Polarization of the optical response of opals
which consist of silica spheres ordered in a close packed face-centred-cubic (fcc) lattice. This work shows that the symmetry of the photonic bands provides valuable information to predict the result of polarised light interacting with opals in optical experiments.
Here is experimental evidences on how light with different polarization can only couple to certain bands in the lowest energy range of the band structure of artificial opals. We analyse the four bands determining the first pseudogap in the vicinity of the L point of the Brillouin zone. This can be thought of as the pseudogap associated with the diffraction in the (111) set of crystalline planes. Angle and polarization resolved transmission measurements were carried out by Tarhanand Watson for dilute colloidal crystals,
The four lowest bands in the vicinity of the L point for an opal structure are identical irrespective of the direction along the surface of the Brillouin zone for an angular range going from normal incidence to an internal angle of about 34 degrees and slightly departing from a common behavior only for higher angles. This is equivalent to saying that within a circle inscribed in the hexagonal face of the Brillouin zone the photonic bands are isotropic. So, for the sake of simplicity, we can take the ΓLU triangle as representative and assume incident light with its wave vector k contained in that plane, the tip of k lying on the LUnd and 3rd) and anti-symmetric (1st and 4th) bands are to be found bounding the first Bragg peak. segment (see Fig. 1). This will permit us classifying the bands by their behavior under mirror reflection with respect to that plane, which coincides with the diffraction plane. Both symmetric (2
Consider the symmetry properties of the incident field for any wave vector k contained in the ΓLU plane. The electric field E can always be written as a linear combination of two base vectors contained in a plane perpendicular to k. In particular, it is convenient to choose such vectors as parallel (ep) and perpendicular (es) to the diffraction plane (see Fig. 1). Then, in the p-polarized configuration the E field is contained in the diffraction plane, and mirror reflection with respect to that plane leaves the field vector unchanged. However for s-polarization the E field is perpendicular to the mirror symmetry plane so that the symmetry operation changes E into -E. Thus, the symmetric (A’) eigenstates along the LU direction can only be excited by symmetric p-polarized incident fields, whereas the s-polarized fields can only couple with the anti-symmetric (A’’) states. So, a polarization sensitivity of the first Bragg peak width for oblique incidence is expected, according to band structure calculations.
In transmittance experiments information is gathered on all possible diffraction processes since these are revealed as reductions in transmitted intensity. By the nature of the experiment, the full thickness of the sample is probed, which ineluctably involves diffuse scattering, loss of signal and other unwanted effects. Reflectance experiments offer information on selected diffraction planes determined by the geometry of the scattering, which makes it suitable for our purposes.
FIG. 1. Geometry of the experiment. The diffraction plane is represented in light gray. The sample surface plane is represented in dark gray. The inset shows the first Brillouin zone for a fcc structure. High symmetry points are indicated
The samples employed here are opals of 297 nm spheres grown by natural sedimentation. A detailed description of the synthesis can be found elsewhere. A typical sample presents a surface of 9 mm22 in cross section. Angle resolved measurements were performed with linearly polarized light with its electric field perpendicular (s-polarization) or parallel (p-polarization) to the diffraction plane, as shown in Fig. 1.
and a thickness of 0.5 mm. Reflectivity measurements were carried out using a collimated beam from a tungsten lamp 1 mm
FIG. 2. (a) Reflectivity spectra as a function of external angle of incidence for s polarized light for θext= 6º, 14º, 23º, 29º and 34º. (b) Reflectivity spectra as a function of external angle of incidence for p polarized light for θext= 8º, 14º, 21º, 25º, 30º and 35º. (c) Normalized reflectivity spectra are shown in solid (dashed) line for s- (p-) polarized light, for an external angle of incidence of 39º degrees.
Reflectivity spectra are presented in fig. 2. In both cases, The shape of the spectra for s and p polarized light as a function of angle deviate from the flat-top-peak with 100% reflectivity predicted by theory for perfect infinite crystals.
Extinction caused by diffuse scattering taking place at defects in the bulk and surface of the crystal is known to affect the intensity and shape of the peak rounding the edges but not affecting the full width at half maximum (FWHM).
In the presence of a mosaic spread, besides an additional decrease in reflected intensity due to non-specular reflections, inhomogeneous broadening could take place. Another source of peak broadening could be the finite size of crystallites in the sample.
Two normalised spectra for s and p polarized light, for an external angle of incidence θext of 39 degrees, are presented in fig. 2c where we can already appreciate the difference in width between both peaks, indicating a strong polarization sensitivity.
In order to compare our experimental results with numerical calculations, several issues have been taken into account. Because of the geometry of the sample in our experiment we are mapping the bands for k in the vicinity of the hexagonal face of the Brillouin zone (see fig. 1a). As the angle of incidence increases, the wave vector k will approach the boundaries of the hexagon centered in L in the direction of one of the high symmetry points located at the corners (W) and centers (U, K) of the sides of the hexagon. This will be determined by the orientation of the sample surface relative to the diffraction plane. For each possible orientation of the sample, different results are expected.
Determining the orientation of the samples is not trivial since they are composed by a mosaic of monodomains typically 20-50 microns in size. While all domains show a preferential out-of-plane orientation with the surface having a (111) orientation, they present a random in-plane orientation. The size of the probe beam being larger than a typical domain will cause averaging over many domains. To avoid this uncertainty in our measurements, reflectivity spectra were collected from normal incidence up to an internal angle of 32 degrees. As pointed above, for this angular range the four lowest lying energy bands are identical irrespective of the direction of tilting. The correspondence between internal and external angles can be obtained from Snell’s law using an effective refractive index neff which is calculated by fitting the experimental angle dependence to Bragg’s law: , where d111 is the interplanar distance for the (111) family of planes (parallel to the crystal surface), λmax is the peak center and qext is the angle formed by the normal to the crystal surface and the incident beam. This expression has proven valid for artificial opals
at angles below the avoided crossing region near the U, K and Wneff=1.336 and d111=244 nm. If we estimate the diameter of the spheres from this value of d111 we obtain 299 nm, in agreement with the value obtained from SEM characterization.
points in the first Brillouin zone. The values obtained from this fit were:
FIG. 3. Calculated photonic bands (lines) and measured band edges from reflectance spectra (circles). Continuous (dashed) lines represent bands anti-symmetric (symmetric) with respect to mirror symmetry. Experimental data correspond to light plane polarized parallel (p) and perpendicular (s) to the diffraction plane.
We have assumed the widely used criterion of associating the full width at half maximum of reflectance peaks
with the edges of a stop band. In the absence of finite size effects or mosaic spread, this approach would be correct as explained above. Following this criterion, fig. 3 shows the experimental results for p-polarized and s-polarized light respectively (symbols), plotted on top of the calculated bands (lines). The fact that the FWHM of the peak for small angles of incidence matches the predicted width implies that finite size effects are negligible and inhomogeneous broadening due to a mosaic spread, if present, will be small. As a matter of fact the broadening observed for large angles of incidence could be due to a mosaic spread, since its effect on the peak width would become more pronounced as the angle of incidence increases.
According to the previous analysis, the outer bands (1 and 4) couple to s-polarized light, while the inner bands (2 and 3), defining a narrower stop gap, couple to p-polarized light. This explains the difference in intensity for s and p polarized light reflectivity as a function of angle shown in fig. 2. In the case of p-polarized light, the stop band defined by the symmetric bands to which this polarization couples dramatically narrows as we move away from normal incidence, therefore the penetration length of the incident light into the crystal increases
and the effect of extinction by bulk defects will increase, while for s-polarized light the width of the stop band suffers a much smaller narrowing and therefore the change in intensity is smaller. Let us remark that in an experiment using unpolarized light, the outer bands would be probed while the inner ones would remain hidden since the broader peaks contain the narrower.
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