Traditional schematic eye models are generic in the sense that they represent average anatomic and optical properties of adult eyes. This type of model is used to understand the optics and vision of the human eye and to design common visual optics. However, an individual eye can be very different from any models. All of the generic models are highly symmetric. They have ideal rotationally symmetric, centered, and aligned surfaces, whereas real eyes show degrees of irregularities with no well-defined optical or symmetry axes. Among the current published generic eye models, the wide-angle Navarro model, based on anatomical data, has been demonstrated to produce on-axis image quality as well as off- axis aberrations that are well aligned with human measurements [Escudero-Sanz 1999]. After evaluating many published eye models in my earlier work [Chen 2003], this model has been selected and used as the base model in majority of the recent CLA eye modeling research work.
The wide-angle Navarro model was built by incorporating published conic constant anatomical values into the Gullstrand-Le Grand spherical surfaces [Le Grand 1956] and by updating the values of the anterior radius and refractive index of the cornea using more recent anatomical data. Other refractive indices in Ref. [Le Grand 1956] were preserved for the standard D-line, 589.3 nm wavelength. Refractive indices for other wavelengths were estimated, departing from experimental data of chromatic dispersions, and adjusting experimental values of longitudinal chromatic aberration (Ref. [Navarro 1985] for details).
Most of current generic eye modeling research requires the assistance of optical design software such as ZEMAX, Code V, and OSLO for both the construction of models and the extension to applications in optical engineering. I have been using ZEMAX for the eye modeling procedure and this dissertation research work is utilized with only this program. ZEMAX is a program that assists the design of optical systems by providing optical modeling and analysis that is based on the ray tracing technology. The optical parameters of an eye model or an optical system are entered in a spread sheet format. Table 3.1 shows the lens data editor in ZEMAX with input parameters of Navarro eye model. The rows describe, from top to bottom, the object (OBJ), the surfaces of cornea (surfaces 1 and 2), pupil (STO; aperture stop), crystalline lens (surfaces 4, 5), and the imaging surface of retina (surface IMA).
The first column ―Surf: Type‖ shows a selected surface type from ZEMAX. The most commonly used optical surface is an aspherical surface named ―Standard Surface‖. Standard surface required 2 specified parameters: radius and conic constant. ZEMAX treats planes as a special case of the sphere (i.e. a sphere with infinite radius of curvature). The surface is centered on the ―current‖ optical axis, with the vertex located at the ―current‖ Z-axis position. The "sag" or z-value of the standard surface is given by
2 2 2
1
1 1(1
)
cr
z
Q c r
, (3-1)where c is the curvature (the reciprocal of the radius), r is the radial coordinate in the lens unit and Q is the conic constant. The radius of the surface vertex curvature is entered in the second column, ―Radius‖, in mm. The conic constant, Q, is assigned at the sixth column. The conic constant of less than -1 describes a hyperbolas surface, -1 describes parabolas, between -1 and 0 is ellipses, 0 defines spheres, and greater than 0 depicts an oblate ellipsoids. As shown in Figure 3.1, the colored lines illustrate the anterior corneal surfaces for different conic constants with the same cornea radius of curvature, R =7.72 mm. Shown on the right is the zoom-in area of 5 mm (corneal radius direction) by 2.5 mm (thickness in z-direction).The effect of conic constant is more observable at the periphery of cornea. Although the human corneal surface extends about 5.5 mm in radius, the most effective visual zone falls inside the center 2mm of radius due to the limitation of the pupil stop. Although the conic constant doesn‘t seem to cause much variation inside 2 mm visual zone, in general it produces significant spherical aberration (SA) and impacts the imaging quality appreciably.
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The third column ―Thickness‖ expresses the distance from the vertex of the present surface to the vertex of the next surface in mm. The fourth column ―Glass‖ is the refractive index data of the material between the current surface and the next surface. For each ―glass‖ name, the glass name entered must be in one of the currently loaded glass catalogs. The parameters of the refractive index should have been added to that glass catalog. If the optical computation considers multiple wavelengths, the data should include dispersion information over the spectral range. The fifth column ―Semi-Diameter‖ (diameter/2) describes the aperture size of each surface. Columns after the sixth describe the decentering of the apex and the tilting parameters of the surface. Since all the surfaces in Navarro model are centered and symmetric to the optical axis as well as most optical system, they are not shown in Table 3.1.
After the data input in the lens data editor, analysis tools of ZEMAX can be used to illustrate the result. Figure 3.2 shows a typical 3-D layout of an eye model in ZEMAX. With an eye model constructed in ZEMAX, light-rays can be traced from the object space (OBJ) sequentially through system to the image plane (IMA), i.e. the retina, in Snell Law. Optical analysis, including point spread function (PSF), wavefront aberration (WFA), Spot diagram, etc. are available in ZEMAX for examining the optical performance. With specified merit functions, ZEMAX uses a mathematical algorithm to perform the Optical Optimization iteration until the specified target criteria are met. The following sections describe these optical optimizations in the approach to the application to real human eyes.
The step-by-step general eye modelling procedures is described in Appendix A. In ZEMAX website [Tocci 2007], and there is also the step-by-step procedure of modelling Liou 1997 model [Liou 1997], which uses gradient refractive index for lens. In addition, a forward, a backward, and a non-sequential eye model module can be downloaded in website [Watkins 2007].
After an eye model is constructed, validation is required. This is normally done by comparing the optical performance of the model with real human eyes. In general optical system design, analysis is performed on the aberrations and final result is examined with Spot Diagram (SPD), Point Spread function (PSF), and Modulation Transfer function (MTF). Agreement with mean ocular aberrations confirms the final general eye model.
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Figure 3.1 (Left): Anterior corneal surface diagram of different conic constants with same cornea curvature of radius=7.72 mm. (Right): The zoom-in block, 5 mm radius by 2.5 mm thick, as indicated in
the left picture.
Figure 3.2 A 3-D layout of eye model in ZEMAX program. The left most plane surface is a dummy surface for illustration, which is not included in Table 3.1.