Band structure and optical properties of opal photonic crystals

A theoretical approach for the interpretation of reflectance spectra of opal photonic crystals with fcc structure and (111) surface orientation is presented. It is based on the calculation of photonic bands and density of states corresponding to a specified angle of incidence in air. The results yield a clear distinction between diffraction in the direction of light propagation by (111) family planes (leading to the formation of a stop band) and diffraction in other directions by higher-order planes (corresponding to the excitation of photonic modes in the crystal). Reflectance measurements on artificial opals made of self-assembled polystyrene spheres are analyzed according to the theoretical scheme and give evidence of diffraction by higher-order crystalline planes in the photonic structure.

Figure 1

Energy bands for an opal photonic crystal (fcc lattice of close-packed dielectric spheres with ${ϵ}_1=2.53$ in air): (a) along symmetry directions in the whole Brillouin zone, and (b) along symmetry directions on the hexagonal face of the Brillouin zone.

Figure 2

Symmetry points in the Brillouin zone for a fcc lattice.

Figure 3

Reduced density of states for several values of the incidence angle from $\theta =5°$ (bottom curve) to $\theta =80°$ (top curve) in steps of 5°. The three figures correspond to different values of the azimuthal angle $\varphi$: (a) $\varphi =0°+2m\pi ∕6$ $(m∊\mathcal{Z})$, or $LW$ direction; (b) $\varphi =30°+2m\pi ∕3$, or $LU$ direction; and (c) $\varphi =-30°+2m\pi ∕3$, or $LK$ direction.

Figure 4

(a) Reduced bands, (b) reduced DOS, (c) reflectance, and (d) diffraction spectra for $\varphi =0°$ ($LW$ direction) and three values $\theta =5°$, 40°, and 70° of the angle of incidence in air. The bands in (a) are plotted in the interval $(0,k_m)$, where $k_m\equiv \sqrt{3}\pi ∕a$ is the $\Gamma L$ distance. In the limit $\theta =0$ the point $k_m$ is identical to $L$. The bands are displayed only for $k_z$ inside the first Brillouin zone.

Figure 5

SEM image of the opal surface $(18\times 22\mu {\mathrm{m}}^2)$.

Figure 6

(a)–(c). Theoretical positions of the gap edges (squares) and of the lowest energy peak (stars) for the $LW$ (a) and the $LK$ (c) direction. The results are for a close-packed fcc structure of polystyrene spheres $(ϵ=2.53)$ with a lattice constant $a=314\mathrm{nm}$. (b)–(d): Experimental reflectance data for several values of $\theta =j\Delta \theta$, with $\Delta \theta =5°$ for the $LW$ (b) and the $LK$ (d) direction; curves from left to right correspond to $j=1,\dots ,16$ and are shifted by the value $0.2(j-1)$.

Figure 7

Enlarged view of experimental reflectance spectra for large incidence angles: $\theta =65°$, 70°, 75°, and 80° (from left to right). Arrows indicate the calculated spectral positions of the lower energy Van Hove singularity [Figs. 6a, 6c]. Directions are $LW$ (a) and $LK$ (b).

Figure 8

Scaling of the spectral features, in dimensionless units, as a function of sphere diameter: stop band central position (triangles) and stop band edges (bars) at near-normal incidence; Van Hove singularities at $\theta =70°$ for $LK$ (circles) and $LW$ (squares) orientations.