Homogenizing spatially periodic materials with respect to Maxwell equations: Chiral materials by mixing simple ones

The following general problem is discussed: given a composite, made from at least two different materials with each its own (scalar, and possibly complex- valued) e and ∝, distributed in space in a regular, crystal-like, pattern, find the equivalent permittivity and permeability (they will, in general, be tensors). A computation in which such repetitive composites are present can then be done by simply replacing these with their equivalent, homogenized, materials. When one does all this, it may come as a surprise to see "cross-dependencies": B, e.g., depends not only on H but on E as well, and D depends both on E and H. This kind of behavior, which is characteristic of the so-called chiral materials (because they rotate the polarization plane of waves), may result from homogenization, even with non-chiral components. Some symmetry arguments are used to find a necessary condition for this to happen: geometrical chirality at the cell level, that is, the cell should fail to be congruent with its image by central symmetry.