## Abstract

We report a theoretical and experimental study of a channel drop filter with two cascaded point-defects between two line-defects in a two-dimensional photonic-crystal slab. Using coupled-mode analysis and a three-dimensional finite-difference time-domain method, we design a filter to engineer the line shape of the drop spectrum. A flat-top and sharp roll-off response is theoretically and experimentally achieved by the designed and fabricated filters. Furthermore, we theoretically demonstrate that drop efficiency is increased dramatically, up to 93%, by introducing hetero-photonic crystals. We also describe a method to modify the bandwidth of the spectrum.

©2005 Optical Society of America

## 1. Introduction

Channel add/drop filters are very important components that can be utilized not only as multi-channel multiplexers/demultiplexers (Mux/Demux) and optical add/drop modules (OADMs) for wavelength-division multiplexing (WDM) systems but also as chemical sensors, etc. In these components, it is important to access the signal on one particular channel without disturbing the signals on the other channels.

Another requirement is to make these components more compact, so that they will find uses in a wider variety of applications. Photonic-crystal (PC) filters [1–3] are especially attractive in this regard since they are extremely small; in addition, they are also capable of implementing multiple functions and are suitable for large-scale integration. Most PC filters are based on two-dimensional (2D) PC slabs including line-defect waveguides and point-defect cavities; among such filters, a device operating as a multi-channel filter has been reported [4].

PC filters belong to the class of resonator-type filters, which generally use coupling between resonators and waveguides. When one resonator is used for one channel in resonator-type filters [5–7], the filter responses become Lorentzian in principle. On the other hand, a filter response with a flattened resonant peak and sharp roll-off from the passband to the stop band, i.e., a flat-top response, is necessary for achieving low crosstalk from other channels and high robustness against fluctuations in input-signal wavelength. Higher order filters, that is, filters consisting of multiple cascaded resonators, are effective in achieving such a flat-top response [6,8]. A theoretical study on PC filters with higher order response has already been made [9]. However, this study is an investigation of a 2D PC of infinite thickness, and physically feasible PC structures with flat-top responses have yet to be investigated.

In this study, we choose a filter based on a 2D PC slab as a feasible structure. Using coupled-mode theory (CMT) in time, we carry out analysis on a simplified model where light can leak into free space. We first design a concrete filter structure based on the analysis, before investigating a fabricated sample experimentally. In addition, a theoretical study to drastically increase the drop efficiency is also carried out. This paper is separated into five sections: Section 2 describes the theoretical analysis, the filter design, and the simulation results. Section 3 describes the sample fabrication and shows the measured results. In Section 4, a strategy for increasing the drop efficiency further is presented, together with simulated results. Finally, Section 5 provides some conclusions.

## 2. Theory

#### 2.1 Coupled mode theory

In order to make the normally Lorentzian transfer response of PC slab filters into a flat-top one, a higher-order filter structure composed of multiple resonators is investigated. The filter structure that we propose is shown in Fig. 1(a). The filter consists of two cascaded point-defect cavities between two line-defect waveguides in a 2D PC slab. This structure serves as an in-plane-type filter where one waveguide (upper) acts as a signal bus and the other (lower) as a receiver. In order to analyze the filter characteristics of this structure using CMT, we first consider a simplified model shown in Fig. 1(b), where 1/*τ*
_{in} is the decay rate from the point-defect cavity into the adjacent waveguide (the bus or the receiver), 1/*τ*
_{v} is the decay rate from the cavity to free space, and *µ* is the mutual coupling coefficient between the two cavities.

These decay rates are related to the in-plane *Q* factor *Q*
_{in} and the vertical *Q* factor *Q*
_{v} by *Q*
_{in}=*τ*
_{in}
*ω*
_{0}/2 and *Q*
_{v}=*τ*
_{v}
*ω*
_{0}/2, where *Q*
_{in} is determined by the coupling loss between the cavity and the waveguide, and *Q*
_{v} is determined by the coupling loss between the cavity and free space and is equivalent to the intrinsic *Q* of the cavity. The amplitude of the incident wave from port 1 is denoted by *S*
_{+1}, and those of the outgoing waves in the bus waveguide to ports 1 and 2 by *S*
_{-1} and *S*
_{-2}, respectively. The amplitudes of the outgoing waves in the receiver waveguide bypassing two point-defect cavities to ports 3 and 4 are denoted by *S*
_{-3} and *S*
_{-4}, respectively. The parameter *β* is the propagation constant in both waveguides. To simplify the model, all the reflectances at the waveguide facets are assumed to be 0. The amplitudes of the point-defect cavities adjacent to the bus and the receiver waveguides are denoted by *a*
_{1} and *a*
_{2}, respectively. Then, the equations for the evolution of the cavity modes in time and the outgoing waves are given as follows [7]:

Based on these equations, the drop efficiency *η* from ports 3 and 4 can be expressed by:

$$=\frac{1}{\frac{{{\tau}_{\mathrm{in}}}^{2}}{{\mu}^{2}}\left[{{{(\omega -{\omega}_{0})}^{4}+2\{{(\frac{1}{{\tau}_{v}}+\frac{1}{{\tau}_{\mathrm{in}}})}^{2}-{\mu}^{2}\}(\omega -{\omega}_{0})}^{2}+\{{(\frac{1}{{\tau}_{v}}+\frac{1}{{\tau}_{\mathrm{in}}})}^{2}+{\mu}^{2}\}}^{2}\right]}$$

Note that the term *ω*
^{4} appears in the denominator. This means that the filter has a higher order response. On the other hand, when the number of bypassed point-defect cavities for one channel in the filter is one, the term *ω*
^{4} does not appear in the denominator; instead, the denominator function is comprised of *ω*
^{2} and constant terms, and the filter response thus becomes Lorentzian. The term *ω*
^{4} in the denominator clearly suggests that at a frequency far from *ω*
_{0}, the drop efficiency in the response of Eq. (7) falls more sharply than in a Lorentzian response. In addition, it also suggests that at a frequency near *ω*
_{0}, the efficiency in the response of Eq. (7) varies more slowly than in a Lorentzian response when the term (*ω-ω*
_{0})^{2} does not exist. This means that the filter response becomes flat-top when the coefficient of the term (*ω-ω*_{0}
)^{2} in Eq. (7) is approximately equal to 0, that is, when the following Eq. (8) is approximately satisfied. This is equivalent to the so-called maximally flat response in the field of microwave filters.

#### 2.2 Design of filter structure

The design problem to achieve a filter that gives a flat-top response is essentially to search for a structure satisfying Eq. (8). Under the condition of Eq. (8), the full-width at half maximum (FWHM) Δ*ω* and the efficiency *η* at *ω*=*ω*
_{0} can be expressed by the following equations:

As can be seen in Eq. (9), *µ* is determined by the required bandwidth Δω. Here the target bandwidth is set to 0.6 nm (FWHM), which is suitable for the 100-GHz spacing (~0.8 nm) of the C-band commonly used in WDM systems. When the resonant wavelength 2*πc*/*ω*_{0}
is 1.55 µm, the FWHM bandwidth Δ*ω* and the required absolute value of *µ* become 3.8×10^{-4}
*ω*
_{0} and 1.3×10^{-4}
*ω*
_{0}, respectively. Equation (10) shows that if *τ*
_{in}/*τ*
_{v}→0 then the efficiency *η* becomes the maximum (25%). This means that when *τ*
_{in}/*τ*
_{v}→0, that is, the coupling between the point-defect cavity and the waveguide is quite significant compared with the coupling between the cavity and free space, light can transfer from one waveguide to the other via the two cavities without leaking out of the cavities into free space. In this study we use a cavity with *Q*=45,000, which was reported in ref. 10 as being a cavity with large *τ*
_{v}. The cavity structure is shown in Fig. 2. When *Q*
_{v}=45,000 (i.e., *τ*
_{v}=90,000${\omega}_{0}^{-1}$), then *τ*
_{in} becomes 8,100${\omega}_{0}^{-1}$ (i.e., *Q*
_{in}=4,050) under flat-top conditions (Eq. (8)), and then a drop efficiency of 21% can be expected from Eq. (10). This efficiency is close to the maximum value (25%).

To design a concrete structure of a flat-top response filter, we have to find a structure having *µ* and *τ*
_{in} derived as above. The parameter *µ* can be estimated from the following equation:

where **e** is the electric field distribution of the cavity and *ε* is the dielectric constant distribution of the structure. The subscripts 1 and 2 refer to the 2D-PC slab structures having point-defect cavities 1 and 2, respectively. On the other hand, no subscript indicates that the structure has both point-defect cavities. We can derive Eq. (11) from CMT by assuming a filter structure without any waveguides. A detailed derivation is found in the Appendix. Using the three-dimensional (3D) finite-difference time-domain (FDTD) method, we calculated the electric field distribution in an isolated point-defect cavity and we substituted it into Eq. (11). The base 2D-PC slab is assumed to be composed of silicon with a triangular lattice of air holes having lattice constant *a*. The parameters used here were a slab thickness of 0.6*a*, air hole radius of 0.29*a*, slab index of 3.4, and an air clad index of 1. The calculated results for *µ* are shown in Table 1. These results were calculated as a function of distance between the two cavities, where *g* is the distance perpendicular to the same row as the defect and *h* is the distance parallel to the row, as shown in Fig. 2. Table 1 shows that *µ* can be changed drastically by changing either distance (*g* or *h*) and becomes the target value (±1.3×10^{-4}
*ω*
_{0}) when the distance (*g, h*)=(7 rows, 1.5*a*). Therefore, we chose to use this distance.

Next, we investigate a structure whose τ_{in} and *Q*
_{in} values satisfy the flat-top condition (Eq. (8)). Since *Q*
_{in} is determined by the coupling loss between the cavity and the waveguide, as described above, *Q*
_{in} can be adjusted by changing the distance between the cavity and the waveguide. Simultaneously, *Q*
_{v} changes a little depending on the distance between the cavity and the waveguide. Therefore, we calculated both *Q*
_{in} and *Q*
_{v} of the structures for a range of distances *d*
_{0} between the cavity and the waveguide, and we investigated the structure satisfying Eq. (8). We performed 3D-FDTD simulations using structures consisting of one cavity and one waveguide in a 2D PC slab, like the simulations used in refs. 11 and 12. The calculated results are shown in Fig. 3(a). As can be seen in this figure, *Q*
_{v} is almost constant, while *Q*
_{in} has a strong dependence on the distance *d*
_{0} between the cavity and the waveguide. Using these *Q* factors we estimated the values of 1/τ_{v}+1/*τ*
_{in}, as shown in Fig. 3(b). This figure shows that, when the distance *d*
_{0} is equal to 5 rows, the value of 1/*τ*
_{v}+1/*τ*
_{in} becomes closest to the target *µ* value (±1.3×10^{-4}
*ω*
_{0}), that is, the flat-top condition (Eq. (8)) is most likely to be satisfied.

Then, using Eq. (7), derived from CMT, we evaluated the drop spectrum of the designed filter. The obtained spectrum is shown as a solid line in Fig. 4.

As mentioned above, when the number of bypassed cavities for one channel in a filter is one, the filter response becomes Lorentzian in principle. Therefore, for comparison, a Lorentzian curve whose FWHM is equal to that of the obtained spectrum is also shown as a dashed line in this figure. As can be seen in this figure, the spectrum obtained for the designed filter exhibits a flat-top response that has a flattened resonant peak and sharp roll-off from the passband compared with the Lorentzian response. Thus, we succeeded in obtaining a structure design for the filter with a flat-top response.

We then performed numerical simulation of this obtained filter structure in order to check whether the filter actually behaves as designed. The 3D-FDTD method was used for the simulation. Specifically, the system was excited using a source located at the bus waveguide with the spatial profile of the fundamental waveguide mode and a Gaussian spectrum centered at the resonant frequency of the cavities. The frequency response of the filter can be obtained by calculating the Fourier transform of the observed time-dependence of the electro-magnetic field strength at each port of the receiver waveguide. As a reference for normalization, we used the Fourier transform of the time-dependence of the electro-magnetic field strength transmitted through the waveguide in a structure without the point-defect cavities. The result is shown as open circles in Fig. 4. As shown in this figure, the simulated result was found to be in good agreement with the theoretical spectrum expected by CMT (solid line). The FWHM and the efficiency in the simulated result are 0.54 nm and 20%, respectively. Thus, we have demonstrated that we can design a PC filter with a flat-top response using two cascaded point-defect cavities between two waveguides in a 2D PC slab.

## 3. Experiment

Encouraged by the above analysis, we fabricated a sample filter with the designed structure and measured its drop spectrum. The fabrication process was as follows: Initially, a resist mask (ZEP-520) was coated onto a silicon-on-insulator (SOI) substrate. PC patterns were drawn on this resist mask by electron-beam lithography. The resist patterns were then transferred to the upper silicon layer using inductively-coupled plasma reactive-ion etching (ICP/RIE). After the dry-etching procedure, the resist was removed using O_{2} plasma. Finally, the SiO_{2} layer under the PC layer was selectively etched away using hydrofluoric (HF) acid to form an air-bridge structure. We chose a lattice constant *a* of 420 nm and fabricated the filter with the structure calculated in Fig. 4. A scanning electron microscope (SEM) image near the point-defect cavities of the fabricated sample is shown in Fig. 5.

Photons with a broad range of wavelengths were injected from one facet of one waveguide using a tunable c.w. laser, while photons coming out of the other facet of each waveguide were observed to check whether they were transmitted through the bus waveguide or dropped from the receiver waveguide. The photons of the only specific wavelengths were found to drop from the receiver waveguide. The observed appearance of the dropped and transmitted light at the resonant frequency is shown in Fig. 6(a). In addition, the observed appearance of the transmitted light only from the bus waveguide at off-resonant frequencies is shown in Fig. 6(b) for comparison. As shown in Fig. 6(a) and (b), we succeeded in realizing in-plane drop via the two point-defect cavities.

The measured drop spectrum is shown as open circles in Fig. 7(a). The drop intensity in this figure is normalized with respect to the observed peak intensity. For comparison, a Lorentzian curve whose FWHM is equal to that of the obtained spectrum is also shown as a dashed line in this figure. As can be seen in this figure, the obtained spectrum has a flattened resonant peak and sharp roll-off from the passband compared with the Lorentzian curve. On the other hand, Fig. 7(b) shows one of the spectra obtained in filters consisting of one point-defect cavity and one waveguide in a 2D PC slab [4,13]. A Lorentzian curve with the same FWHM as the obtained spectrum is shown as a dashed line in this figure. This figure demonstrates that a filter based on one point-defect cavity in a 2D PC slab exhibits nearly Lorentzian response, as expected. The experimental data in Fig. 7(a) and (b) clearly suggest that filters consisting of two cascaded point-defect cavities between two waveguides in 2D PC slabs are effective in giving flat-top and sharp roll-off responses.

Next, we investigate whether the experimental result can be accounted for by CMT, i.e., Eq. (7). Using the experimental result of the filter consisting of one point-defect cavity and one waveguide, we can evaluate *Q*
_{v} and *Q*
_{in} experimentally [4,13,14]. As a result, the values of *Q*
_{v} and *Q*
_{in} were estimated to be 45,000 and 8,900, respectively. This value of *Q*
_{v} is in good agreement with the designed value (47,000), while the value of *Q*
_{in} is significantly different from the designed value (3,600). This discrepancy of *Q*
_{in} is most likely due to the following: Since Qin is determined by the coupling between the point-defect cavity and the waveguide, it is closely related to the electric field distribution in the waveguide mode. The field distribution depends on how the light is confined in the waveguide. Light in a waveguide mode with large group velocity, which corresponds to a refractive-like mode, is confined by total internal reflection, while light in a waveguide mode with small group velocity, which corresponds to a diffractive-like mode, is confined by the photonic bandgap effect [15]. Therefore, it is clear that the electric field distribution in the waveguide mode is strongly dependent on the group velocity. In this case, around the resonant frequency of this point-defect cavity, the group velocity in the waveguide mode is strongly dependent on the frequency. Therefore, the fluctuation of the resonant frequency of the cavity mode results in a significant change of the group velocity in the waveguide mode, the electric field distribution, and *Q*
_{in}. Therefore, the discrepancy between the measured and calculated results of *Q*
_{in} is believed to be due to the fluctuation of the resonant frequency of the cavity mode, for example, imperfections in the fabricated samples, etc. Now, substituting these experimental values of *Q*
_{v} and *Q*
_{in} into Eq. (7), we fitted the equation to the experimental result in Fig. 7(a) by changing *µ*. The fitted curve is shown as a solid line in this figure. As is the case with the experimental result, the intensity is normalized with respect to the peak intensity. As can be seen in this figure, the fitted curve is in good agreement with the experimental result when *µ* is equal to 1.1×10^{-4}ω_{0}. This value of µ is close to the value 1.3×10^{-4}ω_{0} calculated using Eq. (11). At the same time, we can see that the spectrum obtained experimentally is a little different from the designed spectrum shown in Fig. 4. This is most likely due to the difference between the experimental and theoretical values of *Q*
_{in}, as mentioned above. The good agreement between the experimental result and the theoretical result estimated from CMT clearly suggests that we succeeded in realizing a filter with a flat-top and sharp roll-off response by using a filter structure consisting of cascaded cavities between two waveguides in a 2D PC slab.

## 4. Further increase in drop efficiency

#### 4.1 Coupled mode theory

As described above, we proposed a 2D-PC-slab filter consisting of two cascaded point-defect cavities between two waveguides and demonstrated theoretically and experimentally that the drop response becomes flat-top and sharp roll-off. This means that we can design higher-order filters based on the PC-slab type as well as the conventional resonator type. Accordingly, the transfer response of the filter consisting of more than two cavities can be expected to have a more flat peak and sharper roll-off than the filter consisting of two cavities. As for the drop efficiency, however, the value designed in Sec. 2 is 20% at most. As can be seen in Eq. (10), it is theoretically impossible to obtain an efficiency of more than 25%. However, a further increase in drop efficiency is desired for practical application of this filter. Therefore, we now investigate a method to increase the drop efficiency of this filter. We have already proposed a structure to increase the drop efficiency of the original filter utilizing one point-defect cavity in a 2D PC slab for one channel [4,16,17]. The structure is an in-plane array of multiple PCs with different lattice constants. We call this structure an in-plane hetero-photonic crystal (IP-HPC). We demonstrated that the hetero-interface formed between adjacent PCs becomes a perfect but selective mirror for photons with specific frequencies and reflection by the interface can increase the drop efficiency drastically [4]. In addition, we showed that in both out-of-plane-type and in-plane-type filters, a drop efficiency close to 100% could be obtained theoretically [16,17].

In this section we investigate theoretically whether we can increase the drop efficiency of a filter consisting of cascaded cavities between two waveguides in a 2D PC slab by introducing the IP-HPC structure. Figure 8(a) shows a schematic diagram of the investigated filter structure, in which an IP-HPC is introduced into the structure shown in Fig. 1(a). The individual regions PC_{1} and PC_{2} have different lattice constants (*a*
_{1} and *a*
_{2}, respectively). The wavelength ranges of photons transmitted through the waveguides in PC_{1} and PC_{2} are approximately proportional to their lattice constants. Therefore, photons in a certain range of wavelengths cannot propagate along the waveguide in PC_{2}, while those photons can propagate along the waveguide in PC_{1}. The photons in this range of wavelengths are therefore expected to be reflected at the hetero-interface between PC_{1} and PC_{2}. In order to utilize the reflection, a resonant wavelength of the point-defect cavity in the filter is designed to be in this wavelength range. On the other hand, the rest of photons that propagates along the waveguide in PC_{1} can pass through the hetero-interface. Therefore, multiple-wavelength extraction can be also achieved by serially connecting similar structures with decreasing lattice constants. In addition, when the difference between the lattice constants of PC_{1} and PC_{2} is small, the radiation loss from the hetero-interface becomes negligibly small [18]. In this configuration, photons in a resonant wavelength of a point-defect cavity that are not trapped via the cavity are reflected by this hetero-interface and are directed back towards the incident side of the waveguide. These photons not only pass through the cavity but also are reflected by the cavity and directed towards the hetero-interface again. Thus, these reflections on the cavity and the hetero-interface are repeated until the system reaches steady state. These all photons directed back towards the incident side of the waveguide interact with one another and can be cancelled by adjusting the phase difference between those photons. Thus, this interaction can be expected to increase the drop efficiency of the filter shown in Fig. 8(a).

To investigate the drop response of this filter using CMT analysis, we considered a simplified model (Fig. 8(b)). Although this model resembles the model shown in Fig. 1(b), it differs from the model in Fig. 1(b) by the presence of a perfect mirror corresponding to the hetero-interface instead of ports 2 and 4. In Fig. 8(b), the change of phase due to reflection at this perfect mirror is denoted by Δ, and it is assumed that *d*
_{3}=*d*
_{1} and *d*
_{4}=*d*
_{2} to simplify the following analysis. Then, the equations for the evolution of the cavity modes in time and the amplitudes of waveguide modes at all ports are given by [7]:

The drop efficiency *η* from port 3 derived from these equations can be expressed by

$$=\frac{4}{\frac{{\tau}_{\mathrm{in}}^{{\mathrm{system}}^{2}}}{{\mu}^{2}}\left[{{{(\omega -{\omega}_{0}^{\mathrm{system}})}^{4}+2\{{(\frac{1}{{\tau}_{v}}+\frac{1}{{\tau}_{\mathrm{in}}^{\mathrm{system}}})}^{2}-{\mu}^{2}\}(\omega -{\omega}_{0}^{\mathrm{system}})}^{2}+\{{(\frac{1}{{\tau}_{v}}+\frac{1}{{\tau}_{\mathrm{in}}^{\mathrm{system}}})}^{2}+{\mu}^{2}\}}^{2}\right]}$$

where *θ* is the phase difference between photons reflected by the hetero-interface and photons reflected directly by the point-defect cavity and is defined by

Although Eq. (20) resembles Eq. (7) for the filter without the IP-HPC, it differs in three ways from Eq. (7). In Eq. (20), ${\tau}_{\text{in}}^{\text{system}}$ and ${\omega}_{0}^{\text{system}}$ substitute for *τ*
_{in} and ω_{0}, respectively. This merely means that by introducing the IP-HPC, *τ*
_{in} and *ω*
_{0} appear to vary due to the interaction between photons reflected by the hetero-interface and photons reflected directly by the pointdefect cavity. This variance has no influence on the flat-top response of the filter. Note that the numerator of Eq. (20) is 4 times larger than that of Eq. (7). This means that by introducing the IP-HPC, the drop efficiency of the filter can be increased by a factor of four. Qualitatively, this can be understood as follows: When the filter without the IP-HPC has the maximum drop efficiency, the photon energy, except for the radiation loss from the point-defect cavities, is divided equally among all four ports. On the other hand, when the IP-HPC is introduced into the filter, we can gather the photon energy, except for the radiation loss, into port 3 by completely shutting off the energy streams from ports 1, 2, and 4. Therefore, by introducing the IP-HPC, we can increase the drop efficiency by a factor of four. In addition, note that since Eq. (20) resembles Eq. (7), we can also maintain the flat-top response. Thus, it is expected that the drop efficiency of a filter based on cascaded cavities between two waveguides in a 2D PC slab can be increased by introducing an IP-HPC, while maintaining the flat-top response characteristic.

#### 4.2 Design of filter structure

Next, we consider whether both the conditions for the flat-top response and for the maximum drop efficiency can be satisfied simultaneously. When *ω*=${\omega}_{0}^{\text{system}}$, from Eq. (20) the drop efficiency *η* can be expressed by

As can be seen in this equation, it was found that the drop efficiency is maximized when ${\tau}_{\text{in}}^{\text{system}}$/τ_{v}→0. This is because the radiation loss from the point-defect cavities can be suppressed, as is the case with the filter without the IP-HPC. When ${\tau}_{\text{in}}^{\text{system}}$/τ_{v}→0, Eq. (24) becomes

When ${\mathit{\mu}\tau}_{\text{in}}^{\text{system}}$=±1, Eq. (25) becomes maximum (equal to 1). In other words, when ${\tau}_{\text{in}}^{\text{system}}$/τ_{v}→0, the condition for the maximum drop efficiency is ${\mathit{\mu}\tau}_{\text{in}}^{\text{system}}$=±1. On the other hand, from Eq. (20) the flat-top condition can be expressed by

When ${\tau}_{\text{in}}^{\text{system}}$/τ_{v}→0, we can see that Eq. (26) becomes ${\mathit{\mu}\tau}_{\text{in}}^{\text{system}}$=±1. Therefore, it is found that when ${\tau}_{\text{in}}^{\text{system}}$/τ_{v}→0, that is, *Q*
_{v} is much higher than ${Q}_{\text{in}}^{\text{system}}$(=${\mathrm{\tau}}_{\text{in}}^{\text{system}}$ω_{0}/2), the condition for the flat-top response is consistent with that for the maximum drop efficiency. However, we do not know whether both conditions can be satisfied simultaneously when *Q*_{v}
is not much higher than ${Q}_{\text{in}}^{\text{system}}$.

We then investigate whether both conditions can be satisfied simultaneously in some instances of *Q*
_{v}. Here, *µ* is assumed to be constant and equal to 1.3×10^{-4}ω_{0}, which is the value used in Sec. 2. The drop efficiencies at each *Q*
_{v} (or τ_{v}) were evaluated depending on ${\tau}_{\text{in}}^{\text{system}}$ using Eq. (24). The evaluated results are shown in Fig. 9. When the horizontal coordinate in this figure is equal to 1 (see Eq. (26)), the condition for the flat-top response is satisfied perfectly. This figure suggests that, when *Q*
_{v} is no more than 30,000, the conditions for the flat-top response and for the maximum drop efficiency cannot both be satisfied simultaneously. On the other hand, it was found that, when *Q*
_{v} is no less than 90,000, both conditions can be satisfied almost simultaneously, and the drop efficiency under the flat-top condition becomes approximately maximum. This figure also suggests that the efficiency increases with increasing *Q*
_{v}. Thus, we can see that in order to satisfy both conditions for flat-top response and high drop efficiency simultaneously, *Q*
_{v} of the point-defect cavity needs to be no less than 90,000 and is desired to be as high as possible.

Recently, we succeeded in realizing a point-defect cavity with an extremely high *Q*
_{v} of 260,000 [14]. This cavity was obtained by analogy with the cavity used in Sec. 2 and Sec. 3 and specifically by the displacement of six air holes including the neighbors closest to the cavity edges, as shown in Fig. 10. In this Section, using this ultrahigh-*Q* cavity, we design a filter structure that satisfies both conditions for flat-top response and high drop efficiency, and we calculate its characteristic. In order to design such a filter structure, we first evaluated the mutual coupling coefficient *µ* between these two ultrahigh-*Q* cavities using the same calculation as in Sec. 2. The calculated results of *µ* are shown in Table 2. The results in Table 2 were found to be very similar to the results in Table 1. Meanwhile, the FWHM of the drop spectrum is also given by Eq. (9) when the flat-top condition is satisfied even for the filter with the IP-HPC. This means that µ of approximately 1.3×10^{-4}ω_{0} is necessary for obtaining a filter structure with the same FWHM as the spectrum obtained in Sec. 2 (~0.6 nm). Table 2 shows that the distance between the two cavities (*g, h*) of (7 rows, 1.5*a*) is most suitable for the filter structure with such an FWHM value. However, as mentioned above and as shown in Fig. 8(b), it is assumed that *d*
_{4}=*d*
_{2}, which means that the distance *h* parallel to the defect should be 0. Therefore, we use an alternative distance (*g, h*)=(8 rows, 0*a*), where *µ* and FWHM of this drop spectrum are equivalent to 6.7×10^{-5}ω_{0} and 0.30 nm, respectively, as can be seen in Table 2 and Eq. (9). How to tune the FWHM of the drop spectrum will be described in Sec. 4.3. Here, we explain the reason for making *d*
_{4}=*d*
_{2} more specifically. Since the hetero-interface completely reflects the photons of drop frequency, the filter structure is equivalent to a two-port device. In order to obtain the maximum drop efficiency of a two-port device, the coupling between the bus waveguide and the adjacent cavity must equal the coupling between the receiver and the adjacent cavity [17]. The coupling is determined by ${\tau}_{\text{in}}^{\text{system}}$. When *d*
_{2}=*d*
_{4}, the two ${\tau}_{\text{in}}^{\text{system}}$ as well as the two couplings are invariably identical. Therefore, we assume that *d*
_{4}=*d*
_{2}.

Next we calculate τ_{v} and ${\tau}_{\text{in}}^{\text{system}}$ to search for a filter structure satisfying the flat-top condition (Eq. (26)). In the calculation in Sec. 2.2, we merely varied the distance *d*
_{0} between the point-defect cavity and the waveguide. In this section, we also vary the distance *d*
_{2} between the cavity and the hetero-interface, as well as *d*
_{0}, because ${\tau}_{\text{in}}^{\text{system}}$ is also dependent on the phase *θ* determined by the distance *d*
_{2}, as seen in Eqs. (21) and (23). In the calculation, we set *a*
_{1} and *a*
_{2} to 420 nm and 415 nm, respectively. The calculated results of 1/*τ*
_{v}+1/${\tau}_{\text{in}}^{\text{system}}$ are shown in Fig. 11. It is clear that the value of 1/*τ*
_{v}+1/${\mathrm{\tau}}_{\text{in}}^{\text{system}}$ is strongly dependent on the distance *d*
_{2}. This is because ${\tau}_{\text{in}}^{\text{system}}$ significantly varies from *τ*
_{in}/2 to infinity depending on the phase *θ*, as seen in Eq. (21). From Fig. 11 and the target *µ* value of 6.7×10^{-5}ω_{0}, it was found that three structures with distances (*d*_{0}*, d*_{2}
) of (5 rows, 11*a*
_{1}), (6 rows, 7*a*
_{1}), and (6 rows, 9*a*
_{1}) can be considered as candidates satisfying the flat-top condition. Then, we evaluated the drop spectra of these structures using the 3D-FDTD simulation and the same method as described in Sec. 2.2. The calculated results are shown as open circles in Fig. 12(a)–(c). In addition, Lorentzian curves whose FWHMs are equal to those of the calculated results are shown as dashed lines in Fig. 12(a)–(c) for comparison. These figures clearly suggest that spectra with both very high drop efficiencies and flat-top (and sharp roll-off) responses can be obtained by introducing IP-HPCs into the filters based on the cascaded cavities between two waveguides in 2D-PC slabs. We succeeded in realizing a high efficiency of up to 93%, especially in Fig. 12(c).

#### 4.3 Tuning of bandwidth

The FWHM of the drop spectrum shown in Fig. 12(c) is estimated to be about 0.24 nm. For the purpose of tuning the FWHM while maintaining the flat-top response, we need to tune both *µ* and 1/*τ*
_{v}+1/${\tau}_{\text{in}}^{\text{system}}$ simultaneously to satisfy both the conditions of eqs. (9) and (26). In this case, *τ*_{v}
is almost constant and cannot be tuned, since *τ*_{v}
depends mainly on the point-defect geometry. In addition, the FWHM of the spectrum is not influenced much by *τ*_{v}
, since *τ*_{v}
is much larger than ${\tau}_{\text{in}}^{\text{system}}$ in the filter structure of Fig. 12. On the other hand, as can be seen in Fig. 11, ${\tau}_{\text{in}}^{\text{system}}$ is highly and finely tunable, since it varies depending not only on the distance *d*
_{0} between the cavity and the waveguide but also on the distance *d*
_{2} between the cavity and the hetero-interface. On the contrary, *µ* is difficult to tune finely, since it varies just exponentially depending on the distance *g* between the two cavities. Apparently, the fine-tuning of *µ* is crucial in tuning the bandwidth of the spectrum while retaining the flat-top response.

To realize a filter with an FWHM of about 0.6 nm (close to the FWHM obtained in Sec. 2) while maintaining the flat-top response and the high drop efficiency, we now consider a method to tune *µ* finely. Modifying the coupling strength between the two cavities is necessary for tuning *µ* finely. For this purpose, we tried to modify the size of some holes located in the center between the two cavities. The calculated results of *µ* are shown in Fig. 13. As shown in this figure, it was found that *µ* could be increased by enlarging the radius of these air holes. Then, the 3D-FDTD simulation was performed on the filter where the radius of two of these holes was set to 0.49a and the drop spectrum was evaluated. In order to satisfy the condition for the flat-top response, 1/*τ*
_{v}+1/${\tau}_{\text{in}}^{\text{system}}$ was also re-adjusted depending on the change of *µ*, where (6 rows, 8*a*) was chosen as (*d*
_{0}, *d*
_{2}) from the results shown in Fig. 11. The calculated result of the drop spectrum is shown in Fig. 14. As can be seen in this figure, the FWHM of the spectrum is increased to 0.45 nm, about twice the original FWHM shown in Fig. 12(c), while maintaining high efficiency. Thus, we succeeded in finely tuning the bandwidth of the drop spectra while maintaining both high efficiency and a flat-top (and sharp roll-off) response, by modifying the size of air holes located in the center between the two cavities.

## 5. Conclusions

We have theoretically and experimentally investigated a channel drop filter consisting of two cascaded cavities between two waveguides in a 2D photonic crystal (PC) slab, where the *Q* factor of these cavities is high enough to suppress radiation loss. Using three-dimensional finite-difference time domain (3D-FDTD) simulation and measuring the optical characteristic of the fabricated sample, we have succeeded in demonstrating that the response of the designed filter can be engineered to be flat-top and sharp roll-off. In addition, we have theoretically shown that drop efficiency can be increased drastically to 93% by introducing an in-plane hetero-photonic crystal (IP-HPC) structure. We have also succeeded in finely tuning the bandwidth of the drop spectrum by modifying the size of air holes located between the two cavities. We believe that this structure will become a platform for PC-type filters with flat-top response and high drop efficiency. If the number of cavities for each channel in the filter is increased, improved flat-top characteristics can be expected by analogy with higher-order filters.

## Appendix

Here we derive Eq. (11) using coupled mode theory (CMT). We consider a simplified model in which the filter consists of only two cavities and they are coupled to only each other. The electric field distributions in the cavities are assumed to be *â*
_{1}(*t*)e_{1}(*x,y,z,t*)[=â_{1}(*t*)*e*
_{1}(*x,y,z*)exp(*jωt*)] and â1(*t*)*e*
_{2}(*x,y,z,t*)[=*â*
_{2}(*t*)e_{2}(*x,y,z*)exp(*jωt*)], where e_{1}(*x,y,z,t*) and e_{2}(*x,y,z,t*) are the uncoupled cavity mode fields normalized with respect to unit energy, and they satisfy the following Maxwell equations:

where *ε*
_{1} and *ε*
_{2} are the dielectric constants of the cavities, and *µ*
_{0} is permeability of free space. We then suppose that the linear combination **E** of the electric field distributions of the cavities satisfies the following Maxwell equation:

where *ε* is a dielectric constant in the case where two point-defects exist simultaneously. This equation (29) can be expressed as follows:

$$=0$$

We consider the case where the two cavities are identical. When we define *A, B, C*, and *D* as shown in the following Eqs. (31)–(34), respectively, we get Eqs. (35) and (36) from Eq. (30).

When the angular frequency of *â*
_{1} and *â*
_{2} is relatively low compared with *ω*, the second order derivatives in Eqs. (35) and (36) become negligibly small compared with the first order derivatives. Substituting *â*
_{1} and *â*
_{2} from the following Eqs. (37) and (38) into Eqs. (35) and (36) yields the following Eq. (39):

where *ω*
_{0} is defined as follows:

The second term on the right side of Eq. (39) describes the influence from the neighboring cavity. Therefore, we can express the mutual coupling coefficient *µ* as follows:

In the last approximation of this equation we used the relation of *A*≪*B*<*D*≪*C*, which is obvious from Eqs. (31)–(34). For example, in the ultrahigh-*Q* cavity used in Sec. 4 and the distance (*g, h*) of (8 rows, 0*a*), calculated relative values of *A, B, C*, and *D* are 2.68×10^{-2}, -11.5, 8.56×10^{4}, and 215, respectively. From these values, the second and the third terms of Eq. (41) are estimated to have same values (-6.72×10^{-5}
*ω*
_{0}), which illustrates that the last approximation of Eq. (41) is valid. This equation (41) is equivalent to Eq. (11).

## Acknowledgments

We would like to thank W. Kunishi and M. Nakamura for assistance with measurements of the filter characteristics and M. Okano for helpful discussions. This work was partly supported under the Core Research for Evolution Science and Technology (CREST) program of the Japan Science and Technology Agency (JST); and also by a Grant-in-Aid for Scientific Research and Information Technology from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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