One-dimensional surface-plasmonic nanobeam cavities are proposed as a means to confine surface plasmons to a subwavelength-scale mode volume, while maintaining a relatively high Q-factor. By bonding one-dimensional photonic-crystal nanobeam structures to a low-loss metallic substrate, a clear plasmonic TM bandgap can be formed. The introduction of a single-cell defect alongside the engineering of side-air-hole shifts to this plasmonic-crystal nanobeam provides subwavelength-scale plasmonic mode localization within the plasmonic TM bandgap. This suppresses radiation and scattering loss to render a maximum Q-factor of 413 and a modal volume of 3.67 × 10−3 μm3 at room temperature. The possibility of further reduction in the intrinsic loss of the cavity is investigated by lowering the operating temperature and the Q-factor of 1.34 × 104 is predicted at a temperature of 20K for the optimistic case.
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The demand for the control and manipulation of photons on a microscale have been the main thrust for the development of ultra-compact and high-quality integrated photonic circuits during the last several decades. As the dielectric or semiconductor integrated photonic circuits have proven their feasibility down to a wavelength-scale regime, it has now become imperative to scale the footprint of the photonic devices further to the deep subwavelength-scale with a similar performance metric to those of a macroscale photonic device. Surface-plasmon-polaritons (SPPs) are the waves propagating along the interface of metal and dielectric materials. [1,2] SPPs can provide a viable solution to subwavelength-scale photon confinement, by coupling the energy of an optical wave to these short-effective-wavelength electron density oscillations. From this standpoint, a variety of subwavelength-scale SPP devices, such as waveguides, [3-9] modulators, [10-13] cavities, [14-22] and lasers, [23,24] have been actively investigated and successfully demonstrated. Among these, surface-plasmonic cavities have provided a theoretical and experimental platform that can be explored to test the viability of SPPs in densely-packed photonic circuits, as well as in the fundamental optical sciences, such as cavity quantum electrodynamics. Different types of SPP cavities of diverse structural variations have been demonstrated experimentally and theoretically. These demonstrations include a metallic nanowire Fabry-Perot (FP) resonator,  a metal-dielectric-metal FP resonator,  distributed-feedback (DFB) SPP cavity, [16,17] a metallic-fin FP resonator,  a ring/disk-type plasmon resonator, [19-21] and a metal-coated heterostructure nanowire resonator.
In this study, we propose a highly-confined, low-loss surface-plasmonic nanobeam cavity that can simply be constructed by bonding the 1-D photonic-crystal nanobeam cavity to a low-loss metallic substrate. The 1-D plasmonic-crystal has a pronounced plasmonic TM bandgap and can be fashioned into a plasmonic cavity by introducing a defect along a chain of air-holes. By engineering the position of the side air-holes near the single-cell defect and by applying a gradual mode-mismatching to the plasmonic bandgap mirrors, the radiation and scattering loss can be minimized for the lowest-order SPP cavity mode. As a result, the intrinsic SPP Q-factor of 413 can be achieved with a subwavelength-scale modal volume of 3.67 × 10−3 μm3 (V ~10−3 λ 3) at the 1550 nm wavelength band. Furthermore, it is shown that the efficient coupling of the confined SPP mode to the built-in SPP waveguide mode can be achieved by slightly decreasing the reflectivity of a photonic bandgap mirror at one end of the defect cavity.
2. The band structure of the 1-D plasmonic-crystal
The 1-D plasmonic-crystal waveguide structure is schematically presented in Fig. 1(a) . The plasmonic-crystal waveguide can be constructed by bonding the 1-D semiconductor (εd = 11.56 or nd = 3.4) photonic-crystal nanobeam on top of a silver substrate (εm). The complex permittivity of silver was adapted from Johnson and Christy’s experimental study  and fitted with the Drude model to conduct the calculations of the dispersion relationship and the subsequent simulations of the cavity mode by the finite-difference time-domain method (Drude model fitting parameters: background dielectric constant ε ∞ = 3.3651, plasma frequency ωp = 1.3779 × 1016 rad/s, and collision frequency γ = 4.9719 × 1012 s−1). The photonic-crystal nanobeam structure was prepared by periodically truncating air-holes along the beam waveguide with a lattice constant a of 330 nm, a beam-width w of 400 nm, and a beam-height h of 250 nm, as shown in Fig. 1(a). In the absence of the metallic substrate, photonic-crystal modes of purely dielectric origin were observed due to the vertical symmetry of the slab waveguide. With the photonic-crystal nanobeam parameters discussed above, the photonic bandgap (PBG) was found to be observed only for the TE-like slab guided mode around the center wavelength of 1.1 μm.
With the introduction of a metal substrate on the bottom of the 1-D photonic crystal, the vertical symmetry will break and the complicated asymmetric guided modes will become excited along the nanobeam direction (x-direction). The dispersion relationship was calculated in the first Brilluin zone for the proposed 1-D plasmonic-crystal structure and plotted in Fig. 1(b). To calculate the dispersion diagram, we performed the finite-difference time-domain (FDTD) simulations for the unit cell structure. By imposing a prescribed phase advance condition in the boundaries (that are normal to the x-direction) and exciting the structure with a dipole source along with monitors in the low symmetry locations, corresponding Bloch eigenmodes to the prescribed phase advance were extracted. In Fig. 1(b), two lower-frequency modes were observed below the lowest-order photonic mode (blue line). In addition, a bandgap similar to the photonic bandgap (PBG) was clearly opened between the lower-frequency modes for the wavelength range from 1500 to 1605 nm.
To confirm the types of modes being excited at this low-frequency range, the electric-field distributions for each mode were examined. Figures 1(c) and 1(d) illustrate the vertical electric-field (Ey) and the squared electric field (|E|2) for the two lowest modes (designated as PlC1 and PlC2) and the photonic mode (PhC), respectively. In Figs. 1(c) and 1(d), the direction and the magnitude of the electric-field are indicated by the arrows in the plot. For the cases of PlC1 (the black line in Fig. 1(b)) and the PlC2 (red line) modes, the vertical electric-fields (TM-like mode) were found to be dominant in the nanobeam structure. In addition, the maximum of the squared electric field was located at the interface between the dielectric and the silver substrate.
These two observations prove that the PlC1 and PlC2 mode can be classified as having SPP origins. On the other hand, for the PhC mode (black line), most of the electric-field was found to be lying horizontally with respect to the substrate (TE-like mode). In this case, the electric field maximum was not positioned at the metal interface, but within the dielectric medium. The complicated field distribution of the photonic mode (compared to those of a pure dielectric PhC in the absence of a metallic substrate) was caused by the perturbation from the metal substrate on the bottom of the nanobeam.
From the electric-field distributions of the PlC1 and PlC2 modes, as illustrated in Figs. 1(c) and 1(d), it can be confirmed that the appearance of the plasmonic bandgap originated from the different energy densities of PlC1 and PlC2 modes distinctively distributed over the dielectric and air region in the plasmonic-crystal nanobeam structure. The size of the plasmonic bandgap (Δλ/λ) was calculated to be 0.089 with the above-mentioned geometrical parameters. Furthermore, both the PlC1 and PlC2 modes in the plasmonic-crystal were positioned at a lower frequency, compared to the lowest PhC mode, which can be understood by the larger field penetration of the SPP modes into the silver substrate. Since the dielectric photonic modes (PhC modes) are present at a much higher frequency than those of the SPP modes (PlC1 and PlC2 modes), it is possible to only excite the plasmonic modes at the pre-designed frequencies in the plasmonic-crystal waveguide without interfering with the photonic modes.
3. Characteristics of the 1-D surface-plasmonic nanobeam cavity
By incorporating a single-cell defect in the center of the plasmonic-crystal waveguide structure (Fig. 2(a) ), a compact surface-plasmonic nanobeam cavity can be constructed via a defect SPP mode excitation within the plasmonic TM bandgap. This illustrates the analogy with the creation of defect modes in photonic-crystal defect cavities. Figures 2(b)-2(d) illustrate the squared electric fields for the 1-D surface-plasmonic nanobeam cavity. Here, we intentionally engineered the sizes and positions of the nearest-neighbor air-holes around the defect to create a gradual mode-change. This change reduced the scattering-induced radiation losses arising from the abrupt mode-mismatches (Fig. 2(b)). These figures illustrate that the SPP mode was well confined to the single-cell defect region located in the middle of nanobeam structure. As clearly illustrated in the squared electric field distributions from the cross-sectional side-view, most of the electric energy was found to be highly concentrated at the interface between the metal and the dielectric (Figs. 2(c) and 2(d)). Considering the field profiles distributed only in the semiconductor nanobeam,  the vertical modal length was calculated to be 0.48 × (λ/2nd), which confirms the subwavelength confinement in the vertical direction (Fig. 2(d)). As a result of the wavelength-scale longitudinal length of the cavity, only the single plasmonic-crystal defect mode was observed within the plasmonic bandgap, as illustrated in the spectrum in Fig. 2(e). This is one of the attractive characteristics of the proposed cavity in applications for coherent SPP light sources because the co-existence of a photonic mode, or higher order, SPP cavity mode will diminish the spontaneous emissions coupled into the desired SPP mode and complicate the experimental investigation of the SPP mode.
Figure 3 illustrates the characteristics of the proposed surface-plasmonic nanobeam cavity as a function of the shift in the side air-holes near the defect cavity. Since the 1-D nanobeam structure is relatively sensitive to the scattering loss caused by the mode-mismatches for the guided direction (x-direction), we engineered the position of the side air-holes for the gradual mode-change, as shown in the inset of Fig. 3(a). Here, the radius of the side air-holes was fixed at 0.25a. Figure 3(a) indicates that the cavity resonances were positioned within the plasmonic TM bandgap (1442-1577 nm) for the range of hole-shifts from 0 to 50 nm.
To check the degree of field localization, the cavity mode volume (V) was estimated as a function of the air-hole shift. For this evaluation, the dispersive nature of the silver permittivity was included by introducing the effective permittivity in the mode energy density calculations. [8,20,22] The modal volumes were not found to be a sensitive function of the shift of the side air-holes, but were calculated to be in the range of 2.64 × 10−3 - 4.02 × 10−3 μm3 (in the order of ~0.001 × λ 3), as illustrated in Fig. 4(b) . On the other hand, the intrinsic Q-factors (Q int)  were strongly dependent on the side air-hole shift, as they were more vulnerable to the mode-mismatches. Due to the large ohmic loss in the metallic substrate for the frequency ranges considered, the intrinsic Q-factors were ultimately limited by the absorption loss of silver. These absorption Q-factors (Q abs) were found to be nearly invariant around the value of 420, irrespective of the side air-hole shifts. The absorption loss dominantly depended on the portion of energy guided within the metal substrate, which was not strongly influenced by the air-hole shifts.
The intrinsic Q-factors (Q int −1 = Q abs −1 + Q rad −1) were calculated to be increasing gradually, as the shift of air-holes increased from 0 to 40 nm, with a maximum value of 413 for the side air-hole shift of 40 nm. As compared to the absorption Q-factor of 425 (for the 40 nm shift), the radiation/scattering loss was estimated to contribute to 3% of the total loss of the cavity with the optimized parameters. In comparison, the radiation/scattering Q-factor in the absence of the metallic absorption was estimated to be 24800. When the holes were shifted by more than 40 nm, the intrinsic Q-factor again decreased due to the increase in the scattering/radiation loss. Figures 3(c) and 3(d) illustrate the squared electric field profiles for the cases of the 0 and 40 nm shifts, respectively. These two figures clearly show the difference in radiation/scattering losses, and also clearly illustrate that the radiation/scattering loss can be minimized by slightly engineering the near-field distribution.
Since the intrinsic Q-factor of the surface-plasmonic nanobeam cavity is limited by the absorption loss of the metal with the minimization of the scattering/absorption loss, the Q-factor can be increased further by operating the cavity at a cryogenic temperature, at which the collision frequency of the free electrons in the metal is lowered.[17,22] For the estimation of the temperature-dependent intrinsic Q-factors, temperature-dependent conductivity values were used to scale the collision frequency of the free electrons. By decreasing the temperature, the Q-factor can be increased up to 13400 at 20K, with the absorption Q-factor of 1.70 × 105, as shown in Fig. 3(e). For this case, the ratio of absorption to total loss was estimated to be only 8%, which means that the total Q-factor was dominantly determined by a radiation/scattering loss, unlike the room-temperature case. The structural configuration of the 1-D plasmonic-crystal may have an advantage, especially for an experiment at a low-temperature, by simply putting it on the cold finger of a cryostat, thereby providing the easy circumstance of vertical pumping and detection. However, the values of achievable Q-factors should be tested by future experiments, since one of the recent experimental results indicates a minimal temperature effect in lowering the metallic loss  as opposed to several theoretical predictions. [17,22]
4. Coupling of the cavity to a SPP waveguide
With the given simplicity and high-quality of the proposed surface-plasmonic nanobeam cavity, we further investigated the coupling properties of the surface-plasmonic nanobeam cavity using a built-in SPP waveguide. As noted from Fig. 2, the surface-plasmonic nanobeam cavity mode was originated from the confined pure SPP mode, distinct from the plasmonic-crystal waveguide mode. By the mode-mismatch with the plasmonic-crystal mode and the utilization of the plasmonic bandgap, the SPP mode can be confined in the defect region and becomes the cavity mode (Fig. 2(b)). This implies that no air-hole defect mode inside the cavity has an advantage in the coupling to the original SPP waveguide mode by satisfying the mode-matching condition.
To confirm the argument, the coupling of the surface-plasmonic nanobeam cavity mode to the pure SPP waveguide mode was investigated by changing the reflectivity of the plasmonic bandgap mirror with the modification of the number of air-holes at one side of the cavity. Figure 4 illustrates the saturated squared electric field profiles, when only three air-holes were introduced, to decrease the reflectivity of a plasmonic bandgap mirror. As shown in this saturated figure, the scattering loss at the junction between the cavity and the SPP waveguide was not shown and appears to be completely coupled. Here, the scattering loss was calculated as less than 1% of the total loss, estimated by comparing the vertical scattering/radiation losses near the junction and the propagation losses along the SPP waveguide. The coupling efficiency was calculated to be 98%, with a loaded Q-factor of 256 at 20K. At room temperature (300K), the coupling efficiency was found to be 61%, with a loaded Q-factor of 160, when the ohmic absorption loss was considered. As has been demonstrated by this numerical calculation, the surface-plasmonic nanobeam cavity can be integrated with the pure SPP waveguide resulting in an excellent coupling efficiency by the mode-matching. This promises a possibility for the on-chip plasmonic circuit integration.
In this study, we numerically demonstrated the feasibility of a low-loss, small-mode-volume 1-D surface-plasmonic nanobeam cavity that can be created by bonding the 1-D PhC nanobeam defect cavity structure on top of the low-loss metallic substrate. By the periodicity of the PhC nanobeam structure, the plasmonic TM-bandgap is clearly formed. This bandgap is subsequently utilized to construct a defect-type plasmonic cavity. The introduction of a defect along with the engineering of side-hole shifts provides a subwavelength-scale plasmonic mode localization within the plasmonic TM bandgap, which minimizes the radiation loss as low as 3%, relative to the total loss, resulting in a relatively high SPP Q-factor of 413 and a subwavelength-scale mode volume of 3.67 × 10−3 μm3 at room temperature. We expect the proposed surface-plasmonic nanobeam cavities to be utilized as an ultra-compact on-chip cavity and a plasmonic light source that can be directly coupled with a plasmonic waveguide, or a curved tapered fiber waveguide. With the incorporation of adequate gain material, such as an InGaAsP quantum well for the PhC nanobeam structure, the proposed cavity may be configured to an efficient plasmonic light source, or possibly a laser with the help of increased gain coefficient of quantum wells and the lowered metallic loss at cryogenic temperature.
This work was supported by the National Research Foundation of Korea (MEST) grant funded by the Korea government (MEST) (grant number: 2009-0069459). N. P acknowledges the support of the Korea Foundation for International Cooperation of Science & Technology (Global Research Laboratory project K20815000003).
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