This article is also available in Japanese.

Progressive addition lenses (PALs) are used to provide eyeglass wearers with a spectacle lens in which optical power varies smoothly as the user looks through different regions of the lens. For example, when looking straight ahead there may be little added power, and when looking down at some angle there may be significant power. This means the same eyeglass can be used for both driving and reading.

PALs are a specific example of the more general case of free-form optics. In free-form optical design, the shape of an optical surface is not constrained to a simple expression like a conic asphere or even asphere, but is allowed to take any shape necessary to provide the optical performance needed. A free-form optic can add optical power wherever it is needed in order to provide the required correction.

Free-from optics therefore require different analysis and optimization techniques compared to classical lenses. For example, OPD and ray fans may not be so useful when power can vary across a surface in an arbitrary manner. This article describes a very simple PAL lens, and shows how to construct such systems, analyze their performance and optimize them.

An ideal free-form optical surface would simply be a set of data points. However, in order to be able to optimize such a surface, there must be some method to perturb the free-from surface so as to evaluate how to add or subtract power appropriately. Therefore, purely data-based surfaces, like the Grid Sag surfaces or imported CAD objects may be useful for characterizing system performance, but are not so useful for the initial design stage, which is when we want to be able to change the surface smoothly under the control of the optimizer.

The surface types most useful for initial design include:

• Cubic Spline and Extended Cubic Spline
• Radial and Toroidal NURBS
• Polynomial and Extended Polynomial
• Zernike Sag

The spline and NURBS surfaces take sag data directly as their defining parameters, and then fit multiple low-order polynomials through the data to provide a smooth surface suitable for ray-tracing. The Polynomial and Zernike surfaces use very general polynomials of arbitrary order to provide a similar capability.

We will use the Extended Polynomial surface to demonstrate the design considerations required when designing a Progressive Addition Lens. The file progressive_starting_point.zmx may be downloaded from the link on the last page of this article. It is a simple file, and looks like so:

The file uses three configurations, like so:



Note the use of the Maximum solve to make the semi-diameter equal to whichever configuration requires it to be the largest. Note also under General -> Miscellaneous that a margin of 3 mm is added to this, so that the lens is 3mm larger than is needed for the light to get through. This margin is useful to allow space for the frame.

On-axis light comes from infinity, light at 10 degrees comes from an object 1000 mm away, and light at 20 degrees comes from an object 500 mm away. This represents a user looking at different parts of the three-dimensional field of view and seeing objects at different distances. The file is afocal, and so the following spot diagram is in angular units (milliradians). The angular deviation of the light as a function of field can be clearly seen:



The correction lens is made of Polycarbonate and has an Extended Polynomial front surface and a Standard rear surface:



The sag equation of the Extended Polynomial is given by:



so it has a base conic asphere (standard) surface sag upon which the polynomial terms are added. The base standard surface is very helpful, as paraxial rays can interact with it and so paraxial concepts like EFFL are still useful. The polynomials are of the form x^myⁿ, where m and n are integers and x and y are the coordinates of a point on the surafce. In the Extra Data Editor I have chosen to use 40 terms, and I have made all terms up to x⁴y⁴ variable. Any higher order terms I have set to zero and am not optimizing. This is purely a choice on my part: if I wanted to I could produce all polynomial terms to order 230.

I have then built a merit function to optimize the angular radius of the beam at each configuration, plus give reasonable boundaries for glass center and edge thickness. This was built using the default merit function tool. Therefore, our goal is to get the best collimated light out from the lens, and the object's position in the field of view and distance from the lens varies.

Press Tools -> Optimize -> Automatic and ZEMAX quickly optimizes the 24 variables to give us the freeform surface that gives the best collimated output:

The Shaded Model of the lens shows that the surface sag is very complex:



Note that the lens tends to "run away" at the edges where there are no rays to provide control. This is typical of free-form design: either a ray or some other form of constraint needs to be applied over the whole surface to prevent unrealistic sags from being produced.

Now with such a complex surface, simple fans like ray-fan and OPD plots are not enough to describe the performance of the free-form optic. For this optic, we use Analysis -> Miscellaneous -> Power Field Map. Set to show contours at an interval of 0.25 diopters, we can see the spherical and cylindrical power added by this surface over the whole field of view:





Now this plot uses different definitions of power and EFL than the strictly paraxial concepts used elsewhere in ZEMAX. This feature computes optical power or focal length as a function of field coordinate. The power or focal length is determined for the optical system as a whole up to and including refraction from any surface. The method used is to trace a ring of real rays around the entrance pupil at each point in the field. The ray data are used to determine the focal length for each field position. This focal length can then be used to compute the optical power in units of diopters (inverse meters). In the general case, the focal length is a function of orientation in the entrance pupil. By tracing a ring of rays, the average, maximum, and minimum optical power or focal length around the pupil can be determined. From this data, several types of optical power can be computed. The feature can display:

• spherical power,
• cylinder power
• maximum and minimum power
• tangential and sagittal power
• x or y direction optical power

in diopters. Additionally it can display the same data as effective focal length (EFL) in lens units.

These plots are extremely useful in understanding how power is distributed over a freeform surface.

In addition the POWF optimization operand allows direct optimization of any of the terms computed by the Power Field Map at any point. This is vital when a known desired power map is required on a surface.

Designing freeform or progressive lenses is in principle no different to optimizing traditional surfaces. However, because power can be added or subtracted easily at any point on the freeform surface, additional analysis plots an optimization controls are needed. The Power Field Map and POWF operand give designers of freeform optics this control.