Spectacle lens design from mathematical modelling

Clearer vision for millions of people worldwide who wear progressive lenses is the result of over 14 years of research by CSIRO’s Dr Tony Miller. Dr Miller, a mathematician, developed algorithms and software for the optimal design of spectacle lenses. These include the TOROPT design package for atoric and aspheric single vision lenses and the IRIS and GOPT packages for progressive lenses. Progressive lenses are worn by people with ‘presbyopia’, the gradual crystallisation of the lens in the eye which impacts on quality of vision and affects most people from middle age on. Progressive lenses offer wearers both distance and close up vision correction in a single lens, delivering clear continuous vision without the line found in bifocals. His work has also led to the development of sun lenses optimised for wrapped frames, a particularly challenging design problem.

His pioneering work on spectacle lens design has contributed to nearly $1 billion worth of lens sales a year for SOLA International, a world-leading spectacle lens company which originated in Adelaide in 1960. In recognition of his contributions to spectacle lens design Tony Miller received the Sir Ian McLennan Achievement for Industry Award in 2000.

Progressive lenses – an improvement on bifocals

Progressive, or graduated lenses, are a modern alternative to bifocal lenses for the correction of presbyopia, the gradual decline in the ability of the eye to focus at short distances with age. Instead of having a discontinuous jump from a ‘distance’ zone to a ‘near’ reading zone, a progressive lens gives a continuous transition between these two zones. This helps to reduce many of the optical (and cosmetic) problems that are associated with bifocals.

Tony Miller has described the mathematical challenge as follows:

Traditionally the surface shapes of spectacle lenses were defined in terms of spheres, cylinders, or combinations of these which are sections of the surface of a torus. The reason for this was that these shapes could be easily fabricated by traditional lens surfacing machinery. That was fine for single vision lenses and even bifocals where the idea was to take a toric lens and add a segment of another sphere onto it. The optics of these simple kinds of lenses were understood technically in a traditional way. This was adequate in most circumstances, but it was beginning to be a struggle once you got to some of the more sophisticated traditional lenses. Once you got into progressive lenses the traditional way of thinking just wasn’t up to it.

What had happened was that people who were working on the practical side, who didn’t understand or want to know the mathematics, developed simplified rules of thumb and shortcuts for calculating and understanding things. Those rules of thumb persisted and were fine for the purpose, so there was no real incentive to develop more sophisticated ways of looking at it.

With the advent of progressive lenses there was obviously a reason to look at things in a more sophisticated way and also at the same time the advent of readily available computing facilities made the calculations more practical.

Progressive lenses began as a sort of curiosity in the mid 1950s in France, though they didn’t really become a marketable idea till the late 1970s or early 1980s. SOLA recognised that there was an opportunity but was struggling to see how they could develop it. This was one of the reasons they went looking for some way to – as they described it to me – manipulate complex shapes. What they were saying was not being expressed mathematically. So I had to learn those traditional terms and understand them mathematically. Then I could interpret what they were asking for in both mathematical and traditional terms.

I think they didn’t really understand what they were asking for. The technical terms they use are sphere power and cylinder power. You can imagine what a spherical surface is like, it has the same radius of curvature in every direction. A cylinder is curved in one direction but, in the perpendicular direction, it is flat. So the traditional way of thinking of lenses is that they are in some vague, undefined way a combination of a sphere and a cylinder.

In an everyday practical sense that is a good operational way of thinking. It gets you through the daily grind [pun intended] of making lenses, but it is a simplification. What you are really looking at is curvature of a general surface. Instead of talking about spheres and cylinders you talk about a general surface and any point on the surface has a general curvature. To understand and interpret that was really the challenge. It was taking traditional processes, recognising them to be simplifications and trying to fit them into a more general, mathematical interpretation. If you looked at the literature, at the more sophisticated text books, they were heading towards this direction too, but I had to work it out for myself, in my own way.

What I did in this case was to do it the way which seemed most natural to me. I suppose this was influenced by coming from a background in numerical solution of partial differential equations. My natural response was to try to turn this problem into a partial differential equation which was related to the curvature. The approach I took might not have been the most efficient way to do it, but I did it in a way which was familiar to me.

The IRIS progressive lens design system

From 1987 to 1990 Tony Miller developed the mathematical basis for the IRIS progressive lens design system. This involved a novel application of finite element techniques to solve some surface optimisation problems which arise in the design of progressive lenses. As part of this work it was necessary to formulate some measures of optical performance that could be used to compare and rank prospective designs. One important such measure, or merit function, that was developed was RMS blur. Tony Miller assisted SOLA International personnel to implement these techniques as the IRIS lens design software package. This became a vital tool for SOLA throughout the 1990s, and made possible the efficient design of a large series of lenses.

The collaboration with SOLA arose in a round about manner. The initial contact was with a local software development company called Finite Development who were looking for some assistance in manipulating shapes. Finite Development had a contract with SOLA to develop lens design visualisations. As Tony Miller recalled:

Finite Development originally intended to use some software packages which they distributed, but they had quickly realised that this wasn’t going to work. I had a chat with them and they were interested in what I could do and it developed from there.

In those days the Division [Mathematics and Statistics] was specifically not permitted to do external work – it had to be channelled through a company called SIROMath – and so the initial contract was between SIROMath and Finite Development, who in turn were contracted to SOLA.

It was strange because in the first few discussions with Finite Development they didn’t tell me what the problem really was. They said that they just wanted to manipulate shapes and visualise the results. I said, “Well why do you want to do this?” They were a little bit coy at first. However after a few meetings (they must have gone back to SOLA and discussed it with them in the meantime), I was told that the company they were working with was in fact SOLA, though I had to sign a Confidentiality Agreement before being told this. I hadn’t heard of SOLA before in my life, but I soon started meeting the SOLA people.

An improved approach to progressive lens design – the GOPT lens design package

In around 1997, Tony Miller started the development of an improved approach to progressive lens design. Although IRIS had proved very useful to SOLA International, it lacked the ability to easily specify the optical characteristics of the lens along the ‘eyepath’. (The ‘eyepath’ is that portion of the lens that is most frequently used for sharp vision.) The new approach however allowed the direct specification of the eyepath characteristics by a lens designer. This approach has been implemented in the GOPT lens design package, which is now the chief design tool used for progressive lens design by SOLA.

The TOROPT design package for single vision lenses

Single vision lenses are designed to correct refractive errors such as short sightedness and long sightedness. Traditionally, the surfaces of single vision lens were shaped like sections of spheres, cylinders, or some combination of these. This was primarily due to the relative simplicity with which such surfaces could be manufactured. However, with the increasing use of plastic lens materials in the last few decades of the 20th century, other manufacturing techniques such as moulding and casting are now used. These techniques allow a wider class of surface shapes to be readily manufactured. Today, almost all single vision lenses have some form of aspheric, or atoric shape so as to optimise optical performance. Aspheric lenses have a profile that is rotationally symmetric but not completely spherical. They are flatter and slimmer than conventional lenses (which have the same curve across their entire surface – like a ball). Atoric lenses are an extension of aspheric design technology, and are lenses that depart from being an exact circular toric. This allows lens designers to optimise for both the sphere and cylinder powers of a lens.

Around 1989, Tony Miller started to develop, for SOLA International, a technique for optimal aspheric lens design, along with associated software. Over the 1990s, this approach was extended to atoric lens surfaces. Additionally, a more accurate representation of the ‘as worn’ lens configuration was used in the calculations. The result has been the TOROPT lens design software package which has gradually evolved over the 1990s. It is the primary design tool used by SOLA for the design of single vision lenses.

In recent years these tools have enabled SOLA International to launch a range of completely new product categories – such as prescription lenses for wrap sunglass frames. As Tony Miller comments:

Shaping a corrective lens for wrap-around prescription sunglasses without creating blur and distortion can be a design problem. In the past, designers used more of a ‘trial and error’ approach. The new design tools provide a method for quickly finding the best possible design.

Theoretical optical analysis tools for assessing lens design performance

Starting in 1988, Tony Miller developed a raytracing software package for SOLA International which could be used to theoretically analyse the optical performance of lens designs. Since the early 1990s, the mathematical formulation and techniques that were first used in this package, have been repeatedly employed by SOLA’s software developers in a variety of lens analysis and lens calculation software packages that they have developed. Such packages are widely used both within the SOLA organisation and by their customers.

Commercial impact

Over a billion dollars worth of progressive lenses are sold each year to people with presbyopia, the gradual crystallisation of the lens in the eye which impacts on quality of vision and affects most people from middle age on. The impact of this technology has enabled the Australian company SOLA International to consolidate its position in the marketplace, as well as improving the quality of life for millions of spectacle lens wearers.

SOLA International was rated as one of Australia’s top two innovative companies by BRW, and is one of the largest Optical Lens Companies in the world. With manufacturing sites spanning the globe in fourteen different countries, over one hundred million people go about their daily lives wearing SOLA lenses. Starting from humble beginnings SOLA has developed into a true South Australian success story through its technologically innovative products designed and manufactured by its specialised staff.

The ability of SOLA to remain at the forefront of this highly competitive industry is dependant on delivering innovative new products to the market place. This very much relies on a strong tradition and capability in the design and development of ophthalmic lenses using the tools developed by CSIRO. As Tony Miller comments:

Creating a progressive lens is, in a way, attempting the impossible. It involves combining two or more different lenses, each with a different shape, and trying to get a seamless transition from one shape to the next. As some wearers of progressive lenses know, there can be unpleasant distortion and blurring at the transition between the near and distance zones of the lens; a physical problem that can’t be solved perfectly. What you have to do is find the best possible solution – the optimal lens shape. These design tools provide a systematic way for finding the smoothest possible transition and therefore the least possible distortion.

Partial derivatives and other mathematical concepts mightn’t immediately spring to mind when people think about spectacle lenses. But mathematics is an ideal way to think about and describe the subtle shapes that are involved in lens design. If you can express in mathematical terms what you want the lens to do, then you are a long way towards your design goal.

The tools, continuously developed by CSIRO over a 14-year period, have enabled SOLA International to deliver a stream of technically innovative new lens designs. As Dugald Rose, Research Manager, Lens Design, of SOLA International commented in November 2000 at the presentation of the Sir Ian McLennan Award:

With these tools, we’ve been able to design progressive lenses with clear vision over a greater part of the visual field. They are more visually comfortable and more attractive to wear. The tools enable us to continually develop and reduce the time necessary to design and market superior new lens products.

The impact of this technology on the spectacle lens market has been enormous. Last year [1999], SOLA’s lens sales totalled more than US$500 million and a large proportion of the lenses sold were designed using Dr Miller’s software tools.

Sources

• Miller AD, 2010, Personal communication.
• Interview with Tony Miller, 2001, following his receipt of the Sir Ian McLennan Award. [DOC 58 KB]