Paraxial equivalent of the gradient-index lens of the human eye

Abstract

The lens of the eye has a refractive index gradient that changes as the lens grows throughout life. These changes play a key role in the optics of the eye. Yet, the lens is generally simulated using a homogeneous model with an equivalent index that does not accurately represent the gradient. We present an analytical paraxial model of the gradient lens of the eye that gives the direct relation between refractive index distribution and paraxial characteristics. The model accurately simulates the changes in lens power with age and accommodation. It predicts that a decrease in equivalent index with age is associated with a flattening of the axial refractive index profile and that changes in lens power with accommodation are due primarily to changes in the axial variation of the iso-indicial curvature, consistent with Gullstrand's intracapsular theory of accommodation. The iso-indicial curvature gradient causes a shift of the principal planes compared to the homogeneous equivalent model. This shift introduces a clinically significant error in eye models that implement a homogenous lens. Our gradient lens model can be used in eye models to better predict the optics of the eye and the changes with age and accommodation.

Fig. 1.

Coordinate system and notation. The optical axis is taken to be the z-axis with light propagating in the positive z direction and ray height is measured along the y-axis.

Fig. 2.

Relative error between truncated polynomial series and numerical solutions of the paraxial ray equation. For these simulations, y(0) =  1 and y’(0) =  -0.1. The error decreases as the value of the power coefficient b increases. The plot shows the error for b = 2.

Fig. 3.

Power coefficient (b) calculated using Eq. (14) from the age-dependence of the equivalent index of Atchison et al [30] (black squares) using n_s= 1.410 and n_e = 1.379 [11]. For comparison, the values obtained using the equivalent index data of Dubbelman and van der Heijde [33] (red dots) and Chang et al [34] (green triangles) and from in vivo MRI imaging [11] and in vitro X-ray phase contrast tomography [35] are also shown. The three datasets produce comparable values of the power coefficient in younger eyes, but the age dependence is variable.

Fig. 4.

Age-dependence of the exact and approximate ray-transfer matrix coefficients for the lens, using the data of Table 1. For convenience, the power is shown instead of the C-coefficient (P= - n_a * C_L).

Fig. 5.

Age-dependence of the principal plane locations for the exact gradient lens, approximate gradient lens and homogeneous equivalent lens, using the data of Table 1. The plots show the distance between the anterior vertex (V₁) and the object principal point (H) and the posterior vertex (V₂) and the image principal point (H’). Left: Absolute distance. Right: Distances relative to lens thickness. The relative position of the principal planes of the gradient lens is approximately constant with age. There is a significant difference in the position of the principal planes between the gradient model and the equivalent homogeneous model.

Fig. 6.

Contribution of the lens gradient to total lens power and accommodation. Left: The model predicts that the gradient contribution to lens power progressively decreases with age from 61% at age 18 to 53% at age 68. The slope of the decrease in gradient power is larger than the slope of the increase in surface power, resulting in a decrease in total lens power with age. Right: The model predicts that for a 28.5 year subject, the gradient is responsible for 60% (= 0.779 / 1.297) of the total lens power, a fraction that remains approximately constant with accommodation.

Fig. 7.

Value of the power coefficient in terms of the equivalent index, surface index and equatorial index. Left: The coefficient b is less sensitive to variations in the surface index than variations in the equatorial index. The model predicts that the power coefficient will increase with age as the equivalent index decreases with age. The variation of the power coefficient is more pronounced for lower values of the equivalent index, consistent with the plot of Fig. 3. Right: The power coefficient b is strongly dependent on the difference n_G-n_e (Eq. (19)).

Fig. 8.

Solution of the ray equation for a fractional value of the power coefficient b. The plots show the axial refractive index profile (left) and the ray slope (right) for b = 1.8. The ray slope was obtained using a numerical solution of the ray differential equation and from the derivative of Eq. (A8) for y(0) = 1 and y’(0)= -0.1. The numerical solution was obtained using exactly the same approach as for the analysis of Fig. 2.