It has already been suggested that the principle of energy requires that the general expression for I² in (2) when integrated over the whole of the plane ξ, η should be equal to A, where A is the area of the aperture. A general analytical verification has been given by Sir G. G. Stokes (Edin. Trans., 1853, 20, p. 317). Analytically expressed—
| ∫∫ | +∞ | I² dξdη = ∫∫ dxdy = A (9). |
| −∞ |
We have seen that I0² (the intensity at the focal point) was equal to A²/λ²f². If A′ be the area over which the intensity must be I0² in order to give the actual total intensity in accordance with
| A′ I0² = ∫∫ | +∞ | I² dξdη, |
| −∞ |
the relation between A and A′ is AA′ = λ²f². Since A′ is in some sense the area of the diffraction pattern, it may be considered to be a rough criterion of the definition, and we infer that the definition of a point depends principally upon the area of the aperture, and only in a very secondary degree upon the shape when the area is maintained constant.
4. Theory of Circular Aperture.—We will now consider the important case where the form of the aperture is circular.
Writing for brevity
kξ/f = p, kη/f = q, (1),
we have for the general expression (§ 11) of the intensity
λ²f²I² = S² + C² (2),