## Abstract

Since one hundred years it is known that some scarab beetles reflect elliptically and
near-circular polarized light as demonstrated by Michelson for the beetle *Chrysina
resplendens*. The handedness of the polarization is in a majority of cases left-handed but
also right-handed polarization has been found. In addition, brilliant colors with metallic shine are
observed. The polarization and color effects are generated in the beetle exoskeleton, the so-called
cuticle. The objective of this work is to demonstrate that structural parameters and materials
optical functions of these photonic structures can be extracted by advanced modeling of spectral
multi-angle Mueller-matrix data recorded from beetle cuticles. A dual-rotating compensator
ellipsometer is used to record normalized Mueller-matrix data in the spectral range 400 –
800 nm at angles of incidence in the range 25–75°. Analysis of data measured on the
scarab beetle *Cetonia aurata* are presented in detail. The model used in the
analysis mimics a chiral nanostructure and is based on a twisted layered structure. Given the
complexity of the nanostructure, an excellent fit between experimental and model data is achieved.
The obtained model parameters are the spectral variation of the refractive indices of the cuticle
layers and structural parameters of the chiral structure.

© 2013 OSA

## 1. Introduction

The astonishing colors of many beetles have fascinated people since the early times. Of
particular interest are beetles with so-called structural colors, i.e. when the reflection phenomena
are interference based. One example of a scarab beetle which exhibits beautiful colors, metallic
lustre and in addition very interesting polarization phenomena is shown in Fig. 1. It belongs to the subfamily Cetoniinae (Leach, 1815) in the family
Scarabaeidae and are named *Cetonia aurata* (Linnaeus, 1758). Analysis of structural
and optical properties of the exoskeleton of *C. aurata* using Mueller-matrix
ellipsometric (MME) data are the objectives of this report.

Structural colors of beetles are very stable and can last for hundreds of years. The oldest known beetle reflector is estimated to be around 50 million years [1]. Various reflection mechanisms have been identified. In the early times the explanation of these optical phenomena was a matter of controversy as most colors were considered to originate from dyes. However, their angular dependence and inspiration by Newton’s color theory initiated the discussion that the colors are interference-based as reviewed by Lord Rayleigh [2]. Recently Seago et al. [3] have classified them in three major types: multilayer reflectors; three-dimensional photonic crystals; and diffraction gratings. In this report we address the polarization properties of multilayer cuticle reflectors. Michelson did some early work on this topic more than 100 years ago [4] and showed that, e.g. the beetle *Plusiotis* (now *Chrysina*) *resplendens*, exhibits interesting polarization phenomena. In particular he demonstrated that the polarization can be near circular and left-handed. He also discussed that right-handed polarization could be observed. For some time it was considered that Michelson was wrong due to experimental errors and that all observed polarizations were left-handed [5]. However, later Caveney [6] proved that Michelson was correct by performing optical rotatory dispersion measurements on cuticles of scarab beetles whereby he observed both negative and positive rotations. More recent Goldstein measured normal incidence Mueller matrices on the same species as Michelson and found right-handed polarization [7]. Mueller-matrix ellipsometry by Hodgkinson et al. [8] and by our group [9] have further proven this. It should be mentioned that near-circular polarization effects in beetles are rather wide spread. Pye [10] has examined 19 000 species of scarab beetles and found the effect in nine subfamilies of Scarabaeidae but the number of subfamilies is under discussion [11].

The near-circular polarization in reflected light originates from a complex nanostructure in the exoskeleton of the beetles. It is believed that in many cases there is a twisted layered structure in the near-surface region of the cuticle. This type of structure is called a Bouligand structure [12] and can be imaged using electron microscopy techniques. However, to distinguish this structure from an achiral multilayer it is necessary to prepare an oblique cut during sample preparation. Examples of images of Bouligand structures are found in the literature, e.g. in the review of cuticle structures by Lenau and Barfoed [13]. There may be other possible structures causing the polarization phenomena. This includes molecular chirality like in liquid crystals and chiral photonic crystals as has been observed in beetles as well as in butterflies [3, 14, 15].

The determination of the refractive index *n* of the materials constituting cuticles in beetles or wing scales in butterflies is not a simple measurement problem. When using reflection-based methods, one normally has to deal with samples which are small, curved and inhomogeneous. In addition specimen to specimen variations can be large. The constituents of cuticles are chitin and proteins and are expected to be dielectric with a refractive index typical for organic polymers. Values in the range 1.4 to 1.8 in visible spectral region can be anticipated. A wavelength dispersion with an increase of the index at shorter wavelengths is also expected. Thin films of cuticle constituents like chitin can be prepared and their refractive indices can be measured accurately [16]. However, the relevance of these data for real natural structure may be questioned due to differences in density and crystallinity and presence of other cuticle constituents. Chitin also forms crystals which in turn can self-assemble in fibrils as reviewed by Lenau and Barfoed [13]. These crystals and fibrils are birefringent with uniaxial or biaxial refractive indices and complex multilayer structures may arise such as those studied in this investigation. Cuticle structures are rarely homogeneous and density deficiencies may reduce the refractive index. Inhomogeneities may also scatter light, which in a specular reflection experiment is observed as an effective extinction coefficient *k*. The optical properties of the cuticle constituents is therefore described by a complex-valued refractive index *N* = *n* + *ik*.

Some early determinations of *n* and *k* of reflectors on insects using single-wavelength ellipsometry were performed by Brink and Lee [17]. They found *N* = 1.55 + *i*0.02 and *N* = 1.54 + *i*0.56 at a wavelength of 633 nm and 488 nm, respectively, in the metallic-like yellow moth *Trichoplusia orichalcea*. Later, the same group determined the low and high index in a thin film multilayer stack in the moth *Chrysiridia croesus* which exhibits green, purple and orange-pink colors [18]. Using the same technique as in [17], they found indices of the order of 1.63 and 1.74 for the low- and high-index layers, respectively, and also concluded that these layers essentially were non-absorbing (*k* = 0). These data were used with neglected dispersion to simulate reflectance spectra. Berthier et al. [19] used reflectance data and effective medium theory to extract the anisotropic refractive index of wing scales of the butterfly *Morpho menelaus*. These results were then used to deduce the anisotropic (uniaxial) index of the cuticular material in the spectral range 350–800 nm. However, these indices exhibit anomalous effects, i.e. they decrease towards shorter wavelengths and these findings should be further examined.

Also for beetles, there are very few reliable determinations of cuticle refractive indices. Among
the first data in the literature appear to be those determined on the layered system in the corneas
of beetles [20]. An average index
*n* = 1.548 was determined from which indices of alternate layers in a
multilayered system were determined. Several other approaches to determine the index of beetle
cuticles have been presented as reviewed by Noyes et al. [21]. They used reflectance data at several angles of incidence and polarizations and
found *N* = 1.68+*i*0.03 and *N*
= 1.55 + *i*0.14 for the two layers in multilayer stacks in
*Chrysochroa rajah*. This work represent a considerable progress as the investigators
directly address the indices of the two layers instead of first determining an average index.
However, dispersion is neglected as well as anisotropic effects. Recently Yoshioka and Kinoshita
[22] demonstrated that it is possible to
perform direct determination of individual layer indices in the beetle *Chrysochroa
fulgidissima* using a microspectrophotometer on a cross section of the beetle cuticle. They
found that *n* varies from 1.65 to 1.80 when the wavelength changes from 750 nm to
400 nm for the high-index layer and from 1.55 to 1.60 for the low-index layer. The
*k*-value is 0.1 or lower for the high-index layer and very small for the low-index
layer. So far, structural and optical analysis of beetle cuticles mainly relies on indirect
approaches like simulations and qualitative comparisons and presumes structural dimensions
determined from microscopy images.

In this report we have the objective to parameterize the structure of beetle cuticles using spectral Mueller-matrix ellipsometry data recorded at multiple angles of incidence. We apply this methodology to the scarab beetle *C. aurata*. The outcome of the analysis is both spectral refractive indices of the cuticle materials and details about its nanostructure in terms of layer thicknesses and pitch of the chiral structure. This approach differs from previous investigations which present simulations to mimic the reflection properties of cuticles. Here we perform direct regression analysis using experimental data.

## 2. Experimental

Specimens of *C. aurata* were collected locally. *Cetonia aurata* exhibits a variation in color from specimen to specimen. Most of them are green-colored, but some are red and blue-colored specimens can also be found. Several beetles were studied but in this investigation data are presented only from a green-colored specimen shown in Fig. 1. To facilitate comparison with cuticle polarizing features presented in an earlier report [9], we have chosen to use data from the same specimen as in [9].

Cross sections for scanning electron microscopy (SEM) were prepared using a four-step technique including fixation, dehydration, critical point drying and finally cutting with an ultra-microtome. The sample was fixated in 2% glutaraldehyde at 10°C during 24 h, rinsed in buffer and dehydrated using increasing fractions of ethanol and finally dried in a critical point dryer (Polaron E3000). The sample was then embedded in plastic and cut at an angle of 45° using an ultramicrotome. The fixation and drying steps are important to preserve the native microstructure in vacuum. The cross-sections were examined in a Leo 1550 FEG SEM (Gemini, Carl Zeiss, Germany) operating in secondary electron mode and at an accelerating voltage of 4 kV. The sample was covered with a 50 nm thick platinum layer to avoid charging effects.

Measurements of normalized Mueller matrices were performed with a dual-rotating compensator ellipsometer (RC2, J. A. Woollam Co., Inc.) in the spectral range 245 – 1700 nm at incidence angles *θ* in the range 25°–75°. Only data in the spectral range 400 – 800 nm are reported here. With focusing lenses (standard long-focus optics, J. A. Woollam Co., Inc.) the spot size was reduced to an ellipse of size 50×50sec*θ μ*m. The errors in ellipticity introduced by the focusing optics are determined in a calibration procedure and corrections are automatically done in the data acquisition routines. Simple control measurements verify that residual errors after correction can be neglected. With the long-focus lenses used, the beam spread is around 2°. A beam spread introduces depolarization but due to its symmetry around the nominal angle of incidence, the systematic errors average out to first approximation. Simulations shows that a beam spread up to 5° has a very small influence on the analysis. A motorized xy-stage and a camera allowed positioning of the beam with *μ*m resolution to a position free from surface defects. An image of a scutellum is shown in Fig. 1 where the illuminated area, i.e. the measurement spot, can be seen due to some scattered light. Analysis was performed with the software CompleteEASE (J. A. Woollam Co., Inc.).

To describe the optical response we use the Stokes-Mueller formalism in which a sample is represented with a 4×4 Mueller matrix **M** with elements *m _{ij}*(

*i*,

*j*= 1..4) which modifies the Stokes vector

**S**

*= [*

_{i}*S*

_{i0},

*S*

_{i1},

*S*

_{i2},

*S*

_{i3}]

^{T}(T indicates transpose) of incident light to generate a Stokes vector

**S**

*= [*

_{o}*S*

_{o0},

*S*

_{o1},

*S*

_{o2},

*S*

_{o3}]

^{T}of emerging light according to

*m*

_{11}= 1) and normalized Stokes vectors for the incoming light (

*S*

_{i0}= 1) as seen in Eq. (2).

From a Stokes vector **S** = [*S*_{0}, *S*_{1}, *S*_{2}, *S*_{3}]^{T} we can calculate the degree of polarization *P* of light from
$P=\sqrt{{S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}}/{S}_{0}$. In particular, for normalized and unpolarized incident light **S _{i}** = [1, 0, 0, 0]

^{T}, we find from

**S**in Eq. (2) that the degree of polarization

_{o}*P*for the reflected light is

*m*carry information about the polarizing properties of the cuticle as recently presented for several beetles [7, 8, 9]. Here we make further use of

_{ij}**M**in regression analysis to model the cuticle structure with objective to extract parameters like the refractive indices of cuticle materials, layer thicknesses, chirality parameters, etc. The Levenberg-Marquardt algorithm is used to minimize the mean squared error

*L*=

*L*is the number of Mueller matrices, i.e. number wavelengths

_{λ}L_{θ}*L*multiplied by number of angles of incidence

_{λ}*L*,

_{θ}*M*is the number of fit parameters in the parameter vector

**x**and ${m}_{ij,l}^{\mathit{exp}}$ and ${m}_{ij,l}^{\mathit{mod}}$ are experimental and model calculated Mueller-matrix elements, respectively. The fitting algorithm also delivers 90% confidence limits for the fit parameters. The confidence limits are defined as $1.65\sqrt{{C}_{ii}}\sqrt{MSE}$, where

*C*is the value of covariance matrix for the

_{ii}*i*fit parameter. If these limits become to large, there are parameter correlation and some parameters have to be assumed or coupled to other parameters as discussed below.

^{th}## 3. Model considerations

The SEM image in Fig. 2 shows a *C. aurata*
cuticle in cross section. The beetle cuticle has an outer epicuticle mainly composed of wax
[13]. It is multilayered and normally has a
thickness of a micrometer or smaller. The epicuticle can be seen as the top layer in Fig. 2 and at this location it has a thickness of less than 0.5
*μ*m for this specimen. Under the epicuticle, the thicker exocuticle is found
which is the color-generating part of the exoskeleton. The bottom (inner) part is the endocuticle
which is more soft. It can further be seen that the exocuticle has a layered structure but the
sublayer thickness varies throughout the exocuticle. The exocuticle is around 6–8
*μ*m thick if the effect of an oblique cut is taken into account.

To analyze the MME data, an optical model as shown in Fig. 3 was developed. We use a Cartesian *xyz*-coordinate system with the *x*-axis along the interface and lying in the plane of incidence, the *y*-axis in the surface plane and perpendicular to the plane of incidence and the *z*-axis perpendicular to the surface. The top layer, the epicuticle, is modeled as a uniaxial layer with its optical axis in the *z*-direction and with a Cauchy dispersion
${n}_{z}^{\mathit{epi}}={A}_{z}^{\mathit{epi}}+{B}^{\mathit{epi}}/{\lambda}^{2}$ and
${n}_{xy}^{\mathit{epi}}={A}_{xy}^{\mathit{epi}}+{B}^{\mathit{epi}}/{\lambda}^{2}$, where *λ* is the wavelength. Notice that we assume *B ^{epi}* to be the same for the

*z*- and

*xy*-directions due to lower sensitivity for the out-of-plane index leading to correlation between the

*B*-terms in ${n}_{z}^{\mathit{epi}}$ and ${n}_{xy}^{\mathit{epi}}$ if the

*B*-terms are decoupled. Thus ${A}_{z}^{\mathit{epi}}$, ${A}_{xy}^{\mathit{epi}}$ and

*B*are fit parameters. To account for scattering losses a small isotropic imaginary part

^{epi}*k*of the refractive index of the epicuticle is included in the model. An Urbach-tail model was employed with ${k}^{\mathit{epi}}={A}_{U}{e}^{{B}_{U}h{c}_{0}\left({\lambda}^{-1}-{\lambda}_{0}^{-1}\right)}$, where

^{epi}*A*and

_{U}*B*are fit parameters,

_{U}*h*Plancks constant,

*c*

_{0}vacuum speed of light and

*λ*

_{0}a band edge parameter set to 400 nm. In addition the epicuticle thickness

*d*is a fit parameter.

_{epi}The exocuticle is modeled as a twisted layered structure with pitch Λ. Our SEM images do not reveal any Bouligand structure so it is not proven that *C. aurata* has such a structure. However, our model with a twisted layered structure has general applicability including Bouligand structures. The layered structure is divided in a sufficiently large number of biaxial sublayers. These sublayers are assigned three wavelength depending refractive indices
${n}_{\alpha}^{\mathit{exo}}$,
${n}_{\beta}^{\mathit{exo}}$ and *n _{z}* which are assumed to be the same for all sublayers. Each sublayer is rotated an angle Δ

*ϕ*with respect to its adjacent sublayer. For the orthogonal in-plane (

*α*,

*β*) optical axes, the azimuths in each consecutive sublayer vary linearly from the bottom to the top of the exocuticle, whereas the axis with index ${n}_{z}^{\mathit{exo}}$ always is in the

*z*-direction.

Cauchy dispersions are also used for the sublayers in the exocuticle and
${n}_{\alpha}^{\mathit{exo}}={A}_{\alpha}^{\mathit{exo}}+{B}^{\mathit{exo}}/{\lambda}^{2}$,
${n}_{\beta}^{\mathit{exo}}={A}_{\beta}^{\mathit{exo}}+{B}^{\mathit{exo}}/{\lambda}^{2}$ and
${n}_{z}={A}_{z}^{\mathit{exo}}+{B}^{\mathit{exo}}/{\lambda}^{2}$. Similar to the epicuticle we assume *B ^{exo}* to be equal for the three directions. Four parameters are thus introduced:
${A}_{\alpha}^{\mathit{exo}}$,

*B*, ${A}_{\beta}^{\mathit{exo}}$ and ${A}_{z}^{\mathit{exo}}$. Small wavelength-independent imaginary parts

^{exo}*k*=

^{exo}*k*

_{1}=

*k*

_{2}=

*k*of the exocuticle refractive index are introduced to account for scattering/absorption in the exocuticle. There is not sufficient sensitivity to model a wavelength variation in

_{z}*k*.

^{exo}An effect of introducing a small value on *k ^{exo}* is that the chiral structure effectively becomes semiinfinite and the exocuticle thickness is arbitrarily set to

*d*= 8

_{exo}*μ*m and the number of turns

*T*and its distribution Δ

*T*(smearing) are introduced as fit parameters. This smearing is necessary to include to take into account lateral or in-depth variations in pitch. The distribution used is rectangular and calculations for eight MME data sets for

*T*between

*T*− Δ

*T*and

*T*+ Δ

*T*are averaged in the regression procedure. The pitch Λ and its distribution ΔΛ are then determined from

*s*= 360 birefringent sublayers is used to represent a continuous variation. The rotation angle between each sublayer will then be Δ

*ϕ*= 360

*T/s*[°]. Using more than 360 sublayers does not improve the model fit but unnecessary increases computation time. If too few layers are used, a dependence on the number of sublayers chosen is observed. Due to the near-opaque exocuticle, the value of refractive index of the endocuticle is of minor importance. It is set to 1.5 + 0.01/

*λ*

^{2}+

*i*0.01 as an approximation for this type of organic material. In total, the fit parameter vector for the regression is

**x**= (

*d*, ${A}_{z}^{\mathit{epi}}$, ${A}_{xy}^{\mathit{epi}}$

_{epi}*, B*,

^{epi}*A*,

_{U}*B*, ${A}_{\alpha}^{\mathit{exo}}$, ${A}_{\beta}^{\mathit{exo}}$, ${A}_{z}^{\mathit{exo}}$,

_{U}*B*,

^{exo}*k*,

^{exo}*T*, Δ

*T*).

A representative model should provide a good fit to data at all wavelengths and angles of incidence. However, the optical response due to the chiral structure is more pronounced at smaller *θ* and a small *θ* also implies that fewer surface inhomogeneities are present within the measured area. We therefore only use data for *θ* ≤ 60° in the regression analysis.

## 4. Results

A representative experimental result in terms of a contour plot of spectral Mueller-matrix ellipsometry (MME) data measured on a green-colored *C. aurata* is presented in Fig. 4. Figure 5 shows measured Mueller-matrix spectra at *θ* = 25°, 40° and 60° in the spectral range 400 – 800 nm as a subset from the complete data set in Fig. 4. The blue-shift with increasing angle is clearly seen and also that the spectral features are smaller for larger *θ*. Several of the off-diagonal elements, *m*_{13}, *m*_{23}, *m*_{24}, *m*_{31}, *m*_{32}, *m*_{34}, *m*_{42} and *m*_{43}, are small but have some features in the green-blue spectral region. The noise level in the data is of the order of 0.01. We thus conclude that these features stem from the sample which is further supported by the more or less perfect symmetries *m*_{14} = *m*_{41}, *m*_{13} = −*m*_{31}, *m*_{23} = −*m*_{32}, *m*_{24} = *m*_{42} and *m*_{34} = −*m*_{43}. We also observe that *m*_{31} = *m*_{43} and *m*_{13} = *m*_{34}.

Of interest is also to determine the degree of polarization of the reflected light under unpolarized illumination as derived from Eq. (3) and shown in Fig. 6. Below and above the spectral region of color generation, *P* is more or less independent on *λ* and is of the order of 0.25 for *θ* = 25° and 0.55 for *θ* = 75°. Figure 6 also shows *P* versus *θ* at *λ* = 400 nm and we see that *P* has a maximum around *θ* = 55°. Below 450 nm and above 650 nm, *m*_{13}, *m*_{14}, *m*_{23}, *m*_{24}, *m*_{31}, *m*_{32}, *m*_{41} and *m*_{42} are close to zero and the remaining elements fulfill the symmetry criteria [23] for an ideal isotropic sample described by the parameters *N* = −*m*_{12} = −*m*_{21} = cos2Ψ, *S* = *m*_{34} = −*m*_{43} = sin Ψ sinΔ and *C* = *m*_{33} = *m*_{44} = sin Ψ cosΔ where Ψ and Δ are the ellipsometric parameters. As *N* = cos2Ψ ≠ 1 we learn that Ψ has a finite value which together with the observation that *S* ≈ 0 implies that Δ is small which is typical for a low-absorbing dielectric material. If Ψ and Δ are transformed to the pseudo-refractive index <*N*> [9, 23] we find a value of its real part <*n*> in the range 1.42–1.47 depending on wavelength and specimen. In addition a small imaginary part <*k*> is found to be 0.03 or smaller. We conclude that the exocuticle of *C. aurata* from an optical point of view mimics a near-dielectric mirror except in the spectral regions corresponding to the observed colors, i.e. 500–600 nm for small *θ* and 450–500 nm for large *θ*. The maximum in *P* seen around 55° in Fig. 6 is then simply the polarizing angle for a near-dielectric material. We may compare with a pseudo-Brewster angle *θ _{B}* = arctan <

*n*> which for <

*n*>= 1.44 also is around 55°.

The objective is to extract structural and optical parameters from experimental data using the
optical model in Fig. 3. The fit to the experimental data is
excellent as seen in Fig. 5. For the structural parameters
the regression analysis results in *d _{epi}* = 544 ± 2 nm,

*T*= 21.13 ± 0.01 and Δ

*T*= 0.864 ± 0.002 and we find from Eqs. (5) a pitch Λ = 379 nm and its distribution ΔΛ = 15.5 nm. Table 1 summarizes the parameter values of the refractive indices. The refractive indices corresponding to these parameters for the uniaxial epicuticle and the biaxial sublayers constituting the exocuticle are shown in Fig. 7.

## 5. Discussion

At first glance, the results do not appear to be particularly novel as several investigators have addressed the relation between structure and reflection properties of beetle cuticles [8, 18, 19, 21, 24, 25, 26]. However, in all previously reports, forward calculations (simulations) are reported with objective to mimic experimental data demonstrating similarities and general features. In many cases only reflectance spectra are recorded and often only at near-normal incidence. The approach here is more detailed as we perform Mueller-matrix spectroscopy, i.e. we include both polarization and reflectance, and in addition at multiple angles of incidence over a wide spectral range. Furthermore, which is a main progress, instead of simulations, we perform regression analysis to extract specific structural and optical details of the sample under study.

A twisted layered structure is assumed in the model. This approach is chosen as it has been found that this is the most common type of structure, which exhibits chirality in exoskeletons in beetles [13]. However, we do not know the exact number of lamellae or their thicknesses so in the implementation of the model we use a sufficiently large number (360) of sublayers to mimic a continuous rather than discrete structure. The SEM-images may give a hint on the lamellae thicknesses but the thickness varies from position to position on the cuticle and may not be representative for the actual spot of the ellipsometric measurements. Neville and Caveney [5] found that the twist angle Δ*ϕ* was 7.2° which also was used in simulations by Lowrey et al. [27]. In our case Δ*ϕ* is coupled to the number of turns *T* in the exocuticle and has a value of 20.7° and is not found to be critical. Increasing the number of sublayers from 360 to 1035 (Δ*ϕ* = 7.2°) does not affect the fit. We have chosen to use a sufficiently large number of sublayers to mimic a continuous model. This has the advantage that our approach also includes a model with a molecular-based chirality like in a liquid crystal [5]. As long as it is not proven that the *C. aurata* has a Bouligand structure it can not be excluded that the origin of the chirality is molecular.

A distribution in the numbers of turns, presented as a distribution in pitch according to Eq. (5), is included in the model. A rectangular distribution is assumed. In this way a lateral variation in pitch over the beam spot is taken into account. However, a more specific distribution could be used like a chirped structure [27], a double structure [6] or multiple structures [28] but none of these approaches improve the fit. We conclude that the origin of the distribution in the current data can not be resolved and we therefore apply a general distribution.

In some of the Mueller-matrix elements and also in *P* (Fig. 6), interference oscillations are observed. These are related to the total thickness of the exocuticle including the epicuticle. In some specimens these oscillations are more pronounced which indicate low absorption/scattering in the exocuticle. Other scarab beetles like *Chrysina argenteola* and *Anaplognatheus aureus* exhibit very clear oscillations [9]. In this work we have selected specimens with small oscillations because otherwise the complexity in modeling would increase and take focus from modeling the chiral structure of the exocuticle. The cuticle thickness *d _{exo}* can be estimated from the expression for wavelength separation Δ

*λ*of adjacent peaks at

*λ*

_{1}and

*λ*

_{2}in thin film interference spectra as given by

*n*is the average refractive index of the exocuticle and

_{av}*θ*

_{1}is the angle of refraction in the layer. If

*n*is estimated to 1.55 from Fig. 7(b) and Δ

_{av}*λ*to be 11 nm from

*m*

_{43}around

*λ*= 550 nm at

*θ*= 40° (

*θ*

_{1}= 24.5° from Snell’s law), we find

*d*≈ 10

_{exo}*μ*m from Eq. (6). Further discussion about and strategies for determination of cuticle thickness from oscillations in MME-data will be presented elsewhere.

Specimens of *C. aurata* with different colors including almost black, blue, green-blue, green and red are found. Also on a single beetle, the color may vary and most of the green-colored specimens are red-, copper- or bronze-colored on the abdominal side. The blue shift due to the layered structure makes a red-colored beetle look green and a green-colored beetle look blue at grazing incidence. Preliminary analysis of a red specimen shows that the model is applicable also for this color but we found that the distribution Δ*T* in number of turns is larger. Work is in progress to investigate the generality of the proposed model with respect to different colors of *C. aurata*. The variation in *N ^{epi}* and

*N*from specimen to specimen will also be addressed. Each beetle is unique and the refractive indices presented in Figs. 7 are indicative and we do not claim that they represent

^{exo}*the*refractive index for

*C. aurata*.

## 6. Concluding remarks

Mueller-matrix ellipsometry data reveal complex reflection properties of the exoskeletons of beetles and are very rich in information about cuticle nanostructure. An optical model with a chiral color- and polarization-generating structure with a uniaxial dielectric overlayer provides a good description of the Mueller-matrix elements measured on *C. aurata*. By using regression analysis, quantification of refractive indices, layer thickness, pitch and additional parameters is possible.

## Acknowledgments

Specimens of *C. aurata* were kindly provided by Jan Landin. Jeff Hale is
acknowledged for new implementations in CompleteEASE and Arturo Mendoza for valuable comments on the
manuscript. This work is supported by a grant from the Swedish Research
Council. Knut and Alice Wallenberg foundation is acknowledged for support to
instrumentation.

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