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         "id"   : "https://puma.ub.uni-stuttgart.de/bibtex/28369a5b0b9c1d5b9bd78f25cc22c6d13/florianbienert",         
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         "label" : "Theoretical investigation on the elimination of the period chirp by deliberate substrate deformations",
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         "volume": "30","number": "13","pages": "22410","abstract": "We present a theoretical investigation on the approach of deliberately bending the substrate during the exposure within laser interference lithography to compensate for the period chirp. It is shown that the yet undiscovered function of the surface geometry, necessary to achieve the zero-chirp case (i.e. having a perfectly constant period over the whole substrate) is determined by a first-order differential equation. As the direct analytical solution of this differential equation is difficult, a numerical approach is developed, based on the optimization of pre-defined functions towards the unknown analytical solution of the differential equation by means of a Nelder-Mead simplex algorithm. By applying this method to a concrete example, we show that an off-center placement of the substrate with respect to the point sources is advantageous both in terms of achievable period and substrate curvature and that a fourth-order polynomial can greatly satisfy the differential equation leading to a root-mean-square deviation of only 1.4 pm with respect to the targeted period of 610\u2005nm.",
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         "volume": "30","number": "13","pages": "22410","abstract": "We present a theoretical investigation on the approach of deliberately bending the substrate during the exposure within laser interference lithography to compensate for the period chirp. It is shown that the yet undiscovered function of the surface geometry, necessary to achieve the zero-chirp case (i.e. having a perfectly constant period over the whole substrate) is determined by a first-order differential equation. As the direct analytical solution of this differential equation is difficult, a numerical approach is developed, based on the optimization of pre-defined functions towards the unknown analytical solution of the differential equation by means of a Nelder-Mead simplex algorithm. By applying this method to a concrete example, we show that an off-center placement of the substrate with respect to the point sources is advantageous both in terms of achievable period and substrate curvature and that a fourth-order polynomial can greatly satisfy the differential equation leading to a root-mean-square deviation of only 1.4 pm with respect to the targeted period of 610\u2005nm.",
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         "volume": "61","number": "9","pages": "2313--2326","abstract": "We present a theoretical investigation on laser interference lithography used for the exposure of linear gratings. The focus is on the geometry of the arising interference lines on the substrate, in particular on their period and orientation, depending on the illumination geometry as determined by the setup. The common approach with point sources emitting spherical wavefronts is considered for the illumination. Three different cases are discussed, namely the interference between two point sources with either two convex, two concave or mixed, i.e., convex and concave wavefronts. General equations focusing mainly on the calculation of the period and the orientation of the grating lines are derived for each of the three exposure cases considering arbitrarily positioned point sources and arbitrarily shaped substrates. Additionally, the interference of symmetrically positioned point sources illuminating plane substrates is investigated, as these boundary conditions significantly simplify the derived equations.",
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