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Abstract
Virtual image lenses play essential roles in various optical devices and applications, including vision correction, photography, and scientific instruments. Here, we introduce an approach for creating virtual image Luneburg lenses (LL) on graphene. Remarkably, the graphene plasmonic lens (GPL) exhibits electrically tunable virtual focusing capabilities. The design principle of the tunability is based on the nonlinear relationship between surface plasmon polariton (SPP) wave mode index and chemical potential of graphene. By controlling the gate voltage of graphene, we can achieve continuous tuning of virtual focus. A ray-tracing technique is employed to determine the required gate voltages for various virtual focal lengths. The proposed GPL facilitates adjustable virtual focusing, promising advancements in highly adaptive and transformative nanophotonic devices.
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1. Introduction
The concept of the generalized Luneburg lens, as initially formulated by Luneburg in 1944 [1], describes a spherically symmetric refractive structure with a continuously varying index, capable of precisely projecting geometrical images of two concentric spheres onto each other. Eaton later expanded this concept to include scenarios in which one of the spheres resides within the lens itself, and the point source along with its image does not align with the lens’s center [2]. The Gutman lens represents another variant, functioning similarly to a Luneburg lens (LL) but with its focal surface (or arc, in planar designs) positioned inside the lens [3]. These formulations mentioned above inherently lead to lenses that produce real images.
Contrastingly, Nelson introduced a more versatile transformation in 2021, which involves finding a spherically symmetric inhomogeneous lens that can transform a given sphere outside of the lens into a virtual conjugate sphere, thus forming a virtual image from a real source [4]. They defined this problem as the complement to the generalized Luneburg lens problem. The essence of this complementary relationship lies in the focal areas of concern. The generalized Luneburg lens problem deals solely with real image focusing, where the image point and the source point are located on opposite sides of the lens. While the complementary problem defines lenses where the image point and the source point are on the same side of the lens, in which case the image point is a virtual image. In this article, given the association between these two problems, the complementary problem is referred to the virtual image Luneburg lens problem. A virtual image lens is a type of lens that can create a virtual image from a real source. The virtual image appears to be at the location from which the diverging rays seem to originate after passing through the lens or reflecting off a mirror [5]. One possible technology to implement the virtual image lens is to use a metalens, which is a flat surface that uses nanostructures to manipulate out-plane light [6–10].
Surface Plasmon Polaritons (SPPs) are a type of electromagnetic surface wave that propagates along the interface between a dielectric and a conductor, typically a metal [11–14]. Graphene SPPs combine the unique electronic properties of graphene, offering unprecedented opportunities for manipulating light at the nanoscale [15–17]. Their advantages of high confinement, tunability, and relatively low losses open the door to a new realm of possibilities in sensing, imaging, and photonic devices [18–22]. Vakil and Engheta theoretically showed that by designing and manipulating spatially inhomogeneous, nonuniform conductivity patterns across a flake of graphene, one can achieve a one-atom-thick SPP platform for infrared metamaterials and transformation optical devices [23]. Since then, plentiful works for graphene plasmonic lens (GPL) based on this approach have been reported [24–31]. However, these GPLs are exclusively associated with real images, and the tunability of the focusing properties of these lenses is relatively limited [32–35]. This is mainly because these devices were designed to work in the almost linear region in which SPP wave index (or graphene conductivity) changes very slowly with chemical potential when adjusting the gate voltage of graphene. Recently, we reported a method to exploit the nonlinear relationship between SPP wave index and chemical potential, achieving a transformation between a Maxwell Fisheye lens and a Luneburg lens [36]. This nonlinear relationship significantly enhances the tunability of graphene-based devices.
In this article, we introduce a systematical design procedure for an electrically tunable virtual image Luneburg lens on graphene. The proposed lens is designed to work in a nonlinear region in which the SPP wave index changes appreciably with the chemical potential of graphene. By adjusting the gate voltage of graphene, the tunable virtual focusing property can be achieved. A ray-tracing technique is employed to examine the focusing performance, and full-wave EM simulations based on the finite element method software COMSOL are also given for verifications.
2. Theoretical analysis and structure
The general solution to the virtual image Luneburg lens problem was first formulated in [4], based on a variation of the inverse Radon Transform introduced by Luneburg. For a nominal case, where the point source, the point image, and the lens center are aligned, the refractive index $n(r)$ of the complementary lens problem can be described in the form of $(n(\rho ), r(\rho ))$:
Fig. 1. Schematic representations of the spherically symmetric inhomogeneous virtual image Luneburg lens problems. Two cases are illustrated: (a) $r_s < r_v$ and (b) $r_s > r_v$. The circles delimit the boundaries between the inhomogeneous medium $n(r)$, and the homogeneous medium $n = 1$. The notations $n(r)>1$ and $n(r)<1$ respectively indicate that the refractive index distribution within the lens is either entirely greater than $1$ or entirely less than $1$. The dotted line is the virtual portion of the ray, as it appears to emerge from the virtual image in the homogeneous medium.
To realize a tunable Luneburg lens on a graphene platform, here, we first investigate the electrical tunability of graphene. The effective mode index $n_{spp}$ for TM-polarized SPP waves propagating on an isolated graphene sheet is given by [23]:
where $\eta _0$ denotes the intrinsic impedance, and $\sigma _{g}$ is the surface conductivity of graphene, which can be modeled from the Kubo formula [38]. For graphene’s conductivity tuning, one simple way is to modify the chemical potential using the back-gated structure, which is directly dependent on the applied gate voltage [39–41]:In this article, a back-gated structure [23] is utilized to manipulate SPP wave propagation on graphene. This can be realized by employing a single-layer graphene sheet on top of a continuously shaped dielectric spacer, which is grown on a gold ground plane, as shown in Fig. 2. For a given gate voltage, the SPP index will vary with dielectric thickness $t$, leading to a plasmonic gradient index (GRIN) lens. It is worth pointing out that, in the $r_s < r_v$ case, the SPP index profile exhibits a maximum value at the center of the lens and decreases monotonically along the radius, as shown in Fig. 2(b), leading to a pool-shaped ground. In the $r_s > r_v$ case, the SPP index profile exhibits a minimum value at the center of the lens and increases monotonically along the radius, leading to a hill-shaped ground, as shown in Fig. 2(d). In this article, the GRIN lens with a continuously shaped dielectric spacer is employed to simplify the theoretical calculations, and the conclusions drawn from this structure will be applied to all other fabrication schemes [23,27,30], which are based on concentric ring or quasicrystal metasurfaces.
Fig. 2. (a) SPP index profile and (b) cross-section view of the GPL structure in the case $r_s<r_v$, where the gradient SPP index distribution decreases radially. (c) SPP index profile and (d) cross-section view of the GPL structure in the case $r_s > r_v$, where the gradient SPP index distribution rises radially. In (a), (c), a darker color represents a larger SPP index, while a lighter color represents a smaller SPP index.
3. Numerical results
3.1 Virtual image lens with a source closer to the lens than the image
To evaluate the electrical tunability of the GPL, we employ a metric that examines the ratio of the lens’s central refractive index ($n_c$) to the background refractive index ($n_b$), observing the variation of this ratio in response to the gate voltage. The refractive index ratio $n_c/n_b$ should be able to be reduced from $\sqrt {2}$ to $1.31, 1.28, 1.20$ in order to achieve the transformation from the LL ($r_s=\mathrm {R}, r_v=\infty$) to the virtual image lens with $r_v=4\mathrm {R}, 3\mathrm {R}, 2\mathrm {R}$, respectively, as calculated from Eqs. (1–2). For this purpose, a LL with a diameter of $2\mathrm {R}=1.5\, \mathrm {\mu}\textrm{m}$ is first implemented on a graphene sheet using the back-gate structure, as shown in Fig. 2(a,b). The initial gate voltage is chosen as $V_g=10.0\,\mathrm {V}$. Basically, this choice will not affect the propagation characteristics of the lens. On the other hand, the choice of the initial background chemical potential $\mu _{c0}$ is critical for lens tunability, and can be chosen from the intermediate region $0.10 \leq \mu _{c0} \leq 0.15\,\mathrm {eV}$ as discussed in [36]. With the given $V_g$ and $\mu _{c0}$, the thickness distribution of the dielectric spacer can then be calculated to realize a LL according to Eqs. (3–4). It should be noted that, when increasing the gate voltage $V_g$, the index distribution of the lens will no longer be maintained as a LL, because of the nonlinear relationship between the SPP wave index and the gate voltage.
Following the above procedure, Fig. 3 plots the calculated index ratio $n_c/n_b$ as a function of gate voltage $V_g$ for different choices of the initial background chemical potential $\mu _{c0}$. It can be observed that when $\mu _{c0} > 0.13\, \mathrm {eV}$, the index ratio cannot be reduced to $1.20$, which implies that the initially constructed LL will not be able to be transformed to the virtual image lens with $r_v=2\mathrm {R}$, no matter how to adjust the gate voltage. Therefore, a good choice of range for $\mu _{c0}$ can be found between $0.10\,\mathrm {eV}$ and $0.12\,\mathrm {eV}$. In this article, we choose the initial chemical potential of background $\mu _{c0}=0.12\,\mathrm {eV}$ as a design example. To satisfy the index ratio $n_c/n_b=1.20$ for the case of $r_v=2\mathrm {R}$, the corresponding gate voltage is found to be $42.0\,\mathrm {V}$.
Fig. 3. The index ratio $n_c/n_b$ as a function of gate voltage $V_g$ for $r_s=\mathrm {R}$. The green, red and black dotted lines represent the theoretical index ratio of $1.31$, $1.28$ and $1.20$ for $r_v = 4\mathrm {R}, 3\mathrm {R}$ and $2\mathrm {R}$, respectively.
Figure 4(a) shows the ray trajectories through the designed LL, in which a point source located at the circumference of the lens is imaged at infinity (i.e. $r_v=\infty$) at $V_g=10\,\mathrm {V}$. For this purpose, a MATLAB code is programmed to realize ray tracing through the lens. The designed LL exhibits a central SPP index of $n_c=141.4$ and a background index of $n_b=100$, respectively. When the gate voltage increases to $42.0\,\mathrm {V}$, the central index and the background index decrease to $44.4$ and $37.0$, respectively. The outgoing rays become apparently divergent with reverse extended rays focused at the point $x=-1.5\,\mathrm {\mu}\textrm{m}$ (i.e. $r_v=2\mathrm {R}$), which can be regarded as the virtual image of the point source, as illustrated in Fig. 4(b).
Fig. 4. Ray trajectories through the lens ($r_s=\mathrm {R}$) with gate voltage of (a) $10.0\, \mathrm {V}$ and (b) $42.0\, \mathrm {V}$.
Furthermore, it would be expected that $r_v$ can be continuously tuned between $\infty$ and $2\mathrm {R}$, if adjusting the gate voltage from $10.0\, \mathrm {V}$ to $42.0\, \mathrm {V}$, while fixing the source point at $r_s=\mathrm {R}$. To find the corresponding gate voltage for achieving optimal virtual focusing performance at different point $\mathrm {V}$, we can define a defocus error function $\xi \left (\mu _{c0}, V_g\right )=\sqrt { \int _{0}^{\theta _{max}} h^{2}(\mu _{c0},V_g,\theta )\cos ^{-1}(\theta )d\theta }$ using ray-tracing technique, where $\theta$ is the incident angle for a particular ray trajectory, and $h$ denotes the distance between the reverse extended ray and the virtual image point $\mathrm {V}$, as shown in Fig. 5(a). Especially, we add a factor $\cos ^{-1}(\theta )$ to enhance the weight of the rays with large incident angles, since the position of the virtual image is more sensitive to rays with large incident angles.
Fig. 5. Error function for continuous virtual focus tuning. A point source is located at point $\mathrm {S}$, with a corresponding virtual image at point $\mathrm {V}$. (a) Illustration of ray-tracing for the defocus error function. (b)-(d) 2D plots of numerical results of the defocus error function for $r_v=4\mathrm {R}, 3\mathrm {R}$ and s$2\mathrm {R}$, respectively.
Figure 5(b)-(d) demonstrates 2D plots of numerical results of the defocus error function $\xi \left (\mu _{c0},V_g\right )$ from ray-tracing. As shown, a curve of minimum defocus error can then be plotted by the white circle line for each given value of $r_v$. It can be seen that the optimal gate voltage continuously increases with the $\mu _{c0}$ for a given $r_v$. For the case $r_v=2\, \mathrm {R}$, the initial background chemical potential should be chosen below $0.125\,\mathrm {eV}$ to realize a good virtual image focusing, which agrees with the conclusion drawn from Fig. 3. Based on these plots, one can find the required gate voltage for a particular virtual focal length.
To further validate the theory, a full-wave model of the graphene-based virtual image lens has been implemented utilizing COMSOL. For this purpose, the same initial background chemical potential $\mu _{c0}=0.12\,\mathrm {eV}$ is chosen. It can be obtained from Fig. 5(b-d) that the optimal gate voltage for achieving virtual focal length $r_v$ of $4\mathrm {R}$, $3\mathrm {R}$, $2\mathrm {R}$ are $13.9\,\mathrm {V}$, $17.1\,\mathrm {V}$ and $42.0\,\mathrm {V}$, respectively. Figure 6(a-d) gives the corresponding field distributions for different gate voltages. The virtual image is marked as a red dot, and the corresponding wavefront from the virtual image without the lens is illustrated as a black arc. To facilitate a clear comparison, the positions of the wavefront arcs from different virtual images are fixed at the same location, approximately $1.5\mathrm {R}$ from the lens circumference. The equiphase wavefronts arising from the point source and transformed by the lens show good agreement with the black arcs. In this way, the transformed wavefronts from the lens appear to be emitted from the virtual image. In addition, the realized index profiles at different gate voltages agree very well with the theoretical refractive index profiles calculated from Eqs. (1–2) of the corresponding virtual image lens for all four cases, especially around the lens center area, as shown in Fig. 6(e). To enhance the clarity of this comparison, the data on the x-axis uses a normalized radius $r=\lvert x \rvert /\mathrm {R}$.
Fig. 6. Full-wave simulation results of a virtual image lens with $r_s=\mathrm {R}$. (a)-(d) Field distributions for $r_v=\infty, 4\mathrm {R}$, $3\mathrm {R}$ and $2\mathrm {R}$, respectively. The virtual image is illustrated as a red dot and the corresponding wavefront is represented as a black arc, showing good agreement with the equiphase wavefront transformed by the lens at all gate voltages. (e) The index profile comparison between theoretical solutions and realized results.
Moreover, the design process for the tunable virtual image lens with $r_s = \mathrm {R}$ can be extended to the case for $r_s > \mathrm {R}$. In this article, we choose $r_s = 1.5\mathrm {R}$ as an illustrative example. The lens can be designed following the same steps as described above. In this way, the initially designed lens would be a generalized Lunebueg lens with $r_s =1.5 \mathrm {R}, r_v=\infty$. The index ratio $n_c/n_b$ should be tunable from $1.24$ (for $r_v=\infty$) to $1.15$ (for $r_v=4\mathrm {R}$) and $1.12$ (for $r_v=3\mathrm {R}$), according to Eqs. (1–2). This is also achievable by choosing the initial chemical potential of background as $0.12\, \mathrm {eV}$, as indicated in Fig. 7. Figure 8 shows the simulated field distributions for $r_v=\infty, 4\mathrm {R}$ and $3\mathrm {R}$ with the corresponding gate voltage of $10.0\,\mathrm {V}$, $23.0\,\mathrm {V}$ and $60.0\,\mathrm {V}$, respectively. It can be seen that by tuning the gate voltage, the position of the virtual image moves accordingly. The comparison between the realized and theoretical refractive index distributions, as shown in Fig. 8(d), also exhibits a good agreement.
Fig. 7. The index ratio $n_c/n_b$ as a function of the gate voltage $V_g$ for $r_s=1.5\mathrm {R}$. The red and black dotted lines represent the theoretical index ratio of $1.15$ and $1.12$ for $r_v =4\mathrm {R}$ and $3\mathrm {R}$, respectively.
Fig. 8. Full-wave simulation results of a virtual image lens with $r_s=1.5\mathrm {R}$. (a)-(c) Field distributions for $r_v=\infty$, $4\mathrm {R}$ and $3\mathrm {R}$, respectively. The black arcs centered on the virtual image agree well with the equiphase wavefront transformed by the lens at all gate voltages. (d) The index profile comparison between theoretical solutions and realized results.
3.2 Virtual image lens with a source farther from the lens than the image
In this section, we will further explore the tuning characteristics for the second type of virtual image Luneburg lens, in which the source is located farther from the lens than the image, as depicted in Fig. 1(b). For this type of lens, the normalized refractive index distribution will fall below $1$ in the lens region, according to Eq. (1). Here, we select two typical values of $r_s$, namely $r_s = \infty$ and $3\mathrm {R}$ for demonstration. Again, the same ray-tracing technique is applied to examine such continuously focusing performance by following the design steps described in Section 3.1. For both cases, $r_v=\mathrm {R}$ is chosen as the initial image position, and the initial gate voltage is maintained as $10.0\,\mathrm {V}$. Figure 9 gives the calculated $n_c/n_b$ as a function of gate voltage $V_g$ for various initial background chemical potentials $\mu _{c0}$. For the case of $r_s=\infty$, the index ratio $n_c/n_b$ should be able to increase from $0.71$ to $0.81$, $0.85$ in order to realize the transformation from $r_v=\mathrm {R}$ to $1.5\mathrm {R}$, $2\mathrm {R}$, respectively, according to Eqs. (1–2). This is achievable by choosing the initial chemical potential $\mu _{c0}\leq 0.10\,\mathrm {eV}$, as observed from Fig. 9(a). Under the same condition, for the case of $r_v=3\mathrm {R}$, the index ratio $n_c/n_b$ is also found to be able to increase from $0.79$ to $0.86$, $0.89$ to realize the transformation from $r_v=\mathrm {R}$ to $1.25\mathrm {R}$, $1.5\mathrm {R}$, respectively, as shown in Fig. 9(b). For this reason, the initial background chemical potential is chosen as $0.10\,\mathrm {eV}$ for both $r_s=\infty$ and $r_s=3\mathrm {R}$ cases, and the conversion voltage values associated with different $r_v$ can be extracted from Fig. 9.
Fig. 9. The index ratio $n_c/n_b$ as a function of the gate voltage $V_g$ for $r_v=\mathrm {R}$, (a) $r_s=\infty$ and (b) $r_s=3\mathrm {R}$. The red and black dotted lines represent the theoretical index ratio of lenses with different $r_v$ respectively.
A ray-tracing representation is provided in Fig. 10 for a lens with $r_s=\infty$. Initially, with $V_g=10.0\,\mathrm {V}$, incident rays become reversely converging at $r_v=\mathrm {R}$. When tuning the gate voltage to $60.0\,\mathrm {V}$, the reverse extension of the outgoing rays will meet around the point $x=-1.5\, \mathrm {\mu}\textrm{m}$, corresponding to $r_v=2\mathrm {R}$. Field distributions for cases with $r_s=\infty$ and $3\mathrm {R}$ are illustrated in Figs. 11 and 12. These visuals confirm that the location of the virtual image is adjustable via the gate voltage. The black arcs, centered on the virtual image, align well with the lens-transformed equiphase wavefronts at all gate voltages. The realized index profiles under different gate voltages exhibit strong concordance with the theoretical profiles, particularly in the vicinity of the lens central region. These results confirm the proposed tunable GPL design’s efficacy and versatility in various virtual image Luneburg lens scenarios.
Fig. 10. Ray trajectories through the lens ($r_s=\infty$) with gate voltage of (a) $10.0\, \mathrm {V}$ and (b) $60.0\, \mathrm {V}$.
Fig. 11. Full-wave simulations of a virtual image lens with $r_s=\infty$. (a)-(c) Field distributions for $r_v=\mathrm {R}$, $1.5\mathrm {R}$ and $2\mathrm {R}$, respectively. The black arcs centered on the virtual image agree well with the lens-transformed equiphase wavefronts at all gate voltages. (d) The index profile comparison between theoretical solutions and realized results.
Fig. 12. Full-wave simulations of a virtual image lens with $r_s=3\mathrm {R}$. (a)-(c) Field distributions for $r_v=\mathrm {R}, 1.25\mathrm {R}$ and $1.5\mathrm {R}$, respectively. The black arcs centered on the virtual image agree well with the lens-transformed equiphase wavefronts at all gate voltages. (d) The index profile comparison between theoretical and realized results.
4. Conclusions
We proposed and numerically demonstrated a flexible design of spherically and rotationally symmetric GPL with electrically tunable virtual focal length, offering a practical realization of the virtual image Luneburg lens problem. The foundation of this tunability lies in the nonlinear relationship between SPP wave index and graphene chemical potential. A ray-tracing model was used to illustrate the focusing properties of the proposed virtual image lens and to determine the proper gate voltage for a particular position of the virtual image. The tunable virtual image lens is further validated using full-wave EM modeling, where the lens-transformed equiphase wavefronts agree with the corresponding theoretical wavefronts from the virtual image. A comparison between the theoretical refractive index profiles and the realized results confirmed the expected virtual image-focusing properties. The presented results may find opportunities in designing novel tunable transformational plasmonic devices.
Funding
National Natural Science Foundation of China (62071422); National Key Research and Development Program of China (2020YFB1805700).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this article are not publicly available at this time but may be obtained from the authors upon reasonable request.
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