Optimization method for designing optical elements with an extended light source

A method for designing an optical element with two free-form surfaces generating a prescribed illuminance distribution in the case of an extended light source is considered. The method is based on the representation of the optical element surfaces by bicubic splines and on the subsequent optimization of their parameters using a quasi-Newton method implemented in the Matlab software. To calculate the merit function, a version of the ray tracing method is proposed. Using the proposed method, an optical element with record characteristics was designed: the ratio of the element height to the source size is 1.6; luminous efficiency is 89.1 %; uniformity of the generated distribution (the ratio of the minimum and average illuminance) in a given square region is 0.92.

Optimal Approximation of Biquartic Polynomials by Bicubic Splines

Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question.The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

Bicubic splines and biquartic polynomials

The paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.