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 Ψreact. and SPP mode. Here td is the thickness of the dielectric (SiO2) and (tGaAs, w) is the waveguide thickness and width. Further, (a+ (tGaAs, w, td) is the field amplitude of the rectangular waveguide and (b+(w,td) is the field amplitude of the SPP mode. So, the wave function of the symmetric eigen mode Ψ+ can be written as [24][17]
Ψ+(tGaAs, w, td) = a+(hGaAs, w, td)Ψreactangular waveguide(td) + b+(tGaAs, w, td)Ψspp. (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+(tGaAs,w,td)|2, it shows us degree to which the guided mode behave 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
|a+(tGaAs, w,td)|2 = 0.5 (2) [17]
And the field amplitude of SPP mode is given as
b+(w,td)=\(\sqrt{{\left(1-|{a}_{+ }\left({t}_{GaAs, }w, {t}_{d}\right)\right)|}^{2}}\) (3)
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,
|a+(tGaAs, w,td)|2 = \(\frac{\left({n}_{hyb }\left(w, {t}_{d}\right)\right)- {n}_{spp })}{\left({n}_{hyb }\left(w, {t}_{d}\right)\right)- {n}_{rect.\left(w\right)})+\left({n}_{hyb }\left(w, {t}_{d}\right)\right)- {n}_{spp })}\) (4)
Here nhyb is the effective index of the hybrid mode, nSPP is the effective index of the SPP mode and nrect. is the effective index at GaAs/SiO2 interface. In next step, formula for nSPP and nrect 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
κ(tGaAs, w,td) = \(\sqrt{\left({n}_{eff}-{n}_{spp}\right)\left({n}_{eff}-{n}_{rect.}\right)}\) (5)[17]
where neff is the effective index of the waveguide mode.and nspp is the effective index of the surface plasmon mode
nSPP =\(\sqrt{\left({\text{Ԑ}}_{m}{\text{Ԑ}}_{d}\right)\left({\text{Ԑ}}_{m}{+\text{Ԑ}}_{d}\right) }\) (6)
nSPP = = \(\sqrt{\left({\text{Ԑ}}_{m}{\text{Ԑ}}_{d}\right)\left({\text{Ԑ}}_{m}{+\text{Ԑ}}_{d}\right)}\) (7)
where Ԑm, Ԑd is the permittivity of the metal (Ag) and dielectric (SiO2), 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
nreact = \(\sqrt{\left({\text{Ԑ}}_{c}{\text{Ԑ}}_{d}\right)\left({\text{Ԑ}}_{c}{+\text{Ԑ}}_{d}\right)}\) (8)
where, Ԑc, Ԑd is the permittivity of the conductor (GaAs) and dielectric, where effective index help to calculate the coupling strength between the Si/SiO2/Au and GaAs/SiO2/Ag material system. Figure 2 dedicates the coupling strength between plasmonic and photonic mode for Si/SiO2/Au and GaAs/SiO2/Ag material system with variation of td from 2 nm to 15 nm. The device helped to maintain the tradeoff between loss and confinement and to obtain the best value of coupling strength (κ) at w = 200 nm and at td = 10 nm. The enhanced value of the κ for the Ag/GaAs has been obtained as 0.62 in comparison to Au/Si it is 0.51.
As in fig, 2 we showed the variation of coupling strength with thickness of dielectric from 2 nm to 15 nm. Further calculation of the mode character has been carried out by following equations. The reduced equation helped to predict the mode character of the hybrid mode and SPP mode as
|a+(tGaAs, w,td)|2 = \(\frac{{\left({\kappa }(\text{w},{t}_{d}\right)}^{2}}{{n}_{+}\left(\text{w},{t}_{d}\right)- {\left({n}_{react}(\text{w}\right)}^{2}+{\left({\kappa }(\text{w},{t}_{d}\right)}^{2}}\)(9)
The reduced form of the equation from coupled mode theory is given by
n+ = n(w,td)+∆(w,td) (10)
n(w, td) = \(\frac{{n}_{react}\left(\text{w}\right)+{n}_{spp}}{2}\) (11)
∆(w,td) = \(\frac{{n}_{react}\left(\text{w}\right)- {n}_{spp}}{2}\) + κ(w,td)2 (12)
Now putting these values in the Eq. 8, we get
|a+(tGaAs, w,td)|2 =
$$\frac{\left(\left({n}_{eff}-{n}_{spp}\right)\left({n}_{eff}-{n}_{rect.}\right)\right)}{\begin{array}{c}\frac{\left({n}_{react\left(w\right)}+ {n}_{spp}\right)}{2} + \frac{\left({n}_{react\left(w\right)}- {n}_{spp}\right)}{2}+\left(\left({n}_{eff}-{n}_{spp}\right)\left({n}_{eff}-{n}_{rect.}\right)\right)-{{n}_{rect.}\left(w\right)]}^{2}\\ + \left(\left({n}_{eff}-{n}_{spp}\right)\left({n}_{eff}-{n}_{rect.}\right)\right)\end{array}}$$
From Eqs. (13) and (6) we reached to the conclusion that then+(w,td) = nhyb.(w,td) as per correct it [17]. The simulation and analysis using the FEM method leads to neff which helped to obtain the mode hybridization of the optical and SPP mode. The obtained mode character of HPM approximate equal to
Figure 3. Variation of the Mode character with the td at constant w = 200nm. The higher value of mode character in the GaAs/Ag as compared to Si/Au.
|a+(w,td)|2 = 0.48 at td = 10nm for GaAs/Ag with very low modal propagation loss as 0.021db/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 Ag/GaAs is observed as 48% while it is 47% for Si/Au at tradeoff td.