We have investigated the localized design of heterostructure photonic crystal nanocavities in order to make them more suitable for integration. While retaining theoretical quality factors of more than ten million, the total length of the heterostructure nanocavity can be reduced to ~5 μm and the shifted air holes comprising the heterostructure can be restricted to the two rows nearest the nanocavity on each side. Though the area for the heterostructure nanocavity investigated thus far was larger than 10 × 10 μm2 in the photonic crystal slab, heterostructure nanocavities of this new design have sizes of approximately 3 × 5 μm2, thus allowing dense integration.
©2009 Optical Society of America
Nanocavities in two-dimensional (2D) photonic crystal (PC) slabs can have high quality (Q) factors and small modal volumes (V) approaching one cubic wavelength [1,2]. They are currently attracting much attention in various fields of research such as wavelength-selective filters [3–5], optical pulse memories [6,7], slow light [8,9], low-threshold nanolasers [10–12], single-photon emitters [13–15], environmental sensors [16,17], and quantum information processing [18,19]. Dense photonic integration is desirable in all these applications, requiring not only high Q factors but also small nanocavity sizes.
In 2005, we proposed a photonic heterostructure nanocavity that utilizes the photonic mode gap (PMG) effect to confine light in the cavity . This type of cavity consists of a line-defect waveguide formed by a row of missing air holes with gradual variation of the lattice constant. Theoretical Q factors (Q ideal) exceeding 10 million can be achieved, a value that was not attainable in other types of photonic crystal cavities with small V. Various designs for nanocavities that utilize the PMG effect have recently been proposed in order to realize high Q ideal values [20–24]. These nanocavities have been intensively studied with the expectation of realizing the novel optical devices mentioned above, but their total length (the length of the line defect forming the nanocavity) is significantly longer than that of point-defect types of nanocavities. For example, the so-called L3 cavity consists of three missing air holes with a length of only ~2 μm . In our previous reports, we formed our heterostructures by applying shifts of the lattice constant to many rows of air holes running parallel to the line-defect nanocavity. This procedure results in the area comprising the heterostructure nanocavity being larger than 10 × 10 μm2 in the PC slab. This is much larger than the L3 cavity and hinders the integration of these nanocavities, limiting their usefulness in a wide range of research areas. Nevertheless, large nanocavities have been widely employed thus far because of their simple structural design and similarity to the original model . Furthermore, we have always considered the heterostructure design to be capable of dense localization and therefore high flexibility.
In this work, we describe a modified design for heterostructure nanocavities that makes them more suitable for photonic integration. We have investigated both experimentally and theoretically how long the line defect forming the nanocavities must be to retain Q ideal factors of over ten million and how many rows of air holes with lattice constant shifts are required. We demonstrate that the nanocavity length can be shortened to ~5 μm and the shifted air holes can be restricted to the two rows neighboring the line defect on each side, while retaining values of Q ideal that are as high as in the original structure. A heterostructure nanocavity of this new design can be integrated into an area of approximately 3 × 5 μm2 in the 2D PC slab.
2. Nanocavity with a two-step heterostructure
Figure 1(a) shows the basic structure of the nanocavity with a two-step heterostructure studied in this letter. The PC consists of a triangular lattice of circular air holes with radii of 117 nm formed in a 247-nm-thick silicon (Si) slab. These dimensions were determined by scanning electron microscopy from the samples investigated in Sections 3 and 4. The nanocavity consists of a line defect of missing air holes and the lattice constant in the x-direction increases by 5 nm every two periods as the center of the nanocavity is approached. The lattice constants of the central (PC2), intermediate (PC2’), and outer regions (PC1) of the nanocavity are a 2 = 420 nm, a 2’ = 415 nm, and a 1 = 410 nm, respectively. The lattice constant in the y-direction is 710 nm in all regions in order to satisfy the lattice-matching condition. In such a line defect, the transmission and mode-gap band frequencies vary according to the x-direction lattice constant, as shown in Fig. 1(b). In this band structure, the resonant mode in the ground state is confined in the PC2 and PC2’ regions due to the PMG of the PC1 regions; this is the high-Q nanocavity mode and is indicated by the dashed curve. The other higher-order resonant modes have been studied in . The color image superimposed on Fig. 1(a) represents the y-component of the electric field distribution (Ey) for the mode, calculated using the three-dimensional (3D) finite difference time domain (FDTD) method. This mode has a Q ideal exceeding 10 million when the structure’s total area is ~30 lattice constants in the x-direction and 20 rows in the y-direction. This is the minimum size that can still form a photonic band gap (PBG) and the PMG structures capable of confining light in the nanocavity .
In this paper, we define the total length (L) of the heterostructure nanocavity to be the distance between the two air holes defining the edges of the line defect, as indicated in Fig. 1(a). Because little attention was paid to the number of missing air holes in the PC1 region in our previous studies, L was typically much longer than the length of point-defect type nanocavities; for example, L was over 100 μm in the study of , 16.44 μm in , and 11.52 μm in . Such large values of L prohibit the dense alignment of multiple nanocavities in the x-direction. Furthermore, the PC2 and PC2’ regions with lattice constant shifts were extended to the entire device in the y-direction, a width of more than 10 μm. This design does not allow other photonic components to be easily placed in the direction lateral to the heterostructure nanocavity . Therefore, it is desirable both to minimize L and to localize the width of the heterostructure (the PC2 and PC2’ regions) around the line defect. We experimentally and theoretically examine the effect of reducing L in Section 3 and the effect of localizing the heterostructure around the line defect in Section 4.
3. Effect of reducing total length of heterostructure nanocavity
In Fig. 1(a), the line defect in the PC1 regions simply acts as a barrier for the nanocavity mode and hence the number of missing air holes in the PC1 regions can be reduced while maintaining a high Q factor. We investigated five different nanocavities with 10, 5, 3, 2, and 1 missing air holes in the PC1 region. We refer to these structures as NC10, NC5, NC3, NC2, and NC1 with corresponding lengths L of 11.52, 7.42, 5.78, 4.96, and 4.14 μm, respectively (Fig. 1 shows NC5). We note that to date, the highest reported experimental Q factor (Q exp) of 2.5 million was achieved for NC10 .
Figure 2 shows a sketch of the sample design and an outline of the time-domain measurement used to determine Q exp for the five nanocavities. All five nanocavities were formed parallel to an excitation waveguide consisting of a row of missing air holes. The cavity measurements are independent of this device design provided that their resonant wavelengths do not overlap with each other. It has previously been demonstrated that random fluctuations of the resonant wavelengths of nanocavities are introduced by inevitable variations in the radii and positions of the air holes due to the limited precision of nanofabrication techniques . Because these wavelength fluctuations are on a sub-nanometer scale in our samples, whereas the spectral widths of the high-Q resonant modes are less than 1 pm, the probability of overlap among the five nanocavities is negligible. The excitation waveguide is 10% wider than the nanocavities in order to allow excitation of the nanocavities via evanescent mode coupling. The separation between the five nanocavities and the excitation waveguide was set to 6 rows of air holes, which provides sufficient coupling for the dropped light from the nanocavities to be measurable while keeping the influence of the excitation waveguide on Q exp negligible.
Firstly, we investigated the five nanocavities using conventional transmission spectroscopy, obtaining the spectra for dropped light, the spectra for transmitted light, and near-field images for the dropped light. These measurements confirmed that the spectral peaks corresponding to the dropped light from the five nanocavities did not overlap and that the transmittances were greater than 90% at the resonant wavelengths, confirming that the excitation waveguide had little effect on Q exp. Details of this measurement have been presented previously [27,30]. Secondly, we performed the time-domain measurement outlined in Fig. 2 in order to evaluate the lifetimes of the photons (τ) trapped in the nanocavities. Both measurements were performed at room temperature and standard pressure. The tunable, continuous-wave laser was set to the resonant wavelength for each nanocavity and the polarization was controlled to produce pulsed light with a high ON/OFF ratio. A Mach-Zehnder LiNbO3 modulator was used to produce rectangular light pulses with widths of approximately 4 ns, while the electrical pulses were supplied by a pulse generator. The ON/OFF ratio in the pulses was kept above 4000 throughout the experiment. We confirmed using mode-coupled theory that this value was high enough to accurately measure photon lifetimes of the order of a nanosecond. The pulsed light was coupled to the excitation waveguide via a single-mode lensed fiber, where piezoelectric actuators were used to adjust the position of the fiber tip. The photons dropped from the nanocavities in the direction vertical to the slab were collimated using a 0.26-numerical-aperture objective lens, and coupled using another lens to an optical fiber with a core diameter of 80 μm. This configuration helped to eliminate stray light. Finally, the time-domain evolution of emission from the nanocavities was measured using a photomultiplier tube (PMT) by applying the time-correlated single-photon counting (TCSPC) method, where trigger signals were supplied by the pulse generator. The photocathode material was InP/InGaAs and the temperature of the PMT module was 213 K.
Figures 3(a) −(d) present the time-resolved signals obtained from the above experiment, where photon counts are plotted on a logarithmic scale. The measurement time was four minutes for all data. Figure 3(a) corresponds to the scattered light at the input waveguide facet, which confirms the shape of the input pulse. The step near 7 ns is due to the afterpulsing effect in the PMT. Because this effect occurs between 0.5 ns and 2 ns after the start of pulse decay and is more than three orders of magnitude smaller than that of the main peak, the contribution to the decay signal on the order of a nanosecond is negligible. The decay time τ in the fitted exponential curve exp(–t/τ) is 140 ps for the rectangular pulse edge and is hence the measurement resolution. Figures 3(b)–(d) show the emission signals for NC10, NC2, and NC1, respectively. The experimentally determined resonant wavelengths λ, photon lifetimes τ, and Q exp factors for each nanocavity are summarized in Table 1 . The values of λ are distributed randomly between 1600.87 and 1601.40 nm. The values of τ for all nanocavities except NC1 lie in the range 1.8−2.2 ns, giving similar values of Q exp to the highest previously reported . This suggests that the theoretical Q ideal factor could be as high for NC2 as for NC10. The value of τ for NC1 is considerably smaller than those of the other nanocavities, as apparent in Fig. 3(d). This gives a Q exp of less than one million, compared to more than two million for the larger nanocavities. This substantial difference suggests that Q ideal is smaller for NC1 than for the other nanocavities.
Table 2 summarizes the values of λ, Q ideal, and V for the five nanocavities calculated using the 3D FDTD method. The refractive index of the Si slab was set to 3.46 in the calculations. The calculated wavelengths are in good agreement with the experimental results. The Q ideal value for NC1 is considerably smaller than for the other nanocavities, which is also in good agreement with experiment. We note that the calculated values of λ, Q ideal, and V are constant for NC10, NC5, and NC3 and only change for the smaller nanocavities NC2 and NC1. The difference in λ between NC10 and NC1 is only 0.19 nm, which is too small to distinguish experimentally due to the residual variations of the air holes as explained in the sample structure description. In contrast, the reduction of Q ideal with nanocavity size is rather large, being ten times smaller for NC1 than for NC10. The values of Q exp are determined not only by Q ideal but also by the parameter Q loss according to the following relationship: 1/Q exp = 1/Q ideal + 1/Q loss. Here, Q loss is an additional Q factor originating from imperfections in the fabricated samples . This formula shows that when Q ideal is much larger than Q loss, Q exp is heavily influenced by Q loss. The values of Q loss for our five cavities are all of the order of 3.0 × 106, as shown in Table 1. This is smaller than Q ideal for all nanocavities except NC1. Therefore, the reduction of Q exp is significant only for NC1. We suspect that the main origin of Q loss in our samples is sub-nanometer-scale variations in the radii and positions of the air holes. These variations apply randomly to all air holes and thus could induce fluctuations of λ and Q exp even in cavities with the same structure . This is the reason for the variation in Q exp between NC10, NC5, and NC3.
The rapid decrease of Q ideal for NC2 and NC1 can be explained using the same concepts previously reported for the L3 cavity . The out-of-slab radiation leakage from the nanocavities is mainly caused by the abrupt change of the electric field Ey along the line defect, which is due to the sudden decrease of the refractive index at the cavity edge air holes. This phenomenon does not occur in heterostructure nanocavities with a long line defect, because the electric field is sufficiently reduced at the cavity edges by the PMG effect, as shown in Fig. 1(a). Therefore, a smooth Ey profile is obtained and exceptionally high values of Q ideal can be achieved. When the number of missing air holes in the line defect is reduced by too much, the air holes at the cavity edges significantly change the profile of Ey and cause the large reduction of Q ideal. This boundary lies between NC3 and NC2 for the nanocavity shown in Fig. 1. The gradual blue-shift of the wavelength and the decrease of V for NC2 and NC1 are also due to the cavity-edge air holes, which decrease the effective refractive index relative to the nanocavity mode field and make the electric field profile tighter.
The emission images from the nanocavities clearly demonstrate the validity of the above explanation. Figures 4(a) −(c) and (d)−(f) show the experimental and calculated near-field images for NC3, NC2, and NC1, respectively, taken through a 0.4 numerical aperture objective lens. The dashed line superimposed on each image indicates the region of missing air holes. The calculated image for NC3 has a single-lobed spot at the center of the nanocavity, which is the same as that for NC10. However, the calculated image for NC2 has additional side lobes near the cavity edges. Moreover, for NC1 the main spot at the center of the nanocavity vanishes and the side lobes represent the principal radiation. The experimental and calculated images are in reasonable agreement. The larger spot sizes in the experimental images are due to the additional convolution in optical microscopy. The other differences between the images might originate from the random nature of Q loss and the reflection of backwardly emitted radiation from the substrate. Nevertheless, these images confirm that optical scattering loss due to the air holes at the line-defect edges is the main reason for the decreased Q factors in NC1.
We conclude from these results that the total length of the nanocavity with a two-step heterostructure can be reduced to ~5 μm, or thirteen missing air holes, which allows Q ideal to be maintained at more than ten million and the modal volume to be kept below 1.4(λ/n)3. We have thus experimentally confirmed that heterostructure nanocavities can be independently integrated in the x-direction with a separation of ten lattice constants or less than 5 μm.
4. Effect of localizing heterostructure surrounding nanocavity
Next, we consider the effect of localizing the heterostructure that surrounds the nanocavity. In the structure of Fig. 1(a), regions PC2 and PC2’ with lattice constant shifts are introduced along the y-direction throughout the whole device. This design is generally unfavorable for photonic integration. For example, in the device shown in Fig. 2, the heterostructure is also introduced into the excitation waveguide parallel to the nanocavities, complicating evanescent mode coupling between them and causing scattering of the transmitted light due to the cavity modes originating from the odd mode.
It is apparent in Fig. 1(a) that the electric field of the cavity mode rapidly decreases in the y-direction. Therefore, the number of rows of shifted air holes can be reduced while maintaining the characteristics of the high-Q nanocavity. Figure 5 shows a cavity with a localized heterostructure, in which the air hole shifts are restricted to a small number of rows on each side of the cavity. The basic structural parameters for this cavity are the same as those in Fig. 1(a). However, this design consists of four PC regions: PC0, PC1, PC2, and PC2’. The lattice constants in the x-direction for each region are a 0 = 410nm, a 1 = 407.5 nm, a 2 = 417.5 nm, and a 2’ = 412.5 nm, respectively, whereas the lattice constant of 710 nm in the y-direction is the same for each region. The outer region of the cavity, PC0, has the basic PC structure. The three regions PC1−PC2 are now restricted to three rows on each side of the line defect. PC1 is the barrier region for the high-Q cavity mode, in which the number of missing air holes is set to four (L = 6.15 μm). PC2 and PC2’ are the central and intermediate regions of the cavity, respectively. In order to minimize lattice mismatch in the x-direction at the interfaces of PC0 with the other regions, the values of a 1, a 2, and a 2’ have been reduced by 2.5 nm compared to those in Fig. 1(a). These changes are important in order to avoid the formation of defect modes and consequent optical scattering loss at the boundary between PC0 and the heterostructure. We fabricated and performed calculations on four cavities, in which the y-widths of regions PC1−PC2 were 9, 7, 5, and 3 rows. We refer to these structures as NC4-W9, NC4-W7, NC4-W5, and NC4-W3, respectively (Fig. 5 shows the structure of NC4-W7). The experimental procedure used to investigate the characteristics of the four cavities was the same as that shown in Fig. 2. The separation between the excitation waveguide and the cavities was set to 8 rows in order to obtain appropriate optical coupling. The sample was fabricated on the same chip as the sample investigated in Section 3, close to the device shown in Fig. 2. We performed the same experiments as described in Section 3 and observed the fundamental high-Q cavity mode at wavelengths close to that of NC10.
The experimentally determined values of λ, τ, Q exp, and Q loss for the four nanocavities with localized heterostructures are summarized in Table 3 . The resonant wavelengths are shorter than those of NC10−NC1 because the smaller lattice constants a 1−a 2’ result in smaller effective refractive indices relative to the cavity modes. The value of λ for NC4-W3 is significantly shorter than those for the other three cavities, which are almost identical. The values of Q exp for all four cavities are greater than 2 million and Q loss is similar to that of the samples investigated in Section 3. These results suggest that Q ideal for all four cavities might be as high as that for NC10. The somewhat smaller Q loss for NC4-W3 is within the range expected for random variation of this parameter. Unlike the samples studied in Section 3, we did not observe any side lobes at the cavity edges in the near-field images, and no defect modes or scattered light around the cavities were apparent.
The corresponding calculated values obtained using the 3D FDTD method are summarized in Table 4 . We include a reference nanocavity NC10, in which the lattice constants a 1−a 2’ are 2.5 nm smaller than those of NC10 in Section 3. The values of λ obtained from the calculations are in good agreement with the experimental results. The gradual blue-shift from NC4-W9 to NC4-W3 is caused by the reduction in the effective refractive index of the mode. The electric field components in the PC2 (a 2 = 417.5 nm) and PC2’ (a 2’ = 412.5 nm) regions decrease with reduction of the heterostructure y-width while those in the PC0 (a 0 = 410.0 nm) region increase. The large shift in λ for NC4-W3 indicates that the electric field is confined to a short distance in the y-direction. The reference nanocavity has similar values of Q ideal and V to NC10 in Section 3, and little change in these values is seen for samples NC4-W9, NC4-W7 and NC4-W5. This observation indicates that the optical characteristics of heterostructure nanocavities do not change as the number of shifted air holes rows is decreased to four. The color image superimposed on Fig. 5 represents the Ey for NC4-W7, which is almost identical to that in Fig. 1(a). In contrast, for NC4-W3 the values of V and Q ideal change significantly, as well as λ. Nanocavities with larger V tend to have larger Q ideal because the spreading of components for the mode is suppressed in wavevector space, which should reduce the components in the light cone . However, we were unable to confirm this increase of Q ideal experimentally due to the effect of Q loss.
The considerations above demonstrate that this localized heterostructure can be used in conjunction with high-Q nanocavities possessing Q ideal of more than ten million and V of less than 1.5(λ/n)3. Furthermore, the shifted air holes forming the heterostructure can be limited to the two rows nearest the line defect on each side. In other words, this type of nanocavity can be integrated in both the x-direction and y-direction, covering an area of approximately 5 × 3 μm2 in the PC slab. This size is at most fifty percent larger than the L3 nanocavity in terms of the electric field distribution of the nanocavity mode.
It is important to state that nanocavities with a localized heterostructure possess a degree of flexibility. For example, a Q ideal value of 2.8 × 107 with λ = 1593.60 nm is obtained in the case of NC4-W3 where a 0 = 410nm, a 1 = 408 nm, a 2 = 416 nm, and a 2’ = 412 nm, while a Q ideal value of 2.7 × 107 with λ = 1594.56 nm is obtained for NC4-W3 where a 0 = 410nm, a 1 = 407 nm, a 2 = 415 nm, and a 2’ = 415 nm. Because controlling the air hole positions on a sub-nanometer precision is possible using the current electron beam lithography techniques, various combinations of several lattice constants can exist that retain Q ideal over 10 million. This structural flexibility while retaining a high Q ideal is convenient for the fine adjustment of λ. We have also confirmed that these cavities have Q ideal values of more than ten million even when the number of missing air holes in the PC1 region is as little as three. Based on these merits, we expect that high-Q nanocavities with localized heterostructures will prove to be useful in various fields of research.
We have investigated the localized design of photonic crystal nanocavities with a two-step heterostructure in order to make them more suitable for photonic integration. While retaining theoretical quality factors of more than ten million and modal volumes of less than 1.5(λ/n)3, the total length of the heterostructure nanocavity can be reduced to ~5 μm and the lattice constant shifts can be localized to as few as two neighboring rows of air holes on each side of the line defect. Accordingly, nanocavities with two-step heterostructures can be integrated in an area of approximately 3 × 5 μm2 in a PC slab. This area is approximately fifty percent larger than that of the L3 cavity. Our new designs demonstrate the high flexibility of heterostructure cavities. Furthermore, we expect that nanocavities with localized heterostructures will be utilized for various applications including ultra-small wavelength filters, compact optical-buffer memories, environmental sensors, and novel emitters.
This work was partly supported by Special Coordination Funds for Promoting Science and Technology, by Core Research for Evolutional Science and Technology of the Japan Science and Technology Agency, by the Global COE Program, and by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology- Japan.
References and links
2. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-High-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]
7. J. Upham, Y. Tanaka, T. Asano, and S. Noda, “Dynamic increase and decrease of photonic crystal nanocavity Q factors for optical pulse control,” Opt. Express 16(26), 21721–21730 (2008). [CrossRef] [PubMed]
9. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]
10. M. Nomura, S. Iwamoto, K. Watanabe, N. Kumagai, Y. Nakata, S. Ishida, and Y. Arakawa, “Room temperature continuous-wave lasing in photonic crystal nanocavity,” Opt. Express 14(13), 6308–6315 (2006). [CrossRef] [PubMed]
12. M. Yamaguchi, T. Asano, and S. Noda, “Photon emission by nanocavity-enhanced quantum anti-Zeno effect in solid-state cavity quantum-electrodynamics,” Opt. Express 22(22), 18067–18081 (2008). [CrossRef]
13. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005). [CrossRef] [PubMed]
14. W. H. Chang, W. Y. Chen, H. S. Chang, T. P. Hsieh, J. I. Chyi, and T. M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006). [CrossRef] [PubMed]
15. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]
18. A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quautum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83(20), 4204–4207 (1999). [CrossRef]
19. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]
20. 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(4), 041112 (2006). [CrossRef]
21. S. Tomljenovic-Hanic, C. M. de Sterke, and M. J. Steel, “Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration,” Opt. Express 14(25), 12451–12456 (2006). [CrossRef] [PubMed]
23. S. Tomljenovic-Hanic, C. M. de Sterke, M. J. Steel, B. J. Eggleton, Y. Tanaka, and S. Noda, “High-Q cavities in multilayer photonic crystal slabs,” Opt. Express 15(25), 17248–17253 (2007). [CrossRef] [PubMed]
24. S. H. Kwon, T. Sünner, M. Kamp, and A. Forchel, “Ultrahigh-Q photonic crystal cavity created by modulating air hole radius of a waveguide,” Opt. Express 16(7), 4605–4614 (2008). [CrossRef] [PubMed]
25. Y. Takahashi, Y. Tanaka, H. Hagino, T. Asano, and S. Noda, “Higher-order resonant modes in a photonic heterostructure nanocavity,” Appl. Phys. Lett. 92(24), 241910 (2008). [CrossRef]
26. E. Kuramochi, H. Taniyama, T. Tanabe, A. Shinya, and M. Notomi, “Ultrahigh-Q two-dimensional photonic crystal slab nanocavities in very thin barriers,” Appl. Phys. Lett. 93(11), 111112 (2008). [CrossRef]
28. Y. Akahane, T. Asano, H. Takano, B. S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005). [CrossRef] [PubMed]
29. H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79(8), 085112 (2009). [CrossRef]
30. T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006). [CrossRef]