Fluidic Shaping over arbitrary domains: theory and high order finite-elements solver

TL;DR

This paper develops a theoretical framework and a high-order finite element solver for Fluidic Shaping on arbitrary domains, enabling precise modeling of complex optical liquid surfaces and their curvatures for advanced optical component fabrication.

Contribution

It introduces a high-order (quintic) finite element method tailored for arbitrary domains, improving accuracy in shape and curvature predictions over previous low-order or symmetric solutions.

Findings

High-order finite elements outperform low-order in accuracy.

The solver accurately captures complex boundary conditions.

Curvature predictions enable better optical surface design.

Abstract

Fluidic Shaping is a novel method for fabrication of optical components based on the equilibrium state of liquid volumes in neutral buoyancy, subjected to geometrical constraints. The underlying physics of this method is described by a highly nonlinear partial differential equation with Dirichlet boundary conditions and an integral constraint. To date, useful solutions for such optical liquid surfaces could be obtained analytically only for the linearized equations and only on circular or elliptical domains. A numerical solution for the non-linear equation was suggested, but only for the axi-symmetric case. Such solutions are, however, insufficient as they do not capture the full range of optical surfaces. Arbitrary domains offer an important degree of freedom for creating complex optical surfaces, and the nonlinear terms are essential for high quality solutions. Moreover, in the context of optics, it is not sufficient to resolve the shape of the surface, and it is essential to obtain accurate solutions for its curvature, which governs its optical properties. We here present the theoretical foundation for the Fluidic Shaping method over arbitrary domains, and the development of a high order (quintic) finite element numerical solver, capable of accurately resolving the topography and curvature of liquid interfaces on arbitrary domains. The code is based on reduced quintic finite elements, which we have modified to capture curved boundaries. We compare the results against low order finite elements and non-deformed high order elements, demonstrating the importance of high order approximations of both the solution and the domain. We also show the usability of the code for the prediction of optical surfaces derived from complex boundary conditions.