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Stops, Apertures, Pupils and Diffraction 189
At an extreme, when the wave front is limited by an aperture to a
very small size (say, to the order of a half wavelength), the new wave
front becomes spherical about the aperture. Figure 9.12 shows a plane
wavefront incident on a slit AC, which is in front of a perfect lens.
The lens is focused on a screen, EF. We wish to determine the nature
of the illumination on the screen. Since the lens of Fig. 9.12 is assumed
perfect, the optical path lengths AE, BE, and CE are all equal and the
waves will arrive in phase at E, reinforcing each other to produce a
bright area. For Huygen’s wavelets starting from the plane wave front
in a direction indicated by angle , the paths are different; path AF dif-
fers from path CF by the distance CD. If CD is an integral number of
wavelengths, the wavelets from A and C will reinforce at point F. If CD
is an odd number of half wavelengths, a cancellation will occur. The
illumination at F will be the summation of the contributions from each
incremental segment of the slit, taking the phase relationships into
account. It can be readily demonstrated that when CD is an integral
number of wavelengths, the illumination at F is zero, as follows: if CD
is one wavelength, then BG is one-half wavelength and the wavelets
from A and B cancel. Similarly, the wavelets from the points just below
A and B cancel and so on down the width of the slit. If CD is N wave-
lengths, we divide the slit into 2N parts (instead of two parts) and
apply the same reasoning. Thus, there is a dark zone at F when
N
sin
w
where N any integer
the wavelength of the light
w the width of the slit
Thus, the illumination in the plane EF is a series of light and dark
bands. The central bright band is the most intense, and the bands on
either side are successively less intense. One can realize that the
intensity should diminish by considering the situation when CD is
Figure 9.12 Plane wave front
incident on a slit in front of a
perfect lens.
