Numerical Analysis of a GaAs-Based Hybrid Plasmonic Waveguide with Nanoscale Optical Confinement and Low Losses

A numerical analysis of a hybrid plasmonic waveguide (HPW) for deep subwavelength optical confinement and long-range propagation with low loss is presented here. Two types of material platforms, namely, Si/SiO2/Au and GaAs/SiO2/Ag, were analyzed to optimize the HP waveguide. The mode character, an important and crucial design parameter for HP waveguides, was calculated based on the coupled mode theory, providing the coupling strength between the SPP and optical mode. As for the Si/SiO2/Au HP waveguide, the coupling strength varied from 0.47 to 0.60 with a mode area ranging from 0.0002/μm2 to 0.001/μm2 and mode character near to the SPP mode character (i.e., |a+(tsi, w, td)|= 0.47). While for the GaAs/SiO2/Ag HP waveguide, the coupling strength varied from 0.57 to 0.69 with a mode area ranging from 0.0002/μm2 to 0.0004/μm2 and mode character to SPP approached as |a+(tGaAs, w, td)|= 0.49. Finally, a finite element method (FEM) model was used to investigate the modal properties. The simulation analysis shows that at td = 10 nm the GaAs/SiO2/Ag waveguide gives 50% larger propagation length (205 μm), ten times smaller mode area (0.0002) with 60% lower modal propagation loss (0.021 db/μm), and 20% stronger coupling strength (0.62) with HPP mode character as 0.48 as compared to Si/SiO2/Au.


Introduction
The recent progress in the field of Si photonics leads to advancements of photonic devices in various applications, such as grating couplers, narrowband reflectors, and optical interconnects [1][2][3]. The emerging area of nanophotonics helps to address the critical challenges to realize the nanoscale dimension of the optical devices. The advancements in the area of nanofabrication have advantage to control the properties and dimensions of the nanophotonics devices to desired level. But the photonics devices cannot be miniaturized beyond the diffraction limit of the light and this issue has been addressed with proposal of various novel waveguides [4,5]. The plasmonics waveguide-based components are attracting an ever-increasing interest in recent years [6][7][8], these waveguide-based devices lead to an integration of nanoscale confinement and controlling light beyond the diffraction limit.
However, such desirable features come at a cost, namely, high propagation losses associated with field penetration in metal regions. Thus, significant efforts have been directed in reducing or compensating the inherent resistive losses of plasmonic waveguides through elaborate geometrical/material configurations [9][10][11][12]. Plasmonics as a major part of an emerging field of nanophotonics [13][14][15] explore how electromagnetic field can be confined on a scale much smaller than the wavelength.
The confinement and guiding of the light at the nanoscale can be dealt with the mode hybridization of photonic and surface plasmon polariton (SPP) mode analytically with the help of the parameter known as mode character, which tells about the nature of the hybrid mode resulting from coupling of the optical and SPP mode.
Further to address the issue with plasmonics waveguide of high loss and low propagation length, a new type of plasmonic waveguide is proposed called hybrid plasmonic waveguide (HPW) which have the capability to provide both low propagation loss and strong field confinement [16][17][18][19][20][21][22][23][24][25]. The HPW mechanism first time proposed [25] and explained the phenomena for the super mode propagation in low index medium. In this proposed work, the mode coupling and propagation characteristics of HPW waveguide 1 3 with two different material combinations as GaAs/SiO 2 / Ag and Si/SiO 2 /Au have been presented which results in strong coupling with low loss at the subwavelength scale.
The design and simulation analysis of HPW for the achievement of nanoscale confinement of HP mode has been done using the finite element method (FEM) [21][22][23][24][25]. The photonic mode at the semiconductor and dielectric interface can be coupled with SPP mode at dielectric and metal interface to give us HP mode with longer propagation length because of the leaky nature of the modes confined in rectangular geometries. In addition to this, the evanescent nature of the field confined in a high-index region under dielectric region has been introduced to understand the coupling of surface plasmon and optical field.
The HPW is a promising integrated platform for making photonic devices at the real nanoscale. The coupling and propagation analysis of the SPP and photonics mode of the HP waveguide are presented with the two material systems in this paper as GaAs/SiO 2 /Ag and Si/SiO 2 /Au. The HP mode is formed by the coupling of the field confined in the semiconductor region (Ga/Si) with the SPP mode controlled through the low-index dielectric region sandwiched between the metal and semiconductor region. The choice of material is very important for the obtainment of longer propagation lengths with nanoscale confinement. As previously reported material system Si/SiO 2 /metals (Au, Ag) in [15,16,23], the noble metals (Au, Ag) support the longer propagation length (L p ) and low loss with SiO 2 as the confinement region of the HP mode. This helps in the induction of the gain, compensation of the loss in metal, and increases the SPP signal propagation length with the Si. However, the Si-based HPW system has not used as laser action purposes due to the limitation of the Si in terms of the low band gap and operation wavelength range up to only infrared. The wider transparency range (0.9-17 μm) of the GaAs material with a high laser damage threshold with low absorption motivates us to use the GaAs for the HPW [26]. The choice of the GaAs/SiO 2 /Ag material system has the achievement of the longer L p and tight mode area, as GaAs have higher permittivity as compared to Si which leads to better mode coupling between plasmonic and photonic mode. The GaAs/ SiO 2 /Ag material system not only has advantages in coupling and low loss but is also used for the realization of the nanoscale lasers [27]. This mode hybridization is required to predict the nature of the mode character which is denoted as "a." This mode character ("a") helps us to predict the nature of the mode whether it is pure plasmonic or photonics mode and when it becomes the hybrid plasmonic photonic (HPP) mode. The coupled mode theory has used to calculate the "a" which tells the coupling between the SPP and photonic mode, which leads to the formation of HPP mode [16,17].
The proposed devices are easy to fabricate with the available nanofabrication technology. The coupling strength obtained with GaAs/SiO 2 /Ag material system is 0.62 with larger propagation length (L p ) 205 µm and low modal propagation loss (L m ) of 0.021 db/µm. The proposed HPW devices are used for potential application in the high-level photonic integration.

Proposed Waveguide Design
The proposed waveguide design of hybrid plasmonic waveguide is shown in Fig. 1. It consists of GaAs/SiO 2 /Ag material system to guide and confine the HP mode in SiO 2 . The thickness of the GaAs is shown in Fig. 1 as t GaAs = 200 nm, and thickness of the Ag layer as t Ag = 200 nm. The thickness of SiO 2 is denoted as t d , and width of the waveguide is denoted as w.
As per previous work for the [17,24] single mode operation of the waveguide, the width of the waveguide should remain below 240 nm at operating wavelength of 1550 nm for low loss and strong field confinement, with good coupling strength and larger propagation length. The minimum value of t d = 10 nm below which the rectangular geometry of the SiO 2 does not support the HP mode and the mode becomes the pure SPP [17]. So, the subwavelength optical confinement with strong mode confinement and lesser modal propagation loss is achieved at t d = 10 nm.

Numerical Analysis for the Subwavelength Optical Confinement of Proposed Hybrid Plasmonic Waveguide
The numerical analysis of the proposed HPW has been carried out with the help of FEM and coupled mode theory. The optical energy is always confined inside the dielectric waveguide core to form the photonic mode [17]. The coupled mode theory [17] gives the eigen mode Ψ supported in the coupled waveguide system can be described as the superposition of the rectangular mode Ψ react and SPP mode. Here t d is the thickness of the dielectric (SiO 2 ) and (t GaAs , w) is the waveguide thickness and width. Further, (a + (t GaAs , w, t d )) is the field amplitude of the rectangular waveguide and (b + (w, t d )) is the field amplitude of the SPP mode. So, the wave function of the symmetric eigen mode Ψ + can be written as [17,24] (1) The square form of the rectangular mode amplitude is the measure of the character of hybrid mode, known as mode character given by |a + (t GaAs , w, t d )| 2 , it shows the degree to which the guided mode behaves like an optical or SPP mode. As we know from [6], for a mode to be HPM, it should show the possible mode character as [17] And the field amplitude of SPP mode is given as, The simulation of the proposed waveguide design denotes that the hybrid mode effective index is always larger than that of underlying rectangular and SPP waveguide mode. This indicates the behavior of the typical coupled mode system, so the mode character is given as, Here n hyb is the effective index of the hybrid mode, n SPP is the effective index of the SPP mode, and n rect . is the effective index at GaAs/SiO 2 interface. In next step, formula for n SPP and n rect. has been obtained. As we know from [17], the coupling strength plays most important role in dragging the optical mode to metal dielectric interface, and the coupling strength is detonated as κ. It is the coupling strength between SPP and optical mode and given as [17] where n eff is the effective index of the waveguide mode, and n spp is the effective index of the surface plasmon mode, where ε m , ε d is the permittivity of the metal (Ag) and dielectric (SiO 2 ), and from Eq. (6), the value of effective index of plasmonic mode has been obtained. The permittivity has great effect on the propagation length and mode area, as GaAs has higher permittivity as compared to Si which leads to better mode coupling between plasmonic and photonic mode. Similarly, the formula for rectangular waveguide which gives the effective refractive index of the photonic mode has been obtained as where ε c , ε d is the permittivity of the conductor (GaAs) and dielectric, where effective index helps to calculate the coupling strength between the Si/SiO 2 /Au and GaAs/SiO 2 /Ag material system. Figure 2 indicates the coupling strength between plasmonic and photonic mode for Si/SiO 2 /Au and GaAs/SiO 2 /Ag material system with variation of t d from (5) κ(t GaAs , w, t d ) = √ n eff − n spp n eff − n rect. From Eqs. (12) and (6), we reached to the conclusion that the n + (w, t d ) = n hyb .(w, t d ) as reported [17]. The simulation and analysis using the FEM method leads to n eff which helped to obtain the mode hybridization of the optical and SPP mode. The obtained mode character of HPM approximate equal to |a +(w, td) | 2 = 0.48 at t d = 10 nm for GaAs/SiO 2 / Ag with very low modal propagation loss as 0.021 db/μm, larger propagation length as 205 μm, and tight field confinement. Figure 3 shows the variation of the mode character with thickness of the dielectric at constant w. The value of "a" for the GaAs/SiO 2 /Ag is observed as 48% while it is 47% for Si/SiO 2 /Au at trade-off t d .

Effect of Material Parameter on Modal Characteristics of the Hybrid Plasmonic Waveguide
The mode character and coupling strength of the proposed HPW is presented in Figs. 2 and 3. In this section, geometrical optimization of waveguide is carried out which maintain the trade-off between modal propagation characteristics. The choice of material plays a very crucial role on the modal properties of the waveguide. In our proposed design, we chose two different types of the material, which are Si/SiO 2 /Au and GaAs/SiO 2 /Ag. By choosing these two types of material, we obtained very good difference in propagation length, coupling strength, modal propagation loss, and mode area. The wavelength we chose for this study is 1550 nm to target the possible application of the waveguide in telecommunication. The choice of noble metal tells us the longer propagation length characteristic, and appropriate choice of the dielectric enables us the induction of gain, compensation of the loss in metal, and increases the SPP signal propagation length. We chose Ag as highest conductivity material among metal materials in the wavelength range. The metal with highest conductivity has highest propagation length and lesser modal propagation loss. In GaAs and Si, GaAs have higher value of relative permittivity which helps it to enhance more electric field in the dielectric region. The material loss is reduced to some extent by introduction of the GaAs/SiO 2 /Ag as compared to Si/SiO 2 /Au. The real part of the effective index (n eff ) is 2.64 in case of Si/SiO 2 /Au and 2.74 for GaAs/SiO 2 /Ag, where w is 200 nm and t d is 10 nm. So, the field will travel longer in the second case of the GaAs/ SiO 2 /Ag. The strong field enhancement in the SiO 2 is due to the combination of SPP at Ag and SiO 2 interface and the discontinuity in the field confinement E y from at GaAs-SiO 2 interface [20]. Geometrical parameters of the proposed waveguide design can be significantly controlling the guiding performance of the HPW by observing the effect of the thickness and width of the SiO 2 . The effective index n eff of the guided mode rises as width increases or t d decreases. The simulation of the guiding characteristic is done using FEM. The propagation length for the guided HP mode is given by [24] and modal propagation loss is given by The purpose of the optimization is to maintain the tradeoff between the modal propagation loss and mode area. Figure 4a, b show the variation of the L p (µm) with width for Si/SiO 2 /Au and GaAs/SiO 2 /Ag at different t d . By varying the wand t d of the SiO 2 in both the cases, i.e., for GaAs/ SiO 2 /Ag and Si/SiO 2 /Au, it is observed that strong mode confinement with lesser L m (db/µm) and larger L p (µm) is obtained at w < 240 nm at t d = 10 nm. In Fig. 4a, b, it is observed that with the increases of the w and decreases of the t d , the L p (µm) is going to be decreased. This happens in both the cases and similarly in Fig. 4c, d as we increase the width and decreases the thickness, the modal propagation loss is going to be increased in both the cases. In Fig. 4ad, w is varied from 100 to 250 nm with t d varied from 2 to 15 nm. As carried out previously [17] beyond the w of 100 nm, there is plasmonic mode and after > 240 nm, there will be photonic mode. So, from this optimization, the best value of w is 200 nm which is further used for the optimization of the t d at contact w. Figure 5 shows the variation of the L p and modal L m with t d at contact w 200 nm for both the material systems. Figure 5a shows the variation of the L p with t d , where t d varies from 2 to 15 nm, it has been observed that as the t d increases, L p increases for Si/SiO 2 / Au as well as GaAs/SiO 2 /Ag. The larger value of L p 205 µm (14) L m (dB∕um) = (2K 0 (img.)n ef f ) × 4.34  Fig. 4d. Similarly, the lesser value of Lp 123 μm with higher modal Lm 0.035 dB/μm is shown in Fig. 4a, c is obtained with the GaAs material system as compared to Si material system the value of L p is 123 µm at t d = 10 nm. The t d = 10 nm is the optimized thickness of the confinement region which maintains the trade-off between confinement and loss. Similarly in Fig. 5b, the lowest value of the L m is obtained as 0.021 dB/µm at t d = 10 nm for GaAs material system as compared to L m value of 0.035 dB/µm for Si material system. After obtaining the optimized value for the w and t d to maintain low L p and L m , the next step is to calculate the mode area which is able to maintain the nanoscale optical confinement. The nature of the guided mode can be evidenced by modal area of the field confinement [24]. The effective mode area of the guided fundamental mode will be a measure of the nature whether the localized field is optical, pure plasmonic, or hybrid plasmonic [24] and it would be the merit for confinement ability which is given by variable t d , a different type of the behavior is found in case of the mode area in Fig. 6. It is observed that at w = 200 nm and t d = 10 nm in both the cases, we have proper trade-off between the modal propagation loss, mode area, and propagation length. The mode area at these values is 0.001/μm 2 for Si/SiO 2 /Au and 0.0002/μm 2 for GaAs/SiO 2 /Ag. This variation in the nanoscale confinement between two material system leads to selection of the t d = 10 nm as best for nanoscale confinement with low loss. The GaAs material system shows better confinement as compared to the Si material system. As the variation of w from 200 to 220 nm, the L p starts decreasing, and L m starts increasing with increase of A m . The coupling strength is 0.51 in case of Si/SiO 2 /Au and 0.62 in case of GaAs/SiO 2 /Ag and mode character is very near to HPP = 0.48 as compared to other value of the wavelength width and height. After obtaining the best-optimized HPW modal characteristic of the GaAs/SiO 2 /Ag material combination, a comparison table of the previously reported HPW is shown in Table 1, which covers the best-obtained value of the HPW modal characteristic in terms of the loss, confinement, and length.  Table 1 Comparison of previously reported HPW for the best-optimized parametric value for the L p (μm), L m (dB/μm), and A m (/μm 2 ) Reference no.

Conclusion
The analytical design of the subwavelength optical confinement with lesser modal propagation loss and long-range propagation length is proposed. This introduction and comparison of the two materials, and their optimized study by varying the waveguide width and t d , leads to vary lesser modal propagation loss, tight field confinement compared to the hybrid mode confinement to the SiO 2 in case of Si/SiO 2 /Au we have small mode area in Ag and GaAs based system as A m = 0.0002/μm 2 and the lesser modal propagation loss = 0.02 dB/μm, and larger propagation length L p = 205 μm. The proposed easy to fabricate waveguide design of the waveguide will be useful for the realization of the integrated nanophotonics devices with acceptably small loss.