Optimization design of progressive addition lenses based on the toric equation

ZHANG Huixing， WU Quanying， ZHANG Haiping， TANG Yunhai，CHEN Xiaoyi， GONG Haohan， YIN Lidong

（1.School of Physical Science and Technology，SUST，Suzhou 215009，China；2.Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application，Suzhou 215009，China；3.Graduate Practice Station，Soochow Mason-optics Co.，Ltd.，Suzhou 215007，China；4.Suzhou Mason Optical Co.，Ltd.，Suzhou 215007，China；5.Suzhou FOIF Co.，Ltd.，Suzhou 215006，China；6.Zhangjiagang Optical Instrument Co.，Ltd.，Suzhou 215600，China）

Abstract： A method for designing and optimizing progressive addition lenses （PALs） was proposed.The vector height of surface was calculated by the toric equation.Compared with the lens designed by the spherical equation，the curvature of the lens in x direction and y direction can be separately controlled.The optimization method can effectively reduce the astigmatism of the lens.Spherical and toric equations were used to design the lenses respectively，with the same lens parameters.Then lenses processing was performed，and the samples were obtained and tested.The results show that the addition power （ADD） of the PALs designed by the two methods is basically the same.The width of the area with astigmatism greater than 1.50 D on the left side of the lens designed by the toric equation is reduced by 1.70 mm.And the one on the right side is reduced to 0 as well.The width of the near vision zone with astigmatism less than 0.50 D is increased by 3.60 mm.Therefore，the design method by using the toric equation can effectively reduce the range of high astigmatism of the PALs and expand the wide range of the near zone visible area of the PALs simutaneously.

Key words： optical design；progressive addition lenses；toric；power；astigmatism

1 Introduction

PALs provide a natural and convenient way to correct vision problems.It has become the main direction of scientists' research and market popularization.Up to now，many methods of designing and optimizing PALs have been researched out.They are mainly divided into two categories：direct and indirect design methods.In the direct methods，a cost function with power and astigmatism is preset.The power and astigmatism are controlled by weight factors.PALs are designed by minimizing the cost function[1-2].These methods are effective and powerful.However，setting a reasonable cost function and selecting an appropriate numerical calculation method are great challenges for these designs[3].Compared with the direct design methods，the indirect design methods divide thedesign process into several steps.The meridian of PAL is solved firstly.Then the distribution of the power on the meridian is mapped to the entire lens surface through the appropriate contour line，and finally the vector height of the lens is calculated.These methods generally include Maitenaz method，Steele method，Winthrop method，etc[4-6].PALs with better optical performance can be obtained by optimizing the meridian，the contour line，or the center of curvature and so on.For example，the genetic algorithm is used to optimize the meridian，and the chromosomes are used to represent the polynomial coefficients describing the meridian.The power and astigmatism are used as the objective function.A new generation of chromosome populations is generated through the selection，crossover and mutation of the objective function，and the optimal coefficients of the meridian are searched cyclically[7].There is a method for obtaining the contour line by solving the Laplace equation as well.The designed lenses have the characteristics of large visual range in the near zone and short channel length，which can effectively reduce the wearer's vision fatigue[8].In the above indirect design methods，the spherical equation is usually used to obtain the vector height of the lens，but the spherical equation design has insufficient ability to optimize astigmatism of the lens.The toric equation is used to independently control the curvature in the x direction and y direction to get the vector height of the lens.The toric equation has higher degrees of freedom，which can flexibly adjust the surface astigmatism of PALs and further improve the optical performance of PALs.

2 Design steps of PALs

As shown in Figure 1，a rectangular coordinate system is established with the origin at the center of the lens，the x-axis is vertically downward and coincides with the meridian of the lens，the y-axis is horizontally to the right，and the z-axis points to the reader's eyes.The PAL is divided into four areas according to different functions.Among them，area 1 has stable and small power，which is the distance zone.It is responsible for looking at the distance，such as the scenery.Area 2 has large optical power，and in general，this area is smaller than area 1，which is the near zone.It is responsible for closerange work such as reading and writing.Area 3 is progressive channel.Its characteristic is that the power changes continuously in this area.This area is mainly used to focus on medium distance things.Areas 4 are the astigmatism zones，which affect visual imaging.Therefore，the area and the maximum value of astigmatism of this area should be reduced as much as possible during the design process[9].

2.1 Meridian design

The meridian is a line that bisects the lens vertically downward as shown in Figure 2.The positions of the reference points in the distance and near zone and their corresponding power are determined according to the personalized data of the PALs wearers.The meridian design is to describe the change in the power of the lens from the distance to near reference point.The meridian is usually represented by a high-order polynomial[10].

Figure 1 Areas of progressive addition lenses

Figure 2 Meridian and counter line of PALs

2.2 Contour line design

The contour line is a cluster of curves orthogonal to the meridian.The value of the power on the contour line is equal to that of the power at the point of intersection with the meridian.In this way，the power on the meridian is mapped to the entire lens surface.Counter line has many forms.This paper chooses a counter line as follows[11]

Here k=l2+h2；l is the distance from the reference point of the distance zone to the origin of the coordinates.h is adjustment factor，its value range is 10

2.3 Construct surface of PALs with the toric equation

After obtaining power of the entire lens surface.The curvature radius r of the lens surface can be calculated.The relationship between curvature radius r and power P is

where n is the refractive index of the lens.

The calculation formula for the coordinates of the center of curvature corresponding to the radius of curvature（ξ，η，ζ） is as follows[12]

where u is the position on the x-axis.

After knowing the radius of curvature r and the corresponding coordinates of the center of curvature （ξ，η，ζ），the toric equation is used to construct the surface shape of the PALs.The calculation formula of vector height is as follows

where c=1/r is the surface curvature；Ax，Ayare the curvature adjustment elements in the x and y directions，respectively.

Ax，Aycan be expressed as the following

where Cxand Cyis the preset astigmatism value.In the process of reducing astigmatism，the adjustment factors σxand σyare the width adjustment parameters，b is the position adjustment parameter.

3 The influence of the toric equation on power and astigmatism

To simplify the derivation process，we limit the area of discussion to a circular area with a radius of 20 mm on the PAL.The refractive index n of the lens is 1.56.The back surface of the lens is the design surface，and its power is -6 D（diopter）.According to Formula（2），the radius of curvature of the surface r=93.3 mm and the curvature c=10.7 m-1.At y=20 mm caliber （assuming Ax=0.1 m-1），(c+Ay)y=0.047，and the value of (c+Ax)(x-ξ) is also almost zero.Therefore，the following formula can be obtained

Formula （4） can be simplified as follows

In order to show the difference between the toric Equation and the spherical Equation，we ignore the astigmatism caused by the curvature c change in the neighborhood of a point on the surface.In other words，in the neighborhood of this point，we regard the curvature c and the center of curvature as constant in Formula （7）.

Calculating the power and astigmatism distribution of the lens.The maximum radius of curvature R1and the minimum radius of curvature R2of a certain point on the lens satisfy the binary linear Formula[13]

where

According to the formula for calculating the power and astigmatism of PALs，the power and astigmatism caused by the curvature adjustment elements Axand Ayin the toric equation （4） are given by

It can be seen from Formula (10) that compared with the spherical Equation，the toric equation increases the surface power by (n-1)[(Ax+Ay)/2] and produces the extra astigmatism by at least (n-1)|Ax-Ay|.What we should pay attention to that the toric equation with the curvature changes has inherent astigmatism.By adjusting Axand Ayto appropriate values，the extra astigmatism of the toric surface can cancel the inherent astigmatism.The tolerance of power is required to be less than 0.12 D.The preset astigmatism Cxand Cyis generally less than 0.12 D[14].

4 Design results and analysis

In order to explain the advantages of the toric design method more intuitively，two PALs with the same power are designed.The refractive index n of the two lenses is 1.56.The power in the distance zone is -2.50 D，ADD is 2.00 D，therefore，the power in the near zone is -0.50 D.The radius R of two lenses is 30.00 mm.

Formula （5） allows the value of Axat certain locations to a preset value，and smoothly decreases from the position of x=b to both sides.If the astigmatism in the distance zone of the initial lens is small enough，no adjustment needs to be set.While the astigmatism in the astigmatism zone is too large，the adjustment is needed.To reduce the astigmatism，we can set the parameter b，which makes the maximum of the absolute value of Axat the position of the maximum astigmatism.The parameter σxmakes Axclose to zero in the distance zone.If Cxis -0.10 D and σxis 50.00 mm，the curve of Axin the x direction is shown in the Figure 3.

Figure 3 Change of Ax in the x direction

Formula （6） can make the absolute value of Ayzero at y=0 mm and smoothly become larger on both sides.The rate of change of Ayis controlled by σy.For example，if the astigmatism in the progressive channel of the initial lens is small enough，it does not need to be adjusted.While the astigmatism in the astigmatism zone on both sides is large，we can set σyto make the absolute value of Ayin the progressive channel small enough，and larger on both sides.The astigmatism on both sides of the progressive channel can be adjusted without affecting the progressive channel in this way.If Cyis-0.10 D and σyis 15.00 mm，the curve of Ayin the y direction is shown in the Figure 4.

According to the design steps in Section 2，two kinds of lenses are designed，and the lens samples are prepared after a series of steps such as carving and polishing.The PAL obtained from the spherical equation is named PAL（a） and the PAL obtained from the toric equation is named PAL（b）.The power and astigmatism distribution on the entire surface are measured by using a Rotlex Free Form Verifier （FFV）.Figure 5 shows the power distribution of PAL（a） and PAL（b）.The center points of the small circles on the upper and lower parts of the lenses in Figure 5 are the reference points of the distance and near zones，respectively.It can be seen from Figure 5 that ADD of PAL（b） is 0.07 D less than that of PAL（a）.The ADD of the PALs designed by the two methods is basically the same.

Figure 4 Change of Ay in the y direction

Figure 5 Power distribution of PALs

The astigmatism distribution of PAL（a） and PAL（b） are shown in Figure 6.It can be seen from the figure that for PAL（a） and PAL（b），the range of astigmatism which is less than 0.50 D is basically the same in the far vision zone.However，the astigmatism area on both sides of the lenses can be clearly seen that the range of astigmatism of PAL（b） greater than 1.50 D is much smaller than that of PAL（a）.The width of the near zone （the width of astigmatism is less than 0.50 D） of PAL（b） is increased than that of PAL（a）.

Figure 6 Astigmatism distribution of PALs

In order to analyze the astigmatism on both sides of the lenses more accurately，select the power and astig-matism distribution on the horizontal line （x=11.50 mm） of the lens to plot，as shown in Figure 7.

It can be seen from the Figure 7 the width of the astigmatism greater than 1.5 D in the left area of PAL（b）is 7.80 mm，which is 1.7 mm less than that of PAL （b）.The width of the astigmatism greater than 1.25 D in the left area of PAL（b） is 14.80 mm，which is 2.60 mm less than that of PAL（b）.For the right side of the lenses，the width of PAL（b） with astigmatism greater than 1.5 D is 0，which is 8 mm less than PAL（a）.The width of PAL（b）with astigmatism greater than 1.25 D is 2.40 mm less than PAL（a）.

Figure 8 shows the distribution of power and astigmatism in the horizontal direction （x=23.00 mm） of two PALs.At the horizontal line（x=23.00 mm） of PAL（a），the width of astigmatism less than 0.50 D is 11.60 mm.At the same position，the width of astigmatism less than 0.50 D of PAL（b） is 15.20 mm.This means that the width of the near zone of PAL（b） is increased by 3.60 mm compared with PAL（a）.Table 1 shows the comparison of optical performance data of PAL（a） designed by spherical equation and PAL（b） designed by toric equation.

Table 1 Comparison of optical performance

Figure 7 The power and astigmatism distribution on the horizontal line （x=11.50 mm） of the lenses

Figure 8 The power and astigmatism distribution on the horizontal line （x=23.00 mm） of the lenses

5 Conclusion

In this paper，the vector height of PALs is calculated by using the toric equation instead of the spherical equation.The toric equation design can flexibly optimize the astigmatism of the lens.It makes up for the lack of the spherical equation design in the ability to control astigmatism.Processing and test results show that adopting the design of the toric equation can effectively reduce astigmatism on both sides of the PALs and expand the width of near zone visible area under the condition of keeping the distance zone basically unchanged.