Research On The Surface Shape Of Lenses With Non-Uniformly Multi-Point Support Using Plate And Shell Theory

With the increasing demand for integrated circuits in various industries,the development of high-precision optical systems,particularly in China,has gained significant attention.Modern optical instruments impose higher requirements on the design,manufacturing,assembly,and calibration of optical lenses.The assembly and calibration of high-precision lenses are crucial for the overall optical system.In lithographic projection lens structures,the surface shape of the lens is closely related to its support structure.Investigating the relationship between lens models,support structures,and gravitational deformation provides a theoretical basis for the assembly and calibration of the entire lens system.Previous studies have shown that non-uniform support is the main cause of irregular gravitational deformation.Therefore,it is of practical significance to reduce the surface shape errors of optical elements resulting from non-uniform support using plate shell theory.This paper combines plate shell theory and lens structure to analyze the influence of different lens models and support structures on the lens surface shape.The principles of mechanics theory and the Zernike analysis method are employed to investigate the effects and provide theoretical guidance for the assembly and design of lens structures.First,the theory of thin plates and the bending theory of thin circular plates are introduced.The surface deformation of a large-aperture flat lens under non-uniform multi-point support is theoretically derived using the Fourier series expansion method.A novel approach is introduced to summarize the deflection solution under non-uniform multi-point support using the plate’s influence function.The accuracy of the results is validated by comparing them with finite element software simulations.Multiple precision evaluation criteria,such as PV value,RMS value,Zernike coefficients,and deflection,are utilized to ensure the validity of the numerical examples.Comparative verification demonstrates that the lens deformation under non-uniform multi-point support obtained from the derived influence function of the plate has a solution error within 3% when compared to the Fourier and finite element solutions.This confirms the accuracy of the proposed method in accurately solving lens deformation under non-uniform support conditions.Next,the lens surface shape under non-uniform multi-point support is investigated for commonly used large-aperture lens models,including biconcave,biconvex,concave meniscus,and convex meniscus lenses.To solve the lens deformation under non-uniform multi-point support for these four models,an axisymmetric variable-thickness circular plate’s nonaxisymmetric bending equation is established,and the equation is discretized using the stairstep conversion algorithm.To facilitate computation and software calculations,the transfer matrix method is introduced to improve the equation solving process,establishing an overall transfer matrix for deflection,tilt,shear force,and bending moment.Simulations are conducted for 5-point,6-point,8-point,and 10-point support configurations corresponding to practical non-uniform support conditions.The results are compared with analytical solutions derived from theoretical derivations.Under boundary support conditions,the calculated errors of optical parameters,including PV value,RMS value,and key Zernike coefficients,are all less than 10%.This confirms the accuracy of the stair-step conversion algorithm in solving non-uniform mechanical problems.Finally,an analysis and summary of solution errors are provided.Lastly,the study investigates the influence of lens gravity load and frame assembly load on the lens’ s optical surface shape.By utilizing the orthogonality of Zernike polynomials,the lens deflection is transformed into a Fourier series representation,which characterizes the optical surface shape and lens aberrations using Zernike series.The relationship between Zernike series and Fourier series in the context of plate shell theory is explored to determine the influence of mechanical boundaries on Zernike coefficients.This method is then applied to analyze the effects of shim head size,number of support points,and radial displacement on the lens surface shape in the multi-point support lens structure,providing guidance for lens structure design.Furthermore,the method is utilized to analyze the non-uniformity of lens support force.Taking 8-point support as an example,the surface shape and Zernike coefficients for each support point under unit support force are presented,investigating the theoretical approach to compensate for specific Zernike coefficients.This offers guidance for lens structure calibration.In conclusion,this paper provides theoretical guidance for the design and calibration of multi-point support lens structures.The numerical examples demonstrate that the proposed method has a key optical parameter error of no more than 10% when calculating the deformation of variable-thickness lenses under multi-point non-uniform support.In terms of lens structure design,to improve the lens surface shape,the number of support points should be maintained above 8,and the radial width of the shim should not be excessively large.It is advisable to appropriately increase the azimuthal width of the shim and the number of support points while considering the allowance for light transmission holes and allowing for radial movement of the support points.In terms of lens structure calibration,based on the analysis of influence functions,it is theoretically possible to determine the required compensatory force size based on the Zernike coefficients obtained from lens surface shape measurements,thereby improving the non-uniformity of the support force.