Ray vector fields, prismatic effect, and thick astigmatic optical systems

The application of the concept of ray vector fields to optical systems is reexamined. Paraxial or linear optics defines a four-dimensional ray vector field for any optical system: the vector field maps the incident ray vector into the emergent ray vector. In the case of thin systems, including thin astigmatic lenses, one can define a vector field of reduced dimensionality: the vector field is two-dimensional and maps the ray's incident position into the change in reduced direction. When the index of refraction is the same before and after a thin system, the change in reduced direction is the reduced deflection through the system or the reduced prismatic effect. Contrary to what has recently been claimed, this type of two-dimensional vector field does not apply in general to thick systems. However, a number of different types of two-dimensional vector fields can be defined for various particular classes of optical systems. Thick systems differ qualitatively from thin systems. They do not have equivalent thin lenses and cannot generally be replaced by thin lenses. Equations are derived for the change in reduced direction and deflection for a ray through optical systems in general and through separated two- and three-lens systems in particular.