In the optimization design of optomechanical systems, polynomials can not only retain a significant amount of information but also provide a more compact representation of structural deformation and facilitate integration between mechanical structures and optical models. Zernike polynomials have been widely employed due to their orthogonality properties on a unit circle. However, the orthogonality of Zernike polynomials only applies to continuous data on a circular aperture, and it degrades for discrete interpolation and non-circular apertures. Non-orthogonality means that coupling exists among different terms of polynomials, and the number of polynomials cannot be arbitrarily increased or decreased, which can lead to accuracy and stability problems in surface approximation and optimization design. This study aims to propose a conformal orthogonal basis generated by the eigenmodes of the Laplace equation for utilization in the topology optimization of support structures for reflective mirrors, thus avoiding Zernike orthogonality loss. Additionally, due to the conformal properties of the Laplace eigenmodes in the domain, the obtained basis represents the deformed information along the surface normal. As the principal direction of deformation, the surface normal makes the eigenmodes a better fit for surface deformations.

The Laplace characteristic equation and Zernike polynomials both originate from the Sturm-Liouville problem. The solutions on the planar circular domain exhibit similarities with Zernike polynomials, and Trevino et.al.^[10] have compared the characteristic modal functions (Bessel circle polynomials) and Zernike polynomials in eye surface fitting, which indicates that the former provides better fitting. This paper extends the planar domain to surfaces. The finite element solution of the Laplace equation and properties of the eigenvalue problem ensure the discrete orthogonality of the characteristic modal functions. The mathematical properties of this equation guarantee the completeness of analytical solutions of the characteristic modal functions, and the completeness is verified by combining function approximation theory and numerical experiments. In addition, a specific topology optimization example demonstrates that the characteristic modal functions not only yield similar results to Zernike polynomials on circular domains but also can be applied to non-circular apertures where Zernike polynomials are not suitable.

First, based on the Sturm-Liouville decomposition on compact Riemannian manifolds, the completeness of the eigenmodes under analytic conditions is demonstrated. Then, the feasibility of adopting eigenmodes to fit surface deformations is numerically validated by adopting function approximation theory as the basis (Figs. 4 and 5). Additionally, this paper applies the method of surface approximation using eigenmodes to topology optimization of circular mirror support structures and compares it with Zernike polynomial approximation. The comparative results indicate that the objective functions optimized through characteristic modal functions and Zernike polynomials are 4.40% and 4.43% of the original structure respectively. The root mean square (RMS) values are 4.40% and 2.55% of the original structure respectively, and the peak to valley (PV) values are 10.51% and 8.73% of the original structure respectively. Both methods prove comparable optimization effectiveness (Table 1). The curves of the objective and constraint values during the iteration show that both methods have consistent stability and can converge (Figs. 11 and 12). However, there are slight differences in the resulting structures (Figs. 9 and 10). After comparative experiments, this study applies the modal fitting method to a hexagonal mirror, thereby completing the topology optimization design of a hexagonal mirror support structure (Fig. 14) and extending its applicability to non-circular apertures.

This paper proposes to adopt a conformal orthogonal basis, which is the Laplace eigenmodes, for approximating surface deformations, and applies it to topology optimization of optical structures. It also demonstrates analytically and numerically that the Laplace eigenmodes are not only completed on circular domains but also on other irregular shapes. Surface eigenmodes can be employed to approximate smooth mirror surface deformations and achieve topology optimization of optical single mirror support structures with specific modal coefficients being the optimization objectives. Two optimization examples show the applicability of the proposed basis on circular domains and its extensibility on non-circular domains. However, compared to Zernike polynomials, the Laplace eigenmodes studied in this paper only exist in piecewise discrete numerical solutions, which means that the eigenmodes do not have an analytical representation like Zernike polynomials. When solving for the normal of a deformed mirror surface, it is necessary to pay attention to the continuity of the normal vector at the element boundary, which is a field that deserves further exploration in future work.