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We have presented a novel design of a photonic crystal slab (PCS) nanocavity, in which the electric field of the cavity mode is strongly localized in free space. The feature of the cavity is a linear air slot introduced to the center of the mode-gap confined PCS cavity. Owing to the discontinuity of the dielectric constant, the electric field of the cavity mode is strongly enhanced inside the slot, allowing strong matter-field coupling and large interaction volume in free space. Using finite-difference time-domain method, we calculate the properties of the cavity mode as a function of the slot width. The calculated quality factor is still as high as 2×105 and the mode volume is as small as 0.14 of a cubic wavelength in a vacuum, even if 200-nm-wide slot is introduced to the PCS.
©2008 Optical Society of America
1. Introduction
Recently, many sorts of optical cavities with high quality factor (Q) and small mode volume (V) have been developed and fabricated. Proposed cavities include small Fabry-Perot cavities [1], microspheres [2], toroidal resonators [3], and photonic crystal slab (PCS) nanocavities [4–7]. In particular, PCS nanocavities have attracted considerable attention owing to their extremely high Q/V and integratability with other PCS devices. Nowadays some groups have aimed at using PCS nanocavities for the study of matter-field interaction. In 2004, cavity quantum electrodynamics (QED) using a PCS nanocavity and quantum dots (QDs) was realized and the vacuum Rabi splitting with a single QD was observed [8]. In this experiment, QDs were embedded in the PCS nanocavity having electric-field maxima in the cavity material [6], ensuring strong interaction between QDs and light field. In contrast, when gaseous atoms are used as a matter [9–12], PCS nanocavities should be designed in such a way that the electric field of the cavity mode is strong in free space region as proposed in [13].
Here, in this paper, we present a novel and simple design of an air-core PCS nanocavity suitable for the study of cavity QED in free space. The strategy to bring the cavity field maxima in free space is to introduce a narrowlongitudinal air slot onto the center of the width-modulated line-defect cavities that have ultrahigh Q over one million (see, Fig. 1) [7]. Because of the electric-field continuity condition at the dielectric boundaries, the cavity mode can be locally enhanced inside the slot [14]. Furthermore, this field concentration leads to drastic decrease of V by a factor of (ε 0/ε)5/2, where ε 0 and ε are the dielectric permittivity in free space and of the PCS materials, respectively [15]. We numerically investigated this field-concentration effect on the cavity parameters using three-dimensional finite-difference time-domain (FDTD) method and found that attainable Q/V exceeds 108. To the best of our knowledge, this is the greatest value ever reported so far for PCS nanocavities. We also estimated cavity-QED parameters by assuming that a single rubidium (Rb) atom at the first excited state (resonant wavelength of 1.5 µm) was located at the center of the air-slot cavities. The calculated one-photon coupling constant was found to be over 2π×10 GHz, which is greater than all other decay rates in the system, i.e., the atom-field interaction is expected to be in the strong coupling regime.
This paper is organized as follows. In section 2, we present a design of our air-slot PCS nanocavity. Strong light confinement is achieved by gradual width modulation of a PCS waveguide and a sub-wavelength-wide air slot introduced onto it. Section 3 shows our simulation results, such as mode-field patterns and Q, for various slot widths. We also calculate the effective mode volume, which is a measure of the field confinement in free space region. In section 4, we estimate cavity-QED parameters by considering atom-light interaction with a single Rb atom at the cavity center and compare these parameters with those of other cavity-QED systems with neutral atoms. Finally, a summary is given in section 5.
2. Cavity design
Figure 1 shows a schematic of our PCS nanocavity. A linear air slot is introduced to the center of the line defect of a mode-gap confined PCS nanocavity, which is realized by local width modulation of a line-defect width [4, 5]. Without the width modulation, our structure is nothing but a slot waveguide, which is theoretically expected to have a lossless mode with an extremely small modal area [14, 16]. In the original work [15], the cavity was constructed by simply terminating the slot waveguide with a photonic band gap structure. However, Q for this air-slot cavity is as low as 103 because this abrupt termination causes substantial vertical radiation loss. In contrast, the air slot is not terminated in our design but the light confinement is achieved by the gradual width modulation of the PCS waveguide, which can strongly suppress vertical radiation loss. Thus, the mechanism to achieve high Q is basically the same as that for the width-modulated line-defect cavity.
The air slot plays an important role for the realization of strong field localization in free space. The physical mechanism can be simply understood as the local field enhancement at dielectric boundaries. Generally, the electric field of the TE mode in the PCS nanocavity is mainly x-polarized (the coordinate axes are defined in Fig. 1), so that the boundary condition for the electric field
holds at the surfaces of the air-slot walls. Here, r in and r out represent slightly apart positions inside and outside the PCS material, respectively. Therefore, Ex(r out) is enhanced by a factor of ε/ε 0 compared with Ex(r in). Note that, in order to obtain maximum enhancement, the slot width s should be infinitesimally narrow because the new cavity mode after the introduction of such a narrow slot is almost the same as the original one. As a result, the strongest electric field of the original (slot-less) cavity can be enhanced further by introducing the air slot [15]. In contrast, for larger s, the maximum of the electric field is shifted toward the inside of the PCS material, thus decreasing the field strength at the slot boundaries.
To evaluate optical properties of the proposed cavity, we performed a three-dimensional FDTD method. For calculation, we set a lattice constant a of 490 nm, an air-hole radius r of 140 nm, a slab thickness of 204 nm, a slab refractive index of 3.46 (we consider silicon (Si) as a PCS material), and hole shifts d A, d B, d C of δ, 2δ/3, and δ/3, respectively, where δ=14 nm. We showed in the previous work that such a tapered-shift structure along the line defect increased Q to over 107 [4]. The line-defect width, defined as the distance between the center of adjacent holes, was set to be W1 or W1.2, where W1 and W1.2 represent √3 a and 1.2×√3 a, respectively. To select a preferred even mode, we set even symmetry conditions in y-z plane and x-z plane. The perfectly matched layer method with 8 layers was applied for absorbed boundary condition in all directions. Grid spacing was set to 28 nm so that one or more E x components were contained within the air slots, even for the smallest slot width of 49 nm. Q was estimated by the energy decay rate of the cavity mode. A calculation area was as large as 26.5 (x axis)×10.5 (y axis)×58.5 (z axis) lattice periods, above which Q was saturated so that calculated Q was almost equal to the vertical Q.
Fig. 1. Design of an air-slot PCS nanocavity. A linear sub-wavelength-wide air slot is introduced to the center of the line defect of the mode-gap confined PCS nanocavities realized by local line-defect-width modulation. Air holes labeled A, B, and C are shifted outward by d A, d B, and d C, respectively.
Fig. 2. Calculated electric-field (|E|) distributions in the x-z plane. The slot widths are 0 nm (a), 49 nm (b), 84 nm (c), 133 nm (d), and 196 nm (e). PCS structures are depicted by white lines. Each electric field is normalized by total field energy in the whole calculation area.
3. Simulation results and discussion
Figure 2 shows the calculated electric-field (|E|) distributions of the resonant mode for the xz plane. The slot width s (the line-defect width W) was set to be (a) 0 nm (W1), (b) 49 nm (W1), (c) 84 nm (W1.2), (d) 133 nm (W1.2), and (e) 196 nm (W1.2). It can be clearly seen that the mode distribution is quite sensitive to s and the electric field is strongly enhanced in the air slot. Figure 3(a) shows transversal profiles of |E| along the red line indicated in Fig. 2(a). The electric-field maxima are positioned at the walls of the air slot because the enhanced field is originated from polarization charges induced onto the slot walls. Electric-field strengths at x=0 are plotted in Fig. 3(b), showing 4.8-fold enhancement at s=49 nm and monotonic fade out for larger s. It is noted that, even for s=196 nm, the electric field at the slot center is twice as large as that of the no-slot cavity.
Fig. 3. (a) Electric-field profiles along the red arrow indicated in Fig. 2. and (b) electric field at the cavity center for W1 (black) and W1.2 (white).
Electric-field distributions in the x-y plane are shown in Fig. 4. White squares stand for the Si slab. From these results, we found that evanescent fields above and below the air-slot were also enhanced.
Figure 5(a) shows calculated resonant wavelength λ. The effective refractive index of a line-defect region monotonically decreases as the slot width broadens. As a result, the effective length of the cavity becomes shortened and then λ decreases. For the line-defect width of W1, we observed that the cavity mode vanished at s >100 nm, because the waveguide mode goes over the boundary of a photonic-band gap. To widen the air slot further, we changed the line-defect width from W1 to W1.2 to pull the waveguide mode into the photonic band gap. However, the mode becomes close to the band-gap boundary again when the air slot is extended over 220 nm.
Calculated Q is plotted in Fig. 5(b). The dependence of Q on s is explained mainly by the decrease of the effective refractive index, i.e., the condition for total internal reflection (TIR) becomes severe as the slot width increases. An additional insight about the vertical (y-direction) confinement of light is provided by x- and z-directional spatial Fourier transformation (FT) of the cavity field. Figure 6 shows FT spectra of Ex at the middle of the PCS for (a) s=0 nm, (b) 84 nm, and (c) 196 nm, where the leaky regions (for decomposed plane waves with in-plane momentum components (kx,kz) being inside these regions, the TIR condition is broken) are depicted by white circles with radii of 2π/λ. It is apparently shown that the number of momentum components lying inside the leaky region increases as s. It is worth noting that the air-slot cavity with s=196 nm has Q still as high as 2×105, which shows about 100-fold improvement compared with that of the air-slot cavity proposed in [15].
Fig. 4. Calculated electric-field (|E|) distributions in the x-y plane. The slot widths are 0 nm (a), 49 nm (b), 84 nm (c), 133 nm (d), and 196 nm (e).
In order to give a measure of light confinement in the air slot, we define a dimensionless mode volume Ṽc as
where n(r) is the refractive-index distribution of the cavity and r=0 is assumed to be the cavity center. Figure 7 is the dependence of Ṽ c on s. The minimum value of Ṽ c we have obtained in this calculation is 0.017 for s=49 nm, the ratio of which to that of the no-slot cavity is 1.3×10-2. This ratio is larger than the expected value (ε 0/ε)5/2=2.0×10-3 [15] because the narrowest air slot used in our calculation is still so wide that the original cavity mode is considerably modified.
The results obtained above probably have unnegligible uncertainties especially for narrow air slots. To investigate the influence of the grid spacing on the calculation results, we changed the grid spacing to 14 nm (half of the value set above) and recalculated Q and Ṽ c for some narrow-air-slot cavities. These calculation results showed that the uncertainties of Q and Ṽ c for the smallest width of s=49 nm were about 30% and 20%, respectively, and these uncertainties rapidly decreased for larger s.
Fig. 6. Spatial Fourier transformation spectra obtained from Ex in the x-z plane for (a) s=0 nm, (b) 84 nm, and (c) 196 nm. The leaky regions are depicted by white circles.
As described, our cavity exhibits Q=5.7×106 and Ṽ c=0.017 at s=49 nm. The resultant Q/V=3.3×108 (Ṽ c is used as a mode volume V for our air-slot cavity) is the largest value ever reported for PCS nanocavities. Obtained Q and Q/V at s=49 nm are four orders of magnitude better than those of the quasi-one-dimensional microcavity with a 20-nm-wide air slot calculated in [15]. Even at s=196 nm, Q and Q/V are three orders and one order of magnitude better than those in [15], respectively.
4. Application for cavity QED
Compatibility between high Q/V and field concentration in free space is a distinguished feature of our air-slot cavities, which may be suitable for the study of cavity QED with neutral gaseous atoms. Here in this section, we estimate cavity-QED parameters with our cavities and the 5P 3/2 - 4D5/2 transition of 87Rb atoms. The resonant wavelength λ 0 is 1529 nm, and the atomic transverse decay rate γ⊥, which is related to the parallel decay rate as γ‖=2γ⊥, is 2π×0.9 MHz [17]. No additional non-radiative decay is considered here. Although this transition is somewhat unfamiliar compared to the D2 line commonly used for cavity-QED experiments, it lies in a transparent wavelength region of high-index PCS materials (such as Si) and may be practical as a target transition. We assume here that a single 87Rb atom is located at the cavity center. The atom is excited from the 5S1/2 (F=2) state to the 5P3/2 (F=3) state by an external pump field and the transition 5P3/2 (F=3) - 4D5/2 (F=4) is driven by the cavity field. For simplicity, influence of other hyperfine levels in the 5S1/2, 5P3/2, and 4D5/2 states is ignored [18].
For s=49 nm, we obtain the one-photon coupling constant g 0=η(3cγ⊥λ 2 0/4πVc)1/2=2π×13 GHz, which is much larger than 2π×50 MHz of a toroidal microresonator estimated in [11]. The factor η~0.65 is due to averaging over Clebsch-Gordon coefficients for the interaction between a linearly-polarized cavity field and an unpolarized atom. The cavity mode decay rate is κ=ω/2Q=2π×17 MHz, indicating that this atom-cavity system is in the strong coupling regime (g 0≫κ>γ⊥). Note that g 0 calculated here is not the maximum value, which can be obtained at the slot boundaries. We also obtain the critical atomic number N 0=2κ γ⊥/g 2 0=1.8×10-7 and the saturation photon number n 0=γ2⊥/2g 2 0=2.3×10-9 [9], both of which are extremely smaller than unity. Even for the wider air slot of s=96 nm, calculated cavity-QED parameters are [g 0, κ, N 0, n 0]=[2π×4.6 GHz, 2π×0.49 GHz, 4.1×10-5, 1.9×10-8], still allowing the strong coupling. Note that Q of PCS cavities is generally degraded when fabricated experimentally, however, tolerance of degrading Q is as much as the ratio between g 0 and κ~10.
The air-slot cavities have an advantage in that atoms can interact with a nearly maximum field. By contrast, in the case of microspheres and toroidal microresonators [11], atoms interact only with the tails of the evanescent field. Compared to the parameters for graded square-lattice PCS cavities and the D2 transition of caesium atoms [g 0, κ, N 0, n 0]=[2π×17 GHz, 2π×4.4GHz, 8.4×10-5, 1.2×10-8] [10], our air-slot cavities with 87Rb 5P3/2-4D5/2 transition may offer comparable experimental conditions. In addition, the interaction volume of our cavities is greater than that of the square-lattice cavities owing to the wide air slot, which would be a clear advantage to realize cavity-QED experiments with neutral atoms.
5. Conclusion
We have designed a novel PCS nanocavity suitable for light-matter interaction in free space. The main feature is that the electric field of the cavity mode is strongly localized in free space, which is realized by the local field enhancement in a sub-wavelength-wide air slot. In combination with an ultra-high-Q mode-gap confined PCS nanocavity, we have obtained that the calculated quality factor is still as high as 2×105 and the mode volume is as small as 0.1 of a cubic wavelength in a vacuum, even if a 200-nm-wide air slot is introduced to the PCS cavity. We have estimated cavity-QED parameters with a single Rb atom in the first excited state and showed that the atom-light interaction could be in the strong-coupling regime. We also mention that our cavity has many desirable properties for compact optical sensing devices and low threshold lasers [19, 20].
References and links
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