The diffraction grating of Figure 20.9 looks much like a series
of double slits; and, indeed, it is just that. Just as for the
arrangement in Young's double slit experiment, light traveling
from one slit travels a distance greater than the light from
an adjacent slit. Just as with the double slit, this additional
distance determines how one wave combines with or interferes
with the waves from adjacent slits.
Again, just as with the double slit, if this distance is exactly one wavelength-or two, or three or any integral number of wavelengths-waves from two adjacent slits will arrive in phase and constructive interference will occur. From Figure 20.10 you can also see that light from each slit travels an integral number of 's compared to the light from any other slit. If is a wavelength (or an integer times a wavelength, = m ) so that light from two adjacent slits interfere constructively, then light from all the slits interferes constructively and a bright area or maximum will be seen. Just as for the double slit, we find bright regions for
= m (bright)
where m is an integer (m = 0, 1, 2, ...).
So far, this seems just to have repeated the results of the double
slit experiment. The difference-the very useful difference-comes
when we look at the details of the intensity pattern as sketched
in Figure 20.11 which shows a graph of the intensity pattern for
a double slit and for a diffraction grating. For a double slit,
the intensity gradually falls from a maximum to a minimum of zero
and then gradually increases again. But for a diffraction grating,
the intensity drops off far more quickly. In effect, these maxima
become sharp, distinct lines. To understand this, look back at
Figure 20.10. Light from each slit travels a distance farther
than light from its adjacent slit. Let = (1/10) for a moment.
Then light from one slit, such as slit a, will be completely out
of phase with light from another slit five slits away, slit f
in this case, and these will destructively interfere. Light from
each slit will undergo destructive interference with light from
another slit, five slits away. With many slits, any angle other
than those which provide constructive interference will have zero
intensity due to this destructive interference. The bright lines
from a diffraction grating will be very sharp.
Bright lines of constructive interference occur for = m. As the
wavelength varies, so will the angle for which constructive interference
occurs. If we have white light with a full range of wavelengths
from = 400 nm for violet light to =700 nm for red light, each
wavelength-or each color-will have constructive interference at
a different angle. This means that when we look at a white light
source-like the filament of a bulb-we will see each color at a
different position. This produces a spectrum of colors much like
what we saw when white light passed through a prism. This is illustrated
in Figure 20.12.