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Abstract
Surface plasmon polaritons (SPPs) have shown unprecedented potential for miniaturizing photonic devices beyond the optical diffraction limit. However, their high intrinsic loss from metal material has been a major barrier to practical applications. One effective approach to reducing their effective loss is to engineer SPPs coupling by designing hybrid plasmonic waveguides (HPWs). In this work, we first theoretically calculate and analyze the coupling modes in a composite HPW (CHPW), demonstrating that the long-range and short-range supermodes in CHPW correspond to two different SPPs at two metal layer interfaces. Next, we propose and demonstrate a narrow bandwidth perfect absorber based on CHPW structure. The absorber’s bandwidth is only 12.9 nm in the visible range and 6.67 nm in the near-infrared range. Our simulation results demonstrate that the CHPW’s SPPs coupling mechanism can be extended to localized surface plasmons (LSPs) in the proposed perfect absorber, enabling a low-loss mode that contributes to the narrow bandwidth. This suggests that the CHPW configuration could facilitate the development of more advanced low-loss, high-performance plasmonic devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the past two decades, plasmonics metamaterials and metasurfaces have led to the discovery of numerous novel optical phenomena and applications by engineering and controlling the interaction between light and subwavelength-scale nanostructures [1–4]. For example, plasmonic perfect absorbers (PPAs) - which can achieve perfect absorption for specific spectral bands without any reflection and transmission - have gained increasing interest since 2008 due to their wide-range applications [5–13]. Generally, PPAs can be categorized into broadband absorbers [14–24] and narrowband absorbers [25–30]. Broadband PPAs can be developed for applications like solar cells, thermos harvesting and emission manipulation, while narrowband absorbers are used for thermal emitter, sensing and detections [31–33].
Almost all plasmonic nanostructures, including PPAs, inevitably suffer from high metallic loss, leading to broad bandwidth or full width at half maximum (FWHM) [34]. Researchers have designed and developed various structures and strategies to mitigate this issue by engineering and controlling the coupling of surface plasmon polaritons (SPPs) and localized surface plasmons (LSPs), such as forming hybrid plasmonic modes that couple SPPs to normal dielectric waveguide mode to reduce field overlap with metal [35–39], or applying multilayered structures to achieve destructive interference in metal layer [40–43]. These hybrid plasmonic waveguides (HPWs) have been realized in various works and are widely used for integrated photonic devices [44–46], strong light-matter interaction [47–49], nonlinear optics [50,51], nano lasing [52,53] and focusing [54]. Recently, a composite HPW (CHPW) was proposed by coupling HPW mode with another SPPs supported by the same metal layer [55]. Such a CHPW can achieve low-loss and strong field confinement SPPs mode simultaneously, facilitating new integrated photonic devices with higher performance [56,57].
In this work, we first investigate the properties of SPPs modes inside CHPW using the Transfer-matrix method (TMM), a more efficient method for structures with multiple layers. We obtain the dispersions of two types of modes: a low-loss long-range symmetry-like mode and a high-loss short-range antisymmetry-like mode. The dispersions reveals that the low-loss symmetric-like mode is primarily associated with the HWP, while the high-low antisymmetric-like mode is mainly associated with the SPPs on the other side of thin metal layer. Then, based on these findings, we propose a new narrow bandwidth PPA employing the same CHPW structure. Our results demonstrate that the coupling mechanism of SPPs in CHPW can be extended to LSPs, providing a promising platform for the development of more low-loss and high-performance plasmonic applications.
2. Transfer-matrix calculation for CHPW
The properties of SPPs can be studied by directly applying SPPs theory to the corresponding structures, which involves solving Maxwell’s equations with appropriate boundary conditions. However, as the number of layers in an SPPs structure increases, solving the equation set becomes increasingly complex. In such case, the transfer-matrix method (TMM), as another useful and efficient approach for multilayer structures, was also widely used to study coupling properties of SPPs in multilayer structures [58–61]. In this section, we employ TMM to investigate the SPPs modes inside CHPW, and explore how the supermodes in CHPW are related to HPW and other SPPs modes.
The schematic in Fig. 1(a) shows the 2D CHPW structure consisting of Si-Au-SiO2-Si layers on SiO2 substrate [55]. SPPs can be excited at the two dielectric-metal interfaces and propagate along dielectric-metal interfaces with propagating constant ${k_{SPPs}}$, as well as to $+ z$ and $- z$ directions with decay constant $\wedge$, indicated by ‘+’ and ‘-’. Meanwhile, SPPs are also reflected at every interface in $z$ direction. The Au, SiO2 and bottom Si layers form a conventional HPW [55,62], and the SPPs mode in the HPW couples to the SPPs on the other side of metal layer. In our TMM estimation, we only care about the basic behavior of supermodes, so for calculation simplicity, the permittivity of metal is based on the simple Drude model ${\varepsilon _m} = 1 - {{\omega _p^2} / {({{\omega^2} + i\omega {\omega_\tau }} )}}$[63], where ${\omega _p} = 72800\textrm{ }c{m^{ - 1}}$ is the plasma frequency, ${\omega _\tau } = 215\textrm{ }c{m^{ - 1}}$ is the damping, and the index of Si and SiO2 are fixed at 3.5 and 1.44.
Fig. 1. (a) Schematic of composite hybrid plasmonic waveguide and the dispersions of the symmetry-like and antisymmetry-like modes when the metal layer is (b) 10 nm and (c) 6 nm.
For the transverse magnetic (TM) mode, the TMM was built using the magnetic field component Hy whose polarization is parallel to the interfaces and continuous inside the entire structure. As shown in Fig. 1(a), the subscripts indicate the layer number and the interface in the present layer, while the superscripts represent the direction of propagation. For instance, $H_{12}^ + $ represents the $+ z$ going SPPs at the second interface in the first layer. In each layer, the tangential magnetic fields are superposition of the $+ z$ and $- z$ going waves, and thus the magnetic field in each layer is given by $H = {H^ + } + {H^ - }$. Similarly, for the continuous electric field, which is parallel to the interfaces, ${E_x}$, is given by $E = {E^ + } + {E^ - }$. ${E_x}$ and ${H_y}$ are related through ${E_x} ={-} (i/\omega {\varepsilon _0}\varepsilon )({{{\partial {H_y}} / {\partial z}}} )$. Based on these relations, suppose the n-th and m-th layers are two adjacent layers in CHPW structure, the SPPs propagating matrices $M_p^n$ for the n-th layer, and the dynamic matrices $M_d^{n - m}$ connecting the SPPs at n-th and m-th layers’ interface can be built. Equation (1) shows the dynamic matrix, which connects the tangential magnetic field at the interface, and Eq. (2) is the propagating matrix that connects the magnetic field at two interfaces of the n-th layer, where $\varepsilon $ is the permittivity and $\Lambda $ is the SPPs decay constant in each layer, which can be calculated by $k_{SPPs}^2 - \Lambda _{}^2 = \varepsilon k_0^2$, ${k_0}$ is the wavevector in vacuum.
With the dynamic and propagating matrices, we obtain the matrix for the entire CHPW structure:
As shown in Fig. 1(a), SPPs can be only excited at two interfaces of the metal layer, and then extends to $+ z$ and $- z$ directions. In the top air, there are only $- z$ propagating SPPs, while in the bottom substrate, there are only + z propagating SPPs. Hence, the matrix for CHPW is:
Obviously, ${M_{11}} = 0$, and by solving this equation for each frequency $\omega $, we can obtain the corresponding propagating constant ${k_{spp}}$, which are the dispersions of supermodes in CHPW. The coupling of SPPs at the two interfaces of thin metal layer with symmetric dielectric environment forms low-loss long-range SPPs (LRSPPs) with symmetric magnetic field distribution inside metal layer, and high-loss short-range SPPs (SRSPPs) with antisymmetric magnetic field distribution inside metal layer [64–66]. Here in CHPW, due to the asymmetric dielectric environment of metal layer, the SPPs supermodes are symmetry-like and antisymmetry-like, instead of perfectly symmetric and antisymmetric [55]. Due to the multiple interfaces in CHPW, several SPPs modes exits, here we neglect other trivial modes and focus on two dominant modes: long-range symmetry-like mode and short-range antisymmetry-like mode [55]. Although simulations provide some properties of these modes, important information such as dispersions and their relation to the two different metal-dielectric interfaces in CHPW are still missing.
Figures 1(b) and (c) show the dispersions of the symmetry-like and antisymmetry-like modes, respectively, calculated using TMM. The red and blue curves are the dispersions of symmetry-like and antisymmetry-like modes, respectively. The SPPs propagation constants ${k_{SPPs}}$ and frequency $\omega $ are normalized to the SPPs wavevector ${k_{sp1}}$ and frequency at the metal-Si interface ${\omega _{sp1}} = {{{\omega _p}} / {\sqrt {1 + {\varepsilon _{Si}}} }}$, ${k_{sp1}} = {{{\omega _{sp1}}} / c}$, where $c$ is the speed of light in vacuum. In both CHPW and conventional HPW, the mode in Si layers due to the reflection at Si-air and Si-substrate interfaces could impact the final supermodes’ field distribution in the low-index dielectric and the overall the performance of CHPW and HPW [55]. To mitigate this effect, thick top and bottom Si (>100 nm) are used to increase the SPPs’ decay distance. This approach reduces the field distribution and energy stored in the top and bottom Si layers, which is a commonly adopted technique for other hybrid SPP structures as well [37,40,42,43,67,68]. We conducted TMM investigation for a CHPW structure consisting of Si layers with top and bottom thicknesses of t1 = 185 nm, t4 = 220 nm, respectively. For the middle dielectric layer, we choose t3 = 20 nm. Figures 1(b) and (c) show the dispersion curves of the two supermodes for metal thicknesses t2 = 10 nm and t2 = 6 nm, respectively. The blue dotted line is ${\omega _{sp1}}$, and the red dashed line indicates the SPPs frequency ${\omega _{sp2}}$ at metal-SiO2 interface. First of all, we notice that symmetry-like and antisymmetry-like modes’ dispersion curves converge to different SPPs frequencies. This behavior differs from the situation where a thin metal layer is sandwiched by identical dielectric layers. Specifically, the symmetry-like mode’s dispersion curve converges to the metal-SiO2 SPPs frequency ${\omega _{sp2}}$, while the antisymmetry-like mode’s dispersion curve converges to the metal-Si SPPs frequency ${\omega _{sp1}}$. It reveals that the symmetry-like mode is more associated with the metal-SiO2 interface which is a part of HPW, whereas the antisymmetry-like mode is more associated with the metal-Si interface. Furthermore, as shown in Fig. 1(c), due to the stronger coupling for thinner metal layer, when the metal layer’s thickness decreases from 10 nm to 6 nm, the bump in the symmetry-like mode’s dispersion curve becomes more pronounced [65].
Since most interesting applications and phenomena are associated with the electric field of SPPs and LSPs, it is common to focus on studying the distribution and enhancement of the electric field rather than the magnetic field. Figures 2(a) and (b) show the real part of z component of the electric field for supermodes with symmetry-like and antisymmetry-like characteristics, respectively, as calculated using TMM at the wavelength of 1550 nm. The vertical dotted lines indicate $Re ({E_z}) = 0$, and the inset provides a close-up view of the electric field in metal layer. Additionally, we compared our theoretical predictions with numerical simulations at the same wavelength, as depicted in Figs. 2(c) and (d). The 2-dimensional simulation were conducted using Lumerical FDTD with the same material indices as in TMM. Our results demonstrate excellent agreement between theory and simulation.
Fig. 2. Electric field distribution of SPPs in the direction perpendicular to the interfaces at 1550 nm. (a) and (b) are the z component of the electric field of symmetry-like and antisymmetry-like modes calculated via TMM. (c) and (d) are the z component of the electric field of symmetric-like and antisymmetric-like modes obtained via FDTD simulation.
Thus, in this section, by employing TMM and FDTD simulation, we extract both dispersions and electric field of supermodes in CHPW. Our results demonstrate that the low-loss long-range symmetry-like mode is predominantly associated with the low-index (SiO2) dielectric-metal interface, while the high-loss short-range antisymmetry-like mode primarily arises from the high-index (Si) dielectric-metal interface.
3. Perfect absorber based on CHPW structure
Compared to conventional HPW configurations, it has been demonstrated that CHPW offers more advantages [46,55–57]. In this paper, we propose a new application for CHPW structure by designing a perfect absorber with a very narrow working bandwidth. The motivation behind this design is to enhance its potential applications for detection and sensing. Our proposed PPA is schematically shown in Fig. 3(a) and follows the conventional metal-insulator-metal configuration of other PPAs. However, we have replaced the top metal pattern with a CHPW structure. To ensure our PPA is polarization independent, we have designed the CHPW structure as a cylinder in each unit cell, as seen in Figs. 3(b) and (c), which depict the 3D and side views of the unit cell. The period in x and y direction is $\wedge$, the radius of the CHPW is R, and the thickness of the top and bottom Si in the CHPW structure are ${t_t}$ and ${t_b}$, respectively. The thicknesses of the Au and SiO2 layers between the top and bottom Si layers are fixed at 10 nm and 20 nm. To achieve perfect absorption, a tAu = 200 nm metal plate (Au) is used as bottom mirror to enhance the surface plasmon effect and prevent transmission. Additionally, SiO2 with thickness ${t_{Si{O_2}}}$ is used as the spacer, which can be substituted with other dielectrics, such as Al2O3, MgF2, Si3N4.
Fig. 3. (a) Schematic of the proposed plasmonic perfect absorber; (b) 3-dimension view and (c) side view of one unit cell of the perfect absorber structure.
We used Lumerical FDTD to investigate the performance of our proposed PPA. To model an infinitely large periodic structure, we applied periodic boundary conditions to one unit cell in both x and y directions. Perfectly matched layers were implemented in z direction, and a plane wave source was used to illuminate the PPA. To ensure practical application, all material indices were taken from experimental data [69–72]. Our calculation and simulation in the previous section demonstrated that the CHPW configuration supports a SPPs low effective loss mode. Hence, we anticipate that out designed PPA will have lower effective loss and narrower bandwidth compared to other PPAs.
In Fig. 4, we present the reflection spectra of two of our designed PPAs, which exhibit remarkable perfect absorption at 849.5 nm and 1592 nm with absorption rates exceeding 99.98%. The size details of these PPAs are available in Table 2. Notably, the bandwidths of these PPAs are extremely narrow, measuring only about 9.4 nm and 6.67 nm, respectively. This corresponds to quality factors (${\lambda / {\Delta \lambda }}$) 90 and 238.7, which surpass those of most other narrow PPAs and narrow perfect absorbers realized using low intrinsic loss semiconductor ($\Delta \lambda = 15nm$) [73] and complex metal pattern [26,69]. Table 1 highlights the superior performance parameters of our PPA, particularly its high Q factor when compared to previous narrow and ultra-narrow PPAs. Although it is possible to reduce the bandwidth of PPAs even further to a smaller number ($\Delta \lambda \approx 1nm$) in other works, such designs tend to have polarization dependence and/or require complex three-dimensional metal structures [25,27,74–76]. Consequently, they pose significant challenges for fabrication and limit their practical applications.
Fig. 4. Reflection spectra of the perfect absorbers at (a) 849.5 nm and (b) 1592 nm under normal incidence.
Table 1. Comparison of the Designed PPA with other Published Designs
Table 2. Geometry Parameters of Perfect Absorbers at different Wavelengths
The surface plasmons excited in our PPA are LSPs, which differ from the SPPs in CHPW analyzed in section 1. To further investigate the LSPs in our PPA during perfect absorption, we extracted and analyzed both the magnitude |E| and z-component of the electric field Ez, as shown in Fig. 5. The incident light is x-polarized and propagates along -z direction. Figures 5 (a) and (b) illustrate the side view of |E| and real part of Ez. Similar to the results in Fig. 2, the electric field concentrates in the low-index SiO2 layer beneath the thin metal layer. In the metal layer, the magnitude of electric field is very low, which is also similar to the low-loss LRSPPs in CHPW structure. The very small electric field overlap with metal layer leads to low effective loss for PPA, resulting in a narrow bandwidth. Figures. 5(c) and (d) depict the field distribution in the x-y plane, specifically in the middle of the 20 nm SiO2 layer indicated by the horizontal white dotted line in Fig. 5(a). These figures show the characteristics of the fundamental LPSs resonance, where the distribution of |E| is symmetric, while the distribution of Re(Ez) is antisymmetric and dipole-like [78,79]. In addition to the electric filed, we also investigated the free electron current density distribution in the metal layer to confirm it’s the fundamental LSPs resonance. It’s evident that the coupled LSPs in our PPA share similar properties with SPPs in CHPW: the electric field in the metal is low which is critical for minimizing mode effective loss; and the strong field enhancement is all in the dielectric layer instead of just at the metal-dielectric interface.
Fig. 5. Electric field distribution at the resonance wavelength at the perfect absorption wavelength 849.5 nm. (a) and (b) side view of the magnitude and real part of z component of electric field; (c) and (d) magnitude and real part of z component of electric field in x-y plane, indicated by horizontal dotted line in (a).
To validate the coupling similarity between SPPs in CHPW and LSPs in our PPA, we plot the electric field variation along z direction and compare it to the results in Fig. 2, as illustrated in Fig. 6. The real part of Ez (Re(Ez)) is displayed in Fig. 6(a), while Fig. 6(b) shows |E|, both extracted along the vertical white dotted line in Fig. 5(a). The insets in Fig. 6 display zoomed-in view of the field distribution in metal layer. Compared to Fig. 2, it demonstrates that the field distribution in PPA is highly similar to that of the SPPs in CHPW, where the field enhancement is concentrated in the dielectric layer, and the field distribution in metal layer is very low. Furthermore, we observed a symmetric-like profile of the field in the metal layer, resembling the long-range, low-loss symmetric-like mode in CHPW.
Fig. 6. Electric field distributions of the perfect absorber along the vertical dotted white line in Fig. 5(a). (a) Real part of z component; (b) Magnitude of the electric field. The dotted line in (a) indicates Re (Ez) = 0, and the insets show the zoomed-in view of the electric field in metal layer.
Based on our analysis and the results presented in Figs. 5 and 6, we can conclude that the properties and the coupling mechanism of SPPs in CHPW are applicable to LSPs as well. Here we prove this point using a narrow bandwidth PPA, but it is reasonable to expect that our findings can be extended to other LSPs applications.
Additionally, the PPA demonstrated in this work can be easily scaled to different wavelengths by adjusting few geometric parameters, such as the period and radius of cylinders. The performance of PPA varies with different wavelength ranges, because our previous results have shown that the effective loss caused by the thin metal film is suppressed, leaving only metallic losses from the thick bottom metal layer, which is wavelength-dependent due to the normal material dispersion of metal (longer wavelength, larger imaginary part of permittivity, smaller skin depth) [34,80]. As the skin depth in the bottom metal layer decreases with longer wavelengths, we expect the effective loss of PPA to be lower, resulting in a narrower bandwidth. We scale the PPAs that exhibit perfect absorption at 849.5 nm and 1592 nm to other wavelengths, from visible region to near-infrared region, their geometric parameters are summarized in Table 2, by fixing the thicknesses of bottom Si, top Si, SiO2 and Au at 100 nm, 120 nm, 20 nm and 10 nm, respectively. Only three geometric parameters needed to be changed: period, radius and spacer thickness. It is worth nothing that thickness variations in different layers can affect the composite hybrid SPPs and LSPs modes’ profile and their interaction with adjacent units, as discussed in section 2. For example, if we increase the bottom Si cylinder thickness from 100 nm to 105 nm, with corresponding 1000 nm period, 380 nm radius, and 694 nm spacer thickness, we can get a 99.92% perfect absorption with 8.7 nm FWHM at 1569.6 nm. Figure 7(a) shows how the bandwidth and Q factor vary with PPAs’ working wavelength. We observe that the bandwidth narrows, and Q factor increases with longer wavelength, which is consistent with our predictions. Moreover, we investigated the sensing ability of the proposed PPA and demonstrated a sensitivity larger than 200 nm/RIU in the near-infrared range, with absorption remaining higher than 99.5%, as shown in Fig. 7(b).
Fig. 7. (a) Evolution of bandwidth and Q-factor with different PPA configuration. (b) Evolution of reflection spectra of PPA under different environmental indices in near-infrared range.
4. Conclusions
In this work, we have investigated the properties of coupled SPPs modes in the CHPW structure, with a focus on the hybrid coupling of long-range and short-range modes. Our theoretical calculation and numerical simulation demonstrated that the low-loss long-range symmetry-like mode is more associated with HPW, while the high-loss short-range antisymmetry-like mode is more connected to SPPs at the other metal-dielectric interface of the shared metal layer. Additionally, we have demonstrated the potential of the CHPW configuration for developing high-performance devices by realizing a polarization-independent narrow bandwidth PPA, whose working bandwidth are only 12.9 nm and 6.67 nm in the visible and near-infrared range. Importantly, our finding also demonstrated that the coupling mechanism of SPPs in CHPW structure also works for LSPs, enabling the narrow bandwidth of PPA. Moreover, the scalability of the developed PPA to different wavelengths with bitter performance indicates the versatility of the CHPW platform. Overall, we have proved that CHPW as well as the SPPs/LSPs coupling mechanism in it are promising platform for developing low-loss, high-performance plasmonic devices and applications in different fields.
Funding
China Postdoctoral Science Foundation (2020TQ0046).
Acknowledgments
This work was supported by China Postdoctoral Science Foundation (Grant NO. 2020TQ0046).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the finding of this study are available from the corresponding author upon reasonable request.
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