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Designing extended-depth-of-focus wavefronts is required in multiple optical applications. Caustic location and structure analysis offer a powerful tool for designing such wavefronts. An intrinsic limitation of designing extended-depth-of-focus wavefronts is that any smooth surface, with a non-constant mean curvature, unavoidably introduces a separation between caustic sheets, which is proportional to the ratio of change of the mean curvature along a curve embedded in the wavefront. We present a method to obtain extended-depth-of-focus wavefronts where the mean curvature variation ratio is reduced thanks to using a long circle-involute space curve effectively filling the wavefront surface. Additionally, we present a variant of the method in which the wavefront is modified within a small tubular neighborhood of the circle involute in order to partially meet the umbilical condition along that tubular region. Finally, we provide some numerical results showing the potential of our method in an application example.

There is a trend in optical system design toward explicitly considering real-world industrial demands in the metrics to be optimized, from which emerges a cost-performance trade-off. Another relevant recent tendency is the so-called end-to-end design, where the design metric is an expected quality index of the final image, after digital restoration. We propose an integrated approach for analyzing the cost-performance trade-off in end-to-end designs. We exemplify it with a simple optical model where the cost is determined by the inclusion of an aspherical surface. We show that the resulting optimal trade-off configurations when applying an end-to-end design are substantially different from a conventional design. Such differences, as well as the increase in performance, are especially significant for lower-cost configurations.

The Levi-Civita wavefront coupling problem consists of, given two prescribed wavefronts, obtaining a refractive or reflective surface coupling them. We propose a robust numerical method to solve Levi-Civita's problem, whose rationale is to consider that Levi-Civita's solutions are level surfaces of the point characteristic function established between points of incoming and outgoing wavefronts. The method obtains both surface data points and their normals, enabling a more robust surface reconstruction. We carry out a detailed error analysis of our method by means of comparing the surface data estimation with nominal surfaces obtained in reference tests offering analytical solutions to Levi-Civita's problem. The method offers, in computer simulations, highly accurate results with moderate computational cost.

Multifocal and/or extended depth-of-focus designs are widely used in many optical applications. In most of them, the optical configuration has axial symmetry. A usual design strategy consists of exploring the optimal wavefronts that emerging out of the optical system would provide the desired multifocal properties. Those properties are closely related to light concentration on caustic surfaces. We present a systematic analysis of how to obtain those multifocal wavefronts given some prescriptions on the locations of caustics. In particular, we derive several multifocal wavefronts under archetypical prescriptions in the sagittal caustic alone, or combined with the tangential one at certain points, with some emphasis on visual optics applications.

Multifocal lenses comprising progressive power surfaces are commonly used in contact and intraocular lens designs. Given a visual performance metric, a wavefront engineering approach to design such lenses is based on searching for the optimal wavefront at the exit pupil of the eye. Multifocal wavefronts distribute the energy along the different foci thanks to having a varying mean curvature. Therefore, a fundamental step in the wavefront engineering approach is to generate the wavefront from a prescribed mean curvature function. Conventionally, such a thing is done by superimposing spherical wavefront patches and maybe adding a certain component of spherical aberration to each spherical patch in order to increase the depth-of-field associated with each focus. However, such a procedure does not lead to smooth wavefront solutions and also restricts the type of available multifocal wavefronts. We derive a new, to the best of our knowledge, mathematical method to uniquely construct multifocal wavefronts from mean curvature functions (depending on radial and angular coordinates) under certain numerically justified approximations and restrictions. Additionally, our procedure leads to a particular family of wavefronts (line-umbilical multifocal wavefronts) described by 2 conditions: (1) to be smooth multiplicative separable functions in the radial and angular coordinates; (2) to be umbilical along a specific segment connecting the circle center with its edge. We provide several examples of multifocal wavefronts belonging to this family, including a smooth variant of the so-called light sword element.

Progressive addition lenses (PALs) contain a surface of spatially varying curvature, which supplies variable optical power for different viewing areas over the lens. We derive complete compatibility equations providing the exact magnitude of a cylinder along lines of curvature on any arbitrary PAL smooth surface. These equations reveal that, contrary to current knowledge, the cylinder and its derivative depend not only on the principal curvature and its derivatives along the principal line but also on the geodesic curvature and its derivatives along the line orthogonal to the principal line. We quantify the relevance of the geodesic curvature through numerical computations. We also derive an extended and exact Minkwitz theorem restricted only to be applied along lines of curvature, but excluding umbilical points.

We present a comprehensive procedure to simulate real-world scenes viewed through ophthalmic lenses. Such a method enables us to anticipate the effects on image formation of the following combined undesired optical defects typically found in ophthalmic lenses: blur, distortion, and chromatic aberration. Additionally, it helps in comparing the expected scenes seen with different lens designs. The procedure is based on the following steps: (1) to calculate the distortion and local dioptric matrix associated with a set of different gaze directions; (2) to estimate point spread functions (PSF) associated with these matrices; (3) to compute the joint action of distortion, chromatic aberration, and PSF field on the scenes. We illustrate this procedure with two +5D spherical lenses: a moderately good performance lens and a highly degrading one. The method is suitable to evaluate ophthalmic lenses in a virtual reality framework.

Blur and non-uniform magnification are two related undesired effects affecting vision when looking through eyeglasses. We propose a geometrical framework to study the relationship between both effects. Magnification and blur are locally characterized by dioptric and magnification matrices, respectively, which we compute here by using a novel two-ray numerical method. We propose a set of geometrical entities associated with the dioptric and magnification local matrices, which are analyzed in several examples. We prove that there is a strong correlation between such entities (e.g., astigmatism and anamorphic distortion), which, to a certain extent, is maintained even in highly asymmetric lenses. We also show the somewhat anti-intuitive result that the axis of maximal blurring is sometimes close to orthogonal to the axis of maximal distortion. The results provide useful clues for ophthalmic lens design.

In a previous contribution [Appl. Opt.51, 8599 (2012)], a coauthor of this work presented a method for reconstructing the wavefront aberration from tangential refractive power data measured using dynamic skiascopy. Here we propose a new regularized least squares method where the wavefront is reconstructed not only using tangential but also sagittal curvature data. We prove that our new method provides improved quality reconstruction for typical and also for highly aberrated wavefronts, under a wide range of experimental error levels. Our method may be applied to any type of wavefront sensor (not only dynamic skiascopy) able to measure either just tangential or tangential plus sagittal curvature data.

A widespread type of multifocal intraocular lens (MIOL) is based on expanding the depth of focus with specific amounts of spherical aberration. However, knowing the optimal wavefront aberration for multifocality does not directly provide a MIOL geometry. To overcome this issue, we present a new strategy to design MIOLs. The method optimizes directly the IOL surface geometries (aspheres with aspherical coefficients up to tenth order) using a multisurface pseudophakic eye model and a multiconfiguration approach, where the merit function jointly considers the optical quality at different object plane locations. An example of MIOL [22 diopters (D) far distance correction] was designed. For this design, the ocular modulator transfer function (MTF) at 50 cycles per millimeter remained above 0.47 for all object locations. The design provides high optical quality performance for far and intermediate distances and peak optical performance at near distances (MTF>0.57). Additionally, the design shows good performance against pupil changes (3-5 mm pupil diameter range). Finally, when the MIOL was tested on pseudophakic eye models with corneal spherical aberrations within a typical population range, the high multifocal performance was maintained in almost 40% of potential patients (ignoring asymmetric aberrations effects).

The aim of this work was to study, using numerical simulations, the attainable level of accuracy to reconstruct the wavefront aberrations from tangential refractive power data measured with dynamic skiascopy. Two mathematical methods have been implemented. The first one is based on curve integration of the curvature data, previously interpolated with cubic splines. The second one reconstructs the three-dimensional wavefront surface, represented by a Zernike polynomial expansion, using a two-step least-squares method. The different factors affecting the reconstruction--noise, sampling, and wavefront patterns--were quantified. The results provide useful information to design more efficient experimental setups based on spatial dynamic skiascopy.

The accuracy of the reconstruction of the Gradient Refractive Index (GRIN) of the crystalline lens from optimization methods was evaluated. Different input data, including direction cosines of deflected rays, ray impacts obtained from ray tracing and optical path differences from Optical Coherence Tomography (OCT) were studied. Three different GRIN models were analyzed. The experimental errors of the different experimental input data were estimated from comparisons of measurements and simulations using artificial lenses of known geometries. The experimental errors in the surfaces shape measurement and the influence of the number of rays were also considered. OCT-based input data produced the most accurate GRIN reconstructions. We found that optimization methods (combining global and local search algorithms) allow GRIN reconstructions with acceptable accuracies for moderate noise level., (© 2011 Optical Society of America)

A new and complete methodology of monofocal intraocular lens (IOL) design is presented aiming at isoplanatism, i.e. IOLs that provide the eye with optimized optical quality over a wide field of view (typically in a range of ten degrees). The methodology uses a merit function considering dimensional and biomechanical constraints, and a geometrical optical quality metric that is evaluated simultaneously at different field angles. As an example, we present new isoplanatic designs based on different commercial IOL platforms. Aspheric isoplanatic designs improve peripheral quality over current aspheric IOLs. Also, isoplanatic designs provide more stable optical quality across the field and across pupil diameter.

Alvarez and Lohmann lenses are variable focus optical devices based on lateral shifts of two lenses with cubic-type surfaces. I analyzed the optical performance of these types of lenses computing the first order optical properties (applying wavefront refraction and propagation) without the restriction of the thin lens approximation, and the spot diagram using a ray tracing algorithm. I proposed an analytic and numerical method to select the most optimum coefficients and the specific configuration of these lenses. The results show that Lohmann composite lens is slightly superior to Alvarez one because the overall thickness and optical aberrations are smaller.

Wave aberrations in the human eye are usually known with respect to the ideal spherical wavefront in the exit pupil. Using Kirchhoff's diffraction theory, we have derived a diffraction integral to compute the optical field on the retina from the wave aberration data. We have proposed a numerical algorithm based on the Stamnes-Spjelkavik-Pedersen (SSP) method to solve that integral. We have shown which approximations are admissible to reduce the complexity of the diffraction integral. In addition, we have compared our results with those of the conventional procedure used to compute intensities on the retina. We have found significant differences between our results and the conventional ones.

We propose a complete methodology to develop custom monofocal Intraocular Lens (IOL) designs and evaluate their performance on-axis based on an analytical formulation. The analytical formulation was based on Gaussian and primary aberration theory applied to custom (individual biometric data) and realistic (multilayer cornea and thick IOL) pseudoaphakic eye models. Gradient-based optimization algorithms were performed to search for optimal designs. Using two parameters, the best design was obtained by directly minimizing the wavefront variance. We showed, in a case example, that custom designs achieved better final performance than generic IOL designs. Tolerances analysis allowed an evaluation of the implications of the manufacturing errors of the different parameters.

We derive equations for defocus and primary spherical wave aberration coefficients caused by a shift in image plane of a perfect optical system. The spherical aberration equation is accurate at describing changes in the spherical aberration of an aberrated schematic eye.

Optical models of the human cornea and tear film typically employ a single homogeneous cornea with an average refractive index. I propose to use a more realistic multilayer model based on morphological data from the literature. The mathematical methodology to derive the refractive power equation of this model is presented. Special attention is given to the axial gradient index of the refraction structure of the stroma layer because of its optical implications. The importance of considering this multilayer model is quantified in a specific example (orthokeratology) with the help of the derived power equation.

A new wavefront sensing and reconstruction technique is presented. It is possible to measure Laplacian and gradient information of a wavefront with a Hartmann-Shack setup. By simultaneously using the Laplacian and gradient data we reconstruct the wavefront by sequentially solving two partial differential equations.

Computer simulations of alternative LASIK ablation patterns were performed for corneal elevation maps of 13 real myopic corneas (range of myopia, -2.0 to -11.5 D). The computationally simulated ablation patterns were designed with biconic surfaces (standard Munnerlyn pattern, parabolic pattern, and biconic pattern) or with aberrometry measurements (customized pattern). Simulated results were compared with real postoperative outcomes. Standard LASIK refractive surgery for myopia increased corneal asphericity and spherical aberration. Computations with the theoretical Munnerlyn ablation pattern did not increase the corneal asphericity and spherical aberration. The theoretical parabolic pattern induced a slight increase of asphericity and spherical aberration, explaining only 40% of the clinically found increase. The theoretical biconic pattern controlled corneal spherical aberration. Computations showed that the theoretical customized pattern can correct high-order asymmetric aberrations. Simulations of changes in efficiency due to reflection and nonnormal incidence of the laser light showed a further increase in corneal asphericity. Consideration of these effects with a parabolic pattern accounts for 70% of the clinical increase in asphericity.

Corneal and ocular aberrations were measured in a group of eyes before and after cataract surgery with spherical intraocular lens (IOL) implantation by use of well-tested techniques developed in our laboratory. By subtraction of corneal from total aberration maps, we also estimated the optical quality of the intraocular lens in vivo. We found that aberrations in pseudophakic eyes are not significantly different from aberrations in eyes before cataract surgery or from previously reported aberrations in healthy eyes of the same age. However, aberrations in pseudophakic eyes are significantly higher than in young eyes. We found a slight increase of corneal aberrations after surgery. The aberrations of the IOL and the lack of balance of the corneal spherical aberrations by the spherical aberrations of the intraocular lens also degraded the optical quality in pseudophakic eyes. We also measured the aberrations of the IOL in vitro, using an eye cell model, and simulated the aberrations of the IOL on the basis of the IOL's physical parameters. We found a good agreement among in vivo, in vitro, and simulated measures of spherical aberration: Unlike the spherical aberration of the young crystalline lens, which tends to be negative, the spherical aberration of the IOL is positive and increases with lens power. Computer simulations and in vitro measurements show that tilts and decentrations might be contributors to the increased third-order aberrations in vivo in comparison with in vitro measurements.