## 1 Introduction  
Active and passive plasmonic devices are becoming one of the most promising candidates that could overcome the size mismatch between micro scale photonic devices and nanoscale electronic devices [1; 2]. A promising way to achieve electrical control of light could be through plasmon polaritons--Coupled excitations of photons and electrons. Merging nanoelectronics and nano-optics enables the development of active subwavelength scale optics and an abundance of nano-optoelectronic devices and functionalities, such as tunable metamaterials, nanoscale optical processing, and strongly enhanced light-matter interactions for quantum devices and biosensing applications [3]. Recently [4], ultrafast optical switching of infrared plasmon polaritons has been reported in high-mobility graphene. The rapid progress in graphene plasmonics is propelled by their electrical tunability, high field confinement, potentially long lifetimes of plasmons and the strong light-matter interactions in graphene. It is worthy to note that advanced near-field imaging methods allow the plasmon-polariton waves to be efficiently launched and visualized as they travel along van derWaals materials layers (like graphene) or through multilayered heterostructures. This leads to strong nonlinearities, large photonic forces, and enhanced emission and absorption probabilities. A practical approach toward nanoscale light trapping and manipulation is offered by interfaces separating media with permittivities of opposite signs [5]. Graphene plasmons are very promising to be the best alternative to noble-metal plasmons as they exhibit much tighter confinement and relatively long propagation distances, with the advantage of being highly tunable via electrostatic gating [6]. The integration of plasmonic-based devices and photonic devices will eventually lead to an interfacing between plasmonic and photonic guides. Especially, butt-coupling plasmonic and dielectric guides is one of the most efficient ways for light transmission between highly confined modes and low-loss guided wave structures [7]. It is worthy to note that subwavelength optical confinement modes, allows the efficient and beneficial use of compact photonic devices like nanoscale photodetectors; thus, improving noise immunity, response speed and power dissipation in optical communication devices [8; 9]. The versatility of the FFT-BPM in the simulation of wide range of photonic devices is well understood [10; 11; 12; 13; 14]. In this paper, for the sake of simplicity, we shall consider two-dimensional structures invariant in the y-direction. The main essence of the BPM is simple: propagation in a reference homogenous medium over a small distance \(\Delta z\) followed by phase correction that takes into account the spatial dependence of the refractive index of the medium. However, three major problems should be considered carefully when dealing with step discontinuity and plasmonic devices: The transverse magnetic nature of modes, the abrupt index change in the transverse direction and in the propagation direction at the junction plane [15]. Two remedies are used to circumvent the first two problems: the TM nature of the propagated mode is transformed to an equivalent TE field propagating in an equivalent refractive index medium [16; 17]. Concerning the step-like transverse variation of the refractive index of the dielectric and plasmonic guides, we used a sigmoid-like function to approximate such abrupt index changes. It is worthy to note that many smoothing functions could be used [10; 13; 16]. We adopted the sigmoid function because of its simplicity in the analytical and numerical evaluation of the second derivative of the inverse index profile in the TM wave equation. The third problem concerning the reflected field at the junction plane, we adopted a well-established, combined spatial-spectral operator formalism [12; 14]. In this paper, a dielectric waveguide with its end facet terminated by air is considered in two cases: with and without coating layer.