We study the diffraction of electromagnetic plane waves by a three-dimensional photonic crystal, made of a stack of N two-dimensional lossless gratings.
The formalism of transfer-matrices which is widely used in the frame of one-dimensional problems is the basic tool of our study. From the Lorentz reciprocity theorem and using symmetries considerations, it will be shown that the transfer operator of the crystal can be deduced from the transfer operator of a single grating through the use of Chebyshev polynomials. This relation provides a means for explaining the properties of the crystal. In particular, the precise location of the gaps can be predicted from the study of a real function. Outside the gaps, the same function allows us to explain the oscillations of the transmission factor versus the frequency. Numerical examples will illustrate our conclusions.
It is worth noting that our theory provides to someone who is well acquainted with the basic theory of transfer matrices for one-dimensional problems a means to interpret the band gap properties of three-dimensional photonic crystals thanks to the definition of well-adapted unbounded operators (two-dimensional or three-dimensional crystals) instead of complex numbers (one-dimensional problems).