Optical nanoscopy with contact Mie-particles: Resolution analysis Free

Following the success of super-resolution fluorescent microscopy,¹ research efforts are now focused on label-free microscopy,^2–9 where imaging mechanisms are related to nonlinear optical effects or near-to-far-field transformations. In the standard far-field microscopy the Abbe limit defines the resolution at the level of $λ/(2NA)$⁠, where λ is the illumination wavelength and $NA=n0 sin θ$ is the numerical aperture determined by the index n₀ of the object-space and by the half-angle θ of the collection cone.

Over years, various structures have been proposed to bypass the Abbe limit. Examples of such structures include (a) plasmonic lenses,¹⁰ where image magnification takes place in a plasmonic medium with much shorter wavelength followed by scattering into the air; (b) far-field super-lenses,¹¹ where the near field is amplified in a metallic layer and then transformed into propagating waves by a surface grating; (c) hyperlenses,^12,13 where the far fields propagate inside a cylindrically or spherically shaped metamaterial lens with hyperbolic dispersion, and (d) time-reversal mirrors,¹⁴ where scatterers near an object transform its near fields to propagating waves that can be detected and then time-reversed. While these concepts inspired researchers, they have not resulted in many applications yet due to various limitations of these structures, such as fabrication issues, material loss, sensitivity to the operating frequency, or electromagnetic environment.

Super-resolution imaging through dielectric microspheres has emerged as a remarkably simple and broadband imaging technique¹⁵ that overcomes the limitations of the previously proposed structures. Important advancements in this area include the development of imaging through high-index liquid-immersed^16–18 and coverslip-embedded^19,20 microspheres, rigorous experimental quantification of resolution at the $∼λ/7$ level for imaging nanoplasmonic^19–22 and biomedical²³ structures, integration with confocal microscopy,²⁴ and development of surface scanning functionality by locomotion of spheres^25,26 or slabs with embedded spheres.²⁰ Despite all these successes, the physical mechanism of super-resolution in microsperical nanoscopy remains debatable. Initially, it was attributed to the sharper-than-diffraction-limit focusing of light by microspheres, termed photonic nanojet effect.^15,27 It was assumed that sharper focusing is related to higher resolution in imaging. However, the nanojet waist does not represent the optical resolution since the latter requires taking a point source and considering imaging rather than focusing. Numerical modeling of imaging of two point sources near microspheres with moderate refractive indexes $1.3<n<1.46$ and diameters $D∼10λ$ showed that the far-field resolution is generally diffraction-limited at $∼λ/3$ level.^28–30 Somewhat higher resolution but with artificial sidelobes was also reported²⁸ under resonant conditions with the whispering gallery modes (WGMs) in the microspheres.

The motivation for this work comes from the fact that in Mie-particles with the wavelength-scale dimensions and higher index of refraction the fraction of evanescent fields outside the particle can be significantly increased in comparison to that in larger spheres. This property is especially pronounced at the Mie resonances. Such a situation creates a setting for super-resolution imaging, if these resonant states are participating in the image formation. Indeed, our results show that such high-index Mie-particles create both virtual and real images of point sources located in near-field proximity to such particles. It is important that all these images are formed with a strong participation of the evanescent fields containing information about the objects' high spatial frequencies. In addition, all these images appear with a $1.5–2.0$ magnification. We show that this combination of properties results in resonant super-resolution extending slightly beyond $λ/4$ for particles with diameters $λ ≲ D ≲ 2λ$ and refractive index $n∼2$⁠. Our results set theoretical resolution limits for microspherical nanoscopy with incoherent point sources in the air. The resonant super-resolution imaging can be implemented experimentally using bandpass optical filters aligned with the Mie resonances in a microscope setup. On the other hand, significantly better than $λ/4$ resolution recently reported for nanoplasmonic and biomedical structures is likely to be defined by other factors such as coherent illumination of nanoscale objects,^7,31 excitation of anti-symmetric modes in resonantly coupled closely spaced objects,³² or plasmonic contributions.

The imaging geometry is shown schematically in Fig. 1(a). The imaged object is located just below the microsphere. To capture main physical effects, we adopt a two-dimensional model, in which the microsphere becomes a cylinder with refractive index n_c and radius R. The background refractive index is n_b = 1. The classical resolution is defined as the minimal distance between two incoherent point sources that allows one to discern them according to one of the several resolution criteria.³³ Similar to Refs. 7, 20, and 21 we use the Houston criterion. In the spirit of this approach, our object is a pair of two current sources radiating incoherently and separated by δ:

where d is the distance from the cylinder surface to the object plane. Currents (1) excite an electromagnetic field with one magnetic, H_z, and two electric, $Ex,y$⁠, components (TM, transverse magnetic case).

Images produced by two point sources (1) can be found using the field generated by a single point source emitting near the cylinder. The magnetic field in the far-field region has the following form

where $kb=nbk$ and $k=ω/c$⁠. The angular distribution function $f(φ)$ is obtained by solving analytically the Maxwell equations for the radiating current J₁ or J₂.³⁴ The far field can also be represented as

where $H̃z$ is the Fourier component and $g2+h2=kb2$⁠. Evaluating the integral in (3) asymptotically at $ρ→∞$ and comparing with (2) give us the Fourier component

An objective lens collects the far field and forms an image on the detector. The formed image is equivalent to the intensity distribution of the backpropagated far field in the imaged plane for an ideal objective lens with unit magnification and $NA=1$⁠. The knowledge of $H̃z$ allows us to backpropagate the field to any point in space and find the image intensity

The backpropagated field at some location differs from the actual field there since this procedure is equivalent to image formation and is not a solution of an inverse scattering problem. If the far field is generated by some sources in free space, the sharpest image is observed when the position of the imaged plane coincides with the location of the sources. Thus, by scanning over various positions x of the imaged plane we search for the location of the far-field sources as illustrated in Figs. 2, 3, and 4. Generally, such scanning mimics finding the sharpest image in experimental microscopy. Any magnification of the objective lens simply scales the image on the detector, i.e., along y.

We first study the power emitted by a single point current source near cylinders with $nc=1.4$ and 2.0, see Fig. 1(b). The power spectra in both cases show multiple narrow peaks due to the excitation of Mie resonances. For larger cylinders (or spheres), many wavelengths can be accommodated along the perimeter, the Q-factors grow, and the terminology of the Mie resonances is replaced with the WGM terminology.³⁵ The resonances can be characterized by their azymuthal and radial modal numbers. The strongest peaks correspond to the resonances with radial numbers equal to 1. The peak power emission increases with increasing Q-factor and decreasing distance d. The coupling to the cylinders affects greatly the emission of the sources and, therefore, can impact greatly their images. The redistribution of the emission power from the background into the resonant peaks is so marked in Fig. 1(c) that the resonant imaging can be easily realized in the optical microscopy by using a narrow bandpass optical filter transmitting only one emission peak.

Figure 2 compares the images created by a pair of point sources separated by $δ=λ/2$⁠, which approximately corresponds to the smallest distance that can be resolved in free space, near the cylinders with $nc=1.4$ and 2.0. For the $nc=1.4$ cylinder, the sharpest images can be obtained when the imaged plane is below the actual location of the sources. The off-resonance cases produce resolved images with sufficiently high intensity at $−7 ≲ x/λ ≲ −9$⁠, see Fig. 2(b). The on-resonance case can also give similar images but there are some additional images that can be obtained when focusing just below the cylinder at $x/λ∼−4.5$⁠, see Fig. 2(a).

The images obtained with the n_c = 2 cylinder look drastically different. In the off-resonance case, Fig. 2(e), we can also resolve the two sources but now they appear to be located on top of the cylinder, not below it as for $nc=1.4$⁠. In the on-resonance case, Fig. 2(d), the image appears both below and above the cylinder. While imaging of incoherent sources defines the classical limits of resolution, various coherent effects can affect the actual visibility. In particular, the emission of coherent out-of-phase sources produces a zero intensity along the symmetry line that can provide their visibility even at very small separations, see Figs. 2(c) and 2(f).³² Such coherent imaging does not possess conformal properties and needs to be distinguished from incoherent imaging used for the resolution quantification.²² 

Figure 3 shows the images of two incoherent point sources separated by $δ/λ=1/3$ obtained using the n_c = 2 cylinder in the on-resonance and off-resonance cases for various gaps kd. In the off-resonance case (bottom row), we cannot resolve the two points for any values of kd, while such resolution was still possible for $δ=λ/2$⁠, see Fig. 2(e). In the on-resonance case (top row), the resolution becomes dependent strongly on kd. For large kd (kd = 2), the image is similar to the off-resonance case and the point sources are not resolved. For kd = 1 and 1.3 one can clearly resolve the two point sources, both below and above the cylinder. However, as kd becomes progressively smaller, kd = 0.5, the images acquire a significant number of side-lobes that reduce the imaging quality. Apparently, for small kd each point source excites two degenerate Mie modes. The far-field imaging system, therefore, detects the re-radiation of the excited modes. As a result, the image will be determined entirely by the properties of the modes and not by the point sources that excited them. However, for modest $kd∼1$⁠, the resonant excitation seems to enhance the resolution of the two point sources.

In order to investigate the resolution achievable due to the excitation of the Mie resonances in the high-index particle, we show the images of point sources for various distances δ between them, see Fig. 4. It appears that the resolution is close to $λ/4$ or slightly better, according to the Houston resolution criterion. Additional radial construction lines in Figs. 4(a)–4(d) show that the separation of the peaks on the images correlates well with the separation of the point sources. This means that the peaks are the images of the point sources and not some artifacts. However, for large source distances, $δ/λ=2/3$ in Fig. 4(e), we also observe the formation of a minor peak located between the two main peaks.

To conclude, we proposed and demonstrated that a small dielectric particle made of a rather high-refractive-index material and possessing Mie resonances can be used to obtain images of deeply subwavelength objects with resolution significantly higher than in free space. The results are obtained by calculating the fields generated by point sources emitting near a particle and then using the far fields to find the images formed on the detector for various positions of the imaged plane of the microscope lens. In our opinion, this approach gives a more adequate resolution in microsphere-based imaging as compared to modeling a plane wave-incidence and studying the resulting photonic nanojets. Although we used a two-dimensional model, its extension to a three-dimensional one is also possible.

Our results predict that the image resolution can reach $λ/4$ or slightly better for cylinders with n_c = 2 and sizes around $D/λ∼2$⁠. Although this roughly corresponds to the resolution limit of $λ/(2nc)$ that would be achievable if the object and objective lens are immersed in the n_c dielectric, the imaging mechanism is not related to the simple scaling of the wavelength as evidenced by a strongly non-monotonic dependence of resolution on wavelength. In particular, in the off-resonance case the resolution is limited at $λ/3$⁠. In the on-resonance case, there is also an optimal object-to-cylinder distance that provides high-resolution and rather weak artificial side-lobes. Bringing the object closer to the cylinder produces multiple sidelobes that correspond to the fields excited in the particle rather than to the images of the objects. The obtained results set the resolution limits for imaging incoherent point sources by exploring the Mie-resonances in the contact or near-contact dielectric particle. Resonant super-resolution imaging predicted in this work can be realized in experiments with the broadband point emitters such as dye molecules, fluorophores, or nanoplasmonic particles. In the presence of a microsphere, their emission will be strongly redirected into the Mie resonances, as shown in Fig. 1(c). Taking into account large spectral separation between the neighboring peaks, one can use a narrow bandpass interference filter to observe the objects at the frequency of a single Mie resonance. On the other hand, in practical microscope systems the illumination is provided in many cases with coherent wavefronts and the objects often have their own resonances. These factors can dramatically alter the discernibility of nanoplasmonic structures^20,21 or biomedical objects²³ assembled at the top of nanoplasmonic arrays. These mechanisms go beyond the scope of this work but they can be responsible for the higher resolution values $∼λ/7$ reported in recent studies of resolution in microspherical nanoscopy.

This work at UNCC was supported by the Center for Metamaterials, and NSF I/U CRC, Award No. 1068050.