Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: Symmetries and complete photonic band gaps

We present a detailed study of the crystallographic symmetries and band structures of three-dimensional photonic crystals that are amenable to nanofabrication through holographic lithography. For the experimentally preferable umbrella geometry, we identify realistic parameters that lead to structures with complete three-dimensional photonic band gaps. We find a solution, for which the photonic crystals have rhombohedral point symmetry. This solution is a member of the same space group as the celebrated Yablonovite structure and has a similar crystallographic motif. We find a complete photonic band gap with a gap/midgap ratio of 5.8% between the second and third band after silicon inversion at 38% silicon filling fraction. In addition, we identify parameter combinations for which the optimized interference contrast is as large as a factor of 10, rendering the fabrication of this structure possible.

Figure 1

(a) The umbrella configuration: The central beam (No. 1) is along the axis of a cone with apex angle $\gamma$; three side beams (Nos. 2, 3, 4) are equally distributed on the cone. (b) The two-planes geometry: Two pairs of counterpropagating beams in two perpendicular planes.

Figure 2

Dependence of the contrast $C$ of the interference pattern (10) for $(u,v,p,q)=(u,0,0,0)$ on the central beam polarization direction ${\theta }_1-{\theta }_1^{\mathrm{opt}}$ and ${\rho }_1^2∕u\propto I_1∕u$. While ${\theta }_1^{\mathrm{opt}}$ (27) depends on the apex angle $\gamma$, ${\left({\rho }_1^{\mathrm{opt}}\right)}^2∕u=\sqrt{27∕2}$ does not depend on $\gamma$ (29). For sc overall symmetry $\left({\gamma }_{\mathrm{sc}}=70.54°\right)$ we find $\left({\theta }_1^{\mathrm{opt}}=75.52°\right)$ and for rhombohedral overall symmetry with fcc translational symmetry $\left({\gamma }_{\mathrm{fcc}}=38.94°\right){\theta }_1^{\mathrm{opt}}=28.96°$.

Figure 3

Band structure of the rhombohedral photonic crystal corresponding to $(u,0,0,0)$, $\gamma ={\gamma }_{\mathrm{fcc}}$, and $f_{\mathrm{Si}}=38\%$. The underlying Brillouin zone is illustrated in Fig. 6, the real-space crystal in Fig. 5.

Figure 4

Relative width of the gap versus filling fraction corresponding to the parameters used in Fig. 3. The dots refer to the fundamental gap and the open dots to the gap between the seventh and eighth band. The vertical line indicates the optimum filling fraction of $f_{\mathrm{Si}}=38\%$.

Figure 5

Left: Motif of the structure according to Eq. (15) within the rhombohedral unit cell for a filling fraction of $f_{\mathrm{Si}}=38\%$. Right: Fragment of the corresponding crystal structure.

Figure 6

High-symmetry points of the Brillouin zone of rhombohedral structures with fcc translational symmetry (Ref. 36).

Figure 7

Illustration and summary of our symmetry analysis (Ref. 37).