## Abstract

In this study, the whole-surface power distributions in prescription sunglasses at different face form angles were simulated using the coordinate transformation method. The lens was tilted with respect to the optical center by directly transforming its Cartesian coordinate system at the desired face form angle. The results show that prescription sunglasses display different whole-surface power maps compared to plano sunglasses. Moreover, when the tilt angle of the lens is high, a greater power error is observed in the temporal area of prescription sunglass lenses, which makes the power distribution more asymmetric.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Sunglasses and sports glasses have been increasingly used and favored by people who enjoy outdoor activities owing to their appearance and optical protection [1–3]. Usually, sunglasses have a plano (zero power) design. Therefore, people with refractive errors, such as myopia, either wear sunglasses with separately corrected lenses or prescription spectacles covered with a plano or polarized lens. It is difficult for people with such visual requirements to wear sunglasses outdoors. Therefore, designing a pair of prescription sunglasses with corrected lenses, thereby utilizing the advantages of refraction correction and protection from UV radiation, can alleviate this problem.

Most sunglasses or sports glasses have a face form design, whose tilt angle is greater than 30° to produce better aesthetics [4]. If the power of a lens is not zero, the power at the optical center will be different from that specified in standard refraction. This is called oblique central refraction [5–7]. Although the power at the optical center in tilted lenses can be predicted, the off-axis power distribution remains unknown. If the whole-surface effective power becomes clinically significant and strongly affects the vision quality of the wearer [8–10], it may cause discomfort to the wearer because the eye freely rotates behind the lens without fixating at its optical center.

Thus far, the measurement and simulation of the whole-surface power distribution in tilted lenses have not been studied in detail. A previous study reported the measurement and simulation of power error distributions based on the rotation angles of eyes [11]. The measured results showed an uneven power distribution across the lens. The theoretically simulated results obtained using the Coddington-based forward raytracing method showed similar findings. However, it only demonstrated the power profiles in the vertical and horizontal directions instead of the entire surface. Therefore, it is interesting to analyze the whole-surface power profiles of prescription lenses with different face form tilts through simulations.

## 2. Methods

In this study, the whole-surface effective power distributions in face form tilted lenses were simulated using a coordinate transformation method [12]. The effective power profiles, including the spherical equivalent and astigmatism, were compared with the data of non-tilted lenses. Furthermore, the effects of the tilted angle and lens power on the power distribution were determined.

#### 2.1 Specification of the sunglass lenses

Sunglass lenses with three different powers (−2.00 D, plano, and +2.00 D) were simulated in this study. All lenses were made of polycarbonate materials (n = 1.586) with a base curve of +6.00 D and diameter of 60 mm.

#### 2.2 Coordinate transformation method

The coordinates of sunglass lenses can be changed by transforming the Cartesian coordinate system of the lens with respect to the optical center at the desired angle. A general coordinate transformation matrix can be expressed using Eqs. (1a) and (1b). The definitions of the parameters in Eq. (1) are listed in Table 1.

The coordinate transformation process in the three-dimensional Cartesian coordinate system is shown in Fig. 1. The lens is first rotated about the X-axis by θ_{1} with the matrix M_{1} (Fig.1a), followed by θ_{2} about the Y-axis with the matrix M_{2} (Fig.1b), then by θ_{3} about the Z-axis with the matrix M_{3} (Fig.1c).

The final coordinates of the tilted lens surface (x’, y’, z’) was obtained based on Eq. (1). In this study, the face form angle is denoted by θ_{2} rotated about the Y-axis (Fig. 1(b)), whose values are 0°, 10°, and 20°. The other two angles of θ_{1} and θ_{3} were set to zero. A schematic figure for the incident light, sunglass lens, and position of power distribution at D (ϕ=4.5mm) for non-tilted and tilted by θ_{2} about the Y-axis is shown in Fig. 2.

#### 2.3 Power profile

After obtaining the new coordinates of the tilted lens surfaces, a set of Zernike coefficients of the wavefront equations that describe the lens surface height (z) within a pupil region of diameter 4.5 mm can be determined using a polynomial matrix [13,14]. The first row of the matrix consists of the 45 Zernike terms in the Cartesian coordinates (up through the eighth order). The next two rows of the matrix are the first derivatives with respect to x and y. The next three rows of the matrix are the second derivatives, and so on, through to the eighth derivatives. Each element of the matrix is only a function of x and y. Therefore, the set of Zernike coefficients can be obtained by inputting the values of the new coordinates (x’, y’, z’) into the polynomial matrix.

The spherical equivalent (M) and astigmatism (J) of the front and back surfaces were then calculated from the corresponding Zernike coefficients of the wavefront equations [15]. First, the dioptric powers in the horizontal, vertical, and diagonal are determined by Eqs. (2)–4, respectively. The spherical equivalent (M), orthogonal astigmatism (J_{0}), and oblique astigmatism (J_{45}) are then derived from the second-order Zernike terms as shown in Eqs. (5)–8.

^{th}order Zernike coefficient of the meridional frequency m in units of micrometers, n’ is the index of refraction of the material, and r is the pupil radius (2.25 mm). $Z_n^m$ is the Zernike term using the double indexing system. Assuming thin lens conditions, the lens power was determined by the sum of two surface powers. The aforementioned coordinate transformation, Zernike coefficients, power calculation, and contour plots were calculated using MATLAB [16]. In this study, the changes in the power profile of the sunglass lenses of the left eye are presented.

## 3. Results and discussions

The simulated whole-surface power profiles of the spherical equivalent (M) and astigmatism (J) at tilt angles of 0°, 10°, and 20° for the −2.00 D, plano, and +2.00 D sunglass lenses are shown in Figs. 3 and 4, respectively. It can be observed that regardless of the face form angle, the change in power distribution in the sunglass lens with zero power was small. However, the power distributions in the prescription sunglass lenses (−2.00 D and +2.00 D) were strongly affected by the face form angle. Sunglass lenses with either −2.00 D or +2.00 D power exhibited similar power variations in the optical center area, which increased with the face form angle. For example, at the face form angle of 20°, the spherical equivalent (M) and astigmatism (J) at the optical center of the tilted −2.00 D sunglass lenses increased to −2.27 D and +0.14 D, respectively. In contrast, these powers increased to +2.28 D and +0.15 D for the +2.00 D lenses, respectively. The power changes in the optical center of sunglass lenses are consistent with the theoretical calculations from the oblique central refraction, which was 0.21 D and 0.14 D for the spherical equivalent (M) and astigmatism (J), respectively, at a face form angle of 20° [5–7].

Moreover, the power distributions become more asymmetric at higher face form angles in comparison to the zero-tilt angle; this demonstrates the maximum power differences observed in the temporal areas (right side of the profiles) of the tilted prescription sunglasses. Figure 5 shows the effective power variation profiles of the spherical equivalent (M) and astigmatism (J) along the central horizontal line of the −2.00 D, plano, and +2.00 D sunglass lenses with face form angles of 0°, 10°, and 20°. It can be observed that the plano lens exhibits symmetric profiles; however, all prescription sunglass lenses exhibited more asymmetric profiles as the face form angle increased. This uneven increase in the spherical equivalent (M) and astigmatism (J) is consistent with a previous study, which revealed similar profiles for +6.00 D lenses at a pantoscopic tilt of 10° [11].

As shown in Fig. 5, the −2.00 D lens produces more effective powers up to approximately −9.75 D for the spherical equivalent (M) and +5.50 D for the astigmatism (J) in comparison to +4.25 D and +1.25 D, respectively, in the +2.00 D lens at the downward-tilted end (temporal position) with a face form angle of 20°. On the other end of the −2.00 D lens, the M power is approximately −2.36 D, exhibiting a slight change in comparison to −2.00 D, and the J power is approximately +0.34 D. A similar situation was observed for the +2.00 D lens, which exhibits a power of +1.48 D and +0.49 D for M and J, respectively, at the upward-tilted end.

The effect of the tilt angle on the effective power change in the optical center for all lenses and across all lenses for the spherical equivalent (M) and astigmatism (J) for the −2.00 D, plano, and +2.00 D sunglass lenses is shown in Fig. 6. It can be observed that the plano lens does not affect the power; however, all prescription sunglass lenses demonstrated a strong tilt angle effect on the change in power as the faceform angle increased, especially for the maximum difference in M and J across the lens.

Figure 6(a) shows that both the −2.00 D and +2.00 D lenses have similar effective power changes in the optical center area. However, the maximum difference in M and J for the −2.00 D lens is different from that of the +2.00 D lens, as shown in Fig. 6(b). The maximum differences in M and J at the face form angle of 20° were −7.81 D and +5.48 D for the −2.00 D lens and +2.78 D and +1.22 D for the +2.00 D lens. Because the front surface power of the −2.00 D and +2.00 D lenses is the same (+6.00 D), the power difference between M and J for the −2.00 D and +2.00 D lenses can be explained by the different curvature at the back surface of the lens, which was −8.00 D and −4.00 D for the −2.00 D and +2.00 D lenses, respectively. The results of simulations indicate that for higher lens surface powers, the power induced by the tilt angle will be higher.

This study demonstrated that the whole-surface power profiles of prescription sunglass lenses are significantly affected by the face form angle and lens power. However, the effects of lens tilt and power on the whole-surface power changes should be further substantiated by optical measurements using a commercially available focimeter [11] or Hartmann wavefront aberrometer [16]. Moreover, although the power changes in the optical center of sunglass lenses were consistent with the theoretical calculations from the oblique central refraction [5–7], the effective power changes across the lens for the spherical equivalent (M) and astigmatism (J) have not yet been confirmed. This can be verified via theoretical analysis, such as the Zemax software [17], which is a ray tracing method to determine the whole-surface power distributions in tilted prescription sunglass lenses and compare the differences in spherical equivalent (M) and astigmatism (J) using the coordinate transformation method. It is believed that the power distribution profiles obtained from the ray-tracing method will be similar to those obtained from the general coordinate transformation method in this study. Additionally, it is expected that a higher lens power (>2.00 D) and lens tilt (>20°) can also be simulated by the coordinate transformation method to determine the power change profiles of prescription sunglasses. Therefore, it is believed that whole-surface power simulations can help in understanding the visual performance of prescription sunglasses.

## 4. Conclusions

The whole-surface power distributions of prescription sunglasses can be demonstrated by directly transforming the Cartesian coordinate system of the lens at the desired face form angles. According to the results of this study, when the tilt angle of a lens is high, a greater power error is observed in the temporal area of the lens, which makes the power distribution more asymmetric. This may influence the visual performance of prescription sunglasses. Moreover, the sign of the lens power affects the tendency of the power changes; a negative power induces a more significant change in power than a positive power. This may be related to their difference in lens surface design. It is believed that the simulated results can be further substantiated by optical measurements.

## Acknowledgments

Portions of this work were presented at the 12th International Conference on Optics-Photonics Design and Fabrication (ODF ‘20) in 2021, 03S1-15.

## Disclosures

The author declares no conflicts of interest.

## Data availability

The data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the author upon reasonable request.

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