Super-resolution microscopy by microspheres emerged as a simple and broadband imaging technique; however, the mechanisms of imaging are debated in the literature. Furthermore, the resolution values were estimated based on semi-quantitative criteria. The primary goals of this work are threefold: i) to quantify the spatial resolution provided by this method, ii) to compare the resolution of nanoplasmonic structures formed by different metals, and iii) to understand the imaging provided by microfibers. To this end, arrays of Au and Al nanoplasmonic dimers with very similar geometry were imaged using confocal laser scanning microscopy at λ = 405 nm through high-index (n~1.9-2.2) liquid-immersed BaTiO3 microspheres and through etched silica microfibers. We developed a treatment of super-resolved images in label-free microscopy based on using point-spread functions with subdiffraction-limited widths. It is applicable to objects with arbitrary shapes and can be viewed as an integral form of the super-resolution quantification widely accepted in fluorescent microscopy. In the case of imaging through microspheres, the resolution ~λ/6-λ/7 is demonstrated for Au and Al nanoplasmonic arrays. In the case of imaging through microfibers, the resolution ~λ/6 with magnification M~2.1 is demonstrated in the direction perpendicular to the fiber with hundreds of times larger field-of-view in comparison to microspheres.
© 2015 Optical Society of America
Recently, microsphere-assisted optical super-resolution imaging emerged as a surprisingly simple method for visualizing nanometric-scale structures [1–13]. Many developments occurred in this area since the first observation of optical super-resolution by silica microspheres with index n = 1.46 by Wang et al. in 2011 . These include the demonstration of improved imaging by semi-immersed silica spheres , demonstration of the resolution advantage provided by high-index (n>1.8) spheres totally submerged in a liquid [3–7], demonstration of the confocal mode of imaging through microspheres , developing the sample surface scanning capability by locomotion of microspheres [9,10] and by using movable polydimethylsiloxane (PDMS) slabs with embedded the high-index spheres [11–13].
The concept of diffraction-limited resolution plays an extremely important role in microscopy. The image formed in the far-field is degraded due to diffraction effects. For two point objects of equal intensity to be considered resolved, the minimum separation distance is, r = Kλ/nosinθ, where K = 0.473, 0.5, 0.515, and 0.61, as defined by Sparrow , Abbe , Houston , and Rayleigh , respectively. In thus defined resolution, λ is the illumination wavelength, no is the refractive index of the object-space, and θ is the half-angle of the objective's acceptance cone. Thus, the resolution in air is limited by ~λ/2. The resolution can be further improved by using a concept of solid immersion lens (SIL) [18,19]. Using analogy of high-index microsphere with SIL, the maximal diffraction-limited resolution can be estimated as ~λ/2n, which is consistent with recent numerical studies of imaging by microspheres based on classical optics [20,21].
Surprisingly, virtual images of nanoscale objects obtained through microspheres using visible light (400-700 nm) have been found to display features with characteristic dimensions smaller than 100 nm [1–13]. The fundamental question about why the resolution of this method can exceed the resolution limit is debated in the literature.
Another problem is related to experimental quantification of spatial resolution provided by this method. It should be noted that in fluorescence super-resolution microscopy this problem can be solved due to the availability of bright “point”-sources in a form of dye-molecules, fluorophores and quantum dots. The resolution is determined by the width of the system’s point-spread function (PSF). The PSF width can be reduced below the diffraction limit by using the “point”-source nature of the light sources and additional nonlinear effects, like in the case of STED microscopy . However, the microsphere-assisted microscopy is usually used for a label-free imaging where such bright “point”-sources are not available. The mechanisms of label-free imaging rely on much more subtle light-scattering processes in nanoplasmonic or biomedical objects that result in lower effective contrast of images. Previously, the resolution of microsphere-assisted imaging has been estimated using semi-quantitative criteria based on observation of the small features in the optical images that resulted in broad range of resolution claims from λ/6 to λ/17. More rigorous quantification of spatial resolution of this method is required.
The mechanisms of super-resolution imaging by microspheres represent an active area of research. It has been suggested  that it is related to the ability of microspheres or microcylinders to produce tightly focused sub-diffraction-limited output beams termed photonic nanojets . Such nanojets are used in surgical laser scalpels , optical tweezers , and nanolithography tools . In chains of spheres or cylinders, the tightly focused beams are periodically reproduced along the chain giving rise to novel optical modes [27–29] with applications in waveguides  and polarizers . If the direction of propagation is reversed for the photonic nanojet, on the basis of optical reciprocity, these sharply focused beams could potentially lead to high-resolution imaging, slightly beyond λ/2n.
Another possibility is connected with the plasmonic properties of metallic nanostructured objects. For the same frequency, plasmonic excitations have a much shorter wavelength compared to the waves propagating in air. Thus, plasmonic near-fields contain extraordinary detailed information about the fine structure of the object. Conversion of plasmonic near-fields into propagating modes might result in resolution beyond the diffraction limit; however these processes are suppressed by phase matching conditions. In periodical structures, however, the reciprocal lattice vectors would Bragg scatter surface plasmon-polariton modes with large k-vectors into the escape cone near the origin of k-space, making possible for light to eventually escape. This mechanism has been used for studying guided mode resonances in photonic crystal waveguides [32,33] and nanoplasmonic arrays . It can be assumed that a similar mechanism can provide optical super-resolution of periodic structures. In this case, the surface modes can be represented by the plasmon polaritons  and surface states .
One more possibility is represented by imaging performed at the frequencies resonant with the whispering gallery modes (WGMs) in microspheres. Using backscattering, it has been demonstrated that metallic nanoparticles can be detected with ~λ/3n subdiffraction resolution when the microsphere is illuminated at a Mie resonance [37,38]. Thus, a precision enhancement of roughly one and a half compared to the nonresonantly formed photonic nanojets has been demonstrated in the case of resonant illumination.
In this work, we showed that the image treatment techniques based on the object’s convolution with the system’s PSF, which are widely accepted in diffraction-limited optics [39,40], can be extended in the super-resolution area. Our approach is applicable to objects with arbitrary shapes and can be viewed as an integral form of the super-resolution treatment widely accepted in fluorescent microscopy. We studied microsphere-assisted imaging of nanostructures formed by different metals, such as gold and aluminum, with almost identical geometry, but different plasmon resonant properties. It is shown that imaging of these structures through high-index microspheres can be performed with ~λ/6-λ/7 resolution in a laser scanning confocal microscope operating at 405 nm. The etched microfibers represent another type of contact lens compared to microspheres, which have been used for near-field illumination of nanoscale objects  and for imaging of Blu-ray® disk (BD) . In this work, we developed a procedure of fitting images obtained through a cylindrical lens based on a combination of magnification and super-resolution in the direction perpendicular to the lens axis. We demonstrated super-resolution imaging of golden nanoplasmonic arrays with the resolution ~λ/6 across the microfiber and a field of view (FOV) extending submillimeter distances along the microfiber, hundreds of times larger FOV in comparison to microspheres.
2. Experimental method and samples
As illustrated in Fig. 1(a), it is possible to capture the virtual image produced by a sphere in contact with the sample by looking through the sphere at some depth into the sample substrate [1–13]. We used high-index (nsph~1.9-2.2) BaTiO3 glass microspheres with diameters from 2 to 50 µm which were immersed in isopropyl alcohol (IPA) with the refractive index 1.37, as illustrated in Fig. 1(b). The positioning of microspheres was achieved either by self-assembly or by using micromanipulation with a tapered fiber probe.
It should be noted that recently we developed a different approach to developing sample scanning capability based on using movable polydimethylsiloxane (PDMS) slabs with embedded high-index microspheres [11–13]. Such coverslips can be translated along the sample surface to align different spheres with various surface nanostructures. The super-resolution imaging through such coverslips is in many ways similar to the case of liquid infiltration [Fig. 1(b)] used in this work.
Nanoplasmonic arrays made from Au and Al were fabricated and patterned with the same layout at the Air Force Research Laboratory (AFRL) on a sapphire substrate using a bi-layer PMMA/ZEP520A e-beam photoresist lift-off mask exposed in a JEOL JBX-6300FS e-beam lithography system. The developed pattern was subjected to an oxygen plasma ash prior to e-beam metal deposition of 10 nm Ti/40 nm of Au or Al. Lift-off and photoresist removal sample cleaning were completed prior to critical dimension analysis by SEM imaging. The unit cell of the array was represented by a metallic dimer formed by two cylinders with the diameters (D) and distances (d) between their centers varying in different arrays. The dimers had periods of 700 nm and 350 nm along the x-axis and y-axis, respectively. An example of such array made from Au with D = 175 nm and d = 200 nm is illustrated in Fig. 1(c).
Confocal microscopy (without microspheres) using Olympus LEXT-OLS4000 with the 100 × (NA = 0.95) objective lens at λ = 405 nm does not allow resolving the internal structure of individual dimers illustrated by SEM images in Fig. 1(c). However, the confocal microscopy through the microsphere with 20 × (NA = 0.6) objective allows resolving the irradiance minimum at the center of each dimer, as shown in Fig. 1(d).
3. Mapping of plasmonic near-fields
Previous super-resolution modeling results have been mainly restricted to point dipole objects [20,21]. Our experiments were performed with the nanostructured metallic objects where collective excitations can give rise to localized surface plasmon resonances (LSPRs) . It should be noted, however, that the illumination at λ = 405 nm was at shorter wavelength compared to expected positions  of LSPR bands in Au or Al. It has been theoretically predicted that metallo-dielectric antennas formed by metallic nanoparticles coupled to high-index microspheres possess resonantly enhanced rate and directionality of emission [45–47]. These properties can be important for imaging applications.
As shown in Fig. 2(a), we performed 3-D finite difference time domain (FDTD) modeling of plasmonic near-fields in such structures using Lumerical’s “FDTD Solutions” software by placing the frequency-domain field monitor in a xy plane in close vicinity to the dimer array. The illumination was provided from the top down along the z-axis through a dielectric microsphere with a diameter Dsph = 10 μm with a refractive index nsph = 1.9, which was placed in contact with the dimer array. The dimers were comprised of two cylinders made from metals with the material parameters of gold (Au) and aluminum (Al) with a diameter (D) of 100 nm separated by 50 nm gaps, as illustrated in Figs. 2 (b) and 2(c), respectively. The height of the cylinders was 45 nm. The cylinders were placed on a sapphire substrate with a thin 10 nm titanium buffer layer. We used a Cartesian mesh, with a maximum mesh size of ~20 nm in the plane of the dimer array and a local mesh size of ~1 nm in the vicinity of the dimers for accurate representation of the plasmonic near-fields.
The field distributions obtained by a Fourier analysis at 590 nm and 400 nm are shown in Figs. 2(b) and 2(c), respectively. They illustrate several properties. The first is a photonic nanojet formation visible due to a bright spot at the center of symmetry of calculated distributions (the sphere touches the nanoplasmonic array at a slightly right-shifted location relative to the array’s symmetry center). The second is a formation of a concentric pattern of bright spots centered with the photonic nanojet. The reflection of light between a spherical surface and an adjacent flat surface leads to formation of interference Newton rings. At the same time, the diffraction of incident light on the periodical array of dimers leads to formation of bright spots. Finally, strong near-field enhancement takes place inside the plasmonic dimers. If the dimers are overlapped with the interference maxima, the near-field enhancement is especially pronounced, as illustrated by the rightmost dimers in Fig. 2(b).
The plasmonic local field enhancement inside the metallic cylinders occurs in extremely small areas with typical dimensions well below 50 nm. We hypothesize that due to significant perturbation of local density of photonic states [45–47] such near-field maxima determined by collective plasmonic excitations can be efficiently coupled to the microsphere playing a part of dielectric antenna launching of the image in the far-field. Resonantly enhanced emission rate and directionality of the emission [45–47] can result in imaging with greatly improved signal-to-noise ratio and resolution. However, the mechanism of such imaging based on the antenna effect requires further theoretical studies that go beyond the scope of this work.
4. Experimental definition of super-resolution
Textbook definitions of resolution often refer to point objects [39,40]. Such objects with the dimensions much smaller than the wavelength of light are readily available in fluorescent microscopy in a form of dye molecules, fluorophores or quantum dots. In this work, however, the fluorescent markers are not used and the imaging is provided due to relatively weak light scattering mechanisms. In the case of such label-free microscopy the realization of point-objects is complicated since reduction of the size of the objects diminishes their optical contrast and visibility . Therefore, we need to establish a procedure which would allow determining resolution based on images of extended objects (which cannot be approximated as point-sources) with the arbitrary shape.
Such procedure is well-known for the diffraction-limited optics where the image (I) formed is a direct result of the convolution of the object (O) with the PSF of the imaging system [39,40]. This can be mathematically expressed in the integral form as:
In what follows we develop a phenomenological approach to the experimental definition of the optical super-resolution. This approach is based on the assumption that the super-resolved image can be reconstructed by Eq. (1) where the PSF width can take values smaller than λ/2n. In principle, such approach can be applied to the resolution analysis of images obtained by different label-free super-resolution methods such as near-field scanning optical microscope , imaging using plasmon gratings [50,51], hyperbolic metamaterials [52,53], super-oscillatory lens  or two-photon microscopy . In contrast to the previous work , where the resolution analysis was limited by a 1-D treatment with rectangular functions, we generalized this approach for 2-D PSF and for objects with arbitrary shape. Similar approach was used for quantifying resolution by high-index microspheres embedded in movable PDMS slabs [11–13].
As illustrated in Fig. 3, to determine the resolution of the imaging system, we implemented an iterative 2-D convolution procedure aimed at finding the PSF width providing the best fit to the experimentally obtained image.
As an object, we used two circles with the dimensions matching the SEM image. A virtual image of Al dimers with D = 110 nm and d = 150 nm was obtained through an 4.5-μm BaTiO3 glass sphere totally submerged into IPA. This image is shown in the insets of Figs. 3(b), 3(d), and 3(f) for comparison with the calculated images.
As shown in Fig. 3(b), the convolution with the Dirac delta function restores the shape of our object formed by two perfect circles. The rectangular shape of the intensity profile is illustrated by the dashed line in Fig. 3(c). The convolution with PSF with FWHM = λ/4.8 leads to much wider intensity profile with weakly pronounced saddle point compared to that observed experimentally, as shown in Figs. 3(d,e). To achieve better agreement with the experiment, the convolution with the Gaussian PSF with the narrower widths was performed, as illustrated in Figs. 3(f,g) for FWHM = λ/7.2 corresponding to the resolution of 56 nm according to Houston’s criterion.
5. Super-resolution by microspheres
The microsphere-assisted imaging was performed on nanoplasmonic arrays formed by different metals. The two metals chosen, Au and Al, constitute an excellent model system, where the LSPRs peak positions are shifted over the entire visible spectrum due to the size-confinement quantum effects . As an example, for cylinders with D~100 nm and 20 nm height, the LSPR is expected at ~670 nm for Au and at ~460 nm for Al. It should be noted that due to damping, the LSPR peaks have typical widths on the order of tens of nanometers.
The fabrication goal was to make nanoplasmonic arrays from Au and Al with identical layout and dimensions. In practice, however, this was a difficult task because of the different calibrations of the sources of different metals and inevitable variation in the processing parameters. For this reason, series of Au and Al arrays were fabricated for different metal deposition parameters and e-beam exposures. The selection of the dimers for imaging experiments was based on their actual dimensions determined by SEM. Due to fabrication imperfections it was not always possible to find arrays with identical parameters. In this situation, we tried to identify arrays with similar properties. This is illustrated in Figs. 4(a) and 4(e) by the SEM images of Au dimer with circular shape and elliptically deformed Al dimer, respectively. Despite the shape difference, the dimensions of Au and Al dimers are rather close and they can be used for semi-quantitative resolution comparison.
The imaging was performed by the upright Olympus LEXT-OLS4000 microscope with a 100 × (NA = 0.95) objective lens at λ = 405 nm. The images of the dimers shown in Figs. 4(a) and 4(e) were obtained through BaTiO3 glass microspheres with 8 µm and 10 µm diameters, respectively, during their full liquid-immersion in IPA. Their super-resolution images are shown in Figs. 4(b) and 4(f), respectively. The saddle point is better pronounced in Fig. 4(b) in comparison with that in Fig. 3 because of the wider gap (80 nm) between the cylinders.
To calculate the image of Au dimers [Fig. 4(c)], we used a drawn object formed by two circles with D = 100 nm and d = 180 nm. To calculate the image of Al dimers [Fig. 4(g)], we used an object formed by two ellipses with 135 nm major and 100 nm minor axes and 180 nm center-to-center separation. The convolution with a PSF with increasing width blurs the edges of the images and increases the saddle-to-peak ratio in the calculated intensity profiles. The comparison of the calculated and measured intensity profiles is presented in Fig. 4(d) and 4(h) for Au and Al dimers, respectively. The best fit to the experimental profiles was obtained for the PSF widths of λ/6.6 and λ/7.3 for Au and Al dimers, respectively.
The accuracy of determining the resolution (~10%) was limited by the uncontrollable variations of the shape and size of the dimers. This means that although we observed slightly higher resolution value for Al dimer (λ/7.3), it is still within the standard deviation of the experimental data. Based on reproducibility of our results, we can claim that for both metals the super-resolution was found to be in the λ/6-λ/7 range, well beyond the diffraction limit.
6. Super-resolution by microfibers
The field of view (FOV) of imaging through the microsphere is limited by a quarter of its diameter (Dsph) . Taking into account that the super-resolution is expected only for small spheres with the size parameter q = πDsph/λ<100 [1,4], the FOV is limited by the area on the order of few square microns. The spheres can be translated along the surface of investigated structures using different locomotion techniques [10–13], however developing the surface scanning functionality requires additional tools which detract from either generality or simplicity of this method.
In contrast, the microcylindrical lens obtained by etching or pulling the fiber can be easily fabricated with the millimeter length (l) and extremely uniform micron-scale diameter [41,42]. If such microlens is placed at the top of the investigated sample, it provides super-resolution imaging in a narrow, near-contact, region extended through the entire length of the fiber. It results in FOV~0.25Dsph × l which can exceed the FOV for spheres by hundreds of times. The microfiber can be translated and aligned with the objects by various micromanipulation techniques. In principle, high-index microfibers can be embedded inside movable PDMS slabs, similar to microspheres [11–13], to realize the sample scanning functionality.
It should be noted, however, that in contrast to microspheres, the microcylinders function as lenses in only one direction, perpendicular to the cylindrical axis, where better-than-diffraction-limit resolution can be realized.
Multiple pieces of a single-mode SMF-28 fiber were etched down to several micron diameters in acidic solution, 0.75 H20 and 0.25 HF, as illustrated in Fig. 5(a). Similar to the case of microspheres, the microscopy was performed in reflection mode using confocal Olympus LEXT-OLS4000 microscope with the 100 × (NA = 0.95) objective lens at λ = 405 nm. We aligned the fiber along the y-axis of the Au nanoplasmonic dimers (D = 100 nm and d = 150 nm), as illustrated in Figs. 5(b) and 5(c). The index of silica microfiber (ncyl = 1.47) is smaller compared to BaTiO3 microspheres, and the complete submersion in a liquid does not allow observing the virtual image. The imaging through the fiber without the liquid is possible, however similar to the case of silica microspheres [2,7], we found that it results in the suboptimal image quality. To improve the image quality, we used a droplet of acetone deposited at the top of the fiber. In the process of evaporation, the liquid tend to aggregate close to the region where the cylinder touches the object. The presence of a fraction of liquid with n = 1.36 reduces the effective index contrast (neff) for a lens, but it also produces a “healing” effect for the defects at the cylindrical surface resulting in a crisper image with higher contrast and somewhat smaller magnification compared to the case of dry lens. Typical image observed in the course of acetone evaporation is shown in Fig. 5(d). Previously, a saddle point in the irradiance profile perpendicular to the microfiber axes was observed in the image of an array of Au dimers with the 150 nm edge-to-edge separations . (In this work, the separation is 50 nm). However, the adequate resolution treatment taking into account 2-D nature of the objects combined with the convolution with the PSF of cylindrical lens has not been developed previously.
The fitting of the experimental image was achieved using a two-step procedure. First, we took into account that the virtual image is magnified along x-axis. We expanded the drawn object (Fig. 5(e)) along the x-axis by the transverse magnification (M) of the cylindrical lens. As a result, the circles become ellipses elongated along x-axis. The magnification can be estimated using geometrical optics as |M|~|neff/(2-neff)|~2.1, where we used neff = 1.36. Second, we convoluted thus obtained virtual image with the highly asymmetric PSF illustrated in Fig. 5(f). This was achieved by using Eq. (1) with different widths of Gaussian PSF along x- and y-axis. As a result of this convolution, the usual consequences of finite resolution such as smoothening of the edges of the objects and increasing the saddle-to-peak ratio were observed, as illustrated in Fig. 5(g). The image in Fig. 5(g) looks like an array of circular rather than elliptical intensity maxima because the width of PSF along x-axis (λ/6) was much smaller than the width of PSF along y-axis (λ/2).
The image intensity profiles compressed along x-axis by the factor of M are presented in Fig. 5(h). The experimental profile (on red background) is obtained using the image in Fig. 5(d), whereas the theoretical profile (blue dashed curve) is obtained from the calculated image in Fig. 5(g). As seen in Fig. 5(d), in the selected dimer the intensity of the right bright spot was higher due to imperfect fiber alignment. In our modeling, we attenuated the irradiance maxima located in the left column of Fig. 5(g). Very good agreement between the calculated and measured intensity profiles in Fig. 5(h) supports our treatment of the microfiber resolution.
7. Conclusions and outlook
In our work, we use a Houston resolution criterion and show that the image treatment techniques widely accepted in diffraction-limited optics can be extended in the super-resolution area providing a phenomenological tool to determine the super-resolution values in practical situations. Using this approach, we demonstrated that imaging of Au and Al arrays can be achieved with the resolution ~λ/6-λ/7 which is likely to be explained by the plasmonic contributions. We show, however, that obtaining such resolution does not require resonant excitation of localized surface plasmon resonances.
It is shown that imaging by microfibers has an advantage of large FOV compared to microspheres. However, the super-resolution and magnification take place only perpendicular to the cylindrical axis. This makes the interpretation of such images more difficult in comparison with microspheres. It is shown that the resolution analysis for cylindrical lens can be performed using highly anisotropic PSF. For nanoplasmonic dimer array made of gold, the resolution ~λ/6 with the magnification ~2.1 across the silica fiber is demonstrated.
The mechanisms of super-resolution imaging by mesoscale (with D on the order of several wavelengths) spheres and cylinders are a subject of active research including such effects as formation of photonic nanojets [23–26], scattering into small-k plasmon-polariton states in periodic structures [35,36,41], LSPR [43,44] or WGM involvement [37,38]. This paper generally supports the notion that plasmonic contribution can be responsible for the observed super-resolution properties. In particular, the interplay of radiative and nonradiative relaxation may play extremely important role in imaging, especially taking into account that due to strong coupling to high-index microspherical antenna, the radiative life times can be significantly reduced [45–47]. This factor can improve the visibility of nanoscale metallic objects, increase the signal-to-noise ratio, and, potentially, play some role in the super-resolution.
Despite the fact that our understanding of super-resolution imaging mechanisms by microspheres and microcylinders is incomplete, these methods promise to become very important tools in microscopy. They can lead to development of novel optical components such as attachable and movable thin films with embedded spheres which will boost the resolution of conventional microscopes beyond the diffraction limit [11–13]. Alternatively, the microspheres can be moved by the optical tweezers or other techniques [9,10], leading to developing novel microprobes with the sample scanning functionality similar to the probes of near-field scanning optical microscopes , but providing many orders of magnitude higher optical throughput.
Another opportunity is connected with using shorter illumination wavelengths. The diffraction-limited resolution scales with the wavelength. This property, however, does not necessarily hold for super-resolution, especially if some LSPR mechanisms can be activated at certain frequency ranges such as for example interband excitation conditions for different metals. These properties require further studies which can be performed by the methods developed in this work (see also ), but by using excitation at different wavelengths. It seems that the most promising direction is represented by UV and deep UV excitation  which can result in further increasing the resolution beyond current limit of 50-60 nm.
The authors thank Dr. Alexey Maslov for stimulating discussions. The authors also thank Dr. Glen Marrs for his help with the use of Zeiss LSM 710 microscope and Drs. Didier Dreau and Robert Peterson for their help with the use of Olympus FV1000 microscope. This work was supported by the U.S. Army Research Office through Dr. J. T. Prater under Contract No. W911NF-09-1-0450 and by Center for Metamaterials, an NSF I/U CRC, award number 1068050. Also, this work was sponsored by the Air Force Research Laboratory (AFRL/RYD, AFRL/RXC) through the AMMTIAC contract with Alion Science and Technology and the MCF II contract with UES, Inc.
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