# Improved Analytical Theory of Ophthalmic Lens Design
## 3 Results
When expressions (20)-(22) are applied to Equations (18) and (19), the problem is converted into a second-order linear ordinary differential equation, for which finding an explicit solution is not generally possible. However, if the surface has a Taylor series representation, \(z(x)=\sum\limits_{i>0\ even}c_{i}x^{i}\), it can be inserted into Equations (20) and (21), and the uniqueness of the Taylor series yields similar expansions for the surface powers \(F_{2T}\) and \(F_{2S}\),  
\[ \begin{array}{l}F_{2T}(x)=-(n-1)\sum\limits_{i\geq 0\ even}(i+2)(i+1)c_{i+2}x^ {i},\\ F_{2S}(x)=-(n-1)\sum\limits_{i\geq 0\ even}(i+2)c_{i+2}x^{i}.\end{array}  \tag{25}\]  
Now, using these expansions in Equations (18) and (19) delivers the oblique powers also as a power series in \(x\). Finally, we can use them in condition (22) and solve for the coefficients \(c_{i}\) that fulfill the condition. The first two coefficients are  
\[c_{2}=\frac{B-P}{2(n-1)}, \tag{26}\]  
\[c_{4}=\frac{P}{8n\left(u+3v\right)\left(n-1\right)^{3}}\Delta, \tag{27}\]  
where \(\Delta\) is a shorthand description for the quantity  
\[ \begin{array}{l}\Delta=B^{2}\left[u\left(2\,n+1\right)+v\left(4\,n+5\right) \right]+\left[P+L_{2}^{\prime}\left(n-1\right)\right]^{2}\left(u+v+2\,n\,v \right)\\ -B\,P\left[u\left(-n^{2}+2\,n+2\right)+v\left(-n^{2}+4\,n+6\right)\right]\\ -2\,B\,L_{2}^{\prime}\left(n^{2}-1\right)\left(u+3\,v\right).\end{array}  \tag{28}\]  
The coefficients for \(i>4\) can be obtained from the recurrence relation \[c_{i}=-c_{i-2}\frac{(i-2)[u+(i-3)v+2(i-3)nv][B-L_{2}^{\prime}(n-1)-P]^{2}}{2\;i\;n[u +(i-1)v](n-1)^{2}}, \tag{29}\]  
from which any higher-order coefficient can be obtained. For example, the sixth- and eighth-order coefficients are  
\[c_{6}=-\frac{P\left(u+3v+6\,n\,v\right)\left[B-L_{2}^{\prime}(n-1)-P\right]^{2 }}{24\,n^{2}\left(u+3v\right)\left(u+5\,v\right)\left(n-1\right)^{5}}\Delta, \tag{30}\]  
\[c_{8}=\frac{P\left(u+3v+6\,n\,v\right)\left(u+5v+10\,n\,v\right)\left[B-L_{2} ^{\prime}(n-1)-P\right]^{4}}{64\,n^{3}\left(u+3v\right)\left(u+5v\right)\left( u+7\,v\right)\left(n-1\right)^{7}}\Delta. \tag{31}\]  
The term \(c_{2}\) gives the curvature of the second surface at \(x=0\). Indeed, this result for \(c_{2}\) is nothing but the lens having power \(P\). The term \(c_{4}\) provides the second degree of freedom after the base curve. We call this term \"fourth-order asphericity\" of the surface; it is related with the conic constant when the surface is a conicoid. The terms \(c_{6}\), \(c_{8}\), etc., are higher-order asphericity terms. Let us check the accuracy of the presented approach when the back surface contains terms up to order four. Once we choose values for \(P\), \(B\), \(n\), and \(L_{2}^{\prime}\), we can compute the coefficients \(c_{2}\) and \(c_{4}\) by means of (26) and (27). With these coefficients, condition (22) should be satisfied within the approximations used to obtain (18) and (19) and within the fourth-order approximation for the back surface. The results are compared with those of a numerical computation performed as follows. A computer model of the lens is constructed using the values \(P\), \(B\), \(n\), and the thickness that would be used by default in actual lens manufacturing. A fourth-order coefficient \(c_{4}=0\) is chosen as seed. The oblique powers are computed by exact ray-tracing through the lens assuming an inverse distance to the rotation center of the eye \(L_{2}^{\prime}\)[6; 22]. The tracing is repeated in an optimization loop that searches the value of \(c_{4}\) for which condition (22) is better fulfilled. The comparison is presented in Figure 5 as curves of tangential (blue) and sagittal (red) power vs the transverse coordinate. Lens data are \(P=5\) D, \(B=6\) D, \(n=1.5\), \(L_{2}^{\prime}=37\) D, and the optimization was repeated for nine values of \(u\) ranging from \(-\sqrt{2}/2\) to \(+\sqrt{2}/2\). Solid lines represent the analytical predictions (up to order four), and the dashed lines represent the optimal solution obtained by numerical methods.