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Super-Resolution Imaging of a Dielectric Microsphere Is Governed by
the Waist of Its Photonic Nanojet
Hui Yang,
†
Raphaël Trouillon, Gergely Huszka, and Martin A. M. Gijs*
Laboratory of Microsystems, École Polytechnique Fé
dé
rale de Lausanne, 1015 Lausanne, Switzerland
*
SSupporting Information
ABSTRACT: Dielectric microspheres with appropriate refractive index can image
objects with super-resolution, that is, with a precision well beyond the classical
diffraction limit. A microsphere is also known to generate upon illumination a
photonic nanojet, which is a scattered beam of light with a high-intensity main lobe
and very narrow waist. Here, we report a systematic study of the imaging of water-
immersed nanostructures by barium titanate glass microspheres of different size. A
numerical study of the light propagation through a microsphere points out the light
focusing capability of microspheres of different size and the waist of their photonic
nanojet. The former correlates to the magnification factor of the virtual images
obtained from linear test nanostructures, the biggest magnification being obtained
with microspheres of ∼6−7μm in size. Analyzing the light intensity distribution of
microscopy images allows determining analytically the point spread function of the
optical system and thereby quantifies its resolution. We find that the super-resolution
imaging of a microsphere is dependent on the waist of its photonic nanojet, the best
resolution being obtained with a 6 μm Ø microsphere, which generates the nanojet with the minimum waist. This comparison
allows elucidating the super-resolution imaging mechanism.
KEYWORDS: Microsphere, super-resolution imaging, photonic nanojet, optical microscopy, point spread function
Conventional optical microscopes are limited by the so-
called diffraction limit and can resolve features of around
half of the wavelength of illumination λ, as they are only
capable of transmitting the propagating wave components
emanating from the illuminated object.
1,2
The evanescent
components that carry the fine information about the object
decay exponentially in a medium with positive permittivity and
permeability and are lost before reaching the image plane.
Several efforts to this date have been pursued to overcome the
diffraction limit using various approaches including near-field
scanning probes,
3−6
super/hyper-lenses driven by surface-
plasmon excitation,
2,7−12
and fluorescence microscopy with
molecular excitation.
13−17
However, the applications of these
methods have been limited in part due to their sophisticated
engineering designs or pretreatment steps. Alternatively, it has
been demonstrated that the use of nanoscale lenses,
18−20
polymer microdroplets,
21
and especially dielectric micro-
spheres
22−30
on top of the objects can achieve near-field
focusing and magnification, which in turn results in the
capability to resolve features beyond the diffraction limit. Wang
et al. first proposed the use of fused silica beads with refractive
index n∼1.46 and diameter from 2 to 9 μm in combination
with a conventional optical microscope to achieve a resolution
of 50 nm in air with a white light source.
22
Later on, it was
reported that large polystyrene microspheres (above 30 μm)
can also achieve super-resolution imaging with large field-of-
view in air without immersion liquid.
24
Moreover, it was
demonstrated that high-index microspheres (n∼1.9−2.1), fully
immersed in liquid, actually allow enhanced imaging with
minimum resolved feature sizes of ∼λ/7 with white-light
illumination for imaging of nanofeatures
23
and adenoviruses,
25
as well as with fluorescent microscopic setup for resolving the
structures of subcellular organelles.
26
More recently, micro-
spheres were used in combination with confocal microscopy for
achieving a 25 nm lateral resolution under 408 nm wavelength
illumination.
27
In these works, the microspheres are placed on
top of the sample object, where they collect the underlying
sample’s near-field nanofeatures and subsequently transform
the near-field evanescent waves into far-field propagating waves,
creating a magnified image in the far-field, which is collected by
a conventional optical microscope.
22,26,30
The super-resolution
capability of microspheres is linked to two factors: (i) the
microsphere performs as a solid-immersion lens and provides
local enhancement of the refractive index and a reduction of the
illumination wavelength; (ii) the development of the photonic
nanojet.
26,30
A substantial literature has developed regarding
the existence, properties, and potential applications of the
photonic nanojet.
31−56
In principle, a photonic nanojet is a
narrow light beam with high optical intensity that can be
generated by a transparent dielectric symmetric body, like a
microcylinders or microsphere, upon illumination. The nanojet
that emerges from a microsphere locates in the immediate
Received: March 24, 2016
Revised: July 5, 2016
Published: July 11, 2016
Letter
pubs.acs.org/NanoLett
© 2016 American Chemical Society 4862 DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
Figure 1. Mechanism of nanojet generation and imaging of a dielectric microsphere. (a) Focusing of a plane wave light beam into a nanojet at the
rear surface of a free-standing microsphere. At the front surface of the microsphere, the light is refracted at low incidence angle, while at higher
incidence angle it gets mostly reflected. (b) FEM simulation of the light propagation through a 6 μm Ø barium titanate microsphere in water. The
linear region where substantial refracted light enters the microsphere at its front surface is referred to as L, while the width of the exiting beam at the
rear surface is denoted as l; the waist of the nanojet is referred to as w. (c,d) FEM simulation of the light propagation through 2 and 16 μm Ø barium
titanate microspheres in water medium, respectively. (e) When a microsphere is positioned on a grating structure with line width dand illuminated
from the front, the light reflected by the grating allows detecting a virtual image with magnification factor M. When the distance hbetween the
microsphere and the grating is small enough (of order of the illumination wavelength λ), the near-field evanescent wave carrying the fine details of
Nano Letters Letter
DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
4863
vicinity of the rear-surface of the sphere. It is a nonresonant
phenomenon that appears for spheres with a diameter of the
order of 10−100 λand with a ratio of their refractive index nms
to the refractive index of the background medium (water in this
study) nwthat is smaller than about 2. The nanojet can
maintain a subwavelength full width at half-maximum (fwhm)
transverse beam width along a path that can extend more than
∼2λbeyond the sphere, and the minimum fwhm beam width,
referred to as “waist”in this paper, can be smaller than the
classical diffraction limit, in fact as small as ∼λ/3.
The imaging resolution of a classical microscope depends on
the size of the spot that is generated by focusing the incident
propagating wave in the far-field and is therefore limited by
diffraction. For microsphere-assisted optical microscopy,
evanescent waves close to the surface of the microsphere play
a significant role. Even though it was suggested that the
development of the photonic nanojet is essential to the super-
resolution imaging capability of a microsphere,
26,30
the exact
link remained unclear. Here we study, in a quantitative way the
role of the photonic nanojet for super-resolution imaging. First
a systematic numerical study of the light propagation through
microspheres of different size using the finite element method
(FEM) is performed. This allows characterizing the photonic
nanojet at the rear-surface of a microsphere and relating the
microsphere’s theoretical magnification factor to the light
focusing capability of the photonic nanojet. Second, we perform
an experimental study in which a systematic series of barium
titanate glass microspheres with diameter from 3 to 21 μm are
used to image linear test nanostructures that are immersed in
water. The experimental magnification factor and the point
spread function that is analytically determined from the images
allow evaluating the resolution of the optical system, which is
shown to directly correlate with the calculated properties of a
microsphere’s photonic nanojet.
As schematically illustrated in Figure 1a, when a microsphere
is illuminated by a propagating light beam from the far-field, the
light is mostly refracted on its frontal surface at small incident
angle and reflected at higher incident angle, the limiting angle
between the two regions given approximately by the Brewster
angle of the optical interface. When the size of the microsphere
is much bigger than the illumination wavelength, it is a good
approximation to use ray optics to explain how the light that is
incident on the top-surface of the microsphere propagates. The
refracted light propagates through the microsphere and
generates a photonic nanojet near the rear-surface of the
microsphere (details described in the Supporting Information).
An FEM study of the light wave propagation through a barium
titanate glass microsphere (nms = 1.92), with diameter Ø
ranging from 2 to 20 μm in surrounding water medium (nw=
1.33) is performed (details described in Methods). An
electromagnetic wave with wavelength λ= 600 nm is applied
to a boundary with a length that is the same as the size of the
microsphere and that is far away from its top-surface. As the
size of the boundary is much longer than the wavelength, a
plane-wave incident light beam is a good approximation for
studying the photonic nanojet. Figure 1b−d shows the light
intensity distributions near a 6, 2, and 16 μm microsphere,
respectively. When the microsphere is 6 μm in size (Figure 1b),
the photonic nanojet directly emerges from the microsphere
surface. When decreasing the microsphere’s size, the focal point
of the nanojet moves into the microsphere, as seen for the
simulation result of a 2 μm Ø microsphere (Figure 1c). For a
microsphere size bigger than 6 μm, the external focal point of
the nanojet moves away from the rear-surface of the
microsphere and Figure 1d shows the simulation result
obtained for a 16 μm Ø microsphere. To quantitatively study
how light is focused by the microsphere into the photonic
nanojet, we determined from the model calculation the linear
region where substantial refracted light enters the microsphere
at its front surface, referred to as Lin Figure 1b−d, while the
width of the exiting beam at the rear surface is denoted as l. The
waist of the nanojet is referred to as w(see Figure 1b), which is
the fwhm of the nanojet along the x-axis at the peak intensity of
the y-axis. It should be noted that the light source in the FEM
study is actually coherent, while the light source in conventional
optical microscopy is incoherent. The comparison on the light
intensity distribution in the photonic nanojet under coherent
and incoherent illumination is further discussed in the
Supporting Information. The study shows that the intensity
profile along the nanojet waist does not significantly change,
when using either an incoherent or a coherent light source,
indicating that a coherent light-based simulation can provide
sufficient information to study the resolution of the imaging
system.
The imaging mechanism of a dielectric microsphere is
schematically illustrated in Figure 1e. When the focused light
that exits the microsphere illuminates the sample object (in this
case a linear grating structure with feature size d), no nanojet is
generated but instead the reflected light follows a reflection-
symmetric optical path, while evanescent waves that contain the
high spatial-frequency information on the object are converted
into propagating waves within the microsphere (see Supporting
Information) when the distance hbetween the microsphere and
the grating is small enough (of order of the illumination
wavelength λ). In the meanwhile, a magnified virtual image is
generated in the far-field with magnification factor M. The
imaging capability of the microsphere directly correlates with
the formation of the focused photonic nanojet as obtained from
the numerical study, as the same optical paths are involved.
Figure 1f,g shows the calculated light focusing capability of a
microsphere, expressed by the ratio L/l, and the waist of the
photonic nanojet normalized by the illumination wavelength w/
λas a function of the microsphere diameter, respectively. A
bigger L/lindicates a better light focusing by the microsphere
and corresponds to a smaller waist of the nanojet. According to
our simulations, the microsphere with diameter of 6 μm shows
the best light focusing capability and smallest waist of the
photonic nanojet.
To experimentally study the super-resolution imaging effect,
grating structures consisting of 120 nm wide lines with 100 nm
Figure 1. continued
the grating can become propagating in the high refractive index sphere, and later in the medium where it is to be collected by the microscope
objective. (f) FEM simulation results of the light focusing capability of a microsphere, expressed by the ratio L/l, as a function of the microsphere
diameter. The dots are obtained from the simulation, while the red dotted line is a guide to the eye. (g) FEM simulation results of the normalized
waist of the photonic nanojet w/λ, as a function of the microsphere diameter. The dots are obtained from the simulation, while the red dotted line is
a guide to the eye.
Nano Letters Letter
DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
4864
interspacing (see Figure 2a) immersed in water are used as test
samples to be imaged with transparent barium titanate glass
microspheres that are loosely positioned on top of them. A
conventional reflection microscope (Zeiss Axioplan micro-
scope) equipped with a CCD camera (AxioCam MRm camera)
and a 40×water immersion objective with numerical aperture
(NA) of 0.8 is used to take images. A halogen lamp is used as
the white-light illumination source with a wide-band spectrum,
which is from ∼400 to 700 nm and the peak appears at ∼λ=
600 nm (more details are shown in the Supporting
Information). Figure 2b is a microcopy image of the gratings
taken by the 40×objective along, the inset is a 5×magnified
image, clearly showing that the conventional optical microscope
cannot resolve the nanopatterns with a feature size of ∼100 nm.
However, when a microsphere is placed on top of the nano-
objects, the subdiffraction features become observable. Figure
2c,e,g shows microspheres with diameters of 4.2 μm, 7.1, and
11.8 μm positioned on the sample, respectively. The focal plane
of the microscope coincides with the plane of maximum sphere
diameter, hence, on these pictures, the grating nanostructures
are out-of-focus. When the focal plane of the microscope is set
to the image plane of the microsphere, images obtained with
the microspheres of Figure 2c,e,g are shown in Figure 2d,f,h,
respectively. Compared to the size of the objects, the feature
sizes of the line structures in the images are clearly magnified.
As already schematically illustrated in Figure 1e, comparing the
size of the object with that of the image allows determining the
magnification factor M. The experimental Mvalues versus the
microsphere diameter are plotted in Figure 2i. We can see that
the biggest magnification is obtained with microspheres of ∼6−
7μm in size. When we compare the experimental magnification
factor Mwith the theoretical light focusing capability L/l,as
obtained from the simulations, a positive correlation with a
Pearson’s correlation coefficient of 0.91 is obtained, indicating
that a better light focusing capability of a microsphere logically
results in a higher magnification factor.
A remaining issue is whether the light focusing capability of a
microsphere has any impact on its super resolution imaging.
For solving this question, linear test nanostructures with 300
nm wide lines and 900 nm interspacing are imaged through the
microspheres with different size by using the same optical
microscope setup. As illustrated in Figure 3a, the microsphere is
Figure 2. Imaging of a grating nanostructure using different size microspheres and a water-immersion objective. (a) Scanning electron microscopy
image of the silicon grating test structure containing 120 nm wide lines with an interspacing of 100 nm. (b) Optical microscopy image of the
nanostructure taken by a 40×water-immersion objective with NA of 0.8. The insert is a 5×magnified image, clearly showing that conventional
microscopy cannot resolve the nanostructures with this feature size. (c−h) Optical microscopy images obtained by positioning on the grating
microspheres with sizes of (c,d) 4.2 μm, (e,f) 7.1 μm, and (g,h) 11.8 μm, respectively. The images of (c,e,g) are focused on the microspheres’center
plane, while the corresponding images (d,f,h) are focused on the virtual image plane, showing that the grating nanostructure is imaged with a
different magnification factor Mfor microspheres of different sizes. (i) The dots represent the experimental magnification factor as a function of the
microsphere diameter, while the solid line is a guide to the eye. (j) The experimental magnification factor Mas a function of the light focusing
capability L/lobtained from the simulations. The solid line represents a linear fitting curve with a Pearson’s correlation coefficient of 0.91.
Nano Letters Letter
DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
4865
positioned in the middle of two lines and the magnified image
can be obtained when the distance hbetween the microsphere
and the grating is of order of the illumination wavelength λ(see
Figure 1e), therefore only the interspacing at the center of the
microsphere (darker zone in the image) and its neighboring
two lines (clearer zones in the image) are visible in the far-field.
Figure 3b,c shows the microscopic images obtained with the
40×water-immersion objective (NA = 0.8) focusing on a 6.4
μm microsphere and on the image plane, respectively. The light
intensity profile along the dashed line in Figure 3c is shown in
Figure 3f. Moreover, similarly taken images obtained from a 9.9
μm microsphere are shown in Figure 3d,e, while the light
Figure 3. Quantification of the experimental resolution using the analytical point spread function model and correlation with the waist of the
photonic nanojet. (a) A microsphere is positioned specifically in the middle of two lines of a dedicated test grating (line width of 300 nm and
interspacing of 900 nm) to characterize the sharpness of line/interspacing boundary in the virtual image. (b−e) Optical microscopy images obtained
by positioning on the grating microspheres with sizes of (b,c) 6.4 and (d,e) 9.9 μm, respectively. The images of (b,d) are focused on the
microspheres’center plane, while the corresponding images (c,e) are focused on the virtual image plane. Each dashed line indicates where the
intensity profile will be taken that is to be fitted with the analytical point spread function model. (f) Intensity distribution along the dashed line in
panel c with x1and x2the positions of the descending and ascending steps obtained from the fit using eq 5. (g) Intensity distribution along the
dashed line in panel e. (h) The actual image standard deviation σthat is obtained from the fit, related to the true resolution of the system, as a
function of microsphere size. The dots are obtained from the fits with the analytical model, while the solid curve is a guide to the eye. (i) The
correlation between σand the normalized waist of the photonic nanojet w/λ. The solid line represents a linear fitting curve with a Pearson’s
correlation coefficient of 0.88.
Nano Letters Letter
DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
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intensity profile along the dashed line in Figure 3e is shown in
Figure 3g. Comparing Figure 3fwithFigure 3g, clear
differences on the peak intensities and on the intensity profiles
are observed.
This type of image is further analyzed using an analytical
point spread function (PSF) model to study the resolution in a
quantitative way. The PSF is an important factor characterizing
the mechanisms underlying the formation of an image in an
optical system, defined as the response of the imaging system to
a point object.
57−59
This three-dimensional function is
characteristic of the imaging system and can have a rather
complicated analytical expression, hence sometimes requiring
simplifications or approximations to facilitate its use. However,
the shape of the PSF is directly related to the resolution of the
system, as the narrower the PSF, the better the resolution. The
imaging process described here is based on the collimation of
light that is incident on top of the microsphere into the nanojet.
For detection of an object, the light is reflected back from the
object into the detector, essentially following the same mirror-
symmetric trajectory as during incidence. The width of the
nanojet land more precisely the quantity L/ldetailed in Figure
1b is a measure of the focusing capability of a microsphere.
Indeed, the smaller land the larger L/lis, the easier it will be
for an initially evanescent wave, when it is reflected from the
object, to be converted to a wave with higher spatial frequency
that becomes propagating inside the microsphere. If higher
spatial frequencies can be detected, this will result in a sharper
image, hence in a better resolution and a narrower PSF. That is
why the theoretical simulations of the nanojet profiles,
provided, for instance, in Figure 1, are indicative of the
diffraction processes in the optical system and can be useful to
compare with the experimental PSF. Additionally, simulations
hint that the illumination profile partially describes the blurring
introduced by the system and is at least indicative of the PSF in
the absence of severe mismatches in the refractive indices.
60
As
a consequence, these illumination profiles are used below to
establish assumptions on the overall PSF, and are justified a
posteriori.
Typically, the PSF can be approximated through models and
numerical simulations. However, the purpose of this section is
to find an experimental approach to evaluate the lateral
resolution of the system through the PSF and to compare it to
some of the results predicted by the simulations. In the case
considered in this work, that is, where a plane is imaged, only
the x−yprofile of the PSF at the waist of the nanojet is relevant
to the resulting image. Moreover, because of the cylindrical
geometry of the problem, as shown in Figure 1b−d, only the
expression of the radial component at the waist (that is, along
the x-axis in Figure S2c) is required to describe the PSF.
Furthermore, only sections of the images located at the center
of the microsphere are analyzed. This guarantees that
distortions due to the spherical shape of the microlens are
limited, and that small translations along the x- and y-axes at
the vicinity of the central axis of the system do not dramatically
alter the image. This is supported experimentally by the images
shown in Figure 2d,f,h where the gratings are clearly resolved at
the center of the microsphere and are parallel, hence ensuring
that no dramatic radial distortion occurs. As a consequence, at
the center of the microsphere the imaging device can be
assumed to be largely shift independent in the x−yplane.
Combined with the linearity of the system, this fact suggests
that the image can be expected to be solely determined by the
intensity profile of the PSF.
61
Furthermore, as shown by the
shape of the illumination profiles discussed in the Section S6 of
the Supporting Information, the shape of the nanojet is
independent of phase shifts. As the final image is formed by the
illumination light reflected by the object, one can assume that
the whole system and therefore the PSF are not dependent on
phase shifts. Briefly, it allows to relate the recorded two-
dimensional (2D) image Im to the input object Ob through a
convolution operation (denoted as *)
62,63
≡*xy xyIm( , ) (PSF Ob)( , ) (1)
The PSF can be defined as the impulse response function of
the system, that is, the image obtained from the imaging of a
point. Mathematically, this impulse, or point, can be expressed
as the Dirac function δ, the function returning 0 for x≠0 and y
≠0, and whose integral is 1 over
2. More intuitively, δcan be
approximated as an infinitely high, infinitely sharp peak
centered over (0, 0), the function being equal to 0 anywhere
else. Furthermore, δis the unit element for convolution, hence
δ=*xy xy
P
SF( , ) (PSF )( , ) (2)
Direct measurement of the PSF can be challenging, as
obtaining a pure point as the object to image is impossible. It
can be approximated by imaging a very small disk for instance,
but the result can be distorted if the object is below the imaging
capabilities of the device. A numerical simulation, as shown in
Figure 1, or an exact analytical solution can be used, but this is
not always available. Moreover, the purpose of this analysis is to
confirm the results of the numerical simulation (Figure 1f,g)
with an experimental approach. To experimentally characterize
the PSF, it has been suggested to image a step, corresponding
to a Heavyside function Halong one of the two dimensions of
the image, here, for instance, along the x-axis. This is a more
rigorous and elegant way to evaluate the PSF and also the
reason why the grating nanostructure is used as sample in this
work. Indeed, δis the derivative of Halong the x-axis,
64
and the
convolution operation is stable through differentiation, leading
to
=∂
∂*xxHx
P
SF( ) (PSF )( )
(3)
It is commonly assumed that the PSF is accurately described
with a 2D Gaussian,
65−71
which can be reduced to one-
dimension (1D) in the case of imaging a step along the x-axis.
Even though an Airy function can be considered, as it describes
the diffraction pattern generated by a small circular aperture on
the xy-plane, numerical investigations show that a Gaussian is
also a very common fit in the xy-plane for the PSF of a confocal
microscope.
72
To confirm the validity of using a Gaussian
approximation in contrast to an Airy approximation, the
intensity profilesatthewaistalongthex-axis for the
simulations of the nanojet profiles shown in Figure 1b−d
were fit with a Gaussian and an Airy function. In both cases,
these fittings were found to be imperfect (Figure S5) but
nevertheless resulted in comparably good fits (R2> 0.966). As
the use of a Gaussian facilitates considerably the calculations
and the numerical analysis, this function was chosen to
approximate the overall PSF of the system in the rest of this
study.
Thanks to this approximation, the image can now be
obtained by convoluting the step profile (the object) with a 1D
Gaussian (the PSF) characterized by a standard deviation σ
(see eq S4). By integrating eq 3 along the x-axis, the image
Nano Letters Letter
DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
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actually results in the integral of a Gaussian (the PSF) and is by
definition described by the error function erf
∫
π
≡−
xe
t
erf( ) 2d
xt
0
2
(4)
By fitting the profile of the image with erf, we can extract the
σassociated with the PSF, which is directly associated with the
lateral resolution of the imaging system.
73
Experimentally, the microsphere sits in the middle of two
lines. As shown in Figure 3c,e, two stripes are observed for each
microsphere, resulting in two opposite steps. In agreement with
eq 4, the obtained image profile should be described by the
function f
α
σσ
=−
′+−
′+
⎛
⎝
⎜⎛
⎝
⎜⎞
⎠
⎟⎛
⎝
⎜⎞
⎠
⎟⎞
⎠
⎟
f
xxx xx
() erfc ()
2erf ()
2constant
12
(5)
where erfc is the complementary error function, αis a constant
and x1and x2are the positions of the descending and ascending
steps, respectively (shown in Figure 3f), which together define
the magnification of the image. The image standard deviation σ′
is obtained from the fitting, and has to be divided by the
amplification factor associated with the microsphere size (as
shown in Figure 2i) to obtained the actual σ. A description of a
typical experimental image, along with the fitted f, is shown on
Figure 3f. From this fitting, the characteristic σfor each
microsphere size can be computed and is presented on Figure
3h. The other fitting parameters are discussed in the Supporting
Information and support the analysis presented here with x2−x1
∼900 nm, which is in good agreement with the geometry of
the sample, and α∼100 for all the beads considered. In
particular, the fact that the distance x2−x1was correctly
evaluated further validates the shift invariance assumption. If
this was not the case, as this parameter is measured over a large
part of the field of view appearing on the microsphere, a
significant distortion would have occurred thus preventing and
accurate measurement of the intergrating distance. The best σis
obtained with 6 μm size microspheres and is below 100 nm.
The experimental σis also in good agreement with the nanojet
waist values derived from the simulations. The actual σfitted
from the experimental data as a function of w/λobtained from
the simulations is plotted in Figure 3i: a positive correlation
with the Pearson’sefficiency of 0.88 is obtained, indicating that
the imaging resolution of the microsphere is clearly dependent
on the waist of the photonic nanojet. Additionally, wis defined
at the fwhm of the nanojet along x, which in a Gaussian
approximation is σ
σ
≈
2
2 ln(2) 2.355 . As the simulated wis
∼240 nm (for λ= 600 nm) and the experimental σis ∼100 nm,
the correspondence between the simulation and the exper-
imental analysis is excellent. This good agreement validates a
posteriori the assumptions on the overall PSF shape drawn
from considering the simulated nanojet profiles. It also supports
the validity of the simulations, and that the resolution of the
system is largely controlled by the illumination pattern, that is,
the width of the waist of the nanojet.
Considering that the illumination source has a spectral range
from ∼400 to 700 nm (see Supporting Information) and a 100
nm structure is resolved in the experiments, the resolution res is
therefore in between λ/4 and λ/7 with λthe illumination
wavelength in vacuum. According to the Rayleigh criterion, the
minimum feature size that can be resolved by a water-
immersion objective with NA = 0.8 is 188 nm under an
illumination wavelength of 400 nm. The 100 nm resolution is
therefore obtained thanks to the use of microspheres, which
convert the evanescent waves with the high spatial-frequency
information on the object into propagating waves within the
microsphere (see Supporting Information).
Moreover, the minimum waist of the photonic nanojet is
∼0.4 λ(Figure 3i). The relationship between the resolution and
the nanojet waist can hence be written as res ∼0.36w−0.63w.
The diffraction limit of an objective or lens is defined as λ/
(2NA). Our method can resolve an object with feature size of
100 nm under a wide-band illumination with spectrum from
400 to 700 nm, so that the NA of the imaging system is
obtained as 2−3.5 when considering the minimum and
maximum illumination wavelength, respectively. This means
that in our study the use of the 40×microscope objective
together with a 6 μm Ø microsphere would permit a resolution
that would be provided by a hypothetical (as nonexisting)
microscope objective with NA = 2−3.5.
In conclusion, we reported imaging of a sample’s nano-
features beyond the classical diffraction limit by using a
conventional optical microscope in combination with a series of
barium titanate glass dielectric microspheres of different sizes. A
FEM study on light propagation revealed the light focusing
capability of a microsphere of a given size and the generation of
the photonic nanojet. By comparing the experimental imaging
results with the numerical study, we found that the
magnification factor obtained from the virtual images is highly
correlated to the calculated light focusing capability of a
microsphere. Moreover, we quantitatively studied the reso-
lution of the microspheres of different sizes by analyzing the
images and fitting the results with a mathematical model based
on the PSF. Our work demonstrated the intimate link of the
super-resolution imaging mechanism of a dielectric micro-
sphere with its light focusing capability and the development of
the photonic nanojet. Indeed, the combination of refractive and
interferometric effects of the incident light produce the narrow-
waist photonic nanojet that exits the microsphere. In the
imaging mode of the microsphere, identical optical paths are
used for generating the magnified image and, therefore, the
degree of focusing of the incident light into a nanojet is closely
related to the possibility of a microsphere to transform a high
spatial frequency evanescent wave generated by the object into
a propagating wave that becomes detectable in the far field. We
believe that due to these physical insights, dielectric micro-
spheres will be increasingly used in the future, providing a
straightforward and robust tool to be integrated with a
conventional microscope for super-resolution optical micros-
copy. This will allow affordable super-resolution imaging of a
whole range of samples and biological objects, such as virus
particles, labeled nucleic acids and molecules.
Methods. Numerical Simulation. The numerical study on
light propagation through the microspheres and surrounding
water medium is carried out by FEM in COMSOL Multi-
physics software. A scalar equation is used to study transverse
electric waves in a 2D model. A light source with wavelength of
600 nm and the same width of the microsphere size is set away
from the front-surface of the microspheres, 600 nm
corresponding to the peak of the halogen lamp that is used
as the white-light illumination source in the experiments. After
meshing of the model, the element size is ∼22 nm, that is, 1/30
of the wavelength, which is sufficiently small to obtain a precise
solution. Microspheres with a size ranging from 2 to 20 μm are
analyzed in individual models. After each model is solved, Lis
Nano Letters Letter
DOI: 10.1021/acs.nanolett.6b01255
Nano Lett. 2016, 16, 4862−4870
4868
obtained by measuring the distance between the two points
with maximum light intensity on the front-surface of the
microsphere incident with the illumination light, lis obtained
by measuring the distance between the two points with
maximum light intensity on the rear-surface where light exits
the microsphere, and wis the fwhm of the nanojet along the x-
axis at the peak intensity of the y-axis.
Experimental Section. The silicon grating structures
consisting of 120 nm wide lines with 100 nm interspacing (in
Figure 2) are on a MetroChip microscope calibration target,
which is obtained from Pelco (Redding, CA, U.S.A.). The
grating structure with 300 nm wide lines and 900 nm
interspacing (in Figure 3) is made by chromium on a glass
substrate by photolithography. The optical microscopic images
are obtained by using a Zeiss Axioplan microscope mounted
with a AxioCam MRm camera (Carl Zeiss GmbH,
Oberkochen, Germany) and a 40×water immersion objective
with NA of 0.8. A halogen lamp is used as the white-light
illumination source with a peak at λ= 600 nm.
■ASSOCIATED CONTENT
*
SSupporting Information
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acs.nano-
lett.6b01255.
Additional information, figures, and table(PDF)
■AUTHOR INFORMATION
Corresponding Author
*E-mail: martin.gijs@epfl.ch.
Present Address
†
(H.Y.) IMEC, Leuven, Belgium
Notes
The authors declare no competing financial interest.
■ACKNOWLEDGMENTS
The authors would like to thank the European Research
Council (ERC-2012-AdG-320404) and the Swiss National
Science Foundation (200020-152948) for providing funding of
this work.
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