## 1. Introduction

*Calculus *is a mathematical analysis of some essential patents, which mark the progress in progressive lens design from the first commercial PPL, Varilux 1, until Varilux Comfort, the most sold progressive lens brand ever. The choice of the patents after the Varilux invention is subjective in the sens, that I selected the patents of products where I had occasion to participate in the development. There are other outstanding concepts by Carl Zeiss, Sola, Hoya or from other manufacturers which characterize this era. All the calculations use only information from the patents or other published material, accessible to any person interested in this topic. Even if all these patents can be linked to a commercial product, we have to be cautious, when comparing the calculated design with the features of the market product. The patent claims cover for evident reasons a broad variety of geometries and parameters and the given examples do not necessarily represent the product chosen for the market launch.

Some chapters of *Calculus* are dedicated to specific aspects of the progressive surface concept, as the chapters concerning the Minkwitz theorem or Orthoscopy. The first and principal goal writing this scientific/technical history is documenting the evolution of research and technical work in progressive surface design at the end of last century. So it will be of interest to the people, who have worked in the spectacle lens business during this period and generally to any technically minded person looking for information about the history of progressive surface design. The paper will also be useful for the student of optical engineering or physiological optics to understand the basics of progressive lens computation. *Calculus* depicts the history of surface design between the 1950’s and 1990’s in successive chapters, each dedicated to a particular product or a specific characteristic. Each chapter is available as individual pdf document and subdivided into paragraphs. This structure of chapters and paragraphs is listed in the menu on the left.

The computations have been executed with the help of rather simple tools as the commercial sofware *Mathcad* and *Matlab* installed on a laptop. The computer programs written for the calculations are either integrated in the general text (*Mathcad*) or given in a separate paragraph (*Matlab*). *Mathcad* was used for the computation of surfaces defined by analytical functions. For the construction of more sophisticated designs, as Varilux Comfort, the surface representation required tensor product splines and it was advantageous to have recourse to the *Matlab* language.

In order to simplify the method, some calculations are based on approximation models. Their use is discussed in the respective context. There are two short chapters treating definitions and notions of differential geometry and ophthalmic optics / progressive lenses. Nevertheless we presuppose the basic scientific knowledge in these domains of mathematics and spectacle lens optics. In order to understand the calculations completely, you need the patents referenced in the different chapters. You will find them, without problem, on the Internet with the help of *Google Patent Search*.

**Acknowledgement**

One problem for the author was the fact, that he is no expert in computer programming and calculations. So particularly for the last chapter, the surface optimization applying tensor product B-splines, help was very welcome. My special thanks go to Professor Günther Greiner and Matthias Innmann , M. Sc. (computer science), Friedrich-Alexander-University , Erlangen, for writing the program for the minimization of the merit function.

A heartfelt „thank you“ to my wife Bärbel for her patience and understanding accompanying this work, that lasted longer than I thought.

Some basic features of progressive lenses are defined. The backbone of this lens type is the principal meridian and the power progression along the meridian provides clear vision into all distances. A definition of the progression length is given. In the reference points the optical powers are controlled with the focimeter and by means of the fitting cross the lens is centered correctly in front of the wearer’s eye.

Another part of this chapter is dedicated to the presentation of the computation results. Isopower plots depict the power increase over the total suface. The inevitable aberrations are described by isoastigmatism lines as regards the surface astigmatism and by the deformation of a rectangular grid as regards the optical distortion.

This chapter assembles the most important definitions and formulas of differential geometry, which are necessary for the calculation of an aspherical, asymmetrical design.

So for a parametric C2 surface the first and second fundamental form are derived and the formulas for the principal curvatures and the principal directions of the surface are calculated. For a simple and complete description of the surface we derive finally the terms for the mean and Gauss curvature, the mean power and surface astigmatism

The Minkwitz theorem links the lateral astigmatism increase in the direct neighbourhood of an umbilical principal meridian with the power increase along this meridian. The meridian may be situated in a symmetry plane or may be curved, both cases are analyzed. Using a simple model the evolution of the peripheral astigmatism for a non umbilical meridian is discussed.

This simple design is sometimes mentioned in the litterature as elephant’s trunk. It consists of an involute of a circle as umbilical main meridian and circles as orthogonal sections. So the surface can be totally represented by analytical functions and give an umbilical line and isolines- graphs with high precision and regularity.

The design with a continuous power increase from the top to the bottom of the lens offers almost horizontal isopower-lines, but also a rather high amount of peripheral astigmatism deteriorating the optical quality of the far vision (FV)- and near vision (NV)-zones. This underlines the need for a main meridian with stabilized sections in the FV- and NV-part. The Minkwitz theorem is demonstrated qualitatively and quantitatively.

**Contents**

5.1 The coordinate system

5.2 The principal meridian

5.3 Geometry and equations of the progressive surface

5.4 Analysis of design and calculations

The key of success of the Varilux project was to link the mathematical research on progressive surfaces inseparably to the engineering of the manufacturing process. Different manufacturing methods using circles and conic sections as generating surfaces are discussed (patents US 2 869 422, US 2 915 856) . In order to get rid of the constraints of the production process based on especially shaped cams, the „point by point“ manufacturing had been developped (patents US 2 982 058, US 3 041 789 ).

The calculated Varilux 1- type design in this chapter is built from an umbilical principal meridian with stabilized zones for FV and NV, together with circles as orthogonal sections. The equation for the meridian is obtained by numerical integration of its curvature. Several possibilities for a stabilized meridian are considered. The progression length is rather short with 14mm. The obtained surface is distinguished by large, clear far- and near- vision zones corresponding to the target, a progressive with optical characteristics close to a bifocal lens. In the periphery of the progression and near vision zones the design shows a rather high amount of aberrations (astigmatism and distortion).

**Contents**

6.1 Focus on the manufacturing process

6.2 The coordinate system

6.3 The principal meridian

6.4 Geometry and equations of the progressive surface

6.5 Analysis of design and calculations

Varilux 1 was a design trying to optimize the conditions for the static, foveal vision. The Varilux 2 concept takes into account the global vision process including peripheral and dynamic visual perception. The consequence was a totally aspherical surface with an umbilical meridian and orthogonal sections, which in a first approximation, are conic sections with varying parameters. The calculations of the designs are based on the examples given in US 3 687 528 and US 3 910 691.The meridian shows a sinus-like shape, which is reflected in a periodic pattern of the isolines- plots. The design includes two horizontal umbilical lines and two vertical isoprismatic lines in order to reduce the lateral astigmatism and to achieve orthoscopy, which means minimum distortion of a rectangular grid.

The deviation of the orthogonal sections‘ geometry from evolutive conic sections is illustrated. The difference between the Varilux 2 launched on the market and the „patent design“ is briefly discussed.

**Contents**

7.1 The coordinate system

7.2 The principal meridian

7.3 The characteristical data of the conic sections

7.4 Geometry and equations of the progressive surface

7.5 Calculation of the half-axis h(z0) of the conic sections

7.6 Analysis of design and calculations

The distortion of vertical and horizontal lines has a strong influence on the adaptation of progressive lenses. B. Maitenaz describes in US 3 687 528 and US 3 910 691 quantitatively the phenomenon of static and dynamic distortion. A mathematical model for the the prismatic deformation of a rectangular grid by a progressive surface is defined. The characteristics of non-orthoscopic designs of lenses of the first generation are presented.

Based on results of chapter 7 the distortion of horizontal and vertical object lines by the Varilux 2 geometry is calculated and represented graphically. The isoprismatic lines of the design characterize the structure of the grid image. By the concept of Orthoscopy the Varilux 2 design maintains largely the orientation of vertical and horizontal lines.

**Contents**

8.1 Distortion, definition and calculation model

8.2 Non orthoscopic designs

8.3 Distortion of the Varilux 2 design

In patent US 4 315 673 Günther Guilino and Rudolf Barth represent a progressive surface in a curvilinear cylinder coordinate system by a Fourier series of degree one. The obtained surfaces are called surfaces with variable periodicity. The main meridian is an umbilic with stabilized power zones, the periodicity k(z) of the orthogonal section changes as a function of the intersection point with the principal meridian. Based on the formulas and the figures of the patent for the auxiliary coordinate system and the variable periodicity, the design examples of the patent are reconstructed. These designs are different from the design plots published for the marketed Progressif R product. The parameters of the formulas for the main meridian and orthogonal sections are modified to obtain the market product. Progressif R is characterized by an excellent far vision quality, a relatively broad near vision part and a maximum of the peripheral astigmatism little above the add power. The reasons for the choice of the auxiliary curvilinear cylinder cordinate system and the variable periodicity are discussed.

**Contents**

9.1 The coordinate system

9.2 Description of the progressive surface

9.3 The patent

9.4 Progressiv R, the commercial product

9.5 The structure of the progressive surface with variable periodicity

> 9. Progressiv R, the „refraction-correct“ progressive lens

Following the invention of Varilux, Essilor tested and launched a series of new designs. One major conclusion from the ample wearer feedback was, that there is no single specific design, which is optimum for all add powers. Maurice Dufour, a close collaborator of Bernard Maitenaz in the Varilux development, was the inventor of the patent US 4 838 674, Werner Köppen developed the Multi-Design concept. The power and astigmatism plots for the different addition powers of Mono-Design surfaces are homothetic. The patent describes how the variation of certain parameters d1, d2, ΔC allow to optimize the design characteristics per add power (vertical optical modulation). The Multi-Designs for the adds 1 and 3 of a model surface are calculated and discussed. After the Essilor VMD (Varilux Infinity) introduction the development of other Multi-Design lenses using different segmentation criteria, as for example the type of ametropia, prepared the way to the personalized progressive lens manufactured by CNC machines.

**Contents**

10.1 The Mono-Design lenses

10.2 The Multi-Design Patent

10.3 Calculating a Multi-Design model surface

10.4 Vertical Optical Modulation

10.5 Analysis of designs and calculations

> 10. Varilux Multi Design, the first step to a personalized design

Varilux Comfort is characterized by a geometry combining the advantages of a „hard“ and a „soft“ progressive design. The research work to create an ergonomically correct meridian is laid down in the patents US 5 270 745 and US 5 272 495, which define the meridian consisting of three line segments. US 5 488 442 contains the conditions for the surface with a short progression zone and a soft periphery for good peripheral and dynamic vision.

In order to design a geometry, combining these almost contradictory conditions, bicubic tensor B-spline surfaces offer the necessary flexibility. The control points of this spline surface are determined by the minimization of an error functional, with the surface function as input argument. Assuming, that in a first approximation the first derivatives are constant, the problem is reduced to the minimization of a quadratic functional, which is equivalent to the solution of a linear equation system. The input parameters are the weight functions and for penalizing surface astigmatism and deviation from the prescribed power and the prescribed power H_{soll}. These parameters are calculated from the data of US 5 488 442.

The program and the process steps of an approximate optimization are disccussed. The isoplots for power increase and surface astigmatism show a design with large viewing zones for far and near vision, a short power progression and low peripheral astigmatism and gradients.

**Contents**

11.1 Freeform surfaces built with splines

11.2 Establishing the Merit Function

11.3 Minimizing the error functional

11.4 The bicubic tensor product B-spline surface

11.5 Establishing the linear equation system (LES)

11.6 Optimization Method

11.7 Results and analysis

Annex : Matlab program for minimizing the error functional

> 11. Varilux Comfort, natural vision by large viewing zones and a soft periphery