Open Access
Add to BibSonomy
Abstract
Due to the excellent ability to break the diffraction limit in the subwavelength range, metamaterial-based hyperlens has received extensive attention. Unfortunately, radial resolution of most current hyperlens is not high enough, which is a huge obstacle to the application in 3D super-resolution imaging. In this paper, we propose a theoretical solution to this issue by cascading a graded structure outside the conventional Ag-TiO2 spherical hyperlens. The product of the thickness and the refractive index (RI) of the dielectric layer in the graded structure is fixed to 19.8 while RI increases linearly from 1.38 to 3.54 along the radial direction. By reducing the asymptote slope of the dispersion curve, the coupling of the wave vectors to the hyperlens is enhanced and thus radial resolution is significantly improved to 5 nm while ensuring that the focus is still detectable in the far-field. This design paves the way to high-performance hyperlens for 3D imaging and biosensing in the future.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Benefiting from the developments of fluorescent molecules [1,2] and metamaterials [3], super-resolution imaging has attracted worldwide interest due to its excellent versatility in biomedicine, lithography, nanoscale sensing and other fields [4–7]. Among them, three-dimensional (3D) super-resolution methods has been widely studied including the double-ring-modulated selective plane illumination microscopy [8], the method using fluorescence quantum coherence in fluorescence microscopy [9] and new tools to quantify the colocalization of fluorescent signals at single molecule level [10]. However, since those 3D fluorescence microscopy-based methods may change the properties of samples and even produce destructive effects [11], super-resolution imaging that does not rely on fluorescent is in great demand.
The emergence of thin metal-dielectric multilayer metamaterials has provided an effective path to fluorescence-independent super-resolution imaging. Due to the unique non-closed hyperbolic dispersion relation, the evanescent waves could propagate along the radial direction to the far-field, thus breaking diffraction limit [12]. At present, there are many ways to achieve super-resolution imaging by hyperlens. Spherical hyperlens array [13], metamaterial composed of a silver–silicon dioxide composite [14], pyramid-shaped hyperlens [15] and the polymer dendrimer-based silver structure [16] have been successively proposed. Unfortunately, the current research on the super-resolution performance of hyperlens has been mainly discussed in the lateral direction while the attention paid to radial resolution is not sufficient, which may hinder the applications for 3D imaging and nanoscale sensing.
In 2015, Liu’s group first reported radial resolution of hyperlens, proposed a spherical hyperlens with extremely high amount of layers (∼144-layer) and achieved unprecedented 5 nm radial resolution via surface plasmon polaritons (SPPs) at the interface between the metal and dielectric layers, showing admirable application values for both 3D imaging and lithography [17]. However, the accumulation of defects such as cracks and bubbles along with the layer increment would lead to change in effective properties of metamaterials [18], finally resulting in increased losses and a decrease in resolution. Therefore, although increasing the layer amount of hyperlens can improve radial resolution, the realization in practical fabrication is still a challenge. How to reduce the amount of layers added to the hyperlens while keeping ultrahigh radial resolution has become the key issue. Making the hyperbolic dispersion curve of the hyperlens as flat as possible is one effective candidate, regretfully, it would cause the focus approximate outer wall or even inside the hyperlens, which is detrimental to the detectability of the image. Consequently, the dispersion curve of hyperlens should remain a certain degree of steepness to balance the focus to the far field and high radial resolution.
Therefore, we proposed a graded structure cascaded outside a 44-layer conventional hyperlens which is formed by alternately stacked 9 nm Ag layers and 22 nm TiO2 layers. Via simple far-field detection, the small size structure of our design also exhibits an amazing resolution capability down to 5 nm along the radial direction. In detail, 10 units are adopted by the graded structure and each unit is composed of an Ag layer and a dielectric layer with the same thickness. The product of the thickness and the refractive index (RI) of the dielectric layers in the graded structure is fixed to 19.8. The RI of the initial dielectric layer is 1.38 with the corresponding permittivity of 1.9, which is much closer to that of Ag than to TiO2, thus making the asymptote slope of the dispersion curve decreased and the dispersion curve flattened, effectively improving radial resolution. Simultaneously, as the RI of the dielectric layers in the graded structure linearly increases to 3.54 along the radial direction, the thickness of Ag and dielectric layers in the graded structure gradually decrease to the minimum value, guaranteeing that the asymptote slope of the dispersion curve rises again and the focus is finally formed at the outer space of the hyperlens. The high radial resolution and extremely less layer quantity of our design is of great significance for the future fabrication of high-performance hyperlens and realization of 3D super-resolution imaging.
2. Theories and design
According to effective medium theory (EMT) [19], when the incident light is transverse magnetic (TM) polarized electromagnetic wave, the dispersion relation of the curved anisotropic structure like hyperlens could be described in the r-θ plane as:
For a subwavelength hyperlens, the permittivity components of the multilayer metal/ dielectric structure could be determined by [20,21]:
Fig. 1. (a) Isofrequency surfaces of the metamaterials with εr < 0 and εθ >0. (b) Hyperlens based on multilayer metamaterials with curved geometries. The corresponding dispersion curve in r-θ plane is a hyperbola.
The minimum radial movement distance of point source, which could cause change in the far-field focus position, can be determined as radial resolution [17]. To explain how to break the diffraction limit by hyperlens and improve radial resolution, we introduce the Poynting vector $\overrightarrow S = \overrightarrow E \times \overrightarrow H$, which represents energy per unit time through a vertical unit area [22]. It is well known that the Poynting vector $\overrightarrow S$ is parallel to the wave vector $\overrightarrow K$ in vacuum while perpendicular to the isofrequency surfaces in low energy loss scenarios such as hyperlens. Therefore, as shown in Fig. 2(a), when light enters the hyperlens from vacuum, the Poynting vector $\overrightarrow {{S^{\prime}}}$ corresponding to the evanescent waves of the long wave vectors $\overrightarrow {{K^{\prime}}}$ could propagate to the far-field. The high-frequency detail information carried by the evanescent waves that decays exponentially in free space can be restored together with the low-frequency contour information carried by the propagating waves, thus breaking the diffraction limit. However, as shown in Fig. 2(b), when the dispersion curve is steep, the angle between the Poynting vectors $\overrightarrow {{S^{^{\prime\prime}}}}$ (green arrow) and the vertical axis is large, which shortens the propagation distance of the evanescent waves and is finally adverse to super-resolution imaging. Consequently, the dispersion curve should be as flat as possible to improve radial resolution by increasing the propagation distance of the evanescent waves along the radial direction.
Fig. 2. Schematic diagram of the relationship between hyperbolic dispersion curve and imaging. (a) Poynting vector $\overrightarrow {{S^{\prime}}}$ (red arrow) and wave vector $\overrightarrow {{K^{\prime}}}$ when light enters a hyperlens (blue isofrequency curve) from vacuum (grey isofrequency curve). (b) Poynting vectors corresponding to hyperbolic dispersion curves with different degrees of steepness. (c) Poynting vectors and wave vectors for the light propagating in the hyperlens. Due to material losses and curved multilayer geometry, $\overrightarrow {{K^{\prime}}}$ and $\overrightarrow {{S^{\prime}}}$ change to $\overrightarrow {{K^\ast }}$ and $\overrightarrow {{S^\ast }}$ owing to the altered direction of light.
According to geometric knowledge, we know that the curvature of a hyperbola could be described by the slope of its asymptote. The asymptote slope of the dispersion curve of the hyperlens is determined by [23]:
It is worth noting that Liu’s group also proved that the focus position corresponding to the quite flat dispersion curve is very close to the outer wall, which is disadvantageous for photodetectors such as charge-coupled devices (CCD) to collect the far-field propagating light converted from evanescent waves by hyperlens [17]. Therefore, for the purpose of guaranteeing that the focus is still formed in the outer space of the hyperlens, the asymptote slope of dispersion curve needs to reach a certain value. It can be seen that in order to improve radial resolution while ensuring that the focus remains outside the hyperlens at the expense of lesser amount of layers, the hyperlens need to have the advantages of both flat and steep dispersion curves. Consequently, we design a functional structure to regulate the dispersion curve of the hyperlens.
Due to the curved multilayer geometry of the hyperlens and the losses from metamaterials, the orientation of rays would change during the propagation. As shown in Fig. 2(c), the Poynting vector $\overrightarrow {{S^{\prime}}}$ is biased towards the vertical axis at an angle and becomes $\overrightarrow {{S^\ast }}$ depicted with the brown arrow, which is propagated closer along the radial direction. It is well known that the energy losses in metamaterials mainly originates from metals and increase with the augment of layer thickness. When the energy propagates along the radial direction of the hyperlens, the losses will accumulate layer by layer. Therefore, it is necessary to make the thickness of the metal layers close to the outer wall of the hyperlens as thin as possible. The functional structure should consist of two subparts, one subpart flattens the dispersion curve to improve radial resolution, and the other subpart raises the asymptote slope of the dispersion curve to make the focus outside the whole structure. It should be noted that the outer thin subpart should correspond to the steep dispersion curve to ensure the focus could successfully propagate through the hyperlens and prevent dropping too much of radial resolution through reducing losses to diminish the degree of variation in the orientation of rays.
Considering the above analysis, the configuration of the functional structure has been determined. Take the 44-layer conventional spherical hyperlens formed by alternately stacking 9 nm Ag layers and 22 nm TiO2 layers as an example, Fig. 3 shows the far-field focus formations of the conventional spherical hyperlens and the hyperlens cascaded with functional structure. As shown by the red line, incident light from an isotropic material (air) propagates through a multilayer anisotropic conventional hyperlens composed of single kind metal and dielectric, eventually emerges into the air. Captured by hyperlens, the incident light finally converges and form a point in the far-field. However, due to the small amount of layers, the formed focus is also relatively close to the outer wall of the conventional hyperlens. Independent of increasing the layer amount to strengthen SPPs, the cascaded functional structure directly regulates the dispersion curve by changing the dielectric layer material.
Fig. 3. Schematic diagram of the focus formations. The red lines represent the focus formation of the conventional hyperlens while the green lines are the rays propagating in the functional structure and eventually refocusing further away from the outer wall of the hyperlens.
The subpart of functional structure, which is directly neighbored to the conventional hyperlens, is composed of Ag layers and dielectric layers with RI smaller than that of TiO2, and the asymptote slope of the dispersion curve is smaller than that of the Ag-TiO2 hyperlens, which could effectively improve radial resolution. The other subpart is composed of Ag layers and dielectric layers with RI greater than that of TiO2, to make the focus to the position that could be detected outside the whole structure.
Based on the above discussions, we propose a graded structure as the functional structure. For the convenience of description, a layer of metal and a layer of dielectric could be abbreviated as a period in the hyperlens. The above 44-layer spherical hyperlens can be considered as 22 periods and shown in Fig. 4(a), where the grey part is a brief representation of the 18 alternating periods. Under the incident light of 365 nm wavelength, the permittivity of Ag can be obtained: εm = -2.4-0.25i [17] while the permittivity of TiO2 is εd = 8.29. For this case of p = 9/31, the permittivity in θ and r directions are εθ =5.18-0.07i and εr = -25.9-9.47i, which successfully correspond to a hyperbolic dispersion relation.
Fig. 4. (a) 3D structure diagram and (b) cross-sectional simulation diagram of the hyperlens which is cascaded with a multilayer graded structure. The entire simulation area is set to a square with a side length of 6.5 µm, and the outermost 300 nm thick structure is a perfect matching layer boundary for absorbing light incident. The hyperlens with an inner radius of 200 nm is located at the center of the entire simulation area.
A multilayer graded structure with Ag as the metal material is cascaded outside the above 22-period conventional hyperlens. In the graded structure, a layer of Ag and an adjacent layer of dielectric is called a unit. Here, we set the filling ratio of Ag in each unit of the graded structure to 0.5 to ensure the hyperbolic dispersion condition of εr <0 and εθ >0. Furthermore, in our work, the product of the thickness and RI of the dielectric layer in each unit of the graded structure is fixed to 19.8, while RI of the dielectric layer increases linearly along the radial direction from the initial value of 1.38. To better investigate radial resolution of hyperlens, two-dimensional full-wave simulations are conducted using finite difference time domain and finite element method. As shown in Fig. 4(b), the simulation area is 6.5 × 6.5 µm2, and the boundary of which is a perfect matching layer (PML) with 300 nm thickness. A TM-polarized point source is placed several nanometers away from the inner surface of the hyperlens. Propagation medium outside the hyperlens is air.
3. Results and discussion
To better demonstrate the advantages of the hyperlens with graded structure, the comparisons on radial resolution and far-field focus position to conventional hyperlens need to be presented. As shown in Fig. 5, the radial distance between the point source and the inner wall of the conventional hyperlens is set to d1 =29 nm, d2 = 15 nm and d3 = 1 nm. In those cases, the long wave vectors are compressed when the rays propagating along the radial direction, and the emitted light could be successfully refocused at a point outside the hyperlens. In order to indicate the focus position more prominently, we intentionally mark it with a white point.
Fig. 5. Schematic diagram of 14 nm radial resolution of a 22-period conventional hyperlens. The point source which is placed at three different positions (d1 = 29 nm, d2 = 15 nm, d3 = 1 nm) are shown in (a), (b) and (c) respectively. Far-field focus positions marked with white points are shown in (d), (e) and (f), which are corresponding with (a)-(c). The grey dash line represents the focus position of (e) while the focus positions of (d) and (f) are on different sides of the dash line.
The distance between the outer wall of the hyperlens and the focus is defined as f. For radial distance of 29, 15 and 1nm, the corresponding focus position is located at f1 = 111 nm, f2 = 168 nm and f3 = 216 nm, respectively. It could be seen that the focus position significantly changes as the point source moves. The change in f is magnified by a factor of nearly 3.75 relative to the change in d, which is beneficial for converting radial resolution to far-field measurement. It is remarkable that when the movement of the point source is less than 14 nm, the far-field focus position does not alter.
Although the graded structure has above-mentioned advantages, it is only suitable to be used as an auxiliary structure for conventional hyperlens as it cannot break the diffraction limit independently. To design the optimal graded structure for the above 44-layer conventional hyperlens, the relevant parameters including the unit amount and the linearly-increased gradient of RI of dielectric layers have been comprehensively investigated.
In Fig. 6(a), the linearly-increased gradient of RI of dielectric layers is 0.24, and radial resolution is continuously improved with increasing unit amount of the graded structure and reaches the best value of 5 nm when the unit amount is 10. After that, radial resolution of the hyperlens shows a downward trend. It can be calculated that the difference between the overall thickness of the graded structures with 10-unit and 6-unit is only 57 nm, while the achieved radial resolution differs by 8 nm, meaning that radial resolution of the whole structure is very sensitive to the unit amount in the graded structure. In essence, the effect of increasing the unit amount is to continuously cascade the components that could make the dispersion curve steep. Therefore, as shown in Fig. 6(b), the far-field focus position shows a monotonously upward trend as the unit amount increases, and reaches the maximum value of 534 nm when the unit amount of the graded structure is 12. In the 10-unit graded structure which achieves the best radial resolution, according to our design guideline, it can be easily calculated that the RI of the dielectric layers in the former 7 units is smaller than that of TiO2 (RI of TiO2 is 2.88), thus dispersion curve is quite flat and could successfully increase the propagation distance of the evanescent waves to improve radial resolution. The latter 3 units correspond to the steep dispersion curve as RI of the dielectric layers becomes larger than that of TiO2, eventually make the focus 484 nm away from the outer wall of the whole structure. The simulation results validate our originally-proposed two-subpart configuration assumption for functional structures. Although the far-field focus position obtained by the 10-unit graded structure only reaches 484 nm (less than 534 nm), the increment of focus position compared to that of the 9-unit graded structure reached the most significant value of 60 nm. So, we think a graded structure with the unit amount of 10 is optimal for 44-layer Ag-TiO2 hyperlens.
Fig. 6. (a) Radial resolution and (b) far-field focus position as a function of unit amount of the graded structure. The product of the thickness and the RI of the dielectric layer possesses a fixed value of 19.8, the linearly-increased gradient of RI of dielectric layers is 0.24. Other parameters keep the same with those in Fig. 5.
Figure 7 shows the key performance of hyperlens with various linearly-increased gradients adopted by the dielectric layers in the 10-unit graded structure. The initial value of the gradient is set to 0.15, and the following 5 gradient values ramp up with the 0.03 step. The gradient value plays a critical role in the configuration of the graded structure. If the gradient makes the graded structure contain only Ag layers and dielectric layers with RI greater than that of TiO2, radial resolution would be degraded rather than improved compared with that of conventional hyperlens, e.g., initial gradient value of 0.15 will make the radial resolution deteriorate from original 14 nm to 16 nm. Due to the equal thickness setting of Ag and dielectric of each unit in the graded structure, the smaller gradient and the thicker overall thickness of the metal, which causes serious increment in the energy losses and ultimately have a negative impact on radial resolution. Meanwhile, if the necessary subpart composed of Ag layers and dielectric layers with RI greater than that of TiO2 is missing, the focus would also be dragged closer to the outer wall of whole structure. Therefore, suitable linearly-increased gradient of dielectric layers should be selected to make the graded structure have both subparts that can make the dispersion curve flat and steep respectively, eventually achieving simultaneous enhancement of radial resolution and far-field focus position.
Fig. 7. (a) Radial resolution and (b) far-field focus position as a function of linearly-increased gradient of RI of dielectric layers in graded structure. The unit amount of the graded structure is fixed to 10. Other parameters keep the same as those in Fig. 6.
In order to improve the radial resolution effectively, it is necessary to make the dispersion curve flat while taking into account the effect of loss caused by thickness. Consequently, when the total unit amount of the graded structure keeps 10, it is necessary not only to make the unit amount in the subpart where the RI of dielectric layers is smaller than that of TiO2 higher than the remained unit amount in the other subpart as much as possible, but also to ensure that the thickness of the subpart with a quite flat dispersion curve would not cause serious energy losses due to excessive shrinkage of the gradient. With adoption of the linearly-increased gradient of 0.24 in 10-unit dielectric layers, radial resolution achieves optimal 5 nm while the far-field focus position is 484 nm. The gradient of 0.24 is high enough to make the RI of dielectric layers of the outermost three units in the graded structure larger than that of TiO2, thus the rays could be successfully propagated to position in the outer space of the hyperlens. If linearly-increased gradient is increased further, radial resolution will drop rather than improve. Therefore, linearly-increased gradient of RI of dielectric layers is finally selected to be 0.24 in our design.
Taking all factors into consideration, we think that the optimal graded structure for a 22-period Ag-TiO2 conventional hyperlens should contain 10 units while keeping its linearly-increased gradient of RI of dielectric layers 0.24. As-formed new hyperlens still can realize super-resolution imaging in the lateral direction, which can be demonstrated by Fig. 8, as the light flaps formed by the two point sources (two dipoles with wavelength of 365 nm and distance of 130 nm) are successfully separated, i.e., the diffraction limit of half wavelength (182.5 nm) has been broken.
Fig. 8. (a) Schematic diagram of the location of the two point sources. The lateral distance between the two point sources is h1 = 130 nm while the radial seperation between point sources and the inner wall of the hyperlens is 1 nm. (b) Contour map of the magnetic field distribution. (c) The normalized intensity along the black intersecting line in the magnetic field distribution diagram.
As shown in the Fig. 9, the point source is placed at d1 =22 nm, d2 = 17 nm, d3 = 12 nm and d4 = 7 nm away from the inner wall and the corresponding far-field focus positions are at f1 =484 nm, f2 =417 nm, f3 =345 nm and f4 =306 nm, respectively. The moving step of 5 nm of point source can cause significant movement of far-field focus position, which is helpful to easy detection via mechanical translation stages. Compared with the conventional hyperlens, both radial resolution and the relative moving distance of the far-field focus position have been significantly improved. Moreover, the hyperlens as well as the graded structures in our study possess totally 64 layers, have the advantage over 144-layer conventional hyperlens that also shows the astonishing 5 nm radial resolution [17] and is highly achievable in practical fabrication.
Fig. 9. Schematic diagram of 5 nm radial resolution achieved by a 22-period conventional hyperlens cascaded with a 10-unit multilayer graded structure. Linearly-increased gradient of RI of dielectric layers in the graded structure is 0.24. Four different positions (d1 = 22 nm, d2 = 17 nm, d3 = 12 nm and d4 = 7 nm) where the point source is placed at are shown in (a), (b), (c) and (d) respectively. Far-field focus positions marked with white points are shown in (e), (f), (g) and (h), which correspond to (a), (b), (c) and (d). Other parameters keep the same as those in Fig. 7. Two grey dash lines represent the focus horizontal positions of (f) and (h), respectively. The focus position of (e) is lower than both dash lines while the focus position of (h) is between the dash lines.
4. Conclusion
The spherical hyperlens cascaded with the graded structure provides an effective way to improve radial resolution. Via cascading a multilayer graded structure, we demonstrate from simulations that the hyperlens can break diffraction limit and has an ultra-high radial resolution of sub-10 nm (5 nm), which shows an incredible 64% improvement when compared with conventional hyperlens counterpart. To the best of our knowledge, we achieve the extremely high radial resolution as same as the previously-reported 144-layer spherical hyperlens but hold the advantage of much lesser layer amount, which is more advantageous in practical fabrication. At the same time, the far-field focus could still be detected outside the outer wall of the hyperlens and even pushed further away from the outer wall of the whole structure. We believe this structure is of great importance for nano 3D imaging and could effectively facilitate the development of high-performance hyperlens. Moreover, its ultra-sensitive characteristics and extremely small size are expected to play an important role in the biochemical perception and dynamic monitoring.
Funding
National Natural Science Foundation of China (No. 61474008); The Funds for Creative Research Groups of China (No. 62021005); Program for New Century Excellent Talents in University (No. NCET-13-0686); China Postdoctoral Science Foundation (No. 2020M680462); State Key Laboratory of Information Photonics and Optical Communications (BUPT) (No. IPOC2021ZZ03, No. IPOC2020ZZ01).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. M. Valm, S. Cohen, W. R. Legant, J. Melunis, U. Hershberg, E. Wait, A. R. Cohen, M. W. Davidson, E. Betzig, and J. Lippincott-Schwartz, “Applying systems-level spectral imaging and analysis to reveal the organelle interactome,” Nature 546(7656), 162–167 (2017). [CrossRef]
2. L. Möckl, D. C. Lamb, and C. Bräuchle, “Super-resolved fluorescence microscopy: nobel prize in chemistry 2014 for eric betzig, stefan hell, and william e. mourner,” Angew. Chem., Int. Ed. 53(51), 13972–13977 (2014). [CrossRef]
3. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013). [CrossRef]
4. S. J. Sahl, S. W. Hell, and S. Jakobs, “Fluorescence nanoscopy in cell biology,” Nat. Rev. Mol. Cell Biol. 18(11), 685–701 (2017). [CrossRef]
5. G. Zheng, Y. Chen, L. Bu, L. Xu, and W. Su, “Waveguide-coupled surface phonon resonance sensors with super-resolution in the mid-infrared region,” Opt. Lett. 41(7), 1582–1585 (2016). [CrossRef]
6. X. Fu and T. J. Cui, “Recent progress on metamaterials: From effective medium model to real-time information processing system,” Prog. Quantum Electron. 67(9), 100223 (2019). [CrossRef]
7. S. Moon, R. Yan, S. J. Samuel, Y. Shyu, L. Xiang, W. Li, and K. Xu, “Spectrally resolved, functional super-resolution microscopy reveals nanoscale compositional heterogeneity in live-cell membranes,” J. Am. Chem. Soc. 139(32), 10944–10947 (2017). [CrossRef]
8. Y. Zhao, M. Zhang, W. Zhang, Y. Zhou, L. Chen, Q. Liu, P. Wang, R. Chen, X. Duan, F. Chen, H. Deng, Y. Wei, P. Fei, and Y. Zhang, “Isotropic super-resolution light-sheet microscopy of dynamic intracellular structures at subsecond timescales,” Nat. Methods 19(3), 359–369 (2022). [CrossRef]
9. W. Li and Z. Wang, “Breaking the diffraction limit using fluorescence quantum coherence,” Opt. Express 30(8), 12684–12694 (2022). [CrossRef]
10. S. Pelicc, L. Furia, M. Scanarini, P. G. Pelicci, L. Lanzanò, and M. Faretta, “Novel Tools to Measure Single Molecules Colocalization in Fluorescence Nanoscopy by Image Cross Correlation Spectroscopy,” Nanomaterials 12(4), 686 (2022). [CrossRef]
11. K. Abrashitova and L. V. Amitonova, “High-speed label-free multimode-fiber-based compressive imaging beyond the diffraction limit,” Opt. Express 30(7), 10456–10469 (2022). [CrossRef]
12. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef]
13. D. Lee, Y. D. Kim, M. Kim, S. So, H. Choi, J. Mun, D. M. Nguyen, T. Badloe, J. G. Ok, K. Kim, H. Lee, and J. Rho, “Realization of Wafer-Scale Hyperlens Device for Sub-diffractional Biomolecular Imaging,” ACS Photonics 5(7), 2549–2554 (2018). [CrossRef]
14. W. Zhang and C. Yan, “Super-resolution scanning imaging based on metal–dielectric composite metamaterials,” Appl. Opt. 61(1), 196–201 (2022). [CrossRef]
15. L. Chen and G. P. Wang, “Pyramid-shaped hyperlenses for three-dimensional subdiffraction optical imaging,” Opt. Express 17(5), 3903–3912 (2009). [CrossRef]
16. X. Wu, Z. Li, Y. Zhao, C. Yang, W. Zhao, and X. Zhao, “Abnormal optical response of PAMAM dendrimer-based silver nanocomposite metamaterials,” Photonics Res. 10(4), 965–972 (2022). [CrossRef]
17. W. Wan, J. L. Ponsetto, and Z. Liu, “Numerical study of hyperlenses for three-dimensional imaging and lithography,” Opt. Express 23(14), 18501–18510 (2015). [CrossRef]
18. J. Sun and N. M. Litchinitser, “Toward practical, subwavelength, visible-light photolithography with hyperlens,” ACS Nano 12(1), 542–548 (2018). [CrossRef]
19. R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76(2), 026606 (2007). [CrossRef]
20. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects,” Science 315(5819), 1686 (2007). [CrossRef]
21. J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Commun. 1(1), 143 (2010). [CrossRef]
22. O. Takayama and A. V. Lavrinenko, “Optics with hyperbolic materials [Invited],” J. Opt. Soc. Am. B 36(8), F38–F48 (2019). [CrossRef]
23. D. Su, X. Zhang, Y. Ma, F. Shan, J. Wu, X. Fu, L. Zhang, K. Yuan, and T. Zhang, “Real-time electro-optical tunable hyperlens under subwavelength scale,” IEEE Photonics J. 10(1), 1–9 (2018). [CrossRef]
