The problem of using the α,0 and the α, β-quasi periodic transformations within a finite element method in studying electromagnetic waves in a periodic space is addressed. We investigate an a priori error estimate for both transformations which allows us to solve our problem numerically on a uniform mesh. We also analyse the Dual Weighted Residual (DWR) method with the α,0-quasi periodic transformation to derive an a posteriori error estimate. This error estimate is later used to compute efficiently the numerical solution using an adaptive method. We then implement the above finite element methods. It is shown numerically that our numerical results are in good agreement with those in the literature, the α, β-quasi periodic method converges at a far lower number of degrees of freedom than the α,0-quasi periodic method and the DWR method converges faster and requires fewer degrees of freedom than the global a posteriori error estimate or the uniform mesh. We also explore the geometrical freedom given by the finite element method and examine wave scattering by the Morpho butterfly wing.