1. Introduction

2. Integration and Excitation of Plasmonic Structures

2.1. 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 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 1 (a,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]:

$$k_{sp}\left(\omega \right)=\frac{\omega }{c}{n}_{eff}=\frac{\omega }{c}\sqrt{\frac{{\epsilon }_{ext}.{\epsilon }_m}{{\epsilon }_{ext}+{\epsilon }_m}}$$

(1)

where k_sp is the wavevector of the SPP, n_eff represents the effective refractive index of the mode, and ε_ext and ε_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 ω=2πf) and the wavevector k=(ω/c)×n_eff of a specific mode within a given observation range. It's 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. 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 [52]. 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 Drude model [53,54] and ω_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_sp=ω_sp/2π, 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.

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. 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. (f-k) electric field intensity of the wave in the case of propagation occurs along the z-direction (out-of-plane with respect to the figure plane). Structures inspired by [55].

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. 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. (f-k) electric field intensity of the wave in the case of propagation occurs along the z-direction (out-of-plane with respect to the figure plane). Structures inspired by [55].

Several types of propagative plasmonic waveguides have been proposed, as schematized in Figure 1(c-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 1(d,g)), or in a metallic slot waveguide (Figure 1(e,h,i,j)). The LR-SPP is the supermode having a symmetrical phase profile and reduced losses that arises from the coupling of the two SPP modes propagating on the 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 [55].

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 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, [56] offers poor performance because of the significant mode shape mismatch between the plasmonic and the photonic modes (Figure 3(a)) and will not be discussed here.

The underlying physical mechanism harnessed through evanescent coupling (depicted in Figure 3(b)) or grating coupling (shown in Figure 3(c)) can be described by examining the dispersion curves of the involved modes (illustrated in Figure 3(d,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_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_sp. In the integrated configuration where the plasmonic waveguide is placed in proximity to a dielectric waveguide (i.e. via a multilayer configuration as depicted in Figure 3(b)), the effective index n_d of the dielectric waveguide mode is higher than the refractive index of the surrounding dielectric material, by construction (n_d>n_ext): this property can be used to directly excite the SPP when k_d≅k_sp (phase matching), since both dispersion curves cross each other (see the dispersion diagram in Figure 3(d)). In this case, when the polarization of the dielectric guided mode supplies the necessary z- and x- electric field components for SPP excitation, modes coupling occurs. This results in the generation of two supermodes having hybrid photonic-plasmonic nature, featuring field symmetric and antisymmetric distributions and effective indices n_e and n_o (as shown in Figure 3(b)), 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:

$$Lc=\left|\frac{1}{2}\frac{\lambda }{n_e-n_o}\right|.$$

(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 mode coupling is called “evanescent coupling” because it involves the evanescent tails of the guided modes. The required condition (n_d>n_ext) is more easily achieved when the targeted SPP is on the metallic film opposite side with respect to the dielectric waveguide, as shown in Figure 3(b): 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 case of high contrast between n_ext and 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 indices difference, the shorter the coupling length. Thus, such mechanism is particularly efficient when the contrast between n_ext and n_d is high. Delacour et al [20] have realized this for example in the case of an SOI waveguide and a Cu plasmonic slot waveguide at 1.55µm. The experimental coupling length equals 0.9µm and the coupling efficiency is estimated at 70%. In another example involving polymer waveguides, Magno et al [57] demonstrated numerically the coupling between a SU8 on glass waveguide and a buried plasmonic waveguide at 633 nm, with L_c=5.3µm 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 [58].

If the condition k_d≅k_sp cannot be directly fulfilled, the guided dielectric mode must interact with a complementary structure (Figure 3(c)) to increase its wavevector module up to k_sp (k_sp=k_d+δk, see Figure 3(e)): the interaction of the dielectric waveguide mode with a periodic grating (period Λ) serves this purpose since it generates spatial harmonics whose constant propagations equal k=k_d+p 2π/Λ, where p is a relative integer. The first harmonic (p=1) is the most intense: by choosing Λ so that k_sp-k_d=2π/Λ the grating provides the dielectric waveguide mode with the missing component δk to excite the SPP at the dielectric/metal interface (each surface of the metal can be targeted, by proper choice of δk). In the case of strong mismatch of the dielectric and plasmonic waveguides (i.e., if the targeted mode has a strong plasmonic character, with very high k_sp), successive gratings with decreasing period can be used for a progressive adaptation, with the risk of long and lossy transition. Tetienne et al [59] have shown an integrated coupler made of a metallic grating for the transition between a semiconductor waveguide and a plasmonic gold film at 1.3µm: the mode transfers along a 5.5µm transition, with a global excitation efficiency of 24%.

2.2. Localized Surface Plasmon

Localized surface plasmons (LSP) result from the plasmonic excitation in low-dimensional metallic structure 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 homogeneous surrounding medium [60].

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: such a way, plasmonic nanostructures assembly may support collective resonances, and/or propagate energy from one to the next [61]. LSP chain dispersion curve can be thus established, by using analytical model in a homogeneous medium [62], or by numerical methods in the general case. For instance, in Figure 4 is reported the dispersion curve of a plasmonic chain constituted of gold nanocylinders with elliptical cross-section placed on top of a semi-infinite Si substrate, calculated by means of the FDTD method. The LSP chain dispersion curve displays a similar overall behavior as 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 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) [63,64,65,66,67], bound states in the continuum [68,69,70] and surface lattice resonances [71,72,73,74].

LSP in a subwavelength structure enables 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 [75]). 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 the plasmonic metallic film [76]. The typical structure is made of a MNP metallic nanoparticles (MNPs) 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 strongly coupled structure has specific properties, as described below:

• Coupling efficiency:

Figure 5 shows FDTD calculations of a TE mode in a dielectric waveguide evanescently coupled to a LSP chain. The (infinite) LSP chain is directly deposited on a SOI waveguide, and Figure 5 (b,c,d) reveals the anticrossing of both dispersion curves which generates the resulting odd and even supermodes. The strong coupling between both waveguides, in addition to the low slope of the LSP chain curve induces supermodes with very different indices and thus a very short coupling length. Depending on the wavelength, the fundamental TE mode of a SOI waveguide can be totally transferred in the 4^th or 5^th MNP of the finite chain [76], which corresponds to a ~600 nm coupling length (Figure 5(f)). The coupling efficiency is near 99% in the case of a 5 MNP chain (Figure 5(e)). Due to a lower dielectric effective index, the optimal configuration in the case of a Si₃N₄ waveguide at 633 nm enables a total transfer of the mode energy in an 8 MNP chain with a coupling length ~800 nm.

In fact, the energy transfer efficiency is improved with respect to a single MNP case as soon as a second MNP is involved (‘dimer’ chain). In return, however, the chain length increase implies ohmic losses increase. In order to exploit the high electromagnetic power density at the surface of a nanoparticle, the best compromise must be thus identified between the efficient energy transfer induced by the collective resonance in the chain and the ohmic losses limitation.

• Strong coupling:

In the configuration presented in Figure 5(a), the strong coupling [77] between the plasmonic and SOI waveguides induces additional distinctive characteristics specific to this system. Firstly, over a wide range of frequencies, the supermodes propagate in a vortex-like manner. These vortices are related to slow modes [78] and can be also characterized by their phase profile along the chain [79]. Secondly, due to strong coupling, supermodes can be excited also beyond the light lines of the different materials surrounding the plasmonic chain. This entails that they can be excited as radiative modes above the light line of the confinement layer (silica here), and/or below the highest index material light line (silicon here), in the non-guided mode region (see the dispersion diagrams in Figure 5 (b) and (c)). In consequence the plasmonic mode hybridization can modify their radiative or guided nature, or even extend their possible frequency range.

Figure 5. Localised plasmon waveguide integrated on SOI (TE mode) (a) Sketch of the structure. (b,c) FDTD calculated dispersion curves for d = 150 nm, r_x = 42.5 nm, r_y = 100 nm, t = 30 nm (the permittivity of gold is obtained using a Drude model fitted to ellipsometric measurements). (d) A typical spatial distribution of the two supermodes. (e,f) Electric field intensity in the case of a finite chain of (e) 5 or (f) 20 nanoparticles. Inspired by [78].

Figure 5. Localised plasmon waveguide integrated on SOI (TE mode) (a) Sketch of the structure. (b,c) FDTD calculated dispersion curves for d = 150 nm, r_x = 42.5 nm, r_y = 100 nm, t = 30 nm (the permittivity of gold is obtained using a Drude model fitted to ellipsometric measurements). (d) A typical spatial distribution of the two supermodes. (e,f) Electric field intensity in the case of a finite chain of (e) 5 or (f) 20 nanoparticles. Inspired by [78].

Moreover, nanoantenna arrays integrated on waveguides can take advantage of tapering their dimensions within the integration plane to achieve enhanced control over coupling characteristics. For instance, tapering the period of the plasmonic array enables efficient mode conversion and excitation of higher-order modes in the waveguide [80].

3. Surface Plasmons in Photonic Integrated Circuits

3.4. Integrated Magnetoplasmonics

Plasmonics is particularly relevant for enhancing perturbative physical effects such as magneto-optics (MO) [114]. 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, in the case of otherwise isotropic material:

$$\epsilon =\left(\begin{array}{ccc}{\epsilon }_{iso}& +i{g}_z& -i{g}_y\\ -i{g}_z& {\epsilon }_{iso}& +i{g}_x\\ +i{g}_y& -i{g}_x& {\epsilon }_{iso}\end{array}\right)$$

(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 a MO material impacts its electric field through the complex coupling of its components (E_x, E_y, E_z), which can modify the wave polarisation, amplitude and/or phase. Since the permittivity tensor is antisymmetric, this interaction is non-reciprocal: it is inversed by inversion of the propagation direction or the magnetisation 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-guided configurations such as single nanostructures or hybrid membrane [115], array of nanostructures [116,117], alternating silver and garnet layers metamaterials [118], or reconfigurable metasurface or array of nanoantenna [119,120]. Magneto-optical interaction with propagative plasmons have been also studied in grating structuration [121,122].

Nevertheless only a few (theoretical) magnetoplasmonic structures were proposed in waveguiding configurations [123,124]. The main interest of these MO or magneto-plasmonic structures is to realize integrated non-reciprocal transmission of the 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 decibel as 10Log(T⁺ / T^- ). A figure of merit (FoM) Δ can be defined to account the trade-off between non-reciprocity and insertion losses:

$$∆=\left\{\begin{array}{c}\frac{T^{+}-T^{-}}{T^{+}} if T^{+}>T^{-}\\ \frac{T^{+}-T^{-}}{T^{-}} if T^{+}<T^{-}\end{array}\right.$$

(4)

Several MO effects have been identified, all related to the permittivity tensor and gyrotropies, but considering different configurations of light polarisation, magnetization orientations, and interaction nature (bulk or at interfaces) [125]. Among all these configurations, the most relevant for integrated guided structures is the TMOKE (Transverse Magneto-Optical Effect): indeed, TMOKE preserves the light polarisation, it only modifies the propagation wavevector of the light and it occurs at interfaces between MO and dielectric or metallic materials. Such a configuration is easily obtained in multi-layered planar structures, and spatial symmetry breaking is also just a matter of appropriate geometry (Figure 15).

Many guided structures based on TMOKE have been demonstrated, based on: MO metallic layer deposited on dielectric or semiconductor waveguide, interacting with TE [126] or TM [127] mode; MO garnet transferred on a semiconductor interferometer [128] or on a ring resonator [129]; garnet interferometer including a thin silver film [124]; gold grating on a garnet waveguide [130], In the latter structure, the non-reciprocal effect is enhanced by plasmonic light concentration, and in addition its sign is controlled by the grating slit modes coupling with the propagative surface plasmon. Such a way, non-reciprocity can be inversed by proper grating design, while keeping unchanged the magnetization direction.

Such a structure is represented in Figure 16: the gold grating is placed on the top of a MO garnet (BIG) waveguide, whose magnetization is in z direction. The guided TM mode is injected in the structure and interacts with the plasmonic top-grating and with the Fabry-Perot modes inside the grating slits. Their coupling induces anticrossing in the dispersion curves (Figure 16(b)) and non-reciprocity enhancement on each side of this anticrossing. Because of the resonant character of the structure, the isolation is limited to a narrow bandwidth, and the FoM reaches 40%.

Following the same principle, by positioning the plasmonic grating on the side of the waveguide, TE mode undergoes isolation ratio higher than 20dB (8 dB), with possible sign inversion and FoM above 1 (0.2) [131], with a gyrotropy of 0.1 (0.01).

More generally, in this kind of structure, plasmonics may also modify the global properties of the MO material: indeed, in the case of a purely real gyrotropy MO material, TMOKE only induces non-reciprocal phase shift on the propagating wave; thanks to the plasmonic interaction at the TMOKE interface, the global non-reciprocity also applies on the amplitude of the propagating wave [132].

Table 1. Comparison of coupling efficiency and length obtained or evaluated for different plasmonic waveguides or nanostructures excited by dielectric waveguide.

Table 1. Comparison of coupling efficiency and length obtained or evaluated for different plasmonic waveguides or nanostructures excited by dielectric waveguide.

	Structure	Mechanism	Coupling efficiency	λ [µm]	Coupling length Lc (Footprint)	ref	year
SPP waveguide	Cu slot waveguide/SOI waveguide	Evanescent coupling	70% (LRSPP)	1.55	0.9 µm	[20]	2010
	‘buried plasmonic waveguide’/SU8 waveguide/glass	Evanescent coupling	88% (LRSPP)	0.633	5.3 µm	[57]	2012
	Si waveguide/buried plasmonic waveguide/glass	Evanescent coupling	95.5% (LRSPP)	1.55	2.85 µm	[133]	2013
	Gold film/semiconductor waveguide	Grating based coupling	24%	1.3	5.5 µm	[59]	2011
	SOI waveguide/plasmonic slot waveguide	Butt-coupling/end-fire coupling	61% numerical/43% measured	1.55	(>10 µm)	[134]	2010
	InP membrane waveguide/plasmonic slot waveguide	Butt-coupling/end-fire coupling	~55% numerical/~13% measured	1.55	(~5µm)	[56]	2020
LSP, LSP chain waveguide	1 single NP/Si3N4 waveguide	Evanescent excitation	9.7 %	0.850µm	NP width (<100nm)	[75]	2013
	LSP subwavelength chain/SOI waveguide	Evanescent coupling	99%	1.55	0.6 µm	[76]	2012
	LSP subwavelength chain/Si3N4 waveguide /SiO2	Evanescent coupling	/	1.4	1.635 µm	[78]	2017
Antenna on waveguide	Vivaldi antenna /SOI waveguide	Evanescent coupling	91% (numerical)	1.55	1.63 µm	[108]	2017
	Yagi-Uda antenna/SOI waveguide	Evanescent excitation	20% in-coupling	0.85	(~1 µm)	[109]	2012
	LSP nanoantenna/silicon	Evanescent excitation	~4%	1.55	(400 nm)	[110]	2017
	LSP gratings (antenna) aside Si3N4 waveguide	Evanescent coupling	Tunable, from 10% to 50%	0.633	~0.8 µm	[113]	2018

4. Conclusions

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