## Design of photonic crystal microcavities in diamond films

Optics Express, Vol. 16, Issue 3, pp. 1632-1644 (2008)

http://dx.doi.org/10.1364/OE.16.001632

Acrobat PDF (565 KB)

### Abstract

We design photonic crystal microcavities in diamond films for applications in quantum information. Optimization of the cavity design by “gentle confinement” yields a high quality factor *Q*>66000 and small mode volume *V*≈1.1(*λ*/*n*)^{3}. In view of experimental applications we consider the influence of material absorption on the cavity *Q* factors and present a simple interpretation in the framework of a one-dimensional cavity model.

© 2008 Optical Society of America

## 1. Introduction

1. J. Wrachtrup and F. Jelezko, “Processing quantum information in diamond,” J. Phys.: Condens. Matter **18**, S807–S824 (2006). [CrossRef]

2. C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. **85**, 290–293 (2000). [CrossRef] [PubMed]

3. R. Brouri, A. Beveratos, J.-Ph. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. **25**, 1294–1296 (2000). [CrossRef]

4. F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. **92**, 076401 (2004). [CrossRef] [PubMed]

5. F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. **93**, 130501 (2004). [CrossRef] [PubMed]

6. C. Santori, D. Fattal, S.M. Spillane, M. Fiorentino, R.G. Beausoleil, A.D. Greentree, P. Olivero, M. Draganski, J.R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, D.N. Jamieson, and S. Prawer, “Coherent population trapping in diamond N-V centers at zero magnetic field,” Opt. Express **14**, 7986–7994 (2006). [CrossRef] [PubMed]

8. T. Gaebel, I. Popa, A. Gruber, M. Domhan, F. Jelezko, and J. Wrachtrup, “Stable single-photon source in the near infrared,” New J. Phys. **6**, 98 (2004). [CrossRef]

9. J.R. Rabeau, Y.L. Chin, S. Prawer, F. Jelezko, T. Gaebel, and J. Wrachtrup, “Fabrication of single nickel-nitrogen defects in diamond by chemical vapor deposition,” Appl. Phys. Lett. **86**, 131926 (2005). [CrossRef]

10. E. Wu, V. Jacques, F. Treussart, H. Zeng, P. Grangier, and J.-F. Roch, “Single-photon emission in the near infrared from diamond colour centre,” J. Lumin. **119–120**, 19–23 (2006). [CrossRef]

11. C. Wang, C. Kurtsiefer, H. Weinfurter, and B. Burchard, “Single photon emission from SiV centres in diamond produced by ion implantation,” J. Phys. B: At. Mol. Opt. Phys. **39**, 37–41 (2006). [CrossRef]

8. T. Gaebel, I. Popa, A. Gruber, M. Domhan, F. Jelezko, and J. Wrachtrup, “Stable single-photon source in the near infrared,” New J. Phys. **6**, 98 (2004). [CrossRef]

11. C. Wang, C. Kurtsiefer, H. Weinfurter, and B. Burchard, “Single photon emission from SiV centres in diamond produced by ion implantation,” J. Phys. B: At. Mol. Opt. Phys. **39**, 37–41 (2006). [CrossRef]

12. L. Childress, J.M. Taylor, A.S. Sørensen, and M.D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emitters,” Phys. Rev. Lett. **96**, 070504 (2006). [CrossRef] [PubMed]

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

14. Y.L. Lim, A. Beige, and L.C. Kwek, “Repeat-until-success linear optics distributed quantum computing,” Phys. Rev. Lett. **95**, 030505 (2005). [CrossRef] [PubMed]

15. Y.L. Lim, S.D. Barrett, A. Beige, P. Kok, and L.C. Kwek, “Repeat-until-success quantum computing using stationary and flying qubits,” Phys. Rev. A **73**, 012304 (2006). [CrossRef]

16. A.D. Greentree, C. Tahan, J.H. Cole, and L.C.L. Hollenberg, “Quantum phase transitions of light,” Nature Physics **2**, 856–861 (2006). [CrossRef]

*Q*and small mode volume

*V*, enabling the transfer of quantum information between internal electronic or spin states and photonic quantum bits. The emitter-cavity coupling can be further qualified depending on the application: the weak-coupling regime with

*g*≪(

*γ*

_{⊥},

*κ*) suffices e.g. for realizations of quantum networks whereas quantum gates and quantum phase transitions of light require a strong coupling

*g*≫(

*γ*

_{⊥},

*κ*), where

*g*is the emitter-field coupling constant (vacuum Rabi frequency),

*γ*

_{⊥}is the emitter dipole decay rate and κ the cavity field decay rate. The figure of merit for the weak-coupling regime is the Purcell factor, i.e. the enhancement factor of the spontaneous emission rate, given by

*λ*is the cavity mode wavelength and

*n*the material refractive index. An optimization of the Purcell factor thus requires a maximum ratio

*Q*/

*V*. The cavity decay rate is given by

*κ*=

*ω*/(4

_{c}*πQ*), with the cavity mode frequency

*ω*, and the vacuum Rabi frequency is defined as

_{c}*V*

_{0}=

*c*λ

^{2}/(8

*πγ*

_{⊥}) [17

17. J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E **65**, 016608 (2001). [CrossRef]

*g*≫

*κ*and

*g*≫

*⊥ thus demand maximization of*

_{γ}*Q*/√

*V*and 1/√

*V*, respectively. With these equations we can estimate the boundary conditions for photonic crystal cavity design: At room temperature, the phonon-broadened emission lines allow for spontaneous emission enhancement due to cavity coupling if the cavity linewidth 2

*κ*is larger than the emission bandwidth Δ

*ν*. For the NE8 center, Δ

*ν*=1.2 nm (at

*λ*≈800 nm) puts an upper limit on the cavity quality factor

*Q*<670. However, as the mode volume of photonic crystal cavities can be extremely small, i.e. on the order of

*V*≈1(

*λ*/

*n*)

^{3}, we still obtain a Purcell factor of

*F*≈50. On the other hand, strong emitter-cavity coupling might only be observed at low temperatures (≈4K) where lifetime-limited linewidths are attainable. To give an example, using the radiative lifetime of SiV centers,

*τ*=19 ns [18

_{r}18. A. V. Turukhin, C.-H. Liu, A.A. Gorokhovsky, R.R. Alfano, and W. Phillips, “Picosecond photoluminescence decay of Si-doped chemical-vapor-deposited diamond films,” Phys. Rev. B **54**, 16448–16451 (1996). [CrossRef]

*γ*

_{⊥}=2

*π*×4.2 MHz, the conditions for strong coupling are (i)

*V*and

*V*

_{0}are given in units of (

*λ*/

*n*)

^{3}, and (ii)

*Q*≫2.65×10

^{3}√

*V*. Condition (i) is easy to fulfill for a photonic crystal cavity. Taking again

*V*≈1(

*λ*/

*n*)

^{3}, (ii) reduces to

*Q*>2650, which is fairly reasonable as we show below. One should keep in mind, however, that the given values for

*g*are valid for optimum coupling of the emitter to the absolute maximum of the cavity field only. Non-ideal placement of the emitter and fabrication tolerances of the photonic crystal structure both reduce the effective coupling. Technically, the realization of an atom-photon-interface requires technologies for positioning defect centers at predefined locations and creating photonic micro-structures in diamond [19

19. A.D. Greentree, P. Olivero, M. Draganski, E. Trajkov, J.R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, S.T. Huntington, D.N. Jamieson, and S. Prawer, “Critical components for diamond-based quantum coherent devices,” J. Phys.: Condens. Matter **18**, S825–S842 (2006). [CrossRef]

20. S. Tomljenovic-Hanic, M.J. Steel, C. Martijn de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express **14**, 3556–3562 (2006). [CrossRef] [PubMed]

21. 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]

22. C.F. Wang, Y-S. Choi, J.C. Lee, E.L. Hu, J. Yang, and J.E. Butler, “Observation of whispering gallery modes in nanocrystalline diamond microdisks,” Appl. Phys. Lett. **90**, 081110 (2007). [CrossRef]

23. 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]

*Q*and small mode volumes

*V*together with the possibility of scalable architectures. Recent investigations of microcavities in two-dimensional photonic crystal slab structures in diamond have yielded theoretical

*Q*factors of up to 7×10

^{4}[21

21. 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]

20. S. Tomljenovic-Hanic, M.J. Steel, C. Martijn de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express **14**, 3556–3562 (2006). [CrossRef] [PubMed]

20. S. Tomljenovic-Hanic, M.J. Steel, C. Martijn de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express **14**, 3556–3562 (2006). [CrossRef] [PubMed]

24. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003). [CrossRef] [PubMed]

25. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express **13**1202–1214 (2005). [CrossRef] [PubMed]

*Q*factor can be further increased. Second, for practical realizations of photonic crystals in diamond one has to take into account the material absorption. For creating two-dimensional slab structures, thin diamond films are required as base material, e.g. nano-crystalline chemical vapor deposition (CVD) diamond. CVD diamond films, however, show strong absorption in the spectral range of color center emission [26

26. P. Achatz, J.A. Garrido, M. Stutzmann, O.A. Williams, D.M. Gruen, A. Kromka, and D. Steinmüller, “Optical properties of nanocrystalline diamond thin films,” Appl. Phys. Lett. **88**, 101908 (2006). [CrossRef]

*sp*

^{2}-bonded carbon. We here show the influence of material absorption on the cavity

*Q*factors and present a simple interpretation in the framework of a one-dimensional cavity model.

## 2. Cavity design

*xy*) and by total internal reflection in the perpendicular plane (

*z*). Microcavities are formed by point or line defects in the air hole lattice. Analogous PhC slab structures have been successfully employed in semiconductor quantum optics yielding strong coupling of single semiconductor quantum dots [27

27. J.P. Reithmaier, G. Şk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L.V. Keldysh, V.D. Kulakovskii, T.L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature (London) **432**, 197–200 (2004). [CrossRef] [PubMed]

28. 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 (London) **432**, 200–203 (2004). [CrossRef] [PubMed]

*n*=2.4 [7], leading to smaller PhC bandgaps and weaker mode localization. It is thus important to investigate whether one can still realize small mode volume high-

*Q*cavities.

### 2.1. Lattice type

*z*-direction as function of the air hole radius

*R*in units of the lattice constant

*a*. The PhC band diagrams were calculated using a plane wave expansion method [29

29. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*R*are as follows: a wide band gap supports confinement of modes in the plane of the PhC (

*xy*). For finite height slab structures as discussed in the next section, low mid gap frequencies allow for good confinement in vertical direction as a larger fraction of the mode’s k-space components lies below the light line [30

30. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express **10**, 670–684 (2002). [PubMed]

*ω*/

*ω*

_{midgap}of the band gap is only 2.6% and mid gap frequencies are very high. A triangular lattice, on the other hand, offers a relative gap width of 30% at a mid gap frequency of

*ω*

_{midgap}/(2

*πc*/

*a*))≈0.4 for an optimum air hole radius

*R*=0.4

*a*.

### 2.2. PhC slab

*h*suspended in air and patterned with a triangular array of air holes with radius

*R*.

*R*, the slab height

*h*determines the properties of the photonic band gap: for increasing

*h*the mid-gap frequency monotonically decreases which allows for better vertical confinement. On the other hand, there exists an optimum for the band gap width. Very thin slabs exhibit band gaps at high frequencies where the mode spacing is small, very thick slabs pull down additional modes below the light line, i.e. they are no longer single-mode waveguides. Both effects lead to a decrease of the band gap width. For

*R*=0.4

*a*we find the largest band gap for an optimum slab thickness of

*h*=0.75

*a*as displayed in Fig. 2(a).

*Q*factors would be expected for cavity modes lying close to the mid gap frequency of the optimized band gap. Figure 2(a) shows the mode frequency of a M1 cavity (see Sec. 2.3) in the optimized slab structure (

*R*=0.4

*a*,

*h*=0.75

*a*) which indeed lies close to mid-gap. However, the calculated quality factor of this mode is just

*Q*≈500. For comparison, Fig. 2(b) displays band gap and mode frequency of a PhC defect cavity with slab parameters which are derived from a systematic optimization of the cavity

*Q*factor (

*R*=0.29

*a*and

*h*=0.91

*a*, see Sec. 2.4). Despite a smaller band gap width and the fact that the mode frequency lies close to the upper band gap edge, this cavity mode exhibits

*Q*factors in excess of 60000. Part of this effect stems from the lower mid-gap frequency and the larger fraction of the mode wavevectors below the light line. The major part of the

*Q*factor increase, however, is due to a reduction of radiation losses of guided modes following the principle of “gentle confinement” [24

24. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003). [CrossRef] [PubMed]

### 2.3. M1 cavity

*Q*cavities with small mode volume

*V*we choose a cavity design based on a single defect, i.e. one missing air hole. We here adopt the design of Refs. [31

31. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Qmicrocavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

**14**, 3556–3562 (2006). [CrossRef] [PubMed]

21. 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*=2.4×10

^{7}in silicon at the expense of slightly higher mode volumes [32

32. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials **4**, 207–210 (2005). [CrossRef]

**37**, 19–24 (2007). [CrossRef]

*Q*-factors are limited to

*Q*≈7×10

^{4}and further improvement seems difficult as the cavity mode lies very close to the band edge. As the mode volume of double-heterostructure type cavities in diamond is about twice as large as for M1 type cavities, but

*Q*-factors turn out to be almost identical, we adhere to the M1 design.

*Q*without delocalizing the mode, i.e. increasing the mode volume

*V*. The quality factor

*Q*can be separated into 1/

*Q*=1/

*Q*

_{‖}+1/

*Q*⊥ [31

31. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Qmicrocavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

33. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

*Q*

_{‖}describes the confinement due to distributed Bragg reflection in the

*xy*-plane, and

*Q*

_{⊥}refers to the out-of-plane loss due to mode k-space components not matching the conditions for total internal reflection (TIR) at the slab boundaries. In theory, ideal in-plane confinement (large

*Q*

_{‖}) can be achieved by increasing the number of air hole layers surrounding the cavity. Out-of-plane (vertical) radiation losses, on the other hand, appear if the mode wavevectors have vertical components |

*k*

_{⊥}|≥0, or equivalently, in-plane components |

*k*

_{‖}|≤

*k*

_{0}=2

*π*/

*λ*

_{0}, where λ

_{0}is the mode wavelength in air [24

24. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003). [CrossRef] [PubMed]

*k*

_{‖}|=

*k*

_{0}defines the “light line”, corresponding to the dispersion relation of light in air, whereas all wave vectors with |

*k*

_{‖}|<

*k*

_{0}define the “light cone” of radiating modes. Fourier-space analysis of the cavity field distribution reveals that a gentle variation of the mode field envelope function suppresses vertical radiation losses where the ideal envelope is a Gaussian function [24

**425**, 944–947 (2003). [CrossRef] [PubMed]

33. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

31. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Qmicrocavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988–3995 (2004). [CrossRef] [PubMed]

34. D. Englund and J. Vučković, “A direct analysis of photonic nanostructures,” Opt. Express **14**, 3472–3483 (2006). [CrossRef] [PubMed]

*x*-direction by a distance

*d*and change their radius from

*R*to

*R*. The four next-neighboring holes in Γ-

_{B}*K*direction (C) are displaced along the lattice vector by a distance

*m*; their radius is modified to

*R*. Finally, we vary the radius

_{C}*R*of the four second-next neighbor holes (D).

_{D}### 2.4. Optimization of quality factor

35. 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**, 2972–2974 (2006). [CrossRef] [PubMed]

*a*×20

*a*super cell with height 8

*a*, surrounded by perfectly matching layers as absorbing boundaries. We use a resolution of 16 points per lattice period. For each simulation we start with excitation of the PhC structure with a broadband source to find resonant modes and symmetries of field distributions. For calculating mode

*Q*factors we then make use of the mode symmetries and employ a narrow-band source. Resonant mode frequencies and decay rates are extracted by a filter diagonalization method [35

35. 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**, 2972–2974 (2006). [CrossRef] [PubMed]

*V*is calculated by [17

17. J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E **65**, 016608 (2001). [CrossRef]

*E*,

_{x}*E*,

_{y}*H*,

_{x}*H*in the air layer at a small distance (Δ

_{y}*z*=

*λ*/4) above the slab surface [36

36. J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Optimization of the *Q* factor in photonic crystal micro-cavities,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

*P*is proportional to:

*k*-vectors inside the light cone,

*FT*

_{2}denotes the 2D Fourier transform. For all steps of the cavity design optimization, confinement and radiation losses of the cavity mode can be evaluated by calculating the fraction of Fourier components contained within the light cone.

**14**, 3556–3562 (2006). [CrossRef] [PubMed]

*Q*cavity mode at

*ω*=0.384(2

_{c}*πc*/

*a*) with

*Q*≈27000 and mode volume

*V*=1.23(

*λ*/

*n*)

^{3}, close to the results of Ref. [20

**14**, 3556–3562 (2006). [CrossRef] [PubMed]

*R*of air holes of the regular lattice. The optimum value

*R*=0.29

*a*has been found in [20

**14**, 3556–3562 (2006). [CrossRef] [PubMed]

**37**, 19–24 (2007). [CrossRef]

*h*=0.91

*a*and

*h*=0.75

*a*, respectively, and seems to be independent of

*h*. We verify this choice of

*R*by calculating

*Q*vs.

*R*for yet a different slab height of

*h*=0.77

*a*and find again an optimum for

*R*=0.29

*a*. Thus we keep

*R*=0.29

*a*for the remainder of the paper.

*R*and

_{B}*R*of the next-neighbor holes

_{C}*B*and

*C*. Keeping all other parameters constant we find a maximum

*Q*=60420 and

*V*=1.15(

*λ*/

*n*)

^{3}for

*R*=0.27

_{B}*a*and

*R*=0.23

_{C}*a*, arriving at a second set of parameters (see Tab. 2). These small variations of

*R*and

_{B}*R*already more than double the

_{C}*Q*-factor found in first investigations [20

**14**, 3556–3562 (2006). [CrossRef] [PubMed]

*h*=0.91

*a*[20

**14**, 3556–3562 (2006). [CrossRef] [PubMed]

*h*within the interval

*h*=[0.85

*a*,0.96

*a*] we find another increase in

*Q*from

*Q*=60420 for

*h*=0.91

*a*to

*Q*=62700 for

*h*=0.93

*a*. The relative mode volume

*V*=1.15(

*λ*/

*n*)

^{3}does not increase for the thicker slab as the mode frequency decreases at the same time. Thus we here adopt

*h*=0.93

*a*as optimum slab height.

*R*and

_{B}*R*, now together with the displacements

_{C}*d*and

*m*of the respective air holes, yielding a gentle confinement of the cavity mode. Figure 4 shows the quality factor

*Q*of the M1 cavity for variations of the parameters

*R*,

_{B}*R*,

_{C}*d*and

*m*.

*d*and the radius

*R*of holes

_{B}*B*leads to higher quality factors whereas further variations of hole

*C*parameters just degrades

*Q*. We find a maximum quality factor

*Q*=66300 with a small mode volume

*V*=1.11(

*λ*/

*n*)

^{3}at frequency

*R*of second next-neighbor holes

_{D}*D*within the range

*R*=[0.27

_{D}*a*..0.31

*a*] does not further improve

*Q*but rather leads to smaller quality factors.

*Q*very critically depends on very small variations of next-neighbor hole radii and displacements. This critical dependence makes fabricatin of high quality factor photonic crystal M1 cavities challenging.

*E*and

_{x}*H*, respectively, at the center of the slab (

_{z}*z*=0 plane) are shown in Fig. 5, revealing the confinement of the mode within the defect cavity.

*k*-space intensity

*I*defined by eq. (3) for the optimized M1 cavity (a) and a non-optimized cavity with

*Q*≈25000 (b).

*k*-space intensity distribution reveals the effect of gentle confinement: for the optimized cavity mode the intensity within the leaky region (light cone) is greatly reduced and the high-intensity components lie farther away from the light cone as required for a high-

*Q*cavity mode [33

33. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

## 3. Absorption losses

26. P. Achatz, J.A. Garrido, M. Stutzmann, O.A. Williams, D.M. Gruen, A. Kromka, and D. Steinmüller, “Optical properties of nanocrystalline diamond thin films,” Appl. Phys. Lett. **88**, 101908 (2006). [CrossRef]

*sp*

^{2}-bonded carbon. As the absorption is due to sub-gap electronic transitions it only changes very weakly with temperature and does not vanish at cryogenic temperatures. The absorption strength critically depends on the diamond grain size and is generally highest for nano-crystallineCVD diamond (grain sizes <10 nm) and smallest for poly-crystalline CVD diamond (grain sizes >100 nm) and single-crystal diamond films. With increasing grain sizes, however, optical scattering losses become more prominent. Single-crystal diamond films exhibit the best overall optical properties but are very difficult to process into membrane structures required for photonic crystal slabs [19

19. A.D. Greentree, P. Olivero, M. Draganski, E. Trajkov, J.R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, S.T. Huntington, D.N. Jamieson, and S. Prawer, “Critical components for diamond-based quantum coherent devices,” J. Phys.: Condens. Matter **18**, S825–S842 (2006). [CrossRef]

*ω*:

_{c}*ε*(

*ω*,

*r⃗*) denotes the material dielectric constant,

*ε*

_{∞}the background dielectric constant for

*ω*→∞,

*γ*and Δ

*ε*the width (FWHM) and amplitude of the dielectric resonance, respectively. In the limit of weakly absorbing media, i.e.

*ε*≪

_{i}*ε*, with the real and imaginary part of the dielectric constant,

_{r}*ε*and

_{r}*ε*, respectively, the absorption coefficient

_{i}*α*is calculated as:

*α*≈10

^{4}cm

^{-1}at the SiV center wavelength of 740 nm, which still yields

*ε*=0.283≪

_{i}*ε*=5.76. Fig. 7 shows real and imaginary part of an example dielectric function.

_{r}*α*at the cavity mode frequency while at the same time keeping the change Δ

*ε*of the real part of the dielectric function to a minimum. To this end we generally choose a large value of

_{r}*γ*together with a small Δ

*ε*. To give an example we use the M1 cavity parameter set II yielding

*Q*=60420 at

*ω*=0.384. For the choice of parameters in Fig. 7, i.e. ε

_{c}_{∞}=5.76,

*γ*=0.4, Δ

*ε*=0.005 and an emission wavelength of

*λ*

_{0}≈740 nm for the SiV center, we get an absorption coefficient of

*α*=170 cm

^{-1}at the cavity mode frequency

*ω*. The real part of the dielectric function at

_{c}*ω*remains at the background value

_{c}*ε*(

_{r}*ω*)=

_{c}*ε*

_{∞}=5.76 as desired. However, we have to examine the consequences of the variation of

*ε*in the vicinity of the cavity mode frequency. For this purpose, we consider the frequency interval Δ

_{r}*ω*=0.02(2

_{S}*πc*/

*a*) around

*ω*covered by the source used to excite the cavity mode in the FDTD calculations (see Fig. 7). Within the interval [

_{c}*ω*-Δ

_{c}*ω*/2,

_{S}*ω*+Δ

_{c}*ω*/2],

_{S}*ε*(

_{r}*ω*) only changes by

*δε*=0.006, i.e. the relative variation

_{r}*δε*/

_{r}*ε*(

_{r}*ω*)≈10

_{c}^{-3}. We now calculate the cavity

*Q*-factors using a constant set of geometry parameters (parameter set II), but taking into account the different values of the dielectric constant’s real part

*ε*(

_{r}*ω*) introduced by the dielectric resonance in the frequency interval of the FDTD excitation source. The deviation of

*Q*-factors due to the variations of

*ε*(

_{r}*ω*), Δ

*Q*=max|

*Q*(

*ε*(

_{r}*ω*))-

_{c}*Q*(

*ε*(

_{r}*ω*±Δ

_{c}*ω*/2))|, is very small: Δ

_{S}*Q*=20 or Δ

*Q*/

*Q*=3.3×10

^{-4}. By inspection of various parameter combinations for the dielectric function of eq. (4) we find an upper boundary for Δ

*Q*/

*Q*<2×10

^{-3}. Thus, to a very good approximation, we have artificially introduced absorption losses or an imaginary dielectric constant into the FDTD calculations without affecting the real part of the dielectric constant.

*Q*-factors for a lossy material, using the M1 cavity design as given by parameter set II. Fig. 8 shows the calculated quality factor

*Q*as function of the absorption coefficient

*α*.

*Q*with increasing absorption losses displayed in Fig. 8 fundamentally alters predictions for photonic crystal microcavities in diamond. For ultra-nanocrystalline diamond films of high quality we measure absorption coefficients of ≈150±20 cm

^{-1}at the SiV center emission wavelength of 740 nm, yielding a reduction of the predicted quality factor from

*Q*

_{0}=60422 to

*Q*

_{real}≈1350. We arrive at about the same value of the realistic

*Q*-factor

*Q*

_{real}for the optimized M1 cavity (parameter set III) with

*Q*

_{0}=66300 as the quality factor again mainly is determined by absorption losses. This strong dependence of

*Q*-factors on material absorption might be the major reason besides scattering losses for low

*Q*-factors experimentally observed so far in diamond photonic crystal cavities [23

23. 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]

*Q*on

*α*we find a simple law following [37

37. I. Alvarado-Rodriguez and E. Yablonovitch, “Separation of radiation and absorption losses in two-dimensional photonic crystal single defect cavities,” J. Appl. Phys. **92**, 6399–6401 (2002). [CrossRef]

38. T. Asano, B.-S. Song, and S. Noda, “Analysis of the experimental *Q* factors (~1 million) of photonic crystal nanocavities,” Opt. Express **14**, 1996–2002 (2006). [CrossRef] [PubMed]

39. T. Xu, S. Yang, S. Selvakumar, V. Nair, and H.E. Ruda, “Nanowire-array-based photonic crystal cavity by finite difference time-domain calculations,” Phys. Rev. B **75**, 125104 (2007). [CrossRef]

*Q*-factor in presence of material losses can be decomposed into:

*Q*

_{0}is the cavity

*Q*-factor for an ideal, lossless material and

*Q*

_{abs}denotes a quality factor due to absorption losses. In order to define

*Q*

_{abs}we remember the definition of the quality factor

*Q*by the decay of the intracavity photon number

*ν*(

*t*), with

*τ*=1/Δ

_{c}*ω*denotes the photon lifetime in the cavity mode and Δ

_{c}*ω*the cavity bandwidth. From these equations one derives the usual expression for the quality factor:

_{c}*Q*=ω

*τ*=

_{c}*ω*/Δ

*ω*. The “absorption

_{c}*Q*-factor”

*Q*

_{abs}is defined analogously to eq. (7) by the decay of the intracavity photon number due to absorption:

*Φ*

_{abs}(

*t*)=exp(-

*t*/

*τ*

_{abs})

*Φ*(0), introducing

*τ*

_{abs}as average time after which a photon is removed from the cavity mode by absorption.

*τ*

_{abs}, on the other hand, can be expressed by the usual absorption length

*L*

_{abs}=

*α*

^{-1}as

*τ*

_{abs}=

*L*

_{abs}/

*c*=1/(

_{m}*αc*), where

_{m}*c*is the phase velocity in the material. In a simplified approach we substitute

_{m}*c*using a linear dispersion relation

_{m}*ω*=

*c*and arrive at:

_{m}k*k*=2

*πn*/

_{r}*λ*and

*α*=4

*πn*/

_{i}*λ*(for weakly absorbing media), where

*n*and

_{r}*n*are real and imaginary part of the material refractive index, respectively, we obtain a definition for

_{i}*Q*

_{abs}(cf. [39

39. T. Xu, S. Yang, S. Selvakumar, V. Nair, and H.E. Ruda, “Nanowire-array-based photonic crystal cavity by finite difference time-domain calculations,” Phys. Rev. B **75**, 125104 (2007). [CrossRef]

*Q*factors derived from eq. (6) and the

*Q*-factors calculated from FDTD simulations including material losses is very well. Thus, the simple equation (6) allows for easy prediction of cavity quality factors in the presence of absorption. From the simulation of a single photonic crystal cavity and the use of a linear dispersion relation it is not clear, however, whether eq. (6) holds as a general law and whether its predictions are valid for a larger range of cavities. This question is subject of current investigations.

## 4. Conclusion

*Q*-factors can be improved over previous designs. Our calculations for an optimized cavity design yield maximum

*Q*-factors of 66300 with a mode volume

*V*=1.1(

*λ*/

*n*)

^{3}. Due to these high quality factors

*Q*and small mode volumes

*V*together with the possibility of scalable architectures, photonic crystal microcavities are well suited for applications in quantum information.

*Q*-factors and find a strong decrease of

*Q*with increasing absorption coefficient. We also demonstrated that the prediction of

*Q*-factors in the presence of losses is greatly simplified by a one-dimensional cavity model.

*Q*and Purcell factor

*F*for the optimized cavity reduce to

*Q*≈1350 and

*F*≈92. Thus, even for a lossy material demonstrations of modified spontaneous emission and emitter-cavity coupling at roomtemperature should be feasible. For applications in quantum information processing demanding strong emitter-cavity coupling, however, our estimations show that for the coupling of SiV centers at cryogenic temperatures a minimum quality factor

*Q*>2800 is required. As this limit is only a factor of two higher than our calculated results, a realistic solution to this problem could be further work on improvement of purity of nano-crystalline diamond films or the use of single-crystal diamond exhibiting much better optical properties.

## Acknowledgements

## References and links

1. | J. Wrachtrup and F. Jelezko, “Processing quantum information in diamond,” J. Phys.: Condens. Matter |

2. | C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. |

3. | R. Brouri, A. Beveratos, J.-Ph. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. |

4. | F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. |

5. | F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. |

6. | C. Santori, D. Fattal, S.M. Spillane, M. Fiorentino, R.G. Beausoleil, A.D. Greentree, P. Olivero, M. Draganski, J.R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, D.N. Jamieson, and S. Prawer, “Coherent population trapping in diamond N-V centers at zero magnetic field,” Opt. Express |

7. | A.M. Zaitsev, |

8. | T. Gaebel, I. Popa, A. Gruber, M. Domhan, F. Jelezko, and J. Wrachtrup, “Stable single-photon source in the near infrared,” New J. Phys. |

9. | J.R. Rabeau, Y.L. Chin, S. Prawer, F. Jelezko, T. Gaebel, and J. Wrachtrup, “Fabrication of single nickel-nitrogen defects in diamond by chemical vapor deposition,” Appl. Phys. Lett. |

10. | E. Wu, V. Jacques, F. Treussart, H. Zeng, P. Grangier, and J.-F. Roch, “Single-photon emission in the near infrared from diamond colour centre,” J. Lumin. |

11. | C. Wang, C. Kurtsiefer, H. Weinfurter, and B. Burchard, “Single photon emission from SiV centres in diamond produced by ion implantation,” J. Phys. B: At. Mol. Opt. Phys. |

12. | L. Childress, J.M. Taylor, A.S. Sørensen, and M.D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emitters,” Phys. Rev. Lett. |

13. | A.D. Greentree, J. Salzman, S. Prawer, and L.C.L. Hollenberg, “Quantum gate for Q-switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A |

14. | Y.L. Lim, A. Beige, and L.C. Kwek, “Repeat-until-success linear optics distributed quantum computing,” Phys. Rev. Lett. |

15. | Y.L. Lim, S.D. Barrett, A. Beige, P. Kok, and L.C. Kwek, “Repeat-until-success quantum computing using stationary and flying qubits,” Phys. Rev. A |

16. | A.D. Greentree, C. Tahan, J.H. Cole, and L.C.L. Hollenberg, “Quantum phase transitions of light,” Nature Physics |

17. | J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E |

18. | A. V. Turukhin, C.-H. Liu, A.A. Gorokhovsky, R.R. Alfano, and W. Phillips, “Picosecond photoluminescence decay of Si-doped chemical-vapor-deposited diamond films,” Phys. Rev. B |

19. | A.D. Greentree, P. Olivero, M. Draganski, E. Trajkov, J.R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, S.T. Huntington, D.N. Jamieson, and S. Prawer, “Critical components for diamond-based quantum coherent devices,” J. Phys.: Condens. Matter |

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

21. | I. Bayn and J. Salzman, “High-Q photonic crystal nanocavities on diamond for quantum electrodynamics,” Eur. Phys. J. Appl. Phys. |

22. | C.F. Wang, Y-S. Choi, J.C. Lee, E.L. Hu, J. Yang, and J.E. Butler, “Observation of whispering gallery modes in nanocrystalline diamond microdisks,” Appl. Phys. Lett. |

23. | 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. |

24. | Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) |

25. | Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express |

26. | P. Achatz, J.A. Garrido, M. Stutzmann, O.A. Williams, D.M. Gruen, A. Kromka, and D. Steinmüller, “Optical properties of nanocrystalline diamond thin films,” Appl. Phys. Lett. |

27. | J.P. Reithmaier, G. Şk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L.V. Keldysh, V.D. Kulakovskii, T.L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature (London) |

28. | 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 (London) |

29. | S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

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

31. | Z. Zhang and M. Qiu, “Small-volume waveguide-section high Qmicrocavities in 2D photonic crystal slabs,” Opt. Express |

32. | B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials |

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

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

35. | 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. |

36. | J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Optimization of the |

37. | I. Alvarado-Rodriguez and E. Yablonovitch, “Separation of radiation and absorption losses in two-dimensional photonic crystal single defect cavities,” J. Appl. Phys. |

38. | T. Asano, B.-S. Song, and S. Noda, “Analysis of the experimental |

39. | T. Xu, S. Yang, S. Selvakumar, V. Nair, and H.E. Ruda, “Nanowire-array-based photonic crystal cavity by finite difference time-domain calculations,” Phys. Rev. B |

**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: December 18, 2007

Revised Manuscript: January 17, 2008

Manuscript Accepted: January 18, 2008

Published: January 23, 2008

**Citation**

Christine Kreuzer, Janine Riedrich-Möller, Elke Neu, and Christoph Becher, "Design of Photonic Crystal Microcavities in Diamond Films," Opt. Express **16**, 1632-1644 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1632

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### References

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- C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, "Stable solid-state source of single photons," Phys. Rev. Lett. 85, 290-293 (2000). [CrossRef] [PubMed]
- R. Brouri, A. Beveratos, J.-Ph. Poizat, and P. Grangier, "Photon antibunching in the fluorescence of individual color centers in diamond," Opt. Lett. 25, 1294-1296 (2000). [CrossRef]
- F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, "Observation of coherent oscillations in a single electron spin," Phys. Rev. Lett. 92, 076401 (2004). [CrossRef] [PubMed]
- F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, "Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate," Phys. Rev. Lett. 93, 130501 (2004). [CrossRef] [PubMed]
- C. Santori, D. Fattal, S. M. Spillane, M. Fiorentino, R. G. Beausoleil, A. D. Greentree, P. Olivero, M. Draganski, J. R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, D. N. Jamieson, and S. Prawer, "Coherent population trapping in diamond N-V centers at zero magnetic field," Opt. Express 14, 7986-7994 (2006). [CrossRef] [PubMed]
- A. M. Zaitsev, Optical Properties of Diamond: A Data Handbook (Berlin: Springer, 2001).
- T. Gaebel, I. Popa, A. Gruber, M. Domhan, F. Jelezko, and J. Wrachtrup, "Stable single-photon source in the near infrared," New J. Phys. 6, 98 (2004). [CrossRef]
- J. R. Rabeau, Y. L. Chin, S. Prawer, F. Jelezko, T. Gaebel, and J. Wrachtrup, "Fabrication of single nickel-nitrogen defects in diamond by chemical vapor deposition," Appl. Phys. Lett. 86, 131926 (2005). [CrossRef]
- E. Wu, V. Jacques, F. Treussart, H. Zeng, P. Grangier, and J.-F. Roch, "Single-photon emission in the near infrared from diamond colour centre," J. Lumin. 119-120, 19-23 (2006). [CrossRef]
- C. Wang, C. Kurtsiefer, H. Weinfurter, and B. Burchard, "Single photon emission from SiV centres in diamond produced by ion implantation," J. Phys. B: At. Mol. Opt. Phys. 39, 37-41 (2006). [CrossRef]
- L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, "Fault-tolerant quantum communication based on solid-state photon emitters," Phys. Rev. Lett. 96, 070504 (2006). [CrossRef] [PubMed]
- A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, "Quantum gate for Q-switching in monolithic photonic-band-gap cavities containing two-level atoms," Phys. Rev. A 73, 013818 (2006). [CrossRef]
- Y. L. Lim, A. Beige, and L. C. Kwek, "Repeat-until-success linear optics distributed quantum computing," Phys. Rev. Lett. 95, 030505 (2005). [CrossRef] [PubMed]
- Y. L. Lim, S. D. Barrett, A. Beige, P. Kok, and L. C. Kwek, "Repeat-until-success quantum computing using stationary and flying qubits," Phys. Rev. A 73, 012304 (2006). [CrossRef]
- A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, "Quantum phase transitions of light," Nat. Phys. 2, 856-861 (2006). [CrossRef]
- J. Vuickovic, M . Lonicar, H . Mabuchi, and A . Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608 (2001). [CrossRef]
- A.V. Turukhin, C.-H. Liu, A. A. Gorokhovsky, R. R. Alfano, and W. Phillips, "Picosecond photoluminescence decay of Si-doped chemical-vapor-deposited diamond films," Phys. Rev. B 54, 16448-16451 (1996). [CrossRef]
- A. D. Greentree, P. Olivero, M. Draganski, E. Trajkov, J. R. Rabeau, P. Reichart, B. C. Gibson, S. Rubanov, S. T. Huntington, D. N. Jamieson, and S. Prawer, "Critical components for diamond-based quantum coherent devices," J. Phys.: Condens. Matter 18, S825-S842 (2006). [CrossRef]
- S. Tomljenovic-Hanic, M. J. Steel, C. Martijn de Sterke, and J. Salzman, "Diamond based photonic crystal microcavities," Opt. Express 14, 3556-3562 (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, Y-S. Choi, J. C. Lee, E. L. Hu, J. Yang, and J. E. Butler, "Observation of whispering gallery modes in nanocrystalline diamond microdisks," Appl. Phys. Lett. 90, 081110 (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]
- Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 131202-1214 (2005). [CrossRef] [PubMed]
- P. Achatz, J. A. Garrido, M. Stutzmann, O. A. Williams, D. M. Gruen, A. Kromka, and D. Steinmuller, "Optical properties of nanocrystalline diamond thin films," Appl. Phys. Lett. 88, 101908 (2006). [CrossRef]
- J. P. Reithmaier, G. Se¸ k, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot semiconductor microcavity system," Nature (London) 432, 197-200 (2004). [CrossRef] [PubMed]
- 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 (London) 432, 200-203 (2004). [CrossRef] [PubMed]
- S. Johnson and J. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
- K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002). [PubMed]
- Z. Zhang and M. Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express 12, 3988-3995 (2004). [CrossRef] [PubMed]
- B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Maters. 4, 207-210 (2005). [CrossRef]
- D. Englund, I. Fushman, and J. Vuickoviic, "General recipe for designing photonic crystal cavities," Opt. Express 13, 5961-5975 (2005). [CrossRef] [PubMed]
- D. Englund and J. Vuickovic, "A direct analysis of photonic nanostructures," Opt. Express 14, 3472-3483 (2006). [CrossRef] [PubMed]
- 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, 2972-2974 (2006). [CrossRef] [PubMed]
- J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Optimization of the Q factor in photonic crystal microcavities," IEEE J. Quantum Electron. 38, 850-856 (2002). [CrossRef]
- I. Alvarado-Rodriguez and E. Yablonovitch, "Separation of radiation and absorption losses in two-dimensional photonic crystal single defect cavities," J. Appl. Phys. 92, 6399-6401 (2002). [CrossRef]
- T. Asano, B.-S. Song, and S. Noda, "Analysis of the experimental Q factors (~1 million) of photonic crystal nanocavities," Opt. Express 14, 1996-2002 (2006). [CrossRef] [PubMed]
- T. Xu, S. Yang, S. Selvakumar, V. Nair, and H.E. Ruda, "Nanowire-array-based photonic crystal cavity by finitedifference time-domain calculations," Phys. Rev. B 75, 125104 (2007). [CrossRef]

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