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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9371–9377
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High-Q/V air-mode photonic crystal cavities at microwave frequencies

Yinan Zhang, Irfan Bulu, Wai-Ming Tam, Ben Levitt, Jagdish Shah, Tancredi Botto, and Marko Loncar  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9371-9377 (2011)
http://dx.doi.org/10.1364/OE.19.009371


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Abstract

We present results for a photonic microwave resonator designed and fabricated at 17.4GHz with a record high Quality factor (Q = 26,400) at room temperature over a mode volume smaller than one cubic wavelength. The cavity is uniquely designed to have its electric field concentrated in air, which allows for efficient coupling to free space and facilitates interactions with gaseous atomic systems and fluids.

© 2011 OSA

1. Introduction

Photonic crystal cavities with high Quality factors (Qs) are capable of extending the lifetime of cavity photons confined within mode volumes (V) of the order of a cubic-wavelength, and hence can greatly enhance the interaction between electromagnetic fields and matter. Recently, high Q/V photonic crystal cavities have been extensively investigated with various geometries at optical frequencies, acting as a powerful platform for studying cavity quantum electrodynamics (cQED) [1

1. 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). [CrossRef] [PubMed]

] and developing a myriad of nano-photonic devices. Yet few photonic crystal cavities have been reported at microwave frequencies [2

2. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67(24), 3380–3383 (1991). [CrossRef] [PubMed]

7

7. C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region,” Phys. Rev. B 61(16), 10762–10767 (2000). [CrossRef]

].

A high Q/V microwave cavity is demanded for a number of applications. In atomic physics, a single Rydberg atom coupled to a single photon is the fundamental system for cavity QED studies, especially when one reaches the interesting strong coupling regime where energy can be coherently exchanged between the two. For strong coupling to occur, the atom-photon coupling rate g (inversely proportional to square root of V) must exceed both the photon leakage rate (inversely proportional to Q) and the atom decoherence rate. Since the frequency difference of Rydberg states is in the microwave region [8

8. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]

], a high Q/V microwave cavity is crucial to achieve this type of coupling. Strong coupling in Rydberg atoms has been previously observed in large-scale Fabry-Perot cavities and superconducting circuits [8

8. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]

,9

9. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004). [CrossRef] [PubMed]

]. Photonic crystal cavities, in contrast, offer an alternative platform that is more compact and can be operated at room temperature.

Secondly, given the cavity’s intrinsic field enhancement factor, there are also significant advantages in practical applications such as tunable microwave filters and antennas, where the efficiency of a radiator can be significantly improved by coupling with the cavity mode [10

10. A. Petosa, A. Ittipiboon, Y. M. M. Antar, D. Roscoe, and M. Cuhaci, “Recent advances in dielectric-resonator antenna technology,” IEEE Antennas Propag. Mag. 40(3), 35–48 (1998). [CrossRef]

]; fluorescence microscopy for biological systems, where a high Q/V microwave cavity can be used to control the temperature and drive the biological reactions [11

11. M. J. R. Previte and C. D. Geddes, “Fluorescence microscopy in a microwave cavity,” Opt. Express 15(18), 11640–11649 (2007). [CrossRef] [PubMed]

]; resonance-enhanced microwave detectors [3

3. B. Temelkuran, E. Ozbay, J. P. Kavanaugh, G. Tuttle, and K. M. Ho, “Resonant cavity enhanced detectors embedded in photonic crystals,” Appl. Phys. Lett. 72(19), 2376–2378 (1998). [CrossRef]

]; particle accelerators [5

5. E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, “Demonstration of a 17-GHz, high-gradient accelerator with a photonic-band-gap structure,” Phys. Rev. Lett. 95(7), 074801 (2005). [CrossRef] [PubMed]

,6

6. E. Di Gennaro, S. Savo, A. Andreone, V. Galdi, G. Castaldi, V. Pierro, and M. R. Masullo, “Mode confinement in photonic quasicrystal point-defect cavities for particle accelerators,” Appl. Phys. Lett. 93(16), 164102 (2008). [CrossRef]

]; and refractive index sensors.

In this work, we have designed and experimentally demonstrated an all-dielectric photonic crystal microwave cavity. The cavity can be fabricated with a variety of materials using conventional machining techniques. We report a cavity mode at 17.4GHz with a very small mode volume limited to about one cubic-wavelength and a record-high Q-factor of 26,400 at room temperature. Besides the high Q/V factor, the defect mode has a TM-polarized electric field concentrated in the air region, as opposed to most photonic crystal cavities designs with TE-polarized electric field concentrated in the host dielectric. These features are crucial for effectively coupling microwaves to matter in a number of applications. Moreover, when a center-fed antenna is placed inside the cavity acting as a radiating dipole, we observe a strong signature of the cavity’s Purcell enhancement factor. In summary, we believe this type of device offers great promise for studying cavity QED phenomena as well as for enabling novel applications at microwave frequencies.

2. Cavity design

Next, we introduce a defect region into the cavity by gradually increasing the periodicity (hole-to-hole distance) and hole diameter for each segment starting from a pair of outer holes and symmetrically moving towards the center. When the feature size of a segment is enlarged, the band-gap is red-shifted, resulting in a graded photonic band, as shown in Fig. 1(b). This allows confining an air-band mode in the defect region: the air-band mode is coupled to the evanescent Bloch modes within the band-gaps at each end, effectively trapping it between a pair of Bragg mirrors.

A three-dimensional finite-difference time-domain (FDTD) code was used to calculate the resonant frequency and the Q factor of the cavity mode [17

17. Rsoft Inc, Rsoft Fullwave v6.0.2, http://www.rsoftdesign.com.

]. On each side of the cavity we introduce 10 segments with progressive tapering of the hole diameter and spacing, in order to adiabatically couple the cavity mode to the evanescent Bloch mode. This adiabatic process is tuned to suppress the scattering loss resulting from the effective index mismatch between the two modes [18

18. Y. Zhang and M. Loncar, “Ultra-high quality factor optical resonators based on semiconductor nanowires,” Opt. Express 16(22), 17400–17409 (2008). [CrossRef] [PubMed]

,19

19. P. Lalanne and J. P. Hugonin, ““Bloch-wave engineering for high-Q, small-V microcavities,” IEEE. J. Quantum Electron. 39(11), 1430–1438 (2003). [CrossRef]

]. Without material losses, the cavity mode has an ultra-high Q factor of 2,500,000, limited by scattering alone (Q = Q sc). The cavity mode profiles are shown in Fig. 1(d), where the Ey component is plotted at the mirror plane of y-axis and x-axis, respectively. The tapered section’s parameters are tailored to localize an air-band mode resonating at 17.4GHz, which is closer to the air band-edge. At this frequency, the photonic bandgap has maximized confinement along the z-axis, which results in a small cavity mode volume. As expected from the general features of air-band modes, the electric field concentrates in the air-region. However, the maximum of the electric field density [ε|E| 2] is not in the air region, so we define the effective cavity mode volume as,

Veff=ε|E|2dV|Emax,air|2
(1)

where E max, air is the electric field maximum in the air region, located at the center of the cavity. For our cavity we found an effective mode volume of 0.99λ 3.

The cavity mode is normally excited from an external waveguide port. Our device can be modeled accurately with temporal coupled-mode theory [20

20. J. D. Joannopoulos and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

,21

21. S. Fan, “Photonic crystal theory: temporal coupled-mode formalism,” in Optical Fiber Communications V A: Components and Subsystems, I. P. Kaminow, T. Li, and A. E. Willner, eds. (Elsevier 2008), Vol. 1, pp. 431–454.

]. The amount of energy stored (U) is proportional to the launched power (P0).

U=2ω0Q2QwP0
(2)

In Eq. (2) Qw denotes the quality factor responsible for the field energy leakage from the cavity section via the dielectric waveguide. The total cavity quality factor Q, including material losses, can be written as

1Q=1Qsc+1Qm+1Qw
(3)

where Qsc accounts for scattering losses due to mode mismatch in the tapered section and Qm for material losses in the dielectric. Substituting Eq. (3) into Eq. (2), we have

U=Umax=P0w0Q,  when  Qw=QscQmQsc+Qm
(4)

Therefore, for given Qsc and Qm, it is important to tune Qw by fixing the number of Bragg mirror pairs on each side of the resonator so as to achieve optimal critical coupling conditions by satisfying Eq. (4).

3. Fabrication and measurement

The resonator was fabricated with ultra-high purity alumina from Coorstek (AD998 PlasmaPure). This material has a relatively high refractive index of 3.1 at microwave frequencies and very small dielectric losses. Dielectric losses are important as they ultimately limit the quality factor of the resonator. The material that we use has a nominal loss factor of tan(δ) = 2.5 × 10−5 at 6GHz, which results in Qm = 44,000. For our device, we put 14 Bragg mirror pairs at each side, which corresponds to Qw = 210,000. Therefore, the total Q factor is expected to be 36,000. Fabrication was done with standard computer-numerical-control (CNC) milling techniques, with a nominal positional accuracy of ~10 μm for each hole.

The fabricated structure is shown in Fig. 2(a)
Fig. 2 (a) Images of the fabricated alumina cavity with slant sections at both ends to facilitate coupling to metallic waveguides. (b) Setup for transmittance measurement. (c) Coupling components between the metallic waveguide and the dielectric rod.
, and the setup used for our two-port transmittance measurements in Fig. 2(b). The device was connected to a network analyzer via conventional WR62 microwave waveguides placed at each end. Because of the geometrical mismatch between the metallic waveguide (15.8mm × 7.9mm) and our dielectric waveguide resonator (4.72mm × 9.43mm), we designed slant sections at both ends of the resonator to allow partial insertion into the WR62 waveguide and thereby facilitate coupling [Fig. 2(c)]. However, the coupling efficiency of the experimental setup is still limited by the coupling between the metallic waveguide and the dielectric waveguide. In Fig. 3(a)
Fig. 3 (a) Amplitude spectrum with a large frequency range from 15GHz to 18GHz, showing the bandgap of the structure. (b) Amplitude and phase spectra of the cavity mode from the transmittance measurement. The dashed curves are fitted with a Fano resonance model.
, we show the amplitude spectrum from the transmission measurement. It demonstrates a large bandgap ranging from 15.4GHz to 17.7GHz, which is consistent with theory.

The transmission coefficient of the cavity mode can be generally fitted using a Fano model [22

22. M. Galli, S. L. Portalupi, M. Belotti, L. C. Andreani, L. O'Faolain, and T. F. Krauss, “Light scattering and Fano resonances in high-Q photonic crystal nanocavities,” Appl. Phys. Lett. 94(7), 071101 (2009). [CrossRef]

,23

23. Lj. Babić and M. J. A. de Dood, “Interpretation of Fano lineshape reversal in the reflectivity spectra of photonic crystal slabs,” Opt. Express 18(25), 26569–26582 (2010). [CrossRef] [PubMed]

]. Unlike most photonic crystal cavity experiments at optical frequencies, here both amplitude and phase of the transmittance can be obtained. We plot the resulting spectra in Fig. 3(b), where the cavity resonance is clearly seen at 17.405GHz, in good agreement with the design value. We obtain a full-width half-maximum linewidth ΔωFWHM of 0.659MHz, corresponding to a Q factor of 26,400, which is slightly smaller than the expected Q value of 36,000. The observed discrepancy of Q factor can be attributed to higher dielectric losses in the bulk material at this frequency, to fabrication tolerances on the holes position and size and to resonant losses at the transition between waveguide and dielectric. In general, a higher Q factor can be obtained with less lossy materials or at cryogenic temperatures.

Next, we insert a small antenna inside the cavity in order to study the interaction between the cavity mode and a radiating dipole. The antenna is constructed by stripping both the outer conductor and the dielectric core of a coaxial cable (Megaphase Corp., model ClearPath-A06), exposing the inner conductor for a total length of 9 mm. The relatively small diameter (0.305 mm) of the inner conductor was chosen to minimize any perturbation of the cavity mode.

In Fig. 4(a)
Fig. 4 (a) Reflectance measurement of the center-fed antenna, with and without the cavity. (b) Reflectance depth as a function of the z-position. Also shown (dashed line), the simulated electric-field amplitude along the resonator.
, we plot the reflectance spectrum measured with a network analyzer connected to the above mentioned center-fed antenna, when this is placed in free space (red) and in the center hole of cavity (blue). A clearly visible reflectance dip with a 2.2dB depth at ~17.43GHz is observed, which we interpret as a convincing signature of the Purcell effect of the cavity. When the antenna is coupled to the cavity mode, the radiation rate is enhanced, and thus the load impedance is modified. This results in a variation of the reflectance at the analyzer port, due to the impedance mismatch between the antenna load and the cable. From Fig. 4(a), we also note that there is a change in the resonant frequency of the system, from 17.40GHz to 17.43GHz. This is due to the small but non-negligible perturbation of the metallic antenna, which blue-shifts the resonance. The free-space background in Fig. 4(a) (red-solid curve) arises from the small reflectance between the network analyzer/cable connection, and the cable/antenna connection. This background prevents us from accurately quantifying the load impedance variation and subsequently extracting a numerical value for the Purcell enhancement factor and the perturbed Q factor.

Next, we scanned the antenna in z-direction along the length of the dielectric rod to probe the electric field distribution of the cavity mode. The height of the antenna and the distance between the antenna and the photonic crystal (0.5 mm) were kept constant during the measurements. This is analogous to near-field scanning optical microscopy (NSOM) [24

24. U. Dürig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59(10), 3318–3327 (1986). [CrossRef]

,25

25. L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B 76(4), 041102 (2007). [CrossRef]

]. In Fig. 4(b), we plot the measured reflectance depth as a function of the z-position (blue-solid). In comparison, we also plot the electric field amplitude (black-dash). It can be seen that the two curves follow the same z-dependence, which verifies that the S11 dip we observe is due to coupling to the cavity mode.

4. Summary

In summary, a high-Q microwave resonator with a mode volume smaller than one cubic wavelength has been designed and fabricated. A record high Quality factor (Q = 26,400) for photonic crystal cavities at microwave frequencies has been measured at room temperature. In addition to its ultra-high Q/V factor, the cavity is uniquely designed to have its electric field concentrated in air. Coupling to this cavity mode is enhanced by the Purcell factor.

The photonic crystal cavity that we demonstrated has unique advantages when compared to other types of microwave cavities such as sapphire ring resonators. First of all, it supports a mode confined in the air region. Moreover, due to the integrated dielectric waveguides better coupling efficiencies can be realized. Finally, it has a very small mode volume on order of one cubic wavelength. We believe that this device is well suited for conducting microwave cavity QED experiments and for developing a variety of novel microwave devices.

Acknowledgments

Y. Zhang would like to dedicate this paper to Mughees Khan and his wife. The authors would also like to acknowledge the generous support from Schlumberger-Doll Research Center at Cambridge, Massachusetts.

References and links

1.

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). [CrossRef] [PubMed]

2.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67(24), 3380–3383 (1991). [CrossRef] [PubMed]

3.

B. Temelkuran, E. Ozbay, J. P. Kavanaugh, G. Tuttle, and K. M. Ho, “Resonant cavity enhanced detectors embedded in photonic crystals,” Appl. Phys. Lett. 72(19), 2376–2378 (1998). [CrossRef]

4.

M. A. Shapiro, W. J. Brown, I. Mastovsky, J. R. Sirigiri, and R. J. Temkin, “17 GHz photonic bandgap cavity with improved input coupling,” Phys. Rev. Spec Top. Accelerators Beams 4, 042201 (2001).

5.

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, “Demonstration of a 17-GHz, high-gradient accelerator with a photonic-band-gap structure,” Phys. Rev. Lett. 95(7), 074801 (2005). [CrossRef] [PubMed]

6.

E. Di Gennaro, S. Savo, A. Andreone, V. Galdi, G. Castaldi, V. Pierro, and M. R. Masullo, “Mode confinement in photonic quasicrystal point-defect cavities for particle accelerators,” Appl. Phys. Lett. 93(16), 164102 (2008). [CrossRef]

7.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region,” Phys. Rev. B 61(16), 10762–10767 (2000). [CrossRef]

8.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]

9.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004). [CrossRef] [PubMed]

10.

A. Petosa, A. Ittipiboon, Y. M. M. Antar, D. Roscoe, and M. Cuhaci, “Recent advances in dielectric-resonator antenna technology,” IEEE Antennas Propag. Mag. 40(3), 35–48 (1998). [CrossRef]

11.

M. J. R. Previte and C. D. Geddes, “Fluorescence microscopy in a microwave cavity,” Opt. Express 15(18), 11640–11649 (2007). [CrossRef] [PubMed]

12.

Y. Zhang, M. W. McCutcheon, I. B. Burgess, and M. Loncar, “Ultra-high-Q TE/TM dual-polarized photonic crystal nanocavities,” Opt. Lett. 34(17), 2694–2696 (2009). [CrossRef] [PubMed]

13.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997). [CrossRef]

14.

A. R Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 12084–12089 (2008). [CrossRef]

15.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009). [CrossRef]

16.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, and M. Loncar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010). [CrossRef]

17.

Rsoft Inc, Rsoft Fullwave v6.0.2, http://www.rsoftdesign.com.

18.

Y. Zhang and M. Loncar, “Ultra-high quality factor optical resonators based on semiconductor nanowires,” Opt. Express 16(22), 17400–17409 (2008). [CrossRef] [PubMed]

19.

P. Lalanne and J. P. Hugonin, ““Bloch-wave engineering for high-Q, small-V microcavities,” IEEE. J. Quantum Electron. 39(11), 1430–1438 (2003). [CrossRef]

20.

J. D. Joannopoulos and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

21.

S. Fan, “Photonic crystal theory: temporal coupled-mode formalism,” in Optical Fiber Communications V A: Components and Subsystems, I. P. Kaminow, T. Li, and A. E. Willner, eds. (Elsevier 2008), Vol. 1, pp. 431–454.

22.

M. Galli, S. L. Portalupi, M. Belotti, L. C. Andreani, L. O'Faolain, and T. F. Krauss, “Light scattering and Fano resonances in high-Q photonic crystal nanocavities,” Appl. Phys. Lett. 94(7), 071101 (2009). [CrossRef]

23.

Lj. Babić and M. J. A. de Dood, “Interpretation of Fano lineshape reversal in the reflectivity spectra of photonic crystal slabs,” Opt. Express 18(25), 26569–26582 (2010). [CrossRef] [PubMed]

24.

U. Dürig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59(10), 3318–3327 (1986). [CrossRef]

25.

L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B 76(4), 041102 (2007). [CrossRef]

OCIS Codes
(230.5750) Optical devices : Resonators
(350.4010) Other areas of optics : Microwaves
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: March 9, 2011
Revised Manuscript: April 22, 2011
Manuscript Accepted: April 25, 2011
Published: April 28, 2011

Citation
Yinan Zhang, Irfan Bulu, Wai-Ming Tam, Ben Levitt, Jagdish Shah, Tancredi Botto, and Marko Loncar, "High-Q/V air-mode photonic crystal cavities at microwave frequencies," Opt. Express 19, 9371-9377 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9371


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References

  1. 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). [CrossRef] [PubMed]
  2. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67(24), 3380–3383 (1991). [CrossRef] [PubMed]
  3. B. Temelkuran, E. Ozbay, J. P. Kavanaugh, G. Tuttle, and K. M. Ho, “Resonant cavity enhanced detectors embedded in photonic crystals,” Appl. Phys. Lett. 72(19), 2376–2378 (1998). [CrossRef]
  4. M. A. Shapiro, W. J. Brown, I. Mastovsky, J. R. Sirigiri, and R. J. Temkin, “17 GHz photonic bandgap cavity with improved input coupling,” Phys. Rev. Spec Top. Accelerators Beams 4, 042201 (2001).
  5. E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, “Demonstration of a 17-GHz, high-gradient accelerator with a photonic-band-gap structure,” Phys. Rev. Lett. 95(7), 074801 (2005). [CrossRef] [PubMed]
  6. E. Di Gennaro, S. Savo, A. Andreone, V. Galdi, G. Castaldi, V. Pierro, and M. R. Masullo, “Mode confinement in photonic quasicrystal point-defect cavities for particle accelerators,” Appl. Phys. Lett. 93(16), 164102 (2008). [CrossRef]
  7. C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region,” Phys. Rev. B 61(16), 10762–10767 (2000). [CrossRef]
  8. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001). [CrossRef]
  9. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004). [CrossRef] [PubMed]
  10. A. Petosa, A. Ittipiboon, Y. M. M. Antar, D. Roscoe, and M. Cuhaci, “Recent advances in dielectric-resonator antenna technology,” IEEE Antennas Propag. Mag. 40(3), 35–48 (1998). [CrossRef]
  11. M. J. R. Previte and C. D. Geddes, “Fluorescence microscopy in a microwave cavity,” Opt. Express 15(18), 11640–11649 (2007). [CrossRef] [PubMed]
  12. Y. Zhang, M. W. McCutcheon, I. B. Burgess, and M. Loncar, “Ultra-high-Q TE/TM dual-polarized photonic crystal nanocavities,” Opt. Lett. 34(17), 2694–2696 (2009). [CrossRef] [PubMed]
  13. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997). [CrossRef]
  14. A. R Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 12084–12089 (2008). [CrossRef]
  15. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009). [CrossRef]
  16. Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, and M. Loncar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010). [CrossRef]
  17. Rsoft Inc, Rsoft Fullwave v6.0.2, http://www.rsoftdesign.com .
  18. Y. Zhang and M. Loncar, “Ultra-high quality factor optical resonators based on semiconductor nanowires,” Opt. Express 16(22), 17400–17409 (2008). [CrossRef] [PubMed]
  19. P. Lalanne and J. P. Hugonin, ““Bloch-wave engineering for high-Q, small-V microcavities,” IEEE. J. Quantum Electron. 39(11), 1430–1438 (2003). [CrossRef]
  20. J. D. Joannopoulos and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).
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