## Designing a Progressive Addition Lens

A. Ryndina^{1} PhD, V. Kirzhner^{2} PhD, A. Lisichki^{3 }BSc

^{1 }M. S. Optical Industry Ltd., Israel, R&D Manager

^{2} Haifa University, Israel, Advisor of M.S. Optical Ind. Ltd.

^{3 }M. S. Optical Industry Ltd., Israel, General Manager

M.S. Optical Ltd. Presents its new optimization algorithm of progressive addition lens (*PAL*) including different methodical techniques. The algorithm allows the creation of different *PAL* designs within several working hours on a P.C. (For more detailed information, see the full article on our web site: www.msoptica.com)

**Introduction**

We present new lens optimization methods for the design of ophthalmic lenses, called Progressive Addition Lenses, or *PAL* for short. These lenses are designed for correcting presbyopia (Greek for "old eye"). Presbyopia is the normal condition that happens at a certain age, when the eye loses the function of accommodation (focusing) as the eye lens naturally increases in size and becomes more rigid, it loses the ability to change curvature to focus on near distances.

Such Progressive lenses have different curvatures or ophthalmic powers in various areas of the lens surface called zones. In *PAL* design both far vision and near vision are stable zones. By removing the sharp dividing lines in the bifocal and trifocal lenses, the PAL design uses a smooth transitional area called the corridor or intermediate zone.

This is achieved by continuously decreasing the radius of the curvature along the median line of the lens (see Fig. 1).

Both Mathematical optimizations of progressive surface and complex technologies are the reasons for many different progressive addition lens designs on the market today. The optical surface does not have a closed formula. The differences among the lenses are mainly in the width of the corridor, form and size of the far and near zones and distortion of peripheral areas. However, improving one of the parameters causes changes in at least one of the other parameters. When viewing the different designs it stands out that the main criteria of an “acceptable lens” is the ease of adaptation of the wearer.

**1. Definition of the mathematical problem **

In general, the surface of the lens is divided into a number of the domains *D _{i}*

*.*The geometrical forms of these domains are defined by physiological properties of the eye, international standards, the current fashion, new technologies etc. The design is evaluated by the value of cylinder and power of each domain. Thus, cylinders in the vision zones and corridor, formally speaking, take on a 0 value. The power of the far and near vision zones take fixed values

*P*and

_{FV}*P*respectfully. In the corridor zone, power takes on a number of intermediate values between

_{NV }*P*and

_{FV}*P*. The cylinder value of the side domains should be minimized. Note, the limitation on the cylinder value is posed by inequality (for example, the maximal value of the side cylinder should be less then the addition power value). Therefore, there are many different solutions to the design.

_{NV}**2. The optimization method **

**2.1 **** The generation of the initial surface **

The first step is a formation of the central line or in other words, corridor section. It is a power curve that defines z-coordinates (sag) along the central vertical axis. This curve formally consists of three areas. Two of them are spheres with different radii *R* and *r*, where *R*>*r*, that corresponds to far vision and near vision zones. The intermediate area, connecting these spheres has a smooth gradually changing power. Our initial surface has “ideal” vision zones and corridor: there are no “surface” astigmatism, maximal reduction of visual astigmatism, and constant power in the wide diapason of the vision zones. As a result, the side cylinder value is very large. Our next step is to minimize the cylinder value while maintaining the “ideal” zones.

**2.2 The masking **

The far vision, near vision and intermediate zones are not subjected to change during optimization. They are protected by masking. The masking is achieved by any geometrical shape. Its form depends on the area to be maintained. It may be elliptic, rectangular or any other free complex shape.

**2.3 Optimization function**

We optimize locally any small surface domain *S* by special complex function. For example we used a sum of the cylinder values and the mean value of cylinders of the domain. By changing the location coordinates *x* (from left to right and vice versa), and y (down) by one step, the optimization domain *S* moves on the defined domain *D _{i}*.

However, there are more complex types of movement: skipping some points, random choosing of a location, etc. On the border of the masking area a band of small cylinders is created.

After optimization of the cylinder value as described above, we evaluate the finished result by the number of small cylinders. If the number is not reduced enough after optimization, we return to the previous state and take an adaptive step.

**2.4 Adaptive step **

The adaptive step is the changing of certain parameters concluded on the optimization function. What is the reason to do the adaptive step? It is a well-known fact, that all optimization methods find a local optimum. There are many different algorithms that are devoted to solving the problem, originating from the local optimum point. Generally, the algorithm will now test different points. For example, the gradient methods change the starting point, or assume a combination of previous points. For our specific surface the new point means another design and the loss of the previous one. Our method of changing the optimization function makes it possible to both overcome impassable local minimum and to maintain the former obtained design.

**3 Different designs obtained by algorithm**

The “quality” of the computer designs must be reevaluated on a real mold/lens. Beside the analysis of the ophthalmic surface by a lens mapping system and Lensmeter by accepted standard methods, we also use a cross section parallel to the 0^{0}-180^{0} line passing through the far vision zone on the checking level. This graph demonstrates the changes of the astigmatism and power on the full diapason far vision zone.

By changing the algorithm parameters, we obtain different types of the design. Figure 2 demonstrates computer designs obtained by the algorithm. Note, that these cylinder maps differ a little from the real ophthalmic surface, because at this stage the algorithm does not include engineering parameters such as the tolerance of the CNC machines, the thickness of the glass blanks, the different radii of the glass before and after slumping process and so on. Let us consider the following:

The simplest design is shown Fig. 2a. This is optimization of all initial surfaces without masking. Wide near vision zone, narrow far vision zone and high distortion areas characterize this type of design. The longer the computer program runs, the smaller the side cylinder but the vision zones will become “cloudier”.

The working time is controlled by maximal cylinder of the domain *MC*,

*MC*=*(addition)***m*, *m*=1, 2, …

At different stages of the optimization, we can change a weight-function to obtain another design. Thus, for *m*=10-12 we divide the work area into right and left halves by corridor sections and optimize them separately. By changing the parameters of the optimization function we move maximal number of cylinders from the corridor to the sides (nasal and temporal). After that, for *m*=5-6, we move maximal number of cylinders down. We obtain design Fig. 2b. At this stage the reduced side cylinders lead to the narrowing of the near vision zone. In order to maintain it wide and long enough, we put the mask on the near vision zone. If the lens will be used only for small frames, we can move the mask up, as the outer edges will be cut off anyway. The near vision zone takes a form of a “drop” (Fig. 2c).

At this stage, we repeat the process on the far vision zone and clear it. But in the optimization process the width of the corridor is decreased. Finally, we obtain a far vision zone in the shape of a “mushroom” (Fig. 2d).

All the design types described above have a common disadvantage: in the cross section of the far vision zone we obtain a very small constant area. It means, that even a small movement will bring the eye into a distorted area. (see Fig. 3a).

The following design takes into account all the above-mentioned advantages and disadvantages. The first requirement is to maximally clear the far vision zone with constant power. On the strength of asymmetry we afford more distortion of the nasal adjacent area then the temple adjacent area (See Fig. 3b). This requirement was taken into account in formatting an initial surface. We assume, that the typical modern patient, as a rule, works on the computer. In other words, he often uses the intermediate zone. Hence, we define a wide corridor that leads to wide top part of the near vision zone. In order to use our design not only for small frames, the next step is to plan the clearing of the whole near vision zone (Fig. 2e).

This is a design we have perfected at our firm (for detailed technical data see our web site: www.msoptica.com). Fig.1 shows the cylinder map; power map and cross section of the far vision zone are shown in Fig.3b.

We also used this algorithm for designing a new progressive lens with a 10mm corridor and obtained a very successful experiment.

**Conclusions**

The new system of optimization presented by M.S. Optical includes different methodical techniques for formatting the progressive ophthalmic surface. The method of “masking” and the special initial configuration makes it possible to get a high quality lens namely in the vision areas. The wide library of weight-functions allows leaving the local minimums and keeping an obtained lens configuration. Our dynamic system can be easily adapted to all non-symmetrical designs.

**Remark**: The method described above allows us to take the data-result (z-coordinates) and feed it into the new generation free form cutting machines and CNC 3D machines (like ASM, 100 SPF 80 or HSG 100). This process could be used for the production of Injection Lenses and Mold manufacturing.

**List of figures:**

Fig.1 The cylinder map of a progressive addition lens; base 4, addition 2.

# Fig.2 The different *PAL* design obtained by algorithm

# Fig.3 The comparative map of the far vision zone: power and cylinder cross sections, power map.