1. Introduction

Photonic crystals (PCs) possess refractive-index variations with periodicities of the order of light wavelengths, giving photons unique dispersion characteristics. In particular, three-dimensional (3D) PCs enable photons to be controlled in all directions by modifying the 3D photonic band structure and utilizing the resulting complete photonic bandgap (PBG). Such photonic control has been demonstrated by embedding artificial defects and light emitters inside 3D PCs [1–9]. Recently, it was shown that photons can also be controlled using the surface of a 3D PC, which lies at the boundary of the 3D periodicity [10]. For example, a surface nanocavity with a quality factor exceeding 9,000 was realized by appropriate design of the surface structure. Photons with any polarization can be controlled using the 3D PC surface [10], in similar fashion to the bulk crystal. However, the surface can be accessed in more straightforward fashion than structures formed inside 3D PCs, hence utilization of the surface promises to establish a new field of photonic control. For surface applications, the preparation of high-quality uniform surfaces over large areas is important. For this reason, even though the formation of surface modes has theoretically been predicted for a 3D PC with a different type of structure [11], surface modes have been clearly observed only in 3D PCs constructed by stacking striped semiconductor units [10], which provided a uniform surface. However, the accurate construction of such 3D PCs requires complicated and time-consuming fabrication procedures [1]. We have recently developed a new method for creating high-quality 3D PCs based on the double-angled plasma etching of semiconductor wafers and we have demonstrated clear PBG effects at optical communication wavelengths [8]. These 3D PCs have great potential for a wide range of functionalities, including surface applications, because they can easily be constructed using complementary metal–oxide semiconductor (CMOS) compatible processing. In this report, we numerically and experimentally study the surface states of 3D PCs fabricated using the double-angled etching method. Furthermore, we investigate the polarization characteristics of the surface modes and the dependence of these modes on the surface structure.

2. Simulations

Figure 1(a) shows a schematic image of the type of 3D PC examined here: this PC is described by the center-to-center distance between parallel dielectric rods a; the lattice constants a_x, a_y and a_z along the x, y and z directions, respectively; the width of the air holes L_x; and the etching angle θ. Neighboring holes are etched at angles of θ and −θ to the surface. Figures 1(b) and (c) show the crystal structure in real and reciprocal space, respectively, where a_x = a_y = a. When only the surface structure shown in Fig. 1(b) is considered, point K is equivalent to point K′. However, because the etching directions of neighboring holes are different, point K′ must be distinguished from point K.

 

Fig. 1 (a) Schematic picture of a 3D PC constructed using the double-angled etching technique. The surface structure is shown in (b) real and (c) reciprocal space.

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The surface band diagrams of the 3D PC in Fig. 1 were simulated using the 3D finite-difference time-domain (FDTD) calculation using Bloch periodic boundary conditions [12]. The refractive index of the dielectric was set to 3.48, corresponding to the refractive index of silicon in the optical communications wavelength range. The structural parameters θ and L_x were set to 45° and 0.6a_x, respectively. Figure 2(a) shows the projected in-plane band diagram of the 3D PC assuming infinite periodicity in all directions, calculated by adopting Bloch periodic boundary condition along all three axes. A complete PBG is formed in the frequency range between 0.295 and 0.343 c/a. The corresponding band diagram for the case of finite periodicity in the z-direction (perpendicular to the surface) is shown in Fig. 2(b). Here we adopted Mur’s absorbing boundary condition in the z-direction and Bloch periodic boundary condition in the in-plane directions. The modes depicted by red lines occur within the original complete PBG range. Figure 2(c) shows the distribution of the electric field amplitude of the surface mode at the blue point in Fig. 2(b). It is evident that the electric field remains confined to the surface of 3D PCs that are constructed by the double-angled etching method.

 

Fig. 2 FDTD simulation results. (a) Band diagram of the 3D PC with infinite periodicity in all directions. Optical modes exist only within the gray regions. (b) Band diagram of the 3D PC with finite periodicity in the z-direction. The surface modes are denoted by red lines. (c) Distribution of the electric field amplitude at the blue point in (b).

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We next investigated how the surface modes depend on the structural phase of the surface. We define the surface phase to be zero for the surface structure shown in Fig. 3(a) . Here, the first and second sets of etching holes share the same x-coordinates. When the positions of the second set of etching holes are shifted by −a_x in the x-direction with respect to the first set, the surface phase is defined to be 1. Figures 3(b)–(d) show 3D PCs with surface phases of 1/4, 2/4 and 3/4, respectively.

 

Fig. 3 Dependence of surface modes on the surface structure. Schematic pictures of PCs are shown with surface phases of (a) 0, (b) 1/4, (c) 2/4, and (d) 3/4. Corresponding surface band diagrams are shown for surface phases of (e) 0, (f) 1/4, (g) 2/4, and (h) 3/4. Red lines denote the surface modes.

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Figures 3(e)–(h) show the corresponding surface band diagrams for two representative directions Γ−X and Γ−K. The surface modes are shifted to higher frequency as the surface phase increases, and alternative modes appear at low frequencies. The surface band diagrams are the same in the Γ−X direction for the surface phases 0 and 2/4 due to the presence of a mirror plane parallel to the y-axis. Similarly, the band diagram in the Γ−X direction is identical for the surface phases 1/4 and 3/4. In the Γ−K direction, the surface structure and the resulting surface bands are the same only when the surface phases are 0 and 1. We note that two modes are apparent in the Γ−K band diagram for the surface phase 1/4. We discuss the properties of these modes below with respect to polarization.

3. Experiments

We also investigated the surface modes of these 3D PCs experimentally. Because light that is coupled to the surface states does not escape to free space, we measured the surface modes using an evanescent-coupling method with a glass prism (Fig. 4(a) ) [10]. If there is no 3D PC next to the prism, light that is incident at angles within the total internal reflection condition is completely reflected at the lower face of the prism. When a 3D PC is placed under the prism, the portion of the incident light that propagates evanescently can couple to the surface modes; the wavenumber of the evanescent light is larger than in free space. This coupling occurs only when both the frequency and in-plane wavenumber of the incident and surface-propagating light are matched. Therefore, the surface modes can be investigated by measuring the reflection spectra.

 

Fig. 4 Evanescent-coupling measurement. (a) Measurement setup. (b) Top-view SEM image of the 3D PC structure with surface phase zero. (c), (d) Reflection spectra for incident light with P- and S-polarizations, respectively, at different angles of incidence. The blue lines indicate the wavelengths of the spectral minima.

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We fabricated 3D PCs from single-crystalline silicon wafers using the double-angled etching technique [8,13] and with the following structural parameters: a_x = a_y = 670 nm, θ = 45°, l_x = 0.6a_x, and a vertical depth of ~1 period. We prepared samples with surface phases of 0, 1/4, 2/4, and 3/4. First, we prepared fiducial marks for two rounds of etchings. We then formed masks for the first etching at 45° to the surface of the silicon wafer and removed them after the etching process was complete. We then formed masks for the second etching at the opposite angle by using the fiducial marks and removed them to obtain the 3D PCs. The surface phase was varied by shifting the positions of the second set of angled-etching holes in the x-direction. For high aspect-ratio etching, we adopted a reactive-ion etching technique based on an inductively-coupled plasma system using a mixture of SF₆ and O₂ gases at cryogenic temperatures [8,13,14]. Figure 4(b) shows a scanning electron microscope (SEM) image of the top of a 3D PC with a surface phase of zero. The alignment errors in the y-direction of our fabricated structures were estimated to be less than 0.03a_y. In the case of errors less than 0.05a_y, numerical simulations suggest that the complete PBG should remain intact. We used optical transmission measurements to confirm that a complete PBG was formed in the wavelength range between 1.35 and 1.60 μm.

In the evanescent-coupling measurements, the gap between the prism and the 3D PCs was set to ~500 nm. White incident light was separated into P- and S-polarized components using a polarizer. Here, P-polarization is parallel to the plane of incidence and S-polarization is perpendicular to the plane. To change the in-plane wavenumber, the angle of incidence was varied from 43.0° to 64.5°. Figures 4(c) and (d) show the reflection spectra obtained in the Γ−X direction for the 3D PC with a surface phase of zero. Minima were observed in each spectrum, indicating coupling to the surface modes. Although the minima were small in magnitude at large angles of incidence due to the short tunneling length of the evanescent light, the wavelengths of the minima clearly decreased when the angle of incidence was increased. Because the minima were observed at the same wavelengths for both polarizations, we can assume that the modes in the two sets of spectra are the same. We also performed measurements along the Γ−K direction for the structure with surface phase zero and mapped the band diagram, as shown in Fig. 5(a) ; the reflectance is represented by color coding. The results for the two polarizations overlap, and the experimental band diagram agrees well with the simulation in Fig. 3(e). To investigate the surface modes systematically, we evaluated the surface bands for structures with surface phases of 1/4, 2/4 and 3/4, as presented in Figs. 5(b) − (d). When the surface phase was varied, the frequency range changed in similar fashion to the numerical simulations shown in Figs. 3(f)–(h). In particular, we measured two surface modes in the Γ−K band diagram for the structure with surface phase 1/4. These results demonstrate the existence of surface modes in 3D PCs constructed using the double-angled etching technique. They also indicate that our fabrication method provides high surface quality despite the straightforward processing involved.

 

Fig. 5 Measured surface band diagrams for 3D PCs with surface phases of (a) 0, (b) 1/4, (c) 2/4, and (d) 3/4. Reflectance is represented by color coding.

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Finally, we discuss the polarization characteristics of the surface modes. For the structure with a surface phase of 1/4, two modes with different frequencies appeared in the Γ−K direction both in the experiment and simulation, as shown in Figs. 6(a) and (b) . As illustrated by the color-coded measured band diagram in Fig. 6(a), it was found that the two modes were coupled to differently polarized light. To evaluate these polarization characteristics, we calculated the electric field distributions in planes parallel and perpendicular (Fig. 6(c)) to the Γ−K direction, as shown in Figs. 6(d)–(g). The higher-frequency mode has electric field components mainly in the parallel plane. This is consistent with the experimental results, because the P-polarized incident light possesses an electric field within the plane of incidence, which is the plane parallel to the Γ−K direction. In contrast, the lower-frequency mode has electric field components mainly in the plane perpendicular to the Γ−K direction, which is consistent with coupling to S-polarized incident light.

 

Fig. 6 Coupling to incident light of different polarizations for the structure with surface phase 1/4. Measured (a) and simulated (b) surface band diagrams are shown, where modes corresponding to P- and S-polarized light are red and green, respectively. (c) Schematic picture of the 3D PC showing cross-sections parallel and perpendicular to the Γ−K direction. Electric field distributions are presented for the high-frequency (d, e) and low-frequency (f, g) modes in the parallel and perpendicular planes. The electric field amplitude is represented by color coding. The arrows show electric field vectors parallel to the images.

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4. Conclusions

We have investigated the surface modes formed on 3D PCs constructed using the double-angled etching method. Numerical simulations revealed the existence and dispersion characteristics of these modes for a variety of easily realizable surface structures. Experimental investigations performed on fabricated 3D PCs confirmed the formation of surface modes, the dispersion curves of which agreed well with the simulations. We also numerically evaluated and experimentally confirmed the polarization characteristics of the surface modes. Our results suggest that 3D PCs constructed using the simple double-angled etching method possess high surface quality and promise to open up an easy route toward surface photonic control.