Graded pitch profile for the helicoidal broadband reflector and left-handed circularly polarizing cuticle of the scarab beetle Chrysina chrysargyrea

The cuticle of the beetle Chrysina chrysargyrea reflects left-handed polarized light in the broad spectral range from 340 to 1000 nm. Interference oscillations in the experimental Mueller-matrix spectroscopic ellipsometry data reveal that transparent materials comprise the cuticle. A spectral analysis of the interference oscillations makes evident that the pitch profile across the cuticle is graded. The graded pitch and effective refractive indices are determined through non-linear regression analysis of the experimental Mueller matrix by using a cuticle model based on twisted biaxial dielectric slices. Non-uniformity in cuticle thickness as well as in pitch profile near the cuticle surface account for depolarizance of the Mueller matrix. Transmission electron microscopy supports the reliability of the results.


Results and Discussion
Overview of experimental Mueller-matrix data. The polarization properties of light reflected from the cuticle of the beetle C. chrysargyrea for unpolarized incident light have been discussed before 26 but are here briefly recalled. Let us consider incident unpolarized light with Stokes vector [1, 0, 0, 0] T where T means transpose (see equation (8) in Materials and methods). The cuticle of the beetle is represented by its 4 × 4 normalized Mueller matrix (M) and the specularly reflected beam is given by the Stokes vector [1, m 21 , m 31 , m 41 ] T as obtained from equation (9) in Material and methods. Figure 2 shows elements m 21   light. The later result qualitatively explains the silvery appearance of the beetle in Fig. 1(b) and why the beetle looks dark when a right-handed polarizer is placed in front of the camera as seen in Fig. 1(d). The elements m 21 and m 31 are components related to linear polarization, and their oscillatory behaviour is of major importance as will be discussed in the next section. Furthermore, symmetries among the elements of M, proper of chiral systems 30 leave only nine independent elements as is shown in Supplementary Fig. S1. At normal incidence there are further relationships among the nine independent elements that still are fulfilled at low angles of incidence as shown in Supplementary Fig. S2 for θ = 20°. Those further relationships show that the spectral information contained in the oscillatory behaviour is equivalent among the elements and m 21 was chosen.
Effective penetration depth from interference fringes. The clear oscillations observed in the spectra indicate that the electromagnetic waves superpose with enough degree of coherency. That is, light being reflected at different interfaces in the multilayer structure interfere constructively or destructively at specific wavelengths. This interpretation leads us to hypothesize that transparent materials comprise the cuticle of C. chrysargyrea. If absorption was strong enough, the strength of oscillations would be smeared out or damped. Our first objective is to determine which type of information the interference oscillations contain. For this purpose, we follow the procedure previously applied to data from the cuticle of the beetle Cotinis mutabilis 31,32 . Maxima and minima appear in optical spectra of films at discrete wavelengths λ m where the phase factor equals an integer multiple (m) of π, that is, where d is the thickness of the film and n a is the refractive index of the ambient (air). Because the reciprocal relationship between wavelength and interference order in equation (1), a more convenient representation of data is shown in Fig. 2(c) where m 21 is plotted as function of photon energy E. According to equation (1), an integer can be associated to each maxima and minima. However, the actual value of m has an unknown integer offset. We therefore use a temporary index m starting with m = 1 at the first maximum observed at the low energy end of the m 21 spectrum as shown in the insert of Fig. 2(c). By determining the energy position of each maxima and minima, the spectral dependence of m as shown in Fig. 2(d) is found. A slope decreasing with photon energy is observed and attains an asymptotic behaviour at high photon energies. In Fig. 2(d) we observe changes in the slope of m at certain energies which allow identification of four spectral regions denoted I, II, III, and IV to be discussed below. The behaviour of m in Fig. 2(d) is different to the one found in data from the cuticle of the beetle C. mutabilis 31,32 which only showed discrete pitch regions.
The negative values of m 41 seen in Fig. 2(b) shows that C. chrysargyrea reflects left-handed polarized light at θ = 20° in the spectral range where interference oscillations appear in m 21 . These oscillations indicate that there is a fictitious resonance cavity corresponding to the penetration depth of the left-handed polarization. This is an attribute of a circular Bragg reflector which we will refer to as an attenuation length in the discussion below. Therefore, d in equation (1) can be interpreted as an effective penetration depth η and we set d = <η>. Refractive indices of transparent materials are real-valued and normally show wavelength dispersion. However, for simplicity we neglect any dispersion in n av in the preliminary analysis but will introduce it later. With this assumption and by using E m = 1240/λ m for E m in eV and λ m in nm, a linear relationship between m and E m is found from equation (1). By taking the photon energy derivative of m from equation (1)  where η is given in nm. The insert in Fig. 2(d) shows a linear fit (dm/dE m constant) at low photon energies (region IV) and considering n av = 1.6 as typical values for Chrysina beetles 9 , we obtain η = 12.9 μm from equation (2). The objective below is to refine the analysis and determine the spectral variation of η .
Spectral location of the minimum penetration depth and in-depth variation of pitch. Figure 3 shows η versus λ m in more detail and for the full spectral range as obtained from equation (2) by numerically taking the photon energy derivative of m in Fig. 2(d). A general increase with wavelength is observed but step-like changes in η can be identified at certain wavelengths and are marked with vertical dash-dot lines separating the four spectral regions denoted I, II, III, and IV introduced already in Fig. 2(d). On the other hand, for a left-handed Bouligand structure it is known that in a band of selective Bragg reflection, the wave vector component || K LH parallel to the helix axis of the selectively reflected left-handed mode becomes complex-valued and its imaginary part defines the characteristic field attenuation length according to η = || { } K 1/Im LH with the in-depth variation of the field proportional to e −ζ/η where ζ is the distance from the surface 7 . The wave vector || K RH for the right-handed mode is however real-valued and propagation of the right-handed component occurs without attenuation. Outside a band of selective Bragg reflection both right-and left-handed modes propagate without attenuation. In Fig. 3, the U-shaped curves correspond to η calculated for selected pitch values with n av = 1.6 and two values of Δn. These calculations were performed at θ = 20° within the two-wave approximation for light propagation in a semi-infinite chiral media at oblique incidence [33][34][35] . Previously, we have applied this approach to analyse data measured on the cuticle of the scarab beetle C. mutabilis 31,32 . In Fig. 3 it can be noticed that η min , the minimum value of η, increases with the pitch and that a larger in-plane birefringence broadens the selective Bragg reflection and decreases η min .
Clearly, in Fig. 3 the smaller values of η are found in region I and are coincident with η min for Λ I = 240 nm and Δn = 0.1. In region II both, left-and right-handed modes, propagate without attenuation through the Bouligand structure with pitch Λ I near the cuticle surface but the left-handed mode is selectively reflected from helicoidal structures with a continuously increasing pitch. Because region II is centred at wavelength 500 nm, the average pitch is Λ II = 320 nm as determined from λ θ = Λ n cos m av t , where θ t is the angle of wave propagation inside the helicoidal structure determined from Snell's law θ θ = n n sin s in a a v t 36 . Passing from region II to region III <η> steeply increases and electromagnetic waves of wavelength between 600 and 800 nm travel without attenuation about 7 μm through the cuticle. However, in region III selective reflection is produced by Bouligand structures with average pitch Λ III = 435 nm as calculated from λ III = 680 nm. The largest penetration depth is found in region IV centred at λ IV = 890 nm where selective reflection comes from helicoidal structures with average pitch Λ IV = 569 nm located more than 10 μm deep in the cuticle. The cuticle model in Fig. 3 is a schematic representation of the increasing pitch with depth in the cuticle deduced in this paragraph from the analysis of <η> versus λ m . In summary, light with wavelength 350 to 400 nm only penetrates roughly 1 μm at λ = 375 nm into the exocuticle. In the spectral region 400 to 600 nm, the top 2-3 μm of the cuticle is transparent, but the left-handed mode is then selectively reflected with a penetration depth of roughly 1.5 μm into region II at λ = 500 nm (the U-formed curve in Fig. 3). Similar phenomena occur in regions III and IV.
Mathematical representation of the graded helicoidal pitch profile. Further insight into the pitch profile in the cuticle is given in Fig. 4 which is Fig. 3 redrawn with the vertical and horizontal axes interchanged and rescaled. In Fig. 4

the horizontal axis is renamed to
where d is the outer exocuticle thickness and z is the distance from the bottom of the outer exocuticle as defined in Fig. 3. The wavelength axis in Fig. 3 is transformed to the Λ-axis in Fig. 4 according to In this way Fig. 4 illustrates pitch versus cuticle depth d-z. As can be noticed, Λ shows an increasing step-like dependence with depth which can be described with a logistic function given by, where ΔΛ j , η 0j , and γ j are, respectively, the strength, centre, and broadening of the steps between pitch Λ j-1 and Λ j . The graded pitch (dashed line) in Fig. 4 was calculated with equation (4) and parameters values as given in Supplementary Table S1. These values were obtained by trial and error, but they could also be obtained by fitting if more accuracy is needed. Three depth zones with nearly constant values of the pitch (Λ 1 , Λ 2 and Λ 3 ) are identified in Fig. 4 and a fourth zone is estimated (Λ 4 ). In each of these zones the orientation of nanofibrils comprising the Bouligand structure changes at a constant ratio (2π/Λ j ) with depth as illustrated in Fig. 1(a). On the other hand, in between the cuticle zones where Λ continuously increases (e.g. at the transition Λ → Λ 1 2 ), the orientation of nanofibrils changes in a decreasing ratio with depth. Apparently, the pitch profile in the cuticle of C. chrysargyrea is described by Fig. 4. However, as the interference observed in data of Fig. 2 does not originate from physical cavities but is due to penetration lengths of the left-handed mode, we need to consider effects of light propagation in samples of finite thickness and the finite bandwidth of selective Bragg reflection. By doing this, it is possible to determine the actual location of the steps between pitches.
Consider a zone where Λ j is nearly constant extending in the cuticle over a length L j and characterized by the corresponding η min, j . According to data in Figs 3 and 4 we have L j ∼ η min, j . Thus, incident light of wavelength in the band of selective Bragg reflection due to Λ j probes not only the full length L j , but also deeper zones of slightly larger pitch because the finite bandwidth of selective reflection. On the other hand, the continuous change of pitch in zones at the transition Λ → Λ 1 2 , in a first approximation can be linear with depth. The selective Bragg reflection of a chiral structure with a linear variation of the pitch as a function of sample thickness has been investigated by other authors 37 . In that work, numerical calculations were performed at normal incidence for a structure with an in-plane birefringence of Δn = 0.225 which is larger than what is used here. In the present work, consider a value of Λ at the transition Λ → Λ   (4) with the parameters in Supplementary  Table S1 whereas an assumed pitch profile calculated from equation (4)  light of wavelength fulfilling the condition λ θ = Λ n cos av t will penetrate further a distance l where the pitch has increased to the value Λ + ΔΛ. The value of l is limited by the overlap of bandwidths of selective Bragg reflection due to Λ and Λ + ΔΛ. For the first and second steps, we assume l to have a value corresponding to about twice the value of Λ because a sample of two periods in thickness is enough to produce (a broad and weak) selective Bragg reflection in a single period chiral structure of in-plane birefringence Δn = 0.13 as reported by other authors 38 . However, the third step at 12 μm was shifted by a smaller quantity because it is close to the estimated outer exocuticle thickness of 13.5 μm which leaves two to three times the pitch at this depth where Λ IV = 569 nm. Therefore, the assumed pitch profile was shifted as shown with the dash-dot line in Fig. 4 with the parameters given in parenthesis in Supplementary Table S1. In summary, the analysis of spectral positions of maxima and minima in the spectrum of the Mueller-matrix element m 21 has provided evidence for the pitch profile across the cuticle of C. chrysargyrea. In the next section, we use this result to perform a fit to the experimental Mueller matrix to determine structural parameters and optical functions in the cuticle more precisely.
Electromagnetic modelling of the graded pitch profile in the cuticle of C. chrysargyrea.
Modelling the Bouligand structure is accomplished by subdividing the cuticle in biaxial slices with equal thickness and with a step-wise rotation around the z-axis of the laboratory coordinate frame defined in Fig. 3. Each slice is characterized with effective refractive indices (n 1 , n 2 , n 3 ) referred to the principal axes, i.e. n 1 and n 2 are the in-plane refractive indices and n 3 is the index in the z-direction. Referring to the Bouligand structure of Fig. 1(a), n 1 and n 2 are located parallel and perpendicular to the fibrils and n 3 along the helical axis. The direction of n 1 -axis defines an orientation angle φ (in degrees) with respect to the x-axis and has a value φ 0 at z = 0. We assign a graded z-variation in φ z ( ) as described below. The cumulated number of periods, i.e. full 360° turns, is given by In the special case of a linear variation of φ with z, the pitch is given by Λ = z/N p (z), i.e. the inverse of the (constant) slope of N p (z). Generally, we have variation in pitch with z given by For electromagnetic modelling of the cuticle structure we consider a variation in φ(z) such that equation (5) produces a sigmoid-like variation with z at the transitions in pitch in accordance with the qualitative analysis in the previous section. We use where T is the number of turns for the single-pitch case, d is the cuticle thickness, a j , z 0j , and b j have, respectively, equivalent meaning to ΔΛ j , η 0j , and γ j in equation (4). The initial values of the parameters a j , b j , and z 0j , were those reproducing the dash-dot pitch profile in Fig. 4 and determined as described in Supplementary Fig. S3. First, the layer thickness was set to d = 13.5 μm which is approximately the limit value of η in Fig. 3. From the values η 0j given in parenthesis in Supplementary Table S1 we  Determining the cuticle structure of C. chrysargyrea by regression analysis with non-uniform exocuticle thickness. The model described in the previous section accounts for a non-depolarizing system.
However, Mueller matrices measured on the cuticle of beetles show depolarization. Indeed, attempts to fitting the experimental data with the ideal model were unsuccessful manifested as to large amplitude on the oscillations. This is evidencing the importance of considering deviations from an ideal system. Because non-coherent superposition of light reflected from different parts of the cuticle with different thickness produce depolarization, we consider non-uniformity in cuticle thickness. This type of non-ideal feature in the analysis is implemented in the commercial software. Forward calculations show that the major effect of a non-uniform thickness is a damping of the amplitude of the oscillations. A non-uniformity in thickness of 1.7% was found to produce oscillations with amplitude similar to those in the experimental data. Leaving the non-uniformity in thickness as a fitting parameter, the fitting was statistically equivalent. The regression analysis was performed in the same fit procedure using data from measurements at θ = 20 and 50°. Data from larger angles of incidence probe a larger area and the effects of non-uniformity are stronger and are not included. The starting value of the epicuticle thickness was d epi = 350 nm corresponding to values reported for other Chrysina beetles 20 , whereas the starting value of φ 0 was chosen to match the oscillations in the spectrum of m 21 . More details on the regression procedure are given in the Materials and Methods section. The experimental and best-fit Mueller matrices are in very good agreement as is shown in Supplementary Fig. S4.
To investigate how well the non-uniformity in thickness accounts for the deviation from an ideal non-depolarizing Mueller matrix, we calculated the depolarizance D of M, which is an average measure of the depolarization produced by a system for all incident pure states and is given by 39  where P Δ is the degree of polarimetric purity (also called depolarization index), T stands for transpose, and tr stands for trace. Figure 5 shows the depolarizance of the experimental and modelled Mueller matrices at θ = 20 and 50°. First, we observe that with a uniform thickness the depolarization is zero. For reference Mueller matrices for such ideal system are shown in Supplementary Fig. S5. However, by implementing a non-uniformity in thickness, the difference between the experimental and model-calculated depolarizance for λ > 550 nm is considerably reduced. However, a thickness non-uniformity does not explain the depolarizance of the measured M at shorter wavelengths, which corresponds to the same spectral range where the experimental and fitted data differ in amplitude as shown in the Supplementary Fig. S4. Below we investigate the source of depolarization at shorter wavelengths.

Non-uniformity in pitch near the surface of the cuticle of C. chrysargyrea.
To identify the source of depolarization in the wavelength range 245-550 nm, we first notice that this spectral range overlaps with regions I and II in Fig. 3 where the penetration depth is about 4 μm. Therefore, an additional non-ideal feature in the chiral structure producing depolarization below wavelength 550 nm should be introduced. This was accomplished by smearing the position of the step accounting for pitch change at z 03 . Smearing of z 03 means that the parameter is varied within values in a width Δz 03 . This variation introduces non-uniformity in pitch near the cuticle surface. Thus, the non-coherent superposition of light reflected from regions with different pitch values produces depolarization. The parameters of the epicuticle and anisotropic slabs were fixed to those determined before. We comment here that smearing of pitch was previously successfully applied in modelling the cuticle structure of the beetle C. aurata 29 . The best fit was obtained with Δz 03 = 0.25 μm. As shown in the Supplementary  Fig. S6, the description of the data in M below 550 nm was improved. The improvement of the model is confirmed by calculating the depolarizance. As seen in Fig. 6, the non-uniformity in pitch near the cuticle surface gives an excellent description of the experimental data.
Graded pitch profile in beetle cuticle of C. chrysargyrea. The optimized depth profile of pitch found by regression analysis is shown in Fig. 7(a). The epicuticle and outer exocuticle thicknesses are 361 nm and 13.6 μm, respectively, and the number of turns T = 23.6. Values of all fitted parameters and confidence limits are given in Supplementary Table S3 and the correlation matrix is shown in Supplementary Table S4. More than 90% of the correlation values are |c ij | < 0.6. As can be noticed in Fig. 7(a), the pitch ranges between 250 and 600 nm. This variation is in accordance with that reported for the beetle C. strasseni determined with TEM 20 . It is here also relevant to quote Neville, "…from a constructional viewpoint a broad band reflector demands less accuracy of layer spacing control than does a narrow bandwidth reflector…" 12 . Other authors reported a graded pitch ranging between 240 and 360 nm from the analysis of SEM images taken on the cuticle of silver-like C. chrysargyrea 18 . This variation of pitch is not consistent with the reflection of left-handed polarized light in the near infrared. For example, taking the largest pitch (360 nm) and 1.6 as an estimate of the refractive index used by the authors in ref. 18 . the wavelength of Bragg reflection is 576 nm. The change of pitch in the near surface region is shown in the insert in Fig. 7(a) with dashed lines. This type of non-uniformity was observed by other authors in TEM images of the cuticle of C. strasseni 20 . Quoting MacDonald et al., "…in C. strasseni the exocuticle extends to a depth of approximately 15.5 μm, and generally comprises 40 helicoidal pitches, although sometimes an additional helicoid or two are observed as the structure accounts for the curvature of the elytron…" 20 . Here, we have arrived at a similar result by modelling the cuticle Optical functions of beetle cuticle. The effective refractive indices determined from the regression analysis are shown in Fig. 7(b). They are in the range of previously reported data from other beetles 9,29 . At λ = 1000 nm the in-plane birefringence is maximum Δ = − = . n n n 0 16 1 2 and decreases for shorter wavelengths becoming zero at 320 nm. Some authors associate the relative large birefringence to the presence of uric acid 9,18 . In ref. 18 , the in-plane effective birefringence is determined as Δ = − Δ + Δ n f n f n (1 ) c u a where Δn ua = 0.31 and Δn c = 0.04 are the birefringence values of uric acid and chitin, respectively, and f = 0.59 is the volume fraction of uric acid. However, X-ray diffraction data and infrared spectroscopy as shown in Supplementary Figs S7 and S8, respectively, do not show presence of uric acid but further investigation is in progress.
Transmission electron microscopy. TEM was performed to corroborate the results obtained from the regression analysis of Mueller-matrix data. Figure 8 shows a TEM image taken on the cuticle of a specimen of C. chrysargyrea. For the specimen analysed, the epicuticle and outer exocuticle thicknesses are approximately 740 nm and 15.8 µm, respectively. Although the size of the area of the cuticle imaged (≈8 µm) is smaller than the dimension of the probe light beam (≈100 µm), non-uniformity in thickness is observed which is in agreement with the model accounting for depolarization. Clearly, a multilayer structure is found in the outer exocuticle as revealed by dark and light lamellae. In the image, the pitch corresponds to the distance between two dark (bright) lamellae. In Fig. 8, at positions 1 to 6 at depths of about 1.1, 4.1, 7.5, 9.8, 12.3, and 14.3 µm, the pitch was evaluated and found to be 318, 416, 516, 577, 600, and 605 nm, respectively. In summary, the regression analysis performed provides reliable details on the graded pitch profile in beetle cuticle.
We have determined microstructural parameters and optical functions producing the polarization and depolarization properties of the cuticle of C. chrysargyrea. For optical biomimetism appropriate selection of materials  and processing is required to obtain broadband reflectors 40 . The results of this work could serve as a guide to reproduce the outstanding optical performance of the cuticle of C. chrysargyrea. Furthermore, this could inspire the development of applications for security issues by encoding depolarization through precise control of the structure during materials processing.

Conclusions
The cuticle of the beetle C. chrysargyrea is a broadband reflector of left-handed polarized light. Interference oscillations in Mueller-matrix data contain information about the graded pitch profile across the cuticle. The pitch monotonically increases from the surface towards the inner exocuticle. The depolarizing experimental Mueller matrix was reproduced by modelling the cuticle as a stack of biaxial slabs with a step-wise rotation of the azimuth angle of the in-plane principal refractive indices and with non-uniformity of cuticle thickness. An additional non-uniformity in pitch in the near-surface region accounts for a second source of depolarization in the Bragg reflection at short wavelengths.

Materials and Methods
Mueller-matrix measurements. Mueller-matrix measurements were performed using a dual rotating compensator ellipsometer (RC2, J. A. Woollam Co., Inc.) that allows determination of the 15 elements of a normalized Mueller matrix. Focusing probes were used reducing the size of the beam to less than 100 µm. More details on the instrument are found in [23][24][25][26][27]29 . The specular measurements were carried out at angles of incidence (θ) (measured from the normal to the surface) from 20 to 75° in steps of 5° in the wavelength (λ) range of 245 to 1000 nm. The acquisition time was 30 s at each θ. The measurements were done on the elytra, which are the wing covers, of the scarab beetle Chrysina chrysargyrea (Salle, 1874) which is found in Costa Rica and Panama. The specimen studied was collected in Monteverdi, Costa Rica, in 2011 and kindly provided by Dr. Parrish Brady (University of Texas at Austin). Crab chitin and uric acid (Sigma Aldrich) were used as received.

Basics of Stokes-Mueller formalism.
This formalism provides a complete description of the polarization and depolarization properties of light-matter interaction 28,41 . Light beams are described by Stokes vectors with components, where I = I p + I s accounts for the total irradiance. Q = I p − I s and U = I +45° − I −45° are irradiances describing linear polarization. Here, p is parallel to and s is perpendicular to the plane of incidence and + 45° and −45° are measured from the plane of incidence. The fourth component V = I R − I L accounts for circular polarization where R and L stands for right-and left-handed, respectively. It holds that ≥ + + I Q U V Mueller matrices from samples reflecting completely polarized beams for any incident completely polarized beam are non-depolarizing (ideal). In contradistinction, if the reflected light is partially polarized for some incident polarization states, the Mueller matrix is depolarizing. In general, depolarization arises from the incoherent superposition of light beams. In this work, we use Mueller matrices normalized to M 11 (m ij = M ij /M 11 ) and incident Stokes vectors with I = 1.

Cuticle refractive indices.
Because electronic transitions in the UV range and molecular vibrations in the mid IR range are outside of the spectral range of measurements, only normal dispersion is expected. In particular, the effects of IR resonances can be neglected. The three indices n 1 , n 2 , and n 3 of the biaxial outer exocuticle may therefore be modelled with Cauchy expressions, where A α , B α and C α are fitting parameters and λ is expressed in μm. As a good initial choice, it is required that n 1 = n 3 >n 2 to account properly for the sign of m 31 which is noticed mostly at large angles of incidence. The epicuticle is assumed isotropic with refractive index λ λ = + + n A B C / / epi e pi epi 2 epi 4 and for the inner exocuticle we used the same refractive index dispersion as was done for the beetle Cetonia aurata 29 . Initially we set the same value for all B and C parameters with the exemption of C 2 which was set larger to account for a decreasing birefringence at shorter wavelengths. To avoid unphysical refractive indices, all Cauchy parameters were defined to be positive.
Electromagnetic modelling. In summary, the set of model parameters for the regression analysis is {X} = {T, φ 0 , d, d epi , A epi , B epi , C epi , A α , B α , C α , a j , b j , and z 0j }, i.e. parameters in equations (6) and (10) as well as epicuticle thickness (d epi ) and Cauchy coefficients of the refractive index n epi . With such large number of fitting parameters, one would assume that correlation effects become severe. However, the data base is large with full Mueller matrices recorded at multiple angles in a wide spectral range. Furthermore, the dispersion parameters are spectrally global, and the pitch parameters are independent among each transition as they are spectrally local. Optimization of model parameters was carried out in three steps. (i) We can only within certain limits guess values on the model parameters {φ 0 , d epi , A epi , B epi , C epi , A α , B α , C α } and this sub-set was fitted first using the values of parameters in the sub-set {T, d, a j , b j , z 0j } which were estimated from experimental data. (ii) A fitting was then performed on the sub-set {T, φ 0 , d, a j , b j , z 0j } keeping the values fixed of the parameters optimized in step (i). (iii) Finally, the full set {X} was optimized. It should be mentioned that in step (i) we obtained B 1 = B 3 = 0 and these parameters were not further varied in step (iii) where it was also found that B 2 = 0. Non-linear regression where performed by fitting parameters in the model representing the sample to minimize the difference between experimental and model-generated Mueller-matrix data using the Levenberg-Marquardt algorithm as described in more detail elsewhere 29 . The regression was performed with the CompleteEASE software (J. A. Woollam Co., Inc.).
Complementary characterization. X-ray diffraction data were acquired with a Rigaku/Dmax2100 equipment equipped with Cu radiation (K α1 = 1.5406 Å). Infrared spectra were measured with a Spectrum GX system/ Perkin Elmer using a resolution of 4 cm −1 and averaging over 64 scans. Spectra from the cuticle of C. Chrysargyrea and crab chitin were measured in attenuated total reflection mode and for uric acid the KBr pellet technique was employed. For transmission electron microscopy the sample was first fixed in 3% glutaraldehyde for 2 h, then rinsed in sodium cacodylate buffer, immersed in osmium tetroxide for 1 h, rinsed in deionized water, immersed in 1% uranyl acetate for 1 h, rinsed again in deionized water, followed by a dehydration series starting with 30% ethanol and ending with 100% ethanol, and embedding in a resin. The sample was imaged in JEOL 100 S TEM instrument.