
GAUSSIAN BEAMS 287
beam diameter, but now our goal is to achieve a waist at a specic distance, and we want to
know what lens to use and how large the waist will be. We decide to solve the second part of
Equation 9.9 for b:
b
=
b
2
+z
2
b
b =
b
±
√
b
2
−4z
2
2
,
which will have a solution (or two) provided that b
> 2z. In fact, from the previous section,
and Figure 9.3, we know that we cannot generate a waist at a distance beyond half the con-
focal parameter, b
of the beam diameter, d. Once we have solved this equation, we can then
determine ρ, which will be negative in this case, and the required focal length, f =−ρ.For
example, suppose that for an endoscopic