[1] [cor1]Corresponding author

# Non-resonant subwavelength imaging by dielectric microparticles

Reza Heydarian<EMAIL_ADDRESS>Department of Electronics and Nano-
Engineering, Aalto University, P.O. Box 15500, FI-00076 Aalto, Finland
Constantin Simovski  Faculty of Physics and Engineering, ITMO University,
199034, Birzhevaya line 16, Saint-Petersburg, Russia

###### Abstract

Recently a hypothesis explaining the non-resonant mechanism of subwavelength
imaging granted by a dielectric microsphere has been suggested. In accordance
to the hypothesis, the far-field image of a subwavelength scatterer strongly
coupled to a microsphere by near fields is offered by the scatterer
polarization normal to the sphere surface. The radiation of a closely located
normally oriented dipole is shaped by the microsphere so that the transmitted
wave beam has a practically flat phase front. Then this beam turns out to be
imaging i.e. keeps the subwavelength information about the dipole location to
a long distance from it. However, this mechanism of subwavelength imaging was
only supposed in our previous paper. In this paper, we present a theoretical
study which confirms this hypothesis and explain the underlying physics of the
imaging which may hold in several scenarios when either a flat or a slightly
diverging phase front of the wave beam formed by a microsphere enables the
subwavelength resolution of two dipole sources in presence of different
focusing lenses. We numerically simulate one of these scenarios – that when
the focusing lens is located rather closely to the beam-forming microsphere –
at the distance of the order of $100\lambda$ – and represents a microsphere
itself. In our simulations we replace a 3D microsphere by a 2D "sphere"
(microcylinder) so that to use an available electromagnetic solver for a
dielectric microparticle of large optical sizes. The physical mechanism of the
imaging does not suffer of this replacement.

###### keywords:

Diffraction free beam diffraction limit microsphere

##  1 Introduction

Nine years ago, capacity of a simple dielectric microsphere to offer a label-
free nanoimaging has been experimentally revealed[1]. If the object is
complex, e.g. represents a pair of closely located dipole scatterers, a deeply
subwavelength resolution $\delta\sim\lambda/5-\lambda/10$ was obtained for a
range of refractive index and a broad range of the sphere radii. Since the
image of an object comprising the subwavelength details is magnified it can be
recorded without post-processing or developed by a usual microscope [2, 3, 4,
5, 6, 7, 8, 9].

Before 2011 the functionality of the magnified subwavelength imaging without
fluorescent labels was known only for the so-called hyperlens – a
tapered/curved nanostructure with alternating plasmonic and dielectric
constituents forming the so-called hyperbolic meta-material [10, 11, 12, 13].
Since a simple microsphere is incomparably cheaper, grants much finer
resolution and higher magnification than any available hyper-lens, a spherical
micro-lens has been commonly recognized the most promising device for in-vivo
label-free nanoimaging. It attracted a lot of attention of researchers.
However, the mechanisms of its operation have not been fully understood up to
now.

Attempts to refer the subwavelength imaging functionality of the microsphere
to the resonances of the whispering gallery and multiple Mie-resonances of a
spherical cavity were done in works [2, 3, 6, 7, 8]. However, these works
referred to special cases and could not explain the majority of experimental
results. Those theoretical works which pretended to explain the hyperlens
functionality for non-resonant microspheres based on the photonic nanojet
phenomenon were later claimed erroneous [14].

In our recent work [15] the hyperlens-like operation of a dielectric
microsphere beyond the resonances was presumably related to the normal
polarization of the scattering object. Two mechanisms of the subwavelength
imaging of a dipole polarized normally with respect to the sphere were
assumed. In both these mechanisms the sphere forms the so-called imaging beam
which carries the information about the source exact location to a long
distance from the source. The difference between two mechanisms assumed in
[15] is in two very different types of the imaging beam. One type of the
imaging beam implied the paraxial focusing of the beam by the objective lens
and the coherent imaging of the object, another one – the tight focusing and
the non-coherent imaging. In the first scenario, the imaging beam was ejected
from the equatorial plane of the sphere and represented either one or few
(2-3) conical beams. This scenario was supposed in [15] for a sphere whose
radius $R$ is of the order of $(2-5)\lambda$. In this case, a normally
polarized dipole excites in a sphere few creeping waves propagating along the
sphere surface and forming under it a standing-wave pattern that can be
referred as the interference pattern of few TM-polarized leaky modes. The
leakage of these modes is the ejection of the corresponding creeping waves
from the equator of the sphere. Our numerical simulations in [15] have shown:
these ejected rays form a divergent beam with TM-polarization that has the
alternating phase versus the polar angle and the angular distribution of the
intensity resembles the Bessel function. It is not a usual Bessel beam, since
it is not the Bessel function of the radial coordinate. However, when we
analyzed its evolution up to the distances $500-600\lambda$ from the
microparticle center we saw that the angular distribution of the intensity
kept the same at all distances from the sphere.Also, in [15] we have shown
that two mutually coherent dipoles with the phase shift proportional to the
gap between them, create an interference pattern having two phase centers
(virtual sources) separated by a gap which is optically large and proportional
to the gap between the real sources. Unfortunately, we could not simulate the
focusing of these two imaging beams and this our concept remained not
completely proved.

Note that in [15] we have studied not a microsphere, but a microcylinder. We
could not find a reliable tool to simulate optically large spheres. However,
in the 2D problem the underlying physics keeps the same, because we do not
study the resonant phenomena and all effects in which we concentrate are
related solely to the curvature of the dielectric interface. Sure, the
cylindrical cross section operates similarly to the spherical one. We plan to
finalize the study of the subwavelength imaging by such modestly large spheres
in our next works where we will explicitly show the imaging.

In the present work we concentrate on the second scenario of the imaging
supposed in [15]. It is based on the creation of a non-divergent or very
slightly divergent imaging beam that is possible if the microsphere as more
substantial i.e. $R>10\lambda$. For so large spheres the Mie resonances
vanish, the leaky modes form a quasi-continuum, and the formation of the
imaging beam can be governed by the laws of geometrical optics (GO). Namely,
GO can be applied to describe the refraction of the wave on the back side of
the sphere. As to the penetration of the dipole radiation into the sphere, it
is, on the contrary, a quasi-static effect. A dipole source located on the
dielectric interface creates a strong local polarization in its vicinity. A
subwavelength domain of maximal dielectric polarization can be united with a
dipole into an effective source. It is also a dipole effectively located
inside the sphere. Therefore we do not need to consider the diffraction of the
dipole radiation by the front side of the sphere. Of course, the diffraction
effects would be important if we considered the scattered fields. They would
be important also for the transmitted field if we located the primary source
at a substantial distance from the sphere. However, for the dipole source
located just on the front surface of a large dielectric sphere, the
transmitted wave beam is formed by the back half of the spherical surface and
it is possible within the frameworks of GO. The paths of all rays to the back
side of the sphere are optically large and the surface curvature is small, so,
the prerequisites of the GO approximation are respected. The problem arises
only at the frequencies of whispering-gallery resonances. However, in the
range $R>10\lambda$ these resonances are sparse over the frequency axis and we
do not consider them. These speculations allowed us to claim that the
collimated transmitted beam is possible in the transmission region [15].

##  2 Governing idea and its discussion

Figure 1: A normally polarized dipole produces behind the sphere a transmitted
beam depicted in the approximation of GO for two values of the refractive
index. The rays propagating from the dipole source to the back half of the
sphere have $\theta<\theta_{TIR}$ and do not experience total internal
refraction. In the case (a) $n=1.44$, and they all refract in parallel to the
beam axis $y$ forming a tubular beam. The phase front of this beam is nearly
flat. In the case (b) $n=1.38$, these rays are slightly diverging, and the
beam represents a spherical wave with annular pattern. The phase center of
this wave is located on the optical axis far behind the real source.

For the convenience of readers let us repeat the speculations of [15]. In the
approximation of GO the rays of an internal dipole source refract on the back
interface as if it were flat. Using the Snell’s law it is easy to calculate
the direction of rays in the transmitted beam. For a specific value of the
refractive index ($n=1.44$) all these rays are parallel to the axis $y$ and
the critical angle of total internal reflection, as depicted in Fig. 1(a), is
$\theta_{TIR}\approx\pi/4$. In the GO approximation, it is easy to calculate
the intensity distribution across the beam. For a normal (radial to the sphere
centre) dipole and a tangential one the intensity distributions will be
different due to the dipole angular pattern. The result for the intensity
distribution in the case of a radial dipole is presented in Fig. 1(a). In
accordance to this result, the transmitted beam is tubular. Its polarization
is radial with respect to the beam axis – axis $y$ due to the axial symmetry.
In the case when $n<1.44$ the transmitted beam in the GO approximation is not
exactly parallel but is slightly divergent. For this case the intensity
distribution is shown in Fig. 1(b) as a function of the ray tilt angle
$\alpha$. The distribution of $I/I_{max}$ versus $\alpha$ resembles the
intensity distribution for a tubular beam versus the radial coordinate $\rho$.

It is also easy to calculate the phase distribution in these beams. Let the
coordinate origin $x=y=z=0$ be the center of the sphere. For an internal ray
which impinges the back interface of the sphere under the incidence angle
$\theta$ the optical path from the source point $x=z=0,\,y=-R$ to the plane
$y=R$ equals $\Pi=R[2n\cos\theta+R(1-\cos 2\theta)]$. The transverse
coordinate of the transmitted ray $\rho=\sqrt{x^{2}+z^{2}}$ is equal $R\sin
2\theta$, and it is clear that for the case $n=1.44$ (when in accordance to
the Snell’s law all rays transmit with the zero tilt angle $\alpha=0$ to the
axis $y$) the value $k\Pi$ is not exactly constant versus $\rho<R$. However,
the phase front of the transmitted beam can be with a suitable accuracy
adopted flat because the beam is tubular. The intensity of the beam is
effectively concentrated in a ring $R/2<\rho<R$, and inside this ring the
phases of all rays in the plane $y=R$ are nearly equal to one another. For
example, for $R=10\lambda$ the phase difference between the rays having
$\rho=R$ and $\rho=R/2$ is close to $\pi/20$. This difference, though nonzero,
is, definitely, more tiny than the error of the GO model. In other words, the
GO model of the transmitted beam sketched in Fig. 1(a) is not controversial.
When all rays are parallel to one another, the phase font across the beam is
almost flat. Similarly, when $n=1.81$ the phase front turns out to be exactly
flat in the GO model, whereas the rays inside the ring $R/2<\rho<R$ at the
plane $y=R$ will be almost parallel to the beam axis. This self-consistency of
the model allowed us to hope that such a parallel transmitted beam is really
possible.

Next, for the case $n=1.38$ the GO model delivers the phase distribution which
corresponds to the spherical wave and similarly matches the intensity
distribution depicted in Fig. 1(b). The surface to which all rays are
orthogonal in accordance to the Snell’s law is the surface on which all rays
of this hollow conical beam have nearly the same phase in accordance to their
optical path. In other words, this conical beam is a spherical wave having a
point-wise phase center. It is clear that the phase center of this wave is
located on the beam axis $y$ very far behind the source. Now we may formulate
the governing idea of this work as follows: a large microsphere excited by a
point dipole (located on its front surface and polarized normally) forms two
types of the wave beam in the region of the transmitted field. The first type
of the beam is a hollow non-divergent beam (tubular one), the second type is a
spherical wave with annular pattern and very distant phase center. We believe
that both these transmitted beams should offer a magnified subwavelength image
of a tiny object located on the sphere. We have called both these beams as
imaging beams.

Figure 2: Two point dipoles (shown for better visibility by different colors)
are located with a subwavelength lateral gap $\delta$ on the front surface of
the sphere. The radiations of dipoles are not mutually coherent. They
separately produce two non-divergent wave beams whose cross sections are equal
to $2R$. These beams create a magnified image of the dual object in the focal
plane of the objective lens. Since $\Delta\gg\delta$ the subwavelength size
$w<\lambda$ of the focal spot is not needed for superresolution.

Let us consider the first scenario of imaging, when a tubular beam with flat
phase front propagates along the axis $y$ to the distance much larger than the
sphere diameter $2R$. The propagation of such beams in free space and their
focusing by lenses are described in works [16, 17, 18]. In [17] it is pointed
out that such beams should offer the spatial superresolution because being
focused by a lens with short focal distance (this distance $F$ should be then
the value of the order of the beam effective width) this beam converges into a
spot of subwavelength size. For this subwavelength focusing the beam should be
nearly parallel i.e. we need to place the lens at the lower distance from the
beam waist than the Rayleigh distance $D_{R}$, also called the diffraction
length. It is commonly known that $D_{R}=\pi R^{2}/\lambda$, where $R$ is the
effective radius of the beam waist (in our case $R$ is the sphere radius). The
requirement of the tight focusing is essential because a long-focus lens will
collect our tubular beam into a ring [18]. Only a tight focusing grants the
strong tilt of the electric field vector in the converging rays with respect
to the beam axis and results spot-wise image. In this case the maximum of the
intensity is achieved on the beam axis where $\bf E$ is polarized
longitudinally.

Though for the first scenario of imaging the tight focusing is required, in
this scenario we do not need to obtain the subwavelength focal spot. Two non-
divergent beams corresponding to two point dipoles with a small lateral gap
$\delta$ between them will create two spots in the focal plane of the
objective lens as it is shown in Fig. 2. The gap $\delta$ can be deeply
subwavelength, the focal spot diameter $w$ can, on the contrary, exceed
$\lambda$, but still we will obtain the resolution of two point sources
because the distance between the images of our dipoles is drastically enlarged
compared to $\delta$. In accordance to GO $\Delta\gg\delta$ if $L\gg 2R$. The
maximal spot size $w=\Delta\approx L\delta/R$ corresponds to the ultimate case
of the spatial resolution when it is possible that $\delta\ll\lambda\ll w$.
All we need for the far-field subwavelength imaging is the negligibly small
beam divergence (that holds for $L<D_{R}$) and the tight focusing so that to
image a point by a small spot and not by a ring [18].

As to the imaging in the second scenario, when the transmitted beam is
slightly divergent, the superresolution property is evident even without an
illustration. Really, let a slightly divergent imaging beam have a point-wise
phase center on the axis $y$ where the continuations of the rays shown in Fig.
1(b) cross out. Then a normally polarized dipole located on the front surface
of the sphere produces a virtual source located at the left from the sphere at
a big distance $Y$ from it ($Y\gg R$). Two dipoles with a tiny gap $\delta$
between them will produce two virtual sources on their axes. The distance
$\delta_{v}$ between them will be proportional to $\delta$:
$\delta_{v}=Y\delta/R$. There is no problem to resolve these virtual sources
with a microscope if $\delta_{v}>\lambda$.

Of course, all these speculations are valid if the GO model is adequate for
two types of the imaging beam, but the absence of the internal contradictions
in our GO models was inspiring. We made a lot of full-wave numerical
simulations for different values of the refractive index $n$ and different
size parameters $kR$ of the sphere. In these simulations we found the ranges
of $n$ and the ranges of $kR$ which offer both claimed regimes – that of the
tubular beam and that of the hollow conical beam with a far phase center.
These simulations are discussed below.

##  3 Preliminary simulations

We performed extensive simulations of 2D "spheres" using COMSOL Multiphysics,
which is a software much more rapid and reliable than the other
electromagnetic solvers for the 2D problems. We have calculated the intensity
plots and instantaneous wave pictures of the transmitted beam varying $R$ in
the interval $(10-40)\lambda$, where we put $\lambda=550$ nm. We varied $n$ in
the interval $1.4-1.7$ typical for optical glasses. We also varied the gap
between the "dipole" and the "sphere" from zero to $\lambda/10$ and found that
this gap does not influence the field distributions and only may change the
magnitudes.

Though the simplistic model represented by the plot in Fig. 1(a) does not give
a quantitative agreement with our full-wave simulations, it granted a
qualitative insight and useful estimates. The parallel transmitted beam was
obtained for $n=1.49-1.7$ when $R=15-25\lambda$. For each $R$ from this range
there is band of $n$ for which the transmitted beam is parallel, e.g. for .
$R=20\lambda$ the parallel beam is formed if $n=1.55-1.62$. When
$R=25-30\lambda$ the beam for the same values of $n$ is slightly diverging and
corresponds to the type of the imaging beam depicted in Fig. 1(b). For
$n=1.40-1.48$ the parallel transmitted beam was not obtained for any radius
from the whole range $R=10-40\lambda$. In this case, when $R>20\lambda$ the
beam is slightly diverging as in Fig. 1(b). When $10\lambda<R<20\lambda$ and
$n=1.40-1.48$ the phase center of the diverging beam is not point-wise and the
second scenario of the imaging is not possible.

Further, we have checked how the imaging beams of both types vary at the long
distances from the sphere. The replacement of a 3D sphere by a 2D "sphere"
(cylinder), when then the point dipole is replaced by a dipole line, allowed
us to reliably simulate not only very large microparticles ($R=10-40\lambda$),
but also to follow the transmitted beams at very large distances – hundreds of
$\lambda$. Moreover, this replacement allowed us also to simulate the
practical case when the dipole source is sandwiched between the sphere and the
substrate. Let us stress that this replacement is justified: we study the non-
resonant regimes, when the substitution of the 3D sphere by a 2D "sphere"
keeps valid the underlying physics. Even numerically there is the good
correlation with the 3D case. For example in our 2D simulations the beam
effective width calculated for $R=20\lambda$ increases by nearly $10$% at the
distance $y\approx 200\lambda$. The same result corresponds to the evolution
of the axially symmetric radially polarized beam of the same waist radius $R$
in accordance to formulas of [18].

Unfortunately, we could not implement in our simulations neither the first
scenario of the imaging illustrated by Fig. 2 nor the second scenario. The
first scenario cannot be implemented with a realistic microscope. The spheres
with the radius $R>30-40\lambda$ do not offer the subwavelength imaging in the
experiments and do not form imaging beams in our 2D simulations. However, a
microscope objective lens cannot have the focus as short as $F<30-40\lambda$.
As to the second imaging scenario, its implementation would require to
simulate the beam evolution on the paths exceeding $1000\lambda$, that is not
possible even in the 2D variant for any available electromagnetic solver.

Figure 3: Two point dipoles (shown for better visibility by different colors)
are located with a subwavelength lateral gap $\delta$ on the front surface of
the sphere. The dipoles produce two wave beams whose cross sections are
initially equal to $2R$ but at the distance $D_{R}$ become nearly spherical
waves. Though the phase centers of these waves are spread to the area of the
size $2R$, these areas do not intersect because $\delta_{v}\gg 2R$. These
phase centers can be guessed in the insets (COMSOL simulations) and treated as
virtual images of the real sources because $\delta_{v}$ is proportional to
$\delta$.

However, the analysis of the parallel beam evolution at large distances $y\sim
D_{R}$ (Rayleigh range) allowed us to reveal the third scenario of
subwavelength imaging granted by the microsphere. At these distances the
intensity starts to oscillate across the beam (i.e. the tubular beam splits
onto several hollow beams) and simultaneously the tubular beams start to
diverge becoming conical ones. At the distance $y\approx D_{R}$ the beam
produced by the sphere $R=20\lambda$ with $n=1.6$ (which looks non-divergent
up to 200$\lambda$) transforms into a nearly spherical wave with the annular
pattern. It is not an ideal spherical wave – its phase center is not point-
wise. However, it is an area whose size is smaller than $R$. Therefore, even
this spread phase center grants the superresolution. This scenario of imaging
is illustrated by Fig. 3. The size of the spread phase center can be guessed
on the insets (the wave picture of the initially parallel transmitted beam at
the Rayleigh range of distances). It can be easily estimated that the wave
really diverges from the area whose longitudinal and transverse sizes are
smaller than the initial beam radius $R$. These areas can be then treated as
two virtual sources formed in the places where two initially parallel beams
created by two dipole sources transform into two spherical waves. These
virtual sources do not overlap if their lateral size is smaller than the
distance $\delta_{v}$ between them, where $\delta_{v}=\delta D_{R}/R=\pi\delta
R/\lambda$. The resolution will be finer than $\lambda/\pi$ because the
lateral size of a virtual source is smaller than $R$ and because a partial
overlapping of the virtual sources still allows the microscope to resolve
them.

Figure 4: A tangentially polarized dipole produces behind the sphere a
transmitted beam depicted in the approximation of GO for $n=1.44$. The
parallelism of the rays contradicts to the curvature of the phase front. In
the inset the instantaneous wave picture of the beam with the smallest
obtained front curvature is shown. It corresponds to $n=1.7$, $R=5\lambda$

.

Now, let us discuss the role of the normal polarization of the dipole in the
imaging. If a normal dipole can create the imaging beam why a tangential
dipole cannot? The answer becomes clear inspecting Fig. 4. The pattern of the
tangential dipole results in the intensity distribution having the maximum on
the beam axis. For a hollow beam we could neglect the curvature of the phase
front, for a cylindrical beam it is not allowed. From one side, for $n=1.44$
the rays of the transmitted beam are orthogonal to the plane $(x-z)$, from
other side, the phase front has the non-negligible curvature over the beam
cross section. For $R=20\lambda$ the phase of the rays with the radial
coordinate $\rho=R$ differs from the phase of the central ray in the plane
$y=R$ by nearly $\pi/4$ that is not a negligible phase shift. If we change $n$
so that the GO model delivers the exactly flat phase front ($n\approx 1.81$),
the same GO model gives the transmitted beam of intersecting rays noticeably
tilted with respect to the axis $y$. So, in this case the GO model turns out
to be not self-consistent. Therefore we had a little hope to form a parallel
beam with a tangential dipole. In our simulations any combinations of $n$ and
$kR$ resulted neither in a parallel beam like Fig. 1(a) nor a diverging beam
like Fig. 1(b). In the inset of Fig. 4 we show the typical wave picture of the
transmitted wave beam for the tangential dipole. The curvature of the phase
front is varying across the beam and does not correspond to a spherical wave
with point-wise or at least subwavelength small phase center. This our result
fits that of [14] where the simulations were performed using the exact
solution of the 2D problem (dipole line field diffraction by a cylinder).
Though we have a little hope to achieve the subwavelength imaging of a
tangentially polarized object, these speculations are, of course, not strict
and the principal possibility to obtain the imaging beam cannot be proved
involving the GO model and simulations of some particular cases. He we may
claim only that in our numerical studies we have not found the regime of the
imaging beam for a tangential dipole and this negative result fits our
theoretical expectations.

##  4 Fourth scenario of the subwavelength imaging by a microsphere

We believe: what an experimenter sees in the microscope looking at two point-
wise scatterers through a glass microsphere is the image of the virtual
sources created by the imaging beams as it is shown in Fig. 3. However, the
doubts are possible even after these our simulations because we have not
simulated the whole system depicted in this Figure i.e. have not proved our
claim completely. A fully convincing explanation of the non-resonant
hyperlens-like action of a glass microsphere is a challenging task. In order
to publish a scientific paper treating this phenomenon we should have shown
namely the superresolution, at least in the numerical simulations. What we
have shown were only prerequisites of the superresolution.

Since, we are unable to simulate the beam evolution in the scale of thousands
of wavelengths as it implied in Fig. 3, we decided to study the fourth
scenario of the superresolution granted by a microsphere. This is a
modification of our first scenario with the replacement of the macroscopic
focusing lens by a microlens. The center of the microlens is located at a
modest distance $y<100\lambda$ from the object and the microlens has also a
modest optical size. This restriction of the simulation area makes the
simulation rapid and allows us to perform extensive simulations and
optimization of the structure. Unfortunately at the distances $y<100\lambda$
two imaging beams with the waist widths $2R=20-60\lambda$ created by two point
dipoles separated by a small gap $\delta<0.5\lambda$ still intersect.
Therefore, we cannot obtain a strong magnification of the image – our $\Delta$
turns our to be of the order of $2\delta$. However, in order to obtain the
subwavelength resolution we may utilize the property of a radially polarized
non-divergent beam – the possibility to be focused into a subwavelength spot
[17, 18]. In our conceptual proof, the microlens is a glass sphere, similar to
that which forms the imaging beam. We have seen in our simulations that it may
really focus the axially symmetric beam (a parallel one and even a slightly
diverging one) into a subwavelength spot.

In order to focus the parallel or slightly diverging beam we did not need a
large microlens. It is the "sphere" of the same radius whose center is located
at the distances $50-100\lambda$ from the center of the first "sphere". On the
first stage we simulate the subwavelength image of a single dipole source
located on the front size of the first "sphere". On the next stage, we replace
one dipole source by two non-interfering dipoles with a subwavelength gap
between them. Finally, we add a large dielectric block emulating the substrate
so that the imaged object is sandwiched between it and the sphere. Repeating
all the steps we study the impact of the substrate to the imaging and find
that this impact is positive: the substrate plays an important role in the
proper formation of the imaging beam.

##  5 Subwavelength imaging of a single point dipole

Figure 5: Color map of electric intensity in the imaging system
$R_{1}=R_{2}=20\lambda$ excited by a dipole located at a point
$x=0,\,y=-R-\lambda/10$.

All calculations reported below refer to the gap $\lambda/10$ between the
normally oriented Hertzian dipole and the first "sphere". On the stage
reported in this section we simulate the image of one dipole source. Recall
that in our current scenario we need the subwavelength size of the focal spot
$d$. Note, that the subwavelength focus in work [17] was obtained for the 3D
case. In [19] we have theoretically obtained the similar result for the 2D
case. After extensive simulations we have seen that the optimal radius of the
second "sphere" which would grant us the minimal possible $d$ for the case of
the parallel imaging beam is equal $R$. Thus our imaging system is a pair of
microcylinders of the same radius $R_{1}=R_{2}=R$. In Fig. 5 we see an example
of such imaging system and the color map of electric field intensity. The
imaging system consists of two glass "spheres" of radius $R=20\lambda$. The
refractive index of the first "sphere" is $n_{1}=1.6$ and that of the second
"sphere" is $n_{2}=1.52$.

This color map represents the intensity distribution that in general
corresponds to Fig. 1(a). The differences are weak intensity maxima located
between two "spheres" in the area $|x|<R$ and weak interference pattern in the
imaging beam. Both these features result from the multiple reflection of the
imaging beam between two "spheres". If the distance between the "spheres"
increases from $(50-100)\lambda$ to $(150-200)\lambda$, these features vanish.
However, the multiple reflection of the imaging beam between two "spheres"
whose centers are distanced by $60\lambda$, is a weak effect, and we saw that
it is not harmful for the imaging. Therefore, in this paper we report namely
this compact structure.

In the color map we see the image of a point dipole. It is a small bright spot
slightly but noticeably distant from the rear edge of the second "sphere". The
$y$-coordinate of this spot center determines the focal plane of the second
"sphere" operating as a microlens. Analysing the intensity distribution $I(x)$
in this plane we find the effective width $w$ of this focal spot. According to
the Rayleigh criterion ($0.7$ of the intensity at the local maximum),
effective width for the present case is $w=0.37\lambda$.

Figure 6:  Color map of the intensity in the area of the second "sphere"
(microlens) for $n_{2}=1.55$ (a) and $n_{2}=1.49$ (b). White dashed line marks
the microlens focal plane.

The ratio $w/\lambda$ can be reduced for given $R$ and $n_{1}$ if we vary the
refractive index $n_{2}$ of the microlens. To understand the impact of $n_{2}$
we inspected zoomed color maps of intensity in the area of the second
"sphere". In Fig. 6 we depict these maps for two values of $n_{2}$. In both
cases, there are noticeable side-lobes. However, for $n_{2}=1.55$ the central
maximum is more pronounced than in the case $n_{2}=1.49$. It shows that for
given $R$ and $n_{1}$ there is an optimal interval of $n_{2}$. How broad is
this band?

Figure 7: Intensity plots in the focal plane of the second "sphere" for four
values of the refractive index $n_{2}$.

In Fig. 7 we have depicted the intensity plots calculated in the focal plane
of the second "sphere" for several values of $n_{2}$ varying from $1.49$ to
$1.58$ with the step $0.3$. The effective width of the central spot varies
from $0.22\lambda$ for $n_{2}=1.58$ to $0.41\lambda$ for $n_{2}=1.49$. For
smaller $n_{2}$ the subwavelength imaging of the point source is lost – the
central spot exceeds $\lambda/2$. For $n_{2}>1.58$ the ratio $w/\lambda$ keeps
close to $\lambda/5$, i.e. is deeply subwavelength. For $R=20\lambda$ and
$n_{1}=1.6$ the optimal values of $n_{2}$ lie within the interval
$n_{2}=1.55-1.7$.

To illustrate the impact of the optical size of the second (imaging) "sphere",
we show our best result obtained for the case when the imaging beam is formed
by a "sphere" of radius $R=30\lambda$. In this case our imaging system also
comprises two "spheres" of the same radius. The optimal refractive index of
the imaging "sphere" is $n_{1}=1.6-1.7$ and the same refers to the optimal
values of $n_{2}$. The imaging beam produced by the first "sphere" in the case
$n_{1}=n_{2}=1.6$ is slightly diverging and $I(\alpha)$, calculated in the
absence of the second "sphere", qualitatively mimics that shown in Fig. 1(b).

If the second "sphere" is added at the center-to-center distance $90\lambda$
from the first one, the features of the parasitic interference are more
pronounced than those we have observed in the case of the parallel beams for
the same distance between the spheres. Increasing this distance we have
checked that this parasitic effect has no impact to our imaging. Therefore, we
may center the second sphere at the modest distance $90\lambda$ and show the
whole imaging system in the same color map as we did for the parallel beam.
This color map for the slightly diverging beam is presented in Fig. 8. In this
map we see the similar image as we observed in the case of the parallel beam.
In accordance to the intensity plot, calculated in the focal plane and
depicted in Fig. 8(b), in the present case we have $w=0.308\lambda$. This
result cannot be noticeably improved under restriction $n_{1}=n_{2}<1.7$.

Figure 8: Color map of the electric intensity in the imaging system
$R_{1}=R_{2}=30\lambda$, $n_{1}=n_{2}=1.6$ (a) and intensity plot in the focal
plane (b).

For a glass "sphere" there is an interval of $R/\lambda$ for which the
achievable ratio $w/\lambda$ is deeply subwavelength. In our study, when the
dielectric material is an optical glass the optimal values of the ratio
$R/\lambda$ lie around $20$. Our study explains why the non-resonant hyperlens
functionality cannot exist for very large dielectric spheres as well for
submicron spheres. There is the range of size parameters for which the
phenomenon exists. It is rather broad but not ultra-broad. This our result
fits the experimental observations of [1]: in the incoherent illumination the
best lateral resolution of two point scatterers ($\lambda/6-\lambda/8$) was
achieved for a glass sphere ($n=1.46$) whose the radius was within the
interval $2.5\lambda\leq R\leq 12\lambda$. In our 2D case the corresponding
interval is $10\lambda\leq R\leq 30\lambda$.

From these studies we may conclude that both regimes of the parallel and
slightly diverging imaging beams are suitable for the fourth scenario of
subwavelength imaging – that by two microsphere. How fine is this image,
depends on the optical size of the imaging "sphere" and on its refractive
index.

Returning to our simulations, we should point out that the interval
$R/\lambda=(15-25)$ is optimal for sobwavelength imaging of a point source. In
this interval the condition $n_{1}=1.49-1.7$ allows us to observe the parallel
beam similar to that depicted in Fig. 1(a). For smaller $n_{1}$ ($n<1.49$) the
imaging beam becomes slightly divergent. In this case the image spot is still
subwavelength for the first "sphere" with $1.42<n_{1}<1.49$ and second sphere
$n_{2}>1.5$ but $w/\lambda$ increases until $w=0.4\lambda$. The material with
$n_{1}<1.42$ does not allow us to simulate the subwavelength spot even for
$R/\lambda=(15-25)$. For $R/\lambda=30$ we need $n_{1}\geq 1.55$ and
$n_{2}\geq 1.6$ so that to obtain $w/\lambda<0.5$.

##  6 Subwavelength resolution of two dipoles

Figure 9: Color map of the electric intensity in the imaging system repeating
that from Fig. 5 with added block of silicon distanced by $\lambda/10$ from
the dipole source.

Spatial resolution of two dipole sources is not always determined by the
effective focal spot ($w$) corresponding to single point source. It is proved
in physical optics that the lateral resolution $\delta_{\rm min}$ of two
mutually incoherent dipoles and the width of the single source imaging spot
$w$ are equivalent. However, this identity refers only to the case of two
dipoles in free space, when one may express $w$ through the angular size of
the Airy disk. When two point sources, separated by a small gap $\delta$, are
coupled to a microsphere or any other body, there is no equivalence of $w$ and
$\delta_{\rm min}$. Though these values are somehow related, we even cannot be
sure that $w$ and $\delta_{\rm min}$ are values of the same order.

Therefore, it is necessary to perform the simulations with two mutually
incoherent dipoles. We have repeated the above simulations replacing one
dipole source located at the point $y=0$, $x=-R-\lambda/10$ by two point
dipoles stretched along $y$ and located at the points $y=\pm\delta/2$,
$x=-R-\lambda/10$. The gap $\delta$ is incomparably smaller than $R$, and
these dipoles can be considered polarized radially to the imaging "sphere"
center. The absence of mutual coherence of the dipoles is the same as the
suppression of their interference. Since in our simulations, the oscillations
are monochromatic, we could not mimic the mutual incoherence of two dipoles
exactly. We emulate it by the phase shift $\pi/2$ between the dipole moments
per unit length of our dipole lines. The fields of such dipoles do not
interfere at least on the axis $y$, and in the region of our interest their
interference is maximally suppressed.

Our simulations have shown that $\delta\geq w$ in all cases when the imaging
beam is non-divergent. For the most part of the explored ranges of two main
parameters ($R=10-40\lambda$, $n=1.4-1.7$) the subwavelength resolution is not
achieved, though the subwavelength imaging spot was obtained for an individual
dipole. This is so, due to two reasons. First, the interference of two sources
is suppressed not completely even in the paraxial region. Second, the GO
approximation is not fully adequate for our "spheres" due to the impact of
creeping waves, especially when $R<20\lambda$. Though the patterns of creeping
waves are not visible in the simulated color maps, for $R<20\lambda$ their
presence can be detected in the data files.

However, in the presence of the substrate, the creeping waves are suppressed,
and the GO model becomes much more adequate. It explains why the lower bound
of the optimal interval $R/\lambda$ in the experimental studies is as low as
$2.5\lambda$. The substrate broadens the range of $R/\lambda$ suitable for the
formation of the imaging beam.

Figure 10: Color map (a) and intensity plot in the focal plane (b) for the
imaging system from Fig. 9 excited by two dipoles with a gap
$\delta=0.27\lambda$ between them. In the image this gap is magnified
($\Delta=1.93\delta$).

We could not simulate the imaging structure with a macroscopic substrate. In
the present work, we replace a realistic substrate by a substantial dielectric
block of non-resonant sizes. The resonances of the block are excluded adding
the optical absorption. With such a block, we have managed to achieve a
subwavelength resolution $\delta_{\rm min}\ll\lambda$ in the most part of
simulated cases. In presence of the block, the dipoles are sandwiched between
its flat surface and the sphere. The minimal distance between the sphere and
the block is adopted to be equal $\lambda/10$.

First, let us see how the block changes the point dipole image size $w$. Fig.
9 corresponds to the same imaging system as that depicted in Fig. 5 with one
difference – a block of amorphous silicon added so that the dipole source is
sandwiched between the block and the "sphere". In this case the intensity in
the imaging beam mimics much better the intensity distribution calculated
using the GO, because the block completely eliminates the cavity modes. With
the block we have achieved $w/\lambda=0.186$ instead of $w/\lambda=0.220$.

Figure 11:  Intensity plots with and without the silicon block for the imaging
system $R_{1}=R_{2}=30\lambda$, $n_{1}=n_{2}=1.6$ (a) and the same for the
case $n_{1}=n_{2}=1.5$. In the last case, the resolution is not subwavelength
because the refractive index of second sphere is not sufficient for a so large
imaging "sphere".

Second, the presence of the block reduces $\delta_{\rm min}$, approaching it
to $w$. In Fig. 10(a) we present the intensity color map in the area of the
focal plane of the system $R_{1}=R_{2}=20\lambda$, $n_{1}=1.525$ and
$n_{2}=1.575$, when the system is excited by two point dipoles located and
polarized as it is explained above. The gap between them is as small as
$\delta=0.27\lambda$. The same dielectric block as in Fig. 9 results in the
clear separation of two intensity maxima in the focal plane. Note, the
asymmetry of these maxima (different length along $y$) results from non-
negligible interference of the fields produced by our dipoles beyond the axis
$y$. However, in the focal plane, the intensity plot is almost symmetric. This
plot is presented in Fig. 10(b) for both cases – with and without the block.
We see that in absence of the block, our dipoles are not resolved according to
the Rayleigh criterion, but are resolved in presence of it. The gap $\delta$
is imaged in the focal plane by the distance $\Delta$ between two maxima. The
image magnification $\Delta/\delta\approx 1.93$. Note that it is much smaller
than the magnifications reported in the above-cited experiments with the
imaging microspheres. We see two evident reasons for it. First, we simulate
not a 3D microsphere but its 2D analogue. Second, we use a microlens located
at a sub-mm distance from the sources and not a lens system of a microscope
located much farther. The fact that the image magnification has been achieved,
in principle, is more important. It shows that our second "sphere" operates as
a lens, indeed.

The impact of the block is also positive for the regime of slightly diverging
imaging beam. If we add a block to the imaging system $R_{1}=R_{2}=30\lambda$,
$n_{1}=n_{2}=1.6$, excited with only one source, the imaging spot shrinks from
$w=0.308\lambda$ to $w=0.275\lambda$. If we consider two sources with a
lateral gap $\delta=0.327\lambda$ and out of phase ($\pi/2$ shift), we can
resolve them respecting the Rayleigh criterion only in presence of the block.
Magnification in this case is $\Delta/\delta\approx 1.52$. In Fig. 11(a) we
present the intensity plots in the focal plane of the imaging system
$R_{1}=R_{2}=30\lambda$, $n_{1}=n_{2}=1.6$ with and without the block. If we
remove the block, the images of the similarly distanced point dipoles are not
resolvable. However, for a so large "sphere" the subwavelength resolution even
with the block holds only for high refractive indices $n_{1}=n_{2}\geq 1.6$.
In Fig. 11(b) we depict the similar intensity distributions for the case
$n_{1}=n_{2}=1.5$. In this case two sources are distanced by
$\delta=0.545\lambda$ and even a so substantial gap is not resolvable if we
remove the block. Even though the block plays the positive role the truly
subwavelength resolution is not achievable for a so small refractive index and
large radius. These observations qualitatively fit the data from [1] if we
assume that in what concerns the formation of the imaging beams there is the
correspondence between our 2D "spheres" of radius $R$ and 3D spheres of triply
smaller radius. In [1] the spheres with $R=10-12\lambda$ offered the deeply
subwavelength image ($\delta=\lambda/8$) when their refractive indices were as
high as $n=1.7-1.8$, whereas the glass sphere ($n=1.46$) with so large radius
did not offer the deeply subwavelength image. Meanwhile the glass spheres of
smaller radius $R=5-7\lambda$ offered in these experiments the deeply
subwavelength images.

##  7 Discussion and conclusions

In this article, we have assumed and partially validated (by exact numerical
simulations) several scenarios of the non-resonant and non-coherent far-field
magnified subwavelength imaging enabled by a simple glass microsphere. We have
shown that this functionality demands the object polarization normally to the
sphere surface. The phenomenon is mainly based on the geometrical optics and
the dipole radiation pattern, but it may involve also the beam Abbe
diffraction if this diffraction transforms the beam into a nearly spherical
wave. Besides of the partial theoretical validation of this scenario, we
completely proved by exact numerical simulations that the imaging beam formed
in accordance to the laws of GO keeps the subwavelength information on the
object (located on the surface of a 2D microcylinder) at least up to the
distances of the order of $100\lambda$. We have shown that such the object
(consisting in our study of two non-coherent point dipoles) can be imaged by
the second 2D microsphere with the subwavelength resolution and modest
magnification.

In this work we have found a range of the microsphere parameters for which the
GO model is qualitatively adequate. We have determined the suitable intervals
of these parameters for two scenarios of the far-field magnified imaging which
supposedly correspond to the known experiments with the microspheres. We have
shown: if the object is sandwiched between the substrate and the microsphere,
the imaging functionality of the system of two microspheres improves.

Some readers may by puzzled how the GO model may result in the
superresolution, if this phenomenon is presumably related with the evanescent
waves. However, evanescent waves are not necessary for label-free
subwavelength imaging, it is a scientific myth which was demystified by a
hyperlens. Inside the hyperlens an imaging beam propagates from a point-wise
object towards the outer interface. The imaging beam keeps the fine
information about the object location because it is thin compared to the
wavelength in free space, however, this beam does not contain evanescent
waves, it is formed by spatial harmonics which are all propagating in the
hyperbolic medium. The imaging beam transmitted from the hyperlens to free
space also does not contain evanescent waves, it is wider than $\lambda$,
however, its spatial harmonics (propagating in free space) nicely keep the
subwavelength information on the location of the object.

The same physics is responsible for the imaging by a simple glass microsphere.
Our imaging beam also carries the fine information of the exact location of a
point-wise source. Though our imaging beam is not subwavelength thin even in
the zone of its waist, the information of the exact location of the object is
in the direction of our beam. It offers the subwavelength accuracy for the
dipole source location not due to the small thickness of our beam but due to
the pronounced beam axis. This one exactly points out the location of the
radially polarized dipole on the microsphere surface. This information is
evidently finer compared to that offered by a subwavelength small thickness of
the imaging beam propagating across the hyperlens. The axis of the hyperlens
imaging beam is not as pronounced as the axis of our hollow imaging beam. This
is one of the reasons why the imaging functionality of a simple glass
microsphere is much better than that of the hemispherical hyperlens. Another
reason is the presence of plasmonic losses in the hyperlens, and the practical
absence of losses in the optical glass. And the third reason is the narrow
operation band of the hyperbolic metamaterial, whereas the glass keeps
transparent in a very broad band of wavelengths. Though the hyperlens is made
of a super-expensive nanostrucrtured metamaterial, the glass microsphere
excited by a radially polarized object gains this contest.

To conclude: we think that the transition from the 2D "sphere" to a 3D sphere
promises a finer subwavelength resolution than that we have obtained in the
simulations of this paper. And we believe that this resolution and
magnification predicted by our model will be much higher when the macroscopic
objective lens is used instead of the microlens and the distance from the
imaging microsphere to the objective is realistic – corresponding to the cited
experiments. We believe that our work is important as a conceptual proof of
the non-resonant far-field subwavelength imaging functionality of the dipole
radiation transmitted through the microsphere in accordance to the simplest
geometric-optical model.

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