Freeform geometrical optics I: principles

Fractioned-pattern radiation mapping, Part I: modeling

Here, we present a technique that predicts the radiation's distribution in any optical system. It is based on decomposing the emitting source power by assigning a fraction of the total power to each emitted ray. All kinds of power losses in the rays' optical paths are considered. Fractioned radiation patterns are created in the last optical system surface, each associated with a single ray. We refer to fractioned patterns as those that conform to a whole radiating pattern. Thus, the irradiance of the completely illuminated surface is calculated by adding the optical system's fractioned radiation maps. This method is non-zero étendue. The result presented here allows for predicting the radiation patterns accurately with a handful of equations and can help design any image and non-image-forming optical systems.

Fractioned-pattern radiation mapping, Part II: assessment

In Part I, the authors proposed a theoretical background for predicting the radiation distribution in any optical system based on decomposing the emitting source power. Here, we describe the validity of this decomposition through a practical example that uses a radiating source and a single surface optical system. This source is calibrated in a metrology testbed that guarantees its traceability to the candela (cd), the International System (SI) base unit for luminous intensity I v. A second example, this time numerical, shows the method's performance in a multisurface optical system.

Freeform concentrator design for IR wireless-to-fiber link communications

Airplanes use heavy wired harnesses to provide multimedia services to the seats. Optical wireless communications (OWC) are a natural choice to reduce the amount of weight, reduce the wiring complexity, and avoid possible spurious electromagnetic radiation that risks affecting the airplane's navigation systems. The light's dual use as lighting and optical communications functionalities allows for providing light and multimedia content through the reading lamp. Thus, an optical system using optical fibers to replace wires and a reading lamp can provide a cabin seat with lighting and onboard connectivity. However, changing shielded harnesses by optical fibers is-from an optical design point of view-a challenging task as the reading lamp must also meet the stringent requirements to link the optical wireless transmissions to the optical fiber. The difficulty up to now lies in injecting the light emitted from the passenger's device into the optical fiber using the reading lamp as the receiving antenna and light injector. Here, we describe a proof-of-concept device that experimentally allowed for establishing a link between a transmitter and a photodetector coupled to an optical fiber-end, i.e., the link consisted of an optical wireless communication and the launching of the light modulated signal into an optical fiber. Additionally, from the experimental experience, we will describe the optical design strategies permitting designing a compound freeform concentrator to allow optical free space-to-fiber links.

Achromatic stigmatism: achromatic Cartesian ovoid

Monochromatic and chromatic aberrations are imaging defects mainly studied from a geometrical optics point of view. These defects are treated through optimization and minimization methods to achieve acceptable performance in optical imaging systems, where the correct choice of glass materials is one of the main challenges. The selection of glass materials is a complex issue that requires a large amount of computing power within sophisticated computational algorithms and enough professional experience in the area. However, in this work, we propose a new methodology to treat the chromatic and geometrical aberrations simultaneously by taking advantage of the relationship between form parameters of Cartesian surfaces and wavelength in the material. From this relationship, we obtain an achromatism principle that establishes the conditions for refracting systems to present a strictly achromatic stigmatism.

Spatial Three-Mirror Off-Axis Freeform Optical System without Any Symmetry

In this manuscript, we have launched a study on the completely nonsymmetric freeform optical system with neither rotational symmetry nor planar symmetry. An off-axis three-mirror freeform optical system with nonsymmetric geometry is proposed and a direct design method is developed for the nonsymmetric freeform optical system. The design field points are sampled across the full FOV to control the imaging quality and object–image relationship. In this system, the center of the image plane is greatly away from the plane determined by the centers of the three mirrors. This nonsymmetric system with F/1.3, a focal length of 50 mm, and an 8° × 6° field of view can achieve imaging quality close to the diffraction limit. This work provides a feasible nonsymmetric system design idea for the optical community.

Comprehensive design method of a soft multifocal contact lens with NURBS

This study presents a comprehensive method for designing a multifocal contact lens (MCL) with Snell's law and non-uniform rational B-spline (NURBS) curves. Instead of using thin lens approximation, general mathematical formulas have been developed to achieve the accurate coordinates of points on the anterior lens surface profile of the MCL to meet various given optical power distributions. Then the NURBS curve is adjusted to fit these data points to obtain the smooth front lens surface profile. This method not only improves the accuracy of the optical power profiles of MCLs but also reduces the spherical aberration in near/distance optical zones. The experimental results show that the power profiles of soft MCLs agree with those of the simulation results and original design requirements. The proposed method has been proven for the MCL design, and it can be feasibly applied in complex optical lens designs.

On-axis diffraction-limited bi-conical lenses

A diffraction-limited lens having both surfaces conic is shown. The analytical and numerical calculation for all possible solutions of the conical front and back surfaces is presented. Object and image distances, lens thickness, and refractive index are prescribed. The process to obtain on-axis diffraction-limited images with bi-conic lenses and the proof of the method, corroborated through an example in Oslo, are described here.

Refracting and reflecting interfaces transforming a given wavefront into another one

The aim of this work is threefold. First, following Luneburg and using our own notation, we review the Cartesian ovals. Second, we obtain analytical expressions for the reflecting and refracting surfaces that transform a prescribed smooth two-dimensional wavefront into a spherical one. These results are applied to show that the reflecting surface that connects a plane wavefront to a spherical one is a parabolical surface, and we design a lens, with two freeform surfaces, that transforms a spherical wavefront into another spherical one. These examples show that our equations provide the well-known solution for these problems, which is given by the Cartesian ovals method. Third, we present a procedure to obtain exact expressions for the refracting and reflecting surfaces that connect two given arbitrary wavefronts; that is, by assuming that the optical path length between two points on the prescribed wavefronts is given by the designer the refracting and reflecting surfaces we are looking for are determined by a set of two algebraic equations, which in the general case have to be solved in a numerical way. These general results are applied to compute the analytical expressions for the reflecting and refracting surfaces that transform a parabolical initial wavefront into a plane one.

Aplanatism in stigmatic optical systems

The minimization of spherical and coma aberrations in optical imaging systems is currently accomplished through the use of corrective aspheric optical surfaces. In this work, we develop a new, to the best of our knowledge, theory for the design of rigorously aplanatic optical systems, considering as a starting point the rigorous stigmatism theory of optical systems composed of Cartesian surfaces. The main characteristic of these surfaces is their, a priori, zero spherical aberration. In a general parametric formulation for systems made up of a set of these surfaces, the Abbe sine condition is adapted to simultaneously obtain the stigmatism and aplanatism conditions. Thus, we achieved the design of optical systems that in theory are immune to both coma and spherical aberrations.

Calculation of a lens system with one or two aspherical surfaces having corrected spherical aberration

The paper presents a detailed theoretical analysis of characteristics of a rotationally symmetric lens system with one or two aspherical surfaces having corrected spherical aberration and reduced coma aberration for a given position of the object and the image. Formulas for surface shape optimization are derived, and the procedure for calculating the aspherical system is shown. The presented formulas are verified with examples of ray tracing.

Superconical aplanatic ovoid singlet lenses

In this work, we return to Descartes's idea to develop a formalism to construct rigorously stigmatic singlet lenses comprising two Cartesian surfaces. Optical systems are built using a considerable number of spherical surfaces, presenting in most cases spherical aberration. Wasermann and Wolf proposed eliminating spherical aberration and minimizing third-order coma by using two adjacent aspherical surfaces. That is why, using a parametric formulation for Cartesian ovals, we propose the design of singlet lenses where the condition of rigorous stigmatism is guaranteed for each surface, and therefore, strictly speaking, in the pair of stigmatic points, the lens becomes an optical system free of spherical aberration. This formulation is unified to both refractive and reflective optical surfaces. Therefore, within the framework of the theory of rigorously stigmatic optical systems, making use of Cartesian surfaces for the construction of stigmatic ovoid singlet lenses, we achieve the same functionality of optical systems involving a set of spherical lenses. These lenses have the advantage of being formulated according to a generalized shape factor associated with the Coddington shape factor, allowing an easy classification of these stigmatic lenses. The ideal imaging is carried out by applying an exact ray-tracing method through these ovoid singlet lenses.

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.

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

In their paper [Appl. Opt.58, 1010 (2019)APOPAI0003-693510.1364/AO.58.001010] González-Acuña et al. claimed: "an analytical closed-form formula for the design of freeform lenses free of spherical aberration and astigmatism." However, as we show here, their formula can only be applied when the object and image are both real, and the image is inversed; additionally, the refractive index in the object and image media is the same. Here, we present the complete solution of this particular formula.

Roadmap for geometrical optics based on Taylor series expansion of skew ray

The skew ray R¯_n on the image plane of an optical system possessing n boundary surfaces has the form of an n-layered deep composite function. It is hence difficult to evaluate the system performance using ray tracing alone. The present study therefore uses the Taylor series expansion to expand R¯_n with respect to the source ray variable vector. It is shown that the paraxial ray tracing equations, point spread function, caustic surfaces and modulation transfer function can all be explored using the first-order expansion. Furthermore, the primary and secondary ray aberrations of an axis-symmetrical system can be determined from the third- and fifth-order expansions, respectively. It is thus proposed that the Taylor series expansion of the skew ray serves as a useful basis for exploring a wide variety of problems in geometrical optics.

Explicit Cartesian oval as a superconic surface for stigmatic imaging optical systems with real or virtual source or image

Cartesian ovals, also known as rigorously stigmatic surfaces, are the simplest optical systems capable of producing a perfect point image. Exist both implicit and explicit expressions to represent these surfaces, but they treat both refractive and reflective surfaces independently. Because of the complexity of explicit expressions, the ray-tracing techniques for these surfaces are implemented using third-party software. In this paper, we express Cartesian ovals as a degenerated superconic curve and get a new explicit formulation for Cartesian ovals capable of treating image formation using both object and image points, either real or virtual, and in this formulation can deal with both reflective and refractive rigorously stigmatic surfaces. Finally, using the resultant expressions and the vector Snell–Descartes Law, we propose a self-contained analytical ray-tracing technique for all these surfaces.

Freeform geometrical optics II: from parametric representation to CAD/CAM

Freeform optical surfaces are of great importance because of two main properties. The first is their ability to enhance the image quality of image-forming optical systems; the second is their inherent reduction in the number of surfaces in image and nonimage-forming optical systems. However, the main characteristic of freeform surfaces is that they lack symmetry about any spatial axis. This attribute allows describing freeform surfaces with a mathematical parametric representation. Unfortunately, parametric representation can be extremely extended. On the other hand, when describing freeform surfaces, the explicit representation is commonly preferred because of its compactness and CAD-format exportable easiness. Parametrically represented freeform surfaces can nonetheless be exported to a CAD format, with no significant departure of surface shape, as shown here. The vector method presented here guarantees that the surface's sampling density be proportional to the irradiance on the surface.