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Abstract
The diffraction limit of light due to the loss of evanescent waves that carry high spatial frequency information in the far field restricts the practical applications of terahertz imaging technology. In order to break the diffraction limit, we investigate the super-resolution capability of different types of multilayered graphene-dielectric hyperbolic metamaterials. A super-resolution of λ/10 is achieved for both the cylindrical and planar structures. A prominent advantage of graphene-dielectric hyperbolic structures is the dynamic tunability of the dispersion and super-resolution performance by adjusting the chemical potential of graphene through conveniently changing the gate voltage without modifying the geometry of the hyperbolic structures. Furthermore, we have investigated the influence of bilayer thickness variations on the super-resolution performance. Finally, we apply the planar hyperbolic structures for the super-resolution imaging and a roughly five-fold lateral resolution enhancement is realized in our approach. Due to many prominent advantages including super-resolution over broad spectral range, dynamic tunability, good stability and robustness, we believe this work could contribute to the improvement of the resolutions of terahertz imaging systems and the development of hyperbolic metamaterial modulation devices in the terahertz band.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
Shixuan Hao, Jicheng Wang, Ivan Fanayev, Sergei Khakhomov, and Jingwen Li, "Hyperbolic metamaterial structures based on graphene for THz super-resolution imaging applications: publisher’s note," Opt. Mater. Express 13, 1059-1059 (2023)https://opg.optica.org/ome/abstract.cfm?uri=ome-13-4-1059
10 March 2023: A correction was made to Figure 4.
1. Introduction
THz imaging technologies have a broad range of applications in the fields of non-destructive material inspection [1–3], quality control [4–6], medical diagnosis [7,8], and defense and security [9–11], due to the numerous advantages of terahertz waves, such as low photon energy, high penetration to non-polar materials and rich spectral information [12–15]. Compared to X-rays, terahertz waves do not cause harmful ionization or damage to the samples, which is, therefore, especially relevant for non-destructive imaging of live tissues and biological samples. Furthermore, since the wavelengths of terahertz waves are shorter than those of microwaves, THz imaging systems possess inherent advantages of higher spatial resolution and capability of resolving small features of the target object. Finally, terahertz waves are able to detect the object hidden inside packages owing to the penetration ability to many non-polar materials. This is beyond what can be achieved with imaging systems operating in infrared or visible range. Currently, terahertz imaging can be considered as a complementary method to imaging techniques operating in other frequency bands, providing a powerful method for both scientific research and industrial applications.
Spatial resolution is one of the most critical parameters when evaluating the performance of THz imaging systems. Generally, the resolution of an imaging system at a certain frequency is constrained by the diffraction limit, which inevitably restricts the practical applications when resolving small features of the target is required. In order to break the diffraction limit and further extend the application fields, terahertz modulation and super-resolution techniques have attracted significant research attention. Up to date, a variety of methodologies and configurations have been proposed and developed [16]. One typical example of these techniques is near-field scanning optical microscopy (NSOM) [17,18]. Generally, NSOM utilizes a high-resolution probe to achieve super-resolution imaging by point-by-point scanning in the near field, nevertheless, the mechanical scanning is time-consuming, and the distance between the probe and the target must be strictly controlled. Another alternative which has been demonstrated for sub-wavelength imaging uses spatially controlled light with a spatial light modulator, where the reflected or transmitted light is registered via a single-element detector [19,20]. In one typical example, Stantchev et al. demonstrate a proof-of-concept imaging application for the inspection of a printed circuit board, and a subwavelength resolution of ∼λ/4 is achieved. Such an implementation forgoes the need of mechanical scanning or expensive array detector and facilitates system integration, which has important implications towards practical applications. In addition, this kind of imaging configuration is compatible with compressed sensing methodology, which offers great versatility in various imaging scenarios. The drawbacks of this approach is that it generally requires manually switching or dynamically changing of spatial encoding masks, which not only significantly deteriorates the signal-to-noise ratio, but also causes delays in the image reconstruction.
The advent of metamaterials provides a promising solution for real-time super-resolution imaging. Metamaterials are artificial subwavelength-structures possessing unusual properties that do not exist in natural materials [21,22], such as negative refractive index [23], ultra-high refractive index [24] and infinite anisotropy [25]. The dynamically controllable and reconfigurable properties of engineering metamaterials significantly facilitate novel opportunities for THz wave manipulation and modulation. As an important branch of metamaterials, hyperbolic metamaterials (HMMs) are highly anisotropic materials with hyperbolic dispersion [26,27], which can magnify evanescent waves carrying sub-diffraction information and transform them into propagating waves, thus enabling super-resolution imaging. Generally, there are two typical approaches to construct hyperbolic metamaterials: one is based on a multilayer film structure in which metal and dielectric are alternately stacked [28–31], and the other is an array composed with metallic nanowires [32,33]. Compared with the metallic wire structures, the multilayered metal-dielectric structures are generally simpler in the structure design and fabrication. In fact, in some occasions, fabricating and arranging thin and mental wires into the required pattern is beyond the capabilities of the state of art nanofabrication technologies. Furthermore, multilayered structures are also relatively stable and resilient to changes of the environmental conditions, which is especially important to build imaging platform with better uniformity and lower distortions in the far field. Multilayer hyperbolic metamaterials have been investigated in the UV and visible bands for imaging applications such as nanolithography [34], dark-field hyperlens [35,36] and photoluminescence enhancement [37]. For example, Liu et al. achieved 1.8x sub-diffraction demagnification imaging lithography at 365 nm using Ag/SiO2 hyperlens and introduced the plasmonic reflector layer to improve imaging contrast.
To date, most of the hyperbolic structures reported in literatures are based on metallic materials [38,39]. However, one common drawback of all metal-based metamaterials is high attenuation losses at terahertz frequencies [40]. In contrast, graphene has prominent advantages including smaller attenuation loss (the imaginary part of the permittivity of single-layer graphene at 3 THz is ∼3.5 × 103 [41]), excellent optoelectronic properties and dynamic tunability [42,43], which, potentially, constitutes a better alternative for building hyperbolic metamaterials. Therefore, graphene-based hyperbolic metamaterials have been considered as one of the most promising candidates for achieving super-resolution imaging. For example, Taubner et al. reported a near-field super-resolution of λ/7 using a bilayered graphene structure [44]. Andryieuski et al. constructed a fan-shaped hyperbolic lens using a stack of graphene wires embedded into a dielectric medium, and obtained a subwavelength resolution of λ/5 at 6 THz [45].
Due to these promising advantages of graphene, in this paper, we investigate different types of graphene-dielectric hyperbolic metamaterials for subwavelength imaging applications. The hyperbolic structures feature a periodic sequence of deeply sub-wavelength graphene/dielectric bilayers. Both cylindrical and planar structures have been comparatively studied, and their advantages and limitations have been discussed, respectively. We start by analyzing the hyperbolic dispersion property of graphene-dielectric hyperbolic metamaterials and then compare the super-resolution performance of two structures at different terahertz wavelengths. It is noted that by simply adjusting the chemical potential of graphene without changing the structure of the hyperbolic metamaterial, subwavelength resolutions of λ/10, λ/7.5, and λ/6 have been achieved under the transverse magnetic wave illumination of 100 µm (3 THz), 75 µm (4 THz), and 60 µm (5 THz), respectively. Furthermore, in order to evaluate the robustness of the graphene-based hyperbolic structures, we study the influence of the fluctuations of the multilayer thicknesses on the imaging performance. Finally, as a potential practical application, we apply the hyperbolic metamaterials to super-resolution of an object featuring a periodic grating and randomly distributed particles.
2. Theory and design of hyperbolic metamaterials
2.1 Theory of Hyperbolic metamaterials
Due to the diffraction limit, the resolution of conventional optical imaging systems is fundamentally limited to about half of the incident wavelength. This is because that the evanescent waves carrying sub-diffraction information of the imaging target are exponentially decaying during propagation and cannot reach the image plane. One possible approach to circumvent the diffraction limit is to develop a special lens or structure that allows the evanescent waves to be transmitted from the object plane to the image plane without exponential attenuation. Hyperbolic metamaterials are highly anisotropic materials with hyperbolic dispersion, one of their most remarkable properties is the strong ability to transmit evanescent waves to the far field, which makes them a highly attractive solution to achieve super-resolution imaging. Especially, in such a highly anisotropic media, the parallel and perpendicular components of permittivity have opposite signs and the dispersion relation is hyperbolic.
The dispersion relation of the transverse magnetic (TM) wave is as follows [46]
Fig. 1. (a) The isofrequency contour of natural materials. (b) The isofrequency contour of hyperbolic metamaterials with ${\varepsilon _ \bot } = {\varepsilon _x} = {\varepsilon _y} < 0$ and ${\varepsilon _\parallel } = {\varepsilon _z} > 0$. (c) The isofrequency contour of hyperbolic metamaterials with ${\varepsilon _ \bot } = {\varepsilon _x} = {\varepsilon _y} > 0$ and ${\varepsilon _\parallel } = {\varepsilon _z} < 0$.
2.2 Dynamic tunability of the chemical potential and surface conductivity of Graphene
Graphene is a honeycomb-like two-dimensional material made of a single layer of carbon atoms, which has attracted considerable attention for its excellent optoelectronic properties, especially the dynamic tunability. The chemical potential of graphene can be adjusted by chemical doping or applying electric or magnetic fields. Graphene-based hyperbolic metamaterials are superior to metal-based hyperbolic metamaterials because the former have lower transmission loss, smaller structure size, easier optoelectronic integration and flexible tuning performance after preparation.
The optical parameters of graphene are mainly characterized by its surface conductivity ${\sigma _g}$, which can be calculated using the Kubo formula [41]. The Kubo formula describes the surface conductivity of graphene as the sum of intraband and interband transitions:
Fig. 2. (a) The relationship between the chemical potential ${\mu _c}$ and the applied electric field ${E_0}$. (b) The relationship between the relative permittivity of graphene ${\varepsilon _g}$ and its chemical potential ${\mu _c}$ when the frequency is 3 THz.
2.3 Design of the hyperbolic structures
In what follows, we construct and perform a comparative study of two types of graphene-based multilayered hyperbolic metamaterial structures (i.e., cylindrical and planar geometries).
The schematics of the two hyperbolic metamaterial structures are shown in Fig. 3, both of which are composed of 10 pairs of alternating graphene and dielectric on a TiO2 substrate. We then apply a layer of TiO2 with a thickness of 0.5 µm on the inner surface of the hyperbolic metamaterial structure, and inscribe a pair of lines with a central spacing of 10 µm and a width of 3 µm on it to act as the sub-diffraction imaging object.
Fig. 3. Schematics of the multilayered (a) cylindrical and (b) planar hyperbolic metamaterial structures. The thickness of single-layer graphene ${t_g}$ and dielectric film ${t_d}$ are set to be 0.334 nm and 1 µm, respectively. The permittivity of the dielectric ${\varepsilon _d}$ is set to be 10.2.
For multilayer graphene-dielectric hyperbolic metamaterial structures, when the thickness of each layer is much smaller than the wavelength of the incident light, the effective permittivity can be described as [51]:
Fig. 4. The isofrequency dispersion contours of the graphene-dielectric structure when chemical potentials are 0.4 eV, 0.6 eV, 0.8 eV and 1.0 eV for (a) $f$=3 THz, (b) $f$=4 THz and (c) $f$=5 THz under the TM modes.
3. Investigation of the resolution and dynamic tunability of hyperbolic metamaterials
3.1 Imaging resolution of hyperbolic metamaterials
We then use COMSOL Multiphysics 5.6 software to investigate the resolution of both the cylindrical and planar hyperbolic metamaterial structures. As an preliminary demonstration, transverse magnetic (TM) plane waves with a frequency of 3 THz (corresponding wavelength: 100 µm) are incident perpendicular to the structures in Fig. 3, and the magnetic field direction is along the y-axis. Subsequently, by adjusting the chemical potential of graphene and optimizing the dispersion curves of the hyperbolic structures, we observe super-resolution phenomenon for both cylindrical and planar structures. As shown in Fig. 5(a-b), we present the normalized intensity distributions of the magnetic field mode at the output boundaries of the imaging planes with (blue lines) and without (yellow lines) HMM structures, and the imaging planes are the x-y planes marked in Fig. 3. It can be seen that structures with HMM are able to distinguish the two-line objects, while those without HMM cannot. Figure 5(c) and (d) represent the normalized magnetic field distribution in the y-z plane of the cylindrical and planar HMM structures, respectively. For both cases with HMMs, the two-line object with a separation of 10 µm can be clearly resolved, which suggests a super-resolution of one-tenth wavelength. Especially, the separation of the two lines after passing through the graphene-dielectric hyperbolic structure is 22.61 µm and a magnification effect of ∼2.26x is well observed. The magnification mechanism is can be approximately estimated by the ratio of the radii at the inner and outer boundaries, which can be explained through transformation optics and wave-vector compression theory [52]. We note that the cylindrical structure can amplify the sub-diffraction feature, which provides an effective approach when resolving of finer features of the target object is required. In comparison, the separation of the two-line object at the output boundary of the planar structure is around 11.20 µm, which can be easily rationalized that the planar structure does not possess the amplification mechanism.
Fig. 5. The normalized magnetic field mode intensity distribution at the output boundary with and without HMM for (a) the cylindrical structure and (b) the planar structure. (c-d) The normalized magnetic field distributions of the cylindrical and planar structures are shown in 2D simulations. The chemical potential of graphene is 0.5 eV for the cylindrical structure and 0.3 eV for the planar structure.
3.2 Dynamic tunability of the hyperbolic structures
Apart from the low transmission loss, another advantage of the graphene-dielectric hyperbolic structures is the dynamic tunability of the dispersion and super-resolution performance by adjusting the chemical potential of graphene by conveniently changing the gate voltage without modifying the geometry of the hyperbolic structures. This is especially relevant when super-resolution operation over a broadband frequency range is required. In this section, we present a particular demonstration of super-resolution operation of the hyperbolic structure over a broadband range (i.e., 3-5 THz) by simply adjusting the chemical potential of the graphene. As shown in Fig. 6, we plot the cross sections of a two-line object after passing through the hyperbolic structures when operating at different frequencies. The normalized magnetic field intensity distributions at the output boundaries of the cylindrical structures and planar structures are presented in Figs. 6(a) and (b), respectively. It is well observed that by proper setting of the chemical potential of the graphene layers, one could achieve consistent and comparable super-resolution performance at different frequencies. The dynamic tunability of graphene can potentially forgo the need of complicated geometry optimizations and sophisticated device fabrication, which can be considered as a prominent advantage over other types of materials and has important application aspects in practical implementations.
Fig. 6. The normalized magnetic field mode intensity distributions at the output boundary of (a) the cylindrical hyperbolic metamaterial structure and (b) the planar hyperbolic metamaterial structure when the incident frequency is 3 THz, 4 THz and 5 THz.
3.3 Influence of the bilayer thickness variations on the super resolution imaging capability
We have so far investigated the hyperbolic structures for subwavelength imaging assuming the multilayers are perfectly uniform and periodic, nevertheless, in practical fabrication of thus-designed devices, there might be fluctuations in the thickness of the bilayers in both radial and longitudinal directions. Such influence is even prominent for devices operation in THz ranges due to the relative longer wavelength. In order to verify the influence of the multilayer geometry variation on the imaging performance, we then construct the multilayer structured with randomly generated dielectric thicknesses within a range of ±20% around a specified value (e.g., 1 µm), and repeat the simulations, while all other parameters of the hyperbolic structures remain unchanged. The randomly generated thicknesses of dielectric are illustrated in Fig. 7(a). As a reference, we also present the normalized magnetic field mode intensity distributions at 3 THz with randomly generated bilayers and perfectly periodic bilayers based on cylindrical and planar geometries in Figs. 7(b) and (c), respectively, from which we can make the following observations. First, in both cases, super-resolution imaging capability has been well maintained when certain fluctuations of the structure geometry are introduced. Second, the shape of the resolution curve of the hyperbolic structures with random generated bilayer thicknesses matches well with that based on periodic alternating multilayers, and an object with 10 µm separation can still be clearly resolved. A thickness fluctuation of ±20% in the multilayers only results in a less than 7% variation in the imaging resolution, which suggests great stability and robustness of both the graphene-dielectric cylindrical and planar hyperbolic metamaterial structures.
Fig. 7. (a) The randomly generated thicknesses of dielectric from inner to outer layers. The normalized magnetic field mode intensity distribution of (b) the cylindrical structure and (c) the planar structure with fixed/random dielectric thickness when the incident frequency is 3 THz. In both the insets of (b) and (c), the corresponding geometries are presented. The chemical potential of graphene is 0.5 eV for the cylindrical structure and 0.3 eV for the planar structure. The imaging interval at the output boundary of the cylindrical structure with random thickness is 22.84 µm, which is only 1.02% changed compared to the 22.61 µm with fixed thickness. For the planar structure with random thickness, the sub-diffraction features at the output boundary are spaced by 10.47 µm, which varies by only 6.52% compared to 11.20 µm for the fixed thickness.
4. Demonstration of hyperbolic structures for super-resolution applications
4.1 Super-resolution imaging of a grating structure
As a demonstration, we then apply the planar hyperbolic structure for the imaging of an object featuring as a periodic grating structure. A transverse magnetic wave of 100 µm wavelength is incident perpendicularly to a TiO2 grating with a period of 20 µm, as shown in Fig. 8(a). A thin layer of TiO2 underneath the grating is applied to enhance the coupling between the diffraction grating and the hyperbolic metamaterial. The hyperbolic metamaterial consists of 6 pairs of graphene/Topas bilayers. The Topas layer has thickness of 1 µm and a permittivity of 2.34.
Fig. 8. (a) The schematic of the hyperbolic metamaterial structure for super-resolution imaging. The distributions of the grating interference electric field at the output boundary of the HMM structure when (b) ${\mu _{c}}$ = 0.4 eV, (c) ${\mu _{c}}$ = 0.6 eV and (d) ${\mu _{c}}$= 0.8 eV. (e)-(g) are the OTF passband windows corresponding to (b)-(d). An optimized interference pattern is achieved when the graphene chemical potential is 0.8 eV. The incident frequency is 3 THz.
Figure 8 presents the intensity distributions at the output boundary of the hyperbolic structure and the corresponding optical transfer functions (OTF) calculated by rigorous coupled wave analysis (RCWA) [53,54] when the graphene layer is setting at different chemical potentials. OTF is an index to characterize the imaging quality of diffraction-limited systems under incoherent light illumination, and here to characterize the transmission characteristics of diffraction waves and the periodic grating interference imaging effect.
Under the vertical incidence of TM waves, the diffraction equation can be written as
where $\Lambda $ is the grating period, m is the diffraction order, and ${k_x}$ is the transmitted transverse wave vector. The structure acts as a filter for specific wave vectors, allowing a pair of symmetric diffraction orders of light to pass through the hyperbolic metamaterial and form a uniform grating interference electric field.As shown in Fig. 8(b), when the chemical potential of graphene is 0.4 eV, the period of the interference pattern at the output boundary of the hyperbolic metamaterial structure is ∼8.38 µm, which corresponds to a resolution of λ/11.93, although the imaging pattern is nonuniform. By adjusting the chemical potential of graphene to be 0.6 eV, better uniformity is obtained while the resolution (the period is ∼12.57 µm, λ/7.96) is compromised. Furthermore, when the chemical potential of graphene is set at 0.8 eV, both the uniformity and resolution (period: ∼8.38 µm, λ/11.93) have been optimized, as shown in Fig. 8(d). The modulation mechanism on the pattern uniformity of the structure can be explained by noting that a broader passband of OTF generally introduces diffraction waves of other orders and therefore affects the uniformity of the interference pattern. On the contrary, the structure with narrower passband in the OTF typically shows better uniformity, as is well observed in Fig. 8(e-g).
4.2 Super-resolution imaging of micron-sized particles
THz sensing and imaging of powder analytes has been extensively applied for the discovery of illicit drugs [10], and explosive or hazardous powders [55] via their fingerprint THz spectra. The presented THz super-resolution technique could not only resolve micron-sized powders but also potentially distinguish various powders by performing a multi-spectral imaging in the vicinity of the spectral “fingerprint” absorption lines, which provides dual-modality characterizations (i.e., imaging and identification) of particles. As a preliminary demonstration, we present some simulation results of using the planar hyperbolic structures for super-resolution imaging of micron-sized particles. The full width at half maximum of a single particle and the center-to-center distance between two close spaced particles are then used to evaluate the resolution. As shown in Fig. 9(a), multiple powders (diameter: 3 µm) with random distributions are used as the object for imaging. The diffraction-limited image of Fig. 9(a) using a conventional imaging system is shown in Fig. 9(b), and the super-resolution image reconstructed with current method is shown in Fig. 9(c). For a more quantitative illustration, the normalized intensity profile of a single particle and two closely spaced particles are show in Fig. 9(d-e), from which we find that the FWHM resolutions of a single particle are 9.45 µm and 46.55 µm for super resolution image and diffracted limited image, which indicates that the planar hyperbolic structure realizes a roughly 5-fold improvement compared to diffracted limited systems. Furthermore, closely spaced particles, which is not distinguishable for conventional diffract limited method, can be clearly resolved when employing the developed hyperbolic structure.
Fig. 9. Imaging simulation for randomly distributed particles. (a) 3 µm sized particles. (b) Diffraction-limited imaging simulation. (c) Super-resolution imaging simulation based on the planar structure. (d) Normalized intensity profile along the blue dashed line in (b) and (c). (e) Normalized intensity profile along the red dashed line in (b) and (c). Scale bar: 20 µm.
5. Discussion
In this work, we have studied hyperbolic metamaterial structures for super-resolution imaging and applications. It is demonstrated that both the cylindrical and planar structures can achieve a resolution of λ/10 over a broad spectral range. The main difference is that the cylindrical structure can magnify the sub-diffraction feature with ratio defined by the outer-inner radii, which is beyond what can be achieve with its planar counterpart. Apart from increasing the magnification with higher ratio of the outer-inner radii, the resolution could be further enhanced through appropriate optimization of the filling ratios of the graphene and dielectric components, in order to achieve a flatter dispersion and thereby transporting even higher spatial-frequency information [56].
In fact, the resolution of hyperbolic structures could also be influenced by the geometry, attenuation loss and quality of the multilayers. In particular, by manipulating the geometry of the multilayers, one could further modify the magnification ratio. For example, by employing a multilayer structure with an elliptical configuration as shown in Fig. 10, one could achieve a separation of ∼18 µm for a two-line object with a gap of 10 µm. A magnification ratio of ∼1.8 is obtained, which falls between that of the cylindrical and planar structures. This can be considered as an independent “knob”, which enables tailoring of the resolution and magnification ratio of hyperbolic structures in order to suit for various applications. In fact, there are a few tradeoffs when designing hyperbolic metamaterials. One main tradeoff is the loss of the multilayers and the imaging performance (e.g., magnification, contrast, signal-to-noise ratio) of the metamaterial. On the one hand, constructing a thick multilayer structure can enhance the super-resolution performance and improve the imaging contrast. Nevertheless, increased pairs/thicknesses of multilayers would introduce additional losses and attenuate the useful signal. The intrinsic material loss is unavoidable, the influence is especially pronounced for metallic materials. We note that losses in the hyperbolic materials mainly comes from the graphene, which is dependent on the temperature, the electron relaxation time, the chemical potential, and the operation frequency. By adopting graphene with much smaller attenuation loss and increasing the amount of the dielectric component, one could also optimize the performance of the structures. For example, the thickness of the multilayers can be optimized by combining particle swarm algorithm with rigorous coupled wave analysis [51,57]. Another promising optimization direction is to analyze the causes of the attenuation mathematically across the structure, and develop correction algorithm in order to reduce the adverse effect of the attenuation artifact and improve the signal-to-noise ratio [58]. Another tradeoff is the magnification ratio and geometry simplicity. We note that although the concentric bending layers of the cylindrical hyperbolic metamaterial structure are capable of forming an enlarged image at the image plane by compressing the tangential wave vectors, an inherent disadvantage of such a geometry is that such a configuration is relatively difficult to fabricate. In contrast, the planar structures have advantages of simplicity in geometry design and fabrication, but it does not have a true magnification mechanism. Therefore, these two types of hyperbolic structures can be considered complementary to each other, which enables controlling and targeting specific magnification ratio in predictable way. One interesting direction is to develop “Hybrid-Super-Hyperlens” which combines planar multilayer stacks and spherically curved multilayer stacks [59]. With such a configuration, the former structure performs preliminary magnification and the latter carries out the main magnification. Another interesting implementation is designing of planar hyperlens in association with coordinate transformation and conformal transformation theory, which can realize super-resolution imaging with magnification [60] or reduction [61].
Fig. 10. (a) Schematic of multilayered elliptical HMM structure. The imaging plane is the x-y plane marked by the white dashed line. (b) The normalized magnetic field mode intensity distribution at the output boundary of the elliptical HMM when the incident frequency is 3 THz. (c) The normalized magnetic field distribution in the y-z plane of the elliptical structure is shown in 2D simulation. The chemical potential of graphene is set to 0.7 eV. The sub-diffraction features are spaced at 17.50 µm and magnified by a factor of 1.75, which falls between that of the cylindrical and planar structures.
Furthermore, the dynamic tunability of the dispersion of graphene through conveniently adjusting the chemical potential, can be considered as another independent “knob”, which can be used to tailor the spectral properties in intuitively predictable ways. In fact, another approach to improve the transmission is employing resonant structures [62,63]. Finally, as for the quality of multilayers, in Section 3.3, we have investigated the influence of bilayer thickness variations on the super-resolution performance. It is noteworthy to mention that since the multilayers are operating in THz regimes, one could also resort to rolled-up technology to fabricate the multilayer structures [64–66], which could significantly simplify the manufacturing process and reduce the overall cost.
While these hyperbolic structures offer those prominent advantages detailed above, there are still limitations and problems requiring to be addressed and discussed. Firstly, the cylindrical configuration only provides one dimensional resolution enhancement and a spherical geometry with two-dimensional super-resolution capability is therefore more desirable for practical applications. Nevertheless, the results obtained in this work can be easily extended to two dimensions due to spherical symmetry of the structures. Secondly, another concern is the feasibility of experimental verification of the presented numerical demonstration. We note that the current simulation results only illustrate the proof-of-principle of hyperbolic materials for super-resolution imaging, it is therefore expected experimental realization would further push multilayer-structured metamaterials towards industrial-strength THz imaging applications. In what follows, we then discuss the feasibility of transforming the numerical results to experimental ones. First, we note that the permittivity of dielectric in hyperbolic metamaterials is set to be 10.2 in our simulations, and this value is referenced from the research in Ref. 67 [67]. In further experimental demonstration, sapphire can be considered as a candidate, which is mainly composed of Al2O3 and has a permittivity of 9.3-11.5. In fact, the feasibility of using Al2O3 as the dielectric layer to construct multilayer hyperbolic metamaterials has been experimentally demonstrated in Ref. 28. Finally, we note that the current work mainly focuses on the proof-of-concept of hyperbolic metamaterials and optimization directions for super-resolution imaging applications, and further experimental demonstration and verification will be a subject of our future studies.
6. Conclusion
In conclusion, we investigate the subwavelength imaging performance of graphene-dielectric hyperbolic metamaterials with different structures. These structures all consist of a periodic sequence of graphene/dielectric bilayers. We first analyze the hyperbolic dispersion properties of graphene-dielectric hyperbolic metamaterials, and then compare the super-resolution performance of cylindrical and planar structures at different terahertz wavelengths, and discuss the advantages and limitations of both structures. Due to the excellent dynamic tunability of graphene, by modulating its chemical potential without changing the structure of the hyperbolic metamaterial, resolutions of λ/10, λ/7.5, and λ/6 are achieved at TM wave incidence of 100 µm, 75 µm, and 60 µm, respectively. In addition, we demonstrate the stability and robustness of the cylindrical and planar structures by introducing fluctuations to the dielectric thickness. Finally, we apply the planar hyperbolic metamaterial structure to super-resolution imaging of objects with a periodic grating, and obtain a relatively uniform grating interference resolution of ∼λ/12 by optimizing the chemical potential of graphene. As another demonstration, we employ the hyperbolic structure for the imaging of randomly distributed particles, and five-fold lateral resolution enhancement is achieved. We believe this work will be especially relevant to the resolution improvement of terahertz imaging systems and the development of hyperbolic metamaterial modulation devices in the terahertz band.
Funding
NSFC-BRFFR (12111530284); China Postdoctoral Science Foundation (2021M700039); National Natural Science Foundation of China (11904135); China Scholarship Council (202106795004); Open Fund of State Key Laboratory of Millimeter Waves (K202229).
Acknowledgments
The authors would like to thank Mr. Jinglei Hu and Mr. Kaixiang Chen for fruitful discussions and the support from the Belarusian Republican Foundation for Fundamental Research (Project No. F22KI-016).
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.
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