How is the diffraction pattern of the Airy Disk formed?

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.  This phenomenon is best understood by an examination of Huygens' Principle. In 1678, the Dutch pysicist Christiaan Huygens wrote a treatise on the wave theory of light in which he presented a theory now known as Huygens' Principle.  It states that every point on a wave front can be thought of as a new point source for waves generated in the direction the wave is traveling or being propagated.  OR--  the wavefront of a propagating wave of light at any instant conforms to the envelope of spherical wavelets emanating from every point on the wavefront at the prior instant (with the understanding that the wavelets have the same speed as the overall wave).  Fresnel later elaborated on Huygens' Principle by stating that the amplitude of the wave at any given point equals the superposition of the amplitudes of all the secondary wavelets at that point (with the understanding that the wavelets have the same frequency as the original wave). These are termed Huygens’ wavelets. 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., Si). 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.  The 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 M0.

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 can also be seen that the distances between the bright and dark rings will change with changes in the wavelength of light.

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.

*Diagrams redrawn from Francon, M. 1961. Progress in Microscopy. Pergamon Press: London  (also Row, Peterson and Co.: Elmsford, NY).