A new, rigorous, field-based, seminumerical analysis method is presented to obtain the reflection and transmission coefficients of 2D planar periodic structures with arbitrarily shaped metallization patterns for both normal and oblique incidence conditions. It is useful for the analysis, design, and optimization of many single-layer and multilayer planar structures, such as frequency-selective surfaces (FSSs), artificial magnetic conductor (AMC) surfaces, electromagnetic bandgap (EBG) structures, some metamaterials and high-impedance surfaces. In this coupled-field expansion method (CFEM), the x- and y-components of the vector magnetic potential in each homogeneous region in a unit cell are expanded in terms of Bloch-Floquet modes and the solution to the coupled-field problem is formulated. The unique, analytical formulation presented here leads to a linear system with reasonably simple matrix elements. By cascading the matrices representing each interface, multilayer periodic structures are analyzed in a very flexible way. Being field based, CFEM does not require substrate Green's functions to analyze surfaces printed on dielectric substrates. The method was validated by analyzing one single-layer periodic surface (a printed AMC on a dielectric substrate) and one multilayer periodic surface (a circular polarizer) and comparing CFEM results with HFSS results.