Colomer, Jean-François

Deparis, Olivier

Mouchet, Sébastien

Vigneron, Jean-Pol

English

An original method for computing the reflectance of photonic crystal whose unit cell is composed of form-birefringence anisotropic elements (e.g., cylinders) is proposed. This method, based on the layer homogenization of the photonic crystal, is particularly useful when the reflectance is calculated for different crystal orientations. The far-field color due to an additive color effect in a natural photonic polycrystal found on Entimus imperialis weevil was investigated.An original method for computing the reflectance of photonic crystal whose unit cell is composed of form-birefringence anisotropic elements (e.g., cylinders) is proposed. This method, based on the layer homogenization of the photonic crystal, is particularly useful when the reflectance is calculated for different crystal orientations. The far-field color due to an additive color effect in a natural photonic polycrystal found on Entimus imperialis weevil was investigated

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Jun. 2020Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 43Layer Homogenization Method for Modeling Additive Color Effectin Natural Photonic Polycrystals — The Case of Entimusimperialis WeevilSe´bastien Mouchet, Jean-Pol Vigneron, Jean-Franc¸ois Colomer, and Olivier DeparisDepartment of Physics, University of Namur, 61 rue de Bruxelles, B-5000 Namur, BelgiumAbstract— An original method for computing the reflectance of photonic crystal whose unitcell is composed of form-birefringence anisotropic elements (e.g., cylindres) is proposed. Thismethod, based on the layer homogenization of the photonic crystal, is particularly useful whenthe reflectance is calculated for different crystal orientations. The far-field color due to an additivecolor effect in a natural photonic polycrystal found on Entimus imperialis weevil was investigated.1. INTRODUCTIONModeling the optical properties of photonic polycrystals is crucial for the investigation of naturalphotonic structures as well as for the design and fabrication of bioinspired devices [1, 2]. Thecomputation of reflectance (or transmittance) of photonic crystals (PCs) by the Rigorous CoupledWave Analysis (RCWA) method [1, 3] is sometimes cumbersome to perform due to the presence ofform-birefringence anisotropic elements in the crystal unit cell (e.g., cylinders or parallelepipeds),particularly when it has to be calculated for various crystal orientations. The Layer Homogenization(LH) method proposed here solves this issue specifically for crystals with periodicities such thatthere is only specular reflection and no other diffraction orders [4]. For a single crystal orientation,the dielectric function is homogenized within the layers in which the crystal is sliced. A standardthin film solver is then used for the calculation of the reflectance. The disorder in the crystalorientations of the photonic polycrystal is taken into account by averaging reflectance spectracomputed for various crystal orientations and incidence angles.We demonstrate the usefulness of this method by applying it to the study of a natural photonicpolycrystal. Polycrystals found in living organisms are known to give rise to multiscale visual effectssuch as additive colors. A nice example is found in cavities on the cuticle of Entimus imperialis [2, 4–8]. This photonic polycristal is made of one single PC structure with different orientations [5], whichis relevant for developing bioinspired applications. For instance, the E. imperialis structure hasbeen recently replicated in order to serve as biomimetic devices such as gas, temperature or pHsensors [9].2. PHOTONIC CRYSTAL WITH FORM ISOTROPIC AND ANISOTROPIC ELEMENTSOne of the most common method used to calculate the reflectance spectrum of 3D PC films isthe RCWA method [1, 3] which is based on Fourier expansions of the dielectric function and ofthe electric and magnetic fields. In this method, a PC film is treated as a stratified medium withlateral periodicities. The layers of the PC film are perpendicular to the crystal orientation ~l andtaken homogeneous along that orientation. Elements in the crystal unit cell (defined by translationvectors ~ax, ~ay and ~az) are therefore discretized into sets of non overlapping islands of differentmaterials, embedded in a host and having simple shapes such as cylinders or parallelipipeds withaxes parallel to ~l. For instance, a spherical element is approximated by a set of cylinders (Fig. 1).Calculation of the reflectance according to different crystal orientations is easily performed due tothe form isotropy of the sphere: the discretization is the same whatever the crystal orientationis [1].However, when form anisotropic elements are found in the unit cell, calculation of the reflectanceaccording to various crystal orientations is more complicated to perform (Fig. 2): The discretizationis not the same for the different orientations and may involve islands with non simple shapes [4].In this case, the LH method is needed to solve the problem.3. LAYER HOMOGENIZATION (LH) METHOD AND POLYCRYSTAL MODELINGASPECTSBefore modeling the additive color effect in the polycrystal, the various crystal orientations weretreated separately. In order to compute the reflectance spectrum of a PC with form anisotropic44 PIERS Proceedings, Stockholm, Sweden, Aug. 12–15, 2013Figure 1: In the framework of RCWA, any elementin the unit cell is discretized by simple shapes suchas cylinders or parallelepipeds. A sphere, in thiscase, is approximated by a set of cylinders whoseaxes are parallel to the crystal orientation ~l understudy. Whatever the crystal orientation is, the dis-cretization of a sphere is the same due to its formisotropy.Figure 2: A cylinder cannot easily be discretized bya set of simple shapes when studied under any crys-tal orientation ~l that is neither perpendicular norparallel to the axis of the cylinder. The computa-tion of reflectance or transmittance is therefore morecomplicated to perform.elements in the crystal unit cell whatever the orientation is, we developped the LH method [4] inwhich the crystal was sliced into a set of layers for each orientation under study. The dielectricfunction was homogenized within each layer by a simple spatial averaging (Fig. 3). Thanks to thismethod, the resolution of Maxwell’s equations is highly simplified and the reflectance of a singlecrystal orientation can be computed using a standard thin film solver. However, the averaging ofthe dielectric function can be performed only if the refractive index contrast within each layer isweak. Such a weak contrast is found in photonic structures occuring in insects since they are madeof chitin (nchitin = 1.56 [10]) and air (nair = 1).Since the PC is approximated by a stratified medium with homogenized layers (Fig. 3(b)), atranslational invariance appears along any direction parallel to the layer interface. That impliesthat only specular reflection occurs in that layer-homogenized medium. In order to apply ourmethod, the lateral period of the PC has to be small in comparison with the incident wavelength.This feature is rather common in natural PCs, such as Entimus imperialis investigated hereafter.The reflectance spectrum due to an additive color effect can be modeled by averaging reflectancespectra computed for various incidence angles and crystal orientations. This procedure is often usedfor computing optical properties of multiscale structures [11, 12].(a) (b)Figure 3: Layer Homogenization (LH) method. (a) A photonic crystal is first defined for a given crystalorientation ~l as a stratified medium with 2D lattices of islands having simple shapes (cylinders or paral-lelepipeds). The unit cell is described by ~ax, ~ay and ~az vectors. ~az is parallel to the crystal orientation ~l.(b) The photonic crystal is redescribed by a laterally homogeneous stratified medium with an orientation ~l′that is not parallel to ~az. A new crystal unit cell is defined by other translation vectors ~a′x, ~a′y and ~a′z (~a′zbeing parallel to ~l′).Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 454. ADDITIVE COLOR EFFECTS IN ENTIMUS IMPERIALISThe polycrystal found on the cuticle of E. imperialis is a noticeable example of additive photoniccolor [2, 5] (Fig. 4(a)). A photonic polycrystal with different crystal domains is found in the scaleswhich cover the cuticle. In each domain, the same photonic structure is found but with a differentcrystal orientation. Any PC domain with a given orientation gives rise to a single bright iridescentcolor (from blue to orange) (Fig. 4(b)). The unit cell of the PC can be modeled by perforated chitinlayers on which a lattice of protrusions is found (Figs. 4(c), (d)) [2, 5]. Analysis of this morphologyby a method presented in [12] showed that this structure corresponded to a FCC crystal with alattice parameter equal to 396 nm [5]. We demonstrated in previous studies that this structuresolely reflected the incident light in the specular direction [4, 5]. In spite of the very directional andbright color produced by each photonic domain, the color perceived in the far-field by human eyeis a non-iridescent matt green color (Fig. 4(a)). As we will see hereafter, this matt color is due tothe disorder in the crystal orientations.The LH method was used to calculate the reflectance according to the (111) and (100) crystalorientations at different incidence angles (Fig. 5). For the (111) orientation at normal incidence, thespecular reflectance peak was found at 577 nm (i.e., yellow) and an harmonic was found at 297 nm.For the sake of comparison, the reflectance spectra in this crystal orientation were simulated byRCWA as well. We note that for this orientation, the elements composing the crystal unit cell couldbe easily described by islands with simple shapes. The RCWA spectra are in good agreement withthose computed by the LH method. In the case of the (100) orientation, the main reflectance peakwas located at 530 nm (i.e., green). For that orientation, the computation could be performed onlyby the LH method because the unit cell is made of form anisotropic elements. When the incidenceangle θ is increased, the spectrum is blue-shifted for both crystal orientations. For instance, a violetcolor is obtained at an incidence angle of 45◦ for both orientations.The diversity of colors due to the polycristalline structure is better highlighted on a chromaticitydiagram. Chromaticity coordinates were computed for the two investigated crystal orientations, atincidence angles ranging from 0◦ to 75◦ by step of 15◦. They were found to be distributed acrossthe diagram (Fig. 6(a)): Different domains gave rise to yellow, green and blue colorations, as weobserved in single scales (Fig. 4(b)).In order to take into account the disorder in the crystal orientations and to simulate its far-fieldcoloration, the reflectance spectra calculated at various incidence angles and for various crystal ori-entations were averaged. The chromaticity coordinates calculated from the average of the spectrawere found in the faint green central area of the diagram (Fig. 6(a)). They represent the colorperceived by the naked eye, which appeared here to be a faint green color. A previous study [5]provided us with the reflectance spectra measured for different crystal domains by microspectropho-(a)(d)(c)(b)Figure 4: (a) Entimus imperialis is a remarkable example of additive color effect. (b) A photonic polycrystalmade of different colorful domains covers its cuticle. (c) The photonic structure found in each domain wasevidenced by SEM. (d) This structure was modeled by 2D lattices of protursions covering chitin layers withcylindrical perforations. This model had a FCC symmetry.46 PIERS Proceedings, Stockholm, Sweden, Aug. 12–15, 2013300 400 500 600 700 800 900050100  300 400 500 600 700 800 900050100Reflectance [%]  300 400 500 600 700 800 900050100Wavelength [nm]  RCWALH(1 1 1)θ = 0°(1 1 1)θ = 15°(1 0 0)θ = 0°Figure 5: Reflectance spectra in the (1 1 1) and (1 0 0) orientations calculated using two computationalmethods at different incidence angles θ. Curves labelled RCWA were computed by RCWA method andcurves labelled LH were simulated by the Layer Homogenization method. The reflectance spectrum in the(1 0 0) orientation could only be calculated by the LH method because of the form anisotropy of the elementsin the unit cell (i.e., cylinders).(a) (b)Figure 6: (a) Chromaticity coordinates computed at different incidence angles θ (from 0◦ to 75◦ by stepof 15◦) in the cases of the (1 1 1) orientation (©) and (1 0 0) orientation (¨). There is a great diversity ofcolorations produced by the crystal domains. The average of the reflectance spectra (¤) leads to coordinatesin the faint green central area. (b) The chromaticity coordinates calculated from microspectrophotometrymeasurements (©) [5] are distributed across the whole diagram. The coordinates from far-field reflectancemeasurements (¤) [5] also fall in the faint green area.tometry and in the far-field by spectrophotometry (Fig. 6(b)). The diversity of colors produced bythe crystal domains was reproduced numerically by the spectra simulated for different incidenceangles and two specific crystal orientations. The faint green color measured in the far-field [5] iswell reproduced by the average of the simulated spectra due to an additive color effect.5. CONCLUSIONThe additive colors effect occuring in polycrystals made of one single photonic structure was in-vestigated. The calculation of reflectance (or transmittance) spectra according to various crystalorientations often turns out to be problematic when form anisotropic elements are present in theunit cell. The method we developed relied on layer homogenization and solved this problem bycomputing the reflectance thanks to a standard thin film computation code. The method assumedthat the photonic crystal reflected the incident light only in the specular direction (at least in theProgress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 47visible range). The additive color effect of the photonic polycrystal found on the cuticle of Entimusimperialis was predicted by averaging reflectance spectra computed for different incidence anglesand crystal orientations.ACKNOWLEDGMENTThe authors thank Ce´dric Vandenbem for fruitful and helpful discussions. S.M. and J.-F.C. weresupported by the Belgian National Fund for Scientific Research (F.R.S.-FNRS) as Research Fellowand Research Associate, respectively. The authors acknowledge the use of resources of the ElectronMicroscopy Facilities of the University of Namur, Belgium and the Interuniversity Scientific Com-puting Facility (iSCF) located at the University of Namur, which is supported by the F.R.S.-FNRSunder convention No. 2.4617.07.REFERENCES1. Vigneron, J.-P. and V. Lousse, “Variation of a photonic crystal color with the Miller indices ofthe exposed surface,” Proceedings of SPIE, Vol. 6128, 61281G, San Jose, CA, United States,January 2006.2. Deparis, O. and J.-P. Vigneron, “Modeling the photonic response of biological nanostructuresusing the concept of stratified medium: The case of a natural three-dimensional photoniccrystal,” Mater. Sci. Eng. B — Adv., Vol. 169, No. 1–3, 12–15, 2010.3. Moharam, M. G. and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffrac-tion,” J. Opt. Soc. Am., Vol. 71, No. 7, 811–818, 1981.4. Mouchet, S., J.-F. Colomer, C. Vandenbem, O. Deparis, and J.-P. Vigneron, “Method formodeling additive color effect in photonic polycrystals with form anisotropic elements: Thecase of Entimus imperialis weevil,” Opt. Express, Vol. 21, No. 11, 13228–13240, 2013.5. Mouchet, S., J.-P. Vigneron, J.-F. Colomer, C. Vandenbem, and O. Deparis, “Additivephotonic colors in the Brazilian diamond weevil, Entimus imperialis,” Proceedings of SPIE,Vol. 8480, 848003, San Diego, CA, United States, August 2012.6. Wilts, B. D., K. Michielsen, J. Kuipers, H. De Raedt, and D. G. Stavenga, “Brilliant camou-flage: photonic crystals in the diamond weevil, Entimus imperialis,” P. Roy. Soc. B — Biol.Sci., Vol. 279, No. 1738, 2524–2530, 2012.7. Wilts, B. D., K. Michielsen, H. De Raedt, and D. G. Stavenga, “Hemispherical Brillouin zoneimaging of a diamond-type biological photonic crystal,” J. Roy. Soc. Interface, Vol. 9, No. 72,1609–1614, 2012.8. Wu, X., A. Erbe, D. Raabe, and H.-O. Fabritius, “Extreme optical properties tuned throughphase subtitution in a structurally optimized biological photonic polycrystal,” Adv. Funct.Mater., 1–6, 2013.9. Van Opdenbosch, D., M. Johannes, X. Wu, H.-O. Fabritius, and C. Zollfrank, “Fabricationof three-dimensional photonic crystals with tunable photonic properties by biotemplating,”Photonics Nanostruct., Vol. 10, No. 4, 516–522, 2012.10. Sollas, I. B. J., “On the identification of chitin by its physical constants,” P. Roy. Soc. B —Biol. Sci., Vol. 79, No. 354, 474–481, 1907.11. Rassart, M., J.-F. Colomer, T. Tabarrant, and J.-P. Vigneron, “Diffractive hygrochromic effectin the cuticle of the hercules beetle Dynastes hercules,” New J. Phys., Vol. 10, No. 033014,1–14, 2008.12. Welch, V. L., V. Lousse, O. Deparis, A. R. Parker, and J.-P. Vigneron, “Orange reflectionfrom a three-dimensional photonic crystal in the scales of the weevil Pachyrrhynchus congestuspavonius (Curculionidae),” Phys. Rev. E , Vol. 75, No. 4, 041919-1–9, 2007.

Layer Homogenization Method for Modeling Additive Color Effect in Natural Photonic Polycrystals --- The Case of Entimus imperialis Weevil

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Layer Homogenization Method for Modeling Additive Color Effect in Natural Photonic Polycrystals --- The Case of Entimus imperialis Weevil

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