1. Introduction

Photonic crystals are nanostructures with light-wavelength-scale periodicity of the refractive index of the media, typically comprising high-index-dielectric materials. Because the periodicity in the structures can create potential for light similar to electronic potentials in solids, the photonic potential is known as a photonic bandgap (PBG) where light is not allowed to exist or propagate through the photonic crystal [1]. Further, the introduction of designed defects into the periodic structure can create allowed modes in the PBG region. Therefore, point and line defects are engineered for realizing optical devices such as cavities and waveguides, respectively. Moreover, due to advancement in semiconductor-based nanofabrication techniques, semiconductor-based photonic crystals have been developed for ultra-high quality (Q) factor nanocavities [2,3] and low-loss, wide-band waveguides [4,5] at telecommunication ranges.

Semiconductor-based two-dimensional (2-D) photonic crystal slab (thin-film) structures with a thickness of a few wavelengths have been extensively studied because they can strongly confine light within the film, providing simple in-plane integration of various nanocavities and waveguides [6–9]. However, because most thin-film photonic crystals only exhibit PBGs for transverse electric (TE)-like or transverse magnetic (TM)-like polarization modes [2–10], the applications have been limited. For example, thin-film photonic-crystal-based add/drop filters [6–9] only function for one type of polarized light. Though a polarization converter has been proposed for the structure [11], the resulting filters are complex and have some optical losses. Moreover, because semiconductor-based emitters such as quantum wells and intersubband transitions are polarization dependent, optimization of the design is required for each TE and TM PBG structure [12–14]. Furthermore, thin-film photonic crystal structures can seriously degrade waveguides and nanocavities because of the additional optical loss of TE–TM coupling in nonvertical structures [15] with geometrical imperfections. Therefore, thin-film photonic crystals with complete PBGs (which imply PBGs exist for both TE and TM polarizations along the in-plane directions) have been required for overcoming the above limitations. There have been reports of thin-film photonic crystals with complete PBGs [16–18]. However, these reports showed narrow complete PBGs, reaching a few percent and the photonic crystals were not suited for realizing photonic devices such as broad band waveguides and high-Q cavities. Moreover, polarization-free devices such as waveguides or cavities are somewhat difficult to realize in these structures because the dispersion relations of the TE and TM bands are different from each other despite the complete PBG overlapping both bands [16–21].

In this work, to form a complete PBG and design polarization-free devices in a thin-film structure, we investigate a thin-film photonic crystal structure made up of three stacked layers of photonic crystal slabs. We show that a wide complete PBG can be obtained in the telecommunication range by optimizing the stacked photonic crystal slabs. Furthermore, we design a waveguide with a wide bandwidth and nanocavities with high Q factors operating for both TE and TM polarizations.

2. Design of thin-film photonic crystals with a complete PBG

Before achieving a complete PBG in a thin-film photonic crystal structure, we review 2-D photonic crystal slabs. Most TE- and TM-like PBGs can be obtained in hole- and rod-type 2-D photonic crystal slab structures, respectively. Here, we describe the TE and TM modes by TE- and TM-like modes because they mostly have parallel and perpendicular electric fields to the mirror z = 0 (slab center). We utilize the characteristics of 2-D photonic crystal slabs for achieving a complete PBG and construct a three-layer stacked structure consisting of hole- and rod-type 2-D photonic crystal slabs with a triangular lattice (a), as shown in the schematics in Fig. 1. We designed this layered structure because the direction of electric field is different between the center and the near surface of the 2-D photonic crystal slab [22]. In this case, the middle layer (rod-type) acts as the center layer, and the top and bottom layers (hole-type) act as the surfaces. Therefore, the main electric field (E_z) and partial fields (E_x and E_y) of the TM-like mode, in conjunction with the TE-like mode, can be localized in the center of the slab and near the surface, respectively [23]. The hole-type photonic crystal slabs with a hole radius (r) of 0.3a (r₁ = r₃ = 0.3a) and thickness (t) of 0.29a (t₁ = t₃ = 0.29a) are positioned at the top and bottom layers (indicated by subscripts 1 and 3 in the figure). In addition, the middle-layer rod-type photonic crystal slab (indicated by subscript 2) has r₂ = 0.2a and t₂ = 0.21a. Although the photonic crystal has a stacked structure, it is still of the thin-film type with a total thickness of 0.79a (smaller than one wavelength) and can employ the advantages of 2-D photonic crystals. This stacked structure can be fabricated with conventional high alignment wafer-bond techniques for multilayer devices [24].

 

Fig. 1 Schematics of (a) vertically symmetric and (c) asymmetric thin-film three-layer photonic crystals (here, r₁ = 0.3a, r₂ = 0.2a, r₃ = 0.3a, t₁ = 0.29a, t₂ = 0.21a, t₃ = 0.29a) (b) and (d) Calculated band diagrams of photonic crystal structures depicted in (a) and (c), respectively. (e) Respective H_z and E_z distributions of main components for TE and TM polarizations at the position of the dashed line in (c) and at the J point (marked by a circle) of the bandgap-edge modes in (d).

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We can design two arrangements for this photonic crystal structure: vertically symmetric and asymmetric structures as shown in Figs. 1(a) and 1(c), respectively. The vertically symmetric structure has the same position of the top and bottom layers, but the middle layer is shifted by $-a/\sqrt{3}$ along the y-direction with respect to the bottom layer. On the other hand, the vertically asymmetric structure has the middle and top layers with $-a/\sqrt{3}$ and $+a/\sqrt{3}$ shifts along the y-direction with the bottom layer as the center, respectively. The structures are made of silicon (n_Si = 3.4) surrounded by air (n_air = 1) and we calculate the PBG of each stacked structure via the three-dimensional (3-D) finite-difference time-domain (FDTD) method. Here, the band diagrams were obtained separately through excited lights by TE and TM polarizations and monitoring lights. The results of symmetric and asymmetric structures are shown in Figs. 1(b) and 1(d). The symmetric structure exhibits TE and TM PBGs, but there is no overlapping frequency region between the two PBGs. Here, it is noteworthy that the dispersion curves of TE- and TM-like modes in the symmetric structure behave slightly differently because TE and TM polarized lights have different boundary conditions for the holes and rods. On the other hand, the asymmetric structure shows the same dispersion curves for two polarization of TE and TM modes for any in-plane direction, and those two modes are no longer distinguishable with respect to z = 0 in the asymmetric structure-it is a hybrid mode of the superposed two polarizations as shown in Fig. 1(e). This optical characteristic comes from the symmetry in the diamond-like structure and the asymmetry in the direction perpendicular to the slab. Thus, an asymmetric structure is required for achieving a complete PBG and same dispersion relations for two polarization of TE and TM modes. Finally, we achieved a complete PBG in the range of 0.31–0.36 (c/a), corresponding to normalized band width (Δω/ω_{mid-frequency}) of 14.9% wider than those [16,17] and comparable to [18], as shown in Fig. 1(d). The band gap corresponds to a wide (telecommunication) range of 1400–1610 nm at a = 500 nm.

3. Design of waveguides within a complete PBG

Based on the above results, we design polarization-free devices such as waveguides and cavities in the three-layer photonic crystal with a complete PBG. Because the photonic crystal has the stacked (holes-rods-holes) structure of 2-D photonic crystal slabs, we can adopt the design rules of 2-D photonic crystal waveguides and nanocavities in the stacked structure.

First, we introduced a line-defect waveguide with a row of missing holes (width = $\sqrt{3}a$ = W1) in the top layer of the hole-type photonic crystal, as shown in Fig. 2(a). To investigate the polarization-free characteristics of the waveguide, we excited both TE and TM polarizations using the 3-D FDTD method. The calculated dispersion relation of this waveguide is shown in Fig. 2(b). As indicated by black dots, the lossless waveguide band of 0.320–0.325 (c/a) within a complete PBG was obtained under the light line. This corresponds to a bandwidth of 24 nm at the central wavelength of 1,550 nm. Additionally, structural modifications in a line defect were conducted to expand the bandwidth of the waveguide mode. The dispersion relation of waveguides can be primarily controlled by adjusting the width of the waveguides [25]. However, to conserve the arrangement of the three-layer structure, we introduced a line defect using holes with an altered radius [26,27]. When we set the radius of line defect to r’ = 0.32a, a wider bandwidth of 0.323–0.334 (c/a) corresponding to the normalized bandwidth of 3.3% was obtained, as depicted by red dots in Fig. 2(b). This result corresponds to a bandwidth of 54 nm and is about a two-fold increase compared to the previous result. We also introduced a line defect in the middle layer of rods, as shown in Fig. 2(c). The radius of the rods in the row was reduced to 0.15a. The calculated dispersion relation (black dots) in Fig. 2(d) shows that a lossless waveguide band of 0.313–0.318 (c/a) can be obtained. To obtain a waveguide with a broader bandwidth, we additionally shifted the row of line defects by as much as + 0.5a in the x-direction. The calculated result (red dots) in Fig. 2(b) shows that wide waveguide band of 0.322–0.346 (c/a) corresponding to normalized bandwidth of 7.1% larger than [19], can be achieved, which is bandwidth of 100 nm when a = 500 nm. Although we do not consider the coupling of the external light to the waveguide it can be somewhat difficult to couple the external light to the waveguide since the waveguide has an asymmetric structure and multiple layers; it may require more sophisticated design of efficient coupling of light to the waveguide compared to those of conventional waveguides with a single layer.

 

Fig. 2 (a) Schematic of a waveguide with a row of air holes with altered radius (r’) in the top-layer. (b) Calculated band diagrams for a W1 waveguide (black dots) and a waveguide with an altered radius line defect (red dots). (c) Schematic of a waveguide with a row of reduced rods in the middle-layer photonic crystal. (d) Calculated band diagrams for a waveguide (black dots) and a waveguide with shifted rods (red dots).

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4. Design of photonic nanocavities within a complete PBG

Next, we design photonic nanocavities in the thin-film three-layer photonic crystal structure. We can again utilize the 2-D design rules of nanocavities for the three-layer stacked structure. Moreover, 2-D hole-type photonic crystal nanocavities have been widely used for their high performance and easy fabrication. Because these advantages have encouraged diverse studies of 2-D hole-type photonic crystal nanocavities, we investigate the feasibility of adopting these advanced nanocavities to the thin-film three-layer photonic crystal structure.

First, we designed a nanocavity in the top layer by introducing point defects consisting of three missing holes (referred to as the L3 nanocavity), as shown in Fig. 3(a). We then exited both TE (H_z) and TM (E_z) polarizations at the center of the nanocavity. In the calculated field distributions shown in Fig. 3(c), the light is strongly localized in the nanocavity and resonates for both polarizations. Therefore, the light can be trapped and resonate in the thin-film three-layer photonic crystals regardless of the light polarization. The calculated Q factor was 2,100 at the resonant frequency of 0.324 (c/a) within the complete PBG. To investigate the possibility of increasing the Q factor, we designed a two-step hetero-structured nanocavity [2] with three different lattice constants (a₁ = 1.017a₃, a₂ = 1.007a₃, and a₃) in the top layer, as shown in Fig. 3(b). Surprisingly, compared to the previous result, a Q factor of 3 × 10⁵, which was 150 times greater than that for the L3 nanocavity, was obtained with a resonant frequency of 0.316 (c/a) within a complete PBG. The calculated field distributions for the cavity in Fig. 3(d) show that the light is strongly localized for both polarizations in the cavity. Furthermore, to design a higher Q nanocavity, we introduced a multistep-hetero-structure that consists of photonic crystals with seven different lattice constants (a₁ = 1.058a₇, a₂ = 1.056a₇, a₃ = 1.051a₇, a₄ = 1.042a₇, a₅ = 1.031a₇, a₆ = 1.017a₇, and a₇) in the top layer. A maximum Q factor of 3.7 × 10⁷ was theoretically achieved. This result is comparable to those of conventional photonic crystal nanocavities [28,29] and implies that remarkable optical performances can be realized in thin-film three-layer photonic crystals. Consequently, the 2-D photonic-crystal-based high-Q-factor nanocavity designs are applicable in three-layer photonic crystals, and strong light–matter interactions can be realized in the three-layer photonic crystals.

 

Fig. 3 (a) Schematics of (a) L3 nanocavity and (b) heterostructure nanocavity with three different lattice constants (a₁ = 1.017a₃, a₂ = 1.007a₃, and a₃). Calculated field distributions of (c) L3 nanocavity and (d) heterostructure nanocavity.

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5. Conclusion

In summary, we designed thin-film photonic crystals with an asymmetric three-layer structure and theoretically investigated the optical properties. Wide, complete PBGs in the 0.31–0.36 (c/a) range could be achieved regardless of the light polarization. The asymmetric structure shows identical dispersion relations for both TE- and TM-like polarizations for any in-plane direction, which implies that the thin-film three-layer photonic crystals could be applied to polarization-free optical devices. Furthermore, we investigated the optical characteristics of waveguides and nanocavities designed in the thin-film three-layer photonic crystals, and we found that wide waveguide modes of 100 nm and high Q factors of 3.7 × 10⁷ could be achieved. This research will promote further development in the control of unpolarized light and contribute to the advancement of photonic-crystal-based applications.