Equations to design an aplanatic catadioptric freeform optical system.

The present paper introduces a set of equations to design an aplanatic catadioptric freeform optical system. These equations form a partial differential equation system, in which a numerical solution defines the first and last surfaces of the catadioptric freeform optical system, composed of an arbitrary number of reflective/refractive surfaces with arbitrary shapes and orientations. The solution of the equation can serve as an initial setup of a more complex design that can be optimized. An illustrative example is presented to show the methodology introduced in this paper.

General equation of the achromatic stigmatic singlet.

The merge of two models is presented: the achromatic principle and the general equation of the stigmatic lens [J. Opt. Soc. Am. A39, 1524 (2022)JOAOD60740-323210.1364/JOSAA.460993; Appl. Opt.57, 9341 (2018)APOPAI0003-693510.1364/AO.57.009341]. The achromatic principle expresses the bases to design an optical imaging system rigorously free of chromatic aberrations for a given spectrum of electromagnetic waves. It has been applied only to Cartesian ovals. On the other hand, the equation of the stigmatic lens gives a perfect point image for a single wavelength. The merge consists of generalizing the equation of the stigmatic lens to be stigmatic for a given spectrum of electromagnetic waves based on the achromatic principle. The constraints to rigorously achieve the achromatic stigmatic singlet are studied in detail. No paraxial concepts are used, and no iteration process is required. An example of an achromatic stigmatic singlet is presented, and the results are as expected by theory.

Power set of stigmatic freeform catadioptric systems.

A method to design catadioptric systems from scratch based on optimizing an element of the power set of stigmatic catadioptric systems is presented. This set contains all possible stigmatic catadioptric systems. The deduction of the set is also presented in this paper, and its derivation is totally analytical. Additionally, an illustrative example of optimization of an element of the mentioned set is presented. The results are as expected.

Aplanatic freeform-mirror-based optical systems.

The exact partial differential equation to design aplanatic freeform-mirror-based optical systems is presented. The partial differential equation is not limited by the number of freeform surfaces or their orientations. The solutions of this partial differential equation can be useful as initial setups that can be optimized to meet higher criteria. One of these solutions is tested as an example of the initial setup, and the results are as expected by the theory.

Power set of mirror-based non-symmetric stigmatic optical systems.

The set of all possible stigmatic systems made by mirrors is presented. The derivation of the set is analytical, and it is based on the Fermat principle. The properties of the set are properties that all possible stigmatic systems made by mirrors share. The set is tested here with a practical example of optical design, and the results are as expected by theory. This example works with a large field of view rather than a single field, and the volume of the example is several times less than similar systems reported in the literature.

Exact vectorial model for nonparaxial focusing of freeform wavefronts.

We present a new formalism, based on Richards-Wolf theory, to rigorously model nonparaxial focusing of radially polarized electromagnetic beams with freeform wavefront. The beams can be expressed in terms of Zernike polynomials. Our approach is validated by comparing known results obtained by Richards-Wolf theory. Our integral representation is compliant with diffraction theory, is thoroughly discussed and solved for various freeform wavefront that, so far, have not been treated analytically. The extension of the method to other polarization states is straightforward.

Analytical equations for a nonconfocal stigmatic three-freeform-mirror system.

This paper presents a novel method, to the best of our knowledge, to design three-freeform-mirror systems from scratch. The technique consists of getting an initial setup, before optimization, which is directly obtained from the set of all possible stigmatic three-freeform-mirror systems. Then, deformation coefficients are added to each surface and optimized to reduce aberration produced by additional fields. The method has been tested and the results are as expected.

Set of all possible stigmatic pairs of mirrors.

Here, closed-form equations that express a pair of mirrors such that it forms a stigmatic optical system are presented. The mentioned equations are general enough to express the set of all possible pairs of stigmatic mirrors. Several examples for pairs of stigmatic mirrors are given and numerically tested with ray tracing, showing that their optical performance is, as expected, free of spherical aberration. Finally, the limitations and potential applications of stigmatic pairs are discussed.

Design of a pair of aplanatic mirrors.

In this paper, a new, to the best of our knowledge, differential equation for designing a pair of aplanatic mirrors is introduced. The differential equation is a direct consequence of the Fermat principle and Abbe sine condition. If it is solved, the solution expresses the shape of a pair of mirrors such that they form an aplanatic system. The differential equation has been solved numerically. We have also tested the performance of the pair of mirrors, which is as predicted by the theory.

Stigmatic singlet with a user-defined apodization pupil function.

Here we present a method to design a stigmatic lens with a user-defined apodization pupil function. The motive is that the apodization pupil function is required by Richards-Wolf diffraction integrals to compute non-paraxial diffraction patterns. Then, the user-defined apodization pupil function can be chosen such that the focus spot obtained by the stigmatic lens is smaller. The mentioned method is based on numerically solving a non-linear differential equation.

Exact equations to design aplanatic sequential optical systems.

We present the exact differential equations to design an aplanatic sequential optical system, a system that is free of spherical aberration and linear coma. We get the exact set of equations from the Fermat principle and the Abbe sine condition. We solve the mentioned set of equations by implementing the Runge-Kutta algorithm. We test the solutions using commercial ray-tracing software and confirm the expected behavior of the optical system.

Surface solution to correct a freeform wavefront.

In this paper we present the equation to design a refractive surface such that, given an arbitrary wavefront, the surface refracts it into a perfect spherical wave. The equation that computes these refractive surfaces is exhaustively tested using ray-tracing techniques, and the performance is as expected.

On the diffraction of a high-NA aplanatic and stigmatic singlet.

We present a study of the diffraction pattern according to Richards-Wolf for an aplanatic and stigmatic singlet based on an exact analytical equation. We are able to put emphasis on the maximum diameter and illumination pattern, which are the two parameters that influence the diffraction pattern and how to compute it. Designs of low- and high-NA aplanatic and stigmatic lenses are implemented to display these effects.

General stigmatic surfaces.

Given an arbitrary input wavefront, we derive the analytical refractive surface that refracts the wavefront into a single image point. The derivation of the surface is fully analytical without paraxial or numerical approximations. We evaluate the performance of the surface with several cases, and the results were as expected.

General mirror formula for adaptive optics.

We present the general formula to design a mirror such that it reflects the light of a given arbitrary wavefront as a plane wave for two and three dimensions. The formula is fully analytical and close-form. We test the mentioned equations with ray tracing techniques. The results were as expected. We do not use any paraxial concepts or numerical approximations during its derivation.

Two-mirror system for tunable apodization.

Here we present an optical system composed of two mirrors such that at the input/output, the light is a plane wave but with a user-defined apodization factor. The model presented is an analytic closed form with no numerical approximations or iterations. We test the model with illustrative scenarios, and the results are as expected; the system is stigmatic with the desired apodization factor. Thus, this system has several potential applications in high contrast imaging.

Uniqueness of stigmatic solutions.

We compare two analytical methods for designing stigmatic lenses that are based on very different paradigms published recently [Appl. Opt.57, 9341 (2018)APOPAI0003-693510.1364/AO.57.009341; J. Opt. Soc. Am. A37, 1155 (2020)JOAOD60740-323210.1364/JOSAA.392795]. In the process, we derive a third hybrid approach, which is the result of combining the two original methods. Given the same initial conditions, an accurate numerical analysis shows that the three methods yield the same results. This is clear evidence that the problem of designing a stigmatic lens for a known boundary condition has a unique solution independent of the formalism used.

General formula to design a freeform singlet free of spherical aberration and astigmatism: reply.

The comment made by Valencia-Estrada and García-Márquez [Appl. Opt.59, 3422 (2020)APOPAI0003-693510.1364/AO.379238] to our paper [Appl. Opt.58, 1010 (2019)APOPAI0003-693510.1364/AO.58.001010] consists of a trivial generalization of our formulation.

Analytic design of a spherochromatic singlet.

We derive the analytic formula of the output surface of a spherochromatic lens. The analytic solution ensures that all the rays for a wide range of wavelengths fall inside the Airy disk. So, its amount of spherical aberration is small enough to consider the lens as diffracted limited. We test the singlet lens using ray-tracing methods and find satisfactory results, including spot diagram analysis for three different Abbe wavelengths.

Singlet lens for generating aberration-free patterns on deformed surfaces.

We introduce a general closed-form analytic formula to design special lenses that generate spherical aberration-free extended images specified previously by the user. The formula considers arbitrary and non-conventional patterns. The formalism is tested with well-established ray tracing techniques.