Finite element modeling of the radiative ... - APS Link Manager

PHYSICAL REVIEW E 87, 022705 (2013)

Finite element modeling of the radiative properties of Morpho butterfly wing scales A. Mejdoubi,* C. Andraud, S. Berthier, and J. Lafait† Institut des Nanosciences de Paris, UMR 7588 CNRS - Universit´e Pierre et Marie Curie, Paris 6, Case 840, Campus Jussieu, 4 place Jussieu, 75252 Paris Cedex 05, France

J. Boulenguez HEI (Ecole des Hautes Etudes d’Ing´enieur), 13 rue de Toul, 59046 Lille Cedex, France

E. Richalot ESYCOM, Universit´e Paris-Est, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vall´ee Cedex 2, France (Received 17 December 2011; revised manuscript received 21 December 2012; published 13 February 2013) With the aim of furthering the explanation of iridescence in Morpho butterflies, we developed an optical model based on the finite-element (FE) method, taking more accurately into account the exact morphology of the wing, origin of iridescence. We modeled the photonic structure of a basal scale of the Morpho rhetenor wing as a three-dimensional object, infinite in one direction, with a shape copied from a TEM image, and made out of a slightly absorbing dielectric material. Periodic boundary conditions were used in the FE method to model the wing periodic structure and perfectly matched layers permitted the free-space scattering computation. Our results are twofold: first, we verified on a simpler structure, that our model yields the same result as the rigorous coupled wave analysis (RCWA), and second, we demonstrated that it is necessary to assume an absorption gradient in the true structure, to account for experimental reflectivity measured on a real wing. DOI: 10.1103/PhysRevE.87.022705

PACS number(s): 42.66.−p, 42.70.Qs, 78.20.Bh

I. INTRODUCTION

Photonic crystals, or photonic band gap materials, have long occurred naturally and have been widely studied over the past twenty years. They presently constitute a fascinating source of inspiration for physicists and engineers. Their native structures are complex, when compared to man made photonic crystals, and, as such, are of both theoretical and applied interest. These natural photonic structures generally have multiple functions: optical, thermodynamical, hydrological, mechanical. . . . The understanding of this multifunctionality is of great interest in a bioinspired designing approach. Different length scales are implied: photonic crystal structure, butterfly scale, wing region, and whole wing. These size scales all together lead to the wing multifunctionality. At every length scale, the geometry is nearly perfectly ordered; the limited disorder is the origin of the robustness of the optical response—it may contribute to the robustness of the other functions too. A good example for these natural photonic structures is given by some blue species of the Morphidae family (Lepidoptera), such as Morpho rhetenor It is the reason why the literature concerning the optical properties of this genus is abundant since the end of the 1990s with the early works of notably Vukusik [1] and of Berthier [2], following the qualitative pioneering works of Ghiradella [3] on similar structures in the 1970s. Nevertheless, articles devoted to the modeling of these optical properties are not so frequent (without completeness [4–10]). Some of these papers predict the overall optical behavior of the butterfly or of a

*

Present address: CEA-SPINTEC, Bat 10.05, 17 rue des Martyrs, 38056 Grenoble Cedex 9, France. † [email protected] 1539-3755/2013/87(2)/022705(7)

part of its multiscale structure under the point of view of color [4,6,7], or/and the spatial dispersion of the reflected light [5,7]. Some of the recent ones take into account the disorder in the periodic structures, essentially at the level of the fine structure of the scales covering the wing, i.e., the multilayered ridges [7–9]. Nevertheless, it is often general spectral behaviors or reflectance coefficients in arbitrary units which are calculated, even if presented next to experimental results. It is therefore difficult to check in detail the validity of the optical model proposed and more precisely the relevance of the optical index used in these simulations. It seems that the values of reflectance calculated within these models are, in general, much weaker than the values measured on a Morpho wing (except [6]). It was therefore our aim to propose a calculation which takes into account as closely as possible the true microstructure of the scales of the wing with its irregularities and its periodicity, together with a critical analysis of the optical index put in the model, especially concerning its absorption part, with the final aim of a comparison with measurements on the wing of the same specie. Several computational approaches can be employed to model the optical properties of complex structures: the Harrington method of moment [11], boundary-integral equations (BIE) [12], Monte Carlo (MC) [13] (MC methods are a class of computational algorithms that rely on repeated random sampling; they tend to be used when it is impossible to compute an exact result with a deterministic algorithm), Fourier expansion [14], finite-difference time domain (FDTD) [15], and the finite element method (FE) [16–20]. Among these different computational approaches, the FDTD and FE methods appear particularly suitable, due to their precision in the description of geometry and material properties. FDTD is recognized as a versatile and efficient technique for the electromagnetic modeling of reflection by a given structure.

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Among recent studies, important steps forward have been made by Banerjee et al. [6], who modeled the color of a Morpho butterfly wing scale using high accuracy nonstandard FDTD [two-dimensional (2D) model], followed by Lee et al. [7] (3D model), Zhu et al. [8] (2D model), and Saito et al. [9] (2D model). The main problem in this technique is that calculations are relatively long, especially in three dimensions. The challenge is thus to find a reasonable balance between the choice of method, the desired accuracy, and computational expense. In Morpho rhetenor, the photonic structure is complemented by pigments located at the bottom of the structure, both acting to generate vivid blue coloration as a sexual message. Absorption in a broad part of the solar spectrum is also vital for insect metabolism. Its role is therefore crucial in the optical properties of the wing. The distribution of electromagnetic fields in the structure is thus important to understand the various functions of the system. The finite element method is particularly adapted, here. It takes into account true disorder and allows an accurate description of the fields inside and outside, i.e., transmission and reflection spectra. The FE method is flexible, as arbitrary shapes can be modeled. However, calculation times are longer, due to linear equation calculations at each time step, and larger memory is needed, compared to FDTD, when used in the time domain. Consequently, the FE method is mainly used in the frequency domain or for eigenmode analysis. Direct fields measurements could constitute an experimental validation of the simulation results; however, they are no longer possible in the optical domain, with typical dimensions of the structure varying from a few nanometers to a few micrometers. In our study, the precise fit between the experimental and simulated optical properties (mainly reflectivity) will be considered as a test of validity for the field calculations inside. With such background, the rest of this article proceeds as follows: in Sec. III we summarize the methodology used for calculating spectral reflectance. Details related to algorithm implementation are also presented. We follow (Sec. IV A) by reporting detailed comparisons of numerical data with those obtained by Boris Gralak et al. [4] using a modal method. In Sec. IV B we present and discuss FEM results, accurately accounting for the true photonic structure. A conclusion is provided in Sec. V.

FIG. 1. A lateral (a) and transversal (b) schematic representation of striae in the basal scales of Morphidae.

accurate determination of the arrangement and thickness of the lamellae (Fig. 2). The ledges forming the lateral margins of the lamellae appear in juxtaposition, on both sides of the median plane of the ridge. It has been shown that the optical properties of these scales are due to their photonic structure and to the presence of pigments [21] consequently playing a role in solar energy absorption of the wing [22]. A melaninlike pigment is situated in the basal membrane of the scale and in the lower lamellae. But the exact distribution and relative location remain poorly known and difficult to determine. One of our objectives, with the modeling of the radiative properties of such a structure, was to take into account, as accurately as possible, the exact morphology of the structure and to localize the electromagnetic energy inside the structure itself, to elucidate efficiency both as color reflector and solar absorber. III. COMPUTATIONAL METHOD

In order to fully understand, we need to elaborate on the modeling method and the FE approach. The reflectivity, which we are interested in, is measured as the ratio between the flux of the reflected Poynting vector and the flux of the incident Poynting vector. The analysis was performed using the High Frequency Structure Simulator commercial computer software package (Ansoft HFSS, version 11), which is based on FE. The procedure sorted out reflectivity on a personal computer (PC) with core 2 duo processor (4 GHz). HFSS is an interactive simulation system whose basic mesh element is a tetrahedron. This allows resolution of any arbitrary 3D geometry, especially

II. MORPHO WING SCALES

The males of six of the nine subgenera of Morphidae (Lepidoptera) present a vivid blue iridescent coloration, due to the nanostructure of their wing scales. This one is highly specific, but follows a single schematic organization. In most of the species, the color originates in a second scale layer: the basal scales. Upper laminae are very tightly striated (500–600 nm apart). The main distance  [Fig. 1(b)] between two neighboring ridges of the basal scales, directly responsible for the diffraction angle width, varies from 300 to 750 nm, depending on species. The overall thickness is generally 1.5–2.0 μm. The longitudinal ridges have a laminated structure consisting of approximately ten overlapping lamellae, anchored by a fine netting of trabeculae [Fig. 1(a)]. A cross section of the scale observed in scanning electron microscopy (SEM) allows an

FIG. 2. SEM image of the cross section of a Morpho rhetenor basal scale, from a specimen of our collection.

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and bottom faces are modeled using PML, which need to be placed at a distance of at least λ/4 from radiating devices. This paper treats a general class of phenomena, encountered in structural biology, involving a dielectric plate surrounded by air. Incident harmonic plane waves are of the form Einc = E0 ei(−kx x−ky y−kz z+ωt) , where kx ,ky and kz are the wave number components defining the direction K of propagation, K = kx i + ky j + kz k, and ω is the angular frequency in rad/sec. The main value to be obtained is the reflected field amplitude. However, in the presence of repeated inclusions, as in Fig. 3, since only one single cell is modeled, a strategy for computing the reflected power is needed. Different methods can be used to determine reflectance, for instance, by integrating the Poynting vector on a surface placed above the scattering structure (but not inside the PML), thus obtaining   1 (Einc × H∗inc ) ds, Pinc = (2) surf 2   Pref = surf

FIG. 3. (Color online) Schematic view of the unit cell of the model. Reference system: Cartesian coordinate system (x,y,z), −z is the direction of propagation of the plane wave, d is the period of the slab, the structure is infinite in the x direction.

with complex curves and shapes. We give a brief description of the procedure used to calculate reflectivity. For reasons of parsimony, in a first step, the photonic structure of Morpho rhetenor is represented by a single material of refractive index n = 1.56 + i0.06, embedded in a matrix of index nmat = 1. Instead of modeling the entire array of striae composing a basal scale, we use the periodic boundary conditions to restrict the studied domain to a unit cell which contains sufficient information on the structure. Careful examination of the TEM images (like Fig. 2) led us to model a unit cell composed of two striae (Fig. 3), assumed to be infinite in the x direction. The periodic boundary conditions link the fields at the left and right cut vertical faces (y = 0 and y = d = 0.15 μm), as well as at the back and front cut vertical faces (x = 0 and x = 0.15 μm), following Etarget = eiφ Esource ,

(1)

where Esource is the field on the source boundary, Etarget is the field on the target boundary, φ is the phase difference between the two linked boundaries, function of the incident wave, i.e., the wavelength and the angle of incidence. Since we are evaluating a radiating structure, we need to create a free space environment for the device to operate in. This can be achieved by using either the radiation boundary condition [23] or a perfectly matched layer (PML) [15]. In this study, the top

1 (Eref × H∗ref ) ds, 2

(3)

where surf is the evaluation area used for each calculation. The calculator is used to extract two quantities: (i) incident intensity Pinc (computed using the incident field); (ii) reflected intensity Pref (computed using the scattered field solution). These quantities are then used to compute reflectance, R = Pref /Pinc . In this paper, this reflection coefficient is calculated by computing the scattering parameters (S parameters). They are complexvalued, frequency dependent, matrix elements, which describe the reflection and transmission of the electromagnetic energy measured at different ports of devices such as transmission lines. They originate from transmission-line theory and are defined in terms of transmitted and reflected voltage waves. For structures with radiation boundaries, S parameters account for the effects of radiation loss. When a PML is included in a structure, far-field calculations are performed as part of the simulation. The wavelength dependent S11 and S21 parameters are calculated, while the hemispherical reflectance and transmittance are deduced: R(λ) = |S11 |2 and T (λ) = |S21 |2 . Such outputs allow us to also compute absorbance, through energy conservation A(λ) = 1 − R(λ) − T (λ). IV. RESULTS A. Validation of the method

In order to validate the method, we compared our results with calculations made using the RCWA method. We used the structure described by Gralak et al. [4], considered to be close to that of Morpho rhetenor (see Fig. 4 for the detailed scheme). Gralak et al. used a modal method, classically used to simulate synthetic diffraction gratings. In an effort to reproduce their results, we used a very similar method, the RCWA, whose principle is as follows [24]. In the grating region the permittivity and the field are represented by their Fourier series expansions, i.e., the field is expanded on a plane waves basis. In the regions upon and under the grating, the field is represented by the Rayleigh series, i.e., it is written in reflected and transmitted propagative diffracted orders (plane waves). In the grating region, Maxwell

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FIG. 5. Calculated hemispherical reflectance, (a) TE polarization, (b) TM polarization: FE results (crosses), RCWA results (solid line), following the scale geometry of Fig. 4. FIG. 4. (Color online) A schematic representation of the transverse cross section of the modeled scale (in nanometers).

equations are written. In the regions upon and under the grating, the diffraction grating law is applied. Between the different regions, the continuity of tangential fields is assumed. If the grating is layered, transmittance matrices describe the continuity of the field across the interface of contiguous layers. Such a method allows calculation of reflectance spectra and diffraction maps with polarization characteristics. Figure 5 shows the hemispherical reflection of the structure, illuminated by a plane wave under normal incidence, for the two fundamental polarizations, denoted TE (transverse electric, the electric field is orthogonal to the ridges) and TM (transverse magnetic, the electric field is parallel to the ridges). The spectral region of interest is the visible region, i.e., between 380 and 780 nm. Following Gralak [4] and Vukusic [1], we used as an optical index of the structure n = 1.56 + i0.06 and for the air matrix nmat = 1, both assumed independent of the wavelength. Two important points should be made: (i) FE and RCWA data match exactly. (ii) The trends in data are in very good agreement with reported values [4]; a pronounced peak of reflectivity at short wavelengths gives rise to the blue wing color. B. More realistic scale modeling

Simulations were carried out on a structure close to the geometry of the scales of Morpho rhetenor, as measured by TEM images (Figs. 2 and 3), in order to reproduce the results of optical measurements performed on the sample. With this

more complex geometry, the RCWA method cannot be easily employed and simulations had to be performed using the FE method. At first, we used the same values of the optical index of the structure and of the matrix as in previous calculations. Experimentally, the reflectance of the wing was measured under TE and TM polarized light using a Cary 5 spectrometer equipped with an integrating sphere. These measurements were then averaged to reproduce the effects of unpolarized light and normalized with a lambertian diffuse reflectance standard. The spot size was 12 mm × 4 mm, therefore the signal was averaged over a large number of scales. Figure 6 presents the numerical results compared to experimental data. Focusing first on the curves, related to the structure with homogeneous absorption, several remarks can be made: (a) The shapes of the experimental and calculated reflectance curves follow the same trend, both at the main extremum of 475 nm, and, more surprisingly, for secondary peaks around 410 and 350 nm (the last was not graphed). (b) The main peak around 475 nm results in the blue wing color. (c) The magnitude of the calculated reflection coefficient is much lower than observed experimentally. The magnitude was much higher in the previous calculation using the same optical inputs but different morphological parameters. This leads to some skepticism in regard to the (schematic) approaches usually reported (as illustrated in Fig. 4). Several reasons can be invoked to explain the strong difference between numerical and experimental results: (a) The value of the imaginary part of the optical index of the structure is highly debatable (and consequently, but for

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FIG. 6. Hemispherical reflectance spectra: measurements (crosses), calculation by FE on a homogeneously absorbing structure (dashed line) and on a heterogeneously absorbing one (solid line). TE and TM polarizations are averaged in the simulation to reproduce unpolarized natural light.

less, as is the real part). A study conducted by Berthier [25] showed that this value is lower than 0.06 and wavelength dependent (increasing from the blue to red). Chitin, as a pure macromolecule, is generally considered as nonabsorbant, however, in nature it is often linked to other proteins or pigments such as melanin which can influence absorption properties. A careful analysis of the TEM images of the structure of the ridges, supported by other earlier observations, shows (Fig. 7) that melanin in the Morpho wings accumulates at the base of the ridges of the ground scales [26], leading us to hypothesize that absorption increases from the top to the bottom of the multilayer composing each ridge.

FIG. 7. (Color online) TEM image of the structure of a ridge and line profile of its density in the depth of the ridge.

(b) The dispersion in optical index was not taken into account in the calculation. According to Berthier [25], dispersion is relatively weak and essentially concerns the longest visible wavelengths, where reflectivity is still low. (c) The measurement is performed on a large portion of the wing, thus integrating the effect of a large number of scales presenting themselves in a relative disorder. One can therefore expect that the measurement smooths and broadens the spectrum predicted by the calculation. (d) The absorption by the wing membrane and ventral side wing scales is not taken into account in the simulations, due to the weakness of the light flux reaching this region of the wing. We thus assume that the absorption follows a linear gradient from the top of the multilayer (no absorption, imaginary part of the optical index equal to zero) to the bottom (k = 0.06) of the ridge, as suggested by the density line profile analysis shown in Fig. 7. The third curve (solid line) of Fig. 6 presents the numerical results obtained with these new optical inputs, maintaining the same realistic description of morphology. Our new simulations show remarkable increases in magnitude of reflectance, while keeping the same position for peaks. Due to the gradient of absorption, light penetrates the multilayer deeper and is therefore reflected upon a larger number of interfaces, thus increasing global reflectance. Moreover, oscillations appear in the highest part of the visible spectrum due to interferences by the different layers. As mentioned earlier (c), during measurement this effect is smoothed at by integration over a large number of scales. We can therefore conclude that the absorption coefficient of the structure (related to the presence of melanin) plays an important role in reflection of the wing. It is currently difficult to precisely confirm the shape of this gradient in concentration of melanin at the depth of the ridges. Nevertheless, other profiles (steps, different slopes) were tested which do not fit as satisfactorily the experimental response. Therefore, the qualitative and semiquantitative agreement that we observe between simulation and experimentation provides strong evidence of the gradient. Electric field maps, which are extremely rich in information, show that the domains of higher energy concentration in the structure are related to melanin density. Using our model (Fig. 3) with the absorption gradient, we drew field maps of the structure (Fig. 8), illuminated under normal incidence, for two wavelengths, 475 and 590 nm, respectively corresponding to the maximum and the minimum in experimental

FIG. 8. (Color online) (a) Maps of the calculated (finite elements) total electric field (average of TE and TM polarizations) for (a) λ = 475 nm (high reflectance) (b) λ = 590 nm (low reflectance).

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wing reflectance. TE and TM polarizations were averaged to coincide with the visual observations. At 475 nm, the electromagnetic field, due to interference conditions, does not deeply penetrate the structure and is therefore reflected by the first exposed layers, producing high reflectance. At 590 nm, the field penetrates deeper into the structure where it is absorbed (by melanin), thus lowering the reflectance. V. CONCLUSION

This averaging will notably smooth the oscillations of the reflectance due to coherent effects of interference occurring between unit cells (solid line curve in Fig. 6). This study confirms that intimate wing structure is at the origin of blue iridescence in Morpho butterflies. Models devoted to the prediction of such optical properties indicate that the wing behaves like a 3D slightly disordered photonic crystal. Optical properties and multifunction structure appear to be optimized. A more detailed discussion of the role of disorder is subject of a future article. This study, considered as a first step in a more correct account of the complexity of the structure of the wing of Morpho butterflies in its optical properties of bright blue iridescence, has shown that these optical properties (a) are very sensitive to the intimate nanostructure of the wing through its basal scales and ridges and (b) need to take carefully and precisely into account the presence and the localization of melanin in the structure. This simple analysis shows how the presence of an arbitrarily shaped inclusion in a two-phase composite structure can induce changes in the reflectivity, which in turn translate into changes of the optical properties of the various biological species. A more refined model, under development, considering the third dimension of the structure as finite, will allow the analysis of the effect of the disorder of the structure which will be rich in new information. Eventually, our study points toward the possibility of employing these photonic structures to build relevant structures for engineering applications, e.g., photonic crystals.

Herein, we developed a model based on the FE method, to further our studies of Morpho rhetenor basal scale structure. We validated our FE method on a simplified model of this structure (Fig. 4) as previously described [4]. MET images (Fig. 2) of Morpho rhetenor wing basal scales suggest a more complex shape. We applied the FE method on a more accurate geometry described by a unit cell (Fig. 3), direct replication of STEM images. This unit cell introduces nanoscale relative disorder. Although we observed slightly different unit cells, this aspect was ignored for simplicity purposes, and simulations were performed using periodic boundary limit conditions that correspond to a periodic reproduction of the same unit cell. We thus account for both the irregular shape of each individual feature and the large scale periodicity. Moreover, this difference between the first simplified shape and the more precise one already led to considerable differences in optical answer, when using the same optical indices for the media. It was notably impossible to get a high (as high as 70%) level of reflection in the blue part of the visible spectrum (Fig. 6). One way for reproducing the overall shape and the amount of the experimental spectral reflectance of a real wing was to assume a gradient of absorption in the depth of the structure as already suggested by other TEM observations [26], which showed the presence of melanin in the chitin structure of the wing, mostly at the base of the ridges of the basal scales. The final results of our FE model still needed to be averaged for giving a correct account of the optical response of the whole wing in which the basal scales present a relative disorder in their orientation.

This work was supported by the European BioPhot (NEST) project under Contract No. 12915. Thanks to J. P. Hugonin and P. Lalanne, Reticolo software for grating analysis, Institut d’Optique, Orsay, France. Thanks to David Perry for a careful rereading of the English.

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ACKNOWLEDGMENTS

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