1. Introduction

The properties shared between the cuticle (exoskeleton) of some species of beetles and chiral nematic (or cholesteric) liquid crystals (ChNLC) were identified some decades ago [1]: helical structure, large optical rotation, and reflection of near-circularly polarized light. Regarding the shared structure, in ChNLC systems rod-like molecules are preferentially oriented in pseudo-planes along a certain unit vector (the director). In the helical structure, the director of adjacent pseudo-planes is rotated a small constant angle about the helix axis. Depending upon the rotation direction, clockwise or counter-clockwise, the helical structure is right- or left-handed. Additionally, single-domain structures are characterized by the presence of a single helical structure, otherwise the structure is of a multidomain type. The pitch of the structure corresponds to the distance separating two pseudo-planes when the director has completed a full rotation of 360°. In beetle cuticles, chitin-protein fibrils play the same role as rod-like molecules do in the ChNLC phase. Remarkably, electron microscopy images of beetle cuticles as well as of many other natural chiral systems show helical structures with the axis perpendicular to their surfaces [2]. From these observations it was hypothesized that the formation of single domain helical structures arises from an initial constraining layer [2]. Later, such hypothesis was qualitatively confirmed by numerical simulations using the Landau-de Gennes theory [3]. Owing to the helical structure of ChNLC and beetle cuticles, light incident at normal incidence is selectively reflected with the same handedness as that of the chiral structure. This type of selective reflection, commonly referred to as Bragg reflection, takes place in a spectral band centered at a wavelength equaling the product of the in-plane average refractive index and the pitch [1]. Even though reflection of light from beetle cuticles has been extensively studied over the past century, a renewed interest on the subject appeared during the latest decade. Thus, the comparison of experimental and simulated data at near-normal incidence using reflectance [4,5], ellipsometric [6], and Mueller-matrix spectra [7,8] have provided a qualitative description of beetle cuticles in terms of their thin-film analogue systems. In fact, a recent review shows how structures in living matter have inspired the improvement of optical performance of liquid crystals for the normal incidence case [9].

Bragg reflection at oblique incidence of single and multidomain ChNLC is also well understood from experimental and theoretical studies [10–13] as well as from calculations of the dispersion relation of the optical modes [14–17]. In contrast, research on the polarization properties of beetle cuticles at oblique incidence is at its beginning. Recently this issue has been addressed by our research group through the use of variable angle Mueller-matrix spectroscopic ellipsometry [18–22]. This multi-angle approach does not only account for the angle and spectral shift characteristic of all-dielectric reflective systems, but also for the appearance of interesting phenomena like the reflection of right-handed polarized light at increasing angles of incidence in some species [18,21]. Furthermore, structural parameters and refractive indices of beetle cuticles have been extracted by regression analysis of spectral multi-angle Muller-matrix data [19,20].

A close inspection of the available optical data from different species of beetle cuticles, at near-normal or oblique incidence, shows clear differences among the spectra. Some of them are rather smooth and broad, whereas modulation and interference oscillations are observed in data from other species. Particularly, in a previous work we reported the structure of the Mueller matrix of the scarab beetle Cotinis mutabilis (Gory and Percheron, 1833) and the presence of strong interference oscillations in the spectra was highlighted [21]. In this work, we focus on such interference oscillations with a twofold objective: i) to extract the structural and optical information therein contained and; ii) to investigate their relation with the polarization states of optical modes in ChNLC at oblique incidence. In order to achieve these objectives, this contribution is organized as follows. The experimental details and basics of Mueller matrices are presented in Sec. 2. In Sec. 3.1 the microstructure of the cuticle of C. mutabilis is briefly described. The polarization properties of reflected light from incident unpolarized light are discussed in Sec. 3.2. Evidence for a dispersion relation of optical modes in beetle cuticles is provided in Sec. 3.3 by analyzing the spectral dependence of maxima and minima appearing in the spectra of Mueller-matrix elements of C. mutabilis. This interpretation is supported in Sec. 3.4 with calculated dispersion relations for light propagation in cholesteric liquid crystals within a two-wave approximation [17]. In Sec. 3.5 a structural model for beetle cuticle is proposed on the basis of features in the experimental data and theoretical considerations for light propagation. Finally, the conclusions of this work are summarized.

2. Experimental

The beetle C. mutabilis is found in Mexico and the southwestern part of the United States [23]. It presents a green but not shiny color on its dorsal side with orange-brown stripes in the elytra. Its abdominal side presents a segmented structure and shows a shiny metallic-like color. The specimen under study was collected at Querétaro, Mexico. For the study, areas as flat as possible were selected from the segments in the abdomen and a small piece of about 2 × 2 mm² was cut using a sharp knife and tweezers. The piece of the cuticle was mounted on a glass slide with double-sided tape for the Mueller-matrix spectroscopic ellipsometry measurements which were performed with a dual rotating compensator ellipsometer (J. A. Woollam Co., Inc.). Since the cuticle is curved, focusing probes were used to achieve a beam spot with size below 100 μm. More details about the instrument can be found in references [18–21]. The measurements were performed at angles of incidence between 25 and 75° in steps of 5° in the wavelength range of 245 to 1000 nm. Scanning electron microscopy (SEM) images of the cuticle cross-section were acquired with a LEO 1550 Gemini system. The samples were mounted on double-side copper tape and coated with a thin platinum layer of around 2 nm to obtain a conductive surface.

The most general description of the polarization and depolarization capability of light induced by reflection from a sample, is given by its 4 × 4 reflection Mueller matrix,

In the particular case of unpolarized incident light $S_i={\left[1,0,0,0\right]}^{\text{T}}$the reflected light beam is given by $S_r={\left[1,m_{21},m_{31},m_{41}\right]}^{\text{T}}={\left[1,P\right]}^{\text{T}}$, where P = [m₂₁,m₃₁,m₄₁]^T is the so-called polarizance vector of M and T indicates transpose. Since the reflected light in general will be partially polarized, the degree of polarization P for incident unpolarized light can be calculated from [18],

3. Results and discussion

3.1 Cuticle microstructure

The cuticle of the beetle is comprised of the epicuticle, the exocuticle and the endocuticle. Figure 1 shows a SEM image of a cross section of the beetle cuticle revealing the epicuticle and exocuticle comprising a layer around 15 μm thick. The thickness of the epicuticle (Ep) is about 100 nm for the specimen under study. Two regions can be differentiated in the exocuticle: the outer exocuticle (o-Ex) with a fine multilayer structure and the inner exocuticle (i-Ex) also with a multilayer structure. The microstructure observed in Fig. 1 is representative of other species where the selective Bragg reflection is located at the outer exocuticle [1]. In some of them, the inner exocuticle is tanned with a black/brown pigment as in C. mutabilis. Beneath the inner exocuticle is the endocuticle which provides mechanical support.

 

Fig. 1 Green specimen of C. mutabilis and SEM image of a cross section of its cuticle on the abdominal side. The epicuticle (Ep) as well as the outer (o-Ex) and inner (i-Ex) parts of the exocuticle are shown.

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3.2 Polarization properties of the cuticle of C. mutabilis

The features of the Mueller matrix (not shown) of the green specimen studied in this work is similar to that previously reported for a red specimen, i.e. it exhibits the symmetry properties of chiral systems [21]. Here, we first focus on the polarization properties (P, e, φ) of light reflected from the beetle cuticle under unpolarized incident light calculated from Eqs. (3)-(5) and the depolarizance of the experimental Mueller matrix from Eq. (6). These four quantities are shown in Fig. 2 in contour polar representations where the radial and angular coordinates correspond to the wavelength λ and angle of incidence θ, respectively. Noteworthy in Fig. 2 is the behavior of P, e, φ, and D in a narrow spectral range which corresponds to the band of selective Bragg reflection. Referring to Figs. 2(a) and 2(b), we observe that at small angles of incidence (θ<40°) and in the spectral range of 500 to 600 nm, unpolarized incident light is reflected with a relatively high degree of polarization (P>0.8), left-handed polarized and with elliptical to near-circular character (e<-0.5). In Fig. 2(c), |φ| shows a strong variation with wavelength and angle of incidence. It can be noticed that the Bragg reflection band shifts to shorter wavelengths as θ increases. Outside the spectral range of Bragg reflection, e≈0 and |φ|≈90° for all angles of incidence. Since in addition, the degree of polarization P is high for 45°<θ<70°, linear s-polarization is dominating the light reflected at these conditions. Indeed, a maximum in P≈1 is noticed around θ = 55° which presumably indicates a pseudo-Brewster’s angle. Strictly, at the Brewster angle e = 0, |φ| = 90° and P = 1.

 

Fig. 2 Properties of polarized light reflected from a green C. mutabilis for unpolarized incident light: (a) degree of polarization, (b) ellipticity, and (c) azimuth. The depolarizance of the Mueller matrix is shown in (d).

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The depolarizance of the measured Mueller matrix is shown in Fig. 2(d). According to its definition, D is interpreted as unity minus the polarimetric purity of M [24]. Thus, for an ideal depolarizer D = 1 whereas the minimum value D = 0 corresponds to a non-depolarizing system and M is termed a Mueller-Jones matrix. In the spectral range of Bragg-like reflection in Fig. 2(d), D is a rapidly varying function of wavelength showing sharp maxima but, as can be noticed, on the average D≈0.2 and M moderately deviates from a non-depolarizing system. Nevertheless, in the complementary spectral range D<0.05 and M represents a nearly non-depolarizing system. Non-uniformity in cuticle thickness in the measured area, surface and volume irregularities, and lateral inhomogeneities in the cuticle are examples of possible causes of light depolarization.

3.3 Dispersion relation and Bragg-like reflection

In the Mueller matrix spectra of C. mutabilis it is noticeable that near and at the spectral range of selective reflection, the elements of M present strong interference oscillations as was previously reported for a red specimen [21]. For the green specimen studied in this work, such characteristic is exemplified in Fig. 3 with the elements of the polarizance vector at three angles of incidence. For the analysis, the dependence on photon energy is used here. The presence of these interference oscillations is indicative of highly transparent materials comprising the beetle cuticle. Considering the spectra of m₂₁ and m₃₁ at θ = 25° in Fig. 3, four spectral regions can be distinguished from low to high photon energies: I) high frequency interference oscillations, II) slowdown of frequency, III) oscillations of low frequency with decreasing amplitude, and IV) quenching of interference oscillations. At θ = 25 and 50°, features in m₄₁ characterize the three boundaries separating these four spectral regions: I-II onset of selective reflection of left-handed polarized light (m₄₁<0); II-III a peak in m₄₁ appears; III-IV cut-off of the selective reflection band. It can be noticed that increasing the angle of incidence the amplitude of the interference oscillations decreases (mostly in region I) and the boundaries shift to higher photon energies. At large angles of incidence and for incident unpolarized light, right-handed polarized light (m₄₁>0) is reflected from the beetle cuticle in a narrow spectral range as shown for θ = 75°. Previously, it was reported that m₄₁>0 in the beetle Chrysina argenteola at θ>45° [18].

 

Fig. 3 Spectral dependence of the last three elements (the polarizance vector) in the first column in the Mueller matrix of a green specimen of C. mutabilis at three angles of incidence. Notice that scales differ among the graphs.

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In order to extract quantitative information from the interference patterns in the elements of M, we consider the procedure commonly used in optical analysis of isotropic thin films. For incident light at an angle θ from an ambient with refractive index n_i, multiple reflections inside a non-absorbing film of thickness d and refractive index n_av produce maxima or minima in the optical measurements when the phase factor β equals a multiple integer of π [25], that is:

 

Fig. 4 (a) Spectral position of maxima and minima in m₂₁ of a green specimen for θ between 25° and 75° in steps of 10° from left to right. Dispersion relations in Eq. (7) for θ = 25° and 75° after off-set correction for (b) the green specimen studied in this work and (c) the red specimen previously studied [21].

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The arbitrariness in the choice of the first maximum or minimum, that is, corresponding to m = 1 can be removed by an off-set correction. For that, the apparent linear behavior of the data with E in Fig. 4(a) at low photon energies was fitted with a linear relationship m(E) = c₁ + c₂E obtaining the two constants c₁ and c₂. Thus, the off-set correction shown in Fig. 4(b) implies m = m-1-m₀, where m₀ is the closest integer to c₁. For clarity, only data for the smallest and largest angles of incidence are shown in Fig. 4(b); the dashed lines correspond to extrapolation to zero photon energy using m(E) = c₂E. According to Eq. (7), the results shown in Fig. 4(b) provide evidence for the dispersion relation of K_|| in the cuticle of the beetle C. mutabilis. The data in Fig. 4(c) are the results obtained by applying the same procedure to data from a red specimen whose Mueller matrix was previously analyzed [21]. As is shown in the next section, the data in Figs. 4(b) and 4(c) resemble the dispersion relation of optical modes found in ChNLC structures. Some structural parameters of the beetle cuticle can now be estimated. First, by using the data at low photon energies in Fig. 4(b), the thickness of the region in the cuticle producing the interference oscillations can be estimated from Eq. (7) where n_av is interpreted as the effective refractive index of the beetle cuticle at long wavelengths. From the data at θ = 25°, with air as ambient (n_i = 1) and for n_av = 1.54 [19], the estimated thickness is d = 10.3 μm in agreement with the thickness of the outer exocuticle determined from the SEM image in Fig. 1. Furthermore, in Fig. 3 the band where m₄₁<0 is centered around λ₀ = 560 nm which can be used to estimate the pitch Λ to 394 nm according to Λ = λ₀/(n_avcosθ) [11]. Thus, the number of periods in the outer exocuticle is d/Λ = 26.1. For the red specimen the estimated thickness was found to be 7.5 μm [21]. This difference in thickness explains the different slopes observed in Figs. 4(b) and 4(c).

3.4 Dispersion relation of optical modes in ChNLC structures

In this section we provide support that evidences the data in Fig. 4(b) as a dispersion relation. For that, we adopt the framework developed for the propagation and polarization states of optical modes in ChNLC established some decades ago [15–17]. Briefly, the ChNLC is considered to be comprised of anisotropic non-absorbing layers with a local dielectric tensor diag[ε₁,ε₂,ε₃], as illustrated in Fig. 5(a).Alternatively, the principal refractive indices n_j = (ε_j)^1/2 (j = 1,2,3) can be used. The continuous rotation of the principal axes ε₁ and ε₂ defines the pitch Λ after completing 360°, as is schematically shown in Fig. 5(b) for ε₁. Let us consider the oblique incidence of an electromagnetic wave of wavector K_i to the ChNLC from a medium of refractive index n_i as shown in Fig. 5(b). The plane of incidence coincides with the xz-plane. Inside the ChNLC structure, the wave vector K has components parallel (K_║) and perpendicular (K_⊥) to the axis of the helix (z-axis). The former is normalized according to $k=K_{\left|\right|}/\left(4\pi /\Lambda \right)$ whereas the latter is given by $K_{\perp }/\left(2\pi /\lambda \right)=n_i\sin \theta$. The periodicity (Λ/2) along the helix allows looking for solutions represented as an infinite sum of plane waves having wave vectors$K_l=K+2lq$, where l is an integer and q is a reciprocal lattice vector parallel to the helix of magnitude q = 2π/Λ. The substitution of that representation in Maxwell equations leads to certain recurrent relations for the amplitudes of the plane waves. Thus, a set of homogeneous equations is obtained and the roots of the characteristic equation determine the solutions for k (and hence for K_║). Only two solutions (k_1,2) are independent because waves propagating in the forward (k_i⁺) and backward (k_i^-) directions are simply related by $k_{1,2}^{\pm }=\pm k_{1,2}+l$. Three general types of solutions are found: A) The wave vector k is real-valued and the waves propagate without attenuation; B) The wave vector is complex k = k´ + ik´´ indicating damped waves with attenuation length η = Λ/(4πk´´) and k´ equaling a half integer (n/2) which expresses the condition for selective Bragg reflection of order n (K_||´ = 2πn/Λ); C) Two roots of the characteristic equation are conjugate each other and the real part does not satisfy the Bragg condition. Particularly, in the long-wave limit the solutions are real-valued and can be expressed as [15–17],

 

Fig. 5 (a) Anisotropic layers with local principal components ε₁, ε₂, and ε₃ comprising the ChNCL structure; the counter-clockwise rotation of ε₁ (and ε₂) around the axis of the helix (z) generates a left-handed chiral structure. (b) Wave vectors geometry for oblique incidence on a ChNLC structure; the full rotation of 360° of the principal axes defines the pitch Λ.

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The properties of the first-order reflection band can be determined within a two-wave approximation through the solutions of the characteristic equation [17],

Figure 6(a) shows the dispersion relation of the normalized complex wave vector k = k´ + ik´´ in a ChNLC structure calculated from Eq. (9) for θ = 25 and 75°. The incident medium is air (n_i = 1) and the ChNLC parameters are Λ = 390 nm, n₁ = 1.58 and n₂ = n₃ = 1.5, typical values found in beetle cuticle [19]. For simplicity, the dispersion in the refractive indexes was not considered. As can be observed in Fig. 6(a), in most parts of the spectral range the solutions are real-valued indicating modes propagating without attenuation (type A). At low photon energies, the two real-valued solutions k₁→k_a and k₂→k_b, i.e. linearly p- and s- polarized waves, respectively, are obtained, whereas at higher photon energies the reverse situation holds k₁→k_b and k₂→k_a. For each angle of incidence there is a narrow band where at least one of the solutions (k₂) is complex and the Bragg condition k₂´ = 1/2 is satisfied (type B). In other words, the effective wavelength λ_eff inside the ChNLC structure associated to K_║ is such that λ_eff = Λ. Regarding the imaginary part of the wave vector, we see in Fig. 6(a) that k₂´´ peaks at 2.15 and 2.67 eV for θ = 25 and 75°, respectively. For θ = 75° both k₁´´ and k₂´´ present a maximum located approximately at the same photon energy.

 

Fig. 6 (a) Calculated dispersion relation of optical modes for a ChNLC structure of pitch Λ = 390 nm at two angles of incidence and refractive indices n₁ = 1.58, n₂ = n₃ = 1.5; (b) magnification near the spectral range of Bragg-like reflection for three angles of incidence; (c) attenuation lengths for the mode k₂ producing selective reflection. The two modes of the normalized complex wave vector k = k´ + ik´´ are represented with the continuous and dashed lines.

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Figure 6(b) shows a magnification of k´ for θ = 25, 50 and 75°. It can be observed that as θ increases the first-order characteristic reflection band widens and shifts toward higher photon energies [14,15]. Other characteristics of the reflection band largely depend on the refractive indices. For simplicity, we restrict the discussion to uniaxial systems and for a given θ the anisotropy parameter Δn = n₁-n₂ determines the spectral width of the reflection band [9,13]. Solutions of type C for the wavector appear at large angles of incidence when locally biaxial slabs are considered as those deduced from regression analysis of Mueller-matrix data of Cetonia aurata [19]. If normal dispersion in the refractive indices is considered, the reflection bands at large angles of incidence shift to lower photon energies. Instead of the imaginary part of the wavector k₂, the attenuation length is shown in Fig. 6(c) which clearly diverges at band edges. The minimum attenuation lengths for θ = 25, 50, and 75° are η_min = 2.3, 2.1, and 1.94 μm, respectively. Larger values of Δn produce smaller values of η_min.

The polarization states of light waves in ChNCL structures for oblique incidence are in general elliptic [16]. The nature of eigenmodes ranges from near circular (left- and right-handed) at small angles of incidence, to near linear (p- and s- polarized) at medium and large angles. The latter result is in agreement with the polarization properties of the light reflected from the beetle cuticle for unpolarized incident light as was shown in panels (a)-(c) of Fig. 2; a decreased ellipticity of left-handed polarized light in the reflection band (Fig. 2(b)) with increase of θ and a high degree of linear s-polarization for 45°<θ<70° and wavelengths outside the Bragg-reflection band. Since at the latter conditions the cross-polarized reflectance R_ps has been shown to be near zero [21], unpolarized incident light excites the mode k₂ (k₁) at photon energies lower (higher) than the Bragg-reflection band.

The comparison of the data in Fig. 4 with the dispersion of k₂´ in Fig. 6 is an evidence for the similarities of the propagation of optical modes in the cuticle of C. mutabilis to those in ChNLC structures. However, two important differences can be noticed. First, the width of the left-handed reflection band in Fig. 3 for θ = 25° is about 0.4 eV (Δλ = 70 nm) whereas in Fig. 6(b) it is approximately only 0.1 eV. The width can be used to estimate the associated birefringence Δn = Δλ/Λ = 0.18, which is unreasonably high for the natural materials comprising the beetle cuticle. The second difference refers to the slopes of the dispersion relation curves outside the Bragg-reflection band. In the calculated data of Fig. 6(a), the slopes in the low and high photon energies sides are approximately the same whereas in Figs. 4(b) and 4(c), a smaller slope is noticed in the high photon energy side than in the lower one. Obviously, these differences arise from the fact that beetle cuticles do not represent an ideal system as shown in Fig. 5. These issues are addressed in the next section.

3.5 Structural model for the exocuticle of C. mutabilis

The broadening of the selective reflection band of ChNLC has been achieved by using structures of multiple pitches (discrete or graded) inspired by ideas from nature [9]. Indeed, structures with a jump in the pitch have been imaged for beetle cuticles [4,5]. Following this idea, let us consider the helical structure shown in Fig. 7(a) which is comprised of slabs organized in two helical stacks of pitches Λ₁ and Λ₂ and thicknesses d₁ and d₂. The outer exocuticle of C. mutabilis (thickness d₁ + d₂) is represented by these two helical stacks. Since the same composition is expected through the whole outer exocuticle, the same local dielectric tensor is assumed for slabs in these two chiral stacks. The substrate in Fig. 7(a) represents the (sclerotized) inner exocuticle which in C. mutabilis is tanned with a black-brown pigment.

 

Fig. 7 (a) Schematics of wave propagation in a ChNLC-like structure with two chiral stacks of pitches Λ₁ and Λ₂ and of thicknesses d₁ and d₂. (b) and (c) Real part of the wave vector of the optical mode producing Bragg-like reflection at angles of incidence 25° and 75°, respectively; (d) and (e) show the corresponding attenuation lengths. The dispersion relations were calculated for semi-infinite uniaxial structures with pitches Λ = 390 and 360 nm and refractive indices n₁ = 1.58 and n₂ = n₃ = 1.5.

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In order to discuss the propagation of light in the stacked ChNCL-like structures of Fig. 7(a), the dispersion relation of optical modes for each semi-infinite structure was calculated. The pitches and refractive indexes considered were Λ₁ = 390 nm, Λ₂ = 360 nm, n₁ = 1.58, and n₂ = n₃ = 1.5. For clarity, in Figs. 7(b) and 7(c) only the dispersion relation for the real part of the optical mode with wave vector satisfying the Bragg condition (k₂) is shown for θ = 25° and 75°. The corresponding attenuation lengths η₁ and η₂ are shown in Figs. 7(d) and 7(e). As expected, a smaller pitch shifts the first-order reflection band to higher photon energies. Four spectral regions can be identified in correspondence with those in Fig. 3.

In the spectral region I, k₂ is real-valued in both chiral stacks and electromagnetic waves of effective wavelength λ_eff >Λ_1,2 propagate without attenuation throughout the whole helical double-structure. In this case, the total phase change of light reflected is given by,

Finally, in Fig. 3 it was noted that m₄₁>0 at large angles of incidence in region III. Two possibilities are suggested to explain the reflection of right-handed polarized light with ellipticity e>0 from a left-handed structure. The first explanation is similar to that given for reflection of right-handed polarized light in Plusiotis (now Chrysina) resplendens [26]. As was mentioned above, at large θ the polarization of the optical modes acquires a more linear character and stack 1 might behave as a half-wave plate for some wavelengths. Thus, during the propagation through stack 1 incident right-handed polarized light could be transformed into the left-handed which would be reflected from stack 2. As it travels back through stack 1 the emerging light would be observed as right-handed. The effectiveness of stack 1 as a half-wave plate would depend on the relative orientation of the plane of incidence with respect to the segments comprising the abdomen of C. mutabilis (Fig. 1) because the strength of positive values of m₄₁ showed that dependence [21]. The other possibility is related to the incidence of light from a low refractive index medium (air) to beetle cuticle. In liquid crystals cells the chiral nematic phase is sandwiched between symmetric media (glass) typically with a refractive index similar to the average of the liquid crystal. Nevertheless, calculations have shown that changing the external media from glass to air, e shows opposite signs below and above the band of selective reflection at large angles of incidence [27].

4. Conclusions

The polarization properties of the cuticle of the scarab beetle C. mutabilis and allowed and forbidden optical modes in the cuticle have been studied using Mueller matrix spectroscopic ellipsometry. It was shown that for unpolarized incident light, the reflected beam is left-handed polarized with a high degree of circular polarization in a narrow spectral band at small angles of incidence. At intermediate angles of incidence and outside of the Bragg reflection band, the reflected beam has a strong linear s-polarized character. Clear interference oscillations in the spectra are due to the coherent superposition of reflected light from the cuticle interfaces. Structural parameters for the outer exocuticle were estimated from features in the spectra: a thickness of 10.3 μm, a pitch of 394 nm and 26 periods in the helix were found. It was also found that the spectral position of maxima and minima of interference oscillations gives insight into the dispersion relation of the optical modes in the beetle cuticle. A dispersion relation, calculated within a two-wave approximation originally developed for ChNCL structures, supports the experimental evidence. A structural model comprised of two chiral stacks of uniaxial slabs was derived from a theoretical analysis of wave propagation in the beetle cuticle.