Page 55 - Modern Optical Engineering The Design of Optical Systems

38 Chapter Three

Although paraxial calculations are often used in rough preliminary
work on optical systems and in approximate calculations (indeed, the
term “paraxial approximation” is often used), the reader should bear
in mind that the paraxial equations are perfectly exact for the paraxial
region and that, as an exact limiting case, they are used in aberration
determination as a basis of comparison to indicate how far a trigono-
metrically computed ray departs from its ideal location.
The simplest way of deriving a set of equations for the paraxial
region is to substitute the angle itself for its sine in the equations
derived in the preceding section. Thus we get
from Eq. 3.1 ca (R l)u (3.8)

from Eq. 3.2 i ca/R (3.9)

from Eq. 3.3 i′ ni/n′ (3.10)

from Eq. 3.4 u′ u i i′ (3.11)

from Eq. 3.6 ca′ n ca/n′ (3.12)

from Eq. 3.7 l′ R ca′/u′ (3.13)

Notice that the paraxial equations are distinguished from the trigono-
metric equations by the use of lowercase letters for the paraxial values.
This is a widespread convention and will be observed throughout this
text. Note also that the angles are in radian measure, not degrees.
Equations 3.8 through 3.13 may be materially simplified. Indeed,
since they apply exactly only to a region in which angles and heights
are infinitesimal, we can totally eliminate i, u, and ca from the
expressions without any loss of validity. Thus, if we substitute into
Eq. 3.13, Eq. 3.12 for ca′ and Eq. 3.11 for u′, and continue the substi-
tution with Eqs. 3.8, 3.9, and 3.10, the following simple expression
for l′ is found:

ln9R n9R
l9 5 c 5 if l 5` d (3.14)
sn9 2 ndl 1 nR sn9 2 nd

By rearranging we can get an expression which bears a marked simi-
larity to Eq. 2.4 and Eq. 2.11 (relating the object and image distances
for a complete lens system):
n′ (n′ n) n
(3.15a)
l′ R l