## 1.

## Introduction

Recent advances in new materials facilitate the application of gradient refractive index (GRIN) lenses in a variety of optical devices, especially in the development of bio-inspired lenses^{1} and optical systems. Employing optical elements with a spatially variable index of refraction is a powerful way to achieve improved imaging. The best example of such a GRIN lens is the well-known Luneburg lens,^{2} which is free from all monochromatic aberrations. The crystalline lens in the human eye is another example of a GRIN lens. In the present paper we explore a new mathematical model describing the crystalline GRIN lens. The gradual variation of the refractive index of the crystalline lens has been known for a long time and several models have been developed to account for the GRIN structure.^{3}^{–}^{7} Advances in ocular aberration measurements,^{8} magnetic resonance imaging,^{9}^{,}^{10} optical tomography,^{11} optical coherence tomography imaging,^{12} and X-ray Talbot interferometry^{13} have enabled researchers to improve existing eye models. Using this new data, several research groups have attempted to construct more realistic models of the GRIN lens. Navarro et al. proposed a GRIN lens model with concentric iso-indical contours mimicking the external conic surfaces of the lens.^{14} The GRIN spatial distribution of this model follows the experimental age-dependent formula suggested in earlier work.^{15} For the first time, a GRIN lens model features a curved equatorial plane, where anterior and posterior hemispheres meet. Using a different approach, Goncharov and Dainty introduced a wide-field schematic eye model with a GRIN lens, which uses a fourth-order polynomial describing the refractive structure of the lens.^{16} Similar to the Navarro model, the external shape of the lens defines its GRIN structure. By estimating a parabolic path for the rays in the human GRIN lens^{17} and using Sands’ third-order aberrations study in inhomogeneous lenses,^{18} this model presents approximated formulas for the power of the lens and its spherical aberration. Another recent model proposed by Díaz et al. uses a combination of polynomials and trigonometric functions for describing the refractive index distribution.^{19} The coefficients of the refractive index of the lens are given as a linear function of age. Both models, Goncharov and Dainty and Díaz et al., are complete eye models providing age-dependent equations for the curvatures of the cornea and lens. Following the Navarro et al. model for the GRIN lens *in vitro*,^{14} in a recent work by Castro et al., the power law of the GRIN lens profile has been modified to account for a possible toricity of the lens surface.^{12} The variety of eye models featuring different GRIN profiles shows the great interest in lens structure and its effect on optical performance. In spite of the apparent progress made in this area, there is no simple GRIN lens model providing exact paraxial equations for the path of the rays inside the GRIN structure. It would be beneficial to have an analytical way to calculate the power and the third-order aberrations for the lens. Analytical solutions can help researchers gain a better understanding of the GRIN structure role in image formation and simplify the optical analysis of the lens. In addition, if such a model could also provide a more realistic (continuous) geometry of the GRIN lens’s iso-indicial contours, it would become a valuable tool for reconstructing the human eye and modeling the accommodation mechanism. In the following section we introduce such a GRIN lens model and outline its main geometrical properties.

## 2.

## Parametric Model of the GRIN Lens

## 2.1.

### Refractive Index Equation Based on Experimental Data

There are many experimental studies focusing on the distribution of refractive index in the crystalline lens. In 1969 Nakao et al. suggested a parabolic distribution for the refractive index in all directions:^{20}

^{6}introduced more terms in Eq. (1) to get a better fit to experimental data.

^{21}Later, Smith et al.

^{15}proposed power-law to describe the distribution of refractive index along the optical axis as: where the parameter $p$ in the exponent is used to account for age-related changes in the GRIN lens. Equation (2) was used by Navarro et al. as a starting point for modeling GRIN lenses

*in vitro*.

^{14}For clarity we rewrite Eq. (2) as where $\zeta $ is the normalized distance from the center of the lens, ${n}_{c}$ and ${n}_{s}$ are the refractive indices at the center and at the surface of the GRIN lens, respectively. Here, $\zeta $ changes between $-1$ to $+1$ to cover both anterior and posterior hemispheres of the lens; also we avoid introducing complex numbers by using the form ${({\zeta}^{2})}^{p}$.

## 2.2.

### Geometry of Iso-Indicial Contours

From the optical design point of view, it is convenient to describe the external surfaces of the GRIN lens as a conicoid of revolution:

where $c$ and $k$ are respectively the curvature and the conic constant of the surface, and $\rho $ is the distance from the optical $z$ axis. There are other possible mathematical representations for the geometry GRIN lens, for example hyperbolic cosines^{22}or Fourier series of cosines.

^{23}However these alternative representations do not have a straight forward connection with the radius of curvature and conic constant of the lens surface. On the other hand, using Eq. (4) greatly simplifies the parameterization of the surface. Following the idea of constructing the lens with conic surfaces on both sides,

^{14}one might get discontinuity of iso-indicial contours in equatorial interface joining two hemispheres. To avoid this problem, one could add an additional term on the right side of Eq. (4). Before we derive the continuity condition for iso-indicial contours at the equatorial interface, it is more convenient to rewrite Eq. (4) as a function of surface sag: where $r$ is the radius of curvature of the surface. Now introducing an additional term on the right side will help achieve the continuity condition. The surface equation becomes as: where $b$ is a constant, which is used to satisfy the continuity condition by making the first derivative $d\rho /dz=0$ at the equatorial interface connecting the posterior and anterior hemispheres. Based on this approach, Eq. (7) represents our new description for the surface of iso-indical contours:

## Eq. (7a)

$${\rho}_{a}^{2}=2{r}_{a}({t}_{a}+z)-(1+{k}_{a}){({t}_{a}+z)}^{2}+{b}_{a}{({t}_{a}+z)}^{3},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0ex}{0ex}}-{t}_{a}\le z<0$$## Eq. (7b)

$${\rho}_{p}^{2}=2{r}_{p}({t}_{p}-z)-(1+{k}_{p}){({t}_{p}-z)}^{2}+{b}_{p}{({t}_{p}-z)}^{3},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0ex}{0ex}}0\le z\le {t}_{p}$$^{9}show that the center of curvature of the inner contours gradually shifts toward the center $O$ as a result of their steepening. This effect is more obvious in younger eyes, where central contours are still distinguishable. The simplest way to account for such a gradual change in curvature with depth is to define $r$ as a linear function of the normalized distance from the center, $r=R\zeta $. It is worth noting that for both anterior and posterior hemispheres $r$, $R$, $t$, and $T$ are numerically positive quantities; see Fig. 1.

By using Eq. (3), now we shall derive the continuity condition and find the corresponding refractive index for each iso-indicial contour. To satisfy the continuity condition we have to fulfill two constraints for an iso-indicial contour: zero derivative, $d\rho /dz=0$, and equal heights, ${\rho}_{a}({z}_{c})={\rho}_{p}({z}_{c})$, at the joining point ${z}_{c}$, as shown in Fig. 1. Using the first constraint, we determine ${b}_{a}$ and ${b}_{p}$. As a result, for both hemispheres of the lens we have:

## Eq. (8a)

$${b}_{a}=\frac{2}{3}\frac{(1+{k}_{a})({t}_{a}+{z}_{c})-{r}_{a}}{{({t}_{a}+{z}_{c})}^{2}},$$## Eq. (8b)

$${b}_{p}=\frac{2}{3}\frac{(1+{k}_{p})({t}_{p}-{z}_{c})-{r}_{p}}{{({t}_{p}-{z}_{c})}^{2}}.$$## 3.

## Thin Lens Approximation

The optical characteristics of a GRIN lens, such as the optical power and third-order aberrations, are usually not available in analytical form. However, in some cases (e.g., Ref. 24) and for our GRIN lens model it is possible to derive analytical expressions, which are given in Secs. 6 and 7. Although in Sec. 5 we discuss exact paraxial equations, it would be useful to start with a simplified power equation. The optical power of the GRIN lens can be described as the sum of the contributions from three components: the anterior surface of the lens, ${F}_{\mathrm{as}}$; the GRIN structure of the lens, ${F}_{\text{GRIN}}$; and the posterior surface of the lens, ${F}_{\mathrm{ps}}$. The optical power for the anterior and the posterior surfaces are given by a conventional equation^{25}

## Eq. (12)

$${F}_{\mathrm{GRIN}}={\int}_{-1}^{0}\frac{2p({n}_{s}-{n}_{c}){({\zeta}^{2})}^{p-\frac{1}{2}}}{-{R}_{a}\zeta}\mathrm{d}\zeta -{\int}_{0}^{1}\frac{2p({n}_{s}-{n}_{c}){({\zeta}^{2})}^{p-\frac{1}{2}}}{{R}_{p}\zeta}\mathrm{d}\zeta .$$## Eq. (13)

$${F}_{\mathrm{GRIN}}=\frac{2p}{2p-1}({n}_{c}-{n}_{s})(\frac{1}{{R}_{a}}+\frac{1}{{R}_{p}}).$$## Eq. (14)

$${F}_{\text{thin}}=\frac{{n}_{s}-{n}_{\mathrm{aqu}}}{{R}_{a}}+\frac{2p}{2p-1}({n}_{c}-{n}_{s})(\frac{1}{{R}_{a}}+\frac{1}{{R}_{p}})+\frac{{n}_{\mathrm{vit}}-{n}_{s}}{-{R}_{p}},$$## 4.

## Optical Path Length

One other useful characteristic of an optical element is its optical path length (OPL), defined as the product of the geometric length of the light path and the refractive index of the medium.^{25} In a GRIN lens the refractive index gradually changes, then the OPL can be calculated as the sum of the small propagations in each infinitely thin iso-indicial shell. Since the paraxial thickness of these thin shells is simply ${T}_{a}d\zeta $ and ${T}_{p}d\zeta $ for anterior and posterior hemispheres, respectively, using Eq. (3) we can define the paraxial OPL of the presented GRIN model as

## Eq. (15)

$$\mathrm{OPL}={\int}_{-1}^{0}({n}_{c}+({n}_{s}-{n}_{c}){({\zeta}^{2})}^{p}){T}_{a}\mathrm{d}\zeta +{\int}_{0}^{1}({n}_{c}+({n}_{s}-{n}_{c}){({\zeta}^{2})}^{p}){T}_{p}\mathrm{d}\zeta ,$$It is worth mentioning that the geometry of the iso-indicial contours is not contributing to the paraxial OPL of the lens, so Eq. (16) is applicable for any GRIN lens employing the paraxial refractive index distribution in Eq. (3) (e.g., the GRIN lens model proposed by Navarro et al.^{14}).

## 5.

## Analytical Paraxial Ray Tracing

It is notoriously difficult to perform exact ray tracing through a GRIN lens, which is done numerically using optical design software. Even exact paraxial ray tracing equations are not available for GRIN lenses. One could also use an approximate method, where the ray path within the GRIN lens is assumed to be parabolic.^{17} However, it would be desirable to have an exact method for paraxial ray tracing so that all optical characteristics of the lens can be found in closed form. Due to the linear dependence of the iso-indicial contours radius $r$ on the normalized axial distance, $\zeta =z/T$, we are able to derive a closed-form solution for paraxial ray tracing in the geometry-invariant GRIN lens. Paraxial ray tracing is based on two main equations.^{25} According to the first one we have

## Eq. (20)

$$n\left(\frac{z+\delta z}{T}\right)\frac{y(z+2\delta z)-y(z+\delta z)}{\delta z}=n\left(\frac{z}{T}\right)\frac{y(z+\delta z)-y(z)}{\delta z}+\frac{y(z)}{R(z/T)}[n\left(\frac{z+\delta z}{T}\right)-n\left(\frac{z}{T}\right)].$$## Eq. (21)

$$\frac{y(z){n}^{\prime}(z/T)}{Rz}-\frac{{n}^{\prime}(z/T){y}^{\prime}(z)}{T}-n(z/T){y}^{\prime \prime}(z)=0.$$## Eq. (22)

$$y(z)=\{\begin{array}{l}{c}_{1}{}_{2}{F}_{1}[\frac{-1+2p-\alpha}{4p},\frac{-1+2p+\alpha}{4p};1-\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{-z}{{T}_{a}}\right)}^{2p}]\\ +z\frac{{c}_{2}}{{T}_{a}}{}_{2}{F}_{1}[\frac{1+2p-\alpha}{4p},\frac{1+2p+\alpha}{4p};1+\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{-z}{{T}_{a}}\right)}^{2p}]& -{T}_{a}\le z<0\\ \\ {c}_{1}{}_{2}{F}_{1}[\frac{-1+2p-\beta}{4p},\frac{-1+2p+\beta}{4p};1-\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{z}{{T}_{p}}\right)}^{2p}]\\ +z\frac{{c}_{2}}{{T}_{a}}{{}_{2}F}_{1}[\frac{1+2p-\beta}{4p},\frac{1+2p+\beta}{4p};1+\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{z}{{T}_{p}}\right)}^{2p}]& 0\le z\le {T}_{p},\end{array}$$## Eq. (23)

$$u(z)=\{\begin{array}{l}\frac{{c}_{1}}{{T}_{a}}{\gamma}_{2}{\left(\frac{-z}{{T}_{a}}\right)}^{2p-1}{{}_{2}F}_{1}[\frac{-1+6p+\alpha}{4p},\frac{-1+6p-\alpha}{4p};2-\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{-z}{{T}_{a}}\right)}^{2p}]\\ +\frac{{c}_{2}}{{T}_{a}}{\gamma}_{1}{\left(\frac{-z}{{T}_{a}}\right)}^{2p}{{}_{2}F}_{1}[\frac{1+6p-\alpha}{4p},\frac{1+6p+\alpha}{4p};2+\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{-z}{{T}_{a}}\right)}^{2p}]& -{T}_{a}\le z<0\\ +\frac{{c}_{2}}{{T}_{a}}{{}_{2}F}_{1}[\frac{1+2p-\alpha}{4p},\frac{1+2p+\alpha}{4p};1+\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{-z}{{T}_{a}}\right)}^{2p}]\\ \\ \frac{{c}_{1}}{{T}_{p}}{\gamma}_{4}{\left(\frac{z}{{T}_{p}}\right)}^{2p-1}{{}_{2}F}_{1}[\frac{-1+6p+\beta}{4p},\frac{-1+6p-\beta}{4p};2-\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{z}{{T}_{p}}\right)}^{2p}]\\ +\frac{{c}_{2}}{{T}_{a}}{\gamma}_{3}{\left(\frac{z}{{T}_{p}}\right)}^{2p}{{}_{2}F}_{1}[\frac{1+6p-\beta}{4p},\frac{1+6p+\beta}{4p};2+\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{z}{{T}_{p}}\right)}^{2p}]& 0\le z\le {T}_{p}.\\ +\frac{{c}_{2}}{{T}_{a}}{{}_{2}F}_{1}[\frac{1+2p-\beta}{4p},\frac{1+2p+\beta}{4p};1+\frac{1}{2p};\frac{{n}_{c}-{n}_{s}}{{n}_{c}}{\left(\frac{z}{{T}_{p}}\right)}^{2p}]\end{array}$$It is worth mentioning that the tilt or decenter of the lens can be seen as a change in the angle and the height of the incident ray, respectively, and Eqs. (22) and (23) are still applicable.

## 6.

## Analytical Expression for Optical Power

In this section we present an analytical expression for the optical power of the GRIN lens derived with the help of Eqs. (22) and (23). First we consider the power of a homogeneous lens:^{25}

## Eq. (24)

$${F}_{L}=\frac{({n}_{2}-{n}_{1})}{{R}_{1}}-\frac{({n}_{2}-{n}_{3})}{{R}_{2}}+d\frac{({n}_{2}-{n}_{3})({n}_{2}-{n}_{1})}{{n}_{2}{R}_{1}{R}_{2}},$$## Eq. (25)

$$F={A}_{a}\frac{({n}_{s}-{n}_{\mathrm{aqu}})}{{R}_{a}}+{A}_{\mathrm{GRIN}}-{A}_{p}\frac{({n}_{s}-{n}_{\mathrm{vit}})}{-{R}_{p}}+{A}_{d}\frac{({n}_{s}-{n}_{\mathrm{vit}})({n}_{s}-{n}_{\mathrm{aqu}})}{-{n}_{s}{R}_{a}{R}_{p}},$$Using Eq. (25), we can find the focal length, $f$, and the back focal length of the lens, ${f}_{\mathrm{back}}$ as

and where ${B}_{f}$ is defined in the Appendix.We shall stress that the optical power of the lens is not affected by its tilt or decenter and remains one of the fundamental characteristics of the lens.

## 7.

## Third-Order Aberrations

In general, the contribution of a GRIN lens to Seidel aberrations can be divided in two parts. The first part is the surface contribution of the interface between the homogeneous medium and inhomogeneous (GRIN) medium. The second part is the transfer contribution originating inside the GRIN media. For a GRIN lens with iso-indicial contours being coincident with the external surfaces, the surface contribution can be calculated as a conventional contribution from an interface between homogeneous media. Therefore we shall start with a single surface contribution to the primary third-order monochromatic aberrations. The coefficient of third-order spherical aberration is given by^{26}

## Eq. (28)

$${S}_{\mathrm{I}}=-y[{\left(\frac{{u}_{2}-{u}_{1}}{1/{n}_{2}-1/{n}_{1}}\right)}^{2}(\frac{{u}_{2}}{{n}_{2}}-\frac{{u}_{1}}{{n}_{1}})+k\frac{{({n}_{2}{u}_{2}-{n}_{1}{u}_{1})}^{3}}{{({n}_{2}-{n}_{1})}^{2}}],$$## Eq. (29)

$$\delta {S}_{\mathrm{I}}=-y(z)\{\frac{T{n}^{2}(\frac{z}{T}){{y}^{\prime \prime}}^{2}(z)[-{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)]}{{{n}^{\prime}}^{2}(\frac{z}{T})}\phantom{\rule{0ex}{0ex}}+k\frac{{[{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)]}^{3}}{T{{n}^{\prime}}^{2}(\frac{z}{T})}\}\delta z,$$## Eq. (30)

$$\sum {S}_{\mathrm{I}}=-{y}_{0}[{\left(\frac{{u}_{0}-{u}_{a}}{1/{n}_{s}-1/{n}_{\mathrm{aqu}}}\right)}^{2}(\frac{{u}_{0}}{{n}_{s}}-\frac{{u}_{a}}{{n}_{\mathrm{aqu}}})\phantom{\rule{0ex}{0ex}}+{k}_{a}\frac{{({n}_{s}{u}_{0}-{n}_{\mathrm{aqu}}{u}_{a})}^{3}}{{({n}_{s}-{n}_{\mathrm{aqu}})}^{2}}]\phantom{\rule{0ex}{0ex}}+{\int}_{-{T}_{a}}^{{T}_{p}}\mathrm{d}{S}_{\mathrm{I}}-y({T}_{p})\{{\left[\frac{{u}_{p}-u({T}_{p})}{1/{n}_{\mathrm{vit}}-1/{n}_{s}}\right]}^{2}[\frac{{u}_{p}}{{n}_{\mathrm{vit}}}-\frac{u({T}_{p})}{{n}_{s}}]\phantom{\rule{0ex}{0ex}}+{k}_{p}\frac{{[{n}_{\mathrm{vit}}{u}_{p}-{n}_{s}u({T}_{p})]}^{3}}{{({n}_{\mathrm{vit}}-{n}_{s})}^{2}}\},$$## Eq. (31)

$${u}_{p}=\frac{1}{{n}_{\mathrm{vit}}}[{n}_{s}u({T}_{p})+\frac{y({T}_{p})}{{R}_{p}}({n}_{\mathrm{vit}}-{n}_{s})].$$In addition to the marginal ray we also need to trace the chief (principal) ray when calculating coefficients for off-axis aberrations. Using the chief and the marginal rays, the contribution of a single conic surface to the aberration coefficient of third-order coma could be written as^{26}

## Eq. (32)

$${S}_{\mathrm{II}}=-y\left[{\left(\frac{{u}_{2}-{u}_{1}}{1/{n}_{2}-1/{n}_{1}}\right)}^{2}\right(\frac{{u}_{2}}{{n}_{2}}-\frac{{u}_{1}}{{n}_{1}}\left)\right(\frac{{u}_{c2}-{u}_{c1}}{{u}_{2}-{u}_{1}})+k({n}_{2}{u}_{c2}-{n}_{1}{u}_{c1})\frac{{({n}_{2}{u}_{2}-{n}_{1}{u}_{1})}^{2}}{{({n}_{2}-{n}_{1})}^{2}}],$$Similar to our derivation of Eq. (29), we find the contribution to aberration coma from an infinitely thin layer of the GRIN structure:

## Eq. (33)

$$\delta {S}_{\mathrm{II}}=-y(z)\{{y}_{c}^{\prime \prime}(z)\frac{Tn{(\frac{z}{T})}^{2}{y}^{\prime \prime}(z)[-{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)]}{{n}^{\prime}{(\frac{z}{T})}^{2}}\phantom{\rule{0ex}{0ex}}+k\left[{n}^{\prime}\right(\frac{z}{T}){y}_{c}^{\prime}(z)+Tn(\frac{z}{T}){y}_{c}^{\prime \prime}(z)]\phantom{\rule{0ex}{0ex}}\frac{[{{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)}^{2}]}{T{n}^{\prime}{(\frac{z}{T})}^{2}}\}\delta z,$$## Eq. (34)

$$\sum {S}_{\mathrm{II}}=-{y}_{0}\left[{\left(\frac{{u}_{0}-{u}_{a}}{1/{n}_{s}-1/{n}_{\mathrm{aqu}}}\right)}^{2}\right(\frac{{u}_{0}}{{n}_{s}}-\frac{{u}_{a}}{{n}_{\mathrm{aqu}}}\left)\right(\frac{{u}_{c0}-{u}_{ca}}{{u}_{0}-{u}_{a}})\phantom{\rule{0ex}{0ex}}+{k}_{a}({n}_{s}{u}_{c0}-{n}_{\mathrm{aqu}}{u}_{ca})\frac{{({n}_{s}{u}_{0}-{n}_{\mathrm{aqu}}{u}_{a})}^{2}}{{({n}_{s}-{n}_{\mathrm{aqu}})}^{2}}]\phantom{\rule{0ex}{0ex}}+{\int}_{-{T}_{a}}^{{T}_{p}}\mathrm{d}{S}_{\mathrm{II}}-y({T}_{p})\{{\left[\frac{{u}_{p}-u({T}_{p})}{1/{n}_{\mathrm{vit}}-1/{n}_{s}}\right]}^{2}\phantom{\rule{0ex}{0ex}}[\frac{{u}_{p}}{{n}_{\mathrm{vit}}}-\frac{u({T}_{p})}{{n}_{s}}]\left[\frac{{u}_{cp}-{u}_{c}({T}_{p})}{{u}_{a}-{u}_{a}}\right]\phantom{\rule{0ex}{0ex}}+{k}_{p}[{n}_{\mathrm{vit}}{u}_{cp}-{n}_{s}{u}_{c}({T}_{p})]\frac{{[{n}_{\mathrm{vit}}{u}_{p}-{n}_{s}u({T}_{p})]}^{2}}{{({n}_{\mathrm{vit}}-{n}_{s})}^{2}}\},$$Following the same concept we can calculate aberration coefficients for third-order astigmatism, where the contribution of a single surface has the following form

## Eq. (35)

$${S}_{\mathrm{III}}=-y\left[{\left(\frac{{u}_{2}-{u}_{1}}{1/{n}_{2}-1/{n}_{1}}\right)}^{2}\right(\frac{{u}_{2}}{{n}_{2}}-\frac{{u}_{1}}{{n}_{1}}){\left(\frac{{u}_{c2}-{u}_{c1}}{{u}_{2}-{u}_{1}}\right)}^{2}+k{({n}_{2}{u}_{c2}-{n}_{1}{u}_{c1})}^{2}\frac{({n}_{2}{u}_{2}-{n}_{1}{u}_{1})}{{({n}_{2}-{n}_{1})}^{2}}]$$## Eq. (36)

$$\delta {S}_{\mathrm{III}}=-y(z)\{{y}_{c}^{\prime \prime}{(z)}^{2}\frac{Tn{(\frac{z}{T})}^{2}{y}^{\prime \prime}(z)[-{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)]}{{n}^{\prime}{(\frac{z}{T})}^{2}}\phantom{\rule{0ex}{0ex}}+k{[{n}^{\prime}\left(\frac{z}{T}\right){y}_{c}^{\prime}(z)+Tn\left(\frac{z}{T}\right){y}_{c}^{\prime \prime}(z)]}^{2}\phantom{\rule{0ex}{0ex}}\frac{{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)}{T{n}^{\prime}{(\frac{z}{T})}^{2}}\}\delta z,$$## Eq. (37)

$$\sum {S}_{\mathrm{III}}=-{y}_{0}\left[{\left(\frac{{u}_{0}-{u}_{a}}{1/{n}_{s}-1/{n}_{\mathrm{aqu}}}\right)}^{2}\right(\frac{{u}_{0}}{{n}_{s}}-\frac{{u}_{a}}{{n}_{\mathrm{aqu}}}){\left(\frac{{u}_{c0}-{u}_{ca}}{{u}_{0}-{u}_{a}}\right)}^{2}\phantom{\rule{0ex}{0ex}}+{k}_{a}{({n}_{s}{u}_{c0}-{n}_{\mathrm{aqu}}{u}_{ca})}^{2}\frac{({n}_{s}{u}_{0}-{n}_{\mathrm{aqu}}{u}_{a})}{{({n}_{s}-{n}_{\mathrm{aqu}})}^{2}}]\phantom{\rule{0ex}{0ex}}+{\int}_{-{T}_{a}}^{{T}_{p}}\mathrm{d}{S}_{\mathrm{III}}-y({T}_{p})\{{\left[\frac{{u}_{p}-u({T}_{p})}{1/{n}_{\mathrm{vit}}-1/{n}_{s}}\right]}^{2}[\frac{{u}_{p}}{{n}_{\mathrm{vit}}}-\frac{u({T}_{p})}{{n}_{s}}]\phantom{\rule{0ex}{0ex}}{\left[\frac{{u}_{cp}-{u}_{c}({T}_{p})}{{u}_{a}-{u}_{a}}\right]}^{2}+{k}_{p}{[{n}_{\mathrm{vit}}{u}_{cp}-{n}_{s}{u}_{c}({T}_{p})]}^{2}\phantom{\rule{0ex}{0ex}}\frac{[{n}_{\mathrm{vit}}{u}_{p}-{n}_{s}u({T}_{p})]}{{({n}_{\mathrm{vit}}-{n}_{s})}^{2}}\}\mathrm{.}$$In a similar way, the field curvature of a single surface can be achievable as

## Eq. (38)

$${S}_{\mathrm{IV}}=-{n}_{1}{({u}_{c1}y-{u}_{1}{y}_{c})}^{2}\frac{{n}_{2}{u}_{2}-{n}_{1}{u}_{1}}{y{n}_{2}},$$## Eq. (39)

$$\delta {S}_{\mathrm{IV}}=-\frac{{[-{y}_{c}(z){y}^{\prime}(z)+y(z){y}_{c}^{\prime}(z)]}^{2}[{n}^{\prime}(\frac{z}{T}){y}^{\prime}(z)+Tn(\frac{z}{T}){y}^{\prime \prime}(z)]}{Ty(z)}\delta z,\phantom{\rule{0ex}{0ex}}$$## Eq. (40)

$$\sum {S}_{\mathrm{IV}}=-{n}_{\mathrm{aqu}}{y}_{0}{u}_{ca}^{2}\frac{{n}_{s}{u}_{0}-{n}_{\mathrm{aqu}}{u}_{a}}{{n}_{s}}\phantom{\rule{0ex}{0ex}}+{\int}_{-{T}_{a}}^{{T}_{p}}\mathrm{d}{S}_{\mathrm{IV}}-{n}_{s}{[{u}_{c}({T}_{p})y({T}_{p})\phantom{\rule{0ex}{0ex}}-u({T}_{p}){y}_{c}({T}_{p})]}^{2}\frac{{n}_{\mathrm{vit}}{u}_{p}-{n}_{s}u({T}_{p})}{y({T}_{p}){n}_{\mathrm{vit}}}.$$Despite the advantages of the Seidel theory, the third-order aberration calculations are limited to centered, rotationally symmetric systems, and do not support tilted or decentered elements, such as the crystalline lens in the eye. However, deriving the Seidel aberration coefficients of the GRIN lens in closed form is useful for understanding the nature of aberration compensation inside the GRIN structure. In addition to this, *in vitro* studies of the crystalline lens and its reconstruction based on the experimentally measured lenticular aberrations can benefit from the Seidel aberration representation.

It is worth mentioning that the capability of the geometry-invariant GRIN lens model is not limited to paraxial ray tracing and third-order aberration theory. In future work numerical ray tracing will be developed to calculate Zernike coefficients of the GRIN model lens, which can take the tilt and decenter of the lens into account.

## 8.

## Numerical Examples

We present an example of the eye model with the corneal and lenticular shape corresponding to a 40-year-old eye^{16} with GRIN profile exponent $p=3.13$ found in.^{14} Figure 2 shows the main optical characteristics of the GRIN lens including the optical power, focal length, back focal length, as well as Seidel aberration coefficients; the lens geometry and GRIN structure parameters are given on the left side. Figure 2 actually depicts the user interface for the open-source code written by the authors, available at.^{27} This code incorporates all mathematical expressions presented in this paper.

The optical power of the lens shown in Fig. 2 is based on the thin lens approximation in Eq. (14), and the exact power formula in Eq. (25). It is easy to see that the difference in optical power calculation is less than 1.4%, which indicates that Eq. (14) is useful especially if one wants to determine the exponent $p$ for a given optical power. This can be done by solving Eq. (14) for $p$, which leads to

## Eq. (41)

$$p=\frac{{R}_{a}{n}_{\mathrm{vit}}+{R}_{p}({n}_{\mathrm{aqu}}+F{R}_{a})-{n}_{s}({R}_{a}+{R}_{p})}{2[{R}_{a}{n}_{\mathrm{vit}}+{R}_{p}({n}_{\mathrm{aqu}}+F{R}_{a})-{n}_{c}({R}_{a}+{R}_{p})]}\mathrm{.}$$*in vitro*. This approach provides a practical way to approximate the GRIN profile, which defines all optical characteristics of the lens.

The optical power of the crystalline lens and its age-related changes have been a controversial topic for decades. Many studies (e.g., Ref. 28) show that for an unaccommodated lens, its external surfaces become more curved and therefore more powerful with age. On the other hand, measurements of the total optical power of the eye suggest that the power does not change much with age.^{29} This lens paradox might be explained, at least in part, by adjusting the center and surface refractive indices of the GRIN structure (${n}_{c}$ and ${n}_{s}$), the axial position of the peak in the refractive index profile (${T}_{a}$ or ${T}_{p}$), the lens axial thickness (${T}_{a}+{T}_{p}$), and also the exponent $p$.^{16}^{,}^{19}^{,}^{30}^{–}^{32} The latter parameter is the most challenging one to analyze, since calculating the contribution of the GRIN structure to the lens power has not been derived in an easily accessible form.

Pierscionek^{32} suggested that a slight change in the slope of refractive index in the cortex might compensate the increase in lens curvature and prevent the eye from becoming myopic with age. Using Eq. (14) we can calculate the optical power change in the lens due to an age-related increase in the exponent $p$. Following a recent study^{14} we select three age groups (20-, 40-, and 60-year-olds) with corresponding empirical value for $p$, see Table 1. To study the effect of $p$ independently from other variables, such radii and central thickness, all three age groups have identical lens geometry. In Fig. 2 we can see that 1 D change in the optical power can be attributed to GRIN profile steepening alone.

## Table 1

Three age groups (20-, 40-, and 60-year old) with corresponding empirical value for p and corresponding powers.

Age (year) | $p$ | Thin lens power (D) | Exact power (D) |
---|---|---|---|

20 | 2.87 | 20.074 | 19.815 |

40 | 3.13 | 19.884 | 19.629 |

60 | 4.28 | 19.359 | 19.115 |

It can be seen from Eq. (25) that one can easily adjust other parameters of the lens affecting the lens paradox and take into account their effect due to aging on the lens power. To adjust these parameters in a meaningful way, more experimental data on the age-related changes in the GRIN structure is required.

The model presented here is not only useful for human eyes, it can also be beneficial for animal eye studies. For example, Fig. 3 shows the octopus eye model based on the experimental data provided by Jagger et al.,^{33} where a strictly symmetrical lens was modeled. The original experimental data shows some departure from symmetry, which is taken into account in our model, presented in Fig. 3.

## 9.

## Conclusion

The characterization of GRIN lenses by ray-tracing is notoriously difficult and usually requires numerical methods, while only a handful of analytical solutions exist (e.g., Lundberg lens).

In light of this, we introduce and analyze a new class of GRIN lens, which has the following properties. The refractive index distribution is based on the power law defined by the exponent $p$, which can be adjusted in a continuous manner. The mathematical description of the external surfaces is a standard conicoid of revolution with a higher-order term. Iso-inditial contours feature smooth connection between the anterior and posterior hemispheres. Analytical paraxial ray tracing is possible, which provides expressions for all optical characteristics of the lens and its monochromatic aberrations. The description for aberration coefficients of a thin homogeneous layer is useful for a general GRIN lens description.

A few examples are presented to illustrate the advantage of this GRIN mode with special emphasis given to the thin lens approximation formula. The latter is very accurate, and can be used to analyze the role of exponent $p$ in lens paradox. One could also determine the exponent $p$ for a given optical power measured experimentally *in vitro*.

The interior iso-indicial contours mimic the external shape of the lens, which leads to invariant geometry of the GRIN structure. Due to this strong coupling between the external shape of the lens and its GRIN structure, one could study the changes in aberrations with accommodation. A dispersion model and chromatic aberrations of the lens will be derived in future work.

The new GRIN lens model can be used for other types of eyes, even for such an extreme case as the octopus eye. A user-friendly software incorporating all mathematical expressions is available from the authors.^{27}

## Appendices

## Appendix:

### Coefficient Definitions

## Acknowledgments

The authors would like to thank Chris Dainty for his valuable comments. This research was supported by Science Foundation Ireland under grant 07/IN.1/1906.