1. Introduction

Optical antennas based on plasmonic materials are a promising physical concept that aspires to manipulate and control optical radiation at subwavelength scales [1,2].Recently, the study of optical nanoantennas has attracted a growing interest due to the possibility of directing optical emission from systems such as molecules or quantum dots [3–5]. Nanoparticles based on concentric structures with cores made of ordinary dielectrics and shells of plasmonic materials, or vice versa, have been widely studied as plasmonic antennas with resonant frequencies tuned by the ratio of radii of the core and shell in a wide frequency range [6–9]. Concurrently, there has been significant progress in the area of silicon photonics and integrated optics [10–12]. Silicon photonics can provide an alternative for interconnects in electronics and also enables on-chip integration of optical and electronic functions. Optical waveguides can be combined with plasmonics elements such as antennas, metallic nanoparticle chains [13] or plasmon waveguides [14–16] in order to provide new possibilities for subwavelength integrated photonic circuits for telecommunications and optical logic applications. One of the problems in using silicon waveguides is achieving efficient coupling of the waveguide mode to radiation modes in a small footprint and with good directivity. Using methods like grating coupling or gradually reducing the waveguide cross section requires a sizeable area. In the present work, we investigate numerically a method of power extraction from a dielectric waveguide using a core-shell particle with a plasmonic shell in proximity of the waveguide end. Through numerical simulations we perform parametric studies of this method, and demonstrate that, under proper conditions, a core-shell particle placed near the output of the waveguide results in an improvement in the impedance matching between the dielectric waveguide and the free space when the core-shell particle undergoes resonance. Additionally we observe that the reflected energy in the waveguide is strongly dependent on the distance between the particle and the waveguide output. This permits us to sense the distance between the particle and the waveguide by monitoring the reflected energy.

2. Geometry and results

The schematic of the structure is shown in Fig. 1 . We use a silicon carbide (SiC) waveguide (instead of a silicon waveguide since our operating wavelength is set to visible wavelengths and silicon is highly absorbing at these wavelengths). The cross section of the SiC waveguide was set to a square cross section with a series of sizes in our parametric study. The outer radius of the core-shell was set to 50 nm and the radius of the silica (SiO₂) core was taken to be 42 nm. The material of the shell was set to silver (Ag). The dimensions of the core-shell are such that the core shell particle is resonant at a wavelength of 592 nm under a plane wave illumination. The separation, d, between the core-shell and the waveguide output face is varied to perform a parametric study and to obtain the conditions for the best response, i.e. lowest reflected energy in the waveguide. The separation represents the length of the air gap between the waveguide and the core-shell.

 

Fig. 1 Schematic of the Si waveguide and the core-shell particle.

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The simulations were performed using a commercial 3D full-wave electromagnetic wave solver, CST Microwave studio. In CST we used a frequency-domain method based on finite-integration technique (FIT). The relative permittivity of SiO₂ was taken to be 2.25. The permittivity of SiC was set to the tabulated experimental permittivity for beta-SiC [17,18] which also took into account the dispersion in SiC. The relative permittivity of Ag was modeled using the Drude model, i.e. $\epsilon \left(\omega \right)={\epsilon }_{\infty }-\frac{{\omega }_p^2}{\omega \left(\omega +i{\Gamma }_0\right)},$ with realistic losses obtained from experimental data in the literature [19]. The parameter values are as follows, ${\epsilon }_{\infty }=5.0,$ ${\omega }_p=9.2159\text{eV}$(f_p = 2.228 × 10¹⁵ Hz), ${\Gamma }_0=0.0212\text{eV}$(f_Γ = 5.12 × 10¹² Hz). The structure was meshed using tetrahedral mesh elements and adaptive meshing with 6 passes in order to make the mesh finer at locations with higher field gradients. The initial mesh was built based on the material parameters with higher refractive index regions and plasmonic regions getting finer mesh elements. The final mesh consisted of around 50,000 mesh elements with the exact number depending on the geometrical parameters.

With the specified dimensions the waveguide has two orthogonal, hybrid and degenerate modes, one of which was excited in the simulations. The excited mode’s electric field was polarized along the y axis, while the degenerate mode’s electric field was polarized along the z axis. The waveguide supported only these two modes at wavelengths longer than 600 nm as long as the cross section of the waveguide was 200 nm × 200 nm or below. In order to perform a rigorous study of the behavior of the waveguide and core-shell system we did a parameter scan where we varied several parameters. The dimension of the core-shell, permittivity of the core and plasma frequency of the shell material largely determines the resonance wavelength. Since we are not focusing on the tunability of the resonance wavelength we keep these parameters constant. The cross section of the waveguide determines the nature of the modes propagating inside the waveguide including its confinement and dispersion. In our parametric study we vary the waveguide cross-section from 150 nm × 150 nm to 200 nm × 200 nm in steps of 10 nm. The cross section is always kept square. The separation (d) controls the amount of coupling between the waveguide and the coreshell. As the separation is increased the performance of the waveguide should approach that of an isolated waveguide without any coreshell near its output face. In the parameter scan we set the separation to a range of values between 5 nm and 500 nm. Finally, the collision frequency of the shell material (Γ) determines the damping of the coreshell resonance. In the parameter study we vary the collision frequency as a multiple of the collision frequency of silver, i.e. the ratio Γ/Γ₀ is varied between 0.25 and 1.5 in steps of 0.25.

The simulated results were processed to obtain the reflection coefficient for the waveguide mode (S₁₁), the total radiated power in all directions as a fraction of the total input power, also known as total efficiency [20] in the context of antennas, and the losses due to the absorption in the plasmonic shell. Note that the total efficiency accounts for power radiated in all directions. Later in the manuscript we will look into the power radiated in the forward direction. Figure 2 shows the surface plot for the maximum of the total efficiency within the wavelength range of 500 nm to 700 nm. The exact wavelength at which the total efficiency reaches its maximum value depends on the parameter values as shown in further plots. The surface plot is shown for four different values of collision frequency as shown in the corresponding plot titles. From these plots we can see that there is a strong enhancement in the total efficiency when the waveguide-coreshell separation is around 70 nm. The relative enhancement at this optimum separation is stronger when the waveguide cross section is bigger, i.e. the waveguide modes are more confined within the waveguide. With a smaller waveguide cross section the modes are less confined and spread out into the surrounding space. Consequently the smaller cross section waveguides have a fairly large total efficiency even in the absence of the coreshell particles. The enhancement in total efficiency is further increased if the collision frequency is reduced. This is expected since a lower collision frequency implies a stronger resonance in the coreshell particle, which results in a stronger radiation of energy by the coreshell.

 

Fig. 2 Maximum total efficiency in the 500nm to 700 nm wavelength range as a function of the waveguide dimension and the waveguide-coreshell separation for different values of collision frequency. Γ₀ represents the collision frequency of silver.

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Figure 3 shows the plots of reflectance, total efficiency and loss for various separations (d) when the waveguide cross section is set to 200 nm × 200 nm and the collision frequency is set to Γ₀. From these plots it is clear that the presence of the core-shell enhances the radiation into free space for a range of separations between the core-shell and the waveguide. Also not surprisingly the strongest enhancement occurs close to the wavelength of resonance for the core-shell particle under plane wave illumination, i.e. 592 nm. The wavelength of maximum enhancement is red shifted compared to resonance wavelength of the core-shell under plane wave excitation. As pointed out when discussing the parameter sweeps, the strongest effect occurs when the separation is set to 70 nm where we see an enhancement in the radiated energy by a factor of 1.17 at a wavelength of 620 nm compared to the case when there is no core-shell at the waveguide output. More detailed studies of the effect of a change in collision frequency for the same cross section and a 70 nm separation are shown in Fig. 4 . As expected the peak total efficiency rises as the collision frequency is reduced. The minimum in the reflectance plot is fairly low for all values of collision frequency. This implies that the increase in total efficiency is largely caused due to a reduction in the losses. When the collision frequency is set to 0.25Γ₀ the total efficiency reaches a maximum value of 98.4% at a wavelength of 615 nm and is enhanced by a factor of 1.22 compared to the case without the coreshell.

 

Fig. 3 (a) Reflectance of the waveguide mode. (b) Fraction of the input power that is radiated into free space (Total efficiency). (c) Losses due to absorption in Ag. The different curves correspond to different separation between the waveguide and the surface of the core-shell, i.e. d. In all the cases the waveguide cross section is set to 200 nm × 200 nm and the collision frequency of the shell material is set to Γ₀, the collision frequency of silver.

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Fig. 4 (a) Reflectance of the waveguide mode. (b) Fraction of the input power that is radiated into free space (Total efficiency). (c) Losses due to absorption in Ag. The different curves correspond to different values of Γ in the plasmonic shell. Γ₀ represents the collision frequency of silver. In all the cases the waveguide cross section is set to 200 nm × 200 nm and the separation, d, is set to 70 nm.

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In order to observe the presence of resonance in the core-shell particle we plotted the electric field at two different wavelengths corresponding to on-resonance and off-resonance excitations. The field plots are shown in Fig. 5 . We can clearly see that the local electric field around the core-shell shows strong resonance at wavelength of 620 nm whereas the response is more subdued around 565 nm, resulting in a stronger reflection. This in turn leads to a more pronounced standing wave pattern inside the waveguide. On the other hand, due to the resonance at 620 nm the core-shell acquires a strong dipole moment. The resulting dipole strongly radiates energy into the free space.

 

Fig. 5 Plot of the absolute value of the electric field for separation of d = 70 nm and waveguide cross section of 200 nm × 200 nm. (a) Wavelength = 620 nm, (b) Wavelength = 565 nm.

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In order to understand the manner in which the energy is radiated out into the free space we look at the farfield plots at these two wavelengths for a waveguide-coreshell separation of 70 nm. The plots are shown in Fig. 6 . At 620 nm the presence of the resonant coreshell particle results in a forward directed lobe in the radiation pattern. Note that the dipole created by the core-shell is oriented along the y axis, due to the polarization of the exciting waveguide mode, leading to preferential radiation in the xz plane. The pattern is superimposed on the radiation pattern of the waveguide giving rise to the resulting far field plot. The radiation in the –x direction is suppressed by the presence of the waveguide, but there is some enhancement in the directivity in the backward direction at certain angles to the x axis where the radiation is not hindered by the waveguide. On the other hand, at 565 nm the core-shell acts more like an opaque object that suppresses the main radiation lobe of the waveguide and splits it into two small lobes in the xy plane, making an angle of roughly 40° with the x axis.

 

Fig. 6 Far-field patterns of radiation from the waveguide for separation of d = 70 nm and waveguide cross section of 200 nm × 200 nm. (a) Wavelength = 620 nm, (b) Wavelength = 565 nm. The color represents the directivity.

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In Figs. 3 and 4 we looked at the total radiated power without any regard to the direction of radiation. Ideally we would like to maximize radiation in the forward direction. In order to get a quantitative idea of how much energy is radiated in the forward direction we calculated the ratio of the power radiated in the forward half space (P_r,front) to the total radiated power (P_r,total) as shown in the Fig. 7 . At longer wavelengths we observe that the presence of the core-shell particle does not cause a significant change in the fraction of energy that is radiated in the forward direction. Even though the core-shell increases the total radiated power around a wavelength of 620 nm the enhancement is essentially caused through dipolar radiation, which is isotropic in the plane perpendicular to the dipole moment, i.e. the xz plane. Hence around the resonance wavelength the forward radiated power is suppressed compared to the total radiated power. On increasing the separation the ratio approaches the results without the core-shell, i.e. the ratio at shorter wavelengths increases while the ratio at the longer wavelengths tends to stay the same. Changing the collision frequency while keeping the waveguide dimension and the separation fixed has a minimal effect on the ratio since changing the collision frequency results in an isotropic change in the radiated power.

 

Fig. 7 Ratio of the forward radiated power to the total radiated power for various separations between the waveguide and the core-shell (a),and various collision frequency in the plasmonic shell material (b). In both cases the waveguide cross section is set to 200 nm × 200 nm. In (a) the collision frequency of the shell material is set to Γ₀ and in (b) the separation, d, is set to 70 nm.

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3. Distance sensing

From Fig. 3 we can see that the reflectance is particularly sensitive to the separation between the waveguide and the core-shell. This dependence can be exploited to indirectly measure the distance between the waveguide and the core-shell by monitoring the reflectance of the incident mode in the waveguide. To explore this possibility we simulated the reflectance for a series of separation between the waveguide and the coreshell for a range of wavelengths. The waveguide cross section was set to 200 nm × 200 nm and the collision frequency was set to Γ₀ for this study. The result is presented in Fig. 8 in the form of a contour plot. The reflectance shows a peak at a wavelength of around 575 nm and a separation of around 100 nm. This condition corresponds to the state of maximum impedance mismatch with this waveguide and coreshell.

 

Fig. 8 (a) Reflectance and (b) Total Efficiency shown as contour plots as function of wavelength and separation (d).

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On the right half of the contour plot we can see the parameter space where the reflectance is lower and the transmittance is high. Around the wavelength of (615 nm) the region of low reflectance is sensitive to the distance d, but on going towards longer or shorter wavelengths the reflectance is not too sensitive to the distance. In order to sense the distance by monitoring the reflectance we need the reflectance to be highly sensitive to the distance d and hence we should preferably be working around wavelengths where the contour lines are somewhat horizontal and closely packed. Such wavelengths are located around the wavelength with minimum reflectance (615 nm). As an example we show the reflectance as a function of separation for wavelengths of 615 nm and 630 nm in Fig. 9 . The plot is essentially a cross section of the contour plot shown in Fig. 8(a). We can see that the reflectance shows a change of almost 35% over a range of 70 nm for a wavelength of 615 nm. The change is lower as we move away from 615 nm as seen for a wavelength of 630 nm in Fig. 9(b). This provides an ample margin to ensure detectability via simple reflectance monitoring. An obvious problem with this approach is that the reflectance is multi valued and can provide the same reflectance for more than one value of d. Nevertheless, we could use the slope of the reflectance versus d plot to resolve this ambiguity.

 

Fig. 9 Reflectance as a function of separation of core-shell particle from the waveguide.

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4. Conclusions

Using numerical simulations, we have conducted parametric studies and have shown that through the use of resonant core-shell particles with a plasmonic shell we are able to provide a means of improving the impedance matching between a dielectric waveguide and free space. Moreover the radiation characteristics of the waveguide and core-shell system are sensitive to the separation between the two. This may also allow us to control the placement of the core-shell particle by monitoring the reflectance in the waveguide. We performed a detailed parameter studies to show the behavior of such a system when various critical parameters are modified.