How does the diffraction
pattern known as the Airy disk get generated by
shining light through a pinhole? Or more
generally, what causes diffraction of light?
Diffraction of light occurs because of its transverse
wave nature. We have already said that when
light hits an object, it is diffracted.
The formation of the Airy disk can best
be described by looking at how imaging of a luminous
point occurs in a lens system such as is found
in the compound microscope. The following diagram
shows what happens.
If a luminous point
at A is projected through the front lens of an
objective O1, and assuming that the
light is monochromatic, light coming from point
A will define wave surfaces as spheres (e.g.,
So) with their centers at A.
Assuming the objective to be a perfect lens, the
light going through it will also produce wave
surfaces as spheres as well (e.g., So).
The centers of these spheres are at point A'0
which is a geometrical image of A. At any point
on the wave surface of Si according
to Huygen's principle, the image A'0
is formed as if all the points of the wave surface
were actual sources of light with the same vibratory
state. But any point on the wave surface such
as M emits vibrations not only towards A'0,
but also in other directions. In fact all
the points on the wave surface Si diffract
the light which spreads over the image surrounding
the point A'0. The diagram below
at the left shows that all the vibrations emanating
from any point on the wave surface Si
will reach point A'0 in the same vibratory
state. Only two waves from points M and
M0 are shown to keep the figure simple.
As the waves have the same vibration, the amplitudes
are additive and since amplitude is seen by the
eye as brightness, at point A'0, we
have a very bright spot.
diagram on the right shows vibrations going to
a point A'1 from M and M0.
The amplitudes are opposite each other when they
reach the plane (indicated by line P and extending
out from the page) where our diffraction image
is generated. We would now have a dark area
at point A'1 because the luminous amplitudes
cancel each other out and add up to zero. The
same situation would happen if A'1
were on the other side at the same distance from
A'0. And in fact if one
considered the whole plane of line P as shown
by the square in perspective, the image would
be a dark ring with a radius A'1-A'0
with A'0 at the center as shown by
the circle. If the vibrations coming
from points M and M0 were imaged at
a point A'2 on line P twice as
from point A'0 as A'1, the
amplitudes of the vibrations would once again
be additive and one would then see a bright ring
in the plane of line P. It also follows
that the intensities of the vibrations at all
the points on the plane of line P results from
vibrations from all the points on wave surface
Si, not just those from points m and
If all this information
is taken together, then the image seen in the
plane of line P would be a very bright central
circular disk surrounded by alternately bright
and dark rings whose intensity decreases rapidly
as distance increases: the Airy disk.
It must be remembered
that any object observed in the microscope is
subject to the phenomena described here and this
has important consequences for the generation
of enlarged images in the microscope and is why
the concept of numerical aperture is so
important in microscopy.
redrawn from Francon, M. 1961. Progress
in Microscopy. Pergamon Press: London (also
Row, Peterson and Co.: Elmsford, NY).