# Improved Analytical Theory of Ophthalmic Lens Design
## 2 Method  
The first part of the analysis follows the standard procedure to locate tangential and sagittal foci of an ophthalmic lens. Though the reader may find this computation elsewhere [3; 6], we have opted to reproduce the standard development here for completeness and to make clearer the different approximations we later use. First, we consider the refraction of a thin astigmatic pencil whose principal ray lays in a meridional plane of a surface with revolution symmetry (that is, the symmetry axis and the main ray are co-planar). Let \(n\) and \(n^{\prime}\) be the refractive indexes at either side of the surface, as depicted in Figure 1. The tangential and sagittal object points are located at points \(Q_{T}\) and \(Q_{S}\). After the principal ray is refracted at point G, the tangential image for \(Q_{T}\) is \(Q_{T}^{\prime}\), and the sagittal image point for \(Q_{S}\) is \(Q_{S}^{\prime}\). The vector \(\mathbf{N}\) is the normal to the surface S at point G.  
The relationship between the object and image points are given by the Coddington equations  
\[\frac{n^{\prime}}{s^{\prime}}-\frac{n}{s}=\frac{n^{\prime}\cos a-n\cos b}{R_{s}}, \tag{1}\]  
\[\frac{n^{\prime}\cos^{2}a}{t^{\prime}}-\frac{n\cos^{2}b}{t}=\frac{n^{\prime} \cos a-n\cos b}{R_{t}}, \tag{2}\]  
where \(s=\overline{Q_{S}}\)G, \(t=\overline{Q_{T}}\)G, \(s^{\prime}=\overline{GQ_{S}^{\prime}}\), and \(t^{\prime}=\overline{GQ_{T}^{\prime}}\). \(R_{S}\) and \(R_{T}\) are the main radii of curvature at G (sagittal and tangential, respectively). The angles \(a\) and \(b\) are the incidence and refraction angles of the principal ray at G.  
Let us now consider a lens of refractive index \(n\) surrounded by air, as shown in Figure 2. Let the object point be located at Q, so \(t_{1}=s_{1}=\overline{\text{GQ}}\). Consider a thin ray pencil, which hits point G on the front surface, \(S_{1}\). The incidence and refracting angles of its principal ray are called \(a\) and \(b\), respectively. The ray pencil propagates inside the lens up to a point H on the second surface, striking this surface with an incidence angle \(c\) and emerging with refraction angle \(d\). In general, the emerging ray pencil is astigmatic, being \(t_{2}^{\prime}=\overline{\text{HQ}_{T}^{\prime}}\) and \(s_{2}^{\prime}=\overline{\text{HQ}_{S}^{\prime}}\). To obtain the formulas for the sagittal and tangential powers, the Coddington equations are applied to the points G and H. For the first surface the equations are  
\[\frac{n}{s_{1}^{\prime}}-\frac{1}{s_{1}}=\frac{n\cos b-\cos a}{R_{1S}} \tag{3}\]  
\[\frac{n\cos^{2}b}{t_{1}^{\prime}}-\frac{\cos^{2}a}{t_{1}}=\frac{n\cos b-\cos a }{R_{1T}} \tag{4}\]  
and for the second surface  
\[\frac{1}{s_{2}}-\frac{n}{s_{2}}=\frac{\cos d-n\cos c}{R_{2S}}, \tag{5}\]  
Figure 1: Ray refracting through a surface S.  
Figure 2: Ray propagation through a lens. \[\frac{\cos^{2}d}{t_{2}^{\prime}}-\frac{n\cos^{2}c}{t_{2}}=\frac{\cos d-n\cos c}{R_{2T }}. \tag{6}\]  
\(R_{1S}\) and \(R_{2S}\) are the local curvature radii in the sagittal plane of the first and second surfaces at points G and H, respectively. The same convention applies for the radii \(R_{1T}\) and \(R_{2T}\). Next, we will calculate the oblique powers using Equations (3)-(6) to calculate the oblique powers, following the procedure used by [PERSON] [6].