Page 53 - Modern Optical Engineering The Design of Optical Systems

36 Chapter Three

Figure 3.1 Refraction of a ray at a spherical surface.

4. The slope angles (U and U′) are positive if the ray is rotated clock-
wise to reach the axis. (Historical Note: Until the latter part of the
Twentieth Century, the accepted convention for the sign of the slope
in optics was the reverse of the current one, and Fig. 3.1 was an
“all-positive diagram.”)
5. The light travels from left to right.
(In Fig. 3.1 all quantities are positive except U and U′, which are
negative.)
A set of equations which will allow us to trace the path for the ray
may be derived as follows. From right triangle PAC,
CA (R L) sin U (3.1)

and from right triangle QAC,
CA
sin I (3.2)
R
Applying Snell’s law (Eq. 1.3), we get the sine of the angle of refraction,
n
sin I′ sin I (3.3)
n′
The exterior angle QCO of triangle PQC is equal to U I, and, as
the exterior angle of triangle P′QC, it is also equal to U ′ I′. Thus,
U I U′ I′, and
U′ U I I′ (3.4)