## Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs

Optics Express, Vol. 12, Issue 17, pp. 3988-3995 (2004)

http://dx.doi.org/10.1364/OPEX.12.003988

Acrobat PDF (551 KB)

### Abstract

A series of microcavities in 2D hexagonal lattice photonic crystal slabs are studied in this paper. The microcavities are small sections of a photonic crystal waveguide. Finite difference time domain simulations show that these cavities preserve high *Q* modes with similar geometrical parameters and field profile. Effective modal volume is reduced gradually in this series of microcavity modes while maintaining high quality factor. Vertical *Q* value larger than *10 ^{6}
* is obtained for one of these cavity modes with effective modal volume around

*5.40*cubic half wavelengths [(

*λ/2n*)

_{slab}*]. Another cavity mode provides even smaller modal volume around*

^{3}*2.30*cubic half wavelengths, with vertical

*Q*value exceeding10

^{5}.

© 2004 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059 (1987) [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486 (1987) [CrossRef] [PubMed]

4. A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends”, Phys. Rev. B **58**, 4809 (1998) [CrossRef]

5. T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths”, Nature **383**, 699 (1996) [CrossRef]

6. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B , **60**, 5751 (1999) [CrossRef]

*Q*) and small modal volume (

*V*), which make them potentially useful not only in miniaturized photonic devices but also in some quantum optical devices. The former applications include channel drop filter for the wavelength division multiplexer system, high efficiency light emission diodes and low threshold lasers [7

7. S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. **80**, 960 (1998) [CrossRef]

11. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” S. M. , Nature **415**, 621–623 (2002) [CrossRef]

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

13. C. Santori, D. Fattal, J. Vučković, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature **419**, 594 (2002) [CrossRef] [PubMed]

*Q*and small modal volume

*V*. The ratio

*Q*/

*V*is a measure of the strength of various cavity interactions and should be as large as possible.

*Q*value (

*Q*) can be separated into two parts, namely the in-plane

_{tot}*Q*value (

*Q*

*) and the vertical*

_{⊲}*Q*value (

*Q*) by a simple relation

_{⊥}*Q*value (

*Q*). The vertical confinement is due to the standard waveguiding by total internal reflection. In momentum (

_{⊲}*) space analysis, vertical radiation losses occur when the cavity modes have in-plane momentum components (*

**k***) that lie within a region called ‘light cone’ [14*

**k**_{⊲}14. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express **10**, 670 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670 [CrossRef] [PubMed]

15. J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. **38**, 850 (2002) [CrossRef]

*Q*value.

*Q*microcavity in 2D PCS’s is to reduce the

*field components that lie within the light cone and thus enhance the vertical confinement. One group has achieved this by carefully modifying the air hole radii into a graded pattern [16*

**k**_{⊲}16. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express **11**, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

*Q*larger than

_{⊥}*10*. Another group has come up with a design of simpler structure [17

^{5}17. V. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003) [CrossRef] [PubMed]

*0.15 a*, where

*a*is the lattice constant. Experiments have shown that the cavity supports a resonant mode with a high quality factor

*Q*=

*4.5*×

*10*and a small modal volume

^{4}*V*=

*7.0*×

*10*. Recently a group has reported a hexapole cavity mode with

^{-14}cm^{3}*Q*larger than

*10*and small modal volume on the order of cubic wavelengths in material [18

^{6}18. H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. **83**, 4294 (2003) [CrossRef]

*Q*hexapole mode is very sensitive to its surrounding geometrical parameters, which might bring up difficulties for fabrication.

17. V. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003) [CrossRef] [PubMed]

*Q*value varies. Contrary to the acceptor mode in Ref. [17

17. V. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003) [CrossRef] [PubMed]

19. E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. **67**, 3380 (1991) [CrossRef] [PubMed]

*Q*value and effective modal volume. In section 3, we begin with a three missing hole cavity (named as an M3 cavity). We first modify its surrounding geometrical parameters to achieve highest

*Q*value for this mode and then calculate its modal volume. Next, we run the same procedures on two missing hole cavities (M2), one missing hole cavities (M1) and finally cavities with no missing hole at all (M0). In section 4, we give a brief comparison of the cavity modes.

## 2. Calculation methods

*a*×23

*a*for the entire cavity modes calculated in this paper, which is capable of providing sufficiently large

*Q*so that

_{⊲}*Q*is primarily limited by

_{tot}*Q*. The grid size is

_{⊥}*0.05a*in

*x*,

*z*direction and

*0.0433a*in

*y*direction. Perfect matched layers are used as absorbing boundaries [21

21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185 (1994) [CrossRef]

*Q*value of the cavity modes is calculated using a combination of FDTD techniques and Padé approximation with Baker’s algorithm [22–23

_{tot}23. W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,” IEEE Microwave Wireless Components Lett. **11**, 223 (2001) [CrossRef]

*Q*value using a much smaller number of time steps compared to other methods (typically <

*0.5%*numerical errors and ~

*15,000*time steps for

*Q*>

*10*). For comparison reasons, some high

^{5}*Q*values of certain cavity modes are also calculated using conventional method:

*f*is the resonant frequency of the cavity mode,

_{0}*U*is the time averaged energy stored in the cavity, and

*P*is the power loss.

*Q*and

_{⊥}*Q*are calculated separately using this method.

_{⊲}24. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science **284**, 1819(1999) [CrossRef] [PubMed]

*ε*(

*x*,

*y*,

*z*) is the spatial distribution of dielectric constant and

## 3. Waveguide-section microcavities

### 3.1 Three missing hole waveguide-section cavity (M3)

*1.55 µm*. The thickness of the slab is

*0.7a*, where

*a*is the lattice constant. The radius of the air holes is set to be

*R=0.29a*. These two parameters are also the same for M2, M1 and M0 cavities.

*components of the cavity mode that lie inside the leaky region in order to achieve high*

**k**_{⊲}*Q*value. The first step we take is similar to that described in Ref. [16

16. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express **11**, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

*=0 (DC) are eliminated. Since all the cavities in this paper support multiple modes, we confine our discussion only to the donor modes whose*

**k**_{⊥}*field distribution is even in both*

**H**_{𝑧}*x*and

*y*direction. These modes, as we shall see later, can achieve extremely high Q and ultra small modal volume when the waveguide section is well shortened.

**425**, 944 (2003) [CrossRef] [PubMed]

*R*to improve the

_{1}*Q*value.

*d*and

*R*, we find the highest

_{1}*Q*value

*8,000*can be achieved at

*d*=

*0.24a*and

*R*=

_{1}*0.23a*. Figure 2(a) shows the

*field distribution of the M3 mode. The field is spread out into the surrounding lattice of the cavity and since the resonant frequency*

**H**_{𝑧}*0.3077 a/λ*is near to the upper bandedge (~

*0.32 a/λ*), the in-plane quality factor

*Q*is limited to

_{⊲}*35,000*even for a large size (

*23a*×

*23a*).

*space analysis, we take the 2D (*

**k***,*

**k**_{x}*) Fourier transform of*

**k**_{y}*E*,

_{x}*E*and

_{y}*H*,

_{x}*H*field distribution in the air layer one FDTD grid cell above the slab surface, denoted by

_{y}*FT*

_{2}(

*E*),

_{x}*FT*

_{2}(

*E*),

_{y}*FT*

_{2}(

*H*) and

_{x}*FT*

_{2}(

*H*), respectively. The total radiated power

_{y}*P*of the cavity mode can be estimated by the following equations [15

15. J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. **38**, 850 (2002) [CrossRef]

*I*is shown in Fig. 2(b). We can see that there is still a considerable amount of

*component lying inside the light cone, which limits*

**k**_{⊲}*Q*to about

_{⊥}*10,000*. Using Eqs. (1) and (2), the modal volume

*V*is calculated to be

*7.90*(

*λ/2n*)

_{slab}*for the M3 mode.*

^{3}**425**, 944 (2003) [CrossRef] [PubMed]

*Q*=

_{tot}*9.91*×

*10*at resonant frequency

^{4}*0.2618 a/λ*when

*d*=

*0.2a*and slab thickness

*t*=

*0.6a*, according to our simulations. For

*d*=

*0.2a*and

*t*=

*0.7a*,

*Q*equals

_{tot}*2.16*×

*10*with resonant frequency

^{4}*0.2549 a/λ*. However, the donor mode becomes interesting when the waveguide-section is further shortened. Next, we study cavities with only two air holes removed but its surrounding geometrical structures retained.

### 3.2 Two missing hole waveguide-section cavity (M2)

*d*=

*0.23a*and

*R*=

_{1}*0.2a*.

*Q*is calculated to be

_{tot}*5.4*×

*10*, with

^{4}*Q*=

_{⊥}*6.0*×

*10*and

^{4}*Q*=

_{⊲}*4.5*×

*10*. Both

^{5}*Q*and

_{⊥}*Q*are much improved compared to the previous case. The former improvement is due to the reduced

_{⊲}*space components inside the leaky region, as shown in Fig. 3(c) and the latter improvement is due to the reduced resonant frequency at*

**k***0.2955 a/λ*, which is now lying closer to the middle of the bandgap.

*9.52*(

*λ/2n*)

_{slab}*.*

^{3}### 3.3 One missing hole waveguide-section cavity (M1)

*d*and

*R*to search for the setup of the highest

_{1}*Q*donor mode for the M1 cavity. We find when

*d*is tuned to

*0.21a*and

*R*to

_{1}*0.22a*,

*Q*=

_{tot}*5.85*×

*10*can be achieved at resonant frequency

^{5}*0.2902 a/λ*. Since M1 cavity is more compact, we vary more geometrical parameters hoping to improve the

*Q*value. We find that by modifying the radius of four air holes from

*R*=

*0.29a*to

*R*=

_{2}*0.25a*[also shown in Fig. 4(a)], we can largely enhance the

*Q*value for the M1 mode to

*Q*≈

_{tot}*Q*=

_{⊥}*1.06*×

*10*and

^{6}*Q*>

_{⊲}*2*×

*10*. The Resonant frequency is further reduced to

^{7}*0.2881 a/λ*and modal volume reduced to

*5.40*(

*λ/2n*)

_{slab}^{3}.

*Q*value when

*d*and

*R*are tuned to around

_{1}*0.22a*. We can conclude that tuning

*d*to around

*0.22a*allows light to penetrate deeper inside the surrounding lattice in the

*x*direction and be reflected gently. Similarly, tuning

*R*(and also

_{1}*R*for M1 cavity) allows the reflections in the

_{2}*y*direction to be smoother. The combined effect is that the spatial variation of the envelope function is tailored into a Gaussian profile so that the

*components inside the light cone are reduced and thus*

**k**_{⊲}*Q*improved. On the other hand, increasing

_{⊥}*d*and decreasing

*R*helps pull down the resonant frequency closer to the middle of the bandgap [14

_{1}14. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express **10**, 670 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670 [CrossRef] [PubMed]

*Q*to increase.

_{⊲}### 3.4 Zero missing hole waveguide-section cavity (M0)

*x*direction by a distance of

*d*and the radius of the two air holes nearest to the cavity center in the

*y*direction (labeled ‘B’) is modified to

*R*. After careful scanning, we find that

_{1}*d*=

*0.14a*and

*R*=

_{1}*0.27a*gives the highest

*Q*. We also modified other geometric parameters, such as the radius of the ‘A’ holes and the ‘C’ holes, but this only reduces the

*Q*value of the M0 mode, so we have skipped them in this paper. The final

*Q*value is

*1.35*×

*10*with resonant frequency

^{5}*0.2888 a/λ*.

*field distribution is shown in Fig. 5(b). The*

**H**_{z}*field profile is now very simple and the number of the nodal points is much reduced. Figure 5(c) shows the*

**H**_{z}*space electric intensity profile. There are very few*

**k***components inside the leaky region, which accounts for a large*

**k**_{⊲}*Q*(≈

_{⊥}*1.9*×

*10*) of this mode.

^{5}*2.30*cubic half wavelengths. For a lattice spacing of

*450 nm*, which corresponds to resonant wavelength of

*1564 nm*for the M0 mode, the modal volume is only

*2.76*×

*10*

^{-14}*cm*.

^{3}*d*=

*0.14a*and

*R*=

_{1}*R*=

*0.29a*, that is, we only move ‘A’ holes outward by

*0.14a*, without changing any other surrounding lattice. The modal volume is 2.25 cubic half wavelengths, with

*Q*=

_{⊥}*1.12*×

*10*. Since only the position of two air holes needs to be changed for this modified M0 cavity, without affecting any air hole radius, it is relatively easier to fabricate than other cavities discussed above.

^{5}*Q*value of the other modes supported by the M2, M1 and M0 cavity. The donor modes discussed in this paper have the highest

*Q*value for each cavity respectively.

## 4. Comparisons

*x*direction are pulled closer to each other. The number of nodal points becomes smaller and the field gets more compressed in both

*x*and

*y*directions, leading to reduced modal volumes.

*Q*value and the effective modal volume of the four modes. We can clearly see that the M1 mode offers the highest

_{tot}*Q*value and the M0 mode has the smallest modal volume while keeping a relatively large

*Q*. Figure 7(b) compares the

*Q*and the radiation factor

_{z}*RF*of the four modes.

*P*is calculated using equation (3) and (4).

*W*is the total energy stored in the cavity [15

15. J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. **38**, 850 (2002) [CrossRef]

*Q*and

_{⊥}*RF*in Fig. 7(b).

## 5. Summary

*Q*factor larger than

*10*is obtained for the M1 mode by carefully tuning some structural parameters. Modal volume as small as

^{6}*2.30*cubic half wavelengths is achieved for the M0 mode. The profile of the M0 mode is simple, yet with

*Q*factor exceeding

*10*. We believe that M0 and M1 cavities are well suited for large-scale photonic integration and single-photon sources for quantum optics operations.

^{5}## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. |

4. | A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends”, Phys. Rev. B |

5. | T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths”, Nature |

6. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B , |

7. | S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. |

8. | M. Qiu, “Ultra-compact optical filter in two-dimensional photonic crystal,” Electron. Lett. |

9. | M. Qiu and B. Jaskorzynska, “A design of a channel drop filter in a two-dimensional triangular photonic crystal”, Appl. Phys. Lett. |

10. | S. Fan, Proceedings of the SPIE , |

11. | S. M. Spillane, T. J. Kippenberg, and K. J. Vahala “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” S. M. , Nature |

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

13. | C. Santori, D. Fattal, J. Vučković, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature |

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

15. | J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. |

16. | K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express |

17. | V. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

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

19. | E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. |

20. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation,” |

21. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

22. | M. Qiu, “High Q cavities in photonic crystal slabs: determining resonant frequency and quality factor accurately,” submitted for publication (2004) |

23. | W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,” IEEE Microwave Wireless Components Lett. |

24. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science |

25. | J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E. |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 12, 2004

Revised Manuscript: August 9, 2004

Published: August 23, 2004

**Citation**

Ziyang Zhang and Min Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express **12**, 3988-3995 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-3988

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### References

- E. Yablonovitch, ཿInhibited Spontaneous Emission in Solid-State Physics and Electronics,ཿ Phys. Rev. Lett. 58, 2059 (1987) [CrossRef] [PubMed]
- S. John, ཿStrong localization of photons in certain disordered dielectric superlattices,ཿ Phys. Rev. Lett. 58, 2486 (1987) [CrossRef] [PubMed]
- H. Benisty, ཿModal analysis of optical guides with two-dimensional photonic band-gap boundaries,ཿ J. Appl. Phys. 75, 4753 (1994)
- A. Mekis, S. Fan, and J. D. Joannopoulos, ཿBound states in photonic crystal waveguides and waveguide bendsཿ, Phys. Rev. B 58, 4809 (1998) [CrossRef]
- T. F. Krauss, R. M. De La Rue, and S. Brand, ཿTwo-dimensional photonic-bandgap structures operating at near-infrared wavelengthsཿ, Nature 383, 699 (1996) [CrossRef]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, ཿGuided modes in photonic crystal slabs,ཿ Phys. Rev. B, 60, 5751 (1999) [CrossRef]
- S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, ཿChannel Drop Tunneling through Localized States,ཿ Phys. Rev. Lett. 80, 960 (1998) [CrossRef]
- M. Qiu, ཿUltra-compact optical filter in two-dimensional photonic crystal,ཿ Electron. Lett. 40, 539 (2004) [CrossRef]
- M. Qiu and B. Jaskorzynska, ཿA design of a channel drop filter in a two-dimensional triangular photonic crystalཿ, Appl. Phys. Lett. 83, 1074 (2003). [CrossRef]
- S. Fan, Proceedings of the SPIE, v 3002, 1997, p 67-73 [CrossRef]
- Spillane, S. M., Kippenberg, T. J. and Vahala, K. J. ཿUltralow-threshold Raman laser using a spherical dielectric microcavity,ཿ S. M. , Nature 415, 621-623 (2002) [CrossRef]
- C. Santori, D. Fattal, J. Vu¿¡kovi¿ , G. S. Solomon, and Y. Yamamoto, ཿIndistinguishable photons from a single-photon device,ཿ Nature 419, 594 (2002) [CrossRef] [PubMed]
- K. Srinivasan and O. Painter, ཿMomentum space design of high-Q photonic crystal optical cavities,ཿ Opt. Express 10, 670 (2002), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a> [CrossRef] [PubMed]
- J. Vu¿¡kovi¿ , M. lon¿ar, H. Mabuchi and A. Scherer, ཿOptimization of the Q factor in photonic crystal microcavities,ཿ IEEE J. of Quantum Electron. 38, 850 (2002) [CrossRef]
- K. Srinivasan and O. Painter, ཿFourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,ཿ Opt. Express 11, 579 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579</a> [CrossRef] [PubMed]
- V. Akahane, T. Asano, B. S. Song, and S. Noda, ཿHigh-Q photonic nanocavity in a two-dimensional photonic crystal,ཿ Nature 425, 944 (2003) [CrossRef] [PubMed]
- H. Y. Ryu, M. Notomi, and Y. H. Lee, ཿHigh-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,ཿ Appl. Phys. Lett. 83, 4294 (2003) [CrossRef]
- E.Yablonovitch and T. J. Gmitter, ཿDonor and acceptor modes in photonic band structure,ཿ Phys. Rev. Lett. 67, 3380 (1991) [CrossRef] [PubMed]
- K. S. Yee, ཿNumerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,ཿ IEEE Trans. Antennas and Propagation, 14, 302 (1966)
- J. P. Berenger, ཿA perfectly matched layer for the absorption of electromagnetic waves,ཿ J. Comput. Phys. 114, 185 (1994) [CrossRef]
- M. Qiu, ཿHigh Q cavities in photonic crystal slabs: determining resonant frequency and quality factor accurately,ཿ submitted for publication (2004)
- W. H. Guo, W. J. Li, and Y. Z. Huang, ཿComputation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,ཿ IEEE Microwave Wireless Components Lett. 11, 223 (2001) [CrossRef]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. OཿBrien, P. D. Dapkus, and I. Kim, ཿTwo-Dimensional Photonic Band-Gap Defect Mode Laser,ཿ Science 284, 1819 (1999) [CrossRef] [PubMed]
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