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Abstract
A new plasmonic nanofocusing metalens based on aperiodic silica grating arrays was designed and investigated. Assisted by the graphene surface plasmon, the infrared polarized light can be focused in a nanospot with a dynamically controlled focal length by varying the dielectric strip width or the graphene Fermi level Ef. For instance, with λ0 = 8 µm and Ef at 0.3, 0.6 and 0.9 eV, focal lengths of 4.5, 3.8 and 3.5 µm with its corresponding FWHM of 64, 232 and 320 nm, respectively, can be realized. The variation of the focusing efficiency with respect to the incident wavelength and the Fermi level were also investigated. The results of theoretical analysis based on light differential equations agree well with the finite element analysis simulation, which further validate the model.
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1. Introduction
Based on the modulation of the optical properties of metasurfaces, various optical functionalities have been demonstrated, such as focusing [1], absorber [2,3], wave plates [4], beam steering [5], and holograms [6]. Metalic metasurface based plasmonic lenses with varying shapes and geometries including nanohole arrays [7–9], nanoslit arrays [10,11], ring-shaped nanostructures [12], hollow nanocone superstructures [13], multicircular slit apertures [14], dipole antennas [15], etc. have been proposed in previous research.
However, despite the strong field enhancement, the resonance nature of metal structures also renders narrow-band response of the metallic plasmonic devices. In order to achieve broadband or multiband performance of the plasmonic lens, graphene material is applied to the design of plasmonic metalens. Graphene, as a two-dimensional material consisting of a monolayer of carbon atoms, has attracted significant interests in the infrared plasmonic devices due to its gate-controlled Fermi level and the corresponding surface conductivity. Attempts have been made to combine graphene and metallic metasurfaces to achieve active modulation of plasmonic devices [16–21]. For example, Ullah et. al. proposed a metalens consisted of C chapped aperture antenna array covered by monolayer graphene [16]. This metalens shows high efficiency (up to 30.4%) tuning mechanism for linearly polarized THz plane wave [17]. Based on this structure, Huang et. al. used two mutually orthogonal gratings to enhance the focusing efficiency [18]. Another hybrid metalens consisting of a monolayer graphene and a gold film etched with rectangular aperture array were also designed, and its focal tuning mechanism works well at incident angles less than 40° [19]. Instead of the metallic metasurfaces, graphene metasurfaces also have been used as well for the light focusing. Chen et. al. demonstrated a variable-focus reflective metalens based on the strict phase distribution design of nonuniform periodic rectangular graphene sheet arrays [20]. Zhu et. al. realized the polarization-switchable anomalous reflection and focusing with the designed graphene blocks arrays [21]. Other graphene metasurfaces such as graphene monolayer etched with rectangle aperture array or elliptical apertures [22,23], stacked graphene ribbons [24] were also reported for the plane focusing in the infrared at a terahertz range. Particularly, the advancement of graphene or hybrid graphene-metal metalens in focusing are thriving, such as sensitively tunable planar lens [25], dual-band light focusing [24], polarization insensitive focusing metalens [26], gate-controlled terahertz focusing [27], out-of-plane focusing of terahertz beams [28].
In this manuscript, we present a new design of plasmonic metalens which is consisted of monolayer graphene loaded on aperiodic silica grating arrays with varying dielectric strip’s width. Assisted by the graphene surface plasmon, the proposed plasmonic metalens can focus the infrared incident light in a nanoscale spot with a higher focusing efficiency up to 25%. We discuss the proposed plasmonic metalens design as well as the dynamically controlling of the focal length by the incident wavelength, graphene Fermi level and the arrangement of the aperiodic silica grating arrays.
2. Theoretical analysis
2.1 Plasmon dispersion relationship of graphene-loaded periodic grating silica arrays
The theoretical analysis begins with a discussion of the plasmon dispersion relationship of monolayer graphene loaded periodic silica grating arrays. A schematic profile of silica grating periodically loaded with graphene is shown in Fig. 1. The width of the dielectric strip is a, the gap between adjacent dielectric strips is b, and the grating period is $\mathrm{\Lambda } = a + b$. Due to diffraction effect by the grating structure, the incident near-infrared light propagated in the z-axis of the interface can excite the graphene surface plasmon (GSP) with the TM mode.
Fig. 1. Profile of the structure of the graphene loaded periodic silica grating. The light with TM mode is incident in the direction of z-axis.
Now we discuss the propagation of GSP in the x-z plane. Since the structure is uniform in the z direction, the electric field expression [29] on the graphene surface can be written as:
Equation (2) represents the electric field (x+ direction and x- direction) of the propagating GSP, ${a_n}$, ${b_n}$, ${c_n}$ and ${d_n}$ are constants. And ${k_1}$, ${k_2}$ are respectively the propagation constant of the GSP with and without the silica substrate, and can be expressed as:
Here, ${\varepsilon _r} = {\varepsilon _{r2}} = 1$, and the dielectric constant ${\varepsilon _{r1}}$ of silica is 2.25. The conductivity σ of the graphene is given by the Kubo formula [30], ℏ is the reduced Planck constant, the graphene chemical potential ${\mu _c}$ is 0.64 eV, and the relaxation time τ is set to $1 \times {10^{ - 12}}\; s$. Next, applying the boundary conditions at the interface $x = ({n - 1} )\mathrm{\Lambda }$ and $x = n\mathrm{\Lambda } - a$, the electric field components ${E_y}$ and ${E_x}\; $ of the GSP are continuous for the periodic boundary, and the following equations are obtained:
Eliminated the coefficients ${c_n}\; \; $ and $\; \; {d_n}$, we can obtain the matrix equation: $\left( {\begin{array}{{c}} {{a_{n - 1}}}\\ {{b_{n - 1}}} \end{array}} \right) = \left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}} \right]\left( {\begin{array}{{c}} {{a_n}}\\ {{b_n}} \end{array}} \right)$. For a periodic media, we can use Bloch's theorem $\left( {\begin{array}{{c}} {{a_n}}\\ {{b_n}} \end{array}} \right) = {e^{ - i{k_x}\mathrm{\Lambda }}}\left( {\begin{array}{{c}} {{a_{n - 1}}}\\ {{b_{n - 1}}} \end{array}} \right)$, where the phase factor ${e^{ - i{k_x}\mathrm{\Lambda }}}$ represents the eigen values of the matrix $\left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}} \right]$. The final dispersion relation can be written as:
Now we discuss the case of normal incidence in the z+ direction for which the value of ${k_x}$ is 0. The propagation constant ${k_z}$ in the z direction can be obtained by solving the equation of (10). The effective refractive index, ${n_{eff}}$ can be expressed as ${n_{eff}} = {{{k_z}} / {{k_0}}}$, ${k_0}$ is the propagation constant in free space. According to the formulas above, the theoretical value of the ${n_{eff}}$ is $22.89 + 0.101\textrm{i}$. Simulation results were used to verify this value. Figure 2(a) shows the electric field distribution when the incident wavelength is 8 $\; \mathrm{\mu m}$, the values of a and Λ are 50 and 100 nm respectively. The results in Fig. 2(a) demonstrate that the excited SPP’s wavelength is about 350 nm, as shown in Fig. 2(b). According to the expression ${n_{eff}} = {{{\lambda _0}} / {{\lambda _z}}}$ (where ${k_z} = {{2\pi } / {{\lambda _z}}}$ and ${k_0} = {{2\pi } / {{\lambda _0}}}$), the simulation value of ${n_{eff}}$ is 22.85. The simulation value of ${n_{eff}}$ is very close to that obtained through theoretical analysis. Moreover, Fig. 2(c) shows that simulated and theoretical real part of ${n_{eff}}$ dependent on the width of the dielectric strip a. Here, the period Λ is fixed at 100 nm. It is observed that the real part of ${n_{eff}}$ increases with an increase in the dielectric thickness a. It should be noted that the value of the effective refractive index lies between the effective refractive index of the graphene with and without the silica substrate. This indicates that the solution based on Bloch's period theory is accurate within a reasonable range. Thus, the device can be designed based on these results.
Fig. 2. (a) The excited surface plasmon electric field $|\textrm{E} |$ (V/m) of the graphene in the model of Fig. 1. The curve in (b) is the corresponding field intensity obtained from figure (a). (c) The theory and simulated $\textrm{real}({{n_{eff}}} )$ of the propagated surface plasmon in the graphene.
2.2 Principle of the self-focusing gradient index lens
For a transparent medium, the light’s trajectory is related to the spatial distribution of refractive index, which can be expressed by the following differential equations [31]:
Here, the z-component formula in the Eq. (12) is:
Substituting the x-component formula into the Eq. (11), we get:
where, the constant A is related to the position and angle of the incident light. For a light incident perpendicular at $\textrm{x} = {x_0}$, A can be described by $n({{x_0}} )$. Now, we consider a commonly used self-focusing rod, the refractive index function is:Substituting Eq. (12) and solving to get the propagation path of the light:
According to Eq. (17), the propagation path of the light changes periodically with a period of $\frac{{2\pi }}{\alpha }$. For $\frac{{({2n - 1} )\pi }}{{2\alpha }}$ (n = 1,2,3…), the value of x is zero and unrelated with the position of the incident point ${x_0}$. That is, all the light rays converge at one point, resulting in the formation of the phenomenon of self-focusing. In order to facilitate the subsequent discussion, the z coordinate of the first convergence point is defined as the focal length:
2.3 Design of the plasmonic focusing metalens of graphene-loaded aperiodic silica grating arrays
According to the analysis above, an arbitrary effective refractive index of the array structure can be obtained by changing the parameters of the dielectric strip a and the gap distance b. Therefore, if the distance between the dielectric strips can be designed such that the refractive index of the entire structure in the x direction is controllable. However, it should be noted that the gap distance b cannot be made arbitrarily large since this breaks down the Bloch harmonics theory resulting in relatively large errors. Therefore, to obtain the refractive index following the distribution of the self-focusing lens function, we can calculate the width and placement position of each silica dielectric strip according to Bloch's period theory and light differential equation. In order to build the model of the infrared focusing lens, it is necessary to design a self-focusing graded index lens, the refractive index distribution of which in the x-axis direction should satisfy the equation of $n(x )= {n_0}\textrm{sech} ({\alpha x} )$.
Finite element analysis (FEA) model and process simulation are carried out in Comsol Multiphysics software. The refractive index distribution function of the array is initially set with ${n_0} = 25$, $\mathrm{\alpha } = 0.4\mathrm{\;\ \mu }{\textrm{m}^{ - 1}}$, and the incident light wavelength $\lambda = 8\,\,\mathrm{\mu m}$. The array period $\mathrm{\Lambda } = 100\,$ nm, while the value of the dielectric strip width a is variable. According to the theory mentioned in section 2.1 and 2.2, starting from $x = 0$, the width $a$ of each dielectric strip (a1, a2 … an) are set at 72.2, 72.0, 71.3, 70.2, 68.7, 66.7, 64.4, 61.8, 58.9, 55.7, 52.3, 48.7, 45.0, 41.2, 37.3 and 33.3 nm. Moreover, they are distributed symmetrically along the main axis. Setting the parameters to these values, the model of graphene plasmonic metalens for focusing is established, and its schematic is shown in Fig. 3. It consists of a monolayer graphene loaded grating silica aperiodic arrays. The simulated total length in the z-direction is 10 $\mathrm{\mu m}$. The boundary of the simulation area is applied with a perfect matching layer, and the incident light only contains the ${E_y}$ component (TM polarization). In order to make the simulation results as accurate as possible, the grid size is set to 0.1 nm.
3. Discussion and analysis
Figure 4(a) shows the electric field distribution ${|\textrm{E} |^2}$ of the established model shown in Fig. 3(a). The results show that the incident plane wave can successfully excite the GSP, and propagate along the z-axis and finally focus at $z = 3.9\,\mathrm{\mu m}$. This simulated focal length is consistent with the theoretical value of 3.92 $\mathrm{\mu m}$ calculated using Eq. (18). Moreover, the Full Width at Half Maxima (FWHM) of focusing spot is 200 nm, which is approximately equal to ${\lambda _0}/40$. This result demonstrates that the proposed plasmonic metalens achieves the focusing effect beyond the diffraction limit. After focusing, the light field continues to diverge and spread with increasingly smaller electric field intensity which becomes smaller and smaller. Figure 4(b) depicts the light traces which are calculated according to the differential light theory. Obvious comparison of the theoretical results of the light traces are in close proximity with the simulated electric field distribution.
Fig. 4. (a) and (b) are the simulated electric distribution ${|\textrm{E} |^2}$ on x-z section and the calculated light traces, respectively.
According to the light differential equation, different focal lengths can be realized by adjusting the arrangement of the dielectric strips. Three different array structures with different focal lengths were constructed based on the above theory, and the results are shown in Fig. 5, which shows both the theoretical and simulated results for the cases with different focal lengths of 3, 4, and 5 $\mathrm{\;\ \mu m}$. Here, the aperiodic grating array contains a total of 31 dielectric strips, and the width of each dielectric strip ranges from 14 to 73 nm. As demonstrated in Fig. 5(a), the electric field is focused at $z = 3\,\,\mathrm{\mu m}$, and the second focusing spot is located at $z = 9\,\mathrm{\mu m}$. This could be attributed to the fact that the propagation of the electric field of the structure exhibits periodicity. Moreover, this result is consistent with the light traces shown in Fig. 5(d). In addition, the results shown in Fig. 5(b-c) are also consistent with the theoretical results. So, careful arrangement of the silica grating array according to the theory of the light propagation path can effectively achieve the focusing effect. It is can be also observed that the electric field energy is completely consumed after a few cycles of transmission due to the material loss of the graphene itself.
Fig. 5. (a-c) The distribution of the electric field ${|E |^2}$ and (d-f) the calculated light traces correspond to the cases of the focal length of 3, 4 and 5 $\mathrm{\mu m}$, respectively.
It is well established that graphene materials have good electro-optical modulation characteristics, particularly the Fermi level ${E_f}$ of the graphene can be changed by applying a bias voltage [22]. According to the equations (3) to (5), it can be seen that the value of ${E_f}$ will alter the propagation constant of the grating array structure. Thus, we also investigate the influence of the ${E_f}$ on the designed plasmonic focusing metalens. Figure 6(a-c) shows the electric field distribution of the ${E_f}$ with the incident wavelength ${\lambda _0} = 8\,\,\mathrm{\mu m}$. For the ${E_f} = 0.3$, $0.6$ and 0.9 eV, the first focusing spot is respectively located at z = 4.5, 3.8 and 3.5 $\mathrm{\mu m}$ with the spot’s FWHM of 64, 232 and 320 nm. Therefore, as the ${E_f}$ increases, the focal length becomes shorter, and the focal point becomes larger. A comparison of the results in Fig. 6(a-c) shows that an increase ${E_f}$ also reduces the transmission loss. This result can also be explained by the decrease of the imaginary part of the ${n_{eff}}$ of the grating array structure. Besides the Fermi level ${E_f}$, the case of the different incident wavelength was also investigated as shown in Fig. 6(d-f). Here, the ${E_f}$ is fixed at 0.64 eV. For an incident wavelength λ0 of 5, 7 and 9 $\mathrm{\mu m}$, the focal length is respectively 5, 3.9 and 3.6 $\mathrm{\mu m}$, and the FWHM of the focusing spots are respectively 62, 188 and 284 nm. The corresponding value of the ${{FWHM} / {{\lambda _0}}}$ is about 1/80, 1/37, and 1/32, respectively. This result demonstrates that the red-shift of ${\lambda _0}$ would cause a shorter focal length and a larger focusing spot. More systematic relationships of FWHM with Fermi level and incident wavelength is respectively shown in Fig. 6(g) and (h). It is can be observed that the FWHM increases with the increasing of the Fermi level and incident light wavelength. Compared with the reported plasmonic lenses, most of their FWHM are around 0.5 ${\lambda _0}\; $[32,33], and the best effect is observed around 0.1 ${\lambda _{0\; }}$ [12,34]. Thus, the designed plasmonic lenses in this work has a better focusing effect far beyond the diffraction limit.
Fig. 6. (a-f) The distribution of the electric field ${|\textrm{E} |^2}$ correspond to different cases. (a-c) are corresponding to the Fermi level ${E_f}$ of 3, 4 and 5 eV respectively, while (d-f) are corresponding to the incident wavelength of 5, 7 and 9 $\mathrm{\mu m}$, respectively. (g-h) The relationship of FWHM with Fermi level and incident wavelength.
To get a deep insight into the proposed infrared plasmonic focusing lens, we conducted a focusing efficiency analysis on the self-focusing array structure, and defined the focusing efficiency as the ratio of the energy in an area with a radius three times that of the FWHM of the energy of the incident light [35]. As shown in Fig. 7, we calculate the cases with the λ0 and the ${E_f}$ ranging from 6 ∼10 µm and 0.1 −1 eV, respectively. It was found that the focusing efficiency increases with an increase in the incident wavelength, ${\lambda _0}$ or the Fermi level, ${E_f}$. For instance, for ${\lambda _0} = 6\,\,\mathrm{\mu m}$ and ${E_f} = 0.1\,$ eV, the normalized focusing efficiency is only $4.6 \times {10^{ - 4}}{\%}$. However, the focusing efficiency reaches up to 25% as the ${\lambda _0} = 9.8\,\,\mathrm{\mu m}$ and the ${E_f} = 0.85\,$ eV, as demonstrated in the upper right corner of the Fig. 7. This result is because a higher Fermi energy and consequently a lager carrier density will enhance the graphene surface plasmon resonance [22]. As discussed in the published works, the focusing efficiency of the plasmonic metalens is always limited to 25% by the inherent limitations of the maximum coupling efficiency, and the fundamental limits of ultrathin metasurfaces [36,37]. For example, the focusing efficiency of the plasmonic Fresnel Zone Plate (FZP) lens remains constant at ∼0.055%. By combining graphene nanoribbons with FZP architecture, the focusing efficiency is enhanced and limited to 5% [38]. A higher focusing efficiency of 6% was obtained for the terahertz metalenses using graphene coated metallic c-shaped antenna [27]. Various methods have been used to enhance the metalenses’ focusing efficiency of the metalenses using Fabry-Perot cavity [17], multilayer measurface [39], two mutually perpendicular gratings [18], or by adding an interface between two materials in a different layer as the ultrathin metasurface [40]. For instance, a higher focusing efficiency of 66.6% was obtained on graphene metasurfaces etched with rectangle aperture array [22]. However, the FWHM of the focal spot is about 34 µm (0.57 λ). Although the graphene was used in the works in Refs [16–19]., the graphene surface plasmon was not excited. The nanofocusing effect beyond the diffraction limit in this work is due to the graphene surface plasmon. Therefore, considering the nanoscale focusing effect, the focusing efficiency of the proposed plasmonic hybrid metalens is in a reasonable and acceptable range. Moreover, combining the previous simulation results, we found that the more concentrated the focus energy, that is, the smaller the value of FWHM/${\lambda _0}$, the lower the focusing efficiency. Therefore, when designing the focusing device, both factors of focusing effect and efficiency should be considered to meet the application requirements.
Fig. 7. Normalized electric field focusing efficiency $\mathrm{\eta }$ in the range of $6\mathrm{\;\ \mu m} \le {\lambda _0} \le 10\mathrm{\;\ \mu m}\textrm{, }0.1\textrm{eV} \le {E_f} \le 1\; \textrm{eV}$
4 Conclusion
In conclusion, according to Bloch's period theory and light differential equation, a new plasmonic nanofocusing metalens with a hybrid structure is designed. Based on this device, the TM polarized light can be coupled with the excited surface plasmon of the graphene, and focused in a nanoscale spot with a dynamically controlled focal length. We discussed the proposed design based on three different array structures which have different focal lengths, and investigated the effects of both the incident wavelength and the Femi level of the graphene on the focus spot’s size, focal lengths and the focusing efficiency. The results show that reducing the incident wavelength and Fermi level can make the light field energy more focused, but with a lower focusing efficiency. Moreover, the present theoretical results agree well with the simulation results. This new method uses the graphene and silica metasurfaces to realize a wide range of focal length modulation in the infrared band and has broad application prospects in optical communications, real-time imaging and wavefront modulation.
Funding
National Natural Science Foundation of China (11204107, 91750112).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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