Simulation of light scattering from exoskeletons of scarab beetles

An approach for simulation of light scattering from beetles exhibiting structural colors originating from periodic helicoidal structures is presented. Slight irregularities of the periodic structure in the exoskeleton of the beetles are considered as a major cause of light scattering. Two sources of scattering are here taken into account: surface roughness and volume non-uniformity. The Kirchhoff approximation is applied to simulate the effect of surface roughness. To describe volume non-uniformity, the whole structure is modeled as a set of domains distributed in space in different orientations. Each domain is modeled as an ideal uniformly twisted uniaxial medium and differs from each other by the pitch. Distributions of the domain parameters are assumed to be Gaussian. The analysis is performed using the Mueller matrix formalism which, in addition to spectral and spatial characteristics, also provides polarization properties of the scattered light. 2016 Optical Society of America OCIS codes: (160.1585) Chiral media; (050.7330) Volume gratings; (200.4860) Optical vector-matrix systems; (290.0290) Scattering; (310.5448) Polarization, other optical properties; (350.3618) Left-handed materials . References and links 1. A. A. Michelson, “On metallic colouring in birds and insects,” Phil. Mag. 21, 554-567 (1911). 2. C. W. Mason, “Structural colors in Insects. I,” J. Phys. Chem. 30, 383–395 (1926). 3. C. W. Mason, “Structural colors in Insects. II,” J. Phys. Chem. 31, 321–354 (1927). 4. T. F. Anderson, A. G. Richards, “An electron microscope study of some structural colors of insects,” J. Appl. Phys. 13, 748 -758 (1942). 5. H. Arwin, R. Magnusson, J. Landin, K. Järrendahl, “Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson,” Phil. Mag. 92 (12), 1583-1599 (2012). 6. V. Sharma, M. Crne, J. O. Park, M. Srinivasarao, “Structural origin of circularly polarized iridescence in jeweled beetles,” Science 325 (5939), 449-451 (2009). 7. S. A. Jewell, P. Vukusic, N. W. Roberts, “Circularly polarized colour reflection from helicoidal structures in the beetle Plusiotis boucardi,” New J. Phys. 9 (99), 1-10 (2007). 8. A. E. Seago, P. Brady, J.-P. Vigneron, T. D. Schultz, “Gold bugs and beyond: a review of iridescence and structural colour mechanisms in beetles (Coleoptera),” J. R. Soc. Interface, 6 (2), 165–184 (2009). 9. A. R. Parker, D. R. McKenzie, M. C. J. Large, “Multilayer reflectors in animals using green and gold beetles as contrasting examples,” J. Exp. Biol. 201 (9), 1307-1313 (1998). 10. H. Arwin, T. Berlind, B. Johs, K. Järrendahl, "Cuticle structure of the scarab beetle Cetonia aurata analyzed by regression analysis of Mueller-matrix ellipsometric data," Opt. Expr. 21, 22645-22656 (2013). 11. L. Fernandez del Río, H. Arwin, K. Järrendahl, “Polarizing properties and structural characteristics of the cuticle of the scarab Beetle Chrysina gloriosa,” Thin Solid Films. 571, 410-415 (2014). 12. C. Åkerlind, H. Arwin, T. Hallberg, J. Landin, J. Gustafsson, H. Kariis, K. Järrendahl, “Scattering and polarization properties of the scarab beetle Cyphochilus insulanus cuticle,” Appl. Opt. 54, 6037-6045 (2015). 13. D. G. Stavenga, B. D. Wilts, H. L. Leertouwer, T. Hariyama, “Polarized iridescence of the multilayered elytra of the Japanese jewel beetle, Chrysochroa fulgidissima,” Philos. Trans. R. Soc. B 366, 709–723 (2011). 14. V. Sharma, M. Crne, J. O. Park, M. Srinivasarao, “Bouligand Structures Underlie Circularly Polarized Iridescence of Scarab Beetles: A Closer View, ” Materials Today: Proceedings 1S, 161-171 (2014). 15. S. Yoshioka, S. Kinoshita, “Direct determination of the refractive index of natural multilayer systems,” Phys. Rev. E 83 051917 (2011). 16. O. Deparis, C. Vandenbem, M. Rassart, V. L. Welch, J.-P. Vigneron, “Color-selecting reflectors inspired from biological periodic multilayer structures,” Opt. Exp. 14, 3547-3554 (2006). 17. M. Saba, B.D. Wilts, J. Hielschera, G. E. Schroder-Turk, “Absence of circular polarisation in reflections of butterfly wing scales with chiral Gyroid structure,” Materials Today: Proceedings 1S, 193 – 208 ( 2014 ). 18. V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer, 1992). 19. W. D. St. John, W. J. Fritz, Z. J. Lu, D.-K. Yang, “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E 51, 1191–1198 (1995). 20. W. A. Goddard III, D. Brenner, S. E. Lyshevski, G. J. Iafrate, Handbook of Nanoscience, Engineering, and Technology (3ed.) (CRC Press, 2012). 21. M. F. Cohen, J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic Press, 1993). 22. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992). 23. H. Motulsky, Intuitive Biostatistics: A Nonmathematical Guide to Statistical Thinking (Oxford University, 2013). 24. T. G. Mackay, A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (1st ed.) (Singapore: World Scientific, 2010). 25. J. A. Ogilvy, “Wave scattering from rough surfaces,” Rep. Prog. Phys. 50, 1553-1608 (1987). 26. K. Sarabandi, T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profile,” IEEE Trans. Anten. Propag. 45 (9), 1419-1430 (1997). 27. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265-4274 (1996). 28. P. Yeh, Optical Waves in Layered Media (Wiley, 2005). 29. R. Hegedüs, G. Szél, G. Horváth, “Imaging polarimetry of the circularly polarizing cuticle of scarab beetles (Coleoptera: Rutelidae, Cetoniidae),” Vis. Resear. 46, 2786-2797 (2006). 30. I. Hodgkinson, S. Lowrey, L. Bourke, A. Parker, M. W. McCall, "Mueller-matrix characterization of beetle cuticle: polarized and unpolarized reflections from representative architectures," Appl. Opt. 49, 4558-4567 (2010). 31. G. Thullier, M. Herse, D. Labs, T. Foujols, W. Peetermans, D. Gillotay, P.C. Simon, H. Mandel, “The solar spectral irradiance from 200 to 2400 nm as measured by the solspec spectrometer from the atlas and eureka missions,” Sol. Phys. 214, 1–22 (2003). 32. Z. Montiel-González, G. Luna-Bárcenas, A. Mendoza-Galván, "Thermal behaviour of chitosan and chitin thin films studied by spectroscopic ellipsometry," phys. stat. sol. 5, 1434–1437 (2008). 33. L. De Silva, I. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, A. Mcnaughton, “Natural and Nanoengineered Chiral Reflectors: Structural Color of Manuka Beetles and Titania Coatings,” Electromagnetics, 25, 391–408 (2005).


Introduction
Iridescent colors with metallic shine observed in the reflection from beetles such as Chrysina resplendens or Chrysina gloriosa exhibit high visibility and contrast against the background in natural environments.Not surprisingly such beetles have for a long time been used as jewellery and decoration.The mechanisms of color formation in the exoskeleton (cuticle) of such beetles have attracted attention of many researchers in the past [1][2][3][4] and is still of large fundamental scientific interest [5][6][7][8][9][10][11][12][13][14][15][16][17][18].The natural photonic structures of these beetles, that have appeared as a product of a long evolutionary "optimization" process, today inspire engineers and scientists to develop new materials for decorative coatings, display applications, polarizing components to mention a few applications [16,[18][19][20][21].
In contrast to metal surfaces, distinguished by high reflectivity due to a large imaginary part of the refractive index, the relatively high reflectance of iridescent beetle cuticles originates from a complex periodic structure [1][2][3][4][5][6][7][8][9][10][11][13][14][15][16][17].The reflectance with metallic shine can be wideband or confined to a narrow spectral band resulting in a specific color.The reflection depends on the specifics of the structure geometry, illumination conditions and angle of observation.Such a phenomenon is known as Bragg reflection [18], and the corresponding colors observed in the reflected light are termed structural colors.
Besides brilliant metallic appearance, many beetles and particular scarab beetles may possess unusual polarizing properties.For some of them, the polarization of the reflected light is close to circular (usually left-handed) when a beetle is viewed in natural light [1,[5][6][7][8][9][10][11][12][13][14][15][16][17][18].This phenomenon appears to occur mainly in the family Scarabaeidae, for instance in the subfamilies Rutelinae, Scarabaeinae and Cetoniinae [8].Domination of near-circular polarization is an effect of a helicoidal arrangement in the structure interacting with the light.The handedness of the heliocoid coincides with the handedness of the reflected near-circularly polarized light.A similar phenomenon is observed in cholesteric liquid crystals (ChLC).Therefore the mathematical models applied for describing optical features of scarab beetles and ChLC have much in common [6,9,18,19].However, a detailed study of spectral and polarization characteristics of some specimens of scarab beetles, such as Chrysina Argenteola [5] and Chrysina resplendens, shows that these characteristics are not exactly the same as those for typical ChLC's [18,19].For example, many scarab beetles exhibit more broadbanded reflectance spectra than ChLC's.Also, whereas cuticles of e.g.Chrysina resplendens appears to have a gold coating and Chrysina argenteola is often very silver-like it is not common to describe a ChLC as metallic-looking.In addition, a beetle can exhibit both leftand right-handed polarization effects but at different wavelengths and, what also is remarkably, at the same wavelength but at different angles of incidence [5].These examples indicate that structures found in beetle cuticles can be more complicated than those of ChLC's.
The details of the structure can be found by solving the inverse optical problem whereby parameters of the medium, e.g.structure parameters like layer thicknesses and pitch as well as the spectrally and spatially dependent dielectric tensor, can be determined.In the general case, this is a quite complicated numerical task with a large number of iterations.The direct problems, on the other hand, must be based on algorithms for solving Maxwell's equations in two dimensions (2D) or three dimensions (3D) which also requires much computing [20].Due to this difficulty, simplified approaches are applied [6-9, 18, 19].In the majority of cases they are restricted by a one-dimensional model utilizing a helicoidal structure with properties varied along a fixed direction (usually along the surface normal).Parameters of the model usually are defined by utilizing measurements of specularly reflected light [15][16][17][18].The drawback with such an approach is that the model cannot completely describe the light scattering that affects the spectral properties and polarization of the light because non-specular scattering is not included.
A way for overcoming this drawback is considered in the present paper where our objective is to explore an approach that can be applied for fast calculation of scattering from a natural helicoidal structure.This will enable simulation of optical properties of scarab beetles and will be useful for interpretation of experimental data.In the suggested approach, we will consider sources of the light scattering from a helicoidal structure for simulation of specific spectral, polarization and spatial optical characteristics of scarab beetles.The resulting light flux will consist of two components: the first component originates from surface scattering and the second component from volume scattering.The helicoidal structure parameters in our model are implemented with Gaussian distribution functions.By changing the values of the distributions, it is, in principle, possible to simulate any helicoidal medium that can produce scattered light described between a perfect specular reflector and an ideal diffuse scatterer.The third section of the paper presents an example on how the model developed can be applied to simulate scattering from a scarab beetle.

Reflection from a periodic structure
As discussed above, some scarab beetles reflect light with near-circular polarization and with a spectral shift of the reflection maximum towards shorter wavelengths ("blue" shift) at oblique incidence, which are features of reflection from a periodic structure without mirror symmetry.A simple example of such a structure is a twisted uniaxial anisotropic medium, in which the local optical axis of the medium rotates uniformally along one direction whereby its tip forms a helix.Let us choose a Cartesian coordinate system such that its z-axis coincides with the axis of the helix.The dielectric tensor ) ( ˆz ε of the medium can in this case be represented by a sum of two terms [18] as where II ε and ⊥ ε are principal dielectric functions of the uniaxial anisotropic medium in directions parallel and perpendicular, respectively, to its optical axis, p q / 2π = , and p is the pitch.The first term of the sum is position-independent and represents the dielectric tensor of an effective uniaxial medium averaged on a volume with dimensions much larger than the pitch p.This effective medium is uniaxial with its optical axis colinear with the axis of the helix formed by the twisted medium.The second term of the sum represents a traceless tensor describing rotation around the z-axis with the period p/2.The periodic rotation causes Bragg reflection for which the condition can be written in vector form as where i k (2), that the angle of incidence that is formed by i k  and q  is equal to the angle of reflection that is formed by r k  and q  .The wavelength of the maximum reflection in this case depends on both q  and the angle of incidence (or reflection).
If Maxwell's equations for propagation of an electromagnetic wave in a medium with the dielectric tensor in Eq. ( 1) is solved, eigenmodes with elliptical polarizations [18] are found.The ellipticity of the eigenmodes depends on the angle of incidence defined by the product q k r   ⋅ and the ratio q k i   (or q k r   ) .When i k  , r k  , and q  are colinear (i.e.normal incidence) and satisfy Eq. ( 2), the ellipticities of the eigenmodes become ±1 which correspond to circular polarizations.In the opposite case, when i k  and r k  are almost perpendicular to q  , the ellipticities tend to 0 (linear polarizations).The eigenmode with handedness of polarization coinciding with the handedness of the medium helix is forbidden, i.e. light with the corresponding polarization is reflected.
If the helix of a uniformly twisted anisotropic medium in air is perpendicular to the interface, , the spectral band of the Bragg reflection located between , where αo is the angle of incidence relative to the surface normal.A comparison with experimental results obtained for beetles, e.g. in [5], shows that the observed spectral band of the Bragg reflection is wider than what a uniformly twisted anisotropic medium is expected to have, if simulations are done with optical constants similar to those in the beetle cuticles.The discrepancy may be attributed to slight irregularities or imperfectness of the periodic structure as schematically shown in Fig. 1.In support of this assumption, it can be argued that cuticles scatter light rather than causing a specular reflection.Otherwise, the beetles would look like surfaces with flat air-dielectric interfaces, i.e. it would be possible to see images of objects in the reflected light.Such mirror-like reflections can occur in some beetles [17] but is normally not the case Also, if a beetle is illuminated with a collimated light beam of small diameter, the beam after reflection from the cuticle becomes divergent and a slight diffuse scattering, i.e. when the light energy associated with the Lambertian component of the scattered light is small in comparison with specular and haze components [21], appears.The regularities of the periodic structure can be described in terms of surface roughness, variations of the orientation of the helical axis and pitch distributions.These three sources of scattering contribute to the spatial, spectral and polarization characteristics of the scattered light and are described in detail below.

Scattering contributions from roughness
Scanning electron microscopy images of cross sections of layered structures in beetle cuticles demonstrate that their interfaces generally are rough [6,8,13].It is, in fact, possible to find all types of scattering phenomena from cuticles which exhibit very diffuse scattering with no specular reflection to cuticles with a clear specular reflection.In the model developed here, we assume that the resulting scattering caused by the rough interfaces in a cuticle can be represented by scattering from an effective rough interface between the ambient (air) and the exoskeleton.Also, we assume that the surface irregularities represent three types of scattering depending on their dimensions: much smaller than, comparable with and much larger than the wavelength of the light.The first type mainly affects the phase of the specular reflected light, whereas the other two cause non-specular scattering of the reflected and transmitted light as illustrated in Fig. 2. The second type, as well as multireflections between large surface segments, is associated mostly with diffuse scattering, and the influence of the third type can be considered as the sum of specular reflections from the segments comprising the rough surface.respectively, and IR and IL denoting irradiances for right-handed and left-handed circular polarizations, respectively.The reference direction for polarization is here chosen so that 0° lies in the plane of incidence.The transformation of a Stokes vector due to light reflection on a medium is described by S r =M r S i , where S r , S i are the Stokes vectors of the reflected and incident light and M r is a 4x4 matrix known as the Mueller matrix [22].The Stokes vector of the observed light consists of contributions from surface scattering, ) ,

S
, where λ is the wavelength and α is the angle of observation relative to the surface normal.

S
are due to diffusely scattered light.For sake of simplicity, we do not point out that the Stokes vectors are functions of λ.
To determinate the reflection contributions, we introduce a coordinate system with the z-axis coinciding with the cuticle normal.We consider flat segments having characteristic sizes that are much larger than λ reflecting the beam r spec S .The normal of each segment makes the angle γ from the z-axis.In this case γ can be expressed as 2 . Deviations of values characterizing a parameter of a physical system from the mean follow the Gaussian distribution [22].Therefore, it is reasonable to assume that parameters of a cuticles structure that is not perfectly uniform are also described by the Gaussian distribution.Similar assumption was applied for simulation scattering of light from cholesteric liquid crystals [19].Thus, statistics of the surface profile orientation or a fraction of the total area of flat surface segments orientated at γ can be described by the distribution where ξ is a fraction of the diffusely scattered light, )) , ( (

M
is the Mueller matrix of a surface segment reflecting the light, and αo+γ(α,αo) is the incident angle on the surface segment orientated at γ .The notation γ(α,αo) is used to underline that γ depends on α and αo.
Influence of surface non-uniformities, characteristic sizes of which are much less than the wavelength, can be modeled with an effective medium layer.The thickness of this layer is the same as the thickness of the roughness h1.Its refractive index ' n is calculated by using the Bruggeman effective medium approximation [23] for an aggregate structure containing 50% air and 50% of a material specified by the refractive index 3 , where and . Therefore, the reflection from the surface of segments can be considered as reflection from an ideal thin film of an effective medium, with air on one side and a medium with the refractive index n on the opposite side.

M
can be found from the expression coupling the Muller matrix with the Jones matrix J [13]   1 *) ( where ⊗ denotes the Kronecker product and Since the interface air-cuticle is not perfectly flat, there is light scattering caused by the surface roughness.Several models describing behaviour of electromagnetic waves after interaction with random rough surfaces have been developed [25,26].One of the most appropriate of those models is the Kirchhoff approximation the validity of which was tested in many works demonstrated good agreement between theory and experiment [25,26].This fact motivates us to apply the Kirchhoff approximation for simulation of scattering from an aircuticle interface.The Mueller matrix coupling )

S
can be expressed in analogy with the diffuse part of the scattered light in this approximation [25,26] , h2 is the height of the surface non-uniformities that are comparable with the wavelength, and denotes averaging over the area scattering the light [26].It is assumed that the diffuse scattering is unpolarized and can be described with an ideal depolarizer Muller matrix with m11 = 1 and mij = 0 for all other elements as shown in (7).From (4a), (5), and ( 7) follows ( ) By neglecting absorption in the exoskeleton, the Muller matrix coupling t S AC ,

S
with i S can be found by analogy with (5) as where αo, γ and β are related by the Snell's law ) sin( ) sin( )) ( (

M
is obtained from (6), where the diagonal elements Jp and Js of J are the transmission coefficients of the thin film of the effective medium for p and s polarizations, respectively, which are equal to

1+
. Here 12 t and 23 t are the Fresnel transmission coefficients (p-or s-polarization) for the air/film and the film/substrate interface, respectively, at the incident angle αo+γ.
Assuming that the directions and energy of the light rays diffusely scattered from both sides of the interface air-cuticle can be coupled by the Snell and Fresnel laws, from (7) it is possible to get the Mueller matrix for expressing ) (β ), ( ( exp 1 where From (4b), (9), and (11) we have ( )

Scattering contributions from variation in domain orientation
The resulting optical field scattered from an periodic structure having irregularities with smooth but curved interfaces as in Fig. 1 is here considered to be equivalent to scattering from an ensemble of uncorrelated domains, where each domain is an ideal uniformly twisted anisotropic medium specified by its own pitch, p, and the helix axis orientation described by the angle θ that is counted from the axis z as shown in Fig. 3.We assume that the pitches and helix orientations are Gaussian distributed according to

M
can be calculated according to (6) where the elements of J are defined from solutions of the 4x4 matrix formalism developed for layered media [27,28].By applying the 4x4 matrix formalism, we assume that the ambience of the domain has refractive index n and the incident angle with respect to the domain geometry is β+θ.
The Stokes vector of the light ray included in the scattered flux is ) , ( where the subscript "D,V" indicates volume scattering by the domains.

Overall scattering and reflectance
The light scattered from the exoskeleton is a mixture of surface scattering and volume scattering.The resulting Stokes vector is

S
relates to the light flux reflected from the ensemble of the domains after passing the cuticle-air interface (Fig. 3).The Mueller matrix, ) ,

M
is decomposed, are found by analogy with ( 9) and ( 11) as The resulting Muller matrix coupling the Stokes vector of the ray incident on the exoskeleton at αo with the Stokes vector of the ray scattered at α for fixed values of γ and θ is However, since γ and θ varies the final Muller matrix is where the relations between αo, γ, β, α, θ and ' β are described by ( 10), (12), and (15).

Out-of-plane scattering
The above expressions were derived for the special case when the light scattered from the exoskeleton is in the plane of incidence.In general, the incident and scattered rays are not coplanar.On the other hand, the expressions obtained in the previous subsections can be applied for a structure possessing axial symmetry, i.e. when the roughness and domains orientations are characterized as earlier by distributions of the functions containing only one spatial parameter -the angles γ or θ counted from z.The axial symmetry causes a symmetric light-scattering indicatrix with respect to z when the exoskeleton is illuminated along z.Such a situation is true for many scarab beetles.In order to describe out-of-plane scattering, it is necessary to derive general relations between the angles αo, γ, β, α, θ and ' β .
To analyze out-of-plane scattering, it is convenient to operate with vectors that coincide with directions of the light propagation and set orientations of the surfaces segments and domains.Let ( where z  is the basis vector of z.The unit vector d h  coinciding with the helical axis of the domain reflecting the light Finally, the angles θ, β and ' β are defined from the corresponding relations: There are two factors in the proposed model that describe scattering -the surface roughness and non-uniform orientations of the domains.By analyzing the effect of each of them, it is possible to come to the following conclusion.The surface roughness, as well as uniform diffuse illumination, decreases influence of the "blue" shift [18].In other words, surface roughness leads to reduction of the dependence of the spectrum of the observed light on directions of the beetle observation and the light illumination.In the extreme case, when the surface roughness dominates the spectrum of the scattered light is not a function of the mentioned directions and is defined as integration of all spectra for all directions of illuminations and observations.When the surface roughness is negligible small, the situation is similar to the scattering by domains in cholesteric liquid crystals [18,19].

Application of the model
By applying the formalism introduced in the previous section, we can calculate ) , . Therefore, by varying the parameters of the distributions, it is possible to generate results that mimic the observed optical characteristics of scarab beetles in case θ and γ are responsible for scattering.
There are many reports describing structural colors in beetles and butterflies (see e.g. a review by Seago et al. [8]).Related polarization phenomena are addressed using imaging polarimetry [29] as well as using more quantitative analytic methods like Mueller-matrix ellipsometry [5,10,30].In most cases specular reflection is studied and mostly at near-normal incidence.However, rather few reports are found on off-specular scattering [11][12][13] and we therefore have no relevant experimental data to compare our simulations with.However, as an illustration of the generality of our findings, we here briefly discuss some visual observations on a scarab beetle without claiming that we actually simulate scattering from this particular beetle in any detail.As an example of the model introduced in the previous sections, we simulate optical response similar to that of the cuticle of a specimen of the tribe Anomalini of the subfamily Rutelinae shown in Fig. 5.This specimen has a bright green color when illuminated with left-handed circular polarized light and is appears black in right-handed circular polarized light.The brightness of the scattered light decreases with increasing angle of observation counted from the direction of specular reflection.At angles of observation close to specular reflection it is possible to see a bright light spot for which the surface reflection dominates.The green color and the fact that the reflection depends on the type of the polarizer used for observation indicate that volume scattering dominates over surface scattering.However, the bright spot observed at specular reflection is a result of surface reflection including broad-band surface scattering.
In the simulations, the structure mimicking the beetle is illuminated with collimated and unpolarized light characterized by the Stokes vector , where I(λ) is the sun irradiance [31].Let the xz-plane in Fig. 4 be the plane of incidence and with the scattered light observed in the plane yz as described in section 2.4.The incident angle (αo) is set to 45 o , the observation angle (α) is varied from 0 to 90 o .In order to achieve green color, we assume that the principal refractive indices o n and ne are 1.5 and 1.52, respectively, the mean pitch, po, of the domains is 365 nm and 9 = Π nm.The surface distributions were set to Γ=5 o , σ=1, γo=0 o , and ξ=0.1.The parameters responsible for the scattering indicatrix was set to θo=0 o , and T=12 o .There are no experimental values available on these parameters, so the chosen parameter set is selected to illustrate similarities with visual observation.The obtained spectrum of the scattered light intensity in the yz-plane as a function of angle of observation is presented in Fig. 6 (a).

Fig. 2 .
Fig. 2. Scattering in reflection and refraction of light incident at a rough air-cuticle interface.It is convenient to apply the Stokes-Mueller formalism[22] for describing polarization characteristics of partially polarized light.In this formalism, the light is represented by four Stokes parameters arranged in form of the Stokes vector the standard deviation, γo is the mean.

r and 23 r
Jp and Js are the reflection coefficients for p and s polarizations, respectively, of the helicoidal structure with an effective thin film.Both Jp and Js are given by the expression are the Fresnel reflection coefficients (p-or s- polarisation) for the air/film and film/substrate interface, respectively, at the incident angle αo+γ .The phase δ is given by λ po and θo are means of the corresponding distributions, and Π and Τ are the standard deviations.

Fig. 3 .M
Fig. 3. Directions of the scattered light components in the considered model.The Muller matrix of the ensemble of the domains for incident light ray propagating in direction specified by β and light reflected in direction of 'β can be expressed as unit vectors coinciding with the directions of the incident and observed rays, respectively, as shown in Fig.4.Then the normal of the surface segment reflecting the light along α
y  are the basis vectors along the axis x and y.

..
The structure parameters describing scattering are assumed to be the orientation θ of the helix axis and the tilt γ of surface segments.The scattering is introduced in terms of their distribution functions ) the distribution parameters enables us to obtain a wide range of possible optical responses from inhomogeneous media.For example, perfect specular reflection, or the Dirac delta function (δ) of the light-scattering indicatrix, is obtained with The opposite case, Lambertian scattering is obtained when