Research On The Method And Technique For Characterizing Freeform Optical Surface

With the development and advancement of modern precision optical fabrication and measurement,the manufacturing and application of freeform optical elements are gradually realized.Freeform optical surface with the non-rotational symmetry has more abundant degrees of freedom and strong aberration correction capability,making it possible to the direction of high requirements such as miniaturization,light weight,large field of view,low F number and high performance for the optical system.Thus,freeform optics are playing significant roles and having potential applications in the area of modern smart home,advanced industrial manufacturing,green energy and aerospace.The method and technique for characterizing freeform optical surface is a basic and key research content in the field of freeform optics.Its advances enable further improvement for freeform optics.In recent ten years,the research about the method and technique for representing freeform optical surface has become the focus.A few key problems need to be settled.In this dissertation,in-depth research about the representation methods of freeform optical surface is carried out.From two aspects which are the orthogonal and non-orthogonal functions,different kinds of functions for currently representing the freeform surface are summarized.Their advantages and limitations are analyzed.For the orthogonal polynomial such as Zemike circle orthogonal polynomial,it has been extensively applied in the freeform surface representation,wavefront analysis and system aberration evaluation because of its fine mathematical property.For the non-orthogonal function such as XY polynomial,it is usually employed in the design of off-axis and non-rotationally symmetric freeform optical system due to its more strong aberration correction ability.With respect to the issues of the analytical orthogonal function losing its orthogonality in practical applications(such as the obtained discrete data points in real measurement or ray tracing)and having the selectivity of its shaped aperture,numerical orthogonal polynomial that is widely applicable and has high representation accuracy is proposed for representing freeform surface.The deficiencies of existing analytical orthogonal functions are overcome by nmmerical orthogonal polynomials for characterizing the freeform surface.Further,by nmmerical analyses and experiments,the performance of numerical orthogonal polynomials and orthogonal polynomials over the square aperture(namely 2D Chebyshev polynomials,2D Legendre polynomials,Zemike square orthogonal polynomials)for representing freeform surface across the square aperture are compared and evaluated in details.The results demonstrate that numerical orthogonal polynomials have the distinct superiority for characterizing freeform optical surface.Meanwhile,numerical orthogonal polynomials can be employed to represent and analyze freeform surface or wavefront with dynamic varying aperture in real time.For characterizing freeform surface with steep slope in local areas with high accuracy,the representation method by combining Zemike polynomials with radial basis function is proposed.The representation strategy of "breaking up the whole into parts and recombining the parts to the whole" is applied to the presented method.Its representation accuracy can reach nanometer level.The proposed method is able to manifest the local character of complex freeform surface with high accuracy.The limitations of current representation method in a single time across the whole aperture is overcome elegantly.Two key parameters which are the distance of two adjacent sub-apertures and the size of sub-aperture radius are analyzed detailedly for the influence of representation error of freeform surface with steep slope in local areas.The results show that the size of sub-aperture radius has greater impact to the representation accuracy.Therefore,when the distance of two adjacent sub-apertures is chosen and confirmed reasonably,by choosing the size of sub-aperture radius preferentially,the representation accuracy demand can be satisfied for freeform surface with steep slope in local areas in practical testing.With respect to retrieving freeform surface or wavefront from its slope(gradient)discrete data,there are a few limitations about the current zonal approach or modal approach.A numerical orthogonal transformation method by two times is proposed,which is a non-iterative method.The numerical orthogonal gradient polynomials are derived and employed to represent the measured slope data directly.Then according to the relationship between the slope data and the sag data,the freeform surface or wavefront can be retrieved.The presented method can be applied to represent the slope-based freeform surface over the arbitrary shaped aperture or dynamic varying aperture.The results indicate that,when deriving the freeform surface from its discrete slope data by numerical orthogonal transformation method by two times,it has high representation accuracy for the slope-based random freeform surface over the regular shaped aperture such as the circular aperture,square aperture,rectangular aperture,hexagonal aperture or annular aperture.The accuracy is also very high for the irregular aperture containing invalid slope data or dynamic varying aperture.For the slope-based complex freeform surface with steep slope in local areas,the presented method also performs well.At the same time,the proposed method has significant application values and prospects in the field of adaptive optics or ophthalmic optics.