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We demonstrate the operation of two types of waveguides formed in three-dimensional (3D) photonic crystals (PCs). We first created a vertical waveguide by stacking acceptor-type defects, in which near-infrared light propagates in the stacking direction. Light is transmitted independent of polarization in this waveguide because electromagnetic waves couple to a degenerate mode derived from the structural symmetry of the defects. We then connected horizontal and vertical waveguides to form an L-shaped waveguide, which is able to guide near-infrared light from the horizontal to vertical direction in the 3D PC. We envisage the realization of more complex 3D optical interconnections by optimizing the waveguide structures and increasing the PC period in the vertical direction.
©2009 Optical Society of America
1. Introduction
The total control of light in a space of sub-micron dimensions is expected to enable significant development in the field of optoelectronics. Three-dimensional (3D) photonic crystals (PCs) [1–8] have attracted much attention as ideal optical materials, in which the complete photonic band gap (PBG) can control light perfectly without any losses. The operation of 3D PCs has been demonstrated at optical communication wavelengths [1–8], and there have been reports on the investigations of nano-optical components in 3D PCs [2–7,9–17]. In order to allow more flexible control of light by the spatial integration of functional components, several groups have investigated 3D PC waveguide networks either theoretically [14,15] or experimentally [5]. Here we discuss both vertical and L-shaped waveguides in more detail based on the results of both experiments and numerical simulations and demonstrate the realization of 3D PC waveguide networks.
2. Vertical waveguide
Figure 1(a) shows a schematic diagram of a vertical waveguide formed in an eight-layered 3D PC. This waveguide is constructed by stacking two sequences of the four types of layers shown in Figs. 1(a) and 1(b). Layers ‘i’ and ‘iv’ incorporate an acceptor-type defect created by removing part of one rod (length 1.0a, where a denotes the lattice constant of the PC, corresponding to the separation of adjacent stripes). The dispersion relation of the vertical waveguide, assuming infinitely stacked unit structures, was calculated by using the plane wave expansion (PWE) method, which involves solving Maxwell’s equations in the frequency domain. The supercell method was also employed, for which the unit cell dimensions in the x, y, and z directions were set to 8a, 8a, and a, respectively. We defined the z-direction as the propagation direction. The number of used plane waves was 16807. The refractive index of the PC stripes was 3.375. Each rod had a width of 0.26a and a height of 0.30a. Figure 1(c) shows the dispersion relation of the waveguide calculated by the PWE method. The guided mode on the high frequency side (red line) is doubly degenerate due to the structural symmetry; thus, there are three guided modes in the complete PBG. Figure 1(d) shows a scanning electron microscope (SEM) image of the four-layered unit structure used to fabricate such a waveguide. The rods were comprised of GaAs and the stripe period was 700 nm. The eight-layered vertical waveguide was created by stacking two unit structures using the high-precision alignment and bonding method [18].
Fig. 1 (a) Schematic representation of a vertical waveguide composed of acceptor-type defects in a 3D PC. The red and black dashed lines in the lower, cross-sectional diagram respectively indicate acceptor-type defects and rods that are located at a distance near or behind the cross-section. (b) Schematic top views of the four types of stripe layers that comprise the PC structure. (c) Dispersion relation of the vertical waveguide, where az denotes the PC period in the vertical direction, corresponding to 1.2a. (d) Top-view SEM images of the unit structure composed of four layers, with acceptor-type defects incorporated.
We investigated the transmission properties of the eight-layered vertical waveguide both theoretically and experimentally. We defined the polarization of the incident light as either ‘x-polarized’ or ‘y-polarized’, as shown in Fig. 1(a). Numerical analyses were carried out by the 3D finite-difference time-domain (FDTD) method. The lateral dimensions of the structure used in the FDTD model were 17a × 17a, and a vertical waveguide comprised of stacked acceptor-type defects was introduced at the center of the model. Bloch boundary conditions were imposed at the lateral surfaces of the model. The PC was enclosed by uniform dielectric media for consistency with the fabricated waveguide, which was sandwiched by GaAs substrates. Berenger’s perfectly matched layer (PML) boundary conditions were utilized at the top and bottom surfaces, which were outside the uniform media. In the FDTD calculations, the incident plane waves entered the waveguide along the vertical direction, and the transmission spectra were calculated by spatially integrating the time-averaged Poynting vectors, taking into account the numerical aperture of the objective lens used in the experiment. The transmission properties were also measured experimentally by focusing an incident semiconductor laser beam with a tunable wavelength onto the lower side of the vertical waveguide as shown in Fig. 1(a). The transmitted light was collected by an objective lens and detected by an InGaAs photodiode.
Figure 2 shows both the calculated and experimental transmission spectra, in which the transmission intensities are normalized to the maximum values. The transmissivities in the region of 1.47 μm are approximately 14 dB greater than that of an eight-layered PC without any defects. The transmission bandwidths of the measured spectra are approximately 36 nm, as estimated from the full widths at half maximum (FWHM) of the peaks. Our FDTD calculations showed that the complete PBG of the eight-layered PC lies in the wavelength range from 1.41 to 1.69 μm. The increase in transmission intensity in each case is thus attributable to a guided mode within the complete PBG.
Fig. 2 Transmission spectra of the vertical waveguide in an eight-layered 3D PC simulated by FDTD analysis (upper panel) and measured experimentally (lower panel). The transmission intensities are normalized to the maximum values of ‘x-polarized’ light incidence.
We next investigated which of the guided modes contribute to light propagation along the vertical waveguide. We used the FDTD method to calculate the electromagnetic field profiles inside the waveguide for incident light with both polarizations, at the wavelengths corresponding to maximum transmission intensity. The field distributions are in good agreement with those calculated by the PWE method for the degenerate mode, which exists on the high-frequency side of the complete PBG. In contrast, the x and y components of the electric flux density of the other mode, on the low-frequency side of the PBG, have an asymmetrical distribution at the center of the acceptor-type defects. A laser beam incident in the vertical direction is thus unable to couple to this guided mode. We conclude that the measured increase in transmissivity is due to the propagation of light along the vertical waveguide, and that incident light of both polarizations couples to the degenerate mode in the complete PBG.
We also considered why the measured transmission spectrum for ‘y-polarized’ incident light shifts to longer wavelengths compared to the FDTD simulation. The difference in transmission bandwidths between the two polarizations is due to structural imperfections associated with the fabrication accuracy, which breaks the in-plane structural symmetry and lifts the degeneracy. More precisely, the length of the defects in the y-direction was shorter than that in the x-direction. Therefore, the guided mode to which the ‘y-polarized’ light could couple is slightly shifted to the lower frequency side.
We have thus demonstrated the operation of a 3D PC vertical waveguide at optical communication wavelengths. The realization of a 3D PC horizontal waveguide has already been reported [4], and thus the basic functional components for the guiding of light in three dimensions are now available. Consequently, we next fabricated and investigated an L-shaped waveguide created by connecting horizontal and vertical waveguides.
3. L-shaped waveguide
Figure 3(a) shows a schematic cross-sectional representation of the L-shaped waveguide. A horizontal waveguide comprised of an acceptor-type line defect [9] was formed in the center layer of a nine-layered 3D PC. The unit structure of a vertical waveguide was then connected to one end of the horizontal waveguide.
Fig. 3 (a) Schematic cross-sectional representation of an L-shaped waveguide formed by connecting horizontal and vertical waveguides comprised of acceptor-type defects. (b) Cross-sectional SEM image of a fabricated nine-layered 3D PC (upper image) and top view of a four-layered 3D PC (lower image), both containing no defects. (c) Top-view SEM images of the fabricated vertical waveguide (upper image) and horizontal waveguide (lower image).
Figure 3(b) shows a cross-sectional SEM image of a nine-layered 3D PC and a top view of a four-layered 3D PC, neither containing any defects. We achieved a high level of fabrication accuracy. Figure 3(c) shows SEM images of the top four layers of the PC containing the vertical waveguide, and of the bottom five layers containing the horizontal waveguide. We constructed the L-shaped waveguide by stacking these two pieces using the high precision alignment and wafer bonding method [18], giving a structure composed of nine layers.
Figure 4(a) shows the dispersion relations of the waveguides calculated by the PWE method, where the guided modes of the horizontal (blue lines) and vertical waveguides (red lines) are superimposed. The unit cell dimensions for the horizontal waveguide calculations were respectively 6a, a, and 6a in the x, y, and z directions, and the number of used plane waves was 9583. The wave vectors in the direction of propagation were normalized to the PC period a. Because the period in the stacking direction was 1.2a, the dispersion curves of the vertical waveguide fold back at 0.417 (c/a). The transmission bandwidths of the higher-order modes partially overlap at the high-frequency side of the complete PBG, giving rise to the possibility that electromagnetic waves with these frequencies can propagate in the L-shaped waveguide.
Fig. 4 (a) Dispersion relation of the L-shaped waveguide with both the guided modes of the horizontal (blue lines) and vertical waveguides (red lines) superimposed. (b) Ex distributions of the higher-order mode of the horizontal waveguide in the x-y plane (left) and the x-z plane (right).
We note that the fabricated structure had two types of constructional imperfections, as apparent in Fig. 3(c): the center layer containing the horizontal waveguide was shifted by 0.15a in the y-direction and the rod widths of the upper four layers were slightly wider than those of the lower five layers. One may thus expect that the actual guided modes of the vertical waveguide will shift to lower frequencies. The overlapped bandwidths of the two waveguides would then be approximately 0.41 to 0.43 (c/a) in the frequency domain and 0.00 to 0.10 (2π/a) in the wave vector domain.
In order to couple incident light with the above frequencies to the L-shaped waveguide, we irradiated the horizontal waveguide by using a laser beam from the –z direction, as shown in Fig. 3(a). Light in the frequency range from 0.41 to 0.43 (c/a) has a smaller wave vector (<0.10) in the propagation direction at the surface of the PC because the laser beam was focused by an objective lens (N.A. = 0.26). From the point of view of wave vector matching, incident light can easily couple to the higher-order mode. The electric field distribution of the higher-order mode is shown in Fig. 4(b). Because Ex has a symmetry distribution in the x-y plane, the ‘x-polarized’ light can effectively couple to the target guided mode. The incident light was focused at a position located 10 μm away from the vertical waveguide in the x-direction. The propagated light was detected at the upper surface of the vertical waveguide. The laser beam was also irradiated onto the opposite side of the PC, 10 μm to the right of the vertical waveguide, as shown in Fig. 3(a), and the transmitted light was measured in similar way. A comparison of the two transmission spectra allows the intrinsic transmission properties of the L-shaped waveguide to be determined, with the exception of less important factors such as scattered light.
Figures 5(a) and 5(b) show the transmission spectra for ‘x-polarized’ and ‘y-polarized’ incident light, respectively. An increase in transmission intensity was clearly observed in the wavelength range around 1.51 μm when ‘x-polarized’ light was incident on the horizontal waveguide; the bandwidth is approximately 35 nm (FWHM). This value is equal to the bandwidth of the vertical waveguide discussed above because the bandwidth of the horizontal waveguide includes that of the vertical waveguide. Some peaks observed in the ‘x-polarized’ transmission spectrum would be due to the Fabry-Perot interference of the light reflected at both ends of the horizontal waveguide. In contrast, the transmission intensity of the ‘y-polarized’ light does not change because it cannot couple to the higher-order mode of the horizontal waveguide. As expected, no increase in transmission was observed for either polarization when the light was incident on the region of the PC with no horizontal waveguide.
Fig. 5 Transmission spectra of the L-shaped waveguide for ‘x-polarized’ (left panel) and ‘y-polarized’ (right panel) incident light. The red and gray lines correspond to the light incident on the horizontal waveguide and on the PC with no defects, respectively. The transmission intensities are normalized to the maximum value of ‘x-polarized’ light incident on the horizontal waveguide.
The measurements indicate that ‘x-polarized’ light of particular frequencies was propagated through the L-shaped waveguide. The higher-order mode of the horizontal waveguide carried the ‘x-polarized’ light to the 90 degree bend, and we expect that it might couple to the ‘x-polarized’ mode of the doubly degenerate mode in the vertical waveguide. Polarization-maintaining interconnections would be necessary to integrate devices in 3D PCs. The accurate evaluations of the polarization state of output light from the vertical waveguide and the coupling efficiency at the 90 degree bend would be significant to clarify the properties of waveguides in 3D PCs, and we will investigate and report it in the future.
4. Conclusion
We have fabricated a vertical 3D PC waveguide and have clearly observed the propagation of electromagnetic waves in the stacking direction. We have used both experiment and theory to demonstrate that light couples to the degenerate mode formed in the complete PBG. Moreover, we have fabricated an L-shaped waveguide by connecting horizontal and vertical waveguides. We observed the propagation of near-infrared light with a particular polarization in the L-shaped waveguide. This result indicates that it is possible to change the guiding direction of light from horizontal to vertical in a 3D PC. We expect that lossless coupling at the corner of the L-shaped waveguide could be realized by optimizing the structure of the interface between the horizontal and the vertical waveguides [15]. It should also be possible to suppress propagation losses by increasing the period in the stacking direction of the 3D PC [4]. This study represents an advance towards the complex guiding of light in three dimensions between various functional components in 3D PCs, and brings the total control of light in PCs a step closer.
Acknowledgments
This work was partly supported by the Core Research for Evolutional Science and Technology Program from the Japan Science and Technology Agency.
References and links
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