1. Introduction

In the last decade, surface plasmon polaritons (SPP, the electromagnetic waves propagating along metal-dielectric interfaces) have attracted considerable research attention both due to fundamental interest and potential applications including plasmonic lasers [1], nanolithography systems [2], radiation guiding [3] and photovoltaics [4].

Along with absorption losses, SPP scattering losses at interfaces between different media are one of the major mechanisms decreasing the efficiency of various plasmonic optical elements [5,6]. The parasitic scattering is caused by the mismatch of the transverse SPP field profile across the element boundary and can reach up to 30% energy loss at a single interface. In [6–9], approaches for scattering suppression based on the use of anisotropic metamaterials were presented. While this approach allows the scattering losses to be completely eliminated, the design and fabrication of metamaterials with required effective material parameters and their integration into plasmonic elements constitute a complex problem.

In previous papers, we have proposed a simpler approach for SPP parasitic scattering suppression based on the use of insulator-insulator-metal plasmonic waveguides made of isotropic materials [10,11]. It was shown that this two-layer structure allows reducing the scattering losses by an order-of-magnitude (down to 1%–2%). However, only the most basic case (SPP transmission through a one-layer or two-layer dielectric ridge under normal incidence without element examples) was considered in [10,11].

In the present work, we extend the proposed scattering suppression approach to the case of oblique SPP incidence which is essential for real applications (in particular, the ones associated with spatially bounded plasmonic beams [12,13]). We consider SPP transmission through interfaces between two media and through dielectric ridges. We also compare the results of rigorous numerical simulations for SPP with a theoretical model for a TE-polarized plane wave. It is shown that in most cases the conventional Fresnel formulas accurately describe SPP refraction and reflection coefficients when the proposed approach for scattering suppression is used. The discrepancies between the simulation results and the Fresnel formulas are identified and explained. Moreover, we reveal a new feature of the SPP refraction which consists in total scattering suppression above a certain incidence angle regardless of the ridge configuration. Finally, we consider an SPP microlens array as an example of a plasmonic optical element utilizing the proposed scattering suppression technique.

2. Scattering suppression technique and example parameters

Let us first describe the proposed approach for SPP parasitic scattering suppression. The time-coordinate dependence of the electromagnetic field components of an SPP propagating along the interface $z=0$ has the form $\exp \left(-i\omega t+i{k}_{x}x+i{k}_{y}y-{\kappa }_j\left|z\right|\right)$, where $j=d,m$ correspond to dielectric and metal, respectively, and the wave vector components $k_x$, $k_y$, ${\kappa }_z$ satisfy the dispersion relation $k_x^2+k_y^2=k_{spp}^2=k_0^2{\epsilon }_m{\epsilon }_d/\left({\epsilon }_m+{\epsilon }_d\right)$, ${\kappa }_j=\sqrt{k_{spp}^2-k_0^2{\epsilon }_j}$, where $k_0=\omega /c$, ${\epsilon }_m$ and ${\epsilon }_d$ are the dielectric permittivities of the two media. The transmission of the SPP across an interface between the initial dielectric and a dielectric with the permittivity ${\epsilon }_b\ne {\epsilon }_d$ entails the change in the transverse field profile. Thus, the boundary conditions for the tangential field components cannot be satisfied by only three waves (the incident, reflected and transmitted SPP), so the incident SPP is partially scattered into free-space radiation.

To suppress the parasitic scattering, it is necessary to provide the matching of transverse plasmonic mode profiles across the interface by introducing additional degrees of freedom into the corresponding dispersion relation(s). As mentioned above, one of the available options is to change the dielectric permittivity and/or magnetic permeability tensors of the materials of the structure [6–9]. Another approach is based on the modification of the geometry of the plasmonic waveguide. In particular, we have shown that introduction of an additional dielectric layer to a dielectric ridge (Fig. 1) provides partial matching of the field profile of the mode [10, 11]. Assuming that the upper layer thickness $h_2$ (Fig. 1) is large enough, the plasmonic mode of the two-layer structure can be described by the dispersion relation for the TM-polarized modes of the planar waveguide [14,15]:

 

Fig. 1 Geometry of the SPP propagation through a dielectric ridge (the considered structure is invariant in the y-direction). In the paper, also the case of SPP transmission through the interfaces between two media (from medium 1 to medium 2 and from medium 2 to medium 3) is considered.

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In the examples described below, we use the following parameters similar to the example considered in our previous work [11]: free-space wavelength $\lambda =800\text{nm}$, ${\epsilon }_m=-24.06+1.51i$ corresponds to gold [16,17], ${\epsilon }_d=1$, ${\epsilon }_1={1.45}^2$, ${\epsilon }_2={1.7}^2$, $h_2=1.5\mu \text{m}$. In all the examples, we consider two values of the thickness of the first layer: $h_1=0$ corresponding to the conventional single-layer structures, and the value $h_1=62\text{nm}$ minimizing the average parasitic scattering losses found using numerical optimization. The latter value is close to the theoretical estimate $h_1=57\text{nm}$ found from Eq. (1). Let us note that since the studied scattering suppression effect is not resonant, the used parameters are not specific and scattering suppression similar to the results described below can be achieved for different wavelengths and combinations of the materials of the structure provided that the thickness $h_1$ is chosen properly and the thickness $h_2$ is large enough.

For the considered parameters, the normalized propagation constants (effective refractive indices) of the incident (and transmitted) SPP media 1 and media 3 and the plasmonic modes in the medium 2 at $h_1=0$ and $h_1=62\text{nm}$ are $n_{spp}=1.0214+0.0014i$, $n_{mode,0}=1.8118+0.0077i$ and $n_{mode,62}=1.706+0.002i$, respectively. For simplicity of the analysis, we neglect the metal absorption in Sections 3 and 4, but consider realistic (lossy) metal in Section 5.

In all the examples, we also compare the simulation results for the SPP with the analytical model for the refraction of a TE-polarized (transverse electric) plane wave (PW) where the normalized propagation constants of the plasmonic modes are used as the refractive indices of the media (PW model). A similar model was used to calculate the propagation constants of the plasmonic modes of metal stripe waveguides [18].

3. SPP transmission through an interface between two media

Before considering the SPP propagation through a dielectric ridge, let us study the refraction and scattering of an SPP obliquely incident on an interface between two media (semi-infinite in the x-direction), one of which can consist of two layers. As mentioned above, it is instructive to compare the SPP refraction with the refraction of a TE-polarized plane wave.

First, we consider the case when the SPP propagates from the surrounding dielectric to the two-layer medium (i.e. from medium 1 to medium 2 in Fig. 1). The numerical simulations were performed using our in-house implementation of the conical diffraction formulation of the rigorous coupled-wave analysis method (RCWA) [19,20] extended to aperiodic diffraction problems [21]. Reflectance and transmittance values for the PW model were calculated analytically using the conventional Fresnel equations for TE polarization:

Without the scattering suppression ($h_1=0$, Fig. 2(a)), the SPP parasitic scattering losses (and, accordingly, the difference between the simulated data for the SPP and the PW model) reach 0.23, while the average scattering losses amount to 0.19. Figure 2(b) shows that the lower dielectric layer with $h_1=62\text{nm}$ reduces the maximal scattering losses to 0.02, while average losses do not exceed 0.015.

 

Fig. 2 Comparison of the refraction of an SPP and a TE-polarized plane wave on an interface between two media. SPP reflectance (thick solid blue lines), plane wave reflectance (thin solid green lines), SPP transmittance (thick dashed blue lines), plane wave transmittance (thin dashed green lines), and SPP scattering losses (thick dotted blue lines) vs. incidence angle θ are shown. SPP is incident from the surrounding dielectric to the two-layer medium ((a), (b)) or vice versa ((c), (d)). The cases of $h_1=0$ ((a), (c)) and $h_1=62\text{nm}$ ((b), (d)) are considered. The insets in (c) and (d) show the SPP scattering losses (S) near the total internal reflection critical angle ${\theta }_{TIR}$ marked by vertical black dashed lines.

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In Figs. 2(c) and 2(d), similar plots are shown for the case when the plasmonic mode is incident from the two-layer medium (i.e. from medium 2 to medium 3 in terms of Fig. 1). The introduction of the lower layer also reduces both the maximal (from 0.24 to 0.05) and the average (from 0.11 to 0.014) scattering losses. It is worth noting that the scattering losses reach a maximum near the critical angle ${\theta }_{TIR}$ above which the total internal reflectance occurs both with (global maximum) and without (local maximum) scattering suppression (the insets in Figs. 2(c) and 2(d)). This phenomenon is similar to the energy redistribution between the diffraction orders of a diffraction grating near a Rayleigh anomaly when one of the orders becomes evanescent.

Figures 2(b) and 2(d) show that in both incidence geometries, the SPP reflectance and transmittance values with scattering suppression are well described by the analytical expressions for the corresponding PW model (average difference between the PW and the SPP data does not exceed 0.01).

4. SPP transmission through a dielectric ridge and DLSPPW mode excitation

Let us now consider the SPP transmission through a one-layer or two-layer dielectric ridge with finite length l (Fig. 1). In this case, the corresponding model is the PW transmission through a plane-parallel plate. Figure 3 shows the SPP and PW transmittance and reflectance vs. the incidence angle θ and the ridge length l (which becomes the plane-parallel plate thickness for the PW model). The SPP scattering losses vs. the incidence angle and the ridge length are shown in Fig. 4.

 

Fig. 3 Comparison of the reflectance (R) and transmittance (T) of an SPP propagating through a dielectric ridge and a plane wave (PW) propagating through a plane-parallel plate vs. the angle of incidence θ and the ridge length (or plate thickness) l: $h_1=0$ (a), $h_1=62\text{nm}$ (b).

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Fig. 4 SPP scattering losses (S) vs. the angle of incidence θ and the ridge length l: $h_1=0$ (a), $h_1=62\text{nm}$ (b).

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At $h_1=0$ (Fig. 4(a)), average scattering losses are as high as 0.2. As in the previous cases, the utilization of the proposed scattering suppression technique ($h_1=62\text{nm}$, Fig. 4(b)) reduces the average scattering losses by an order-of-magnitude to 0.027.

The SPP plots in Fig. 3 and Fig. 4 have several interesting features both for $h_1=0$ and $h_1=62\text{nm}$. First, above a certain incidence angle ${\theta }_{cr}$ (shown by white dashed lines in Fig. 4(a) and 4(b)) the scattering losses are zero. The critical angle ${\theta }_{cr}$ is defined by the expression $\sin {\theta }_{cr}=k_0\sqrt{{\epsilon }_d}/k_{spp}$. Above this angle the y-axis projection of the wave vector of the incident SPP is greater than the PW wave vector in the surrounding dielectric and therefore the corresponding plane waves outside the ridge are evanescent. Accordingly, the SPP cannot be scattered into propagating free-space radiation and the parasitic scattering is completely eliminated.

Second, the SPP reflectance and transmittance (Fig. 3) experience sharp (resonant) changes for some combinations of the incidence angle and the ridge length. For example, Fig. 5(a) shows the fragment of the SPP transmittance at $h_1=0$ in the region depicted by white dashed rectangle in Fig. 3(a). Along with the wide Fabry-Perot maximum present also in the PW model, there are several regular features (local transmittance maxima (numbered 1, 3, 4, 5) or minima (numbered 2)) which are specific for SPP. These features are associated with the excitation of the leaky modes of the ridge which in this case acts as a dielectric-loaded SPP waveguide (DLSPPW). The dispersion curves of these modes calculated using COMSOL software are shown with black dashed lines in Fig. 5(a) and are in good agreement with the SPP resonances. To further confirm that the excitation of DLSPPW modes is the origin of these transmittance features, we plot the electric field intensity distributions for the SPP diffraction (calculated using RCWA) and the DLSPPW eigenmode (calculated using COMSOL) at the point indicated by a white asterisk in Fig. 5(a). The calculated distributions turn out to be almost identical (Fig. 5(b)).

 

Fig. 5 (a) Fragment of the SPP transmittance distribution at $h_1=0$shown by white dashed rectangle in Fig. 3(a), black dashed lines are the dispersion curves of the DLSPPW modes, white dashed line shows the angle ${\theta }_{cr}$. (b) ${\left|E\right|}^2$ distributions in the dielectric ridge for SPP diffraction (left) and DLSPPW eigenmode (right) at the point depicted by white asterisk in Fig. 5(a). The dielectric ridge and the interface between the metal and the surrounding dielectric are shown by dashed white lines.

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Let us note that in spite of the DLSPPW mode excitation, the mean difference between the SPP transmittance and reflectance with the scattering suppression (at $h_1=62\text{nm}$, Fig. 3(b)) and the corresponding PW model does not exceed 0.015. These similarities between the refraction of SPP with the parasitic scattering suppression and the plane wave refraction suggest that a number of approaches used to improve the efficiency of the conventional optical elements can be extended to the elements of the plasmonic optics. For example, our simulation results (which will be published elsewhere) show that it is possible to create an analog of anti-reflection coatings for the plasmonic elements.

5. Plasmonic element example: a plano-convex microlens array

To confirm that the proposed scattering suppression technique can be used for the design of plasmonic optical elements, we consider as an example a periodic microlens array consisting of plano-convex dielectric lenses with the following parameters: aperture $a=8{\lambda }_{spp}$ (${\lambda }_{spp}=k_0\lambda /\mathrm{Re}\left\{k_{spp}\right\}$ being the wavelength of the incident SPP, and $\mathrm{Re}\left\{t\right\}$ being the real part of t), array period $d=a$, focal length $f=1.2a$. For the considered parameters, ${\lambda }_{spp}=783\text{nm}$, $a=6.27\mu \text{m}$, and $f=7.52\mu \text{m}$. Each of the refractive lenses constituting the array performs the phase modulation of the incident SPP (the phase modulation of SPP by dielectric structures was studied in detail in [22]). The lens profile $x\left(y\right)$ was calculated from the Fermat principle:

As in the previous cases, we compare the SPP results with the PW model. Let us mention once again that in this case the dielectric permittivity of a real (lossy) metal is used in the simulations (absorption losses are taken into account also in the PW model). Figure 6 shows the SPP ${\left|E_y\right|}^2$ distributions 10 nm above the metal surface (Figs. 6(c) and 6(d)) within one period along the y axis and the corresponding distributions for the PW model (Figs. 6(a) and 6(b)). All the field distributions were calculated using the RCWA technique. The complex shape of the focal areas having several local intensity maxima is caused by the lens array periodicity along the y axis.

 

Fig. 6 ${\left|E_y\right|}^2$ distributions for SPP ((c), (d)) and PW ((a), (b)) lens arrays within one period. $h_1=0$ ((a), (c)) or $h_1=62\text{nm}$ ((b), (d)). Lenses are shown with solid white lines.

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Without the scattering suppression (Figs. 6(a) and 6(c)), the SPP focal intensity normalized by the PW focal intensity is only 0.8 due to scattering losses. With scattering suppression (Figs. 6(b) and 6(d)), the normalized SPP focal intensity is 0.97 which indicates that the scattering losses are almost completely eliminated. In addition, the absolute value of the SPP focal intensity increases by 30% when $h_1$ is increased from 0 nm (Fig. 6(c)) to 62 nm (Fig. 6(d)). Let us emphasize that this intensity enhancement is primarily due to the parasitic scattering suppression, and not due to the decrease in the absorption inside the lens (although the latter also takes place since $\mathrm{Im}\left\{n_{mode,62}\right\}<\mathrm{Im}\left\{n_{mode,0}\right\}$, where $\mathrm{Im}\left\{t\right\}$ is the imaginary part of t). Indeed, if we consider the lossless case for comparison, the SPP focal intensity still increases by more than 20% when the scattering suppression layer is introduced.

Thus, the presented scattering suppression approach indeed provides a way to improve the efficiency of plasmonic elements.

6. Conclusion

In the present work, we have demonstrated that a two-layer isotropic dielectric structure can be used for SPP parasitic scattering suppression for arbitrary incidence angle. The average scattering losses are reduced by an order-of-magnitude down to 1–3%. With the scattering suppression, the SPP refraction can be in most cases accurately described by an analytical model based on the Fresnel equations. As an example of a plasmonic element, a plano-convex lens array was considered. The proposed approach can also be used for the design of other plasmonic elements such as reflectors, plasmonic crystals and plasmonic laser cavities.

Acknowledgments