Research on Off-axis Aberration Optimization Model for Bifocal Glass Lens

Presbyopia entails the gradual deterioration of the eye’s ability to focus, and bifocal lenses have become a common remedy to alleviate blurred near vision and counteract eye strain induced by prolonged usage. The refractive index and Abbe number of these lenses play a crucial role in precisely correcting focal points within the near vision range. Nevertheless, when light rays approach the lens at off-axis angles or with tilts, aberrations occur, often with uneven distribution across real glass materials. The amalgamation of bifocal lenses further compounds the complexity of these aberrations. To enhance the correction of aberrations in bifocal glass lenses, distortion and astigmatism aberration serve as input parameters to calculate the optical quality assessment of the lenses. This study introduces a novel approach to optimize the pairing of bifocal glass lenses through the Taguchi method and fuzzy inference. The experimental design involves the application of the LS9 orthogonal array, streamlining the number of experiments while upholding the reliability of experimental data. The objective is to refine the selection of glass lenses and successfully devise an optimized design model that can be effectively applied to the configuration of bifocal glass lenses.


Introduction
For single glass lenses, the refractive index and Abbe number determine the basic optical characteristics and magnitude of aberrations.However, for bifocal glass lenses, in addition to considering the optical characteristics of individual lenses, it is necessary to calculate the interaction between the two lenses and the aberrations caused by off-axis light rays in the near vision area.The distortion aberration may be magnified or reduced, and the astigmatism aberration corresponds to the horizontal and vertical focal lines, making it difficult to predict the aberrations after the combination of bifocal glass lenses.Hence, the pursuit of aiding optical designers in promptly and effectively identifying the best arrangement of aberrations in bifocal glass lenses has emerged as a compelling research area.Prior scholarly investigations have predominantly centered on refining the optical design concerning lens aberrations, while there exists a dearth of real-world instances illustrating the utilization of aberrations in bifocal

2.
Method and Experiment

Lens aberration
The ideal optical system is also called Gaussian optical system, paraxial optics or first-order optics.The image formed by it is an ideal image point, but in an actual optical system, to meet the criteria for illumination and the scope of observation, the light rays are not all paraxial trajectories.Therefore, there will be differences from the ideal optical system, resulting in aberrations.Mathematically, the aberration can be represented by sinθ expansion.When θ is a small angle, the series converges quickly, so the items after the second term can be omitted, namely sinθ is approximately equal to θ (In an optical system with a larger aperture, the second and subsequent items cannot be completely omitted.),if the first and second items of the expansion are taken, which is called the third-order optical theory, there will be 7 kinds of aberrations, 5 of which are monochromatic aberrations are called Seidel aberrations (single wavelength), and they are of the following five types: (1)Spherical aberration This is an inherent distortion associated with spherical lenses.As parallel light traverses the lens periphery, its focal point draws nearer to the lens, whereas for light traversing the lens center, the focal point recedes farther from the lens.The displacement between the point of focus along the optical axis and the perpendicular optical axis is termed spherical aberration.
(2)Coma aberration: When the light outside the optical axis enters the lens obliquely, it cannot be gathered into a single point on the imaging plane, and a comatic flare is formed toward the center of the imaging plane or in the opposite direction.
(3)Astigmatism: On the optical axis, the object point can become an image point, but the point in the off-axis region cannot become an image point, but becomes an ellipse or a line.This aberration is called astigmatic aberration.
(4)Curvature of field The imaging is not a plane but forms a concave image plane with the center of curvature on the left.Therefore, when the point of focus aligns with the center of the imaging plane, it is sharp, and conversely, the focal point becomes blurred when it is around the edge. (

5)Distortion
The ideal lens is that the image of the object and the lens must be the same shape, but in fact there is a phenomenon called distortion aberration when the image is enlarged or reduced.
A light ray emitted from an off-axis object 'P' intersects the spherical surface at 'O' and the refracted ray finally intersects the image plane at 'P'' .In this case the axis of symmetry is not [OCS] any more, it becomes [PBC].When an arbitrary point 'Q' on the surface is concerned, the aberration could be described similarly to Eq. 1 and as shown in Figure 1.
'( ) ( ) ) Herein, we assign a constant coefficient in front of each term and the suffix of each coefficient indicates the power of 'h1', r and cosθ.Eq. 4 comprises 5 primary aberrations and each one has certain attributes.as follows:

Taguchi method
The Taguchi method [7][8] is a statistical-based approach to quality optimization using experimental design.It has the advantage of controlling the interaction between control parameters and noise factors to identify the optimal combination of parameter levels.The steps of the Taguchi method are shown below.
(2)Determinant factor levels (3)Choose the appropriate orthographic (4)Design orthogonal table coefficients (5)Run product experiments (6)Analyzing experimental results.To employ the Taguchi approach and engage in experimental design, it becomes imperative to define the quality attributes concerning the problem.These attributes encompass those with a preferred lower value, those with a preferred higher value, and those aiming for a specific target value.Additionally, quality metrics like the signal-to-noise ratio (S/N ratio) are considered.In this investigation, our focus is on the quality attribute striving for a reduced value, which is a non-negative quantity situated closer to the lower specification limit (LSL).An example of such an attribute is the distortion and astigmatism aberration in bifocal glass lenses.The minimum value for these quality characteristics is zero.The formula for the signal-to-noise ratio (S/N ratio) for this quality attribute is represented by Eq. 5, where "y" symbolizes the distortion and astigmatism aberration in bifocal glass lenses.The supplementary sub-lens of the bifocal glass lens is positioned at the front of the lens, with the primary lens located at the rear.The curvature radii of the front and rear surfaces of the supplementary sub-lens are 500mm and -500mm, respectively.Meanwhile, the primary lens boasts a front surface curvature radius of 500mm and a rear surface curvature radius of 100mm.The thickness of the sub-lens measures 2mm, while the main lens has a thickness of 10mm.A comprehensive table detailing the dimensions and optical parameters of the bifocal glass lens can be found in Table 1, along with an optical simulation software design using CODE V11 (with examples for the sub-lens (KF1), main lens (BK1), Aspheric(0th), and Aspheric(0th)), as presented in Table 2. Pooling data from 9 different sets of bifocal lenses, we identify and individually compute the more prominent aberrations, such as the astigmatic aberration linked to the ambiguity circle attribute and the distortion aberration related to the magnification attribute.For experimental design, we employ the  [9][10][11], have seen subsequent developments, including the introduction of "fuzzy relationships" and "logical control," among others.The fuzzy inference system consists of a sequence of control rule sequences, denoted by "IF" and "THEN," which are expressed using various programming languages.The control action set, obtained following the inference, is then looped back into the controlled process.This control action set can be considered as a fuzzy language programming that encompasses multiple sets of conditions, forming a range of fuzzy linguistic terms such as maximum, large, medium, small, and minimum.The construction of a fuzzy inference system is typically divided into segments, including the fuzzy rule base, database, defuzzification [12], with its structure depicted in Figure 2.
In order to translate the input data into fuzzy information that can be interpreted by the fuzzy control system, the input values need to be transformed using a membership function that yields values between [0, 1].This study employs the "triangle" membership function, as described in Eq. 6.You can feed nine sets of standardized data into MATLAB's Fuzzy Toolbox as input parameters [13][14][15].The fuzzy rule base is computed nine times and subsequently subjected to the defuzzification process, yielding nine sets of MPCI outputs, which are presented in Table 7.As for the other factors like A2, A3, A4, and so forth, the MPCI values for each of the three levels pertaining to these factors (A1, A2, A3, A4) were sequentially computed, as demonstrated in Table 8.The Range column signifies the variance between the maximum MPCI values (Max.-Min.).The larger the discrepancy among MPCI values, the more significant the impact of the particular factor.Following the computation of the MPCI response table, significant factors affecting the quality attributes can be pinpointed through the application of MPCI ANOVA (analysis of variance), DOF (degrees of freedom), sum of squares, variance, and contribution percentages.The contribution percentages of the four factors (A1, A2, A3, and A4) are as follows: factor A1 (14.6100%), factor A2 (34.7628%), factor A3 (7.5098%), and factor A4 (43.1174%), as detailed in Table 9. Upon conducting statistical analysis, it becomes evident that the influence of factors A2 and A4 is more pronounced.Making adjustments to A2 and A4 can lead to a swift convergence toward the target quality characteristics.In contrast, the impact of factors A1 and A3 is comparatively marginal.When fine-tuning the configuration of bifocal glass lenses, primary focus should be placed on A2 and A4, with A1 and A3 serving as supplementary considerations.This approach can significantly enhance the optical quality of the composite, bifocal glass lens.The optimal combination, corresponding to the MPCI index derived from the multi-objective optimization approach, is A1 1 A2 3 A3 2 A4 1 .

Conclusion
This paper integrates theory and experimentation, employing the orthogonal array (LS9) to compute astigmatism and distortion aberrations in various groups of bifocal eyeglass lenses.This calculation is based on the Taguchi experiment and fuzzy rule principles, followed by the acquisition and standardization of signal-to-noise ratio quality.Subsequently, it leads to the realization of a fuzzy system.The Fuzzy Toolbox MATLAB is employed for constructing the fuzzy system, formulating fuzzy regulations, utilizing fuzzy databases, and implementing fuzzy controllers.The comprehensive MPCI (multi-function index) is then established, and statistical analysis (ANOVA) is conducted on the MPCI response values to determine the optimal bifocal eyeglass lens configuration in terms of factors and levels.
The application of LS9 orthogonal tables significantly diminishes the need for an extensive array of experimental designs, and it harnesses programming to incorporate novel applications that encompass fuzzy systems.The derived configuration for bifocal eyeglass lenses is tailored to optimize the composite model, with the research findings serving as a resource for swift bifocal eyeglass lens selection.

Figure 1 .
Figure 1.Imaging of an off-axis object point 'P' .The geometrical relationship among the points 'B', 'O' and 'Q' is assumed to lie in the same vertical plane.The detail configuration is illustrated in the Figure 1 on top.The Figure 1 shows a side view from the left side of spherical refracting surface.According to the configuration, Eq. 1 would be expressed.In a similar way, the aberration at point O referred to axis [PBC].The off-axis aberration could be expressed as the difference between a'(Q) and a'(O) Applying cosine law to triangle [OBC], 'ρ' in Eq. 2 could be expressed as function of 'r', 'b' and 'θ'.From similar triangles [BOC] and [OSP'], the distance OB (OB=b=kh1) is proportional to the height of image 'P''.Replacing 'b' in Eq. 3 by the relation will yield the general 3 rd order off-axis aberration function.

Fuzzy rules 1 -
Rule In cases where [Astigmatism is in a Small range] and [Distortion falls within the Small category], the outcome is [MPCI in the Small realm].2-Rule When [Astigmatism is of Small magnitude] and [Distortion falls into the Medium range], the result becomes [MPCI characterized as Small-Medium].3-Rule If [Astigmatism is Small in magnitude] and [Distortion exceeds the Large threshold], the consequent [MPCI shifts to the Medium scale].4-Rule In instances where [Astigmatism falls within the Medium spectrum] and [Distortion is within the Small category], the resulting [MPCI is classified as Small-Medium].5-Rule If [Astigmatism is moderately sized] and [Distortion is at the Medium level], the output is [MPCI categorized as Medium].6-Rule When [Astigmatism is Medium] and [Distortion exceeds the Large range], the resulting [MPCI steps into the Medium-Large range].7-Rule In scenarios where [Astigmatism is significantly Large] and [Distortion is in the Small category], the outcome is [MPCI assessed as Medium].8-Rule If [Astigmatism is notably Large] and [Distortion falls into the Medium range], the end result becomes [MPCI positioned in the Medium-Large spectrum].9-Rule When [Astigmatism reaches the Large scale] and [Distortion is within the Large range], the resulting [MPCI is classified as Large].Once you've established the rule database for the fuzzy system, you can harness MATLAB's integrated Fuzzy Toolbox to generate the program, including specifications for input and output variables, fuzzy rules, and databases.

Table 1 .
Bifocal glass lens parameter configuration.

Table 2 .
Bifocal glass lens optical design.

Table 6 .
Extracted from the fuzzy rule repository presented in Table6, a total of 9 consolidated rules are provided below: Rules 1 to 9: Fuzzy rules.
6) Figure 2. Structural Layout of Fuzzy System Inference Procedure