We present a complete and rigorous mathematical analysis of the diffraction by a volume grating with periodically modulated refractive index. The analysis is entirely derived from the electromagnetic theory of continuous media, or in other words, from: a) the macroscopic Maxwell equations; b) the material equations; c) the electromagnetic boundary conditions. We propose an exact resolution of this mathematical problem, and then we give explicit methods for numerical computations. The theory presented is an alternative form of the characteristic mode theory. From the given geometrical parameters of the grating :
— thickness of the grating,
— angle of slant,
— Fourier coefficients of the modulation profile,
it is possible to compute the diffraction efficiency at each order for any angle of incidence.
In the present paper, after an outline of the theory, we develop the detailed calculations in order to obtain explicit formulas, which can then be converted into programs for computation.
This paper is only theoretical; experimental tests are not discussed.