Jacobi Circle Polynomials For Wavefront Expansion And Modal Phase Retrieval

Wavefront function have complex influence on optical system when aberration exist.A appropriate way of analysis is dividing the wavefront function as basic modes.The set of orthogonal functions frequently used in such methods are Zernike circle polynomials,which are mutually orthogonal on the unit circle,the low-order terms of Zernike modes are corresponding to Seidel aberrations.However,the gradients of Zernike circle polynomials are not orthogonal.Therefore,modal cross coupling occurs for the modal approaches from wavefront gradient.In addition,the unified explicit power function expression of the Zernike annular polynomials has not been given in previous studies,the radial polynomials are obtained recursively from Zernike radial polynomials through the Gram–Schmidt orthogonalization process,possibly because of the complexity of the recursive formula of a typical polynomial.To tackle this problem,we provide a set of orthogonal polynomials on the unit circle,with orthogonal radial derivatives,as a choice of wavefront expansion and modal wavefront reconstruction.The remainder of this paper is organized as follows:Firstly,we provide a set of polynomials that are mutually orthogonal on the unit circle(unit disc),for which the radial polynomials are well-known Jacobi polynomials,the set of polynomials may be refer to as Jacobi circle polynomials.The relationship between Zernike circle polynomials and Jacobi circle polynomials is discussed.Any classical Zernike mode and linear combination of Zernike modes can be represented as a finite linear combination of Jacobi modes,theoretically providing the possibility of reconstructing Zernike wavefront modes by Jacobi modes.The wavefront expansion by Jacobi circle polynomials and first several order of Jacobi modes is shown.The order of the Jacobi and Zernike modes is adjusted for ease of comparison.Secondly,the modal approaches for wavefront reconstruction from wavefront gradient data using Zernike and Jacobi modes are given.We introduce the modal approaches incorporating the Gram matrix and then derive the interpretations of “aliasing”and “cross coupling” of the Gram matrix method.The correction for zero-order Jacobi coefficients over the unit circle is discussed.The modal approach incorporating the Gram matrix for high sampling is considered.It is shown that cross coupling is avoided when Jacobi circle polynomials are selected as the basic function.Thus,the Gram matrix method has potential application in lateral shearing interferometers and could be considered for high-sampling Shack-Hartmann wavefront sensors.A comparison of the Jacobi and Zernike modes using least squares estimation is presented,which indicates that Jacobi modes exhibit similar performance to the Zernike modes.In addition,the construction of Jacobi annular polynomials is derived.A unified definition of the Jacobi circle polynomials over the unit circle,concentric scaled pupils and annular pupils is given.This formula is expressed as an explicit power function and can be conveniently obtained without use of a Gram–Schmidt process.Further,the outer radius can be freely selected;thus,any concentric circle can be selected as the unit circle for central symmetry optical systems.The wavefront expansion by Jacobi annular polynomials and first several order of Jacobi modes is shown.Any classical Zernike mode and linear combination of Zernike circle polynomials and Zernike annular polynomials can be represented as a finite linear combination of Jacobi modes over annular pupils,theoretically providing the possibility of reconstructing Zernike wavefront modes by Jacobi modes.Jacobi annular polynomials and Jacobi radial derivatives retain orthogonality over the annulus.Hence,the modal approach incorporating the Gram matrix and Jacobi modes for high sampling is extended to annular pupils.Finally,we brief describe the Zernike modal phase retrieval method by extended Nijboer–Zernike theory.We provide the explicit result of far field Debye diffraction integral for Jacobi modes,and discuss the potential application of Jacobi modes in the modal phase retrieval method by extended Nijboer–Zernike theory.