# Improved Analytical Theory of Ophthalmic Lens Design
## 1 Introduction
Because of the form of the previous equation, cancellation will take place for all values of \(x\); that is, third-order theory applied to spherical lenses predicts that a given oblique error will be cancelled at full field if the base curve is selected so that \(f_{sph}(B,P)=0\). Finally, as each oblique error has its own function \(f_{sph}(B,P)\), only one type of error can be cancelled. Lenses with aspherical surfaces but still keeping revolution symmetry have also been thoroughly studied. The extension of the third-order theory to include lenses with aspherical surfaces has usually been based on a temporary fourth-order approximation to concioids [2; 8; 18]. This procedure provides one extra parameter per aspheric surface: the conic constant. Once the asphericity is incorporated into the equations, a new second-order approximation on the transverse coordinate \(x\) yields a result similar to that of spherical lenses,  
\[E=x^{2}f_{sph}(B,P,Q_{1},Q_{2},n,L)\]  
where \(Q_{1}\) and \(Q_{2}\) are, respectively, the conic constants of the front and back lens surfaces, and \(f_{sph}\) is no longer a quadratic equation on \(B\) and \(P\). The extra degrees of freedom provided by the asphericity allows for extending the range of lens powers for which there is a solution \(f_{sph}=0\), which is infinite now. Moreover, a solution \(f_{sph}=0\) can be obtained for any base curve, provided we select the conic constants adequately. Finally, the fact that \(f_{sph}\) does not depend on \(x\) implies that correction of the oblique error is uniform across the whole field. However, when aspheric lenses with conicoid surfaces are studied by means of exact ray tracing, it is observed that full field compensation of the oblique error cannot be achieved [9; 17]. Numerical computations reveal that, indeed, any lens power can be corrected of one oblique error for any value of the base curve, but correction takes place at a single viewing direction (a single value of the x coordinate) and then, typically, performance rapidly deteriorates as the viewing angle increases [9]. The reason for this disparity between the two computation approaches is that although local curvatures of the conicioids are incorporated into the Coddington equations through a fourth-order approximation (which is the bare minimum to introduce the conic constants), the resulting equations for the oblique errors are finally simplified by just retaining up to second-order terms in \(x\). One important conclusion that can be deduced from the third-order approach is that, assuming one surface of the lens is kept spherical, it is not possible to simultaneously correct the astigmatism error and mean power error. That means all spectacle lenses (with at least one spherical surface) will have some uncorrected errors that might affect the visual quality. Therefore, a compromise is needed in how the unwanted power error is distributed between the mean power and the astigmatism. A usual strategy involves the definition of a merit function that can include terms for the oblique astigmatism, lens thickness, distortion, or higher-order aberrations. One example of such merit function is the LeGrand criterion described by [PERSON] et al. [12], which uses a weight of \(\sqrt{2}/2\) on the astigmatism relative to the mean power. Another example could be the minimization of the blur as defined by [PERSON] [19], which implies a relative weight of \(1/2\) on the astigmatism relative to the mean power. [PERSON] et al. [1] proposed a merit function that combines oblique astigmatism, distortion, and a restriction to avoid inflection points on the surface. As we have previously described, third-order theory has been applied to demonstrate the existence of solutions for merit functions that correct either oblique astigmatism, mean power error, or tangential error, but the question about the existence of solutions for general merit functions is still unanswered. [PERSON] et al. [8] described results for aspherical lenses that eliminates oblique astigmatism while reducing high-order aberrations.