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Abstract
We propose an evaluation method to judge the fitting extent to ophthalmic lens for the individual wearer. An Eye-Lens-Object optical system is set according to wearer's visual performance and the characteristic of ophthalmic lens assembly. A visual reference surface is proposed to calculate the object distance. The RMS radius of the spot diagram and MTF average value from optical design software Zemax are regarded as the criterion of assessing the image quality on the retina. Three cases are simulated to verify that our method is effective. The wearers can experience a comfortable wearing feeling when the evaluation method is used during the design of ophthalmic lens. The validity of our method is demonstrated to instruct designing the progressive addition lens with the freeform surface.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The task of the eye's refractive parts is to create an image of the external world on the photoreceptor layer of the retina. The imaging quality of a real object is, however, affected by refractive errors, dispersion, diffraction effects, and scattering [1]. The ophthalmic lens is used to solve the problems caused by these errors.
There are several methods to evaluate the quality of the ophthalmic lenses. They are calculating power and astigmatism based on the vector heights of the surface [2–6], using automated focimeter [7], measuring the power of ophthalmic lenses by a deflectometric technique [8,9], and evaluating the properties of addition progressive lenses by the wavefront, etc [10–12]. The Lens-Eye-Object optical system has been set up in some evaluation methods to assess the image quality by the optical design software [13,14], but there are few measuring points. Furthermore, the calculation method of the object distance is not given. In real scenes, when the object distance changes the direction of the eye's axis also changes. The optical power of the eyes varies with the object distances and the direction of the eye's visual axis. This shows that the object distance is important on the evaluation of ophthalmic lens. And therefore, We propose a new Eye-Lens-Object optical system model based on the object distance and the habit of the wearer. The azimuth angles and the object coordinates corresponding to rays on different places of the ophthalmic lens are calculated from the offset and the tilt of the ophthalmic lens during the process of lens fitting. We can thus estimate the image quality of the ophthalmic lens in the design process, which is related to different diopter, face characteristic, sight habit, ophthalmic lens and ophthalmic lens’ frame of the individual. We use our novel method to assess the ophthalmic lens’ parameters before manufactured. Therefore, we can improve the comfort level of the wearer, promote the development efficiency, and reduce product costs. The method is particularly effective to help us to design the progressive addition lens with the freeform surface.
2. Evaluation method of the Eye-Lens-Object optical system
The degree of clarity of the object observed by the wearer depends on the refractive power adjustment ability of the eyes, the power of the ophthalmic lens and the distance of the observed object. The method we proposed combines various factors to evaluate the imaging performance of the object through the ophthalmic lens and the eye.
2.1 The model of the human eye
The human eye has limited focal power adjusting ability. We adopt the Liou-Brennan model of the human eye shown in Fig. 1(a). The field angle is zero degree. The parameters are obtained from [1,15].
Fig. 1. Schematic diagram of the model of the human eye: (a) Scheme of the relaxed Liou–Brennan Eye model. (b) Schematic representation of the eye model when observing distant objects and observing near objects.
The far point distance Sfar is defined as the distance between the principal surface p and far point Qfar of the naked-eye. The near point distance Snear is the distance between the principal surface p and near point Qnear of the naked-eye. The inverse distances are called far point refraction Afar=1/Sfar (Sfar<0) and near point refraction Anear=1/Snear (Snear<0). The difference between the far and near point refraction is referred to as the amplitude of accommodation ΔAmax= Afar- Anear [1]. In the human eye the accommodation of refraction power is realized by the contraction and relaxation of the ciliary muscle and the zonular fibers respectively. It is a complex and ingenious mechanism of accommodation. Only when the axial length and the refraction power of eye match each other, can a clear image be obtained on the retina. In the visual optics, the axial length and the refraction power are two aspects of optical imaging of eyes. In our model the variation of axial length is employed to reflect ocular accommodation process, since a clear image can be obtained when the refractive power matches the axial length [16]. The distance lr from the posterior surface of the crystalline lens to retina is defined as axial length of the eye. Here the lr_min and lr_max present the amplitude of accommodation, shown in Fig. 1(b).
When the human eye is turned to the observed object, the eyeball rotates around the center of rotation O, and the optical axis in the eye model rotates with the same angle. Generally, the head is cooperatively deflected with one's sight. The angle of deflection of sight is the summation of the head's and eye's rotation angles. The relationship between the head and eye's rotation angle is achieved as the Eq. (1) [17–25]
2.2 The model of Eye-Lens-Object optical system
The model of Eye-Lens-Object optical system is set to evaluate the image quality on the retina when one wearer observes the object through ophthalmic lens. The position of the optical axis of the eye changes as the eye rotates, as shown in Fig. 2.
The coordinate system O-xyz for Eye-Lens-Object is adopted. The origin of coordinate is the rotating center of the eye. The axis z is through the assembly center OL0, and it consists with direct-vision axis. The axis y is perpendicular to the plane O-xz as shown in Fig. 3. The coordinate system O-xyz shifts and rotates while the head rotates around the atlanto-occipital joint, which is the head's rotating center [23]. Each point on the front and back surfaces of the lens is represented using the coordinate of O-xyz. In our simulation, the angle between the left and right lenses, the offset of the lens assembly center, the vertical camber angle of the wearing, and the distance between the lens and the rotation center of the eye, are taken into account [2]. The coordinate (xb,yb,zb) of an arbitrary point Pb on the ophthalmic lens is defined in the coordinate system O-xyz. When the wearer observes the object through the point Pb, the optical axis of the eye also passes the point Pb. αe and βe could be determined by the Eq. (2).
Here αe and βe is the vertical and horizontal deflect angles of the eyes’ axis, respectively.
2.3 The location of the object
2.3.1 The visual reference surface
A visual reference surface must be built based on the wearer's vision habit. The reference coordinate system O'-x'y'z’ is static relative to the ground. When the head of the wearer does not rotate, the O-xyz coordinate system coincides with O'-x'y'z’. The visual reference surface is perpendicular to the y'O'z’ plane and extends infinitely along the x’ axis. All object points P are on the visual reference surface. The key gaze points at the direct-vision direction, including the far distance point, middle distance point and near distance point of the wearer's view are adopted to represent the vision habit. According to the key gaze points the curve where the visual reference surface intersects the y'O'z’ plane is fitted by Piecewise cubic Bezier curves [26,27]. The schematic diagram of the visual reference surface is shown in Fig. 4. This method of fitting maintains the continuity of the first derivative between the various piecewise curves. The parameter equation of the visual reference surface is the same as the curve's formula as following.
2.3.2 The calculation of the object coordination
The intersection point of the sight and front surface on the lens is Pg, and Pb is on the back surface. The position vector of Pg is rg={xg, yg, zg } and the direction cosine vector of the sight eg={egx, egy, egz }, respectively. The vertical and horizontal deflection angles are αg and βg. The shift and rotation of the O-xyz coordinate system arise owing to the head rotating. The position vector of Pg and direction cosine vector of sight in O-xyz are changed to in O'-x'y'z’ by coordinate transformation in accordance with the position of head's the rotating center [18,28] . The position vector of Pg in the O'-x'y'z’ is r'g={ x'g, y'g, z'g }.
The vertical and horizontal deflection angles are α'g and β'g.
2.4 The image evaluation
A visual reference surface for an individual is simulated based on section 2.3.1. For obtaining the limit of the distance lr for the individual the naked-eye model is built in the optical design software Zemax at first. The parameters of the eye model are presented in Table 1. The distance lr (lr >0) from the posterior surface of the crystalline lens to retina is set as a variable and the RMS radius of the spot diagram is set as objective function. We can get lr_min and lr_max by optimizing while object distances are set as Snear and Sfar. Next, an Eye-Lens-Object optical system model is set in the optical design software Zemax by inserting the lens in front of the naked-eye. When the eye is looking forward, the optical axis of the eye passes through the assembly point OL0 of the lens and the distance from OL0 to the center of rotation of the eye is q. The position of OL0, the value of q, and the vertical and horizontal inclination angles of the lens are suitable for individual characteristics matching the spectacle frame.
In the established Eye-Lens-Object optical system model the coordinates of the visual ray through one location at the ophthalmic lens are achieved by ray tracing. The position vector of object point P is obtained by means of the method described in section 2.3.2. Given an object distance, the optimal image on the retina is searched by optical design software. During the search process, the distance lr is set as a variable with the constraint condition lr_min ≤ lr ≤ lr_max and the RMS radius of the spot diagram is set as objective function. The MTF average value can be calculated simultaneously. A series of RMS radius are obtained by ray tracing all points corresponding to the whole ophthalmic lens during the process. The RMS radius of the spot diagram contour and the average MTF contour are thus obtained. These contours reflect the image quality on the retina of a lens wearer.
The RMS radius of the spot diagram and MTF are used to evaluate the image quality of the human eyes, which is verified by the experiments for the young eyes and older eyes [13,14]. The MTFs of the tested young eyes and older eyes embody their comfortable feeling [14].
3. Results and discussion
Three cases are simulated by applying the proposed method to demonstrate how to evaluate the appropriateness of ophthalmic lens for the individual wearer.
3.1 Myopic eye wearing the single focal lens
The diameter of the ophthalmic lens is set as 48 mm. The radii of the front and back spherical surface of the ophthalmic lens are 292.5 mm and 146.25 mm, respectively. The central thickness is 1 mm. The angle between the left and right lenses is 10°, and the vertical camber angle of the wearing is 5°. The pupil height is 3 mm. The distance q from the back surface of the lens to the center of rotation of the eye is 25 mm. The focal power is −2.0 D. The far point distance and near point distance of the eye are −0.5 m and −0.2 m, respectively. The amplitude of accommodation is 3.0 D. kα and kβ are 0.20 based on “mixed” type classified participants in the literature, respectively [25]. The horizontal (vertical) distance from the eyes’ rotating center to atlanto-occipital joint is approximately 80 mm (40 mm) [23].
The following discussions are based on the O'-x'y'z’ coordinate system. When the wearer reads or writes, the paper's center is defined as P1. The centers of the keyboard and screen of the computer are defined as P2 and P3, respectively. The observed point clinging onto one's body is defined as P0, which has the same height as the paper. The location of 5 m far away from the wearer is defined as P4.
Table 1. The coordinates and object distance of the personalized vision points of the wearer
All personalized data are listed in Table 1. The visual reference surface is simulated based on the locations of the key points of the wearer. The intersect curve between the visual reference surface and the x'O'z’ plane is shown in Fig. 5. The fitting coefficients of the equation are listed in Table 2.
Fig. 5. The critical gaze point and the curve of intersection of the visual reference surface with the x'O'z’ plane for the ophthalmic glasses wearer. (a)Schematic diagram of sight passing through visual key points, (b) the intersect curve between the visual reference surface and x'O'z’ plane.
Table 2. The intersect curve coefficients between the visual reference surface and x'O'z’ plane
The lr_min and lr_max values are found to be 17.007 mm and 18.354 mm by optimizing through the Zemax. The coordinates of rays through the lens are achieved by ray tracing. The RMS radius of the spot diagram contours of the Eye-Lens-Object optical system and the average MTF contours at 10 cycles/mm are shown in Fig. 6 and Fig. 7.
In Fig. 6, the solid line shows the RMS radius of the spot diagram being 4 µm. It means that the RMS radius on the retina does not exceed 4 µm when the ray passes through a circle with a radius of about 17 mm on the ophthalmic lens. It is smaller than the visual resolution. Figure 7 shows the MTF contours at 10 lp/mm. It is larger than 0.95 (0.925) over a radius of 10 mm (17 mm). The wearer with a −2.0 D spherical lens feels comfortable in observing both far and near objects. It is because that the amplitude of accommodation of the wearer's eye reaches to 3.0 D, the near-point diopter is −3 D after wearing the lens with −2.0 D, and the effective near-point distance is 0.3 m. As seen from Fig. 6 and Fig. 7, the profile is almost circular although asymmetrical in the x and y directions. The asymmetry is more obvious at the edge of the lens. It might result from the top of the lens tilting outwards and the noticeable angle between the left and right lens. From Fig. 6 to Fig. 7, the image quality reduces when the ray passes through the peripheral portion of the lens, which might originate from the larger aberration because of lens imaging at a wide field angle when the wearer does not look straight ahead. Fortunately, the edge of the lens is not in need for use when one looks forward in the case of near reading and writing. Therefore, this kind of decline of image quality has no influence on the reading and writing.
3.2 Myopic eye with presbyopia wearing single focal lens
Consider a myopic wearer with the same refractive power who is a presbyopia with 1.3 D amplitude of accommodation. The far point distance and near point distance of the eye are −0.5 m and −0.3 m, respectively. The minimum distance lr_min and maximum distance lr_max are found to be 17.007 mm and 17.757 mm by optimizing using the Zemax. The RMS radius of the spot diagram contours of the Eye-Lens-Object system and the average MTF contours at 10 cycles/mm are obtained by optimizing the radius of the spot diagram. The counterpart contours are shown in Fig. 8 and Fig. 9.
The results show that in the upper and middle portions of the lens, the RMS radius of the spot diagram is less than 4 µm, and the MTF is larger than 0.925 at 10 lp / mm. On these areas, the image on the retina is clear. When the sight passes through the part of 9 mm below the lens’ center, the RMS radius of the spot diagram becomes larger than 4 µm and average MTF is smaller than 0.90 at 10 lp/mm. When the sight passes through the 17 mm below the center of the lens, the RMS radius radius is 16 µm and average MTF at 10 lp/mm is reduced to 0.75. This ophthalmic lens is suitable for observing objects at distant and intermediate distances. Let's look into whether the ophthalmic lens is suitable for myopic wearer with presbyopia. After wearing a single focal lens with −2.0 D, the near-point diopter of −3.3 D turns into −1.3 D and the effective near-point distance is 0.77 m. It can only guarantee one see the medium-distance objects, but not near objects. Since the adjustment ability of the patient wearer is limited, the ophthalmic lens does not meet the reading and writing needs of −2.98 D.
3.3 Myopic eye with presbyopia wearing progressive addition lens
The above difficulty could be solved by using a progressive addition lenses (PAL) with a distance zone of −2.0 D and an addition focal power of 2.0 D. The focal power and the astigmatism calculated by differential geometry method are presented in Fig. 10 and Fig. 11. The PAL's vectors height data are taken for the Zernike standard coefficients, which are key inputs for the Zemax software. The contours of RMS spot diagram and MTF at 10 lp/mm are therefore obtained as shown in Fig. 12 and Fig. 13. The RMS radius of the spot diagram is approximately 5 µm and the MTF is larger than 0.9 in all of the distance, progressive and near zones. It indicates that the wearer could have clear vision in observing distant objects or reading. It is because that after wearing a progressive addition lens with addition focal power of 2.0 D, the near-point diopter still keeps −3.3 D due to the focal power of 0 D at the reading zone of the progressive addition lens, the effective near-point distance is 0.3 m. Compared the contours of Fig. 12 and Fig. 13 with the astigmatism contours of Fig. 11, there are similarities and there are also differences. The distance area achieved by our method is smaller in Fig. 12 and Fig. 13 than that calculated by differential geometry method in Fig. 11. The astigmatism areas are moved up in Fig. 13. The width of MTF with 0.95 at 10 lp/mm in Fig. 13 is wider than that of the astigmatism with 0.5 in Fig. 11. It is shown that our ophthalmic lens evaluation could provide useful information to help improve the design quality of the PAL.
4. Conclusion
In this paper, an ophthalmic lens evaluation method based on the Eye-Lens-Object optical system model is proposed. In this method, we consider a lot of factors such as the distance of the observed object and the observation habit of the ophthalmic lens wearer. We set a visual reference surface based on key points of observing to solve the difficulty of determining object distance. We set an Eye-Lens-Object optical system model and obtain the RMS radius of the spot diagram and MTF average value through the optical design software Zemax. Three cases are simulated for three types of eyes, respectively. The RMS radius of the spot diagram and MTF average value can be considered as the criterion of assessing the image quality on the retina. The key advantage of our method lies in the quantitative description, which is objective and be able to reflect the practical feeling of a wearer. The method could further give quite meaningful guide to design the PAL with freeform surface.
Funding
National Natural Science Foundation of China (61875145, 11804243); Jiangsu Province Key Discipline of China's 13th five-year plan (20168765); Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions (17KJA140001); Six Talent Peaks Project in Jiangsu Province (DZXX-026).
Acknowledgments
The authors are also grateful to Professor Lin Qian of Soochow University for valuable advice.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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