## Abstract

We have derived the general condition to achieve 100% drop efficiency in the resonant tunneling-based channel drop filters of a three-port system with reflection feedback. According to our theoretical modeling based on the coupled mode theory in time, the condition is that the Q-factor due to coupling to a bus port should be twice as large as the Q-factor due to coupling to a drop port and the phase retardation occurring in the round trip between a resonator and a reflector should be a multiple of 2*π*. The theoretical modeling also shows that the reflection feedback in the three-port channel drop filters brings about relaxed sensitivity to the design parameters, such as the ratio between those two Q-factors and the phase retardation in the reflection path. Based on the theoretical modeling, a five-channel drop filter has been designed in a two-dimensional photonic crystal, in which only a single reflector is placed at the end of the bus waveguide. The performance of the designed filter has been numerically calculated using the finite-difference time domain method. In the designed filter, drop efficiencies larger than 96% in all channels have been achieved.

©2004 Optical Society of America

## 1. Introduction

Photonic crystals (PCs) or photonic bandgap (PBG) structures enable us to control photons or, in general, electromagnetic waves in a dielectric medium. The means to control photons is mainly obtained by introducing defects in PCs. Micro-cavities or resonators formed by point defects and waveguides formed by line defects in PCs have been subjects for plenty of research because of their capability to confine photons within a small volume, and they are expected to be key building blocks for miniature photonic functional devices and photonic integrated circuits (PICs). Among various PC-based devices, ultra-compact channel drop filters based on resonant coupling between cavity modes of point defects and waveguide modes of line defects have drawn primary interest due to their substantial demand in wavelength division multiplexed (WDM) optical communication systems.

So far, several structures of channel drop filters based on two-dimensional (2D) PCs have been proposed [1–6]. These structures may be classified into two categories: four-port system and three-port system. The channel drop filters of four-port system make use of resonant tunneling through a resonator between two parallel waveguides [1,2]. In these structures, the resonator should support two degenerate modes of different symmetry in order to attain 100% drop efficiency, but enforcing degeneracy between two resonant modes of different symmetry requires a very sophisticated resonator design. The channel drop filters of three-port system also have been proposed by several groups. Noda *et al*. proposed a channel drop filter with a simple structure, in which a defect resonator is placed on the side of a bus waveguide in a 2D slab PC [3]. The resonator traps photons of its own resonance frequency from the bus waveguide through evanescent coupling and emits some of them vertically into the air through the weak confinement of the slab. In this case, the coupling between the resonator and the air plays a role of one port. Another type of three-port channel drop filter has been proposed, where the photons trapped by a resonator are coupled to an in-plane waveguide through direct coupling [4,5]. This kind of three-port structure can be easily extended to multi-channel drop filters by using a set of resonators of different sizes, i.e., different resonance frequencies, or by using the concept of the hetero PC [5,6]. So, in general, the multi-channel drop filters based on the three-port system are much easier to design than those based on the four-port system. In the three-port system-based filters, however, channel drop efficiency is inherently less than 50% since a part of the photons trapped in a resonator are also emitted back to the bus waveguide when channel drop occurs. As for this limitation, the possibility to enhance the drop efficiency using reflection at the interface in the hetero PCs has been suggested [5,7]. The exact mechanism and the general conditions for the drop efficiency enhancement, however, have not been analyzed so far. On the other hand, the similar approach of using controlled reflection to cancel the overall reflection in a full demultiplexer, which is based on coupling among an ultra-low-Q-factor microcavity and resonators with high Q-factor, also has been proposed [8].

In this work, using the coupled mode theory in time [9], we have derived the general conditions to achieve 100% drop efficiency in three-port system-based channel drop filters by introducing reflection in a bus waveguide. The results of the theoretical modeling have been applied to design a 2D PC-based five-channel drop filter with a simple structure. The performance of the designed filter has been calculated using the finite-difference time-domain (FDTD) method.

## 2. Theoretical modeling

Figure 1 shows a general structure of the channel drop filter based on a three-port system. The temporal change of the normalized mode amplitude of the resonator, *a* is described by [9]

where *ω*
_{o} is the resonance frequency, 1/*τ _{b}* and 1/

*τ*are the decay rates into the bus and the drop waveguides, respectively, and

_{d}*s*

_{+i}and

*s*are the amplitudes of the incoming and the outgoing waves, respectively, in each port as depicted in Fig. 1. A resonant mode is assumed to be symmetric along the bus waveguide direction. The mode amplitude,

_{-i}*a*is normalized such that |

*a*|

^{2}is equal to the energy stored in the resonator, and the wave amplitude,

*s*is normalized such that |

*s*|

^{2}is equal to the power of the wave. The phases of the coupling coefficient from the incoming wave to the resonator mode depend on the choices of the reference planes for the incoming waves. In this work, the reference plane for the waves in the bus waveguide is picked at the center plane of the resonator as depicted in Fig. 1, which results in

*θ*

_{1}=

*θ*

_{2}.

From the power conservation and the time-reversal symmetry, the relationship among the incoming/outgoing wave amplitudes in the waveguides and the resonator mode amplitude can be derived as follows [9,10]:

On the other hand, the reflection at the end of the bus waveguide results in *s*
_{+2}=*e*
^{-jϕ}s_{-2}, where *ϕ* is the phase retardation that occurs during the round trip. When wave is launched only into the bus waveguide (*s*
_{+3}=0), the transmission from the waveguide to the drop waveguide and reflection can be calculated from the Fourier transform of the above equations. The resulting expressions are given by

and

where *s*̃ represents the Fourier transform of *s*. Equations (3a) and (3b) show that *T*(*ω*
_{o})=1 and *R*(*ω _{o}*)=0 occur if and only if

*ϕ*=2

*mπ*(m=integer) and

*τ*/

_{b}*τ*=4. This indicates that in the three-port channel drop filter, 100% channel drop efficiency can be achieved if the resonators are properly located from the reflector and if they are designed in such a way that the Q-factor due to the coupling into the bus waveguide is twice as large as the Q-factor due to the coupling into the drop waveguide. As for the phase condition for 100% drop efficiency, it could be rather counter-intuitive at first glance. However, if we understand that the resonator works as a partial reflector with time delay and the resonator and the reflector form a Fabry-Perot (F-P) etalon, it is quite straightforward. Although the phase condition for reflection to be zero is the same with the F-P etalon case, the way of interference between reflected waves in this case is different from the F-P etalon case. Equations (3a) and (3b) also indicate that the waves reflected from the resonator and the reflector in the bus waveguide interfere not only directly, but also through the resonator. Unless the reflection in the bus waveguide is introduced, in the case of

_{d}*s*

_{+2}=

*s*

_{+3}=0, the transmission to the drop waveguide calculated from the same equations above becomes

In this case, one can see that the maximum transmission of 50% is obtained when *τ _{b}*/

*τ*=2 and the transmission is 4/9(≈44.4%) when

_{d}*τ*/

_{b}*τ*=4.

_{d}Figure 2(a) shows the analytically calculated peak transmissions of the three-port channel drop filters with and without reflection as functions of *τ _{b}*/

*τ*. For the filter with reflection,

_{d}*ϕ*=2

*π*is assumed. Figure 2(b) plots the peak transmission of the filter with reflection as a function of

*ϕ*for

*τ*/

_{b}*τ*=4. It can be noted that the performance of the three-port filter with reflection is a slowly varying function near the optimum with respect to both the decay rate ratio and the phase retardation. It seems that the indirect interference through the resonator makes the performance of the filter less sensitive to

_{d}*ϕ*. As seen in (2c), the dropped wave amplitude is proportional to the resonator mode amplitude,

*a*when 3

_{+3}=0. The resonator mode is excited by both the incoming wave,

*s*

_{+1}and the reflected wave,

*s*

_{+2}(=

*e*

^{-jϕ}

*s*

_{-2}). The reflected wave is given by ${s}_{+2}={e}^{-j\varphi}\left({s}_{+1}-\sqrt{2\u2044{\tau}_{b}}a\right)$ from (2b). Note that the reflected wave is negatively proportional to the resonator mode amplitude. When phase error occurs, the excitation of the resonator mode decreases due to the phase mismatch between

*s*

_{+1}and

*s*

_{+2}. This decreased mode excitation, however, results in increment of the reflected wave and consequently, increment of the resonator mode amplitude. As a result, the amplitudes of the resonator mode and the dropped wave become less sensitive to the phase error, and the interference through the resonator works as sort of negative feedback with respect to the phase error. Besides, reflection in the bus waveguide itself seems to naturally work as negative feedback with respect to the variation of the Q-factor ratio,

*τ*/

_{b}*τ*. This can be understood in the picture of multiple reflections. From the theoretical calculation in the non-reflection case, one can find that the wave amplitude transmitted to the reflector is an increasing function of

_{d}*τ*/

_{b}*τ*near the optimum, while the dropped wave amplitude is a decreasing function. Therefore, if the first dropped wave amplitude increases (decreases) for a non-optimal Q-factor ratio (

_{d}*τ*/

_{b}*τ*≠4), compared to the value for the optimal case, the wave amplitude transmitted to the reflector decreases (increases). As a result, the dropped wave amplitude in the second coupling decreases (increases), so that the total transmission varies less sensitively with respect to

_{d}*τ*near the optimum.

_{b}/τ_{d}## 3. Design and numerical calculations

Based on the theoretical calculations presented above, a five-channel drop filter was designed in a 2D photonic crystal made of square lattice of dielectric rods in air. In this 2D photonic crystal, PBG exists only for the transverse magnetic (TM) mode whose electric field is oriented along the rods [11]. Thus, in this work, all calculations were carried out for the TM mode. Figure 3 shows the filter structure to be designed, in which the point defect resonators of different resonance frequencies are located on the side of the bus waveguide, and the drop waveguides are directly coupled to the resonators in perpendicular direction to the bus waveguide. The structure is basically the same as the previously proposed structure in [4]. The only difference is that the bus waveguide in the designed structure is closed at one end for 100% reflection.

In the design presented in this paper, the radius of the rods is 0.2a, where a is the lattice constant, and the dielectric constant of the rods is 11.56. For the resonators, five point defects with radii of 0, 0.05a, 0.065a, 0.08a, and 0.1a were chosen to cover a wide range of the PBG. The distance between the defect and the respective waveguide determines the corresponding decay rate. The decay rate also depends on the defect size and the radius of the rods near the defects. Therefore, in order to satisfy *τ _{b}/τ_{d}*=4, all these parameters should be properly determined. In order to obtain the optimal ratio between those two the decay rates, the resonators may be designed in such a way that the peak transmissions of the filters without reflection feedback become 4/9(≈44.4%). From the numerical calculation using the 2D FDTD method with the perfectly matched layer absorbing boundary condition for photonic crystal waveguide structures [12], it was found that all the five resonators exhibited transmissions between about 38% and 39% when two rods of r=0.2a, which is just the same with other rods in the structure, were located between the defect and the bus/drop waveguides. According to the theoretical calculation presented above, transmissions will be larger than 96.5% if reflection with

*ϕ*=2

*π*is introduced. For more optimal resonators, the sizes of the rods between the defect and the bus/drop waveguides can be trimmed, although further trimming of the resonators were not performed in this work.

After the resonators have been designed, they should be properly located to satisfy *ϕ _{i}*=2

*mπ*, where

*i*represents the index for the different resonators. The phase retardation can be expressed by

*ϕ*=4

_{i}*πd*̄

_{i}n*/*

_{i}*λ*+

_{i}*δ*, where

_{i}*d*is the distance from the closed end of the bus waveguide as depicted in Fig. 3,

_{i}*n*̄

*is the effective index at*

_{i}*ω*=

*ω*, and

_{i}*δ*is the phase shift occurring in the reflection. The retardation due to the propagation in the bus waveguide can be easily calculated, but

_{i}*δ*is not easy to calculate since it includes the phase shift due to the penetration of the reflected wave into the PC. Therefore, we calculated the transmission for each resonator separately as changing the location to find out the proper location in which the peak transmission is close to 100%. Figure 4 shows the peak transmission calculated using the 2D FDTD method for the resonators with defect sizes of r=0, 0.05a, 0.065a, and 0.08a at different locations. As seen in the figure, for each resonator, there exist several locations where transmission is very close to 100% although only discrete variation of location is allowed in the structure. This stems from the fact that the peak transmission of the proposed filter is a slowly varying function of

_{i}*ϕ*near the optimum as seen in Fig. 2(b). Due to this relaxed sensitivity to the phase retardation, it seems that multi-channel drop filters with a performance good enough for practical purposes are always obtainable with the structure in Fig. 3.

According to the results shown in Fig. 4, the proper location of each resonator was determined. In order to guarantee the negligible coupling between the resonators or the drop waveguides, the separation between the resonators were made larger than 4a. Figure 5(a) shows the resulting structure of the designed five-channel drop filter, and its numerically calculated transmission spectrum from the bus waveguide to each drop waveguide is plotted in Fig. 5(b). As seen in Fig. 5(a), the bus waveguide is terminated with another channel drop filter composed of a directly coupled resonator rather than the closed end to save the device area. When a resonator is directly coupled to the bus/drop waveguide, resonance transmission of 100% is naturally obtained [10], and the filter at the end of the bus waveguide works as a reflector for the other channels. In order to use the space more efficiently, channel drop filters are placed on both sides of the bus waveguide. As seen in Fig. 5(b), channel drop efficiencies larger than 96% were achieved in all channels. Figure 6(a) and Fig. 6(b) show the wave propagations at the resonance frequencies of the drop filters at port C (with the defect of r=0.065a) and port B (with the defect of r=0.08a), respectively.

## 4. Conclusion

In this work, we have derived general conditions for obtaining 100% drop efficiency in a three-port channel drop filter with reflection feedback. According to the theoretical modeling using the coupled mode theory in time, the performance of the three-port channel drop filter with reflection feedback shows very relaxed sensitivity to variation of design parameters near the optima. Based on the theoretical modeling, a five-channel drop filter with a simple structure has been designed. The performance of the designed filter has been calculated using the 2D FDTD method. The calculation shows that for all channels in the designed filter, channel drop efficiencies larger than 96% have been achieved. The design of the five-channel drop filter with such a simple structure confirms the relaxed sensitivity of the three-port channel drop filter with reflection feedback.

In the designed 2D PC-based five-channel drop filter, however, vertical confinement of photons is not considered. Therefore, for practical filters, the proposed structure should be realized in different forms, such as PC slabs, in which vertical confinement is provided and 3D calculation is required for design. In such practical structures, performance of the channel drop filter will be degraded by imperfect vertical confinement. The theoretical modeling presented in this paper can be easily extended for the practical structures by including the vertical loss of the resonator.

In the designed filter structure, the channels that are not dropped will be reflected back to the input port. This will cause problems if the designed structure is incorporated in PICs. However, if the designed structure is used as a single component, the problem can be easily solved by using a circulator at the input port like in most channel drop filters based on fiber Bragg gratings. Moreover, the designed structure has no problem whether it is incorporated in PICs or not, if used for a full demultiplexing operation.

Finally, the structure of multi-channel filter designed in this work can be easily combined with the hetero PC.

## Acknowledgments

This work was performed with the support by the KISTEP through the Nano Research and Development program. This work was supported in part by the KOSEF through the Advanced Basic Research Laboratory (ABRL) program.

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