  We theoretically investigate the homogenization of the dielectric response to
transverse electric waves of a transverse grating characterized by the Kapitza
condition, i.e. the permittivity is rapidly modulated with a modulation depth
scaling as the large wavelength to modulation period ratio. We show that the
resulting effective dielectric permittivity, in addition to the standard
average of the underlying dielectric profile, has a further contribution
arising from the fast and deep dielectric modulation. Such contribution turns
out to be comparable with the other one and hence can provide an additional way
for designing dielectric metamaterials. As an example, we discuss an effective
metal-to-dielectric transition produced by the Kapitza contribution obtained by
changing the grating depth.
