## 3 Surface Plasmons in Photonic Integrated Circuits
### Integrated Magneto-Plasmonics  
Plasmonics is particularly relevant for enhancing perturbative physical effects such as magneto-optics (MOs) [121]. Magneto-optical effects are related to the action of the magnetization of a material on the propagation of an optical wave through it. This action is expressed by the non-diagonal elements of the permittivity tensor of the MO material, which is as follows for an otherwise isotropic material:  
\[\epsilon= \begin{pmatrix}\varepsilon_{iso}&+ig_{z}&-ig_{y}\\ -ig_{z}&\varepsilon_{iso}&+ig_{x}\\ +ig_{y}&-ig_{x}&\varepsilon_{iso}\end{pmatrix}  \tag{3}\]  
where \(g_{u}\) (\(u=x\),\(y\),\(z\)) are the gyrotropy constants, which are complex values in the most general case. The interaction of an electromagnetic wave with an MO material impacts its electric field through the complex coupling of its components (E\({}_{\rm x}\), E\({}_{\rm y}\), E\({}_{\rm z}\)), which can modify the wave polarization, amplitude, and/or phase. Since the permittivity tensor is antisymmetric, this interaction is non-reciprocal; it is reversed by inversion of the propagation direction or the magnetization direction. In other words, this effect induces time-symmetry breaking. Usual MO materials are either metallic (Fe, Co, Ni, and alloys) or dielectric like ferrite or garnet oxides (BIG, YIG, Ce:YIG). Combined with a plasmonic metal, these materials induce magneto-plasmonic effects, with enhanced effective complex gyrotropies. Magneto-plasmonic multilayered structures have been mainly studied in non  
Figure 14: Array of integrated plasmonic nanoantennas dedicated to the independent control of directivity, direction, and intensity of radiation, inspired by [120]. a\({}_{\rm x}\) and a\({}_{y}\) represent the radii of the ellipsoidal sections of the MNPs, t is the thickness of the MNPs, N is the number of MNPs per chain, d is the center-to-center distance, and D is the smallest distance between the MNPs and the dielectric guide. The red arrow indicates the direction of light injection. guided configurations such as single nanostructures or hybrid membranes [122], arrays of nanostructures [123; 124], alternating silver and garnet layers metamaterials [125], or reconfigurable metasurfaces or arrays of nanoantennas [126; 127]. Magneto-optical interactions with propagative plasmons have been also studied in grating structuration [128; 129]. Nevertheless, only a few (theoretical) magneto-plasmonic structures were proposed in waveguiding configurations [130; 131]. The main interest in these MO or magneto-plasmonic structures is to realize integrated non-reciprocal transmission of light, like optical isolation or circulation. Such functions can considerably enrich the architecture of photonic circuits. The non-reciprocal transmission is obtained only if all the spatial symmetries are also broken, in addition to the time-symmetry breaking. The non-reciprocity of an integrated isolator is defined as the ratio between its forward and backward transmissions, denoted as \(T^{+}\) et \(T^{-}\), often expressed in decibels as 10 Log(\(T^{+}\) /\(T^{-}\)). A figure of merit (FoM) \(\Delta\) can be defined to account for the trade-off between non-reciprocity and insertion losses:  
\[\Delta= \begin{cases}\frac{T^{+}-T^{-}}{T^{+}}\;\;\;\text{if}\;\;\;T^{+}>T^{-} \\ \frac{T^{+}-T^{-}}{T^{-}}\;\;\;\text{if}\;\;\;T^{+}<T^{-}\end{cases}  \tag{4}\]  
Several MO effects have been identified, all related to the permittivity tensor and gyrotropies, but considering the different configurations of light polarization, magnetization orientations, and interaction nature (bulk or at interfaces) [132]. Among all these configurations, the most relevant for integrated guided structures is the TMOKE (transverse magneto-optical effect); indeed, TMOKE occurs at interfaces between MO and dielectric or metallic materials, it preserves light polarization and only modifies the propagation wavevector of the light.