Multiple Bragg diffraction in opal-based photonic crystals: Spectral and spatial dispersion

We present an experimental and theoretical study of multiple Bragg diffraction from synthetic opals. An original setup permits us to overcome the problem of the total internal light reflection in an opal film and to investigate the diffraction from both the $\left(111\right)$ and $\left(\overline{1}11\right)$ systems of planes responsible for the effect. As a result, angle- and frequency-resolved diffraction and transmission measurements create a picture of multiple Bragg diffraction that includes general agreement between dips in the transmission spectra and diffraction peaks for each incident white light angle and a twin-peak structure at frequencies of the photonic stop band edges. Two opposite cases of the interference are discussed: an interference of two narrow Bragg bands that leads to multiple Bragg diffraction with anticrossing regime for dispersion photonic branches and an interference of a narrow Bragg band and broad disorder-induced Mie background that results in a Fano resonance. A good quantitative agreement between the experimental data and calculated photonic band structure has been obtained.

Figure 1

Low-energy photonic band structure of opal-based PhCs. The dependencies ${\lambda }_{hkl}\left({\theta }^{\mathrm{opal}}\right)$ are calculated from Eq. (1). The symmetry points of the surface of the first BZ are shown in the lower abscissa. The corresponding angles of light incidence $\theta$ from the quartz hemisphere onto an opal film calculated from Snell's law are shown in the upper abscissa. The points of the MBD are marked by cycles. (Insert) Cross-section of the BZ of the fcc lattice made by the scanning plane $\Gamma X{L}_{g}L$.

Figure 2

(a) The sample holder consisting from two quartz hemispheres with thin opal film in between. (b) The scheme shows the relation between ${\mathbf{k}}^i$ (the incident wave vector) and ${\mathbf{k}}_T$, ${\mathbf{k}}_{111}^s$, and ${\mathbf{k}}_{\overline{1}11}^s$ (the wave vectors of transmitted and diffracted light) and correspondent transmission $T$ and diffraction $D_{111}$, $D_{\overline{1}11}$ spectra. The normal vectors ${\mathbf{n}}_{111}$, ${\mathbf{n}}_{\overline{1}11}$ to the $\left(111\right)$, $\left(\overline{1}11\right)$ sets of planes denote the $\left[111\right]$ and $\left[\overline{1}11\right]$ directions.

Figure 3

The integral spectra $D_{\left(\overline{1}11\right)}$ of the white light diffracted from the $\left(\overline{1}11\right)$ planes of an opal film as a function of the incident angle ${\theta }^{\mathrm{opal}}$. The spectra obtained in the $\Gamma L_g{K}_{g}L$ scan of the fcc BZ. The thickness of the film is 24 layers of $a$-SiO${}_2$ particles with the diameter $D=350$ nm. The opal filler is air. The curves are shifted vertically by the constant value of 0.3. The cursors show “twin peaks”, i.e., the traces of the $\left(111\right)$ diffraction band. The insert ($\times 6$) shows the twin peaks in the enlarged scale.

Figure 4

Diffraction spectra $D_{\overline{1}11}$ of an opal film registered at various scattering angles ${\theta }^s$ in the region of the MBD for white light incident angle ${\theta }^{i,\text{opal}}={34}^{\circ }$ (thin black curves). The thick blue curve represents the normalized integral spectra. The film has 24 layers of $a$-SiO${}_2$ particles with the diameter $D=350$ nm and the filler is air.

Figure 5

The angle-resolved optical spectra of the 24 layer thick opal film with air as a filler ($a$-SiO${}_2$ particles diameter $D=350$ nm). The spectra obtained in the $\Gamma L_g{K}_{g}L$ scan of the fcc BZ as a function of the angle ${\theta }^{\mathrm{opal}}$ in the region of the MBD. The black solid curves show transmission (T) spectra; the red dashed curves show $D_{111}$ diffraction spectra; the blue dot-and-dashed curves show $D_{\overline{1}11}$ diffraction spectra.

Figure 6

Experimental and calculated spectral positions of $\left(111\right)$ and $\left(\overline{1}11\right)$ bands in the transmission and reflection spectra and of the twin peaks structure in the $D_{\left(\overline{1}11\right)}$ spectra. The sample is the 24-layer-thick opal film with air as a filler ($a$-SiO${}_2$ particles diameter $D=350$ nm). The green dash lines represent best fits to the data using the phenomenological approach described in Ref. [38]. The black circles show the data obtained from transmission ($T$) spectra; the red triangles are related to the data obtained from $D_{\left(111\right)}$ diffraction spectra; the blue squares indicate the data obtained from $D_{\left(\overline{1}11\right)}$ diffraction spectra; the blue stars point out the traces of the $\left(111\right)$ diffraction band in $D_{\left(\overline{1}11\right)}$ diffraction spectra; the cyan symbols denote position of the peaks in calculated reflection spectra.

Figure 7

Photonic band structure with account of three reciprocal lattice vectors 0, ${\mathbf{g}}_{111}$, and ${\mathbf{g}}_{\overline{1}11}$. Thick lines represent solutions with real wave vectors. Thin and dashed lines represent real and imaginary parts of $k_z$ for evanescent solutions. (a) Normal incident $k_x=0$. (b) Multiple Bragg diffraction regime. Imaginary parts within photonic band gaps (shaded areas) are multiplied by ten. Dispersion bands are assigned to $\left(hkl\right)$ reciprocal lattice vectors: black curve for $\left(000\right)$, red curve for $\left(111\right)$, and blue curve for $\left(\overline{1}11\right)$. Curves for the real and imaginary parts of wave vector are signed by Re and Im, respectively.