Wavefront Aberration Function

Abstract

In this paper we obtain equations for the two aperture second derivatives of the wavefront aberration function in terms of the position of the local sagittal and tangential foci. The derivatives presented here are valid for meridional rays. The local foci positions for any tangential aperture zone are calculated by tracing one ray that corresponds to that zone and using Coddington's equations. The second derivative equations can be incorporated in the usual procedure of aberration computation, increasing from two to four the number of equations available for each tangential ray traced.     

Derivation of the propagation equations for higher order aberrations of local wavefronts

From the literature the analytical calculation of local power and astigmatism of a wavefront after refraction and propagation is well known; it is, e.g., performed by the Coddington equation for refraction and the classical vertex correction formula for propagation. Recently the authors succeeded in extending the Coddington equation to higher order aberrations (HOA). However, equivalent analytical propagation equations for HOA do not exist. Since HOA play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the propagation equations of power and astigmatism to the case of HOA (e.g., coma and spherical aberration). This is achieved by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the aberrations of a propagated wavefront directly from the aberrations of the original wavefront containing both low-order and high-order aberrations.     

Generalized Coddington equations found via an operator method

Generalized Coddington equations allow the optical properties of an arbitrarily oriented incoming astigmatic ray bundle to be found following refraction by an arbitrary surface. Generalized Coddington equations are developed using the abstract concept of vergence and refraction operators. After suitable incoming vergence and refraction operators have been formed, these operators are re-expressed in a common coordinate system via similarity transformations created from the series of space rotations necessary to align the coordinate systems. The transformed operators are then added together to produce the vergence operator of the refracted ray bundle. When properly applied, these generalized Coddington equations may be used with complex wavefronts and complex refracting surfaces if local surface curvature properties are known for both where the two intersect. The generalized Coddington equations are given in matrix form so that they may be easily implemented.     

Aberration properties in a chirped grating for coarse wavelength division demultiplexing

We studied the imaging performance of a chirped grating for a demultiplexer designed for coarse wavelength division multiplexing using a wavefront aberration analysis and the ray tracing simulation. The demultiplexer was composed of a chirped grating, cylindrical lenses, and a waveguide. The best image point and the spot shape focused by the chirped grating were effectively calculated with the wavefront aberration. We applied the aberration analysis to design a waveguide to connect branched beams to photodetectors, and we confirmed the demultiplexing performance experimentally.     

Alternative uses of coddington's equations in optical design

Abstract

Considering a second-order patch surrounding an axial reference object point, formulas for the second field derivatives of the wavefront aberration function, corresponding to rays both in the tangential and sagittal sections, are given. To obtain these derivatives, Coddingtons equations are used in a way alternative to that employed to calculate the second aperture derivatives. The wavefront aberration function for any point in the patch is written in terms of data acquired tracing tangential rays from the axial point alone. The effectiveness of the procedures is tested numerically in two photographic objectives. The plots for the field derivatives can be incorporated to the traditional ones to improve the global optimization of the optical system.     

Paraboloid–aspheric lenses free of spherical aberration

A method to design singlet paraboloid-aspheric lenses free of all orders of spherical aberration with maximum aperture is described. This work includes all parametric formulas to describe paraboloid-aspheric or aspheric-paraboloid lenses for any finite conjugated planes. It also includes the Schwarzchilds approximations (which can be used to calculate one rigorous propagation of light waves in physic optics) to design convex paraboloid-aspheric lenses for imaging an object at infinity, with explicit formulas to calculate thicknesses easily. The results were verified with software through ray tracing.     

Variant of the anastigmatic telescope with three mirrors for back focal length

In this paper, an optical design is presented for an anastigmatic telescope with back focal length corrected with exact ray tracing to eliminate spherical, coma, and astigmatism aberrations. The telescope is formed of three conical mirrors, two of them polished on the same substratum. The optical design is divided into three stages: we began the design obtaining the Gaussian parameters in a first-order solution; posteriorly, were obtained analytically the three mirrors' asphericity in a third-order design. The final design stage consists of the implementation of the Fermat's principle, the Abbe sine condition, and the Coddington equations for the exact correction for the three aforementioned aberrations.     

Active optics null test system based on a liquid crystal programmable spatial light modulator

We present an active null test system adapted to test lenses and wavefronts with complex shapes and strong local deformations. This system provides greater flexibility than conventional static null tests that match only a precisely positioned, individual wavefront. The system is based on a cylindrical Shack-Hartmann wavefront sensor, a commercial liquid crystal programmable phase modulator (PPM), which acts as the active null corrector, enabling the compensation of large strokes with high fidelity in a single iteration, and a spatial filter to remove unmodulated light when steep phase changes are compensated. We have evaluated the PPM's phase response at 635 nm and checked its performance by measuring its capability to generate different amounts of defocus aberration, finding root mean squared errors below λ/18 for spherical wavefronts with peak-to-valley heights of up to 78.7λ, which stands as the limit from which diffractive artifacts created by the PPM have been found to be critical under no spatial filtering. Results of a null test for a complex lens (an ophthalmic customized progressive addition lens) are presented and discussed.     

Optical characterization of ophthalmic lenses by means of modulation transfer function determination from a laser speckle pattern

A method is presented for measuring the modulation transfer function of ophthalmic lenses by use of the generation of laser speckle with an integrating sphere. The measurements are performed with a rectangular double-slit aperture positioned at the output port of the integrating sphere. The distance between the lens and the detector determines the spatial frequency being tested; therefore high frequencies are tested close to the lens and low frequencies are tested far away from the lens. We can conclude that the double-slit method can be a versatile technique for comparing the optical quality of ophthalmic lenses from different makers.     

Aberrations introduced by a lens made from a birefringent material

We incorporate an algorithm for ray tracing in birefringent media into a full ray-tracing package based on the Mathematica software application. To validate the package, we compare the calculated and observed wave-front aberration introduced by an optical system that comprises lenses fabricated from birefringent material. Using the package, we calculate the influence of the lens shape factor on the aberrations associated with the e-ray polarization and show that it differs significantly from that of the o-ray polarization.     

Dynamic eye model for adaptive optics testing

An artificial dynamic eye model is proposed. The prototype enabled us to introduce temporal variations in defocus and spherical aberration, resembling those typically found in the human eye. The eye model consisted of a meniscus lens together with a modal liquid crystal lens with controllable focus. A diffuser placed at a fixed distance from the lenses acted as the artificial retina. Developed software allowed the user to precisely control the dynamic generation of aberrations. In addition, different refractive errors could simultaneously be emulated by varying the distance between the components of the model. The artificial eye was first used as a dynamic generator of both spherical aberration and defocus, imitating the behavior of a real eye. The artificial eye was implemented in an adaptive optics system designed for the human eye. The system incorporated an electrostatic deformable mirror and a Hartmann-Shack wavefront sensor. Results with and without real time closed-loop aberration correction were obtained. The use of the dynamic artificial eye could be quite useful for testing and evaluating adaptive optics instruments for ophthalmic applications.     

Microfluidic Contact Lenses

Contact lens is a ubiquitous technology used for vision correction and cosmetics. Sensing in contact lenses has emerged as a potential platform for minimally invasive point‐of‐care diagnostics. Here, a microlithography method is developed to fabricate microconcavities and microchannels in a hydrogel‐based contact lens via a combination of laser patterning and embedded templating. Optical microlithography parameters influencing the formation of microconcavities including ablation power (4.3 W) and beam speed (50 mm s^?1) are optimized to control the microconcavity depth (100 µm) and diameter (1.5 mm). The fiber templating method allows the production of microchannels having a diameter range of 100–150 µm. Leak‐proof microchannel and microconcavity connections in contact lenses are validated through flow testing of artificial tear containing fluorescent microbeads (Ø = 1–2 µm). The microconcavities of contact lenses are functionalized with multiplexed fluorophores (2 µL) to demonstrate optical excitation and emission capability within the visible spectrum. The fabricated microfluidic contact lenses may have applications in ophthalmic monitoring of metabolic disorders at point‐of‐care settings and controlled drug release for therapeutics.     

Dual camera calibration for 3-D machine vision metrology

A dual-camera calibration algorithm and procedure are presented. The algorithm is based on the method of direct linear transformation and results in an overdetermined set of linear algebraic equations that can be utilized in determining the coordinate position of some identifiable feature in a scene relative to any set of orthogonal axes. Both radial and tangential lens distortions are considered in a second step of the calibration algorithm to improve the accuracy of the position sensor. Since the algorithm incorporates both types of lens distortions, it can be applied to a camera with any combinations of lenses, e.g. a lens system with a supplementary lens for close-up observations     

Modelling the first- and third-order properties of conceptual thick lenses by using three air-spaced thin lenses

A virtual thick-lens module comprising three air-spaced thin lenses is proposed, which is able to have the identical first-order quantities and third-order Seidel aberrations for just real thick lenses, or groups or components within any thick lens, hence it is capable of studying the aberration behaviours of conceptual lenses without detailed structures. The three thin-lens powers are first evaluated to match the required first-order quantities. Then the aberrations of each thin lens are solved to satisfy the given total aberrations. When the incident rays are changed, each thin lens will induce new aberrations according to the thin-lens formulae. The new system aberrations of the thick module are obtained by adding individual thin-lens aberrations. The module can be directly applied to finite and infinite conjugates, focal and afocal lenses, as well as telecentric and non-telecentric lenses. Examples are given to simulate two real lenses and to optimise the balanced aberrations of a conceptual zoom lens.     

Optical tolerances of an actively tilted high-numerical-aperture two-lens objective for optical recording

Analytical expressions for the primary wave-front aberrations of an actively tilted two-lens objective are derived, and expressions for the higher-order wave-front aberrations for disk tilt of this lens system are presented. This analysis is important because the two-lens objective opens the way to achieving higher-numerical-aperture systems for optical recording with acceptable tolerances that cannot be achieved with a single-lens objective. To test whether the conclusions drawn from the analytically derived results remain valid for high numerical apertures, we compare the results with those obtained by ray tracing: It is shown not only that the two-lens system is tolerant of disk-thickness variations and decentering of the lenses but that it can also be made tolerant of disk tilt when the lens facing the disk is actively tilted.     

Calculation of optical imbalances in ophthalmic lenses using a new algorithm based on the local dioptric power matrix formalism.

We present a new algorithm to calculate the optical imbalances and differential prismatic effects that appear when two eyes look at an object through correcting eyeglasses. These are important magnitudes in ophthalmic optics because large amounts of them will disturb the binocular vision of the spectacle wearer. As a practical application of our algorithm, the distribution of optical imbalances and differential prismatic powers for a pair of progressive addition lenses has been calculated, and we obtain information about the effects of this kind of lens on the binocular vision of the wearer.     

Designing lenses to correct peripheral refractive errors of the eye.

The purpose of this work was to design ophthalmic lenses that correct peripheral refractive errors of human eyes along a meridian. We designed lenses with the tangential section of one surface based on a figured spheroid but figured in the tangential section only. The curvature of the sagittal section of this surface was adjusted separately. A merit function was used to modify these surfaces until the lenses had power errors that corrected the eye. Examples are presented of lenses that correct a schematic eye. They do excellent jobs of correcting the peripheral power errors of the eye and are relatively insensitive to small changes in fitting distance. We conclude that it is theoretically feasible to design lenses to correct peripheral refractive errors.     

Reflection and refraction of narrow Gaussian beams with general astigmatism at tilted optical surfaces: a derivation oriented toward lens design

The formulas for the reflection and refraction of a narrow Gaussian beam with general astigmatism at a tilted optical surface are derived by ray-tracing techniques. The propagation direction of the reflected and refracted beams is computed by tracing the central ray of the incident beam, and the characteristic parameters of the respective wavefronts are worked out by applying the formulas developed for the generalized ray tracing. Moreover, the Gaussian form of the reflected and refracted amplitude distributions along the transverse coordinates is determined by requiring the matching of the incident, reflected, and refracted light spots on the optical surface. No limiting assumptions are made regarding the form of the optical interface or the orientation of the incident astigmatic wavefront. In the end, to illustrate a simple application of these formulas, the reflection of a Gaussian beam at a conicoid is considered, and a simple property of the conicoidal mirrors is reported.     

Modelling the primary chromatic aberrations and secondary spectrum of conceptual thick-lens modules

A virtual thick-lens module comprising three air-spaced thin lenses is proposed, which is able to have the identical primary chromatic aberrations and secondary spectrum for just real thick lenses, or groups or components within any thick lens, hence it is capable of studying the chromatic aberration behaviours of conceptual lenses without their detailed structures. The three thin-lens powers are first evaluated to match the required first-order quantities. Then the chromatic aberration coefficients of each thin lens are solved to satisfy the given system aberrations associated with the reference optical configuration. When the incident rays are changed, each thin lens will induce new chromatic aberrations according to the thin-lens formulae. The new system aberrations are obtained by adding those individual thin-lens aberrations together. The module can be directly applied to finite and infinite conjugates, focal and afocal lenses, as well as telecentric and non-telecentric lenses. Evaluations of the chromatic aberration variations of three different types of lenses are demonstrated.     

Ray tracing of Fresnel systems

In this study based on an original program of aberration calculations and automated optical lens design we try to increase ray tracing of a Fresnel system so that the original program not only contains the initial functions but also satisfies the calculations of a Fresnel or mixed system (both general and Fresnel lenses).