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• Abstract

SECTION 1

## Introduction

Recently, slow light [1], [2], [3] has attracted much attention for realizing optical delay line based devices, for example, the optical buffer, memory, and router. Moreover, slow light can enhance nonlinear effects because effective interaction length between light and matter becomes longer. In order to realize slow-light waveguide, a lot of construction methods are being considered, such as the photonic crystal coupled waveguide (PCCW) [4], [5], the coupled resonator optical waveguide (CROW) [6], [7], and the dispersion engineered photonic crystal waveguide [8], [9]. CROW is constructed by cascaded resonators and can control delay time and bandwidth by tuning the number of resonators and the structure. In addition, zero group velocity dispersion (GVD) can be obtained at the center of transmission miniband. Among configurations relying on 1-D to 3-D photonic crystal (PC) nanocavities, 1-D PC-CROW based on photonic wires is the most attractive in terms of simplicity, while keeping the advantages of PCs such as compactness and integrability. So far, 1-D PC-CROW constructed by removing air-holes in 1-D PCs periodically (let us call it the coupled-defect-type 1-D PC-CROW) is considered; however, it was shown that this 1-D PC-CROW has large leakage losses [10], [11], [12]. Leakage losses can be decreased by getting the envelope of the electromagnetic field distribution at the core/cladding boundary close to Gaussian function [13], [14]. In order to realize high- $Q$ factor, nanocavity structure based on mode-gap confinement was proposed in 2-D PCs [15]. In addition, it was shown that 2-D PC-CROW constructed by cascading such high-$Q$ nanocavities can achieve small group velocity and low leakage losses [16], [17]. On the other hand, 1-D PC high- $Q$ nanocavity with modulated Bragg mirrors was also reported [18], [19], [20], [21]. One-dimensional PC high-$Q$ nanocavity is constructed by connecting two kinds of PCs with different waveguide parameters, as shown in Fig. 1. PC has a mode-gap in a specific frequency range, and the mode-gap can be shifted by changing waveguide parameters. Therefore, by connecting two kinds of PCs with different waveguide parameters, a mode-gap barrier appears, and local confinement of light in this frequency is realized. By connecting these 1-D PC high-$Q$ nanocavities, it is expected that low-leakage-loss 1-D PC-CROW can be realized. However, transmission characteristics of 1-D PC-CROW with modulated mode-gap barriers have not been reported to the best of our knowledge. In this paper, we evaluate transmission characteristics of 1-D PC-CROW with modulated mode-gap barriers by using the 3-D vector finite element method (VFEM) for periodic waveguide analysis [22]. In order to shift the mode-gap, we consider 1-D PC-CROW with changed air-hole distance and 1-D PC-CROW with changed air-hole size. We show that 1-D PC-CROW with modulated mode-gap barriers can realize small group velocity and low leakage losses simultaneously compared with conventional coupled-defect-type 1-D PC-CROW.

Fig. 1. Structure and operation principle of 1-D PC nanocavity with modulated mode-gap.
SECTION 2

## Evaluation of Group Velocity and Leakage Losses in 1-D PC-CROW

### 2.1. 1-D PC-CROW With Changed Air-Hole Distance

In Fig. 2, we show a structure of 1-D PC-CROW with changed air-hole distance. Structural parameters are set as follows. The waveguide width $w$ is 540 nm, the waveguide height $h$ is 200 nm, and the air-hole radius $r = 120\ \hbox{nm}$, respectively. Each air-hole distance is labeled as shown in Fig. 2, and the distance $a(i)$ is determined by a parabolic function [20], as long as $a(i)$ is smaller than a maximum value $a_{\max}$ TeX Source $$a(i) = a_{0} \left\{1 + (i/M_{a})^{2} \right\}\eqno{\hbox{(1)}}$$ where $a_{0} = 400\ \hbox{nm}$, and $M_{a}$ is integer. $a_{\max}$ is set as 420 nm. If $a(i)$ calculated by (1) is larger than $a_{\max}$, $a(i)$ is set as $a_{\max}$. The length of one period $\Lambda$ becomes $2(a_{0}/2 + a_{1} + a_{2} + \cdots + a_{m-1}\ + a_{m}/2)$. We assume Si, SiO2, and air as a core material, under cladding, and over cladding materials, respectively. We set the refractive index of 3.5 as Si and 1.45 as SiO2. To evaluate dispersion characteristic of 1-D PC-CROW, we apply 3-D VFEM for periodic waveguide analysis [22]. Fig. 3(a)(c) show dispersion curves, group velocity, and leakage losses of 1-D PC-CROW with $m = 7$ for quasi-TE mode, where $\lambda$ is the operating wavelength, $\beta$ is the propagation constant along the propagation direction, $v_{g}$ is the group velocity, and $c$ is the velocity of light in a vacuum. Compared with conventional coupled-defect-type 1-D PC-CROW [12], group velocity and leakage losses of mode-gap confinement based 1-D PC-CROW at the zero GVD frequency are small. As $M_{a}$ becomes large, group velocity becomes larger. When $M_{a}$ is set as 30, the air-hole distance defined by $a_{\max}$ is only $a(7)$. On the other hand, when $M_{a}$ is set as 15, $a(4)$, $a(5)$, $a(6)$, and $a(7)$ are defined by $a_{\max}$. Therefore, the strength of refractive index modulation is weak in 1-D PC-CROW with $M_{a} = 30$. Then, the electromagnetic field spreads throughout the resonator. As a result, the coupling between resonators becomes strong and group velocity becomes larger. Fig. 4(a)(c) show the field distribution of 1-D PC-CROW with $M_{a} = 15$, 25, and 30, respectively. It can be confirmed that field spreads throughout the resonator as $M_{a}$ becomes large. Leakage losses can be decreased by increasing the $M_{a}$. If the electromagnetic field at the core/cladding boundary changes with Gaussian function, the leaky component of the wave vector, which is obtained by computing Fourier transform of the field profile, can be suppressed [11], [13], [14]. When the $M_{a}$ increases, the changing ratio of air-hole distance becomes smaller, therefore, the electromagnetic field distribution in the resonator changes gently and gets close to Gaussian function. Black lines in Fig. 4(d) and (e) show field profile of 1-D PC-CROW with $M_{a} = 15$ and 25 at the $\hbox{Si/SiO}_{2}$ boundary. Red lines show fitted curve by using Gaussian function. We can see that envelope of the field profile of 1-D PC-CROW with $M_{a} = 25$ is similar with Gaussian function, while large difference exists at the edge of the resonator between field profile of 1-D PC-CROW with $M_{a} = 15$ and the fitted curve. The $M_{a}$ should be less than 30 because when we set $M_{a}\ >\ 30$, $a(7)$ becomes smaller than $a_{\max}$. Of course, mode-gap barrier appears, even if we set $M_{a}\ >\ 30$. However, the frequency range where mode-gap barrier appears changes in such 1-D PC-CROW. Here, in order to compare transmission characteristics of 1-D PC-CROW with the same mode-gap barrier frequency, $M_{a}$ is set as $M_{a} = 15$, 20, 25, and 30 for $m = 7$.

Fig. 2. Structure of 1-D PC-CROW with changed air-hole distance (1 period).
Fig. 3. (a) Dispersion curves, (b) normalized group velocity, and (c) leakage losses of 1-D PC-CROW with changed air-hole distance, where $r$, $m$, and $a_{\max}$ are fixed as 120 nm, 7, and 420 nm, respectively.
Fig. 4. Electric field $(E_{x})$ distribution of 1-D PC-CROW with (a) $M_{a} = 15$, (b) $M_{a} = 25$, and (c) $M_{a} = 30$ and field profile at the $\hbox{Si/SiO}_{2}$ boundary and fitted curve of 1-D PC-CROW with (d) $M_{a} = 15$ and (e) $M_{a} = 25$, where $r$, $m$, and $a_{\max}$ are fixed as 120 nm, 7, and 420 nm, respectively.

Next, we examine structural dependence of group velocity and leakage losses of 1-D PC-CROW. Fig. 5(a) and (b) show structural dependence of group velocity and leakage losses at the zero GVD frequency. As the $m$ becomes small, group velocity becomes large because the strength of coupling between resonators becomes strong. We can confirm the tendency that leakage losses becomes small as the $M_{a}$ becomes large. We note that leakage loss is determined by the field distribution in the resonator; therefore, there is an optimum $M_{a}$ for the each $m$ value [see Fig. 4(d) and (e)]. If we choose $m = 7$ and $M_{a} = 25$, normalized group velocity and leakage loss are 0.0287 and 0.148 dB/mm. Normalized group velocity and leakage loss of coupled-defect-type 1-D PC-CROW are 0.12 and 90 dB/mm [12]; therefore, we can see that mode-gap confinement based 1-D PC-CROW can achieve small group velocity and low leakage loss simultaneously. Generally, leakage loss is expressed as a loss per unit length; however, in order to specify the optimum structure for the optical delay devices, leakage losses expressed as a per unit time is more suitable because we also have to consider the group velocity in order to evaluate the delay time. Fig. 5(c) shows structural dependence of leakage losses per unit time. The leakage losses per unit time are defined by the product of leakage losses per unit length and group velocity. The smallest leakage loss per unit length is obtained by choosing $m = 7$ and $M_{a} = 25$; however, the smallest leakage loss expressed per unit time is obtained by choosing $m = 9$ and $M_{a} = 35$ because the group velocity of 1-D PC-CROW with $m = 9$ and $M_{a} = 35$ is smaller than that with $m = 7$ and $M_{a} = 25$ while leakage losses per unit length are of similar value. Delay time in CROW is determined by two factors: One is group velocity, and another is loss. In terms of group velocity, CROW with $m = 9$ and $M_{a} = 35$ is better. On the other hand, delay time can be obtained is proportional to the number of resonators to be cascaded. The number of resonators to be cascaded is limited by losses. In this respect, CROW with $m = 7$ and $M_{a} = 25$ is preferable. Losses expressed as per unit time include these two factors; therefore, when the maximum permissive loss value is decided, the largest delay time can be obtained in the 1-D PC-CROW with $m = 9$ and $M_{a} = 35$.

Fig. 5. Structural dependence of (a) normalized group velocity, (b) leakage losses per unit length, and (c) leakage losses per unit time of 1-D PC-CROW with changed air-hole distance at the zero GVD frequency, where $r$ and $a_{\max}$ are fixed as 120 nm and 420 nm, respectively.

So far, transmission characteristics of 1-D PC-CROW with the same frequency range of mode-gap barriers are investigated. Therefore, we examine transmission characteristics of 1-D PC-CROW with different frequency range of mode-gap barriers. The frequency range of mode-gap barriers can be shifted by changing $a_{\max}$. Fig. 6(a)(c) show structural dependence of group velocity, leakage losses per unit length, and leakage losses per unit time, respectively. In this investigation, $m$ is fixed as 7. We can see that as the $a_{\max}$ becomes small, group velocity becomes large. Fig. 7(a) and (b) show electric field distributions of 1-D PC-CROW with $a_{\max} = 410\ \hbox{nm}$ and 425 nm, respectively. As $a_{\max}$ becomes small, the strength of refractive index modulation becomes weak and the electromagnetic field spread throughout the resonator, resulting in strengthening the coupling between resonators and increment of group velocity. Leakage losses less than 1 dB/mm can be realized by choosing $M_{a}\ >\ 25$. There is a tradeoff between group velocity and bandwidth; therefore, 1-D PC-CROW with small $a_{\max}$ can broaden bandwidth keeping leakage losses per unit time small. Then, we evaluate structural dependence of transmission characteristics of 1-D PC-CROW with $a_{\max} = 410\ \hbox{nm}$. Fig. 8(a)(c) show structural dependence of group velocity, leakage losses per unit length, and leakage losses per unit time of 1-D PC-CROW with $a_{\max} = 410\ \hbox{nm}$, respectively. We can see that small leakage losses can be realized even if we choose small $m$ value. Hence, we can miniaturize the length of one period and broaden bandwidth by choosing small $a_{\max}$ value.

Fig. 6. Structural dependence of (a) normalized group velocity, (b) leakage losses per unit length, and (c) leakage losses per unit time of 1-D PC-CROW with changed air-hole distance at the zero GVD frequency, where $r$ and $m$ are fixed as 120 nm and 7, respectively.
Fig. 7. Electric field $(E_{x})$ distribution of 1-D PC-CROW with (a) $a_{\max} = 410\ \hbox{nm}$ and (b) $a_{\max} = 425\ \hbox{nm}$.
Fig. 8. Structural dependence of (a) normalized group velocity, (b) leakage losses per unit length, and (c) leakage losses per unit time of 1-D PC-CROW with changed air-hole distance at the zero GVD frequency, where $r$ and $a_{\max}$ are fixed as 120 nm and 410 nm, respectively.

### 2.2. 1-D PC-CROW With Changed Air-Hole Size

In Fig. 9, we show a structure of 1-D PC-CROW with changed air-hole size. Structural parameters are set as follows. The waveguide width $w$ is 540 nm, the waveguide height $h$ is 200 nm, and the lattice constant of 1-D PC $a = 400\ \hbox{nm}$, respectively. Each air-hole in the resonator is numbered as shown in Fig. 9, and the air-hole radius $r(i)$ is determined by a parabolic function [20] as long as $r(i)$ is larger than a minimum value $r_{\min}$ TeX Source $$r(i) = r_{0} \left\{1 - (i/M_{r})^{2} \right\}\eqno{\hbox{(2)}}$$ where $r_{0} = 120\ \hbox{nm}$, and $M_{r}$ is an integer. In order to locate the mode-gap barriers in the same frequency range with the previous structure ($a_{0} = 400\ \hbox{nm}$ and $a_{\max} = 420\ \hbox{nm}$), $r_{\min}$ is set as 88 nm. If $r(i)$ calculated by (2) is smaller than $r_{\min}$, $r(i)$ is set as $r_{\min}$. The length of one period $\Lambda$ becomes $(2m + 1)a$. Fig. 10(a)(c) show structural dependence of group velocity, leakage losses per unit length, and leakage losses per unit time at the zero GVD frequency, respectively. As the $m$ becomes large, group velocity becomes small. We can confirm the tendency that leakage losses become small as the $M_{r}$ becomes large. They are similar characteristics with 1-D PC-CROW with changed air-hole distance. The minimum leakage loss per unit length can be obtained by choosing $m = 8$ and $M_{r} = 15$; on the other hand, the minimum leakage loss per unit time is obtained by choosing $m = 9$ and $M_{r} = 13$ because the group velocity of 1-D PC-CROW with $m = 9$ and $M_{r} = 13$ is smaller than that with $m = 8$ and $M_{r} = 15$, while leakage losses per unit length have similar value. These results are similar with them of 1-D PC-CROW with changed air-hole distance. Leakage losses of 1-D PC-CROW with changed air-hole size are comparable with 1-D PC-CROW with changed air-hole distance.

Fig. 9. Structure of 1-D PC-CROW with changed air-hole size (1 period).
Fig. 10. Structural dependence of (a) normalized group velocity, (b) leakage losses per unit length, and (c) leakage losses per unit time of 1-D PC-CROW with changed air-hole size at the zero GVD frequency, where $a$ and $r_{\min}$ are fixed as 400 nm and 88 nm, respectively.

Next, we examine transmission characteristics of 1-D PC-CROW with different frequency range of mode-gap barriers. The frequency range of mode-gap barriers can be shifted by changing $r_{\min}$. Fig. 11(a)(c) show structural dependence of group velocity, leakage losses per unit length, and leakage losses per unit time, respectively. In this investigation, $m$ is fixed as 6. We can see that as the $r_{\min}$ becomes large, group velocity becomes large. As $r_{\min}$ becomes large, the strength of refractive index modulation becomes weak, and the electromagnetic field is spread throughout the resonator, resulting in strengthening the coupling between resonators and increment of group velocity. Leakage losses less than 1 dB/mm can be realized by choosing $M_{r}\ >\ 15$. Finally, we evaluate structural dependence of transmission characteristics of 1-D PC-CROW with $r_{\min} = 115\ \hbox{nm}$. Fig. 12(a)(c) show structural dependence of group velocity, leakage losses per unit length, and leakage losses per unit time of 1-D PC-CROW with $r_{\min} = 115\ \hbox{nm}$, respectively. We can see that small leakage losses can be realized, even if we choose small $m$ value. Hence, we can miniaturize the length of one period and broaden bandwidth by choosing small $r_{\min}$ value. At last, we summarize transmission characteristics of 1-D PC-CROW in Table 1. We can see that 1-D PC-CROW with modulated mode-gap barriers can realize small group velocity and low loss simultaneously compared with conventional coupled-defect-type 1-D PC-CROW. If $m$ is set as 9, group velocity becomes one order of magnitude smaller, and on the other hand, one period length becomes longer. The length of one period can be shorten by choosing $a_{\max}$ or $r_{\min}$ adequately, keeping leakage losses per unit length small enough.

Fig. 11. Structural dependence of (a) normalized group velocity, (b) leakage losses per unit length, and (c) leakage losses per unit time of 1-D PC-CROW with changed air-hole size at the zero GVD frequency, where $a$ and $m$ are fixed as 400 nm and 6, respectively.
Fig. 12. Structural dependence of (a) normalized group velocity, (b) leakage losses per unit length, and (c) leakage losses per unit time of 1-D PC-CROW with changed air-hole size at the zero GVD frequency, where $a$ and $r_{\min}$ are fixed as 400 nm and 115 nm, respectively.
TABLE 1 Comparison of transmission characteristics at the zero GVD frequency of 1-D PC-CROW
SECTION 3

## Conclusion

We have investigated structural dependence of group velocity and leakage loss in 1-D PC-CROW with modulated mode-gap using 3-D VFEM for periodic waveguide analysis. In order to shift mode-gap, we considered 1-D PC-CROW with changed air-hole distance and 1-D PC-CROW with changed air-hole size. We have shown that 1-D PC-CROW with modulated mode-gap barriers can realize small group velocity and low leakage losses simultaneously compared with conventional coupled-defect-type 1-D PC-CROW. We can expect that large delay time can be realized by using such 1-D PC-CROW; however, we may need to design a taper waveguide for high-efficiency coupling between silicon wire and 1-D PC-CROW. Designing a taper waveguide and measuring delay time by performing time domain analysis are next issues.

## Footnotes

This work was supported by the Japan Society for the Promotion of Science. Corresponding author: Y. Kawaguchi (e-mail: kawaguchi@icp.ist.hokudai.ac.jp).

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