Direct determination of the refractive index of natural multilayer systems

It is well known that the metal-like strong reflection observed in the elytra of some kinds of beetles is produced by multilayer thin-film interference. For the quantitative analyses of the structural colors in these elytra, it is necessary to know accurate values of the refractive indices of the materials that comprise the multilayer structure. However, index determination is not an easy task: The elytron surface is not flat but curved and usually contains many irregular bumps, which cause scattering loss. These structural characteristics prevent us from directly applying conventional optical techniques for index determination, such as ellipsometry, since these techniques require a perfectly specular surface. In this paper, we report a new experimental procedure that can directly determine the refractive indices of individual layers in natural multilayer systems. This procedure involves semi-frontal thin-sectioning of the sample and subsequent optical examinations using a microspectrophotometer. We demonstrate that the complex refractive index and its wavelength dependence can be successfully determined for one kind of beetle.

Figure 1

(a) Jewel beetle Chrysochroa fulgidissima and (b) an electron micrograph of the multilayer system beneath the elytron surface [the red striped part in (a)]. Scale bar: 500 nm.

Figure 2

Reflectance ($R$) and transmittance ($T$) spectra of a test thin-film sample made of a 45.5-nm-thick silicon layer sputtered on quartz glass substrate. Rough curves are experimentally determined using the assembled microspectrophotometer employed in this study. Smooth curves are theoretically drawn according to the thin-film interference model in which the wavelength-dependent complex refractive index and thickness of the silicon layer are predetermined by using a spectroscopic ellipsometer. It is noted that $R$ and $1-T$ are plotted so that the vertical distance between the two curves corresponds to absorptance.

Figure 3

Semi-frontal thin section of the elytron of the jewel beetle. (a) TEM image, (b) optical image under transmission illumination, and (c) optical image under epi-illumination. The rectangle in (b) is tilted for comparison with (a). Note that the upper right region is the epoxy resin. Scale bar: 10 $\mu$m.

Figure 4

Results of a line scan measurement. (a) Reflectance in air and (b) transmittance in immersion oil, measured at a wavelength of 470 nm. Three broken lines, denoted by $d$, $e$, and $f$, indicate the positions of the spectra shown in Fig. 5; position $d$ is in the region of the epoxy resin, while $e$ and $f$ correspond to electron-lucent and electron-dense stripes, respectively. (c) Optical image under epi-illumimation. The black line approximately indicates the scanned line where reflectance and transmittance are measured. Scale bar: 10 $\mu$m.

Figure 5

Reflectance and transmittance spectra for small regions of the thin section, determined using a microspectrophotometer. (a) Reflectance at the three positions indicated by three broken lines marked with $d$, $e$ and $f$ in Fig. 4. From top to bottom, the spectra are for the electron-dense stripes ($f$) and the electron-lucent stripes ($e$), and the epoxy resin region ($d$). A theoretical curve is also drawn according to the thin-film interference model to fit the spectrum of the epoxy resin part. This fitting analysis determines the thickness of the thin section to be 65 nm (see text for details). (b) Transmittance spectra for the electron-lucent ($e$) and electron-dense ($f$) stripes.

Figure 6

Refractive indices of the epoxy resin used for embedding the sample elytron and the immersion oil. The filled circles and squares denote the experimental results for a prism-shaped resin block and the immersion oil contained in a triangular glass cell, respectively, determined by measuring the minimum deviation angle between the incident and refracted light at several wavelengths of lasers. Curves are drawn according to Cauchy’s equation, expressed in Eq. (1), which give the best fit to the experimental results, with parameters $A=1.501$ and $B=5.271\times {10}^{-3}$ ($\mu$m${}^2$) for the epoxy resin and $A=1.498$ and $B=6.520\times {10}^{-3}$ ($\mu$m${}^2$) for the immersion oil.

Figure 7

The real (a) and imaginary (b) parts of the wavelength-dependent complex refractive indices for the two types of the materials that comprise the jewel beetle’s multilayer structure, denoted by ${\mathrm{n̂}}_{\text{h}}={\eta }_{\text{h}}+i{\kappa }_{\text{h}}$ and ${\mathrm{n̂}}_{\text{l}}={\eta }_{\text{l}}+i{\kappa }_{\text{l}}$ for high- and low-index layers, respectively. Six gray curves are obtained from six different pairs of reflectance and transmittance spectra. The black curves are drawn according to Eqs. (1) and (6) that approximate the experimentally determined index values (see text for details).

Figure 8

Wavelength of the reflectance peak of the elytron of the jewel beetle. The filled circles represent the experimental results for the red striped part (upper) and green part (lower) of the elytron; these results are obtained by the so-called $\theta$-$2\theta$ scan measurement, except for the data at 0${}^{°}$, which are obtained by using a microspectrophotometer. It has been experimentally confirmed that the reflection from the small area of the elytron is almost specular, but not perfectly. Curves are theoretically drawn according to the interference condition, ${\lambda }_{\text{p}}=2({\eta }_{\text{h}}{d}_{\text{h}}\cos {\theta }_{\text{h}}+{\eta }_{\text{l}}{d}_{\text{l}}\cos {\theta }_{\text{l}})$, where ${\eta }_{\text{h}}$ (${\eta }_{\text{l}}$), $d_{\text{h}}$ ($d_{\text{l}}$), and ${\theta }_{\text{h}}$ (${\theta }_{\text{l}}$) are the real part of the refractive index, thickness, and refraction angle of the high-index (low-index) layer. The index value is obtained using Eq. (1) and the thicknesses of the two types of layers, $d_{\text{h}}$ and $d_{\text{l}}$, are chosen to be 108 and 139 nm for the red striped part and 75 and 96 nm for the green part, respectively; these values are consistent with the TEM observations. The refraction angles are calculated according to Snell’s law. Since the index value and refraction angle are dependent on the wavelength, ${\lambda }_{\text{p}}$ is numerically obtained by self-consistently solving the above interference condition. We confirmed that the inclusion of the imaginary part of the refractive index induces only a small shift of a few nanometers in the peak wavelength, by theoretically calculating the reflectance spectrum of the multilayer thin-film model of the elytron. It is noted that when a refractive index difference becomes very large, the wavelength at the maximum reflectance can deviate from ${\lambda }_{\text{p}}$ because the spectral peak comes to have a flat-top shape [6]. However, such an effect is negligible in the present case with a small refractive index difference.

Figure 9

Schematic illustration of semi-frontal thin-sectioning of the multilayer structure. $d$: thickness of the section; $D_{\mathrm{h}}$ and $D_{\mathrm{l}}$: thicknesses of the high- and low-index layers, respectively; ${\Lambda }_{\mathrm{h}}$ and ${\Lambda }_{\mathrm{l}}$: lengths of the regions where the section comprises only high- or low-index layers, respectively; $\psi$: the tilt angle between the section and the multilayer; and $l$: period of the stripe pattern in the semi-frontal thin section.