Page 207 - Modern Optical Engineering The Design of Optical Systems

190 Chapter Nine

1.5 , 2.5 , etc. When CD is 1.5 , the wavelets from two-thirds of the
slit can be shown (as in the preceding paragraph) to interfere and can-
cel out, leaving the wavelets from one-third of the aperture; when CD
is 2.5 , only one-fifth of the slit is uncanceled. Since the “uncanceled”
wavelets are neither exactly in nor exactly out of phase, the illumina-
tion at the corresponding points on the screen will be less than one-
third or one-fifth of that in the central band.
For a more rigorous mathematical development of the subject, the
reader is referred to the references following this chapter. The mathe-
matical approach is one of integration over the aperture, combined
with a suitable technique for the addition of the wavelets which are
neither exactly in nor exactly out of phase. This approach can be
applied to rectangular and circular apertures as well as to slits.
For a rectangular aperture, the illumination on the screen is given by
2
2
sin m sin m
I I 1 2 (9.12)
0 m 2 m 2
1 2
w sin
i
m i i 1,2 (9.13)
i

In these expressions is the wavelength, w the width of the exit aper-
ture, the angle subtended by the point on the screen, m 1 and m 2
correspond to the two principal dimensions, w 1 and w 2 , of the rectan-
gular aperture and I 0 is the illumination at the center of the pattern.
When the aperture is circular, the illumination is given by
1 m 2 1 m 2 2 1 m 3 2 1 m 4 2
2
...

I I 1
3
4
2
0 2 2 3 2 2! 4 2 3! 5 2 4!
2J (m)
1
I 2 (9.14)
0
m
where m is given by Eq. 9.13 with the obvious substitution of the
diameter of the circular exit aperture for the width, w, and J 1 (m) is the
first order Bessel function. The illumination pattern consists of a
bright central spot of light surrounded by concentric rings of rapidly
decreasing intensity. The bright central spot of this pattern is called
the Airy disk.
We can convert from angle to Z, the radial distance from the center
of the pattern, by reference to Fig. 9.13. If the optical system is rea-
sonably aberration-free, then
w
l′
2 sin U′