## 2 Integration and Excitation of Plasmonic Structures
### Propagative Surface Plasmon  
Propagative surface plasmon is generated at the interface between a metal and a dielectric material. It can propagate at several wavelengths' distance along this interface. It has the characteristic profile of a transverse magnetic (TM) wave, featuring its primary electric field component perpendicular to the interface and a secondary, smaller component in the propagation direction. Its profile also exhibits an exponential decrease, forming an evanescent profile on each interface side, as shown in Figure 1a,b. Propagative SPP is analytically described by its dispersion curve, which is established by solving the Maxwell equations in the case of the double exponential profile [6]:  
\[k_{sp}(\omega)=\frac{\omega}{c}n_{eff}=\frac{\omega}{c}\sqrt{\frac{\epsilon_ {ext}.\epsilon_{m}}{\epsilon_{ext}+\epsilon_{m}}} \tag{1}\]  
where \(k_{sp}\) is the wavevector of the SPP, \(n_{eff}\) represents the effective refractive index of the mode, and \(\epsilon_{ext}\) and \(\epsilon_{m}\) are the permittivity of the dielectric and the metal materials, respectively. Dispersion curves and dispersion diagrams will be extensively used in this manuscript as a powerful tool to identify the behavior of plasmonics and hybridized plasmonic-photonic modes. This tool helps to identify and engineer various coupling mechanisms. A dispersion diagram shows the relationship between the frequency f (or angular frequency \(\omega=2\pi\)f) and the wavevector \(k=(\omega/c)\times n_{eff}\) of a specific mode within a given observation range. It is important to note that dispersion curves for non-guided waves propagating through homogeneous media exhibit a straight-line shape that passes through the origin of the axes. The slope of these curves indicates the refractive index of the medium. An interaction between two modes with compatible polarization is expected near the crossing point of the corresponding two non-interacting dispersion curves. The result of the interaction will be the formation of two supermodes whose dispersion curves will manifest a gap (anticrossing) in the vicinity of the aforementioned crossing point. The width of the gap formed is representative of the \"strength\" of the interaction between the two original modes. For more details on reading and using band diagrams, see [59]. In Figure 2, the SPP dispersion curve is represented in the case of air-gold and InP-gold plasmonic modes, where gold permittivity spectrum is approximated by the Drude model [60; 61] and \(\omega_{\rm p}\) represents the plasma resonance angular frequency in the metal. These curves reveal the primary properties of SPPs: (1) the SPP dispersion curve consistently remains below the light line of the surrounding dielectric medium, indicating that SPPs propagate with a higher wavevector (or equivalently a higher effective index) than the electromagnetic wave in this medium; (2) as the SPP frequency approaches the plasmonic resonance, i.e., \(f_{\rm sp}=\omega_{\rm sp}/2\pi\), the dispersion curve becomes quasi-horizontal. This indicates that SPPs propagate very slowly, with a near-zero group velocity, and becomes highly confined near the metal (featuring a high k). (3) The plasmonic resonance frequency exhibits a strong dependence on the refractive index of the surrounding medium. Several types of propagative plasmonic waveguides have been proposed, as schematized in Figure 1c-k. Some of these are designed for long-range propagation (LR) by minimizing the direct interaction between the wave energy and the metal. This is achieved through coupled SPPs on each side of a thin metallic film (Figure 1d,g) or in a metallic slot waveguide (Figure 1e,h-j). The LR-SPP is the supermode with a symmetrical phase profile and reduced losses that arises from the coupling of the two SPP modes propagating on the  
Figure 1: (**a,b**) Profile of a surface plasmon polariton (SPP) on an air–Au interface: (**a**) The color scale corresponds to the normalized real part of the x-component of the electric field.