## Abstract

An ultra-narrow angular optical transparency window based on photonic topological transition (PTT) is theoretically and numerically investigated in a low-loss hyperbolic metamaterial (HMM) platform, which consists of aligned metallic nanowires embedded indielectric host matrices. Our results indicate that, near the transition point of PTT, the designed system exhibits high-efficiency optical angular selectivity close to normal incidence by tailoring the topology of metamaterial’s equi-frequency surface (EFS). Moreover, the operating wavelength (*λ*_{0}) is flexibly tunable by selecting appropriate material and geometrical parameters, which provides significant guidance for the later experimental design. Our method is further applied to super-resolution imaging, with a resolution of at least ${\lambda}_{0}/4$ and over a significant distances (>12*λ*_{0}). The HMM-supported angularly selective system could find promising applications for high-efficiency light manipulation and lensless on-chip imaging.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Metamaterials, which are artificial optical composites engineered on subwavelength scales that cannot be found in nature, have unprecedented flexibility in manipulating electromagnetic waves and generating various novel optical applications, such as biosensors [1, 2], optical detectors [3–5], modulators [6–8], and perfect absorbers [9, 10]. More recently, with the development of nanofabrication technologies, hyperbolic metamaterials (HMMs) have generated considerable research interest, thanks to their extremely anisotropic dielectric tensors. Until now, a large number of HMMs-based electromagnetic properties have been investigated numerically and experimentally, such as optical forces [11, 12], negative index [13, 14], and sub-wavelength imaging [15, 16]. As we will show, photonic topological transition (PTT) in HMMs have drawn considerable attention [17–20]. Unlike the topological phase transition in previously designed systems [17, 18], here, the PTT only refers to the optical topological transition of HMMs with equi-frequency surface (EFS), which can directly manipulate the nature of electromagnetic waves inside the metamaterials. It is well known that EFS maps the angular dependence of wave vectors and indicates the direction of energy flow [19]. Therefore, the topology of EFS can not only directly modulate the propagation direction of light, but also effectively manipulate light-matter interactions in metamaterials. And it is worth noting that near the PTT frequency point, namely the transition point of EFS of metamaterials from closed ellipsoid to open hyperboloid, HMMs can significantly inhibit the diffraction and scattering of incident light, which opens up a new route to effectively detect and manipulate sub-wavelength information at nanoscales.

In this work, we introduce the basic principle of engineering an ultra-narrow angular optical transparency window based on PTT in metamaterial, in which HMM consists of high aspect ratio metallic nanowires embedded in dielectric host matrices. As a result, near the transition point of PTT, we achieves high-efficiency optical angular selectivity, that is, only the incident light very close to the vertical incidence can effectively pass through the metamaterial, while the incident light from other directions is completely blocked. Here, for p-polarized incident light with near-infrared wavelength (*λ*_{0}), by tailoring the topology of metamaterial’s EFS, we can achieve an ultra-narrow angular transparency window less than 2° and an ultra-high transparency (up to 98%) close to normal incidence. Compared with previous angularly selective systems, such as metallic gratings [21, 22], photonic crystals [23, 24] and epsilon-near-zero (ENZ) media [25], the designed metamaterial in this paper has much narrower angular selectivity and higher optical transparency. To show the feasibility of the above method, we theoretically proposed the application of ultra-narrow angular transparency effect to super-resolution imaging. Furthermore, we numerically demonstrate the imaging of 2D object with subwavelength characteristics, having a resolution of at least a quarter of the excitation wavelengths (${\lambda}_{0}/4$) and over a significant distance (or HMM thickness) corresponding roughly to more than twelve wavelengths (>12*λ*_{0}). For the previous designed super-resolution imaging systems, there are still some defects such as small thickness of the lens (poor man’s lens [26]), limited imaging area (hyperlens [27]), image distortions (photonic crystals [28]), lack of supporting propagation waves (poor man’s lens/subwavelength plates [29]), etc., these deficiencies have been greatly improved in the present design of metamaterial. In addition, our method has many other attractive advantages, including simplicity, extensibility of working wavelength and large-scale repeatability, which can be used for a range of angle-dependent optical applications, such as privacy protection [30] and lensless on-chip imaging [31].

## 2. Theories and analyses

#### 2.1. Ultra-narrow angular optical transparency in the HMM based on PTT

As shown in Fig. 1(a), the nanowire-based HMM is realized by embedding array of gold (Au) nanorods with hexagonal lattice in a porous alumina (Al${}_{2}$O${}_{3}$) host matrix of thickness *h*. The lattice constant and diameter of the nanowiresare donated by *a* and *d*, respectively. The permittivity of Al${}_{2}$O${}_{3}$ is ${\epsilon}_{A{l}_{2}{O}_{3}}=3.12$ for optical region of our interest [32], and the complex permittivity of Au can be defined by the Drude model ${\epsilon}_{Au}(\omega )={\epsilon}_{\infty}-{\omega}_{p}^{2}/({\omega}^{2}+i\gamma \omega )$ with ${\omega}_{p}=1.37\times {10}^{16}$ rad/s, and $\gamma =4.08\times {10}^{13}$ rad/s, respectively [33]. Given the above geometrical parameters, including nanowires diameter (*d*) and lattice constant (*a*), are much smaller than the excitation wavelength (*λ*_{0}), so the HMM system can be regarded as an effective uniaxial medium. And we utilize the effective medium theory (EMT) to describe the permittivity parallel to nanowires (${\epsilon}_{\Vert}$) and perpendicular to nanowires (${\epsilon}_{\perp}$), which are calculated by the following equations [33]:

*f*is the filling ratio of Au, which is deduced as $f=2\pi {(d/2)}^{2}/(\sqrt{3}{a}^{2})$. Figure 1(b) depicts the effective complex permittivities, ${\epsilon}_{\Vert}$(

*ε*) and ${\epsilon}_{\perp}$(

_{z}*ε*and

_{x}*ε*), of the HMM with $f=0.0243$ (corresponding to $d=18$ nm and $a=110$ nm) in the near-infrared region. It can be seen that Re(${\epsilon}_{\Vert}$) continuously changes from positive to negative as the wavelength increases, where has Re$({\epsilon}_{\Vert})=0$ at ${\lambda}_{0}=1551$ nm. In contrast, Re(${\epsilon}_{\perp}$) remains positive throughout the entire investigated wavelength range. To demonstrate the basic physics principle at work here, westudy the EFS of extraordinary (p-polarized) waves in this medium:

_{y}*k*,

_{x}*k*, and

_{y}*k*are respectively the wavevector components along

_{z}**-,**

*X***-, and**

*Y***-directions, and**

*Z**c*is the speed of light in vacuum. Based on Eq. (3), we calculate the complex wavevector ${k}_{z}/{k}_{0}$ of metamaterial’s EFS at the wavelength of 1551 nm, as shown in Figs. 1(c)-1(d). The transition point of PTT denotes the wavelength at which Re$({\epsilon}_{\Vert})\approx 0$ characterizing the transition from an ellipsoid to a hyperboloid, and corresponding to the metamaterial’s EFS degenerates to two points on the

**-axis. In contrast to the ideal loss-free medium, the topology of EFS at**

*Z**λ*

_{0}maintains a narrow hyperboloid in Fig. 1(c), which means that the electromagnetic waves with small horizontal wavevectors (

*k*and

_{x}*k*) are also allowed to propagate inside the metamaterial. Meanwhile, as shown in Fig. 1(d), wavevector diagram for Im$({k}_{z}/{k}_{0})$ as a function of ${k}_{x}/{k}_{0}$ and ${k}_{y}/{k}_{0}$ achieves a degenerate state at the origin. And the inset of Fig. 1(d) shows that the electromagnetic waves with wavevector along

_{y}**-axis has Im$({k}_{z}/{k}_{0})=0$, which indicates that there is no absorption lossin the medium at this moment. On the other hand, for the electromagnetic waves with small horizontal wavevectors (**

*Z**k*and

_{x}*k*), the components Im$({k}_{x}/{k}_{0}$) and Im$({k}_{y}/{k}_{0})$ are not equal to zero, which means they will be affected by the material’s inherent losses and the energy will be attenuated with further propagation in the medium. In other words, near the transition point of PTT, although the electromagnetic waves have a small horizontal wavevectors (

_{y}*k*and

_{x}*k*) components in the propagation directions, its inherent losses inhibit the electromagnetic waves propagation away from

_{y}**-direction and maintain a ultra-narrow angular transparent window.**

*Z*#### 2.2. Methodology

As depicted in Fig. 2(a), a p-polarized light at wavelength *λ*_{0} is coming from the surrounding medium (air) onto the metamaterial slabs at an incident angle *θ*. The propagation features can be theoretically analyzed using the transfer matrix method (TMM) [34]. And here, in order to verify the theoretically calculated results, the finite-difference time-domain (FDTD) method-based commercial software package (Lumerical FDTD solutions) is also utilized to numerically simulate the light wave propagation inside the structure. In the FDTD simulations, the perfectly matched layers (PML) are applied along the ** Z** direction, while Bloch boundary conditions are used in the

**and**

*X***directions. In order to balance the simulation time and accuracy, the mesh size along the**

*Y***-,**

*X***-, and**

*Y***-axes is set to $3\phantom{\rule{0.2em}{0ex}}nm\times 3\phantom{\rule{0.2em}{0ex}}nm\times 30\phantom{\rule{0.2em}{0ex}}nm$, respectively. According to source-free Maxwell’s equations and boundary conditions for magnetic field (**

*Z**H*) between adjacent layers, the wave equation in the metamaterial can be expressed as

**-axis, and**

*X**ω*is the light wave frequency. By simple algebraic derivation, the transfer matrix of metamaterial can be obtained as follows

*k*corresponds to the propagation of light in the

*k*-th layer with a thickness of

*d*. The resultant total transfer matrix ($M(\lambda )$) is the quadrature of the transfer matrices of all layers of metamaterial, and therefore can be expressed as

_{k}*n*represents the total number of layers of metamaterials. By means of the TMM, we can obtain the transmission coefficient $t(\lambda )$ for p-polarized light as

## 3. Results and discussions

A schematic illustration of the proposed model structure is depicted in Fig. 2(a). Based on the above mechanism of angular optical transparency effect, we can find that the incident light is almost completely pass through the metamaterial at normal incidence, while for oblique incidence at other angles, the incident light is completely blocked due to structural reflection and absorption losses. It is noteworthy that, for the normal incidence ($\theta ={0}^{\circ}$), the silicon substrate with high refractive index is selected, which is helpful to reduce the reflection of the system. Figure 2(b) gives the TMM-calculated and FDTD-simulated optical transmission of p-polarized light as function of incident angles (*θ*) for a HMM-based structure surrounded by the air at ${\lambda}_{0}=1551$ nm, where the thickness of the HMM with $f=0.0243$ (correspond to $d=18$ nm, $a=110$ nm) is set to $h=18.6\text{}\mu $m. The narrowest transparency window has a full-width at half-maximum (FWHM) of $\sim {0.7}^{\circ}(-{0.35}^{\circ}<\theta <{0.35}^{\circ})$ and an ultra-high optical transparency up to 98% at $\theta ={0}^{\circ}$. And the inset of Fig. 2(b) shows that the numerical simulations (FDTD) agree well with the theoretical calculations (TMM). To further verify the ultra-narrow angular transparency effect, we plot the electric field distributions (|*E*|) at the HMM-air interface when the incidence angle $\theta ={0}^{\circ}$ and $\theta ={2}^{\circ}$, respectively. As depicted in Figs. 2(c)-2(d), when $\theta ={0}^{\circ}$, the electromagnetic modes excited in HMM cannot effectively penetrate the gold nanowires and most of the energy is transmitted in the host alumina dielectric between the gold nanowires. Therefore, the influence of the metallic nanowires losses can be ignored and the incident light almost completely penetrates the structure in this situation. On the contrary, when $\theta ={2}^{\circ}$, due to the influence of structural reflection and absorption losses, all incident light is completely blocked by the structure [36].

Furthermore, the transition point of PTT discussed previously just corresponds to a single working wavelength (${\lambda}_{0}=1551$ nm), but actually the structure shows a broadband angular transparency effect. Figure 3(a) displays that the numerically simulated angular optical transmission (*T*) as a function of incident angles ($-{4}^{\circ}<\theta <{4}^{\circ}$) and excitation wavelengths (*λ*_{0} ranging from 1530 nm to 1570 nm), in which the narrowest angular transparency window with $FWHM\approx {0.7}^{\circ}(-{0.35}^{\circ}<\theta <{0.35}^{\circ})$ and the maximum transmission $T=98\%$ occur around the PTT wavelength of 1551 nm. In addition, Fig. 3(b) summaries the angular FWHM and optical transmission *T* of transparency window extracted from Fig. 3(a). And we can find that, in our whole investigated wavelength range, the angular FWHM of transparency window does not exceed ${1.1}^{\circ}$ and its peak transmittance *T* maintains more than 70%, mainly in that the structure exhibits a nearly flat equi-frequency dispersion characteristics near the PTT wavelength [37].

Subsequently, we investigate the influence of structural parameters of HMM on angular optical transparency window. The results in Fig. 4(a) illustrate that the angular optical transparency window becomes narrower with the increase of the HMM thickness *h*, while the maximum transmission under the same conditions hardly changes at normal incidence ($\theta ={0}^{\circ}$). This is because, as the HMM thickness increases, the absorption loss of the vertical component remains almost the same, while the loss of the horizontal component increases accordingly, thus inhibiting the electromagnetic waves propagation away from ** Z**-direction and maintaining a much narrower angular transparency window. In addition, we depict the impact of filling ratio (

*f*) of Au in HMM, as can be seen in Fig. 4(b). Here, we achieve different filling ratios by changing the diameter of nanowires (

*d*) while the lattice size (

*a*) is kept constant. From Fig. 4(b) we can find that the width of the angular optical transparency windows change with varying

*f*, which is mainly due to the fact that the transition point of PTT has shifted to a different wavelength. As depicted in the inset of Fig. 4(b), for a filling ratio of $0.0263$, the structure works best at 1486 nm, whereas for the filling ratio of $0.0223$ the best result is observed at around 1619 nm. Therefore, the system can be operated at different working wavelengths by choosing proper filling ratio. The FDTD simulations agree well with the theoretical calculations by the TMM methods. And these theoretical results will provide significant guidance for the later experimental design.

## 4. Imaging application

The mechanism above provides a good angular optical transparency performance close to normal incidence, and is therefore useful in many applications, such as super-resolution imaging. Here, we perform numerical simulations of an 18.6 *μ*m thick HMM slab to assess its imaging properties for a subwavelength two-slit object, as shown in Fig. 5(a). As object to be imaged two 600 nm (*S*) wide slits in an opaque Au film are assumed, spaced by a gap of 400 nm (*D*). From Fig. 5(b), we find that the two individual slits can be clearly resolved by the HMM slab, but in the absence of the HMM, the object with the same conditions cannot be resolved. To further verify this result, as depicted in Figs. 5(c)-5(d), we plot the electric field and phase distribution of the object and its optical imaging on the HMM-air interface with or without HMM coating, respectively. By comparison, it is illustrated that the diffraction and scattering of light waves from the object with HMM coating are remarkably suppressed due to the ultra-narrow angular transparency effect. In other words, the PTT occurs at the excitation wavelength of 1551 nm, which renders the dispersion curve flat for Re$({k}_{\perp})\approx 0$, similar to the dispersion of epsilon-near-zero media [25, 38], so that almost all spatial frequency components of subwavelength details of an object field are transported along ** Z**-direction with the same propagation wavevector Re$({k}_{\Vert})$ [37]. The above results clearly establish that the HMM slab behaves as a super-resolution medium, which can reconstruct, with low-loss, subwavelength details of an object down to about ${\lambda}_{0}/4$ length scales, over long distances (>12

*λ*

_{0}).

## 5. Conclusion

In summary, we introduce and fully characterize the photonic topological transitions in near-infrared HMM platform made of metallic nanowires array embedded in the dielectric matrices, and propose a general method to achieve ultra-narrow angular optical transparency near the transition point. Both the theoretical calculations and numerical simulations have demonstrated that, by tailoring the topology of metamaterial’s EFS, the designed structure can exhibit an ultra-narrow angular transparency window less than 2° and an ultra-high transparency (up to 98%) close to normal incidence. Furthermore, the concepts described in this paper can be extended such that the structure can be operated over a broad spectral range by suitable choice of material and geometrical parameters, thus providing significant guidance for the later experimental design. Apart from these, we further explore its potential application in imaging, and the results clearly establish that the HMM slab behaves as a low-loss super-resolution medium at near-infrared wavelengths, which can transport subwavelength details of an object down to about ${\lambda}_{0}/4$ length scales, over a significant distances (>12*λ*_{0}). The superior optical properties of the proposed metamaterial, together with large scale and low-cost manufacturing, open new opportunities for manipulating light propagation in various angle-dependent applications, including high-efficiency solar energy conversion, privacy protection and detectors with ultra-high signal-to-noise ratios.

## Funding

National Natural Science Foundation of China (NSFC) (61775064); Fundamental Research Funds for the Central Universities (HUST: 2016YXMS024).

## Acknowledgments

The author Xiaoyun Jiang (XYJIANG) expresses her deepest gratitude to her PhD advisor Tao Wang for providing guidance during this project.

## References

**1. **A. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. Wurtz, R. Atkinson, R. Pollard, V. Podolskiy, and A. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nat. Mater. **8**, 867 (2009). [CrossRef] [PubMed]

**2. **D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. G. De Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science **349**, 165–168 (2015). [CrossRef] [PubMed]

**3. **J. Zhang, Z. Zhu, W. Liu, X. Yuan, and S. Qin, “Towards photodetection with high efficiency and tunable spectral selectivity: graphene plasmonics for light trapping and absorption engineering,” Nanoscale **7**, 13530–13536 (2015). [CrossRef] [PubMed]

**4. **S. Xiao, T. Wang, Y. Liu, C. Xu, X. Han, and X. Yan, “Tunable light trapping and absorption enhancement with graphene ring arrays,” Phys. Chem. Chem. Phys. **18**, 26661–26669 (2016). [CrossRef] [PubMed]

**5. **X. He, F. Liu, F. Lin, G. Xiao, and W. Shi, “Tunable MoS${}_{2}$ modified hybrid surface plasmons waveguides,” Nanotechnology **30**, 125201 (2019). [CrossRef]

**6. **X. He, Z. Y. Zhao, and W. Shi, “Graphene-supported tunable near-IR metamaterials,” Opt. Lett. **40**, 178–181 (2015). [CrossRef] [PubMed]

**7. **C. Shi, X. He, J. Peng, G. Xiao, F. Liu, F. Lin, and H. Zhang, “Tunable terahertz hybrid graphene-metal patterns metamaterials,” Opt. Laser Technol. **114**, 28–34 (2019). [CrossRef]

**8. **X. He, G. Xiao, F. Liu, F. Lin, and W. Shi, “Flexible properties of THz graphene bowtie metamaterials structures,” Opt. Mater. Express **9**, 44–55 (2019). [CrossRef]

**9. **H. Lu, X. Gan, D. Mao, Y. Fan, D. Yang, and J. Zhao, “Nearly perfect absorption of light in monolayer molybdenum disulfide supported by multilayer structures,” Opt. Express **25**, 21630–21636 (2017). [CrossRef] [PubMed]

**10. **X. Jiang, T. Wang, S. Xiao, X. Yan, and L. Cheng, “Tunable ultra-high-efficiency light absorption of monolayer graphene using critical coupling with guided resonance,” Opt. Express **25**, 27028–27036 (2017). [CrossRef] [PubMed]

**11. **Y. He, S. He, J. Gao, and X. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express **20**, 22372–22382 (2012). [CrossRef] [PubMed]

**12. **P. Ginzburg, A. V. Krasavin, A. N. Poddubny, P. A. Belov, Y. S. Kivshar, and A. V. Zayats, “Self-induced torque in hyperbolic metamaterials,” Phys. Rev. Lett. **111**, 036804 (2013). [CrossRef] [PubMed]

**13. **J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science **321**, 930 (2008). [CrossRef] [PubMed]

**14. **J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. **98**, 101901 (2011). [CrossRef]

**15. **Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

**16. **S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. **7**, 265–271 (2013). [CrossRef]

**17. **L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics **8**, 821 (2014). [CrossRef]

**18. **W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. **114**, 037402 (2015). [CrossRef] [PubMed]

**19. **P. Huo, Y. Liang, S. Zhang, Y. Lu, and T. Xu, “Angular optical transparency induced by photonic topological transitions in metamaterials,” Laser Photonics Rev. **12**, 1700309 (2018). [CrossRef]

**20. **X. Wu, “Angular optical transparency induced by photonic topological transition in hexagonal boron nitride,” https://doi.org/10.1007/s11468-018-0882-4 (2018).

**21. **A. Alu, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, “Plasmonic brewster angle: broadband extraordinary transmission through optical gratings,” Phys. Rev. Lett. **106**, 123902 (2011). [CrossRef] [PubMed]

**22. **C. Argyropoulos, K. Q. Le, N. Mattiucci, G. D’Aguanno, and A. Alu, “Broadband absorbers and selective emitters based on plasmonic brewster metasurfaces,” Phys. Rev. B **87**, 205112 (2013). [CrossRef]

**23. **Y. Lu, L. Yan, Y. Guo, Y. Pan, W. Pan, and B. Luo, “Elevation-azimuth angular selectivity and angle-frequency filtering in asymmetric photonic crystal,” Opt. Express **24**, 24473–24482 (2016). [CrossRef] [PubMed]

**24. **J. Guo, S. Chen, and S. Jiang, “Optical broadband angular filters based on staggered photonic structures,” J. Mod. Opt. **65**, 928–936 (2018). [CrossRef]

**25. **L. Alekseyev, E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. **97**, 131107 (2010). [CrossRef]

**26. **T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science **313**, 1595 (2006). [CrossRef] [PubMed]

**27. **Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**, 1686 (2007). [CrossRef] [PubMed]

**28. **B. Casse, W. Lu, R. Banyal, Y. Huang, S. Selvarasah, M. Dokmeci, C. Perry, and S. Sridhar, “Imaging with subwavelength resolution by a generalized superlens at infrared wavelengths,” Opt. Lett. **34**, 1994–1996 (2009). [CrossRef] [PubMed]

**29. **R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science **317**, 927–929 (2007). [CrossRef] [PubMed]

**30. **Y. Qu, Y. Shen, K. Yin, Y. Yang, Q. Li, M. Qiu, and M. Soljačić, “Polarization-independent optical broadband angular selectivity,” ACS Photonics **5**, 4125–4131 (2018). [CrossRef]

**31. **A. Greenbaum, W. Luo, T. W. Su, Z. Göröcs, L. Xue, S. O. Isikman, A. F. Coskun, O. Mudanyali, and A. Ozcan, “Imaging without lenses: achievements and remaining challenges of wide-field on-chip microscopy,” Nat. Methods **9**,889 (2012). [CrossRef] [PubMed]

**32. **D. P. Edward and I. Palik, “Handbook of optical constants of solids,” (Elsevier, 1985).

**33. **Y. He, H. Deng, X. Jiao, S. He, J. Gao, and X. Yang, “Infrared perfect absorber based on nanowire metamaterial cavities,” Opt. Lett. **38**, 1179–1181 (2013). [CrossRef] [PubMed]

**34. **N. h. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E **65**, 046607 (2002). [CrossRef]

**35. **T. Pan, G. Xu, T. Zang, and L. Gao, “Goos–hänchen shift in one-dimensional photonic crystals containing uniaxial indefinite medium,” Phys. Status Solidi B **246**, 1088–1093 (2009). [CrossRef]

**36. **B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. **96**, 023114 (2010). [CrossRef]

**37. **M. Esslinger, R. Vogelgesang, N. Talebi, W. Khunsin, P. Gehring, S. De Zuani, B. Gompf, and K. Kern, “Tetradymites as natural hyperbolic materials for the near-infrared to visible,” ACS Photonics **1**, 1285–1289 (2014). [CrossRef]

**38. **I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics **11**, 149 (2017). [CrossRef]