## Ultra high-Q photonic crystal nanocavity design: The effect of a low-ε slab material

Optics Express, Vol. 16, Issue 7, pp. 4972-4980 (2008)

http://dx.doi.org/10.1364/OE.16.004972

Acrobat PDF (476 KB)

### Abstract

We analyze the influence of the dielectric constant of the slab on the quality factor (*Q*) in slab photonic crystal cavities with a minimized vertical losses model. The higher value of *Q* in high-ε cavity is attributed to the lower mode frequency. The *Q* ratio in a high-ε (silicon) vs. low-ε (diamond) slab is examined as a function of mode volume (*V _{m}
*). The mode volume compensation technique is discussed. Finally, diamond cavity design is addressed. The analytical results are compared to 3D FDTD calculations. In a double heterostructure design, a

*Q*≈

*2.6×10*is obtained. The highest

^{5}*Q*≈1.3×10

^{6}with

*V*=

_{m}*1.77×(λ/n)*in a local width modulation design is derived.

^{3}© 2008 Optical Society of America

## 1. Introduction

2. A. D. Greentre, J. Salzman, S. Prawer, and L. C. Hollenberg, “Quantum gate for Q-switching photonic bandgap cavities containing two level atoms” Phys. Rev. A **73**, 013818 (2006). [CrossRef]

3. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express **14**, 3556 (2006). [CrossRef] [PubMed]

*Q*≈

*7*×

*10*[4

^{4}4. I. Bayn and J. Salzman, “High-Q photonic crystal nanocavities on diamond for Quantum Electrodynamics,” Eur. Phys. J. Appl. Phys. **37**, 19–24 (2007). [CrossRef]

7. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nat. Mater. **4**, 207 (2005). [CrossRef]

*Q*) raises the question: what is the physical mechanism responsible for the dielectric constant influence on the

*Q*value in a photonic crystal cavity?

5. C. F. Wang, R. Hanson, D. D. Awschalom, E. L. Hu, T. Feygelson, J. Yang, and J. E. Butler, “Fabrication and characterization of two-dimensional photonic crystal microcavities in nanocrystalline diamond,” Appl. Phys. Lett. **91**, 201112 (2007). [CrossRef]

*10*) is one order of magnitude lower than the predicted value. This demonstrates that high-Q PC realization requires single-crystalline diamond implementation, where scattering is minimized. Recently, we have reported [10,11] the first single crystalline PC fabrication for optical characterization. This process will allow realization of higher

^{3}*Q*in diamond. Thus, further effort in high-Q cavity design on diamond is justified. In addition, more deep insight in the influence of a low dielectric constant on cavity quality factor for low-ε materials is required. Thus, the main goals of the current article are to improve design for a high-

*Q*, low mode volume cavity in diamond, and to uncover the physical basis for the influence of ε on

*Q*.

8. D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express **12**, 5961 (2005). [CrossRef]

4. I. Bayn and J. Salzman, “High-Q photonic crystal nanocavities on diamond for Quantum Electrodynamics,” Eur. Phys. J. Appl. Phys. **37**, 19–24 (2007). [CrossRef]

*Q*for both slabs, is introduced. A simple rule for the required mode volume increase for different

*ε*’s is derived. In Section 3, we present 3D-FDTD analysis of PC cavities in diamond. We start with applying the mode volume compensation to previously designed DH cavities, reaching a significant improvement in

*Q*(

*Q*≈

*2.6*×

*10*), with a mode volume

^{5}*V*≈

_{m}*1.78*×

*(λ/n)*. Then, the limitations of this technique are discussed. Finally, we calculate local width modulated cavities and demonstrate the highest reported quality factor in diamond

^{3}*Q*≈

*1.3*×

*10*with

^{6}*V*=

_{m}*1.77*×

*(λ/n)*. A detailed comparison of these two methods to the results of Section 2 is given. Quantum Information Applications are discussed.

^{3}## 2. The influence of a low ε: Semi-Analytical approach

^{st}and the 2

^{nd}TE-like modes in silicon [7–9

7. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nat. Mater. **4**, 207 (2005). [CrossRef]

3. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express **14**, 3556 (2006). [CrossRef] [PubMed]

*Q*(

_{l}*Q*) for the lateral (vertical) losses [8

_{v}8. D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express **12**, 5961 (2005). [CrossRef]

*Q*is determined by the number of PC periods around the cavity and its increase leads to an unbound increase in

_{l}*Q*. Thus, by embedding the cavity in a sufficiently large PC, the value of

_{l}*Q*can be made arbitrarily large, and the total quality factor is limited only by

_{l}*Q*. In the following analysis we assume that

_{v}*Q*=

*Q*. (This equality requires an increased number of PC periods around the cavity for the low-ε material, compared to that of the high-ε one [12

_{v}12.
In the waveguide based cavities, the number of PC periods required for a similar *Q _{l}* in slabs of different ε’s is different in the

*x*and

*z*directions (different confinement mechanism). Qualitatively, a diamond-based PC will reach a

*Q*similar to that of Si with

_{l}*1.5*–

*2*times more PC periods than that of a Si-based PC, In the x - z direction, respectively.

### 2.1 Optimized k-distribution: Ideal Gaussian Envelope cavity [88. D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express **12**, 5961 (2005). [CrossRef]

]

**12**, 5961 (2005). [CrossRef]

7. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nat. Mater. **4**, 207 (2005). [CrossRef]

9. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

*y*=

*0*plane, the waveguide field is defined by (

*E*), forming TE polarization. The cavity design is based on the lowest waveguide with odd (even) symmetry of the

_{x}, E_{z}, H_{y}*H*along

_{y}*x*(

*z*) axis. The vertically radiated power losses are given by [8

**12**, 5961 (2005). [CrossRef]

*η*=

*(µ/ε)*,

^{1/2}*λ*is the cavity mode wavelength,

_{0}*k*=

*2π/λ*=

_{0}*[k*+

^{2}_{l}*k*,

^{2}_{y}]*=*

**k**_{l}*(k*=

_{x}k_{z})*k(sinθcosφ sinθsinφ)*and

*k*=

_{y}*kcosθ*. Note that

*k*<

_{l}*k*defines the light cone (see Fig. 1 inset).

**12**, 5961 (2005). [CrossRef]

*σ*and

_{x}*σ*are the modal widths in real space in the

_{z}*x*and

*z*directions respectively, and (±

*k*±

_{x0}*k*) are the J points coordinates given by

_{z0}*k*=

_{x0}*π/a*and

*k*=

_{z0}*2π/√3a*.

*Q*cavity field to compare different designs in materials with different dielectric constant.

### 2.2 Quality factor vs. high mode volume

*J*points, i.e.

*ω*, where

_{0}≈ω_{w}*ω*and

_{0}*ω*are the cavity and the waveguide edge frequencies. Since materials with higher

_{w}*ε*will lead to a lower

*ω*, resulting in the lower light cone (

_{w}*k*=

*ω*) [3

_{0}/c3. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express **14**, 3556 (2006). [CrossRef] [PubMed]

*k*-components crossing the light cone boundary decreases

*exponentially*with

*ω*. As a result, the vertical losses in a slab with a high-

_{0}*ε*are substantially lower than those in a low-

*ε*one, which explains the different

*Q*factors.

*Q*values for a similar cavity design in high-

*ε*and low-

*ε*materials, we present a calculation of the vertical quality factors ratio in silicon and diamond (

*Q*) as a function of the modal width in the

_{vSi}/Q_{vD}*x*direction (

*σ*). Note that the inverse problem design [8

_{x}**12**, 5961 (2005). [CrossRef]

*ε*materials. Therefore, in the following analysis we first assume:

*σ*=

_{xD}*σ*. The mode widths in the

_{xSi}*z*direction is defined by the PC bandgap confinement, and is obtained from the Gaussian fit to the best reported DH cavities in diamond [4

4. I. Bayn and J. Salzman, “High-Q photonic crystal nanocavities on diamond for Quantum Electrodynamics,” Eur. Phys. J. Appl. Phys. **37**, 19–24 (2007). [CrossRef]

5. C. F. Wang, R. Hanson, D. D. Awschalom, E. L. Hu, T. Feygelson, J. Yang, and J. E. Butler, “Fabrication and characterization of two-dimensional photonic crystal microcavities in nanocrystalline diamond,” Appl. Phys. Lett. **91**, 201112 (2007). [CrossRef]

*σ*=

_{zD}*1a*in diamond, and

*σ*=

_{zSi}*0.85a*in silicon.

*Q*) versus mode width in

_{vSi}/Q_{vD}*x*direction is presented. The

*σ*increase results in further localization of field distribution in the

_{x}*k*-space, thus decreasing vertical power losses through the light cone. Since the vertical power in

*k*-space decreases nearly as a Gaussian from the J points towards Γ, a similar increase in

*Q*as a function of

_{v}*σ*is expected for both silicon and diamond cavities. The fast increase in

_{x}*Q*displayed in Fig. 2 is understood since this ratio reflects the difference of |

_{vSi}/Q_{vD}*FT*| at the light cone edges in both materials. Qualitatively, the ratio of radiation losses will behave as

_{2}(H_{y})*exp[(k*, and increase nearly exponentially with the mode volume

^{2}_{Si}-k^{2}_{D})σ_{x}^{2}]*V*.

_{m}∝σ_{x}### 2.3 Mode volume compensation technique

*k*-space: for any cavity design in a high-

*ε*material, a similar design in a lower-

*ε*material can be adjusted to produce a similar

*Q*-value by a modest increase in the mode volume (

*V*).

_{m}∝σ_{x}*σ*of ideal Gaussian modes in cavities with slab refractive indices

_{xn}*n*are presented,

*versus*the mode-width of a Silicon based cavity (

*σ*), that provides

_{xSi}*the same value of Q*[13]. As seen in the figure, an accurate fit to the

*σ*-

_{xn}*σ*relation is given

_{xSi}*by σ*=

^{2}_{xn}*A*+

*B(σ*, where

_{xSi}-C)^{2}*A, B,*and

*C*are constants for each

*n*. Again, vertical power losses are dominated by the |

*FT*| value at the light cone edge (Eq. (1)), which is defined by the near J point Gaussian field (Eq. (2)). Therefore, by equating the field value in an arbitrary point on the light cone, a similar

_{2}(H_{y})*σ*-

_{xn}*σ*relation is derived. Moreover, for

_{xSi}*σ*>

_{xSi}*0.9a*, we can approximate:

*σ*, with the coefficients

_{xn}≈ασ_{xSi}+β*α*and

*β*depending on

*ε*only, since

*ε*defines both

*σ*and

_{z}*ω*.

_{0}*n*=

_{D}*2.4*):

*σ*≈

_{xD}*1.57σ*-

_{xSi}*0.51*. This

*phenomenological*result is important since it allows us to evaluate the increase in mode volume required in order to obtain a diamond PC cavity

*with the same vertical quality*factor as one with a similar design in silicon. As an upper bound we can state that for the same

*Q*, the mode width in the

_{v}*x*direction for the diamond cavity (

*σ*) should be ~

_{xD}*1.6*times larger than in silicon (

*σ*). Taking into account the mode width in

_{xSi}*z*direction, the upper bound for the mode volume in diamond (to obtain the same

*Q*-value) is nearly ~

_{v}*1.9*times higher than in silicon! (This result is consistent, but significantly extends a previous comparison [3

**14**, 3556 (2006). [CrossRef] [PubMed]

## 3. Waveguide based PC cavities in Diamond

**4**, 207 (2005). [CrossRef]

9. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

14. M. Qiu, “Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately,” Microwave Opt. Technol. Lett. **45**, 381–385 (2005). [CrossRef]

*24*×

*16*PC periods in

*xz*plane, with the height of

*2a*in the

*y*direction. This cell is surrounded by the PML and mirror boundaries are applied in all directions. The calculations were initially performed with the discretization of

*0.04a*[15

15. D. Englund and J Vuckovic, “A direct analysis of photonic nanostructures,” Opt. Express **14**, 3472 (2006). [CrossRef] [PubMed]

*0.03a*exhibited similar

*Q*. The results convergence for a larger structure exhibited similar

*Q*.

### 3.1 Modified Double Heterostructure (DH) design

*a*,

_{2}*a*elongated in the

_{21}*x*-direction (

*a*>

_{2}*a*>

_{21}*a*) into the one missing-hole waveguide [7

**4**, 207 (2005). [CrossRef]

*z*direction is unchanged. In this way, waveguide confinement is obtained in the

*x*direction and band-gap confinement in the

*z*direction. The number of elongated periods for

*x*>

*0*is

*N*

_{2}and

*N*for

_{21}*a*,

_{2}*a*respectively (see Fig. 4(a)). The detailed cavity geometry is summarized in Table 1.

_{21}*N*. As one can observe in Fig. 4(b), the

_{21}*1.2*times increase in the

*V*results in the

_{m}*4*times higher

*Q*(

*Q(N*=

_{21}*0*→

*4*)=

*6.4*×

*10*→

^{4}*2.53*×

*10*). Further increase in

^{5}*N*exhibits saturation in the increase of

_{21}*Q*(

*Q(N*), regardless of the higher

_{21}=8)≈2.6×10^{5}*V*. This behavior is not predicted by the (approximated) analysis given in Section 2. The difference between the DH and the Ideal Gaussian mode distribution is responsible for it (along

_{m}*x*axis the fit between them is nearly perfect, while for

*z*≠

*0*, it is not). As a result, the

*H*distribution in the

_{y}*k*≠

_{z}*0*, results in significant vertical losses, even for the increased mode volume cavities (

*N*=

_{21}*4*÷

*8*). These losses prevent further improvement in

*Q*with the DH design.

### 3.2 Local width modulation design

*W1*) [9

9. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

*z*-shifted away from the waveguide. These holes are divided in 3 groups:

*A*,

*B*,

*C*and each is shifted by

*D*,

_{A}*D*,

_{B}*D*, respectively (see Fig. 5 (a,b)). In this way a more gentle confinement is obtained in both x and z directions. The hole’s radii are

_{C}*r*=

_{A}*0.2875a*,

*r*=

_{B,C}*r*=

*0.275a*. The slab thickness is

*h*=

*0.96a*.

*Q*is optimized by the various hole’s shifts

*D*,

_{A}*D*,

_{B}*D*. The results are shown in Fig. 5(d), while detailed cavity geometry is specified in Table 2. The best quality factor

_{C}*Q*≈

*1.3*×

*10*is obtained for

^{6}*D*=

_{A}*0.056a*,

*D*=

_{B}*0.031a*,

*D*=

_{C}*0.019a*, at the frequency

*ω*=

_{0}*0.323(2πc/a)*and exhibit

*V*=

_{m}*1.775*×

*(λ/n)*. Its magnetic field profile (

^{3}*H*) at the plane

_{y}*y*=

*0*is shown in Fig. 5(c). This is an optimal value, while for higher and lower mode volumes the values of

*Q*are lower.

*V*, the better

_{m}*Q*is expected. Since the increase in

*V*subsequently delocalize the mode in space via the increase in the mode widths in

_{m}*x*and

*z*directions, the compression in

*k*-space results in a decrease in vertical power losses, and in a higher

*Q*. In the width modulation design,

*V*deviation from its optimal causes a subsequent increase in the vertical power losses that degrade the

_{m}*Q*. One can observe this behavior from the |

*H*| distributions presented in Fig. 6. A detailed inspection of Fig. 6 reveals that any departure from the “optimal” obtained point (central part in Fig. 6), bring higher intensity peaks of the field within the light cone. The reason for this behavior is that the mode cannot be defined by two uncoupled widths (

_{y}*σ*and

_{x}*σ*

_{z}*no separation of variables*). The effect of changing

*D*results in a change in the mode width in the whole

_{A,B,C}*xz*plane, and while the mode is localized in

*x*direction it is at the same time delocalized in

*z*. Thus, a maximal effect is obtained, beyond which, the

*k-space*distribution is no longer affected in the same way. It seems that, due to the more “flexible” control over the mode distribution in the whole

*xz*plane, (compared to the modified DH design), a higher

*Q*is attainable in the local width modulation cavities.

*D*may improve the

_{C}*Q*. However, this further effort, considering the operating wavelength of the diamond cavity (

*λ*~

*637.3nm*) would require sub nanometer fabrication resolution (currently beyond experimental access). Further work is needed to provide a full parameter-space variation towards an “optimal” diamond-based cavity.

### 3.3 Quantum Information Applications

*Q*~

*10*) allow significant decrease of decoherence, while impose serious challenge in photon information access. We assume that this dilemma can be solved by cavity integration via the

^{6}*Q*switching architecture [2

2. A. D. Greentre, J. Salzman, S. Prawer, and L. C. Hollenberg, “Quantum gate for Q-switching photonic bandgap cavities containing two level atoms” Phys. Rev. A **73**, 013818 (2006). [CrossRef]

*Q*>

*10*). In this way, both long decoherence time and controlled light coupling to PC waveguide are preserved. The coupling between two cavities, forming two qubits, can be realized via PC waveguide as proposed in [2

^{5}2. A. D. Greentre, J. Salzman, S. Prawer, and L. C. Hollenberg, “Quantum gate for Q-switching photonic bandgap cavities containing two level atoms” Phys. Rev. A **73**, 013818 (2006). [CrossRef]

## 4 Conclusions

*Q*, by applying an ideal Gaussian field model [8

**12**, 5961 (2005). [CrossRef]

*Q*in the waveguide based PC cavities in low dielectric constant material is explained in the terms of the mode frequency. The influence of mode volume increase via mode width elongation is given and a mode compensation technique is described. Simple analytical rules for the required mode volume compensation are derived.

*Q*≈

*2.6*×

*10*with

^{5}*V*≈

_{m}*1.8*×

*(λ/n)*is obtained, while further improvement via mode compensation is impossible due to the

^{3}*Q*saturation. We have shown that with a local-width modified-cavity design the

*Q*≈

*1.3*×

*10*with

^{6}*V*=

_{m}*1.775*×

*(λ/n)*is obtained. As far as we know, this is the best ultra-high Q cavity design obtained in diamond. Further work in a diamond high-Q cavity design might improve the

^{3}*Q*, but will require sub-nanometer precision in the device fabrication.

## Acknowledgment

## References and Links

1. | S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond Based Photonic Crystal Microcavities,” 11th Micro-optics Conference (MOC’05), Tokyo, Oct 30- Nov 2, 2005. |

2. | A. D. Greentre, J. Salzman, S. Prawer, and L. C. Hollenberg, “Quantum gate for Q-switching photonic bandgap cavities containing two level atoms” Phys. Rev. A |

3. | S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express |

4. | I. Bayn and J. Salzman, “High-Q photonic crystal nanocavities on diamond for Quantum Electrodynamics,” Eur. Phys. J. Appl. Phys. |

5. | C. F. Wang, R. Hanson, D. D. Awschalom, E. L. Hu, T. Feygelson, J. Yang, and J. E. Butler, “Fabrication and characterization of two-dimensional photonic crystal microcavities in nanocrystalline diamond,” Appl. Phys. Lett. |

6. | C. Kreuzer, J. Riedrich-Möller, E. Neu, and C. Becher, “Design of Photonic Crystal Microcavities in Diamond Films,” Opt. Express |

7. | B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nat. Mater. |

8. | D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express |

9. | E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. |

10. | J. Salzman, “Photonic Crystals in Diamond for Quantum Information Technology,” PIERS Proceedings, 1331 – 1334, March 26–30, Beijing, China, 2007. |

11. | I. Bayn, B. Meyler, A. Lahav, J. Salzman, Paolo Olivero, Barbara Fairchild, and Steven Prawer, “First photonic crystal devices on single crystal diamond,” Ib-2, IMEC-13, 9–10 on December, Haifa Israel. |

12. |
In the waveguide based cavities, the number of PC periods required for a similar x and z directions (different confinement mechanism). Qualitatively, a diamond-based PC will reach a Q similar to that of Si with _{l}1.5–2 times more PC periods than that of a Si-based PC, In the x - z direction, respectively. |

13. | These calculations are based on the characteristic mode frequencies and widths of DH cavities with refractive index n. |

14. | M. Qiu, “Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately,” Microwave Opt. Technol. Lett. |

15. | D. Englund and J Vuckovic, “A direct analysis of photonic nanostructures,” Opt. Express |

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

**OCIS Codes**

(230.5750) Optical devices : Resonators

(230.5298) Optical devices : Photonic crystals

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: February 5, 2008

Revised Manuscript: March 16, 2008

Manuscript Accepted: March 16, 2008

Published: March 27, 2008

**Citation**

Igal Bayn and Joseph Salzman, "Ultra high-Q photonic crystal nanocavity design: The effect of a low-ε slab material," Opt. Express **16**, 4972-4980 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4972

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### References

- S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, "Diamond Based Photonic Crystal Microcavities," 11th Micro-optics Conference (MOC’05), Tokyo, Oct 30- Nov 2, 2005.
- A. D. Greentre, J. Salzman, S. Prawer, and L. C. Hollenberg, "Quantum gate for Q-switching photonic band-gap cavities containing two level atoms" Phys. Rev. A 73, 013818 (2006). [CrossRef]
- S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, "Diamond based photonic crystal microcavities," Opt. Express 14, 3556 (2006). [CrossRef] [PubMed]
- I. Bayn and J. Salzman, "High-Q photonic crystal nanocavities on diamond for Quantum Electrodynamics," Eur. Phys. J. Appl. Phys. 37, 19-24 (2007). [CrossRef]
- C. F. Wang, R. Hanson, D. D. Awschalom, E. L. Hu, T. Feygelson, J. Yang, and J. E. Butler, "Fabrication and characterization of two-dimensional photonic crystal microcavities in nanocrystalline diamond," Appl. Phys. Lett. 91, 201112 (2007). [CrossRef]
- C. Kreuzer, J. Riedrich-Möller, E. Neu, and C. Becher, "Design of Photonic Crystal Microcavities in Diamond Films," Opt. Express 16, 1632-1644 (2008). [CrossRef] [PubMed]
- B. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nat. Mater. 4, 207 (2005). [CrossRef]
- D. Englund, I. Fushman, and J. Vuckovic, "General recipe for designing photonic crystal cavities," Opt. Express 12, 5961 (2005). [CrossRef]
- E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
- J. Salzman, "Photonic Crystals in Diamond for Quantum Information Technology," PIERS Proceedings, 1331-1334, March 26-30, Beijing, China, 2007.
- I. Bayn, B. Meyler, A. Lahav, J. Salzman, P. Olivero, B. Fairchild, and S. Prawer, "First photonic crystal devices on single crystal diamond," Ib-2, IMEC-13, 9-10 on December, Haifa Israel.
- In the waveguide based cavities, the number of PC periods required for a similar Ql in slabs of different ε’s is different in the x and z directions (different confinement mechanism). Qualitatively, a diamond-based PC will reach a Ql similar to that of Si with 1.5-2 times more PC periods than that of a Si-based PC, In the x - z direction, respectively.
- These calculations are based on the characteristic mode frequencies and widths of DH cavities with refractive index n.
- M. Qiu, "Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately," Microwave Opt. Technol. Lett. 45, 381-385 (2005). [CrossRef]
- D. Englund and J Vuckovic, "A direct analysis of photonic nanostructures," Opt. Express 14, 3472 (2006). [CrossRef] [PubMed]
- J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016, 608 (2002).

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