1. Introduction

Hyperbolic metamaterials (HMMs) have revolutionized the field of nanophotonics exhibiting applications for cloaking, spontaneous emission engineering, or supporting propagation of large wave vectors that carry the information of the subwavelength features [1,2]. Although the high wave vector waves turn evanescent after leaving the hyperbolic medium, hyperlenses aid to compress wave vectors and make them propagate in the far field [3]. The magnified subwavelength features of the inner hyperlens surface fulfill the Abbe criterion to be resolved by conventional optics. Typically, type I HMMs with transverse effective permittivity of marginally above zero and axial effective permittivity of significantly below zero yield a nearly flat isofrequency surface, therefore, all waves propagate in the radial directions within the hyperlens. This condition called the ‘canalization regime’ is nearly a must for optimal hyperlens performance [4,5]. Cylindrical [6–8] and spherical hyperlenses [9] with alternating dielectric/metal layers have been portrayed as efficient experimental platforms for one- and two-dimensional super-resolution imaging, respectively. Numerically demonstrated flat design versions, e.g. trapezoid-shaped hyperlenses with alternating layers stacked in the desired direction of the Poynting vector [10] or cylindrical hyperlenses with polished input and output planes [11], appear excessively complex for conventional nanofabrication solutions. The appropriate selection of metal and dielectric materials opens hyperlens functionalities in the near UV and blue spectral range [12–15]. The utility of dielectric materials with elevated permittivities such as silicon or gallium arsenide has been theoretically predicted to extend hyperlens performance in the entire visible range [16]. In contrast, hyperlenses can tailor light propagation as a demagnifying lens by decompressing propagative wave vectors for subwavelength light focusing [17,18]. Although the applications of demagnifying hyperlens have been mainly destined for enhanced UV lithography [19,20], the subwavelength optical resolution in the hyperlenses has the potential to reveal nanoscale dynamics hidden beyond the diffraction limit [21,22] or to excite single fluorescent molecules at physiological micromolar concentrations [23].

Here, we propose to apply spherical hyperlens with alternating Au/Si layers to yield extremely low focal spots and enable sub-diffraction fluorescence correlation spectroscopy (FCS). The alternating Au/Si structure features design simplicity for experimental implementation and operates up to the far red wavelengths covering spectral excitation bands of a majority of conventional fluorophores and avoiding unwanted characteristic phototoxicity of the UV or blue range [24,25]. FCS is a powerful and effective tool to monitor single molecule dynamics in aqueous solutions [26]. On one hand, stimulated emission depletion (STED) microscopy, near-field scanning optical microscopy (NSOM), localization super-resolved techniques are capable to unlock sub-diffraction FCS [27–30]. Nevertheless, they require advanced and complex optical systems for operation, e.g. a vortex beam with two spatially overlapped lasers or near-field probes. Additionally, non-invasive NSOM FCS may be flawed as the fluidity of cell membranes hampers keeping the probe in their proximity [29]. Localization super-resolved FCS techniques are associated with complex algorithms for image reconstruction and reduced time resolution, which make the FCS autocorrelation function (ACF) construction cumbersome. On the other hand, plasmonic structures have been shown to yield confined detection volumes to reveal hidden fine cell membrane organization using a conventional confocal microscope [31–33]. However, despite an enhanced signal and resolution, the FCS of molecule diffusion through nanoantenna excitation volumes succumbs to single-point measurements without spatial information of the diffusion properties of nearby molecules. Our proposed hyperlens diminishes the transmission of the excitation/fluorescence light intensity, nevertheless, the subwavelength focusing spots of λ/17 can be achieved at the 635 nm excitation. Moreover, we show that micrometer-scale hyperlenses enable spatial resolution of λ/4 and a large enough field of view (FoV) to map live cell membranes [34]. Using Monte-Carlo simulations of FCS correlation functions of diffusing Atto647N molecules we numerically prove FCS feasibility with hyperlens nanoscale focusing and demonstrate the diffusion time reduction up to 2 orders of magnitude. By modeling lipid analogue diffusion in a 2D membrane with nanoscale transient trapping sites we illustrate the hyperlens applicability to reveal nanoscale dynamics, e.g. location-dependent hindered diffusion of sphingolipid analogues in cholesterol-enriched nanodomains in live cell membranes. Lastly, high-quality scalable fabrication of spherical hyperlenses is accessible via the nanoimprinting technique [34,35] and the super-resolved spatiotemporal imaging can be implemented with solely a conventional confocal microscope and a time correlator.

2. Results and discussion

Figure 1(a) shows the concept of the spherical hyperlens subwavelength focusing and molecule diffusion through the detection volume. The hyperlens composed of alternating layers of gold and silicon with thicknesses of 10 nm further focuses a Gaussian beam (λ=635 nm) incident on the outer surface of the hyperlens. Atto647N molecules are selected as suitable organic fluorophores for the source wavelength (635 nm) that have been routinely studied for subwavelength FCS in the past [29,36,37]. We compute the effective tangential (${\varepsilon _\theta }$) and radial permittivities (${\varepsilon _r}$) of the hyperlens as follows:

 

Fig. 1. Gold/silicon hyperlens for nanoscale single-molecule excitation. (a) The scheme of the proposed hyperlens model with diffusing fluorescent molecules. The excitation light propagates through the hyperlens from the bottom. The fluorescence collected from the top is deprived of Ohmic losses. (b) The real and imaginary parts of effective permittivities of the hyperlens with the filling ratio of 0.5. The red vertical line marks the light source wavelength. (c) Wavelength regions of HMM, dielectric, and metal properties of the multilayered Au/Si structure as a function of the filling ratio. The red marker points to the structure/source wavelength configuration employed in this work. (d) and (e) Intensity profile in the axial and lateral planes of the hyperlens, respectively (scale bars 200 nm). The transverse intensity profile illustrated in Panel (e) is picked up along the plane 10 nm above the inner surface bottom. The shaded semicircles mark the gold layers in the hyperlens. (f) Normalized intensity of the Gaussian beam in the focal spot and the hyperlens along the inner surface in both dimensions.

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Here, $f \equiv 0.5$ is the metal filling ratio, while ${\varepsilon _m}$ and ${\varepsilon _d}$ denote metal and dielectric complex permittivities, respectively. Figure 1(b) demonstrates that the real part of the effective permittivity resides in the ‘canalization regime’ with highly negative Re(${\varepsilon _r}$), and slightly positive Re(${\varepsilon _\theta }$) at the excitation wavelength. At the same time, the source wavelength is set away from Im(${\varepsilon _r}$) peak maximum avoiding extremely high Ohmic losses in the radial direction. Figure 1(c) shows the range of hyperbolic dispersion of the hyperlens as a function of the filling ratio. The type I HMM turns to type II HMM above 665 nm, where the hyperlens no longer supports propagation of low wave vector waves. The main reason for choosing Au instead of Ag as a noble metal is the fact the real part of Ag permittivity is significantly lower than Au. Due to this highly negative permittivity value, conventional dielectrics such as Si or GaAs cannot provide Type I hyperbolic dispersions in multilayered structures with Ag in the wavelength range above 600 nm, where numerous experimental sub-diffraction FCS studies are conducted.

To demonstrate nanoscale energy localization at the hyperlens inner surface we perform FDTD numerical simulations of light propagation of a circularly polarized Gaussian beam focused onto the outer surface of the hyperlens (Fig. 1(d)). The shown hyperlens is composed of 24 Au/Si layer pairs of total thickness h = 520 nm with the inner diameter of D_inner= 200 nm. The simulated Gaussian beam FWHM amounts to 320 nm which corresponds to a diffraction-limited spot of high NA > 1 objective lenses that are commonly leveraged for FCS experiments in aqueous solutions. The nanofocusing occurs within the hyperlens medium towards the inner surface of the hyperlens, while the field intensity evanescently decays above the inner surface. A hyperlens of this geometry downsizes the focal spot without the substantial loss of peak intensity as compared to diffraction-limited excitation emphasizing the potential for practical application. Since transverse electric (TE) waves propagate through hyperlens without notable magnification, we employ a circularly polarized light to ensure the symmetric subwavelength focusing in both lateral dimensions (Fig. 1(e)). The Gaussian field intensity profile is observed in the inner surface of the hyperlens which constitutes the lateral dimensions of the FCS detection volume. Figure 1(f) displays the deeply subwavelength beam FWHM of 61 nm after the propagation of the diffraction-limited beam through the hyperlens.

Bearing in mind that the single-molecule excitation necessitates significant peak intensity, we have performed numerical simulations of nanoscale focusing for a wide range of hyperlens thicknesses (number of Au/Si layer pairs) and inner diameters (Fig. 2(a)-(c)). The beam spot contracts more as it propagates through a thicker hyperlens medium. Figure 2(d) displays the sub-diffraction resolution dependence map as a function of hyperlens thickness and inner diameter. The focusing beam spot FWHM is reduced the most in the case of micrometer-thick hyperlenses with small inner diameters down to 37 nm. Then, expanding the inner diameter inherently leads to a large available FoV at a price of poorer spatial resolution. Despite the large inner diameter subverts the nanoscale focusing, establishing FoV to micrometer scale is a prerequisite for practical hyperlens bioimaging. We mimic wide-field illumination conditions by simulating plane wave propagation through hyperlens. The transmitted intensity profile on the inner surface (Supplement 1, Fig. S1) shows that 92% of the inner surface area receives 60% of maximal transmitted intensity, indicating that nearly all hyperlens inner surface contributes to the imaging FoV. The FWHM map displays marginally higher values as compared to the computed FWHM based on the conventional hyperlens demagnification factor of (D_inner+ 2 h)/D_inner (Supplement 1, Fig. S2a). In contrast, a linearly polarized (TM) Gaussian beam nearly precisely follows the demagnification factor-based FWHM map (Supplement 1, Fig. S2b). The presence of the TE modes in the circularly polarized light that are not demagnified elucidates the inferior resolution. Nonetheless, the circularly-polarized beam FWHM downsizes below 80 nm by thick hyperlenses with an inner diameter below 650 nm. For instance, at this spatial resolution, STED-FCS starts visibly revealing nanoscale biological heterogeneities such as sphingolipid analogues transiently trapped in cholesterol-mediated molecular complexes [38]. Focusing the beam spot further by growing the hyperlens thickness leads to an exponential drop in transmitted power $T_{635}^{hyperlens}$ (Fig. 2(e)). The peak intensity descents by more than 2 orders of magnitude upon propagation through a 1 µm thick hyperlens which causes a theoretical limitation of hyperlens performance for single molecule excitation. The compensation of the intrinsic loss in the hyperbolic medium can be theoretically carried out by integrating an active medium into the dielectric material, e.g. an organic dye [39], which would yield a negative imaginary part of permittivity in a working wavelength range. Introducing the gain into hyperbolic multilayered structures could be a promising approach to enhance FCS correlation function quality.

 

Fig. 2. Hyperlens nanoscale focusing as a function of inner diameter and thickness. (a-c) Axial plane intensity profiles of hyperlenses with an increasing inner diameter and reducing thickness, respectively (scale bars 200 nm). The reduced thickness leads to higher transmission, although the enlarged inner surface hampers achieving deep sub-diffraction resolution. The insets represent intensity decays between the incident and exit surfaces along the z-axis. The normalized exponential decay rates amount to 2.7·10⁻³nm^-1_, 3.4·10⁻³nm^-1, 3.8·10⁻³nm^-1 for geometries shown in (a), (b), and (c), respectively. (d) FWHM of the intensity profiles at the inner surface as a function of inner diameter and hyperlens thickness. (e) Transmission through the hyperlens as a function of inner diameter and hyperlens thickness. A bicubic spline smoothing [44] is applied for image clarity of (d) and (e). The marked letters in Panels (d) and (e) are referred to corresponding numerical simulations depicted in Panels (a-c).

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The fabrication of proposed hyperlenses is an experimentally feasible task using nanoimprinting technique [35] or isotropic wet etching [9] followed by alternating layer deposition processes. Individual hyperlens geometry can be formed by isotropic wet etching in a glass substrate covered with an etching mask. Then, metal and dielectric layers are deposited in an alternating manner using an electron beam evaporator or atomic layer deposition. Hyperlens arrays could be fabricated by the nanoimprinting process. The reverse pattern of a PDMS mold can be coated by a photoresist and then pressed to a glass substrate by the photoresist side. The PDMS could be then withdrawn, and the nanopatterns are cured via UV light exposure and thermal annealing followed by the deposition of metal and dielectric layers.

Our proposed hyperlens nanoscale focusing relies on bending the rays within co-centrally curved layers. However, various hyperbolic metalenses with a phase compensation mechanism to focus a plane wave could be potentially considered for the sub-diffraction FCS analysis [40]. Gradient-index (GRIN) metalenses [41] have been shown numerically to yield nanoscale focusing ∼λ/6 (or better) in the air at the wavelength of 830 nm (or higher), which appears to be of similar performance as the values reported for the hyperlens in our work: λ/17 to λ/3 at 635 nm. However, a majority of single-molecule FCS studies employ fluorophores with excitation wavelengths in the visible range, therefore the nanoscale focusing by a GRIN metalens has to be adjusted for a desired working wavelength. Concave geometry hyperbolic metalenses [42] enable plane wave focusing inside metamaterial medium down to 70 nm or ∼λ/9 at 633 nm wavelength. Similarly, plasmonic waveguide coupler (PWC) integrated metalenses enable nanoscale focusing down to 90 nm or λ/7 [43]. Indeed, the resolution of state-of-the-art hyperbolic metalenses would be applicable for the spatiotemporal mapping of cell membranes as long as the hyperbolic metalens is truncated to make the focal volume accessible externally. The transmitted incident power per unit area has to be carefully controlled to maintain sufficiently high excitation rate for single molecule excitation.

Spherical hyperlens enables the sharpening of the Gaussian beam at other wavelengths as well by modifying the material of metal/dielectric layers and the filling ratio. We have conducted FDTD numerical simulations of the nanoscale focusing after propagation through hyperlenses composed of alternating layers (filling ratio 0.5) of Ag/Ti₃O₅ and Ag/Si at 410 nm and 532 nm wavelengths, respectively (Supplement 1, Fig. S3). The hyperlens inner radius and thickness are maintained as for the geometry from Fig. 2(b). Both these structures exhibit highly negative radial effective permittivity values and positive tangential effective permittivity values in the selected wavelengths and are able to downsize Gaussian beam (corresponding to a NA∼1 lens) beyond the diffraction limit. The inner surface beam FWHM reduces to 99 nm and 116 nm at λ=410 nm and λ=532 nm, respectively. FCS technique can be potentially extended to these hyperlens structures as high transmitted intensity on the inner surface can be provided. Moreover, by reducing the filling ratio of Ag/Si hyperlens one may improve the beam nanofocusing at 410 nm as the epsilon-near-pole resonance approaches the latter working wavelength (Supplement 1, Figure S4).

Reaching sub-diffraction FCS without increasing the excitation power requires taking into account the focal spot peak intensity in the hyperlens inner surface that is extracted from FDTD numerical simulations. Here, we employ a model of freely diffusing Atto647N fluorescent molecules, as Atto647N exhibits the absorption band within the working wavelength range of the hyperlens. An autocorrelation function (ACF) of lag time ($\tau $) from a fluorescence time trace is calculated as follows:

Here, the angular brackets denote time averaging, F is the time-dependent total fluorescence intensity, and $\delta F$ is the fluorescence intensity fluctuations. The collected fluorescence photon count rate from single molecules (CRM) is given by the following law [45]:

Here, $\sigma ({G(\tau )} )$ is the standard deviation of the ACF, N is the average number of molecules present in the detection volume, T is the total duration of data acquisition, and B denotes the background fluorescence intensity. We fix the FCS model with a single molecule in the detection volume (N = 1), channel width $\Delta \tau $ = 1 µs, and total acquisition duration T = 10 s. The selected T value provides 2 or 3 orders of magnitude longer time scales compared to lipid diffusion in live cells [50–53] that would empower the correct FCS operation at the characteristic time scales [54,55]. SNR values above 3 would result in around 99.7% probability [56] that $G{(\tau )_{\tau \to 0}}$ values are above non-correlating background. The SNR threshold allows us to provide a metric of the satisfactory signal level for FCS recording, although higher SNR values must be prioritized for accurate dynamics parameter determination. The SNR equal to 3 translates to a normalized peak intensity value of I_hyperlens/I₀= 0.09, where I_hyperlens/I₀ is defined as $FWHM_0^2/FWH{M^2}\,\cdot T_{635}^{hyperlens}$, where I₀ and FWHM₀ are the peak intensity of the beam and the focal spot size in the absence of the hyperlens. A 10 nm thick inner silicon layer sufficiently separates the aqueous environment with fluorophores from the nearest gold layer, therefore the metal-induced fluorescence quenching is omitted [57].

Figure 3(a) displays a map of I_hyperlens/I₀ as a function of hyperlens inner diameter and thickness. Based on the found excitation intensity threshold, we find the hyperlens geometries that make microsecond-scale FCS feasible, and we correlate those geometries with nanofocusing performance and field of view (FoV) (Fig. 2(d)). Thanks to field confinement, micrometer-thick hyperlenses with low inner diameters deliver enough excitation power to the molecules. The microsecond FCS measurements of Atto647N fluorescence fluctuations at inner diameters above 1 micrometer are expected to fail for thicknesses above 600 nm illustrating hyperlens geometry constraints of FoV size with deeply sub-diffraction resolution. Next, we extract the axial intensity decay from the FDTD simulations above the inner surface of the hyperlens and retrieve the axial waist ω_z at which intensity drops below 1/e² (Supplement 1, Fig. S5). Expanding inner hyperlens surface leads to the axial waist increase approaching that of the free-space excitation beam (Supplement 1, Fig. S6). The focal volume aspect ratio $\kappa = {\omega _z}/{\omega _{tr}}$ is well confined in small inner diameter hyperlenses, similar to zero-mode waveguides [58], and then slowly approaches the free-space excitation values in large hyperlenses. The Gaussian volume approximation has been previously evidenced as a qualitatively correct model to interpolate FCS diffusion data at the substrate interface and the optical antenna nanostructures [59–63]. Monte Carlo simulations provide reliable FCS correlation functions resembling experimental reality [64–68]. We model FCS of single Atto647N molecule diffusion in an aqueous solution according to a standard 3D-Brownian diffusion model :

 

Fig. 3. Feasibility of single molecule diffusion FCS using the Au/Si hyperlens. (a) The normalized Gaussian beam peak intensity on the inner surface of the hyperlens as a function of hyperlens inner diameter and thickness. The dashed curve marks I_hyperlens/I₀= 0.09 which corresponds to SNR = 3 of the FCS correlation amplitude. The letters in circles correspond to numerical simulations of FCS correlation functions of Atto647N diffusion depicted in Panels (b-d). ACFs featuring SNR close to 3 for hyperlenses with (b) D_inner = 200 nm; (c) D_inner = 740 nm; and (d) D_inner = 1400 nm. The diffusion times τ_d determined from the ACF fit are shown in Table 1.

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Table 1. 3D diffusion FCS model parameters for different hyperlens geometries

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As the transverse waist is around tenfold shorter than the curvilinear inner surface size for selected geometries, the influence of the spherical inner surface shape on the focal volume is assumed imperceptible. The diffusion time reduction in the hyperlens (Table 1) competes well with that of the scanning STED-FCS technique [36] which is capable to reveal trapping sites of hindered lipid diffusion within cell membrane environments. By limiting the inner diameter, the diffusion time reduces by 48x at the price of disabling multipoint imaging of nanoscale diffusion heterogeneities. In contrast, by keeping the inner diameter at the micrometer scale for practical nanoscale imaging, the diffusion time reduction of 3-4x is within the reach. In our modeling, we fix the background value at zero, as Si and Au nanostructures show imperceptible background in the red spectral range [70], moreover the total real fluorescence intensity $N\cdot CRM$ can be boosted to mitigate background noise influence in real experimental conditions via increasing the fluorophore number of molecules in the detection volume. Nevertheless, in the real experimental conditions B can increase the noise of the hyperlens-enabled autocorrelation function (Supplement 1, Fig S7).

We numerically demonstrate the hyperlens applicability to resolve hindered diffusion in nanoscale sites of cell membranes. We model 2D diffusion of lipids in a cell membrane with a fixed diffusion coefficient D₁= 0.5 µm²/s corresponding to the sphingomyelin (SM) diffusion coefficient in cholesterol-depleted cell membranes [33,36]. Additionally, we include heterogeneities, i.e. nanoscale hindered diffusion sites of 50 nm and spaced from each other by 300 nm in a similar fashion as the experimental study of SM trapping nanodomain mapping using STED-FCS [36]. This model resembles independent experimentally observed characteristics of cholesterol-enriched nanodomains that have size from a few to a hundred nm [21,36,71,72] and are separated by 200-300 nm [36]. We vary the diffusion coefficient of nanoscale sites (D₂) between D₁ and 0.1D₁ as the diffusion coefficient of sphingolipid analogues in cholesterol-enriched nanoassemblies can reduce by 10 times [33]. The cell membrane is assumed to conformally lay on the inner hyperlens surface, therefore the molecules are confined in a 2D space at the focal plane within the FCS numerical simulation settings. The probability to enter the heterogeneity nanodomain is set to unity. Within the FCS simulation, the delayed transit of a molecule that ends up in the trapping site increases the autocorrelation function decay time due to hindered diffusion. Figure 4(a) shows that the diffusion retardation in the nanodomains is nearly imperceptible for a diffraction-limited FCS. In contrast, the hyperlens sub-diffraction FCS (Fig. 4(b)) enables distinguishing even insignificant heterogeneity diffusion coefficient changes based on the characteristic decay time. It’s worth noting that the real-world biological applications of sub-diffraction FCS reside in millisecond range where the ACF SNR remains significant even in the presence of the excitation intensity losses from hyperlens.

 

Fig. 4. Nanoscale diffusion heterogeneities revealed by hyperlens sub-diffraction FCS. 2D diffusion FCS correlation functions are obtained from (a) the diffraction-limited excitation and (b) the hyperlens (D_inner= 200 nm, h = 920 nm). The schemes above show the modeled membrane 2D diffusion where the molecule diffusion coefficient D₁ is fixed to 0.5 µm²/s. Modeled D₂ values attributed to heterogeneities of 50 nm that are spaced by 300 nm from each other amount to D₁ and 0.1D₁. The central gray and red circles illustrate the source Gaussian beam diameter and the hyperlens focal beam diameter, respectively.

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Lastly, the proposed methodology implies the operation of a trans-illumination fluorescence microscope to preserve the collected single molecule fluorescence intensity. Moreover, the excitation losses quantified in Fig. 3(a) can be technically compensated by pumping up the excitation source power. In contrast, an epi-fluorescence photon collection would imply an additional signal decline proportional to the transmitted fluorescence intensity loss (Supplement 1, Fig. S8). We assess that the SNR of a $G{(\tau )_{\tau \to 0}}$ in the epi-fluorescence mode drops additionally by a factor of ${\sim} 3.6/T_{635}^{hyperlens}$ as compared to the trans-fluorescence mode. Although this dramatic reduction of performance limits the range of available FoV/resolution options, high demagnification hyperlenses would still enable FCS analysis with focal spot FWHM down to 110 nm.

3. Conclusion

We have numerically demonstrated a perspective approach of using hyperlenses with spatial resolution below 40 nm for fluorescence correlation spectroscopy far beyond the reach of the diffraction limit. The nanoscale focusing supported by the hyperlens enables accelerating FCS diffusion time by more than an order of magnitude. A trade-off of the nanoscale focusing, inner diameter, and field confinement has been quantified to demonstrate the sub-diffraction FCS feasibility of diffusing Atto647N fluorophores even for micrometer-size inner diameter opening applicability for nanoscale dynamics mapping. The hyperlens light confinement aids to reveal 2D hindered diffusion of lipid analogues in cell membrane nanodomains of 50 nm which is elusive for diffraction-limited optical systems. The FCS correlation function SNR demonstrated here is subjected to improvement via increasing acquisition time, diffusion time scales, and excitation source power upon technical availability, which would extend the hyperlens performance to FoV of several micrometers with deep subwavelength resolution. The sub-diffraction hyperlens FCS concept is fully generalized to other excitation wavelengths and fluorophores in the visible range thanks to hyperlens design tunability via the metal/dielectric material and filling ratio. Altogether, we believe that the proposed platform opens opportunities to investigate hidden nanoscale cell membrane organization and study individual molecules at physiological micromolar concentrations using conventional optics.

4. Methods

Ansys Lumerical 2022 R1.4 has been used for FDTD numerical simulations. The gold and silicon dispersions have been adopted from Rioux et al. [73] and Aspnes & Studna [74], respectively. The silver dispersion has been adopted from Johnson and Christy [75], while T₃O₅ permittivity at λ=410 nm has been adopted from [9]. The conformal meshing around the hyperlens structure of size of 1 nm has been set to all FDTD simulations. The simulation region boundary conditions are set to a perfectly matched layer in all dimensions. The FWHM and transverse waists (${\omega _{tr}} = {\raise0.7ex\hbox{${FWHM}$} \!\mathord{\left/ {\vphantom {{FWHM} {\sqrt {2ln2} }}}\right.}\!\lower0.7ex\hbox{${\sqrt {2ln2} }$}}$) of the hyperlens focal spot have been extracted from Gaussian function fits of field intensity profiles along the inner surface. The data analysis and plotting have been performed in a Jupyther notebook (Python 3).

We leverage SimFCS 3 program http://www.lfd.uci.edu/globals/ for FCS Monte-Carlo simulations. The diffusion coefficient of Atto647N molecules in water is set to 156 µm²/s [37] for 3D-free Brownian diffusion Monte-Carlo simulations. Input CRM values in the FCS simulations are computed as $15\cdot {\raise0.7ex\hbox{${{I_{hyperlens}}}$} \!\mathord{\left/ {\vphantom {{{I_{hyperlens}}} {{I_0}}}} \right.}\!\lower0.7ex\hbox{${{I_0}}$}}$ kHz.The size of the box with reflecting boundaries around the focal volume has been set close to the inner hyperlens diameter. As for the 2D membrane diffusion simulations, we adopt the diffusion coefficient of phosphoethanolamine lipid labeled with Atto647N dye (D₁= 0.5 µm²/s) [29,38]. The probability to enter the nanoscale transient trapping site is set to unity. The acquisition time of the diffraction-limited 2D diffusion FCS has been set to 60 s instead of 10 s for the other simulations, as the characteristic transit time scales approach 100 ms in that case.