Page 213 - Modern Optical Engineering The Design of Optical Systems

196 Chapter Nine

amplitude were uniform over the aperture. Any change in the intensity
distribution in the beam will change the diffraction pattern from that
described above. Obviously, a similar change in the transmission of the
aperture will produce the same effects.
A “gaussian beam” is one the intensity cross section of which follows
the equation of a gaussian, y e x 2 . Laser output beams closely approx-
imate gaussian beams. From mathematics we know that exponential
functions, such as the gaussian are extremely resistant to transforma-
x
tions (consider, for example, the integral or differential of e ).
Similarly, a gaussian beam tends to remain a gaussian beam (as long as
it is “handled” by reasonably aberration-free optics) and the diffraction
image of a point source also has a gaussian distribution of illumination.
The distribution of intensity in a gaussian beam is illustrated in
Fig. 9.18 and can be described by Eq. 9.22.
I (r) I e 2r 2 /w 2 (9.22)
o
where I (r) the beam intensity at a distance r from the beam axis
I 0 the intensity on axis
r the radial distance
e 2.718.…
w the radial distance at which the intensity falls to I 0 /e , i.e.,
2
to 13.5 percent of its central value. This is usually
referred to as the beam width, although it is a semi-
diameter. It encompasses 86.5 percent of the beam power.

Beam power
By integration of Eq. 9.22 we find the total power in the beam to be
given by

1
P / I w 2 (9.23)
tot 2 0

Figure 9.18 Gaussian beam intensity profile.