As an optical simulation base, the established human eye model of Liou and Brennan was used [9]. The optical setup for this optical system for an aperture diameter of 3 mm is summarized in Table .

Table 1: Liou Brennan eye model.

This basic model includes aspheric shapes for all optical surfaces, which can be derived from Eq. (1), a gradient refractive index distribution for the crystalline lens [see Eq. (2)], and dispersion properties of the optical components [10], [11].

$$z\mathrm{(}r\mathrm{)}=\frac{\begin{array}{c}{r}^{2}\end{array}}{\begin{array}{c}R+\sqrt{R-\mathrm{(}1+k\mathrm{)}{c}^{2}{r}^{2}}\end{array}}.$$(1)

With *c* as the curvature value, 1/Radius, (mm^{−1}), *k* as the conic constant for the surface, *r* as the distance from the optical axis (mm). The gradient index distribution (GRAD A and GRAD P) within the eye lens for a special wavelength of 555 nm can be derived according to the following Eq. (2) from Ref. [9], with *z* as the distance along the optical axis to the apex and *w* as the distance in the lateral direction:

$$n\mathrm{(}w\mathrm{,}\text{\hspace{0.17em}}z\mathrm{)}={n}_{00}+{n}_{01}z+{n}_{02}{z}^{2}+{n}_{10}{w}^{2},$$(2)

where *n*_{00}=1.368, *n*_{01}=0.049057, *n*_{02}=−0.015427, *n*_{10}= −0.001978 for the anterior part of the eye lens (GRAD A) and *n*_{00}=1.407, *n*_{01}=0.00000, *n*_{02}=−0.006605, *n*_{10}=−0.001978 for the posterior part (GRAD P). The refractive indices *n*_{00}(*λ*) for other wavelengths beside 555 nm can be calculated with the following formula [11]:

$${n}_{00}(\lambda )={n}_{00}(@555\text{nm})+0.0512-0.14555\lambda +0.0961{\lambda}^{2}.$$(3)

In the original Liou-Brennan model, the retinal surface is assumed to be a pure spherical surface with a radius of curvature of 12 mm, and the physiological fact that the light-detecting photoreceptor layer is buried below the interface between the vitreous body and the top retinal layer is neglected.

In our approach, an optically homogeneous tissue with a thickness of 250 μm and a refractive index of n_{retina}=1.370 at 555 nm wavelength with an Abbe’s number of 50.23 was assumed for the retinal layer [1], [12]. Thus, the length of the vitreous was reduced accordingly. Although the retinal structure consists of several layers, the assumption of a homogeneous layer is reasonable, as the variation of the inter-retinal refractive index is much smaller compared to the index step at the transition layer at the vitreo-retinal boundary [13]. The foveal shape was obtained from measurements of the retinal topography by Optical Coherence Tomography (OCT) (Heidelberg Spectralis SD-OCT, Germany).

Figure 1 shows, on the left side, the OCT cross-section of a healthy left human eye and, on the right side, the corresponding mesh-diagram of the three-dimensional foveal region. The depicted topography covers an extension of 4 mm in both lateral dimensions and a height variation of about 140 μm.

Figure 1: Cross-section of the fovea of a healthy eye (left image) obtained by an optical coherent tomography system (Heidelberg Spectralis SD-OCT). For a pronounced presentation, the axial direction is magnified with respect to the lateral direction. The inclination on the right side (left image) indicates the optic nerve head (ONH). On the right image, a mesh-diagram of a three-dimensional shape of the fovea region is shown.

In previous researches the foveal shape was limited to simple geometrical descriptions such as spheres and paraboloids. This approach is restricted for distinct foveal zones and simple shapes. For our optical analysis, both an average and an extraordinary foveal form were used, which are depicted in Figure 2, as well as a pure spherical reference foveal shape with a radius of curvature of 11.63 mm and a thickness of 350 μm. The extraordinary form was transferred from Ref. [14]. For all three cases, the thickness of the vitreous was adapted such that the overall length of the optical system remained constant.

Figure 2: Comparison of different foveal shapes; in blue, the cross-section from the OCT measurements of an average foveal shape and, in red, a cross-section of an extraordinary foveal shape found in literature [14].

To use the measured topographical data of the fovea for optical analysis, we fitted the cross-section to a rotation-symmetric aspheric equation [15], with *c* as the curvature value (mm−1), *k* as the conic constant for the surface, *r* as the distance from the optical axis (mm), and *A*_{2}_{i} as the aspheric coefficients according to the following formula:

$$z\mathrm{(}r\mathrm{)}=\frac{\begin{array}{c}c{r}^{2}\end{array}}{\begin{array}{c}1+\sqrt{1-\mathrm{(}1+k\mathrm{)}{c}^{2}{r}^{2}}\end{array}}+{\displaystyle \sum _{i=1}^{N}{A}_{2i}{r}^{i}}.$$(4)

For the two different foveal shapes, the following parameters were determined (Table ):

Table 2: Aspheric parameter.

The quality of the fit for the average foveal shape is not significantly improved for aspheric coefficients of higher order than *A*_{12}; thus, they can be ignored. The presented fits are valid up to a diameter of 3 mm, which covers the main area of the fovea. As we are primarily interested in the influence of the foveal shape on image formation, this rotation-symmetric approach is reasonable.

From both model contributions, namely, the model of the anterior part of the eye and the optical model of the foveal shape, a combined overall model is now composed. For the comprehensive optical analysis, the dependency on several parameters, such as wavelength, aperture diameter, or the angle of incident ray bundles, were investigated with the optical simulation program ZEMAX (Zemax LLC, Seattle, WA, USA) [15]. The essential evaluation criteria were the extent of the image spots and their centroid position for different imaging situations.

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