In this section, we calculate and compare band structures for different periodic structures. Two different types of models have appeared in the literature: those that model the lattices by skeletons of interconnecting struts and those that use smooth surfaces (such as the minimal surfaces formed by lipid–water mixtures;

Hyde *et al*. 1997). Different models that share the same leading terms in their Fourier series will exhibit very similar optical properties; in particular, skeletal and minimal surface models differ only in their higher order Fourier harmonics. The MIT photonic bands program does allow both types of models but the strut models are slightly easier to implement.

We have constructed skeletal strut models of three common periodic structures, and one primitive unit of each of these structures is shown in . Note that the outermost struts in each structure occur as three pairs of corresponding struts that point in the same direction; these pairs of struts join adjacent primitive units together to create a three-dimensional periodic structure.

Note also that the P-structure has nodes where six struts connect at mutual angles of 90°; the D-structure has nodes where four struts connect at mutual angles of 109.47°; and the G-structure has nodes where three struts connect at mutual angles of 120°.

The Fourier series of the refractive indices all have the form

where

is the average index of the structure; Δ is proportional to the index contrast; and

*F*(

*x*,

*y*,

*z*) represents the leading terms in the series. Both

and Δ will vary with the volume fraction of chitin in butterfly wing scales. Chitin exhibits some absorption that could be modelled by allowing

and Δ

*n* to have a small imaginary part. This small effect has been ignored since it does not alter any of the conclusions made in this paper.

The form of the function

*F* for the three different structures is

Note that elsewhere in the literature these expressions may have been written in different but mathematically equivalent forms. These same trigonometric functions can also be used to approximate minimal surfaces. In such models, the boundary between chitin and air is defined by an equation of the form

or, in other words, a level set. For strut-based models, the volume fraction of chitin can be chosen by varying the thickness of the struts. For minimal surface models, the volume fraction can be chosen by varying the parameter

*c*. For

*c*=0, a 50 per cent volume fraction is obtained.

The mathematically inclined will note that the leading terms precisely capture the rotation and reflection symmetries of their respective structures. Importantly, *F*_{P} and *F*_{D} have inversion symmetry, whereas *F*_{G} does not and thus corresponds to a chiral structure (i.e. it is not equivalent to its mirror image). Furthermore, we would like to point out how the Fourier series relate to the BZ. The three terms in *F*_{P} correspond to the three pairs of opposite faces of the BZ in *a*. Likewise, the six terms in *F*_{G} correspond to the six pairs of opposite faces of the BZ in *b*. Finally, the four terms in *F*_{D} correspond to the four pairs of opposite hexagonal faces of the BZ in *c*. The smaller square faces on this BZ do not have any Fourier terms associated with this type of tetrahedral geometry (though there are more complicated higher connectivity minimal surfaces, such as the C(P) structure, that do contain such terms). This lack of leading terms in some directions is important; thus, in *c*, we expect to see wider bandgaps in the direction of the hexagonal faces (which reflect in the green) and much narrower gaps in the direction of the square faces (which reflect blue).

The band structure of these geometries has been investigated previously (

Michielsen & Kole 2003) in a technological context, where large refractive index contrasts were assumed and the authors were interested in complete photonic bandgaps. Here, we focus on the iridescence and the polarization properties, which are more relevant for low-contrast structures.

For low index contrast systems (such as chitin–air), the location and width of the bandgaps are approximately proportional to the relevant Fourier coefficients. Thus, the qualitative structure of the bandgap diagram will change in a systematic way with volume fraction and index contrast: increasing the index contrast will increase the width of the bandgaps and increasing the average index (either by changing the contrast or changing the volume fraction) will systematically shift the mid-gap frequency of all the bandgaps. Thus, it is sufficient to compare different lattice types for just one typical combination of contrast and volume fraction to understand their qualitative behaviour.

The photonic bandgaps were calculated for a volume fraction of 50 per cent and refractive index of 1.5 and are shown in . For the purpose of making comparisons between different lattices, the overall lattice scale sizes were chosen to yield the same lowest frequency bandgaps. The vertical scale on each chart is frequency and the bandgap (where no propagating solutions exist) has been coloured with the corresponding hue. Frequencies in the UV are shown by progressively darker shades of violet. In addition, there are polarization bandgaps: these are frequency ranges where only one polarization state can propagate and are shaded in grey. In such cases, one polarization state can propagate through the structure, but the other polarization state is reflected. The horizontal axis shows the conventional letters used in solid-state physics to denote special directions associated with each BZ. The corresponding orientations of the BZ are shown pictorially across the top for those unfamiliar with this notation. The angles between these directions (in degrees) are also indicated in . The *slope* of the bands can immediately reveal the degree of iridescence as can the variation of the depicted hues. Structures with steep bands are more iridescent than structures with flatter band diagrams.

Let us first focus on generic properties. The relative width of the bandgap at its widest is approximately 10 per cent for all three structures. This width would vary with the volume fraction of chitin, but would vary in about the same way for all three structures. Also note that, in all cases, the colour changes from green to blue over a 30°–40° tilt. Although the amount of colour change depends on index contrast and volume fraction, the angular tilt is a fixed property of the lattice and does not change with volume fraction.

Note that for the D-structure (*b*), the width of the bandgap in the X-direction (square face) is much narrower than that in the L-direction (hexagon face), as expected from the presence and absence of leading terms in the Fourier series. Under white light illumination, a narrower bandgap will reflect less light than a wider gap. This effect will act to partially suppress iridescence since in directions which reflect blue/violet, there will not be as much reflected light as in the directions that reflect green/yellow. Also as expected, the P-structure (*a*) is most iridescent but, in addition, at the shorter wavelengths, this reflected light should be highly polarized. Likewise, the G-structure (*c*) shows strong polarization effects in the H-direction where it is reflecting in the violet and UV. Thus, while some iridescence features are common to all these structures and there are quantitative differences between the structures, the most striking difference is in the polarization properties (we return to this point later in this paper).

In addition to the three regular models in , the photonic crystal in

*T. imperialis* was also reconstructed using electron tomography (

Argyros *et al*. 2002). The crystal was measured to be a distorted version of the D-structure in

*b* where the angles between struts are no longer equal and the struts are of different lengths. The corresponding distorted BZ is shown in . The approximate colours in suggest that although the structure should look green from certain directions, it should also reflect reds, blues, violets and even UV from other directions.

We calculated the bandgaps of the structure using the published data and description of the structure (

Argyros *et al*. 2002). The dimensions and coordinates defining the structure and the various special points on the surface of the BZ are all available in a format compatible with the MIT photonic bandgaps program in the electronic supplementary material. The optical properties of the distorted D-structure closely resemble the regular D-structure. For example, the bandgaps in directions corresponding to the six-sided faces are large (relative widths of approx. 10%), whereas those in the directions corresponding to the four-sided faces are small. The results of a quantitative calculation of the central bandgap wavelength in

*all* possible directions are shown in .

The most obvious difference between and the iridescence maps in is the presence of reflections in the longer wavelength (orange). We were able to see some visual evidence of this by looking at the underside of some scales from *T. imperialis* as shown in . The differences between and might be due to the periodic crystals in two butterflies having different structures, or possibly the same crystal aligned differently.