Design of Nanoscale Hybrid Insulator-Metal-Insulator Plasmonic Waveguide

Optical properties of the fundamental hybrid mode of hybrid insulator-metal-insulator plasmonic waveguide (HIMIPW), consists of insulator-metal-insulator sandwiched between two dielectric waveguides, have been investigated to achieve the relatively high propagation length and large normalized intensity at 1.55 μm of working wavelength. The main aim of the current work is to settle the issues of high power loss and size of waveguide dimension. The optimum waveguide dimension of 0.2 μm × 0.02 μm, has obtained the propagation length of ~ 289.26 μm. The normalized intensity in the low-index region of the HIMIPW has been achieved as ~ 67.50 μm−2, which is due to the electric field enhancement in this region. It is beneficial for the design of applications, such as bio-sensing, optical manipulations, etc. The electric field intensity has attained the highest value at the wavelength of 1.55 μm, for the optimum dimension of HIMIPW (i.e., w = 0.2 μm, th = 0.2 μm, and tl = 0.02 μm), which is due to the highly localized surface plasmon resonance at the metal-dielectric interfaces. The investigation of the coupling length between the two identically parallel HIMIPWs, with a separation distance, has been done. Further to improve the coupling length, a metallic strip has been inserted between them, keeping the separation distance unchanged. The higher coupling length leads to lower crosstalk between two parallel hybrid plasmonic waveguides, which can be highly useful to achieve the larger integration over the photonic chip.


Introduction
Nowadays, the technology requires higher bandwidth communication services, such as high quality-video/image transfer, cloud computing, etc., which can be achieved efficiently by the systems/networks based on optical technology. For the optical technologies, the optical waveguides are one of the essential devices. In order to achieve high optical integrations, optical devices have to be improved with miniaturization of size and smaller losses. The propagation loss in the photonic/ dielectric waveguides is almost negligible; however, it suffers from the diffraction limit [1,2]. Due to the coupling between photons and free electron density oscillation of the metal, the localized surface plasmon resonance is excited at the surface of metal-dielectric interfaces, produces extremely localized enrichment. It can be offer the tight light confinement and guidance of electromagnetic waves (EWs) below the diffraction limit [3][4][5][6][7], and hence, resolves the issue of diffraction limit. Plasmonic mechanism allows the optical signal/data to propagate through the true-nano scale regions (<100 nm) [8][9][10][11][12], but suffers from the high propagation losses due to the presence of metal. The recently proposed hybrid plasmonic based wave guiding mechanism, which combines the wave guiding properties of dielectric and plasmonic waveguides [13][14][15], is capable to resolve their respective issues of diffraction limit and large propagation loss. In these waveguides, the surface plasmonic (SP) mode supported by plasmonic waveguide (PW) combines with the dielectric mode supported by dielectric waveguide, and the resulting mode is called as the hybrid mode. It essentially provides a common platform to interconnect the electronics and photonics devices. To achieve ultra-small waveguide dimension, various hybrid plasmonic waveguides (HPWs) have been proposed in literature, such as hybrid metal-insulator-metal [14][15][16][17], hybrid dielectric loaded plasmonic structure [18][19][20], hybrid metal cap plasmonic waveguide [21], hybrid insulator-metal-insulator [22,23], etc. Out of these possible structures of HPWs, the HIMIPW offers the least propagation loss with tight light confinement at true nano-scale regions. The optical devices based on hybrid plasmonic mechanism, such as all-optical logic gates [24][25][26], power splitter [27][28][29][30], ring resonator [31], etc. have been recently discussed in literature. The flow of light has been significantly controlled by the hybrid insulator-metalinsulator plasmonic waveguide structures. Therefore, this mechanism will be helpful to design the solar cells/energy harvesting applications by engineering the structures. The sensitivity of a bio-sensor is higher than a conventional plasmonic sensor and it is capable of obtaining more information about biological molecules as compared to conventional plasmonic sensors [32,33]. The HPWs based on silicon-on-insulator (SOI) are useful to achieve the dense monolithic photonic integration with passive devices, due to its ultra-small dimension and adaptability with CMOS technology.
In the current work, the optical properties of fundamental hybrid mode profile in HIMIPW have been studied, with the help of finite element method (FEM). The investigations have been done by varying the dimension of the waveguide, to achieve the small propagation loss with high normalized intensity. The HIMIPW gives a large normalized intensity in the low-index/spacer region, due to the field enrichment caused by the surface plasmon polaritons and the discontinuity of the electric field in the spacer region. In order to comprehend the dense hybrid plasmonic integrated circuits, the coupling length between two identically parallel HIMIPWs have been estimated by varying an edge-to-edge separation distance. The smaller coupling length leads high crosstalk between two HPWs. Further, to improve the coupling length, a metallic strip has been inserted between the two parallel HIMIPWs, and the impact of different metals, as the inserted metallic strip, on the coupling length has been investigated.

Waveguide Modeling and Numerical Method
The cross sectional view of HIMIPW, consists of insulatormetal-insulator (SiO 2 -Ag -SiO 2 ) layers, inserted between two dielectric regions of silicon (Si), has been depicted in Fig. 1(a). Figure 1(b) indicates that the normalized power is mainly concentrated in the low-index regions. The dielectric materials, Silica (SiO 2 ) and Silicon (Si) are used respectively as the low-and high-index materials, having their corresponding refractive indices of 1.44 and 3.48 [34]. However, the Drude model can be used to calculate the permittivity of metal (silver, Ag), and can be expressed as [29], where, Ɛ ∞ , w p and γ are dielectric constant at infinite angular frequency, bulk plasma frequency and damping frequency of silver respectively, with their respective values of 3.7, 1.3826 × 10 16 Hz and 2.7438 × 10 13 per sec. Hence, the permittivity of silver can be extracted as −129 + 3.3i at 1.55 μm of wavelength. Silver (Ag) has been used as the plasmonic material, as it offers the low propagation loss and high field enhancement for plasmonic based devices; whereas, air and silica are respectively used as cladding and substrate materials. The thickness of low-index regions are assumed as t l1 and t l2 , and the same for the high-index regions are t h1 and t h2 . Moreover, the width of the HIMIPW has been considered as w. For the design and analysis of optical properties of

Optical Properties of Hybrid Mode of the HIMIPW
The investigations on optical properties of fundamental hybrid mode of HIMIPW have been done by changing the waveguide width (w) and thickness of dielectric regions. The metal thickness is taken as 0.03 μm to guarantee no effect on the plasmonic mode. The skin depth in plasmonic metal (Ag) remains roughly uniform at 0.02 μm in the near-infrared regions [34]. The modal properties of the quasi-TM mode of HIMIPW have been explored in terms of mode effective index, light confinement and normalized intensity, in the spacer regions. The real part of mode effective index can be presented as (Re(N eff ) = Re (β)/k), where, k and β are respectively, the wave number of free-space and propagation constant. Furthermore, the propagation length (L p ) of the waveguide is an important feature to design and analyze HPWs and its related devices. It can be described as a length (distance) over which the guided optical power is decreased to 1 e of the initial optical power and mathematically, it can be denoted as below [14], where, Im(N eff ) represents the imaginary part of mode effective index and λ is the working wavelength. On the other side, confinement factor (CF) is another key aspect of the optimal design of HIMIPWs. It can be defined as the ratio of amount of total optical power inside the spacer regions to the input optical power of HPW and can be expressed as [35], where, P zi (x, y) and P z (x, y) are time-averaged Poynting's vector along the positive z-axis, respectively in spacer region and whole waveguide geometry. Parallel to the CF, the normalized intensity (I) in the low-index region of the HIMIPW can be described as the ratio of normalized power (i.e., confinement factor) to the area of the low-index region [36].

Effect of Waveguide Width on Fundamental Hybrid Mode Profile
In this subsection, the impact of w on the fundamental hybrid mode profile has been examined by considering the thicknesses of dielectric regions as, t h1 = t h2 = t h = 0.2 μm, and t l1 = t l2 = t l = 0.02 μm, and w has been varied from 0.1 μm to 0.9 μm at working wavelength 1.55 μm. With respect to the variations in w, the behavior of real-part of effective index and propagation length have been depicted in Fig. 2 (a), whereas, the same for the confinement factor and normalized intensity have been shown in Fig. 2 (b). From Fig. 2 (a), it is clear that the propagation length (line in red color), and the real part of effective index (dotted line in black color) are initially decreasing, and increasing, respectively with the increase in value of w, and then, saturates for w > 0.7 μm. Similarly, the percentage of light confinement (dotted line in black color) in the spacer region, is initially increases and then, saturates with w > 0.7 μm, as depicted in Fig. 2(b). Moreover, the confinement factor of light wave is mainly dependent upon the power in the spacer region, which is directly reliant  Figure 2 (b) indicates that the normalized intensity (line in red color) in the low-index region, is first increases and goes to a maximum value, and then, decreases with the increasing w. It is mainly due to the fact that for larger values of w, the CF is almost constant, but at the same time the area of spacer region increases with increase in w. This leads to decrease in the value of normalized intensity and at w = 0.2 μm, the maximum value of the normalized intensity has been obtained as 67.50 μm −2 . Therefore, for the further investigations of the optical properties of HIMIPW, w is considered as 0.2 μm.

Effect of Dielectric Thicknesses on Fundamental Hybrid Mode Profile
As discussed in the last subsection, w has been fixed as 0.2 μm, to investigate the impact of thicknesses of dielectric layers on optical properties of the fundamental mode. Figure 3 shows the variations in the real part of effective index and propagation length, in terms of different thicknesses of low-and high-index regions of HIMIPW. From Fig. 3 (a), it is clear that Re (N eff ) decreases with the increasing t l , and it increases with the increasing t h values. If t h is high and t l is low, then, HPW behaves as conventional plasmonic nature (due to Ag-Si). Further, if both t h and t l are high, then also, the HPW turns towards the conventional plasmonic waveguide (due to Ag-SiO 2 ). The relationship between the propagation length and the thicknesses of the low-and high-index layers have been depicted in Fig. 3 (b), where the increase in propagation length has been observed with the increasing t h . This is mainly due to the dominating nature of dielectric waveguide (due to Si-SiO 2 ) on the HPW. However, if the t l is increasing, then the nature of propagation length is dependent on the t h . For example, if t h1 = t h2 = t h < 0.15 μm, propagation length is initially increasing, then saturates with increasing value of t l ; however, for t h > 0.15 μm, propagation length is first increasing, up to a maximum value, and then, it decreases with increasing t l . This nature of propagation length is due to the change in nature of HPW towards the plasmonic and dielectric behaviors, with However, if t h > 0.2 μm, at the low value of t l , the nature of mode is like conventional plasmonic mode (due to Ag-Si) and with increasing value of t l , the conventional plasmonic mode turns towards hybrid plasmonic mode, and further, for t l > 0.06 μm, the hybrid plasmonic mode turns toward plasmonic mode (due to Ag-SiO 2 ).
Further, Fig. 4 shows the relation of CF and normalized intensity with the thicknesses of dielectric layers. From Fig.  4(a), it is clear that with the increase in t h , the confinement of light initially increasing; moreover, after t h > 0.2 μm, CF is decreasing, as light starts to concentrate towards the high-index region. At t h = 0.1 μm, confinement factor is firstly increasing with t l and after some values (t l > 0.06 μm), the behavior of confinement factor is linear with a small constant slope. The confinement factor at t h = 0.1 μm is small as compared to other values. Further, with increase in t l , the light confinement factor first increases and after a certain value of t l , it is decreases with increasing t l . Therefore, it can be concluded that the maximum confinement factor can be achieved at the waveguide dimension of t h = 0.2 μm, with t l is in between 0.02 to 0.04 μm. The variations in normalized intensity can be visualized in Fig. 4 (b), which shows that with the increase in t l , the normalized intensity decreases. Moreover, in terms of the thickness of the high index layer, the normalized intensity is quite lesser, for t h < 0.1 μm and t h > 0.2 μm, which is mainly due to the lesser confinement of optical power in the spacer region. The maximum value of normalized intensity has been obtained at t h = 0.2 μm, with the lower values of t l .

Effect of Upper Dielectric Thicknesses on Fundamental Hybrid Mode Profile
In this subsection, the design of asymmetrical HIMIPW has been explored by keeping the width and thicknesses of lower dielectric layers fixed as, w = 0.2 μm, t h2 = 0.2 μm and t l2 = 0.02 μm. The designs of asymmetric HIMIPW have been investigated, first by varying t l1 , from 0.02 μm to 0.1 μm, while keeping t h1 fixed as 0.2 μm, and illustrated in Fig. 5 (a). The figure indicates that the propagation length and light confinement of the quasi-TM modes are inversely proportional. The propagation length is initially decreases to a minimum value and then, increases with the increasing t l1 . This is mainly due to the fact that for lower t l1 values, the hybrid plasmonic mode turns towards the conventional plasmonic mode (due to Ag-SiO 2 ). Moreover, after t l1 > 0.06 μm, the propagation length increases, as the conventional plasmonic mode turns toward dielectric mode, which is caused by the lesser or negligible impact of metal, in presence of thicker spacer region. The variations in CF illustrate just opposite nature as that of the propagation length. The second approach for design and investigations is by considering t l1 = 0.02 μm and varying the values of t h1 from 0.1 to 0.3 μm, as depicted in Fig. 5 (b). From figure, it can be observed that the propagation length increases to a maximum value (370 μm) at t h1 (= 0.225 μm), and then, it reduces gradually as the hybrid mode turns toward the conventional plasmonic mode (due to Ag-Si). However, the maximum CF in the spacer region has been observed as~27.5%, nearly at t h1 = 0.175 − 0.2 μm. Therefore, from the above analysis, it can be predicted that to achieve the miniaturized optical waveguide structure, the symmetrical HIMIPW is more beneficial than its asymmetrical counterpart. The optimized parameters for the symmetrical HIMIPW are w = 0.2 μm, t h1 = t h2 = t h = 0.2 μm, and t l1 = t l2 = t l = 0.02 μm and the optimized values of propagation length, confinement factor (%) and normalized intensity are 289.26 μm, 28% and 67.5 μm −2 , respectively. The comparison of the modal properties (propagation length and normalized intensity) of fundamental mode of the current work with the recently reported works has been depicted in Table 1. It clearly shows that the propagation length achieved in the current work is significantly better than that reported in recent literature. Also, in terms of the normalized intensity, the current work exhibits the considerably improved performance. In order to analyze the behavior of the localized surface plasmon resonance at the surface of the silver-silica interfaces by varying of wavelength, the optimum waveguide dimension has been considered as, w = 0.2 μm, t h = 0.2 μm, and t l = 0.02 μm. The electric field intensity is first increases up to the maximum value, and then decreases with the increasing wavelength, as shown in Fig. 6. It provides the highly localized electric field intensity at around 1.55 μm. Hence, the performance of the HIMIPW structure, at the working wavelength (1.55 μm), is more efficient.

Crosstalk between Adjacent Identical Parallel HIMIPWs
In order to search the possibilities to realize the highly dense monolithic integration of the presented HIMIPWs, the crosstalk performance between two identical parallel HIMIPWs has been done, for a separation distance (d). Figure 7 shows the cross-sectional view of coupling arrangement between two parallel HIMIPWs. To analyze the crosstalk between two HPWs, the coupling length has to be determined, as provided in Eq. (4) [14,16]. For the minimum/ negligible crosstalk between the two HPWs, the coupling length must be significantly larger [37]. The coupling length (L c ) is defined as the distance (length) at which the optical power is transferred from one HIMIPW to another and it can be expressed as [19], where, β s and β a are the propagation constants of symmetric and antisymmetric quasi-TM modes, respectively. To examine the modal properties of the symmetric and antisymmetric modes, the separation distance (d) has been varied from 0.1 μm to 0.6 μm. Figure 8 clearly indicates the mode profiles for the symmetric and antisymmetric quasi-TM mode. Further, Table 2 presents the values of coupling length and effective index of symmetric and antisymmetric modes for a wide range of d. From the table, it has been observed that if d increases, the real part of effective index of symmetric mode decreases and that of the antisymmetric mode increases. With further increase in d, at d = 0.80 μm, the effective index of both modes achieves the value of 2.41, which is known as the index of decoupled hybrid plasmonic mode. From the imaginary part of effective index, it can be established that the propagation loss of antisymmetric mode is larger than that of the symmetric mode. On the other hand, with the increases in d, the propagation loss of symmetric mode increases and that of antisymmetric mode decreases and both converges at d = 0.8 μm. The relationship between the coupling length and separation distance has been plotted in Fig. 9, which shows the  Further, to reduce the crosstalk and hence, to increase coupling length between two HIMIPWs, a metallic strip between the two HPWs has been inserted [35], as depicted in Fig. 11. Due to the presence of middle metallic strip, the increase in coupling length can be anticipated. This is mainly based on the fact that in the dielectric region the field attenuation is much slower than that in the metal region. To analyze the impact of metallic strip on the coupling length, the separation distance (d) has been fixed at 0.30 μm, whereas the width (W m ), and height (h) of metallic strip have been varied respectively, from 0.04 μm to 0.20 μm, and 0.10 μm to 0.60 μm, for different considered metals for metallic strip, such as Aluminium (Al), Copper (Cu), Gold (Au), and Silver (Ag). The refractive indices of Al, Cu and Au have been considered respectively as, 1.5785 + 15.658i, 0.7158 + 10.655i, and 0.5241 + 10.745i, at 1.55 μm of wavelength [38]. Figure 12 clearly indicates that the electromagnetic waves propagate through one waveguide to other in the presence of metallic strip. The mode profiles for the symmetric and antisymmetric quasi-TM mode with inserted metallic strip between two waveguides have shown in Figs. 12 (a) and (b), respectively. Figure 13 illustrates that the variations in coupling length    Table 3 indicates that the coupling length with Al as metallic strip, is greater than that with other metals. This is mainly due to the fact that Al has higher values of imaginary part of the refractive index for both the symmetric and antisymmetric modes, which further causes to the higher propagation loss. From Table 4, it has been observed that coupling length increases with the increase in height of inserted metal; however, for h = 0.27 to 0.37 μm, coupling length has both decreasing and increasing nature for all the considered metals. This is mainly due to the fact that with this range of metallic strip height, the impact of the upper part of HIMIPW is quite lesser. However, with further increase in h, the coupling length increases. Moreover, the presence of the metallic strip between two identical parallel HPWs, may cause the inconvenience for the fabrication of such optical devices.

Conclusion
In the present work, the analysis of fundamental hybrid mode properties in HIMIPW has been done to obtain the significant light confinement of optical power in nanoscale low-index region (~20 nm) and high propagation length (~289 μm). The normalized intensity of roughly 67.50 μm −2 , at the working wavelength of 1.55 μm has been obtained, which is mainly due to the field enrichment in the spacer regions of HIMIPW. However, the normalized intensity can be further improved by reducing the thickness of spacer region, up tõ 10 nm. This is beneficial for many applications, such as nonlinear optics, optical bio-sensing, etc. The electric field intensity has achieved the highest value at working wavelength of 1.55 μm, and it indicates that the performance of the current design of HIMIPW is more efficient at the target wavelength. The investigations, to achieve the suitable coupling length between two identical parallel HIMIPWs, has been done with different separation distances, which is further improved by inserting a metallic strip between the two parallel hybrid plasmonic waveguides at the same separation distance. Higher coupling length leads to low crosstalk between two hybrid plasmonic waveguides and it is useful to achieve the larger integration on photonic chip. The maximum values of the coupling length have been attained, with the use of wider and thicker metallic strip at the center between two waveguides. The SOI-based HPWs are advantageous for the realization of monolithic integration of passive devices and adaptable with CMOS technology.