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In this paper, we propose a reconfigurable surface plasmon polariton (SPP) wave adapter designed by transformation optics, which can control the propagation of SPP waves on un-even surfaces. The proposed plasmonic device is constructed using homogeneously tunable materials (e.g. liquid crystals) so that the corresponding SPP wave transmission can be reconfigured by applying different voltages. Additionally, modified designs optimized for practical fabrication parameters are investigated. Their performance is verified by full-wave simulations. The proposed devices will pave the way towards developing tunable plasmonic devices.
©2012 Optical Society of America
1. Introduction
Surface plasmon polaritons (SPPs) are induced by the interaction between free electrons in the metal and electromagnetic radiation. Since SPP modes feature the capability to control light at sub-wavelength scale (e.g. nano-scale) [1–5], SPP devices including plasmonic modulators [6, 7], plasmonic waveguides [8–13], and plasmonic couplers [14, 15] can be used to propagate optical signals. Therefore, it will find wide applications in future optical chips, where the dimension mismatch between the optical circuit (small size) and the optical waveguide (large size) becomes unacceptable. It is found that most of the current SPP devices [16–24] can only propagate on an even surface. This greatly limits their applications in real systems. Recently, transformation optics based techniques [25–29] have been applied to address this issue. However, all of these devices have fixed performance. To realize the fully controllable optical signal propagation, SPP devices with electrically tunable performance are needed, which is the goal of this paper. With such reconfigurable transformation optics devices, we can electrically control the SPP wave transmission over the un-even surface.
In this paper, the reconfigurability of the designed SPP device is achieved by employing liquid crystals. Quasi-conformal mapping (QCM) [30] is applied to design the material property of the proposed SPP devices in the region around the un-even bump to ensure desired SPP signal propagation. In the past, the QCM technique has been successfully applied to design many different functional devices [31–36]. In our design process of the proposed devices, the finite difference method along with successive over-relaxation (SOR) and Gauss-Seidel relaxation scheme is applied. To demonstrate our design concept, three-dimensional SPP devices with un-even surfaces will be studied. Specifically, the impact of device design parameters including material property, and physical dimension is studied. An algorithm is developed to optimize the SPP device performance. Finally, full-wave simulations are used to verify the performance of these devices. It is found that we can electrically control the SPP propagation on the un-even surface for different scattering ratios using the proposed devices.
2. Device design and simulation
SPP propagating on an even surface, SPP propagating on an un-even surface, and general schematic of the proposed transformation optics based SPP wave adapter (uniform in z dimension) are shown in Fig. 1 (cross-section view). As shown in Fig. 1(a), when the p-polarized SPP wave (excited by 500 THz Gaussian beam) is launched from the left, it will initially propagate along the interface without any interference. Once it reaches the bump as depicted in Fig. 1(b), strong wave scattering is observed, which indicates large propagation loss. Here we have assumed that the whole device is embedded in a background material with εref = 2.4025. The input SPP waves are propagating on a dielectric-silver interface (referring to Fig. 1(a)-1(c)) with a cosine-shaped bump at the center. The bump topology follows Eq. (1):
where A is the normalized maximum distance of the convex part bulging from the y = 0 surface (here A = 0.2µm), w = 4µm, B = w-0.4µm. The permittivity of silver (Ag) is described by the Drude model. To control SPP wave propagation over this type of un-even surface, transformation optics technique (i.e. quasi-conformal mapping technique [30–36]) is applied to the area surrounding the bump (as shown in Fig. 1 (c)).
Fig. 1 E-field of SPP wave: (a) On even surface. (b) On un-even surface. (c) On un-even surface with transformation optics based SPP wave adapter. (d) Calculated refractive index distribution of the optic wave adapter shown in Fig. 1(c).
By applying QCM, the original rectangular region for which the length and width are 8 x 3.2µm2 is distorted to an irregular shape and is decomposed into 20 x 8 unit cells. After the transformation, each cell can have different isotropic dielectric property and the relative permittivity of each unit cell is calculated as ε = εref /sqrt(det g) (g is the 2x2 covariant metric of each cell [30, 31]). With the help of such a transformation, the un-even surface appears flat for the SPP waves. This is confirmed by the results plotted in Fig. 1(c), where the wave scattering is minimized. In Fig. 1(d), the calculated refractive index distribution of the ideally transformed SPP wave adapter is illustrated. It is found that the required refractive index values range from 1.448 to 2.009 (i.e. permittivity varies from 2.0967 to 4.03608).
In order to add tunability to the transformation optics based SPP wave adapter, we will use liquid crystals (non-cholesteric type) as the materials to realize the different unit cells (as shown in Fig. 1(d)). The general structure of the proposed device is shown in Fig. 2 . The SPP wave is propagating laterally on the metal surface (xz-plane) along the x-direction as shown in Fig. 2(a). The liquid crystal and transformation optics based gradient index structures are designed in xy-plane. Such planar layout will not only be favored for the fabrication aspect but also provide spaces for other functional devices for the coupling of SPP with photonic band gap [37] and nonlinear optical materials [38]. As illustrated in the inset of Fig. 2, biasing voltages can be applied to each cell to electronically control its refractive index. In practice, there are two ways to tune the refractive indexes of liquid crystal cells. Under the first scenario, the biasing voltage is fixed for all cells and different cell heights are applied. For the second case, the uniform substrate height is applied and different biasing voltages are used. Since the second method will impose additional difficulties in fabricating the multiple biasing lines within the proposed device, the first configuration is employed in this paper. As a result, different cell heights are introduced for the proposed reconfigurable SPP wave adapter as shown in Fig. 2(b). In this way, we can apply single biasing voltage to switch the performance of the whole device. To practically implement the biasing voltage, a polymer based stepped spacer can be fabricated using nano-fabrication technique such as spatial light modulator (SLM) based projection lithography [39]. A conducting layer will be formed on top of this spacer, serving as the biasing pad [40]. The alignment of the liquid crystal directors by the applied voltage relative to the E-field of the SPP changes the refractive index of the liquid crystal for SPP signals.
Fig. 2 (a) 2D side view of the proposed reconfigurable SPP device. (b) 3D view of the proposed reconfigurable SPP device.
Next, the height of each liquid crystal cell needs to be designed. A high birefringence liquid crystal can have δn~0.4 [41]. In our design and numerical analysis processes, we have assumed that the liquid crystal will reach its maximum refractive index (i.e. nmax = 1.788) when the applied electric field is 25 V/µm, and it will exhibit the minimum refractive index of 1.48 (i.e. nmin = 1.48) when the applied electric field is 0 V/µm. More liquid crystal materials can be found in [41, 42]. One possible liquid crystal that can be used is TL-216. Within this range, a linear relation between the biasing voltage and the refractive index will hold. Following the above assumptions, Eq. (2) is derived to calculate the liquid crystal cell’s height.
where D is the height of each unit cell, Vmax = 25V, Ediff = Emax-Emin = 25V/µm, and n is the refractive index of the specific transformed unit cell, ndiff = nmax - nmin = 0.308.For the proper operation of the proposed reconfigurable device, we need to determine its corresponding control voltages under different states (e.g. we need to decide, under which biasing voltage, it will allow the SPP wave to completely pass.). In principle, different combinations of these control voltages can be used. For the purpose of demonstration, we have specifically assumed in our simulations that, when the biasing voltage is 25V, the proposed reconfigurable SPP wave adapter will act as the ideal case shown in Fig. 1(c). Based on this condition along with Eq. (2), the unit cell height can be calculated (note: to eliminate the negative calculation values when applying Eq. (2), we have used nmin = 1.4476 during our calculations for Fig. 3(a) . For all of the other cases shown in Fig. 3, nmin = 1.48 is applied). The resulting refractive index distribution of the proposed device under 25V biasing voltage is depicted in Fig. 3(a), which matches with the results shown in Fig. 1(d). Under other biasing voltages, the refractive index distribution within the proposed device will deviate from the ideal case, leading to different extent of scattering. In this way, we can use the proposed device to electrically control the transmissions of SPP waves over un-even surfaces.
Fig. 3 Calculated refractive index distribution: (a) Originally proposed SPP wave adapter. (b) Truncated SPP wave adapter with refractive index 1.48< n ≤1.788. (c) Truncated SPP wave adapter with both refractive index and cell height confined within a certain range. (d) Modified SPP wave adapter with SSLF method applied to the design in (c).
To further improve the design of proposed SPP wave adapter and make it practical for real implementation, several modifications have been made to the ideal design shown in Fig. 3(a). First, for the ideal design, the required refractive index varies from 1.448 to 2.009, which is beyond the liquid crystal’s index variation range in the market. To address this issue, we have simply truncated the original design to confine the refractive index within the range of 1.48< n ≤1.788. The corresponding SPP wavelength range is from 0.3 to 0.38 µm. The results are shown in Fig. 3(b). Secondly, the calculated cell height of the ideal design has large variations, which is not suitable for practical fabrication. For example, for the structure shown in Fig. 3(a), the calculated cell height could vary from 1µm to 81µm or even larger, showing very large height variations. Also, since the size of each cell is around 0.4µm x 0.4µm, the large cell height (e.g. cell 1 and cell 2 as labeled in Fig. 3(a)) will lead to a very large aspect ratio, which is difficult to fabricate. Therefore, another modified design is given in Fig. 3(c), where we have also truncated the height of each cell to be within the range from 1 to 25µm.
Furthermore, based on the design presented in Fig. 3(c), we have employed the so-called sectional-straight-line-fitting (SSLF) method [43] to optimize the whole device performance and minimize the cell height variations. As an example, we have shown the calculated heights for the cells in the top row before (the DLC row in Table 1 ) and after (the DSSLF row in Table 1) the modifications and the results are given in Table 1 (note: here we list the parameters for the left half row (10 cells), the right half is the same due to the symmetry.). To better explain the implementation procedure of the SSLF method, the refractive indices of the cells before (the nLC row) and after (the nSSLF row) the modifications are also given in Table 1. During the calculation, the refractive index difference (∆n) between the neighboring cells in the original design is calculated as ∆n = nLC(i + 1)-nLC(i) and listed in the fourth row of Table 1. By examining these index differences, if ∆n≤0.0071 (truncation threshold), the corresponding neighboring cells will be combined to share the same height. If ∆n>0.0071, the cell heights will keep the original values. After applying this iterative process to all cells, the modified cell refractive indices and heights are given (the last two rows in Table 1). It is observed that, after applying the SSLF method, the cells 1 to 6 in the top row are reformed to have the same height (therefore they together form a single, large cell). Following the same procedure, we have re-designed the whole structure. Figure 3(d) illustrates the proposed device optimized by the recursive SSLF algorithm, where all of the reformed bigger cells have been highlighted with closed red lines. With this modification, the height variations within the proposed device can be reduced to the minimum level, which makes it easy to fabricate.
Table 1. Heights and Refractive Indices of Top Row Cells Before and After the Application of SSLF Methoda
To verify the performance of the original and modified reconfigurable SPP wave adapters proposed in this paper, full-wave electromagnetic simulations are conducted [44]. First, the behaviors of proposed devices under the maximum transmission condition (i.e. biasing voltage = 25V) are investigated. The simulation results are plotted in Fig. 4 , where the SPP wave is induced by a Gaussian beam with a field magnitude of 0.5V/m. In Fig. 4(a), the performance of the proposed reconfigurable SPP device with ideal parameters (as shown in Fig. 3(a)) is shown. As desired, the incident SPP wave propagates along the un-even surface smoothly. Similar results are observed in Fig. 4(b), where the modified design with practical refractive index values is simulated. When both the refractive index and the cell height are confined within a certain range for practical realization, small scattering appears as shown in Fig. 4(c) (the structure shown in Fig. 3(c) is considered here). Such un-wanted wave scattering can be eliminated after we apply the sectional-straight-line-fitting method to optimize the whole structure. Figure 4(d) illustrates the results of this device.
Fig. 4 Simulated SPP wave propagation within the proposed 3D SPP devices: (a) Proposed SPP wave adapter with ideal parameters. (b) Modified SPP wave adapter with refractive index 1.48< n ≤1.788. (c) Modified SPP wave adapter with both refractive index and cell height confined within a certain range. (d) Modified SPP wave adapter with SSLF method applied.
Finally, the electrical tunings of the proposed liquid crystal based SPP devices are modeled and the results are given in Fig. 5 . For the purpose of demonstration, three typical scenarios are considered, among which the biasing voltage changes from 25V to 22V. Correspondingly, the refractive index distributions within the proposed device under these different biasing voltages are shown in Fig. 5(a)-5(c). Apparently, the color of the refractive index map fades from Fig. 5(a) to Fig. 5(c), which will introduce the desired different levels of SPP wave transmissions. This is confirmed by the simulation results shown in Fig. 5(d)-5(f). As seen from Fig. 5(d), when biasing voltage of 25V is applied to the proposed SPP device, there is almost no wave scattering. When the control voltage decreases to 24V, a small extent of SPP wave scattering is observed as reflected in Fig. 5(e). By decreasing the biasing voltage to 22V, large SPP wave scattering is observed (Fig. 5(f)). In principle, we can use the proposed device to fully control the SPP wave transmission over un-even surface by applying different biasing voltages, which is convenient for practical implementation and can provide fast tuning speed.
Fig. 5 Calculated refractive index distribution of the proposed SPP device under different biasing voltages: (a) 25V. (b) 24V. (c) 22V. E-field of SPP wave propagation within the proposed device under different biasing voltages: (d) 25V. (e) 24V. (f) 22V.
3. Further discussion on the proposed devices
To clearly illustrate and compare the performance of different types of proposed designs for SPP wave transmission (e.g. the structures discussed in Fig. 3), we have sampled and plotted the magnitude of scattered electric field intensity at a fixed point on the SPP wave transmitting route. Figure 6 (left) shows the plotted field intensity at x = 6µm and 0.5µm≤ y ≤3µm (referring to Fig. 3) for different cases. As desired, under the complete transmission condition (i.e. biasing voltage = 25V), the proposed transformation optics based SPP wave adapters can reduce the wave scattering very well. This can be further illustrated by the figure shown in the right part of Fig. 6, where the trapezoidal numerical integrations of the curves shown in the left part of Fig. 6 have been calculated. It is clear that all of the proposed SPP devices work properly for guiding SPP waves over un-even surfaces (with reduced scattering field level).
In Fig. 7 , similar curves are drawn for the proposed device with different biasing voltages. From the left part of Fig. 7, it is clearly observed that, by changing the biasing voltage from 25V to 22V, the level of the SPP wave scattering is increased. Moreover, from the right part of Fig. 7, the magnitude of the calculated trapezoidal integral at 22V biasing voltage is almost twice of that when the biasing voltage is 25V. All of these results demonstrate that, with different biasing voltages, the SPP wave scattering level can be controlled, leading to the desired reconfigurable SPP wave adapter performance.
Fig. 7 Distribution of scattered electric field intensity for the proposed reconfigurable SPP device under different biasing voltages.
4. Conclusion
A new type of reconfigurable transformation optics based SPP device is presented, where liquid crystals are used to realize the tuning. To practically implement the proposed devices, several modified and optimized designs are investigated. Their performance is verified by full-wave electromagnetic simulations. Therefore, the new class of reconfigurable SPP device permits fast tuning of the whole device with realizable materials and fabrication procedures. It is expected that the proposed device can be used to realize plasmonic circuits with the capability to transmit signals at the nano-scale arbitrarily.
Acknowledgments
This work is supported by research grants from the U.S. National Science Foundation under Grant Nos. ECCS-1128099, CMMI-1109971, and DMR-0934157.
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