Page 39 - Modern Optical Engineering The Design of Optical Systems

22 Chapter Two

image plane as a function of h, the position of the ray in the object
plane, and y, the position of the ray in the aperture of the optical system.
If the system is symmetrical about an axis (called the optical axis), the
power series expansion has only odd power terms (in which the sum of
the exponents of h and y add up to 1, 3, 5, etc.) The first-order terms
of this expansion effectively describe the position and size of the
image. (See Eqs. 5.1 and 5.2.)
First-order (or gaussian or paraxial) optics is often referred to as
the optics of perfect optical systems. The first-order equations can be
derived by reducing the exact trigonometrical expressions for ray
paths to the limit when the angles and ray heights involved approach
zero. As indicated in Chap. 3, these equations are completely accu-
rate for an infinitesimal threadlike region about the optical axis,
known as the paraxial region. The value of first-order expressions
lies in the fact that a well-corrected optical system will follow the
first-order expressions almost exactly, and also that the first-order
image positions and sizes provide a convenient reference from which
to measure departures from perfection. In addition, the paraxial
expressions are linear and are much easier to use than the trigono-
metrical equations.
We shall begin this topic by considering the manner in which a “per-
fect” optical system forms an image, and we will discuss the expres-
sions which allow the location and size of the image to be found when
the basic characteristics of the optical system are known. In a subse-
quent chapter we will take up the determination of these basic charac-
teristics from the constructional parameters of an optical system. And
finally, methods of image calculation by paraxial ray-tracing as well as
compound optical systems will be discussed.

2.2 Cardinal Points of an Optical System
The Mathematician Gauss discovered that the imagery of an optical
system (i.e., the location of an image, the size of the image, and the
orientation of the image) could easily be calculated if one knew the
location of a few points on the optical axis of the system. Henceforth,
in addition to assuming isotropic media, we will also assume that an
optical system is one of axial symmetry, in that all surfaces are figures
of rotation about a common axis, called the optical axis.
A well-corrected optical system can be treated as a “black box” the
characteristics of which are defined by its cardinal points, which are its
first and second focal points, its first and second principal points, and
its first and second nodal points. The focal points are those points at
which light rays (from an infinitely distant axial object point) parallel