.020998


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Abstract

We study circular grating resonators (CGRs) which are formed by a central defect surrounded by concentric rings composing a grating and which display perfect azimuthal modal-symmetry. Because of their radial symmetry they exhibit a complete band gap for a minimal index contrast. However, as is the case for all 2D resonators their quality factors are limited by vertical losses. To reduce the vertical losses we introduce a chirp of the grating period by reducing it towards the central defect. The chirped CGRs exhibit drastically improved quality factors of up to tens of millions with a modal volume of a few cubic wavelengths.

© 2009 Optical Society of America

1. Introduction

For future integrated optical components one of the most critical parameters is the footprint. Smaller footprints can be achieved by employing resonant structures that lead to longer effective propagation paths. Applications for such resonant structures are filters [1

1. J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microringresonator,” Opt. Lett. 29(24), 2861–2863 ( 2004), http://ol.osa.org/abstract.cfm?URI=ol-29-24-2861. [CrossRef]

], delay-lines [2

2. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon. 1(1), 65–71 ( 2007), http://dx.doi.org/10.1038/nphoton.2006.42. [CrossRef]

], electro-optical modulators [3

3. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 ( 2005), http://dx.doi.org/10.1038/nature03569. [CrossRef]

], all-optical switches [4

4. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 ( 2004), http://dx.doi.org/10.1038/nature02921. [CrossRef]

], lasers [5

5. K. Baumann, T. Stöferle, N. Moll, R. F. Mahrt, T. Wahlbrink, J. Bolten, T. Mollenhauer, C. Moormann, and U. Scherf, “Organic mixed-order photonic crystal lasers with ultrasmall footprint,” Appl. Phys. Lett. 91(17), 171,108–3 ( 2007), http://link.aip.org/link/?APL/91/171108/1.

], as well as sensing [6

6. C. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 ( 2003), http://link.aip.org/link/?APL/83/1527/1. [CrossRef]

] and spectroscopy applications [7

7. A. Nitkowski, L. Chen, and M. Lipson, “Cavity-enhanced on-chip absorption spectroscopy using microring resonators,” Opt. Express 16(16), 930–936 ( 2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-16-11930.

]. Furthermore, beyond classical optics, resonant structures with a very high quality factor and small mode volume could pave the way towards integrated devices that harness cavity quantum electrodynamics effects [8

8. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 ( 2004), http://dx.doi.org/10.1038/nature03119. [CrossRef]

]. In this letter we focus on one type of resonator structure, namely circular grating resonators (CGR) [9

9. N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, “Evidence for band-edge lasing in a two-dimensional photonic bandgap polymer laser,” Appl. Phys. Lett. 80(5), 734–736 ( 2002), http://link.aip.org/link/?APL/80/734/1. [CrossRef]

, 10

10. A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G. Bona, and W. Bächtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96(6), 3043–3049 ( 2004), http://link.aip.org/link/?JAP/96/3043/1. [CrossRef]

, 11

11. S. Schönenberger, N. Moll, T. Stöferle, R. F. Mahrt, B. J. Offrein, S. Götzinger, V. Sandoghdar, J. Bolten, T. Wahlbrink, T. Plötzing, M. Waldow, and M. Först, “Circular Grating Resonators as Small Mode-Volume Microcavities for Switching,” Opt. Express 17(8), 5953–5964 ( 2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-8-5953. [CrossRef]

]. Similar cavity structures have shown much promise and already theoretical quality factors of a couple ten thousand [12

12. D. Chang, J. Scheuer, and A. Yariv, “Optimization of circular photonic crystal cavities - beyond coupled mode theory,” Opt. Express 13(23), 9272–9279 ( 2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-23-9272. [CrossRef]

, 13

13. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “Lasing from a circular Bragg nanocavity with an ultrasmall modal volume,” Appl. Phys. Lett. 86(25), 251,101–3 ( 2005), http://link.aip.org/link/?APL/86/251101/1.

, 14

14. X. Sun, J. Scheuer, and A. Yariv, “Optimal Design and Reduced Threshold in Vertically Emitting Circular Bragg Disk Resonator Lasers,” IEEE J. of Sel. Top. in Quantum Electron. 13(2), 359–366 ( 2007). [CrossRef]

]. Such structures and CGRs enable cavities with an ultra-small footprint and at the same time a high quality factor. Compared with linear structures, they offer enhanced photonic confinement due to the photonic band gap in the two lateral dimensions. To achieve a complete band gap for radial wavevectors, CGRs require only a minimal refractive index contrast of the materials used to fabricate the structure. This is an inherent advantage over photonic crystal cavities where the photonic band gap is induced by a lattice with high index contrast, and thus, the choice of suitable materials is very limited. Furthermore, because their radial symmetry they enable mode patterns with perfect azimuthal symmetry, e.g. dipole-like resonances. Hence, CGRs could lead to highly integrated resonant devices with an extended choice of materials and mode patterns while exhibiting extremely small optically active volumes.

Fig. 1. (a) Schematic top view of a CGR consisting of a central defect with a defect radius rc surrounded by concentric rings 1 to 7. (b) Cross section of the CGR where the duty cycle D is given by the ratio between the width of trench q and the grating period a (D=q/a). The height of the CGR is h. The CGR consists of Si as high-index material and is surrounded by SiO2 as low-index material.

2. Circular grating resonators with periodic rings

As the quality factor is largely determined by losses in the vertical direction, the configuration with the largest band gap that provides the strongest photonic confinement is not necessarily the one with the highest quality factor. Starting from this circular Bragg geometry we simulate a CGR and maximize its quality factor by varying its three free geometric parameters: its defect radius rc, its duty cycle D, and its height h. The quality factors are calculated using the finite-difference time-domain (FDTD) method using the MEEP code [16

16. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 ( 2006), http://ol.osa.org/abstract.cfm?URI=ol-31-20-2972. [CrossRef]

]. The cylindrical symmetry was exploited in such a way that the calculations were reduced to 2D. A uniform mesh with 40 grid points per grating period a was used and perfectly matched layer boundary conditions were employed on a calculation cell size of 44a×13a, in the r- and z-directions, respectively. To generalize the results, they are expressed in units normalized to the grating period a. Frequencies are reported in units of c/a.

3. Chirped circular grating resonators with high quality factors

qN=DaΔNandΔN+1=ΔNΓ.
(1)

In the limit of large N the trench width approaches the trench width without a chirp.

We introduced two additional geometric parameters to describe the chirped CGRs: the shift of the first ring Δ1 and the decay constant Γ. We now vary all five geometric parameters to maximize the quality factor. For three different azimuthal orders m=0, m=1, and m=3 the results are shown in Table 1. For the azimuthal order m=0 we obtain a maximum quality factor of approx. 6.6×107. Increasing the azimuthal order lowers the maximum quality factor to 5.6×106 for m=1 and to 2.2×106 for m=3. The defect radius rc decreases from 1.60a for m=0 to 1.53a for m=1 and to 1.10a for m=3. It turns out that all three azimuthal orders have the same duty cycle of D=0.34, same height of h=0.80a, and same shift of the first ring

Table 1. For three different azimuthal orders m the geometric parameters of the chirped CGRs are shown for which a maximized quality factor Q is achieved.

table-icon
View This Table

of Δ1=-0.29a. Only for the azimuthal order m=0 is the decay constant is larger (Γ=0.88) compared to the other two configurations. All three resonance frequencies are located well within the band gap which extends from 0.182c/a to 0.239c/a. This corresponds to a band gap of Δν/ν=27%.

Fig. 2. (a) The quality factor Q, (b) the frequency ν, and (c) the modal volume Vm as a function of the defect radius rc for a CGR without chirp and an azimuthal order of m=0. The fixed geometric parameters are the duty cycle D=0.34 and the height h=0.85a. In (b), the lower band edge of the corresponding infinite linear grating is shown as dashed line.

Fig. 3. (a) The quality factor Q, (b) the frequency ν, and (c) the modal volume Vm as a function of the shift of the first ring Δ1 for a chirped CGR and an azimuthal order of m=0. The fixed geometric parameters are the defect radius rc=1.60a, the duty cycle D=0.34, the height h=0.80a, and the decay constant Γ=0.88.

The mechanism of the chirp can be understood by plotting the effective band gap as function of the ring number N as shown in Fig. 4. This is analogous to the local bands used to describe multiheterostructure 2D photonic-crystal resonators [17

17. Y. Tanaka, T. Asano, and S. Noda, “Design of Photonic Crystal Nanocavity With Q-Factor of ~109,” J. Light-wave Technol. 26(11), 1532–1539 ( 2008). [CrossRef]

]. For chirped CGR, the local band edges are modified by the smaller effective lattice constant and the larger effective duty cycle towards the central defect. The band gap therefore decreases towards the central defect. In the inner part of the CGR (N≤2) no band gap is present. Starting from the resonance frequency, a band gap gradually opens up for N>2 and increases towards the outer rings of the CGR. Therefore the resonant mode can adapt from a very delocalized, high-quality-factor, band-edge-like mode in the center of the CGR to the large band-gap region in the outer parts of the CGR. This enables a very high quality factor at only a small compromise on the strong confinement. In contrast, CGR with periodic gratings achieve high quality factors by approaching the resonance frequency to the band edge.

Fig. 4. Effective local band edges for the optimum chirped configuration from Fig. 3 (m=0, rc=1.60a, D=0.34, h=0.80a, Δ1=-0.29, Γ=0.88). The lower (squares) and upper (circles) band edges are shown as function of the ring number N. The solid lines are single-exponential fits to the respective numerically calculated bands, which exhibit slight numerical noise due to aliasing effects resulting from finite grid resolution of the calculation. The resonance frequency (dotted line) of this CGR is ν=0.221c/a, and the lower and upper band edges (dashed lines) of the corresponding infinite periodic linear grating are shown for comparision.

The functionality of the chirp strongly depends on the decay constant Γ. In Fig. 4 the upper band edge increases and lower band edge decreases exponentially with the ring number. The decay constant defines the rate the band gap decreases towards the central defect. Using a too small decay constant would lead to a too fast decrease of the band gap, whereas using a too large decay constant would lead to a too slow decrease. The quality factor shows a sharp maximum at the optimum decay constant. All in all, the quality factor varies with the decay constant in very similar fashion as it varies with the shift of the first ring.

To further explore the nature of the extraordinarily high quality factor of the chirped CGR we investigate the resonant field distributions. In Fig. 5 we plot the cross section of the electric field component Eφ for three different shifts of the first ring Δ1. The color scales are used in such a way that even smaller electric field values further away from the center defect are visible and they are almost everywhere supersaturated. The near-field patterns are visually indistinguishable in the three cases. However, the far-fields pattern differ drastically. An indication of nodal planes appear precisely at the maximum quality factor configuration (Δ1=-0.29a), indicating the cancelation of the lowest-order multi-pole moment (i. e. dipole) [20

20. S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 ( 2001), http://link.aip.org/link/?APL/78/3388/1. [CrossRef]

]. The two other configurations show dipole-like radial far-field patterns which are an indication of large vertical losses and therefore low quality factors. This mechanism offers an additional explanation for the high quality factors.

Fig. 5. Cross section of the electric field component Eφ at resonance for three different chirped CGRs where the shift of the first ring Δ1 is varied: (a) Δ1=-0.26a, which exhibits a Q=356000; (b) Δ1=-0.29a, which exhibits a Q=66200000, and (c) Δ1=-0.32,a which exhibits a Q=1270000. The other geometric parameters are the same as in Fig. 3 (m=0, rc=1.60a, D=0.34, h=0.80a, Γ=0.88)

Finally, in order to assess the quality of our chirp relation (Eq. 1) we start from this optimal configuration and we varied the positions of all Si/SiO2 interfaces fully independently. We found, however, that the assumed chirp relation for this configuration seems to yield the maximum quality factor. For other configurations than m=0 small shifts in the positions of the Si/SiO2 interfaces lead to slightly larger quality factors. However, these shifts are on the order of the calculation grid resolution and therefore at the limits of numerical significance.

4. Conclusion

Acknowledgement

The authors gratefully acknowledge financial support from EU within the Circles of Light project (FP6-034883).

References and links

1.

J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microringresonator,” Opt. Lett. 29(24), 2861–2863 ( 2004), http://ol.osa.org/abstract.cfm?URI=ol-29-24-2861. [CrossRef]

2.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon. 1(1), 65–71 ( 2007), http://dx.doi.org/10.1038/nphoton.2006.42. [CrossRef]

3.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 ( 2005), http://dx.doi.org/10.1038/nature03569. [CrossRef]

4.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 ( 2004), http://dx.doi.org/10.1038/nature02921. [CrossRef]

5.

K. Baumann, T. Stöferle, N. Moll, R. F. Mahrt, T. Wahlbrink, J. Bolten, T. Mollenhauer, C. Moormann, and U. Scherf, “Organic mixed-order photonic crystal lasers with ultrasmall footprint,” Appl. Phys. Lett. 91(17), 171,108–3 ( 2007), http://link.aip.org/link/?APL/91/171108/1.

6.

C. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 ( 2003), http://link.aip.org/link/?APL/83/1527/1. [CrossRef]

7.

A. Nitkowski, L. Chen, and M. Lipson, “Cavity-enhanced on-chip absorption spectroscopy using microring resonators,” Opt. Express 16(16), 930–936 ( 2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-16-11930.

8.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 ( 2004), http://dx.doi.org/10.1038/nature03119. [CrossRef]

9.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, “Evidence for band-edge lasing in a two-dimensional photonic bandgap polymer laser,” Appl. Phys. Lett. 80(5), 734–736 ( 2002), http://link.aip.org/link/?APL/80/734/1. [CrossRef]

10.

A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G. Bona, and W. Bächtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96(6), 3043–3049 ( 2004), http://link.aip.org/link/?JAP/96/3043/1. [CrossRef]

11.

S. Schönenberger, N. Moll, T. Stöferle, R. F. Mahrt, B. J. Offrein, S. Götzinger, V. Sandoghdar, J. Bolten, T. Wahlbrink, T. Plötzing, M. Waldow, and M. Först, “Circular Grating Resonators as Small Mode-Volume Microcavities for Switching,” Opt. Express 17(8), 5953–5964 ( 2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-8-5953. [CrossRef]

12.

D. Chang, J. Scheuer, and A. Yariv, “Optimization of circular photonic crystal cavities - beyond coupled mode theory,” Opt. Express 13(23), 9272–9279 ( 2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-23-9272. [CrossRef]

13.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “Lasing from a circular Bragg nanocavity with an ultrasmall modal volume,” Appl. Phys. Lett. 86(25), 251,101–3 ( 2005), http://link.aip.org/link/?APL/86/251101/1.

14.

X. Sun, J. Scheuer, and A. Yariv, “Optimal Design and Reduced Threshold in Vertically Emitting Circular Bragg Disk Resonator Lasers,” IEEE J. of Sel. Top. in Quantum Electron. 13(2), 359–366 ( 2007). [CrossRef]

15.

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 ( 2001), http://www.opticsexpress.org/abstract.cfm?URI=oe-8-3-173. [CrossRef]

16.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 ( 2006), http://ol.osa.org/abstract.cfm?URI=ol-31-20-2972. [CrossRef]

17.

Y. Tanaka, T. Asano, and S. Noda, “Design of Photonic Crystal Nanocavity With Q-Factor of ~109,” J. Light-wave Technol. 26(11), 1532–1539 ( 2008). [CrossRef]

18.

M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q Nanocavity with 1D Photonic Gap,” Opt. Express 16(15), 11,095–11,102 ( 2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-15-11095.

19.

Y. Takahashi, H. Hagino, Y. Tanaka, B. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17,206–17,213 ( 2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-25-17206.

20.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 ( 2001), http://link.aip.org/link/?APL/78/3388/1. [CrossRef]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(230.5750) Optical devices : Resonators

ToC Category:
Integrated Optics

History
Original Manuscript: September 2, 2009
Revised Manuscript: October 28, 2009
Manuscript Accepted: October 28, 2009
Published: November 3, 2009

Citation
Nikolaj Moll, Thilo Stöferle, Sophie Schönenberger, and Rainer F. Mahrt, "Ultra-high quality-factor resonators with perfect azimuthal modal-symmetry," Opt. Express 17, 20998-21006 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-20998


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References

  1. J. Niehusmann, A. Vorckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, "Ultrahigh quality-factor silicon-on-insulator microringresonator," Opt. Lett. 29(24), 2861-2863 (2004), http://ol.osa.org/abstract.cfm?URI=ol-29-24-2861. [CrossRef]
  2. F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nature Photon. 1(1), 65-71 (2007), http://dx.doi.org/10.1038/nphoton.2006.42. [CrossRef]
  3. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435(7040), 325-327 (2005), http://dx.doi.org/10.1038/nature03569. [CrossRef]
  4. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431(7012), 1081-1084 (2004), http://dx.doi.org/10.1038/nature02921. [CrossRef]
  5. K. Baumann, T. Stoferle, N. Moll, R. F. Mahrt, T. Wahlbrink, J. Bolten, T. Mollenhauer, C. Moormann, and U. Scherf, "Organic mixed-order photonic crystal lasers with ultrasmall footprint," Appl. Phys. Lett. 91(17), 171,108-3 (2007), http://link.aip.org/link/?APL/91/171108/1.
  6. C. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83(8), 1527-1529 (2003), http://link.aip.org/link/?APL/83/1527/1. [CrossRef]
  7. A. Nitkowski, L. Chen, and M. Lipson, "Cavity-enhanced on-chip absorption spectroscopy using microring resonators," Opt. Express 16(16), 930-936 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-16-11930.
  8. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432(7014), 200-203 (2004), http://dx.doi.org/10.1038/nature03119. [CrossRef]
  9. N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80(5), 734-736 (2002), http://link.aip.org/link/?APL/80/734/1. [CrossRef]
  10. A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G. Bona, and W. Bachtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96(6), 3043-3049 (2004), http://link.aip.org/link/?JAP/96/3043/1. [CrossRef]
  11. S. Schonenberger, N. Moll, T. Stoferle, R. F. Mahrt, B. J. Offrein, S. Gotzinger, V. Sandoghdar, J. Bolten, T. Wahlbrink, T. Plotzing, M. Waldow, and M. Forst, "Circular Grating Resonators as Small Mode-Volume Microcavities for Switching," Opt. Express 17(8), 5953-5964 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-8-5953. [CrossRef]
  12. D. Chang, J. Scheuer, and A. Yariv, "Optimization of circular photonic crystal cavities- beyond coupled mode theory," Opt. Express 13(23), 9272-9279 (2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-23-9272. [CrossRef]
  13. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86(25), 251,101-3 (2005), http://link.aip.org/link/?APL/86/251101/1.
  14. X. Sun, J. Scheuer, and A. Yariv, "Optimal Design and Reduced Threshold in Vertically Emitting Circular Bragg Disk Resonator Lasers," IEEE J. Sel.Top. in Quantum Electron. 13(2), 359-366 (2007). [CrossRef]
  15. S. Johnson and J. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8(3), 173-190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=oe-8-3-173. [CrossRef]
  16. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31(20), 2972-2974 (2006), http://ol.osa.org/abstract.cfm?URI=ol-31-20-2972. [CrossRef]
  17. Y. Tanaka, T. Asano, and S. Noda, "Design of Photonic Crystal Nanocavity With Q-Factor of ∼ 109," J. Lightwave Technol. 26(11), 1532-1539 (2008). [CrossRef]
  18. M. Notomi, E. Kuramochi, and H. Taniyama, "Ultrahigh-Q Nanocavity with 1D Photonic Gap," Opt. Express 16(15), 11,095-11,102 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-15-11095.
  19. Y. Takahashi, H. Hagino, Y. Tanaka, B. Song, T. Asano, and S. Noda, "High-Q nanocavity with a 2-ns photon lifetime," Opt. Express 15(25), 17,206-17,213 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-25-17206.
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