Analysis of Coupled Surface Plasmon in LHM Mediated Dielectric Gap Multilayer Structure

The purpose of this work is to investigate theoretically the characteristics of confined electromagnetic modes propagating along the interfaces of a multilayer device. This one dimensional (ID) sensor is formed by stacking a left-handed material (LHM) layer between a SiCt2-glass prism and a dielectric gap layer in contact with gold (Au). The results indicate that the total thickness of the LHM layer and dielectric gap, in optimum conditions, give the ability of tuning significantly the characteristics of the resonant modes correlated to surface plasmons (SPs) propagation along the interfaces of the designed device. By considering two arrangements between LHM and Au, two opposite resonant behaviors observed in p-reflectance spectra are analyzed in the angular interrogation mode and discussed thoroughly.


Introduction
Recently, metal-dielectric multilayer nanostructures supporting the propagation of surface plasmons (SPs) in appropriate conditions are currently investigated in the interest to assess particularly the limit of their specific sensitivity enhancement [1][2][3]. Generally, this performance parameter, which is requested to be high, is evaluated on the shift of the resonance condition with respect to any changes introduced in the refractive index (RI) of a sensing medium [4]. Generally, an interface performed between an infinite active metallic layer, such as silver (Ag), gold (Au), aluminum (Al), and a dielectric medium, once analyzed with p-polarized light beam via the attenuated total reflection (ATR) technique, exhibits an SP propagation. It should be noted that the angular resonance condition, of the generated SP mode, depends critically on both RIs and thicknesses of the whole associated media of the designed multilayer configuration [5][6][7]. Furthermore, the shift produced on the resonance peak and its line width evaluated on an interfacial SP profile were adopted as references to highlight the temperature effect [8], the carriers concentration of doped Si [9], percentages in nano-composite materials [10], the control of interface stability [11], and so on. Based on both the changes in resonance angles, 1 S P R    of the reflectance spectrum and its width, numerous types of geometries with appropriate choices of active materials have been reported in [7,9,[12][13][14][15] and applied to wide uses such as gas detection [16,17], medical diagnosis [18], and photonic devices [19]. By the use of graphene multilayer/Ag, Verma et al. [20] and

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Szunerits et al. [6] who also employed other strategies, succeeded in designing highly sensitive SPR sensors. Generally, in the design of SPR sensors, the sensing properties may be tuned in different limits that depend mainly on the whole of the parameters of involved active materials and the surrounding dielectrics [20][21][22][23][24]. In order to further improve the relative sensitivity, Ouyang et al. [25] used the advantages of stacking Si/MoS 2 under the Au layer exhibiting a single SPR mode whose full width at half maximum (FWHM) termed, dq 0.5 at 50 % of reflectivity, is significantly wider when increasing the number of MoS 2 layers. To produce a sharp and sensitive peak ascribed to coupled modes, Kullab et al. [26] proposed a novel four-layer scheme where a metamaterial (left-handed material and right-handed material) is included as a core layer to demonstrate that the designed structure presents an advantage over other optical sensors. It should be noted that the largest peak of the resonance condition, measured on angular reflectance spectra, constitutes a disadvantage for using the resonance angle as a key parameter in characterizing RI of aqueous solutions on the SPR sensors' surface [27,28].
To manipulate the sensitivity in acceptable limits, recent research [29] adopted the strategies including the porosity effect in an SPR sensor to estimate the angular SPR curve and detect the accuracy (D.A=1/ 0.5 ) of 1.098/°. In order to maximize the sensing sensitivity, a slab waveguide structure, based on the air gap enclosed between anisotropic LHM layers operating in the microwave frequency range, has been explored for refractometry applications [30].
Here, in the purpose of improving the sensitivity in a different way, we investigate the sensing response of a planar multilayer structure disposed on a prism-coupler. We discuss the influence on both RI and the thickness of an inner core (sensing medium) included between the left-handed material and Au. Clear evidence of ultra-high sensitivity and specific behaviors are shown on the profiles of confined electromagnetic modes, depending on the incident angle, 1  of an excitation light, which has not been yet observed in metal-insulator-metal structures.
This paper is organized as follows. First, we describe the essential formulas of electromagnetic fields that propagate within the configuration under study and the enhancement factors (EFs). Second, we present the obtained results with significant interpretations. Finally, the paper ends with conclusions. The transfer matrix method (TMM), leading to simulate the p-reflectance profile, is repeated in the appendix section.

Field equations near flat interfaces of an SPR sensor
The multilayer structure under study is schematically illustrated in Fig. 1. The one dimensional (1D) planar nanocavity, deposited on a SiO 2 glass prism, is made up of a dielectric layer bounded by two active materials, LHM and Au. In the present work, the optical response was calculated for Au and LHM as an outermost layer. Besides, it is assumed that the in-and outer-most media extend to infinity. All the parameters of the multilayered structure are specified in Fig. 1 are the reflection coefficients of the structure interfaces.
In a similar form of (1)-(4), the associated magnetic fields B propagating through the proposed four-layer sensor, can be easily determined by the usual Maxwell equation: The above set of (1)-(4) is the starting point to carry out a comprehensive analysis on the propagation of the guided resonant modes according to both directions, u x and u z , when a monochromatic wave irradiates the structure under an incidence angle  1 beyond  cr . In this condition, and for an adjusted LHM layer thickness, d LHM , most of the incident energy is absorbed by the layered materials and generates SPs at the boundary LHM-dielectric layer interface. As indicated in the example geometry of Fig. 1, the supported SP modes will be studied in the two possible arrangements between LHM and Au separated by a dielectric layer (sensing medium). Thus, in our theoretical simulation, the chosen parameters of the active material layers are described by the complex permittivity, e Au (l)= -21.3+i1.35 for gold, which is taken from [31], and     whose characteristics, therefore, are considered for the analytical resolution in terms of sensitivity (extracted from angular reflectance spectra) of the proposed SPR sensor on both the thicknesses of LHM (d LHM ) and dielectric layer d D .

Physics of surface plasmons
Surface plasmons are basically collective oscillating charges that propagate along a thin metallic layer bounded by a dielectric medium with electric field decaying exponentially, i.e., the evanescent character, in both the media. Otherwise, SPs occur when an incident p-polarized light of a given wavelength strikes the metallic surface at an angle, 1  > cr  through the coupling prism (or metallic grating). Under these conditions, the light reflectivity from the prism base reduces to the minimum due to the plasmonic absorption mechanism. The position of this particular angle, 1  , called the angular resonance of SP and the width at 50% of the reflected light profile, can be substantially affected with little variation in the optical constants of associated media of the proposed device. This effect is due to the presence of the evanescent field surrounding the active metamaterial (LHM). So, due to these effects, it should be necessary to investigate the sensitivity by combining different electromagnetic properties of conducting materials as LHM and Au with dielectric substrates. It is to be mentioned here that the choice of Au comes from the fact that it presents favorable properties over other metals.

Results and discussion
First, to investigate the potential of the proposed SPR nanocavity formed with a dielectric layer of RI   D D n   and surrounded by a LHM layer (e 2 , m 2 ) and a bulk-LHM, we plot the reflectance curve versus incidence angle, as shown in Fig. 2. This interfacial response is calculated from the transfer matrix method (TMM) which was previously detailed elsewhere [22]. The formalism of TMM, applied to the proposed SPR nanocavity with a prism-coupler, is repeated in the appendix section of the paper. For the stacking shown in the inset of Fig. 2(a), by considering the thicknesses, 780 nm and 330 nm of LHM layer and dielectric, respectively, the (1D)-nanocavity exhibits a single SPR mode excited at 49.23°. These . The resonant modes sustained by the interfaces within the dielectric layer (sensing layer) can be coupled to each other and make more energy stored on the outermost medium (Au). This effect, being resulted from the optical coupling process generated between electromagnetic modes inside the designed (1D)-nanocavity, has been recently observed experimentally by Hayashi et al. [33] and Goswami et al. [34] in other variant SPR systems. According to the above quantitative characteristics related to the SP' excitation, the proposed device exhibits a significant application as a tunable filter. Next, we discuss theoretically the effects induced in the resonance (ATR) intensity on both the thicknesses of LHM, d LHM and the dielectric layer, d D . The structure being considered here concerns the arrangement as: SiO 2 prism/LHM(d LHM )/dielectric gap (d D )/Au-bulk. As a result, with a fixed thickness, d D =330 nm [see Fig. 3(a)], by tuning the thickness, d LHM from 779.6 nm to 780.4 nm, a dominant amplification of ~9.8510 4 arises at 40.51° for d LHM =780 nm. Taking this condition, the above amplification value may be further optimized [see Fig. 3(b)] by increasing against the thickness d D around the range of 329 nm to 333 nm. The p-reflectance peak of the exhibited SPR mode gets a value in the order of 2.1310 6 estimated at the resonance angle of 40.74°. Such an optimization fulfilling to the condition, d D /(d D +d LHM )~0.3 leads to highlight that the thickness of the LHM layer, and the one of the dielectric layer of a fixed RI, n D , plays an essential way for producing a sharper optical amplification that can be exploited in an SPR sensing purpose. It is worth noting that the effect on RI, n D of the dielectric layer, was discussed in our previous work [22].

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In this part, we put now emphasis on the characteristics evaluated on the enhancement factors (EFs) curves, f  and f  previously defined in Figs. 7(a) and 7(b). Therefore, in the optimized condition of Fig. 3(a), i.e., with d D = 330 nm and d LHM =780 nm, the angular dependencies of f  and f  in the large angular range from 33° to 73° are depicted in Figs. 4(a) and 4(b), respectively. On the angular profile of each curve, the total thickness, d = (d D + d LHM )=1110 nm, gives the ability of generating simultaneously two sharper confined modes (associated to the supported SPs) at the resonance angles 33.83°(first-mode), and 72.22° (second-mode). Moreover, it should be noted that the two EFs' profiles through the structure present the same stop-band defined between the successive confined modes. The narrowest peak intensities of these confined modes for f  are slightly different, i.e., 0.810 6 and 1.3510 6 , except for the ones of f  , there is rather a notable difference, 3.0910 6 and 0.1510 6 . So, the multilayer structure: SiO 2 prism/LHM (780 nm)/dielectric gap (330 nm)/ Au-bulk sustains a giant amplification as reported in Fig. 2(b), and consequently, the structure characterized by the highest EFs is very promising to develop bio-sensing applications. For a comparison, recently, Sekkat et al. [24] proposed Ag/Cytop interfaces as a plasmonic planar structure leading to achieving a giant amplification~1.310 6 , which remains much less than the one we obtained here.
Following the above analysis, we finally focus on simulating the relationships between the angular resonances,  SPR with the variation of thicknesses, d D and d LHM of the EFs intensities as shown in Fig. 4. Thus, based on typical curves as depicted in Fig. 4(a), when we consider the thickness, d D fixed at 330 nm, it is observed that the resonance condition, SPR  prism-LHM-dielectric gap layer-Au outermost medium, with an LHM layer set to be 780 nm., and dielectric layer is set to be 330 nm. The other medium parameters are the same as those reported in the inset of Fig. 2  A similar study has been conducted on the evolution of each mode in the EF curve of f // versus d LHM and d d . By taking into account typical curves as shown in Fig. (4)

Conclusions
In this paper, a plasmonic sensor, with an active LHM layer of simultaneous negative permittivity and permeability, stacked on a dielectric layer and Au, has been investigated theoretically. This SPR sensor, based on gold (Au) taken as an outermost medium, has the capability of exhibiting substantial electromagnetic modes confined on the structure' interfaces. However, the resulting optical amplification predicted on both the longitudinal and perpendicular factors due to the existence of resonant modes with narrower angular widths, depends critically on the ratio evaluated between the structure' thicknesses. It has been observed that the characteristics (peak intensity, resonance angle, and line width) of the SPR modes probed in an angular interrogation method can be highly controlled on the change of the dielectric gap thickness of fixed refractive index. Based on the electromagnetic calculations, the giant optical amplification ~3.0910 6 of exhibited mode on the interfaces of the proposed device, is highly sensitive to the ratio defined between the structure thicknesses d D /(d D +d LHM ) and it can be constituted as a new limit which is even greater than the one of conventional SPR sensors for developing photonic applications.