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
Photoni c crystals (PhCs) with photonic band gaps (PBGs) are expected to be key platforms for future large-scale optical integrated circuits [1,2]. 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 . 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,  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 .
2. PBG waveguides
2-1. Management of radiative, scattering, and coupling loss
Definitely, “loss” is the most important issue as regards PBG waveguides. The most significant loss in PBG waveguides in 2D PC slabs is radiative loss, which occurs when the mode lies above the light line of the cladding material. Typically, this loss is larger than 100 dB/mm. To eliminate this loss, we have to tune the mode below the light line, which can be accomplished in various ways related to the structural design of waveguides. For example, if we vary the width of the line-defect waveguides, the mode can be easily tuned. Hereafter, we refer the width of waveguides as a unit of the width of a normal single-missing-hole-line defect, that is, W=sqrt(3) a. (a is the lattice constant). We have undertaken a detailed study of how this strategy works for PBG waveguides with SiO2 cladding [6,7].
If we are working below the light line of the cladding by employing an appropriate choice of structure, we do not have to worry about radiative loss, but there is still much to concern us. Recently, we measured the propagation loss of single-mode PBG waveguides at wavelengths below the light line in a SiO2-clad PhC slab and air-clad PhC slabs, as shown in Fig. 1(a). The obtained loss (1.5 dB/mm for the SiO2-clad PhC and 0.6 dB/mm for the air-clad PhC) is better than that of our previous reports (~6 dB/mm) for SiO2-clad PhC [6,7]. This loss is the lowest value reported for single-mode PhC waveguides operating within the PBG. We believe that the measured propagation loss mainly originates from extrinsic scattering loss due to some structural disorder (roughness and size fluctuation), and the improvement from the previous result is due to improvements in fabrication accuracy and homogeneity. A quantitative explanation using roughness data and cladding difference is now being sought, and will be reported elsewhere .
Another issue related to loss is coupling loss. The connection loss between a single-mode fiber (SMF) and a PBG waveguide must be huge (~30 dB) because of the large difference in their spot size. To reduce this connection loss, we have reported the successful fabrication of an adiabatic mode connector that utilizes a taper and a continuously tuned PhC to connect the mode in a single-mode fiber to a mode in a PBG waveguide, as shown in Fig. 1(b) [9,10]. We measured the conversion loss by butt-coupling an SMF. The measured conversion loss was 3–4 dB per port, which is significantly better than 30 dB. Recently, we applied this technique to various kinds of PhC samples (waveguides and resonators), which will be described later in this paper.
2-2. Group delay in slow light mode
The dispersion is tunable over a wide range in PBG waveguides, which is one of their most important features. A large dispersion over a wide wavelength range cannot be achieved by other conventional waveguides. In addition to simple applications to dispersion-controlled waveguides, the large dispersion controllability of group velocity may prove important for various applications related to optical nonlinear phenomena.
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 . 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.
Figure 2 shows the result we obtained with the time-domain method. We observed a significantly large time delay of up to ~100–200 psec for the 1-mm sample. When we compare it with the transmission spectrum, it is apparent that the large delay is observed near the mode gap edge. This large delay corresponds to a group index of 30–60. As far as we know, this is the largest time delay ever reported in PhCs. The observation of such a large time delay indicates the high quality of our samples because the propagation loss is generally more pronounced for slower light modes and a large time delay requires longer samples. In addition, the observed group delay dispersion is very similar to the group index dispersion in the frequency-domain measurement in Ref . Thus, we confirmed that the large group dispersion is the intrinsic dispersion of the PBG waveguides, which can be observed in frequency- or time-domain phenomena. It should be noted that, although Fabry-Perot resonance is still visible in the group delay curves in Fig. 2 (relatively large noise-like fluctuation is due to Fabry-Perot resonance), this time-domain experiment does not need intrinsic Fabry-Perot resonance, unlike the previous frequency-domain measurement.
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. ), 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. ), 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  and coherent population modulation , 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.
Recently, however, there has been considerable success regarding this matter in 2D PhC slabs. The approaches that have been adopted are all intended to minimize the k distribution inside the radiation cone of the 2D PhC slabs (which is basically the same as the light cone of the cladding material discussed in 2.2). Johnson et al. proposed  the use of multi-nodal modes in which the radiation in the vertical direction is reduced due to destructive interference. Ryu et al. extended their idea  and used whispering gallery modes (quadrupole modes) in square PhC slabs. Vuckovic et al. showed  that the introduction of edge dislocation in the center of the dipole modes in PhC slabs increases Q.
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. . 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.
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 , and we have independently studied this scenario  to realize a resonant-tunneling-type resonator-waveguide coupled system, which will be described later.
The proposed structure is shown in Fig. 4(a, b). The cavity is composed of a two-(or more) point defect, in which we optimized the position and radius of the innermost holes to realize a high Q. The high Q mode in this cavity is related to a dipole mode in a conventional single-point cavity and will converge into line-defect modes with a normal width (1.0 W) when the length becomes large. From a 3D FDTD calculation, we found that Q increases greatly as the length becomes longer after an appropriate modification of the innermost holes. The calculated Qs are 8.2×104, 1.57×105, 2.34×105, and 3.45x105 for 2-, 3-, 4-, and 5-point cavities, respectively. The estimated mode volumes are 7.06, 7.59, 9.39, and 10.56 (λ/2n)3 for 2-, 3-, 4-, and 5-point cavities, respectively.
4. Coupled elements
4-1. Coupling between resonators and waveguides
In the previous section, we showed that high-Q and ultrasmall resonators can be realized by PBG resonators in PhC slabs. These resonators can be effectively coupled to waveguides in various ways. This is one of the most important advantages of PBG resonators compared with other micro-resonators. Resonator-waveguide coupled systems are starting points for various applications, and will be key components in future PhC-based optical circuits. Here we discuss how we implement resonators in functional coupled systems. First, we investigate a simple two-port design in which there are only single input and output waveguides. Using this design, we investigate a resonant-tunneling transmission filter. Then, we show four-port and three-port designs in which there are two (or one) input and two output waveguides. A channel-drop filter can be implemented with this design.
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 . 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.  Thus, in order to realize a high QT and a high T simultaneously, we must carefully design QV and QH to achieve optimization.
We have already fabricated a simple version of this resonant tunneling filter, which is a combination of a single-missing-hole point defect coupled to single-missing-hole line defects . Although we confirmed the resonant tunneling operation, the observed Q and T were low (~1000 and a few %, respectively). The reason for the poor performance was a low Qv and inappropriate QH design. To realize a large QT and high T in the resonant tunneling filters, first, we have to use high QV resonators, and then appropriately design QH according to the relation 1/QT=1/QV+1/QH and T=(QT/QH)2. From this design rule, it is clear that QV should be as high as possible. This naturally leads to the use of the high-Q resonators discussed in the previous section..
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%.
4.3. Two-port design for SiO2-clad PhC: resonant tunneling filter using mode gap
In this section, we study another design for resonant tunneling filters, as shown in Fig. 6(a). This structure is basically a variation of the width-reduced waveguides that can operate below the light line of SiO2 cladding (not air). It should be remembered that the filters in the previous section are designed for so-called air-clad PhC slabs and do not work for SiO2-clad PhC slabs. This is because the previous structure is a variation of 1.0 W waveguides, which do not have a meaningful transmission window below the light line of SiO2. By contrast, the present structure can be implemented for SiO2-clad PhC slabs.
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. .
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.
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 , 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.
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. 
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.
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.
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