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A method that simply improves the quality (Q) factors of two-dimensional photonic crystal nanocavities using a three-dimensional finite-difference time domain calculation is described. The leaky area for a high-Q nanocavity mode is visualized in a real cavity structure by extracting the leaky components within a light cone in momentum space and by transferring them back into real space using an inverse Fourier transformation. The Q factor is remarkably improved by appropriately shifting the positions of air holes at the leaky area. We design three-missing-air-hole and zero-cell-defect nanocavities with Q factors of 5,000,000 and 1,700,000, respectively, for demonstration.
© 2016 Optical Society of America
1. Introduction
Nanocavities in two-dimensional (2D) photonic crystal (PC) slabs are exceptional optical resonators possessing both high quality (Q) factors and small modal volumes (V) [1–4]. Their extremely high Q/V values provide various benefits for optical devices including a high resolution, a high sensitivity, a low operating energy, and enhancements in nonlinear optical phenomena. High-Q nanocavities have been applied to various applications such as wavelength-selective filters [5,6], biosensors [7,8], optical pulse manipulation devices [9–13], solid-state cavity quantum electrodynamics [14–16], and low-threshold lasers [17–20]. So far, various types of nanocavities have been examined for these applications, e.g., line-defect, H0, and heterostructure nanocavities. A method that easily improves the design Q factors (Qdesign), which is simply applicable to any nanocavity structure, is important for extending the potential application areas of nanocavities.
In 2003, we reported an important design rule to increase Qdesign of nanocavities; the electric-field distribution of the cavity should slowly vary in order to suppress the out-of-slab photon leakage [2]. Subsequently, we designed an L3 nanocavity consisting of three missing air holes with Qdesign of 260,000 by appropriately shifting the positions of six air holes near the cavity edges [21], where the shifted holes were determined through a trial-and-error process using a three dimensional (3D) finite-difference time-domain (FDTD) calculation. In 2005, we proposed heterostructure nanocavities with Qdesign of more than 10 million [3]. Recently, the experimental Q factors of silicon heterostructure nanocavities have been increased to nine million by improvements in fabrication [4,22–26]. However, the Q values of more than a million are still limited in a few nanocavity structures, in spite of the fact that there have been several theoretical works about designing high-Q nanocavities [27–30]. For the further development of nanocavity research, it is necessary to develop these fundamental analyses to a more usable method.
In previous studies, we briefly reported a direct visualization of the leaky components of nanocavities, which clearly indicate the significant area causing the out-of-slab photon leakage in a real cavity structure [28,31]. Qdesign was remarkably improved by shifting the positions of the air holes at the leaky area. It is noted that this method can be applied to any nanocavity structure with only the 3D FDTD method. Here, we detail the method using L3 and zero-cell (H0) nanocavities for demonstration. Qdesign is increased up to 5.0 million and 1.7 million for the L3 and H0 cavities, respectively, while the increases in V are suppressed within 10%.
2. Visualization method to improve Qdesign
High-Q nanocavity modes are confined in 2D PC slabs by two main mechanisms: the photonic band-gap effect in the x–y plane and the total internal reflection (TIR) at the slab–air interface in the z direction. The main factor determining the Qdesign value is the imperfections in TIR confinement [2,32,33]. The resonant modes formed in small cavities consist of various wavevector components. Therefore, they include the components that do not fulfill the condition for TIR, i.e., the components within the light cone in momentum space. Such leaky components easily radiate in the z direction from the nanocavity, resulting in reduced Qdesign.
It has been demonstrated that modification of the air-hole positions is effective for reducing the leaky components, thereby retaining the small V [2,21,30]. Here, we have to know which air holes should be shifted to increase Qdesign most effectively. To this end, the visualization of the leaky components works wonderfully. Qdesign for any nanocavity can be increased by repeating the following steps.
Step 1: Calculation of the electric-field distribution by the 3D FDTD method
Figure 1(a) shows the x- and y-polarized electric-field distributions (Ex and Ey) of the fundamental resonant mode for a normal L3 nanocavity. The parameters for the 3D FDTD method are set as follows: the lattice constant a = 410 nm, the hole radius r = 108 nm, the slab thickness t = 220 nm, and the refractive index n = 3.46 by assuming operation in the optical communication band (~1.55 μm). The cell size is dx = 0.1a, dy = 3a/16, and dz = 0.0894a ( = t/6). Qdesign of 5,600 is obtained in a normal L3 cavity.
Fig. 1 (a) Electric-field distributions (Ex and Ey) of the fundamental resonant mode of an L3 nanocavity. The left (right) image corresponds to Ex (Ey). (b) 2D FT spectra of Ex and Ey in momentum space plotted on a logarithmic scale. (c) Real-space image of the leaky components obtained by inverse FT of the wavevector components within the light cone. (d) Calculated Qdesign with different shifts in the two air holes at the cavity edges.
Step 2: Fourier transformation of the electric-field distribution
Figure 1(b) shows the 2D Fourier transformation (FT) spectra for Ex and Ey. The components inside the light cone are leaky, which do not fulfill the TIR condition. It is well known that the leaky components arise from the rapid change in the electric-field distribution at the cavity edges [2,28]. In order to gently change the form of the electric-field distribution of the cavity, we have to select several holes near the cavity and then have to move their positions appropriately.
Step 3: Visualization of the leaky components by inverse Fourier transformation
The real-space images of the leaky components can be obtained by inverse FT only for the components inside the light core. Figure 1(c) shows the results. It is clearly seen that the leaky components of Ey2 are concentrated at the two air holes nearest to the cavity edges, whereas the components of Ex2 are much smaller. This fact indicates that the electric-field distribution of Ey rapidly changes at these locations. Therefore, these two air holes should be shifted prior to the other holes to effectively reduce the leaky components.
Step. 4: Modification of the air-hole positions (radii) at the leaky area
Figure 1(d) presents Qdesign calculated by the 3D FDTD method when the positions of the two air holes at the leaky area are shifted along the x direction. A maximum Qdesign of 139,000 is obtained when we shift the holes outward by 0.20a. This result is in good agreement with previous reports [2,21]. It is noted that an increase in Qdesign can be also obtained by a change in the air-hole radius. However, shifting the air hole is better in terms of fabrication.
3. Optimization of an L3 nanocavity
We can further increase Qdesign for an L3 nanocavity by repeating the optimization round from Step 1 to Step 4. Figure 2 shows a summary of eight optimization rounds. The red areas show the total leaky components (Ex2 + Ey2), and the blue, green, and yellow circles indicate the shifted air holes. The magnitude of the shift and Qdesign are displayed. Figure 2(a) indicates the result for round 1 obtained in Fig. 1. Figure 2(b) shows the cavity for round 2, where the main leaky components exist at the third air holes from the edges. The maximum improvement in Qdesign of 254,000 is obtained when the holes are shifted outward by 0.23a. Repeating the optimization round, Qdesign is increased to as high as 5,020,000 in round 8, as shown in Fig. 2(h). This is comparable to the highest value reported for an L3 cavity [30]. Figure 2(i) shows the leaky area and all shifted holes after round 8. The symmetry of the structure should be maintained. The leaky area expands into a wide area, some of which appears at the positions of the holes that have already been shifted. Therefore, further optimization might be insignificant because the tolerance to the structural disorder might be decreased by repeating the optimization round [24,34]. In addition, the difference in the refractive index assumed in the calculation and the actual value in fabricated samples would affect the experimental Q. Thus, experimental studies is necessary. In fact, the average Q value of 1.2 million have been experimentally reported for the L3 cavity with Qdesign of 5 million, designed using genetically optimization method [35].
Fig. 2 Improvement in Qdesign for a normal L3 nanocavity. (a)–(h) Visualization of the total leaky components (Ex2 + Ey2) and the shifted air holes for each round. The color intensity is normalized in each round. Arrows indicate the direction of the shift. (i) Leaky area for the cavity after optimization round 8. All shifted holes are colored.
Figure 3 presents the evolution of Qdesign and V during the eight rounds. The increase in V from the normal L3 cavity is only 7% after the eight optimization rounds, whereas the Q factor is increased by three orders of magnitude. The resonant wavelengths and free spectral range rarely change from the round 1 to the round 8. Figure 4(a) shows the profiles for Ey in round 0 and round 8 along the centerline of the cavity in the x direction. It is seen that the difference is prominent at the cavity edges. Figure 4(b) shows a comparison between round 4 and round 8, where the difference is almost invisible, whereas Qdesign is increased by a factor of four.
Fig. 3 Evolution of the Q factor and modal volume during the eight optimization rounds of the L3 nanocavity shown in Fig. 2.
Fig. 4 Profile of Ey along the centerline of the cavity in the x direction. (a) Comparison between round 0 and round 8. (b) Comparison between round 4 and round 8.
With the visualization method, it is possible to regularly increase Qdesign to the highest value reported so far for an L3 nanocavity. Furthermore, the positions of any shifted holes are not readjusted during the eight optimization rounds. This clearly indicates that the visualization method resolves the question of how to select air holes to improve Qdesign most effectively. Only the air holes aligned on the centerline of the L3 cavity are shifted in the x direction in previous reports [2,21,30]. However, Fig. 2(i) indicates that the modifications of the other holes are also important. It is also noted that the 2nd, 4th, and 6th nearest holes to the cavity edges are not modified in Fig. 2(i). This feature indicates that the visualization approach is essentially different from the previous exhaustive search method [21,30]. How the air holes at the leaky area should be shifted—left or right and up or down—is still a question for further study. It is noted that most of the shifted air holes in Fig. 2(i) are moved in the x-direction. This feature may originate from the fact that major part of the leaky components arise from Ey.
4. Optimization of an H0 nanocavity
Finally, we apply the optimization method to an H0 nanocavity formed by a defect with a shift in two adjacent air holes, as shown in Fig. 5(a). Here, we use the following parameters: r = 0.26a, t = 0.6a, and n = 3.4 with FDTD cell sizes of dx = 0.1a, dy = , and dz = 0.1a. The shift in the initial structure is 0.14a, and Qdesign is 173,000, as presented in Fig. 5(a). These values are good agreement with previous reports [36]. The slight difference in Qdesign originates from the differences in the calculation parameters. Three optimization rounds are shown in Figs. 5(b)–5(d), which increase Qdesign to 1,670,000. Modifications in the radii of the air holes are utilized in round 2 for demonstration. The air-hole shift is also able to increase Qdesign. Figure 6 shows the evolution of Qdesign and V during the optimization rounds. V is almost constant. Qdesign of 8 million has been reported for an H0 cavity [30]. Thus, it will be important to investigate how the experimental Q values can be increased using these structures.
Fig. 5 Improvement in Qdesign for an H0 nanocavity through three optimization rounds. The red areas show the total leaky components (Ex2 + Ey2), and the green and blue circles indicate the modified air holes.
5. Conclusion
We have demonstrated a method to increase Qdesign of a nanocavity via visualization of the leaky components. Qdesign is dramatically improved by shifting the positions of the air holes at the leaky area while retaining a small V. The visualization method is intuitively comprehensible. Furthermore, this method can be applied to any PC cavity and any resonant mode such as higher-order modes that are useful for Si emitters [20,37]. In fact, we have used the visualization method in order to improve Qdesign of a heterostructure nanocavity reported in a previous work [4]. Some works also reported an improved Q inspired by the visualization method [12,13,38]. It is also advantageous that this method requires only the 3D FDTD method. It will be possible to install the optimization steps described in Section 2 into commercial FDTD software. How the air holes at the leaky area should be shifted—left or right and up or down—is still a question for further study. However, the calculations for finding the optimum values in each optimization round are not labor-intensive because of the development of the FDTD calculation speed. Experimental work on L3 nanocavities with improved Qdesign values will be reported soon.
Acknowledgments
This work was partly supported by Future Pioneering Projects, the CPHoST program, and Project for Developing Innovation Systems of MEXT.
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