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A periodic strip grating on a grounded dielectric slab is used to rotate or twist the polarization of the incident plane wave after reflection in a scanned reflector antenna system, and is used to reduce the cross-polarization and sidelobe level in horn antenna as artificially soft and hard surfaces. Analytical expressions for the reflection coefficients of the structure as artificially soft and hard surfaces were derived by using the variational method. However, these expressions were valid only for the small period case. The paper proposes a rigorous and complete analysis method without these constraints and obtains analytical expressions for the reflection coefficients that are approximate solutions for the small period case.