1. Introduction

Structural colors in nature, such as those on butterfly wings, beetle cuticles, and bird feathers, have received considerable attention in a variety of research areas [1–23]. Unlike pigmentation, structural colors result from the interaction of light with microstructures that have a featured size comparable to visible wavelengths [1,2]. Using various microstructures, light can be reflected with selective wavelengths [3–9] and polarizations [10,11], in particular directions [12–17], and even in a tunable manner [18–20]. As a result, brilliant structural colors can be achieved for special bio-functions, such as signal communications and camouflage [2,14,15]. Compared with equivalent manmade optical devices, the biophotonic structures often possess higher complexity [8–21] and render better performance in some cases [20,21]. Hence, the study of natural structural colors is important in exploring novel optical effects and designing new photonic devices [20–23].

Conventionally, when light is normally incident on a periodic structure with in-plane period a larger than the wavelength λ, it can be partially reflected into an angle θ ≠ 0 [24]. Such a diffraction phenomenon is described by a grating equation $\sin \theta =m\lambda /a$. For the same order m of diffraction, the diffraction angle of green light θ is smaller than that of yellow light [Fig. 1(a) ]. The angular dispersion ($d\theta /d\lambda =\tan \theta /\lambda$) is much larger than that of a prism so that this phenomenon is widely used to decompose light in modern optical technologies [25].

 

Fig. 1 (a) When a green light beam is normally incident on a mirror with periodic grooves on the surface, it is diffracted into an angle θ smaller than that of a yellow one (for the first order). (b)-(e) When a white light beam with a diameter of 1 mm is normally incident on an elytron of beetle H. sexmaculata, green light is diffracted into an angle larger than that of yellow light and the angular dispersion dθ/dλ is eight times the value in (a). (b) Schematic of the experimental setup for (c). For more vivid photos of the beetle, see http://www.dannesdjur.com/bilder/heterorrhina_sexmaculata_sexmaculata_1.jpg (c) Diffraction pattern on the screen. (d) and (e) Reflectivity of the elytron as a function of the wavelength λ and emergence angle θ. R(λ,θ) is defined as the ratio of the measured reflected intensity from the elytron to the value of specular reflection from a mirror.

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In this paper, we study the structural and optical properties of the elytra of beetle Heterorrhina sexmaculata by electron microscopy and optical spectroscopy. We find that the elytra are composed of a three-dimensional (3D) quasiordered photonic structure which can diffract light in a “wrong” way. In addition, the angular dispersion is about one order of magnitude higher than that of a conventional diffraction grating. Based on theoretical derivation and modeling, we identify that an interesting diffraction photonic bandgap (from an anticrossing of longitudinal and transverse modes) and additional disorder effect play important roles in this phenomenon.

2. Experimental results

Beetle H. sexmaculata belongs to a family of Cetoniinae (flower beetles), found in the rainforests of Indonesia and Malaysia. The elytra of the beetle display mainly green and orange colors, depending on both the incident and viewing angles. Here, we focus on the diffraction at normal incidence. When a white light beam is normally incident on an elytron [Fig. 1(b)], a diffraction pattern of colorful rings is formed on the screen above the elytron [Fig. 1(c)]. In contrast to conventional diffraction phenomena, yellow light is found to be diffracted into an angle smaller than that of green light. From measured spectra of reflectivity [Figs. 1(d) and 1(e)], a large angular dispersion (0.013 rad/nm) is found where the value is much higher than that in a conventional diffraction grating (tan42°/580nm = 0.0016 rad/nm). Hence, a narrow band of light (564.2 nm < λ < 596.5 nm) is diffracted into a wide angle range (56.3° > θ > 32.5°). In this frequency range, a high efficiency of diffraction is found where R_D/R_all > 90% with R_D and R_all being obtained by integrating R(λ, θ) sinθ over |θ | > 15° and > 0°, respectively [sinθ is used in the integration because the reflection is the same for different angles φ, giving rise to uniform colorful rings in Fig. 1(c)]. Due to a bandgap effect (shown below), this diffraction efficiency is higher than that of surface gratings [15,17]. We notice that although a small peak occurs at wavelength of 610 nm, the specular reflectivity is almost uniform in the whole visible frequency range [Fig. 1(e)], giving rise to a white spot at the center of the screen.

Figures 2(a) , 2(b) and 2(c) show the transmission electron micrographs (TEM) for the microstructures of elytra, where the light and dark areas represent chitin (A) and melanoprotein (B), respectively. The top part (I) has three layers of large thickness (ABA, about 400 nm in each layer) and thus contributes slightly to the reflection of visible light [5]. In the middle part (II), the stacking layers (AB)_L have a small layer thickness (93 nm each) and are inserted by melanoprotein rods. Due to the existence of the rod array, light can be diffracted. The layers below part II have a layer thickness chirping from 50 nm to 500 nm, which can present a specular reflection for all visible wavelengths [6]. This explains the white spot observed in Fig. 1(c).

 

Fig. 2 (a)-(b) Transmission electron micrographs of the elytron of beetle H. sexmaculata, where the light and dark areas represent chitin (A) and melanoprotein (B), respectively. (a) Transverse cross section of the elytron. (b) Longitudinal cross section of the elytron showing that the middle part II is pierced by an array of rods. Inset is the Fourier Transform of (b). (c) Enlarged plot of (b). (d) Radial distribution function of the rod array in (b), indicating an average rod spacing of 850 nm. (e) Part II is simulated by a periodic layered structure (AB)₁₀₀ which is inserted by a triangular lattice of rods with diameter D = 500 nm and lattice constant a. The layer thickness d_A = d_B = d/2 = 93 nm. The refractive index n_A = 1.56 and n_B = n_rod = 1.68. A triangular lattice was used because it has a Fourier transform image (with 6 nearest neighbor points for the origin) more close to the ring the inset to (b) than a square lattice (with 4 nearest neighbor points for the origin).

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Figure 2(d) shows the radial distribution function (RDF) of the rod array in part II. Here, the RDF is defined as f(r) = Σ _i N_i(r)/Σ_i N₀, where N_i(r) is the average rod density in the circular ring (r, r + dr) around rod i and N ₀ is the total rod density [26,27]. When the array is fully random, f(r) should be a constant of 1. In Fig. 2(d), the first peak at r = 0.85 μm and the second peak at r = 1.62 μm can be clearly seen. This indicates that the rod array has weak order and the average rod spacing is 0.85 μm. From Fourier transform of the rod array [3], a circle can also be obtained, indicating the isotropy of the array (see the inset to Fig. 2(b)).

3. Theoretical analysis

To understand the observed diffraction phenomenon, part II in the elytra is simplified as periodic dielectric layers (AB)₁₀₀ that are inserted by a triangular lattice of rods [Fig. 2(e)]. The layer thicknesses are d _A = d _B = d/2 = 93 nm. The refractive index n _A = = 1.56 and n_B = n _rod = $\sqrt{{\epsilon }_B}$ = 1.68 [7]. The diameter of rods is 500 nm. Figure 3(a) shows the band structure (k _x = k _y = 0) calculated by a plane-wave method [28] for such a 3D photonic crystal with a = 0.85 μm [29–31]. Here curves A₁ (with $n_{e}2\pi /\lambda =k_z$) are the lowest frequency band, curves A₂ (with $n_{e}2\pi /\lambda =-k_z+2\pi /d$) are the folding band of A₁, and curves B₁ (with ${\left(n_{e}2\pi /\lambda \right)}^2=k_z^2+{\left(2\pi /a\text{'}\right)}^2$) have a nonzero parallel wavenumber $\left|q\right|=2\pi /a\text{'}$ with $a\text{'}=\sqrt{3}a/2$. Due to the periodicity along the z direction, a gap occurs between bands A₁ and A₂ and it remains when the rods are eliminated from the structure. The wavelength of gap center can be analytically obtained by:

 

Fig. 3 (a) Photonic band structure (k_x = k_y = 0) and (b) reflectance at normal incidence for the structure in Fig. 2(d) (a = 850 nm). The results in (a) and (b) are calculated by plane-wave and scattering-matrix methods, respectively. (c) Dependence of the two gaps in (a) on the rod spacing a. (d) Emergence angles for the diffraction gap in (b), compared with the experimental results in Fig. 1(d). The blue line in (d) is the result for a periodic structure with a = 850 nm. The lines and gray areas in (c) and (d) are calculated by a plane-wave method and the dots are obtained from Eqs. (1)-(3).

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Above the lowest bandgap, another gap is formed by an anticrossing of bands A₂ and B₁. This gap does not exist for layered structures without rods and its central wavelength can be expressed as:

For a = 850 nm, a narrow band (572 nm < λ < 577 nm) of light is diffracted into a small angle range (51° < θ < 51.6°). Such a kind of diffraction can be found from the measured reflectivity in Fig. 1(d).

To fully understand the experimental results, we consider part II in the elytra as a combination of a series of 3D photonic crystals with the same volume fraction of chitin f_A and different rod spacings a. When the rod spacing a is increasing, bands A₁ and A₂ do not move while bands B₁ shift down in frequencies. As a result, the Bragg gap persists while the diffraction gap redshifts [Fig. 3(c)]. Since the lattice constant is ranging in a wide range (from 0.75 μm to 1.35 μm) for the quasiordered rod array, the central wavelength of the diffraction gap changes from 565 nm to 596 nm, giving rise to a wide range of diffraction angle (31° < θ < 60°). According to Eq. (3), the diffraction angle θ decreases with increasing λ₁, which explains the unusual experimental results in Fig. 3(d). At a certain emergence angle, the observed diffraction peak has a width (25 nm) wider than the theoretical value (5 nm) [Fig. 1(e)]. From integrating sphere experiments, the measured total reflectance is found to have a maximum of 36% (at λ = 569 nm, not shown) smaller than the ideal case. The discrepancies may result from the structural imperfection in the layers and rods and the absorption from the melanoprotein [7].

4. Discussion

It is interesting to note that the disorder of rods can induce an ultranegative angular dispersion in diffraction. Disordered photonic structures have been studied extensively due to intriguing phenomena such as Anderson localization and random lasing [30,34]. Due to the absence of periodicity, such systems are difficult to handle in theory and a supercell is usually needed in numerical simulations [26,27,35]. Here, we present an interesting approach that considers a quasiordered structure (which is nonperiodic and disordered, but has short-range order) as a combination of some periodic ones. The good agreement between theoretical and experimental results suggests that this method is valid at least to random weak scattering systems (n_A/n_B ~1). We note that this method (using RDF in Fig. 2) does not include particular structural symmetry such as those in quasiperiodic photonic crystals (QPCs, which are nonperiodic but has long-range order) [35]. If this method is applied to QPCs, the total bandgap could be estimated while special results such as various defect modes may not be predicted by this method.

The unusual diffraction relies not only on the disorder of rods, but also on the fact that the diffraction gap of the related periodic structure has a narrow width and red-shifts with increasing the spacing of rods. We note that the diffraction gaps can also be achieved in two-dimensional (2D) photonic crystals that are invariant in the y direction and have a high ratio for the lattice constants in the x and z directions (a_x > 4d). The gap width becomes small when the components have a small contrast of refractive index. This requires that air cannot be included in the biological structures since biological materials, such as chitin, keratin, and melanin, have a refractive index ranging from 1.5 to 2 [7]. For this reason, the unusual diffraction has not been observed in the Morpho butterfly wings [13] and Dynastes beetle cuticles [18]. We also note that conventional one-dimensional (1D) gratings diffract light in a wide frequency range [25] and thus cannot support the phenomenon even when disorder is introduced.

5. Conclusion

In summary, we have discovered that the elytra of beetle H. sexmaculata display unusual diffractive colors, where the angular dispersion is abnormal and has amplitude about one order larger than the value of a conventional diffraction grating. A novel diffraction photonic bandgap (from an anticrossing of longitudinal modes A₂ and transverse modes B₁) of the 3D structures in the elytra and additional disorder effects are found to be responsible for the intriguing phenomenon. This ultralarge dispersive ability could be advantageous for camouflage [2,14] and mimicking the structures may lead to novel optical devices [20–23].