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Abstract
A graphene plasmonic lens with an electrically tunable focal length is proposed and numerically investigated. The design philosophy of the proposed tunable lens is based on the nonlinear relationship of surface plasmon polariton (SPP) wave index with respect to chemical potential of graphene. By controlling the gate voltage of graphene, the proposed lens can be continuously tuned from a Maxwell Fisheye lens to a Luneburg lens. A ray-tracing method is employed to find out the corresponding gate voltages for various focal lengths. Full-wave EM simulations using COMSOL show that excellent focusing performances can be achieved. This work offers a new way in exploiting active transformational plasmonic elements in the mid-infrared region.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are electromagnetic surface waves originating from the interaction of electromagnetic field to the collective plasma excitations of metal. Due to the abilities to beyond the Abbe’s diffraction limit and implement tight field confinement, SPPs have promoted a large number of applications at subwavelength scale [1,2]. Transformation plasmonics provides a collection of techniques toward realization of on-chip transformation nanophotonic devices [3,4].
Achieving tunable optical elements that can adapt optical response to meet various application requirements has been a long-standing goal. In recent years, there have been considerable efforts dedicated to actively shaping out-of-plane wavefronts based on metasurface technology, both in theory and experimentation [5,6]. Traditionally, micro-electro-mechanical deformation or displacement has been employed to tune the optical response of metasurface devices [7]. Additionally, non-mechanical actuation mechanisms such as phase-change materials [8], thermo-optics [9], electro-optics [10,11], and all-optical effects [12] have been demonstrated as effective means of creating active metasurface devices.
In addition to wavefront shaping in free space, active control of in-plane plasmonic waves propagating along conducting surface holds significant promise for minimizing photonic integrated circuits. Optofluidics has been used to create reconfigurable lenses for metal SPP waves [13]. SPP waves traveling on graphene have attracted wide interest among researchers due to their two-dimensional manipulation of light and absence of geometrical aberrations [14,15]. Vakil and Engheta proposed that a one-atom-thick platform for infrared metamaterials and transformation optical devices could be achieved by designing and manipulating spatially inhomogeneous, nonuniform conductivity patterns across a graphene flake [16]. They suggested two prevailing scenarios for attaining nonuniform conductivity distributions, namely applying different gate voltages and designing the profile of the ground plane beneath the dielectric spacer holding the graphene. Since then, plentiful works for graphene plasmonic (GP) lens based on this approach have been reported [17–19]. Because of easy fabrication, graded index metasurfaces can be realized using uneven ground biased structure [20–22]. For instance, Zeng et al., Wu et al. and Xiong et al. designed different patterns of graded index metasurface by shaping the dielectrics underneath monolayer graphene into specific photonic crystals [23–25].
It is worth noting that the focusing performances of the GP lenses proposed in ref. [22,23] were found to be unaffected when varying the gate voltage of graphene. This is mainly because the proposed lenses were designed to work in the almost linear region in which SPP wave index (or graphene conductivity) changes very slowly with the chemical potential when adjusting the gate voltage of graphene. Besides, there are other works on active in-plane wavefront shaping by using graphene [26,27]. However, there is little research reported on the characteristics of the drastic change of graphene conductivity with respect to chemical potential to date.
In this article, the nonlinear relationship of the SPP wave index is investigated, and a systematical design procedure for an electrically tunable GP lens is presented. 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 designed Maxwell Fisheye Lens (MFL) can be transformed to a Luneburg Lens (LL) with the focus continuously tuned from lens circumference to infinity. For this purpose, a ray-tracing technique is employed to examine the focusing performance. The choice criterion for the initial chemical potential of the GP lens is discussed in terms of tuning ability. Full-wave EM simulations using commercial software COMSOL are also given for verification.
2. Theory and background
As has been proved by Luneburg in 1944, the refractive index distributions for the Generalized Luneburg lens (GLL) which form perfect images of two given spheres on each other may be described by a set of integral equations [28]. The two well-known solutions can be derived as:
Maxwell Fisheye Lens ($f_1=f_2=1$):
Luneburg Lens ($f_1=1, f_2=\infty$):
For TM-polarized SPP waves propagating on an isolated graphene sheet, the dispersion relation is given by [16]
Here $K_B$ is the Boltzmann constant, $e$ is the electron charge, $\hbar$ is the reduced Planck constant, $\omega$ 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 is predictable with a graphene gate voltage [30–32]:
The dependence of SPP wave index on the chemical potential of graphene, calculated from Eqs. (3–5), is given in Fig. 1. As shown, both the real part and the imaginary part of the SPP wave index decrease nonlinearly with the chemical potential of graphene. The variations of the SPP wave index can then be divided into three regions, i.e., the steep region ($\mu _{c} < 0.10\, \mathrm {eV}$), the intermediate region ($0.10 \leq \mu _{c} \leq 0.15\, \mathrm {eV}$), and the flat region ($\mu _{c} > 0.15\, \mathrm {eV}$). Most of the previous research works on GP lens were designed to operate in the flat region in which the SPP wave index only slightly changes with the chemical potential of graphene, and therefore exhibits quite constant focusing performances. On the other hand, the SPP wave index changes dramatically in the steep region and therefore exhibits a strong nonlinearity. However, this region should be avoided because of its high propagation losses. Between these two regions, there exists an intermediate region, in which the SPP wave index exhibits an appreciable nonlinearity as well as acceptable propagation losses ($<-15\, \mathrm {dB}/ \mathrm {\mu}\mathrm {m}$). Therefore, we can take advantage of this intermediate region to design an electrically tunable lens.
Fig. 1. Variations of the real and imaginary part of SPP wave index with respect to chemical potential of graphene at $\lambda_0 =10\,\mathrm{\mu}\mathrm{m}$ and $T=300\,\mathrm {K}$. A log-$10$ scale is used for the x-axis.
As shown in Fig. 2, a back-gated structure [16] is utilized to manipulate SPP waves propagating on a graphene sheet. The structure model incorporates a single-layer graphene sheet on top of a continuously shaped dielectric spacer (e.g. a silicon spacer), which is grown on a gold ground plane. When a static gate voltage is applied between the graphene sheet and the gold ground plane, the SPP wave index will vary with dielectric thickness $t$, thus a spherically symmetric gradient index (GRIN) plasmonic lens can be realized. It should be mentioned that a number of fabrication schemes have been reported for GRIN lenses, such as concentric rings [16], quasicrystal metasurfaces [23] and Dot-Density-Renderer structure [24]. 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 schemes.
Fig. 2. (a) Top and (b) Cross-section view of the GP lens structure. The silicon layer is grown on the gold substrate, with a monolayer of graphene on the top. The surface conductivity of graphene can be tuned via an external gate voltage.
3. GP lens transformation
3.1 Transformation from MFL to LL
Here, we will demonstrate a GP lens that can be easily transformed from a MFL to a LL by adjusting the gate voltage of the graphene sheet. For this purpose, a MFL with a diameter of $2\mathrm {R}=1.5\, \mathrm {um}$ is first implemented on a graphene sheet, as shown in Fig. 2. The detailed procedure can be described as follows:
- 1) Choose an initial gate voltage, e.g., $V_g=10.0\, \mathrm {V}$. Theoretically, this value will not affect the propagation characteristics of the lens.
- 2) Choose an initial chemical potential $\mu _{c0}$ of the background graphene from the intermediate region which is located between $0.10$ and $0.15\, \mathrm {eV}$. Calculate the corresponding background SPP wave index $n_b$. It should be noted that choice of the initial chemical potential $\mu _{c0}$ is critical for the lens transformation from a MFL to a LL, and we will discuss the details later in this article.
- 4) Increase the gate voltage $V_g$, the index distribution of the lens will no longer be maintained as a MFL distribution because of the nonlinear relationship between the SPP wave index and the gate voltage. For each different index distribution, calculate the index ratio between the lens center and the background graphene $n_c/n_b$. Note that the index ratio $n_c/n_b$ will decrease with the gate voltage. This is because the value of $n_c$ decreases faster than $n_b$ with increased gate voltages, due to the fact that the SPP wave index exhibits a convex function of the chemical potential as shown in Fig. 1.
This index ratio $n_c/n_b$ is an important measurement with regard to the lens tunability. In order to realize lens transformation from a MFL to a LL, the index ratio $n_c/n_b$ should be tunable between $2$ (for MFL) and $\sqrt {2}$ (for LL). As shown in Fig. 3, the index ratio decreases with the gate voltage faster for a smaller initial chemical potential $\mu _{c0}$. This is simply because the nonlinearity becomes stronger for a smaller initial chemical potential, as shown in Fig. 1. It can be also observed that when $\mu _{c0} \geq 0.13\, \mathrm {eV}$, the index ratio cannot be reduced to $\sqrt {2}$, which implies that the initially constructed MFL will not be able to be transformed to a LL, no matter how to adjust the gate voltage. On the other hand, too strong nonlinearity should be carefully avoided, since it will lead to high propagation losses. Considering all of the above criteria, a good choice of range for $\mu _{c0}$ can be found between $0.10\,\mathrm {eV}$ and $0.13\,\mathrm {eV}$. In particular, to satisfy index ratio $n_c/n_b=\sqrt {2}$, the corresponding gate voltages are found to be $16.0\,\mathrm {V}$, $20.5\,\mathrm {V}$, $30.0\,\mathrm {V}$, $60.0\,\mathrm {V}$ for initial chemical potential $\mu _{c0}$ of $0.10\,\mathrm {eV}$, $0.11\,\mathrm {eV}$, $0.12\,\mathrm {eV}$, and $0.13\,\mathrm {eV}$, respectively.
In this article, we choose an initial chemical potential $\mu _{c0}=0.12\, \mathrm {eV}$ for demonstrating lens transformation from a MFL to a LL. For this purpose, the initial gate voltage remains as the same $V_g=10.0\, \mathrm {V}$. Figure 4(a) plots the calculated lateral profile of SPP wave index and the corresponding thickness profile of the dielectric spacer for the constructed MFL according to Eqs. (1–4). As shown, the calculated SPP wave indices are 200.0 and 100.0 for lens center and graphene background ($n_c/n_b$=2). When the gate voltage is increased to $30.0\,\mathrm {V}$, the two SPP wave indices are found to decrease to $64.0$ and $45.2$ ($n_c/n_b$=$\sqrt {2}$). A ray-tracing technique is then employed to depict SPP waves propagating through the lens. Figure 4(b) gives the exit angles $\theta _e$ for rays with various incident angles $\theta _i$. It clearly shows that the exit angles $\theta _e$ approach to almost $0^\circ$ at the gate voltage of $30.0\, \mathrm {V}$. In addition, the ray trajectories in Fig. 4(c) show that a point source located at the circumference of the lens can be successfully imaged at the symmetrical position of the lens at $V_g=10.0\, \mathrm {V}$, and from Fig. 4(d) the rays will exit the lens almost in parallel at $V_g=30.0\, \mathrm {V}$. Here, only incident angles within $\pm 60^\circ$ are considered because most of incident energy is within this range. Therefore, the designed MFL ($f_1=f_2=\mathrm {R}$) is transformed to a LL ($f_1=\mathrm {R}, f_2=\infty$) only by adjusting the gate voltage from $10.0\, \mathrm {V}$ to $30.0\, \mathrm {V}$. Full-wave simulations using commercial software COMSOL based on finite element methods (FEM) are employed to verify the propagation features. The corresponding field distributions are presented in Fig. 4(e-f), which clearly proves this transformation.
Fig. 4. Transformation from MFL to LL by adjusting the gate voltage of graphene. (a) Thickness profile of the designed MFL (green line) with $\mu _{c0}=0.12\, \mathrm {eV}$, the corresponding SPP wave index at $V_g=10.0\, \mathrm {V}$ (blue line) and $30.0\, \mathrm {V}$ (red line). (b) The exit angle $\theta _e$ as a function of the incident angle $\theta _i$ with gate voltage of $10.0\, \mathrm {V}$ and $30.0\, \mathrm {V}$. (c, d) Ray trajectories in the lens at $V_g=10.0\, \mathrm {V}$ and $30.0\, \mathrm {V}$. (e, f) Field distributions at $V_g=10.0\, \mathrm {V}$ and $30.0\, \mathrm {V}$.
It should be noted that the focusing performances and the electrical tunability of the proposed GP lens remain unaffected when including the propagation loss in the simulation. However, the field intensity at the focal point will become weaker because of the propagation loss. To quantitatively characterize the reduction of field intensity at the focal point, we compute the propagation loss by using a path integral method for the case of $V_g=10\,\mathrm {V}, 30\,\mathrm {V}$, respectively. The calculated propagation loss along the lens diameter is about $-18.0\,\mathrm {dB}$ and $-4.5\,\mathrm {dB}$ for $V_g=10\,\mathrm {V}, 30\,\mathrm {V}$, respectively.
3.2 Continuous focusing from MFL to LL
It would be expected that $f_2$ can be continuously tuned between $\mathrm {R}$ and $\infty$ if adjusting the gate voltage of graphene sheet between $10.0\, \mathrm {V}$ and $30.0\, \mathrm {V}$ while fixing $f_1=\mathrm {R}$. Again, a ray-tracing technique can be used here to examine such continuous focusing performance. For this purpose, it will be more convenient to place a point source at point $\mathrm {S}$ with a distance $f_2$ in ray-tracing. To find the required gate voltage of graphene sheet for achieving optimal focusing at point $\mathrm {F}$, we define a defocus error function $\xi \left (\mu _{c0}, V_g\right )=\sqrt { \int _{0}^{\theta _{max}} h^{2}(\mu _{c0},V_g,\theta )\cos ^{2}(\theta )d\theta }$, where $\theta$ is the incident angle, $\theta _{max}$ is set to $60^\circ$ and $h$ denotes the distance between the exit ray and the point $\mathrm {F}$, as shown in Fig. 5(a). In particular, a factor $\cos (\theta )$ is added to enhance the weight of the rays in the center region.
Fig. 5. Error function for continuous focus tuning. (a) Illustration of ray-tracing. (b) The plot of $\mathrm {R}/f_2$ as a function of gate voltage for different $\mu _{c0}$. (c-f) The error functions for $f_2=2\mathrm {R}, 3\mathrm {R}, 4\mathrm {R}$ and $\infty$.
Figure 5(c-f) gives the 2D plot of the numerical results of the defocus error function $\xi \left (\mu _{c0},V_g\right )$ from ray-tracing. The calculation procedure can be described as follows:
- 1) Choose the same initial gate voltage $V_g=10.0\, \mathrm {V}$.
- 2) Choose an initial chemical potential of the background graphene $\mu _{c0}$. Calculate the thickness profile of the dielectric spacer to realize the index distribution of a MFL.
- 3) Place a point source at point $\mathrm {S}$, with a distance $f_2$ equal to $2\mathrm {R}$, $3\mathrm {R}$, $4\mathrm {R}$ and $\infty$, respectively. Calculate the corresponding error functions $\xi \left (\mu _{c0},V_g\right )$.
- 4) Sweep the gate voltage from $10-60\, \mathrm {V}$, and repeat step 3.
- 5) Sweep the initial chemical potential of the background graphene $\mu _{c0}$ from $0.10\, \mathrm {eV}$ to $0.16\, \mathrm {eV}$, and repeat steps 1-4.
For each given value of $f_2$, a curve of minimum defocus error can then be plotted by the white circle line, as shown in Fig. 5(c-f). This curve represents the optimal focusing at point $\mathrm {F}$. It can be seen that the optimal gate voltage continuously increases with the $\mu _{c0}$ for a given $f_2$. In particular, there will be no appropriate gate voltage which can lead to a good focusing when the $\mu _{c0}$ is larger than $0.13\, \mathrm {eV}$ for the case $f_2=\infty$ (see Fig. 5(f)). In this way, the designed MFL will not be able to be transformed to a LL, which agrees with the conclusion drawn from Fig. 3. Figure 5(b) plots the normalized focal power $\mathrm {R}/f_2$ as a function of the gate voltage. As shown, the normalized focal power $\mathrm {R}/f_2$ continuously varies with the gate voltage. Therefore, the focal point of the designed lens can be continuously tuned from $f_2=\mathrm {R}$ (MFL) to $f_2=\infty$ (LL).
For full-wave EM verifications, the same initial chemical potential $\mu _{c0}=0.12\, \mathrm {eV}$ is used. According to Fig. 5(b), the optimal gate voltage at focal length of $2\mathrm {R}, 3\mathrm {R}, 4\mathrm {R}$ and $\infty$ are $13.4\, \mathrm {V}, 15.4\, \mathrm {V}, 17.0\, \mathrm {V}$ and $30.0\, \mathrm {V}$, respectively. Figure 6(a-e) plots the corresponding field distributions simulated by using COMSOL. As shown, a point source with various focal lengths $f_2$ can be successfully imaged at the circumference of the lens with the corresponding optimal gate voltages. Figure 6(f) presents the normalized intensities along the dot lines in 6(a-e) with respect to the maximum amplitudes of the focusing spot. The full-widths at half-maximum (FWHM) of the focusing spot for MFL and LL are $\lambda _0/238$ and $\lambda _0/105$ respectively. Considering the corresponding effective wavelength of SPP waves at the circumference of lens (e.g., $\lambda _p=\lambda _0/100$ for MFL and $\lambda _p=\lambda _0/45.2$ for LL), the focusing spot sizes are measured between $0.43\lambda _p-0.47\lambda _p$ for all the cases. In addition, Fig. 6(g) compares the realized index distributions of the GP lens with the theoretical index distributions of the corresponding GLL. As shown, the realized index distributions agree quite well with the theoretical results, especially around the lens center area. This partly verifies the continuous focusing from MFL to LL observed in our full-wave simulations.
Fig. 6. Full-wave simulation results for $\mu _{c0}=0.12\, \mathrm {eV}$. (a-e) Field distributions with the gate voltage of $10.0\, \mathrm {V}$, $13.4\, \mathrm {V}$, $15.4\, \mathrm {V}$, $17.0\, \mathrm {V}$ and $30.0\, \mathrm {V}$. (f) The FWHM of the focusing spot. (g) The index profile comparison between theoretical solutions and realized results.
3.3 More GLL cases
The above design procedure for the tunable lens with $f_1= \mathrm {R}$ can be further extended to the case of $f_1 < \mathrm {R}$, or $f_1 > \mathrm {R}$. For a lens with two external foci ($f_1 > \mathrm {R}$, $f_2 \geq \mathrm {R}$), the refractive index distribution $n(r)$ of the generalized Luneburg lens can be described by the following parametric equations:
For a lens with one internal focus and one external focus ($f_1 < \mathrm {R}$, $f_2 \geq \mathrm {R}$), the index distribution may be discontinuous at the interface of $r=f_1$. To avoid such discontinuity, we choose Brown’s solution [33] for the index distribution $n(r)$ in the region $f_1 \leq r \leq 1$:
In this article, we choose $f_1=0.8 \mathrm {R}$ (internal focus) and $f_1=2 \mathrm {R}$ (external focus) for demonstration. The lens can be designed following the same steps in Section 3.1, and a similar ray-tracing procedure as described in Section 3.2 can be used, with the same initial gate voltage of $10\, \mathrm {V}$. For the internal focus case of $f_1=0.8 \mathrm {R}$, the index ratio $n_c/n_b$ should be tunable between $2.25$ (for $f_2=1$) and $1.59$ (for $f_2=\infty$), according to Eqs. (9–11). This is achievable by choosing the initial chemical potential of background as $0.12\, \mathrm {eV}$, as shown in Fig. 7(a). However, for the external focus case of $f_1=2 \mathrm {R}$, a tuning range of the index ratio $n_c/n_b$ between $1.66$ and $1.17$ is required, according to Eqs. (7,8). For this purpose, we choose a smaller chemical potential of background $0.10\, \mathrm {eV}$ to enhance the lens tunability, as shown in Fig. 7(b).
Fig. 7. The calculated index ratio $n_c/n_b$ as a function of the gate voltage $V_g$, (a) $f_1=0.8\, \mathrm {R}$ and (b) $f_1=2\, \mathrm {R}$.
Figure 8 shows the simulated field distributions with $f_2=\mathrm {R}, 2\, \mathrm {R}, 3\, \mathrm {R}$ and $\infty$ for $f_1=0.8 \mathrm {R}$, and $f_1=2 \mathrm {R}$, respectively. It can be seen that by tuning the gate voltage, the source point located at a distance $f_2=\mathrm {R}, 2\, \mathrm {R}, 3\, \mathrm {R}$ and $\infty$ can be well imaged at the focal point $f_1= 0.8\, \mathrm {R}$, or $2\, \mathrm {R}$. The corresponding gate voltages can be determined from Fig. 7(a) and (b). These results show that the proposed design method for GP lens is applicable in realizing lens transformation for various cases of the GLL.
Fig. 8. Full-wave simulation results for GP lens. (a-d) Field distributions of $f_1=0.8\, \mathrm {R}$ with the gate voltage of $10.0\, \mathrm {V}$, $12.4\, \mathrm {V}$, $13.5\, \mathrm {V}$, and $18.5\, \mathrm {V}$. (e-h) Field distributions of $f_1=2\, \mathrm {R}$ with the gate voltage of $10.0\, \mathrm {V}$, $13.3\, \mathrm {V}$, $15.9\, \mathrm {V}$, and $55.0\, \mathrm {V}$.
4. Discussion
This article proposed an electrically tunable GP lens, which is realized based on the nonlinear relationship between the SPP wave index and the chemical potential of graphene. By adjusting the gate voltage of graphene, the proposed GP lens can be continuously tuned from a MFL to a LL. A ray-tracing technique is employed to find out the required gate voltages for achieving various focal lengths, and full-wave simulations are carried out to verify the focusing performance. Both numerical calculations and full-wave simulations show that the proposed GP lens own remarkable tunability within an ultra-compact structure. The presented results may find opportunities in designing novel tunable transformational plasmonic devices.
Funding
National Key Research and Development Program of China (2020YFB1805700); National Natural Science Foundation of China (62071422).
Disclosures
The authors declare no conflicts of interest.
Data Availability
The 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.
Supplemental document
See Supplement 1 for supporting content.
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