Study on Design and Diamond Turning of Optical Freeform Surface for Progressive Addition Lenses

Optical freeform surface components have attracted much attention due to their high degree of design freedom and small size. However, the design and processing difficulty of such components limit its wide application in optics industry. In recent years, diamond turning has been considered an efficient method for processing optical freeform surfaces, but the research on tool path generation of this processing method is not systematic. Progressive addition lens (PAL) is a typical optical freeform surface and is widely used to correct people’s vision problems. Firstly, this paper introduces a method of designing PAL.+en, an optimized tool path generation method for diamond turning of the optical freeform surface is proposed, the equal angle method is used to select the discrete points, and a tool nose radius compensationmethod suitable for both slow slide servo (SSS) and fast tool servo (FTS) is adopted. Finally, the turning experiment is carried out with a single point diamond lathe, and a PAL surface with a roughness of 0.087 μm was obtained. +e power and astigmatism distributions were measured using a Rotlex freeform verifier to verify the rationality of the optical design.


Introduction
An optical freeform surface is defined as any nonrotationally symmetric surface or a symmetrical surface that rotates around an asymmetrical axis [1]. Such surfaces can be designed in any shape, and in most cases, these surfaces have submicron profile accuracy and nanoscale surface quality [2]. e advent of the freeform surface provides designers with wider design space, which can achieve higher optical utilization through smaller optical elements, reduce the number of optical components in the optical system, and improve the optical performance. ese components have a large number of requirements in the aerospace, biomedical, and scientific industries. It also has a lot of daily applications, such as automotive lighting systems, F-theta lenses in laser printers, and mobile phone cameras [3,4].
At present, the main methods of fabricating optical freeform surfaces include injection molded [5,6], grinding and polishing [7], raster milling [8,9], and diamond turning (or tool servo turning) [10][11][12][13]. In practical applications, appropriate processing methods are selected according to the comprehensive factors such as the material and size of the workpiece. Recently, slow slide servo (SSS) and fast tool servo (FTS) have been extensively studied by scientific research institutes and applied to optical freeform surface fabricating. Compared with other methods, diamond turning has its unique advantages: the workpiece can usually be formed in a single process, low cost, and good form accuracy and surface finish. As early as 1983, Douglas of the University of Tennessee in the United States used an FTS device based on air-floating guides and linear motors to machine off-axis parabolic mirrors [14], which also made FTS technology truly used in optical freeform surface machining.
is study focused on the design and fabrication of typical optical freeform surface PAL. e curvature of the lens changes continuously from top to bottom, which can physically help the eye focus and provide a continuous vision from far to near. is kind of lens was originally used to correct presbyopia, and now it is also used to treat myopia of teenagers. e design of PAL is very complicated and difficult.
ere are two main design methods, one is to model in CAD software and simplify the lens design as a functional minimization problem [15,16], and the other is a mathematical method that can directly express the surface with a series of formulas [17][18][19][20][21]. e first design method is relatively simple, but the processing of subsequent data is more complicated, and it is usually necessary to use the spline interpolation to obtain the machining coordinates. Compared with the first method, the second method has less accurate control of astigmatism distribution, but its modeling is more accurate, so this method is adopted in this study.
In this paper, the design method of PAL is firstly discussed, which is divided into meridian line design and surface construction. On this basis, a design example is given and used for subsequent machining. en, a general tool path generation method for tool servo turning of optical freeform surfaces is proposed, which can provide more stable cutting motion. Finally, the tool path was generated for the designed lens surface, and the turning experiment was carried out in the way of point-to-point with a single point diamond lathe. e experimental results are tested and analysed, which indicate that the optical distribution of the fabricated lens was conformed to the design principle, and the surface quality was good.

Design of the PAL
As people age, the crystalline lens becomes rigid and the flexibility of the ciliary muscle decreases. In order to see the close target clearly, they have to move far away, so presbyopic glasses need to be worn. Traditional presbyopic glasses cannot take into account both far and near sightedness. At this time, the bifocal lens was invented, but the visual effect of the glasses is not good because of its special structure. PAL can meet the needs of distant and near vision and provide a continuous field of view, so it is more comfortable to wear. e functional partition of the PAL is shown in Figure 1. Area A is the far-view zone, which is mainly responsible for distant vision. Area C is the near-view zone, which mainly meets the needs of reading and other near-sighted objects. Area B is a progressive corridor, which is the middle area connecting the far and near-sighted areas.
is part can provide smooth and continuous vision, which is the focus of the design. Area D is the blending zone, in which there is inevitable astigmatism. In this study, the design of the meridian line determines the vertex power distribution. en, the designed surface was constructed according to the distribution of curvature along the line. is method can choose a large number of parameter combinations, and different parameters will generate different optical power and astigmatism distribution to meet the different needs of wearers.

Design of Meridian Line.
e refractive ability of ophthalmic lens is generally expressed by the vertex power, which is the reciprocal of the paraxial focal length of the lens measured in meters. Its unit is diopter and its symbol is "D". It can be expressed by the following formula: where n is the refractive index of the material and r 1 and r 2 , respectively, represent the curvature radius of the front and back surfaces. From this formula, we can know that the vertex power of the glasses is the sum of the vertex power of the front and back surfaces. e difference between PAL and traditional lenses is that the curvature of one of its surfaces is variable. If the surface of curvature change is the front surface, it is called the exterior progressive addition lens, and if it is the back surface, it is the interior progressive addition lens. As shown in Figure 2, pp ′ is the meridian line. e change in vertex power along the meridian line is called the addition power curve. In order to ensure smooth and continuous visual effect, the meridian line design should ensure the following: firstly, stable optical effect around distant-sighted point F and near-sighted point N; secondly, smooth and gradual change of the power addition curve D(x, 0) in the whole power-adding range. e region where the optical effect is stable means the region where the vertex power does not change significantly. In the optical design, the size of the stable region near F or N depends on the curvature of the surface (the vertex power is proportional to the surface curvature). e slower the curvature growth rate is, the larger the stable far-sighted or nearsighted area is.
In order to meet the abovementioned conditions, we need the curvature of the surface along the meridian line to change as smoothly as possible. As a criterion of smoothness, we can require that the mean square gradient of D(x, 0) is the minimum, or in other words, that D(x, 0) minimizes the Dirichlet integral [21]: where m is the first order to make ( In order to obtain the minimum value of the abovementioned formula, according to Euler-Lagrange equation, the following results are obtained: In order to make the curvature change slowly at the F and N point, the order of the first nonzero order derivative should be as high as possible. Solving equation (3) with boundary conditions e following formula can be obtained: Substituting D(h − L, 0) � D N into equation (5), we can obtain According to the definition of m, we can obtain Equation (7) can also be written as follows: and equations (5) and (6) can be written as follows: According to the definition of l, the following formula can be obtained: e coefficient c n can be obtained by solving the linear equations with the combination of equations (10) and (11), and the power addition curve can be obtained by choosing the appropriate parameters.

Construction of PAL Surface.
After determining the vertex power distribution along the meridian line, we should extend the obtained power to the whole lens surface, and then calculate the surface vector height according to the principle of differential geometry. ere are many ways to extend the power of the meridian to the whole lens surface. Generally, the contour line of equal power which is orthogonal to the meridian line is chosen [22]. e contour line can be straight line, circle, hyperbola, ellipse, and so on. e designer can select the contour line according to the actual needs.
e PAL shown in Figure 3 is externally progressive, meaning that this lens has a freeform front surface and a spherical back surface. In order to obtain the lens height, we assume that the lens is tangent to the vertical plane at point O. Make a horizontal plane perpendicular to the vertical plane and through point O, dividing the progressive surface into two parts along the meridian.
We define a Cartesian coordinate system in which the progressive surface is tangent to the XOY plane at point O, the X-axis is vertically downward, and the Z-axis points to the eye. As shown in Figure 4, let Q be any point on the meridian line, and the curvature radius of this point is r. Take a point of q in the normal direction of the Q point and make Qq equal to r. e q point is the curvature centre of the Q point. e trajectory of curvature centre at each point in the meridian is called the evolute of the meridian line, which is represented by mm′ in Figure 4.
Let the coordinate of point q be (ε, η, ζ), and the coordinate of point Q on the meridian line PP ′ be (u, 0, z), where u represents the X coordinate of Q point. en, the coordinates and curvature radius of q point can be expressed by the equation containing u: According to the geometric relationship shown in Figure 4, the height Z of Q point relative to the XOY plane can be derived from the following formula: where Mathematical Problems in Engineering 3 According to the formula for calculating the vertex power of Section 2.1, if the externally progressive design is adopted and the curvature radius of the back surface of the lens is selected, combined with equation (9), the curvature radius r(u) of any point on the meridian line can be expressed as follows: To sum up, the appropriate parameters (including m, l, h, and L) are selected according to the design requirements, and then the coefficient c i is solved by the linear equations, and the equation of the power addition curve can be obtained. e vector height equation of the surface can be deduced by selecting the appropriate contour curve u(x, y) and substituting it into equations (13)-(15).

Design Example.
A PAL surface was designed by the abovementioned method, assuming that the diameter of the lens was 70 mm. e type of the contour curve that is orthogonal to the meridian line is selected as hyperbola. And the remaining design parameters can be seen in Table 1. e final image of lens vector height is shown in Figure 5.

Tool Path Generation for Diamond Turning
e generation method of ultraprecision tool paths depends on the surface description. However, no matter the surface is modeled in CAD software or expressed directly by the mathematical formula, it must be transformed into point coordinates for processing. From Section 2, we have obtained the vector height formula of the PAL surface, and the next step is to generate the tool path suitable for diamond turning. Figure 6. e workpiece is mounted on the spindle and rotates with the spindle. e spindle is mounted on a working table with X direction guide rail, and Z direction guide rail drives the diamond cutting tool to complete reciprocating cutting motion. Sometimes, a fiber-linked probe was integrated onto the machine tool for the purpose of alignment [23]. e machining principles of FTS and SSS are similar,the difference is that the Z direction cutting motion in the FTS technology is realized by the FTS device. is type of machine tool is distinguished from a conventional machine that can only fabricate rotationally symmetrical surfaces due to its special configuration. Any three-dimensional tool path of nonrotational symmetric freeform surface can be realized when the X-axis, Z-axis, and C-axis move simultaneously according to a given set of numerical motion commands [24].

Diamond Turning Principle. A typical machine tool configuration for SSS technology is shown in
It can be known from the above that diamond turning has three processing parameters: spindle angle, X direction displacement, and Z direction displacement. ese three parameters, respectively, correspond to the machine tool's spindle, X-slide and Z-slide (or FTS). In order to generate motion commands for machining, we need to switch from the Cartesian coordinate system to the cylindrical coordinate system. Specifically, the Z coordinates are the same in both coordinate systems, and Cartesian (x, y) are converted to cylindrical (ρ, θ) by the following formula: x � ρ × cos(θ),

Selection of Discrete Points.
e turning path of the freeform surface is a spiral in space. First, we need to select a set of discrete points on the XOY plane, and then map these points to the freeform surface to get the Z direction coordinates, so as to obtain all the required processing data.
ere are three main methods to select discrete points: equal angle method, equal arc length method, and equal chord length method. In recent years, some scientific research institutes have put forward some variable feed methods, such as equal residual height method and equal load method, but the application is not extensive. e equal angle method means that, at the same turn of a plane helix, the central angle between any adjacent two points is equal. Similarly, the equal chord length method means that the two adjacent points have the same chord length; equal arc length rule means equal radians. However, the time interval between cutting points selected by the equal chord length method and equal arc length method is not uniform. is leads to the need to configure time parameters at each cutting point in the turning process, which will greatly increase the amount of data in the machining file. Besides, these two methods will cause unnecessary spindle acceleration in the cutting process, which will affect the processing quality. erefore, these two methods are not commonly used in the current research. In contrast, the equal angle method is simple to calculate, and the rotation angles of the spindle at the same time interval are the same, which does not produce redundant acceleration. It has been widely used in FTS and SSS processing, and often achieved better results.
In this paper, the equal angle method is used to select discrete points. e PAL surface designed in Section 2.3 was planned to generate discrete point graphs. In order to facilitate the demonstration, one point is taken at 10°intervals and the feed rate is 5 mm/r, as shown in Figure 7. e Z coordinates corresponding to each cutting point can be obtained by mapping the points on the XOY plane to the freeform surface.

Tool Nose Radius Compensation.
Optical freeform machining generally adopts the diamond arc edge turning tool, which can process workpiece of various shapes and obtain high surface quality, and has been widely used in precision machining. However, in the process of machining, the radius of tool nose will lead to over cut and affect the surface fidelity. For the optical freeform surface, the precision of the surface shape directly affects its optical capability, so the compensation of the tool radius cannot be ignored. e influence of tool nose radius is shown in Figure 8. e dotted line represents the contour of the ideal surface and the solid black arc represents the contour of cutting edge. When the radius of the tool nose is not compensated, the cutting point in the tool path is determined by the position of the tool tip. It can be seen from the figure that the round tool nose geometry will lead to over cut.
At present, the tool nose radius compensation method adopted by most scientific research institutions is the twoway tool radius compensation [11,25], in which the cutting point position is determined in advance, and the radius compensation is then carried out in the normal direction of the cutting point. e feature of this method is that at different cutting points on the freeform surface, the shift value is not constant as the slope angle changes [24]. Hence, it can make the tool vibrate in the radial direction, which requires high performance of X-slide. It may not be able to compensate when machining very complex microstructures, and it is not suitable for single-degree-of-freedom FTS processing. In this paper, a one-way tool nose radius compensation method is adopted, which only compensates in Z direction and makes X feed free of vibration. Moreover, it is suitable for both SSS and FTS technology.
According to the diamond turning principle of the freeform surface, the motion of the tool in the cutting process is completed on the XOZ plane, which can be expressed as ρ � x in the cylindrical coordinate system. When using the one-way compensation method, the radial coordinate only depends on the cutting radius and the feed rate. is tool nose radius compensation method is illustrated in Figure 9. e curve L is the transversal of the freeform surface on the XOZ plane, C is the tool contact point, S is the tool location point, and a f points to the direction of tool feed. In the process of machining, the tool location point S is uniformly fed, so we can determine the radial coordinates of this   e mathematical expression of the freeform surface in cylindrical coordinates can be expressed as follows: As can be seen from Figure 9, the coordinates of tool location point S can be expressed as follows: where R is the cutting radius (mm), a f is the feed rate (mm/ r), r is the tool nose radius (mm), and a is the angle between the tangent of the tool contact point C and the ρ axis, which can be obtained from the following implicit equation: According to the abovementioned formulas, the coordinates of all tool location point S can be obtained, which completes the tool nose radius compensation.

Generating Tool Path.
After determining the method of selecting discrete points and the method of tool nose radius compensation, we can generate the tool path. e mathematical expression for PAL is available in Section sec2.2 of this article. We set M as the number of spindle revolutions and N as the sampling number per rotation. en, the spindle angle can be expressed as follows: where i and j are cycle variables, and θ(i, j) corresponds to the jth pulse of the ith turn of the spindle. Substituting equation (21) into equations (18) and (20), we can obtain tan a p − f ′ ρ s (i, j) + r sin a, θ(i, j) � 0.
e value of acorresponding to each cutting point can be obtained by solving equation (23), then the Z coordinate of the tool location point S can be expressed as follows: Using the abovementioned method, the tool path of the PAL surface is generated. Figure 10 is a sketch of spiral tool path generation. In order to display the diagram conveniently, a larger feed speed is adopted. en, according to the requirements of the NC system, the coordinates of tool location points are arranged in the form of θ − ρ − z, and the machining program can be obtained.

Experiments and Discussion
To verify the proposed method, experiments were carried out with Moore's 250UPL single point diamond lathe. e processing mode of the equipment is SSS. e X-axis and Zaxis of the machine tool are both linear axis and the stroke is 200 mm. e spindle is fixed on the worktable and can move along the X-axis. e turning tool moves along the Z-axis to achieve vibration cutting. Specific machine layout can be seen in Figure 6.    Because the surface of precision components cannot be pressurized, the workpiece is fixed on the machine by the way of sucker and bonding. e material of the workpiece is resin with a diameter of 70 mm. e specific optical design parameters are shown in Table 1. Machining process is divided into roughing and finishing, and processing parameters can be seen in Table 2. e processing results are shown in Figure 11. e surface roughness was measured by the Dektak XT probe surface profiler. In order to make the measurement   Mathematical Problems in Engineering more accurate, the surface roughness was measured in four directions (0, 90, 180, 270 degrees), twice in each direction, and then the average value was calculated. e sampling length is 0.8 mm and the evaluation length is 4 mm. e average roughness (Ra) obtained was 0.087 μm. e power and astigmatism (or called cylinder) distribution of the lens were measured with the Rotlex freeform verifier, as shown in Figure 12. It is less than 0.18 D that the different of the addition power between the designed and manufactured PAL. e results show that the power of the far-and nearsighted zone of the lens changes slowly, the transition zone (progressive corridor) changes smoothly and gradually, and the astigmatism mainly concentrates on both sides of the lens, which meets the design requirements.

Conclusions
is paper presents a PAL design method based on mathematical principles, which is different from the freeform surface model formed by spline interpolation to obtain more accurate calculation data. Based on the PAL design method, an optimized tool path generation method which can avoid unnecessary vibration of machine tool is proposed. e surface of PAL is obtained by turning experiment, and the rationality of optical design is verified by the Rotlex freeform verifier. Some conclusions are summarized as follows: (1) According to the method described in Section 2, different parameters can be selected for individualized design of lenses. e design parameters can be selected including the values of m and l, the length of light adding channel, the type of orthogonal contour curve, and the power of far-view zone and near-view zone. e combination of different parameters can meet the different needs of wearers. (2) e tool path generation method described in this paper is suitable for both FTS and SSS technology. It can provide reference for diamond turning of optical freeform surface components. A more stable tool nose radius compensation method is adopted. It can make the X-slide feed uniformly throughout the processing, avoid unnecessary acceleration, and obtain better surface quality. (3) e designed PAL was fabricated by a single point diamond lathe. e results show that the optical distribution of the lens is consistent with the design, and the power error is within an acceptable range. e surface quality is good and the roughness is 0.087 μm.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.