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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 (Vm). 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 Vm=1.77×(λ/n)3 in a local width modulation design is derived.
©2008 Optical Society of America
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 [1,2]. 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], while the best result is obtained for the double heterostructure (DH) cavity Q≈7×104 [4]. This result is more than one order of magnitude lower than that of similar designs in silicon [7–9]. 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]. 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 [10,11] 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]. This discussion is an extension of a qualitative analysis presented in our previous work [4]. 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 Vm≈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 Vm=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 1st and the 2nd TE-like modes in silicon [7–9] and diamond [3–6]. 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: Ql (Qv) for the lateral (vertical) losses [8]. Note that Ql is determined by the number of PC periods around the cavity and its increase leads to an unbound increase in Ql. Thus, by embedding the cavity in a sufficiently large PC, the value of Ql can be made arbitrarily large, and the total quality factor is limited only by Qv. In the following analysis we assume that Q=Qv. (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]).
Fig. 1. The notation of the linear PC waveguide in relation to the reciprocal lattice directions. In the inset the 1st 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]
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] and the local width modulated cavities [9]. At y=0 plane, the waveguide field is defined by (Ex, Ez, Hy), forming TE polarization. The cavity design is based on the lowest waveguide with odd (even) symmetry of the Hy along x (z) axis. The vertically radiated power losses are given by [8]:
where: η=(µ/ε)1/2, λ0 is the cavity mode wavelength, k=2π/λ0=[k2l+k2y], kl=(kx kz)=k(sinθcosφ sinθsinφ) and ky=kcosθ. Note that kl<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]:
where: σx and σz are the modal widths in real space in the x and z directions respectively, and (±kx0 ±kz0) are the J points coordinates given by kx0=π/a and kz0=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≈ωw, where ω0 and ωw are the cavity and the waveguide edge frequencies. Since materials with higher ε will lead to a lower ωw, resulting in the lower light cone (k=ω0/c) [3], 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 (QvSi/QvD) as a function of the modal width in the x direction (σx). Note that the inverse problem design [8] enables an equal mode-width along the waveguide direction for both high and low-ε materials. Therefore, in the following analysis we first assume: σxD=σxSi. 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] and in silicon [5]. This fit results are σzD=1a in diamond, and σzSi=0.85a in silicon.
In Fig. 2 the vertical quality factor ratio (QvSi/QvD) versus mode width in x direction is presented. The σx 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 Qv as a function of σx is expected for both silicon and diamond cavities. The fast increase in QvSi/QvD displayed in Fig. 2 is understood since this ratio reflects the difference of |FT2(Hy)| at the light cone edges in both materials. Qualitatively, the ratio of radiation losses will behave as exp[(k2Si-k2D)σx2], and increase nearly exponentially with the mode volume Vm∝σx.
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 (Vm∝σx).
In Fig. 3, the mode-width σxn of ideal Gaussian modes in cavities with slab refractive indices n are presented, versus the mode-width of a Silicon based cavity (σxSi), that provides the same value of Q [13]. As seen in the figure, an accurate fit to the σxn-σxSi relation is given by σ2xn=A+B(σxSi-C)2, where A, B, and C are constants for each n. Again, vertical power losses are dominated by the |FT2(Hy)| 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 σxn-σxSi relation is derived. Moreover, for σxSi>0.9a, we can approximate: σxn≈ασxSi+β, with the coefficients α and β depending on ε only, since ε defines both σz and ω0.
Fig. 3. Mode volume compensation: σxn vs. σxSi, σxn is mode width along x direction for slab refractive index n, that provides QvSi(σxSi)/Qvn (σxn)≈1 and σxSi is mode width in silicon. In solid lines the calculated σxn is shown. Hollow circles denote the analytic fit of the form σxn 2=A+B(σxSi-C)2, where A, B and C are functions of ε=n 2. The x marks denote linear fit of the form σxn≈ασxSi+β. The green line is diamond (n=2.4).
For the case of diamond PC (green line: nD=2.4): σxD≈1.57σxSi-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 Qv, the mode width in the x direction for the diamond cavity (σxD) should be ~1.6 times larger than in silicon (σxSi). Taking into account the mode width in z direction, the upper bound for the mode volume in diamond (to obtain the same Qv-value) is nearly ~1.9 times higher than in silicon! (This result is consistent, but significantly extends a previous comparison [3]).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] 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]. 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], 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]. 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, Vm vs. N21.
Table 1. Modified/simple Double Heterostructure cavity parameters.
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 Vm 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 Vm. 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 Hy distribution in the kz≠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 DA, DB, DC, 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 rA=0.2875a, rB,C=r=0.275a. The slab thickness is h=0.96a.
The cavity Q is optimized by the various hole’s shifts DA, DB, DC. 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 DA=0.056a, DB=0.031a, DC=0.019a, at the frequency ω0=0.323(2πc/a) and exhibit Vm=1.775×(λ/n)3. Its magnetic field profile (Hy) 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 DA, DB, DC. (c) Hy in the plane y=0 (d) Q, Vm vs. DA. The detailed holes shifts are summarized in Table 2.
Table 2. Local width modulation summary for the type A1 cavities in a units.
On a first glance, based on the Ideal Gaussian (approximated) model, the higher Vm, the better Q is expected. Since the increase in Vm 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, Vm deviation from its optimal causes a subsequent increase in the vertical power losses that degrade the Q. One can observe this behavior from the |Hy| 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 (σx and σz no separation of variables). The effect of changing DA,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.
We would like to stress that further optimization of the cavity design is possible. For instance, a gentle modulation in DC 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/Vm)1/2, where V0=cλ2/(8πγ⊥) [16]. Adopting γ⊥=32Mhz [2], and following similar arguments to [6], we obtain that g≫κ requires Q/V1/2m≫2.5×103. The mode width modulation Q/V1/2m 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 Vm for PC designs (V1/2m≪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], 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].
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]. 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 Vm≈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 Vm=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
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12. 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.
13. These calculations are based on the characteristic mode frequencies and widths of DH cavities with refractive index n.
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