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

doi:10.1364/OE.16.004972

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

Igal Bayn and Joseph Salzman »View Author Affiliations

Optics Express, Vol. 16, Issue 7, pp. 4972-4980 (2008)
http://dx.doi.org/10.1364/OE.16.004972

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▸ Article Outline
– Abstract
1. Introduction
2. The influence of a low ε: Semi-Analytical approach
3. Waveguide based PC cavities in Diamond
4 Conclusions
– References and links

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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≈2.6×10⁵ is obtained. The highest Q≈1.3×10⁶ with V_m =1.77×(λ/n)³ in a local width modulation design is derived.

1. Introduction

Slab 2D photonic crystal (PC) structures in diamond are being considered as an attractive architecture for the control and manipulation of light-matter interaction in the strong coupling regime for quantum information processing devices in the solid state [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.

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]

]. The implementation of this architecture requires the capability to couple the optical emission of a diamond color center (NV center) to a cavity with a sharp spectral resonance. Several designs for the high-Q PC cavities on diamond were recently reported [3–6

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]

], while the best result is obtained for the double heterostructure (DH) cavity Q≈7×10⁴ [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]

]. This result is more than one order of magnitude lower than that of similar designs in silicon [7–9

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

]. This major discrepancy in cavity quality factor (Q) raises the question: what is the physical mechanism responsible for the dielectric constant influence on the Q value in a photonic crystal cavity?

Significant progress is made in diamond fabrication processing. The first nano-crystal diamond PC cavities were produced and optically characterized [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]

]. Due to high scattering the measured quality factor (10³ ) 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

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.

] the first single crystalline PC fabrication for optical characterization. This process will allow realization of higher 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.

In Section 2 we describe the analytic model of an ideal high-Q PC cavity, derived by the inverse problem method [8

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

]. This discussion is an extension of a qualitative analysis presented in our previous work [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]

]. The ratio of quality factors in high-ε and low-ε materials is analyzed as a function of mode volume. The mode volume compensation technique, which allows reaching similar 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≈2.6×10⁵ ), with a mode volume V_m ≈1.78×(λ/n)³ . Then, the limitations of this technique are discussed. Finally, we calculate local width modulated cavities and demonstrate the highest reported quality factor in diamond Q≈1.3×10⁶ with V_m =1.77×(λ/n)³ . A detailed comparison of these two methods to the results of Section 2 is given. Quantum Information Applications are discussed.

2. The influence of a low ε: Semi-Analytical approach

The PC cavities considered in this work are formed by a membrane suspended in air and periodically modulated by a triangular array of cylindrical air holes. In this structure the maximum bandgap is obtained between the 1^st and the 2^nd TE-like modes in silicon [7–9

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

] and diamond [3–6

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]

]. Thus, we focus on the cavities with resonant frequency within this bandgap. Cavity mode is confined laterally by the Bragg reflection and vertically by total internal reflection (TIR). Therefore, cavity quality factor can be divided into two major components: Q_l (Q_v ) for the lateral (vertical) losses [8

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

]. Note that Q_l is determined by the number of PC periods around the cavity and its increase leads to an unbound increase in Q_l . Thus, by embedding the cavity in a sufficiently large PC, the value of Q_l can be made arbitrarily large, and the total quality factor is limited only by Q_v . In the following analysis we assume that Q=Q_v . (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

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_l 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.

]).

Fig. 1. The notation of the linear PC waveguide in relation to the reciprocal lattice directions. In the inset the 1^st Brillouin zone of the triangular lattice (blue) and the first irreducible Brillouin zone (cyan) with high symmetry points are depicted.

2.1 Optimized k-distribution: Ideal Gaussian Envelope cavity [8
D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express 12, 5961 (2005). [CrossRef]
]

We start by reviewing an analytic model for cavities formed as a spatial modulation of a PC waveguide. The waveguide is defined by one row of holes in the ΓJ direction (see Fig. 1), similarly to the DH design [7

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

] and the local width modulated cavities [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]

]. At y=0 plane, the waveguide field is defined by (E_x, E_z, H_y ), forming TE polarization. The cavity design is based on the lowest waveguide with odd (even) symmetry of the H_y along x (z) axis. The vertically radiated power losses are given by [8

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

P_{ν} \approx \frac{η}{2 λ_{0}^{2} k} \int_{k_{l} \leq k} \frac{d k_{x} d k_{z}}{k_{l}^{2}} k_{z} {∣ F T_{2} (H_{y}) ∣}^{2}

(1)

where: η=(µ/ε)^1/2 , λ₀ is the cavity mode wavelength, k=2π/λ₀ =[k² _l +k² _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).

Following the inverse problem approach to minimize vertical power losses an ideal Gaussian field envelope was introduced at [8

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

F T_{2} (H_{y}) = \sum_{k_{x_{0}}, k_{z_{0}}} sign (k_{x_{0}}) \exp {(- (k_{x} - k_{x_{0}}) \frac{σ_{x}}{\sqrt{2}})}^{2} \exp {(- (k_{z} - k_{z_{0}}) \frac{σ_{z}}{\sqrt{2}})}^{2}

(2)

where: σ_x and σ_z are the modal widths in real space in the x and z directions respectively, and (±k_x0 ±k_z0 ) are the J points coordinates given by k_x0 =π/a and k_z0 =2π/√3a.

This Ideal Gaussian envelope function is regarded as an optimal and it is used as an analytical model that approximates the real high-Q cavity field to compare different designs in materials with different dielectric constant.

2.2 Quality factor vs. high mode volume

We assume that the PC waveguide modulation, to obtain an ideal Gaussian envelope, preserves the cavity resonance at the close vicinity to the waveguide edge at the J points, i.e. ω₀≈ω_w , where ω₀ and ω_w are the cavity and the waveguide edge frequencies. Since materials with higher ε will lead to a lower ω_w , resulting in the lower light cone (k=ω₀/c) [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]

], the magnitude of the k-components crossing the light cone boundary decreases exponentially with ω₀ . As a result, the vertical losses in a slab with a high-ε are substantially lower than those in a low-ε one, which explains the different Q factors.

As an example of the 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_vSi/Q_vD ) as a function of the modal width in the x direction (σ_x ). Note that the inverse problem design [8

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

] enables an equal mode-width along the waveguide direction for both high and low-ε materials. Therefore, in the following analysis we first assume: σ_xD =σ_xSi . The mode widths in the 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

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]

] and in silicon [5

]. This fit results are σ_zD =1a in diamond, and σ_zSi =0.85a in silicon.

Fig. 2. The mode width σ_x influence in silicon and diamond: Q_vSi/Q_vD vs. σ_x , while V_m~σ_x .

In Fig. 2 the vertical quality factor ratio (Q_vSi/Q_vD ) versus mode width in x direction is presented. The σ_x increase results in further localization of field distribution in the 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_v as a function of σ_x is expected for both silicon and diamond cavities. The fast increase in Q_vSi/Q_vD displayed in Fig. 2 is understood since this ratio reflects the difference of |FT₂(H_y)| at the light cone edges in both materials. Qualitatively, the ratio of radiation losses will behave as exp[(k² _Si-k² _D)σ_x ²], and increase nearly exponentially with the mode volume V_m∝σ_x .

2.3 Mode volume compensation technique

By slightly enlarging the mode volume of the cavity in the low-ε material, the higher vertical losses can be compensated by a better localization in the 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 ).

In Fig. 3, the mode-width σ_xn of ideal Gaussian modes in cavities with slab refractive indices n are presented, versus the mode-width of a Silicon based cavity (σ_xSi ), that provides the same value of Q [13

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

]. As seen in the figure, an accurate fit to the σ_xn -σ_xSi relation is given by σ² _xn =A+B(σ_xSi-C)² , where A, B, and C are constants for each n. Again, vertical power losses are dominated by the |FT₂(H_y)| 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 σ_xn -σ_xSi relation is derived. Moreover, for σ_xSi >0.9a, we can approximate: σ_xn≈ασ_xSi+β, with the coefficients α and β depending on ε only, since ε defines both σ_z and ω₀ .

Fig. 3. Mode volume compensation: σ_xn vs. σ_xSi , σ_xn is mode width along x direction for slab refractive index n, that provides Q_vSi (σxSi)/Q_vn (σxn)≈1 and σ_xSi is mode width in silicon. In solid lines the calculated σ_xn is shown. Hollow circles denote the analytic fit of the form σ_xn ²=A+B(σ_xSi -C)², where A, B and C are functions of ε=n ². The x marks denote linear fit of the form σ_xn ≈ασ_xSi +β. The green line is diamond (n=2.4).

For the case of diamond PC (green line: 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_v , the mode width in the x direction for the diamond cavity (σ_xD ) should be ~1.6 times larger than in silicon (σ_xSi ). Taking into account the mode width in z direction, the upper bound for the mode volume in diamond (to obtain the same Q_v -value) is nearly ~1.9 times higher than in silicon! (This result is consistent, but significantly extends a previous comparison [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]

]).As will shown later (Section 3), this result has limited validity.

3. Waveguide based PC cavities in Diamond

In this section we discuss the mode compensation technique and its limitations by analyzing two diamond PC cavity designs: the modified double heterostructure [7

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

] and the waveguide local width modulation [9

]. Several modifications to the original structures are introduced to accommodate for the diamond low dielectric constants. The common feature of these two designs is that they both rely on a local dielectric constant modulation in the vicinity of PC waveguide.

Cavity modes were calculated with 3D-FDTD, whereas for quality factors and resonant frequencies, the Pade approximation technique is used [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]

]. The computation cell consists of 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

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

], while a higher discretization of 0.03a exhibited similar Q. The results convergence for a larger structure exhibited similar Q.

3.1 Modified Double Heterostructure (DH) design

This structure is formed by introducing two different lattice constants: a₂ , a₂₁ elongated in the x-direction (a₂ >a₂₁ >a) into the one missing-hole waveguide [7

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

]. The lattice constant in the 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 N₂₁ for a₂ , a₂₁ respectively (see Fig. 4(a)). The detailed cavity geometry is summarized in Table 1.

Fig. 4. Modified Double Heterostructure cavity: (a) Geometrical parameters for N₂₁ =4 cavity, with schematic of waveguide confinement drawn below. (b) Q, V_m vs. N₂₁ .

Table 1. Modified/simple Double Heterostructure cavity parameters.

h [a]	r [a]	W [a]	a₁ [a]	a₂₁ [a]	N₂	N₂₁
0.96	0.275	0.977√3	0.05	0.025	1	0÷8

Here, the mode volume compensation is implemented by varying N₂₁ . As one can observe in Fig. 4(b), the 1.2 times increase in the V_m results in the 4 times higher Q (Q(N₂₁ =0→4)=6.4×10⁴ →2.53×10⁵ ). Further increase in N₂₁ exhibits saturation in the increase of Q (Q(N₂₁=8)≈2.6×10⁵ ), regardless of the higher V_m . 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 x axis the fit between them is nearly perfect, while for z≠0, it is not). As a result, the H_y distribution in the k_z ≠0, results in significant vertical losses, even for the increased mode volume cavities (N₂₁ =4÷8). These losses prevent further improvement in Q with the DH design.

3.2 Local width modulation design

This cavity is also formed by local index variation in the vicinity of the one missing-hole waveguide (W1) [9

]. Here, adjacent holes are 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_C , 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 r_A =0.2875a, r_B,C =r=0.275a. The slab thickness is h=0.96a.

The cavity Q is optimized by the various hole’s shifts D_A , D_B , D_C . The results are shown in Fig. 5(d), while detailed cavity geometry is specified in Table 2. The best quality factor Q≈1.3×10⁶ is obtained for D_A =0.056a, D_B =0.031a, D_C =0.019a, at the frequency ω₀ =0.323(2πc/a) and exhibit V_m =1.775×(λ/n)³ . Its magnetic field profile (H_y ) at the plane 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.

Fig. 5. Local width modulation cavities (A1): (a) A1 cavity computational domain. (b) Detailed geometry. The holes A, B, C are shifted by D_A , D_B , D_C . (c) H_y in the plane y=0 (d) Q, V_m vs. D_A . The detailed holes shifts are summarized in Table 2.

Table 2. Local width modulation summary for the type A1 cavities in a units.

Cavity #	D_A	D_B	D_C
1	0.04	0.026	0.013
2	0.05	0.03	0.016
3	0.056	0.031	0.019
4	0.062	0.042	0.021
5	0.07	0.047	0.023

On a first glance, based on the Ideal Gaussian (approximated) model, the higher V_m , the better Q is expected. Since the increase in V_m subsequently delocalize the mode in space via the increase in the mode widths in 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_m deviation from its optimal causes a subsequent increase in the vertical power losses that degrade the Q. One can observe this behavior from the |H_y | 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 (σ_x and σ_z no separation of variables). The effect of changing D_A,B,C results in a change in the mode width in the whole 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.

Fig. 6. log(|FT(Hy)|² ) in a.u. for the A1 cavities with different D_A shifts (see Table. 2).

We would like to stress that further optimization of the cavity design is possible. For instance, a gentle modulation in D_C may improve the 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

In order to facilitate an evaluation of the above results in practical implementations we examine (review) the parameter space of cavity Quantum Electrodynamics (QED) experiments in this design. Strong coupling regime between a single NV center and a photon in a cavity imposes that g≫κ, γ_⊥, where g is the Rabi frequency, κ is the cavity mode decay rate and γ _⊥ is the NV dipole decay rate. While κ is given by the cavity Q (κ=ω₀ /4πQ)), the g and γ _⊥ are related via g=γ_⊥ V₀ /V_m)_1/2 , where V₀ =cλ² /(8πγ_⊥) [16
J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).
]. Adopting γ _⊥=32Mhz [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]
], and following similar arguments to [6
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]
], we obtain that g≫κ requires Q/V^1/2 _m ≫2.5×10³ . The mode width modulation Q/V^1/2 _m is 9.76×10⁵ , which is ~5 times higher than in the DH cavity at Section 3.1 (both cavities have similar mode volumes). Therefore, g≫κ is satisfied. As in [6], g≫γ⊥, does not impose serious limitation on V_m for PC designs (V^1/2 _m ≪2.8×10³).

The ultra-high-Q cavities (Q~10⁶ ) 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 Q switching architecture [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]

], controlled by Stark shift. (This requires a cavity design with Q>10⁵ ). 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

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

We have analyzed the influence of varying the material dielectric constant on PC cavity Q, by applying an ideal Gaussian field model [8

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

]. The physical mechanism behind the reduced 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.

In diamond, the mode volume compensation technique is applied to the modified double heterostructure deign. The best value of Q≈2.6×10⁵ with V_m ≈1.8×(λ/n)³ is obtained, while further improvement via mode compensation is impossible due to the Q saturation. We have shown that with a local-width modified-cavity design the Q≈1.3×10⁶ with V_m =1.775×(λ/n)³ 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 Q, but will require sub-nanometer precision in the device fabrication.

Acknowledgment

Igal Bayn would like to thank to Dr. Anne Weill-Zrahia, project manager of the Technion NANCO Cluster for her devotion and indispensable help that made this work possible. Support from the Russell Berrie Nanotechnology Institute is acknowledged.

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 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]
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]
6.	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]
7.	B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nat. Mater. 4, 207 (2005). [CrossRef]
8.	D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express 12, 5961 (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]
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 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_l 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.
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. 45, 381–385 (2005). [CrossRef]
15.	D. Englund and J Vuckovic, “A direct analysis of photonic nanostructures,” Opt. Express 14, 3472 (2006). [CrossRef] [PubMed]
16.	J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).

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 ?’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).

Appl. Phys. Lett.

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]
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]

Eur. Phys. J. Appl. Phys.

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]

Microwave Opt. Technol. Lett.

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]

Nat. Mater.

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

Opt. Express

D. Englund, I. Fushman, and J. Vuckovic, "General recipe for designing photonic crystal cavities," Opt. Express 12, 5961 (2005). [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]
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]
D. Englund and J Vuckovic, "A direct analysis of photonic nanostructures," Opt. Express 14, 3472 (2006). [CrossRef] [PubMed]

Phys. Rev. A

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]

Phys. Rev. E

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016, 608 (2002).

Other

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 ?’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.
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.

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