Progressive addition lenses (PAL) have very wide application in the modern glasses market. The unique progressive surface can make a lens have progressive refractive power, which can meet the human eye’s different needs for distance-vision and near-vision. According to the national glasses fabrication standard, the difference between actual optical power after fabrication and nominal design value should be less than 0.1D over the lens effective area. The optical power distribution of PAL is determined directly by the surface. Consequently, the surface processing accuracy requirement is proposed. Beginning from the surface expressions of progressive addition lenses, the relationship equations between the surface sag and optical power distribution are derived. They are demonstrated through tolerance analysis and test of an example progressive addition lens with addition of 2.09D (5.46D-7.55D). The example addition surface is fabricated under given accuracy by a single-point diamond ultra-precision machine. The optical power of the PAL example is tested with a focal-meter after fabrication. The optical power addition difference between test result and design nominal value is 0.09D, which is less than 0.1D. The derived relationship between the surface error and optical power is verified from the PAL example simulation and test result. It can provide theoretical tolerance analysis proof for the PAL surface fabricating process.
The Progressive Addition Lens (PAL) is fit to myopia and presbyopia at the same time. It is shown from a large amount of investigations that PAL is helpful for teenaged myopia [1-3]. PAL is popular in modern countries, and tends to substitute for traditional glasses. There is great progress on design and performance evaluation for PAL. The main design methods of PAL are Winthrop [4], Steele [5], mixed design [6, 7], and divide and combine compensation optimization [8]. However there are few studies related to its surface error analysis. The optical power distribution of PAL is a key factor for sharpness and comfort for the glasses consumer. The surface error determines the optical power of PAL directly. The relationship between the optical power and surface error will be studied in detail and validated with simulation and experiment test in this paper.
II. THE BASIC STRUCTURE OF PAL
The PAL is mainly composed of five parts as shown in Fig. 1. The upper BASE is the far-sight area, the downside ADD is the near-sight area and the IRs at the edge are astigmatism areas, and the middle IC is the transition area. Points A and B are the reference points of the far-sight and near-sight areas, respectively. The main progressive meridional line MM' crosses the BASE, IC and ADD area. Points A and B are also on this meridional line. The meridional line is the track of points of intersection between the sight line and the progressive surface [9].
III. DESIGN AND EVALUATION OF PAL EXAMPLE
There are two main factors that affect the lens performance during the design of PAL. They are the design of the meridional curvature variant curve and the surface profile distribution [10]. According to different consumers, the power addition and the position of far-sight and near-sight are required. And the optical power distribution along with the meridional line is designed [11].
A PAL with progressive addition of 2.09D (0.46D-2.00D) is designed as an example. The material of the lens is PMMA, which is with index of refraction of
The effective aperture of the PAL example is 60 mm,
In Eq. (1) [12], m is the order of first non-zero at
The meridional line design rule is that the curvature of far-sight point and near-sight point should be changed slowly, i.e. the curvature variant curve of
From the boundary conditions, we get the following equations.
M is the minimum order of polynomial (1). The first non-zero order of
Based on Eq. (4), the augmentation matrix of linear Eq. (5) is acquired.
From matrix (5), the efficient
After the meridional line is fixed, the optical power is generally distributed with copy of power on meridional line by secondary curve perpendicular to the meridional addition curve [13].
The gradient direction of the addition power curve is determined by
(1) when m = 4, l = 3, result of
(2) when m = 3, l = 4, result
(3) when m = 3, l = 4, result .
The meridional addition power curves are plotted in Fig. 3. The No. 3 blue curve is very stable and is selected in this sample PAL design.
The meridional addition power curve is distributed on the whole lens in accordance with an ellipse transform method, which is shown in Fig. 4.
When u is changed from −L to −L+h, the series intersect cross lines between the spherical surface and cylinder surface build the progressive addition surface, substitute expression (7) in Eq. (8), the sag of the surface is solved [11].
Where (
The optical power and astigmatism distribution are the main evaluation indexes of PAL. The performance of PAL is evaluated with Eq. (9). P is spherical power and C is cylinder power.
H and K are the average curvature and Gaussian curvature of this point respectively,
p, q, r, s, t is the differential of surface sag.
The cylinder power is given in Figs. 5(a) and 5(b), and the spherical power is given in Figs. 6(a) and 6(b). The optical power is distributed from 5.46D to 7.55D, which meets the design requirements. The astigmatism is mainly distributed at the edge, which is not the usual visual area, and thus has small effect on vision comfort.
IV. SURFACE ERROR AND OPTICAL POWER RELATIONSHIP OF PAL
During the fabricating process, the difference between the actual optical power and designed result will be great, which will affect the visual sharpness and comfort. According to the national glasses fabrication standard [14], the difference limit is 0.1D between the actual optical power and the nominal value. The PAL surface error and optical power relationship are built.
Firstly, differential of Eq. (8), the surface sag error is expressed as Eqs. (11) and (12).
The relationship between Δ
With equivalent substitution, the relationship between the optical power variant and surface sag error is acquired.
According to the limit Δ
With solutions of the above equations, the surface error tolerance can be analyzed. With assumption that the aperture of the lens is D, the x and y are from −D/2~D/2.
According to the surface tolerance analysis method of Eq. (15), the surface sag error is calculated with optical power difference of 0.1D. The tolerance distribution on the lens surface along the x and y direction is shown as Figs. 7(a) and 7(b) respectively. Since the process accuracy larger than 0.1 mm is easier to control on the process instrument, the data points with surface sag error larger than 0.1 mm are omitted in surface error Figs.
[FIG. 7.] Surface error distribution along with X and Y direction: (a) X direction; (b) Y direction.
The most sensible district surface errors are shown in Table 1. The minimum surface error is 0.006 mm, which determine the accuracy during fabrication process.
[TABLE 1.] data of the most sensible surface error tolerance/(mm)
data of the most sensible surface error tolerance/(mm)
To verify the relationship, the optical power difference is simulated with surface error of 0.006 mm. It is shown from Fig. 8 that the maximum optical power difference is less than 0.1D, which meets the national standard requirements. Consequently, the relationship between the surface error and optical power is verified.
To validate the relationship in further processing procedure, the PAL example is fabricated with a single-point diamond ultra-precision machining equipment Nano-form 700 ultra. The vertex radius of far-sight and near-sight is tested with a Taylor-Hobson profile-meter respectively after processing. The test data is shown in Table 2.
[TABLE 2.] Vertex radius test result
Vertex radius test result
The optical power and addition are tested by focal-meter LM-1800P [15]. The nominal designed optical power of the PAL example is 0.46D-2.55D. Its test result is 0.55D-2.65D, which is shown in Table 3 The nominal optical power addition is 2.09D, and the tested result of power addition is 2.1D, which is shown in Fig. 9. The optical power addition difference between the nominal design and test result is 0.01D. The designed nominal optical power of far point is +0.46D, and its tested result is +0.55D. The optical power of far point difference between the designed nominal and tested result is 0.09D, which meets the standard requirements.
[TABLE 3.] Tested result data with NIDEK LM1800P
Tested result data with NIDEK LM1800P
The relationship between the surface error and optical power is derived based on surface expression of PAL. A PAL example with progressive addition of 2.09D (5.46D-7.55D) is designed and evaluated. With the example, the surface error is analyzed based on the built relationship model with optical power. And the optical power of PAL example is simulated according to the given surface error. The simulation result verified the relationship model. The PAL example is processed and tested. From the tested result, the data calculated from relationship model and tested with the sample processed lens is consistent each other. The relationship model is validated both from simulation and experiment test.
