Simulation of Corneal imaging properties for near objects

Using raytracing simulation to study the effect of corneal imaging metrics for different aperture sizes as a function of object distances with different schematic model eyes.


INTRODUC TION
In recent decades, a range of new intraocular lens (IOL) designs have been proposed.In addition to classical lenses with a spherical optic design on each side, aspheric lenses that address the spherical aberration of the cornea have been introduced.Some of these, with low negative asphericity on one or both optical surfaces, aim to correct the intrinsic spherical aberrations of the IOL (so-called aberration-free IOL) when positioned in a collimated or a convergent light beam; others, with a larger amount of negative asphericity (so-called aberration correcting IOL), aim for negative spherical aberration in the lens, in order to correct for the positive spherical aberration of the cornea.
Multifocal lenses, which promise spectacle independence by simultaneously generating images from objects positioned at varying distances from the retina, have also been launched. 1Different design concepts have been proposed, such as refractive or diffractive designs, or maintaining 2 (bifocal), 3 (trifocal) or more distinct foci.Some have a high near addition for reading (multifocal IOL) or a low addition for maintaining both far and intermediate distance vision (Enhanced depth of focus, EDOF).Multifocal and EDOF lens designs typically correct some degree of spherical aberration, either the intrinsic spherical aberration of the IOL or even the spherical aberration of the cornea. 1 However, given that such multifocal or EDOF lens designs are used to maintain intermediate or near vision, the corneal aberrations when imaging objects at near distances must be considered. 2,3For example, a cornea based on the Liou-Brennan schematic model eye 4 shows around 0.26 µm of spherical aberration over a 6 mm central zone.With other model eyes, such as the Gullstrand schematic eye 5 with spherical surfaces, the spherical aberration is even higher. 6,7But for objects at near distances this magnitude cannot be assumed to remain constant.When viewing near objects, optical aberrations change with viewing distance, and especially with multifocal lenses designed to image both far and near objects.This is also true for EDOF designs used for far and intermediate vision; here, the optical aberrations of the cornea when imaging far and near or far and intermediate objects must be considered. 3he purpose of this simulation study was to analyse the imaging properties of the cornea using several schematic model eyes, including the Liou-Brennan, Atchison, Gullstrand and Navarro.The terms of the analysis are best focus and best wavefront position, and to derive the change in root-mean-squared (rms) spot size, rms wavefront error and spherical aberration with various pupil sizes while imaging objects from infinity to intermediate and near distances using raytracing strategies.

METHODS
For this simulation study, a raytracing setup was implemented for the anterior segment of four different schematic model eyes: The Liou Brennan model eye (LBME 4 ), the Atchison model eye (ATCHME 8,9 ), the Gullstrand model eye (GULLME 5 ) and the Navarro model eye (NAVME 10,11 ).The anterior segment specifications of the model eyes tested are listed in Table 1.
The LBME is characterised by four coaxial refracting surfaces, an aperture stop located at the lens front surface which is nasally displaced by 0.5 mm, and an incident beam tilted 5° nasally to account for the eccentric location of the fovea.Both corneal surfaces are considered aspherical. 4he ATCHME, the GULLME and the NAVME are all designed with coaxial optical elements and a coaxial incident beam.The cornea of the ATCHME has an aspherical front and back surface, 8 whereas the cornea of the NAVME has an aspherical front surface and spherical back surface. 11The cornea of the GULLME is characterised by spherical front and back surfaces. 5he aperture stop was located at the front lens plane of each model eye.Aperture size varied from 2 to 5 mm in 1 mm steps, which equates to a visible (entrance) pupil size from around 2.4 to 6.0 mm.For each model eye and aperture stop diameter, 10,000 rays were initialised from a point light source (divergent ray bundle with an equallyspaced grid distribution of 7 mm at the corneal front apex plane), with object distances varying from 10 cm to 10 m in a uniform logarithmic scale, and a collimated beam starting from a plane 1 cm in front of the cornea with equally spaced parallel rays within a circular pupil of 7 mm.In total, 4 × 4 × 31 scenarios were considered.
Raytracing was implemented in MATLAB version 2019b (MathWorks, mathworks.com).For each scenario (i.e., model eye, aperture size and object distance) the best focus was determined in terms of minimising the rms ray scatter at the focal plane. 12In addition, the best wavefront focus was calculated in terms of rms variation of the optical path length from the object (a point source for objects at finite distances or a plane for the Key points collimated beam) to the focal point.The location of best focus and best wavefront focus was determined based on a nonlinear search algorithm 13,14 for each schematic model eye for aperture sizes from 2 to 5 mm (in 1 mm steps) as a function of object distance.In addition to the location of the best focus and best wavefront focus, we analysed the rms ray scatter and the spherical aberration term Z 4 0 from a Zernike decomposition of the wavefront referenced to a 6mm circle considered at the corneal front surface plane.All simulations were performed in a monochromatic fashion using the nominal refractive indices listed in Table 1.  Figure 2 shows the axial position of the best focus (in blue) and the best wavefront focus (in green) for the four model eyes (2a: LBME, 2b: ATCHME, 2c: GULLME, 2d: NAVME) with respect to the paraxial focus calculated using classical vergence formulae for 2-5 mm aperture sizes and object distances from 10 cm to 10 m.As a result of the positive spherical aberration of the cornea with all schematic model eyes, this focal length is systematically shorter than the focal length from the paraxial calculations.For large pupil sizes and near objects, the deviation of the best focus and best wavefront focus from the paraxial focus is more pronounced compared with small aperture sizes and distant objects.As an example, for a 3 mm aperture, the best focus/best wavefront focus is 0.2855/0.2296mm, 0.2095/0.1571mm, 0.2767/0.2074mm and 0.1218/0.0914mm closer to the cornea compared with the paraxial focus for distant objects using the LBME, ATCHME, GULLME and NAVME models, respectively.For objects at 10 cm, the respective best focus/best wavefront focus is even further from the paraxial focus and closer to the cornea (0.7083/0.5741 mm, 0.5051/0.3789mm, 0.6090/0.4566mm and 0.3611/0.2711mm).In Figure 3, the rms ray scatter at the focal plane is plotted for all four schematic model eyes (3a: LBME, 3b: ATCHME, 3c: GULLME, 3d: NAVME), for 2-5 mm aperture sizes as a function of object distance.The ray scatter at the best focus plane is depicted in blue, and for the best wavefront focal plane in green.In general, ray scatter increases with the size of the aperture stop and decreases with object distance.As a consequence of the best focus being determined from an optimisation for the least ray scatter, the ray scatter is systematically smaller compared with that at the best wavefront focus plane.As an example, for a 3 mm aperture, the rms ray scatter at the best focus plane/best wavefront focus plane is 3.0876/3.7596µm, 2.8425/3.4828µm, 3.8170/4.6794µm and 1.6558/2.0275µm for distant objects with the LBME, ATCHME, GULLME and NAVME models.For objects at 10 cm, the respective ray scatter at best focus/ best wavefront focus plane is systematically larger, namely 10.4469/11.1282µm, 5.0971/6.2558µm, 6.2743/7.7059µm and 3.6778/4.5097µm, respectively.
Figure 4 shows the rms wavefront error at the focal plane for all four schematic model eyes (4a: LBME, 4b: ATCHME, 4c: GULLME, 4d: NAVME) for 2-5 mm aperture sizes as a function of object distance from 10 cm to 10 m, respectively.The rms wavefront error at the best focus plane is depicted in blue, and the best wavefront focal plane in green.As a consequence of the best wavefront focus being determined by minimising the wavefront error, the rms wavefront error is systematically smaller compared with the wavefront error at the best focus plane.As an example, for a 3 mm aperture, the rms wavefront error at the best focus plane/best wavefront focus plane is 0.0416/0.0271µm, 0.0377/0.0231µm, 0.0515/0.0314µm and 0.0220/0.0135µm for distant objects with the LBME, ATCHME, GULLME and NAVME models, respectively.For objects at 10 cm, the respective wavefront error at best focus/best wavefront focus plane is systematically larger at 0.1335/0.1264µm, 0.0507/0.0309µm, 0.0638/0.0388µm and 0.0369/0.0226µm.The spherical aberration term Z 4 0 extracted from the wavefront error within a reference circle of 6 mm at the corneal front surface plane is shown in Figure 5 for all four schematic model eyes (5a: LBME, 5b: ATCHME, 5c: GULLME, 5d: NAVME) and for 2-5 mm aperture sizes as a function of object distance from 10 cm to 10 m, respectively.With increasing aperture size and smaller object distances, the spherical aberration term increases systematically.The spherical aberration term for the best focus plane/the best wavefront focus plane is depicted in blue/green.As the Gullstrand schematic model eye uses spherical surfaces for the cornea, this model yields the highest values of corneal spherical aberration, whereas the Navarro schematic model eye yields the lowest corneal spherical aberration.As an example, for a 3 mm aperture, the spherical aberration (referenced to a 6 mm zone corneal front surface) for best focus plane/best wavefront focus plane is 0.2732/0.2741µm, 0.2539/0.2547µm, 0.3461/0.3472µm and 0.1506/0.1511µm for distant objects with the LBME, ATCHME, GULLME and NAVME models, respectively.For objects at 10 cm, the respective spherical aberration for best focus/best wavefront focus plane is systematically larger at 0.4016/0.4025µm, 0.3768/0.3777µm, 0.4730/0.4741µm and 0.2748/0.2755µm.

DISCUSSION
There is much discussion regarding the imaging performance of IOLs, especially in an era of aspheric, aberration free and aberration correcting lens designs, with multifocal and enhanced depth of focus lenses. 1 Most of the controversial discussions on aspheric lenses concern the amount of spherical aberration that should be corrected for comfortable vision; some researchers postulate that corneal spherical aberration should be fully corrected, while others propose a moderate correction of corneal aberration to maintain the robustness of the lens to displacement and tilt.However, after cataract surgery, the resulting (spherical) aberration of the pseudophakic eye results from corneal aberration, lens aberration and the alignment of the cornea and lens relative to the visual axis and the outline of the pupil.Schematic model eyes reflect an average behaviour of ocular imaging, and the description of the cornea varies across model eyes. 7For example, with the GULLME all surfaces are spherical and coaxial. 5,15n other schematic model eyes such as the ATCHME or the NAVME eyes, the surfaces are aligned but have, at least in part, an aspherical shape. 8,6,11In the LBME, the surfaces are aspherical, the lens shows a gradient index structure, the aperture stop is defined explicitly and the incident ray is slanted towards the axis of symmetry of the refracting surfaces in order to account for the eccentric position of the fovea in the human eye. 4As a consequence, the aperture stop is shifted by 0.5 mm in the nasal direction so that an incident ray bundle passes through the aperture stop and is focused on the fovea.However, multifocal and EDOF lenses are designed to maintain 'pseudoaccommodation'.There are different options to achieve multifocality, either with diffractive structures or with refractive designs which are rotationally symmetric or use sectors or zones to allow near vision. 1 When considering lenses which are designed for simultaneously imaging both far and near objects onto the retina, the imaging properties of the cornea for both far and near distance objects have to be considered.However, most studies focus on the imaging properties of the cornea for distant objects, and ignore the change in imaging behaviour when viewing near objects.
In the present investigation we analysed imaging properties of the corneas of four schematic model eyes in a simulation setup for 2-5 mm apertures as a function of the object distance.This simulation was performed with numerical raytracing using the curvature, asphericity and refractive index data of the LBME, ATCHME, GULLME, and NAVME as shown in Table 1.For the LBME we considered the incident ray to be slanted by 5° towards the axis of symmetry as defined by the refracting surfaces, and the aperture stop to be decentred 0.5 mm nasally.The ATCHME, GULLME, and NAVME are all strictly coaxial and centred.For simulation of distant objects we defined a collimated beam with a diameter of 7 mm, and for the simulation of near objects we used a point light source located at the respective object distance, with a divergent beam having an equidistant ray pattern and a diameter of 7 mm at the corneal front apex plane.The rays were followed through the cornea with the aperture stop located at the front lens apex plane.There is no unique definition of best focus, and the choice of which plane is defined as best focus will depend on the application.In this study we used two different measures for best focus; the best focus plane was defined as the focus with the least rms scatter, 12 and the best wavefront focus plane was defined as the plane with the least rms wavefront error.For each schematic model eye and aperture size, we defined the location of the best focus and the best wavefront focus for 30 object distances in a logscale from 10 cm to 10 m plus infinity, and analysed the ray scatter together with the wavefront error in terms of optical path length differences from the source to the image.
The best focus and the best wavefront focus are located coaxially for the ATCHME, the GULLME and the NAVME, but off-axis for the LBME.The axial distance with respect to the anterior corneal apex increased from around 31 mm for objects at infinity to approximately 40 mm for objects positioned 10 cm in front of the eye for aperture sizes of 2 -5 mm (Figure 1).Comparing the best focus and best wavefront focus position with the respective paraxial focus derived from vergence calculations, we find that as a consequence of spherical aberration, both best focus positions are systematically closer to the cornea, and the offset increases with aperture size and decreases with object distance as shown in Figure 2. It is not surprising that the ray scatter increases systematically with aperture size as shown in Figure 3, but it also decreases with object distance.In addition, we find that the ray scatter is systematically larger with the best wavefront focus compared to the best focus plane, meaning that the target criterion for searching the focus position has a large impact on the ray scatter properties.It is also not surprising that for a larger aperture size the rms wavefront error increases for all four model eyes as shown in Figure 4.The wavefront error also decreases systematically with the object distance, and is systematically larger at best focus compared to best wavefront focus, which again supports the conclusion that the target criterion for searching the focus position affects the imaging metrics.
The most interesting finding of this study is that the spherical aberration term Z 4 0 of the Zernike decomposition of wavefront error referenced to a 6 mm zone (often used for selecting the appropriate aspherical lens design) shows a large decrease with object distance.For instance, in the LBME, the spherical aberration which is known to be around 0.26 µm (depending on the aperture size) for objects located at infinity, increases to 0.4 µm or more for object distances of 10 cm.With the GULLME with spherical surfaces, this increases from around 0.36 µm for objects at infinity to values of around 0.46-0.50µm for objects at 10 cm.For corneas of model eyes with larger spherical aberration (e.g., the GULLME with spherical surfaces), the effect of aperture size is much more pronounced compared with corneas of model eyes having low spherical aberration (e.g., the NAVME or LBME).
As a clinical consequence, especially for multifocal and EDOF lenses, even a so-called fully-correcting lens design intended to compensate for the total spherical aberration of an average cornea (e.g., the LBME) for distant objects will not correct the spherical aberration for near vision; this lens design will only partly correct spherical aberration at near.This image deterioration due to the residual (uncorrected) spherical aberration for near objects must be taken into account.
In conclusion, the imaging characteristics of the cornea for near objects are frequently ignored.Using a raytracing simulation study on the Liou-Brennan, Atchison, Gullstrand and Navarro schematic model eyes, we found that: 1. the focal distance of the cornea increases significantly with a decrease in object distance from around 31 mm for objects at infinity to approximately 40 mm for objects at 10 cm; 2. paraxial calculation systematically overestimates the focal distance, especially for larger aperture sizes and smaller object distances; 3. rms ray scatter at the focal plane is systematically increased with larger aperture sizes and shorter object distances; 4. rms wavefront error is systematically increased with larger aperture sizes and shorter object distances; 5. the spherical aberration term of the wavefront error is slightly increased for larger aperture sizes (depending on the model eye) and systematically greater for small object distances and 6. rms ray scatter and wavefront error differ significantly if the focus is defined as best focus (least rms ray scatter at the focal plane) or best wavefront focus (least rms wavefront error with respect to the central 6 mm zone at the corneal front surface).This means that the optimisation of focus on ray scatter is strictly at the cost of wavefront error and vice versa.

AC K N OW L E D G E M E N T
This work was supported in part by the Dr Rolf M Schwiete Foundation, Mannheim, Germany

F I G U R E 1
The axial position of the aperture stop is provided as the distance from the corneal back surface apex.TheLiou-Brennan model eye considers a decentred aperture stop and a tilted incident beam; all other model eyes are strictly centred and coaxial.Position of the best focus plane (blue) and best wavefront focus plane (green) for the model cornea and aperture sizes 2-5 mm as a function of object distance varied from 10 cm to 10 m and for infinity (Inf).a (upper left), b (upper right), c (lower left), and d (lower right) shows the situation for the cornea of the Liou-Brennan/Atchison/Gullstrand/Navarro schematic model eyes, respectively [Colour figure can be viewed at wileyonlinelibrary.com]RESULTSFor a collimated incident beam from an object at infinity, the location of the best focus with respect to the corneal front apex plane for the LBME/ATCHME/GULLME/NAVME for a 3 mm aperture was 31.2790(temporal: 2.0443 mm )/31.2170mm/30.7041mm/31.0091mm.The location of the best wavefront focus was 31.3348mm (temporal: 2.0 495 mm)/31.2694mm/30.7734mm/31.0395mm, respectively.The axial location of the best focus (in blue) and best wavefront focus (in green) for object distances from 10 cm to 10 m, and for a collimated beam (object at infinity) and 2-5 mm aperture sizes are shown in Figure1(1a: LBME, 1b: ATCHME, 1c: GULLME, 1d: NAVME).The focus moved from approximately 31 mm for objects at infinity to around 40 mm for objects at 10 cm for all model eyes and aperture sizes.

F
I G U R E 2 Position of the best focus plane (blue) and best wavefront focus plane (green) with respect to the paraxial focus derived from a vergence calculation for the model cornea and aperture sizes 2-5 mm as a function of object distance varied from 10 cm to 10 m and for infinity (Inf).a (upper left), b (upper right), c (lower left), and d (lower right) shows the situation for the cornea of the Liou-Brennan/Atchison/Gullstrand/Navarro schematic model eyes, respectively [Colour figure can be viewed at wileyonlinelibrary.com]

F I G U R E 3
Root-mean-squared (rms) ray scatter at best focus plane (blue) and at best wavefront focus plane (green) for the model cornea and aperture sizes 2-5 mm as a function of object distance varied from 10 cm to 10 m and for infinity (Inf).a (upper left), b (upper right), c (lower left), and d (lower right) shows the situation for the cornea of the Liou-Brennan/Atchison/Gullstrand/Navarro schematic model eyes, respectively [Colour figure can be viewed at wileyonlinelibrary.com]

F I G U R E 4
Root-mean-squared (rms) wavefront error at best focus plane (blue) and at best wavefront focus plane (green) for the model cornea and aperture sizes 2-5 mm as a function of object distance varied from 10 cm to 10 m and for infinity (Inf).a (upper left), b (upper right), c (lower left), and d (lower right) shows the situation for the cornea of the Liou-Brennan/Atchison/Gullstrand/Navarro schematic model eyes, respectively [Colour figure can be viewed at wileyonlinelibrary.com]

F I G U R E 5
Spherical aberration term Z40 extracted from the wavefront error referenced to a circle with diameter of 6 mm.Spherical aberration is determined for the best focus plane (blue) and the best wavefront focus plane (green) for the model cornea and aperture sizes 2-5 mm as a function of object distance varied from 10 cm to 10 m and for infinity (Inf).a (upper left), b (upper right), c (lower left), and d (lower right) shows the situation for the cornea of the Liou-Brennan/Atchison/Gullstrand/Navarro schematic model eyes, respectively [Colour figure can be viewed at wileyonlinelibrary.com] Specifications of the four schematic model eyes under test: R a /R p refer to the central corneal front and back surface curvature, Q a /Q p to the corneal front and back surface asphericity, d C to the central corneal thickness, n C and n A to the refractive index of cornea and aqueous humour T A B L E 1