On the equivalence of the summation and transfer-matrix methods in wave propagation through multilayers of lossless and lossy media

We explicitly show that the well-known transmission and reflection amplitudes of planar slabs, obtained via an algebraic summation of Fresnel amplitudes, are completely equivalent to those obtained from transfer matrices in the scattering approach. This equivalence makes the finite periodic systems theory a powerful alternative to the cumbersome Fresnel coefficients in the analysis of superlattices and photonic crystals. The versatile transfer matrices, with their compact formulae, can be useful in calculating transmission properties not only of superlattices made of dielectrics but also of left-handed materials and conducting media