1. Introduction

A photonic crystal (PhC) with a photonic band gap is a promising candidate as a platform on which to construct devices with dimensions of several wavelengths for future photonic integrated circuits. Recently, ultrasmall resonators with high quality factors and very low loss waveguides (WGs) in a PhC slab with a triangular air-hole lattice have been demonstrated and many functional devices have been proposed [1–9]. In particular, a resonator-waveguide coupled system in which single-mode waveguides are effectively coupled to an ultrasmall resonator with a high quality factor is expected to be used as a channel add/drop filter [4–7] or all-optical switching device [8, 9]. However, it is not easy to design a composite system because the coupling characteristic between a resonator and a WG in a PhC is different from that of a conventional system. For example, a conventional structure in which a ring resonator is arranged between parallel WGs can function as a channel drop filter, and its dropping efficiency is almost 100 %. By contrast, the theoretical limitation of the dropping efficiency of a similar structure in a PhC is only 25 % because the light output from the resonator is divided into four ports [4, 10].

Some structures have been reported that successfully solve this problem [4–6, 10]. Manolatou et al. [10] and S. Fan et al [4] proposed a structure in which two isolated single-mode resonators are positioned between parallel WGs to achieve a dropping efficiency of 100%. The distance between the two isolated resonators is optimized so that the output light from these two resonators destructively interferes in the undesired ports and the light exits from only one port. But this distance optimization is difficult because the positions of the air holes that are removed when constructing a resonator and WG are discretely determined in the PhC. Moreover, when we use a short distance to allow us to reduce the device size, the direct coupling between the two resonant modes disturbs the destructive interference.

Song et al. [5] and Noda et al. [6] proposed an excellent way to eliminate the direct coupling mentioned above. One resonator is positioned between parallel WGs. This system is composed of two kinds of PhCs (heterostructure) so that the light reflected back from the hetero-interface and the light coupled out from the resonator destructively interfere to eliminate any output to undesired ports. Moreover, a multi-channel-drop-filter can be constructed by cascading these structures. Nevertheless, the difficulty of optimizing the distance between the resonator and the hetero-interface remains.

These filters using destructive interference necessitate careful handling of the phase relation and take a long time to design. To solve this problem, we devised a new channel-drop filter that does not require careful consideration of the interference process.

2. Three-port resonant tunneling filter

Our coupled resonator-waveguide system is based on a two-port resonant tunneling filter that can automatically eliminate the output toward the input port and achieve highly efficient dropping from the output port. This makes the design much simpler than that of a conventional dropping filter. However, a two-port system cannot be used to realize a multiport filter, which has a lot of drop WGs, because in principle all the light that does not resonate with the resonator is reflected back to the input port.

So we first develop a three-port-resonant-tunneling filter that has a through WG. The through WG (P2) and an additional WG (P4) are attached to the input WG (P1) and the drop WG (P3), respectively, of a two-port system as shown in Fig. 1. The structures of P1 and P2 are the same as those of P3 and P4 to keep the device structure symmetric. Moreover the resonant frequency of the resonator is tuned in a frequency band where the light can propagate in P1 and P3 but not in P2 and P4 due to the difference between the transmission bands of WGs. As a result, the system can be regarded as a two-port system and the light is output only from P3 when the frequency is resonant. Moreover, the light is output only from P2 when the frequency is in a common band where the light can propagate in both P1 and P2, because there is no resonating mode in the resonator and the light cannot reach P3 or P4. This is the mechanism of our three-port resonant tunneling filter.

 

Fig. 1. Three-port resonant tunneling filter. (a), (b), (c) are schematic diagrams of three kinds of three-port resonant tunneling filters. P1, P2, P3 and P4 are input, through, drop, and additional waveguides, respectively. Resonance modes are shown simply by a circle and four arrows.

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Figure 2 shows an example of a three-port resonant tunneling filter based on a 2D-PhC with a triangular air-hole lattice. Here, we employed the structure shown in Fig. 1(a) and calculated its transmission spectra by 2D-FDTD with the following parameters: lattice constant a=400 nm, air hole radius=0.275a, effective refractive index of the slab=2.78. These parameters are used in all the calculations. Figure 2(c) is a schematic diagram of the three-port resonant tunneling filter composed of the width-tuned WGs [11, 12] shown in Fig. 2(a) and the four-point defect resonator shown in Fig. 2(b). The brackets indicate the WG boundaries.

 

Fig. 2. Three-port resonant tunneling filter. (a) Width-tuned waveguide. (b) Four-point defect resonator with a width of W_c=W₀. (c) Schematic diagram of the three-port resonant tunneling filter. P1, P2, P3, and P4 are width-tuned waveguides with widths of W₀, 0.95W₀, W₀, 0.95W₀, respectively. The brackets indicate the waveguide boundaries. (d) The dotted lines are the transmission spectra of the width-tuned waveguides. The solid lines are the output signals of the three-port resonant tunneling filter at P2 and P3. (e) Magnetic field profile (H_z) in three-port resonant tunneling filter when the frequency is resonant. (f) Magnetic field profile when the frequency is in the common band of P1 and P2.

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The dotted lines in Fig. 2(d) are the transmission spectra of the width-tuned WGs with widths (W) of W₀=a×√3 (P1 and P3) and 0.95W₀ (P2 and P4). The lines of holes on both sides of the cores are shifted to reduce the core width and the holes adjacent to the shifted lines are elongated. The coupling loss between two different WGs is very small, because their mode profiles are very similar [13]. The transmission band shifts towards a shorter wavelength when the core is narrowed because of the reduction in the effective refractive index of the core [11, 12]. As a result, a gap appears between two transmission band edges. And the resonant frequency of the resonator is in the gap region. The solid lines in Fig. 2(d) are the transmission spectra of the three-port resonant tunneling filter. When the frequency is resonant, the input light from P1 is output only from P3 and the transmittance (T) is almost 100%. When the frequency is in the common band of P1 and P2, the light is output from P2. The magnetic field (H_z) profiles are shown in Fig. 2(e) and (f). They show that the light tunnels into the WG boundary barrier and reaches P3 when the frequency is resonant, and that the light easily passes the WG boundary and reaches P2 when the frequency is in the common band.

Next, we investigated the way in which the resonator position affects the dropping efficiency. Here, we used width-tuned WGs with widths of W₀ (P1 and P3) and 0.90W₀ (P2 and P4) and a three-point defect resonator with a width of W₀. The distance between P1 and P3 in the y-direction is 4.0 W₀, which is made narrower than the 5.0 W₀ value of Fig. 2 to increase the coupling efficiency between the resonator and the WG modes and thus reduce the calculation time.

Figure 3(a) shows the Δ dependence of the transmission spectra at P3, where Δ is the distance from the WG boundary to the center of the resonator in the x direction. We can divide the coupling situations into four types because the resonant mode of the resonator expands in four directions as shown in Fig. 2(e) and it overlaps with the WG mode in the limited region indicated by the circles as shown in Fig. 3(b). Type 1: Δ<-6a. All the coupling regions are on the left hand side of the WG boundary. This condition is almost the same as that described in the literature [5, 6]. Type 2: Δ=-6a~3a. Since half of the coupling regions are on the left hand side of the WG boundary, the coupling efficiency is smaller than that of type 1 and the widths of the transmission spectra of the resonating modes are narrower than those of type 1. Transmission spectra marked by × in both types 1 and 2 indicate that the light coupled out from the resonator destructively interferes with the light reflected back from the WG boundary. This interference disturbs the shape of the transmission spectra but it is not severe when the quality factor (Q) is low. However, it becomes a cause of low dropping efficiency when higher Q is required. In this case, we must take into account the resonant frequency and the dispersion of the WG and Δ, and optimize the phase relation for high dropping efficiency. Type 3: Δ=3a~6a. The coupling regions are on the WG boundary. In this case, the phases of the light coupled out from the resonator and the light reflected back from the WG boundary are always opposite and the output toward P1 is automatically eliminated. This condition is the same as that of a two-port resonant tunneling filter. Therefore, this structure does not require a careful consideration of the interference process. Type 4: Δ>6a. The coupling efficiency between the resonator and the WGs is very small because the overlap of the resonator and waveguide fields is very small, this results in very sharp transmission spectra. This structure can also function as a two-port resonant tunneling filter when the coupling efficiency between the resonator and the WG is much larger than that of direct coupling between WGs.

 

Fig. 3. (a) Resonator position dependence of the dropping efficiency of the three-port resonant tunneling filter. P1, P2, P3 and P4 are width-tuned WGs with widths of W₀, 0.90W₀, W₀, 0.90W0, respectively. The resonator is a three-point-defect cavity with a width of W₀. The symbols O and × indicate conditions of high and low transmittance, respectively. Δ is the distance from the WG-boundary to the center of the resonator in the x-direction. (b) Four coupling conditions. The circles indicate coupling points.

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Figure 4(a) shows the scanning electron micrograph of a three-port resonant tunneling filter fabricated on a silicon-on-insulator (SOI) substrate by a combination of electron beam lithography and ECR dry etching. The thicknesses of the Si core and SiO₂ cladding are about 200 nm and 3 µm, respectively. The hole diameter is about 210 nm and a=420 nm. We used width-tuned WGs with widths of W₀ (P1 and P3) and 0.90W₀ (P2 and P4). We employed a two-point defect resonator whose innermost holes are reduced in size and pushed away from the center to suppress the radiation loss and the center hole size is varied to tune the resonant frequency [2]. Figure 4(b) is the transmission spectra at P2 and P3 and the spectrum of the reference WG (W=W0). One signal is dropped from P3 in the gap region. The coupling loss between P1 and P2 is less than 2dB. The detailed spectra are shown in Fig. 4(c). They are normalized by a peek power of the reference spectrum. The dotted line is an averaged line of the reference. Since the peek power of P3 normalized by the dotted line is about 65 % and the error bar of the reference is about 40%, the transmittance is estimated to be 65 ± 20 %, which agrees with the calculated dropping efficiency (T) of 68.4 % and is much larger than 25 %, which is the theoretical limitation of a simple coupled system. Here, T=(1-Q_V/Q_T)² [3] for two-port resonant tunneling filter, the intrinsic Q factor of the resonator (QV=53800) and the total Q factor of the coupled system (Q_T=9300) are calculated by 3D-FDTD.

 

Fig. 4. Experimental result for three-port resonant tunneling filter. (a) Scanning electron micrograph of the fabricated filter structure on a silicon-on-insulator substrate. The thicknesses of the Si-core and the SiO₂ cladding are about 200 nm and 3 µm, respectively. The hole diameter is about 210 nm and a=420 nm. P1, P2, P3 and P4 are width-tuned waveguides with widths of W₀, 0.90W₀, W₀, 0.90W₀, respectively. The brackets indicate waveguide boundaries. The resonator is a two-point defect cavity with a width of W₀. (b) Transmission spectra at P2 and P3. (c) Transmission spectra at P3 and reference (W=W₀). The estimated Q factor is about 5000 and the transmittance is 65±20%.

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Next, we consider the structure of Fig 1(c). Here, we employed a coupling condition of type 3 and used width-tuned WGs with width of W₀ (P1 and P3) and 0.95W₀ (P2 and P4) and four-point defect resonator with a width of W₀ as shown in Fig. 5(a). The dotted lines in Fig 5(b) are the transmission spectra of the width-tuned WGs with width of W₀ and 0.95 W0. The solid lines are the transmission spectra at P3 and P4. Although this filter is not strictly symmetric because of the short length of P4, the dropping efficiency is almost 100%, which means that the input and output coupling coefficients are almost the same, and this system can be regarded as a symmetric two-port resonant tunneling filter when the frequency is resonant. The magnetic field (H_z) profiles are shown in Fig. 5(c) and (d). They show that the light tunnels into the WG boundary barrier and reaches P3 when the frequency is resonant, and that the light easily passes the WG boundary and reaches P2 when the frequency is in the common band.

 

Fig. 5. Three-port resonant tunneling filter. (a) Schematic diagram of the three-port resonant tunneling filter. P1, P2, P3, and P4 are width-tuned waveguides with widths of W₀, 0.95W₀, W₀, 0.95W0, respectively. The brackets indicate the waveguide boundaries. (b) The dotted lines are the transmission spectra of the width-tuned waveguides. The solid lines are the output signals of the three-port resonant tunneling filter at P2 and P3. (c) Magnetic field profile (Hz) in three-port resonant tunneling filter when the frequency is resonant. (d) Magnetic field profile when the frequency is in the common band of P1 and P2.

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3. Multi-port resonant tunneling filter

Next, we construct a multi-port system. Muti-port systems described in the literature [5, 6] use an excellent way to accomplish constant Q factors and dropping efficiencies at each port. Since their systems use the condition of type1, they require a distance between two coupling regions (L_c) shown in Fig. 3(b) to keep the structure symmetric. In contrast, our filters using the condition of type 3 and 4 do not require Lc. Moreover, the filter in Fig. 1(b) can be used in their system and makes the device shorter than half of their size because L_c is longer than the required length of P4 which determines the length of our multi-port system. This is an advantage of our filter. But such a system is composed of PhCs with different lattice constants and contains many lattice mismatches, which makes the device integration difficult. Since our aim is to integrate many devices in one PhC, we try to use one PhC with one lattice constant.

As shown in Fig. 6(a), we connect two of the three-port filters shown in Fig. 1(c) in series. Of course, the filters shown in Fig. 1(a) and (b) could also be employed in a multi-port system, but they require bent waveguides to cascade them. In contrast, the filter in Fig. 1(c) can be used to construct a multi-port filter simply. Here, three kinds of width-tuned WGs (WG₁, WG₂, WG₃) and two different resonators (R₁, R₂) are used. The resonant frequencies are tuned so that light resonating with R₁ can propagate only in WG₁, and light resonating with R₂ can propagate in both WG₁ and WG₂. As a result, the equivalent filters for each resonant frequency can form a two-port system as shown in Fig. 6(b) and (c). Since the input and drop WGs are symmetrically positioned against the resonator so that the input and output coupling coefficients are almost the same, the dropping efficiency is almost 100 %.

 

Fig. 6. Design of multi-channel drop filter. (a) Structure of two-channel-drop filter. WG₁, WG₂ and WG₃ are three kinds of width-tuned WGs and R₁ and R₂ are two different resonators. (b) and (c) are equivalent filters for resonant frequencies of R₁ and R₂, respectively.

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Figure 7(a) shows a ten-drop-port filter composed of ten three-port filters in the same manner. Here, twelve width-tuned WGs (WG₁~WG₁₂) and ten three-point defect resonators (R₁~R₁₀) are used. The WGs and resonators are in sequence by subscript number from the input to the through port. The WG and resonator widths are tuned so that the equivalent filters for ten resonant frequencies can be regarded as ten two-port systems. The widths of WG₁ ~WG₁₂ are 1.00, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.90, 0.89 × W₀, respectively. The widths of the R₁~R₁₀ are tuned by shifting the four holes on either side of the core to increase or reduce the core width, and their values are 1.09, 1.07, 1.05, 1.03, 1.01, 0.99, 0.97, 0.95, 0.933, 0.915×W₀, respectively. The differences between the WGs are negligible, so only the WG boundaries are indicated by brackets. Figure 7(b) shows the transmission spectra at each drop port. Ten signals with different frequencies are output, the crosstalk between the drop ports is less than -25 dB. Figure 7(c) shows the Q factors and the dropping efficiencies at each port. The fluctuation of the Q factor is due to slight difference of the coupling efficiencies at each resonator. The reason why some efficiencies can not reach 100% is that this structure is not strictly symmetric because the length of P4 is short and the widths of WGs parallel to the x axis and angled WGs are slightly different due to a limitation of the FDTD resolution. But, all the transmittance values are over 80%. So this filter, which contains ten drop ports and functions in the C-band when a=400 nm, can achieve high quality signal separation with a very small device size (L) of 18 µm.

 

Fig. 7. (a) Structure of ten-channel-drop filter composed of twelve width-tuned WGs (WG₁ ~WG₁₂) and ten three-point defect resonators (R₁~R₁₀). The waveguides and resonators are in sequence by subscript number from the input to the through port. The widths of WG₁~WG₁₂ are 1.00, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.90, 0.89 × W₀, respectively. The widths of the R₁~R₁₀ are tuned by shifting the four holes on either side of the core to increase or reduce the core width, and their values are 1.09, 1.07, 1.05, 1.03, 1.01, 0.99, 0.97, 0.95, 0.933, 0.915×W₀, respectively. The device size (L) is 18 µm when a=400 nm. (b) Calculated transmission spectra of ten-channel-drop filter. All the transmittance values are over 80% and the crosstalk between drop ports is less than -25 dB. This filter can function in the C-band when a=400 nm. (c) Dropping efficiencies and Q factors at each port

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4. Summary

We designed new three-port resonant tunneling filters based on a 2D-PhC with a triangular air-hole lattice. They do not require careful consideration of the interference process to achieve a high dropping efficiency from one port. This makes the design much simpler than that of any channel drop filter yet reported. In addition, we fabricated a three-port resonant tunneling filter on an SOI substrate and achieved a high transmittance of 65± 20% with a quality factor of 5000. Moreover, we demonstrated an ultrasmall ten-channel-drop filter with a device size of 18 µm by 2D-FDTD calculation. This work reveals that the coupled resonator-waveguide system is a promising element for future optical integrated circuits.