1. Introduction

Three-dimensional photonic crystals (3D PCs) have an anomalous photonic band structure. A complete photonic band gap (c-PBG) – a frequency range in which the propagation or existence of light waves is forbidden for all wavevectors and all polarizations – can be formed by using appropriately designed 3D structures [1–13]. Defects can be artificially introduced into 3D PCs to make fundamental photonic elements such as optical waveguides and photonic nanocavities, which will allow the fabrication of 3D photonic circuits and optical interconnections for optoelectronic integrated circuits. In 2013, we successfully demonstrated 3D guiding of photons by introducing 45-degree oblique waveguides and horizontal waveguides in 3D PC, as schematically shown in Fig. 1(a). Here, the development of 45-degree oblique waveguide [14] played a key role for the success of 3D guiding of photons, because it has an equivalent dispersion characteristics to those of the horizontal waveguides that had already been established [15, 16].

 

Fig. 1 Schematic view of 3D light guiding structure in stripe-stacked 3D photonic crystal. (a) shows schematic structure of the connected oblique, horizontal and oblique waveguides. (b) shows schematic structure of overhead view of oblique waveguide. (c) The band diagram of oblique waveguide.

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The 45-degree waveguide can be also used as the input/output port of the 3D PC circuits to the free-space. In the successive work [17], we showed that the coupling efficiency to the free-space decreases due to the surface mode of the 3D PC [18, 19], and it can be improved by introducing a 2D grating-like structure to create a surface-mode gap, in which surface modes are prohibited. However, the radiation angle, which is an important parameter for coupling to an external optical system, has not been investigated. Therefore, in this work, first, we study the frequency dependence of the radiation angle of an oblique waveguide. The results show that the output light radiates in a different direction from the oblique-waveguide direction, and the radiation polar angle depends on the frequency. Then, we consider a method of obtaining vertical radiation that does not depend on frequency. We propose introducing a symmetric structure at the end of the oblique waveguide by branching the oblique waveguide into four ports. By using this symmetric end structure, light is radiated vertically and the radiation angle does not depend on the frequency or the direction in which the oblique waveguide is formed.

2. Radiation angle of oblique waveguide

We investigated an oblique waveguide formed in a stripe-stacked 3D PC. The width and height of the rods were set at 0.3a and $\left(\sqrt{2}/4\right)a$, respectively, where a is the period of the stripe. The refractive index was set at 3.4, assuming the use of silicon at near infrared wavelengths. A c-PBG is formed in the frequency range of 0.335–0.399c/a. The oblique waveguide consists of a stacked structure in which rods are partially removed (with a defect width of 0.6a) along the < 011 > direction [14, 17] is shown in Fig. 1(b). The dispersion relationship of this waveguide has a single-mode guiding range in the frequency range of 0.335 – 0.374c/a, as shown in Fig. 1(c). The dominant component of the electric field is the [100]-component (E_[100]) when the waveguide is formed along the [011]-direction.

We analyzed the radiation characteristics of the output light from this oblique waveguide into free space by using the finite-difference time-domain (FDTD) method. Figure 2 shows the calculation results of the electric field pattern in the (001) plane at a frequency f = 0.360c/a (or a wavenumber of 0.313(2π/a)) The results reveal that the guided light in the oblique waveguide is coupled to a free-space propagation mode. To accurately evaluate the radiation angle, the radiation light power is observed at a spherical surface of radius 10a. The light power distribution in the ϕ – θ plane and its stereographic projection are shown in Fig. 2(b) and 2(c), respectively. The light radiated from the oblique waveguide shows a single-peak pattern. The concentration of light intensity in the region where θ ∼ 90° in Fig. 2(b) or the periphery in Fig. 2(c) is due to light localization around the surface due to the existence of a surface mode of the 3D PC [19]. The radiation pattern in the (100) plane is shown in Fig. 2(d). The light is radiated in a direction between the [001]- and $\left[0\overline{1}0\right]$-directions. Here, the radiation polar angle, θ, at which the maximum radiated power occurs is 39° toward the $\left[0\overline{1}1\right]$-direction, which is on the other side from the oblique-waveguide direction (the [011]-direction). This is considered to be due to the wavenumber conservation law, as explained in the next section.

 

Fig. 2 Calculation results of electric field pattern of the oblique waveguide at a frequency of 0.360c/a. (a) Cross-sectional view, (b) radiation power distribution in ϕ – θ plane, (c) radiation power distribution in stereographic projection, and (d) radiation pattern in (100) plane.

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In order to discuss the determinant of the radiation angle, we analyzed the dominant wavevector of the eigenmode of the oblique waveguide in the [011]-direction and estimated the radiation angle based on the wavenumber conservation law. First, the eigenmode of an infinite-length oblique waveguide was calculated by using the FDTD method with periodic boundary conditions. Second, the electric field pattern in the (001) plane was spatially Fourier transformed to obtain the electric field pattern in wavevector space which is projected onto the (001) plane. The calculation results of the electric field pattern (E_[100]) and it Fourier transformation at frequency f = 0.360c/a are shown in Fig. 3(a) and 3(b), respectively. In Fig. 3(b), the components inside the air light line, which can be connected to a free-space mode, have a single peak at $k_{\left[0\overline{1}0\right]}=0.31\left(2\pi /a\right)$, which corresponds to a situation where the radiation polar angle θ = 21° toward the $\left[0\overline{1}1\right]$-direction, considering the wavenumber conservation law.

 

Fig. 3 Calculation results of eigenmode analysis of the oblique waveguide at a frequency of 0.360c/a. (a) Electric field pattern of E_[100] in (001) plane, (b) absolute value of spatial Fourier-transform spectrum of E_[100] (wavevector distribution), and (c) schematic illustration of the wavenumber conservation law.

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The reason for the difference of the radiation polar angle between the results of the finite-space calculation [Fig. 2] and the eigenmode estimation [Fig. 3] is considered to be due to the influence of the structural finiteness. The discussion based on eigenmodes does not consider the influence of the decreasing periodic number of the 3D PC at the upper side of the oblique waveguide (i.e., a part of the electromagnetic field penetrating into the free-space), and the influence of the surface modes of the 3D PC. Furthermore, the origin of the asymmetric wavevector distribution about the k_[010]-direction in wavevector-space is illustrated as the schematic structure of the wavevector of the guided light shown in Fig. 3(c). The wavevector k_[011] in the waveguide direction is determined by the dispersion relationship, so that k_[011] = 0.313(2π/a) at frequency f = 0.360c/a. The wavevectors oriented orthogonally to the waveguide are related to the electric field pattern of the eigenmode, so that k_[100] ~ 0, $k_{\left[0\overline{1}1\right]}\sim \pm 0.5\left(2\pi /a\right)$ in the oblique waveguide. In this situation, the wavevectors projected onto the (001) plane are k_[010] ∼ 0.57, −0.13(2π/a) as shown in Fig. 3(c), and so the distribution of the electric field pattern in wavevector-space becomes asymmetric about the k_[010]-direction.

The above discussion is at a frequency of f = 0.360c/a. The frequency dependence of the radiation polar angle, θ, is shown in Fig. 4. In this frequency range, we confirmed that the output light distribution shows a single-peak radiation pattern constantly. The radiation light and surface propagating light are difficult to differentiate in the frequency region below 0.348c/a, and therefore, the radiation angle is not shown.

 

Fig. 4 Calculated radiation angle of light observed at spherical surface and light estimation by electric field pattern of eigenmode of oblique waveguide.

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In the frequency dependence estimated by using the eigenmode, the radiation polar angle decreases at high frequency. The wavevector oriented parallel to the waveguide direction (k_[011]) varies corresponding to the dispersion relationship shown in Fig. 1(c). On the other hand, the wavevectors oriented orthogonally to the waveguide (k_[100], $k_{\left[0\overline{1}1\right]}$) show little frequency dependence. Then, the wavevector of radiation light is tilted towards k_[001] by increasing the frequency, so that radiation polar angle θ decreases. Moreover, the wavevector distribution explains the reason why the radiation light has a single-peak pattern without frequency dependence. In the photonic crystal waveguide, because light is guided in the waveguide by the Bloch-mode, the radiation light might be radiated in some directions [20–23]. Nevertheless, in the oblique waveguide, since the end of the waveguide is not orthogonal to the output plane, only a single peak is inside the air light line [Fig. 3(c)].

3. Control of radiation angle by introducing symmetric end structure

In the previous section, we showed that the radiation polar angle is inclined and widely varies depending on the frequency due to the asymmetric wavevector distribution in the output (001) plane. When the radiation angle depends on the frequency, it is difficult to effectively couple the radiated light to an external optical system. Therefore, radiated light is required to be perpendicular to output plane and single-peak pattern independent of the frequency. In addition, a suitable end structure should have low reflections at the boundary between the 3D PC and free space, and at the connection of the waveguide and the end structure. To reduce unnecessary reflections, we considered a symmetric end structure that is similar to the original oblique waveguide.

As shown in Fig. 5, we developed a symmetric end structure that is composed of four forked oblique waveguides along the equivalent crystallographic orientation (< 011 >). When the output light propagates through the symmetric end structure, the electromagnetic field pattern is expected to have a symmetric distribution. We expected that the output lights are radiated from each four oblique waveguides in the same phase, so that the radiated light is perpendicular and single-peak pattern. These forked oblique waveguides are formed along equivalent crystallographic directions (i.e., the point Group C_4v), but do not have equivalent structures, so that the symmetric end structure belongs to the point Group C_2v. When the branch length is too short, the characteristics of the symmetric end structure are similar to those of the original oblique waveguide. On the other hand, when the branch length is too long, the characteristics of the symmetric end structure are like four independent original oblique waveguides. Therefore, to couple the light guided in the waveguides, the branch length should be several wavelengths.

 

Fig. 5 Schematic structure of original oblique waveguide (a,b) and the symmetric end structure (c,d).

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In this section, we discuss a structure in which the branch length is 5 layers, for example. Fig. 6 shows the electric field pattern and radiation distribution of the output light radiated from the symmetric end structure into free space at a frequency f = 0.360c/a. In the enlarged cross-sectional view [Fig. 6(b)] of the electric field pattern of the symmetric end structure, the output light is not radiated from four output ports independently, but is radiated as if from a single output port. In the radiation power distribution [Fig. 6(c) and 6(d)], the output light is radiated perpendicularly from the symmetric end structure. The radiation polar angle, θ, is estimated to be 4.5° from Fig. 6(e). Thus, the symmetric end structure based on the original oblique waveguide could make the radiation light perpendicular.

 

Fig. 6 Calculation results of electric field pattern of the symmetric end structure at frequency 0.360c/a. (a),(b) Cross-sectional views, (c) radiation power distribution in ϕ – θ plane, (d) radiation power distribution in stereographic projection, and (e) radiation pattern at (100) plane.

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Finally, Fig. 7 shows the frequency dependence of the radiation polar angle, reflection, and the collection efficiency, which is defined as the light power observed by an objective lens (NA = 0.70) normalized by the guided light power. The out-coupling efficiency of the symmetric end structure is about 0.6 similar to the original waveguide. From the frequency dependence of the radiation polar angle, we find that light is radiated almost perpendicularly (θ < 6°), independent of the frequency. Because the asymmetry of the wavevector distribution is reduced by the symmetric end structure, independent of the propagating wavevector (or the frequency), the output light could be radiated perpendicularly. Slight increases in reflectance compared with that of the original oblique waveguide can be seen, which are caused by reflection at the branch point of the waveguides. The collection efficiency is about 0.3 – 0.4, which is higher than that of the original oblique waveguide, which is 0.2 – 0.3. This is because of the perpendicular radiation light, and therefore, it is expected that effective coupling to an external optical system may be possible.

 

Fig. 7 Frequency dependence of radiation angle and reflectance, and collection efficiency. The solid lines and the dashed lines are results for the symmetric end structure and the original structure, respectively.

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4. Conclusion

We have investigated the light output characteristics of oblique waveguides in a stripe-stacked 3D PC. First, we analyzed the output characteristics of a simple oblique waveguide and showed that the output light is radiated in an inclined direction that is determined by the wavevector distribution in the output plane. Thus, the radiation angle varies depending on the frequency. We then investigated introducing a symmetric end structure at the end of the oblique waveguide to improve the output characteristics. By using this structure, a symmetric wavevector distribution is obtained, so that the output light is radiated perpendicularly from the output plane. Moreover, because the asymmetry of the wavevector distribution is reduced by the symmetric end structure independently of the propagating wavevector, the frequency dependence of the radiation polar angle is smaller than that of the original oblique waveguide. Introducing such an end structure, in which crystallographic symmetry is taken into account, is useful for effective input/output coupling.