## 2 Integration and Excitation of Plasmonic Structures
### Propagative Surface Plasmon
More specifically, the white arrows indicate the orientation of the total E field, while the \(+\) and \(-\) markers indicate the distribution of free charges on the Au surface. (**b**) The color gradient indicates the total electric field intensity. The superposed white line depicts the distribution of the intensity of the electric field x-component. (**c–k**) Plasmonic modes calculated for different waveguide geometries: (**c–e**) real part (red) of the electric field y-component when the propagation occurs along the x-direction (in-plane with respect to the figure plane). The label LR (SR) indicates the symmetric (asymmetric) coupling of two surface plasmons (SPPs) generating a “long-range” (“short-range”) mode. In (**d**) and (**e**) circled numbers 1 and 2 highlight the SPPs of the individual interfaces of the respective waveguides. (**f–k**) Electric field intensity of the wave in the case that propagation occurs along the z-direction (out-of-plane with respect to the figure plane). Structures inspired by [62]. Figure 2: Dispersion curves of propagative surface plasmons at the gold–air interface and gold–InP interface. The solid (dashed) lines represent the contribution solely from the real (imaginary) part of the complex permittivity. two side interfaces of the metallic waveguide. Associated with it is the dual supermode, the SR (short-range) SPP, which features an asymmetrical phase profile and increased losses [62]. To benefit of their high confinement and compactness while limiting the induced losses, such plasmonic waveguides can be selectively integrated into conventional single-mode dielectric photonic waveguides. The typical interfacing schemes (Figure 3) exploit either a progressive transition (such as evanescent coupling or grating coupling) or butt-joint transition between the photonic and the plasmonic waveguides. Butt-joint transition, also called end-fire transition, [63] offers poor performance because of the significant mode shape mismatch between the plasmonic and the photonic modes (Figure 3a) and will not be discussed here. The underlying physical mechanism harnessed through evanescent coupling (depicted in Figure 3b) or grating coupling (shown in Figure 3c) can be described by examining the dispersion curves of the involved modes (illustrated in Figure 3d,e): the dispersion curve of the SPP at the metal-dielectric interface (represented by the red curve) is situated significantly below the light line of the corresponding dielectric (characterized by an index \(n_{\text{ext}}\)). This is especially true for modes with strong plasmonic character, particularly when the SPP dispersion curve becomes quasi-horizontal. The plasmon cannot be excited by an electromagnetic wave with a propagation constant lower than \(k_{\text{sp}}\). In the integrated configuration where the plasmonic waveguide is placed in proximity to a dielectric waveg  
Figure 3: (**a**–**c**) Diagram illustrating three different typical ways of coupling a dielectric waveguide with a plasmonic waveguide: (**a**) end-fire coupling, (**b**) evanescent coupling, and (**c**) coupling via a diffraction grating. (**d**) The wave vector \(k_{\text{d}}\) of the guided mode in the dielectric waveguide is equal to that required to excite the plasmonic mode, \(k_{\text{sp}}\). (**e**) The wave vector of the dielectric waveguide mode is not sufficient and requires an additional contribution \(\delta k\), arising from the diffraction grating, to match that of the plasmonic waveguide mode. uide (i.e., via a multilayer configuration as depicted in Figure 3b), the effective index \(\mathrm{n_{d}}\) of the dielectric waveguide mode is higher than the refractive index of the surrounding dielectric material by construction (\(\mathrm{n_{d}}>\mathrm{n_{ext}}\)); this property can be used to directly excite the SPP when \(\mathrm{k_{d}}\cong\mathrm{k_{sp}}\) (phase matching), since both dispersion curves cross each other (see the dispersion diagram in Figure 3d).