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built to “beat” the Abbe limit. His “gedankenexperiment” to prove this is not altogether
convincing and cannot exceed the resolution obtained with unit NA (in air); however, the
result is correct and can be reached by other means. The interplay between field of view
restriction and resolution has more recently been employed to improve resolution in linear
systems by Shapiro et al.
[7]
and Gazit et al.
[8]
. The new term for these approaches is
“sparsity,” implying that there is known to be relatively little information in the image
(e.g., it fills a small known region of the field of view), which harks back to Lukosz's approach.
In recent decades, the idea that resolution is unlimited for incoherent self-emitting,
particularly fluorescent objects has been underlined by developing the powerful techniques
of STED
[9]
, PALM, and STORM
[10]
, all of which employ inherent features of the
fluorescence itself. They can be considered as coherent-to-incoherent conversion, and,
therefore, are of no relevance to phase imaging. In this chapter, I will have little to say
about these techniques (see the review by Lipson et al.
[11]
) because I want to concentrate
on linear, and in particular phase microscopy, where the images produced are properties of
the object itself and not of stains.
16.2 Where Was Abbe's Theory Incomplete?
It is interesting to ask how Abbe's theory should be modified in the light of subsequent
developments. One view of Abbe's theory is obtained by looking at it as an application of
Heisenberg's uncertainty principle. Actually, the argument below was first used by Heisenberg
as an explanation of the uncertainty principle; he called it the “gamma-ray microscope”
gedankenexperiment. Consider a one-dimensional system. If a photon with momentum
p
0
5
h
/
from a point object at position
x
in the microscope field of view is received by the
imaging detector, it must have passed through the microscope lens. If this lens subtends
semi-angle
λ
α
at the object, the photon can have transverse momentum
p
x
in the range between
6
p
0
sin
α
. Thus, the uncertainty is
δ
p
x
5
2
p
0
sin
α
. It follows from Heisenberg that
δ
x
$
h=p
x
5
λ=
2 sin
α
5
λ=
2NA
(16.1)
where NA
5
n
sin
is the NA and
n
5
1 for imaging in air. This is Abbe's limit. But clearly,
it refers to a single photon, and we can consider the use of
N
independent photons, to which
Poisson statistics apply, so that they impinge on the lens aperture at random positions. Then,
the combined uncertainty in
p
x
is
δ
p
x
5
2
O
Np
0
sin
α
α
, from which it follows that
2NA
p
δ
x
5
λ=ð
Þ
(16.2)
This is the principle behind the stochastic super-resolution techniques PALM and STORM.
Another point of view claims that a basic flaw was that Abbe's theory considered the image
of an
infinite
periodic grating. Abbe's basic idea was that the resulting diffraction orders are
delta-functions in Fourier space (which is represented more or less by the lens aperture) and
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