Semi-analytical finite ray-tracing through the quadratic symmetric GRIN lens.

The propagation of light within a gradient index (GRIN) media can be analyzed with the use of differential equations for a given non-homogenous refractive index profile. Numerical methods are often necessary to perform ray-tracing in GRIN media; however, analytical solutions exist for several types of GRIN lenses. In this paper, paraxial and non-paraxial differential equations are derived to calculate the ray path in a GRIN lens. It is shown that the paraxial equation has an analytical solution for a GRIN media with a quadratic profile within the paraxial region. The analytical solution can be obtained by using Legendre polynomials or by the Frobenius method involving a power series. Using the Legendre or Frobenius solution, one can calculate the refractive indices along the ray path. A new recursive relationship is proposed to map the trajectory of light at finite heights. To illustrate the finite ray-tracing method utilizing a non-paraxial differential equation, two lenses (with spherical and elliptical iso-indicial contours) are considered. The lenses' back focal distances, for rays entering the lenses at varying finite heights, are calculated. For each lens, its spherical aberration is estimated. The effective focal length and the shape of the principal surface are also obtained. The accuracy of the results is then compared to the numerical ray-tracing using an optical design software, Zemax OpticStudio. The predicted spherical aberration for the spherical lens differs from numerical ray-tracing by less than λ14 at the marginal zone, while the error for the effective focal length is less than λ100.

Symplectic numerical methods in optics and imaging: ray tracing in spherical gradient-index lenses and computer-generated image rendering.

This paper provides an introduction to symplectic numerical integration techniques and examines various optical applications. We first outline the fundamentals of Hamiltonian optics and detail the construction of a symplectic method via the splitting technique. Numerical experiments involving a selection of spherically symmetric gradient-index lenses compare the accuracy of various first-, second-, and fourth-order symplectic methods with equivalent nonsymplectic methods. The best-performing methods are then further tested as part of an image rendering task involving nonlinear ray tracing, comparing the trace time required by each method. Future improvements, recommendations, and uses for symplectic ray tracing are also considered.

Aplanatic meniscus lens corrector for Ritchey-Chrétien telescopes.

An analytically designed aplanatic meniscus lens is proposed for increasing the field of view (FoV) of Ritchey-Chrétien (RC) telescopes. Compared to other field correctors, the proposed solution does not affect the initial RC telescope's aplanatic properties, and it can be added or removed from an existing RC telescope without changing the mirrors' shape. The final system features less astigmatism, which is balanced to flatten the image surface.

Development of a prototype active optics system for future space telescopes.

It is envisaged that future large space telescopes will be lightweight and employ active optics to maintain optical quality throughout the mission lifetime. We have proposed a 4 m, two-mirror space telescope with an active optics system based on reimaging the telescope primary mirror onto a small active mirror (110 mm optical pupil). Using Zemax, we demonstrate the feasibility of using this mirror to correct low-order Zernike aberrations and show that the aberration is well corrected across the 2.5 arcmin field of the telescope, operating at 0.55 µm. We describe the modeling carried out to develop the active mirror design. Using end-to-end modeling, a 25-actuator mirror with polar actuator geometry, and a ratio of mechanical to optical pupil diameter of 2 has been chosen. A single-actuator prototype has been manufactured and used to test stroke, linearity, and hysteresis. Finally, we describe the design of a laboratory breadboard that will image phase screens onto an exact replica of the space active mirror and show the results of measuring the phase screen accuracy.

Nonpupil adaptive optics for visual simulation of a customized contact lens.

We present a method for determining the deformable mirror profile to simulate the optical effect of a customized contact lens in the central visual field. Using nonpupil-conjugated adaptive optics allows a wider field simulation compared to traditional pupil-conjugated adaptive optics. For a given contact lens, the mirror shape can be derived analytically using Fermat's principle of the stationary optical path or numerically using optimization in ray-tracing programs. An example of an aspheric contact lens simulation is given to illustrate the method, and the effect of eye misalignment with respect to the deformable mirror position is investigated. The optimal deformable mirror conjugation position is found to be near the posterior corneal surface. Chromatic aberration analysis is also presented, and our findings indicate that the polychromatic simulation quality is similar to that of the monochromatic case, even though the mirror is a reflective component. The limitations of a single continuous surface deformable mirror to mimic a contact lens are outlined, with some recommendations for improving the quality of simulation.

Extending the depth of field with chromatic aberration for dual-wavelength iris imaging.

We propose a method of extending the depth of field to twice that achievable by conventional lenses for the purpose of a low cost iris recognition front-facing camera in mobile phones. By introducing intrinsic primary chromatic aberration in the lens, the depth of field is doubled by means of dual wavelength illumination. The lens parameters (radius of curvature, optical power) can be found analytically by using paraxial raytracing. The effective range of distances covered increases with dispersion of the glass chosen and with larger distance for the near object point.

Crystalline lens paradoxes revisited: significance of age-related restructuring of the GRIN.

The accommodating volume-constant age-dependent optical (AVOCADO) model of the crystalline lens is used to explore the age-related changes in ocular power and spherical aberration. The additional parameter m in the GRIN lens model allows decoupling of the axial and radial GRIN profiles, and is used to stabilise the age-related change in ocular power. Data for age-related changes in ocular geometry and lens parameter P in the axial GRIN profile were taken from published experimental data. In our age-dependent eye model, the ocular refractive power shows behaviour similar to the previously unexplained "lens paradox". Furthermore, ocular spherical aberration agrees with the data average, in contrast to the proposed "spherical aberration paradox". The additional flexibility afforded by parameter m, which controls the ratio of the axial and radial GRIN profile exponents, has allowed us to study the restructuring of the lens GRIN medium with age, resulting in a new interpretation of the origin of the power and spherical aberration paradoxes. Our findings also contradict the conceptual idea that the ageing eye is similar to the accommodating eye.

Opto-mechanical design of a dispersive artificial eye.

We present an opto-mechanical artificial eye that can be used for examining multi-wavelength ophthalmic instruments. Standard off-the-shelf lenses and a refractive-index-matching fluid were used in the creation of the artificial eye. In addition to dispersive properties, the artificial eye can be used to simulate refractive error. To analyze the artificial eye, a multi-wavelength Hartmann-Shack aberrometer was used to measure the longitudinal chromatic aberration and the possibility of inducing refractive error. Off-axis chromatic aberrations were also analyzed by imaging through the artificial eye at two discrete wavelengths. Possible extensions to the dispersive artificial eye are also discussed.

Accommodating volume-constant age-dependent optical (AVOCADO) model of the crystalline GRIN lens.

The purpose of this manuscript is to introduce a new age-dependent model of the human lens with two GRIN power distributions (axial and radial) that allow decoupling of its refractive power and axial optical path length. The aspect ratio of the lens core can be held constant under accommodation, as well as the lens volume by varying the asphericity of the lens external surfaces. The spherical aberration calculated by exact raytracing is shown to be in line with experimental data. The proposed model is compared to previous GRIN models from the literature, and it is concluded that the features of the new model will be useful for GRIN reconstruction in future experimental studies; in particular, studies of the accommodation-dependent properties of the ageing human eye. A proposed logarithmic model of the lens core enables decoupling of three fundamental optical characteristics of the lens, namely axial optical path length, optical power and third-order spherical aberration, without changing the external shape of the lens. Conversely, the near-surface GRIN structure conforms to the external shape of the lens, which is necessary for accommodation modelling.

Geometry-invariant GRIN lens: finite ray tracing.

The refractive index distribution of the geometry-invariant gradient refractive index lens (GIGL) model is derived as a function of Cartesian coordinates. The adjustable external geometry of the GIGL model aims to mimic the shape of the human and animal crystalline lens. The refractive index distribution is based on an adjustable power-law profile, which provides additional flexibility of the model. An analytical method for layer-by-layer finite ray tracing through the GIGL model is developed and used to calculate aberrations of the GIGL model. The result of the finite ray tracing aberrations of the GIGL model are compared to those obtained with paraxial ray tracing. The derived analytical expression for the refractive index distribution can be employed in the reconstruction processes of the eye using the conventional ray tracing methods. The layer-by-layer finite ray tracing approach would be an asset in ray tracing through a modified GIGL model, where the refractive index distribution cannot be described analytically. Using the layer-by-layer finite ray-tracing method, the potential of the GIGL model in representing continuous as well as shell-like layered structures is illustrated and the results for both cases are presented and analysed.

An analytical method for predicting the geometrical and optical properties of the human lens under accommodation.

We present an analytical method to describe the accommodative changes in the human crystalline lens. The method is based on the geometry-invariant lens model, in which the gradient-index (GRIN) iso-indicial contours are coupled to the external shape. This feature ensures that any given number of iso-indicial contours does not change with accommodation, which preserves the optical integrity of the GRIN structure. The coupling also enables us to define the GRIN structure if the radii and asphericities of the external lens surfaces are known. As an example, the accommodative changes in lenticular radii and central thickness were taken from the literature, while the asphericities of the external surfaces were derived analytically by adhering to the basic physical conditions of constant lens volume and its axial position. The resulting changes in lens geometry are consistent with experimental data, and the optical properties are in line with expected values for optical power and spherical aberration. The aim of the paper is to provide an anatomically and optically accurate lens model that is valid for 3 mm pupils and can be used as a new tool for better understanding of accommodation.

Adjustable internal structure for reconstructing gradient index profile of crystalline lens.

Employing advanced technologies in studying the crystalline lens of the eye has improved our understanding of the refractive index gradient of the lens. Reconstructing and studying such a complex structure requires models with adaptable internal geometry that can be altered to simulate geometrical and optical changes of the lens with aging. In this Letter, we introduce an optically well-defined, geometrical structure for modeling the gradient refractive index profile of the crystalline lens with the advantage of an adjustable internal structure that is not available with existing models. The refractive index profile assigned to this rotationally symmetric geometry is calculated numerically, yet it is shown that this does not limit the model. The study provides a basis for developing lens models with sophisticated external and internal structures without the need for analytical solutions to calculate refractive index profiles.

Geometry-invariant GRIN lens: iso-dispersive contours.

A dispersive model of a gradient refractive index (GRIN) lens is introduced based on the idea of iso-dispersive contours. These contours have constant Abbe number and their shape is related to iso-indicial contours of the monochromatic geometry-invariant GRIN lens (GIGL) model. The chromatic GIGL model predicts the dispersion throughout the GRIN structure by using the dispersion curves of the surface and the center of the lens. The analytical approach for paraxial ray tracing and the monochromatic aberration calculations used in the GIGL model is employed here to derive closed-form expressions for the axial and lateral color coefficients of the lens. Expressions for equivalent refractive index and the equivalent Abbe number of the homogeneous equivalent lens are also presented and new aspects of the chromatic aberration change due to aging are discussed. The key derivations and explanations of the GRIN lens optical properties are accompanied with numerical examples for the human and animal eye GRIN lenses.

Subjective evaluation of intraocular lenses by visual acuity measurement using adaptive optics.

We present a new method for subjectively evaluating intraocular lenses (IOLs) without implantation surgery. To illustrate the method, three types of single-piece IOL (equispherical monofocal, rotational symmetric aspheric monofocal, and diffractive bifocal) were assembled into a model eye and evaluated using an ocular adaptive optics system by a single subject. To separate the spherical aberration of the crystalline lens, the subject's corneal topography and wavefront aberrations were measured and modeled. Three levels of Zernike spherical aberration were generated and superposed on the IOLs and the subject's eye. The corrected distance visual acuity was measured by psychophysical visual procedure.

Geometry-invariant gradient refractive index lens: analytical ray tracing.

A new class of gradient refractive index (GRIN) lens is introduced and analyzed. The interior iso-indicial contours mimic the external shape of the lens, which leads to an invariant geometry of the GRIN structure. The lens model employs a conventional surface representation using a coincoid of revolution with a higher-order aspheric term. This model has a unique feature, namely, it allows analytical paraxial ray tracing. The height and the angle of an arbitrary incident ray can be found inside the lens in a closed-form expression, which is used to calculate the main optical characteristics of the lens, including the optical power and third-order monochromatic aberration coefficients. Moreover, due to strong coupling of the external surface shape to the GRIN structure, the proposed GRIN lens is well suited for studying accommodation mechanism in the eye. To show the power of the model, several examples are given emphasizing the usefulness of the analytical solution. The presented geometry-invariant GRIN lens can be used for modeling and reconstructing the crystalline lens of the human eye and other types of eyes featuring a GRIN lens.

Comparison of retinal image quality with spherical and customized aspheric intraocular lenses.

We hypothesize that an intraocular lens (IOL) with higher-order aspheric surfaces customized for an individual eye provides improved retinal image quality, despite the misalignments that accompany cataract surgery. To test this hypothesis, ray-tracing eye models were used to investigate 10 designs of mono-focal single lens IOLs with rotationally symmetric spherical, aspheric, and customized surfaces. Retinal image quality of pseudo-phakic eyes using these IOLs together with individual variations in ocular and IOL parameters, are evaluated using a Monte Carlo analysis. We conclude that customized lenses should give improved retinal image quality despite the random errors resulting from IOL insertion.

Investigation of the isoplanatic patch and wavefront aberration along the pupillary axis compared to the line of sight in the eye.

Conventional optical systems usually provide best image quality on axis, while showing unavoidable gradual decrease in image quality towards the periphery of the field. The optical system of the human eye is not an exception. Within a limiting boundary the image quality can be considered invariant with field angle, and this region is known as the isoplanatic patch. We investigate the isoplanatic patch of eight healthy eyes and measure the wavefront aberration along the pupillary axis compared to the line of sight. The results are used to discuss methods of ocular aberration correction in wide-field retinal imaging with particular application to multi-conjugate adaptive optics systems.

The achromatic design of an atmospheric dispersion corrector for extremely large telescopes.

For off-zenith observations with ground-based astronomical telescopes, the effect of atmospheric dispersion relative to diffraction on image size increases with telescope diameter. Correction of atmospheric dispersion in extremely large telescopes (ELTs) might become critical. A common solution for ELTs is to use linear atmospheric dispersion correctors (ADCs). In spite of their simplicity, the intrinsic chromatic aberrations of linear ADCs could render diffraction-limited imaging impossible when used in a fast focus. The chromatic problems of the linear ADC in ELTs can be resolved by replacing the linear ADC by the achromatic ADC designs presented here, which provide diffraction-limited image quality and offer several opto-mechanical advantages over linear ADCs.

All-spherical catadioptric telescope design for wide-field imaging.

The current trend in building medium-size telescopes for wide-field imaging is to use a Ritchey-Chrétien (RC) design with a multilens corrector near the focus. Our goal is to find a cost-effective alternative design to the RC system for seeing-limited observations. We present an f/4.5 all-spherical catadioptric system with a 1.5° field of view. The system consists of a 0.8 m spherical primary and 0.4 m flat secondary mirror combined with a meniscus lens and followed by a three-lens field corrector. The optical performance is comparable to an equivalent f/4.5 RC system. We conclude that, for telescopes with apertures up to 2 m, the catadioptric design is a good alternative to the RC system.

Reconstruction of the optical system of the human eye with reverse ray-tracing.

We present a practical method for reconstructing the optical system of the human eye from off-axis wavefront measurements. A retinal beacon formed at different locations on the retina allows probing the optical structure of the eye by the outgoing beams that exit the eye through the dilated pupil. A Shack-Hartmann aberrometer measures the amount of wave aberrations in each beam at the exit pupil plane. Wavefront data obtained at different oblique directions is used for tomographic reconstruction by optimizing a generic eye model with reverse ray-tracing. The multi-configuration system is constructed by tracing pre-aberrated beams backwards from each direction through the exit pupil into the optical system of the aberrometer followed by the generic eye model. Matching all wave aberrations measured at each field point is equivalent to minimizing the size of the beacon spots on the retina. The main benefit of having a personalized eye model is the ability to identify the origin of the ocular aberrations and to find the optimal way for their correction.