## Abstract

On the wing of the moth *Trichoplusia orichalcea* a prominent, apparently highly reflective, golden spot can be seen. Scales from this area of the wing exhibit a regular microstructure resembling a submicrometer herringbone pattern. We show that a diffraction process from this structure is responsible for the observed optical properties, such as directionality, brightness variations, polarization, and color.

© 1995 Optical Society of America

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### Equations (12)

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(1)
$$\begin{array}{ll}E\left(P\right)=\hfill & A{\stackrel{\u02c6}{k}}^{d}\times {\displaystyle \underset{S}{\iint}\left[\stackrel{\u02c6}{n}\times E-\eta {\stackrel{\u02c6}{k}}^{d}\times \left(\stackrel{\u02c6}{n}\times H\right)\right]}\hfill \\ \hfill & \times \text{exp}\left[ik\left({\stackrel{\u02c6}{k}}^{d}-{\stackrel{\u02c6}{k}}^{i}\right)\xb7r\right]\mathrm{d}S,\hfill \end{array}$$
(2)
$${H}^{i,d}=\frac{1}{\eta}{\stackrel{\u02c6}{k}}^{i,d}\times {E}^{i,d}.$$
(3)
$$\begin{array}{l}{\stackrel{\u02c6}{u}}_{1}=\left(+\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},+\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},0\right),\\ {\stackrel{\u02c6}{\nu}}_{1}=\left(-\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},+\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},\sqrt{1-{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\beta}\right),\end{array}$$
(4)
$$\begin{array}{l}{\stackrel{\u02c6}{u}}_{2}=-{\stackrel{\u02c6}{u}}_{1},\\ {\stackrel{\u02c6}{\nu}}_{2}=\left(+\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},-\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},\sqrt{1-{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\beta}\right),\end{array}$$
(5)
$$\begin{array}{l}{\stackrel{\u02c6}{u}}_{3}=\left(+\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},-\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},0\right),\\ {\stackrel{\u02c6}{\nu}}_{3}=\left(+\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},+\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},\sqrt{1-{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\beta}\right),\end{array}$$
(6)
$$\begin{array}{l}{\stackrel{\u02c6}{u}}_{4}=-{\stackrel{\u02c6}{u}}_{3},\\ {\stackrel{\u02c6}{\nu}}_{4}=\left(-\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},-\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{b},\sqrt{1-{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\beta}\right),\end{array}$$
(7)
$$\begin{array}{ccc}{\stackrel{\u02c6}{n}}_{j}={\stackrel{\u02c6}{u}}_{j}\times {\stackrel{\u02c6}{\nu}}_{j},& j=1,\dots ,4,& {\stackrel{\u02c6}{n}}_{5}={\stackrel{\u02c6}{n}}_{6}=\left(0,0,1\right)\end{array}.$$
(8)
$$\begin{array}{r}\left[\begin{array}{l}{{n}_{j}}^{{x}^{\prime}}\hfill \\ {{n}_{j}}^{{z}^{\prime}}\hfill \end{array}\right]=\left[\begin{array}{ll}\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{d}\hfill & \pm \text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{d}\hfill \\ \mp \text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{d}\hfill & \text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{d}\hfill \end{array}\right]\phantom{\rule{0.2em}{0ex}}\left[\begin{array}{l}{{n}_{j}}^{x}\hfill \\ {{n}_{j}}^{z}\hfill \end{array}\right],\\ j=1,\dots ,6,\end{array}$$
(9)
$$\begin{array}{r}\left[\begin{array}{l}{{n}_{j}}^{{x}^{\prime}}\hfill \\ {{n}_{j}}^{{z}^{\prime}}\hfill \end{array}\right]=\left[\begin{array}{ll}\text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{r}\hfill & -\text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{r}\hfill \\ \text{sin}\phantom{\rule{0.2em}{0ex}}{\varphi}_{r}\hfill & \text{cos}\phantom{\rule{0.2em}{0ex}}{\varphi}_{r}\hfill \end{array}\right]\phantom{\rule{0.2em}{0ex}}\left[\begin{array}{l}{{n}_{j}}^{x}\hfill \\ {{n}_{j}}^{z}\hfill \end{array}\right],\\ j=1,\dots ,6.\end{array}$$
(10)
$${r}_{pq}=\left[x,y,{z}_{pq}-\frac{{{n}_{j}}^{x}}{{{n}_{j}}^{z}}\left(x-{x}_{p}\right)-\frac{{{n}_{j}}^{y}}{{{n}_{j}}^{z}}\left(y-{y}_{q}\right)\right],$$
(11)
$$\begin{array}{ll}E\left(P\right)=\hfill & A{\displaystyle \sum _{j=1}^{6}{\stackrel{\u02c6}{k}}^{d}\times \left[\left({\stackrel{\u02c6}{n}}_{j}\times {E}_{j}\right)-\eta {\stackrel{\u02c6}{k}}^{d}\times \left({\stackrel{\u02c6}{n}}_{j}\times {H}_{j}\right)\right]}\hfill \\ \hfill & \times {\displaystyle \sum _{p,q}{\displaystyle \iint \text{exp}\left[ik\left({\stackrel{\u02c6}{k}}^{d}-{\stackrel{\u02c6}{k}}^{i}\right)\xb7{r}_{pq}\right]\mathrm{d}x\mathrm{d}y}}.\hfill \end{array}$$
(12)
$$\begin{array}{l}{\displaystyle \iint \text{exp}\left[ik\left({\stackrel{\u02c6}{k}}^{d}-{\stackrel{\u02c6}{k}}^{i}\right)\xb7{r}_{pq}\right]\mathrm{d}x\mathrm{d}y}\\ \phantom{\rule{0.6em}{0ex}}=-lw\left(\frac{\text{exp}\left(i{\theta}_{1}\right)-1}{{\theta}_{1}}\right)\left(\frac{\text{exp}\left(i{\theta}_{2}\right)-1}{{\theta}_{2}}\right)\text{exp}\left(i{\theta}_{3}\right),\end{array}$$