16 Theoretical Foundations of Super-Resolution in Microspherical Nanoscopy
409
In this regard, the miniaturization of SILs that took place around 2009–2011 was
an important step in developing super-resolution methods because it demonstrated
that the miniature contact microlenses capable of broadband imaging can be fabricated by the standard surface patterning techniques [26–28]. The nanoscale lenses
not only can be put in a very close contact with the object, but also give rise to various
near field effects related to their nanosize curvature and small focal distance. The
use of nano-SILs, however, did not provide any significant resolution improvement
over the immersion effect.
In 2011, the imaging through dielectric microspheres in contact with an object
was demonstrated [29]. This can be viewed as a natural development of the SIL
concept in which the microsphere forms a magnified virtual image of the object with
a participation of its near fields. This pioneering work immediately captured the
attention of researchers due to significant improvement in quality, optical contrast,
and resolution of the images. Silica microspheres in air were used in [29]. After
that, it was shown that by using semi-immersed microspheres, one can increase the
image contrast [30]. A significant advancement in this area was a proposal [31] of
using high-index (n > 1.8) microspheres which are fully liquid-immersed [32–34] or
embedded in elastomeric slabs [35–39]. It made possible FL imaging of subcellular
structures, viruses and proteins in a liquid phase [40–44] and confocal imaging
[33, 45] of structures, which cannot be resolved by conventional microscopes. A
comparison of SIL and microsphercial imaging indeed showed significant resolution
improvement in the later regime [33]. Coated high-index microspheres also showed
subwavelength resolution [46].
A particularly important aspect of these studies was an experimental quantification
of resolution. Initially, it was based on discernibility of small features such as gaps
separating metallic nanodisks or nanoholes in plasmonic arrays [29, 32, 47]. In this
approach, the resolution was associated with the minimal discernable feature sizes in
the optical images of such objects. The problem of this approach can be illustrated,
for example, by taking two touching circular objects. Their image calculated by a
convolution with the point spread function (PSF) shows that the intensity profile
through the centers of the circles actually displays a central dip. If such a dip was
observed experimentally, the simplified method based on discernibility of the smallest
feature would have suggested an interpretation that the zero gap between the cylinders
became “resolved” or “discerned”. This clearly illustrates the contradictions and
limitations of this method because it would have meant that the infinite resolution
was achieved. It is clear that this simplified method can be used with extreme caution
and only for semi-quantitative resolution analysis. This was likely the reason for a
broad variation of resolution claims from λ/7 [32] to λ/8 − λ/14 [29] to λ/17 [45],
which were made during the initial development of microspherical nanoscopy.
A more rigorous resolution quantification method for arbitrarily shaped objects
requires taking a convolution with the two-dimensional (2D) PSF. The images need
to be calculated for different PSF widths and compared with the experimental images
to determine the PSF width providing the best fit for the experimentally observed
images. Based on the Houston resolution criterion, such PSF width can be accepted
as the resolution of the system. This approach is well known in the far-field optics, but
409
In this regard, the miniaturization of SILs that took place around 2009–2011 was
an important step in developing super-resolution methods because it demonstrated
that the miniature contact microlenses capable of broadband imaging can be fabricated by the standard surface patterning techniques [26–28]. The nanoscale lenses
not only can be put in a very close contact with the object, but also give rise to various
near field effects related to their nanosize curvature and small focal distance. The
use of nano-SILs, however, did not provide any significant resolution improvement
over the immersion effect.
In 2011, the imaging through dielectric microspheres in contact with an object
was demonstrated [29]. This can be viewed as a natural development of the SIL
concept in which the microsphere forms a magnified virtual image of the object with
a participation of its near fields. This pioneering work immediately captured the
attention of researchers due to significant improvement in quality, optical contrast,
and resolution of the images. Silica microspheres in air were used in [29]. After
that, it was shown that by using semi-immersed microspheres, one can increase the
image contrast [30]. A significant advancement in this area was a proposal [31] of
using high-index (n > 1.8) microspheres which are fully liquid-immersed [32–34] or
embedded in elastomeric slabs [35–39]. It made possible FL imaging of subcellular
structures, viruses and proteins in a liquid phase [40–44] and confocal imaging
[33, 45] of structures, which cannot be resolved by conventional microscopes. A
comparison of SIL and microsphercial imaging indeed showed significant resolution
improvement in the later regime [33]. Coated high-index microspheres also showed
subwavelength resolution [46].
A particularly important aspect of these studies was an experimental quantification
of resolution. Initially, it was based on discernibility of small features such as gaps
separating metallic nanodisks or nanoholes in plasmonic arrays [29, 32, 47]. In this
approach, the resolution was associated with the minimal discernable feature sizes in
the optical images of such objects. The problem of this approach can be illustrated,
for example, by taking two touching circular objects. Their image calculated by a
convolution with the point spread function (PSF) shows that the intensity profile
through the centers of the circles actually displays a central dip. If such a dip was
observed experimentally, the simplified method based on discernibility of the smallest
feature would have suggested an interpretation that the zero gap between the cylinders
became “resolved” or “discerned”. This clearly illustrates the contradictions and
limitations of this method because it would have meant that the infinite resolution
was achieved. It is clear that this simplified method can be used with extreme caution
and only for semi-quantitative resolution analysis. This was likely the reason for a
broad variation of resolution claims from λ/7 [32] to λ/8 − λ/14 [29] to λ/17 [45],
which were made during the initial development of microspherical nanoscopy.
A more rigorous resolution quantification method for arbitrarily shaped objects
requires taking a convolution with the two-dimensional (2D) PSF. The images need
to be calculated for different PSF widths and compared with the experimental images
to determine the PSF width providing the best fit for the experimentally observed
images. Based on the Houston resolution criterion, such PSF width can be accepted
as the resolution of the system. This approach is well known in the far-field optics, but