Liquid Shaping Unlocks Complex Optical Component Designs

Scientists are developing innovative techniques for fabricating optical components, and a new study details a theoretical foundation and computational solver for Fluidic Shaping, a method relying on the equilibrium of liquid volumes under geometrical constraints. Amos A. Hari and Moran Bercovici, both from the Faculty of Mechanical Engineering at the Technion, Israel Institute of Technology, and working with colleagues at the Department of Materials, ETH Z urich, have created a high-order finite element solver capable of accurately modelling liquid interfaces on arbitrarily shaped domains. This represents a significant advance, as previous analytical solutions were limited to linear equations and simple geometries, while existing numerical methods only addressed axi-symmetric cases. The ability to model nonlinear behaviour and complex, arbitrary domains is crucial for achieving high-quality optical designs and accurately predicting curvature, a key factor governing optical properties.

Scientists have devised a new technique for crafting bespoke optical components using the physics of liquids. The method allows for complex designs previously unattainable with existing fabrication processes, promising greater control over light manipulation for applications ranging from imaging to telecommunications. Scientists have obtained analytical solutions for the linearized equations of Fluidic Shaping only on circular or elliptical domains.

Numerical solutions exist for the non-linear equation, but only for the axi-symmetric case. These solutions are insufficient as they do not capture the full range of optical surfaces, and arbitrary domains offer an important degree of freedom for creating complex optical surfaces. The nonlinear terms are essential for high quality solutions. Researchers present the theoretical foundation for the Fluidic Shaping method over arbitrary domains, and the development of a high order (quintic) finite element numerical solver.

This solver accurately resolves the topography and curvature of liquid interfaces on arbitrary domains. The code is based on reduced quintic finite elements, modified to capture curved boundaries. Results are compared 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.

The usability of the code is also shown for the prediction of optical surfaces derived from complex boundary conditions. Fluidic Shaping is a novel fabrication method originally introduced by Frumkin and Bercovici. In this method, a volume of liquid polymer is submerged in an immiscible immersion liquid of equal density, inducing neutral buoyancy. By pinning the liquid polymer to a geometrical boundary condition, a desired optical component shape can be obtained.

Unlike traditional optical manufacturing methods, the Fluidic Shaping approach does not require mechanical grinding or polishing, yet produces components of sub-nanometric surface quality due to the natural smoothness of liquid interfaces. Frumkin and Bercovici introduced the principle concepts of the method and provided a linearized analytical solution for axi-symmetric lenses obtained using a circular bounding frame.

Elgarisi et al. expanded the use of Fluidic Shaping for rapid fabrication of freeform optical components, generalizing the mathematical model and allowing for height variations along the circular bounding frame. Through analytical solutions of the linearized model and experimental validations, they characterised the range of freeform surfaces that can be produced using Fluidic Shaping over circular domains.

In 2024, Na et al. introduced an optimisation methodology for axi-symmetric circular lenses fabricated by Fluidic Shaping, numerically solving the one-dimensional nonlinear axi-symmetric equation. They showed that this representation is more accurate than the approximated analytical solutions, especially when the assumptions of linearization are no longer valid.

They implemented the numerical solution in a ray tracing software and designed the lens to provide the best image using an optimisation technique. In 2025, Elgarisi et al. explored whether Fluidic Shaping is a viable manufacturing approach for common ophthalmic lenses, demonstrating that deviating from a circular bounding frame to an elliptical one allows for the creation of ophthalmic lenses with both astigmatic and spherical corrections.

Variations in the boundary footprint introduce a useful degree of freedom for lens design, motivating the expansion of the Fluidic Shaping theory to account for arbitrary domains. Solving a highly nonlinear boundary value problem on an arbitrary domain requires a numerical approach. In this optical context, the solution should at the very least provide accurate information about the unknown surface and its first and second derivatives, which are required to calculate the curvature of the surface, determining the optical power of the corresponding lens.

For precision optic applications, where the shape of the lens must be accurate to a fraction of the operation wavelength, particular accuracy is necessary. For example, a 1cm diameter lens operating in the visible range (λ ≃500nm) requires shape accuracy on the order of λ/10 = 50nm, which is 6 orders of magnitude smaller than the characteristic geometrical length.

Numerous existing methods struggle to create complex optical shapes with high precision. Traditional techniques often struggle with the freeform designs increasingly demanded by modern optics.

Quintic finite element analysis enables complex optical component fabrication

A high order (quintic) finite element solver underpinned the development of a new approach to fabricating optical components via Fluidic Shaping. This technique relies on the equilibrium state of liquids with neutral buoyancy, constrained by defined geometrical boundaries, to create precise optical surfaces. Previous analytical solutions to the governing nonlinear partial differential equation were limited to simplified scenarios, linearized equations and circular or elliptical shapes, and lacked the capacity to model complex geometries effectively.

Consequently, a numerical method capable of handling arbitrary domains became essential for unlocking the full potential of Fluidic Shaping. Rather than determining the shape of a liquid interface, this work prioritised accurate calculation of its curvature, a critical parameter governing optical properties. The team constructed their solver using reduced quintic finite elements, a numerical technique that divides a complex shape into smaller, simpler elements for analysis.

These elements were specifically modified to accurately represent curved boundaries, a feature often overlooked in standard finite element methods. This adaptation ensures the solver can faithfully reproduce the intricate topography of the liquid surface. Achieving accurate results demanded more than just curved elements; the order of the finite element itself proved vital.

Comparisons against lower order finite elements and non-deformed high order elements demonstrated the importance of high order approximations for both the solution and the domain itself. By employing quintic elements, the solver minimises numerical errors and provides a more precise representation of the liquid surface. Furthermore, the code’s usability was confirmed through its ability to predict optical surfaces derived from complex boundary conditions, suggesting its potential for designing lenses with tailored optical characteristics. This fluidic approach promises sub-nanometric surface quality due to the inherent smoothness of liquid interfaces, offering a pathway to creating high-precision optics without the need for abrasive processes.

Quintic Finite Element Solver Validates High-Resolution Liquid Interface Reconstruction and Curvature Calculation

At the core of this work lies a high order (quintic) finite element numerical solver, developed to accurately resolve the topography and curvature of liquid interfaces on arbitrary domains. This solver achieves a remarkable level of precision, essential for optical applications demanding shape accuracy on the order of λ/10, approximately 50 nanometres for a 1cm diameter lens operating within the visible spectrum.

Such accuracy represents a six order of magnitude improvement over the characteristic geometrical length, a feat previously unattainable. Verification against manufactured and known analytical solutions confirms excellent convergence of the results obtained using this new method. Beyond resolving the shape, the research demonstrates the solver’s ability to accurately calculate surface curvature, a critical parameter governing optical properties.

Comparisons between solutions derived from the nonlinear equation and those from linearized analytical approaches reveal the importance of incorporating nonlinear terms, particularly when assumptions of linearization are invalid. These nonlinearities are essential for achieving high quality solutions and capturing the full range of optical possibilities.

The study highlights the significance of high order approximations, both of the solution and the domain itself. Testing against low order finite elements and non-deformed high order elements clearly demonstrates the benefits of this approach in minimising geometrical mismatch error arising from spatial discretisation. This careful attention to detail allows for the prediction of optical surfaces derived from complex boundary conditions, opening avenues for designing lenses with tailored properties.

The code’s usability is further demonstrated through its application to designing and analysing micro-lens arrays. By predicting nominal optical properties and assessing the impact of manufacturing imperfections, such as frame defects or volume inaccuracies, the solver provides valuable insights for optimising the Fluidic Shaping process. Once implemented, the solver provides a means to accurately model the topography of the liquid interface, and subsequently, its optical properties.