Orthogonal basis with a conicoid first mode for shape specification of optical surfaces

Abstract

Abstract:A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.

Interbasis expansions in the Zernike system

“…The coefficients U n 1 ,n 2 1 , 2 can be calculated again as we did following (20) for ϑ = π/2 − ε on the common rim of the disk and half-sphere. As before, in the limit ε → 0 on both sides of (51), with cos ϑ = cos ϕ , cos ϕ ≈ 1, and sin ϕ ≈ cos ϑ / sin ϕ in (5) and (6), where x = ξ 1 = 0. Using the expressions for Legendre and Gegenbauer polynomials in (14) and (16) for even 1 = 2q 1 and n 2 = 2p 2 , with q 1 , p 2 non-negative integers,…”

“…Finally, we underline the fact that the original Zernike polynomials have a great practical importance in phase-contrast microscopy and in the correction of wavefronts in circular pupils. Recent work [4,5,6,15] has extended this technique to pupils of essentially arbitrary shape through diffeomorphisms that conserve their basic properties. This has been applied to describe wavefronts in sectorial, annular and polygonal-shaped pupils, the latter specifically tailored to the hexagonal components of large astronomial mirrors.…”

“…We first consider the interbasis expansion between the I and II orthonormal basis functions in (11) and (13) for fixed values of the principal quantum number n, where n m=−n (2) indicates that the sum over m is in the range (18), so the radial quantum number n r = 1 2 (n − |m|) ∈ Z + 0 is integer. The relation between the primed and unprimed angles is found equating (4) and (5), as cos ϑ = sin ϑ cos ϕ, cos ϕ = sin ϑ sin ϕ…”

Review of optical freeform surface representation technique and its application

“…50 Tatian, 51 Barakat, 52 Mahajan and Dai, 50 Swantner and Chow, 53 Hasan and Shaker, 54 Díaz and Mahajan, 55 and Díaz and Navarro 56 reported analytical Zernike orthogonal polynomials with differently shaped apertures. Ferreira et al 57 proposed a rigorous and powerful theoretical framework to obtain the orthogonal basis with a conicoid first mode for surface specification. Liu et al 58 applied twodimensional (2-D) Chebyshev polynomials to characterize the "W"-shaped freeform optical elements.…”

Orthogonal basis for the optical transfer function

“…In this sense Forbes proposed a highly successful set of functions, adapted from orthogonal polynomials, to specify aspheres [3] and free-form [4] surfaces. In a recent work we proposed a method to generate systems in which the first basis function is the sphere (or conicoid) and the rest of functions are orthogonal to it [2]. Complex Zernike polynomials were also used in the extended Nijboer-Zernike theory for the computation of optical Point-Spread Functions (PSF) [5], as they can represent the complex pupil function that is both the amplitude and phase of a wavefront at the (circular) pupil plane [10].…”