A General Modal Theory for Reflection Gratings

Abstract

The problem of diffraction from lossless reflection gratings is treated with a modal expansion formulation which incorporates an impedance condition on the mode eigenfunctions along the groove aperture . The theory for both fundamental polarizations is applied to three different grating profiles and is found to yield results which conform well with predictions of an established integral theory over a restricted range of groove depths . The upper bounds on these -depths are approximately 02 periods for the symmetrical triangular profile and 0 . 5 periods for the semi-circular and sinusoidal profiles . Beyond these limits numerical instabilities occur and the results often fail to satisfy standard numerical tests .

Differential theory: application to highly conducting gratings

“…To clarify the problem, we first choose to work on a lamellar grating profile illuminated in TM polarization for which various methods are available to check the results, namely, the modal method, [10][11][12] the rigorous coupled-wave (RCW) method, [13][14][15][16] and the differential theory. 6 We first show that the value n 1 ϭ 0 ϩ i10 produces bad condition numbers 17 for the matrix, which must be inverted when the inverse rule has to be used.…”

The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves

“…Even the case of surfaces having a single protuberance or cavity have been widely treated by several different methods [4-71. Modal methods have been widely used for scattering from periodic gratings with highly symmetric grooves, like rectangular [8], triangular [9], semicircular [lo] or wire gratings [ l l ] . For perfectly conducting infinite periodic gratings, Andrewartha et al developed a modal theory that is suitable for arbitrary shapes of the grooves, but it presents numerical instabilities for deep cavities [12]. A very simple and powerful modal method for periodic gratings with arbitrarily shaped grooves has been presented recently by Li [13].…”

Automatic inspection of a residual resist layer by means of self-organizing map

“…Simulated signatures for a given profile can also be easily calculated by the multilayer modal method by Fourier expansion. [24][25][26] In our work, a large set of simulated signatures are supplied to an SOM during the training step to identify them with the corresponding classes given by the profiles P 1 and P 2 .…”