15 May 1999 Ž . Optics Communications 163 1999 259–269 Full length article Wavefront fitting using Gaussian functions Marcial Montoya-Hernandez 1 , Manuel Servın, Daniel Malacara-Hernandez ) , ´ ´ ´ Gonzalo Paez Centro de InÕestigaciones en Optica, A.C., Apdo. Postal 1-948, C.P. 37 000, Loma de Bosque 115, Col. Lomas del Campestre, Leon, Gto., Mexico Received 17 August 1998; received in revised form 29 January 1999; accepted 1 March 1999 Abstract Often the polynomial description of a wavefront shape is inaccurate because sharp local deformations are difficult to represent. In this case an analytical representation in terms of Gaussians may give better results. We have made a study of properties of Gaussians in fitting wavefronts. We analyzed an specific array of Gaussian functions and show their easy Fourier transformation. In Fourier space some criteria are proposed for setting the Gaussian width, the separation between these functions and making an estimation of the wavefront fitting error. Two simulated wavefronts are fitted and a comparison with a Zernike polynomial is made. It is well known that when fitting using Zernike polynomials one needs to find the optimal number of terms beyond which the errors in the approximation become larger. We show that using Gaussians the accuracy of the fit increases with the number of terms. We demonstrate some interesting properties, such as their facility to fit local deformations and fast parameter determination. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Wavefront approximation; Radial basis function; Interpolation; Adaptive optics 1. Introduction The analytical representation of a wavefront is extremely useful in fields such as GRIN rod analysis, adaptive optics and interferometry. In GRIN rod analysis one needs an analytical representation of the wavefront shape in order to study the propagation of light rays. Also it is necessary to have an analytical wavefront representation to study the correction of deformations by mean of adaptive optics. In interferometry an analytical representation of a wavefront may be useful to manipulate it in the computer and try alternative interferometric procedures that must check with laboratory. These procedures may be used to observe the fringes that arise from lateral or rotational shear, scaling or rotation of the wavefront without the information loss associated with pixel operations. wx Most of these representations have been done using a linear combination of polynomials 1 . The main limitation of the polynomial representation is that each term extends its influence over the entire pupil. This may be a problem given that the Ž . Ž . resulting matrix when non-orthogonal polynomials are used in Least Squares Fitting LSF may be difficult to invert. This problem disappears when orthogonal polynomials such as the Zernikes are used. But some problems remain, one is that Zernike polynomials apply only to circular pupils. Despite that, annular pupils have been represented analytically using also orthogonal polynomials. The fitting error however is distributed over the entire wavefront not only at those zones where the deformation is located. Another problem is that Zernike polynomials become large as their degree is increased. So, when it is necessary to do a fit with high degree Zernike polynomials the process become too slow. ) Corresponding author. E-mail: dmalacara@foton.cio.mx 1 E-mail: montoya@andromeda.cio.mx. 0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0030-4018 99 00120-0