Ester Pérez Sinusía

Generalization of Zernike polynomials for regular portions of circles and ellipses

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.

Multi-point Taylor approximations in one-dimensional linear boundary value problems

Abstract We consider second-order linear differential equations in a real interval I with mixed Dirichlet and Neumann boundary data. We consider a representation of its solution by a multi-point Taylor expansion. The number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent ∀ x ∈ I . We propose several algorithms to approximate the multi-point Taylor polynomials of the solution based on the power series method for initial value problems.

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

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.

New series expansions for the confluent hypergeometric function M(a,b,z)

Abstract Three new series expansions of the confluent hypergeometric function M ( a , b , z ) in terms of elementary functions are given. The starting point is an integral representation of M ( a , b , z ) . Then, different multi-point Taylor expansions of an appropriate function in the integrand are used. The new expansions are used to evaluate M ( a , b , z ) . They provide accurate evaluations of the confluent hypergeometric function, in particular improving the results in the region of small or moderate ∣ z ∣ where the power series definition is recommended for the evaluation of M ( a , b , z ) .

Two-point Taylor expansions and one-dimensional boundary value problems

We consider second-order linear differential equations ϕ(x)y + f(x)y + g(x)y = h(x) in the interval (―1, 1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider ϕ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x = ±1 containing the interval (-1,1). The two-point Taylor expansion of the solution y(x) at the extreme points ±1 is used to give a criterion for the existence and uniqueness of solution of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the solution(s) when it exists.

Zernike-like systems in polygons and polygonal facets.

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Opt. Lett.32, 74 (2007)10.1364/OL.32.000074OPLEDP0146-9592] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both the general form and the explicit expressions for a typical example of telescope optical aperture are provided.

Asymptotic approximation of singularly perturbed convection-diffusion problems with discontinuous derivatives of the Dirichlet data

We consider a singularly perturbed convection-diffusion equation, -∈ A u + υ . ⊇u = 0, defined on two domains: a quarter plane, (x, y) ∈ (0,∞) x (0,oo), and a half plane, (x,y) ∈ (-∞,∞) x (0,∞). We consider for these problems Dirichlet boundary conditions with discontinuous derivatives at some points of the boundary. We obtain for each problem an exact representation of the solution in the form of an integral. From this integral we derive an asymptotic expansion of the solution when the singular parameter e → 0 + (with fixed distance r to the points of discontinuity of the boundary condition). It is shown that, in both problems, the first term of the expansion contains the primitive of an error function. This term characterizes the effect of the discontinuities on the ∈-behaviour of the solution and its derivatives in the boundary or internal layers.

Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter

We consider the second-order linear differential equation

Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions

y=left[frac{Lambda^2}{x^alpha}+g(x)right]!y,