Transmission and reflectance

Optical characterization

In order to optically characterise these structures, light transmission measurements are performed. In transmission spectra, information on every Bragg diffracting plane is contained. As samples have an average thickness of the order of 1 mm, thousands of layers are crossed by the transmitted radiation. The sample surface tested is of several mm2.

FIG. 1. Optical transmission at q =0º for opal like structures made of spheres with different diameter: (1) 535 nm, (2) 480 nm, (3) 415 nm, (4) 350 nm, (5) 305 nm, (6) 245 nm, (7) 220 nm. Spectra have been vertically shifted for the sake of clarity.

In Fig. 1, the spectra recorded for samples made of spheres with different diameter (from 220 nm to 535 nm) are shown. These spectra were measured at normal incidence (q=0º). A clear attenuation band in the optical transmission can be observed in each case due to the Bragg reflection caused by the (111) planes. As the sphere size decreases, the Bragg reflection moves linearly towards shorter wavelengths. This shows the accurate tuning of the optical properties resulting from the sphere size control.

FIG. 2. The Bragg reflection maximum wavelength (lc) plotted against the sphere size and fit to Bragg law (dashed line).

In Fig. 2, the Bragg reflection maximum (l c) has been plotted against the sphere size (f). We can fit l c using Bragg law for normal incidence: l c = 2·n·d , where n is the effective index of refraction of the SiO2/air composite and d=0.816·f the distance between crystalline planes in the direction q =0º. Thus from the slope of the fitted curve (dashed line in Fig. 2) we obtain n=1.349, which is extremely close to n=1.348, the value obtained averaging the dielectric constant e =
(n1)2 f + (n2)2 (1-f),
where f is the filling factor (f=0.74 for a close packed structure).

In order to estimate peak broadening effects, we have compared our experimental results with the analytical expression derived by Tarhan and Watson.(1) In their model (Dw/wc) = 0.054 for the [111] stop band of an fcc photonic crystal. The experimental (Dw/wc) is larger in all cases (0.08 in average), which can in principle be due to the existence of domains.

Transmission measurements of visible and near infrared (NIR) radiation were also performed at different angles q with respect to the surface normal. The results obtained for a sample formed by spheres of 440 nm diameter, (according to AFM), are shown in Fig. 3. No difference was found between TE and TM polarized light.

FIG. 3. Transmission spectra for diferent incidence angles, q with respect to the surface normal in a sample made of 440 nm diameter spheres. From bottom to top, q = 0º, 10º, 15º, 20º, 25º, 30º, 35º, 40º. The vertical bar indicates one decade. Spectra have been vertically shifted for the sake of clarity.

At normal incidence (q =0º), a clear band corresponding to the [111] Bragg reflection can be observed (band 1 centred at 957.5 nm), along with a weaker deep (band 2 centred at 525 nm). As the angle increases, band 1 shifts to shorter wavelengths according to the Bragg law while band 2 shifts to longer wavelength. The latter band can be explained as a [220] reflection. This behaviour is summarized in Fig. 4, where both peaks’ wavelengths are plotted against q along with the calculated angular dependence of the [111] and [220] Bragg reflections (solid and dashed lines respectively) of an fcc structure. We have considered f =440 nm, as measured by AFM,
and n=1.348.

FIG. 4. l c of the (111) band plotted against q (circles) and the calculated angular dependence of the [111] an [220] bands of a cubic close packing (solid and dashed line respectively).

Thus, although the arrangement of layers can follow either fcc or hcp structure, both scanning electron microscopy (SEM) photographs and transmission measurements indicate that the cubic one could be favoured. This tendency to the fcc arrangement has been observed before in colloidal crystals of latex particles slowly grown from dilute suspensions.

1.- I.I. Tarhan, and G.H. Watson, Phys. Rev. B, 54, 7593 (1996).