A metal plate array (MPA) which is a structure complimentary to a metal hole array (MHA), supports spoof surface plasmon polaritons (SSPP) as well as an MHA does. Babinet's principle attributes the phenomenon of duality to transmission characteristics of the complimentary impedance surfaces because of the symmetry of the behaviors of electric and magnetic fields. However, it is also a fact that the complimentary structures do not follow this principle if they have wavelength-size thickness, because electromagnetic waves do not treat such thick structures as a boundary surface but as propagation spaces with the specific boundaries such as a waveguide which shows SSPP modes. If the thickness is so small that it is negligible, Babinet's principle is still valid, while it has been uncertain how the layer thickness works to break the principle as it is increased. The unconfirmed transformation is revealed analytically and experimentally with the use of MPAs and MHAs of varying thicknesses.