1. Introduction

Many butterfly species are known to have sub-micron-sized structures inside their wing scales that produce structural color [1–7]. One of the best-known examples is the $Morpho$ species, whose brilliant blue wings have attracted many researchers to investigate this strong blue reflection [8–13]. The fundamental goal of these researchers is to understand the mechanisms of interesting optical properties, including strong wavelength-selective reflection, angular dependence, and polarization effects, amongst others. It is also expected that deep understanding of these optical phenomena could lead to new applications, possibly by mimicking the biological microstructures.

Wavelength-selective reflection in structural color is usually associated with optical interference. For a periodic multilayer structure, consider light waves reflected from two interfaces separated by one period. The well-known interference condition for constructive interference between these two waves correctly estimates the reflected wavelength [14,15]. The photonic crystal, found in some butterfly wing scales, is an apparently more complex structural type [16–20]. However, the periodicity allows us to predict the wavelength of reflection by calculating the photonic band gap frequency, using the analogy of Bragg diffraction in X-ray scattering.

When the system is not periodic, there is no simple condition for constructive interference, and it becomes difficult to interpret the wavelengths of the spectral peaks. For example, in a multilayer system where layer thickness differs between layers, it is not easy to explain the wavelengths of reflectance peaks [21]. Rajah Brooke’s birdwing butterfly, $Trogonoptera$ $brookiana$, shown in Fig. 1(a), is interesting in this respect. It has been reported that the microstructure in the wing scale, although periodic, is rather complex [22,23]: ridges on the scale are periodically spaced like a diffraction grating (Fig. 1(b)), and in the cross section each ridge has several lamellae on both sides that are regularly spaced in the vertical direction (Fig. 1(c)). Although the structure may appear to be a simple multilayer system with effective layer thicknesses, Wilts $et$ $al.$ interestingly revealed that this may not be so; the reflectance spectrum has several wavelength peaks in the visible range [24]. In addition, the spectral shape is largely dependent on the polarization. Wilts $et$ $al.$ performed a detailed optical simulation of the observed wing scale microstructure, using the finite-difference time-domain method. This confirmed that the melanin pigment in the ridge center largely affects the reflectance spectrum. However, several questions remain with respect to interpretation of the spectral shape: what are the physical processes causing the various spectral peaks? Can we specify the reflection paths for constructive interference at the reflectance peaks? To answer these questions, we performed detailed optical analysis of the microstructure inside the wing scale of the male Rajah Brooke’s birdwing butterfly. We first characterized the scale to enable construction of a realistic model, which could reproduce the experimentally determined reflectance spectrum. Then, we simplified the model to enable interpretation of the spectral shape, and calculated reflectance spectra while varying several structural parameters. For the optical simulation we employed the method of rigorous coupled-wave analysis (RCWA), a common method of analyzing grating diffraction [25,26].

 

Fig. 1. Wing scale structure of $Trogonoptera$ $brookiana$. (a) Photograph of the butterfly. (b) SEM image of the scale surface. (c) TEM image of a cross section of the scale. (d) Structural model with dimensions indicated. Nine lamellae are assumed in this model. Scale bar: (a) 1 cm, (b) 10 $\mu$m, and (c) 1 $\mu$m.

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2. Methods

The microstructure of the scale was observed by scanning electron microscope (SEM, JEOL JCM-6000) and transmission electron microscope (TEM, Hitachi H-7650). For the TEM observations, small pieces of the wing were embedded in epoxy resin after a dehydration process using ethanol. The resin block was thin-sectioned to approximately 70 nm using an ultramicrotome.

Microspectrophotometry was used to determine the reflectance spectrum of a small region within a scale. The experimental system consisted of an optical microscope (Olympus BX51) and a fiber optic spectrometer (Ocean Optics USB2000). The microscope was equipped with a xenon lamp and an objective lens with 50$\times$ magnification (Olympus SLMPlan N, NA 0.35). The fiber diameter was 200 $\mu$m, enabling examination of a 4 $\mu$m diameter region on the scale. The polarization of the epi-illumination was selected using a polarizer (U-PO3). In addition, we placed a 200-$\mu$m-diameter pinhole in the plane of the aperture stop of the microscope, so that the illumination became nearly collimated at the sample position [27]. We ensured that the examined surface was perpendicular to the optical axis. The top flat part of the curved scale was examined after confirming that the focus could be adjusted over the entire region of interest using a high numerical aperture (NA) objective lens (Olympus MPLFLN100x, NA 0.9). The reflectance spectrum was determined as the ratio of the observed spectrum to that of a diffuse reflection standard (Labsphere Spectralon).

In the above microspectrophotometry, only zeroth-order diffraction was detected, because the objective had a relatively small NA of 0.35. The grating period of the wing scale was 866 nm, so for the shortest examined wavelength of 400 nm, first order diffraction occurred at 27.5$^{\circ }$. This angle was larger than the 20.1$^{\circ }$ angular aperture of the objective. Therefore, in the following calculations our main interest is the zeroth-order diffraction efficiency, although other orders are discussed when necessary.

To determine the reflectance spectrum, we used the RCWA method [25,26]. In this method, the coupling constants between the incident and diffracted light can be determined by solving the linear equations obtained from the boundary conditions for electric and magnetic fields. We used a self-developed computer code written in Mathematica (Wolfram Research). All calculations were performed assuming normal incidence.

3. Results

We first investigated the microstructure of the scale using electron microscopy. Figure 1(b) shows the SEM image of the scale surface, where we see periodically-spaced ridges running longitudinally. In cross section, as shown in Fig. 1(c), each ridge is observed to have lamellae on both sides, with lamellae attached to adjacent ridges almost in contact. These observations are consistent with those of previous studies [22–24]. We constructed a structural model, closely resembling, but slightly simplified from, our electron microscope observations, as shown in Fig. 1(d). We note that the center of the ridge appears dark in the transmission electron micrograph, indicating a higher electron density. Because such dark regions have been known to contain melanin pigment, the calculation included an imaginary part for the refractive index in the ridge center.

Next we used the microspectrophotometer to measure the polarization-dependent reflectance spectrum shown in Fig. 2(a). We define the polarization as follows: in TM (TE) polarization the electric field vector is perpendicular (parallel) to the ridges, respectively. In both polarizations, several peaks are observed in the examined wavelength range, similar to previous studies [24]. We examined several other scales and confirmed that the spectral peaks are reproducibly observed, although their magnitudes are not identical.

 

Fig. 2. Polarization-dependent reflectance spectrum. (a) Experimental results determined by the microspectrophotometer. (b) Zeroth-order diffraction efficiency calculated using the structural model shown in Fig. 1(d). Black and red curves correspond to TM and TE polarizations, respectively. Refractive indices of cuticle and ridge center are assumed to be $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.067i$, respectively.

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Applying RCWA to the model structure (Fig. 1(d)), we calculated the reflectance spectrum for the two polarizations, shown in Fig. 2(b). In this calculation, the refractive index of the cuticle was assumed to be $n_{\mbox {c}}=1.575$, the value determined by the Becke line test. For the ridge center, the refractive index was assumed to be $n_{\mbox {rc}}=1.575+0.067i$. The magnitude of the imaginary part was determined from transmission measurements after the wing scale was immersed in a refractive index-matching oil, assuming the pigment is distributed uniformly over the scale. We note that, physically, the real part of $n_{\mbox {rc}}$ should be larger than $n_{\mbox {c}}$, because resonances with electronic states increase both the real and imaginary parts of $n$. However, we used the same value for the real part to make the model simpler. Similarly, we used wavelength-independent refractive index values. In fact, wavelength dependence of the refractive index has been reported for the cuticle and melanin pigment in insect species [28–30]. For example, the real part increases from approximately 1.54 to 1.58 in the wavelength range 400-800 nm [28]. However, the dispersion is not large and only slightly affects the spectral shape.

Comparison of Figs. 2(a) and (b) shows that the calculations reproduce the experimental results well. We note that the reflectance is largely dependent on the polarization. In particular, there is a strong peak for TM polarization in the wavelength range 640-700 nm, while reflectance is low for TE polarization. Polarization-dependent photographs of the wing scale were presented previously [24], and the observed strong polarization dependence may be related to structural birefringence. In this paper, however, as the first stage of our analysis, we focus our attention on TM polarization, and investigate the physical origins of the spectral peaks. Detailed analysis of the polarization dependence will be published elsewhere. For convenience, we name the peaks in the TM polarization reflectance as P1, P2, P3, and P4, in order of decreasing wavelength, as shown in Fig. 3(b).

 

Fig. 3. Structural models and reflectance spectra. (a), (c), and (e): structural models named M0, M1, and M2, respectively. M0 corresponds to the cross section of the model shown in Figs. 1(d). (b), (d), and (f): reflectance spectra for TM polarization corresponding to the models shown in (a), (c), and (e), respectively. In (b) and (d), the zeroth-order diffraction efficiency calculated using RCWA is shown as reflectance. In (f), the red curve is the reflectance calculated using a method for multilayer systems [31], while the black curve is the same spectrum shown in (d). In these calculations, we assume nine lamellae with refractive index $n_{\mbox {c}}=1.575$. The ridge center refractive index $n_{\mbox {rc}}$ is $1.575+0.067i$ in (b) and $1.575+0.01i$ in (d). In (e) (model M2), the two layer types, with thicknesses 125 nm and 96 nm, have refractive indices 1.575 and 1.184, respectively.

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3.1 Analysis

3.1.1 Model simplification and the origin of P1

We started our analysis by simplifying the model structure. Our original model, named M0, has the cross section shown in Fig. 3(a). This was simplified in two ways to produce model M1, shown in Fig. 3(c). Firstly, the triangular structure on top of the ridge center was removed. Secondly, while M0 has a rhombus-shaped structure at the tip of each lamella, in M1 the lamellae are treated as flat layers. The reflectance spectrum calculated for this model is shown in Fig. 3(d), where peaks are seen corresponding to those in Fig. 3(b), although peaks P1 to P3 shift to slightly shorter wavelengths. In contrast, peak P4, located at 530 nm, shifts to a longer wavelength. We therefore consider P4 separately. As another simplification to focus our attention on the effects of structure, we decreased the imaginary part of $n_{\mbox {rc}}$ to 0.01 in this calculation. We did not set it completely to zero, because a zero imaginary part results in many sharp reflectance peaks originating from the phenomenon of guided mode resonance [32] which make the spectrum complicated. With this reduced imaginary part of $n_{\mbox {rc}}$, the reflectance becomes higher; note in Fig. 3(d) the values on the vertical axis are larger than in Fig. 3(b).

We further simplified the structure into a multilayer model, named M2 and shown in Fig. 3(e), consisting of periodically stacked layers of two types. The first type is the cuticle layer with thickness 125 nm and refractive index $n_{\mbox {c}}=1.575$. The second is an air-rich layer, 96 nm thick, for which the refractive index is assumed to be 1.184. This is an estimated average (weighted by volume) of the indices of the ridge material (1.575) and air (1.0) within the layer. In this model, the reflectance spectrum was calculated as shown in Fig. 3(f), which has a broad reflection band in the wavelength range 550 nm to 720 nm. This band originates from first order interference of the periodic multilayer stack; the interference condition is satisfied at wavelength 621 nm (=$2(125\times 1.575+96\times 1.184)$), almost at the center of the reflection band. We note that the band is wide enough to contain P2 and P3. We also note that P1 closely matches one of the sidebands of the multilayer interference spectrum. It is known that the side bands of a main reflection band can be interpreted as due to simple thin-film interference when the entire multilayer structure is modeled as one layer with an appropriate refractive index.

The side band of the multilayer system appears as an oscillation in the reflectance spectrum, the frequency of which is directly related to the total thickness of the system. Using model M2, we increased the number of layer pairs (or periods, where a period includes a high-$n$ and a low-$n$ layer) from 9 to 20, as shown in Fig. 4(a), and re-calculated the reflectance spectrum. As expected, more oscillations are seen at wavelengths greater than 700 nm (Fig. 4(b)), while peaks P2 and P3 become higher and narrower. We performed similar calculations for the more realistic structure shown Fig. 4(c), in which triangular structures are included on the tops of the ridges, and the imaginary part of $n_{\mbox {rc}}$ is assumed to be 0.067. The calculated spectra for 9 and 20 layer pairs, shown in Fig. 4(d), again display rapid oscillations at longer wavelengths for the 20-layer-pair model. From these results, we can interpret P1 as one of the side bands obtained by considering the scale structure as a single layer. For the wavelength range 500 nm to 550 nm, the effect of the number of layer pairs on the spectral shape differs between the two models. For the model shown in Fig. 4(a), the differences between the spectra in Fig. 4(b) are consistent with the expected behavior of the side bands. In contrast, for the more realistic model of Fig. 4(c), the spectrum is largely unaffected by the number of layer pairs (Fig. 4(d)), implying that the origin of P4 is not a side band. The reflectance of the latter model is lower than that of the former (compare the vertical scales of Figs. 4(b) and (d)), mainly because the imaginary part of $n_{\mbox {rc}}$ is larger in the latter model. However, the triangular structures on the tops of the ridges may also be involved, because such structures are known to suppress reflection by the so-called moth-eye effect. We will discuss this, together with the origin of P4, in the Discussion section.

 

Fig. 4. Reflectance spectra of model structures with different numbers of layers. (a) A model structure based on M1. (b) Reflectance spectrum (zeroth-order diffraction efficiency) with 9 (black) and 20 (red) layer pairs in the structure shown in (a). Refractive index values $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.01$i are assumed. (c) A more realistic model including triangular structures on the tops of the ridge centers, where $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.067i$ are assumed. (d) Reflectance spectrum (zeroth-order diffraction efficiency) with 9 (black) and 20 (red) layer pairs in the model structure shown in (c).

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3.1.2 Origins of peaks P2 and P3

Figure 3(f) shows that the wide reflection band of the simpler M2 multilayer model appears to split into peaks P2 and P3 in model M1. Because M1 consists of stacked layers with vertical pillars (ridges), it is natural to consider that the ridges are related to this splitting. We calculated the reflectance spectra obtained by varying the lateral period $\Lambda$ of the ridges (Fig. 5(a)). In these calculations, the ratio of the ridge width $w$ to $\Lambda$ was kept constant. Varying the parameters in this manner keeps the average refractive index constant, so we avoid an unnecessary peak shift in the spectrum. The calculated reflectance (zeroth-order diffraction efficiency) is shown in Fig. 5(b). When $\Lambda$ is much smaller than the wavelength of light, the structure can effectively be treated as a simple multilayer. Thus, in the spectrum with $\Lambda =400$ nm, the broad reflection band ranging from 570 nm to 690 nm is considered to originate from first order interference in the periodic multilayer stack. The long-wavelength edge of this band appears to remain at the same wavelength (around 690 nm) while $\Lambda$ varies. In contrast, an additional peak located at 460 nm for $\Lambda =400$ nm gradually shifts to a longer wavelength with increasing $\Lambda$, and enters the region of the broad reflection band for $\Lambda =866$ nm, resulting in the two peaks P2 and P3.

 

Fig. 5. Reflectance spectra of M1-type structures with different ridge separations. (a) Model structure. The separation $\Lambda$ and the width of the pillar (ridge) $w$ are varied, keeping the ratio $w / \Lambda$ at a constant value of 0.32, obtained using the values $w=277$ nm and $\Lambda =866$ nm (see Fig. 1(d)). The number of layer pairs is assumed to be 20. Assumed refractive index values are $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.01i$. (b) Calculated spectra (zeroth-order diffraction efficiency) for, from top to bottom, $\Lambda$ = 400 nm, 500 nm, 600 nm, and 866 nm. The value of $\Lambda$ is displayed in the top right corner of each plot. The gray vertical lines show the wavelengths calculated by applying the Bragg condition, Eq. (1).

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From the results of the above calculations, we can interpret P2 as the remains of the broad reflection band that originates from the periodic multilayer. The origin of the peak that shifts with $\Lambda$ must be related to the ridges. In fact, we can interpret it as the Bragg diffraction from tilted planes as shown in Fig. 6(a). Here model M1 is expanded to be infinitely large and treated as a two dimensional photonic crystal. Figure 6(b) illustrates this reflection process in reciprocal space, showing the incident and reflected (scattered) wavevectors, $\boldsymbol {k}_{\mbox {i}}$ and $\boldsymbol {k}_{\mbox {s}}$, respectively, and the reciprocal reflection vector $\boldsymbol {G}$. Expressing the spacing between the tilted planes as $\delta$ and the angle of incidence on the planes as $\theta$, the Bragg condition gives the wavelength $\lambda _{\mbox{B}}$ for constructive interference as

 

Fig. 6. Bragg diffraction from tilted planes. (a) Schematic illustration of reflection. $\theta$ is the angle of incidence on the planes, whose separation distance is $\delta$. (b) Reciprocal representation. The incident and scattered vectors are denoted by $\boldsymbol {k}_{\mbox {i}}$ and $\boldsymbol {k}_{\mbox {s}}$, respectively, and $\boldsymbol {G}$ is the reciprocal vector associated with the reflection. The origin of reciprocal space is denoted by $\Gamma$. (c) Schematic illustration explaining normal reflection.

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3.1.3 Reflection direction of peak P3

A question immediately arises regarding the direction of the Bragg reflection illustrated in Fig. 6(a). Here light is apparently reflected in an oblique direction, while experimentally we observe at normal reflection and theoretically calculate the zeroth-order diffraction in Fig. 5(b). The answer is that light can be reflected in both normal and oblique directions (first and higher orders of diffraction), depending on the wavelength $\lambda$ and the spacing $\Lambda$. For the case of $\Lambda =400$ nm, diffraction is limited to zeroth order in the examined wavelength range under normal incidence, because the diffraction grating interference condition, $\Lambda \sin \phi = m \lambda$, can only be satisfied for $m=0$ and $\phi =0$. We can understand the normal reflection under the normal incidence by considering the reflection path illustrated in Fig. 6(c). Here incident light is first reflected by the tilted planes, then undergoes total internal reflection at the interface with air, and finally is reflected again by the oppositely tilted planes. When $\Lambda$ is small, the Bragg planes’ tilt is large, so the reflected light propagates highly obliquely, and the condition for total internal reflection is satisfied. On the other hand, for larger $\Lambda$, total internal reflection does not occur, and some light refracts into air to become the first order diffraction. Figure 7 shows the diffraction efficiencies of the zeroth and first orders for two different $\Lambda$ values. For $\Lambda =500$ nm, the major part of diffraction is limited to the zeroth order, because the diffraction condition can only be satisfied when the wavelength is shorter than 500 nm (Fig. 7(a)). However, for $\Lambda =866$ nm, both the zeroth and first orders can contribute to the total diffraction efficiency, as shown in Fig. 7(b). We note that the first-order efficiency becomes higher in the region around 600 nm where the zeroth-order diffraction efficiency is low. Thus, the trough in the zeroth-order spectrum is somewhat masked in the total diffraction efficiency. Note that the successive reflection illustrated in Fig. 6(c) is for intuitive understanding of how normal reflection occurs. In fact, reflection occurs simultaneously and coherently in these different directions.

 

Fig. 7. Diffraction efficiencies for (a) $\Lambda$=500 nm and (b) 866 nm calculated for the model structure shown in Fig. 5(a). The blue and red curves show the zeroth- and first-order diffraction efficiencies, respectively. The black curve shows the sum of the zeroth- and $\pm$1st-order diffraction efficiencies.

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3.2 Discussion

We have determined the origins of the three spectral peaks. The longest-wavelength peak P1 is one of the sidebands of the broad reflection band. The side band oscillation of the spectrum can be explained by treating the multilayer stack as a single layer. The second peak, P2, is what remains of the broad reflection band of the periodic multilayer. Peak P3 is caused by Bragg diffraction from tilted planes within the scale structure; P3 largely shifts to a longer wavelength as $\Lambda$ increases, and comes close to P2 for $\Lambda =866$ nm, the measured period of the butterfly wing scale. We have not found a specific reflection path for constructive interference to explain peak P4.

To determine the origin of P4, we calculated the photonic band diagram of the model two-dimensional photonic crystal shown in Fig. 8(a). The reciprocal lattice is shown in Fig. 8(b), with the first Brillouin zone indicated, and the zone boundary along the $k_y$ direction denoted by X’. Figure 8(c) shows the photonic band diagram (left) and compares it with (right) the reflectance spectrum for 20 periods shown in Fig. 4(d). This shows the three peaks P2, P3, and P4 in the reduced-frequency range $\omega \Lambda / 2 \pi c$ = 1.2–1.75, which corresponds closely to the gap between the two electromagnetic modes drawn in red. The two modes drawn in blue can also symmetrically couple to the external plane wave. The reflectance becomes low at the frequencies of these modes on the zone boundary X’, causing troughs in the spectrum. This is interpreted as the incident plane wave coupling with these modes and entering the photonic structure. In particular, the density of these modes is high around X’, as the bands become flat. However, if the field pattern of the electromagnetic mode changes from the uniform pattern of the external plane wave as the wavevector moves away from X’, coupling to the external wave weakens, and reflectance increases. We therefore hypothesize that P4 is not caused by optical interference for a specific reflection path, but by weak coupling between internal and external waves.

 

Fig. 8. (a) Model structure for photonic band diagram. The black rectangle shows the unit cell. The assumed refractive index of the cuticle (grey region) is 1.575. (b) Reciprocal space. The rectangle represents the first Brillouin zone. Point X’ denotes the zone boundary along the $k_y$ direction. (c) Photonic band diagram. The lowest four electromagnetic modes that can symmetrically couple to the external plane wave are shown in color (red or blue), while black curves show the modes that cannot couple. For comparison, the reflectance spectrum is shown on the right. This is the spectrum for 20 layer pairs shown in Fig. 4(d).

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The comparison between Figs. 4(b) and (d) indicated that the triangular structure on top of the ridge center affects the spectral shape in the wavelength range 500-550 nm, where P4 is located. It is known that periodic protrusions on a surface reduce reflection, a well-known example being found in the compound eyes of moths. In such a system, the protrusions can be considered as an interface with gradually changing refractive index [33]. The triangular structures on the ridge centers could function in a similar manner. The reflectance decreases substantially when the triangular structure is included, as is shown clearly in Fig. 9. When reflection at the outermost interface is suppressed, the side band oscillation, which can be seen in Fig. 4(b), becomes less prominent, and the reflection from inside, within the photonic crystal structure, becomes more prominent, as shown in Fig. 4(d).

 

Fig. 9. Effects of the triangular structure on top of the ridge center. Red and black curves show the zeroth-order diffraction efficiency for the model structure with and without the triangular structure, respectively. The structural models are similar to those shown in Fig. 4(a) and (c), but for 9 layer pairs. Refractive indices $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}} = 1.575+0.067i$ are assumed in these calculations.

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One unique feature of this butterfly is that the lamellae in the ridges are well developed, and they form almost-connected cuticle layers; many other butterfly species have an air-cuticle multilayer in the scale interior that is the part below the ridges. Thus, the presence of the ridge centers means that the scale structure of this butterfly should be treated as a two-dimensional photonic crystal rather than as a periodic multilayer. As a result, the wing scale exhibits strongly polarization-dependent reflection (Fig. 2). Although such polarization dependent reflection has been previously reported in $Morpho$ butterflies [8,10], the difference in reflectance between the two polarizations is much larger in this butterfly. This suggests we could apply the effect to design optical components such as a polarization-dependent optical filter. Our preliminary analysis of the TE polarization suggests the reflectance peaks can be interpreted similarly to those for the TM polarization, although the peaks are located at different wavelengths. A deep understanding of these reflection mechanisms and polarization effects will be useful in designing photonic structures with desired optical properties.

Our analysis revealed that the reflective properties are largely affected by a few structural factors, such as the distance between the ridges and the triangular structures on their tops. In particular, we note that the distance between the ridges produces two spectral peaks located close to each other (P2 and P3). This may indicate that, rather than broad band reflection, two separated peaks are more functional as a tool for visual communication within this butterfly species. In addition, the triangular structures on the tops of the ridge centers make these peaks well-separated, as shown in Fig. 9. However, this suggestion should be carefully tested, because fine spectral features in reflective properties can be averaged out over a complete wing which contains numerous wing scales.

3.3 Conclusion

We have performed detailed analysis of the photonic structure inside the structurally colored green wing scale of the Rajah Brooke’s birdwing butterfly, $Trogonoptera$ $brookiana$. Using realistic structural models and rigorous coupled-wave analysis, we determined the origins of three of the four peaks in the TM-polarization reflectance spectrum. For the remaining peak, we did not find a specific reflection path, but suggested weak coupling to the external wave as a possible explanation. We believe that such detailed understanding of optical processes in natural photonic structures will be useful in designing bio-inspired optical components.