## 2 Integration and Excitation of Plasmonic Structures
### Propagative Surface Plasmon
In this case, when the polarization of the dielectric guided mode supplies the necessary z- and x- electric field components for SPP excitation, mode coupling occurs. This results in the generation of two supermodes with a hybrid photonic-plasmonic nature, featuring field symmetric and antisymmetric distributions and effective indices \(n_{e}\) and \(n_{o}\) (as shown in Figure 3b), respectively. The half-beating length of these supermodes determines the coupling length \(L_{c}\) of both waveguide modes, which is related to their wavevectors difference:  
\[Lc=\left|\frac{1}{2}\frac{\lambda}{n_{e}-n_{o}}\right|. \tag{2}\]  
After a propagation distance of \(L_{c}\) through the coupling region, the energy of the dielectric guided mode is totally transferred into the SPP. Such a mode coupling is called \"evanescent coupling\" because it involves the evanescent tails of the guided modes. The required condition (\(\mathrm{n_{d}}>\mathrm{n_{ext}}\)) is more easily achieved when the targeted SPP is on the opposite metallic film side with respect to the dielectric waveguide, as shown in Figure 3b. Indeed, the outside material can have a low refractive index, independent of the guided mode effective index. On the dispersion curves map, this interaction induces anticrossing of both curves. In the case of high contrast between \(\mathrm{n_{ext}}\) and \(\mathrm{n_{d}}\), the involved plasmonic mode dispersion curve is almost flat (horizontal), and thus the generated supermodes have very different effective indices: the higher the supermode index difference, the shorter the coupling length. Thus, such a mechanism is particularly efficient when the contrast between \(\mathrm{n_{ext}}\) and \(\mathrm{n_{d}}\) is high. [PERSON] et al. (2018) have realized this for example in the case of an SOI waveguide and a Cu plasmonic slot waveguide at 1.55 \(\mathrm{\SIUnitSymbolMicro m}\). The experimental coupling length equaled 0.9 \(\mathrm{\SIUnitSymbolMicro m}\) and the coupling efficiency was estimated at 70%. In another example involving polymer waveguides, [PERSON] et al. (2018) numerically demonstrated the coupling between a SU8 on a glass waveguide and a buried plasmonic waveguide at 633 nm, with \(\mathrm{L_{c}}=5.3\)\(\mathrm{\SIUnitSymbolMicro m}\) and a coupling efficiency of 88%. The use of SOI waveguides is particularly interesting since, in that case, the dielectric waveguide mode has a high effective index. However, when coupling the fundamental TM mode of a high-contrast dielectric waveguide (with a high effective refractive index) to the LR-SPP mode of a thin-film plasmonic waveguide embedded in a low refractive index medium (with a low effective refractive index), it can be beneficial to employ layers of dielectric material with higher refractive indices that encapsulate the plasmonic waveguide. This helps to satisfy the phase-matching condition, as shown in ([PERSON] et al., 2018). If the condition \(\mathrm{k_{d}}\cong\mathrm{k_{sp}}\) cannot be directly fulfilled, the guided dielectric mode must interact with a complementary structure (Figure 3c) to increase its wavevector module up to \(\mathrm{k_{sp}}\) (\(\mathrm{k_{sp}}=\mathrm{k_{d}}+8\)k, see Figure 3e); the interaction of the dielectric waveguide mode with a periodic grating (period \(\Lambda\)) serves this purpose since it generates spatial harmonics whose constant propagations equal \(\mathrm{k}=\mathrm{k_{d}}+\mathrm{p}\ 2\pi/\Lambda\), where \(\mathrm{p}\) is a relative integer. The first harmonic (\(\mathrm{p}=1\)) is the most intense; by choosing \(\Lambda\) so that \(\mathrm{k_{sp}}-\mathrm{k_{d}}=2\pi/\Lambda\), the grating provides the dielectric waveguide mode with the missing component \(8\)k to excite the SPP at the dielectric/metal interface (each surface of the metal can be targeted, by proper choice of \(\delta\mathrm{k}\)).