Page 88 - Fundamentals of Light Microscopy and Electronic Imaging

DIFFRACTION BY A GRATING AND CALCULATION OF ITS LINE SPACING, D 71

phase with each other and destructively interfere. The light intensity at this position (the
first minimum in the diffraction pattern of the spot) is 0. Indeed, it can be shown that the
sum of contributions from all points in the aperture results in an amplitude of 0 at this
location and nearly so at all other locations in the image plane other than in the central
diffraction spot and surrounding diffraction rings. A geometrical explanation of the
phenomenon is given by Texereau (1963).
In Figure 5-7b we observe that the aperture (closing down the condenser diaphragm
in a conjugate focal plane) reduces the angular aperture of the optical system, which
increases the size of the diffraction spot and the distance P P to the first diffraction
minimum. Therefore, reducing the angular aperture decreases spatial resolution in the
image. If we reexamine the optics of the pinhole camera, it now becomes clear why
viewing a point source through a pinhole aperture held in front of the eye allows per-
ception of an observable diffraction disk (Fig. 5-5). For the fully dilated eye the Airy
disk of a point source covers 2 cone cells on the retina, but with the reduced angular
aperture using a pinhole, the disk diameter expands some 40-fold, stimulates dozens of
receptor cells, and results in the perception of a disk.

DIFFRACTION BY A GRATING AND CALCULATION
OF ITS LINE SPACING, D

We will now examine the diffraction of light at a specimen using a transparent diffraction
grating as a model for demonstration and discussion. We should bear in mind that the
principles of diffraction we observe at a grating on an optical bench resemble the phe-
nomena that occur at specimens in the microscope. A diffraction grating is a planar sub-
strate containing numerous parallel linear grooves or rulings, and like a biological
specimen, light is strongly diffracted when the spacing between grooves is close to the
wavelength of light (Fig. 5-8). If we illuminate the grating with a narrow beam from a
monochromatic light source such as a laser pointer and project the diffracted light onto a
screen 1–2 m distant, a bright, central 0th-order spot is observed, flanked by one or more
higher-order diffraction spots, the 1st-, 2nd-, 3rd-, etc.-order diffraction maxima. The
0th-order spot is formed by waves that do not become diffracted during transmission
through the grating. An imaginary line containing the diffraction spots on the screen is
perpendicular to the orientation of rulings in the grating. The diffraction spots identify
unique directions (diffraction angles) along which waves emitted from the grating are in
the same phase and become reinforced as bright spots due to constructive interference. In
the regions between the spots, the waves are out of phase and destructively interfere.
The diffraction angle of a grating is the angle subtended by the 0th- and 1st-order
spots on the screen as seen from the grating (Fig. 5-9). The right triangle containing at
the screen is congruent with another triangle at the grating defined by the wavelength of
illumination, λ, and the spacing between rulings in the grating, d. Thus, sin λ/d, and
reinforcement of diffraction spots occurs at locations having an integral number of
wavelengths—that is, 1 ,2 ,3 , and so on—because diffracted rays arriving at these
unique locations are in phase, have optical path lengths that differ by an integral num-
ber of wavelengths, and interfere constructively, giving bright diffraction spots. If sin
is calculated from the distance between diffraction spots on the screen and between the
screen and the grating, the spacing d of the rulings in the grating can be determined
using the grating equation

mλ d sin ,