1. Introduction

In recent years, metamaterials have been largely studied due to their exotic material properties, which do not depend only on the intrinsic chemical properties of the materials that constitute them, but on their specific structures, offering the possibility of design material properties at will [1]. These artificial materials allow to confine light beyond the diffraction limit, a capability stemming from light interaction with sub-wavelength metallic nanostructures through the so-called plasmonic resonances [2,3]. In this sense, an ultimate goal is to create miniature integrated optical circuits using enhanced light-matter interaction at nanoscale level and processing data at much higher speeds than standard electronic chips [4]. Research has mostly focused on ensemble of meta-atoms to form metamaterials but, despite enormous promise, metamaterials have suffered from the optical losses. Recently, the focus has turned to two dimensional phased array also known as metasurfaces. Metasurfaces however have mainly been used for free space light control [5–7].

Different works have been reported in order to manipulate light in integrated systems through the excitation of localized surface plasmons (LSP) in complex metallic nanostructures, such as Yagi-Uda antennas [8], periodic arrays of metallic nanoparticles [9], metallic gratings [10], ring resonators [11], and optical modulators [12]. Simpler metallic structures have also been studied, such as dimer nanoantennas [13], metallic taper concentrator [14], single metallic nanorods [15], and metallic strips [16].

However, the large majority of the work done in integrated plasmonics is based on the fabrication of metallic nanostructures on top of the waveguides, where evanescent waves interact with the plasmonic resonances. The study of the interaction of confined propagative modes with metaplasmonic nanostructures remains essentially unexplored, and only few works can be found in literature. Among these works are the study of metallic gratings buried in optical microfibers for in-line rainbow trapping [17], the study of single metallic nanoparticles in silicon waveguide gap [18] and light transmission filters [19].

In this contribution, we systematically analyze the interaction of high-order waveguide modes with the fundamental dipolar resonance of a metallic nanoparticle (NP) embedded in the core of the dielectric waveguide. We first study the evolution of the resonance wavelength of the nanoparticle with its position inside the core of a single-mode waveguide. We then perform a similar analysis for the case of a multi-mode waveguide and show that the dipolar LSP resonance can be excited with higher order modes of the waveguide only if the symmetry of the system is broken by shifting the nanoparticle away from the center of the waveguide. These results shine new light on plasmonics integrated systems and open the way to the investigation of more complex metaphotonic devices.

2. Description of the structure

The schematic in Fig. 1 shows the structure under investigation, which consist of a dielectric waveguide of width w_c, thickness t_c and refractive index n_c = 2 (close to the value of a silicon nitride waveguide). The medium surrounding the waveguide is considered as air (n_s = 1). A gold nanoparticle is embedded in the waveguide. The dimensions of the particle are the width w_x = 100 nm, the thickness t_y = 30 nm and the length l_z = 50 nm, where the subscripts indicate the coordinate system. Light propagates along the z axis. By adjusting the dimensions of the waveguide w_c and t_c it is possible to control the number of modes propagating in a given frequency region [20].

 

Fig. 1 (a) Schematic representation of a gold nanoparticle immersed in the core of a dielectric waveguide of width w_c, height t_c and refractive index n_c = 2. The cladding is considered air (n_s = 1). The particle dimensions are w_x = 100 nm, t_y = 30 nm, and l_z = 50 nm. (b) Cross section of the structure in the xy plane showing the definitions of the vertical (s_y) and lateral (s_x) displacements of the nanoparticle measured from the center of the core.

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The numerical results presented in this paper are obtained using the finite integration technique [21]. The dielectric function of gold was taken from reference [22].

3. Single-mode waveguide

We first consider a waveguide of width w_c = 500 nm, and height t_c = 200 nm, that only supports the propagation of the two fundamental modes (TE0 and TM0) in the range λ = [875-1600] nm. The dispersion curves for this waveguide are plotted in Fig. 2(a). The cut-off wavelength of the TE0 mode is λ_c,TE0 = 1627 nm, while for TM0 it is λ_c,TM0 = 1269 nm. For wavelengths shorter than λ = 875 nm, higher order modes of the waveguide become guided. The insets in Fig. 2(a) show the intensity of the mode profile as well as the orientation of the electric field for the fundamental modes.

 

Fig. 2 (a) Dispersion curves of a waveguide of width w_c = 500 nm and thickness t_c = 200 nm. The waveguide supports the propagation of the fundamental TE0 and TM0 modes in the spectral range λ = [875-1627] nm. Higher order modes appear at shorter wavelengths. The insets show the intensity of the mode profile and orientation of the electric field for the fundamental modes. (b) Transmission and reflection spectra for TE0 and TM0 modes when the particle is at the center of the waveguide. For TE0 the transmittance drops by 82% at λ = 1048 nm. The insets show the intensity of the electromagnetic field and the charges distribution in the nanoparticle.

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The transmission and reflection spectra of Fig. 2(b) show the interaction of the TE0 and TM0 modes of the waveguide with the nanoparticle when this is placed at the center of the core, i.e. when s_x = s_y = 0 nm. The broad resonance observed for the TE0 mode in both transmission and reflection spectra, corresponds to the excitation of a dipolar LSP resonance whose charges distribution is antisymmetric respect to the yz plane [inset in Fig. 2(b)]. The strong interaction between LSP and TE0 modes result in a reduction of the transmittance by 82% at a wavelength λ = 1048 nm. The TM0 mode does not excite the LSP resonance of the nanoparticle because the electric field is vertically oriented and it is not symmetrically compatible with the dipolar resonance of the nanoparticle.

When the nanoparticle is shifted out of the center of the waveguide, the interaction of the TE0 mode with the LSP resonance decreases because the intensity of the mode is reduced when it reaches the edges of the waveguide. In consequence, the transmittance of the mode is increased, as recently reported for the case of a nanoparticle in a Si waveguide [19]. This situation is illustrated in the transmission and reflection spectra of Fig. 3. The results in Fig. 3(a) correspond to vertical displacements of the particle of s_y = 0, 40, 70 and 85 nm, and in Fig. 3(b) to horizontal displacements s_x = 0, 100, 180, 200 nm. In both, vertical and horizontal shifts, the LSP resonance of the particle is blue shifted due to a reduction of the effective index seen by the particle when it approaches the air cladding. It is worth noting that, due to the spatial distribution of the TE0 mode [inset in Fig. 2(a)], the interaction of this mode with the nanoparticle is less sensitive to displacements along the y direction, leading to a smoother variation of transmission minima for vertical shifts.

 

Fig. 3 Transmission and reflection spectra of the TE0 mode for (a) vertical displacements of the nanoparticle (s_y) and (b) for horizontal displacements (s_x). In both cases, the LSP resonance is blue shifted when the particle is shifted out of the center of the core.

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4. Multi-mode waveguide

In order to investigate the interaction of the LSP dipolar mode of the NP with higher-order modes of the waveguide, we now increased the thickness of the core to t_c = 450 nm, maintaining the same width than before (w_c = 500 nm).

The dispersion curves for the TE0, TM0, TE1 and TM1 modes of the waveguide are depicted in Fig. 4(a), as well as the intensity profile and electric field lines distribution for each mode. As can be observed, for TE1 the electric field is symmetric with respect to the yz plane, while the TM1 mode, the electric field rotates around the z axis of the core. The cut-off wavelengths for each mode are λ_c,TE0 = 2052 nm, λ_c,TM0 = 1191 nm, λ_c,TE1 = 1176 nm and λ_c,TM1 = 1154 nm.

 

Fig. 4 (a) Dispersion curves of the first four modes of a waveguide of width w_c = 500 nm and thickness t_c = 450 nm. The insets show the intensity mode profile and distribution of the electric field lines. (b) Transmission and reflection spectra of the four modes when the nanoparticle is at the center of the core. The dipolar LSP resonance is only excited with the TE0 mode, with the resonance leading to a reduction of the transmittance of 73% at λ = 1074 nm.

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When the metallic nanoparticle is at the center, only the fundamental TE0 mode of the waveguide excites the dipolar LSP resonance, as can be seen from the transmission and reflection spectra of Fig. 4(b). For this mode, the transmittance drops by 73% at the resonance wavelength λ = 1074 nm.

The interaction of the nanoparticle with TE1 and TM1 is not observed because there is in each case symmetry incompatibility with the fundamental plasmonic mode of the particle. For example, even if the TE1 mode has horizontally oriented electric fields, this mode is symmetric respect to the yz plane while the plasmonic dipolar mode is antisymmetric with respect to the same plane. This incompatibility leads to the absence of spectral characteristic in the corresponding transmission or reflection spectra. A similar situation occurs for the TM1 mode which is antisymmetric with respect to the xz plane while the plasmonic mode is symmetric with respect to the same plane.

In order to excite the dipolar plasmonic resonance with higher order modes, it is necessary to break the symmetry of the system. We propose a simple solution by simply shifting the particle from the center of the waveguide. The results are shown in the transmission and reflection spectra of Fig. 5.

 

Fig. 5 Transmission and reflection spectra of the four first modes of the waveguide for (a) vertical displacement s_y = 140 nm, and (b) lateral displacement s_x = 160 nm. For a vertical displacement of the nanoparticle, the LSP resonance is excited with TE0 and TM1 modes of the waveguide, while for a lateral displacement it is excited with TE0 and TE1 modes.

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When the nanoparticle is vertically shifted from the center by a distance s_y = 140 nm, the LSP resonance of the nanoparticle is excited with both TE0 and TM1 modes of the waveguide. The transmittance at a wavelength λ = 1063 nm drops by 43% for TE0 and by 37% for TM1 [Fig. 5(a)]. A similar situation is observed when the particle is laterally shifted by a distance s_x = 160 nm along the x axis. In this case, due to the symmetry of the waveguide modes, the dipolar LSP resonance of the particle is excited by both the TE0 mode and the TE1 mode (instead of the TM1 mode for the vertical shift). The transmittance is attenuated by 45% for TE0 and 13% for TE1, at the wavelength λ = 1056 nm.

The plots in Fig. 6 represent the percentage of light that is converted from TE0 to TM1, and from TE0 to TE1 for vertical and horizontal shifts of the particle, respectively. As can be observed, for vertical shifts of the particle, 5% of the TE0 mode can be converted into the TM1 mode for a vertical displacement of s_y = 140 nm, while only a 0.9% is converted from TE0 to TE1 for a horizontal displacement of s_x = 160 nm. Even if the mode efficiency conversion is small, these results serve as proof-of-concept and open new perspectives in the design of integrated structures capable to control guided modes.

 

Fig. 6 (a) Mode conversion from TE0 to TM1 as a function of the vertical displacement of the nanoparticle (s_y) and (b) from TE0 to TE1 for a lateral displacement (s_y). In the first case, 5% of TE0 mode is converted into TM1, while for the second one, only 1% is converted from TE0 to TE1.

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The ideas proposed in this work can be tested experimentally using thin film deposition and processing techniques used in the CMOS industry. For instance, a metallic nanoparticle embedded in the core of a silicon nitride waveguide can be fabricated by making use of multilayered planarization techniques [23]. A symmetric environment can also be created by using a silica substrate and an index matched polymer as superstrate. Fabrication imperfections such as rounded corners will have minimal influence on the presented results as they do not modify the symmetry of the nanoparticle. Other fabrication defects will lead to more complex modes of the particles and our analysis can be applied considering the symmetry of the those modes.

5. Conclusions

We demonstrated that the dipolar LSP resonance of a metallic nanoparticle embedded in the core of a dielectric waveguide, can be excited with confined propagative modes only if the symmetry of these modes is compatible with the charges distribution of the plasmonic resonance. For the case of a multimode waveguide, the dipolar LSP resonance can be excited with higher order modes of the waveguide only by breaking the symmetry of the system, a situation that is accomplished by shifting the nanoparticle from the center of the core. Due to the strong interaction of the guided modes with the metallic nanoparticle, the transmittance of the waveguide can be largely modified with a single particle. The analysis performed in the present contribution is a proof-of-concept that can be applied to more realistic waveguides and using more complex plasmonic particles exhibiting multipolar resonances to observe Fano resonances on chip. We believe that the simple analysis introduced here opens exciting perspectives in the design of new integrated nano/metaphotonic devices with application in linear and non-linear integrated optics for communications and sensing.