Page 54 - Modern Optical Engineering The Design of Optical Systems

Paraxial Optics and Calculations 37

From right triangle QA′C we get
CA′
sin I′ (3.5)
R
and substituting Eqs. 3.2 and 3.5 into Eq. 3.3 gives us

n
CA′ CA (3.6)
n′
Finally, the location of P′ is found by rearranging CA′ (R L′) sin U′
from right triangle P′A′C into
CA′
L′ R (3.7)
sin U′

Thus, beginning with a ray defined by its slope angle U and its inter-
section with the axis L, we can determine the corresponding data, U′
and L′, for the ray after refraction by the surface. Obviously, this
process could be applied surface by surface to trace the path of a ray
through an optical system.

3.2 The Paraxial Region
The paraxial region of an optical system is a thin threadlike region
about the optical axis which is so small that all the angles made by the
rays (i.e., the slope angles and the angles of incidence and refraction)
may be set equal to their sines and tangents. At first glance this con-
cept seems utterly useless, since the region is obviously infinitesimal
and seemingly of value only as a limiting case. However, calculations
of the performance of an optical system based on paraxial relation-
ships are of tremendous utility. Their simplicity makes calculation and
manipulation quick and easy. Since most optical systems of practical
value form good images, it is apparent that most of the light rays origi-
nating at an object point must pass at least reasonably close to the
paraxial image point. The paraxial relationships are the limiting rela-
tionships (as the angles approach zero) of the exact trigonometric rela-
tionships derived in the preceding section, and thus give locations for
image points which serve as an excellent approximation for the
imagery of a well-corrected optical system.
Paradoxically, the paraxial equations are frequently used with rela-
tively large angles and ray heights. This extension of the paraxial region
is useful in estimating the necessary diameters of optical elements and
in approximating the aberrations of the image formed by a lens system,
as we shall demonstrate in later chapters. This works because the
paraxial equations are linear, not trigonometric, and can be scaled.