# Improved Analytical Theory of Ophthalmic Lens Design
## 3 Results
The first conclusion we can draw from Figure 5 is that oblique powers do not behave as expected at full field, that is, for all values of the transverse coordinate used in the computation, up to \(15\) mm (which corresponds to \(\theta_{2}^{\prime}=29\) deg). This is especially easy to spot for the classical solutions, which occupy the main diagonal of the grid in Figure 5. For punctual lenses (\(u=-0.707\)), we expect that \(F_{T}=F_{S}\). For Percival lenses (\(u=0.707\)), \(F_{T}-P\) should equal \(P-F_{S}\). For lenses with zero tangential error (\(u=0\)), \(F_{T}\) should equal \(P\). We indeed observe these behaviors but only for values of \(x\) smaller than around \(5\) mm (\(\theta_{2}^{\prime}=10\) deg). As the transverse coordinate grows larger, the error in meeting condition (22) also grows, more or less uniformly across the different values of the parameter \(u\). The reason for this error is explained next: When the expressions for the oblique powers are inserted into the condition (22), and we solve for the coefficients of the Taylor expansion of \(z(x)\), we find a condition for \(c_{2}\) that gives (26), a special relation between \(c_{2}\) and \(c_{4}\) that gives (27), and the recurrence relation (28) linking \(c_{i}\) with \(c_{i-2}\) starting with \(i=6\). Let us assume that there are just two terms (second- and fourth-orders) in the series expansion for \(z(x)\). The outline procedure then gives two contradictory equations for \(c_{4}\), equation (27) and \(c_{4}=0\), the latter from the recurrence relation (28) with \(c_{6}=0\). The meaning of this is that there is no surface defined with just a quadratic and a quartic coefficient that allows (22) to be fulfilled for all \(x\), at least with the approximations used to reach (18) and (19) and with the approximation giving rise to the sagittal and tangential refractive powers of the surface. Figure 6, using the same color and line type schemes as before. Once again, the match between the analytical results and the numerical computation is remarkably good up to \(x\cong 12\) mm (\(\theta_{2}^{\prime}=24\) deg). Additionally, the two techniques now provide a much better fulfillment of condition (22) for any \(u\). We observe that the analytical solutions start deviating from the numerical one for \(x>12\) mm, especially the tangential power. We suspect that at this value of the transverse coordinate, the approximations made on the angles and the curvatures start showing up. The results presented in Figure 6 strongly suggest that higher-order polynomial terms are indeed needed for a full-field correction of the oblique aberrations, even when the oblique powers have been obtained with the low-angle, low-curvature, and negligible-thickness approximations. It is worth noting that all the coefficients \(c_{i}\) with \(i>2\) are proportional to \(\Delta\). If the lens parameters combine in a way that \(\Delta=0\), all these coefficients will be zero, and condition (22) will be satisfied at full field. A surface with a single second-order term is, within the approximations used, nothing but a spherical surface, and the condition \(\Delta=0\) expresses the relation between the base curve and the lens power that will satisfy condition (22). \(\Delta(u)=0\) is then the equation of the Tschering ellipse for each value of the parameter \(u\). There is a final consideration about the oblique powers shown in Figure 6. Their behavior is smoother and more \"parabolic\" than those in Figure 5, where the curves clearly deviate from the parabolic shape when the fourth-order term dominates. This behavior is not a special feature of the chosen example; we have always found the same pattern in many other cases numerically analyzed. To have a better understanding of this feature, let us assume we start with a back surface defined by a polynomial having \(N\) terms and maximum order \(2N\). Using Equations (26)-(29) we find the coefficients \(\{c_{i}\}_{i=2,\ldots,2N}\) for which the lens satisfies condition (22).