Robust method to retrieve the constitutive effective parameters of metamaterials

We propose an improved method to retrieve the effective constitutive parameters (permittivity and permeability) of a slab of metamaterial from the measurement of $S$ parameters. Improvements over existing methods include the determination of the first boundary and the thickness of the effective slab, the selection of the correct sign of effective impedance, and a mathematical method to choose the correct branch of the real part of the refractive index. The sensitivity of the effective constitutive parameters to the accuracy of the $S$ parameters is also discussed. The method has been applied to various metamaterials and the successful retrieval results prove its effectiveness and robustness.