Page 94 - Fundamentals of Light Microscopy and Electronic Imaging

ABBE’S THEORY FOR IMAGE FORMATION IN THE MICROSCOPE 77

approaches the grating. It is easy to see the linear diffracted rays by darkening the
room and introducing a cloud of chalk or talcum dust between the grating and the
screen.
Finally, examine the diffraction pattern of a sinusoidal (holographic) grating.
The sinusoidal grating channels much of the incident light into the 1st-order dif-
fraction spots, which are much brighter than the spots produced by a conventional
ruled grating. Such gratings are very useful for examining the spectra of filters
and illuminators.

ABBE’S THEORY FOR IMAGE FORMATION IN THE MICROSCOPE

Ernst Abbe (1840–1905) developed the theory for image formation in the light micro-
scope while working in the Carl Zeiss optical workshop in Jena, Germany (Fig. 5-13).
Abbe observed that diffracted light from a periodic specimen produces a diffraction pat-
tern of the object in the back focal plane (the diffraction plane) of the objective lens.
According to Abbe’s theory, interference between 0th- and higher-order diffracted rays
in the image plane generates image contrast and determines the limit of spatial resolu-
tion that can be provided by an objective. For a periodic object such as a diffraction grat-
ing, it is easy to demonstrate that light from at least two orders of diffraction must be
captured by the objective in order to form an image (see Exercise, this Chapter). In the
minimal case, this could include light coming from two adjacent diffraction spots, such
as the 0th- and one 1st-order spot generated by the grating. If light from only a single
diffraction order is collected by the lens (only the 0th order is collected), there is no
interference, and no image is formed. Put another way, there is no image if light dif-
fracted at the specimen is excluded from the lens. Extending the concept further, the
larger the number of diffraction orders collected by the objective, the sharper and better
resolved (the greater the information content) are the details in the image.
Objects that exhibit fine, periodic details (diffraction gratings, diatoms, striated mus-
cle) provide important insights about the roles of diffraction and interference in image
formation. Figure 5-14 shows a diffraction grating with periodic rulings illuminated by a
collimated beam of light having planar wavefronts. Light diffracted by the rulings is col-
lected by a lens, and an image of the rulings is created in the primary image plane.
Note the following:

• A certain amount of incident light does not interact with the specimen and is transmit-
ted as undeviated (nondiffracted) rays. These rays form the central 0th-order diffrac-
tion spot in the diffraction plane and go on to evenly illuminate the entire image plane.
• Each ruling in the grating acts as an independent source of diffracted waves that
radiate outward as a series of spherical wavefronts (Huygens’ wavelets) toward the
objective lens. Note that the aperture of the lens is large enough to capture some of
the diffracted light. The effective NA of the objective is critically dependent on the
setting of the condenser aperture. The diffracted light forms higher-order diffrac-
tion spots flanking the 0th-order spot in the diffraction plane.
• The 0th- and higher-order diffraction spots in the diffraction plane correspond to
locations where there is constructive interference of waves that differ in their path