First-order design of mirror systems with no axial symmetry

Abstract

All-reflective imaging systems that are asymmetrical and eccentric have the advantage of providing more degrees of freedom to improve image quality. A disadvantage of these asymmetrical imaging systems is that they suffer from asymmetric mapping. This asymmetric mapping manifests itself mainly in the presence of keystone distortion and anamorphism. Due to the increase in degrees of freedom, the complexity of such systems escalates; thus, the designer is confronted with the difficult task of determining optimal starting points. This work addresses several first-order aspects of the design and characterisation of asymmetrical, all-reflective, aspherical, eccentric imaging systems. In contrast to the work of Stone and Forbes, which is based upon the theory of Hamiltonian optics and includes both the first- and second-order considerations, this work is based upon the theory of collineation. Because of the inherent simplicity of the collinear mapping, which is a projective transformation, we are able to present a simple but certainly not naive way of designing and characterising such asymmetrical all-reflective imaging systems. The simplicity of this proposition has the advantage that we can gain insights into asymmetrical mapping behaviour. Specifically, we apply the collinear mapping model on all-reflective asymmetrical imaging systems resulting in the description of how the mapping between conjugate planes may be described. First we will define keystone distortion and anamorphism. Then we will introduce and investigate the significance of the Cardinal points and planes, the Scheimpflug condition and the horizon planes and show how they are applied in the designing of imaging systems that are free of both keystone distortion and anamorphism. Having established a first-order layout of the optical system, we will then develop a process for converting the first-order layouts into imaging systems consisting of real aspheric surfaces.