Effective structural chirality of beetle cuticle determined from transmission Mueller matrices using the Tellegen constitutive relations

Several beetle species in the Scarabaeoidea superfamily re ﬂ ect left-handed polarized light due to a circular Bragg structure in their cuticle. The right-handed polarized light is transmitted. The objective here is to evaluate cuticle chiral properties in an e ﬀ ective medium approach using transmission Mueller matrices assuming the cuticle to be a bianisotropic continuum. Both di ﬀ erential decomposition and nonlinear regression were used in the spectral range of 500 – 1690 nm. The former method provides the sample cumulated birefringence and dichroic optical properties and is model-free but requires a homogeneous sample. The materials chirality is deduced from the circular birefringence and circular dichroic spectra obtained. The regression method requires dispersion models for the optical functions but can also be used in more complex structures including multilayered and graded media. It delivers the material properties in terms of model functions of materials ’ permittivity and chirality. The two methods show excellent agreement for the complex-valued chirality spectrum of the cuticle. ©


I. INTRODUCTION
Several beetle species in the Scarabaeoidea superfamily reflect left-handed polarized light due to a circular Bragg structure in their exoskeleton, also called cuticle. 1 The effect is illustrated for the scarab beetle Cetonia aurata (Linnaeus, 1758) in Fig. 1, where it is seen that the beetle appears nonreflecting (black) when viewed through a right-handed polarizer and preserves its color when viewed through a left-handed polarizer.The cuticle is composed of chitin and proteins 2 and is essentially nonabsorbing, and the righthanded polarized light is transmitted as illustrated schematically in Fig. 1.The circular Bragg structure is chiral and is also called a Bouligand structure and has been visualized in several beetles by electron microscopy. 3,46][7][8] A more advanced methodology, compared to reflectance measurements, is Mueller matrix ellipsometry, which has the advantage that it allows polarization and depolarization features to be quantified.Mueller matrix measurements on the chiral beetle structure were pioneered by Goldstein 9 and Hodgkinson et al. 10 Spectral reflection Mueller matrices can be recorded with high accuracy using commercial instruments and are very rich in information about complex electromagnetic structures.Calculations of polarization states including depolarization of reflected light for any incident state of polarization have been demonstrated for several beetles. 11,12y using electromagnetic modeling, structural details like layer thicknesses, pitch of Bragg structures, pitch distributions, and refractive indices of biaxial constituents can be extracted. 13,14A sum decomposition of a Mueller matrix provides a phenomenological description of beetle cuticle reflection in terms of optical elements like retarders and polarizers. 15Transmission Mueller matrices allow quantification of chirality in terms of structural birefringence and dichroism. 16n this work, the objective is to evaluate chiral properties of the beetle cuticle as a bianisotropic continuum in a model-free effective medium approach based on differential decomposition of transmission Mueller matrices and to compare with results from nonlinear regression of Mueller matrices using dispersion models.

II. THEORETICAL BACKGROUND A. Tellegen constitutive relations
The beetle cuticle studied here exhibits chirality.The origin is not molecular but is classified as structural and can be well represented with an electromagnetic model when analyzed using reflection Mueller matrices. 13Here, we employ an effective medium approach to quantify the effective chirality of the cuticle.The full constitutive relations are used, which in the so-called Tellegen representation (also called EH-representation) are where D, E, H, and B are the electric displacement field, electric field, magnetic field, and magnetic flux density, respectively, ε 0 and μ 0 are the vacuum permittivity and permeability, respectively, and c 0 is the speed of light.Material properties are given by the permittivity tensor ε, the permeability tensor μ, and the magnetoelectric tensors ξ and ζ.With a time dependence e Àiωt , it holds that ξ ¼ χ þ iκ, where χ is the nonreciprocity tensor and κ is the chirality tensor.In this case, the medium is reciprocal, 15 which implies 17 χ ¼ 0 and ζ ¼ Àξ T , where T indicates the transpose.We, thus, have ξ ¼ Àζ T ¼ iκ.A dielectric circular Bragg structure probed with a plane wave along its helical axis can be modeled with ε = diag(ε x ,ε y ,ε z ) and Here, an xyz Cartesian coordinate system is used with the z axis along the helical axis.The permittivity elements are related to the complexvalued refractive indices by , where n j is the refractive index and k j is the extinction coefficient.At normal incidence, there is no sensitivity to the absolute values of the refractive indices n x and n y in the used methodology since the absolute phase of the light beam is not accessible.However, phase differences are measured in an ellipsometric measurement, and Δn = n y −n x is accessible.As we use irradiance normalized data in the analysis, there is no sensitivity to isotropic absorption either, but the difference Δk = k y −k x is accessible.Furthermore, there are no fields from the probe beam in the z-direction and ε z can be set arbitrarily.Here, we use ε z ¼ ε x .Similarly, there is no sensitivity to κ z , which also can be assumed arbitrarily.We choose to set κ z ¼ κ x , and κ can, therefore, be set to κ I with the complexvalued scalar κ = κ Re + iκ Im and the identity matrix I. Finally, we set μ = Ι.The relevant constitutive relations can then be written as or in a more compact form

B. Differential decomposition
All birefringent and dichroic properties of a sample are directly obtained by taking the logarithm L = lnM of a transmission Mueller matrix M provided that the sample is homogeneous along the beam path. 18The matrix L can be decomposed as L = L m + L u , where L u carries the depolarization properties and L m is given by where linear (L) and circular (C) birefringence (B) and dichroism (D) are defined by directions along (x/y) and ±45°from the reference coordinate system and for left and right (l/r) circular polarizations.Notice that the elements of L m are cumulated values along the beam path in the sample and, thus, are considered as sample properties as they depend on sample thickness.LB, LB 0 , and CB are given in units of radians, whereas LD, LD 0 , and CD are dimensionless.Different conventions are used for signs in L m and in the definition of its elements.Here, we use L m conventions as in Ref. 19 and conventions for its elements as above.
The relation between κ and CB and CD is given by which relates the effective materials parameter κ and the phenomenological sample parameters CB and CD.Equation ( 5) follows if the e Àiωt convention is used and if κ enters the wave vector q, where q ¼ 2π=λ ffiffiffiffiffi εμ p + κ À Á , with +(−) representing right(left)handed polarization.

III. EXPERIMENT A. Sample preparation
The specimen of the beetle C. aurata was collected locally.The outermost layer of the cuticle is a thin epicuticle consisting mainly of wax 2 and with a thickness around 500 nm in C. aurata. 13Below the epicuticle is the exocuticle that holds the helicoidal structure providing chirality, and that is the main structural element responsible for color and polarization features.Farthest in the cuticle, there is a thick and relatively soft endocuticle.The ellipsometric transmission measurements were performed on one of the elytra (cover wings) that was removed from a beetle and carefully scraped manually with a sharp knife on the inside to remove the endocuticle.In this way, the exocuticle could be exposed in an area of a few square millimeters.The epicuticle, which was left intact, has been found to be uniaxial 12 with its optic axis perpendicular to the surface plane, and thus, it will not affect polarization in normal incidence transmission measurements.

B. Mueller matrix ellipsometry
Transmission Mueller matrices were measured in the spectral range of 300-1690 nm at normal incidence with a dual rotating compensator ellipsometer from J. A. Woollam Co., Inc.However, for λ < 500 nm, the transmittance is too low for determination of a Mueller matrix of sufficient quality, and analysis was restricted to λ > 500 nm.A Mueller matrix is a 4 × 4 matrix providing a complete description of polarizing and depolarizing properties of a sample for light with any input polarization described by a so-called Stokes vector.The Mueller matrix elements M ij (i, j = 1…4) are normalized to the total reflectance, i.e., element M 11 , according to m ij = M ij /M 11 .Further details about the Mueller-Stokes formalism can be found elsewhere. 20ith focusing optics, the spot size was reduced to approximately 50 μm.For lenses with 28 mm focal length, the beam divergence is less than ±3.8°, which results in small z-components from the incident transverse field.The effect of this small component can be neglected.Measurements were performed on the elytron from a C. aurata beetle with results as shown in Fig. 2. The sample azimuth f with respect to the instrument xy frame of reference is unknown and fitted in the nonlinear regression.Optical activity is observed in the antidiagonal with a resonance around a wavelength of λ = 560 nm.However, the noise level is large for λ < 500 nm due to low transmittance, and modeling is restricted to λ > 500 nm.
Nonlinear regression of the data in Fig. 2 was performed with the WVASE software (J. A. Woollam Co., Inc.) using the Levenberg-Marquardt method. 21The results of a fit are values of the best-fit parameters and their 90% confidence limits.A structural model with a single homogenous layer with thickness d = 20 μm was assumed.In normal incidence transmission studies, optical properties perpendicular to the sample surface are not accessible as discussed above.The same holds for in-plane optical functions, and only their differences, i.e., birefringence and dichroism, are modeled in terms of Δn = n y −n x and Δk = k y −k x , respectively.In the analysis, Eq. ( 3) was used with Gaussian dispersion models for κ = κ Re + iκ Im , which is given by where A j , E 0j , and Γ j are amplitude, resonance energy, and broadening of resonance j, respectively; κ 0 is a constant; and KK stands for the Kramers-Kronig transform.Notice that the spectral dependences here are written versus photon energy E = hc 0 /λ (in units of eV), where h is Planck's constant.The linear birefringence Δn and linear dichroism Δk are modeled with a Cauchy dispersion and an Urbach tail, respectively, where A, B, C, A U , and B U are fitting parameters.The band edge C U is not fitted as it correlates with A U .

A. Nonlinear regression
Figure 3 shows the measured Mueller matrix from Fig. 2 in the spectral range used for nonlinear regression.Below 500 nm, the transmission is very low and systematic errors affect the fitting.Above 800 nm, no further information is found, and the data are omitted for clarity.The best-fit model-calculated data are also shown in Fig. 3.The best-fit parameters for the two Gaussian resonances are found in Table I.Moreover κ 0 = −9.2× 10 −5 ± 2 × 10 −6 .The Cauchy and Urbach parameters in Eq. ( 7) are A = 0.0012 ± 0.0001, B = −0.00017± 0.00002 μm 2 , C = −0.000028± 0.000004 μm 4 , A U = 0.00086 ± 0.00003, and B U = −0.00011± 0.00005 μm, and the fitted sample azimuth is f = 128 ± 1°.
For elements m 41 and m 14 , the fits are excellent.In general, the fits for the diagonals are good except for in the range of 500-550 nm for some elements.For the remaining elements, the fits show various degrees of deviation from the data, especially for shorter wavelengths and close to the Bragg resonance.

B. Differential decomposition
Figure 4 shows the parameters in the matrix L m calculated from the data in Fig. 2. The dominating features are the CB and CD elements.The circular Bragg resonance is observed with a peak in CD at 571 nm.The CD spectrum has a shoulder on the longwavelength side, indicating that the structure is more complex than a single helicoidal structure.The CB spectrum is Kramers-Kronig related to the CD spectrum, and together CB and CD are referred to as the Cotton effect. 22CB shows a positive background level as seen for short wavelengths and is referred to as a positive Cotton effect. 22he linear birefringence spectra (LB, LB 0 ) and dichroism spectra (LD, LD 0 ) are one order of magnitude smaller than CB and CD.LB and LD are close to zero, and both LB 0 and LD 0 have small but nonzero values indicating that the sample is in an orientation at which the optical axis is oriented at ±45°.This is consistent with the regression result f = 128°, which due to 180°symmetry is equivalent to −52°, i.e., 7°from −45°.The four elements LB, LD, LB , and LD 0 are nearly nondispersive except for features in the spectral range for the Bragg resonance.

C. Comparison between differential decomposition and nonlinear regression analysis
Figure 5 shows κ = κ Re + iκ Im in Eq. ( 6) according to the best-fit model data in Fig. 3 compared with κ = κ Re + iκ Im calculated from CB and CD from Fig. 4 using Eq. ( 5).
The agreement between the two methods is very good with some deviations in κ Re at short wavelengths.We attribute this to systematic errors due to low sample transmission for λ < 550 nm.The shoulder on the long-wavelength side of the Bragg resonance is well reproduced by the double resonance dispersion model used in the regression.

D. Discussion
The chirality spectra determined with the two methods are very close.The differential decomposition is only a transformation of primary data and all information is preserved, which is very advantageous as all details can be observed.In the present case, the cuticle is not homogeneous along the beam path, which implies that the cuticle effective anisotropic properties are obtained rather than the intrinsic properties of the cuticle materials.This is not a drawback as we compare with nonlinear regression in which a homogeneous effective medium is assumed.However, this is not a requirement in the regression approach, and more complex structures can be analyzed including graded or multilayered structures.For example, chirality of a buried layer may be extracted.The drawback with regression analysis is that if the model used is wrong or incomplete, false results may be obtained.
It may be of interest to determine the maximum specific rotation of the cuticle material.The specific rotation is given by 1=2jCBj=d.At λ = 560 nm, we find |CB| = 0.37 radians (21°) and with a thickness of 20 μm we have 525°/mm compared to 560°/mm earlier found at λ = 562 nm in C. aurata. 15n the related work, 16 we found a cuticle thickness of 20 μm that is used here as an estimate.The actual value is of minor importance as it cancels out.This is due to that it enters as a fixed parameter in the regression.An error in thickness will then cause an error in the model parameter κ.When κ is calculated from CB and CD, the thickness enters in Eq. ( 5) and the effect of an error is eliminated to first order.The effect will be that there is an

Resonance
A j (−) E 0j (eV) Γ j (eV) 1 0.00042 ± 0.00002 2.138 ± 0.003 0.197 ± 0.004 2 0.00081 ± 0.00002 2.173 ± 0.001 0.066 ± 0.002 uncertainty in the absolute values (scale uncertainty) in Fig. 5.The error is estimated to be <10%, but it is of minor importance as the objective with this communication is to compare determination of chirality using a model-free approach with the use of an electromagnetic model-based approach.
The cuticle materials are dielectric and nonabsorbing, but a cuticle is locally inhomogeneous and some scattering may occur leading to apparent absorption.However, due to the use of a normalized Mueller matrix, both electromagnetic modeling and differential decomposition will be unaffected by this.No isotropic absorption is, therefore, included in the electromagnetic modeling.However, the differential decomposition reveals some dichroism that is added with an Urbach tail dispersion to the electromagnetic model to improve the fit.This is not Kramer-Kronig consistent with the Cauchy dispersion for the refractive index.However, the physical origins of LB and LD are most probably different.
The structure is generally believed to be a twisted layered structure composed of biaxial layers arranged in multiple turns.With a pitch of around 400 nm 13 and a cuticle thickness of 20 μm, there are around 50 turns.If the number of half turns is an integer, the linear birefringence and dichroic effects would cancel out in an effective medium approach that is used here.The small but nonzero values of LB 0 and LD 0 are most likely due to that the number of half turns deviates from an integer resulting in net linear anisotropy.
The CD spectrum in Fig. 4 shows a shoulder on the longwavelength side of the Bragg resonance.This indicates that there are two helicoids with different pitches in the cuticle, which also is confirmed by the need of two Gaussian resonances in the regression analysis of the data.][25] From the present data, we cannot determine whether the two resonances stem from two lateral structures or two structures at different depths, i.e., if they are in parallel or in series.In electromagnetic modeling using reflection Mueller matrices, we found that a smearing of the pitch was needed, 13 but in the present case, two distinct resonances are more probable as smearing would broaden the resonance rather than result in a shoulder.
Small resonancelike features are observed in the primary data in Fig. 2 in many of the off-diagonal elements.These features propagate through the differential decomposition and can be seen in LB, LD, LB 0 , and LD 0 in Fig. 4.They occur in the same spectral region as the Bragg resonance.The chiral model with an addition of anisotropic dispersion and absorption can reasonably reproduce baseline effects in the linear properties but not the features related to the Bragg resonance and the small ripples observed in the experimental data in Fig. 3 as well as in the results in Fig. 4.Those features are due to that the cuticle is a multilayer consisting of lamellae, all composed of the same anisotropic material and rotated from lamella to lamella.Consequently, the light experiences small differences in the refractive index when travelling between lamellae, which creates interference effects within the cuticle resulting in the resonance feature and the spectroscopic ripples in the linear properties.The model based on a single layer cannot reproduce these features because of the absence of a stack of layers.Nonetheless, the experimental data can be successfully fitted with the simple chiral layer because the linear effects are much smaller than the circular effects.To further support this explanation of the origin of the features in the linear properties, a differential decomposition was performed on simulated Mueller matrix transmission data generated from a structural model developed for C. aurata. 13s seen in Fig. 6, the features in the linear properties appear at the Bragg resonance wavelength.The ripples are very weak in this simulation but can be seen on an expanded scale.

V. SUMMARY AND CONCLUSIONS
Nonlinear regression and differential decomposition are performed on the same transmission Mueller matrix from a beetle cuticle with objective to determine chirality of a circular Bragg structure.The cuticle medium was assumed to be a homogeneous bianisotropic medium.The two methods gave very similar results.Differential decomposition is model-free and all information is preserved, but the method requires in-depth homogeneity.Nonlinear regression requires dispersion models that may lead to incomplete modeling.However, it can be applied to inhomogeneous systems, e.g., graded layers and multilayered structures.

ACKNOWLEDGMENTS
This work was supported by Knut and Alice Wallenberg Foundation and Swedish Government Strategic Research Area in Materials Science on Advanced Functional Materials at Linköping University (Faculty Grant Nos.SFO-Mat-LiU and 2009-00971).Jan Landin is acknowledged for providing the beetle sample and Jens Birch for providing beetle images.The authors also acknowledge one of the referees for excellent work to point out how to understand oscillations in the linear birefringence and dichroism spectra.

FIG. 4 .FIG. 5 .
FIG. 4. Circular birefringence and dichroism spectra CB and CD, respectively, and linear birefringence spectra LB, LB 0 and dichroism spectra LD, LD 0 as determined by differential decomposition of the transmission Muller matrix in Fig. 2.

FIG. 6 .
FIG. 6. Differential decomposition of transmission Mueller matrix data obtained by forward calculations using a structural model based on twisted anisotropic multilayers used in electromagnetic modeling of reflection Mueller matrix data (Ref.13).
Scarab beetle Cetonia aurata reflects left-handed polarized light as visualized in (a) where the beetle is viewed through a left-handed polarizer.
and CB ¼ 2πd(n l À n r )=λ.Here, λ is the wavelength, d is the sample thickness, and subscripts indicate polarizations with FIG. 1.

TABLE I .
Best-fit values on parameters of the Gaussian resonances in Eq. (6a).