Investigation of structural colors in Morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density

We use the nonstandard-finite-difference time-domain (NS-FDTD) method to investigate the interaction of light with the complicated microstructures in the Morpho butterfly scales, which produce the well-known brilliant blue coloring. The NS-FDTD algorithm is particularly suitable to analyze such complex structures because the calculation can be performed in a short time with high accuracy on a relatively coarse numerical grid. We analyze (1) the microstructure obtained directly by binarizing an electron microgram of the cross section of a scale, (2) the reflection and diffraction properties of three model structures—flat, alternating, and tree-shaped alternating multilayers, and (3) an array of alternating multilayers with random noise superposed on the height of the structures. We found that the actual microstructure well reproduced the reflection spectrum in a blue region by integrating the reflection intensities over all the reflection angles. Under normal incidence, the flat multilayer mainly stresses on multilayer interference except for shorter wavelengths, while alternating multilayer rather enhances the effect of diffraction grating due to longitudinally repeating structure by strongly suppressing the reflection toward the normal direction. In the array of alternating multilayers, the reflection into larger angles is considerably suppressed and the spectral shape becomes different from that expected for a single alternating multilayer. This suppression mainly comes from the scattering of reflected light by adjacent structures, which is particularly prominent for the TM mode. Thus a clear difference between the TE and TM modes is observed with respect to the origin of spectral shape, though the obtained spectra are similar to each other. Finally, the polarization dependence of the reflection and the importance of the alternating multilayer are discussed.

Figure 1

(a) Morpho rhetenor and (b) an enlarged view of a wing covered with many scales. (c) An enlarged view and (d) a transverse cross section of a ground scale [24]. The scale bar in (d) is $3\mu \text{m}$.

Figure 2

Actual microstructure obtained by binarizing the TEM data for the transverse cross section of a M. rhetenor ground scale [24]. Scale bar is $3\mu \text{m}$. Numbers 1–4 are used in Fig. 3.

Figure 3

Angular dependence of the reflection for various wavelengths from 400 to 500 nm in the TM mode. The ridge number shown in each figure corresponds to the ridge in Fig. 2. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 4

Angle-integrated reflection spectra for 12 individual ridges in Fig. 2 in the TM mode. A thick solid line shows an average of 12 spectra. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 5

Angular dependence of the reflection intensities at wavelengths of 400, 420, 450, 480, 500, and 550 nm calculated for the structure shown in Fig. 2 under normal incidence. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 6

Angle-integrated reflection spectrum calculated for the structure shown in Fig. 2 under normal incidence (open circles). The solid line is the angle-integrated reflection spectrum measured using an integrating sphere for M. rhetenor [26]. In order to compare the simulation with the experiment, the calculated results for the TM (dashed line) and TE modes (dotted line) were added equally. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 7

(a) Flat, (b) alternating, and (c) tree-shaped alternating multilayers.

Figure 8

Angular dependence of the reflection intensities from the following three model structures under normal incidence at various incident wavelengths: (a) flat multilayer, (b) alternating multilayer, and (c) tree-shaped alternating multilayer. (d)–(f) Angle-integrated reflection spectra for the three model structures. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 9

(a) Multilayer with an infinite width, (b) flat multilayer model, and (c) diffraction grating.

Figure 10

Angular dependence of the reflection intensities of an alternating multilayer at shorter wavelengths under normal incidence in the (a) TM and (b) TE modes. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 11

(a) Angular dependence of the reflection intensities of a flat multilayer calculated by the NS-FDTD method with a layer width of 300 nm, thickness of 55 nm, spacing between two adjacent layers of 150 nm, number of layers equal to 6, and refractive index of 1.55 from Ref. [26]. (b) Calculated result for a refractive index of 1.01 while the optical path length between adjacent layers is kept constant. (c) Calculated result by using the analytical model and (d) measured result for a M. didius wing after removing the cover scales [26]. The calculations for (a) and (b) were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 12

Model for an array of alternating multilayers with random heights. The deviation from a mean value of the ridge height is denoted as $\Delta y$.

Figure 13

Angular dependence of the reflection intensities (left) and angle-integrated reflection spectra (right) of an array of alternating multilayers with irregularity in the ridge height. The irregularity was introduced by giving a uniform random number within a maximum range of $\pm \Delta y_{\text{max}}$, where $\pm \Delta y_{\text{max}}=\left(\text{a}\right)$ 0, (b) 50, and (c) 205 nm. In (b) and (c), the angle-integrated spectra were calculated for five series. The five curves and their average are plotted as thin and thick curves, respectively. The calculations were performed with a mesh size of $10\times 10{\text{nm}}^2$.

Figure 14

Angular dependence of the reflection intensities of a single ridge (dashed line) and an array of regularly arranged ridges (solid line) with a spacing of 700 nm at the wavelengths of 220, 325, and 400 nm, where the angle-integrated reflection spectrum shows peaks. The diffraction orders corresponding to the peaks of the reflection intensities are indicated as positive or negative numbers.

Figure 15

Angular dependence of the diffraction spots of various orders for a spacing of 700 nm (thin solid lines) and of the reflection peaks for a single alternating multilayer (shaded area). The thick solid line indicates the peak position of the reflection intensity obtained by interpolating the data shown in closed circles, while the thin solid lines indicate the bandwidths evaluated at half of the height of the reflection bands. The density of the shade roughly expresses the reflection intensity. Note that the overlapping regions of these curves give peaks in the angle-integrated reflection spectrum shown in Fig. 13a.

Figure 16

Angular dependence of $\alpha$ for various values of $\Delta y_{\text{max}}$.

Figure 17

Angle-integrated reflection spectra in the (a) TM and (b) TE modes for a single alternating multilayer (solid line) and an array of ten alternating multilayers with the irregularity in height with $\Delta y_{\text{max}}=205\text{nm}$ (dotted line). The reflection spectrum for the array of ten alternating multilayers was obtained by averaging five series of calculations.

Figure 18

Ridge-interval dependence of angle-integrated reflection intensities at 320 nm, which are divided by the total number of model structures (closed circles with dotted lines). The data were obtained for ten alternating multilayers with a height irregularity of $\Delta y_{\text{max}}=205\text{nm}$ and refractive indices for the cuticles of 1.56 and $1.56+0.06i$, respectively. The solid lines were calculated using Eq. (10) with various values of $a$. The unshaded, weakly shaded, and heavily shaded areas indicate that light being emitted toward $\varphi =64°$ suffers from no scattering, scattering from one ridge, and scattering from more than two adjacent ridges, respectively; these were estimated geometrically.