Simultaneous field enhancement and loss inhibition based on surface plasmon polariton mode hybridization

2.1 Illustrations of the designed multilayer trench gratings

The diagram of a unit cell of the compound periodic microstructure is presented in Figure 1A. The stripe waveguide was formed by two asymmetric metal-dielectric-dielectric (MDD) units. Silicon dioxide (SiO₂) and silicon (Si) were respectively categorized as the core and the cladding dielectric layers, and this configuration is capable of effectively binding the light. The SiO₂ layer thickness (h) is the key parameter in this study. The material of the metal layer is aluminum (Al). Different from the common metal-dielectric-metal (MDM) unit, such asymmetric MDD waveguide will effectively inhibit the absorption loss induced by compact metal layers, which will be proved by related calculations below and, eventually, through experimental validation. Figure 1B presents the cross-sectional scanning electron microscope (SEM) image, and the inset exhibits the detail of a fabricated MDD multilayer configuration. In this compound structure, Fabry-Perot (FP) resonances were excited laterally along the width of multilayer grating stripes and coupled with the cavity resonant modes excited within the adjacent trenches, thereby resulting in mode hybridization. The electric field enhancement was optimized by regulating the absorption and radiation quality factors in each mode configuration. Meanwhile, the effect of loss inhibition achieved by hybridization of two SPP modes in the compound structure was clearly presented based on a comparison of the various quality factors related to each part and the entire structure. The loss inhibition was fundamentally ascribed to the mutual recycling of radiation energy between the two adjacent modes. The maximization conditions of field enhancement and loss reduction were both generated at the quasi-static limit of the SiO₂ layer thickness. The working principle of this designed structure is simply exhibited in Figure 1C.

Figure 1:

Schematic and physical illustrations of the designed multilayer trench gratings.

(A) Schematic diagram of a unit cell of the designed structure. (B) Cross-sectional SEM image of the microstructure, and the inset exhibits the detail of fabricated multilayer MDD configuration. (C) Working principle diagram of the designed structure. The normalized color bar ranges from 0 to 1 for gray and red, respectively.

2.2 Electric field enhancement modulation

To illustrate the strong confinement in the MDD waveguide, the simulated electric field intensity distribution and the cross-sectional field plots along the dashed lines of the grating stripe part are presented in Figure 2A. The standing SPP in the multilayer waveguide formed a typical FP resonance mode in the horizontal direction (x direction) [28], [29]. The diffusion of energy was effectively obstructed by the Si layer with a high real part of permittivity in the longitudinal direction (z direction), thereby resulting in an extremely small mode volume [30], [31]. FP modes with the same resonance order exhibited a maximum electric field intensity in the vicinity of point A for all thicknesses. Figure 2B presents the dependence of the electric field enhancement on thickness. According to Seok et al., the interesting relationship between the field enhancement and the structure’s quality factors can be defined as follows [32] (Supporting Information, Section 1):

Figure 2:

Relationships between electric field enhancement, absorption, and radiation characters in the designed structure.

(A) Electric field intensity distribution and the cross-sectional field plots along the dashed lines of the grating stripe part supporting an FP mode. The color bar ranges from 0 to 13 for blue and white, respectively. (B) Dependence of the electric field enhancement of the grating stripe part on h. The inset is the SEM image of the multilayer grating stripe part with h=20 nm. The scale bar is 200 nm. (C) Dependences of QradFP and QabsFP on h. The symbols “■” and “×” respectively represent the simulated and experimental results. The gray areas in (B) and (C) represent the applicable region of quasi-static approximation. (D) Electric field intensity distribution and the cross-sectional field plots along the dashed lines of the trench part supporting a cavity (C) mode. The color bar ranges from 0 to 4. (E) Dependence of the electric field enhancement of the trench part on h. The inset is the SEM image of the trench part, and the scale bar is 500 nm. (F) Dependences of QradC and QabsC on h. The symbols “■” and “×” respectively represent the simulated and experimental results. The solid curves are polynomial fit of simulated data points.

where λ_r is the resonance wavelength; ⎸E_loc⎸_max and E_i are the maximum value of electric field intensity in the concerned area and the incident field amplitude, respectively; A_c is the effective aperture of the microstructure; V_eff is the effective mode volume of the concerned configuration, which is defined by the half-max electric field intensity in the designed structure (Supporting Information, Section 2); Q_rad and Q_abs are the radiation and absorption quality factors of the concerned structure, respectively. Clearly, the field enhancement is closely related to the values of Q_rad and Q_abs, thereby signifying that the maximum electric field enhancement of the designed configuration can be obtained by regulating its radiation and absorption characteristics. In addition, the total quality factor, Q_tot, of the configuration is the inverse summation of Q_rad and Q_abs such that 1/Q_tot=1/Q_rad+1/Q_abs [32]. Consequently, the values of Q_rad and Q_abs can be derived from Q_tot and Eq. (1). Firstly, in this FP waveguide, the total quality factor QtotFP is expressed as follows [33] (Supporting Information, Section 3):

where L is the length of FP waveguide, namely, the width of the grating stripe; β is the complex propagation constant of SPP; and α is the absorption coefficient with the relationship α=2Im(β) [33]. Obviously, the calculation of β is the key to calculating the value of QtotFP. On the basis of the asymmetric MDD waveguide theory, the propagation constant β can be directly solved by the relevant dispersion equations related to the complex permittivity of each material (Supporting Information, Section 3), thereby successively calculating QtotFP,QabsFP, and QradFP. In simulations, we extracted the data of each material from literature [34], [35] and drew their fitting curves to conduct calculations; with regard to the calculations in experiments, the practical material data measured by ellipsometer were adopted. Consequently, the simulated and experimental results can be mutually validated to guarantee the correctness. Both the absorption Q values of such MDD configuration calculated by simulated and experimental data are much higher than that of conventional MDM unit [36], exhibiting an obvious inhibition effect of absorption loss due to the absence of compact metal layers (Supporting Information, Section 4). Notice that Eq. (2) should be modified when h is less than 20 nm (gray area in Figure 2B and C) because the Q factor of the designed waveguide configuration is only determined by the complex permittivity of the Al layer under the following quasi-static approximation [33], [37]:

where ε_Al is the complex permittivity of the Al layer, and ω_r is the resonance frequency. Following the above calculations, each Q factor of the grating stripe part will be precisely solved. Figure 2C presents the simulated and experimental dependences of the QradFP and QabsFP on the SiO₂ layer thickness h. According to Eq. (1), if the field enhancement of the stripe part is considered as a function of QradFP, the maximization condition can be clarified by taking a derivative with respect to QradFP (Supporting Information, Section 1). Seok et al. defined the maximization condition Q_rad=Q_abs because it makes the derivative become 0 [32]. Notice that there is an implied specific condition in their research that for the dielectric layer thickness, their Q_rad is almost a monotonic function in the waveband of interest. Therefore, for the maximum of field enhancement, the zero point of derivative is the unique solution. Recently, Liu et al. demonstrated a beautiful work that an extreme high field enhancement was realized through the combination of localized plasmonic nanoantennas and a photonic crystal microcavity with the identical optimum condition, and the tendency change of Q_rad was still not considered [38]. However, in the present study, the value of QradFP is not monotonic anymore for the selected values of SiO₂ layer thickness h. If merely considering the condition Q_rad=Q_abs, only the local optimum would be obtained instead of the global one. In this case, the inflection points of QradFP must be considered as well. Figure 2B and C indicate that the field enhancement and QradFP have the same monotonicity when Q_abs>Q_rad (Interval II) and are opposite when Q_abs<Q_rad (Interval I), which accords well with the variation trend predicted by Eq. (S3). Obviously, the maximum field enhancement was observed in the vicinity of 20-nm thickness, which is attributed to the sudden change of the monotonicity of QradFP near the quasi-static limit. It provides an original approach to enhance the electric field in the plasmonic waveguide, namely, to engineer the structure’s radiation and absorption characteristics so as to achieve the maximization condition, especially by virtue of the quasi-static limit. Meanwhile, the effective mode volume V_eff in the grating stripe part gradually shrunk when decreasing h from large volume towards the vicinity of quasi-static limit (Supporting Information, Figure S1), signifying that the optimization of V_eff can also be achieved accompanied by the maximization of electric field enhancement. It is worth noting that quasi-static approximation is a concept of dimension which is merely applicable to tiny structures far less than incident wavelength [33]. It means the quasi-static limit position can only be defined within an approximate range rather than by a precise value. To further approach the precise thickness of maximum field enhancement, numerical simulations with high accuracy have to be taken into account.

Similarly, Figure 2D presents the simulated electric field intensity distribution of the cavity resonance mode (C) in the trench, and Figure 2E displays the dependence of the field enhancement on h. The maximum electric field intensity was extracted in the vicinity of point B. As compared to the FP resonance mode, the corresponding absorption quality factor QabsC in the trench part was first derived based on the following equation [39]:

where β is the complex propagation constant of the cavity resonance mode, which can be solved by the dispersion equation at the single interface [39] (Supporting Information, Section 3). Subsequently, its total and radiation Q values were solved simultaneously. Figure 2F presents the simulated and experimental dependences of QradC and QabsC on the SiO₂ layer thickness h. The results satisfy the general rule of the semi-open architecture structure wherein its radiation loss cannot be ignored [40]. When radiation occupies the dominance of loss, QabsC will become extremely large. For simulated and experimental results, any minor difference in QradC or QtotC will generate a significant discrepancy between simulated and tested values of QabsC, as shown in Figure 2F. Because the values of QabsC are always larger than the values of QradC, according to Eq. (S3), the field enhancement and QradC must maintain the same monotonicity in the concerned waveband, and every inflection point of QradC must correspond to a local extreme point of the field enhancement. As expected, this phenomenon can be clearly observed in Figure 2E and F.

2.3 Loss inhibition based on energy mutual recycling

After discussing the quality factors of the grating stripe and trench, respectively, the loss characteristics of the entire configuration were investigated by the coupled mode theory [32], [41]. Directly, the total Q factors of the multilayer trench grating structure were extracted as Q_tot=λ_r/Δλ by fitting a Lorentzian curve to the measured reflectance spectrum (R) [22], [32], as exhibited in Figure 3A, where Δλ is the full width at the half maximum of the resonance peak. The total radiation Q factors were then calculated as follows [32], [41] (Supporting Information, Section 5):

Figure 3:

Comparisons of absorption and radiation characters between FP mode, cavity mode, and the total structure, respectively.

(A) Measured reflectance curves with different h values treated by the Lorentzian fitting. (B) Left column: comparison between the experimental values of QabsFP,QabsC, and Qabstot at different h values. Right column: comparison between the experimental values of QradFP,QradC, and Qradtot at different h values. Majority of Qabstot values are in between QabsFP and QabsC (in the colorful area), whereas the majority of Qradtot values are larger than both QradFP and QradC (beyond the colorful area).

Following above calculations, the total absorption Q factors were also obtained by the relationship 1/Qabstot=1/Qtot−1/Qrodtot.Figure 3B exhibits the comparison between the multiple absorption Q factors, including QabsFP,QabsC, and Qabstot, as well as the radiation Q factors. Interestingly enough, the majority of Qabstot values are in between QabsFP and QabsC, but the majority of Qradtot values are larger than the values of both QradFP and QradC. The latter results indicate that with respect to the characteristic of radiation loss inhibition, the hybrid mode was better than both FP and cavity modes taken individually. Intuitively, this is obviously an unusual situation that the radiation of the entire structure is less than that part of that structure. Therefore, understanding the underlying mechanism requires further examination.

To explain this unusual observed phenomenon, the Poynting vector diagram of the compound configuration, reflecting the strong flow and transmission of the energies between the grating stripe and the trench parts, is presented in Figure 4A. Obviously, when calculating the radiation losses of the individual grating stripe and trench structures, the energy exchange in the modes hybridization process was not taken into account. In fact, the real escaped energy was a small proportion of the radiation, while the major portion was mutually reclaimed by the FP and cavity modes. The essence of the loss inhibition for mode hybridization may be defined by the energy mutual recycling between the two SPP modes, resulting in a reduction in the radiation loss of the entire microstructure, thereby reducing the total loss. As expected, the total loss of the compound configuration was always less than both the values of the two individual parts within all the selected values of SiO₂ layer thickness, as highlighted in Figure 4B. Wang et al. once had a worry that the radiation loss would grow rapidly following an increase in the nanostructure size, and it was then unclear whether the Q factor would increase beyond the quasi-static limit. For larger Q values, strong radiation should be forbidden [37]. The designed compound structure offered in the present study provided a solution in which, although the radiation loss of the grating stripe grew dramatically within the quasi-static limit (see Figure 2C, from 10 nm to 20 nm), it exhibited a decrease in the total loss of the entire configuration due to the mode hybridization. The Q_tot reached a relatively high point when h was about 20 nm (boundary of the gray area in Figure 4B corresponding to the quasi-static limit). Moreover, the maximum field enhancement in the grating stripe was also observed under this condition (see Figure 4B), thereby exhibiting a nice combination of loss inhibition and field convergence. Although the above significant effects are obtained at the expense of additional effective mode volume of cavity mode, the proposed approach for improving the performance of widely used SPP-based composite structures is indeed effective, which is fundamentally distinguished from the LSPR-based work of Seok et al., and therefore worth taking this step forward.

Figure 4:

Loss inhibition is ascribed to the energy recycling between FP and cavity modes.

(A) Poynting vector diagram of the multilayer trench grating structure with h=20 nm. The background is the electric field intensity distribution of designed microstructure, and the color bar ranges from 0 to 14. (B) Dependences of the experimental values of QtotFP,QtotC, and Q_tot on h. The dependence of the electric field enhancement of the grating stripe part on h is also given as reference. The gray area represents the applicable region of the quasi-static approximation.

4.1 Fabrication of multilayer trench gratings

The designed structure was fabricated by laser direct writing (LDW) lithography (Heidelberg, Germany, DWL 4000). The photoresist was first deposited on polished clean Si wafers by spin coating with a thickness of about 450 nm. The samples were then patterned by the LDW method and subsequently developed for 40 s in sodium hydroxide (NaOH) solution. The samples were etched by reactive ion etching in an SF₆ plasma for 40 s so as to obtain the 400-nm-deep trenches with a grating constant of 2.5 μm. Following soaking in the acetone solution to clear the residual photoresist, the samples were then processed in a magnetron sputtering reactor to realize the film coating process, thereby resulting in five alternate layers: Al, SiO₂, Si, SiO₂, and Al. The layer thicknesses of Si and Al were both 50 nm, while the SiO₂ layers had different thicknesses, including 10 nm, 20 nm, 40 nm, 60 nm, 80 nm, and 100 nm. The complex permittivity of each material involved in experimental calculations were measured by ellipsometer. In order to enable excitation of the hybrid mode or the simultaneous excitations of the FP and cavity resonances, the width ratios between the grating stripe and trench were approximately set from 0.67 to 0.79, which were derived from previous simulations.

4.2 Reflectance spectral response measurement

The integrated reflectance spectral responses of the fabricated samples were measured using a spectrometer equipped with a 160-mm integrating sphere in the wavelength range from 500 nm to 2500 nm for the sake of calculating the Q factors of the configurations as well as other relevant parameters. The resolution of the spectrometer is 4 nm, and the normal incidence light in the spectrometer is unpolarized.

4.3 Modeling and simulations

The designed structure was theoretically analyzed through the finite difference-time domain method. In both x and y directions, the periodic condition was adopted as the boundary condition, and the perfectly matched layer condition was used in the z direction. A plane wave with linear polarization perpendicular to the gratings was adopted as the normal incidence source. Through simulations, it can be proved that the calculation results of electric field enhancement and various quality factors will remain the same when changing the incidence to an unpolarized one. The mesh accuracy in models is 10 nm. The material characteristic parameters of Si and SiO₂ such as the permittivity and refractive index were extracted from the database of Palik [34], and the Al parameters were extracted from Shiles et al. [35]. The width ratio of the two parts of the unit cell were optimized by simulations to support the formation of the hybrid mode, where an FP mode is strongly coupled to a cavity mode. The SiO₂ layer thickness was also varied from 10 nm to 100 nm in a series of simulations to carefully examine the relationship between the field enhancement and the quality factors.

Supporting Information accompanies this paper at https://doi.org/10.1515/nanoph-2020-0023.