1. Introduction

Surface plasmon polaritons (SPPs) are electromagnetic excitations being coupled to the surface collective oscillations of free electrons in a metal, which tightly bound to and propagate along metal-insulator interfaces [1]. As we all know, there are many nanostructures designed for guiding SPPs efficiently such as gap/stripe waveguides, trench/slot waveguides, semiconductor nanowires, and photonic band gap waveguides. One of the most favorable is the metal-insulator-metal (MIM) waveguide, which has been keeping its popularity because of the attractive properties and extensive applications. As essential components of the nanophotonic circuits, MIM waveguides do a good balance between light confinement and propagation loss [2]. As a consequence, a variety of functional plasmonic devices based on MIM waveguides have been proposed theoretically and examined experimentally, such as optical filters [3,4], splitters [5], interferometers [6], sensors [7] and all-optical switches [8]. In the research of sensor design and application, plasmonic refractive index (RI) sensor achieves high sensitivity and resolution because the combination of plasmonic waveguide and nanometric resonant cavity (waveguide-cavity system for short) provides a good mechanism for realizing sensing and wavelength selective. In other words, the propagating surface plasmon resonances (PSPRs) and localized surface plasmon resonances (LSPRs) features of waveguide-cavity system pick out the specific wavelength which required exactly [9]. For example, the most common structures used in plasmonic RI sensors are MIM waveguides coupled with a ring cavity [10,11], a nanodisk cavity [12], a rectangle cavity [13,14], a gear-shaped cavity and so on [15,16].

Within the near-infrared region (NIR), plasmonic RI sensors investigated above obtain a sensitivity of 357.1nm/RIU (unit of nm/RIU is defined in section 2.3) based on the nanodisk cavity [11], a sensitivity of 1563nm/RIU based on the hexagonal cavity [16] and a sensitivity of 1576.7nm/RIU based on the ring cavity [17]. However, to assess the sensing performance of a plasmonic RI sensor comprehensively, transmission efficiency and optical resolution are supposed to be considered together, apart from focusing only on improving the sensitivity. Accordingly, the figure of merit (which is explained in section 2.3) is the key point throughout the paper. In addition, when it comes to the transmission efficiency T (T = P2/P1) [18–20], the input power P1 and output power P2 are defined as integral values of energy flux density, which are calculated on the detective boundary lines S1 and S2 being marked in Fig. 1. In consideration of the unrealizable detection in practice, however, the input power P1 which injects into the gap region of wg1 should be set as an exact value. More explanations are discussed in details in the section below.

 

Fig. 1 The two-dimensional schematic view of the plasmonic refractive index sensor based on a fillet cavity coupled with two MIM waveguides symmetrically with d = 30nm, w = 50nm, and a = b = 350nm.

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In this paper, we propose a plasmonic RI sensor which composed of a fillet cavity sandwiched by two MIM waveguides symmetrically. The transmission characteristics including transmission efficiency, wavelength sensitivity, resolution, figure of merit and relevant SPP mode distributions are analyzed by employing the finite element method (FEM). The relevant conclusions obtained by numerical simulation pave the way for further researches of plasmonic sensors and even nanophotonic circuits.

2. Numerical simulations

2.1 Model

To begin with, a two-dimensional schematic view of the plasmonic RI sensor based on waveguide-cavity system is shown in Fig. 1. An annular fillet cavity with edge lengths a = b = 350nm formed by an end-to-end MIM waveguide is sandwiched by two MIM waveguides (wg1 and wg2) symmetrically at a coupling distance of d = 30nm. In view of the integration of nanophotonic circuits, the lengths of wg1 and wg2 are set as 300nm. The gap width and bending radius of the fillet cavity are set as w = 50nm. As shown in the schematic diagram, we identify the fillet cavity from MIM waveguides with different color for stressing the fact that materials to be detected are filled only in this cavity excluding MIM waveguides.

With respect to the definition of transmission efficiency T being mentioned in the section above, we have reason to believe that the total input power P1 calculated on boundary S1 is the superposition of damped incident power after propagating some distances and reflected power back from the terminal of wg1. Accordingly, random numerical values of P1 may lead to errors while the transmission efficiency of the whole system is calculated. Consequently, for such a two-dimensional structure, the input power P1 which injects into the gap region from left is set as a constant value using boundary mode analysis. Whereas, the final output power P2 calculated on boundary S2 is reasonable because we employ perfect matched layers (PMLs) as the absorbing boundary which are even enclosed by scattering boundary layers (SBCs) to let SPPs pass through without reflection (PMLs and SBCs are not plotted in the schematic diagram of Fig. 1). In the process of practical detection, the output power P2 can be easily detected and the transmission spectrum can be analyzed by a near-infrared spectrograph.

2.2 Theory

In our simulation, the TM-polarized incident wave with in-plane electric field components is directly coupled to the fundamental SPP mode [21]. Actually, a number of SPP modes can supported by metal-insulator interfaces when the SPPs of individual metal-insulator interface start interacting with each other. For the gap width between two metals decreased to w = 50nm, SPPs of both interfaces can be coupled into the fundamental SPP mode only, which has an effective refractive index larger than $n_{eff}=\beta /k_0={[{\epsilon }_m{\epsilon }_d/({\epsilon }_m+{\epsilon }_d)]}^{1/2}$. The dispersion relation of the fundamental SPP mode supported in MIM waveguide can be explained by the following expressions [22]:

It goes without saying that nanocavities are crucial elements in PSPRs (LSPRs) sensors. SPP waves are reflected back and forth in the fillet cavity which is considered as a Fabry-Perot cavity [23]. Accordingly, the stable standing wave distributions build up constructively when the following resonant condition is satisfied [24]:

Based on the Eq. (5), the free-space resonant wavelength ${\lambda }_m$ is proportional to $n_{eff}$ and $s$ simultaneously. To know $n_{eff}$ of the MIM waveguide, the $n_{eff}-n_0$ linear relation curves with wavelengths of 900nm, 1550nm and 2500nm are plotted in Fig. 2(a) for instance. That is to say, once the propagation length of SPP waves is determined by the fixed geometric structure, the value of $n_{eff}$ can be learnt according to the linear relation, and finally the resonant wavelength ${\lambda }_m$. In addition, the $n_{eff}$ behavior of silver-silicon-silver waveguides is presented in Fig. 2(b).

 

Fig. 2 (a) The n_eff-n₀ relation between effective index of MIM waveguide and refractive index of material filled in cavity under different incident wavelengths. (b) The real part and imaginary part of n_eff vary with near-infrared wavelength ranging from 750nm to 2500nm.

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For metals, silver is employed because its electromagnetic response within NIR has the smallest imaginary part of relative permittivity, which is known to be less power-consumed than gold and aluminum [25]. Accordingly, we model its frequency-dependent complex relative permittivity by the Drude model $\epsilon (\omega )={\epsilon }_{\infty }-{\omega }_p/[\omega (\omega +i\gamma )]$, where ${\epsilon }_{\infty }=3.7$ is the dielectric constant at infinite angular frequency, ${\omega }_p=9.1$[eV] is the bulk plasma frequency, $\gamma =0.018$[eV] is the damping frequency of oscillations, and $\omega$ is the angular frequency of incident wave [10].

2.3 Evaluation

To assess the sensing performance of a plasmonic RI sensor comprehensively, it is unpersuasive to focus only on the wavelength sensitivity (S) which is defined as $\Delta \lambda /\Delta n$ nanometer per refractive index unit (nm/RIU), another assessment factor namely figure of merit (FOM) should be considered which is defined as the ratio of S to FWHM (full width at half maximum of resonant peak) [26]. Noticeably, larger FOM manifests higher sensitivity and better resolution of a plasmonic RI sensor. In addition, higher transmittance signifies lower power consumption.

3. Results and analysis

As shown in Fig. 3, the desired resonant peak profile of the transmission spectrum due to LSPRs are firstly obtained by optimizing coupling distance d and transmission efficiency T. The narrowed line-width and decreased peak value drops to 0.78 from 0.95 as d changes from 10nm to 30nm are shown in Fig. 3(a). Apparently, the transmittance reaches up to 95% when d = 10nm. This can be explained by the fact that light-matter interaction and power transfer (coupling strength) between waveguides and cavity become more sufficient as d increases, and hence the higher transmittance. However, the narrowest resonant peak with FWHM = 12nm (d = 30nm) is selected for better resolution. In Fig. 3(b), by filling the two MIM waveguides (wg1 and wg2) with air, silica and silicon, the resonant peaks with transmittance of 41%, 66% and 78% are obtained. It is noting that the three peaks are staggered for the sake of distinguishing each other more easily, because they are all corresponding to the first-order resonant mode which in the position of wavelength at 1498nm. The reason why transmission efficiency can reach to about 78% is wg1 and wg2 are filled with silicon while only the cavity is filled with the material under detecting. That is to say, different kinds of dielectric materials used in MIM waveguides are particularly important. The refractive index of air, silica and silicon adopted here are 1.0, 1.5 and 3.48, respectively [27].

 

Fig. 3 The first-order resonant peak profile with (a) different coupling distance d between waveguides and cavity (b) three kinds of dielectric materials filled in wg1 and wg2 without cavity based on the silver-silicon-silver waveguides.

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Under fully consideration of better resolution and acceptable transmittance, the silver-silicon-silver waveguides coupled with a fillet cavity at d = 30nm are used for subsequent studies. The transmission spectrum and corresponding normalized SPP mode distributions of |H_z| are presented in Fig. 4. As presented in Fig. 4(a), the two resonant peaks (peak1 and peak2) generated at wavelength of 1498nm and 716nm are corresponding to the first and second resonant mode, which are marked as mode1 and mode2, respectively. Figure 4(b) shows that there are no SPP waves pass through the wg2, because 1100nm is not a resonant wavelength. However, the SPP mode distributions of first and second resonant mode can be clearly seen in Fig. 4(c) and 4(d).

 

Fig. 4 (a) The transmission spectrum within NIR (750nm-2500nm) of the plasmonic RI sensor based on silver-silicon-silver waveguides coupled with a fillet cavity with d = 30nm. The normalized SPP mode distributions of (b) non-resonant mode at 1100nm (c) first-order resonant mode at 1498nm (d) second-order resonant mode at 716nm, respectively.

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After above researches, the way transmission spectrum changing with different refractive index $n_0$ of the material under detecting is studied. The transmission spectrum plotted in Fig. 5(a) demonstrates that resonant peaks all exhibit a red shift as $n_0$ increased from 1.0 to 1.5. In next research contents, the peak1 is the point of focus because it presents the highest wavelength sensitivity and satisfies our purposed design within the NIR. Though the transmittance declines from 78% to 50%, the line-width of all peaks are kept to FWHM = 12nm, and the sensitivity is calculated as S = 1496nm/RIU. As a result, the FOM is obtained as large as 124.6. To illustrate the sensing principle simply, curves of the linear relation between $n_0$ and ${\lambda }_m$ are plotted in Fig. 5(b) as $n_0$ changes from 1.0 to 1.5. The refractive index $n_0$ of material under detecting can be learnt by reading the matching peak value ${\lambda }_m$. The black curve of mode1 presents a higher sensitivity than mode2 because of its larger slope.

 

Fig. 5 (a) The transmission spectrum of the plasmonic RI sensor under different refractive index n₀ which changes from 1.0 to 1.5. (b) The linear relation between refractive index n₀ of material under detecting and resonant peak value λ_m (mode1 and mode2).

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For better sensing applications, the performance of plasmonic sensor discussed above is further promoted within NIR as far as possible. Therefore, as shown in Fig. 6, we propose a semi-fillet cavity by modifying the inner sides of fillet cavity to square shape while keeping the outside sides filleted. Other geometrical parameters and numerical values remain unchanged.

 

Fig. 6 The two-dimensional schematic view of the optimized plasmonic RI sensor based on a semi-fillet cavity coupled with two silver-silicon-silver waveguides symmetrically with d = 30nm, w = 50nm, and a = b = 350nm (The modified part is encircled by golden dashed line).

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In the same way, the shifting of resonant peak1 and peak2 are investigated by changing $n_0$ from 1.0 to 1.5. The transmission spectrum and relevant linear relation of mode1 and mode2 are shown in Fig. 7(a) and 7(b), respectively. Likely, it is calculated that peak1 shows a sensitivity of 1560nm/RIU while peak2 of 776nm/RIU. The FWHM is about 15nm which ensure a relatively good resolution for RI sensing. However, the calculated value of FOM = 104 is a little lower than 124.6 which is attained by the structure based on a fillet cavity. The reason why transmittance decreases with longer wavelength can be interpreted in two ways. One is the intrinsic absorption loss caused by metal itself, and the other one is the dispersive loss which can be expressed by a dispersive coefficient $\alpha =4\pi \mathrm{Im}\left\{n_{eff}\right\}/{\lambda }_{spp}$ [20]. Figure 2(b) (which is located in the section 2.2) indicates that the imaginary part of $n_{eff}$ is extremely small which can be neglected in the NIR. Consequently, it is simple to learn that longer incident wavelength induces fewer dispersive loss, the reduced transmittance of resonant peaks is mainly due to metals.

 

Fig. 7 (a) The transmission spectrum of plasmonic RI sensor based on silver-silicon-silver waveguides coupled with a semi-fillet cavity under different n₀ which changes from 1.0 to 1.5. (b) The linear relation between refractive index n₀ and resonant peak value λ_m.

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In addition, a comparison of the linear relation between $n_0$ and ${\lambda }_m$ based on the principle of RI sensing is presented in Fig. 8(a). Actually, there is still another approach to improve the sensitivity. On the basis of Eq. (5) deduced in section 2.2, the scale of semi-fillet cavity can be enlarged by increasing the edge length a (b = a). As we can see in Fig. 8(b), the peak1 exhibits a red shift regularly with increasing edge length, and the linear relation between edge length and resonant wavelength is obtained as edge length increases from 350nm to 700nm by an increment of 50nm. That is to say, the resonant wavelength increases predictably as edge length gets larger, which is significant to predict corresponding ${\lambda }_m$ with regard to the different size of cavity. To compare with the sensitivity of 1560nm/RIU based on the semi-fillet cavity with a = b = 350nm, the sensitivity of semi-fillet cavity with doubled edge length a = b = 700nm is calculated as 3476nm/RIU. However, the sensitivity is improved at the cost of increased line-width, which results in lower resolution.

 

Fig. 8 (a) The comparison diagram about linear relation between refractive index n₀ and peak value λ_m concluded from the RI sensors with fillet cavity and semi-fillet cavity, respectively. (b) The peak value λ_m varies linearly with increasing edge length a (b = a) of the semi-fillet cavity.

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In consideration of practical application, unlike the theoretical models discussed above, the perfect symmetry of the structure is likely to get broken for the limitation of precision during the fabrication process. Accordingly, a measurement factor g is introduced to examine, which is denoted as the offset in y direction between wg1and wg2 based on the horizontal center line. After calculation, the numerical simulation results demonstrate that resonant wavelength and peak profile in the transmission spectrum remain unchanged as well as the values of S and FOM with g increases from 1nm to 5nm.

4. Conclusion

In summary, we propose a near-infrared plasmonic refractive index sensor for realizing RI sensing of materials under detecting, which achieves a sensitivity of 1496nm/RIU and a FOM of 124.6. To further improve sensing performance, the fillet cavity is modified to a semi-fillet one, which contributes to a sensitivity of 1560nm/RIU and a FOM of 104. In addition, it is worth emphasizing that both of the two cavities achieve extremely narrow line-width with FWHM = 12nm and FWHM = 15nm, respectively. At the end of this paper, another way for improving the sensitivity (up to 3476nm/RIU) is introduced by enlarging the size of cavity. However, the sensitivity is improved at the cost of increased FWHM, which results in lower resolution. To obtain a larger FOM, it is significant to achieve the trade-off between sensitivity and resolution. As a consequence, more mechanisms such as Fano resonance for narrowing the line-width should be investigated apart from making efforts to improve the sensitivity. In addition to realizing plasmonic sensors, the waveguide-cavity system can also be applied to optical filter applications, which will play an important role in plasmonic circuits and other nanophotonic applications.