Perturbative methods in theory of phase gratings

(Received 15 February 1993, revised 17 September 1993, accepted 19 October 1993)

Abstract
Perturbative methods are generally invoked for problems in which there is a small parameter. In the theory of phase gratings, the small parameter is the modulation amplitude of the refractive index, and the classical perturbative method is then the Born approximation. But it is well-known that the Born approximation fails at the Bragg resonance, however small the modulation amplitude is. In this paper a perturbative method is presented, which is working at Bragg resonance as well. A sequence of numbers (called the eigenvalues of the problem) are introduced; they depend on the geometrical configuration (incidence angle, grating parameters). It is shown that the Bragg resonance occurs if (and only if) two eigenvalues become equal. These eigenvalues - and the corresponding solutions of the equations - can be expanded in powers of the modulation amplitude. The expansions are different according to whether the corresponding eigenvalue is simple or double. Explicit formulae or algorithms are given. Computing programs have been written from them. These programs are efficient.