Foundations of geometrical optics

In geometrical optics, light is described by rays that propagate according to three laws: rectilinear propagation, refraction, and reflection. Their direction of propagation indicates the direction of the flow of light energy. They are normal to a wavefront. They are not a physical entity in the sense that we cannot isolate a ray, yet they are very convenient for describing the process of imaging by a system. We begin this chapter with a brief introduction of the Cartesian sign convention for the distances and heights of the object and image points, and the angles of incidence and refraction or reflection and slope angles of the rays. We discuss Fermat’s principle that the optical path length of a ray from one point to another is stationary, and derive the laws of rectilinear propagation in a homogeneous medium, refraction by a refracting surface, and reflection by a reflecting surface (first in 2D and then in 3D). These laws are used to obtain ray-tracing equations representing the propagation of a ray exactly from a certain point to a point on a refracting or a reflecting surface, or refraction or reflection of the ray by the surface, and propagation of the refracted or reflected ray to the next surface. The purpose of exact ray tracing is to determine the aberrations of a system consisting of a series of refracting and/or reflecting surfaces that generally have a common axis of rotational symmetry called the optical axis. Such a system is called a centered or a rotationally symmetric system. Its surfaces bend light rays from an object according to the three laws to form its image.