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OPTICS LETTERS / Vol. 40, No. 4 / February 15, 2015
Slot-embedded photonic-crystal resonator with enhanced modal confinement Chang Yeong Jeong,1 Chang-Koo Kim,2 and Sangin Kim1,* 1
2
Department of Electrical and Computer Engineering, Ajou University, Suwon 443-749, South Korea Department of Chemical Engineering and Division of Energy Systems Research, Ajou University, Suwon 443-749, South Korea *Corresponding author: [email protected] Received December 12, 2014; revised January 9, 2015; accepted January 12, 2015; posted January 14, 2015 (Doc. ID 230544); published February 9, 2015 A photonic-crystal (PC) resonator has attracted a great deal of attention for the strong light-matter interaction. Many attempts have been made to achieve a high-quality factor of the PC resonator, but they always have accompanied increases of modal volumes. In this work, we propose a novel method to enhance modal confinement of the PC resonator without compromising the quality factor. In the proposed structure, a thin low-index slot layer is embedded in a two-dimensional PC for vertical confinement, which results in a remarkable mode volume reduction without a decrease of the quality factor. By optimizing the slot thickness, a quality factor to mode volume ratio, which is a figure of merit for an optical resonator, could be increased by 8 times. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (350.4238) Nanophotonics and photonic crystals; (140.4780) Optical resonators; (170.4520) Optical confinement and manipulation. http://dx.doi.org/10.1364/OL.40.000554
Strong modal confinement is one of the most important factors for spontaneous emission enhancement [1,2], and a photonic crystal (PC) has attracted a great deal of attention for this purpose owing to its unique characteristic of bandgap [3]. Extensive research was conducted to obtain high-quality factor in two-dimensional (2D) PC resonators for enhanced light-matter interaction at room temperature [4–6]. However, the enhancement of the 2D PC’s quality factor always accompanied a mode volume increase, and there exists a tradeoff between the quality factor and the mode volume [7,8]. Since it was known that mode confinement of slab waveguides could be enhanced by introducing a thin slot [9–14], many attempts have also been made to reduce the mode volume of the 2D PC resonators by introducing an additional confinement means in the horizontal direction [15–18]. In all those works, the vertical direction mode confinement provided by a dielectric slab was rather week, so that a decrease of the quality factor was inevitable. Therefore, in this work, we proposed a novel type of 2D PC resonator structure to improve optical-modal confinement without compromising the quality factor. We demonstrate that by introducing a slot waveguide for the vertical confinement in the conventional 2D PC resonator, the tradeoff between modal confinement and quality factor can be relieved. The proposed slot-embedded PC resonator enables the localization of most of the electromagnetic energy in the thin slot of the point defect region. The resonant mode is confined by the PC in the horizontal direction and by the slot mode coupling in the vertical direction [9–11]. We hope that the novel mechanism of enhancing modal confinement introduced in this work can help achieve strong light-matter interaction at room temperature and realize more efficient light sources in the future. Figure 1 shows a schematic of the slot-embedded PC resonator, which consists of circular dielectric rods arranged in a rectangular lattice and a single-point defect. The lattice constant (a), the radius (r), and the refractive index (nr ) of the regular rods are fixed at 400 nm, 0.2a, and 3.45, respectively, and the radius of the point defect 0146-9592/15/040554-04$15.00/0
(r d ) is 0.16a. The height (h) is fixed at 2a for all rods. In order to consider a practicable low-index slot material inside the dielectric rods, we assume the slot material to be oxidized AlGaAs, which has a refractive index ns ns 1.60. To investigate the bare effect of the embedded slot on the resonant mode properties, we considered airsuspended dielectric rods and obtained the quality factor (Q) and the mode volume (V m ) as functions of the slot thickness (ts ). The quality factor Q and the mode volume V m are defined as the ratio of a resonant wavelength to its full-width at half-maximum and as R
Vm
εrjErj2 d3 r ; maxεrjErj2
(1)
respectively. The quality factor to mode volume ratio (Q∕V m ) is calculated as the performance metric of the proposed resonator. In this work, numerical calculations
Fig. 1. Slot-embedded photonic-crystal resonator, which consists of air-suspended dielectric rods arranged in a rectangular lattice and a single-point defect placed in the center. The lattice constant (a) and the height of the rods are fixed at 400 and 800 nm, respectively. The radii of the regular (r) and defect (r d ) rods are 0.2a and 0.16a, respectively. © 2015 Optical Society of America
February 15, 2015 / Vol. 40, No. 4 / OPTICS LETTERS
were conducted using the three-dimensional (3D) finitedifference time-domain (FDTD) method (Lumerical FDTD Solutions 8.6). A rectangular lattice of 21 by 21 rods was used for the calculation, ensuring that the resonant modes confined in the resonator were not influenced by the outermost boundaries of the perfectly matched layer. A dipole source was used to excite the resonant modes in the resonator and was placed inside the defect rod. Nonuniform mesh ranging from 1 to 20 nm was used for the calculation so that subwavelength structures including a thin slot could be meshed properly. Figures 2(a) and 2(b) show the photonic band structures for ts 0 nm (the bare dielectric rod PC) and ts 400 nm. The proposed PC consists of the dielectric rods and has the photonic bandgap for the transverse magnetic (TM) mode, which has an electric field parallel to the rods [3,19]. A wide bandgap appears for ts 0 nm due to the large index difference between the rods and the surrounding medium (air). However, when the thicker slot (ts 400 nm) is inserted, the effective refractive index of the rods decreases significantly, and thus the bandgap substantially narrows. The single-point defect placed in the PC can act as an efficient optical resonator and enables the single-mode operation [20–22]. Since the resonant-mode frequency is located inside the bandgap region and the bandgap frequency is higher for large ts , the resonant mode frequency increases as ts becomes thicker. Figure 3 depicts the resonant-mode profiles in the proposed resonator. Figures 3(a) and 3(b) show the energy density distributions of the proposed resonator without the slot in the horizontal and the vertical directions, respectively. In these distributions, the energy density (W ) is defined as ½εjEj2 , and the horizontal mode profile is obtained in the middle of the rods. As shown, the monopole mode is formed in the cavity, and most of the energy is confined in the point defect region. Figures 3(c) and 3(d) display the energy density distributions for ts 4 nm in the horizontal and the vertical directions, respectively, and the inset shows a magnified plot of the vertical-mode profile near the slot. The comparison indicates that the horizontal-mode profile is analogous to that of the bare PC resonator. However, in the vertical-mode profile, most of the energy is strongly localized in the thin slot region. This strong energy localization is achieved in the same way observed in dielectric slot waveguides, in which two TM polarized slab modes are coupled in a thin slot region and the field concentration is highly enhanced [9–14]. Overall, with the help of
Fig. 2. Bandstructures of the slot-embedded photonic crystal for (a) ts 0 nm and (b) ts 400 nm. The case for ts 0 nm corresponds to the bare photonic crystal without the slot. The radius of the rods is 0.2a.
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Fig. 3. Energy density distributions of the slot-embedded photonic crystal resonator. Energy density distributions without the slot in (a) the horizontal and (b) the vertical direction. Energy density distributions for the slot of ts 4 nm in (c) the horizontal and (d) the vertical direction. The inset shows the magnified plot of the energy density distribution near the slot.
the bandgap effect of the PC in the horizontal direction and the slot mode coupling effect in the vertical direction, strong 3D modal confinement is achieved in the proposed resonator. For a more systematic analysis, we investigated the resonant mode characteristics of the proposed resonator by varying ts . Figure 4(a) shows the Q and V m of the resonant modes as a function of the slot thickness ts from ts 0 to ts 20 nm. It is noted that the Q of the proposed resonator with small ts is almost identical to that with ts 0 nm (the bare PC resonator). This is because the index difference between the rods and the surrounding medium for those resonators is almost the same. Since the Q of the resonant mode is predominantly determined by the characteristics of the PCs, almost identical bandstructures of the two PCs give rise to similar values of Q. However, with the help of the strong energy localization created by the introduction of the slot, the V m of the resonant mode is significantly reduced, resulting in an almost seven-fold enhancement in the V m for the case of ts 2 nm. Figure 4(b) shows the overall performance Q∕V m of the proposed resonator. It is evident that the resonator’s performance is sharply enhanced by introducing the slot and gradually degraded with increasing
Fig. 4. Resonant mode characteristics of the slot-embedded photonic crystal resonator. (a) Quality factor Q and mode volume V m , and (b) Q∕V m of the resonant mode as functions of ts from 0 to 0.05a (20 nm).
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ts . Since the Q of the resonant modes for ts < ∼10 nm remains almost constant, the performance improvement is mainly attributed to the enhanced modal confinement. The overall resonator performance for a large slot deteriorated with increasing ts , and for ts > 300 nm, a more rapid decline was observed due to the degradation of both Q and V m (not shown here). As ts increased, Q was gradually increased and maximized at ts 300 nm, and for ts > 300 nm, Q was substantially degraded owing to the weak confinement of the PCs. The modal confinement also deteriorated rapidly as ts increased. The rapid degradation was attributed to two factors. First, modal coupling incurred as the slot became very weak for large ts . Previous research on dielectric slot waveguides showed that the two guided modes could not be strongly coupled in the slot for large thicknesses, leading to a large mode area [9–11]. Under the same principle, modal coupling in the proposed resonator was so weak for large ts that the modal confinement in the vertical direction became very poor, thereby increasing V m . Secondly, due to the large ts of the slot, the effective refractive index of the rods was considerably reduced, and it led to a small index difference between the rods and the surrounding air. This caused the resonant mode to be weakly confined in the horizontal direction and increased V m . Next, we briefly discuss the feasibility of a slotembedded PC resonator based on a dielectric slab. PC slab resonators that can support TM polarized-resonant modes have been reported in recent studies [23,24]. It seems that in such resonators, the introduction of a low-index slot enables a significant enhancement of the modal confinement of the resonant mode without lowering its quality factor. Figure 5(a) shows a schematic of a slot-embedded PC resonator based on a dielectric slab. The defects are
Fig. 5. Slot-embedded photonic-crystal resonator based on a dielectric slab with air holes. (a) Schematic of the 3D structure. The air holes are arranged in a triangular lattice. Energy density distributions without the slot in (b) the horizontal and (c) the vertical direction, and those for a slot of 2 nm in (d) the horizontal and (e) the vertical direction.
formed by introducing one missing hole and shrinking the radii of its six nearest holes. The lattice constant (a) of the PC is 510 nm, and the radii of the regular and the six nearest defect holes are 0.41a and 0.28a, respectively. The thicknesses of the slab and the introduced slot are 280 and 2 nm, correspondingly. The refractive indices of the slab and the slot are 3.45 and 1.60, respectively, which are the same values as those in the rod-based PC resonator investigated above. Figures 5(b) and 5(c) show the energy density distributions without the slot in the horizontal and the vertical directions, respectively, and Figs. 5(d) and 5(e) exhibit those for a slot thickness of 2 nm in the horizontal and the vertical directions, respectively. The inset of Fig. 5(e) displays the magnified plot of the vertical energy density distribution near the slot. It is observed that in the vertical direction, the modal energy is more strongly localized in the slot compared to that for the bare PC resonator, while the horizontal energy distributions do not exhibit large differences. The calculated Q and V m of the resonant mode for the bare PC resonator are 1013 and 0.036 μm3 , respectively, and those for the slot-embedded PC resonator are 1015 and 0.0043 μm3 , respectively, exhibiting an eight-fold enhancement of the modal confinement with similar values of Q. Figures 6(a) and 6(b) show the Q; V m , and Q∕V m of the resonant modes as functions of the slot thickness from 0 to 10 nm. As shown, the Q; V m , and Q∕V m of the resonator demonstrate the same trends observed in the rod-based PC resonators. Therefore, even for the dielectric slab-based slot-embedded PC resonator, strong 3D-modal confinement can be achieved. It is expected that with the implementation of full 3D-modal confinement without metal, strong light-matter interaction will be feasible in all-dielectric optical resonators in the future. In conclusion, we proposed a novel PC resonator structure and demonstrated that its modal confinement can be significantly enhanced without compromising the quality factor by introducing a thin low-index slot. We determined that the resonant modes can be strongly confined in the thin slot of a point defect region by the photonic bandgap effect in the horizontal direction and the slot mode coupling effect in the vertical direction, thereby achieving strong 3D-modal confinement. Therefore, the slot-embedded PC resonator can be effectively used to enhance light-matter interaction at room temperature and also be useful for a variety of photonic applications.
Fig. 6. Resonant mode characteristics of a slot-embedded photonic-crystal resonator based on a dielectric slab. (a) The quality factor Q and mode volume V m and (b) the Q∕V m of the resonant modes as functions of slot thickness.
February 15, 2015 / Vol. 40, No. 4 / OPTICS LETTERS
This work was supported by National Research Foundation of Korea Grants (NRF-2014-R1A2A2A01006720, NRF-2009-0094046, and NRF- 2012-R1A2A2A01004416). References 1. E. M. Purcell, Phys. Rev. 69, 37 (1946). 2. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, Phys. Rev. Lett. 95, 013904 (2005). 3. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008). 4. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, Nature 425, 944 (2003). 5. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005). 6. Y.-J. Fu, Y.-S. Lee, and S.-D. Lin, Opt. Lett. 38, 4915 (2013). 7. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron. 145, 391 (1998). 8. K. Nozaki and T. Baba, Appl. Phys. Lett. 88, 211101 (2006). 9. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, Opt. Lett. 29, 1209 (2004). 10. R. Ding, B.-J. Tom, W.-J. Kim, X. Xiong, R. Bojko, J. M. Fedeli, M. Fournier, and M. Hochberg, Opt. Express 18, 25061 (2010). 11. Y. C. Jun, R. M. Briggs, H. A. Atwater, and M. L. Brongersma, Opt. Express 17, 7479 (2009).
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12. Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, Opt. Express 18, 22061 (2010). 13. F. Dell’Olio and V. M. N. Passaro, Opt. Express 15, 4977 (2007). 14. A. Armaroli, A. Morand, P. Benech, G. Bellanca, and S. Trillo, J. Lightwave Technol. 27, 4009 (2009). 15. J. Gao, J. F. McMilan, M. C. Wu, J. Zheng, S. Assefa, and C. W. Wong, Appl. Phys. Lett. 96, 051123 (2010). 16. J. Jágerská, H. Zhang, Z. Diao, N. L. Thomas, and R. Houdré, Opt. Lett. 35, 2523 (2010). 17. J. D. Ryckman and S. M. Weiss, IEEE Photon. J. 3, 986 (2011). 18. M. G. Scullion, A. D. Falco, and T. F. Krauss, Biosens. Bioelectron. 27, 101 (2011). 19. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 328 (1993). 20. H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, Science 305, 1444 (2004). 21. H.-G. Park, S.-H. Kim, M.-K. Seo, Y.-G. Ju, S.-B. Kim, and Y.-H. Lee, IEEE J. Quantum Electron. 41, 1131 (2005). 22. M.-K. Seo, K.-Y. Jeong, J.-K. Yang, Y.-H. Lee, H.-G. Park, and S.-B. Kim, Appl. Phys. Lett. 90, 171122 (2007). 23. Y. Zhang, M. W. McCutcheon, I. B. Burgess, and M. Loncar, Opt. Lett. 34, 2694 (2009). 24. M. W. McCutcheon, P. B. Deotare, Y. Zhang, and M. Loncar, Appl. Phys. Lett. 98, 111117 (2011).
OPTICS LETTERS / Vol. 40, No. 4 / February 15, 2015
Slot-embedded photonic-crystal resonator with enhanced modal confinement Chang Yeong Jeong,1 Chang-Koo Kim,2 and Sangin Kim1,* 1
2
Department of Electrical and Computer Engineering, Ajou University, Suwon 443-749, South Korea Department of Chemical Engineering and Division of Energy Systems Research, Ajou University, Suwon 443-749, South Korea *Corresponding author: [email protected] Received December 12, 2014; revised January 9, 2015; accepted January 12, 2015; posted January 14, 2015 (Doc. ID 230544); published February 9, 2015 A photonic-crystal (PC) resonator has attracted a great deal of attention for the strong light-matter interaction. Many attempts have been made to achieve a high-quality factor of the PC resonator, but they always have accompanied increases of modal volumes. In this work, we propose a novel method to enhance modal confinement of the PC resonator without compromising the quality factor. In the proposed structure, a thin low-index slot layer is embedded in a two-dimensional PC for vertical confinement, which results in a remarkable mode volume reduction without a decrease of the quality factor. By optimizing the slot thickness, a quality factor to mode volume ratio, which is a figure of merit for an optical resonator, could be increased by 8 times. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (350.4238) Nanophotonics and photonic crystals; (140.4780) Optical resonators; (170.4520) Optical confinement and manipulation. http://dx.doi.org/10.1364/OL.40.000554
Strong modal confinement is one of the most important factors for spontaneous emission enhancement [1,2], and a photonic crystal (PC) has attracted a great deal of attention for this purpose owing to its unique characteristic of bandgap [3]. Extensive research was conducted to obtain high-quality factor in two-dimensional (2D) PC resonators for enhanced light-matter interaction at room temperature [4–6]. However, the enhancement of the 2D PC’s quality factor always accompanied a mode volume increase, and there exists a tradeoff between the quality factor and the mode volume [7,8]. Since it was known that mode confinement of slab waveguides could be enhanced by introducing a thin slot [9–14], many attempts have also been made to reduce the mode volume of the 2D PC resonators by introducing an additional confinement means in the horizontal direction [15–18]. In all those works, the vertical direction mode confinement provided by a dielectric slab was rather week, so that a decrease of the quality factor was inevitable. Therefore, in this work, we proposed a novel type of 2D PC resonator structure to improve optical-modal confinement without compromising the quality factor. We demonstrate that by introducing a slot waveguide for the vertical confinement in the conventional 2D PC resonator, the tradeoff between modal confinement and quality factor can be relieved. The proposed slot-embedded PC resonator enables the localization of most of the electromagnetic energy in the thin slot of the point defect region. The resonant mode is confined by the PC in the horizontal direction and by the slot mode coupling in the vertical direction [9–11]. We hope that the novel mechanism of enhancing modal confinement introduced in this work can help achieve strong light-matter interaction at room temperature and realize more efficient light sources in the future. Figure 1 shows a schematic of the slot-embedded PC resonator, which consists of circular dielectric rods arranged in a rectangular lattice and a single-point defect. The lattice constant (a), the radius (r), and the refractive index (nr ) of the regular rods are fixed at 400 nm, 0.2a, and 3.45, respectively, and the radius of the point defect 0146-9592/15/040554-04$15.00/0
(r d ) is 0.16a. The height (h) is fixed at 2a for all rods. In order to consider a practicable low-index slot material inside the dielectric rods, we assume the slot material to be oxidized AlGaAs, which has a refractive index ns ns 1.60. To investigate the bare effect of the embedded slot on the resonant mode properties, we considered airsuspended dielectric rods and obtained the quality factor (Q) and the mode volume (V m ) as functions of the slot thickness (ts ). The quality factor Q and the mode volume V m are defined as the ratio of a resonant wavelength to its full-width at half-maximum and as R
Vm
εrjErj2 d3 r ; maxεrjErj2
(1)
respectively. The quality factor to mode volume ratio (Q∕V m ) is calculated as the performance metric of the proposed resonator. In this work, numerical calculations
Fig. 1. Slot-embedded photonic-crystal resonator, which consists of air-suspended dielectric rods arranged in a rectangular lattice and a single-point defect placed in the center. The lattice constant (a) and the height of the rods are fixed at 400 and 800 nm, respectively. The radii of the regular (r) and defect (r d ) rods are 0.2a and 0.16a, respectively. © 2015 Optical Society of America
February 15, 2015 / Vol. 40, No. 4 / OPTICS LETTERS
were conducted using the three-dimensional (3D) finitedifference time-domain (FDTD) method (Lumerical FDTD Solutions 8.6). A rectangular lattice of 21 by 21 rods was used for the calculation, ensuring that the resonant modes confined in the resonator were not influenced by the outermost boundaries of the perfectly matched layer. A dipole source was used to excite the resonant modes in the resonator and was placed inside the defect rod. Nonuniform mesh ranging from 1 to 20 nm was used for the calculation so that subwavelength structures including a thin slot could be meshed properly. Figures 2(a) and 2(b) show the photonic band structures for ts 0 nm (the bare dielectric rod PC) and ts 400 nm. The proposed PC consists of the dielectric rods and has the photonic bandgap for the transverse magnetic (TM) mode, which has an electric field parallel to the rods [3,19]. A wide bandgap appears for ts 0 nm due to the large index difference between the rods and the surrounding medium (air). However, when the thicker slot (ts 400 nm) is inserted, the effective refractive index of the rods decreases significantly, and thus the bandgap substantially narrows. The single-point defect placed in the PC can act as an efficient optical resonator and enables the single-mode operation [20–22]. Since the resonant-mode frequency is located inside the bandgap region and the bandgap frequency is higher for large ts , the resonant mode frequency increases as ts becomes thicker. Figure 3 depicts the resonant-mode profiles in the proposed resonator. Figures 3(a) and 3(b) show the energy density distributions of the proposed resonator without the slot in the horizontal and the vertical directions, respectively. In these distributions, the energy density (W ) is defined as ½εjEj2 , and the horizontal mode profile is obtained in the middle of the rods. As shown, the monopole mode is formed in the cavity, and most of the energy is confined in the point defect region. Figures 3(c) and 3(d) display the energy density distributions for ts 4 nm in the horizontal and the vertical directions, respectively, and the inset shows a magnified plot of the vertical-mode profile near the slot. The comparison indicates that the horizontal-mode profile is analogous to that of the bare PC resonator. However, in the vertical-mode profile, most of the energy is strongly localized in the thin slot region. This strong energy localization is achieved in the same way observed in dielectric slot waveguides, in which two TM polarized slab modes are coupled in a thin slot region and the field concentration is highly enhanced [9–14]. Overall, with the help of
Fig. 2. Bandstructures of the slot-embedded photonic crystal for (a) ts 0 nm and (b) ts 400 nm. The case for ts 0 nm corresponds to the bare photonic crystal without the slot. The radius of the rods is 0.2a.
555
Fig. 3. Energy density distributions of the slot-embedded photonic crystal resonator. Energy density distributions without the slot in (a) the horizontal and (b) the vertical direction. Energy density distributions for the slot of ts 4 nm in (c) the horizontal and (d) the vertical direction. The inset shows the magnified plot of the energy density distribution near the slot.
the bandgap effect of the PC in the horizontal direction and the slot mode coupling effect in the vertical direction, strong 3D modal confinement is achieved in the proposed resonator. For a more systematic analysis, we investigated the resonant mode characteristics of the proposed resonator by varying ts . Figure 4(a) shows the Q and V m of the resonant modes as a function of the slot thickness ts from ts 0 to ts 20 nm. It is noted that the Q of the proposed resonator with small ts is almost identical to that with ts 0 nm (the bare PC resonator). This is because the index difference between the rods and the surrounding medium for those resonators is almost the same. Since the Q of the resonant mode is predominantly determined by the characteristics of the PCs, almost identical bandstructures of the two PCs give rise to similar values of Q. However, with the help of the strong energy localization created by the introduction of the slot, the V m of the resonant mode is significantly reduced, resulting in an almost seven-fold enhancement in the V m for the case of ts 2 nm. Figure 4(b) shows the overall performance Q∕V m of the proposed resonator. It is evident that the resonator’s performance is sharply enhanced by introducing the slot and gradually degraded with increasing
Fig. 4. Resonant mode characteristics of the slot-embedded photonic crystal resonator. (a) Quality factor Q and mode volume V m , and (b) Q∕V m of the resonant mode as functions of ts from 0 to 0.05a (20 nm).
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ts . Since the Q of the resonant modes for ts < ∼10 nm remains almost constant, the performance improvement is mainly attributed to the enhanced modal confinement. The overall resonator performance for a large slot deteriorated with increasing ts , and for ts > 300 nm, a more rapid decline was observed due to the degradation of both Q and V m (not shown here). As ts increased, Q was gradually increased and maximized at ts 300 nm, and for ts > 300 nm, Q was substantially degraded owing to the weak confinement of the PCs. The modal confinement also deteriorated rapidly as ts increased. The rapid degradation was attributed to two factors. First, modal coupling incurred as the slot became very weak for large ts . Previous research on dielectric slot waveguides showed that the two guided modes could not be strongly coupled in the slot for large thicknesses, leading to a large mode area [9–11]. Under the same principle, modal coupling in the proposed resonator was so weak for large ts that the modal confinement in the vertical direction became very poor, thereby increasing V m . Secondly, due to the large ts of the slot, the effective refractive index of the rods was considerably reduced, and it led to a small index difference between the rods and the surrounding air. This caused the resonant mode to be weakly confined in the horizontal direction and increased V m . Next, we briefly discuss the feasibility of a slotembedded PC resonator based on a dielectric slab. PC slab resonators that can support TM polarized-resonant modes have been reported in recent studies [23,24]. It seems that in such resonators, the introduction of a low-index slot enables a significant enhancement of the modal confinement of the resonant mode without lowering its quality factor. Figure 5(a) shows a schematic of a slot-embedded PC resonator based on a dielectric slab. The defects are
Fig. 5. Slot-embedded photonic-crystal resonator based on a dielectric slab with air holes. (a) Schematic of the 3D structure. The air holes are arranged in a triangular lattice. Energy density distributions without the slot in (b) the horizontal and (c) the vertical direction, and those for a slot of 2 nm in (d) the horizontal and (e) the vertical direction.
formed by introducing one missing hole and shrinking the radii of its six nearest holes. The lattice constant (a) of the PC is 510 nm, and the radii of the regular and the six nearest defect holes are 0.41a and 0.28a, respectively. The thicknesses of the slab and the introduced slot are 280 and 2 nm, correspondingly. The refractive indices of the slab and the slot are 3.45 and 1.60, respectively, which are the same values as those in the rod-based PC resonator investigated above. Figures 5(b) and 5(c) show the energy density distributions without the slot in the horizontal and the vertical directions, respectively, and Figs. 5(d) and 5(e) exhibit those for a slot thickness of 2 nm in the horizontal and the vertical directions, respectively. The inset of Fig. 5(e) displays the magnified plot of the vertical energy density distribution near the slot. It is observed that in the vertical direction, the modal energy is more strongly localized in the slot compared to that for the bare PC resonator, while the horizontal energy distributions do not exhibit large differences. The calculated Q and V m of the resonant mode for the bare PC resonator are 1013 and 0.036 μm3 , respectively, and those for the slot-embedded PC resonator are 1015 and 0.0043 μm3 , respectively, exhibiting an eight-fold enhancement of the modal confinement with similar values of Q. Figures 6(a) and 6(b) show the Q; V m , and Q∕V m of the resonant modes as functions of the slot thickness from 0 to 10 nm. As shown, the Q; V m , and Q∕V m of the resonator demonstrate the same trends observed in the rod-based PC resonators. Therefore, even for the dielectric slab-based slot-embedded PC resonator, strong 3D-modal confinement can be achieved. It is expected that with the implementation of full 3D-modal confinement without metal, strong light-matter interaction will be feasible in all-dielectric optical resonators in the future. In conclusion, we proposed a novel PC resonator structure and demonstrated that its modal confinement can be significantly enhanced without compromising the quality factor by introducing a thin low-index slot. We determined that the resonant modes can be strongly confined in the thin slot of a point defect region by the photonic bandgap effect in the horizontal direction and the slot mode coupling effect in the vertical direction, thereby achieving strong 3D-modal confinement. Therefore, the slot-embedded PC resonator can be effectively used to enhance light-matter interaction at room temperature and also be useful for a variety of photonic applications.
Fig. 6. Resonant mode characteristics of a slot-embedded photonic-crystal resonator based on a dielectric slab. (a) The quality factor Q and mode volume V m and (b) the Q∕V m of the resonant modes as functions of slot thickness.
February 15, 2015 / Vol. 40, No. 4 / OPTICS LETTERS
This work was supported by National Research Foundation of Korea Grants (NRF-2014-R1A2A2A01006720, NRF-2009-0094046, and NRF- 2012-R1A2A2A01004416). References 1. E. M. Purcell, Phys. Rev. 69, 37 (1946). 2. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, Phys. Rev. Lett. 95, 013904 (2005). 3. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008). 4. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, Nature 425, 944 (2003). 5. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005). 6. Y.-J. Fu, Y.-S. Lee, and S.-D. Lin, Opt. Lett. 38, 4915 (2013). 7. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron. 145, 391 (1998). 8. K. Nozaki and T. Baba, Appl. Phys. Lett. 88, 211101 (2006). 9. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, Opt. Lett. 29, 1209 (2004). 10. R. Ding, B.-J. Tom, W.-J. Kim, X. Xiong, R. Bojko, J. M. Fedeli, M. Fournier, and M. Hochberg, Opt. Express 18, 25061 (2010). 11. Y. C. Jun, R. M. Briggs, H. A. Atwater, and M. L. Brongersma, Opt. Express 17, 7479 (2009).
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