Diffraction at a curved grating. 

Toufic  Abboud 
Centre de Mathematiques Appliquees - CNRS URA 756,
Ecole Polytechnique, 91128 Palaiseau Cedex, France.

Habib AMMARI 
Centre de Mathematiques Appliquees - CNRS URA 756,
Ecole Polytechnique, 91128 Palaiseau Cedex, France.


We study the influence of the curvature of a circular cylindrical grating on the diffraction of a 
TE or TM polarized electromagnetic plane wave. For every ratio  of the wave length $\lambda$ 
to the period $d$, we prove the local approximation in the near field by a plane infinite grating, 
when the radius $R$ of the circular grating goes
to infinity. We show the existence of  regions, optically different, where the error is of order
$O(d/R)$, separated by  transition regions where the error is like 
$ \ds O ( d/(\lambda^{2/3} R^{1/3}) )$. In each region, we calculate the first corrector, solution 
of an infinite plane grating diffraction problem.



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