1. Introduction

Conventional lens images objects by exploiting a proper phase transformation to the propagating waves, focusing them at a certain distance away from the lens. The significant factors that debilitate the conventional lenses performance are chromatic and spherical aberrations; due to the material frequency-dependent refractive index, and variation of focus with aperture [1,2]. However, the fundamental intrinsic restriction toward subwavelength imaging comes from the diffraction limit [3]. Due to diffraction, spatial information smaller than one-half of the wavelength cannot propagate to the far filed and, will spread out [4]. One approach to realize subwavelength resolution imaging is superlens [5], which utilizes negative refractive index materials to amplify the evanescent fields [5–8]. However, due to the resonant nature of this amplification, observation of the subwavelength features has not been possible with the proposed lossy materials [9–11]. Addressing this bottleneck, hyperlenses based on the materials with anisotropic optical permittivity can overcome the energy spreading associated with the diffraction by compressing the space into the narrow direction [11–13]. Hyperlens is named for materials with hyperbolic dispersion relation [11], with the negative dielectric permittivity parallel to the optical axis, and positive permittivity perpendicular to it.

3D NMLH materials, PhC, metamaterials and stacks of 2D materials in the forms of the flat slabs, curved shape and obliquely cut slabs have been suggested to implement hyperlenses. Among these attempts, bandwidth of PhC-based hyperlens is heavily limited by the interface structure periodicity size [14]. Besides, it has been shown that 3D homogeneous NMLH materials display hyperbolic dispersion relation between the plasma frequencies corresponding to the differences in effective masses in different directions. Unfortunately, no known 3D homogeneous NMLH material exhibits anisotropy exceeding 30% at the optical or infrared spectral range [15]. Hyperlenses of metamaterials utilizing anisotropic nanoplasmonic of layered system and aligned wire structure [16] have been reported. Obliquely cut [13] and cylindrical shape [11] of layered metamaterial hyperlenses have been successfully implemented with applications in photolithography [17], high-throughput imaging with lens array [18], real time imaging at visible frequency [19], spectroscopy [20], and virtual scanning tip imaging [21]. However, these curved forms of magnifying and demagnifying hyperlenses also have drawback in terms of application and implementation. A simple design of flat hyperlens is polishing the input and output surface of a cylindrical configuration into arbitrary desired shapes [22]. Nonetheless, the required fabrication process seems difficult and leads to deformation in image [23]. Another alternative flat-to-flat imaging structure consists of two geometric parts, designed in two different curvilinear orthogonal coordinate systems [24]. However, the magnification ratio on the entire output plane is not identical [23]. Following that, a complicated mathematical method for uniform demagnifying planar hyperlens through the conformal transformation theory was utilized [25]. Also, metamaterials-based hyperlenses operating at visible wavelengths, such as silver/GaAs hyperlenses [18,26], are partially lossy, and have image distortion due to the layered nature of the metamaterial configuration.

2D materials with anisotropic optical permittivity and hyperbolic dispersion in the UV spectrum, e.g., graphene and h-boron nitride, have been also studied as the basis to implement hyperlenses [12,27]. However, practical fabrication of the suggested magnifying structures goes into question. Chinese fans or cylindrical configurations which have been proposed in several articles [11–13,27], seem not to be practical, due to the fact that the layers have to be aligned into the radial direction. As the radius of curvature increases, more vacuum space will be inducing between these planes. Hence, the optical properties of a Chinese fan configuration change in the radial direction. Besides, since the sheets have to be located with a gradual spacing to produce curvature edge, the fabrication of a Chinese fan with stack of 2D materials cannot be possible.

Here, we utilize two carbon and silicon passivated porous graphene phases, CPG and SiPG, with hyperbolic dispersion relation at the visible spectrum. Exploiting these 2D materials as an oriented hyperlens configuration can resolve impractical flaw related to the radial placement of the 2D layers in Chinese fan. Also, the proposed 2D materials fill the gap of 2D hyperlenses operating at the visible spectrum. In addition, we demonstrate that homogonous 2D materials solve the problem of undesirable image distortion that the metamaterial-based hyperlenses have due to inhomogeneity of the medium. Furthermore, 3D homogeneous NMLH material has change in the sign of permittivity for directions parallel and perpendicular to the optical axis, due to plasmon resonance frequencies in these directions. These changes in the sign of permittivity limits the bandwidth of hyperbolic range. However, our proposed 2D hyperlenses does not suffer from bandwidth limitation, resulting from the change of permittivity sign perpendicular to the optical axis.

We demonstrate oriented hyperlenses with a stack of CPG/SiPG layers at visible wavelength. For the graphene-based materials, transferring the 2D sheets to a macroscopic structure without losing their properties is crucial. For this, we model the CPG fabrication process from a cyclohexa-m-phenylene (CHP) network in molecular-dynamic (MD) framework. The proposed fabrication process of the oriented hyperlens is challenging, but would be achievable by implementing selective graphene-based layers patterning [28,29], and lift-off process of few layers’ graphene phases material [30,31] on the desirable substrate with 45 degrees slop.

The CPG and SiPG geometry parameters have been optimized exploiting density functional theory (DFT). According to the obtained geometric parameters for the monolayers and stacks of CPG and SiPG, the dielectric functions are computed. Then, the obtained optical spectrum is employed to evaluate the designed flat and oriented hyperlenses performance, through the electromagnetic (EM) simulation with a finite element method (FEM) numerical approach. Our design can fulfill real time subwavelength imaging at visible spectrum with potential applications in biology, material science, and chemistry.

2. Methods of analysis

In this section, the numerical calculations and theoretical model exploited in this context are presented. Molecular dynamic (MD) simulation details utilized for the CPG fabrication process are explained. DFT calculations basis, hybrid functionals, and parameters utilized in the geometrical optimization and optical spectrum analysis are presented. Moreover, the theoretical model in the dielectric function investigation is described.

2.1 2D materials structure

The initial structure to fabricate CPG is based on the CHP network (Fig. 1(a)), which its synthesis has been reported recently [32]. Considerable experimental progresses have been achieved in selective dehydrogenation of hydrocarbons. Thermally induced Pt (111) [33], Cu (111) [34] surface assisted dehydrogenation processes, and even a low temperature phosphorus modified Pd (111) [35] surface assisted dehydrogenation reaction have been developed. The same fabrication processes would be also possible for the selective dehydrogenating of the CHP network and substituting one carbon atom instead of each two adjacent hydrogen atoms (Fig. 1(d)). In this regard, utilizing MD simulation, we find out that during the chemical vapor deposition (CVD) process, the methylidyne radical (CH^3• or CH) compounds can passivate dangling bonds (DBs) of the porous graphene (i.e., dehydrogenated CHP) (Fig. 1(b)).

 

Fig. 1. Top view of (a) CHP network optimized structure with lattice constant of 7.508 Å, (b) CHP after dehydrogenation and passivating DBs by CH compounds during the CVD process at 464 °K, (c) optimized structure of configuration b with lattice constant of 7.362 Å, and (d) selective dehydrogenation and optimization of configuration c (namely CPG) with lattice constant of 7.249 Å. Dark blue and cyan balls in geometrical models represent carbon and hydrogen atoms, respectively.

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For this purpose, the Nose Hoover [36] MD simulations is done in the NVT canonical ensemble, where the number of particles (N), the volume (V), and the temperature (T) are conserved. The required potential parameters are taken from the optimized Tersoff and Brenner empirical interatomic potentials models [37,38]. A random distribution of the CH compounds are deposited to the dehydrogenated CHP network. Until then, results indicate that the CHP network should reach equilibrium at a random initial velocity of the atoms resulting from the Maxwell-Boltzmann distribution at the temperature of 460$^\circ \textrm{K}$ to $470^\circ \textrm{K}$. To avoid the gravity and repulsion interactions between the CH molecules, the deposition process is done at a very low pressure (one unbonded CH molecule in the unit cell). This condition provides the situation that CH molecules passivate the decussate DBs (Fig. 1(b)). This happens due to the interatomic forces, and brings about the structure relaxation in the desirable configuration represented in Fig. 1(c). It is worth mentioning that this CVD process is examined for carbide (C^4•), methylene (CH₂^2•) and methyl (CH₃^•); however, none of them have the same behavior as CH.

During the CVD process, passivation of DBs by the CH components occurs, thus the nanostructure comes out of the ground state (see Fig. 1(b)). Geometric optimization has been done by the L-BFGS algorithm [39], to minimize the forces in the system and finding a stationary point (see Fig. 1(c)). This process continues until the values of tolerances for force, stress, step size, and self-consistent field convergence become less than 10⁻³ eV.Å⁻¹, 10⁻⁴ eV.Å⁻³, 10⁻³ Å and 10⁻⁶ eV, respectively. Subsequent selective dehydrogenation of the sample and applying the same geometric optimization as the previous step leads to the corresponding CPG structure (Fig. 1(d)).

2.2 DFT computational details

Atomic evaluation for the derivation of geometrical and optical characteristics is performed with DFT, using the generalized gradient approximation (GGA) function [40]. Hubbard GGA + U standard semi-empirical corrections is applied to modify electron self-interaction and poor description of conduction-band energy levels of the local exchange-correlation functionals [41,42]. The functional of GGA associated with the revised Perdew-Burke-Ernzerh (RPBE) approximation reduces overbinding of the chemisorbed atoms by about a factor of two compared to the PBE mode [43]. The developed SG15 dataset of the optimized norm-conserving Vanderbilt pseudo-potentials [44,45] is used for each element. The SG15 with 100 Hartree density mesh cut-off using for carbon, hydrogen and silicon indicates precision as high as a state-of-the-art all-electron calculation [46]. A Monkhorst-Pack grid [47] k-point Г-centered sampling of $251 \times 251 \times 1$ is chosen, which determines the mesh size used for the integration across the Brillouin zone [12].

2.3 Optical properties

We use the Kubo-Greenwood formula to extract the susceptibility tensor [48,49]:

The calculation of the dielectric function is performed with a k-point sampling of $800 \times 800 \times 4$, 36 valence and 24 conduction bands, and a Gaussian broadening of 0.02 eV.

3. Result and discussion

The unit cells of the CHP network and CPG hyperlens are analogous to a $3 \times 3$ graphene supercell, represented in Fig. 1(a) and 1(d), respectively. These unit cells (in which A and B are the primitive vectors) are located in the xy planes with a 20 nm interval along the C vector (z axis) to prevent their interaction. While CPG structure is planar, C₆H₃ components of the CHP network are tilted 10.47 degree with respect to the sheet plane (Ref. [50] declares 10.45 degree). The optimized lattice constant of graphene, CHP network and CPG are 2.461 Å, 7.508 Å and 7.249 Å, respectively. The obtained lattice parameters of the CHP network are well close to the theoretical results of Ref. [50], while the experimental lattice constant of 7.4 Å is reported [35]. It has been reported recently that the Si atoms can stabilize graphene nanopores by passivating the DBs around the perimeter of the holes (Fig. 2(d) inset) [51]. Experimental and theoretical evidences suggest that this condition is durable in the ambient atmosphere and liquids. This pore stabilization can be also understood based on the fact that the Si atoms incline toward tetrahedral coordination. Thus, those C adatoms that are bonded to the Si atoms would form dendrites outside the graphene plane, effectively preventing the self-healing process [51]. This feature of SiPG and superior optical properties of CPG stimulate our interest to select them for implementing the hyperlens. Applying the same computation basis results in the SiPG lattice parameter of 7.609 Å.

 

Fig. 2. The anisotropy permittivity tensor of graphene, CHP network, CPG and SiPG monolayer in upper row (a-d) and AB-stacking periodic layers in lower row (e-h), respectively. “${\parallel} $” represents x and y components and “$\bot $” represents the z component. The insets show the primitive lattice vector A (red), B (orange) and C (blue) of atomic unit cell considered for dielectric constant calculations. (e) In graphite, the real part of ${\varepsilon _\parallel }$ is negative in the range of 4.1-6.5 eV (302-190 nm) and ${\varepsilon _ \bot }$ remains positive during this range. (f) Similarly, in CHP network-stack the real part of ${\varepsilon _\parallel }$ is negative between 3.5 eV and 5.5 eV (354-225 nm) where ${\varepsilon _ \bot }$ is positive. (g) The same situation for CPG-stack happens in the range of 2-3.9 eV (620-318 nm) and (h) for SiPG-stack in 1.85-2.5 eV (670-495 nm). In the inset images at lower row, A-type layers are depicted brighter than B-type layers for AB-stacking sheets pattern.

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In order to implement a hyperlens, materials with anisotropic dielectric functions are needed. Graphene-based configurations, due to their 2D structures, possess significant anisotropy in three dimensions, which are appropriate for the implementation of the hyperlenses. The dielectric function tensor of these structures has two independent components, ${\varepsilon _\parallel } = Re({{\varepsilon_\parallel }} )+ i\; Im({{\varepsilon_\parallel }} )$ and ${\varepsilon _ \bot } = Re({{\varepsilon_ \bot }} )+ i\; Im({{\varepsilon_ \bot }} )$, parallel and perpendicular to the 2D material sheets, respectively. Due to the symmetric geometry of these systems along the x- and y-axes here (see Fig. 2), the optical spectra are isotropic for the light polarization along these axes. ${\varepsilon _\parallel }\; $ and ${\varepsilon _ \bot } $ related to the graphene, CHP network, CPG, and SiPG single sheet are illustrated in the first row of Fig. 2, which are similar to the results of Ref. [52] for graphene, and Ref. [50] for the CHP network. However, in order to implement the hyperlens, characteristics of a single sheet is not adequate. As mentioned, unit cells of the CHP network, CPG and SiPG 2D materials are similar to the graphene’s unit cell but multiplied three times along A and B primitive vectors. Geometric optimization result of AB-stacking sheets pattern of graphite, CHP network-, CPG- and SiPG-stack with the same procedure described in DFT computational details are depicted in Figs. 2(e)–2(h) insets, respectively. It is worth mentioning that in these figures, A-type layers are depicted brighter than B-type layers for AB-stacking sheets pattern. The 2D layers are periodically located along the z-axis in graphite, CHP network-, CPG- and SiPG-stack with a distance of 6.709, 7.384, 6.472 and 7.238 Å, respectively. As a result, the optical spectrum for these stacks have been shown in Figs. 2(e)–2(h). The optical couplings between the layers change the absolute values of the dielectric functions in stack structures. According to the dielectric function of the CPG-stack in Fig. 2(g), the real part of ${\varepsilon _\parallel }$ is negative between 2 eV and 3.9 eV (corresponds to the wavelengths between 318 nm and 620 nm), while the real part of ${\varepsilon _ \bot }$ is positive in the mentioned range.

Also, from the dielectric function in Fig. 2(h), it can be seen that the SiPG-stack shows the same behavior as CPG-stack in the visible range between 1.85 eV and 2.5 eV (corresponds to the wavelengths between 495 nm and 670 nm). However, in the graphite and the CHP network-stack, the real part of ${\varepsilon _\parallel }$ is negative in the UV spectrum (and the real part of ${\varepsilon _ \bot }$ is positive) (see Figs. 2(e) and 2(f)). This means that the desirable optical behavior of graphite in UV is shifted toward the visible spectrum in the CPG- and SiPG-stack.

The high-frequency dielectric function of a (semi-)conductor material containing a significant amount of free electrons or holes is typically governed by the dynamics of the free charge carriers. The resulting response is plasma-like, with the dielectric function being described by the Drude model $\varepsilon (\omega )= {\varepsilon _\infty } - \omega _p^2/\omega ({\omega + i\Gamma } )$ [15,53,54]. Where the dielectric constant of the background medium ${\varepsilon _\infty }$ is the (frequency-independent) contribution of the bound electrons, and the Drude broadening $\mathrm{\Gamma }$ is responsible for the EM losses due to inelastic processes. The plasma frequency ${\omega _p}$ is described as $\sqrt {N{e^2}/{m_{eff}}} $, where N, e, and ${m_{eff}}$ stand for the free-electron concentration, charge, and effective mass, respectively. It can be derived immediately that the effective permittivity is negative for $\omega < {\omega _p}/\sqrt {{\varepsilon _\infty }} $. Our calculation indicates that the electron concentration of CPG is increased about four times (compared to the graphene) through the porosity process. On the other hand, the same computation shows that the increase of the effective mass is much more; hence, has a greater effect than the electron concentration for the porous graphene-based materials in this study. This justifies the larger/smaller value of ${\omega _p}$ for the graphene/porous materials. These findings show that both CPG and SiPG configurations possess useful anisotropy properties in the visible wavelength spectrum, compared to the other 2D atomic structures, which is appropriate for the implementation of the hyperlenses.

In order to investigate EM response of the hyperlens, we have considered a slab with permittivity tensor $({{\varepsilon_x},{\varepsilon_y},{\varepsilon_z}} )$, illuminated by a transverse-magnetic (TM) polarized plane wave, as shown in Fig. 3. The dispersion relation in an isotropic medium such as a vacuum slab can be described as $k_x^2 + k_z^2 = {\varepsilon _0}{\omega ^2}/{c^2} = k_0^2$, where ${k_x}$ and ${k_z}$ stand for the wave vectors along the x and z axes, respectively, and ${\varepsilon _0}$, $\omega $, c and ${k_0}\; $ are permittivity of free space, angular frequency, speed of the light in vacuum, and free space wave vector, respectively. Thus, regarding to the wave vectors components, the $k$-space dispersion is bounded to a circle (see Fig. 3(b), green curve). When ${k_x}$ is larger than the wave vector ${k_0}$, ${k_z}$ becomes an imaginary number. In this situation, the waves propagating along the z axis decay exponentially and become evanescent. Hence the subwavelength information, which are carried by the waves with ${k_x} > {k_0}$, decays and cannot reach the far field. However, in an anisotropic medium with different (or similar) sign of the permittivity components (i.e., ${\varepsilon _x}$ and ${\varepsilon _z}$), the dispersion relation represents hyperbolic (or elliptical) curve as $k_x^2/{\varepsilon _z} + k_z^2/{\varepsilon _x} = {\omega ^2}/{c^2}$, assuming ${\varepsilon _x} > 0$, ${\varepsilon _z} < 0$ (or ${\varepsilon _x} > 0$, ${\varepsilon _z} > 0$), as shown in Fig. 3. In the hyperbolic curve, for any value of ${k_x}$, there is a real value for ${k_z}$ and consequently the subwavelength information can transfer to the far field. In this case, if ${\varepsilon _z}$ has larger value compared to ${\varepsilon _x}$, the dispersion curve will be roughly flat, i.e., ${k_z}$ is almost constant and the image and the object only differ by a phase [11,12].

 

Fig. 3. (a) Layout of incident plane wave upon a medium slab with thickness d; (b) Dispersion curve for isotropic medium (green circle), anisotropic medium with ${\varepsilon _x} > 0$ and ${\varepsilon _z} > 0$ (red ellipse) and anisotropic medium with ${\varepsilon _x} > 0$ and ${\varepsilon _z} < 0$ (blue hyperbola).

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Recently formation of the folded graphene sheets by mechanical deformation [55], 2D materials transferring to an arbitrary substrate [56], and four different stacking styles for the folded graphene nanoribbons [57], have been reported. Taking into account these techniques to implement CPG- and SiPG-stack hyperlenses, a flat lens EM evaluation is done using FEM. In Fig. 4(a), the field distribution H_y is calculated from Maxwell’s equations in the absence of hyperlens with the vacuum background. Two semicircle sources with TM polarization are located at the top of the designed flat lens with the separation of 118 nm. In Fig. 4(b), CPG sheets are aligned perpendicular to the x-axis in the slab with a thickness of 140 nm. The CPG-stack have the permittivity of ${\varepsilon _\parallel } ={-} 20.755 + i\; 10.091$ and ${\varepsilon _ \bot } = 0.062 + i\; 1.502$ at the operational wavelength of 560 nm. Figure 4(b) demonstrates the field distribution H_y of the designed CPG-stack flat lens. By comparing the field distribution in Figs. 4(a) and 4(b), it can be inferred that the evanescent waves in the vacuum (see Fig. 4(a)) are transformed to the propagating waves in the hyperlens (see Fig. 4(b)). The field intensities at the imaging plane (i.e., the bottom surface of the slab) for the vacuum, and the CPG-stack based hyperlens are depicted in Fig. 4(f). The intensity of two-point sources is discernible at the imaging plane after propagating through the hyperlens (see Fig. 4(b)), while it cannot be distinguished for the vacuum space (see Fig. 4(a)). This effect is also confirmed by the normalized time-average power flow arrows shown in Figs. 4(a)–4(b). These arrows are representative of Poynting vector in which absolute value along negative direction of z-axis have been considered. According to these flow arrows, it can be seen that the optical power in the vacuum is scattered through both x and z axes (see Fig. 4(a)), while it is passed straightly through the z-axis in the CPG-stack lens (see Fig. 4(b)).

 

Fig. 4. (a) H_y field distribution of flat lens configuration with vacuum medium. The thickness of medium is 140 nm and two semicircle sources located along the x-axis on the top of the lens separated with the distance of 118 nm. (b) H_y field distribution with CPG-stack as medium in which ${\varepsilon _\parallel } ={-} 20.755 + i\; 10.091$ and ${\varepsilon _ \bot } = 0.062 + i\; 1.502$. (c) H_y field distribution with a SiPG-stack in medium region with permittivity of ${\varepsilon _\parallel } ={-} 3.453 + i\; 9.407$ and ${\varepsilon _ \bot } = 2.765 + i\; 1.580$. (d) H_y distribution of silver/GaAs metamaterial hyperlens consists of 14 nm thickness of layers with the permittivity of ${\varepsilon _{silver}} ={-} 13.524 + i0.412$ and ${\varepsilon _{GaAs}} = 16.127 + i2.052$ at wavelength 560 nm. (e) H_y distribution of silver/GaAs metamaterial hyperlens consists of 14 nm thickness of layers with the permittivity of ${\varepsilon _{silver}} ={-} 11.046 + i0.332$ and ${\varepsilon _{GaAs}} = 17.418 + i2.695$ at wavelength 520 nm. The black arrows indicate time-average power flow through the system with absolute value along negative direction of z-axis. (f) The comparison of image intensity for vacuum, CPG-stack, SiPG-stack, silver/GaAs metamaterial at wavelength of 560 nm and 520 nm, respectively. (g) and (h) x-component of electric field intensity at the imaging plane of flat hyperlenses at wavelength of 560 nm and 520 nm, respectively.

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The performance of SiPG-stack hyperlens is evaluated in Fig. 4(c) at the wavelength of 560 nm, where the SiPG-stack have the permittivity of ${\varepsilon _\parallel } ={-} 3.453 + i\; 9.407$ and ${\varepsilon _ \bot } = 2.765 + i\; 1.580$. At this wavelength, the permittivity ${\varepsilon _\parallel }$ has a much larger imaginary part compared to its real part, which it means that the wave is partially dissipated during propagation within the SiPG-stack. The x- and z-component of Poynting vector are the product of -E_z${\times} $H_y and E_x${\times} $H_y, respectively. Minor deviation of arrows toward x-axis in Fig. 4(c) stems from the considerable value of E_z in comparison to E_x. This fact indicates that SiPG-stack flat lens has not been successful in completely propagating the signal along the z-axis.

To better understand the advantage of the proposed CPG-stack hyperlenses, we compare its performance with a well-known metamaterial based hyperlens, composed of alternating layers of silver and GaAs with a thickness of 14 nm [18], at the wavelengths of 560 nm and 520 nm, where the dielectric permittivity of silver is $- 13.524 + i0.412$ and $- 11.046 + i0.332$ [58], respectively, and the permittivity of GaAs is $16.127 + i2.052$ and $17.418 + \; i2.695$ [59] respectively. Figures 4(d) and 4(e) show the H_y field distribution respectively at the wavelengths of 560 nm and 520 nm, for the silver/GaAs metamaterial hyperlens. The same explanation for Poynting vector deviation from z-axis as what described for SiPG-stack flat lens is true for both cases of metamaterial-based hyperlenses.

Figure 4(f) compares the field intensity of the CPG-stack, SiPG-stack, and silver/GaAs hyperlenses on the imaging plane. According to the Rayleigh criterion, point sources will be resolved when their field intensity distributions overlap at half their distance with an intensity of ≤75% of the maximum intensity [1,60]. It can be seen that at source resolution of 118 nm, just CPG-stack flat lens follows the Rayleigh criterion and provides super resolution (i.e., has ≥25% intensity drop at sources intensity interval).

Since the metamaterial layers thickness is much less than the wavelength, we can consider this slab as a material with effective medium as follow [11]:

Table 1. Optical Constants of Materials

View Table

It is worth mentioning that both power dissipation, and the ratio of anisotropy (i.e., $real[{{\varepsilon_\parallel }(\omega )} ]/real[{{\varepsilon_ \bot }(\omega )} ]$) [61,62] affect the resolution and performance of hyperlens. Therefore, the power dissipation density in x-z plane should be taken into account. The power dissipation density at any given wavelength and location inside the structure is related to the electric field components (E_x, E_y, and E_z) as follows [63]:

Traditional lens transforms light at the boundary of two media under the Snell’s law, which is a function of refractive index and incident angle [1,4]. As stated in the introduction, in traditional lenses, resolution is limited by geometrical and chromatic aberrations. Geometrical aberration alters the focal point of incident rays at different heights due to refraction angle changes through surface curvature. However, the mechanism of hyperlens is guiding light through the certain direction and surface curvature does not affect the propagation. Chromatic aberration happens due to the wavelength dependency of optical permittivity, which exists in both cases. In hyperlens, anisotropy ratio (i.e., $real[{{\varepsilon_\parallel }(\omega )} ]/real[{{\varepsilon_ \bot }(\omega )} ]$), that affects the guiding direction, is altering with operational wavelength, and limits the resolution.

Figure 4 represents the evaluation of non-magnifying behavior of the different medium in flat slab configuration. What is considered in this case is determining the lowest amount in resolution based on the Rayleigh criterion for the propagating wave in the materials without attention to magnification. In other words, this observation indicates the ultimate resolution between the different cases of materials without magnification that considered to study the propagation response. It should be emphasized that although ${\varepsilon _\parallel }$ of CPG-stack has an imaginary part comparable to that of SiPG-stack (see Table 1), the final field peaks are distinguishable. That is because the ratio of $real[{{\varepsilon_\parallel }(\omega )} ]/real[{{\varepsilon_ \bot }(\omega )} ]$ for CPG-stack is much larger, which is one of the main parameters affecting the resolution quantity as mentioned in this context. On the other hand, the supremacy of CPG-stack over the metamaterial is due to the high imaginary part of metamaterial permittivity through the propagation direction. Which means although this metamaterial has considerable ratio of $real[{{\varepsilon_\parallel }(\omega )} ]/real[{{\varepsilon_ \bot }(\omega )} ]$ and guiding the light mainly toward the z-axis, the final transmission suffers from the huge loss. Figure 4(g) and 4(h) illustrate the x-component of electric field intensity at the image plane, normalized to the maximum case at each operational wavelength, in order to consider the loss effect on the intensity at image plane. Figure 4(h) indicates that CPG-stack does not comply the Rayleigh criterion like its counterpart at 560 nm wavelength. The reason for this observation is that despite the fact that CPG-stack at 520 nm has smaller imaginary part of permittivity regarding to that at 560 nm through the propagation direction (i.e., ${\varepsilon _\parallel }$), it has smaller ratio of $real[{{\varepsilon_\parallel }(\omega )} ]/real[{{\varepsilon_ \bot }(\omega )} ]$. Accordingly, CPG-stack at 520 nm partially disperses the light power to other direction, thus declining the final resolution.

Due to the fact that the flat lens cannot provide magnification, the change in the geometry of the structures is investigated. An approach is to implement oriented stacks with angle α (respect to the z-axis), shown in Figs. 5(b). In Figs. 5(a)–5(e), similar to Figs. 4(a)–4(e), two semicircular sources are located on top of the lens with a distance of 1 nm, at the operational wavelength of 560 nm. Here, the orientation angle α is considered 45°, and the thickness of the lens is assumed to be 350 nm, large enough to create resolution in the order of wavelength on the imaging plane. The H_y field distribution for vacuum, CPG- and SiPG-stack oriented hyperlenses, are shown in Figs. 5(a), 5(b), and 5(c), respectively. For the both CPG- and SiPG-stack hyperlenses, the far-field intensity from the two sources in the imaging plane at the bottom surface of the slab is distinguishable. Despite a single peak image intensity in the case of vacuum, two distinct peaks exist for the CPG- and SiPG-stack hyperlenses, illustrating 1 nm resolution with a high precision. In these cases, high permittivity anisotropy leads to approximately straightforward travelling of signal through the hyperbolic media. Therefore, objects will be magnified without limitation on the resolution.

 

Fig. 5. Simulation results of hyperlens based on CPG-stack and SiPG-stack at the wavelength of 560 nm. The distance between the two semicircular sources is 1 nm. The hyperlens thickness is 350 nm and orientation angle of stacks relative to the z-axis at each side of lens (α) is 45°. (a)-(c) represent H_y field distribution of two sources propagating through vacuum, CPG- and SiPG-stack, respectively. (d) H_y distribution of silver/GaAs metamaterial hyperlens consists of 15 nm thickness layers at wavelength 560 nm. (e) H_y distribution of silver/GaAs metamaterial hyperlens consists of 15 nm thickness layers at wavelength 520 nm. (f) compares the H_y intensity in the imaging plane for these media. (g) and (h) x-component of electric field intensity at the imaging plane of oriented hyperlenses at wavelength of 560 nm and 520 nm, respectively.

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Figures 5(d) and 5(e) show the field distribution H_y at the wavelength of 560 nm and 520 nm, respectively, for the silver/GaAs stack hyperlens with 45° orientation from the z-axis (and 15 nm thickness of the layers). It is apparent that layered structure of metamaterial brings about nonuniformity in EM wave propagation. Figure 5(f) compares the field intensity of the oriented CPG-stack, SiPG-stack, and silver/GaAs stack hyperlenses (with an orientation angle of 45°) on the imaging plane. A distortion in the image field intensity can be seen for the oriented silver/GaAs metamaterial hyperlens. It should be noted that Fig. 5(f) depicts each of the field intensities normalized to unity, in order to examine the Rayleigh criterion, and field intensity drop. However, Fig. 5(g) and 5(h) illustrate the x-component of electric field intensity at the image plane, normalized to the maximum case at each operational wavelength of 560 nm and 520 nm, respectively.

According to Table 1, the metamaterial at wavelength of 520 nm has smaller imaginary part in the propagation direction (i.e., $Im({{\varepsilon_ \bot }} )= 17$) in comparison to its counterpart at wavelength of 560 nm (i.e., $Im({{\varepsilon_ \bot }} )= 75$). Thus, silver/GaAs metamaterial hyperlens shows proper resolution in this case at 520 nm wavelength, compared to that at 560 nm wavelength. However, Fig. 5(h) confirms that metamaterial hyperlens at 520 nm wavelength suffers from higher loss than the other hyperlenses in this wavelength. But due to its relatively high ratio of anisotropy exhibits proper resolution. The main points that hold in this study are that metamaterials are lossy medium due to the metal component (see Table 1) and exhibit field distortion in oriented configuration (see Figs. 5(f-h)). Inhomogeneous nature of the metamaterial causes multiple reflection at the boundaries which brings about field distortion at oriented cut at image plane. Also, it can be inferred from Table 1 that at both operational wavelength of 560 nm and 520 nm, CPG- and SiPG-stack have smaller permittivity imaginary part, which will result in smaller loss (see Fig. 5(g) and 5(h)). Another factor that impresses the field intensity is the amount of reflection at the boundary of sources with permittivity of 1 and hyperlens. While CPG-stack at wavelength of 560 nm has refractive index of 1.07, less light reflects back from the boundary.

4. Conclusions

In summary, we utilized two carbon/silicon passivated porous graphene phases, CPG/SiPG, for hyperlens implementation. The anisotropy permittivity of CPG and SiPG in the case of monolayer and periodic stack layer structures are shown in the visible spectrum range with the framework of density functional theory (DFT), the GGA functional, the RPBE method and the SG15 potential. The porosity in these materials provides capability to overcome the diffraction limit and achieving high resolution in the visible spectrum range through an oriented hyperlens. The results are compared with those of well-known silver/GaAs metamaterial based hyperlenses. The proposed approach toward the homogeneous hyperlens design does not encounter with deficiency of the PhC, 3D NMLH material, metamaterial and other types of the 2D material based hyperlenses.