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Modelling structural colour from helicoidal multi-layer thin films with natural disorder

T. J. Davis, L. Ospina-Rozo, D. Stuart-Fox, and A. Roberts

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Abstract

A coupled mode theory based on Takagi-Taupin equations describing electromagnetic scattering from distorted periodic arrays is applied to the problem of light scattering from beetles. We extend the method to include perturbations in the permittivity tensor to helicoidal arrays seen in many species of scarab beetle and optically anisotropic layered materials more generally. This extension permits analysis of typical dislocations arising from the biological assembly process and the presence of other structures in the elytra. We show that by extracting structural information from transmission electron microscopy data, including characteristic disorder parameters, good agreement with spectral specular and non-specular reflectance measurements is obtained.

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Supplementary Material (1)

NameDescription
Supplement 1       Derivation of main equations

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. a) Sketch of the structure of the hardened wing casing (elytron) of the beetle Anoplognathus parvulus. The elytron is a composite structure in which the trabeculae, cylindrical fibres and endocuticle provide structural support [52]. The exocuticle contains the chiral nanostructures that produce circularly polarized reflectance [53]. The dimensions of the elytron components were obtained from TEM photographs of transversal sections. b)-d) Examples of circular polarisation dependence of structural colour in three scarab species: b) Anoplognathus chloropyrus, c) Anoplognathus parvulus, d) Anoplognathus smaragdinus. The beetles were illuminated with white light and the reflected light filtered by the appropriate polariser.

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Fig. 2.
Fig. 2. The array defect model considers two types of defects: a) regions of the array suffer displacements $\mathbf { u}(s_h)$ with random step changes. The gradients appears like uncorrelated delta functions; b) tilted regions or regions with different periodicities lead to continually changing displacements $\mathbf { u}(s_h)$ that are approximated by linear variations with distance, resulting in gradients changing randomly in steps and the second derivatives appearing like uncorrelated delta functions.

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Fig. 3.
Fig. 3. a) The correlation function Eq. (23) as a function of $\tau /l$ with $2\alpha _h^2 \sigma ^2 l=1$; b) the Fourier transform $\rho (k)$ of the correlation function compared with a Lorentzian $1/(1+7 k^2)$ and a Gaussian $e^{-(7k)^2}$

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Fig. 4.
Fig. 4. The periodicity of layers within a beetle shell. a) Structural colour from Anoplognathus parvulus; b) Transmission electron micrograph (TEM) of a section of the shell. The rectangle shows the region where the image data were Fourier transformed; c) The Fourier transform of the TEM image with three periodicities highlighted that contribute to the optical spectrum. The transform is mirror symmetric in one plane parallel to the layers; d) A profile through the centre of the transform peaks and a sum of three Gaussian functions used to model the transform.

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Fig. 5.
Fig. 5. Comparison between experiment and theory spectra from the Anoplognathus parvulus sample of Fig. 4(c) in different illumination-observation geometry. (a) Two components of the sample’s colouration were measured: Specular reflectance was obtained when the angle of the illumination probe $\theta _i$ and the angle of the detector probe $\theta _r$ were opposite but of the same magnitude; Reflectance in geometries away from the mirror angle was measured by varying the position of the bisector so that the sum $\theta _i+\theta _r$ was always $20^\circ$. (b) Experimental and theoretical spectra at the mirror angle. The experimental data were scaled to give the same background offset. The curves have been offset for clarity; c) Experimental and theoretical spectra at different geometries away from the mirror angle to study the effect of small-scale disturbances in the chiral nanostructures.

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Equations (34)

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δ ϵ ¯ ( r ) = ∑ h ϵ ¯ h e i h ⋅ ( r − u ( r ) ) ,
D ( r ) = ∑ h D h ( r ) e i ( k h ⋅ r − h ⋅ u ( r ) ) ,
d R ¯ d z = − i α h ( M ¯ h + 2 β h R ¯ + ( α 0 / α h ) R ¯ ⋅ M ¯ − h ⋅ R ¯ ) .
β h = h 2 + 2 n b k 0 ⋅ h 2 ϵ b k 0 2 − 1 n b k 0 ∂ ( h ⋅ u ) ∂ s h ,
cos ⁡ θ B = n λ 2 n b d .
cos ⁡ θ h = n λ n b d − cos ⁡ θ 0 = 2 cos ⁡ θ B − cos ⁡ θ 0 .
ϵ ¯ h = ( ϵ h x x ϵ h x y 0 ϵ h y x ϵ h y y 0 0 0 0 ) ,
M ¯ h = ( − cos 2 ⁡ θ h 0 1 2 sin ⁡ 2 θ h 0 − 1 0 1 2 sin ⁡ 2 θ h 0 − sin 2 ⁡ θ h ) ( ϵ h x x ϵ h x y 0 ϵ h y x ϵ h y y 0 0 0 0 ) = ( − ϵ h x x cos 2 ⁡ θ h − ϵ h x y cos 2 ⁡ θ h 0 − ϵ h y x − ϵ h y y 0 1 2 ϵ h x x sin ⁡ 2 θ h 1 2 ϵ h x y sin ⁡ 2 θ h 0 ) .
M ¯ − h = − ( ϵ − h x x cos 2 ⁡ θ 0 ϵ − h x y cos 2 ⁡ θ 0 0 ϵ − h y x ϵ − h y y 0 1 2 ϵ − h x x sin ⁡ 2 θ 0 1 2 ϵ − h x y sin ⁡ 2 θ 0 0 ) .
M ¯ h = ϵ h ( − cos 2 ⁡ θ h 0 0 0 − 1 0 1 2 sin ⁡ 2 θ h 0 0 ) ,
M ¯ − h = − ϵ − h ( cos 2 ⁡ θ 0 0 0 0 1 0 1 2 sin ⁡ 2 θ 0 0 0 ) .
R ¯ = R ¯ p + R ¯ s = R p ( cos ⁡ θ h cos ⁡ θ 0 0 cos ⁡ θ h sin ⁡ θ 0 0 0 0 − sin ⁡ θ h cos ⁡ θ 0 0 − sin ⁡ θ h sin ⁡ θ 0 ) + R s ( 0 0 0 0 1 0 0 0 0 ) ,
d R ¯ d z = − i α h ( M ¯ h + 2 β h R ¯ ) ,
R ¯ ( z ) = − i α h e − 2 i α h ∫ 0 z β h ′ d z ′ ∫ 0 z e 2 i α h ∫ 0 z ′ β h ′ ′ d z ′ ′ M ¯ h ′ d z ′ ,
R ¯ ( z ) = e − i α h β h z sin ⁡ ( α h β h z ) β h M ¯ h .
M ¯ h ⋅ D 0 ( z t ) / D 0 = ( sin ⁡ 2 θ h z ^ / 2 − cos 2 ⁡ θ h x ^ ) ( ϵ h x x ± i ϵ h x y ) − y ^ ( ϵ h y x ± i ϵ h y y ) .
ϵ ¯ ( z ) = ( ϵ α cos 2 ⁡ h z + ϵ β sin 2 ⁡ h z ( ϵ α − ϵ β ) sin ⁡ h z cos ⁡ h z 0 ( ϵ α − ϵ β ) sin ⁡ h z cos ⁡ h z ϵ α sin 2 ⁡ h z + ϵ β cos 2 ⁡ h z 0 0 0 ϵ γ ) ,
ϵ ¯ ( z ) = ϵ γ ( 0 0 0 0 0 0 0 0 1 ) + ϵ α + ϵ β 2 ( 1 0 0 0 1 0 0 0 0 ) + ( ϵ α − ϵ β ) 4 ( 1 − i 0 − i − 1 0 0 0 0 ) e i 2 h z + ( ϵ α − ϵ β ) 4 ( 1 i 0 i − 1 0 0 0 0 ) e − i 2 h z .
l d β ξ ( z ) d z + β ξ ( z ) = σ ξ ( z ) ,
⟨ β ξ ( z ) β ξ ( z + τ ) ⟩ = σ 2 2 l e − | τ | / l .
R ( z ) = − i α h κ h e − 2 i α h ∫ 0 z β ( z ) d z ∫ 0 z e 2 i α h ∫ 0 z ′ β ( z ′ ′ ) d z ′ ′ d z ′ ,
⟨ R ∗ R ⟩ = | α h κ h | 2 ∫ 0 z ∫ 0 z e 2 i α h ∫ z ′ ′ z ′ β h ′ ′ ′ d z ′ ′ ′ ⟨ e 2 i α h ∫ z ′ ′ z ′ β ξ ′ ′ ′ d z ′ ′ ′ ⟩ d z ′ d z ′ ′ .
⟨ e 2 i α h ∫ z ′ ′ z ′ β ξ ′ ′ ′ d z ′ ′ ′ ⟩ = e − 2 α h 2 ∫ z ′ ′ z ′ ∫ z ′ ′ z ′ ⟨ β ξ ( z 1 ) β ξ ( z 2 ) ⟩ d z 1 d z 2 = e − 2 α h 2 σ 2 l [ | τ | / l + exp ⁡ ( − | τ | / l ) − 1 ] ,
e − 2 α h 2 σ 2 l [ | τ | / l + exp ⁡ ( − | τ | / l ) − 1 ] ≡ 1 2 π ∫ − ∞ ∞ ρ ( k ) e − i k τ d k ,
⟨ R ∗ R ⟩ = 1 2 π ∫ − ∞ ∞ ρ ( k ) | R ( 2 α h β h − k ) | 2 d k .
lim l → 0 ρ ( k ) = 4 α h 2 σ d 2 4 α h 4 σ d 4 + k 2 ,
lim l → ∞ ρ ( k ) = π α h σ b exp ⁡ ( − k 2 4 α h 2 σ b 2 ) .
⟨ R ¯ † ⋅ R ¯ ⟩ = 1 2 π ∫ − ∞ ∞ ρ ( k ) [ R ¯ † ( 2 α h β h − k ) ⋅ R ¯ ( 2 α h β h − k ) ] d k .
⟨ R ∗ R ⟩ = | α h κ h | 2 2 π ρ ( 2 α h β h ) ,
k h = n b k 0 + h + n b k 0 β ξ s ^ h = ⟨ k h ⟩ + 2 α h β ξ z ^ ,
⟨ R ∗ R ⟩ ≈ N k 0 2 P ( β h ) ,
⟨ R ¯ † ⋅ R ¯ ⟩ ≈ N k 0 2 [ M ¯ h † ⋅ M ¯ h ] P ( β h ) ,
I h = ⟨ D h † ⋅ D h ⟩ = D 0 † ⋅ ⟨ R ¯ † ⋅ R ¯ ⟩ ⋅ D 0 = N k 0 2 [ D 0 † ⋅ M ¯ h † ⋅ M ¯ h ⋅ D 0 ] P ( β h ) .
D 0 † ⋅ M ¯ h † ⋅ M ¯ h ⋅ D 0 = | ϵ h | 2 ( t s ′ 2 + t p ′ 2 cos 2 ⁡ θ h ) ( cos 2 ⁡ θ 0 t p 2 | D p | 2 + t s 2 | D s | 2 − 2 cos ⁡ θ 0 t p t s Im D p D s ∗ ) ,
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