Page 210 - Modern Optical Engineering The Design of Optical Systems

Stops, Apertures, Pupils and Diffraction 193

Figure 9.16 The dashed lines represent the diffraction
patterns of two point images at various separations. The
solid line indicates the combined diffraction pattern. Case
(b) is the Sparrow criterion for resolution. Case (c) is the
Rayleigh criterion.

the image separation reaches 0.61 /NA, the maximum of one pattern
coincides with the first dark ring of the other and there is a clear indi-
cation of two separate maxima in the combined pattern. This is Lord
Rayleigh’s criterion for resolution and is the most widely used value for
the limiting resolution of an optical system.*
From the tabulation of Fig. 9.14, we find that the distance from the
center of the Airy disk to the first dark ring is given by

0.61 0.61
Z 1.22 (f/#) (9.16)
n′ sin U′ NA

This is the separation of two image points corresponding to the
Rayleigh criterion for resolution. This expression is widely used in
determining the limiting resolution for microscopes and the like.

*The diffraction pattern of two point images will always differ somewhat from the dif-
fraction pattern of a single point. It is thus possible to detect the presence of two points
(as opposed to one) even in cases where the two points cannot be visually resolved or sepa-
rated. This is the source of the occasional claims that a system “exceeds the theoretical
limit of resolution.” In Chap. 15 it is shown that there is a true limit on the resolution of
a sinusoidal line target; the limit on the spatial frequency is v 0 2NA/ 1/ (f/#).