Because of the large number of progressive power lens (PPL) designs and the different technologies available to make them,1–3 vision scientists and vision care professionals may benefit from scoring methods to classify these lenses. The scoring should quantify the lens quality at its three main regions, far, intermediate, and near, as well as the amount of lateral aberrations caused by the power variation.4,5 Once we know the properties (scoring) of each lens, we may be able to select the best design for a particular set of visual needs of the user.
There are already several articles studying the design parameters like the distance and near width and area, the power addition, the corridor width and length, and the unwanted cylinder and the off-axis performance caused by the aberrations.6–8
Zhou et al.9 studied the optical performance of three types of PPL with different philosophies but with the same nominal spherical power and addition. They concluded that the RMS of astigmatism and higher order aberrations of those PPLs had no significant difference in the entire measurement area. This conclusion was also found by Villegas and Artal.7
A well-known scoring technique is due to Sheedy.10,11 It is based on sphere and cylinder contour maps measured with a mapper. Sheedy defines the usable regions of the lens as those in which the errors in sphere and cylinder are below some threshold values. The test also includes the value of the maximum unwanted cylinder caused by the power variation. In a previous work,12 it has been demonstrated that the outcome of the Sheedy test depends, to a large extent, on the way the lenses are designed and measured.
The lens power obtained with a standard focimeter basically matches the sum of the curvatures of the surfaces that make up the lens (times the main refractivity with appropriate sign). The power measured by mappers depends on the technology of the instrument but it is close to the power measured by a focimeter (FP) at the center of the lens; as we move away from the paraxial area, the measured power becomes different in each instrument. Ching-Yao et al.13 have shown in their work that, even in different mappers, the final results have no meaningful differences. However, the user power (UP) not only depends on the curvature of the surfaces but also on the base curve and the position of the lens relative to the eye. When the lens is fitted without tilts, both types of power are identical in a small region around the center of the lens; but as the eye rotates out of this center, both types of power become different. In addition, if the lens is fitted with pantoscopic or wrapping tilts, the two powers get different even at the center of the lens.
The power needed to compensate for the refractive error is known as nominal power and is measured at the prescription room. A lens can be computed so that its FP matches the nominal power; in that case, we say that the lens is a curvature-optimized or focimeter-optimized lens. Ophthalmic lenses made before the arrival of free-form technology could only be computed this way, and henceforth, we refer to them as classical or standard lenses. On the other hand, the lens can be optimized in such a way that the UP matches the nominal power. In this case, the position of the lens with respect to the eye must be known before the surfaces of the lens are optimized. Because user parameters (which define the position of the lens with respect to the eye) enter the optimization, these lenses are known as customized lenses.
In a previous work,12 we demonstrated that the Sheedy scoring of a PPL changes significantly if the power maps are computed with FP or if they are computed with UP both for classical and customized lenses. We also observed that the scoring was very sensitive to the base curve (refractive power of the front surface) with which the lens was made.
The aim of this work is to present a more systematic study of the effect of the base curve on PPL scoring, taking into account whether the lens is classical or customized. This analysis has important practical consequences for lens prescription especially if we are about to compare the performance of classical and customized lenses. In most classical lenses, the progressive surface is the front one, and that implies that the curvature increases in the near region of the lens (it gets more convex). On the contrary, the back surface of a back side PPL has to be flatter in the near region (less concave). This flattening greatly affects the lens performance and, henceforth, the lens scoring when it is computed with the UP maps. This behavior has been reported in a qualitative analysis by Meister and Fisher.2,3 Here, we provide a quantitative verification of the same by means of the Sheedy scoring technique.
Most of the difference between the FP and the real lens performance (UP) is caused by oblique aberrations, and it is a well-known fact that for each prescription value, these aberrations can be minimized by selecting an optimum base curve. For spherical surfaces and no tilts, the optimal base curve can be approximately determined by means of the Tscherning ellipses, but nowadays, we may accurately compute them with real ray tracing. In Fig. 1, we show the functions that relate base curve and prescription to eliminate either oblique astigmatism or oblique power error for spherical lenses with refractive index 1.499 when viewing through a point 15 mm away from the optical center of the lens. According to these curves, a +2.00-diopter (D) lens (the near region of a plano PPL with addition +2.00) would require a base curve +8.00 D to eliminate oblique astigmatism or about +6.50 D to eliminate spherical error. Because of cosmetic and practical reasons (manufacturing range and frame constraints), base curves that steep are rarely used, and most manufacturers use and recommend much flatter base curves. The consequence is that, in lenses with locally spherical surfaces (the classical designs), oblique aberrations get larger, and the difference between the FP and the UP of lenses with too flat base curves gets bigger. We conducted this study to quantify the effect of this difference on the PPL fields of view. Customized designs are optimized with respect to the UP, and to provide the right one, the back surface is locally aspherized during the optimization process. The right amount of asphericity may cancel oblique aberrations.14
For this study, we have generated designs for six back surface progressive lenses. They slightly differ in the targeted profile and power distribution that are set in the optimization process. As a result, the widths of the near and far regions that we get after the optimization will also be slightly different. In addition, there are some small differences in corridor length (defined for the purpose of this study as vertical distance from 10% to 90% of the addition) among the different designs as well as small differences in the vertical distance from the fitting cross to the 10% of the addition. However, the power profile along the umbilical line has been set so that the minimum fitting height is 18 mm for all the designs, which means all the designs require the same amount of eye rotation to get 90% or 95% of the addition. In any case, the variability among the six designs is small enough so they all can be considered as general purpose designs. The particular characteristics of each design are not relevant to this study. Even more, the study is not intended to assess which design is better for a particular situation or which design should be selected for a particular patient. On the contrary, the aim of this study is to understand how the base curve selection may affect the scoring of the lens and eventually its performance. We can use the multiparameter representation introduced by the Guilino5 to easily grasp the Sheedy scores of the designs we have used in the present study. In this multiparameter plot, each lens is represented by a colored disk in an xy frame. The x axis stands for the width of the near region; the y axis stands for the width of the far region. The diameter of the disk is proportional to the width of the intermediate region, and its color encodes the maximum value of astigmatism. We have selected three types of designs with similar near widths (ñ10 mm), but the small differences in corridor among them give three different values of the far widths (ñ8, 10, and 12 mm).
Three designs are customizable and have been optimized in such a way that the widths of the far and near regions match the objective values as perceived by the user (UP). In this optimization, the pantoscopic and wrapping tilts have been set to zero, and the distance from the vertex of the back surface to the rotation center of the eye has been set to 27 mm. We call these designs custom 1 to 3 (cst1, cst2, and cst3). For the other three designs, the power used in the optimization is FP. Because of this, no custom parameters enter the optimization. We call these designs classical 1 to 3 (cls1, cls2, and cls3). The material selected for all the lenses was CR-39, with refractive index 1.499. Designs cst and cls are paired so that they present similar field widths, although they were not designed with exactly the same target power maps.
All the lenses used in the study have the same prescription, plano at distance and addition power +2.00 D. The six designs have been optimized with a set of five different base curves: 2.00, 3.50, 5.00, 6.50, and 8.00 D. The lenses have not been manufactured; instead, all the analysis is carried out on the computed power maps. This way, we avoid any error inherent to the manufacturing process and the subsequent measurement of the lens.
The selected reference base curve is 5.00 D. Lenses cls1 and cst1 (base 5.00) have approximate coordinates (10,12) mm in the multiparametric plot, but the Sheedy scoring has been computed with FP maps for cls1 and with UP maps for cst1. In a similar way, cls2 and cst2 have been paired with coordinates (10,10), as well as cls3 and cst3 with coordinates (8,10).
The Sheedy scoring test is then applied twice to each of the 30 lenses, first using FP maps and second using UP maps. We present the results (the width and the area of the far, intermediate, and near regions, as well as the maximum values of cylinder) for each lens design and each base curve.
The measurements of the field width at the height of the fitting cross are grouped by design in Fig. 2. For each design and each base curve, we provide these widths for both FP maps and UP maps. The FP width of classical designs does not depend on the base curve. The UP widths are also quite constant on cls2 and cls3. In cls1, the variations of the UP width across different base curves are slightly bigger, but yet they are smaller than 0.5 mm.
The differences between the FP and the UP widths are relatively small in the distance region because the obliquity of the rays refracting through it is small and all the lenses are plano at distance vision.
The width of the intermediate region is plotted in Fig. 3. The FP size (width) of the classical designs does not depend on the base curve. On the contrary, the UP size does change, and it gets smaller as the lens gets flatter.
The customized designs behave the other way around. Their UP size is base curve independent, but the FP size grows as the lens gets flatter.
We observe another interesting fact about the UP of any PPL design. The width of the intermediate region is always smaller than the width measured with a focimeter. This is because of the fact that effective UP addition is also bigger than the FP addition. When the corridor length is equal, the Minkwitz theorem15,16 implies that the width of the corridor should be smaller for the former type of power.
Because the UP width of the intermediate region in classical designs is base dependent, we could suggest that steeper base curves should be used to improve intermediate vision fields with this type of lenses. This suggestion is opposed to the typical aesthetic concern that demands flatter base curves. In this sense, the use of customized lenses allows the same lens area at intermediate vision independent of the selected base curve.
The near region is where we find bigger differences between FP and UP, as well as bigger differences in the performance of classical and customized designs. The near region has to be evaluated at different drop distances from the fitting cross to fully characterize the lens behavior. Scoring method measures the widths 14, 16, 18, and 20 mm below the fitting cross.
Although all designs had a recommended minimum fitting height of 18 mm, the measures at different heights provide a more general idea of the functionality of the near region of the design and how fast it delivers addition power as the user gazes downward.
The dependence of the widths with the distance to the fitting cross is presented in Fig. 4. The UP widths of classical lenses change a big deal with the base curve, just the same as the FP widths of customized lenses. On the other side, UP widths of customized lenses are base curve independent, just the same as FP widths of classical lenses. The general trend is that the flatter the base curve, the smaller the widths of the near region. Even the near region may disappear (according to the Sheedy thresholds) for the flatter base curves. This is mainly caused by the high values of oblique astigmatism (perceived by the user) that is present at the near region in flat classical lenses. In addition, the customized lenses have to be aspheric in this region to compensate for this astigmatism, and the consequence is that their FP gets distorted, with high values of astigmatism that limits, even making the useful near region disappear.
The same bar graphs used for the distance and intermediate regions are presented in Fig. 5 for the width of the near region at the minimum fitting height point (18 mm below the fitting cross).
As can be seen in measures of the widths (Fig. 5) in the near zone, the greater similarity between zones of clear vision measured in the FP maps and UP maps occurs for the steeper base curves. This effect is similar in classical designs and in customized designs. This demonstrates that a classical design will have a better performance by choosing a steeper base curve.
Maximum Unwanted Cylinder
The last parameter that gives information in the Sheedy scoring system is the maximum value of unwanted astigmatism. The maximum values are reached both at the nasal and temporal sides of the corridor because of the change of curvature in this area of the lens. The behavior of this parameter with the base curve and the type of design is shown in Fig. 6.
The maximum value of the unwanted cylinder behaves in a similar way with the width and area of the near region. For classical lenses, the FP value remains stable with respect to the base curve, whereas the UP value decreases as the base curve becomes steeper. Once again, the performance of classical lenses clearly improves with the use of steeper base curves.
For customized lenses, we find once again the opposite behavior: the UP value of maximum unwanted astigmatism remains stable, but the FP value increases as the base curve becomes steeper. This behavior is correlated with the add power provided by each design. In customized lenses, the add power perceived by the user is stable and equals the nominal add power, but the focimeter add power is weaker than the nominal, the difference being bigger as the base curve gets flatter. The mechanisms that increase the user add power are the oblique astigmatism and the vertex distance in the near region, not the geometric curvature variation of the lens. The consequence is that a customized lens will provide the nominal addition with a smaller increase of curvature on its progressive surface than a classical lens. As the unwanted astigmatism is closely related to the geometrical change in curvature, the FP value of the unwanted astigmatism will decrease with decreasing base curves in customized lenses. The same argument explains the opposite behavior found on classical lenses.
The Sheedy scoring system is a multiparameter test. In a study like the present one in which we analyze different lens designs with different base curves, it is difficult to grasp the general relationships between test parameters and lens characteristics. For that reason, a multiparameter representation technique was used in a previous work.12 In this technique, each lens is represented by a disk or bubble.
Only the widths at different heights from the fitting cross have been shown in the previous graphs instead of widths and areas as Sheedy did in his articles. This is caused by the similar behavior between the widths and the areas. In this section, both parameters are shown using the multiparameter representation to note that the areas follow the same pattern as the width in all cases.
In the representation, the Cartesian coordinates of its center will be the sizes of the lens areas at the far and near regions. In particular, the x axis represents either the width at 18 mm or the area at 18.5 mm of the near region. The y axis represents either the width or the area of the far region. The diameter of the disk is proportional to either the width or the area of the intermediate region. Finally, the color of the disk may be used to encode any other parameter. In the multiparameter representations shown in the previous work, the color was used to encode the maximum value of unwanted astigmatism. In this case, we will use the color to encode the lens design.
The results shown in the previous figures are condensed in the multiparameter representation shown in Figs. 7 and 8. In Fig. 7, we see the behavior of the widths of the three regions (far, intermediate, and near) of all the analyzed lenses computed with FP maps (Fig. 7A) and UP maps (Fig. 7B).
It is evident that the widths of the classic designs are stable regardless the base curve when they are computed with FP maps (Fig. 7A). On the contrary, there is a clear drift of the widths of the customized designs with respect to the base curve. This drift is bigger for the width of the near region because both the effect of the oblique aberrations and the local power of the lens at the near zone are also bigger. The width of the intermediate region is less dependent on base curve because, at the intermediate region, the effect of the oblique aberration is smaller, and this width is mainly determined by the addition and the length of the corridor, as commanded by the Minkwitz law.8,9
Once again, the behavior of the classic and customized designs interchanges when the widths are computed with UP maps. In this case, the widths are base independent for customized designs, and they significantly drift for the classic designs (Fig. 7B).
The behavior of the areas of the three viewing regions is shown in Fig. 8A and B. The previous description for the width is directly applicable to the areas.
In this article, we have made a comparison between progressive lenses optimized to provide the nominal FP and lenses optimized to provide the nominal power in the position of use. The lenses of this second group are traditionally called custom lenses, which currently can only be made using free-form technology.
For this study, we have selected three designs for each type (all of them back side progressive lenses), with similar scores in the Sheedy test to be evaluated according to the way they have been optimized. After that, the six designs have been calculated at different base curves, and their Sheedy scores were calculated for both FP and UP.
The main result of this work is that the performance of a design will be highly dependent on the base curve if they are evaluated with a different method to measure power than the one used for their optimization.
To illustrate this idea, consider the classic and custom no. 2 designs made in base curve 5.00, a value typically used for a plano lens addition +2.00 D. If we evaluate both designs with a focimeter, the custom design seems clearly worse, with its coordinates being (4,10) in the multiparameter representation compared with the coordinates (10,10) of the classic design. Apparently, the custom design has a quite narrow near area, whereas the rest of its characteristics are similar. If we evaluate the UP maps of these lenses, the results are reversed: the coordinates of the custom design are now (9,8) against (3,10) for the classic design. Although the distance region scoring of the custom design has slightly reduced, its near width matches the design goal, and this remains the same for all base curves.
Focimeter power maps are not a good option to score PPLs because they do not give us good information about their actual performance.
In practice, the most important thing is how a design behaves once it is fitted in its position of use. As we have shown, lenses can be customized to have similar performances regardless of the choice of base curve used for their optimization. Therefore, customized lenses can provide a stable behavior (widths and areas) for any given base curve.
On the other hand, classical lenses have different performances for each of the base curves. We have found that by using steeper base curves, we can improve the behavior (width and area) in the intermediate and near zones. The biggest differences were found in the near area because there is the greatest obliquity of the rays and it is the area of the lens with greater power because lenses were calculated plano for distance and addition +2.00 D.
The steeper base curve selection runs counter to the current aesthetic trend to select base curve as flat as possible. When a patient has problems with the near area, it could be that the error comes from the base curve and not from the design itself.
Because the customized lenses change their FP map depending on the selected base curve map (in addition to other customization parameters), it is necessary to use a double labeling: one to know the nominal prescription and the other to check the expected power with a focimeter.
One of the most important contributions of this team is the introduction of the representation of multiparameter results on simple graphs. We consider the multiparameter representation to be a powerful tool to visualize the benefits and features of progressive lenses. Similar to bifocals, in which one or two numbers classify the size and position of the near area, the Sheedy test assigns four parameters to each progressive lens. The multiparameter representation allows visualizing together these four parameters, making easier the design classification, the comparison with other designs, or the effects that may result from factors such as the change of the base curve.
Indizen Optical Technologies
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This work was partially supported by the Spanish Ministry of Science and Technology under grant DPI2009-09023.
Received March 30, 2012; accepted November 13, 2012.