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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1551–1561
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Waveguides, resonators and their coupled elements in photonic crystal slabs

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1551-1561 (2004)
http://dx.doi.org/10.1364/OPEX.12.001551


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Abstract

The design, fabrication, and measurement of photonic-band-gap (PBG) waveguides, resonators and their coupled elements in two-dimensional photonic crystal (PhC) slabs have been investigated. We have studied various loss mechanisms in PBG waveguides and have achieved a very low propagation loss (~1 dB/mm). For these waveguides, we have observed a large group delay (>100 ps) by time-domain measurement. As regards PBG resonators, we realize very high-Q and small volume resonators in PhC slabs by appropriate design. Finally, we demonstrate various forms of coupled elements of waveguides and resonators: 2-port resonant-tunneling transmission devices, 4-port channel-drop devices using the slow light mode, and 3-port channel-drop devices using the resonant-tunneling process.

© 2004 Optical Society of America

1. Introduction

Photoni c crystals (PhCs) with photonic band gaps (PBGs) are expected to be key platforms for future large-scale optical integrated circuits [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

,2

2. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals (Princeton University Press, 1995).

]. Due to the unique properties of PhCs, the size of most optical components can be greatly reduced and the strength of light-matter interaction can be significantly increased. Initially it seemed that three-dimensional (3D) PhCs with 3D PBGs were essential. However, recent studies show that very good performance can be obtained for 2D PhC slab structures [3

3. S.G. Johnson, S. Fan, P.R. Villeneuve, J. D. Joannopoulos, and L.A. Kolodziejski, “Guided modes in photonic-crystal slabs,” Phys. Rev. B 60, 5751–5780 (1999). [CrossRef]

]. PhC slabs are 2D PhC structures located within high-index-contrast slab waveguides. In PhC slabs, lightwaves are confined by a combination of in-plane PBG confinement and vertical total-internal-reflection (TIR) confinement. Although we still use TIR in the vertical direction, lightwaves can be strongly confined within ultra-small waveguides and resonators in PhC slabs without practical leakage if we design their structures appropriately.

In this paper, we report our recent progress on PBG waveguides and resonators in PhC slabs as regards both design and fabrication. In relation to PBG waveguides, we will describe our recent success in decreasing the scattering and coupling loss, which enables us to measure the group delay dispersion of slow-light modes in PBG waveguides directly by a time-domain method. With regard to PBG resonators, we will demonstrate two types of structure designed to increase quality-factor (Q) significantly while keeping the mode volume small: hexagonal modes and line-defect-like modes. The final topic is coupling among waveguides and resonators. As discussed later, coupled elements will be key components for future PhC applications. We will investigate various forms of coupled elements: 2-port resonanttunneling transmission devices, 4-port channel-drop devices using the slow light mode, and 3-port channel-drop devices using the resonant-tunneling process. To design these devices, we use various unique properties of PBG waveguides and resonators that have been clarified recently, such as the slow light mode, resonant tunneling, and the waveguide mode gap.

Our basic structure in this study is a Si PhC slab based on a silicon-on-insulator (SOI) substrate, known as an SOI PhC slab, [4

4. I. Yokohama, M. Notomi, A. Shinya, C. Takahashi, and T. Tamamura, “Two-dimensional Si photonic crystals with 0.2-µm thickness on oxide using SOI substrates,” in Tech. Digest of Fifth Optoelectronics and Communication Conference (OECC 2000), 42–43 (2000).

] which can operate in the fiber communication wavelength region. The PhC is formed by electron-beam lithography and electron-cyclotron-resonance ion-stream dry etching in the top Si layer whose thickness is approximately half a wavelength in Si. The SiO2 layer works as a lower cladding layer. The upper cladding layer is air. We refer to this as a SiO2-clad PhC slab in this paper. Sometimes we selectively etch off the SiO2 layer to form an air-bridge structure. In such cases, both cladding layers are air. We call this an air-clad type PhC slab. The typical geometrical parameters are the lattice constant a=400 nm, radius r=0.275a, and thickness t=0.5a. The fabrication process has already been reported in detail [5

5. C. Takahashi, J. Takahashi, M. Notomi, and I. Yokohama, “Accurate dry etching with fluorinated gas for two-dimensional Si photonic crystals” in 2000 Fall Meeting of Materials Research Society (MRS Proceedings, vol. 637, E1.8, 2000).

].

2. PBG waveguides

2-1. Management of radiative, scattering, and coupling loss

Fig. 1. (a) Propagation loss measurement for single-mode PBG waveguides in SiO2-clad and air-clad PhCs. (b) Schematic diagram of an adiabatic mode connector, which consists of a spot-size converter and a mode-profile converter.

2-2. Group delay in slow light mode

We previously reported the experimental observation of an extremely large group velocity dispersion and slow group velocity (~1/100c) by using frequency-domain measurements, that is, we used intrinsic Fabry-Perot interference within the sample to deduce the group index [11

11. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]

]. Here, we report a group delay dispersion measurement undertaken using the time-domain method, in which we measure the group delay of the transmitted signal to determine the group index of the PBG waveguide. We input a 3-GHz sinusoidally-modulated optical signal from a tunable laser-diode light source into the PBG waveguide, and measured the phase delay of the transmitted signal by an electronic phase comparator to deduce the delay time. By sweeping the wavelength, we can obtain the wavelength dependence of the group delay. For this experiment, we used 0.65W 0.76-mm-long PBG waveguides in SiO2-and-polymer-clad PhCs with an adiabatic mode connector and 1.0W 1-mm-long PBG waveguides in air-clad PhCs.

Fig. 2. Time-domain measurement of group delay dispersion in PBG waveguides in SOI PhC slabs. (a) 0.65W waveguide in SiO2-clad PhCs. (b) 1.0W waveguides in air-clad PhCs.

We observed ultra-slow modes in the 1.0W waveguide, but not in the 0.65W waveguide. We believe that this can be explained by the fact that the slow-mode region in the 1.0W waveguides is near the anticrossing point between the PBG-guided and TIR-guided modes. Since in this frequency region the PBG-guided and TIR-guided modes have opposite sign of curvature (e.g., see Fig. 3(a) in Ref. [11

11. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]

]), the anticrossing produces a very flat mode in the wide k range. The slow-mode region in the 0.65W waveguides, however, there exists only TIR-guided mode (e.g., see Fig. 3(b) in Ref. [11

11. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]

]), and thus the slow mode is limited only in the vicinity of the zone boundary in the k space. Later, we will make use of this slow light mode in the 1.0W waveguide in channel drop filters.

Recently, slow light modes have been observed in various phenomena including electromagnetically induced transparency [12

12. L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–596 (1999). [CrossRef]

] and coherent population modulation [13

13. M.S. Bigelow, N.N. Lepeshkin, and R.W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

], by controlling the material dispersion. Such slow modes are expected to provide ultimate control of the light-matter interaction with a small light intensity. The origin of the slow light in Fig. 2 is very different (it is purely due to control of the structural dispersion), but this result indicates that the PBG waveguide may be another candidate for enhancing light-matter interaction.

3. PBG resonators

3.1. High-Q small-V resonators in PhC slabs

The Q of optical resonators that rely on TIR confinement is limited by their mode volume Vm. That is, if Vm is reduced to λ-scale size, Q will always decrease to a very small value. This can be intuitively explained as follows. Small-volume modes have a broad distribution in the k-space, but TIR confinement only works for the k component that satisfies the TIR condition. That is, a broad k distribution results in a decrease in Q. However, PBG confinement does not impose such a restriction on the k distribution. Therefore, ultimately small-Vm and high-Q resonators are expected to be realized. Such small and high-Q resonators are required for large cavity quantum electrodynamics effects in a solid state, and are also important for various applications. This is the most important feature of PBG resonators. In this context, a 3D PBG appears to be a fundamental requirement because 2D PBG resonators impose a similar restriction on the k distribution in the out-of-plane direction. In fact, the Q of 2D PBG resonators decreases when the mode size is reduced.

3.2. Hexapole cavity in hexagonal PhC

We investigated multi-nodal modes in hexagonal PhC slabs. Hexagonal PhCs are very important since they have the largest 2D PBG. First, we look for suitable multi-nodal modes in hexagonal PhC slabs. Many kinds of multi-nodal modes can be obtained in a large cavity, but we want to keep the volume as small as possible. The most suitable candidate is the hexapole mode because it is the smallest order of multi-nodal mode whose symmetry matches that of the crystal lattice. To eliminate the vertical radiation loss, a very delicate balance is required. Thus, other modes, such as quadrupole modes, are inferior in terms of obtaining a small Vm and a large Q.

We investigated the Q value of the hexapole modes in air-clad hexagonal PhC slabs by using 3D FDTD, and we optimized the structural parameters to achieve a high Q and a small volume simultaneously. [17

17. H-Y. Ryu, M. Notomi, and Y-H. Lee, “High quality-factor and small mode-volume hexapole modes in photonic crystal slab nano-cavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]

]. We modified the radius (r m) of six nearest-neighbor holes to lower the frequency, as shown in Fig. 3(a). It should be noted that the positions of these six holes shift when r m is varied. In Fig. 3, we plot the electric field distribution of a typical hexapole mode in (b) real space and (c) k-space. The k-space distribution in (c) shows that the Fourier intensity is largely concentrated around six M points. The white circle represents the light cone of air. Note that there is almost no intensity inside the light cone in (c). This clearly shows that this mode (c) exhibits very little radiation into the air.

Figure 3(d) shows Q and V as a function of the resonance frequency (the varied parameter is r m). Here the hole radius r of the crystal is kept constant at 0.35a. A significantly high Q value of 5×105 is obtained when r m=0.26 a. The effective mode volume V is 7.2(λ/2n)3, which is slightly larger than that of the dipole modes in hexagonal PhCs but still a very small value. It should be noted that when Q is at its maximum value, V is almost at its minimum in this case. The high Q and small V mechanisms must be similar to that in the quadrupole modes in square PhCs. The higher Q and smaller V values in hexagonal PhCs compared with those in square PhCs can be attributed to the larger PBG bandwidth of the hexagonal PhCs and the larger azimuthal mode number (=3) of the hexapole modes.

Fig. 3. Hexapole cavity: (a) structural design, (b) field distribution in the real space (r=0.35a, r m=0.26a), (c) field distribution in the k space (r=0.35a, r m=0.26), (d) Q and Vm versus the resonant frequencies.

The achieved Q is already much larger than those of other high-Q cavities in PhC slabs. But we found that further improvement is possible by varying the radius of the crystal. When r=0.275a, a Q of 2×106 is achieved at r m=0.26a. This value is the highest ever reported for PBG cavities. Under this condition, the resonance frequency approaches the bandedge, and therefore the mode delocalization also contributes to improving the Q value.

3.3. Short Fabry-Perot resonator

Another way of increasing Q is to make use of line-defect-like modes. An infinitely long line defect that operates below the light line of the cladding has a theoretically infinite Q. Of course, when the length is finite and becomes shorter, Q will be decreased. But if one is going to modify a line defect operating below the light line into a tiny short line-defect-like cavity, there is still much chance of obtaining a significantly large Q. In other words, if the mode profile of line-defect resonators is not so different from that of infinitely long line-defects, their Fourier amplitude should mostly lie outside the light cone. Similar strategies have been pointed out recently [18

18. Y. Akahane, T. Asano, B.S. Song, and S. Noda, “High-Q photonic nanocavity in two-dimensional photonic crystal,” Nature 425, 944 (2003). [CrossRef] [PubMed]

], and we have independently studied this scenario [19

19. S. Mitsugi, A. Shinya, E. Kuramochi, M. Notomi, T. Tsuchizawa, and T. Watanabe, “Resonant tunneling wavelength filters with high Q and high transmittance based on photonic crystal slabs,” in 2003 IEEE LEOS Annual Meeting Conference Proceedings (LEOS 2003), 214–215 (2003).

] to realize a resonant-tunneling-type resonator-waveguide coupled system, which will be described later.

Fig. 4. Short line-defect-like mode: (a) 2-point cavity (L=2). Shaded holes are shifted in the outward direction to increase Q. (b) 3-point cavity (L=3). (c) Q and Vm versus the defect length.

4. Coupled elements

4-1. Coupling between resonators and waveguides

4-2. Two-port design: resonant tunneling transmission filter

Here we investigate the structure shown in Fig. 5(a). In this device, a single resonator is coupled to two straight waveguides (input and output). Sometimes this coupling configuration is called as “direct coupling”. Light from the input waveguide can pass through the resonator into the output waveguide by the resonant tunneling process. This tunneling occurs only when the light wavelength matches the resonance wavelength of the cavity. That is, this device works as a wavelength filter. We call this a (two-port) resonant tunneling transmission filter.

The performance of this device is represented by the transmission Q and transmittance T. The transmission characteristics of these resonance devices can be analyzed by the coupled-mode theory [20

20. C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. , 35, 1322–1331 (1999). [CrossRef]

]. This analysis confirms that the transmission characteristics can be explained by vertical Q (QV) and horizontal (in-plane) Q (QH). QV is mainly due to decay into the radiation loss from the cavity, that is, the Q of the isolated cavity, if the crystal size is sufficiently large. QH is mainly due to decay into the waveguides as a result of coupling. Then total Q (QT) is given by 1/QT=1/QV+1/QH and simple analysis of the coupled-mode theory leads to energy transmittance (T) expressed as T=(QT/QH)2. [21

21. M. Notomi, A. Shinya, E. Kuramochi, S. Mitsugi, H-Y. Ryu, T. Kawabata, T. Tsuchizawa, T. Watanabe, T. Shoji, and K. Yamada, “Photonic-band-gap waveguides and resonators in SOI Photonic crystal slabs,” to be published in IEICE Trans. Electron. March issue, (2004).

] Thus, in order to realize a high QT and a high T simultaneously, we must carefully design QV and QH to achieve optimization.

As shown in Fig. 5, we used the 2-point cavities (L=2 line-defect-like resonators) described in 3.2. as resonators. The estimated QV of this resonator is ~38500. Using this resonator, we aimed at high-Q and large-T filters. We designed QH~7300, then T~70%.

Fig. 5. Two-port resonant tunneling transmission filter: (a) Structural design. Shaded holes are shifted in the outward direction to increase Q. (b) Measured transmission spectra for a resonant tunneling filter and a reference PhC waveguide without a cavity.

4.3. Two-port design for SiO2-clad PhC: resonant tunneling filter using mode gap

The basic structure consists of a 0.65 W (W=sqrt(3) a) width-reduced waveguide, but the radius of the nearest-neighbor holes varies slightly in the barrier region, as shown in Fig. 6(a). This increase in the hole radius in the barrier part shifts the transmission window, and this part becomes equivalent to a 0.40 W waveguide. Since the mode gap of 0.40 W waveguides is located within the transmission window of 0.60 W waveguides, the light field is confined in this mode gap wavelength region. That is, the center part (cavity part) functions as a resonator although there is no true PBG in the barrier part. Another type of resonator that utilizes the mode gaps of PBG waveguides has also been discussed by Kinoshita et al. [23

23. K. Inoshita and T. Baba, “Lasing at bend, branch and intersection of photonic crystal waveguides,” Electron. Lett. , 39, 844–846 (2003). [CrossRef]

].

We fabricated this mode-gap type of resonant tunneling filter with adiabatic mode connectors as described in section 2.1. This is the first PBG resonator device that can be directly connected to single-mode fibers. Figure 6(b, c) show the transmission spectra of this device. The measured Q and energy transmittance values are 408 and 86% for (b) and 1350 and 12% for (c), respectively. The theoretical transmittance values obtained with the 3D FDTD method are 89% and 18%, which agree well with the experimental results.

Fig. 6. Resonant-tunneling filter using the mode gap in the width-varied waveguides: (a) structural design. (b) Measured transmission spectrum around the resonant wavelength for the barrier width of Nb=2. (c) Measured transmission spectrum for Nb=3. The samples are fabricated in SOI PhC slabs with SiO2 and polymer cladding having adiabatic mode connectors (described in 2.1).

4.4. Four-port design: channel drop filter using slow light mode

Fan et al. proposed using two singly-degenerate resonators coupled to two straight waveguides (bus and drop lines) as a channel drop filter [24

24. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998). [CrossRef]

], in which a particular wavelength signal is selectively dropped from the bus line to the drop line, and other channels are transmitted to the bus line. The key issue is to utilize bonding and anti-bonding modes in the double cavities to cancel out the undesired output channels by the destructive interference. To obtain good performance, the following requirements must be met. 1) The two resonators should be identical. 2) The lifted degeneracy due to the finite coupling effect should be cancelled out by appropriate structural tuning. However, this is not an easy task.

Here, we propose another design for a similar channel drop filter that is shown in Fig. 7(a). In this design, we use only one resonator, but it has a pair of quasi-degenerate even and odd modes. Thus, in practice this functions in a similar way to the channel drop device proposed by Fan et al. The key feature of our device is that our resonator is a type of short line-defect resonator (see 3.3) with a width of 1.0 W and we use the waveguide mode with a very low group velocity region in 1.0 W waveguides (see 2.2) to form the resonator mode. Since the 1.0 W waveguide has an exceptionally large group index near the mode gap, the frequency separation between the lowest even and lowest odd modes is very small. The existence of quasi-degenerate even and odd modes makes the desired channel drop operation possible.

Fig. 7. Four-port channel drop filters using the slow light mode in 1.0 W waveguides: (a) structural design, (b) measured transmission spectra for the drop line and the bus line (through).

2D-FDTD calculation proved that the proposed device exhibits almost 100% dropping efficiency, and recently we have fabricated real devices in SOI PhC slabs with SiO2 lower cladding layers. Figure 7(b) shows the measured transmission spectra which show that dropping efficiency is larger than 80%.

4.5. Three-port design: channel drop filter using resonant tunneling process

The channel drop filter discussed in 4.4 makes use of destructive interference to eliminate undesired channel outputs. To realize appropriate interference, the structural symmetry is very important and the phase relation among different ports must be carefully designed, especially when using two resonators. To solve this problem, we propose another design for channel drop filters, which uses a resonant tunneling process. In 4.2, we investigated two-port devices using the resonant tunneling process. Here, we propose three-port devices using the same process, which function as channel drop filters. The proposed device is shown schematically in Fig. 8(a). The width of the bus-line waveguide is changed from W0 to W1 in a similar way to the resonant tunneling filter using the mode gap described in 4.3. Thus, a signal whose wavelength lies in the mode gap of the W1 waveguide will be reflected. But if the signal wavelength is resonant with the cavity mode, the signal can resonantly tunnel into the drop-line waveguide. Since in this operation we do not use any interference, careful control of the phase relation and symmetry is unnecessary.

Figure 8(b) shows an FDTD simulation of transmission spectra for the drop and bus lines. Note that almost 100% dropping efficiency is achieved. To cascade this device, we propose another design, shown in Fig. 8(c), where the drop-line waveguide is tilted to the resonator, and we install resonators on both sides of the waveguide. We confirmed by numerical calculation that basically the same performance is obtained with this design. This cascaded design functions as a multi-channel drop filter using the resonant tunneling process. The detailed characteristics of this design will be reported elsewhere. [25

25. A. Shinya, M. Notomi, S. Mitsugi, E. Kuramochi, T. Kawabata, S. Kondo, T. Watanabe, and T. Tsuchizawa, “Photonic crystal devices combining width-tuned waveguides and cavities,” International Workshop on Photonic Electromagnetic Structures, Kyoto, 2004.

]

Fig. 8. Three-port resonant-tunneling channel drop filter: (a) basic configuration to explain the operation principle. (b) Simulated transmission of (a). (c) Structural design of the cascaded multi-channel drop filter using the resonant tunneling process.

5. Summary

It is assumed that future PhC-based photonic integrated circuits will be based on a combination of PBG waveguides and PBG resonators. We showed that a significant improvement has been achieved for PBG waveguides and resonators in PhC slab structures, which can be easily fabricated using SOI substrates and mature Si nanofabrication technology. We showed that the propagation loss is reduced to ~1 dB/mm and the poor connection efficiency with SMFs has been overcome by introducing adiabatic mode connectors. Their low loss characteristics resulted in a large group delay time (>100 psec) in 1.0 W PBG waveguides. The limitation due to the vertical radiation loss in the PBG resonators in PhC slabs has been largely overcome by the use of an appropriate design. We described two different methods for improving the Q of PBG resonators in PhC slabs.

Since the performance of PBG waveguides and resonators is being rapidly improved, we are now targeting coupled elements (that is, PBG resonators coupled to PBG waveguides). We showed that it is possible to realize various kinds of coupled elements in PhC slabs: for example 2-port resonant-tunneling transmission filters, 4-port channel-drop filters using the slow light mode, and 3-port channel-drop filters using the resonant tunneling process. Most of these elements have been fabricated and their operation has been confirmed. The unique properties of PBG waveguides and resonators, such as mode gaps in waveguides, resonant tunneling through the cavity, and slow light modes, were utilized for designing these elements.

Acknowledgments

We acknowledge T. Tsuchizawa, T. Watanabe, H. Fukuda, and K. Yamada in NTT Microsystem Integration Laboratories for their collaboration in the course of this work, and T. Kawabata, S. Kondo, G. Kira, D. Takagi, and T. Tamamura for their help. We also thank Y. Hirayama for his continuous encouragement.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

2.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals (Princeton University Press, 1995).

3.

S.G. Johnson, S. Fan, P.R. Villeneuve, J. D. Joannopoulos, and L.A. Kolodziejski, “Guided modes in photonic-crystal slabs,” Phys. Rev. B 60, 5751–5780 (1999). [CrossRef]

4.

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5.

C. Takahashi, J. Takahashi, M. Notomi, and I. Yokohama, “Accurate dry etching with fluorinated gas for two-dimensional Si photonic crystals” in 2000 Fall Meeting of Materials Research Society (MRS Proceedings, vol. 637, E1.8, 2000).

6.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, “Single-mode transmission within the photonic bandgap of width-varied single-line-defect photonic crystal waveguides on SOI substrates,” Electron. Lett. 37, 243–244 (2001). [CrossRef]

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11.

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12.

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–596 (1999). [CrossRef]

13.

M.S. Bigelow, N.N. Lepeshkin, and R.W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

14.

S.G. Johnson, A. Mekis, S. Fan, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001). [CrossRef]

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H-Y. Ryu, S-H. Kim, H-G. Park, J-K. Hwang, Y-H. Lee, and J-S. Kim, “Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,” Appl. Phys. Lett. 80, 3883–3885 (2002). [CrossRef]

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J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002). [CrossRef]

17.

H-Y. Ryu, M. Notomi, and Y-H. Lee, “High quality-factor and small mode-volume hexapole modes in photonic crystal slab nano-cavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]

18.

Y. Akahane, T. Asano, B.S. Song, and S. Noda, “High-Q photonic nanocavity in two-dimensional photonic crystal,” Nature 425, 944 (2003). [CrossRef] [PubMed]

19.

S. Mitsugi, A. Shinya, E. Kuramochi, M. Notomi, T. Tsuchizawa, and T. Watanabe, “Resonant tunneling wavelength filters with high Q and high transmittance based on photonic crystal slabs,” in 2003 IEEE LEOS Annual Meeting Conference Proceedings (LEOS 2003), 214–215 (2003).

20.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. , 35, 1322–1331 (1999). [CrossRef]

21.

M. Notomi, A. Shinya, E. Kuramochi, S. Mitsugi, H-Y. Ryu, T. Kawabata, T. Tsuchizawa, T. Watanabe, T. Shoji, and K. Yamada, “Photonic-band-gap waveguides and resonators in SOI Photonic crystal slabs,” to be published in IEICE Trans. Electron. March issue, (2004).

22.

Y. Xu, Y. Li, R.K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E62, 7389–7404 (2000).

23.

K. Inoshita and T. Baba, “Lasing at bend, branch and intersection of photonic crystal waveguides,” Electron. Lett. , 39, 844–846 (2003). [CrossRef]

24.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998). [CrossRef]

25.

A. Shinya, M. Notomi, S. Mitsugi, E. Kuramochi, T. Kawabata, S. Kondo, T. Watanabe, and T. Tsuchizawa, “Photonic crystal devices combining width-tuned waveguides and cavities,” International Workshop on Photonic Electromagnetic Structures, Kyoto, 2004.

OCIS Codes
(230.5750) Optical devices : Resonators
(230.7370) Optical devices : Waveguides
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Focus Issue: Photonic crystals and holey fibers

History
Original Manuscript: March 2, 2004
Revised Manuscript: March 29, 2004
Published: April 19, 2004

Citation
M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, "Waveguides, resonators and their coupled elements in photonic crystal slabs," Opt. Express 12, 1551-1561 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1551


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References

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  21. M. Notomi, A. Shinya, E. Kuramochi, S. Mitsugi, H-Y. Ryu, T. Kawabata, T. Tsuchizawa, T. Watanabe, T Shoji, and K. Yamada, "Photonic-band-gap waveguides and resonators in SOI Photonic crystal slabs," to be published in IEICE Trans. Electron. March issue, (2004).
  22. Y. Xu, Y. Li, R.K. Lee, and A. Yariv, �??Scattering-theory analysis of waveguide-resonator coupling,�?? Phys. Rev. E62, 7389-7404 (2000).
  23. K. Inoshita and T. Baba, �??Lasing at bend, branch and intersection of photonic crystal waveguides,�?? Electron. Lett., 39, 844-846 (2003). [CrossRef]
  24. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, �??Channel drop tunneling through localized states,�?? Phys. Rev. Lett. 80, 960-963 (1998). [CrossRef]
  25. A. Shinya, M. Notomi, S. Mitsugi, E. Kuramochi, T. Kawabata, S. Kondo, T. Watanabe, and T. Tsuchizawa, "Photonic crystal devices combining width-tuned waveguides and cavities," International Workshop on Photonic Electromagnetic Structures, Kyoto, 2004.

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