Optical design with rotationally symmetric NURBS

The standard aspheric surface, a conic surface figured with a polynomial expansion, eq 1, provides excellent correction in many optical design problems. Z(r) = r(2)/R / 1 + root1-(1-(1+k)((r2) / (R2)) + A.r(4) + B.r(6) + C.r(8) + ... But there are problems where this set of basis functions does not provide the best solution. This paper discusses a parametric curve alternative called Non-Uniform Rational B-Splines (NURBS). NURBS are used extensively in the computer aided geometric design industry (CAGD) because they offer a rational segmented polynomial curve with the flexibility of setting both the order of the B-spline segments and the locations of the knots, or joints, between the B-spline segments providing local curve control. In addition, they are numerically stable and they have geometric intuitive control points for manipulating the curve. The advantages and disadvantages of using NURBS in optical design applications are discussed. Brief explanations of NURBS curve mathematics, properties,and design techniques are presented. The paper concludes. with an optical design example comparing the optical performance between rotationally symmetric NURBS surfaces and standard aspheric surface. Note that all of the aspheric surfaces discussed in this paper are rotationally symmetric.