The basics of crystallography and diffraction (3rd Edition)

7.4 Geometry of diffraction patterns from gratings and nets 175
transmitted light microscopy by showing light passing from the condenser and through
the specimen and objective lens to the eyepiece rather than light passing in two directions
through the condenser/objective lens as it is reflected from the specimen as in reflected
light microscopy.
The conditions for constructive/destructive interference from a set of parallel slits
will, fairly obviously (ignoring end effects), be the same all along their length and the
diffraction pattern will be a series of light/dark bands (corresponding to constructive
and destructive interference conditions, respectively) running parallel to the slits. A net
may be regarded as a two-dimensional grating, with a criss-cross pattern of slits, or two
gratings superimposed upon each other with their slits running in different directions.
Each grating will give rise to its own diffraction pattern of light/dark bands, but the net
effect is that constructive interference will only occur when the light bands intersect and
reinforce—giving rise to the observed diffraction peaks or spots. Hence, consideration
of the, in effect, ‘one-dimensional diffraction’case of a grating will lead us to a complete
analysis of the ‘two-dimensional diffraction’ case of a net.
First we will consider diffraction from a grating made up of very narrow or thin slits
spaced distance a apart in which each slit is the source of just one Huygens’ wavelet
across its width; from this we will determine the conditions for constructive interference
and therefore the directions of the diffracted beams. By simply extending the analysis to
two dimensions we will demonstrate the reciprocal relationship between the diffracting
net and the positions of the spots in the diffraction pattern. Then we shall consider
diffraction from a wide aperture—a circular hole or wide slit which is the source of not
one but many Huygens’ wavelets and we shall see that this modifies the intensities, but
does not change the positions, of the diffracted spots. Finally we shall consider gratings
or nets of limited extent and will show how this leads to the occurrence of subsidiary
peaks or spots.
Figure 7.5 shows just part of a diffraction grating—a one-dimensional net—consisting
of many narrow slits distance a apart. The incident light is shown as parallel beams from
a
A
B
a 1
Huygens’ wavelets
Fig. 7.5. A set of lines in the net acts as a diffraction grating, each slit acting as the source of a single
set of Huygens’ wavelets. The diffracted beam is drawn for a path difference of one wavelength.