Differential Methods

Using advanced simulation to aid microlithography development

“…A third approach based on the differential method [61] was employed by Yeung to model the scattering of light by inhomogeneity in a planar photoresist layer caused by beaching, or the change of the attenuation index, of the photoresist during exposure [43]. This method was extended to a photoresist layer with a singly periodic, nonplanar profile using a coordinate transformation technique [62].…”

Characterization of diffraction gratings by use of a tabletop soft-x-ray laser

“…Taking into account that the differential method is well documented in the literature ͑see Ref. 9 and references therein͒ and that the main idea of its formulation is similar for both polarization cases, we give the relevant details for the E ʈ case only. Recently, a new formulation of the differential method in TM polarization that removes numerical instabilities associated with deep gratings and highly reflecting materials was presented.…”

“…6 Although we observed good agreement between these methods for gratings that support a number of diffracted orders ͑such as those used in the visible region of the electromagnetic spectrum͒ that is not too great, the differential method sketched above was the only one to produce stable results for gratings that support a large number of diffracted orders. Because of this particular capability, which has already been reported in the literature, 9 we chose the differential method to simulate the performance of gratings designed for grazing-incidence operation in the soft-x-ray region. The wavelength-to-period ratios for the gratings considered here are ͞d ϭ 0.01407 ͑300 lines͞mm͒ and ͞d ϭ 0.08442 ͑1800 lines͞mm͒, which give a total of 143 and 24 propagating orders, respectively.…”

“…The code is based on a class of theoretical methods known in the literature as differential. 9 The term "differential" refers to the fact that the rigorous computation of the electromagnetic field diffracted by the grating is reduced in these methods to the resolution of a system of ordinary differential coupled equations.…”

Light-scattering characterization of transparent substrates

“…Far-field light scattering has been extensively studied these past decades [1][2][3][4][5][6][7][8][9][10] as a technique for surfaceroughness characterization. In the case of opaque samples, using both angle-resolved-scattering (ARS) measurements and a specific electromagnetic theory, we can deduce the surface-roughness spectrum, which once integrated provides a root mean square of the roughness in the optical bandwidth.…”