Bienert Florian, Graf Thomas, Ahmed Marwan Abdou
Opt Express. 2022 Jun 20;30(13):22410-22420. doi: 10.1364/OE.458636.
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 nm.
我们对激光干涉光刻曝光过程中故意弯曲衬底以补偿周期啁啾的方法进行了理论研究。结果表明，实现零啁啾情况（即在整个衬底上具有完全恒定的周期）所需的表面几何形状的尚未被发现的函数由一阶微分方程确定。由于该微分方程的直接解析解很困难，因此开发了一种数值方法，该方法基于通过Nelder-Mead单纯形算法朝着微分方程的未知解析解优化预定义函数。通过将此方法应用于一个具体示例，我们表明衬底相对于点源的偏心放置在可实现的周期和衬底曲率方面都是有利的，并且四阶多项式可以极大地满足微分方程，导致相对于目标周期610 nm的均方根偏差仅为1.4 pm。
