# Improved Analytical Theory of Ophthalmic Lens Design
## 1 Introduction  
Spectacle lenses can be designed considering several factors, such as visual performance, aesthetics (thickness or curvature of the front surface, also known as _base curve_), or manufacturing limitations. When visual performance is considered, the usual strategies are primarily focused on the minimization of oblique aberrations, which are the residual defocus and astigmatism appearing at oblique gaze directions or when the lenses are tilted. Secondary objectives can be the reduction of distortion, higher-order aberrations, or even chromatic aberrations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The first complete approaches to the management of oblique aberrations were based on the so-called third-order approximation [6, 9], although trigonometric ray tracing was also used early on in the development of obliquely corrected eyeglasses [11]. Nowadays, computers allow us the construction of elaborate lens-eye models that can be explored with detail by means of exact ray tracing or wavefront tracing [5, 12, 13]. Lens performance can be analyzed in terms of the number of oblique errors or the size of the geometrical blur patch on the retina [6, 14]. The computation of geometrical wavefronts and the corresponding coefficients for higher-order aberrations or optical transfer functions have led many authors to conclude that ophthalmic lens performance is indeed mainly determined by oblique errors [15]. Although modern lens design may involve exact ray tracing and recurrence or optimization algorithms [16, 17], the third-order approach is still useful for getting an initial point for more complex analysis, or to find key relationships and limitations that would guide further design. The classical procedure for obtaining analytical expressions of the oblique errors in lenses with revolution symmetry is based on the use of Coddington's equations, which provide the location of the tangential and sagittal foci for a given surface. Repeated application of these equations to the two lens surfaces, along with the use of third-order and thin-lens approximations to compute incidence and refracted angles, leads to analytical expressions for tangential and sagittal lens versgences or powers. The main and classical result from this analysis, when applied to spherical lenses, are the well-known Tscherning ellipses [6]. They provide the combination of base curve and prescription for which either oblique astigmatism, mean power error, or tangential error are cancelled. In general, the third-order solution takes the form  
\[E=x^{2}f_{sph}(\kappa_{i},P,n,L)\]  
where \(E\) stands for some type of oblique error (classically, oblique astigmatism, oblique mean power error, or tangential error), \(x\) is the radial coordinate, or distance from the optical axis, of the point the viewing line is passing through, and \(f_{sph}\) is a quadratic function of the curvature of either surface, \(\kappa_{1}\) or \(\kappa_{2}\), and the lens power, \(P\); it also depends on the lens refractive index, \(n\), and the object vergence, \(L\). More generally, \(f_{sph}\) will be a function of the curvature of both surfaces and the lens power. For thin lenses, they are related by the lens equation, so only two of them are needed to specify lens power and lens bending. In modern lens design, the front surface is usually spherical, and its refractive power is known as the base curve, \(B\). Setting \(f_{sph}(B,P)=0\) leads to the Tscherning ellipse for the particular oblique error, and it establishes a relationship between lens power and base curve that will cancel the error. With \(f_{sph}(B,P)=0\) being a closed curve, there is a finite range of powers for which the error cancelation can be achieved. Additionally, for any particular power within this range, there will be two values (Ostwald's and Wollaston's solutions) of the base curve that will seat on the ellipse, henceforth cancelling the oblique error.