Ultra-high-Resolution, Label-Free Hyperlens Imaging in the Mid-IR

The hyperbolic phonon polaritons supported in hexagonal boron nitride (hBN) exhibiting long scattering lifetimes are advantageous for applications like super-resolution imaging via hyperlensing. Yet, challenges exist in hyperlens imaging to distinguish individual and closely spaced objects and in correlating the complicated hyperlens fields with the structure of an unknown object underneath the hyperbolic material. Here, we make significant strides to overcome each of these three challenges. For the first two, we demonstrate that monoisotopic h 11 BN (>99% 11 B) provides the ability to experimentally resolve structures as small as 40-nm and those with sub-25-nm spacings, inferring at least 154- and 270- times smaller than free-space wavelength, showing improvements in spatial resolution. We also present an image reconstruction algorithm that provides a structurally accurate, visual representation of the embedded objects using only the hyperbolic dielectric function and thickness as input parameters. Further, we offer additional insights into the frequency dependence for realizing optimal hyperlens performance. Thus, our results significantly advance label-free, high-resolution, spectrally selective hyperlens imaging and image reconstruction methodologies.


Introduction
Sub-diffractional imaging in conventional optical microscopy is not possible due to Abbe's diffraction limit, as light scattered from deeply sub-wavelength objects rapidly decay from the surface, resulting in evanescent fields that do not propagate into the far-field. The hyperlens concept provides a promising approach to overcome this challenge. This approach exploits hyperbolic polaritons (HPs) supported in highly anisotropic materials exhibiting a dielectric permittivity tensor that is opposite in sign along orthogonal directions, e.g., ( > 0, < 0) or ( < 0, > 0), where = = and are tangential and axial permittivities, respectively [1][2][3] . In the anisotropic region, high-k wavevectors can propagate at an angle consistent with the material dispersion, and the angle can be defined to the surface normal (z-axis) 4-6 : The scattering off of a sub-diffractional object can therefore launch HPs into the hyperbolic slab, which will then propagate at this frequency-dependent angle to the opposite surface where they can be directly probed (Fig. 1a). For a curved hyperbolic material 7,8 , these hyperbolic "rays" propagating along the radial directions are expanded, resulting in magnified hyperlens fields that can become resolvable in the far-field when the features expand beyond the diffraction limit. This is also possible through (flat) metalens designs 9,10 , or in the near-field via probes such as scattering-type scanning near-field optical microscopy (s-SNOM) 5,6 , which we have employed here.
Hyperlensing for sub-diffraction-limited imaging was originally demonstrated with artificial hyperbolic metamaterials (HMMs) 7 , i.e., metal and dielectric stacks 7,8 . However, these HMMs suffer from high optical losses inherent in the metallic layers, which limits the transmission efficiency and the maximum wavevectors accessed, restricting the minimum resolvable feature size. In contrast, natural hyperbolic materials 5,6,11,12 , such as hexagonal boron nitride (hBN), have considerablly reduced optical losses. The hyperbolicity of hBN comes from two sets of optic phonons that are spectrally separated, giving rise to two distinct Reststrahlen bands (upper, URB and lower, LRB) where ( ) < 0 along at least one axis. Within these bands hyperbolic phonon polaritons (HPhPs) are supported, enabling hyperlens imaging 5,6,11 .
Although hBN exhibits reduced optical loss compared with HMMs 5,6,11 , the spatial resolution and the HPhPs transmission are still limited. Further, instead of magnified images, the hyperlens fields collected represent complex products of (sub-wavelength) diffraction, posing challenges to identify the imaged objects without prior knowledge.
Here we address these challenges by combining material developments with an image reconstruction algorithm. With the implementation of monoisotopic h 11 BN (>99%) we reduce the optical loss by threefold, resolving objects as small as 44-nmm in diameter and interparticle gaps of < 25-nm. Based on the smallest object imaged, we report a normalized resolution of at least /154, where is the free-space wavelength of the incident light, representing a nearly four-fold improvement compared with previous reports of hyperlens imaging using naturally abundant (NA) hBN 5,6 . Equally promising, the hyperlens fields from monoisotopic hBN are significantly improved due to the higher transmission efficiency, tripling the peak-signal-to-noise ratio (PSNR) over NA hBN for slab of the same thickness. Moreover, we also provide and demonstrate a backpropagation method to identify spatial locations and sizes of unknown objects from the hyperlens fields, using only the dielectric function and thickness of hBN as input parameters enabling the potential for on-the-fly conversion of the collected hyperlens fields into object images. In addition, we also offer insights into the frequency dependence for optimizing hyperlens performance. The combination of monoisotopic h 11 BN and computational image processing highlighted here therefore provides significant advancements towards practical mid-infrared (MIR) hyperlens designs, and lays the foundation for future works investigating the potential for practical spectroscopic hyperlens imaging.

RESULTS
In the simplest case, the frequency-dependent angular propagation of the HPhPs can be visualized using a ray-tracing picture (Fig. 1a). Here, evanescent fields are produced as long-wavelength MIR light is scattered by a sub-diffractional object, which launches propagating HPhPs due to the requisite momentum and the proximity to the hyperbolic slab. These volume-confined HPhPs propagate at the frequency-dictated angle through the slab (Eq. 1) until they reach the opposite surface 1 , where they are reflected (Fig. 1a).
Although sub-wavelength modes cannot propagate into free space, this reflection at the top surface results in evanescent fields that extend just above the surface of the hyperbolic slab that can be probed via external means sensitive to near-fields, such as s-SNOM 5,6 . For frequencies close to that of the longitudinal optic While both 10 B-and 11 B-enriched hBN have been grown with high purity [13][14][15] with enhanced phonon lifetimes and propagation lengths over NA materials 16,17 , h 11 BN (>99% 11 B) was chosen for these studies due to the near-negligible spectral shift in the optic phonons, i.e., (~1359.8,~1608.7 cm -1 ) with respect to NA material (~1360,~1614 cm -1 ). Therefore, this small variation in the spectral response allows direct comparison between the NA and isotopically enriched hyperlenses at the same incident frequencies while preserving roughly equal values for the real part of the permittivity. Here, we have chosen 125 nm flakes for both the h 11 BN and NA hBN slabs, therefore isolating the role that loss reduction plays in improving hyperlens performance.

Hyperlens fields of isolated and closely spaced structures
To compare the imaging performance of NA hBN and h 11 BN, we first measured the hyperlens fields resulting from scattering of a MIR light by a 44-nm diameter disk embedded beneath each of the two hBN slabs, collected via s-SNOM ( Fig. 2a-b). The lower losses of h 11 BN (Fig. 2a) lead to a stronger s-SNOM amplitude contrast than NA hBN at the same frequencies (Fig. 2b). Further, the longer propagation lengths of the h 11 BN HPhPs increase in the number of internal reflections (and thus, more concentric hot-rings) that occur before the HPhPs decay (Fig. 2a,c vs. Fig. 2b,d). To quantitatively compare the transmission efficiency between the two forms of hBN, we benchmark the peak signal-to-noise ratio (PSNR) of the two hyperlens systems. While the noise levels for both s-SNOM images are very similar (details included in SI, section IV; 0.011 for NA hBN versus 0.014 for h 11 BN), the amplitude of the collected signal from the h 11 BN hyperlens is much stronger. Specifically, line scans from this hyperlens field (Fig. S4a) indicate that the signal is far above the noise level, resulting in a PSNR of 11.5. In contrast, the comparable hyperlens fields using NA hBN are just resolvable due to low signal strength, with a corresponding PSNR of only 3.64 (Fig. S3, Fig. S4a, Table S1). As such, the ability to resolve such small structures at ω=1480 cm -1 (λ FS  . 2d). The corresponding hyperlens fields were collected via s-SNOM for both the h 11 BN and NA-hBN flakes with an incident laser at ω = 1480 cm -1 , as provided in Fig. 2c and 2d, respectively. Consistent with the imaging of single disks, the image quality is superior for those collected through the h 11 BN slab. Furthermore, line scans across the disks, collected along the center and diagonal   S4b and Table S1).

Image reconstruction algorithm
While these efforts highlight improvements in the PSNR for the collected hyperlens fields, directly identifying the structure of underlying objects from these measurements is still challenging. Because HPhP rays are launched from all edges of the object into all directions, the resultant concentric patterns give rise to complicated fields even for simple systems such as an isolated disk (Fig. 2a). For more complex objects or multiple closely spaced structures, such patterns highlight extensive interference between the overlapping HPhP rays, making the collected fields complicated and not easily identifiable as the objects they represent (Fig. 2c). In other words, the measured fields are not direct images and thus must be reconstructed into an image of the underlying object for the hyperlens concept to be useful. The right-most image shows a sample measurement of the crystal output, the central image depicts approximate ray paths within the crystal, and the leftmost image represents the input field reconstruction after 200 iterations. We start the algorithm by enforcing the measurement condition; is the measured field, ∠ denotes the phase of the complex field (to be found), and represents the iteration number. The projection operator extracts the propagating modes from ( ) by enforcing the boundary constraints on the input and output faces of the crystal. The angularspectrum propagation operator transports the propagating modes of the input field by a distance . To match the boundary conditions at the bottom of the crystal, we add the evanescent modes (obtained from the previous iteration Ê ( −1) ) to get the total field Ê ( ) . We further apply the small object constraint to get our reconstruction. Subscripts and represent propagating and evanescent components of a given electric field, respectively, ℛ (⋅) represents the real part of (⋅), and | | represents an absolute value operation.
To address this challenge, we use the angular spectrum method to back-propagate the measured fields at the top surface of the hBN slab to the unknown scattering source at the bottom surface.
The algorithm is a modified Gerchberg-Saxton (GS) algorithm 18 , as illustrated in the flowchart provided in To test the reconstruction algorithm, hyperlens fields were collected from the h 11 BN hyperlens over the 2x2, 100-nm diameter dot array with the smallest gap (~25 nm) at various frequencies ( Fig. 4a-d). The acquired hyperlens fields are significantly different at each of these frequencies and illustrate consistency with predicted fields via Finite Element Method (FEM) simulations, as shown in Fig. 4e-h. Note that additional speckling is observed in FEM simulations, which is the result of high-order HPhPs superimposed upon the fundamental HPhPs that are at the heart of our discussions here. To minimize the speckling so that the simulated hyperlens field is easier to read, the simulations in Fig. 4 are performed with artificially increased loss of h 11 BN by three-fold, and FEM simulations with original loss are included in Fig. S9.
Although we focus on the s-SNOM amplitude fields in this study, we note that similar behaviors are also shown in s-SNOM phase maps (Fig. S5).
Despite the complexity in the hyperlens fields from the 2x2 disk arrays (Fig. 4), the reconstruction algorithm successfully retrieves images of the underlying structures featuring accurate measurements of the structure size and alignment, as shown in Fig. 4i-l. The extracted diameter and interparticle gaps of the disks, averaged over the four incident frequencies, are 87 nm and 27 nm, respectively, which are in good agreement with the actual diameter (~100 nm) and gap values (~25 nm). While outside of the scope of this work, by correlating the multi-frequency measurements within the reconstruction protocol (rather than treating each frequency as an independent measurement), this method can be refined further to give an improved estimate of the underlying structures. The data summarized in Table S2 can also be used to extract other information as well, for instance as a means to fine-tune the material dielectric properties, 19 such as the recent use of near-field measurements for optimizing the dielectric function of the biaxial hyperbolic cystal MoO 3 20 . Note that we perform this at different frequencies, i.e., different propagation angles of HPhPs (Eq. 1), which is also of significant importance for resonant materials, e.g., biomaterials, since imaging on/off resonance could provide strong material contrast, potentially useful in object material identification as well 21,22 .

Hyperlens imaging figures of merit
While our image reconstruction algorithm successfully retrieved the underlying image of the embedded objects, realizing far-field IR hyperlenses also requires high image resolution and HPhP transmission. By implementing monoisotopic h 11 BN, we have significantly improved the image quality, which also shows a strong dependence on the incident frequency, as shown in Fig. 4. Thus, understanding how those factors influence the hyperlens field quality is imperative for future applications. As the fields collected by s-SNOM are derived from HPhPs scattered from the metal/air edge underneath that propagate through the hBN hyperlens slab, two factors improve the quality of the imaged hyperlens fields: (a) the higher launching efficiency from metal object underneath hBN, and (b) the lower losses of h 11 BN. The launching efficiency is related to either the polarizability of the launcher (nickle disk here) 23 or the momentum difference between hBN/launcher and suspended hBN 24,25 . As such, the launching efficiency either maintains a constant or slowly decreases with increasing frequnency (SI, section VIII). To describe the attenuation of the HPhPs as they propagate through the hBN slab, the quality factor (Q) is defined as follows 26,27 : where ′ and ′′ are the real and imaginary parts of the wavevector, respectively, and defines the number of cycles before the HPhP wave decays to 1/ . Thus, increasing implies a stronger hyperlens signal, and a higher amplitude of the first and each subsequent hot ring observed within the hyperlens fields. To find  5, and we find that the is highest at around 1490 cm -1 for both NA and h 11 BN. As such, we expect the best image quality for a given hBN-based hyperlens to occur near this peak-frequency, which agrees with our experimental observations (Fig. 4).
Although the analytical solution is not quantitatively accurate when the HPhP wavevectors are not significantly larger than the free-space values (more discussions are in SI, section VIII), this approach still provides qualitative guidance towards optimizing hyperlens operation and design in this regime as well.
While the image quality is clearly influenced by the imaging frequency, the isotopic enrichment is still the dominant driver in the improved image quality we report. By comparing the of h 11 BN and NA hBN, we estimate that h 11 BN should provide an approximate three-fold increase in transmission, causing better image quality over the entire URB, as shown in Fig. 5. Consistent with this, the PSNR of hyperlens fields collected using h 11 BN are indeed three times larger than those using NA hBN, as discussed earlier and presented in Fig. 2, Fig. S5, Table S1. It is important to note that spectral overlaps between the URB of hBN and the molecular fingerprint region of the MIR enables the potential for resonant hyperlens imaging (Fig. 5); that is, imaging of biologically relevant species at frequencies corresponding to the vibrational resonances of the materials for chemical identification. This could be especially beneficial for biological studies as the molecular vibrational bands of many molecules, e.g., the amide I and II bands of proteins 28 , fall within the frequency range of the URB of hBN. For resonant imaging of bio-materials, the largest hyperlens field contrast will occur at the chemical vibrational frequencies 21,22 , and h 11 BN and h 10 BN together cover a broad range of vibrational frequencies of interest, as shown by the overlayed black lines in Fig. 5. Thus, for different purposes, these two monoisotopic hBN materials (and potentially h 10 B 15 N and h 11 B 15 N if grown) could be used to optimize such imaging methodologies.

Conclusion
In conclusion, through the exploitation of ultra-low loss h 11 BN hyperlens devices, we demonstrate the ability to resolve the smallest features we fabricated (~44 nm), showing a record-high imaging resolution of at least /154, which is a nearly four-fold improvement over previously reported data using NA hBN material 5,6 . Additionally, we demonstrate the ability to discern four closely spaced ~100 nm disks, even at interparticle separations of 25 nm, indicating the potential for imaging objects with separations that are on the order of /270. Note that the high PSNR of h 11 BN hyperlens fields suggests that the ultimate resolution limit has yet to be reached. Furthermore, the transmission of h 11 BN is predicted to be three times higher than NA hBN, which is consistent with the three-fold increasement of PSNR measured experimentally, providing significant opportunities in employing thicker and/or curved hyperlenses necessary for far-field hyperlens designs. Further, using a numerical algorithm, we demonstrate precise image reconstruction and accurately retrieve the details of the embedded 2x2, ~100 nm disk array with ~25 nm gaps, validating this as a potential methodology for detecting unknown sub-diffractional objects.
Combining the numerical algorithm and ultra-low loss monoisotopic hBN, we provide the broad potential of imaging modalities, such as resonant hyperlens imaging with incident frequencies coincident with the