Figure 11. Equation to a tyre form toroidal surface
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Dr Prof Mo Jalie, SMSA, FBDO (Hons), Hon FCGI Hon FCOptom, MCMI, is a Visiting Professor of Optometry at the University of Ulster in Coleraine, and at the post-graduate facility at Varilux University. He served for nine years as Head of Department of Applied Optics at City & Islington College, where he taught optics, ophthalmic lenses and dispensing. He is a recognised international authority on spectacle lens design and has written several books including Principles of Ophthalmic Lenses. His most recent book, Ophthalmic Lenses & Dispensing was translated into Russian. He has authored over 200 papers on ophthalmic, contact and intra-ocular lenses, and on dispensing; and is a consultant editor to The Optician (UK) and technical editor to The Indian Optician journal. He holds patents for aspheric spectacle and intra-ocular lenses. Jalie is a past-chairman of the Academic Committee of the Association of British Dispensing Opticians, and was the first Chairman of the Faculty of Dispensing Opticians. He is the ABDO representative on the BSI committees on ophthalmic lenses and spectacle frames and a past member of the Education Committee of the General Optical Council. In 1998 Jalie was thrice honoured: he was made Honorary Fellow of the British College of Optometrists, a Life Fellow of the Association of British Dispensing Opticians, and in December of that year he was granted the Max Wiseman Memorial Research Medal.
A new skill which must be developed by spectacle lens designers is that of converting an ECP’s prescription into a data file of (x,y,z) coordinates which can drive a CNC generator or a 3D printer which actually produces the finished lens. The software which does this is described within the industry as Lens Design Software (LDS) and is incorporated in a typical Lab Management System (LMS).
This talk describes some of the various steps which the LDS must consider and in particular some of the lesser known routines which the system must follow and must be programmed into the LDS computer to produce the data required by the free form equipment. The various steps are illustrated in Figure 1.
The first step is to identify and select the semi-finished blank which is to be used and this choice is based upon the desired off-axis performance of the lens. Usually, the requirement is to select curves, or combinations of curve and asphericity in the case of aspheric lenses, which reduce or eliminate either oblique astigmatism or mean oblique error (power error). The software will have a selection procedure based upon the best form (corrected curve) requirement to address these aberrations and the specific range of semi-finished blanks which the lab actually keeps in stock.
The next routine is the computation of the total prism which must be incorporated in the lens, either to fulfill the ECP’s prescription and/or to shift the optical centre of the finished lens to the desired position, so called “prism-to-cut”. It may also be necessary to combine any prescribed prism with thinning prism required to reduce the thickness difference between the top and bottom edges of the lens.
MINIMUM LESS THICKNESS
The minimum thickness of a minus lens lies at its optical centre. It can be seen in Figure 2 that for a circular minus lens without prism, the minimum thickness lies at the centre of the lens. Furthermore, on edging the lens to a smaller diameter to fit into the frame, the edge thickness, tE, reduces.
Therefore, the centre thickness, tC, can be made a simple function of the lens power such as shown by the expression which works well for glass lenses as well as the more rigid higher-index plastics materials.
tC = 2.0 + 0.2 x power of lens
Thus, we might expect a glass lens series to have the centre thicknesses shown in Figure 2. For lens powers higher than -5.00 D, the minimum centre thickness remains 1.0mm.
Note that the thickness of a CR 39 lens would be increased to avoid any flexure of the lens when it is mounted in a spectacle frame.
When the lens incorporates a prism, as might be necessary through prescribed prism, or because prism-to-cut is required, the centre thickness must be adjusted by the centre thickness of the prism. In the case of minus lenses, the first question to be addressed is “are we dealing with a lens which incorporates prism, or a prism which incorporates a lens?”
Figure 1. Calculation steps to convert ECP’s Rx into a finished lens by CNC or 3D Printer
Figure 2. Determination of the minimum thickness of a minus lens
Figure 3. Thickness addition for prism
Figure 4. Plus lenses – cylindrical element – the minimum thickness is at the edge of the minus axis
Figure 5. Plus lenses – prismatic element – the minimum thickness is at the apex of the prism
Figure 6. Plus lenses – astigmatic prism – the minimum thickness is at the apex of the prism
Figure 7. Plus lenses – astigmatic prism – the minimum thickness no longer lies at the apex of the prism
The answer depends upon the position of the optical centre of the lens. If it lies within the visible lens area we are dealing with a lens which incorporates prism, in which case the power of the lens influences the centre thickness addition. If it lies outside the visible lens area, we are dealing with a prism which incorporates a lens and the thickness addition is a simple function of the prism power. The necessary addition, tG, to the centre thickness to accommodate the prism is shown in Figure 3.
The optimum thickness for a plus lens depends upon its power and the type of mount into which the lens will be edged. For lenses which demonstrate a large variation in thickness between their centre and edges, usually, lenses over the power of +2.00D, the optimum edge thickness for mounting in a plastics frame varies from about 1.0 to 2.0mm, depending upon the power of the lens. For example, a +2.00D lens which is to be mounted in a plastics frame would require an edge thickness of about 1.5mm, whereas a +10.00D lens would only need an edge thickness of 1.0mm at its thinnest point. When the lens is to be mounted in a metal frame, or needs to be drilled for a rimless mount the edge thickness needs to be increased to stop it flaking under pressure from the rim or rimless screw.
The thinnest point on a plus lens lies somewhere round the edge of the lens. In the case of circular spherical lenses, the thinnest point on the edge of the lens lies at the furthest point from the optical centre. If the lens is not decentred (or incorporates no prism), the edge thickness is the same all around the edge of the circular lens.
A good working rule for the edge thickness, tE, of a plus lens whose spherical power is S, where in the case of an astigmatic prescription, S is taken to be the sphere in the minus cylinder transposition, is tE = 2 – 0.2 S down to a minimum of 1.0 mm, which would occur when S = +5.00.
In the case of circular plus astigmatic lenses, the edge thickness is greatest along the minimum power meridian (i.e., along the plus axis meridian) and least along the maximum power meridian (the minus axis meridian). It is, therefore, the minus axis meridian which controls the thickness of the lens. This can be seen in Figure 4 which illustrates a plano-convex cylinder whose plus axis lies at 90° and whose thin edge substance (at 180) is zero, i.e., the cylinder is supposed to be knife-edged. The graph shown in Figure 4 shows the variation in edge thickness around the edge of a 60mm diameter, circular, uncut lens as would be seen by inspecting the edge starting at zero in TABO notation (as shown by the insect who, starting at zero, is moving anti-clockwise around the edge, measuring the edge thickness as it travels).
It goes without saying that when a spherical element is combined with the cylindrical element, the positions of the minimum and maximum thickness on the edge of a circular lens do not change.
In the case of circular prisms, the minimum edge thickness corresponds with the prism apex and the maximum edge thickness with the prism base, which is shown base down in Figure 5.
When a cylindrical element is combined with a prismatic element the position of the minimum edge thickness varies with the angle between the cylinder axis and the base setting of the prism. This can be seen in Figure 6, which shows the result of combining a prism with a cylinder such that the base setting of the prism coincides exactly with the minus cylinder axis.
Since both the cylindrical element and the prismatic element have zero edge thickness at their thinnest points on the edge, the combination also has zero edge thickness at 0°, for a right eye, on the nasal edge of the lens. The centre thickness of this astigmatic prism is the sum of the centre thicknesses of the individual components.
Figure 8. Plus lenses – astigmatic prism – the minimum thickness lies at 20 and 160
Figure 9. Determination of the back curves when the centre thickness is known
Figure 10. Tyre formation toroidal surface
Figure 11. Equation to a tyre form toroidal surface
In Figure 7, the base direction of the prism has been rotated through 90°, the base now lies at 270 and the edge thickness of the combination is no longer zero at any point on the edge! It is clear from Figure 7 that a significant amount of material can be removed from this combination in order to obtain zero edge thickness, and that when this is done, by further surfacing of the component (Figure 8), the thinnest point on the edge of the astigmatic element no longer lies along the 180 meridian. In order to calculate the thickness of this combination it is first necessary to determine the position on the edge of the lens where the minimum substance occurs. The solution to this problem is a quartic equation and it is easy to appreciate why the computer is employed to solve this everyday problem!
In the case of shaped lenses, a frame tracer is employed to determine the (x,y) or (x,y,z) coordinates of points around the edge of the lens and the minimum edge thickness determined by iterative calculation of the thickness around the edge. The data from the frame tracer is input into the LDS which may then determine the centre thickness of the lens, tC, and the compensated surface powers determined once tC is known (Figure 9).
For the astigmatic lens specification shown in Figure 9, the (x,y,z) coordinates must now be determined from the equation to a toroidal surface. Suppose the toroidal surface is to be produced as a tyre-form surface. Figure 10 shows the equatorial, CE, and transverse, CT, centres of curvature of each principal meridian of the surface and their respective radii of curvature, rE and rT. The equation to the surface is shown in Figure 111 together with the solution for the z coordinate which can be programmed into the LDS to produce the SDF file for a generator or the STL file for a 3D Printer. This equation is solved for as many (x,y) coordinates as the generator or 3D printer requires to produce the surface. Figure 12 shows the z-values for (x,y) intervals stepping in 5mm intervals. Naturally much smaller intervals would be chosen to produce the necessary surface.
The form in which the (x,y,z) data should be transferred to the generator is described in The Vision Council of America’s Data Communication Standard2 using the SURFMT record, (see entry 22.214.171.124), which explains how the matrix containing the data should be set out. A typical data file which is sent to a CNC generator is shown in Figure 13. The first two lines simply identify the job. The third line gives the generator information about the file type, the eye, R for right eye, which surface Back or Front, the number of rows and columns of z-data it can expect and whether any partial derivatives or slope data are given for the perimeter of the matrix. Each block of numbers headed ZZ represents one line of data for an 78mm diameter blank, the z-heights being given in millimetres.
Figure 14 illustrates, first, a CNC machine to which an SDF file has been sent, the CNC cutter working on a surface which it is cutting in a spiral motion, the (x,y,z) data having been converted to polar coordinates, followed by a 3D Printer from the pioneering company in this field, LUXEXCEL.
Figure 12. Free form description of the toroidal surface -5.00 DC x 90 / -7.00 DC x 180. The values circled in red (numbered 1 to 4) correspond with the sags or z-values) shown in the inset at the top right of the Figure
Figure 13. Typical SURFMT record
Figure 14. CNC Machining and 3D Printing of lenses
1. Jalie, M. Principles of Ophthalmic Lenses, 5th ed. London: Association of British Dispensing Opticians, 2016.
2. The Vision Council of America. Data Communication Standard Version 3.08. 2010 Mar 22 p30.
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