Page 216 - Modern Optical Engineering The Design of Optical Systems

Stops, Apertures, Pupils and Diffraction 199

Two points regarding the above are well worth emphasizing. First,
laser researchers speak in terms of a “beam waist.” Note that in the
equations above and in common usage it is described as a radial
dimension, not a diameter; the diameter of the waist is 2w. Second, the
waist and the focus are not the same thing, as a comparison of Eqs.
9.28 and 2.3 will indicate. In most circumstances the difference is triv-
ial and gaussian beams may be handled by the usual paraxial equa-
tions. But when the beam convergence is small (i.e., with an f-number
of a hundred or so), it is possible to distinguish both a focus and a sep-
arate beam waist. For example, if we project a 1-in laser beam
(through a focusable beam expander) on a screen about 50 ft away, we
can focus the beam to get the smallest possible spot on the screen. The
focus is now at the screen. However, there is a location a few feet short
of the screen at which a smaller beam diameter exists. This is the
beam waist; it can be demonstrated by moving the screen (or a sheet
of paper) toward the laser and observing the reduction of the spot size.
Note that with the screen now at this beam waist position, the beam
expander can be refocused to get a still smaller spot on the screen.
Then there will be a new waist still closer to the laser, etc., etc., etc.
Note well that the focus is the smallest spot which can be produced
on a surface at a given, fixed distance. The waist is the smallest
diameter in the beam (see Gaskill, p. 435).
Note also that all the phenomena described in this section result
from the gaussian distribution of beam intensity and not from the fact
that the source may be a laser. The same effects could be produced by
a radially graded filter placed over the aperture of the system. (The
temporal and spatial coherence of a laser beam are, of course, what
make it practical to demonstrate these effects.)

9.12 The Fourier Transform Lens
and Spatial Filtering
In Fig. 9.19 we have a transparent object located at the first focal
point of lens A. As indicated by the dashed rays in the figure, lens A
images the object at infinity so that the rays originating at the axial
point of the object are collimated. These rays are brought to a focus
at the second focal plane of lens B, where the image of the object is
located.
Now let us realize that the Fourier theory allows us to consider the
object as comprised of a collection of sinusoidal gratings of different
frequencies, amplitudes, phases, and orientations. If our object is a
simple linear grating with but a single spatial frequency, it will devi-
ate the light through an angle according to Eq. 9.20, except that a