The dielectric lamellar diffraction grating

A rigorous modal theory describing the diffraction properties of a dielectric lamellar grating is presented . The numerical implementation is shown to be suited to modelling the behaviour of high refractive index gratings . This suggests that an approach of this type may be successfully applied to the problem of lossy metallic lamellar gratings . 1 . Introduction The problem of diffraction by a perfectly conducting lamellar grating has long attracted interest [1-4] . The simplicity of the grating profile enabled the specification of modal expansions incorporating explicitly the boundary conditions . Recently Knop [5] has discussed the problem of diffraction by a dielectric lamellar grating . His interest arose from the suggested use of surface-relief phasegratings for storing pictorial information [6] . Knop's method relied upon the solution of an eigenvalue problem derived from the Helmholtz equation and involved the use of a Fourier series to describe the discontinuous function equal to the square of the refractive index in the dielectric, and one in air . The differential technique of Neviere et al. [7] encountered serious numerical difficulties associated with the use of such series for large refractive index transitions . This was one motive which prompted us to seek an alternative method of solution, with the potential of being generalized to treat a lamellar grating made of a metal with a refractive index complex and large in modulus . The treatment proposed here for the dielectric lamellar grating is a generalization of the modal treatments appropriate in the infinite conductivity case . The form of the modes was to some extent suggested by functions occurring in the treatment of the asymmetric slab waveguide [8] . We find the method developed here for the dielectric grating to have all the elegance of the classical treatment for perfectly conducting lamellar gratings . 0030-3909/81/2803 0413 S02-00 i 1981 Taylor & Francis Ltd D ow nl oa de d B y: [T ec hn is ch e U ni ve rs ite it D el ft] A t: 12 :1 7 11 D ec em be r 2 00 7 414 L. C. Botten et al . 2 . Theory 2.1 . Notation and method Initially we shall consider the diffraction of a P-polarized plane wave of free space wavelength ), incident upon the lossless structure (shown in figure 1) at some angle ~ . The grating extends from x = oo to x = + cc and is perfectly periodic of period d. The incident electric field is defined to be where ao = k o sin (p, Xo = k o cos 0 and ko = 2it/), . (In this paper the temporal dependence of exp(-itot) has been suppressed from the specification of all field quantities .) Because of the periodicity of the structure all plane wave fields have the same phase shift across a period as the incident field (i .e . all fields are pseudoperiodic .) E'(x,y)=exp [i(aox JCo(y h/2))]3,