Progressive lenses are an important example of the more general topic of free-form optical design. In free-form design, surfaces are not simple parametric functions like conic asphere or even asphere but are instead free to take whatever shape is required to add optical power and aberration control wherever it is needed.

Such surfaces require analysis features and optimization controls that conventional optical designs do not. This article uses a progressive addition lens as an example of the techniques involved.

**Authored By Mark Nicholson **

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## Introduction

Progressive addition lenses (PALs) are used to provide glasses wearers with a spectacle lens in which optical power varies smoothly as the user looks through different regions of the lens.

PALs are a specific example of free-form optics. In free-form optical design, the shape of a surface is not constrained to a simple expression, but is allowed to take any shape necessary to provide the optical performance needed at a given location. Thus, free-from optics require different analysis and optimization techniques compared to classical lenses that can account for power varying across a surface.

This article describes a simple PAL lens, shows how to construct such a system, analyze its performance, and optimize it.

Surface types

An ideal free-form optical surface would simply be a set of data points. However, 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 attached example file, progressive_starting_point.zmx, is a simple file and looks like so:

The file uses three configurations.

Note the use of the Maximum Solve to make the Clear Semi-Diameter equal to whichever configuration requires it to be the largest. Note also in the **System Explorer...Aperture** the applied Clear Semi Diameter Margin Millimeters of 3 mm so that the lens is 3 mm 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 effective focal length (EFFL) are still useful. The polynomials are of the form x^{m}y^{n}, where m and n are integers and x and y are the coordinates of a point on the surface. In the Lens Data Editor, the Maximum Term # is set to 40 terms, and all terms up to x^{4}y^{4} are variable. Any higher order terms are set to zero and fixed. This is purely designer choice: if desired, you could produce all polynomial terms to order 230.

The Merit Function optimizes for 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 Wizard. Therefore, the design 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.

Call the Local Optimizer by choosing **Optimize...Optimize!...Start**. OpticStudio quickly optimizes the 24 variables to give us the freeform surface that gives the best collimated output.

Analyzing the surfaces

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 the Ray-Fan and OPD plots are not enough to describe the performance of the free-form optic. For this optic, we use **Analyze...PAL/Freeform...Field Map**. Configure the settings to show contours at an interval of 0.25 diopters. From this, 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 OpticStudio. 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 in diopters:

- spherical power
- cylinder power
- maximum and minimum power
- tangential and sagittal power
- X- or Y-direction optical power

Additionally, it can display the same data as effective focal length (EFL) in lens units. These plots are extremely useful, and when coupled with the POWF optimization, allow for very useful visualization and control of the power distribution across a freeform surface.

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