properties of an $N$-layer graphene for light with normal incident on graphene plane. One is a standard method, in which we calculate the dynamical conductivity, $\sigma_N$, by assuming that the electric field is spatially uniform in the layered material, as shown schematically in Fig. \[fig:sl\](a). Though the electronic states form standing wave in the $c$-axis direction due to interlayer coupling, the current operator (in Kubo’s formula) is invariant about the $c$-axis direction. As a result, the optical matrix element is nonzero for only one or two specific final states against an initial state, which is the selection rule of the wavenumber that makes the calculation simple. Meanwhile, the adaptive range of this method using $\sigma_N$ is limited to the case that the resultant electromagnetic field is sufficiently uniform over the layers. It is not straightforward to find a method to take into account the possible spatial variations of the electromagnetic field due to light absorption by $\sigma_N$ itself, irregularities along the $c$-axis caused by defects, cracks and so on.

![(Color Online) (a) The bunched conductivity, $\sigma_N$, is defined when the electromagnetic fields are sufficiently uniform in all layers. The electromagnetic fields are expressed by $E_x$ (red arrow) and $B_y$ (blue). 