## 2 Integration and Excitation of Plasmonic Structures
### Localized Surface Plasmons  
Localized surface plasmons (LSPs) result from plasmonic excitation in low-dimensional metallic structures with typical dimensions lower than the wavelength in the metal. LSP modes are eigenmodes of such a subwavelength metallic structure, which behaves as a dipole or a multipole. The LSP dipolar response of the nanostructure to the electromagnetic excitation is characterized by its polarizability, which has an analytical expression in case of \"simple\" shapes and a homogeneous surrounding medium [67].  
The excited dipole in a plasmonic nanoparticle radiates itself an electromagnetic wave of the same frequency, which can excite another plasmonic nanostructure with similar resonance; in this way, plasmonic nanostructure assembly may support collective resonances and/or propagate energy from nanostructure to the next [68]. An LSP chain dispersion curve can be thus established by using an analytical model in a homogeneous medium [69] or numerical methods in a general case. For instance, Figure 4 shows the dispersion curve, calculated using the FDTD method, of a plasmonic chain composed of gold nanocylinders with elliptical cross-sections placed on top of a semi-infinite Si substrate. The LSP chain dispersion curve displays a similar overall behavior as that of the SPP: it is positioned below the light line and exhibits a nearly horizontal slope at the highest k vectors. In other words, the collective modes of the LSP chain propagate through the chain elements as they do in a waveguide despite the intrinsic discontinuity of the metallic elements.  
Furthermore, chains of plasmonic nanoantennas have the ability to support topologically non-trivial modes (edge states resembling topological insulators) [70; 71; 72; 73; 74], bound states in the continuum [75; 76; 77], and surface lattice resonances [78; 79; 80; 81].  
LSPs in a subwavelength structure enable very high light confinement, with enhanced miniaturization and electromagnetic field concentration with respect to 2D plasmonic films. Nevertheless, the excitation of a single LSP structure is very inefficient because of the high mismatch between its resonant mode size and diffraction-limited electromagnetic waves. The excitation efficiency of such a single nanostructure deposited on a silicon nitride waveguide reaches less than 10% (9.7% in [82]). To transfer all the energy of the dielectric guided mode into an LSP, a viable solution is to exploit the waveguides' evanescent coupling as shown above, considering a plasmonic chain instead of a plasmonic  
Figure 4: **(left)** Sketch of an infinite plasmonic chain of gold ellipsoidal nanocylinders placed on top of a semi-infinite Si substrate. **(right)** FDTD-calculated dispersion curve of the plasmonic transverse chain mode when \(d=150\) nm, \(r_{x}=42.5\) nm, \(r_{y}=100\) nm, and \(t=30\) nm. The red and the green dashed curves highlight the plasmonic chain mode and the Si light line, respectively.  
metallic film [83]. The typical structure is made of a MNP metallic nanoparticle (MNP) chain integrated on or near a dielectric waveguide (Figure 5). The chain can be designed to propagate either TE or TM modes and can be in any position near the waveguide if their relative distance allows the evanescent tail of the dielectric mode to overlap with the LSPs. In addition to enabling the LSP excitation, such a strongly coupled structure has specific properties, as described below.