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Nov 01, 2024

Electrically tunable graded photonic crystal lens based on graphene plasmons | Scientific Reports

Scientific Reports volume 14, Article number: 26169 (2024) Cite this article Metrics details Controlling the nonlinear relationship between surface plasmon polariton (SPP) mode index and chemical

Scientific Reports volume 14, Article number: 26169 (2024) Cite this article

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Controlling the nonlinear relationship between surface plasmon polariton (SPP) mode index and chemical potential of graphene can be used in the field of active transformation optics. Here, we propose an electrically tunable 2D Graded Photonic Crystal (GPC) lens based on graphene SPP platform. Our platform comprises a graphene monolayer integrated into a back-gated structure with nano-patterned gate insulators. When the chemical potential of the graphene surface is designed to operate in the nonlinear region, the designed GPC lens can be continuously transformed between a Maxwell’s fish-eye lens and a Luneburg lens by tuning the gate voltage. The range of the lens background chemical potential for allowing this transformation is systematically studied. To compensate for the significant errors inherent in the conventional effective medium theory (EMT) during the homogenization of photonic crystals (PCs), we propose a generalized effective medium theory (GEMT). The validity and accuracy of this approach are verified through comparisons with true values (based on rigorous eigenvalue solutions) and EMT values. Due to its advantages of on-site controls and easy fabrication characteristics, the proposed graphene GPC provides a new way for practical on-chip light manipulation.

The capability to govern the flow of light energy is fundamental to modern information and communication technologies. Nevertheless, the task of electrically controlling photons, which are uncharged, presents a significant challenge. Surface plasmon polaritons (SPPs), which are hybrid excitations of photons and electrons, are marked by overcoming the Abbe’s diffraction limit and implementing tight field confinement1,2. Hence, they are considered as one of the most promising candidates for nanophotonic components. For these reasons, graphene has been extensively investigated as a one-atom-thick plasmonic platform, resulting from its ability to support and control low-loss SPP waves in terahertz and mid-infrared regimes3,4. In addition, graphene’s most unique and remarkable property is that its optical properties can be dynamically tuned by controlling the chemical potential5,6,7,8.

On the other part, photonic crystals (PCs), characterized by their periodically varying optical properties, are frequently fabricated in discrete media with built-in local electrodynamics responses9. They facilitate precise manipulation of light propagation within integrated optical circuits and serve to emulate advanced physical phenomena10. In 2005, E. Centeno introduced the concept of graded photonic crystal (GPC)11, which are based on the gradual modulation of PC parameters, such as lattice constant, orientation, or filling factor, resulting in a progressive alteration of the effective refractive index. Typically, there are two ways for calculating the effective refractive index of a PC: the effective medium theory (EMT)12,13,14 (including Maxwell–Garnett theory or Bruggeman theory) and the PC method (based on rigorous eigenvalue solutions)15. The results obtained through the PC method are considered to be accurate. The EMT predicts the overall structure’s effective refractive index by considering the volume fractions and refractive indices of different materials within the PC. These two methods demonstrate considerable consistency within the long-wavelength limit, i.e. the normalized frequency \(a/\lambda _b < 0.10\), where a is the lattice constant and \(\lambda _b\) is the operating wavelength in background medium. As the normalized frequency increases (\(0.10< a/\lambda _b < 0.35\)), the disparity between the results obtained from these two approaches gradually enlarges15.

Over the past few years, plasmonic gradient-index (GRIN) lenses, with the advantages of being flat and free of geometrical aberrations, have been extensively investigated theoretically and experimentally16,17,18. However, because the PC effect cannot be readily tuned in common materials, conventional photonic crystal systems exhibit very limited tunability9. Recently, two-dimensional van der Waals heterostructures have emerged as a promising platform for tunable PCs19,20,21,22. Specifically, in monolayer graphene heterostructure, tunable photonic crystals can be realized with electrostatic gating and nano-patterned gate insulators6,19,23,24. For instance, L. Xiong et al. made the first experimental verification on a tunable 2D PC for graphene SPPs19, and near-field imaging results reveal the formation of photonic bandgaps and a significant modulation of the local plasmonic density of states. Zeng et al. and Wu et al. designed different patterns of graded index on graphene metasurfaces by shaping the dielectrics underneath graphene into specific photonic crystals6,23. On the other hand, there have been considerable efforts dedicated to actively shaping out-of-plane wavefronts based on metasurface technology25,26,27,28. For example, Hodjat et al. presented a reprogrammable near-infrared metasurface design using vanadium dioxide (\(\mathrm {VO_2}\)) as a phase-change material, enabling electrical control of the nano-antennas’ metallic and insulating states for efficient beam steering and active light modulation25. Liu et al. present an electrically tunable terahertz meta-lens that utilizes a hybrid structure of a metallic metasurface and monolayer graphene to achieve wide-range focal length tunability through gate voltage modulation28. However, in previous works, the tunability of these graphene plasmon (GP) devices has remained relatively limited. This limitation arises because these devices were engineered to operate in a near-linear region, where the SPP mode index (or graphene conductivity) changes slowly with the chemical potential when adjusting the gate voltage. Recently, we demonstrated an electrically tunable GP lens that can be continuously tuned from a Maxwell’s fish-eye lens (MFL) to a Luneburg lens (LL) by only adjusting the gate voltage29. The design philosophy of the GP lens is based on the nonlinear relationship of SPP mode index with respect to chemical potential of graphene. Nevertheless, the proposed GP lens employs a continuously shaped dielectric spacer to simplify the theoretical calculations, which faces a significant challenge in fabrication.

In this work, we propose a tunable GPC lens structure with graphene SPP platform, which is much more friendly for nano-fabrication. The radially graded mode index is engineered through the variation of the filling factor for each unit. This lens operates within a nonlinear region where SPP mode index exhibits appreciable changes in response to variations in the chemical potential. By adjusting the gate voltage, the designed MFL can be continuously transformed to a LL. The importance of choosing an initial chemical potential of background (\(\mu _{c0}^{b}\)) is discussed, and the range of \(\mu _{c0}^{b}\) that allows focal length to be tuned from lens circumference to infinity is given. The numerical results show that the proposed GPC lens works very well.

In addition, to address the significant inaccuracies encountered when calculating the effective refractive index of PCs using the EMT, we introduce a generalized effective medium theory (GEMT). This new method enhances the precision of effective refractive index calculations and simplifies the design process for GPC lenses, promising greater efficacy in practical applications.

Here, we present the dependency of SPP waves on the chemical potential of graphene. For TM-polarized SPP waves propagating on free-standing isolated graphene, the dispersion relation is given by4:

where \(n_{spp}\) is the effective mode index for the SPP waves, and \(\eta _0\) denotes the intrinsic impedance. \(\sigma _{g}\) is the surface conductivity of graphene, which can be modeled from the well-known Kubo formula30. It is comprised of intraband and interband transitions: \(\sigma _g=\sigma _{g, \text{ intra } }+\sigma _{g, \text{ inter } }\). The expression of the intraband contribution reads31:

The expression of the interband contribution reads:

Here, \(K_B\), e, and \(\hbar\) are universal constants related to the Boltzmann constant, electron charge, and reduced Planck constant, respectively. \(\omega\) is the angular frequency, T is the temperature, \(\tau\) is the electron-photon relaxation time, and \(\mu _c\) is the chemical potential of graphene, which can be calculated using the parallel plate capacitor model32,33, expressed as:

Where \(V_g\) is the gate voltage, \(\alpha _{c}=\varepsilon _{0}\varepsilon _{r} /(e h)\) is estimated from a single capacitor model, \(\varepsilon _0\) is the vacuum permittivity, \(\varepsilon _r\) is the relative permittivity of spacer dielectric, h is the dielectric thickness, and \(V_{Dirac}\) is the voltage offset induced by natural doping, which is assumed to \(0\, \textrm{V}\) in this work.

The relationship between the SPP mode index and the chemical potential of graphene at \(\lambda _0=20\, \mathrm {\upmu m}\) (\(15~\textrm{THz}\)), as calculated from Eqs. (1–4), is depicted in Fig. 1. The figure illustrates a nonlinear decrease in both real and imaginary components of the SPP mode index concurrent with an increase in graphene chemical potential. This variation can be categorized into three distinct regions: a steep region at low chemical potentials (\(\mu _{c} < 0.05\, \textrm{eV}\)), an intermediate region within the range of \(0.05 \le \mu _{c} \le 0.10\, \textrm{eV}\), and a flat region beyond \(0.10\, \textrm{eV}\). Previous investigations23,34,35,36,37,38 predominantly focused on lenses operating in this flat region, where the SPP mode index slightly changes with chemical potential, thereby exhibits very limited tunable focusing performances. In contrast, the SPP mode index in the steep region is subject to substantial variability and pronounced nonlinearity, coupled with elevated propagation losses, rendering this region less desirable. The intermediate region presents a balance, exhibiting appreciable nonlinearity coupled with tolerable propagation losses (\(-6\, \mathrm {dB/ \upmu m}\) at \(\mu _c=0.05\,\textrm{eV}\)). This region provides a favorable platform for the development of highly tunable devices.

Variations of the real and imaginary part of SPP mode index with respect to the chemical potential of graphene at \(\lambda _0=20\, \mathrm {\upmu m}\) and \(T=300\,\textrm{K}\). A log-10 scale is used for the x-axis.

The tuning performance of a GPC lens depends primarily on the tunability of the refractive index ratio between its central unit and the surrounding background. In this section, we introduce a systematic method to investigate the tunability of this critical index ratio, providing a foundational analysis for optimizing its adaptive functionality.

A back-gated configuration is constructed in alignment with Eqs. (1–4), as shown in Fig. 2. The structure model incorporates a single-layer graphene sheet on top of a dielectric spacer, grown on a nano-patterned gold ground plane. Application of a static gate voltage between the graphene and the ground plane modulates the graphene’s chemical potential via the electric-field effect4. This modulation is contingent upon the spacer thickness. By designing the two different thicknesses of the dielectric spacer, we have achieved the formation of two distinct areas: a disk area characterized by a dielectric thickness \(h_d\) and associated chemical potential \(\mu _{c}^d\), a background area with dielectric thickness \(h_b\) and corresponding chemical potential \(\mu _{c}^b\). The SPP mode indices for these two areas are denoted as \(n_d\) and \(n_b\), respectively. With \(h_d >h_b\), it results in \(\mu _{c}^d<\mu _{c}^b\), such that the corresponding SPP mode index \(n_d>n_b\). Based on this unit structure, a 2D PC can be constructed on the graphene surface easily, and the effective mode index of this 2D PC satisfies \(n_b< n_{eff} < n_d\).

Schematic of the unit structure with two different areas of chemical potential: (a) top-view and (b) cross-section view. \(d\) and \(a\) respectively represent the disk diameter and the lattice constant.

In this work, we have chosen high-resistivity silicon as the dielectric spacer, characterized by a relative permittivity of 11.5. The simulations are conducted at a temperature of \(T=300\, \textrm{K}\) with an operating wavelength of \(\lambda _0=20\, \mathrm {\upmu m}\). The momentum relaxation time \(\tau\), representing intrinsic losses, is assumed to be \(1.5\,\textrm{ps}\). This value is conservatively chosen to feature the practical losses in graphene39 and to account for the theoretical estimation of its maximum mobility31. For instance, Dean et al. have reported that a relaxation time as high as \(3\,\textrm{ps}\) was experimentally obtained40. Note that \(\tau\) significantly impacts the imaginary part of the SPP mode index, which could lead to a diminished focusing effect. Practically, it is essential to optimize both carrier mobility and relaxation time to enhance the focusing quality.

To solve the band structure of the 2D PC, a commercial software COMSOL Multiphysics is employed by considering the unit cell surrounded with periodic boundary conditions. The eigenvalue problem can then be solved numerically, which is denoted as the PC method in this paper. The effective refractive index is calculated as \(n_{eff}=kc/\omega (k)\), where c is the speed of light in vacuum, k is the modulus of the wave vector and \(\omega\) refers to the corresponding angular frequency. The results obtained from this method are accurate, however, the process is complex and time-consuming.

To simplify the calculations for effective refractive index, homogenization of periodic structures using analytical formula has been done long ago13,41. Traditionally, the effective refractive index of a PC can be approximated by the EMT formulation23,42:

where \(\gamma =\pi d^2/4a^2\) is the filling factor. However, EMT involves only isolated units and does not consider the effects introduced by periodicity, thus making it an approximate approach.

The band structure of a 2D PC will remain unchanged on the normalized frequency scale, despite changes in the absolute values of the overall refractive indices. For instance, when the normalized frequency \(a/\lambda _b\) and the filling factor \(\gamma\) are held constant, and the values of \(n_d\) and \(n_b\) are both doubled, the effective refractive index \(n_{eff}\) will also be doubled. To comply with such scaling property, it becomes necessary to normalize the effective index and the disk index with the background index, namely \(\tilde{n}_{eff} = n_{eff}/n_b\), \(\tilde{n}_d = n_d/n_b\).

Comparison between GEMT, EMT and PC method. (a,b) The variation of the effective index \(\tilde{n}_{eff}\) (\(a/\lambda _b=0.24\)). (c,d) The variation of the effective index \(\tilde{n}_{eff}\) (\(a/\lambda _b=0.10\)).

Figure 3 gives a comparison between the simulation results obtained via the PC method (green line) and those based on the EMT (blue line). It is observed that the discrepancies between these two results increase with the filling factor \(\gamma\) and disk index \(\tilde{n}_d\). To compensate for such discrepancies between the EMT and the PC results, we propose a generalized effective medium theory (GEMT) to enhance the EMT formula by incorporating higher-order terms of the filling factor and disk index. Therefore, it provides a more accurate representation of the nonlinear relationship between the effective index, the filling factor, and the disk index. The proposed GEMT formula can be expressed as:

where \(a_2\), \(b_2\), \(b_1\), and \(b_0\) are coefficients to be determined. Notably, at a filling factor of zero, \(\tilde{n}_{eff}\) will reduce to 1, thereby conforming to the physical phenomena of this limiting case.

In general, the coefficients \(a_2\), \(b_2\), \(b_1\), and \(b_0\) are dependent on the operating frequency. Given the substantial errors associated with EMT within the normalized frequency range of \(0.10< a/\lambda _b < 0.35\)15, we select two frequencies \(a/\lambda _b = 0.24\) and \(a/\lambda _b = 0.10\) as examples for calculating the corresponding coefficients in the GEMT formula, respectively. The choice of \(a/\lambda _b=0.10\) is based on literature15,42,43 indicating that this frequency point marks the boundary of the long-wavelength limit. To ensure that the normalized effective index of the lens central unit is \(2n_b\), and that the unit operates in the first band, we can obtain the condition \(2n_bk_0a \le \pi\). Given \(\lambda _b = 2\pi /n_bk_0\), we further obtain \(a/\lambda _b \le 0.25\). Considering fabrication constraints, we aim to select the largest possible value for \(a/\lambda _b\). However, due to the potential appearance of bandgaps and strong anisotropy on the Brillouin zone boundary, \(a/\lambda _b\) cannot be set to 0.25. Therefore, we select \(a/\lambda _b = 0.24\) for lens design and to verify the superiority of GEMT.

Utilizing the PC method, datasets of \(\tilde{n}_{eff}\) under different combinations of \(\gamma\) and \(\tilde{n}_d\) can be obtained and then fitted using Eq. (6) for these two frequencies, respectively (detailed fitting procedures can be found in the Supplementary materials file). The final results for the coefficients are listed in Table 1. Figure 3 shows the comparison between the GEMT and EMT with the PC method at the normalized frequency of \(a/\lambda _b = 0.24\) and \(a/\lambda _b = 0.10\). As shown, the GEMT results consistently exhibit much better agreement with the PC method compared to the EMT results, especially in scenarios with larger filling factors and disk indices. It should be noted the GEMT formulation with \(a/\lambda _b = 0.24\) will be extensively used in this work.

The normalized index \(\tilde{n}_{eff}\) of the central unit serves as a crucial metric for assessing the tunability of a GPC lens when modulating the gate voltage. Initially, the lens central unit of \(\tilde{n}_{eff}=2\) is established by adjusting the thickness of the silicon spacer. The detailed design process unfolds as follows:

We commence by selecting an initial gate voltage, for instance, \(V_g=1.0\, \textrm{V}\). Theoretically, this initial value is not expected to influence the propagation characteristics of the SPP wave.

An initial chemical potential \(\mu _{c0}^b\) is chosen from the intermediate region as shown in Fig. 1. The corresponding background SPP mode index \(n_b\) is then calculated, followed by determining the dielectric thickness \(h_b\) using Eq. (4). The selection of \(\mu _{c0}^b\) is pivotal for the lens transformation from a MFL to a LL.

Choose the size of \(d/a = 0.85\) for the central unit, which represents a maximum available disk size (see discussion on Fig. 6a). Under the stipulated condition \(\tilde{n}_{eff} = 2\), the specific value \(n_d=\tilde{n}_{d}n_b\) for the disk area can be easily calculated using the GEMT formula with \(a/\lambda _b\)=0.24. The associated chemical potential \(\mu _{c0}^d\) and dielectric thickness \(h_d\) can then be obtained from Eqs. (1) and (4), respectively.

Upon increasing the gate voltage \(V_g\), the normalized index \(\tilde{n}_{eff}\) will no longer remain as 2 for the above designed lens central unit. This alteration is attributed to the nonlinear relation between the SPP mode index and the gate voltage. It is notable that the normalized index decreases with an increase in gate voltage, primarily because the value of \(n_d\) diminishes more rapidly than \(n_b\).

Figure 4 presents the calculated normalized index as a function of the gate voltage, with the chemical potential \(\mu _{c0}^b\) ranging from \(0.06\,\textrm{eV}\) to \(0.08\, \textrm{eV}\). The data reveal that the normalized index decreases more rapidly with the gate voltage for lower values of \(\mu _{c0}^b\). This trend is attributed to the enhanced nonlinearity at lower chemical potentials, as demonstrated in Fig. 1. It can be observed that the normalized index is able to reduce to \(\sqrt{2}\) when \(\mu _{c0}^b \le 0.07\, \textrm{eV}\). This observation suggests the feasibility of transforming the lens from a MFL (\(\tilde{n}_{eff}=2\)) to a LL (\(\tilde{n}_{eff}=\sqrt{2}\)). However, high nonlinearity should be avoided, as it results in elevated losses. Consequently, an applicable range for \(\mu _{c0}^b\) is identified between \(0.06\,\textrm{eV}\) and \(0.07\,\textrm{eV}\). For \(\mu _{c0}^b=0.07\,\textrm{eV}\), the required transformation gate voltage to achieve \(\tilde{n}_{eff}=1.66, 1.57, \sqrt{2}\) are identified as \(1.37~\textrm{V}\), \(1.70~\textrm{V}\) and \(6.0~\textrm{V}\), which correspond to the generalized Luneburg lenses (GLL) with \(f_1=\textrm{R}\) and \(f_2\) equals \(2~\textrm{R}\), \(3~\textrm{R}\) and \(\infty\), respectively. This analysis underlines the critical influence of background chemical potential (\(\mu _{c0}^b\)) and gate voltage on the tunability of the graphene-based PC.

The normalized index \(\tilde{n}_{eff}\) as a function of gate voltage \(V_g\). The blue, red, and black dotted lines denote the normalized indices of generalized Luneburg lenses (GLL) with \(f_1=\textrm{R}\) and \(f_2\) equal to \(2\textrm{R}\), \(3\textrm{R}\), and \(\infty\), respectively.

As a proof of principle, here we demonstrate a design of a GPC lens on the graphene SPP platform. Figure 5 illustrates the proposed GPC lens structure, featuring a radial GRIN distribution. As discussed above, the lens structure encompasses two distinct areas on the graphene surface: a disk area, created by a prominently etched square lattice dot array on the ground plane, and a lens background area. The effective SPP mode index \(n_{eff}\) inside the lens is dependent on the filling factor \(\gamma\) of the graphene unit, therefore can be written as a function of the lens radius r. Due to the radially decreasing refractive index distribution inside the lens, the diameters of the disks also gradually decrease along the radial direction. Based on the above tunability analysis, we select an initial background chemical potential as \(0.07~\textrm{eV}\). For this purpose, the lattice period and the central unit size are chosen as \(a = 65\, \textrm{nm}\), \(d=55~\textrm{nm}\) (\(d/a = 0.85\)), respectively. The diameter of the MFL is chosen as \(2\textrm{R}=1950~\textrm{nm}\) (\(\textrm{R}=15a\)).

The GPC lens structure: (a) top view, (b) cross-section view. \(\textrm{R}\) denotes the radius of the lens, which is designed as \(\textrm{R} = 15a\).

Details of other design parameters can be determined as follows.

Choose the initial gate voltage as \(1.0~\textrm{V}\).

The initial background mode index can be calculated as \(n_b = 72.6\), and \(\lambda _b=275.5~\textrm{nm}\) (\(a/\lambda _b=0.24\)), according to Eq. (1). The corresponding dielectric thickness of the background area \(h_b = 176~\textrm{nm}\) is obtained from Eq. (4).

For a MFL, the effective mode index of the central unit is \(n_{eff}=2n_b=145.2\). According to the GEMT formulation, the mode index of the disk can be calculated as \(n_d = 165.5\). This leads to a calculated initial chemical potential of \(0.044~\textrm{eV}\) for the disk area, and the corresponding dielectric height of the disk can be calculated as \(h_d = 446~\textrm{nm}\).

Upon increasing the gate voltage to \(6.0~\textrm{V}\), the chemical potentials for both the background and the disk area will change accordingly. The effective mode index of the central unit will reduce to approximately \(\sqrt{2}n_b\). The calculated values of the design parameters are listed in Table 2.

Subsequently, we can determine the diameter of each disk within the lens by engineering the equi-frequency curves (EFCs)44 of the graphene unit at the normalized frequency of \(a/\lambda _b=0.24\). To accomplish this, the band curves and EFCs are numerically computed using COMSOL. The SPP mode index exhibits high dispersion, mainly attributed to the strong dispersion characteristic of graphene surface conductivity. The full-wave analysis of such a unit presents a nonlinear eigenvalue problem, posing significant computational challenges. We employ COMSOL’s stationary solver for iterative self-consistent solving. When the steady-state condition is met, the eigenvalues can be accurately determined.

Figure 6a illustrates the computed band structures along the wave vector \(k_x\) direction for units with four different sizes, covering a substantial range of the lens unit sizes. As observed, with an increase of the ratio \(d/a\), the band curve progressively bends rightward, indicating a corresponding increase in the effective index of the unit. Notably, when the ratio \(d/a\) exceeds \(0.85\), the operational frequency (\(15\, \textrm{THz}\)) will enter the band gap, which suggests inhibition of SPP wave propagation. Figure 6b shows the EFCs for the four unit sizes, revealing that within the range \(0.25 \le d/a \le 0.85\), the EFCs approximate a circular shape, indicative of PC behaving as an isotropic medium. In consideration of the technological constraints in micro-nano fabrication, the minimum diameter ratio \(d/a\) has been limited at \(0.25\), corresponding to a disk diameter of \(16\, \textrm{nm}\).

Band structure engineering for the graphene-based unit. (a) Band diagram for four sizes of unit along the \(k_x\) direction. (b) The corresponding EFCs of these units at the operating frequency of \(15\, \textrm{THz}\) (\(\lambda _0=20\,\mathrm {\upmu m}\)). (c) The GEMT-based (red line), PC-based (green line), and EMT-based (blue line) normalized index as a function of unit size at \(a/\lambda _b=0.24\).

Figure 6c gives the deduced normalized effective mode index as a function of disk diameter from the PC method, compared with the GEMT formulation based on the coefficients (\(a/\lambda _b=0.24\)) given in Table 1. As shown, the GEMT results agree with the PC method quite well. In this way, the gradient index distribution of the MFL can be easily realized with gradient disk diameters according to the GEMT formula.

Following the above design procedure, a MFL is designed at the initial gate voltage of \(1.0~\textrm{V}\). When the gate voltage is increased to \(6.0~\textrm{V}\), the normalized mode index in the lens central area drops to approximately \(\sqrt{2}\), enabling the lens to approximate the functionality of a Luneburg lens. Figure 7 shows the comparison between the normalized mode index distribution realized by the GPC lens and the theoretical distribution obtained from Eqs. 7–10. As shown, when increasing the gate voltage, the overall mode index distribution will decrease. With the gate voltage of \(1.37~\textrm{V}\), \(1.70~\textrm{V}\), and \(6.0~\textrm{V}\), the central mode index will decrease from 2 to 1.66, 1.57 and \(\sqrt{2}\), corresponding to the GLL with \(f_1=\textrm{R}\), \(f_2=2\textrm{R}, 3\textrm{R}, \infty\), respectively. For all four gate voltages, the mode index distribution matches the theory quite well. This indicates that the focusing performance of the lens can be continuously tuned from a MFL to a LL.

Full-wave simulations via COMSOL are employed to verify the propagation features with Perfectly Matched Layer (PML) boundaries applied, as shown in Fig. 8. Similar to the traditional MFL and LL, the plasmonic version of the fish-eye (or Luneburg) lens can focus a point (or plane wave) source to a point image on the rim of the lens12,45,46. The filed distributions (\(E_z\) component) in Fig. 8a show when a point source is placed on the rim of the MFL, the corresponding image can be realized at the symmetrical position of the rim, at the initial gate voltage of \(1.0\,\textrm{V}\). As the gate voltage increases to \(1.37\, \textrm{V}\) and \(1.70\, \textrm{V}\), a point source with focal length of \(2\textrm{R}\), \(3\textrm{R}\) can be successfully imaged at the circumference of the lens, respectively, as shown in Fig. 8b,c. Finally, when the voltage is increased to \(6.0\, \textrm{V}\), the incident plane wave is also well-focused at the edge of the lens (see Fig. 8d). Therefore, by adjusting the voltage, we have achieved a continuous transformation from a MFL to a LL. The full-widths at half-maximum (FWHM) of the focusing spot for the MFL and LL are \(\lambda _0/165\) and \(\lambda _0/55\) respectively, as indicated in Fig. 8e. Considering the corresponding effective wavelength of SPP waves at the circumference of the lens (e.g., \(\lambda _b=\lambda _0/72.6\) for MFL and \(\lambda _b=\lambda _0/25.7\) for LL), the focusing spot sizes are measured between \(0.44\lambda _b-0.47\lambda _b\) for all the cases.

The mode index profile comparison between theoretical solutions and realized results for different gate voltages.

(a–d) Field distributions for the designed GPC lens at \(1.0\, \textrm{V}\), \(1.37\, \textrm{V}\), \(1.70\, \textrm{V}\) and \(6.0\, \textrm{V}\). (e) The FWHM of the focusing spots.

In this work, we proposed an electrically tunable GPC lens based on graphene SPPs, which is constructed on a nano-patterned ground with electrostatic gating. The gradient index distribution is realized by engineering the EFCs for various filling factors. The design philosophy of tunability is to utilize the nonlinear relationship between SPP mode index and chemical potential of graphene. By adjusting the gate voltage, the designed MFL lens can be continuously tuned to LL.

We have introduced the GEMT to simplify the design process of the GPC lens and compensate for the significant errors present in the traditional EMT during the homogenization of PCs. The concept of the GEMT is to enhance the EMT formula by incorporating higher-order terms of the filling factor and disk index. This modification aims to better characterize the nonlinear relationship between the effective refractive index and the filling factor, as well as the disk index. In comparison to the traditional EMT, the GEMT significantly enhances the design precision and can also be extended to other frequency bands.

The generalized Luneburg lens problem, formulated by Luneburg46 in 1944, means that each point on a sphere is perfectly imaged onto another concentric sphere, with both the object and its image located within a homogenous region, either external to or on the surface of a spherically symmetric, inhomogeneous lens. The refractive index distribution n(r) for the GLL problem can be generally described by the following parametric equations47:

where \(\rho\) is the argument for the parametric function, and

It is assumed the radial distance r is normalized with respect to the radius of the lens, \(f_1\) and \(f_2\) are the image distances measured from the center of the lens, such that \(f_1,f_2 \ge 1\), and \(n_0\) is the refractive index of the lens background medium.

Two particular analytical solutions of the set of Eqs. (7–8) are known:

Maxwell’s Fish-eye Lens (\(f_1=f_2=1\)):

and Luneburg Lens (\(f_1=1, f_2=\infty\)):

Data presented in this work are available on request from the corresponding author.

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This work is supported by the National Natural Science Foundation of China (62071422) and the National Key Research and Development Program of China (2020YFB1805700).

College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, 310027, China

Chenglong Wang & Xidong Wu

School of Information and Electronic Engineering, Zhejiang University of Science and Technology, Hangzhou, 310023, China

Xiang Guo

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C.W., X.G. and X.W. conceived, developed, and refined the central idea of the present study. C.W. and X.W. analyzed the results and cured the interpretation, presentation, and visualization of the data. All authors reviewed the manuscript.

Correspondence to Xidong Wu.

The authors declare no competing interests.

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Wang, C., Guo, X. & Wu, X. Electrically tunable graded photonic crystal lens based on graphene plasmons. Sci Rep 14, 26169 (2024). https://doi.org/10.1038/s41598-024-76467-x

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Received: 25 July 2024

Accepted: 14 October 2024

Published: 30 October 2024

DOI: https://doi.org/10.1038/s41598-024-76467-x

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