Cartesian oval

Abstract: The Simultaneous Multiple Surface (SMS) method in 3D geometry is presented. Giving two orthotomic input ray bundles and other two orthotomic output ray bundles, the method provides an optical system with two free-form surfaces that deflects the rays of the input bundles into the rays of the corresponding output bundles and vice versa. In nonimaging applications, the method allows controlling the light emitted by an extended light source much better than single free-form surfaces designs, and also enables the optics contour to be shaped without efficiency losses. The method is expected to find also applications in imaging optics

Abstract: This paper presents a closed-form solution to the sag of the Cartesian oval and an alternate iterative method for obtaining the sag. The emphasis is in providing a methodology for determining the sag and derivatives of a Cartesian surface for optical design, ray-tracing purposes. We verify our results by comparison of our solutions and by real ray tracing.

Abstract: Elementary optical processes, such as the refraction or reflection of light from a point source by a spherical surface, give rise to wavefronts that may be described precisely by algebraic curves and surfaces of rather high degree. Usually, one is content to “sample” these wavefronts through the expedient technique of ray tracing. By propagating the refracted or reflected wavefronts backward in time, however, a particularly simple “initial” form — the anticaustic — may be identified. The refracted or reflected waves can then be regarded as emanating from the anticaustic via Huygens' principle in a homogeneous medium, i.e., they are offsets to the anticaustic. The simplest anticaustics of practical interest are (in two dimensions) the ellipse/hyperbola, the limacon of Pascal, and the Cartesian oval, and (in three dimensions) the corresponding surfaces of revolution — oblate spheroids, hyperboloids of two sheets, and certain cyclides. We use these ideas to derive exact propagating–wavefront equations and to explore some of their interesting geometric properties.

Abstract: In this work, novel imaging designs with a single optical surface (either refractive or reflective) are presented. In some of these designs, both object and image shapes are given but mapping from object to image is obtained as a result of the design. In other designs, not only the mapping is obtained in the design process, but also the shape of the object is found. In the examples considered, the image is virtual and located at infinity and is seen from known pupil, which can emulate a human eye. In the first introductory part, 2D designs have been done using three different design methods: a SMS design, a compound Cartesian oval surface, and a differential equation method for the limit case of small pupil. At the point-size pupil limit, it is proven that these three methods coincide. In the second part, previous 2D designs are extended to 3D by rotation and the astigmatism of the image has been studied. As an advanced variation, the differential equation method is used to provide the freedom to control the tangential rays and sagittal rays simultaneously. As a result, designs without astigmatism (at the small pupil limit) on a curved object surface have been obtained. Finally, this anastigmatic differential equation method has been extended to 3D for the general case, in which freeform surfaces are designed.