We calculate the Sommerfeld precursor that results after transmission of a generic electromagnetic plane wave pulse with transverse electric polarization, through a one-dimensional rectangular N-layer photonic crystal with two slabs per layer. The shape of this precursor equals the shape of the precursor that would result from transmission through a homogeneous medium. However, amplitude and period of the precursor are now influenced by the spatial average of the plasma frequency squared instead of the plasma frequency squared for the homogeneous case.

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The scattering of electromagnetic waves by a slab whose refractive index is changing along its boundary planes is exactly calculated in a closed analytical form. The key feature of the calculation is the introduction of a new set of modes. As a specific example, we calculate the reflected and transmitted fields generated by the interaction of an incoming plane wave with an N-layered medium, the layers of which are perpendicular to the boundary planes of a slab.

The transmission coefficient of a one-dimensional multi-layer medium is obtained by summing the amplitude coefficients of all individual transmitted light-rays. As compared to the transfer-matrix method, the sum-overall light rays derivation results in a very intelligible expression, which is interesting from a theoretical point of view. Exactly as in the case of a single-layer medium, the sum of all light-rays through the multi-layer medium is obtained from a geometric series. The system does not have to be periodic; the layers have arbitrary physical lengths and each layer has an arbitrary electromagnetic response.