# A Lens for Linear Angular Mapping

Lens Mapping

By Yuval Sharon M.Sc.

Introduction
An imaging lens is device that maps the object space onto the image space. For the mostly desired case of scaled image forming, the mapping is linear with the location:
every point p(x,y) in the object, is transformed to a point P(X,Y) in the image so that X=αx and Y=αy
Is there a mapping that is linear with the view angle and not with the location? or regarding the plane of incidence, is there a lens that can transform a viewing angle Θ of the object to a translation Y in the image so that Y=αΘ ?
The following analysis shows that there is.

Selected geometry
Assume an optimal PCX design in which light enters from the flat side (air to dielectric) and is focused by the convex side. Thus most of the distortion is governed by the refraction at the front.

Conclusions
A PCX with n=1.57 (for example Polycarbonate or Schott BAK1) will have a mapping close to linear with regard to the viewing angle, up to a view angle of about 45 deg.
Note that best results will be obtained when the central thickness of the lens is close to its radius of curvature.
Such a lens may be found useful in any device in which the viewing angle is more important than the scaled image, such as a triangulation system. Using a lens that generates directly the desired mapping may prove advantageous in cases where calculation power is limited, and also cost-wise when a conventional imaging lens is an overkill.

About the Author
Yuval is one of the owners of CSTM Ltd:
http://www.cstm.co.il/indexe.htm
a Jerusalem based engineering company, which undertakes projects involving Optics, Mechanics and Electronics.

Yuval has earned his B.Sc. (1984) in Electro-Optics from the JCT, and his M.Sc. (1992) from the Hebrew university.

Yuval has been working in the industry since1983, and has experience in plastic optics, light sources, diffractive optics, QA, manufacturing, process control and electrochemistry. See Yuval Sharon’s profile at: http://il.linkedin.com/in/ysharon