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ISSN : 1229-6457(Print)
ISSN : 2466-040X(Online)

The Korean Journal of Vision Science Vol.27 No.1 pp.29-40
DOI : https://doi.org/10.17337/JMBI.2025.27.1.29

Analysis of the Optical Effects in the Eccentric Lenses Applied with the Myopic Gullstrand Schematic Model Eye: A Ray Tracing Approach without Paraxial Approximation

Bon Yeop Koo¹⁾

, Myoung Hee Lee²⁾

, Young Chul Kim³⁾

¹⁾Dept. of Optometry, Shinsung University, Professor, Dangjin
²⁾Dept. of Optometry, Baekseok Culture University, Professor, Cheonan
³⁾Dept. of Optometry, Eulji University, Professor, Seongnam

^* Address reprint requests to Young-Chul Kim (https://orcid.org/0000-0001-5103-900X) Dept. of Optometry, Eulji University, Seongnam TEL: +82-31-740-7201, E-mail: yckim@eulji.ac.kr

Received March 1, 2025 Review March 29, 2025 Accepted March 29, 2025

Abstract

Purpose : This study aimed to analyze the optical effects with eccentric lenses applied to the Gullstrand schematic model eye, under myopic conditions.

Methods : The Gullstrand schematic model eye was precisely designed using a 3D simulation program. A ray tracing technique, providing accurate analysis without paraxial approximations, was employed to precisely investigate changes in the focal length and refractive power induced by eccentric lenses.

Results : The effects of eccentric lenses was analyzed on focal distance, refractive variations, and optical path differences with a particular focus on parallel rays incident at a specific height from the central axis. As the incident ray height of the rays became extremely large, the effect of eccentricity decreased, and the foci of the emitted rays were located at points close to each other. The total refractive power, which includes the refractive power of the lens and eye, and the prism refractive power effect due to eccentricity, changed greatly as the incident height of the rays decreased.

Conclusion : The findings of this study provide valuable insights into the design and prescription of eccentric lenses for the correction of ocular alignment anomalies, such as strabismus and heterophoria. Additionally, it is expected to ensure optimal visual performance for users by minimizing non-linear optical effects caused by eccentric lenses.

Key Words : Eccentric lenses , Gullstrand schematic sodel sye , Optical simulation , Prentice’s rule , Ray tracing technique

굴스트란드 모형안에서 발생하는 편심 렌즈에 의한 광학적 효과 분석: 광선 추적 기반 분석

구본엽¹⁾, 이명희²⁾, 김영철³⁾

¹⁾신성대학교 안경광학과, 교수, 당진
²⁾백석문화대학교 안경광학과, 교수, 천안
³⁾을지대학교 안경광학과, 교수, 성남

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Ⅰ. Introduction

In the field of visual optics, designing an accurate schematic model of the human eye is essential for study on refractive errors and correction methods. The human eye has a complex optical system, and optical performance is highly variable under various refractive conditions, so sophisticated design is required for accurate analysis. The main purpose of using a schematic model eye in optical study is that the precise realization of refractive errors in real human eyes, which are difficult to recruit subjects for, can be performed in a relatively simple way and that their quantitative analysis is possible.^1-3)

In particular, myopia is one of the refractive errors that frequently occurs worldwide, and is known to provide visual comfort and improved quality of life through early diagnosis and appropriate correction.⁴⁾ Myopia is usually corrected with various methods, such as spectacle lenses, contact lenses, or laser surgery.⁵⁾ However, from a clinical perspective, the environment in which spectacle lenses are used can cause eccentricity in the eye's line of sight and central axis and can affect the passing ray path and optical properties.⁶⁾

Meanwhile, there are also cases where eccentric lenses are intentionally prescribed to correct ocular aliment abnormalities such as heterotropia and heterophoria.⁷⁾ In this case, effective correction of ocular alignment anomalies is possible, but quantitative prescriptions are essential because negative optical effects may occur due to eccentricity.⁸⁾ Statistical results on actual eccentric lens prescriptions help to understand and improve the negative effects such as induced heterophoria, chromatic aberration, and spatial distortion.⁹⁾

Therefore, this study aimed to precisely analyze the effect of lens eccentricity using a ray-tracing technique without paraxial approximation, targeting an optical system using a lens in the Gullstrand schematic model eye of myopia condition. The main purpose was to accurately calculate and compare changes in focal length, refractive power, and ray path that occur due to lens eccentricity. In particular, this study set up a situation where parallel rays are incident from the central axis at a specific height and attempted to closely compare optical properties under various eccentricity conditions. Furthermore, this study was to evaluate the optical effects of the lenses that were intentionally eccentric for the correction of ocular alignment anomalies and to provide basic data to understand and improve the negative effects that may occur when prescribing them.

Through these attempts, it is expected that nonlinear effects that may occur during the lens design and manufacturing process will be analyzed and that it will play an important role in improving visual performance in the actual user environment. In addition, we demonstrated the usefulness of the ray tracing technique as a precise optical analysis tool and intended to use it in various vision correction study in the future.

Ⅱ. Methods

1. The Gullstrand model eye and lens design

The design of the Gullstrand model eye used a 3D optical simulation program, ANSYS SPEOS Ver 2021 (ANSYS Inc., USA), and applied the data values of the ocular medium presented in previous study.¹⁰⁾ Here, the radius of curvature of the front and back surfaces of the cornea was 7.70 mm and 6.80 mm, and the refractive index was 1.376. The radius of curvature of the anterior and posterior surfaces of the lens cortex was 10.00 mm, -6.00 mm, and the refractive index was 1.386, and the radius of curvature of the anterior and posterior surfaces of the lens nucleoplasm was 7.90 mm, -5.70 mm, and the refractive index was 1.406. The refractive index of the aqueous humor and vitreous body was set to 1.336.

The design of the lens partially referred to the results presented in previous studies, and considering the actual state of wearing glasses, the front and back curvature radius was set to 337.65 mm and 100.00 mm, the center thickness was 2.00 mm, and the refractive index was set to 1.520.¹¹⁾ Accordingly, the front and back refractive powers of the lens were +0.99 D and -5.20 D, respectively, and the overall refractive power was -4.20 D.

The refractive error value set in this study was -4.00 D, and the lens refractive power D_S′ at this time represents the value calculated as equation 1.

D_{S} = \frac{{D_{C}}^{'}}{1 - {d_{C}}^{'}}

(1)

Here, D_C ′ means -4.00 D in an uncorrected refractive error condition (unaided visual acuity), and d means the vertex distance as -12.00 mm

2. Ray tracing setting

1) Ray tracing by refraction

In general, when analyzing optical systems such as lenses, paraxial approximation is applied to reduce computational complexity. However, if information about the refractive index, radius of curvature, and arrangement of the optical system is known, the properties of the optical system can be analyzed without approximation through the law of refraction.

Fig. 1 shows the process by which parallel rays enter the lens and are refracted. When a ray reaches the lens surface and is incident on a point (z₁, x₁ ), the angle of incidence can be defined, and the angle of refraction can be calculated according to Snell's law. The refracted ray passes through the lens, enters a point (z₂, x₂) on the back, and is refracted again before passing through the lens and exiting.

In this study, the change in the ray path was accurately calculated without approximation at the two surface points presented in the above process, and the two points (z₁, x₁ ) and (z₂, x₂ ) were calculated by calculating the contact point from the equation of the straight line for the incident ray and the equation of the sphere representing the lens surface.

2) Ray tracing for emmetropic and ammetropic eyes

Fig. 2(a) and 2(b) show ray tracing results for the Gullstrand model eye in emmetropia and ametropia conditions. Each point represents the position where the ray is refracted, and Fig. 2(b) is the ray tracing result of a myopic eye corrected with a -4.00 D lens.

3. Eccentric effects setting

1) Prism effect at the points of decentration

Prentice’s formula for calculating the prism effect at a decentered point in a single lens is, as is well known, proportional to the incident height of the ray, it is defined as equation 2.

P = h_{0} D^{'}

(2)

Where h₀ is the height of the ray converted to centimeters, and D ′ is the refractive power of the lens. Fig. 3 shows the prism power calculated according to Prentice's formula and the prism power calculated by ray tracing for a single lens at –4.00 D. For relative comparison, the prism refractive power is presented as an absolute value. At this time, it was confirmed that there was almost no error in the two results.

2) Eccentric effect of the lens

Fig. 4 shows the ray path in a myopia corrected with a single lens. When the optical axis of the lens and the visual axis of the eye are the same, the path of the ray is indicated by a dotted line. At this time, when the optical axis of the lens is eccentric at a certain height from the visual axis of the eye, the path of the ray changes like a red solid line and it is imaged at a point off the retina due to the influence of the refractive power of the prism according to the eccentricity. In this study, for accurate analysis, the eccentricity of the lens was set to, and the eccentricity in the opposite direction to the incident height was set to a (+) value.

Ⅲ. Results

1. Changes in focal length and refractive power according to ray tracing

Fig. 5 shows the focal length (a) and refractive power (b) of emmetropia and ammetropia calculated using ray tracing techniques.

Since the focal length is the reciprocal of the refractive power, it was confirmed that it gradually decreases non-linearly as the incident height of the ray increases. It was found that there was a slight change without a huge difference, and as the incident height of the ray increased, the difference value gradually increased according to the prism refractive power of the lens.

Fig. 6 shows the change in refractive power between emmetropia and ammetropia calculated using ray tracing techniques. As the incident height of ray increases, the refractive power can be seen to gradually increase due to the prism effect of the lens. It was confirmed that the change in refractive power according to the height of ray incident is not linear but increases rapidly as the height increases.

2. Eccentricity effect due to ray tracing

Fig. 7 is the result of analyzing the light path according to the light incident height.

7(a) refers to the case where the incident ray enters along the optical axis (h₀ = 0), and in this case, the ray diverges rather than converges. This is because refractive power in the (-) direction occurred due to lens eccentricity in the fully corrected state. Therefore, in Fig. 7(a), the focus is not indicated separately due to the large focal distance, and the intersection of each point of the ray, lens, and eye is indicated.

Fig. 7(d) indicates the name of each point. The first two points represent the contact points between the ray and the lens surface at the front and back of the lens. After passing the vertex distance of 12.00 mm, the image is refracted at the front and back of the cornea in the Gullstrand model, and then refracted at the front of the lens cortex, front of the lens nucleoplasm, back of the lens nucleoplasm, and back of the lens cortex, and then imaged at the focus, which is the point where it meets the optical axis. At this time, as the incident height of the light increases, the focus of each light appears to form an image at a position adjacent to each other.

Fig. 8 shows the change in focal length and refractive power according to lens eccentricity at the incident heights of 1.00 mm, 2.00 mm, and 3.00 mm.

In Fig. 8(a), it was confirmed that the focal distance changes almost linearly depending on the amount of eccentricity, and in particular, when it is a (+) value, it shows a perfectly straight line. Even at (-) values, it almost does not change to a straight line, but it was confirmed that the slope changes slightly starting from 0.

Fig. 8(b) shows a curve with a small curvature when the eccentricity is a (-) value at a low incident height of 1.00 mm with the total refractive power calculated as the reciprocal of the focal length. In particular, the lower the incident height of the ray, the relatively larger the change in refractive power. Here, the total refractive power means that it includes both the refractive power of the lens and the eye and the prism refractive power generated by eccentricity.

Fig. 9 shows the coordinates of the ray and the back of the lens. As shown in Fig. 9(b), it can be seen that the coordinate value, that is, the height of the ray at the rear of the lens, has a constant value with little change depending on the amount of eccentricity.

Analyzing this in more detail,

x_{1} = h

(3)

z_{1} = R_{1} (1 - cos ϕ_{1})

(4)

ϕ_{1} = {sin}^{- 1} (\frac{h}{R_{1}})

(5)

Here, the thickness of the center of the lens t is extremely small compared to the radius of curvature and can be ignored, and the contact point with the rear of the lens has a small difference from the contact point with the front of the lens, so the height is approximately,

x_{2} \approx h

(6)

In other words, Fig. 9(b) shows a distribution like parallel lines with almost no change in height.

On the other hand, in Fig. 9(a), the coordinate z is parabolic. When a ray refracted from the front of the lens enters the back of the lens, the point of contact (z₂, x₂ ) between the ray and the back of the lens can be obtained by calculating the point of contact from the straight-line equation of the ray and the circular equation of the back of the lens. The equation of the circle for the back of the lens is

{(z_{2} - (t + R_{2}))}^{2} + x_{2}^{2} = R_{2}^{2}

(7)

Here, the central thickness t of the lens is extremely thin, so the height coordinate x₂ is approximately equal to the incident height of the ray. That is, x₂ ≈ h.

{(z_{2} - (t + R_{2}))}^{2} \approx R_{2}^{2} - h^{2}

(8)

z_{2} - (t + R_{2}) = \pm \sqrt{R_{2}^{2} - h^{2}}

(9)

Here, both (+) and (-) are possible solutions, but since rays are refracted in front of the lens, the appropriate solution is (-). Thus,

\begin{matrix} z_{2} = (t + R_{2}) - \sqrt{R_{2}^{2} - h^{2}} \\ = (t + R_{2}) - {(R_{2}^{2} - h^{2})}^{1 / 2} \\ = (t + R_{2}) - R_{2} {(1 - {(\frac{h}{R_{2}})}^{2})}^{1 / 2} \end{matrix}

(10)

Here, since (h/R₂) ≪ 1, using the binomial distribution Maclaurin series,

{(1 + x)}^{α} = 1 + \frac{α}{1!} x + \frac{α (α - 1)}{2!} x^{2} + \dots

(11)

Expanding the second term on the right-hand side,

z_{2} = (t + R_{2}) - R_{2} [1 - \frac{1}{2} {(\frac{h}{R_{2}})}^{2} + \frac{(1 / 2) (- 1 / 2)}{2!} {(\frac{h}{R_{2}})}^{4} + \dots]

(12)

Since it is (h/R₂) ≪ 1, if we take only the second term in the parentheses and ignore the rest,

\begin{matrix} z_{2} \approx (t + R_{2}) - R_{2} [1 - \frac{1}{2} {(\frac{h}{R_{2}})}^{2}] \\ = t + \frac{1}{2} \frac{h^{2}}{R_{2}} \end{matrix}

(13)

Therefore, z₂ is proportional to h².

If the lens is eccentric, the incident height of the ray becomes h = h₀ + δ. where h₀ is the incident height of the ray and δ is the amount of eccentricity. Therefore, z₂ is also proportional to the square of the eccentricity δ. in other words,

\begin{matrix} z_{2} \approx t + \frac{1}{2} \frac{h^{2}}{R_{2}} \\ = t + \frac{1}{2 R_{2}} {(h_{0} + δ)}^{2} \end{matrix}

(14)

At the back of the lens, the z coordinate of the ray and the lens contact point shows a parabolic distribution, which is a quadratic function of the eccentricity δ, as shown in Fig. 9(a).

Fig. 10 shows the distribution of the exit angle of rays exiting the rear surface of the lens.

In Fig. 1, the ray exit angle at the back of the lens is

{θ_{20}}^{'} = {θ_{2}}^{'} - ϕ_{2}

(15)

Here R₂ is the radius of curvature of the rear surface of the lens. θ₂′ is the angle of refraction at the rear of the lens and is determined by Snell's law. In other words, parallel rays of light enter the front of the lens, are refracted, and then reach the back of the lens. The angles defined by the relationship and geometric distance between the angles are, respectively,

ϕ_{2} = {sin}^{- 1} (\frac{h}{R_{2}})

(16)

θ_{1} = {sin}^{- 1} (\frac{h}{R_{1}})

(17)

{θ_{1}}^{'} = \sin^{- 1} (\frac{n_{1}}{n_{2}} \sin θ_{1})

(18)

θ_{2} = ϕ_{2} - (θ_{1} - {θ_{1}}^{'})

(19)

{θ_{2}}^{'} = \sin^{- 1} (\frac{n_{2}}{n_{3}} \sin θ_{2})

(20)

Additionally, the central thickness t of the lens is extremely thin, so the ray height x₂ at the back of the lens in Fig. 2 is an approximation to

x_{2} \approx h

(21)

Also, using the Taylor electric field for arcsine,

{sin}^{- 1} (x) = x + \frac{1}{2} \frac{x^{3}}{3} + \frac{1}{2} \frac{3}{4} \frac{x^{5}}{5} + \frac{1}{2} \frac{3}{4} \frac{5}{6} \frac{x^{7}}{7} + \dots

(22)

summarize θ₂₀′ in equation (15),

(23)

Here, f_L ′ and D_L ′ are the focal length and refractive power of the thin corrective lens, respectively. θ₂₀′ is the angle of exit of the ray exiting the back of the lens and is the bending angle caused by the lens because the parallel ray is incident on the lens. From formula (23), the absolute value of the bending angle is,

(24)

It can be confirmed that the bending angle is the same as the Prentice formula (2). Here, h is the sum of the incident height h₀ of the ray and the eccentricity δ of the lens. Therefore, the emission angle distribution in Fig. 10 appears as an almost straight line depending on the eccentricity δ.

Ⅳ. Discussion and Conclusion

When analyzing optical systems such as lenses, paraxial approximation is applied to reduce computational complexity. However, it has become possible to calculate highly complex formulas using computers, and now, if information on the refractive index, radius of curvature, position, and arrangement of the optical system is known, it is possible to analyze the characteristics of the optical system in detail through the generally well-known laws of refraction.¹²⁾

In this study, for detailed analysis of optical properties according to lens eccentricity, a ray tracing technique without paraxial approximation was applied to confirm the focal length change, refractive power change, and ray path difference between emmetropia and ammetropia through the Gullstrand model eye implemented in 3D simulation.

The case of ametropia was selected as myopia, which has the highest prevalence worldwide.¹³⁾ Accordingly, a single lens with a refractive index of 1.520, front and back radii of curvature of 526.30 mm and 100.00 mm, respectively, refractive power of +0.99 D and –5.20 D, and total refractive power of –4.20 D could be designed through theoretical calculation.

According to the Prentice formula, which is widely applied in clinical practice and is generally well known to us, the effect of the prism refractive power occurring at points outside the optical axis of the lens was predicted to change linearly as the incident height of the ray increases.¹⁴⁾ However, it was confirmed that the refractive power, which is commonly calculated as the reciprocal of the focal length, changes rapidly non-linearly as the ray incident height increases. Additionally, it was discovered that the greater the incident height of the light beam, the closer the foci are located to each other. This is believed to be because the effect of eccentricity was relatively reduced due to the large incident height. In particular, in the change in focal length according to the incident height of the ray, an almost perfectly straight line was seen in the section where the eccentricity value had a positive value, and this was judged to be a section to which the Prentice formula was well applied.

However, the height of the ray exiting from the back of the lens was found to have little effect on the amount of eccentricity. This is expected to be because the front and back radii of curvature (526.30 mm and 10.00 mm) of the lens designed in this study are extremely large compared to the amount of eccentricity (-6.00 mm ~ +6.00 mm) and the center thickness of the lens (2.00 mm). In the follow-up study, simulation was performed using actual measurement data rather than a theoretically calculated lens design. It is judged necessary to apply it to design.

In this study, the optical properties according to lens eccentricity in the 3D Gullstrand model eye are as follows.

When the incident height of the ray is 0 and follows the optical axis, a negative refractive force occurs due to lens eccentricity, so it does not converge and diverges. As the incident height of the rays became very large, the effect of eccentricity decreased, and the foci of the emitted rays were located at points close to each other. In the section where the eccentricity amount is (+), the focal distance was almost a perfectly straight line, which is considered to be a section where the Prentice formula is well applied. The total refractive power, including the refractive power of the lens and the eye, and the prism refractive power effect due to eccentricity, changed significantly nonlinearly as the incident height of the ray decreased.

Based on the results of this study, it is believed that when clinical experts prescribe eccentric lenses to correct ocular aliment anomalies such as heterotropia and heterophoria, they should optically identify the non-linearly changing sections of the lens and minimize negative visual influences to provide optimal visual performance to the wearer.

Figure

Fig. 1.

Conceptual diagram of ray tracing method in this study.

Fig. 2.

Ray tracing for (a) emmetropia and (b) ametropia (means for –4.00 D corrected with spectacle lenses) by the Gullstrand schematic model eye.

Fig. 3.

Comparison of the prism power calculated with Prentice’s rule and ray tracing.

Fig. 4.

Conceptual diagram of ray path in this study according to the single lens for refractive error correction.

Fig. 5.

Comparison of the focal length (a) and refractive power (b) between emmetropia and ammetropia calculated using ray tracing techniques.

Fig. 6.

Change of the refractive power calculated using ray tracing techniques.

Fig. 7.

Change of the ray path according to incident height. (a) 0.00 mm, (b) 1.00 mm, (c) 2.00 mm and (d) 3.00 mm, respectively.

Fig. 8.

Change of the focal length (a) and total refractive power (b) according to incident height.

Fig. 9.

Intersection of the back surface with the lens and ray according to eccentricity amount (a) z coordinate and (b) x coordinate.

Fig. 10.

Ray emission angle from the rear of the lens according to eccentricity amount.

Table

Reference

Rayamajhi A, Helmer M et al.: The human eye: From Gullstrand’s eye model to ray tracing today. Lux junior 2021: 15. Internationales Forum für den lichttechnischen Nachwuchs, Ilmenau: Tagungsband 04-06, 2021.
Koo BY, Kim YC: Analysis of optical characteristics on the gullstrand schematic eye through 3D simulation. Korean J Vis Sci. 23(3), 247-257, 2021.
Koo BY, Kim YC et al.: A new criterion for retinal image resolution. New Phys Sae Mulli. 74, 408-417, 2024.
Brennan NA, Bullimore MA et al.: Efficacy in myopia control. Prog Retin Eye Res. 83, 100923, 2021.
Bullimore MA, Brennan NA et al.: The risks and benefits of myopia control. Ophthalmology 128(11), 1561-1579, 2021.
Du Toit R, Ramke J et al.: Tolerance to prism induced by ready-made spectacles: Setting and using a standard. Optom Vis Sci. 84(11), 1053-1059, 2007.
Serdiuchenko VI: Use of prisms in ophthalmology: A review Part 1. The use of prisms in strabismus: Historical background, methodologies and their outcomes. Space. 1, 4, 2020. https://www.ozhurnal.com/sites/default/files/2020-4-12.pdf
Press OD, Leonard J: The art of prescribing low amounts of prism: Basic clinical applications. Optom Clin Pract. 5(2), 3, 2023.
Kim DJ: Comparative analysis of the optical characteristics of finished magnifying glasses and eyeglass lenses and the prism error between the optical center of self-prescribed finished magnifying glasses and binocular pupillary distance. J Korean Oph Opt Soc. 29(1), 1-8, 2024.
Koo BY, Lee MH et al.: Analysis on chromatic aberration of the gullstrand schematic eye through 3D simulation. Korean J Vis Sci. 24(2), 155-163, 2022.
Jha GS, Carpena P et al.: Development of high refractive index plastics. e-Polymers 7(1), 120, 2007.
Carpena P, Coronado AV: On the focal point of a lens: Beyond the paraxial approximation. Eur J Phys. 27(2), 231, 2006.
Holden BA, Fricke TR et al.: Global prevalence of myopia and high myopia and temporal trends from 2000 through 2050. Ophthalmology 123(5), 1036-1042, 2016.
Schroth V, Jaschinski W et al.: Effects of prism eyeglasses on objective and subjective fixation disparity. PLoS One 10(10), e0138871, 2015.