Lens and mirror design via the principal surface

For many laser applications, it is desired to focus a collimated beam with a specified transformation of the intensity distribution. The transformation properties of a lens or mirror system can be specified in terms of the principal surface r(alpha), which maps the height of the incident ray parallel to the optic axis onto a given angle at the focus. The intensity distribution at the focus is then given by the relation I(alpha) = I(r)r(dr/dalpha)/ sinalpha. One aspheric surface in an optical system is sufficient to yield diffraction limited focusing. By means of two aspheric surfaces, diffraction limited performance with a specified principal surface can be achieved. The problem of optical design is stated as follows: Given a principal surface r(alpha) and a maximum focal angle alpham, find the pair of optical surfaces for which diffraction limited focusing is achieved. It is shown that specification of r(alpha) and alpham uniquely determines the lens design to within a scale factor, given the refractive index of the lens. It is further shown that one straightforward Runge-Kutta integration routine generates both surfaces for either a lens or a pair of mirror surfaces. The complete family of aplanatic lenses will be described. Deviation from sphericity will be discussed, as will the possibility of realizing the specified lens designs. The family of lenses that map uniform incident intensity into uniform illumination about the focus will also be described. Extension of the method to off-axis aberrations will be considered.