1. Introduction

With the recent development of optical communication systems, in order to realize high-speed and large-capacity photonic network, compact and high-performance optical waveguide devices have been intensively developed. Under these circumstances, several kinds of photonic crystals (PhCs) have been proposed as an artificial material and the photonic devices based on PhCs have attracted a lot of attentions ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]}. PhCs are periodic nanoscale structures made of two or more kinds of dielectric materials and some of them prohibit light propagation with specific frequency into them. This frequency band is referred to as photonic band gap (PBG) and is utilized to realize ultra-compact photonic devices since radiation loss at sharp bends is able to be suppressed. In view of actual fabrication, two-dimensional slab type PhCs are most widely discussed. In slab type PhCs, the PBG usually exists only for one polarization state of two orthogonal polarization states. If an absolute PBG for both polarization states is realized, the applicability of PhCs is further extended. For that reason, several types of PhC structures to realize absolute PBGs have been reported ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]} and photonic devices utilizing an absolute PBG have been also reported ^{[17,18,19,20,21,22]}. However, in some of the PhCs proposed so far, the absolute PBGs have been realized in the higher frequency above light line. Therefore, in the actual application of those PhCs, the out-of-plane radiation loss may occur at waveguide discontinuities. Moreover, the most PhCs related to the absolute PBG has been studied in two-dimensional model. Although some PhCs with two-or three-dimensional structures have absolute PBGs below light line ^{[11,12,13,14,15,16]}, these absolute PBGs do not exist between first and second photonic bands. In addition, these PhCs utilize anisotropic dielectric materials or three kinds of materials with higher refractive index to realize PhC slab with finite slab thickness. Considering to realize PhC devices on SOI platform, Si-based PhC seems to be preferable. In Ref. ^[9], it is demonstrated that the proposed Si-based honeycomb PhCs with finite slab thickness is able to realize an absolute PBG between first and second photonic bands. However, in this report, the approximated two-dimensional analysis is carried out and, in the actual three-dimensional model, the absolute PBG does not exist.

In this study, we propose novel Si-based PhCs with an absolute PBG and the structural parameters are optimized to expand a bandwidth of the absolute PBG. Through the numerical investigation in approximated two-dimensional model, it is shown that the optimized steering type PhC has the absolute PBG with a bandwidth of 164 nm at the center wavelength of 1.55 $\mu$m. Moreover, utilizing the designed PhC, we investigate the possibility of a compact photonic circuit which supports both polarization states. Finally, we analyze the dispersion property of the actual three-dimensional model of the proposed PhC and confirm the existence of an absolute PBG.

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2. PhC with absolute PBG

In this discussion, the slab type PhCs consist of Silicon (Si) sandwiched by silica (SiO$_2$) is assumed and the slab thickness is set to be 240 nm. The refractive indices of Si, SiO$_2$, and air are assumed to be $n_{\rm Si} = 3.4$, $n_{\rm SiO_2} = 1.45$, and $n_{\rm air} = 1$, respectively. For simplicity, by utilizing the effective index method (EIM) ^[27], the approximated two-dimensional models are discussed by using the finite element method (FEM) ^[28,29] developed in our group and the effective indices are, respectively, assumed to be 2.83 and 2.16 for TE and TM wave analysis.

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2.1. Dispersion property of air-ring type PhC

First, we consider an air-ring type PhC ^[22] shown in Figure 1a and discuss optimal parameters to realize absolute PBG. This PhC is considered as a compound PhC of air hole type and rod type PhCs. It becomes a conventional air hole PhC when the rod radius $r_1 = 0$. On the other hand, it becomes a rod type PhC when the air hole radius $r_2 = a/\sqrt{3}$. In general, air hole type PhCs have PBG for TE wave and rod type PhCs have PBG for TM wave. Thus, there seems to be a possibility to realize an absolute PBG by the optimal selection of the parameters. Figure 1b shows the dependence of the PBG band on $r_1$ when $r_2$ is set to be $0.48a$. From this figure, we can see that the overlap of the PBGs for TE and TM waves is maximized at $r_1 = 0.29a$. Figure 1c shows the dispersion properties of the air-ring type PhC with $r_1 = 0.29a$ and $r_2 = 0.48a$. We can see that the absolute PBG exist in the normalized frequency range $a/\lambda$ of $0.362\sim 0.371$, in other words the mid-gap ratio ($\Delta\omega/\omega_0$) is $2.46\%$ where $\Delta\omega$ and $\omega_0$ are, respectively, the bandwidth and central the frequency of the absolute PBG. Although the PBG is realized below the light line, the PBG band is not enough wide compared with PhCs made of anisotropic dielectrics ^{[13,14,15,16]}.

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2.2. Dispersion property of steering type PhC

Next, in order to improve the bandwidth of an absolute PBG, we consider the PhC as shown in Figure 2a. We refer this type of PhC to as a steering type PhC. The steering type PhC is considered to be a structure in which the position of the rod of the air-ring type PhC is shifted by $a/\sqrt{3}$ along $y$ direction. Figure 2b shows the dependence of the PBG band on $r_1$. From this figure, we can see that the overlap of the PBGs for TE and TM waves is maximized at $r_1 = 0.17a$. Figure 2c shows the dispersion properties of the steering type PhC with $r_1 = 0.17a$. We can see that the absolute PBG exist in the normalized frequency range $a/\lambda$ of $0.421\sim 0.468$, in other words the mid-gap ratio is $10.57\%$. This result corresponds the wavelength bandwidth of 164 nm at the center wavelength of $1.55~\mu$m. This PBG exist below light line and its bandwidth is greatly expanded compared with that of air-ring type PhC.

In order to confirm the existence of absolute PBG, we calculate the transmission property of this PhC with finite periods when plane waves are incident along $\Gamma$-$K$ and $\Gamma$-$M$ directions, respectively. The actual computational model shown in Figures 3a and b are considered. Figure 3c shows their transmission and reflection properties for $\Gamma$-$K$ direction incidence case and Figure 3d shows them for $\Gamma$-$M$ direction incidence case. We can see that the transmission and reflection spectra are totally complementary since material absorption losses are ignored in this calculation.

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3. PhC waveguide based on steering type PhC

In the previous section, we discussed the air-ring and the steering type PhCs. Next, we discuss PhC waveguides based on PhC with absolute PBG. In this section, we consider only the steering type PhC whose PBG band is wider than that of air-ring type PhC. In the following discussion, $r_1$ is assumed to be $0.17a$.

Here, we consider two types of PhC waveguides shown in Figure 4. The PhC waveguide in Figure 4a is designed by filling the unit cells by Si. The dispersion properties of this PhC waveguide are shown in Figure 5. From these results, it can be seen that the single mode operation is realized for TM mode. However, the higher order mode of TE mode is observed. Figure 6 shows the propagating fields in the straight PhC at a normalized frequency of $a/\lambda = 0.44$. In this simulation, the PhC-based perfectly matched layer (PML) whose validity is confirmedin Ref. ^[30] is utilized as absorbing boundary conditions at the input and output end. We can see that, for both polarizations, the light is strongly confined in the core region owing to the absolute PBG. Next, we consider the 60 degree bend of this PhC waveguide. Figure 7 shows the structure of 60 degree bend and the propagating field through the bend. For TE mode, the excitation of the higher order mode is observed since this PhC waveguide is a multimode waveguide for TE mode. On the other hand, for TM mode, the higher order mode is not observed owing to the single mode operation although transmittance is fairly low. Thus, in order to improve the transmission property through waveguide bend, we modify the bending structure. Figure 8 shows the improved bend structure and the propagating fields. We can see that, for both modes, the transmission properties are greatly improved. However, for TE mode, higher order mode is still observed in the reflected wave.

Next, we consider the PhC waveguide shown in Figure 4b. Here, $w$ is set to be $w = 0.58a$. Figure 9 shows the dispersion property of this waveguide. In this PhC waveguide, the single mode operation bands exist for both TE and TM modes although these bands do not overlap. Though polarization independent operation is not achieved in this PhC waveguide, polarization dependent devices, such as polarization splitter, is able to be designed using this PhC waveguide ^[18,19]. Figure 10 shows the propagating fields in the straight PhC at a normalized frequency of $a/\lambda = 0.43$. We can see that, for both polarizations, the light is strongly confined in the core region owing to the absolute PBG. Figure 11 shows the structure of 60 degree bend and the propagating field through the bend. We can see that the transmission properties are greatly improved compared with those of the previous examples.

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4. Analysis of actual three-dimensional model

Finally, we analyze an actual three-dimensional structure of the proposed PhC by using three-dimensional full vector FEM ^[28] developed in our group. Although, in the previous two-dimensional model, the slab thickness is assumed to be $d = 240$ nm to use the effective index method, in this analysis, the slab thickness is defined by the normalized thickness and is assumed to be $d/a = 0.5$. Figures 12a and b show the unit cell and the periodic slab of the proposed PhC and Figure 12c shows its dispersion property. We can see that the absolute PBG exist in this PhC although the bandwidth is narrowed compared with the two-dimensional model. This is due to that the parameters are not optimized in the three-dimensional model. By optimizing the structural parameters in an actual three-dimensional model, the bandwidth of an absolute PBG is considered to be expanded.

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5. Conclusions

We discussed novel photonic crystals which can realize absolute PBG. Furthermore, we designed the PhC waveguides based on the proposed PhCs and discussed the dispersion properties and the transmission properties through 60 degree bends. We are now studying the photonic devices utilizing absolute PBG, such as a polarization splitter, a polarization rotator, and an optical isolator. The design of photonic crystals with wider absolute PBG in an actual three-dimensional model is also now under consideration.

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Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 15K06009.

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Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this manuscript.

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