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the end of the optimization, as the system approaches close to the diffraction limit, the optimization is changed to WFA, and a “fine-tune” optimization is performed to get the desired system performance.

The default numbers of Rings and Arms in the ZEMAX optimization are 3 and 6. However, an eye model with a clinically measured corneal topography does not present a well-behaved rotational symmetry as those of the typical industrial optical elements. To prevent falling into local minimum, higher numbers of arms and rings are generally used in the optimization. Both damped least squares and orthogonal descent algorithms are usually performed and compared for the same reason.

13.2. Personalized and Population-Based Eye Modeling

The general eye models that were described were based on average ocular parameters obtained from young emmetropic adults. These models are generally rotationally symmetric, centered, and aligned surfaces, whereas the real eyes show degrees of irregularities with no well-defined optical or symmetry axes. In other words, both monochromatic aberration and transverse chromatic aberration are known to vary widely across subjects. This section considers more specific eye modeling.

13.2.1. Customized Eye Modeling

Customized eye models are based on the clinical measurement of the individual human eye. In the recent years, high precision ophthalmic patient data have become available to characterize anterior and posterior surfaces of the cornea, ocular WFAs, and ocular element biometry accurately. These measurements can be incorporated into the construction of an optically functional and analytical, personal-tailored, eye model. Figure 13.3 shows the Pentacam data of layers of corneal topography that can be used for modeling. In 2004, Navarro published the first construction of 19 personalized eye models of healthy eyes with the optical design software, CodeV.8 The concept of personalized eye modeling is displayed in Fig. 13.4. The patient data may include modeling parameters, such as refractive index and ocular geometric data, and information that evaluates the optical performance, such as total WFA, refractive error measurement, and VA. It is

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Fig. 13.3. Pentacam data of corneal surfaces for eye modeling.

Fig. 13.4. Schematic diagram of the method of personalized eye modeling.

almost guaranteed that a generic eye model is needed as the base model that provides a draft functioning model with a complete set of mean optical parameters. For an adult eye, we have used the emmetropic Navarro model of 1985 that uses constant index refraction for the lens rather than the more computationally burdensome gradient index. The unaccommodated state model was based on the Gullstrand-LeGrand model with modifications in the asphericities (conic constants) of the refractive surfaces as well as the radius of the anterior corneal surface and the dispersions of the refractive

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indexes to fit the longitudinal chromatic aberration. This model varies continuously with accommodation and reproduces the overall average optical performance (aberrations, polychromatic MTF and PSF) of the eye both on-axis and off-axis remarkably well.

As shown in Fig. 13.4, after constructing the base model, the first step in optical modeling is the substitution of the optical and geometric parameters with the clinically measured data. Most research groups that perform eye modeling assume that the refractive indices are constants among individuals and are equal to those of the generic model used. However, with the published research results of the age dependence of refractive indices, it is feasible to adapt age-corrected refractive indices, as well as many other age dependent parameters. The clinically obtained geometric data of the patient’s eye will remain invariable parameters throughout the modeling process. One potentially important aspect is the dependence of the result on the initial generic model used, because the final personalized model will depend on those parameters of the initial model that are not changed (iterated) during the optimization process.

The replacement of the thicknesses data of ocular elements obtained from the ultrasound biometry is straightforward. The topography substitution requires description. The corneal topographic data exported from the clinical instrument require mathematical interpolation and extrapolation to obtain a sufficient area, namely a 10-mm diameter, of an anterior corneal map. The topography map(s) should be presented in elevation instead of power. The corneal map obtained in the clinic often contains missing data points due to the interference of eyelids and eyelashes. C++ can be used to create a ZEMAX-readable grid sag from this corneal map. Before the insertion of the user-defined surface, the grid sag surface should be selected in the anterior corneal surface in the lens-data-editor of the draft model. Misalignment between separate sets of clinical data should be prevented during the measurements and minimized in the modeling.

After the step 1, an initial personalized eye model is formed. The second step is the validation of this model through the optical optimization to approach the expected optical performance of the patient vision. As mentioned earlier, the most commonly acquired clinical eye data that indicates vision quality are the refractive errors and the VA. The VA measurement gives an estimated minimum requirement for the focus size of retinal blur

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(i.e. the size of the PSF). Refractive error is the second-order aberration that corresponds to the three variables that are related to sphero-cylindrical refraction prescription (S, C, X).

13.2.1.1. Optimization to the refractive error

When a specified refractive error of a model eye is requested, unless the ocular axial length is known for the individual eye, the optimization may be performed in two steps. Because the axial length of the human eye is the dominant factor of the spherical refractive error, the first step of optimization may require assigning the axial length as a free variable in the iteration. Anterior chamber depth (ACD) and lens thickness (LT) are generally age related and less dependent on refractive error. ACD and LT may be modified from the base model according to the age if these parameters are not obtained from the clinic. The free variable that is used to achieve the desired spherical equivalent refraction is normally the vitreous chamber depth (VCD). The merit function at this stage is the demand for an exact spherical equivalent. An ideal Gaussian thin lens of the desired refractive correction will be entered in front of the cornea of the model. For example, to achieve a myopic (hyperopic) eye of 5D, a −5D (+5) thin lens will be added. With the correction lens in place, we expect that the typical optical optimization such as the use of default merit function in ZEMAX will bring the final model to the optimized image quality. The object (light source) is set at infinity. The point source at infinity will be best focused on the retinal surface after iteration. In the second optimization step, the cylindrical refractive error will be achieved. The correcting Gaussian thin lens that is anterior to the corneal surface must be replaced with the exact refractive error prescription of all three parameters, including astigmatism. Therefore, at least three free variables need to be assigned for the optimization. For the cylindrical refractive error, both crystalline lens and corneal are the possible sources. Axial length plays no role here. If the corneal topography was assigned from the patient’s clinical record, then the corneal surface parameters are fixed. The free variable can only be placed at lens parameters such as surface shape, de-center, or tilt.Another simple approach to simulate or approximate reality is to add a thin lens with three variables near the crystalline lens position. Axial length remains constant during this stage of optimization. After the

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