Optical Eye Modeling and Applications

13.1.2.Eye Modeling Using Contemporary Optical Design Software

Most of current generic eye modeling research utilizes optical design software, such as ZEMAX, Code V, and OSLO, for both the construction of models and the subsequent applications in optical engineering. These programs assist the design of optical systems by providing optical optimization and analysis that is based on ray tracing technology. The optical parameters of an optical system or an eye model are entered in a spreadsheet format. In this contribution, ZEMAX will be used as an example of the modeling tool. Using parameters of Navarro eye model, Table 13.1 shows the form of the lens data editor in ZEMAX. From top to bottom in this table, the rows describe the light source (OBJ, object), the two surfaces of the cornea (surfaces 1 and 2), the pupil of the iris (STO; aperture stop), the two crystalline lens surfaces (surfaces 4 and 5), and the imaging surface of retina (surface IMA). From left to right, the first column “Surf: Type” allows the user to select many surface types from ZEMAX. The most commonly used optical surface is an aspherical surface named “Standard Surface.” The required two parameters of a “Standard Surface” are radius of the curvature (R) and conic constant (Q). 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 unless otherwise specified. The “sag” or z-value of the standard

Table 13.1. Parameters of Navarro eye model in ZEMAX lens editor.

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surface is given by:

where c is the curvature (the reciprocal of the radius R), r is the radial coordinate in the “lens unit,” and Q is the conic constant. The default “lens unit” is millimeters. 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. If the conic constant is less than −1, then it describes a hyperbolic surface. If it is −1, then it describes parabolas. If it is between −1 and 0, then it describes ellipses. If it is 0, then it defines spheres, and, if it is greater than 0, then it depicts oblate ellipsoids. As shown in Fig. 13.1, the colored lines illustrate the anterior corneal surfaces for different conic constants with the same corneal radius of curvature, R = 7.72 mm. The small influence of conic constant at the periphery of the cornea is not visible to human eye. Though the human corneal surface extends roughly 5.5 mm in radius, the most effective visual zone falls inside the center 2 mm of radius due to the

Fig. 13.1. Standard surfaces that are described with a curvature of radius = 7.72 mm and various conic constants.

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limitation of the pupil stop. Although the conic constant does not seem to cause much variation inside the 2 mm visual zone, it produces significant spherical aberration (SA) and affects the imaging quality appreciably.

The third column in the table, “Thickness,” expresses the distance from the vertex of the present surface to the vertex of the next surface in millimeters. The fourth column, “Glass,” represents the refractive index data of the material between the current surface and the next surface. For each “glass” name that is specified in the table, the corresponding refractive index parameters must be prepared and entered in one of the currently loaded glass catalogs in the ZEMAX program. 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 13.1.

After the data are entered in the lens data editor, the analysis tools of ZEMAX can be used to illustrate the result. Figure 13.2 shows a typical 3D 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 the system to the image plane (IMA), i.e. the retina, according to the Snell Law. Optical analysis, including point-spread function (PSF), WFA, and spot diagram (SPD), are available in ZEMAX for examining the optical

Fig. 13.2. A 3D layout of Navarro eye model in ZEMAX.

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performance. If the optical performance does not meet the required target or purpose, “Optical optimization” can be used to approach the target. With specified merit functions, ZEMAX uses a numerical algorithm to perform the optical optimization iteration until the specified target criteria are met. The validation of an eye model is determined by the closeness of the optical performance of the model to the target eye. In general, analysis is performed on the aberrations, and the final model is examined using the SPD, the pointspread function, and the modulation transfer function.

A systematic general eye modeling procedure can be found on the ZEMAX Web site.6 The ZEMAX modeling of a more complex Liou and Brennan model4 that uses a gradient refractive index lens can also be found on the Web site. In addition, a forward, a backward, and a nonsequential eye model module can be downloaded.7 The sequential forward and backward models allow ray tracing to be performed in one direction, while the nonsequential model treats the eye model as a single optical element. Light rays are allowed to diffract/reflect multiple times on the same surfaces until they exit the element or are absorbed.

13.1.3. Optical Optimization and Merit Function

For more specific or customized eye modeling, ocular parameters must be mathematically tailored in order to describe better the properties of the target eye. “Optical optimization” is the iteration algorithm that takes the initial optical design and changes the values of the assigned parameters (variables) in steps to approach the specified targets. The starting layout should have a suitable number of optical surfaces of appropriate types, since optimization can change only the values of the selected parameters, but not the number or types of surfaces. Optimization can be accomplished in three steps.

1.Construct a reasonable initial layout so that rays can be traced from the object plane to the image surface.

2.Specify free variables to be “optimized” and the corresponding tolerances to prevent unrealistic results or convergence to local minima.

3.Define the merit functions that describe the ultimate goals at the end of iteration.

Many previously published eye models can be used as the initial model. The parameters of a selected base model should be entered first. The

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variables of iteration, which are required for the optimization algorithm to progress, are specified next. Since optics are very precise (even distances of micrometers can make a big difference), we need to determine the values of all our variables at each step of the optimization carefully. The selection of variables is important for optimization. The less rigid parameters are assigned with more tolerance. In the eye modeling, variables are allocated on different ocular components at different modeling stages. After the variables are selected, suitable metrics are used as the indicators of progress of optical optimization. These metrics are defined as the merit functions. A merit function is a numerical representation of how closely the optimization result meets a specified set of goals. Usually, different merit functions will lead to different optimization results. Then, the final values of merit functions after optimization are indicators to evaluate the success of eye modeling. Therefore, the optimization and selection of merit functions are the most important process in the eye modeling procedure.

The optimization feature provided by the contemporary optical software programs is powerful. ZEMAX optimization uses an actively damped least squares or an orthogonal descent algorithm. The algorithms are capable of optimizing a merit function that is composed of weighted target values. ZEMAX has several default merit functions. For the majority of applications, the optical optimization is performed to achieve optimal imaging quality. In the other word, the default merit functions include attempts to minimize optical aberration or obtain the smallest focus spot (or PSF).

In eye modeling applications, the goals of optimizations are to produce a realistic human eye with personal clinically measured or validated ocular measurements. These specific merit functions are assigned using the Merit Function Editor in ZAMAX. If the clinical measured WFA map is available for an eye, the personalized eye modeling will aim to reproduce the exactly measured wavefront on the modeled eye. Since the clinical measured WFA data is typically expressed in Zernike polynomial coefficients, the merit function at the final optimization produces a wavefront of the exact series of Zernike coefficients. The ZEMAX operand, ZERN, which designates the intended set of Zernike coefficients of the target wavefront, will be used for this purpose. However, when the wavefront data are not obtained from the patient, the most common clinical eye examinee record,

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the sphero-cylindrical refraction prescription and the VA, will be the targets of optimizations.

ZEMAX default optimization is applied to improve the performance of wide-ranging optical systems. Generally, the goal is to produce the best imaging quality for the final optical system. The default merit functions are designed to approach either the minimum focus size or the spot radius in the SPD (i.e. the geometric optics approach) or the minimum aberration or root mean square WFA (RMS WFA, i.e. the wave optics approach). In eye modeling, RMS is typically used instead of peak-to-valley (PTV) optimization because of the weakness of PTV. The PTV approach considers only two points, the highest and the lowest, and ignores all points that lie between. Important issues, such as roughness are ignored, while a very small high or low point may be exaggerated beyond their significance. RMS greatly improves the PTV method since it takes into account areas on the optic that may vary when compared to the optic’s general surface characteristics.

The numerical value of the merit function is physically significant when using RMS as the optimization type. If the merit function is RMS- Wavefront-Centroid, then the numerical value of the merit function is the RMS wavefront error in the unit of waves (λ). If the merit function is RMSSpot Radius-Chief, then a value of 0.145 means the RMS spot radius is 0.145 lens units. If the lens units were millimeters, the RMS spot radius will correspond to a focus radius of 145 micrometers RMS. If more than one field or wavelength is defined, then the merit function numerical value is the weighted average of the RMS values for the various fields and wavelengths.

Note that optimization using the RMS spot radius merit function will, in general, yield an optimum design different from the RMS wavefront merit function. The reason for this difference in design is that ray aberrations are proportional to the derivative of the wave aberrations. Therefore, it is unreasonable to expect that the minimum of one aberration corresponds to the minimum of the other aberration. A general rule is to use wavefront error if the system is close to diffraction limited (for instance, a PTV wavefront error of less than two waves). Otherwise, use the spot radius.

In the eye modeling work, a typical focus size is larger than 2 micron for green light (555 nm). This size is derived by 1.22 λ f/d, where f is the equivalent focal length of the eye (17 mm), and d is the pupil diameter ( 6 mm). Therefore, usually, SPD is used first to run optimization. Then at

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