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Abstract
We propose a scheme to achieve controllable nonreciprocal behavior in asymmetric graphene metasurfaces composed of a continuous graphene sheet and a poly crystalline silicon slab with periodic grooves of varying depths on each side. The proposed structure exhibits completely asymmetric reflection in opposite directions in the near-infrared range, which is attributed to the pronounced structural asymmetry and its accompanying nonlinear effects. The obtained nonreciprocal reflection ratio, reaching an impressive value of 21.27 dB, combined with a minimal insertion loss of just -0.76 dB, highlights the remarkable level of nonreciprocal efficiency achieved by this design compared to others in its category. More importantly, the proposed design can achieve dynamic tunability by controlling the incident field intensity and the graphene Fermi level. Our design highlights a potential means for creating miniaturized and integratable nonreciprocal optical components in reflection mode, which can promote the development of the integrated isolators, optical logic circuits, and bias-free nonreciprocal photonics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nonreciprocal optical effects are imperative to a wide range of applications, such as full-duplex communication systems and lasers, so that undesirable parasitic feedback can be prevented [1–3]. Traditionally, the Faraday effect in magneto-optical materials, such as iron garnets, has been employed to break reciprocity, where an asymmetric permittivity tensor is obtained under biasing with static magnetic fields [4–6]. A more recent proposal involves structures with time-dependent modulation of refractive index, where time-reversal symmetry is explicitly broken to achieve nonreciprocity of the investigated system [2,7]. Combined with spatial asymmetries, the nonlinear effect has become a popular research topic to achieve strong nonreciprocal responses, because of the simple material requirement and fabrication processes for these nonreciprocal devices, but with limitations in terms of simultaneous excitation from opposite directions [8–11]. For example, nonreciprocal transmission has been demonstrated by designing bifacial plasmonic metasurfaces, consisting of asymmetric silver nanostripes separated by an ultrathin dielectric spacer layer [12]. In addition, nonlinear bifacial dielectric metasurfaces are also designed to realize nonreciprocal behavior, benefiting from high structural asymmetry and strong field enhancement [13].
However, these nonreciprocal devices by using the materials’ nonlinear effects always require high power of the incident signals, which limits their nonreciprocal efficiency to some extent [14]. To reduce this optical intensity requirement, it is important to develop ingenious resonant structures, with more effective excitation of devices and great enhancement of light-matter interaction and nonlinear generation, such as waveguide devices, cavities, gratings, and so on [15–20]. Another promising solution to this problem is the utilization of desired nonlinear material with large nonlinear indices. Some semiconductors, such as silicon, have high third-order Kerr nonlinearity, which can be widely used in the dielectric metasurfaces to realize nonreciprocal response [13,21–23]. Graphene, a single layer of carbon atoms arranged in a two-dimensional (2D) lattice, has emerged as a promising material for manipulating light-matter interactions due to its exceptional electronic and optical properties. In particular, it has very large nonlinear coefficients [24–27]. The tunability of graphene’s conductivity through electrostatic gating or chemical doping makes it an ideal platform for designing metasurfaces with dynamic tunability. Furthermore, the introduction of graphene is crucial for the advancement of compact nonreciprocal systems that find applications in optical communications, signal processing, and various other fields [2,28–31].
Optical nonreciprocity in reflection mode, which is also known as unidirectional invisibility [32] or one-way mirror [33], has been widely demonstrated in parity-time (PT)-symmetric and non-PT-symmetric systems. In this scenario, systems can reflect waves from a particular direction but not the complementary one. Dual-band nonreciprocal reflection (NR) has been observed at exceptional points in a compact plasmonic waveguide structure [34]; Controllable NR in a non-PT-symmetric metasurface has been demonstrated theoretically and numerically based on the far field coupling [35]; Completely asymmetric reflections in opposite light propagations has been experimentally reported with a nonreciprocal space-time phase modulated nonlinear metasurface, consisting of different resonating dielectric nanoantennas [7]. NR of the semiconductor InSb is explored in an applied magnetic field in Voigt geometry [33,36] Based on this theory, a simple and tunable one-way mirror is experimentally designed, with high isolation power of 35 dB and low insertion loss (IL) of only -6.2 dB, simultaneously [33]. The terahertz NR has been successfully achieved through the implementation of destructive interference between the direct reflection and the multiple reflections in the resonance cavity between the InSb and metasurface structure, where the isolation power can reach 55 dB, while the IL is −3.92 dB, greatly improved compared to the pure InSb [36].
In this study, our objective is to develop simple and tunable structures using graphene metasurfaces, in order to achieve both high nonreciprocity and comparatively low IL in reflection mode. By placing a continuous graphene sheet on a poly crystalline silicon slab with different groove depths on either side, a highly pronounced structural asymmetry is obtained. This asymmetry holds great significance in enhancing nonreciprocal efficiency, particularly when combined with the system’s nonlinear effect. Without physical mechanisms involved, the proposed structure itself can achieve a high nonreciprocity of 21.02 dB, which can be further improved to 21.27 dB by considering its nonlinear response. The obtaining IL with the optimized structure design is as low as -0.76 dB, comparable to other previous works [33,36,37]. Furthermore, dynamic tunability of the nonreciprocal nonlinear effect is highlighted through the manipulation of the incident field intensity and the Fermi level of the graphene layer, without any modifications to the geometry of the proposed structure. This aspect holds tremendous value in achieving flexible manipulation in various applications, including photonics, telecommunications, and display technologies.
2. Design and simulation
We explore to realize high NR with asymmetric graphene metasurface structure. The 3D schematic of our proposed structure is presented in Fig. 1(a) as a continuous graphene sheet placed on a poly crystalline silicon slab with periodic grooves cut into both sides. We utilize grooves on the slab to initiate the release of the bound state, resulting in a resonance in reflection and transmission [23]. In this work, we have deliberately selected varying depths of grooves on opposite sides to enhance the system's asymmetry. The Lorentz reciprocity principle remains intact, ensuring that the wave transmission remains consistent in both transmission directions, irrespective of the system's asymmetry [2]. Figure 1(b) illustrates the frontal view and the structural parameters of the unit cell. Transverse-magnetic (TM)-polarized wave is used to excite the structure, i.e., the incident electric field is oriented along the length of the grooves. By considering the operating wavelength and the obtained nonreciprocal response, the structure is strictly optimized. The period of the unit cell and the groove width are p = 520 nm and w = 40 nm, respectively. The total thickness of the silicon slab is t = 100 nm. The groove depth is selected as h1 = 5 nm and h2 = 10 nm on the top and bottom sides, respectively. Poly-silicon is considered as lossless in the infrared range and its refractive index is ${\varepsilon _l} = 3.4$. Graphene is defined by a conductivity model [38]:
Fig. 1. (a) The 3D schematic of the proposed asymmetric structure, composed of periodic grooves cut into both sides of silicon slab, covered by a continuous graphene layer. (b) The front view of the unit cell, with corresponding geometrical parameters. Port boundary condition is employed in the simulation to define the incident wave.
To realize nonreciprocity, COMSOL Multiphysics, a finite element technique is used to conduct the numerical simulation of the asymmetric graphene metasurface. A 2D model is constructed to accelerate the calculation, since the structure is infinite in z-direction. We use the periodic boundary condition in x-direction and port boundary condition in y-direction to generate incident waves on the structure. For convenience, the top and bottom sides are described as port 1 and port 2, respectively, as shown in Fig. 1(b). The corresponding reflection spectra are expressed as |S11|2 and |S22|2, respectively. Graphene is defined as a surface current, expressed as $J = {\sigma _L}(\omega )E$, where E is the electric field along the graphene monolayer.
3. Results and discussion
3.1 Efficient nonreciprocal response achieved with the proposed design
To achieve a highly efficient nonreciprocal effect, we start by calculating the reflection spectrum of solely silicon slab with asymmetric grooves. It turns out that the identical reflection is obtained under excitation from opposite sides, as the red line shows in Fig. 2(a). This obtained result verifies that the asymmetry caused by slab grooves is insufficient to achieve NR. In the near-IR range, the combination of graphene and resonant structures offers exceptional potential in dynamically manipulating the propagation of electromagnetic waves. Thus, in the next step, graphene monolayer is introduced only on the top side to increase the structural asymmetry. The solid black and blue lines in Fig. 2(a) present the reflection spectra of the proposed graphene metasurface structure under TM polarization with excitations from port 1 and port 2, respectively. It is shown that there is a reflection valley at the resonance wavelength λ = 1.16245 um in the case of port 1 excitation. However, for the port 2 excitation case, quite a different reflection curve is obtained. The reflection keeps relatively high throughout the operating wavelength range. Furthermore, a reflection peak is achieved at the resonance wavelength λ = 1.16256 um. Nonreciprocal reflection ratio (NRR), i.e., the difference between reflection from both excitation directions, can reach 0.763 (=10log10(|S22|2)-10log10(|S11|2) = 21.02 dB) at λ = 1.16245 um, which is particularly high considering that no physical mechanisms are involved, such as dynamic modulation [39], magneto-optic effects [6], and nonlinear effects [12]. The green dashed arrow line in Fig. 2(a) indicates the significant difference in reflection from both sides. This is the intrinsic nonreciprocity of the structure, which can be explained by the electromagnetic modes excited from different excitations.
Fig. 2. (a) The computed reflection response of the proposed graphene metasurface structure under opposite illumination direction. Solid and dashed lines present the reflection spectra in linear and nonlinear regime, respectively. Red line shows the reflection spectrum of the structure without graphene covered. (b), (c) The computed enhancement of electric field distribution in one unit cell at corresponding resonance wavelength for illumination from port 1 and port 2.
We further demonstrate the electric field distribution around the resonance wavelength. Figures 2(b) and (c) present the peak electric field enhancement distribution of an entire unit cell for the illuminations from port 1 and port 2, respectively. The maximum electric field amplification can reach 60 times for the port 1 excitation, while it is less than 20 times for the opposite case. The discovery of a robust asymmetric field enhancement provides a promising foundation for enhancing nonreciprocal efficiency even further. In this context, we include the generation of the nonlinear effects. Under strong laser field, the optical Kerr effect will occur, described by the change of the material’s permittivity, ${\varepsilon _{NL}} = {\varepsilon _L} + {\chi ^{(3)}}{|E |^2}$, where ${\chi ^{(3)}}$ is the third-order nonlinear susceptibility. Obviously, the nonlinear permittivity is related to the local electric field strength. Combined with our current result in Figs. 2(b) and (c), a significant Kerr effect can be produced by the proposed structure. It is important to emphasize that the induced Kerr effect can result in varying reflection responses owing to the different field enhancement distribution, depending on the excitation directions.
Poly-silicon has strong nonlinearity in the near-IR range, with $\chi _{si}^{(3)} = 2.8 \times {10^{ - 18}}{\textrm{m}^\textrm{2}}\textrm{/}{\textrm{V}^\textrm{2}}$. Due to the third-order nonlinear process, the surface current of graphene can be expressed as $J = {\sigma _L}(\omega )E + {\sigma ^{(3)}}(\omega ){E^3}$ [27], where ${\sigma ^{(3)}}(\omega )$ is the nonlinear surface conductivity. In the optical and near-IR regime, ${\sigma ^{(3)}}$ is calculated by the formula [40]:
3.2 Dynamic tunability of the nonreciprocal reflection
The nonlinear behavior exhibited by the material employed, specifically the poly-silicon slab and graphene, is anticipated to contribute to the dynamic tunability of the obtained NR response. The presence of nonlinearity in these materials implies that their optical properties can be altered by varying the intensity of the incident light. To investigate the field intensity’s effect on the nonreciprocal efficiency, reflection response with various incident intensity values is presented in Fig. 3(a). The solid lines represent the reflection spectra excited from port 1, and the dashed lines represent those from port 2. We also show the reflection response in linear regime in Fig. 3 (a) with black lines. For port 2 excitation in nonlinear regime, the reflection resonance and the peak value remain unchanged compared to the linear case, even though the illuminating intensity increases to 120 kW/cm2. For port 1 excitation, we observe that with a very low input intensity value of 20 kW/cm2, the change of reflection spectra is not obvious, since the nonlinear Kerr effect of the structure is not triggered by weak electric field distribution. However, the resonance wavelength shifts obviously with the further increasing of the incident intensity. The central wavelength of the reflection peak can reach 1.16261 um, with a shift of 160 nm for the input intensity of only 100 kW/cm2, which indicates a significant shift compared to the linear regime case [25]. This can be explained by the strong enhanced electric field at the reflection valley, as demonstrated in Fig. 2(b). Figure 3(b) illustrates the computed reflection coefficients of the proposed graphene metasurface structure excited from both directions, as a function of the input field intensity. The fixed wavelength is 1.16255 um, and the selected input intensity range is between 0.01 and 200 kW/cm2. The structure will operate in the linear regime in the case of low-input intensities. With the increase of the input field intensity, the Kerr effect is triggered and the structure will work in the nonlinear regime. We find that at the fixed wavelength λ = 1.16255 um, the NRR reaches a peak value with input intensity around 60 kW/cm2. This result is consistent with the demonstration in Fig. 2(a). To verify how the NRR changes with the input intensity and the operating wavelength simultaneously, we also present the reflection in the contour plot in Supplement 1 [41]. It is demonstrated that a red shift is observed for the wavelength of the NRR peak, as the increasing of the input intensity. In addition, the power range (PR), where a significant difference occurs in the reflection between two illuminating waves from opposite directions, is very large. This phenomenon fully demonstrates the tunability of the working wavelength by modulating the input intensity.
Fig. 3. (a) The reflection spectra of the proposed graphene metasurface structure in nonlinear regime under various incident intensity values. The solid and dashed lines depict the reflection spectra excited from port 1 and port 2, respectively. (b) The computed NR response from opposite direction illumination as a function of the input intensity at the fixed wavelength λ = 1.16255 um. Maximum NRR occurs with the input intensity of 60 kW/cm2.
By applying an external magnetostatic or electrostatic field, graphene’s properties can be simply controlled, allowing the dynamic tunability of graphene-based devices. According to Eq. (1), the conductivity of graphene is affected by many factors, including the temperature, the collision rate, and the Fermi level of graphene. In this work, we investigate the effect of Fermi level on the tunability of the NR response. We choose to vary the Fermi level from 0.3 to 1 eV, and the incident intensity is fixed at 60 kW/cm2. We first calculate the resonant wavelength for the opposite illuminations with various Fermi levels of graphene in the linear regime, as the black lines show in Fig. 4(a). It is found that when the Fermi level is smaller than 0.55 eV, the resonant wavelength shows a redshift in both cases with the increase of the Fermi level. Simultaneously, it is observed that the maximum electric field amplification is relatively low, which is not able to trigger the nonlinear effect of the designed structure efficiently at these resonances, as the blue lines show in Fig. 4(a). However, with the Fermi level larger than 0.55 eV, the maximum field enhancement at the resonant wavelength increases rapidly as the increase of Fermi level. The structure’s nonlinear response can be greatly stimulated and a large NRR is potential in this case. In addition, the modification of the Fermi level leads to different graphene properties, as a result, to a wavelength shift in the resonant response of the graphene plasmons, as well as the electric field enhancement. Next, we perform simulations with various Fermi levels under the nonlinear regime. Here, we consider only the effective EF values, i.e., larger EF than 0.55 eV. As the black line shows in Fig. 4(b), by adjusting EF, the working wavelength to which the NRR peak corresponds, approximately follows the variation of the resonance in the linear regime depicted with black line in Fig. 4(a). As expected, this interesting result could be used to dynamically tune the operation wavelength. The peak NRR value obtained by the proposed structure is also presented with blue line in Fig. 4(b). The obtained NRR is expected to change due to the modification of the graphene’s properties. The highest NRR occurs at EF = 0.7 eV, because we fix the input field intensity at 60 kW/cm2. As we have demonstrated, the input intensity has a considerable effect on the nonreciprocal nonlinear response of the proposed structure. By adjusting the field intensity, higher NRR can be realized at various EF values.
Fig. 4. (a) The calculated resonance wavelength (black lines) and the field enhancement (blue lines) of the proposed structure versus the Fermi level of graphene. The solid lines represent the corresponding results of the proposed structure excited from port 1. The dashed lines represent that from port 2. (b) The obtained working wavelength (black line) of the NRR peak and the NRR peak value (blue line) achieved by the proposed graphene metasurface structure, as a function of the Fermi level of graphene.
In Fig. 5, we investigate the effect of the structure geometry on the reflection response of the structure in a linear regime. Figure 5(a) is the reflection spectra when the depth of the top groove h1 are 7, 5, and 3 nm, depicted with black, blue, and red lines, respectively. The graphene Fermi level is fixed at 0.7 eV. The bottom groove depth h2 keeps being 20 nm. The solid and dashed lines represent the calculated reflection of the proposed graphene metasurface structure excited from port 1 and port 2, respectively. We find that as the top groove depth h1 increases, the resonance wavelength shows a blueshift for both illumination directions. Meanwhile, for the excitation from port 1, the reflection peak is hardly changed. However, for the excitation from port 2, the reflection valley slightly changes. Among all the h1 values, the blue line with h1 being 5 nm in Fig. 5(a) is the optimized result for 0.7 eV. Figure 5(b) presents the reflection spectra with h2 being 22, 20, and 18 nm, while h1 is fixed at 5 nm. Similar to the effect of h1 on the reflection response in Fig. 5 (a), the effect of h2 is obvious on the working wavelength but is trivial on the peak and valley values of the reflection spectra with excitation from port a and port 2, respectively. More simulations have demonstrated that by adjusting the other parameters and the graphene Fermi level, the proposed structure can still achieve a high reflection difference between two excitation directions with various h1 and h2. We investigate the impact of groove depth as the proposed structure features intricate designs with narrow grooves. It is verified that as long as the fabrication error is within 2 nm, we can accurately capture the resonant wavelength by sweeping the same wavelength range from 1.161 to 1.165 um.
Fig. 5. (a) The calculated reflection spectra of the proposed graphene metasurface structures as a function of wavelength for the top groove depth h1 of 7, 5, and 3 nm. In this calculation, the graphene Fermi level and the bottom groove depth h2 are fixed at 0.7 eV and 20 nm, respectively. keeps being 20 nm (b) The calculated reflection spectra of the proposed graphene metasurface structures as a function of wavelength for the top groove depth h2 of 22, 20, and 18 nm. In this calculation, the graphene Fermi level and the bottom groove depth h1 are fixed at 0.7 eV and 5 nm, respectively.
4. Conclusion
In this study, an exceptionally high level of nonreciprocal performance is realized by designing an asymmetric graphene metasurface structure, which consists of a continuous graphene sheet placed on top of a patterned silicon slab. In order to amplify the asymmetry of the proposed structure, we have incorporated periodic grooves with varying depths on both sides of the slab. Without any physical mechanism involved, the proposed structure alone can achieve an NRR of up to 21.02 dB in the linear regime. When delving into the nonlinear regime, an even greater enhancement in the nonreciprocal efficiency of our system is achieved with a relatively low input field intensity of 60 kW/cm2. The NRR can reach an impressive value of 21.27 dB, and IL is as low as -0.76 dB. These values are comparable to or even better than those reported in previous works. This is one important objective in this work to simultaneously pursue a high NRR and a low IL within one simple nonreciprocal device. Note that this impressive nonreciprocal performance is achieved without the need for spatiotemporal modulation of the entire structure, deposition of magneto-optic layers, or high-intensity laser, as used in other methods. The physical mechanism underlying this phenomenon can be elucidated with the nonlinear response of the material and the distinct field distribution that arises with opposite light excitations. By considering the optical Kerr nonlinear effect, it is apparent that an increase in the input intensity results in an augmentation of materials’ permittivity, leading to a shift of the operation wavelength. Combined with the asymmetry of the structure, this shift is different for excitation from opposite sides, enabling higher nonreciprocal efficiency. Achieving dynamic tunability of the nonreciprocal performance is another crucial objective of this study. We have successfully demonstrated this feature by the variation of the incident field intensity and the graphene Fermi level, without alteration of the geometry of the proposed structure. Furthermore, we have conducted a study on how the geometry of the structure affects its reflection spectra and have implied that the nonreciprocal effect can be observed within the same wavelength range as the optimized structure, provided that the fabrication error is within 2 nm. With the simplicity of the proposed structure and the maturity of fabrication technology, the nonreciprocal design can be extended to other frequencies and classical wave systems, and promote the development of integrated isolators, optical logic circuits, and bias-free nonreciprocal photonics.
Funding
National Natural Science Foundation of China (12104203, 12264027, 12274241, 61927813); Jiangxi Provincial Natural Science Foundation (20224BAB211015).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data supporting the findings of this study are available from the corresponding author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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41. See Supplemental Material for a detailed analysis on the dynamic tunability of the nonreciprocal response by the incident intensity.
