Optics of Progressive Lenses - Part 2
Characterizing Progressive Optics

By Darryl Meister, ABOM

In Part 1, we learned the basics of progressive construction. Part 2 and 3, teach the optical construction, advantages and limitations of progressive lenses.

Recall that any arbitrary point on a progressive lens surface contains both a maximum and a minimum surface curvature, which are perpendicular to each other. Essentially, each point is locally toric, similar to a lens with cylinder power. The maximum and minimum surface curvatures at this point produce maximum and minimum surface powers, which we will refer to as the principal surface powers, just as the two planes containing the maximum and minimum powers in a cylinder lens are referred to as the principal meridians.

For example, consider a point on a progressive lens surface that produces a minimum principal power (P₁) of +6.00 D and a maximum principal power (P₂) of +7.50 D. The surface astigmatism at this point is equal to the difference between these two powers, or P₂ - P₁ = 7.50 - 6.00 = 1.50 D. The mean power of the surface at this point is equal to the average of these two powers, or ½ (P₁ + P₂) = ½ (6.00 + 7.50) = +6.75 D. Astigmatism is comparable to cylinder power, while mean power is comparable to sphere power.

Traditionally, the mean power is offset by the value of the Base curve—or offset by the power at the distance reference point—in order to arrive at the net plus power produced by the design once the lens has been surfaced to the desired prescription. This produces a more meaningful distribution, since it shows the change in actual Add power of the progressive surface as experienced by the wearer. If the base curve is +6.00 D in our example, the mean power can be described as 6.75 - 6.00 = +0.75 D at this particular point.

Contour plots are a type of graphical representation of a quantity or optical property that varies with two other variables, such as the vertical and horizontal coordinates of a lens surface. Sampling a quantity over an array of regularly distributed points across the lens usually creates contour plots.

Points of equal value for the quantity are then connected by curves, called contour lines. Generally, only contour lines at regularly spaced intervals are shown (e.g., 0.50, 1.00, 1.50, and so on), and the contour levels may also employ gradations of color. Most commonly, these plots are used to show the distribution of optical properties, particularly surface astigmatism and mean power, across the lens surface.

Astigmatism plots generally indicate regions of blur and image distortion, and can also give some indication as to the size of the distance, intermediate, and near zones. The spacing of the contours in these plots will also indicate how rapidly the unwanted astigmatism is increasing. Mean power plots can be used to determine the location of the full add power, as well as regions of excess plus power that may also result in blur—including unwanted plus power in the distance zone.

It should be noted that, while plots of surface astigmatism and mean power are indicative of performance, they are not necessarily representative of performance. They provide a look at certain convenient physical quantities, but accurate interpretation of these plots requires some skill and experience. There are also other quantities of interest, including measures of binocular performance, image swim, and blur. Furthermore, plots that depict ray- traced quantities, which are calculated using lens-eye modeling of optical performance, generally provide more visually meaningful comparisons than plots of surface quantities.

It is also possible to characterize the optics of a progressive lens surface using wavefront analysis, which considers higher-order aberrations—such as spherical aberration, coma, and trefoil—in addition to mean power and astigmatism. One particularly convenient way of expressing wavefront aberrations is with the use of a Zernike series, which is a series of polynomials whose terms represent quantities such as astigmatism, mean power error, forms of coma, and so on.

The Progressive Umbilic

Recall that a progressive lens surface contains a single vertical meridian that is free from surface astigmatism, known as the umbilic. Since the umbilic is free from surface astigmatism, the surface is locally spherical at each point along this meridian. Specifically, any point along the umbilic can be represented with an osculating (or "kissing") sphere equal in curvature to the curvature of the actual surface at that point, and tangent to the surface.

The umbilic represents the centerline of the progressive corridor—or the channel of relatively clear vision connecting the distance and near viewing zones. The curvature down the umbilic steadily increases in order to produce a progressive increase in mean power from a point producing the lowest mean power (in the distance zone) to a point producing the highest mean power (in the near zone). The vertical distance between these two points is referred to as the corridor length of the lens. The total change in mean power along the umbilic between these two locations represents the specified Add power.

While the horizontal, vertical, and oblique curvatures at any infinitely small point along the umbilic are equal, these curvatures start to differ away from the umbilic. The surface astigmatism, which is equal to the maximum difference between these curvatures at any point, rapidly increases as you move into the lateral blending regions of the lens. In the vicinity of the umbilic, this surface astigmatism actually increases at a very predictable rate away from the umbilic. Minkwitz showed that the rate of change in surface astigmatism (A) away from the umbilic grows at twice the rate of change in mean power (M) along the umbilic. Mathematically, this can be described as:

ΔA/Δx ≈ 2 × ΔM/Δy

Where (ΔA) is the change in astigmatism, (Δx) is the lateral distance away from the umbilic, (ΔM) is the change in mean power, and (Δy) is the vertical distance along the umbilic. If the horizontal distance (Δx) to a point (A) is equal to the vertical distance (Δy) to a point (M), we can assume that the change in surface astigmatism at point (A) is approximately equal to twice the change in mean power at point (M).

It is interesting to note that, although any infinitely small point along the umbilic is locally spherical, the curvatures immediately above and below this point will differ since the mean power of the surface is gradually changing along the umbilic. The pupil of the eye is 2 to 5 mm in diameter under typical lighting conditions, which means that eye samples an area of this size at any point on the surface—including points along the umbilic. Consequently, the refractive power of the surface will differ between the upper and lower margins of the pupil. This variation in refractive power across the pupil results in an imaging defect (or aberration) known as intrinsic coma.

For instance, consider a simple progressive lens that produces an Add power of +2.00 D at a distance of 16 mm below the distance zone. If the umbilic has a linear change in power from distance to near, the mean power of the surface changes roughly 2.00 ÷ 16 = 0.125 D per millimeter. For a 4 mm pupil size, the difference in mean power between the upper and lower edges of the pupil is equal to 0.125 × 4 = 0.50 D.

This 0.50 D variation in refractive power across the pupil creates a vertically oriented patch of blur on the retina. (Note that it does not produce a total power error of 0.50 D; the blur from this aberration is equivalent to a much smaller power error.) Fortunately, the blur produced by the intrinsic coma inherent in a typical progressive lens is generally well within the depth of focus of the eye, so it is not an issue for the wearer.

Minkwitz's theorem provides us with a few useful guidelines regarding the optics of a progressive lens in the vicinity of the umbilic (progressive corridor):

1. The rate of change in surface astigmatism around the umbilic is inversely proportional to the corridor length of the lens. As the corridor length becomes shorter, the unwanted astigmatism must grow more rapidly away from the umbilic.
2. The rate of change in surface astigmatism around the umbilic is proportional to the Add power of the lens. For a simple progressive lens design, this also means that the maximum level of unwanted astigmatism in the periphery of the lens will be proportional to the Add power.
3. Because of the first two points, the size of the progressive corridor will depend upon both the length of the corridor and the Add power. This means that lenses with shorter corridor lengths or higher Add powers will by necessity provide less intermediate vision and mid-range utility.

Distribution of Surface Optics

One of the most fundamental aspects of the basic progressive lens design is the distribution of its surface optics, including power and astigmatism. These features define the gross performance of the lens, and represent a veritable "fingerprint" that distinguishes one progressive lens design from another. The magnitude, distribution, and gradient (or rate of change) of power and astigmatism are factors that affect the performance and wearer acceptance of the lens design. Progressive lenses are often arbitrarily classified into either one of two broad "philosophies" of lens design by the relative magnitude and distribution of their surface optics:

• Harder designs. A "harder" progressive lens design concentrates the unwanted astigmatism into smaller areas of the lens surface, thereby expanding the areas of perfectly clear vision at the expense of higher levels of blur and distortion. Consequently, harder progressive lenses generally exhibit the following characteristics: wider distance zones, wider near zones, and higher, more rapidly increasing levels of surface astigmatism (closely spaced contours).
• Softer designs. A "softer" progressive design spreads the unwanted astigmatism across larger areas of the lens surface, thereby reducing the overall magnitude of blur at the expense of narrowing the zones of perfectly clear vision. The astigmatic error may even encroach well into the distance zone. Consequently, softer progressive lenses generally exhibit these characteristics: narrower distance zones, narrower near zones, and lower, more slowly increasing levels of astigmatism (widely spaced contours).

In general, harder progressive lens designs will provide wider fields of clear vision, at the expense of higher levels of swim, distortion, and blur. These lens designs will provide better central or foveal vision, which is important during tasks that involve critical viewing or that require good visual acuity. Additionally, harder lens designs frequently employ shorter, narrower progressive corridors.

Softer lens designs, on the other hand, will provide reduced levels of astigmatism and swim, while limiting the size of the zones of clear vision.

These lens designs will provide better peripheral or dynamic vision, which is important during tasks that involve active viewing. Softer designs tend to improve "comfort" and adaptation for emerging presbyopes, while harder designs offer more of the kind of utility current bifocal wearers enjoy. Softer lens designs often employ longer, wider progressive corridors as part of their overall design strategy.

Essentially, as you increase the region of the lens used to "blend" the distance and near zones together—effectively spreading the blending region out over a larger area—you decrease the levels of surface astigmatism.

Conversely, as you decrease the region of the lens used to blend these two zones, you increase the levels of surface astigmatism. Modern progressive lenses are seldom absolutely "hard" or absolutely "soft," but rather strive for a balance between the two in order to achieve better overall utility. A manufacturer may also choose to employ the features of a softer design in the distance periphery in order to improve dynamic peripheral vision, while employing the features of a harder design in the near periphery in order to ensure a wide field of near vision. This "hybrid"-like design is another approach that sensibly combines the best features of both philosophies.

Progressive Lens Design

Geometrically, we often describe a progressive lens surface as a "locally toric" surface that is "globally smooth." A smooth surface must have a continuous 2nd derivative over the entire surface, which is related to the rate of change in the height (or sagitta) of the surface. A continuous 2nd derivative implies that the surface has a continous surface height and continuous 1st derivative, and ensures that the surface meets the following criteria:

1. Continuous Surface Height: This results in a surface that is continuous all over with no breaks, ledges, or other discontinuities.
2. Continuous 1st Derivative: This results in a surface with smooth changes in prism and no lines of demarcation (e.g., no sharp peaks, ridges, or valleys).
3. Continuous 2nd Derivative: This results in a surface with smooth changes in power and magnification with no abrupt changes in vision.

Consequently, the primary goal of progressive lens design is to produce a smooth surface that possesses all the necessary structural features, including distance, intermediate, and near zones, with a minimum of aberrations, or optical imaging defects. Often, the first step in progressive lens design is to determine the performance requirements of the final lens design, including the length of the progressive corridor, the configuration of the central viewing zones, the distribution of optics in the periphery, and so on. Once these performance requirements are understood, they can be used to establish the various parameters that fundamentally define the basic progressive lens design:

1. The geometry of the progressive corridor
2. The size of the distance zone
3. The size of the near zone
4. The distribution of optics in the periphery

There are numerous approaches to defining mathematically a progressive lens surface. Depending upon the mathematical functions used by the lens designer to create the initial surface, the fundamental design parameters described above may be directly defined in the equations or arrived at experimentally by adjusting related parameters. If these functions are sophisticated enough to produce a sufficiently adequate and well- behaved progressive lens surface, the surface may be suitable for use without any significant refinements.

However, this "starting" surface is commonly optimized—or optically refined—further in order to maximize performance or to achieve specific performance requirements. This is usually done numerically using lens design software that attempts to find a "real" surface that satisfies the ideal performance requirements of the lens design as closely as possible. Essentially, the software minimizes the differences between an actual smooth surface and a theoretical target surface.

In many cases, this software employs a finite element method, which uses special functions to model surfaces and to find solutions to minimization problems. An initial starting surface is first specified and then modeled mathematically using the finite element method. The computational area of the lens surface is "discretized" by breaking regions of the surface up into square elemen across a reference grid, or mes Each intersection between these square elements represents a position on the lens surface, and is referred to as a node. Each node contains an array of mathematical quantities that can characterize the surface a point, including its local curvatures. These nodes are mathematically connected using basis functions known as splines.

In regions on the surface where optical performance is more critical, or changing more rapidly, the mesh can be subdivided as needed into smaller square elements with additional nodes to allow tighter control of the surface in those regions. The surface characteristics at these nodes are smoothly connected by mathematical spline functions, which ensure that the surface maintains a continuous 2nd derivative between these points.

A target distribution of optical quantities, representing the ideal distribution of quantities such as mean power and astigmatism, is first determined for each node and its corresponding point on the lens surface. Generally, a smooth surface cannot achieve this target distribution—at least for every point. Finite element method seeks to minimize the difference between the desired optical performance at any point on the surface and the actual optical performance possible with a continuously smooth surface. This is accomplished by minimizing merit functions at each node, which are equations of the form:

M = Σ W_i × (A_i - T_i)² = W₁ × (A₁ - T₁)² + W₂ × (A₂ - T₂)² ...

Where (M) is the value of the merit function to minimize at a given node location, (W) is the weighting—or importance factor—of the measurement, (A) is the actual value of the measurement, and (T) is the target value of the measurement at that point. Merit functions are used to find a "least squares" solution. Common measurement quantities to minimize may include mean power errors, astigmatism, gradients of power, and so on.

The weightings for these quantities can vary as a function of the node location, allowing different regions of the lens surface to emphasize different performance attributes (e.g., blur in the central viewing zones and image swim in the periphery). The individual nodes, or even collections of nodes, can be weighted, as well. Some regions of the lens must maintain exact optical specifications. The central viewing zones, for instance, are more heavily weighted so that the analysis achieves the desired target performance in these regions—at the expense of the peripheral regions, if necessary.

The goal of finite element method is to minimize the total sum of all merit functions across the entire mesh. The program repeatedly manipulates the actual surface in order to reduce the sum of the entire range of merit functions across all nodes. These iterations eventually result in a real surface that comes as close as possible to producing the target optical distribution. Essentially, the process seeks to find an actual progressive surface that provides the desired optics in the central viewing zones, while minimizing optical errors elsewhere as much as possible.

The measurement quantities may be calculated directly from surface characteristics, such as the principal surface powers, or they may be derived after first ray tracing a lens with the intended surface characteristics. Ray tracing is the process of modeling the theoretical optical performance of the lens as perceived by the wearer with the lens in its intended position of wear—including vertex distance and pantoscopic tilt—by calculating the refraction of light rays passing through the lens from various object points.

Once the lens design has been computed, a mold must be made. Surface description data are calculated for the lens design and then communicated to expensive milling machines. For metal molds, often used for injection-molded materials like polycarbonate, the mill may cut a progressive surface directly onto the metal mold. The mold is then polished to a high luster. For glass molds, often used for cast materials like hard resin, the mill may cut a progressive surface onto a ceramic former. A glass mold is then heated over the former until it softens and assumes the shape of the former in a process called slumping.

Alternatively, the glass mold may be cut to the desired shape directly using a computer-controlled free-form or digital surfacing process.

Once the mold has been made, some initial prototype lenses are produced. These lenses are then inspected to ensure that they achieve the desired optics. Often, because of lens material shrinkage and other variations in the manufacturing process, the initial lens design may need to be modified slightly in order to produce a manufactured lens blank that achieves the desired surface powers. Once the initial design has been "tweaked," a new mold is made and the inspection is repeated. This entire process must be performed for each Base curve and Add power combination, in every lens material. Since a typical progressive lens series may include up to 72 Base Curve and Add power combinations in up to 10 lens materials, this process requires a massive amount of development work and cost.

Summary

Methods of describing, designing and manufacturing a progressive surface involve calculation and are the result of improved computing power. They also have benefited from the manufacturer’s learnings in vision science, the actual wearer results of previous designs and the clinical trials that are used to develop new ideas and designs. While complex, the optics of progressive lenses provides a simple and sophisticated solution to the needs of the presbyope.

In Part 3, we’ll discuss the effects of the progressive corridor and the characteristics of peripheral design; understand the evolution of mono, multi and design by prescription lenses and “as-worn optimization”.