The effective physical properties of the electromagnetic wave can be controlled by using a metamaterial, which is an artificial material based on a metal structure smaller than a wavelength. In this paper, focusing on the effective impedance of the material through the meta material, the propagation of the electromagnetic wave is controlled. Especially, the optimal theoretical models are proposed to realize the high impedance surface and the anisotropic matched impedance surface. As a result, ultrathin bespoke impedance surfaces can be achieved. The theoretical limit of the high impedance surface that can be used in the microwave region is clarified A design model has been proposed to achieve the high transmittance polarizer through optimization of the anisotropic matched impedance. It has become possible to design the optimum bespoke impedance by the homogeneous medium model and the hyperbolic dispersion principle proposed in this study. The validity of the theoretical model is theoretically explored and numerically analyzed.