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Abstract
Toroidal, spherocylindrical, ellipsoidal, and combined surfaces were used to correct astigmatism in aspheric spectacle lenses, and the differences among the proposed techniques were compared. Four astigmatic spectacle lenses were designed with the same optical parameters. A freeform measuring machine was used to evaluate posterior surfaces of aspheric spectacle lenses, and spherical and cylindrical power maps were generated and compared. The measured data were analyzed via commercial software. The toroidal lens helped to extend the clear vision range around the lens, and the ellipsoidal and spherocylindrical surfaces resulted in a more accurate centering of the lens around the optical axis of the eye, avoided astigmatism, and provided better visual perception.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Astigmatism is a defect of the eye that causes point sources situated at infinite distances to result in images comprising of two small perpendicular light segments. These images are spaced apart from each other along the optical axis by a distance which is a function (or depends on the degree) of astigmatism [1]. The axis of astigmatism is typically classified as a) with-the-rule (WTR) astigmatism, b) against-the-rule (ATR) astigmatism, or c) oblique (OBL) astigmatism [1,2]. The effects of astigmatism on visual function are determined by the induced meridional blur [3–9]. The focal planes of astigmatism are orthogonal. In the case of ATR (negative correction of cylindrical x-axis by 90°), a more pronounced myopic meridian tends to blur the vertical detail, whereas in WTR (negative correcting cylindrical x-axis by 180°) a more pronounced myopic meridian blurs the horizontal details [6,7].
The development of astigmatism in young adults (younger than 40 years) is common, but it is not pronounced [1,5]. With increasing age, a general shift in the axis of astigmatism is found from a predominant WTR astigmatism (in adults younger than 40 years) to a predominant ATR astigmatism (in adults older than 40 years). This shift in the astigmatic axis in older ages appears to be caused by changes in the corneal curvature [1]. The optical correction of astigmatism can take many forms, including spectacles, contact lenses, and refractive surgery [1,3,10]. Spectacle lenses used to correct astigmatism are described as cylindrical or toroidal, and the axes of the maximum and minimum powers of the cylindrical lens are orthogonal [1,3,4,6–11]. The axes of the cylindrical lens are aligned with the axes of the astigmatism of the eye to provide optical correction [6].
In the past, the correction of astigmatism has been achieved using spherocylindrical lenses which correspond functionally to the association of a spherical lens with a cylindrical lens [12]. However, accurate definition of spherocylindrical surfaces suitable for obtaining optimum correction is often difficult and complex [13]. A toric surface is generated by rotating the circular arc about an axis which does not extend through the center of this circular arc. Therefore, it has different radii of curvature in its two principal sections [14]. These toric surfaces, particularly in the case of spectacle lenses with high-vertex dioptric powers, cannot satisfy all the requirements, i.e., they cannot sufficiently correct defective vision of the eye in all the viewing directions [15]. The shape of the cornea of the eye is ellipsoidal [16]. It is for this reason that the central problem of fitting a spectacle lens more accurately to the human eye has not been solved by spherical, toric, or parabolic lenses. It has been found that (a) an ellipsoid represents a better approximation for the form of the surface of the cornea of the human eye, and (b) contact lenses with inner elliptical surfaces represent a marked improvement regarding the comfort and time the lens is worn by the patient. However, manufacturing of these surfaces during the last century has required elaborate machines and processes associated with increased costs [17,18]. Malacara et al. [19] described spherocylindrical, toroidal, and ellipsoidal surfaces, and their associated and specific properties so that a choice could be made during the design, manufacturing, and testing process. Malacara-Doblado et al. [20] analytically studied the separation between toroidal surfaces and showed that the spherocylindrical surfaces outside the central region were singular but not identical. Greynolds [21] compared various representations and generalizations to ascertain how biconic surfaces smoothly blend the profiles together, and designed an anamorphic Ritchey–Chretien telescope. Forbes [22,23] worked with rotationally symmetric surfaces that were either (i) strongly aspheric or (ii) constrained in terms of the slope in comparison to that for a best-fit sphere, and considered an orthogonal basis had significant benefits when characterizing shape regarding its efficiency and numerical robustness. Meanwhile, a method of polynomials was introduced in conjunction with robust and efficient algorithms to compute the surface shape, and the polynomial derivatives were used for the characterization of the shapes of rotationally symmetric aspheres for applications to a wide class of freeform optics. Zalevsky1 et al. [24] presented a method that adapted the special, all-optical extended depth of a focus length from the field of digital imaging to ophthalmology. Based on this paradigm, he provided the required vision solutions to develop special spectacles which were capable of solving common ophthalmic problems, such as myopia, presbyopia, and regular/irregular astigmatism.
In this study, we applied the toroidal, spherocylindrical, ellipsoidal, and combined surfaces to correct astigmatism to design the aspheric spectacle lenses and to compare the differences. Four contemporary aspheric spectacle lenses with the same parameters were designed and manufactured. The rear surfaces of aspheric spectacle lenses were measured and converted to achieve the desired optical properties based on the FMM method. The measurement data were analyzed with commercial software to combine the anterior and posterior surfaces. The four different astigmatic optical surfaces are comparable for measuring the spherical and cylindrical powers of the surface shape, combining actual production results with theoretical results. Section 2 presents a brief description of the different astigmatic optical surface principles. The experimental results and the discussion are outlined in section 3. The conclusions of the study are presented in section 4.
2. Principles of different astigmatic optical surfaces
Aspheric surfaces are described using a right-handed, orthogonal coordinate system in which the Z-axis is the optical axis. The origin of the coordinates is at the vertex of the aspheric surface, as shown in Fig. 1.
We assumed that there is a prescription in the form ${S \mathord{\left/ {\vphantom {S {C \times \theta }}} \right.} {C \times \theta }}$ (relaxing the requirement such that $0\;<\;\theta \le {180^\circ }$) for a spherocylindrical lens [25]. Note that S is used to designate the spherical power, C is used to designate cylindrical power, and $\theta$ is the orientation of the axis when a cylindrical coordinate system is adopted. In this discussion, the spherocylindrical surface will be described in terms of its toroidal equivalent. Thus, the above form would be changed to [26],
In the expression of the ellipsoidal surface, which is generated by rotating an ellipse with respect to one of its axes, the axis of symmetry of the generated ellipsoidal surfaces is a line perpendicular to the optical axis, and is tangent to the origin of coordinate system. If the axis of symmetry is the y-axis, the sagittal surface can be expressed as [19]
We choose the point O as the origin, and the direction along the center of the lens to O as the Z-axis. This is set up as an X-Y-Z Cartesian coordinate system and a cylindrical X’-Y’-Z’ coordinate system, as shown in Fig. 2. The $cc$ curvature for the cylindrical lens can be obtained as
The original surface can be found to be,
If the cylindrical lens is concave or convex, the Z coordinates ${Z_c}$ of the cylindrical lens can be computed using the following formula,
3. Results and discussion
3.1 Optical parameters and simulations
The optical parameters required for the aspherical spectacle lenses are summarized in Table 1.
Table 1. Optical Parameters of Aspherical Spectacle Lenses Used for Astigmatism Correction (unit: D, 1/m).
Figures 3–6 show the simulations for each case based on the use of the freeform verifier software (FFV) (ROTLEX, Israel), which is a commercial software used as a moiré deflectometer with a point source. Moire fringes are produced when a pitch grating is superimposed on another pitch grating at a smaller angle. The angle between the interfering gratings is kept constant in ROTLEX’ systems. The frequency and direction of the resulting fringes are highly sensitive to the difference between the two pitches [27]. The window of observation was a circle with a diameter of 60 mm, which is much larger than a spectacle frame. The power distributions are in line with expectations in the view zones of every simulation. The simulated cylindrical powers and axes adopted in the four tested lenses were −0.97 D, −0.98 D, −0.98 D, −0.98 D, 179°, 179°, 179°, and 29°, respectively. These values were very close to the theoretical cylindrical power and axis. Image changes were smooth in all cases, and the spherical powers were −3.95 D, −3.98 D, −3.98 D, and −3.95 D, respectively.
Fig. 3. Simulation of sphere (left) and cylinder (right) for a toroidal surface. Sph: spherical power; Add: additional power; Cyl: cylindrical power; Axis: the axis of astigmatism.
As shown in Table 2, for the toroidal and combined surfaces, the absolute value of the power distributions gradually decreases from the center to the edge, and changes from 4.0 D to 3.5 D, while those of the spherocylindrical and ellipsoidal surfaces increase gradually, from 4.0 D to 4.75 D. The combined surface has the largest center area of power −4.0 D, followed the toroidal surface, and finally the spherocylindrical and ellipsoidal surfaces, which are approximately equal.
Table 2. Simulation Results of Aspherical Spectacle Lenses Used for Astigmatism Correction (unit: D, 1/m).
These designs were evaluated to be suitable for a wearer with astigmatism. The maximum and minimum astigmatic powers of the toroidal surface are in the horizontal and sagittal directions, respectively. Whereas those of the spherocylindrical and ellipsoidal surfaces are in the sagittal and horizontal directions, respectively. Regarding the combined surface, the axis of the maximum astigmatic power is at an angle of 120° and the power is equal to −1.25 D. For the maximum astigmatism areas, the toroidal surface has the largest area, followed by the spherocylindrical surface, the ellipsoidal surface, and finally the combined surface. For the minimum astigmatism areas, the combined surface has the largest area, followed by the toroidal surface, and the smallest is the spherocylindrical surface.
3.2 Optical properties analysis
The manufacturing process for the posterior surface uses the CNC optical generating machine of Satisloh VFT-orbit (Jiangsu Mingyue Photoelectric Technology Co. Ltd.) for turning, milling, grinding, and polishing. Four lenses are shown in Fig. 7.
Fig. 7. Four different astigmatic lenses: 1. toroidal surface, 2. spherocylindrical surface, 3. ellipsoidal surface, and 4. combined surface.
Surface height measurements were conducted with a precision OptoTech GmbH OMM-60 Freeform Measuring Machine (FMM) (Jiangsu Mingyue Photoelectric Technology Co. Ltd.). The posterior surfaces of the lenses were measured to derive the full refractive properties. Each lens was measured on a grid of points (x, y). The points were spaced approximately 1 mm apart, which led to a measurement accuracy of approximately ± 1 µm. A 60-mm diameter blank lens requires approximately 5329 samples per lens to achieve this measurement density. These data were recorded as a text file. The contour plots of the average spherical and cylindrical powers (astigmatism) of the rear surfaces are shown in Figs. 8–11, respectively. From Fig. 8, it can be seen that the spherical power of the rear surface is 5.99 D, and the cylindrical power is 0.99 D; it gradually decreases from the center to the edge. The maximum and minimum astigmatic powers are in the horizontal and sagittal directions, respectively. Figures 9 and 10 show that the spherical and cylindrical powers are equally distributed throughout the entire surface. The spherical powers of the rear surface are 6.12 D and 6.01 D, gradually increasing in diopters from the center to the edge, while the cylindrical powers are 0.96 D and 0.90 D, respectively. The maximum and minimum astigmatic powers are in the sagittal and horizontal directions, respectively. Figure 11 shows that the maximum and minimum of the sphere and cylinder of the combined ellipsoidal posterior surface are equal to 6.08 D and 0.95 D, respectively, gradually decreasing from the center to the edge. The maximum astigmatic power is at an angle of 120°. The variation trends of the cylindrical distributions shown in Figs. 3, 4, 5, and 6 are the same as those of the simulated results.
Fig. 8. Mean power measurements of sphere (left) and cylinder (right) of the posterior toroidal surface.
Fig. 9. Power measurements of sphere (left) and cylinder (right) of the posterior spherocylindrical surface.
Fig. 10. Power measurements of sphere (left) and cylinder (right) of the posterior ellipsoidal surface.
Fig. 11. Power measurements of sphere (left) and cylinder (right) of the combined posterior surface.
Meanwhile, the tested data were subsequently imported into the FFV for analysis with a 60-mm diameter. The optical properties of all the lenses were calculated for each surface and for the combinations of pairs of surfaces. The contour plots of the average spherical power and cylinder (astigmatism) are shown in Figs. 12, 13, 14, and 15. The diameter of each lens was set to 60 mm.
Fig. 12. Contour plots of toroidal lens for average spherical power (left) and cylinder power (right).
Fig. 13. Contour plots of spherocylindrical lens for average spherical power (left) and cylinder power (right).
Fig. 14. Contour plots of ellipsoidal lens for average spherical power (left) and cylinder power (right).
Fig. 15. Contour plots of combined lens for average spherical power (left) and cylinder power (right).
As shown in Table 3, the cylindrical powers and axes adopted for the four tested lenses were −0.98 D, −0.98 D, −0.97 D, −0.99 D, 179°, 178°, 157°, and 29°, respectively. These values were very close to the theoretical and simulated cylindrical powers and axes. Diopter power and astigmatism changes were smooth for all cases. For the toroidal and combined surfaces, the absolute value of the power of the lens gradually decreased from the center to the edge, changing from 4.0 D to 3.5 D. The powers of the peripheral-cylindrical and ellipsoidal surfaces also gradually increase, changing from 4.0 D to 4.5 D. The combined surface has the largest center area of the power −4.0 D, followed by the toroidal surface, then by the ellipsoidal surface, and finally by the spherocylindrical surface. Regarding the toroidal surface, the maximum and minimum astigmatic powers are in the horizontal and sagittal directions, respectively. The maximum and minimum astigmatic powers for the spherocylindrical and ellipsoidal surfaces are in the sagittal and horizontal directions, respectively. For the combined surface, the axis of the maximum astigmatic power is at an angle of 120° and the power is −1.25 D.
Table 3. Measuring Results of Aspherical Spectacle Lenses Used for Astigmatism Correction (unit: D, 1/m).
According to the measured results, there are respective deviations of approximately 0 D, −0.03 D, −0.03 D, and 0.03 D, for each lens based on the comparison of the power simulations (left) shown in Figs. 3, 4, 5, and 6. The measured results of the cylinder for the toroidal and spherocylindrical lenses exhibit an error margin of approximately 1.25%, compared with the simulated results. The deviations of approximately 0.03D and 0.01D were observed for the ellipsoidal and combined lenses, respectively.
From the measured results and analyses, it can be inferred that although there are some differences in the outcomes of the spherocylindrical, toroidal, ellipsoidal, and combined surfaces, the theoretical design requirements and desired effects were achieved for all astigmatic optical surfaces, which have astigmatism correcting properties.
Toroidal, spherocylindrical, and ellipsoidal lenses typically have different refractive powers or magnifications along two principal axes or meridians separated by 90°. The power and astigmatism characteristics of the spherocylindrical lens are close to those of the ellipsoidal lens, whereas, those of the toroidal lens are close to those of the combined lens. Ellipsoidal surfaces and spherocylindrical surfaces are used for the most practical purposes sufficiently close to a toroid that they can be substituted for a toroid in a lens design program. The functions for the toroidal, spherocylindrical, and ellipsoidal lenses can be reduced to the expression for a spherical surface by making the sagittal and tangential curvatures equal in value.
Compared with the work proposed by Malacara-Doblado [20], wherein the deviations between the toroidal and the spherocylindrical optical surfaces outside the approximately identical central region in the vicinity of the optical axis were studied, for the proposed method, the entire surface power and astigmatic values of four different astigmatic optical surfaces with toroidal and spherocylindrical optical surfaces were compared. In comparison with the research of Zacarías [19], four different astigmatic lenses were manufactured, and the different optical properties of each lens were compared. The use of each of these surfaces was internally defined, and the analytical expressions of the four different astigmatic surfaces were clearly presented. Consequently, the results indicate that the proposed method enables optical design developers to flexibly select any of these different surfaces in a straightforward manner.
4. Conclusions
In this study, we compared the outcomes following applications of spherocylindrical, toroidal, ellipsoidal, and combined surfaces in the effort to correct astigmatism in spectacle lenses. Four lenses were produced with CNC optical generating machine, and were measured by the FMM. Equivalent power contour plots for spherical, cylindrical, and combined front and rear surfaces for each lens were obtained and compared with the FFV. The nonoptical FMM method can be used to evaluate the optical properties of the aspheric spectacle lens by measuring the height of the posterior surface. We observed that all four surfaces had some common characteristics. Specifically,
- 1. According to the test results, four different astigmatic lenses rather than the spherical lenses can be utilized to resolve astigmatism;
- 2. The four tested astigmatic lenses were the optical elements associated with two different powers in two different orientations. Essentially, astigmatic lenses built into the single lenses were associated with the spherical power for the correction of myopia or hyperopia, and with the cylindrical power for the correction of astigmatism;
- 3. Different power levels were obtained with curvatures at different angles and were preferably maintained relative to the eyes;
- 4. The astigmatic lenses used in spectacles were held fixed relative to the eye, and thus always provided optimal vision correction.
- 1. Spherocylindrical, toroidal, and ellipsoidal lenses tend to rotate with respect to the eye, thereby temporarily providing suboptimal vision correction. Combined lenses do not rotate;
- 2. The ellipsoidal and spherocylindrical surfaces resulted in a more accurate centering of the lens around the optical axis of the eye, avoided astigmatism, and provided better visual outcomes.
Funding
National Natural Science Foundation of China (61605114).
Acknowledgments
Manufacturing and measuring equipment was provided by Jiangsu Mingyue Photoelectric Technology Co. Ltd. The authors would like to thank Jiakang Zhang for useful discussions about this problem.
Disclosures
The authors declare no conflicts of interest.
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