# Improved Analytical Theory of Ophthalmic Lens Design
## 4 Discussion
where \(\alpha\) and \(\beta\) provides a weighted average of the oblique powers that are evaluated at points \(x_{k}\), as it is the target power \(P_{tgt}(x_{k})\), and \(W_{k}\) are weights controlling the importance given to the transverse coordinate. The idea behind this approach is that the surface minimizing (36) can be found among the surfaces that comply with (22) and whose coefficients are given by (26) to (29). If that were the case, we would have converted a multi-parametric optimization of the surface coefficients into the minimization of a functional that depends on a single variable. The merit functions are usually constructed to balance the error between sagittal and tangential power or between mean power and astigmatism. One instance of these merit functions consists of the minimization of the defocus defined by [PERSON] [19], which is given by  
\[\delta(x)=\sqrt{\left[F_{S}(x)-P\right]^{2}+\left[F_{T}(x)-P\right]^{2}} \tag{37}\]  
For example, we could choose  
\[f=\int_{0}^{\hat{x}}\delta^{2}(x)dx, \tag{38}\]  
where \(\hat{x}\) is a limiting value for the field of view. Another example is the \"rational lens\" [12]. Let us consider a more general merit function of the form:  
\[ \begin{array}{lcl}f=\int_{0}^{\hat{x}}\{w_{1}\left[F_{S}(x)-P\right]^{2}+&w_ {2}[F_{T}(x)-P]^{2}+w_{3}[F_{S}(x)+F_{T}(x)-2P]^{2}\\ &+w_{4}[F_{S}(x)-F_{T}(x)]^{2}\}dx,\end{array}  \tag{39}\]  
which aims to reduce tangential error, sagittal error, power error, and astigmatism uniformly over the transverse coordinate and with weights that can be organized in a single 4-tuple \(w=(w_{1},w_{2},w_{3},w_{4})\). This general merit function includes the minimization of the Raasch defocus when the weight vector is set to \(w=(1,1,0,0)\). Let \(z_{u}(x)\) be the surfaces that meet condition (22) for the parameter \(u\), and let us try to minimize \(f\) in the set of functions \(\{z_{u}\}\). Now, \(f\) depends on the single variable \(u\) (which will determine the coefficients \(\{c_{i}\}_{i=2,\cdots,2N}\)), and we just must impose \(\partial f/\partial u=0\). The problem can be analytically solved if \(N\) is large enough so that property (33) holds. In that case we use \(F_{T}=P+c_{T2}x^{2}\) and \(F_{S}=P+c_{S2}x^{2}\), where the coefficients are given by  
\[ \begin{array}{lcl}c_{T2}=\frac{L_{2}^{\prime\,2}\,P\,u}{u+3\,v}+\frac{Pu\left( B^{2}-L_{2}^{\prime\,2}+2\,L_{2}^{\prime}\,P-P^{2}\right)}{n\left(u+3\,v \right)}-\frac{Pu\left(B^{2}+B\,P-P^{2}\right)}{\left(u+3\,v\right)\left(n-1 \right)},\\ c_{S2}=-\frac{L_{2}^{\prime\,2}\,P\,v}{u+3\,v}-\frac{P\,v\left(B^{2}-L_{2}^{ \prime\,2}+2\,L_{2}^{\prime\,2}\,P-P^{2}\right)}{n\left(u+3\,v\right)}+\frac{ Pv\left(B^{2}+B\,P-P^{2}\right)}{\left(u+3\,v\right)\left(n-1\right)}\end{array}  \tag{40}\]  
Now, the minimization of (39) can be analytically obtained, with the result  
\[u_{min}=\frac{w_{1}+4w_{3}-2w_{4}}{\sqrt{w_{1}^{2}+8w_{1}w_{3}-4w_{1}w_{4}+9w _{2}^{2}+24w_{2}w_{3}+12w_{2}w_{4}+32w_{3}^{2}+8w_{4}^{2}}}. \tag{41}\]  
For the particular case of the Raasch defocus, the optimum value for \(u\) is \(u_{min}=1/\sqrt{10}\cong 0.316\). This prediction has been validated with the result obtained when the Raasch defocus is numerically minimized with exact ray tracing and multi-parametric optimization of the surface coefficients. Figure 9 shows the tangential and sagittal powers for the lens with parameters \(P=-4\) D, \(B=0.5\) D, \(n=1.5\), and \(L_{2}^{\prime}=37\) D, which satisfies condition (22) with \(u=0.316\), compared with the oblique powers of the lens numerically optimized. The difference between the two approaches is below 0.01 D for the range of transverse coordinates for which the angles and curvature approximations make sense. The prediction fits the numerical result quite well, which indicates that the previous method is adequate for exploring lens performance using nonlinear merit functions without the need for numerical optimization.