Optics of Progressive Lenses - Part 2
Characterizing Progressive Optics
|January 1, 2011
|April 2, 2015
Darryl Meister is a Certified Master Optician, technical marketing manager for Carl Zeiss Vision, technical representative to the VCA and ANSI, has been a key contributor to many important industry initiatives and writes and lectures frequently on ophthalmic optics, lenses and dispensing.
completion of this program the participant should be able to:
- Understand the effects of the progressive corridor’s design considerations.
- Describe the characteristics of peripheral design effects and design symmetry.
- Understand the evolution of mono, multi and design by prescription lenses.
- Be able to define “as-worn optimization”
- Learn the effects of prism thinning on progressive lenses.
This course is approved for one (1) hour of CE credit by the American Board of Opticianry (ABO).
In Part 1, we learned the basics of progressive construction. Part 2 and 3,
teach the optical construction, advantages and limitations of progressive
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
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
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
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
- 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.
- 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
- 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
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
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
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
- Continuous Surface Height: This results in a surface that is
continuous all over with no breaks, ledges, or other discontinuities.
- 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).
- 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:
- The geometry of the
- The size of the distance zone
- The size of the near zone
- 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 Wi × (Ai - Ti)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
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
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.
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
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”.