# Improved Analytical Theory of Ophthalmic Lens Design
## 4 Discussion  
Several authors have pointed out that the fourth-order asphericity is well defined for any pair \((B,P)\) and for the three classical solutions [3; 9; 12]. In other words, in classical third-order theory, for any pair \((B,P)\) we can always find a coefficient \(c_{4}\) for which the lens meets one of the three classical conditions \(E_{A}=0\), \(E_{p}=0\), or \(E_{T}=0\). According to the theory presented in this paper, the fourth-order asphericity is well defined except when \(u+3v=0\), that is, when \(u=-3v\). By inspection of the recurrence relation (29) and examples (30) and (31), we can see that any higher-order term \(c_{i}\) fails to be well defined for all \(u_{j}\), such that \(\big{\{}u_{j}=-(j-1)v\big{\}}_{j=4,6,\cdots,j}\). If we use the normalization condition (23), we can state that any theory of order \(2N\), that is, a theory using monomials up to order \(2N\) in the expansion of \(z(x)\), will fail to meet condition (22) for the set of values of the parameter \(u\) satisfying  
\[u_{j}=\frac{1-j}{\sqrt{1+(1-j)^{2}}},j=4,6,\cdots,2N \tag{34}\]  
Figure 7: Tangential power for a lens with \(P=5\) D, \(B=6\) D, \(n=1.5\), and \(L_{2}^{\prime}=37\) D, computed for \(u=-0707\). It was computed from Equation (18) using an eighth-order expansion for the back surface. The orange line is the tangential power computed with all the terms resulting from (18), while the blue curve has been computed only with the quadratic term. This is a decreasing monotonic sequence with limit \(-1\) and starting at \(u_{4}=3/\sqrt{10}\). We conclude that the general problem stated by condition (22) has a solution for any pair \((P,B)\) when \(u\in\left(-3/\sqrt{10},1\right]\). This interval includes the classical solutions: point focal lenses, Percival lenses, and lenses with zero tangential error. Lenses with zero sagittal error (\(u=-1\)) or with very small sagittal error (\(u\gtrsim-1\)) lie in the complementary interval \(u\in\left[-1,-3/\sqrt{10}\right]\). In this interval, and whenever Equation (36) is satisfied for a given \(j\), all the coefficients with \(i>j\) are not defined. Lenses with identically zero sagittal error can be computed, as all the coefficients are finite. However, convergence of expansions (25) will be slower than for those problems with bigger values of \(u\). Despite the fact that Equations (27)-(30) are not linear on \(u\), we can find the relationships between the asphericity parameters of a given order corresponding to different values of \(u\). For example, let us call \(c_{4p}\), \(c_{4m}\), and \(c_{4t}\) to the fourth-order asphericities corresponding to point-focal lenses, Percival lenses, or zero tangential error lenses, respectively. It can be shown that  
\[c_{4t}=\frac{2}{3}\;c_{4m}+\frac{1}{3}\;c_{4p}, \tag{35}\]  
that is, the fourth-order asphericity coefficient of the lens with zero tangential error is a weighted sum of the coefficients for the Percival lens and the point-focal lens. Among the classical solutions, asphericity for zero tangential errors provides a more balanced solution. Figure 8 shows the curve \(c_{4}(u)\) in the interval \(u=\left(-\sqrt{3}/10,\;1\right]\) for the example lens used in Figures 5-7. The amount of asphericity needed to meet some design criterion, whether it is a determined value of \(u\) or the use of other merit functions, which are related, as we show next, is of great value to the lens designer. In general, smaller values of asphericity may facilitate the lens manufacturing and will make the lens more stable with respect to variation of the wearer's parameters or lens misalignment. We finally discuss the relation between condition (22) and the use of more general merit functions that may balance combinations of the oblique powers across the complete field of view. A standard example of such merit function can be  
\[f=\sum_{k}w_{k}\big{[}(\alpha P_{T}(x_{k})+\beta P_{S}(x_{k})-P_{tgt}(x_{k}) \big{]}^{2} \tag{36}\]  
Figure 8: Fourth-order asphericity \(c_{4}\) as a function of \(u\) for the same lens example used in Figures 5–7.