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

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.

Figures (9)

• Figure 1: Illustration of the Fluidic Shaping configurations that can be described as a minimization of equation (\ref{['eq:energy']}) subjected to the volume constraint equation (\ref{['eq:constraint']}). (a) The basic examined configuration. We submerge a dish, filled with optical liquid of density $\rho$, in a container, filled with immersion liquid of density $\rho_\mathrm{im}$. We assume that the optical liquid is perfectly pinned at height $\bar{u}$ on the dish's edge, and that the shape of the interface $u$ is determined by a balance of surface tension $\sigma$, gravity $g$ and the total optical liquid volume $V$. (b) A multiple interface configuration. For each free surface we imagine a corresponding dish as in the basic configuration. The optical liquid volumes is shared between all dishes, and the direction of gravity may flip with the dish's orientation. (c) A multiple interface configuration with an additional volume constraint. We take the multiple surface configuration and partition the immersion container to create a sealed closure around the bottom surface. By adding additional immersion liquid volume $\Delta V$ to the sealed closure, we introduce residual pressure which displaces the interfaces.
• Figure 2: Treatment of internal and boundary elements in a triangular mesh of an arbitrary domain. (a) The triangular discretization of a general domain gives rise to three types of elements: (b) An element in the bulk of the mesh with no overlap with the boundary. This is the most common element in the mesh and it doesn't warrant any special treatment. (c) An element with one node on the boundary. The additional degrees of freedom of the high order element do not naturally coincide with the prescribed boundary condition, requiring a local transformation. (d) An element with an edge on the boundary. In addition to transforming the nodal degrees of freedom, the edge itself should be deformed to maintain a high order accuracy.
• Figure 3: Accuracy analysis of our deformed elements method. The implementation of our deformed finite elements method (purple) improves solution accuracy compared with classical approaches: linear elements (green) and non-deformed reduced quintic elements (grey). We solve the Fluidic Shaping problem for a spherical lens over a circular domain, compare each numerical solution with the analytical spherical surface, and report the errors and apparent convergence rate for each method. Across the tested meshes, our deformed elements method outperforms the others by orders of magnitude, and its convergence rate suggest even larger gains for finer discretizations. The $H^0$ norm is defined as $\norm{\mathrm{error}}_{H^0} := \sqrt{\int_{\Omega} (u-u_\mathrm{exact})^2 \dd{\Omega}}$, the $H^1$ semi-norm as $\norm{u}_{H^1} := \sqrt{\int_\Omega (u-u_\mathrm{exact})_{,i}(u-u_\mathrm{exact})_{,i} \dd{\Omega}}$, and the $H^2$ semi-norm as $\norm{u}_{H^2} := \sqrt{\int_{\Omega} (u-u_\mathrm{exact})_{,ij}(u-u_\mathrm{exact})_{,ij} \dd{\Omega}}$. To obtain high order derivative for the linear finite element solution, we used the Clough-Tocher interpolation, which yields a continuous differentiable interpolant which approximately minimize surface curvature.CloughTocher2DInterpolatorSciPyV1162
• Figure 4: High precision optics demand nonlinear terms. Elgarisi et al.ElgarisiOptica showed that the shape of a circular lens can be approximated analytically through linearization of the Fluidic Shaping problem, equation (\ref{['eq:repr-prob']}). However, by comparing the numerical results of our method to the analytical results of Elgarisi et al. we show that the linear analytical solution is not accurate enough for high precision optics. (a) An isometric view of the lens as obtained from the numerical solver. (b) The absolute difference in the topography between the analytic and numeric surfaces, showing deviations of 500 nm - beyond what is allowed for precision optics. This lens was obtained for diameter $3.5$ cm, volume $3$ ml, Bond $3$, and height $3 + 0.55\sin(4\theta)$ mm, for $\theta\in[0,2\pi)$.
• Figure 5: Sensitivity analysis of a spherical ophthalmic lens manufactured using the Fluidic Shaping method on an Oakley Slender Satin eyewear frame. (a-c) The computed nominal topography, geometric spherical power, and geometric cylindrical power. We investigate the process' sensitivity to errors in (d-f) the boundary frame height, (g-i) the immersion liquid density (which alters the Bond number to $\mathrm{Bo}=2$ for surface tension of $7.7$$\mathrm{mN/m}$naKaust2024), and (j-l) the injection of excess lens liquid. The geometric spherical power is defined as $-(\kappa_x+\kappa_y)/2$ and the geometric cylindrical power as $(\kappa_x-\kappa_y)$, where $\kappa_x$ and $\kappa_y$ are the surface curvature in the $x$ and $y$ directions.