Analysis Of The Surface Figure Of Point-supported Lens Based On The Thin Plate Theory

The rapid development of integrated circuits has put forward higher requirements on lithography technology and lithography equipment.As an important part of the lithography machine,the objective lens of the lithography machine has a serious impact on the performance of the lithography equipment.The deformation of the objective lens under its own weight will affect the wave front difference of the optical system,which is difficult to effectively eliminate such errors.Since the macroscopic size of the lens thickness and the distribution of the lens support points is much smaller than the lens diameter,its geometric characteristics conform to the definition of thin plates in elastic mechanics.Therefore,it is of great significance to improve the overall accuracy and performance of the lithography machine to study the change of deflection based on the theory of elastic thin plate through mechanical analysis method.First,the deflection problem of the lens under the uniform point support is studied.Based on the reciprocal theorem and plates and shells theory,the Zernike coefficients,PV value and RMS value in the optical field are introduced as indicators to characterize the overall surface accuracy of the mirror surface,through theoretical derivation,the deflection change expression of the uniform thickness circular lens under the condition of several uniform points is obtained,and the Zernike fitting is performed on the deflection analytical solution to obtain 37 Zernike coefficients;at the same time,the finite element simulation software is used to calculate the numerical solution of the mirror deflection under the same working conditions.The Zernike coefficients are obtained by fitting the numerical solution,and the two results are compared and verified.Then,when the reaction force at each support point is non-uniformly distributed,the deformation of the lens is analyzed in depth.The analytical expression of deflection is derived,and the deflection of each point of the mirror is fitted to the Zernike polynomial to obtain the Zernike coefficients.Then,the numerical solution of the deflection under the same working conditions is obtained through finite element simulation.The Zernike fitting coefficients of the numerical solution are again verified by comparison.Finally,based on the software interface written by Matlab,the calculation program of the lens surface shapes under uniform support and non-uniform support are packaged into independent software.The expression of mirror deflection is calculated by using the reciprocal theorem combined with the theory of elastic circular thin plates.The results in this paper show that,for uniform point-support,the typical point deflection error of the mirror deflection is less than 5%,and the error of the optical related metrics such as PV value and RMS value is less than 6%;for non-uniform point-support,the accuracy error of PV value and RMS value is less than 7%.In addition,the calculation method in this paper is also applicable to other complicated working conditions such as interior point support and variable thickness point support.