Fano Resonance between Mie and Bragg Scattering in Photonic Crystals

We report the observation of a Fano resonance between continuum Mie scattering and a narrow Bragg band in synthetic opal photonic crystals. The resonance leads to a transmission spectrum exhibiting a Bragg dip with an asymmetric profile, which can be tunably reversed to a Bragg rise. The Fano asymmetry parameter is linked with the dielectric contrast between the permittivity of the filler and the specific value determined by the opal matrix. The existence of the Fano resonance is directly related to disorder due to nonuniformity of $a\mathrm{\text{−}}{\mathrm{SiO}}_2$ opal spheres. The theoretical “quasi-3D” model produces results in excellent agreement with the experimental data.

Figure 1

(a) The SEM image of the (111) opal plane with $a\mathrm{\text{−}}{\mathrm{SiO}}_2$ particles. (b) The circles and the centers of circles derived by processing the SEM image (a) using a Hough-type procedure. (c) The distributions of the circle diameters $D$ (red circles) and the distances between circle centers $a_{00}$ (blue squares) obtained by processing the image (b).

Figure 2

Transmission spectra of two opal samples for the $\Gamma \to L$ geometry and the results of calculations of the transmission spectra of an disordered ensemble of isolated spherical particles with $D=260\mathrm{nm}$ and $D=316\mathrm{nm}$ with the use of Mie theory.

Figure 3

(a) The transmission spectra of an opal sample ($D=316\mathrm{nm}$, thickness $\approx 0.6\mathrm{mm}$) as a function of the filler permittivity ${\epsilon }_f$ in the region of the (111) photonic band (black dotted curves). The color curves (red solid, blue solid, and green dashed) are the results of fitting with a Fano function. (b) The results of calculations of the transmission spectra using the quasi-3D model of disordered opal structure described in the text: The number of layers was taken to be 1000, and averaging was done over 40 000 realizations for ${\epsilon }_f=1.816$ and 10 000 realizations for other values of ${\epsilon }_f$. The curves are shifted vertically by the values shown.

Figure 4

The Fano asymmetry parameter $q$ that changes continuously from negative to positive values with filler permittivity ${\epsilon }_f$ increasing for three samples with different thickness of 0.6, 0.8, and 2.2 mm, respectively, $D=316\mathrm{nm}$. The solid line is a guide for the eyes only.