Considering the propagation of surface plasmon polaritons (SPPs) supported by a graphene monolayer can be effectively controlled via electrostatic gating, we propose a graphene monolayer on a graded silicon-grating substrate with dielectric spacer as an interlayer for plasmonic rainbow trapping in the infrared domain. Since the dispersive relation of SPPs is dependent on the width of dielectric spacer filling the silicon grating, the guided SPPs at different frequencies can be localized at different positions along the graphene surface, associated with the period of silicon grating. The group velocity of slow SPPs can be made to be several hundred times smaller than light velocity in vacuum. We also predict the capability of completely releasing the trapped SPPs by dynamically tuning the chemical potential of graphene by means of gate voltage. The advantages of such a structure include compact size, wide frequency tunability, and compatibility with current micro/nanofabrication, which holds great promise for applications in graphene-based optoelectronic devices.
© 2013 Optical Society of America
For decades, slow light has attracted intensive research interest because of a broad range of potential applications, including optical buffers, signal processing, and enhanced light-matter interactions. To address the needs of various applications, much effort has been devoted to exploring new mechanisms that can significantly reduce or even stop light . A series of discoveries have shown the capability of achieving ultra-slow propagation of light, including the use of quantum interference effects [2, 3], photonic crystals , stimulated Brillouin scattering [5, 6], and electromagnetically induced transparency . However, conventional slow-light systems can operate within a very limited bandwidth near the resonant frequencies due to the restriction of delay-bandwidth product, so it remains a great challenge to slow down the group velocity (νg) in a wide frequency band.
Metamaterials and plasmonics are cross-disciplinary fields that are emerging into the mainstreams of many scientific areas [8–12]. Recently, “rainbow trapping” has been widely proposed as a scheme for localized storage of wide-band electromagnetic fields in metamaterials and plasmonic structures [13, 14], which reveals the capability of slowing down light waves of different frequencies at different positions. With the ability to produce highly confined and localized optical fields, such a scheme is believed to be capable of promoting new applications in optical physics, such as enhanced light-matter interactions, on-chip light localization, spectrum splitting, broadband absorbers [15–17]. Currently, great effort has been focused on developing various architectures for “rainbow trapping”, including plasmonic grating structures [18–21], insulator-negative-index-insulator [13, 22], insulator-metal-insulator , and metal-insulator-metal waveguide tapers [15, 24, 25]. For the purpose of reducing the νg of surface plasmon polaritons (SPPs) along a metal surface, it is highly required to tailor the dispersion curve by engraving the metal surface with a groove array or covering it with a periodical dielectric grating, which makes the practical implementation relatively difficult [14, 18, 19]. In contrast to metal, graphene, whose optical response is characterized by surface conductivity, has attracted intriguing attention in optoelectronic field [26, 27]. At low frequencies such as mid-infrared range, graphene behaves like a very thin metal layer with negative permittivity as the imaginary part of conductivity can be tuned to be positive via external tunability such as electric field, magnetic field, and gate voltage [28–30]. This unique optical feature has made graphene a promising candidate for constructing novel nanophotonic devices, such as superlens, Luneburg lens, SPP waveguides, and optical hyperlens [28, 31–35]. The plasmonic waves in graphene have very large wavevectors and extremely high confinement, which enable us to build optical devices with dimensions much below the diffraction limit. The highly localized optical field produces strong light-matter interactions and can be used to build sensors with high sensitivity or to build nonlinear optical devices with high efficiency [36, 37].
However, there is no report on graphene plasmonic “rainbow trapping” so far. In this article, we propose a novel scheme by a graphene monolayer on a graded silicon-grating substrate with dielectric spacer as an interlayer for plasmonic rainbow trapping in the infrared domain. We demonstrate that the guided SPPs at different frequencies within a broadband region can be localized at different positions on a graphene sheet. The νg of slow SPPs is estimated to be several hundred times smaller than that of light velocity in vacuum. Compared with the previous graded plasmonic grating structures, the present graphene-based system has the advantage of highly confining light energy within a deep subwavelength region on graphene surface. Additionally, the release of the trapped waves can be made within a wide spectral band by tuning the gate voltage. While it was suggested that the release of trapped rainbow in graded plasmonic grating structures can be made by modulating the effective index of dielectric medium, a complete release for a wide band of frequencies is still challenging due to the limited tunability via temperature or external field [18, 20]. As for the present graphene-based system, the trapped plasmonic waves for different frequencies within a wide-band frequency can be completely released in sequence by dynamically modulating the chemical potential of graphene via bias voltage.
2. A graphene monolayer on a dielectric layer with a uniform silicon-grating substrate
The chemical potential of graphene is determined by the carrier density, which can be controlled by a static electric field via a gate voltage [28, 38]. The chemical potential of graphene is dependent on the gap distance between the graphene and substrate. Therefore, we can create certain desired chemical potential patterns across a graphene by designing the specific profile of the ground plane underneath the dielectric spacer holding the graphene in order to have nonuniform static biasing electric field under the graphene . In our work, we first consider a uniform plasmonic grating structure (UPGS): a graphene monolayer on a uniform silicon-grating substrate with dielectric spacer as an interlayer as schematically shown in Fig. 1(a). By biasing the graphene with a single static voltage, the static electric field is distributed periodically according to the height of d1 and d2, leading to periodical static electric field. This results in periodical chemical potential, μc1 and μc2, and conductivity distributions, σg1 and σg1, under the graphene. Analogous to SPPs guided along a metal surface covered with periodical dielectric gratings [19–21], the graphene layer can be biased to support distinct effective indices of SPPs for different regions, associated with d1 and d2. The chemical potential under the graphene can be estimated by the carrier density, which is related to the thickness and permittivity of dielectric spacer, and bias voltage, Vbias [28, 38]. We have used this method to retrieve the chemical potential for a graphene monolayer supported by a silicon dioxide thin film and the estimated chemical potentials agree with the results very well in Ref. 38. Here we assume the relative dielectric permittivity of dielectric spacer is ε2 = 2.1, d1 = 171nm, and d2 = 29nm. In this case, the chemical potentials under the graphene for the two regions, corresponding to d1 and d2, respectively, are μc1 = 0.2eV, and μc2 = 0.5eV under the assumption of Vbias = 45.6V.
In our work, the graphene layer is treated as an ultra-thin metal film with a thickness of Δ. Note that Δ is not real thickness of graphene, which is typically ~0.33nm for graphene monolayer. The equivalent permittivity of graphene is given by 38]38]. Similar to previous theoretical study on plasmonics and metamaterials based on graphene [28, 32, 33, 35], Kubo formula has been widely employed to calculate graphene's surface conductivity under the assumption of chemical potential. Previous study has shown that the estimated surface conductivity from this formula agrees with the experimental results very well .
Since SPPs is highly confined on the graphene surface, the silicon-grating substrate has very little influence on the photonic properties of SPPs on the graphene surface. By matching the boundary conditions for the air-graphene-dielectric spacer system, the SPP dispersion relation may be derived as 39]. Figure 1(b) shows the dispersion curves of SPPs in the UPGS for different w1 with μc1 = 0.2eV and μc2 = 0.5eV, corresponding to Vbias = 45.6V. It can be seen that the dispersion curve shows a blue-shift of the cut-off wavelength with the increased w1. It should be noted that, if the graphene monolayer is removed from the top of dielectric spacer in Fig. 1(a), the remaining dielectric grating structure does not support any SPP mode. In this case, if light is coupled to this kind of grating structure, although its propagation characteristics will be modulated. However, since the period of the silicon grating is 1~2 orders of magnitude smaller than the light wavelength of our interest in this work, it doesn't have a cutoff frequency for each period. It means that the resultant structure without graphene won't support the propagation of slow light in the frequency range of our interest. From the definition of group velocity vg≡dw/dk, we may conclude that, as light frequency approaches the cut-off frequency, the vg of SPP mode will be significantly reduced. Figure 2 shows the group index c/vg as a function of light frequency for different w1 under different bias voltages, which reveals that the vg of SPP mode shows a significant reduction near the cut-off frequency, associated with w1. We can also see from the figure that the bias voltage has a significant influence on the cut-off frequency of SPP mode, hence the vg of SPP mode is closely related to the bias voltage. For a fixed w1, the cut-off frequency, corresponding to the lowest value of the νg, reduces with the bias voltage, which can be attributed to the fact that a smaller bias voltage will result in a smaller effective refractive index of SPP mode. It can be seen that the vg of SPP mode can be significantly reduced to several hundred times smaller than light velocity in vacuum, which implies practical implementation for slow-light devices in graphene based nanophotonic circuits.
3. A graphene monolayer on a dielectric layer with a graded silicon-grating substrate
However, similar to conventional resonance-based slow-light systems that are limited to delay-bandwidth product for a fixed period, UPGS can only slow down the νg of SPP mode within a rather narrow bandwidth near the cut-off frequency [14, 19, 40]. Adiabatic control of the dispersion removes such a restriction and allows the recent demonstration of optical trapped rainbow [14, 18–21]. In our work, we propose to realize the “trapped rainbow” effect by a graded plasmonic grating structure (GPGS) as schematically shown in Fig. 3, where w1 increases by a fixed step along the propagation direction. UPGS could be approximated by a series of many uniform gratings, each with a constant period of silicon grating. The group index of SPP mode along the propagation direction is gradually increased along the propagation direction, associated with the silicon grating period. As for graded optical structures, one important question has to be addressed: how slowly must the grade vary with position to satisfy the adiabatic condition. Generally, we can employ a well-known eikonal approximation, so-called Wentzel-Kramers-Brillouin (WKB), to evaluate adiabaticity for varying nanostructures Eq. (4) may be rewritten asEq. (5) are shown in Fig. 4 for different incremental steps (0.2, 0.5, 1, and 2nm) with wavelength of 11μm, corresponding to the period ranging from 70nm to 90nm. We can see that δ meets the adiabatic condition represented by Eq. (4) for these four incremental steps. However, a smaller step will ensure the dispersion changes slowly along the propagation direction, which is helpful for coupling light into SPP modes with a very low vg. If we set the adiabatic tolerance limit to be 0.1, the incremental step has to be smaller than 1nm.
To validate the plasmonic rainbow trapping effect described above, two-dimensional finite-element method (FEM) using COMSOL has been performed to simulate SPPs propagation in GPGS shown in Fig. 3. Convergence tests are done to ensure that the numerical boundaries and meshing do not interfere with the simulation results. After simulation, the calculated data can be directly plotted with postprocessing by use of COMSOL. In our calculation, we have directly inputted the equivalent complex permittivity of graphene monolayer, εg, into the FEM. In this way, the dispersive properties of graphene can be involved in our modeling since εg is sensitively dependent on the surface conductivity of graphene, which is related to the chemical potential and light frequency . A light beam of 1W with TM polarization (magnetic field parallels to y direction) is normally incident to the left-hand side of the waveguide for exciting SPPs. Figures 5(a)–5(e) illustrate the electric field distributions in the x-z plane for different excitation wavelengths, i.e. 8, 8.5, 9, 9.5, and 10μm with the incremental step of 0.5nm. It is interesting to note that, considering that the localized field intensity for different incident wavelengths is varied, we have adopted different value for the maximum color in Figs. 5(a)–5(e). It can be seen that SPP modes of different wavelengths are localized at different positions on the graphene surface, associated with the period of the grating. The localized positions for SPPs of different wavelengths are well consistent with those are predicted by the dispersion relation from the secular equation that was used for obtaining Fig. 1(b). From Figs. 5(a)–5(e), we observe clearly the standing-wave characteristics for field magnitude in the GPGS along the propagation direction, which can be attributed to the intermodal coupling between the forward and backward modes [15, 22]. The forward and backward modes interfere coherently with each other, hence forming alternative peaks and nodes of the field magnitude. It should be noted here, the energy reflection arising from intermodal coupling around the band-edge can never be inevitable, no matter how small an incremental step is employed .
4. Release of the “trapped rainbow”
The question arises after trapping graphene plasmonic rainbow is how to release the trapped waves. While it was suggested the trapped waves in the previous metal based plasmonic rainbow trapping systems can be released by modulating the dispersion relation via temperature or external field tunability, a complete release within a wide bandwidth is still challenging because of the limited tunability of refractive index of dielectric medium [18, 20]. According to Ref , only the trapped modes at the output edge can be released. In comparison, graphene's surface conductivity can be easily tuned to be capable of supporting propagation of SPPs, where the effective refractive index can be widely modulated via gate voltage, implying the ability to release the trapped waves within a broad frequency region.
Figure 6(a) shows the dependence of the chemical potential μc1, and μc2 on the bias voltage for d1 = 171nm, and d2 = 29nm, respectively. We can see that both of the chemical potentials increase with the bias voltage. A larger chemical potential will result in a much smaller effective refractive index of SPPs. In section 3, we have demonstrated that SPP modes ranging from 8μm to 10μm can be separately localized along the graphene’s surface with Vbias = 45.6V. If the bias voltage is increased, the cut-off frequencies for different w1 shown in Fig. 1(b) will shift to longer frequencies (corresponding to shorter wavelengths). In other words, the trapped waves of 28.7THz (10.5μm), located at the position associated with w1 = 30nm at Vbias = 45.6V, will no longer be trapped and released for a bias voltage that is slightly larger than 45.6V. Similarly, if the bias voltage is further increased, the trapped waves that are located at the positions associated with w1 less than 30nm at Vbias = 45.6V, will be released. In this way, the trapped waves can be released one by one by tuning the dispersion curve via bias voltage, representing a possible way to realize the optical buffers for future on-a-chip optical communications. It should be noted here, in previous metal based plasmonic trapping systems release of the trapped waves can be made only within a very narrow band spectrum due to the limited tunability of dispersion curves via tuning the refractive index of dielectric medium . As for the present one, the trapped waves can be released within a broad bandwidth due to the large tunability of graphene’s chemical potential via bias voltage. Figure 6(b) shows the dispersion curve for different w1, while the bias voltage shifts to 120V, corresponding to μc1 = 0.33eV and μc2 = 0.81eV. We can see that the cut-off frequencies for w1 ranging from 10nm to 30nm are all excess of that for w1 = 10nm at Vbias = 45.6V. The lowest cut-off frequency for w1 = 30nm with Vbias = 45.6V is 28.8THz (10.4μm), which shifts to 37.6THz (7.98μm) with Vbias = 120V [red solid line in Fig. 6(b)]. As has been shown in Fig. 1(b), the cut-off frequencies, varying from 36.4THz (w1 = 10nm) to 28.8THz (w1 = 30nm) at Vbias = 45.6V, all are lower than those for w1 ranging from 10nm to 30nm at Vbias = 120V. This means that even the trapped waves of 36.4THz (8.24μm), located at the input edge with w1 = 10nm at Vbias = 45.6V, can thoroughly pass through the GPGS from the left-hand to right-hand side. Consequently, the trapped waves with frequencies ranging from 28.8THz (10.4μm) to 36.4THz (8.2μm) can be completely released by shifting the gate voltage from 45.6V to 120V.
For future experimental fabrication of the present structure, we can first employ photoresist to coat the silicon substrate and form the photoresist grating by electron beam lithography (EBL). The photoresist grating is used as a mask. The silicon grating substrate can be etched by fast atom beam (FAB) etching plus lifting off photoresist . Then plasma-enhanced chemical vapor deposition (PECVD) is employed to form the dielectric spacer (e.g., silica) above the silicon grating and flow annealing is then used to form a smooth dielectric spacer on the silicon grating substrate. Finally, the graphene layer above the dielectric spacer can be formed by use of a direct transfer from Cu growth substrates to dielectric spacer . We believe, at least in principle, the present structure can be fabricated by current micro/nanofabrication technology. If the upper surface of dielectric spacer is not as flat as it is designed in the preparation process, we need to reevaluate the dispersion curve of SPP by considering the influence of the surface corrugation on μc1 and μc2, corresponding to w1 and w2. In this case, the cut-off frequency may shift to shorter or longer wavelength as opposed to a graphene monolayer on a flat surface of dielectric spacer, however, plasmonic rainbow trapping could still be expected since the cut-off frequency is still dependent on the w1.
In conclusion, we have proposed a novel plasmonic “trapped rainbow” system based on a graphene monolayer on a graded silicon-grating substrate with dielectric spacer as an interlayer. Similar to SPPs guided along a metal surface covered with periodical dielectric gratings, such a graphene layer can be biased to support distinct effective indices of SPPs for different regions along the propagation direction. The dispersion relation of SPPs is dependent on the width of dielectric spacer filling the silicon grating. Simulation results by finite-element method demonstrate that the guided SPPs at different frequencies can be localized at different positions along the graphene surface, associated with the period of silicon grating. The estimated group velocity of plasmonic modes is several times smaller than light velocity in the air. The trapped waves within a broad frequency band are predicted to be completely released one by one by modulating the dispersion curve via bias voltage. Such a structure has the merit of compact size, wide frequency tunability via bias voltage, and compatibility with current micro/nanofabrication, which enables great potential applications in graphene-based optoelectronic devices, such as enhanced light-matter interactions, on-chip light localization, spectrum splitting, broadband absorbers.
Lin Chen and Tian Zhang are supported by NSFC (Grant No. 11104093), and ‘the Fundamental Research Funds for the Central Universities’, HUST: 2013TS046”. Guoping Wang is supported by 973 Program (2011CB933600) and NSFC (Grant Nos. 60925020 and 11274247).
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