Pitch profile across the cuticle of the scarab beetle Cotinis mutabilis determined by analysis of Mueller matrix measurements

Helicoidal structures of lamellae of nanofibrils constitute the cuticle of some scarab beetles with iridescent metallic-like shine reflecting left-handed polarized light. The spectral and polarization properties of the reflected light depend on the pitch of the helicoidal structures, dispersion of effective refractive indices and thicknesses of layers in the cuticle. By modelling the outer exocuticle of the scarab beetle Cotinis mutabilis as a stack of continuously twisted biaxial slices of transparent materials, we extract optical and structural parameters by nonlinear regression analysis of variable-angle Mueller-matrix spectroscopic data. Inhomogeneities in the beetle cuticle produce depolarization with non-uniformity in cuticle thickness as the dominant effect. The pitch across the cuticle of C. mutabilis decreased with depth in a two-level profile from 380 to 335 nm and from 390 to 361 nm in greenish and reddish specimens, respectively, whereas in a yellowish specimen, the pitch decreased with depth in a three-level profile from 388 to 326 nm.

. Pictures of specimens of Cotinis mutabilis.

References Experimental
The specimens of the beetle C. mutabilis under study were collected at Querétaro, Mexico, No special collecting permit or "Animal Care Protocol" was required at the time. Arturo Mendoza Galván, a Mexican citizen and researcher at Cinvestav, was responsible for collecting the specimens. Figure S1 shows pictures of the three specimens analysed. As can be seen, the abdominal side presents a shiny metallic-like colour (greenish, yellowish, or reddish). For the study, areas as flat as possible were selected from the segments in the abdomen and a small piece of about 23 mm 2 was cut using a sharp knife. The piece of the cuticle was mounted on a glass slide with double-sided tape for the Mueller-matrix spectroscopic ellipsometry measurements performed with a dual rotating compensator ellipsometer (RC2 system, J. A. Woollam Co., Inc.). The measurements were performed at angles of incidence () between 20 and 75° in steps of 5° in the wavelength () range 245 to 1000 nm. Since the cuticle is slightly curved, focusing probes were used to achieve a beam spot with size below 100 m. Non-linear regression analysis to fit model-generated data to experimental data was performed with the CompleteEASE software (J. A. Woollam Co., Inc.) which provides best-fit parameters and 90% confidence intervals. More details about the instrument and modelling can be found in references [1][2][3].
The electron micrographs were taken with a scanning electron microscope LEO 1550 Gemini. For these studies, the samples were mounted on double-side copper tape in such a way that the cross-section was exposed: The samples were coated with a thin platinum layer of approximately 2 nm to obtain a conductive surface. Figure S1. Pictures of greenish, yellowish, and reddish specimens of C. mutabilis. Figure S2 (left) shows polar contour maps of the experimental Mueller matrices of C. mutabilis measured at angles of incidence between 20 and 75° (the polar angle) in the wavelength range 245 to 1000 nm (the radial axis). The features of the variable-angle Mueller matrices are similar to those previously reported for other specimens and exhaustively described before [2,3]. In particular, we emphasize four major features: i) for non-polarized incident light with Si=[1,0,0,0] T the reflected beam represented by Sr=[1,m21,m31,m41] T is left-handed polarized (m41<0) at low angles of incidence in the narrow spectral range of 485 to 650 nm, consistent with figure 1(c); ii) symmetries between the elements describe a chiral system [2], namely, m12=m21, m13=-m31, m14=m41, m23=-m32, m24=m42, and m34=-m43, leaving nine symmetry-independent elements in M; iii) the pseudoisotropic character outside the Bragg reflection band, i.e. the Mueller matrix is nearly block-diagonal; iv) the shift of the selective Bragg reflection to shorter wavelengths at increasing angles of incidence characteristic of an all-dielectric system. The model-

Model details
To calculate the Mueller matrix of the model in figure 5(a) it is necessary specify the values of various parameters: thicknesses (d,depi), refractive indices (nepi,n1,n2,n3,ns+iks) and parameters in equation (9) that determine the pitch profile. The procedure to assign initial values of model parameters is described below.
Thicknesses. As was described in previous reports [3,4] and summarized in section 2.2, the thickness of the outer exocuticle can be estimated from a spectral analysis of maxima and minima of oscillations in m21 at =20° for wavelengths longer than 650 nm (outside the selective Bragg reflection band). We found d=9.8 m for the yellowish specimen analysed in this work. Previously reported outer exocuticle thicknesses for other specimens estimated with the same procedure were 6.5 µm and 8.5 µm. For non-linear regression analysis the number of anisotropic slices should be large enough to describe a continuous variation of the azimuth in equation (6) of the main text. In this work we have used 500 anisotropic slices which fulfil thus the requirement of continuity [1]. For the epicuticle, we set depi=80 nm according to electron microscopy studies of specimens of C. mutabilis [2,3].

Refractive indices.
where Aj, Bj, and Cj are fitting parameters defined as 0. By using wavelengths expressed in units of m in equation (S1), the initial values of (Aj, Bj, Cj) of the various refractive indices in the cuticle were nepi=(1.42, 0.009, 0.003), n1=(1.5, 0.015, 0.001), n2=(1.47, 0.015, 0.001), and n3=n1. The initial choice n3=n1, was appropriate to obtain the correct sign of m31 mostly at large angles of incidence. For the tanned inner exocuticle we used the same complex refractive index (Ns =ns+iks) used before for the beetle C. aurata [1].
Parameterization of a single chiral stack. In this case, the twist of anisotropic slices representing the helicoidal structure is parameterized as a variable azimuth angle (in degrees), which locates the orientation of 1 [1], where z is the position measured from the bottom, d the thickness of the structure, T is the number of turns, and 0 the azimuth offset of 1 with respect to the plane of incidence.
Defining the cumulated number of periods as, where aj, z0j, and bj are, respectively, the strength, position, and broadening of the j-th change of pitch. Note that in equation (S6) aj and bj are dimensionless parameters. From equations (S3), (S4) and (S6) we get the pitch distribution as a function of z as, Number

Non-linear regression analysis
Non-linear regression analysis was performed with the CompleteEASE software (J. A. Woollam Co., Inc.) which provides best-fit parameters and 90% confidence intervals. In

Depolarization caused by the cuticle of C. mutabilis
Because non-uniformity in thickness was introduced as the source of depolarization, it is necessary to determine the validity of this assumption. An appropriate way is to quantify the depolarizance (D) of the experimental and best-fit Mueller matrices according to [6], ( ) where P is the degree of polarimetric purity (also called depolarization index) and tr stands for trace. Thus, D is an average measure of the depolarization produced by a system for all incident pure states. Figure S6 shows the depolarizance of the experimental and bestfit model Mueller matrices at angles of incidence =20 and 75°. In a model without nonuniformity in d, D would be zero. However, by fitting non-uniformity in thickness, a significant improvement in modelling of depolarization is achieved. For data from the     0.0468 ± 0.003 --*These parameters attained value zero and therefore they were fixed for the final regression.