investigation of nacre nanostructure by analyzing its structural color pattern

Produced by light interacting with structures at micrometer or nanometer levels, structural color can be utilized to investigate details of material structures. In this article, we studied the pattern of interference color generated from repetitive aragonite–conchiolin double layers on colorless nacreous pearls. Based on qualitative wave analysis and quantitative electromagnetic computation, we theoretically concluded such patterns are mainly determined by aragonite layer thickness. We also demonstrated how to predict the aragonite layer thickness and estimate the conchiolin refractive index variation on a Tahitian pearl with near-colorless body color and strong iridescence. We believe this approach offers a new perspective to study nanostructures in biology and mineralogy.

In Fig. S1a, we plotted the electromagnetically computed reflectance, which was used to calculate the CIELCH colors in above table. Fig. S1b shows the position of the series colors at cylindrical CIELCH space. With a schematic demonstration on how to fill in the cylindrical space with colors of 8-bit computer graphics (Fig. S1c), we synthesized the series colors in Fig. S1d. As no real color gamut could fit in the numerical CIE color space, Fig. S1c doesn't reflect the real mapping from sRGB to CIELCH. With some inevitable distortions, it is still sufficient for illustrative purpose.

Insight into light waves propagating and interacting with nacre's multiple double layers
Assume a monochromatic wave ̃ incident on two media's boundary, part of it reflects back as ̃, while the remaining portion transmits as ̃ (Fig. S2a). ̃ and ̃ could be expressed as ̃ and ̃, where r and t are amplitude coefficient upon reflection and transmission. r and t could be calculated via Fresnel Equations at p-and s-polarizations (parenthesized in equations S6) [2]. While t is always positive, r could be either positive or negative, with minus sign indicating a π phase shift. The absolute value of r is relatively very low, so we could focus on the light scattering backwards upon 1st reflection at each interface for multiple boundaries (̃1 , ̃2 , …) (Fig.   S2b). The RI and layer thickness are alternative (n1, d1) and (n2, d2) for nacre structure. We set n0=n2 for the simplicity of mathematical treatment, so the amplitude coefficient could be notated as (r12, t12), and (r21, t21) on every other interface. As r12=-r21 due to a π phase shift between the internal and external reflections, the reflected waves could be expressed by Equations (S6), where is the OPD associated with each submicroscopic layer, and λ0 is the wavelength in vacuum. The waves could be grouped by one reflected from conchiolin/aragonite interface, and the other, aragonite/conchiolin interface. The waves at each group sustain the same phase difference associated with the sum OPD of aragonite-conchiolin double layer. As is illustrated in Fig. S1c, the waves would enhance when their phase difference is integer multiple of 2π, otherwise, they average out.

Figure S2. Schematic illustration of (a) light incident at two media's boundary in the form of p-polarized wave, and (b) light incident at consecutive double -layers. (c) Superposition of in-phase & out-of-phase waves. (d) Superposition of two near out-of-phase waves with slightly varying phase difference and amplitude.
Above is a detailed illustration of how interference equations (1) and (2) were derived following Bragg's Law, which simply reveals the relationship between the nanostructure and the hue of interference color. To further analyze how lightness and chroma vary, we focused on the waves reflecting from one layer's top and bottom interfaces, ̃( −1) and ̃. Under Bragg condition, the two waves have a phase difference of either − 2 0 2 or + 2 0 2 ; and the superposition of such consecutive two waves keep in-phase, so the irradiance of enhanced light waves highly correlates with the amplitude of such a superposition. As ̃( −1) and ̃ are near out-of-phase with a small deviation of 2 0 2 , a decrease in d2 results a decrease in amplitude of the superposition (Figs. S2d, left). On the other hand, an increase in n2 also reduces amplitude of the superposition, as the result of decrease in individual amplitude (Figs. S2d, right). In summary, either a decrease in d2 or an increase in n2 would reduce the lightness and chroma of interference color, and vice versa. In addition, such a qualitative analysis could be quantified by the electromagnetic method, with reflections of all orders taken into account.
The discussion so far hasn't taken coherent length into consideration. The coherent length of the light source we generally encounter is unmeasurable, but we can qualitatively estimate when reflecting light becomes incoherent based on an approximate range. This range is several dozens of micrometers according to Young's experiment. For the wave propagating forward, ̃ suffers amplitude decay every time it advances through two media's boundary, as indicated by the transmittance coefficient t. And the decay doubles on the reflecting wave, as expressed at equations (S7). In terms of irradiance, I is square of the amplitude (equations S8). The irradiance of the light reflecting from layer j keeps exponential decay with (t12t21) 2 as the base, which equals [4(n1n2)/(n1+n2) 2 ] 2 at normal incident. Given n1 is constant and n1>n2, a greater conchiolin index means slower energy decay of light when it impinges in the repetitive double layers, so higher proportion of incident light could reflect back from deeper layers. The longer the light waves travel, the more likely they become out of coherent length. As wavelengthindependent reflection mainly adds lightness to the color, a greater conchiolin index possibly results a less saturated color. We plotted the exponential decay associated with as reference (Fig. S3), with it in mind that is slightly lower than the actual irradiance reflecting at each boundary, which includes the energy from higher orders of reflections.

Impact from pigment absorption
Because it is not easy to accurately obtain the absorbance, we set various intensities on a spectrum collected from a golden pearl surface to mimic the different pigment concentrations [1]. Fig. 4S shows the manipulated spectra and the corresponding pattern simulations for 350 nm aragonite thickness case. Figure S4. Interference pattern simulations with assumed absorbance of a golden pearl.