Figure 1

(Color online) Schematic of a $3+1\mathrm{D}$ optical microchip using the slanted pore 3D PBG material. The bottom, intermediate, and top cladding sections of the microchip (consisting of the slanted pore 3D PBG materials) are separated and intercalated with lattice-matched 2D photonic crystal circuit layers. The first (lowest) planar circuit layer, sitting on the bottom cladding section, is inserted at the plane $z=1∕8c$. The planar circuit layer consists of a square lattice of circular rods with a radius of $0.2a$ and a thickness of $0.6a$, with one of the lattice points centered at $(x,y)=(-1.25a,-4.5a)$ for the first layer (dashed arrow). The second planar circuit layer is positioned at the plane $z=(7∕8+n)c$, where $n$ is an integer. The plane $z=0$ is marked by magenta (gray) lines. Note that the coordinates $z$ used here refer to positions in the bulk ${\mathbf{SP}}_{\mathbf{2}}$ material before insertion of 2D photonic crystal circuit layers.Reuse & Permissions

Figure 2

(Color online) Schematic for the $3+1\mathrm{D}$ optical microchip showing only the 2D photonic crystal microchip layers and the vertical waveguides. In-plane waveguides and bends are patterned in the 2D photonic crystal layers by removing certain dielectric rods. Here, new coordinates $(x^{\prime },y^{\prime },z^{\prime })$ are introduced to describe the $3+1\mathrm{D}$ microchip after insertion of 2D circuit layers, where $x^{\prime }=x-0.75a$, $y^{\prime }=y+2.5a$, and the plane $z^{\prime }=0$ refers to the bottom face of the first 2D layer. Consequently, the origin is shifted to the (bottom face) center of one of the rods as shown by the large black arrow. For reference, the rod marked by the dashed arrow in Fig. 1 is also shown by a dashed arrow here. The first $[z^{\prime }=0.6a\text{–}(0.6a+3.75c)]$ and the second $[z^{\prime }=(1.2a+3.75c)\text{–}(1.2a+7.5c)]$ three-dimensional separator sections correspond to $z=0.125c\text{–}3.875c$ and $z=3.875c\text{–}7.625c$, respectively. Vertical waveguides (interconnects) are created by removing dielectric from the ${\mathbf{SP}}_{\mathbf{2}}$ structure within volumes contained by red (dark gray) and green (gray) rectangular boxes. The first (lowest) two red (dark gray) boxes and the lowest green (gray) box are centered at $(0,-1.25a,0.75a)$, $(0.5a,-1.25a,1.35a)$, and $(0.5a,-1.25a,-0.45a)$, respectively.Reuse & Permissions

Figure 3

(Color online) Schematic of the vertical waveguide and its dispersion relation. (a) Within a single unit cell of 3D PBG of size $a\times a\times c$ (black wire box), dielectric material within the two red (dark gray) rectangular boxes is removed. Repeating this process vertically in the ${\mathbf{SP}}_{\mathbf{2}}$ cladding section, extends the length of vertical waveguide as depicted in Fig. 2. One corner of the bottom face of the unit cell at (0,0,0) corresponds to $(x^{\prime },y^{\prime },z^{\prime })=(-0.25a,-2a,0.6a)$ in Fig. 2. The size of the red (dark gray) and green (gray) boxes is $0.5a\times 0.5a\times 0.25c$ in the $x$, $y$, and $z$ directions, respectively. The two red (dark gray) boxes are centered at $(0.25a,0.75a,0.15a)$ and $(0.75a,0.75a,0.75a)$, respectively. Below the first and above the second 2D microchip layers, dielectric material within the green (gray) boxes is also removed. The green (gray) box below the first 2D layer is centered at $(0.75a,0.75a,-1.05a)$. (b) The dispersion relation for the vertical waveguide in the bulk ${\mathbf{SP}}_{\mathbf{2}}$ material created by periodically repeating the unit cell (black wire box) shown in (a). The single mode waveguide dispersion relation that spans the frequency range of $a∕\lambda \sim 0.335–0.376$ is plotted as a solid black curve. This corresponds to a vertical waveguiding bandwidth of $\sim 175\mathrm{nm}$ near the wavelength of $1.5\mu \mathrm{m}$. The inset shows the actual dielectric structure of the vertical waveguides. The blue (dark gray) parts correspond to dielectric material of the ${\mathbf{SP}}_{\mathbf{2}}$ structure. The yellow color (gray) indicates dielectric material removed to create a vertical waveguide, which correspond to the volumes inside the red (dark gray) boxes shown in (a).Reuse & Permissions

Figure 4

(Color online) Poynting vector intensity of light (frequency $a∕\lambda =0.345$) propagating along the combined vertical bend (U-turn) and in-plane 90° bend, for the ${\mathbf{SP}}_{\mathbf{2}}$-based 3D microchip, obtained by FDTD simulation. The blue (black) and magenta (gray) shading corresponds to small and large magnitudes of the Poynting vectors at a particular instant, respectively. Only the structure of the two 2D photonic crystal layers (gray cylinders) and the vertical waveguides (bright rectagular boxes of removed dielectric) are shown. The inset shows transmission and reflection spectra of two vertical U-turn bends, the first with a vertical interconnect of length $3.75c$ [red (solid)] and the second with length $7.75c$ [blue (broken)]. The corresponding spectra for a simple 90° in-plane bend are superimposed (black lines with circles). There is minimal reflection around the U-turn, throughout the entire single-mode bandwidth (of $\sim 100\mathrm{nm}$) of the circuit.Reuse & Permissions