Page 211 - Modern Optical Engineering The Design of Optical Systems

194 Chapter Nine

For resolution at the image, the NA of the image cone is used; for res-
olution at the object, the NA of the object cone is used.
To evaluate the performance limits of telescopes and other systems
working at long object distances, an expression for the angular sepa-
ration of the object points is more useful. Rearranging Eq. 9.15 and
substituting the limiting value of Z from Eq. 9.16, we get, in radian
measure,
1.22
radians (9.17)
w
For ordinary visual instruments, may be taken as 0.55 m, and
using 4.85 10 6 radians for 1 second of arc, we find that
5.5
seconds of arc (9.18)
w
when w is the aperture diameter expressed in inches. By a series of
careful observations, the astronomer Dawes found that two stars of
equal brightness could be visually resolved when their separation was
4.6/w seconds. Notice that if the Sparrow criterion is used instead of
the Rayleigh criterion in Eq. 9.18, the limiting resolution angle is
4.5/w seconds, which is in close agreement with Dawes’ findings.
It is worth emphasizing here that the angular resolution limit is a
direct function of wavelength and an inverse function of the aperture
of the system. Thus, the limiting resolution is improved by reducing
the wavelength or by increasing the aperture. Note that focal length or
working distance do not directly affect the angular resolution. The lin-
ear resolution is governed by the wavelength and the numerical aper-
ture (NA or f-number), and not by the aperture diameter.
In an instrument such as a spectroscope, where it is desired to sep-
arate one wavelength from another, the measure of resolution is the
smallest wavelength difference, d , which can be resolved. This is usu-
ally expressed as /d ; thus, a resolution of 10,000 would indicate that
the smallest detectable difference in wavelength was 1/10,000 of the
wavelength upon which the instrument was set.
For a prism spectroscope, the prism is frequently the limiting aper-
ture, and it can be shown that when the prism is used at minimum
deviation, the resolution is given by
dn
B (9.19)
d d
where B is the length of the base of the prism and dn/d is the dis-
persion of the prism material.
A diffraction grating consists of a series of precisely ruled lines on a
clear (or reflecting) base. Light can pass directly through a grating, but