Theoretical framework to describe the reflection of circularly polarized light by a natural photonic crystal: elytron of a Chrysina resplendens scarab

William E. Vargas

doi:10.1364/OPTCON.449029

1. Introduction

Optical properties of synthetic materials with structural chirality have been considered for decades with the purpose of describing how the different polarization states of incident light are modified by their interaction with this type of media. The growing knowledge of the interaction between light and chiral materials has led to the modelling and/or development of functional optical systems: gratings [1], metamaterials [2], light scattering media [3], doped cholesteric networks for lasing applications [4], and others. Natural polarizers have also attracted the attention of scientists due to their capability of reflecting and transmitting circularly polarized light when illuminated with non-polarized radiation [5,6]. An understanding of the mechanism used to do this is leading to the development of functional materials [7–9]. First report on both components of circular polarization in reflectance spectra by Chrysina resplendens’ elytra dates from 1911. It was based on the work of Michelson [10]. The research carried out by Caveney in 1971 allows to link Michelson’s observation with the morphological structure found through the elytron of a specimen of these beetles, as shown in Fig. 1 [11], as well as with large volume fractions of uric acid crystallites embedded through the chitin fibrils forming the elytron. Since then, optical properties of elytra of C. resplendens and other species of scarabs have been the focus of many research works based on spectrophotometric and ellipsometry measurements, usually accompanied by electron microscopy analysis which provide information about the structural composition of the elytra, within a certain expected degree of variability in the thickness of their different sections due to the natural origin of the involved materials. The theoretical framework presented in this article, although applied to a specific species of scarab beetle, can be easily applied to other species of scarabs, other animals, fruits, and leaves with chiral structures in some parts of their bodies. It can be also applied to synthetic chiral materials like cholesteric liquid crystals. The method is simple but robust enough to obtain a rigorous photonic characterization of these optical systems.

Fig. 1. Image of the cross section of the cuticle of a C. resplendens scarab beetle showing its epicuticle on the top, followed by the first helicoidal arrangement of chitin fibrils of thickness h₁, a unidirectional layer of thickness u, and a second helicoid whose thickness is h₂ (Figure taken from [11], reproduced with permission).

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Starting from the morphological analysis of the elytron, and a determination of the mean value of the volume fraction of uric acid distributed through the helical structure, the method proposed here allows to obtain the variation of the spectral position and width of the photonic band gap (PBG) with the variation in depth throughout the elytra (λ_o and Δλ_o, respectively). Additionally, effective values of the ordinary and extraordinary refractive indices, and the corresponding birefringence, are obtained as a function of depth. These optical constants can be linked with well-known formalisms (radiative transfer matrix [12], finite element [13], and finite integration [14] methods based on numerical solutions to Maxwell’ equations) to evaluate co- and cross-polarized reflectance and transmittance spectra by chiral structured media. Starting from a behavior specified for λ_o and Δλ_o as a function of depth, the formalism developed here allows to retrieve that variation with depth of the structural pitch required to obtain the desired photonic behavior. This last point is also of great relevance for the development of new functional materials.

Regarding 1D structural chiral materials like cholesteric liquid crystals and cuticles of some jewel scarabs, their photonic behavior has been modelled mainly for single-pitch samples or those displaying small jumps in the pitch at certain depths of the cuticle [15,16]. The modelling has been focused on correlating the expected spectral position of the PBG, and its width, with the presence of peaks in the reflection spectra. Both the spectral position of the PBG, and its width, are obtained in terms of the structural pitches from the dispersion relation for frequency and wave number of the circular components of the propagating electric field [17]. For a single-pitch sample, normally illuminated, λ_o= n_avP_o and Δλ_o= ΔnP_o, where P_o as the pitch, n_av is the average refractive index, and Δn is the birefringence [18]. Few works have incorporated graded pitches into the derivation of the dispersion relation, and when carried out, depth dependences of the pitches have been a priori assumed [19]. The explicit formalism to include any depth dependence of the pitch into the dispersion relation was developed by Vargas et al. when considering reflectance spectra and photonic characterization of the cuticle of golden-like and red Chrysina aurigans scarabs [6]. This achievement has allowed to successfully correlate the depth dependence of the PBG and its width with corresponding spectrophotometric measured reflectance spectra and approach the spectral features of these spectra by calculated ones obtained from radiative transfer matrix formalisms, for other species of beetles.

2. Morphology of the optical system and optical measurements

The drawing in Fig. 2 schematizes the structure of the cuticle of a C. resplendens beetle [11]. The left-handed circular polarization (LHCP) component of the incident light is reflected by the first left-handed helicoidal structure, with a small contribution (∼4%) of non-polarized light due to the optically homogenous epicuticle coating on top of the exocuticle. The unidirectional layer acts as a half wave plate (HWP) which changes the polarization of the right-handed circular polarized (RHCP) light transmitted through the first left-handed twisted arrangement. LHCP light is transmitted through the unidirectional layer which is reflected by the left-handed second helicoidal structure. When this reflected radiation is going back through the unidirectional layer, its polarization state is again changed by the retarder layer, and RHCP light will travel through the first twisted structure emerging as reflected light. Both helicoids behave like 1D photonic crystals for left-handed propagating radiation, whose optical anisotropy is enhanced by the presence of uric acid crystallites embedded in the twisted structures of chitin fibrils. For the sample considered by Caveney, h₁= 4.77 ± 0.04 µm and h₂= 16.20 ± 0.06 µm. The total thickness of the optically anisotropic structure is h = 22.77 ± 0.06 µm. The values of h₁, h₂, and h have been estimated separately from Fig. 1 [11]. With the appropriate software to manage images, the scale bar of the figure was vertically placed, with its base coinciding with the bottom of the chiral structure. In this way, a vertical axis given in µm was defined. For each one of the two helical sections and for the whole structure, a set of about 30 measurements of top and bottom positions was obtained, to calculate the average values and corresponding standard deviations. Caveney reported as thickness of the unidirectional layer u = 1.80 ± 0.04 µm. For the total thickness and that of the first helicoid, he reported h = 22 µm and h₁= 5 µm, with no indication of uncertainty in these measurements. Figure 3 shows a transmission electron microscopy (TEM) image of a section of the cuticle of a C. resplendens scarab. It displays the characteristic parabolic patterns of oblique sections of chiral materials [11,20].

Fig. 2. Schematic cross section diagram of the cuticle of a C. resplendens scarab beetle showing its epicuticle on the top, followed by the first helicoidal arrangement of chitin fibrils, a unidirectional layer, and a second helicoid. Each twisted structure is characterized by left-handed chirality [11].

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Fig. 3. TEM image of an oblique section in the cuticle of a C. resplendens scarab showing the unidirectional layer in the middle, and sections of the two helical structures displaying parabolic patterns at the top and bottom. A structural half-pitch distance is indicated by the black non-labeled scale. (Figure taken from [11], reproduced with permission.)

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The chitin fibrils are parallel to the cut of the sample in the darkest regions of the helical sections. A structural half- (complete-) pitch corresponds to two (three) successive darkest regions. Figure 4 shows the reflectance spectra of the circularly polarized light reflected by the cuticle of the specimen considered by Caveney when normally illuminated with non-polarized radiation. The spectrum of the RHCP light displays a reflectance edge close to 550 nm, and that of the LHCP shows a reflectance peak also close to 550 nm. Corresponding RGB colors, R, G, and B values, chromaticity coordinates (x,y), and luminous reflectance Y have been included in the insets of the figures, to be considered in Section 7. Through next Sections, a sequence of steps is followed to characterize the three-layered system consisting of the two helicoids and the unidirectional layer between them. The main spectral features displayed by the reflectance measurements reported by McDonald, from 400 to 850 nm, agree with those observed in the measurements carried out by Caveney [11,21], with more ripple structure probably due to the use of a smaller illumination spot in his experimental setup and to the use of co- and cross-polarizing filters. Similar optical measurements have also been reported by Finlayson et al. in the same spectral range [22], and by Vargas et al. who considered a more extended spectral range, from 300 to 1000 nm [23].

Fig. 4. Visible wavelength spectra for (a) right-handed circularly polarized (R_RHCP) and (b) left-handed circularly polarized (R_LHCP) light reflected by the elytron of the C. resplendens scarab analyzed by Caveney [11], when normally illuminated with non-polarized radiation. The solid lines added for visual aid were obtained by cubic spline. Parameters related with color calculations have been indicated to be used in Section 7. Dots correspond to values obtained from the experimental ones reported by Caveney in Fig. 5(b) of his publication [11].

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3. Unidirectional layer

The unidirectional layer acts as an HWP, introducing a retarder phase in the circularly polarized radiation traveling through it, and changing the polarization of the transmitted light with respect to that of the incoming one. The retarder phase is given by [24]

(1)$$\Gamma = \frac{{2\pi u}}{\lambda }\left\langle {\Delta n} \right\rangle, $$

where <Δn > is the spatial average of the birefringence. The retarder phase is equal to π for the wavelength λ=λ_HWP. The intensity of the light transmitted through a HWP is given by [24]

(2)$$I = \frac{{{I_o}}}{2}{\sin ^2}\left( {\frac{\Gamma }{2}} \right), $$

where I_o is the intensity of the incident radiation at the illuminated interface. Figure 5 displays the relative intensity of the light transmitted through the unidirectional layer in Caveney’s sample, showing the transmission edge from λ_HWP towards shorter wavelengths. The profile of the figure resembles those of the measured right-handed reflectance spectra (see Fig. 6(c) in [23]). The largest amount of RHCP radiation incident on the top of the retarder layer and transmitted through it is characterized by a wavelength λ=λ_HWP. This means that the λ_HWP-value can be estimated from the spectral position of the reflectance edge in the right-handed circularly polarized spectrum, to be used in the determination of the birefringence of the unidirectional layer and its average uric acid volume fraction. This issue will be considered in Section 7. Caveney reports a value of 590 nm for λ_HWP. The average birefringence of the unidirectional layer is then given by <Δn>=λ_HWP/2u = 0.164 ± 0.004. Caveney measures <Δn>=0.166 as birefringence of the retarder layer at 560 nm of wavelength. The average birefringence of the uric acid in the unidirectional layer is given by <Δn>=<f_u(z)>Δn_ua+(1-<f_u(z)>)Δn_c, where Δn_ua and Δn_c are the birefringence of uric acid and chitin, respectively, and f_u(z) is the volume fraction of uric acid at depth z through the unidirectional layer, with < f_u(z)> as its spatial average. The average volume fraction of uric acid in the HWP is

(3)$$\left\langle {{f_u}(z)} \right\rangle = \frac{{\left\langle {\Delta n} \right\rangle - \Delta {n_c}({\lambda _{HWP}})}}{{\Delta {n_{ua}}({\lambda _{HWP}}) - \Delta {n_c}({\lambda _{HWP}})}}.$$

Fig. 5. Profile of the relative intensity of radiation transmitted through a unidirectional layer acting as a half wave plate, with its thickness being 1.80 µm and a birefringence <Δn>=0.164.

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Fig. 6. Sketch of a SEM image showing the sequence of structural half-pitches conforming the helicoidal structure, as well as two labeled half-pitches and a structural one, P_s. The value ϕ(z = 0) = 0 has been assumed.

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The λ-wavelength dependence of the chitin’s birefringence has been taken from the literature, i.e. Δn_c= Δn_c(λ) [25], with an average value equal to 0.0026 through visible and near infrared wavelength ranges. The birefringence of chitin is Δn_c= 0.0025 at 590 nm. The spectral average birefringence of uric acid (<Δn_ua>=0.31) has also been taken from the literature [26]. This value is interpreted as an average over visible wavelengths. The spectral dependence of the uric acid’s birefringence has not been published yet. But, within a modelling approach of the optical properties of C. resplendens’ cuticle, Mendoza-Galván et al. have obtained from ellipsometry analyses the spectral variation of the retarded phase due to the unidirectional layer [27].

By using their reported thickness for the HWP (u = 1.76 µm), the value of < f_u(z)> has been optimized to obtain an average value of <Δn_ua> equal to 0.31 for visible wavelengths. In this way, it is required that < f_u(z)>=0.41. From the average refractive index [n_av(λ) = (n_o+n_e)/2] and birefringence [Δn(λ)=n_e-n_o], the spectral variation of the extraordinary and ordinary refractive indices is calculated for each material [28,29]: n_e(λ)=n_av+Δn/2 and n_o(λ)=n_av-Δn/2, n_o and n_e being the ordinary and extraordinary refractive indices of the anisotropic twisted structure. Corresponding dielectric functions are obtained from them: ε_e= n_e² and ε_o= n_o² for the extraordinary (parallel to the unit vector along the average azimuth orientation of the chitin fibrils) and ordinary (perpendicular to the unit vector) rays, respectively. From this analysis, the birefringence of uric acid is Δn_ua= 0.334 at 590 nm. Then, from evaluation of Eq. (3), <f_u(z)>=0.48 ± 0.01 for Caveney’s sample.

4. Helicoids: description of the structure, effective and structural pitches

As mentioned, the cuticle of C. resplendens consists of two helicoids and a unidirectional layer in between. For the first (second) helicoid, the z-coordinate specifies the depth from the bottom of the epicuticle (unidirectional layer), as seen in Fig. 2. TEM and scanning electron microscopy (SEM) have been utilized to study the morphology of these type of chiral materials. TEM-images show a sequency of black and white pseudo-layers, with the darkest regions coinciding with those planes where the chitin fibrils are parallel to the cut of the sample (see Fig. 3, and Fig. 8 in [30], for example). From electron microscopy images, it is possible to approach the variation of the pitch with z. For SEM-images, the clearest pseudo-layers correspond to those depths where the orientation of the chitin fibrils coincides with the cut (see for example Fig. 2 in [31]). The distance between two successive clear pseudo-layers is a structural half-pitch. A complete structural pitch, P_s, corresponds to three successive clear pseudo-layers. Every two successive clear pseudo-layers, the average orientation of the fibrils has increased by π radians, as indicated in Fig. 6. This orientation is specified by the azimuth angle ϕ which changes with z, i.e., ϕ=ϕ(z). An arbitrary value can be assigned to the value of ϕ at z = 0. This z-dependent angle plays a fundamental role when modelling the optical properties of these chiral materials by finite element methods [32] or Berreman’s radiative transfer matrix formalism [33]. The tensor components of the dielectric function of the chiral material depend on z through ϕ. There are two ways to describe the morphology of each helicoid.

4.1 From the structural pitch to the effective one and the azimuth angle

By using the scale of the electron microscopy image, the structural pitch can be directly obtained at specific z values. A more continuous variation of the structural pitch P_s with z is obtained by interpolation. The z-dependence of ϕ is obtained by integration of its differential relation [dϕ=2πdz/P_s(z)]: namely

(4)$$\phi (z) - \phi (z = 0) = \int\limits_0^z {\frac{{2\pi }}{{{P_s}(x)}}dx}.$$

The integrand in Eq. (4) can be interpreted as a local reciprocal vector G_s(x) = 2π/P_s(x) whose average value up to depth z,

(5)$$\left\langle {{G_s}(z)} \right\rangle = \frac{1}{z}\int\limits_0^z {{G_s}(x)dx \equiv \frac{{2\pi }}{{{P_{eff}}(z)}}}, $$

can be treated as an effective reciprocal vector with its corresponding effective pitch P_eff(z)${\equiv}$P(z). From these definitions, one can see the following relations:

(6a)$$P(z) = {\left[ {\frac{1}{z}\int\limits_0^z {\frac{{dx}}{{{P_s}(x)}}} } \right]^{ - 1}} = {\left\langle {1/{P_s}} \right\rangle ^{ - 1}},$$

(6b)$${P_s}(z) = {\left\{ {\frac{d}{{dz}}\left[ {\frac{z}{{P(z)}}} \right]} \right\}^{ - 1}}.$$

From Eqs. (4) and (5), with ϕ(z = 0) = 0, the relation ϕ(z) = 2πz/P(z) is obtained. It can be used to obtain those ϕ-values required to radiative transfer calculations, or directly from Eq. (4) (see Eq. (8) in [34]). The effective pitch is equal to the inverse of the average density of turns up to depth z, namely P(z)= 1/[N/z] with N = [ϕ(z)-ϕ(0)]/2π being the number of turns of the helical structure up to depth z. The structural pitch is the inverse of the local density of turns (see Eq. (4.3) in [35]), namely, P_s(z) = 1/[dN/dz].

4.2 From the azimuth angle to the effective pitch and the structural one

By using the scale of the electron microscopy image, the azimuth angle can be directly obtained at specific z values. A quasi-continuous variation of ϕ with z is obtained by interpolation, and the effective pitch is calculated from P(z) = 2πz/ϕ(z) [36,37]. Then, the structural pitch is evaluated from Eq. (6b). This second approach was applied to Caveney’s sample [11]. His Fig. 3 displays the structural pitch times the average refractive index, in terms of the number of half turns carried out.

The average refractive index can be obtained from the n_e and n_o values reported in his Table 1: n_e= 1.603 and n_o= 1.700. Therefore, n_av= 1.652. The half-pitches indicated by dots in his figure were used to interpolate for all the half-pitches needed to cover the thickness of the helicoid being considered, with linear extrapolation to the first and second half-pitches because these are not indicated in Caveney’s figure. These two extrapolated values, together those interpolated ones corresponding to an even number of half-pitches define the sequence of structural pitches. They correspond to dots in Fig. 7 and are denoted as the experimental values, P_exp. Yellow bars in this figure point out those regions of linear extrapolation. The consecutive sum of these pitches specifies the corresponding z-values. In this way, the ϕ=ϕ(z) sequence was obtained, and the effective pitch, P(z)=P_eff, was calculated with subsequent interpolation to display a more continuous variation. Solid lines display these effective pitches. Linear extrapolation was assumed to calculated P(z = 0) because microscopy images do not provide a way to determine where the turn of the chitin fibrils starts. From this set of interpolated P(z)-values, also including those contained in the regions of linear extrapolation, the structural pitch was evaluated applying Eq. (6(b). The results are indicated by dashed lines in Fig. 7. The reliability of this procedure is measured by the following merit function:

(7)$$F = \frac{1}{{M - 1}}{\sum\limits_{j = 1}^M {\left|{\frac{{{P_{\textrm{in} }}({z_j}) - {P_{\exp }}({z_j})}}{{{P_{\exp }}({z_j})}}} \right|} ^2}, $$

where the P_in(z_j)-values correspond to those interpolated values of the structural pitch calculated at the same z_j positions of the experimental ones. The values of F are indicated in Fig. 7, with M = 12 for the first helicoid and 37 for the second one.

Fig. 7. Variation with depth of the effective (P_eff) and structural (P_s) pitches of C. resplendens for the first (a) and second (b) helicoidal structures. Dots correspond to values obtained from the experimental ones reported by Caveney in Fig. 3 of his publication [11]. Yellow bars indicate those regions of linear extrapolation.

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The structural pitches displayed in Fig. 7 can be compared with reported ones. The availability of other measurements is scarce. Finlayson et al. have characterized the half-pitch variation of C. resplendens in terms of the number of half-pitches covered [22]. Although the profile of the half-pitch of the second twisted structure resembles the depth dependence displayed in Fig. 7(b), the behavior of the half-pitch of the first twisted arrangement, which decreases with depth instead of increasing, is significantly different to that displayed in Fig. 7(a). According to the results of McDonald, the depth dependence of the half-pitch through the first helicoid looks more like that of a C. cupreomarginata specimen, in the report of Finlayson et al. This is a second species recently discovered as reflecting both polarizations [21]. Only a systematic morphological study, by electron microscopy, using a significant number of specimes of C. resplendens and C. cupreomarginata, will clarify whether the two trends observed for the pitch in the first helical structure can be attributed to biological variability.

As mentioned, when considering helical structures, the azimuth angle ϕ=ϕ(z) enters in models based on finite element methods and in the 4 × 4 matrix formulation of Berreman to calculate the tensor components of the dielectric function. The angle can be obtained directly from integration of 2π/P_s(z) (Eq. (4)), without any mention of the effective pitch. Or one can obtain a small set of effective pitch values from the z-dependence of the azimuth angle. Then, a much larger set of effective pitch values is obtained by interpolation and use in the models to obtain the azimuth angle from the relation ϕ(z) = 2πz/P(z). This second method does not need to use the concept of structural pitch. These are the two methods explained in Sections 4.1 and 4.2. It is not that the structural pitch contains a more precise description of the structure's morphology. One can be obtained from the other by applying Equations (6). What is erroneous is to calculate the azimuth angle from the relation ϕ(z) = 2πz/P_s(z). Finalyson et al. use Berreman’s formalism to evaluate reflectance spectra by a cuticle of C. resplendens [22]. They indicate that “the chirped helicoid was modelled using a spline interpolation fit to the measured lamellar pitch data”. From this expression, one cannot conclude if they use the relation 2πz/P(z) or 2πz/P_s(z) to generate the ϕ(z) values required by the Berreman’s method.

5. Profile of the effective uric acid volume fraction through the whole structure

In previous works [36,37], it has been assumed an inverse dependence of the effective uric acid volume fraction on the effective pitch of the helical structure, with successful results. This assumption has been phenomenologically argued in [37]. The mechanism involved in the incorporation of uric acid crystallites through the chitin matrix of the chiral structure is not completely known yet. The low birefringence of the chitin is by far not enough to replicate the measured reflectance spectra from modeling. The need of a high birefringence to approach from the model the main spectral features of the measured reflectance spectra, indicates to us that the orientation of the crystallites is not random. During the morphogenesis of the cuticle, the protein-chitin matrix was probably permeated by dissolved uric acid, depositing nanocrystals which grow by aggregation to form oriented larger crystals as the chiral structure is being stabilized. The chitin fibers and the proteins that surround them could exert control over the orientation of the uric acid crystals so that they follow the orientation of the fibers [6]. The larger the pitch of the structure is, the larger the available space between planes of chitin nano-fibrils. As the pitch increases with depth, it would be expected to have larger embedded uric acid crystallites. However, beyond a certain crystallite size, around a few tens of nanometers, the larger the size of the crystallites, the larger the amount of space free of uric acid, i.e., the space between crystallites increases. Consequently, the volume fraction of uric acid will decrease as the pitch of the structure increases. It is based on this phenomenological analysis that an inverse relation between volume fraction of uric acid and effective spatial period, or pitch, has been assumed.

This assumption can be justified a little more rigorously in the following way: given a volume V_T inside the helical arrangement, located in the vertical neighborhood of the position z and with height proportional to the effective pitch P(z), it can be written as V_T= αAP(z), where α is a proportionality constant and A is the illuminated area. Within this volume, that occupied by the chitin fibrils can be estimated from V_C= [αP(z)/d][A^½/d][πA^½d²/4] where d∼3 nm is the diameter of the chitin fibrils [38], αP(z)/d is the number of chitin planes through the distance αP(z), A^½/d is the number of lateral chitin fibrils contained in the illuminated area, and πA^½d²/4 is an estimation of the volume of each fibril associated with the illuminated area. The available volume to be occupied by the uric acid nanocrystals is V = V_T-V_C ∼ αAP(z)/4. The effective uric acid volume fraction can be written as F_ua= C/P(z) where C is a constant. As consequence, the effective local volume fraction of uric acid is inversely proportional to the effective pitch. The effective value is defined by F_ua= C/<P(z)>, where < P(z)> is the spatial average value of the effective pitch. In this way, minimum and maximum values are given by F_ua,min= C/P_max and F_ua,max= C/P_min, respectively, with F_ua,min<F_ua<F_ua,max where P_min and P_max are minimum and maximum values of the effective pitch.

In this case, the system has two helical structures with a unidirectional layer in between. Through both twisted arrangements, the uric acid volume fractions are given by F_1,ua(z)=C₁/P₁(z) and F_2,ua(z)=C₂/P₂(z), with corresponding effective values defined as: F_1,ua= C₁/<P₁(z)> and F_2,ua= C₂/<P₂(z) > . The weighted effective uric acid volume fraction of the whole arrangement is F_ua= (h₁F_1,ua+uf_u+h₂F_2,ua)/h where f_u${\equiv}$<f_u(z)> is the average volume fraction of the unidirectional layer. To have a z-dependent continuous variation of the uric acid volume fraction, and assuming a linear variation through the unidirectional layer

(8)$${f_u}(z) = {f_u}(a)[{1 + \Delta ({z - a} )/u} ], $$

where the Δ-parameter determines the deviation from a constant value for f_u, the following identities must be satisfied: f_u(a)=F_1,ua(P₁(a))=C₁/P₁(a) and f_u(b)=F_2,ua(P₂(b))=C₂/P₂(b), where a = h₁ and b = h₁+ u. Therefore,

(9a)$${C_1} = {f_u}(a){P_1}(a) = \frac{{\left\langle {{f_u}(z)} \right\rangle {P_1}(a)}}{{1 + \Delta /2}}$$

and

(9b)$${C_2} = {f_u}(b){P_2}(b) = \frac{{\left\langle {{f_u}(z)} \right\rangle (1 + \Delta ){P_2}(b)}}{{1 + \Delta /2}},$$

with f_u= f_u(a)(1+Δ/2). The values of P₁(a) = 398 nm, P₂(b) = 497 nm, <P₁>=357 nm, and < P₂>=415 nm have been obtained from the analysis reported in the previous Section. By assuming a constant value for the uric acid volume fraction through the unidirectional layer (Δ=0), one obtains F_ua= 0.56. Caveney measures this average volume fraction as 0.70 which correlates with a value for Δ=1.80. This corresponds to a gradient of 0.25 µm^-1 for the effective uric acid volume fraction through the unidirectional layer. The uric acid volume fraction decreases from its deepest side to that at the bottom of the first helicoidal structure.

6. Photonic band gap characterization

Under normal illumination, the spectral position of the PBG is given by λ_o= n_av(λ_o)P_o for a twisted structure with a single pitch value P_o [39]. The larger the depth through the cuticle where the pitch is constant, the larger the height of the reflectance peak because more reflected rays interfering constructively are added. For a graded twisted arrangement of structural pitch P_s(z), the spectral position of the PBG is given by λ_o(z)=n_eff(λ_o,z)P_s(z) where n_eff(λ_o,z) is the effective refractive index of the chiral structure at depth z and for a λ_o-wavelength in the middle of the PBG. This can be probed in the following way: starting from λ_o(z) = [λ₊(z)+λ_-(z)]/2 for the wavelength in the middle of the PBG, the upper (+) and lower (-) limits are given by [6]

(10)$${\lambda _ \pm }(z) = \frac{{2\pi \sqrt {{\varepsilon _e}{\varepsilon _o}/{\varepsilon _{\textrm{av}}}} }}{{Q\sqrt {1 \mp \sqrt {1 - (1 - {K^2})(1 + {R^2}/{Q^4})} } }}$$

where K = (ε_e-ε_o)/(ε_e+ε_o) and ε_av= (ε_e+ε_o)/2. Equation (10) comes from the dispersion relation for the propagation of circularly polarized radiation through anisotropic media [40]. In previous works [6,37,41], Q and R have been written in the following ways:

(11a)$$Q = \frac{{2\pi }}{{P(z)}}\left[ {1 - \frac{z}{{P(z)}} \cdot \frac{{dP}}{{dz}}} \right],$$

(11b)$$R = \frac{q}{{P(z)}}\left[ {\frac{{2z}}{{P(z)}}{{\left( {\frac{{dP}}{{dz}}} \right)}^2} - 2\frac{{dP}}{{dz}} - z\frac{{{d^2}P}}{{d{z^2}}}} \right],$$

with q(z) = 2π/P(z). According with the definitions summarized in Section 4.1, one can recognize that Q = 2π/P_s(z)=G_s(z) (see Eq. (6b)) and R = dG_s/dz. Consequently, R²/Q⁴ is equal to (dP_s/dz)²/4π². The local value of the wavelength in the middle of the PBG is given by

(12)$${\lambda _o}(z) = \frac{1}{2}\sqrt {\frac{{{\varepsilon _e}{\varepsilon _o}}}{{{\varepsilon _{\textrm{av}}}}}} \left[ {\frac{1}{{\sqrt {1 - \Lambda } }} + \frac{1}{{\sqrt {1 + \Lambda } }}} \right]{P_s}(z) \equiv {n_{eff}}({\lambda _o},z){P_s}(z), $$

where $\Lambda = \sqrt {1 - (1 - {K^2})[1 + {{(d{P_s}/dz)}^2}/4{\pi ^2}]}$. If the chiral structure is characterized by a single pitch [P(z)=P_o= P_s], dP_s/dz = 0, and then n_eff(λ_o,z)=n_av= (n_e+n_o)/2, which is the expected result. For a twisted structure with a single pitch value, the width of the PBG is given by Δλ_o= ΔnP_o [39]. For a graded helicoid of structural pitch P_s(z), the width of the PBG is given by Δλ_o(z) = Δn_eff(λ_o,z) P_s(z). The explicit form of Δn_eff is obtained from Δλ_o(z) = λ₊(z)-λ_-(z), making use again of the dispersion relation specified by Eq. (10):

(13)$$\Delta {\lambda _o}(z) = \sqrt {\frac{{{\varepsilon _e}{\varepsilon _o}}}{{{\varepsilon _{\textrm{av}}}}}} \left[ {\frac{1}{{\sqrt {1 - \Lambda } }} - \frac{1}{{\sqrt {1 + \Lambda } }}} \right]{P_s}(z) \equiv \Delta {n_{eff}}({\lambda _o},z){P_s}(z), $$

where ε_e= ε_e(λ_o), ε_o= ε_o(λ_o), and ε_av= ε_av(λ_o). The Eq. (13) is given by Δλ_o= ΔnP_o with Δn = n_e-n_o for a helical structure with a single pitch P_o [39]. For non-normal incidence, the expressions for λ_o and Δλ_o contain the multiplicative factor cosθ, where θ is the refraction angle at the epicuticle-first helicoid interface, which introduces a blue shift in the reflectance spectra and in the depth dependence of the PBG. The larger the birefringence, the larger the width of the reflection peak. In the presence of a chirped structure, the peak becomes a broad reflection band. The depth- and spectral-variations of the extraordinary and ordinary refractive indices at PBG wavelengths can be obtained from that of the effective refractive index and birefringence. Namely, n_e(λ_o,z)=n_eff(λ_o,z)+Δn_eff(λ_o,z)/2 and n_o(λ_o,z)=n_eff(λ_o,z)-Δn_eff(λ_o,z)/2. Since Caveney’s research, recent works summarize the morphological characterization obtained from TEM or SEM analysis by displaying the lamellar pitch in terms of the number of half-turns. One can obtain the structural pitch as function of depth, P_s= P_s(z), from this kind of data representation. A cubic spline can be used to obtain a quasi-continuous variation of P_s(z), and the result will be like to the P_s-broken lines in Fig. 7. In the context of the novel expressions given in Eqs. (12) and (13), the photonic characterization can be carried out directly, provided that the optical constants and birefringence are known at PBG wavelengths.

Fig. 8. Variation with depth z of (a) the average effective refractive index and effective birefringence at wavelengths λ_o in the middle of the PBG, and (b) of the PBG and its spectral position for the first helicoidal structure in the elytron of C. resplendens.

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Figure 8(a) shows the depth dependence of the effective average refractive index n_eff and of the effective birefringence Δn_eff for wavelengths in the middle of PBG, as well as of the optical band gap Δλ, for the first helicoid. The variation with depth of the spectral position of the PBG is also displayed in Fig. 8(b). Propagating radiation with wavelengths contained in the range λ_o-Δλ_o/2 to λ_o+Δλ_o/2 is reflected more effectively due to constructive interference associated to Bragg reflections. Around the PBG, the radiation is reflected less efficiently. One could call this radiation as partially reflected LHCP light. The spectral average value of the PBG is <Δλ>=33.2 nm.

Its spectral position changes with depth z from 513 nm in the visible to 743 nm in the near infrared. The spectral average of the effective refractive index and birefringence are <n_eff>=1.59 and <Δn_eff>=0.082, respectively. Figure 9 displays the corresponding optical parameters for the second helicoid. The spectral average value of the PBG is <Δλ>=103.4 nm. Its spectral position changes with depth z from 449 nm in the visible to 1000 nm in the near infrared. The spectral average of the refractive index and birefringence are <n_eff>=1.66 and <Δn_eff>=0.246. This birefringence is three times that of the first helicoid. As consequence, the PBG through the deepest twisted structure is about three times that of the upper arrangement.

Fig. 9. Variation with depth z of (a) the average effective refractive index and effective birefringence at wavelengths λ_o in the middle of the PBG, and (b) of the PBG and its spectral position for the second helicoidal structure in the elytron of C. resplendens.

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The effect of the birefringence on the reflectance peak and its width has long been considered for helical structures with single pitches: the larger the birefringence, the larger the reflectance peak and its width [42]. A similar behavior is found in chiral materials with graded pitches but involving reflectance bands [43]. The model proposed here to describe the depth dependence of the uric acid through both helical structures and unidirectional layer, leads to values of birefringence which are consistent with the expected behavior observed for the RHCP- and LHCP-reflectance components. As indicated by Caveney [11]: In this species a major portion of the reflected light is right circularly polarized, with a broad peak between 575 and 624 nm; and a minor reflection of left circularly polarized light with a peak at 560 nm. The luminous reflectance values indicated in Fig. 4 are consistent with this assertion: Y_RHCP= 28.0% and Y_LHCP= 25.4%. The second helicoid has large values of the uric acid volume fraction (F_2,ua= 0.72), and the large gradient of this through the unidirectional layer leads to lower values of the uric acid volume fraction through the first helicoid (F_1,ua= 0.24). Diagrams like those displayed in Figs. 8(b) and 9(b) have been obtained in previous works when considering other species of scarabs [6,37,41]. In such cases, the method consists of using Eqs. (10) and (11) which are based on the z-dependence of the effective pitch. The method followed here is based on the z-dependence of the structural pitch.

Once obtained the photonic characterization of both helical structures, a correlation can be made with the reflection spectra. With this purpose, the co-polarized reflectance measurements reported by Finlayson et al. [22] are used. These correspond to samples normally illuminated. The original data were kindly share by E. D. Finlayson, with similar ones also reported in [21]. They used co- and cross-polarizers, and a small illumination sport, to carry out the measurements. These two facts allow to discriminate the presence of a ripple structure in the measured reflectance spectra, for wavelengths larger than 475 nm, due to less dispersion on the phase differences of the rays reflected at different depths. The transparent waxy epicuticle, whose surface contains flat scales with geometrical cross sections close to 50 µm², contributes with a small background of non-polarized reflectance [23]. To highlight the prominent spectral features, Fig. 10 displays the smoothed versions of the measured dominant co-polarized reflectance components of the spectra, whose details include those seen in Caveney’s measurements. The cross-polarized spectra show significant smaller reflectance with values within 0.06 ± 0.02 for R_LR and 0.07 ± 0.02 for R_RL through the whole wavelength range considered. The spectrum corresponding to the LHCP reflected light, R_LL, is characterized by a small shoulder with no ripple structure at short wavelengths, a reflection peak centered close to 590 nm, and a largest peak spectrally located at 740 nm. According to Fig. 8, the shoulder involves wavelengths smaller than those PBGs correlated to Bragg reflections occurring close to the top of the first helicoid. This reflectance shoulder is due to what was called previously as partially reflected LHCP light. Beyond 475 nm, the first reflectance peak increases rapidly with wavelength, involving Bragg reflections for depths from the top of the structure to 2.6 µm with a local minimum in the birefringence at depths close to 2.2 µm. Beyond 2.6 µm in depth, the increase of the structural pitch varies more slowly with depth (see Fig. 7(a)), and the effective refractive index and the PBG show the same behavior. This favors the presence of the largest peak observed in the reflectance spectrum, while the decrease of the birefringence does not. These two effects compete, reaching the maximum of the reflectance peak in 740 nm, as mentioned before. The RHCP reflectance spectrum, R_RR, shows basically a broad reflection band from 500 nm to the limit of measurements, 850 nm. The birefringence through the second helicoid is about three times that of the first structure. Figure 9(b) shows that the reflectance band start at 453 nm, involving Bragg reflections at depths close to 6.3 µm, where the birefringence shows a local minimum (see Fig. 9(a)). The large birefringence, which decreases beyond 8.1 µm, makes the reflectance spectrum less sensitive to variations in the gradient of the structural pitch with depth.

Fig. 10. Smoothed co-polarized reflectance spectra by the cuticle of a C. resplendens specimen. R_LL (R_RR) corresponds to the spectrum with LHCP (RHCP) incident radiation and LHCP (RHCP) reflected light, with the sample being normally illuminated. The curves are smoothed versions of the data displayed in Fig. 3(a) of [22]. The original data were kindly shared by E. D. Finlayson. Similar measurements were reported in Fig. 7.7 of [21].

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Berreman’s formalism is a theoretical matrix model developed from solving Maxwell’s equations, whose implementation is basically computational due to a large quantity of calculations required to describe the propagation of the light through a large set of optically anisotropic quasi-multilayers. Transmittance and reflectance spectra, as well as relative intensities of forward and backward propagating fields can be evaluated from matrix formalisms like Berremańs one and others [44,45]. They do not provide explicit expressions for the z-dependent spectral position of the PBG and corresponding effective refractive index at PBG-wavelengths, or its width and its corresponding effective birefringence at PBG-wavelengths. In previous works carried out in our group, Eqs. (10) and (11) have been used to display the z-dependence of the PBG. Now, a novel concise way to obtain these z-dependences from the variation of the structural pitch (Eqs. (12) and (13)), as well as n_eff(λ_o,z) and Δn_eff(λ_o,z), is reported. The dependence on z of the spectral position of the PBG and birefringence at PBG-wavelengths, for each of the helical structures of the cuticle, are directly related to the presence of reflection peaks or bands in the corresponding reflectance spectra.

The formalism linked with Eqs. (12) and (13) can also be applied in the context of an inversion approach to tune the reflectance spectra of functional materials. Once defined depth dependent profiles for the PBG and its width, λ_o(z) and Δλ_o(z) subjected to the condition Δλ_o(z)/2λ_o(z)$\le$(1-γ)/(1+γ) with γ=((1-K)/K)^½, the required z-dependence of the structural pitch can be obtained by integration of

(14)$$\frac{{d{P_s}}}{{dz}} = sign \cdot 2\pi {\left[ {\frac{{{K^2}{{(1 + {{\rm X}^2})}^2} - {{(1 - {{\rm X}^2})}^2}}}{{(1 - {K^2}){{(1 + {{\rm X}^2})}^2}}}} \right]^{\textrm{ }1/2}}, $$

where X = [1-Δλ_o(z)/2λ_o(z)]/[1+Δλ_o(z)/2λ_o(z)], sign=+1 when dλ_o/dz > 0, and sign=-1 when dλ_o/dz < 0. Equation (14) is obtained from the ratio between Eqs. (13) and (12) after some straightforward arithmetic operations.

7. Color calculations

In concordance with the type of light source used by Caveney, the color calculations reported in Fig. 4 were carried out by assuming an incandescent tungsten lamp as illuminant, whose reference chromaticity coordinates are x´=0.4476 ad y´=0.4074 [46]. The standardized CIE spectral chromaticity matching functions [x(λ), y(λ), and z(λ)] have been used. It is particularly interesting to use this framework to approach λ_HWP from the dominant wavelength λ_d corresponding to the RHCP reflectance spectra. The chromaticity coordinates (x,y) corresponding to the spectra of Fig. 4 were calculated from

(15)$${X_j} = \left[ {\int\limits_{{\lambda_o}}^{{\lambda_1}} {R(\lambda ){S_A}(\lambda ){x_j}(\lambda )d\lambda } } \right]/\left[ {\int\limits_{{\lambda_o}}^{{\lambda_1}} {{S_A}(\lambda )y(\lambda )d\lambda } } \right], $$

where S_A(λ) is the relative spectral power distribution of illuminant A (incandescent tungsten lamp), and j = 1,2,3. In addition, λ_o= 255 nm, λ₁= 740 nm, x₁= x, x₂= y, x₃= z, X₁= X, X₂= Y, and X₃= Z. Namely, x = X/(X + Y+Z) and y = Y/(X + Y+Z). For the reflectance spectrum of Fig. 4(a): x = 0.5338 and y = 0.4287, and x = 0.4697 and y = 0.4722 for that of Fig. 4(b).

The value of λ_d is read from the intersection of the locus in the CIE chromaticity diagram and the line drawn from point (x´,y´) to (x,y). As seen in Fig. 11 where the (x,y) coordinates corresponding to the RHCP reflectance spectrum of Fig. 4(a) were considered, the dominant wavelength, λ_d, is close to 590 nm. By assuming linear variation of wavelength between the dots corresponding to 585 and 590 nm, the estimated value of λ_d is 588.7 nm. By using the subroutine colour.dominant_wavelength provided by Colour, affiliated project of NumFOCUS, within its open-source Python package, the following results: λ_d= 588 nm is obtained for the RHCP reflectance spectra of Fig. 4(a) and λ_d= 576 nm for Fig. 4(b) displaying the LHCP reflectance spectra. As seen, ${\lambda _{HWP}} \cong {\lambda _d}$ for the RHCP reflectance spectrum.

Fig. 11. CIE chromaticity diagram showing the method followed to obtain the dominant wavelength corresponding to the chromaticity coordinates obtained from the RHCP reflectance spectrum displayed in Fig. 4(a).

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8. Summary and conclusions

This paper reports a methodology to analyze the optical properties of natural optically anisotropic uniaxial materials characterized by graded pitches of two corresponding helicoidal structures, positioning between them a unidirectional layer acting as a half wave retarder. The present theoretical framework can be useful when considering the optical properties of natural [47,48] and synthetic functional materials with similar twisted structures [49–52]. The method has been applied to reflectance spectra and electron microscopy characterization of the elytron of a C. resplendens jewel scarab beetle, reported in the pioneering work of Caveney which has been the starting point of much research works recently published [21–23,27,53–55]. The optical and compositional characterization of the structured cuticle have been extended, with description of the relationship between structural and effective pitches, the introduction of a model to describe the variation as a function of depth of the uric acid volume fraction, and novel expressions to obtain the local effective refractive index and birefringence of helicoidal structures at wavelengths of the photonic band gap, are the major contributions of this work. For the application carried out, the correlation between effective birefringence so obtained for each helicoidal structure in the cuticle, is consistent with the relative weight of each RHCP and LHCP reflectance component of the total reflectance spectrum measured by Caveney. The key role that the determination of at least the average value of the uric acid volume fraction has must be highlighted. The application of the present model depends on this fact, which would encourage the researchers to measure this quantity or, even better, to carry out measurement of uric acid volume fraction profiles through the cuticle of this kind of biological systems.

From morphological information obtained by means of electron microscopy and spectrophotometric measurements, a complete characterization of the crystalline photonic behavior of the material can be carried out, thus being able to compare natural or synthetic optical systems that even with the naked eye could appear identical. This capability could be of relevance in the development of chiral materials with specific optical functionalities, and in traditional fields like taxonomic identification in entomology and botanic, as well as in health sciences [56]. In the context of an inversion approach, once desirable profiles of the PBG [λ_o(z)] and its spectral width [Δλ_o(z)] have been defined, the new formalism associated with Eqs. (12) and (13) can allow to determine the dependence on z of the structural pitch profile that each helical structure should have. This could provide a method for the development of functional materials.

The author wishes to emphasize the following issue related with the use of effective and structural pitches in the context of radiative transfer or finite element methods to calculate reflectance or transmittance spectra: the pitch that is directly involved in obtaining interpolated values of ϕ(z), with no integration, is the effective pitch, not the structural one. This latter would be involved through the average value of its inverse which in fact gives the inverse of the effective pitch. This issue was not recognized or clearly established by many authors whose research works have served as valuable references in this field of optical physics [33,43]. As Berreman himself did, they used the word pitch with no distinction between effective or structural one when using the equation ϕ(z) = 2πz/P(z) to evaluate the components of the dielectric function tensor to be used in radiative transfer matrix calculations, or in finite element methods. In the novel equations reported in this work to carry out the photonic characterization of a helical structure (Eqs. (12) and (13)), the structural pitch is directly involved. In Berreman’s formalism and in finite element methods, the effective one must be used.

Acknowledgments

The author thanks the support given by the Universidad de Costa Rica to carry out this research work. The author also thanks to Ewan D. Finlayson, at the University of Exeter, for kindly sharing the original data used to display the Fig. 10 of this article.

Disclosures

The author declares no conflicts of interest.

Data availability

Optical constants and birefringence of the materials involved in this research are available upon reasonable request.

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Theoretical framework to describe the reflection of circularly polarized light by a natural photonic crystal: elytron of a Chrysina resplendens scarab

Abstract

1. Introduction

2. Morphology of the optical system and optical measurements

3. Unidirectional layer

4. Helicoids: description of the structure, effective and structural pitches

4.1 From the structural pitch to the effective one and the azimuth angle

4.2 From the azimuth angle to the effective pitch and the structural one

5. Profile of the effective uric acid volume fraction through the whole structure

6. Photonic band gap characterization

7. Color calculations

8. Summary and conclusions

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Equations (18)

Optics Continuum