1. Introduction

Diffraction gratings are among the most versatile and efficient building blocks used for the design of photonic devices. The development of theoretical, numerical, and fabrication techniques made it possible to design and implement subwavelength resonant gratings, which possess extraordinary optical properties [1]. One of the classes of such gratings that attracted particular attention in the past decade is constituted by high-index dielectric subwavelength structures surrounded by low-index materials, which are usually referred to as high-contrast gratings (HCGs) or high-contrast metastructures [2,3]. Such gratings exhibit remarkable features including ultrabroadband high reflectivity and high-quality-factor resonances.

The vast majority of resonant subwavelength gratings presented in the existing works (both the dielectric gratings discussed above and metallic gratings [4–6]) operate with the free-space radiation (i. e. with the incident plane electromagnetic waves). However, it is of great interest to extend the subwavelength grating concept to the integrated optics platforms, since it may prove promising for the development of novel on-chip optoelectronic systems. One of the actively studied integrated platforms is the plasmonics platform dealing with surface plasmon polaritons (SPPs) — surface electromagnetic waves propagating along the metal-dielectric interfaces. For SPPs, different nanophotonic elements have been proposed, including the structures for focusing [7,8] and localizing [9] the SPPs. In a previous work [10], we proposed and theoretically and numerically studied near-wavelength dielectric diffraction gratings for SPPs. Our proof-of-concept simulations demonstrated that, under certain conditions, the diffraction of the SPPs is remarkably close to the diffraction of transverse-electric (TE-) polarized plane waves. However, the gratings studied in [10] had a relatively low optical contrast and were investigated at a fixed frequency of the incident SPP.

In the present work, we design and numerically study high-contrast gratings for SPPs. We demonstrate that such integrated HCGs can be used as broadband plasmonic mirrors for both the normal and oblique incidence geometries. In order to increase the mirror efficiency and design practically feasible structures, we adopt the SPP parasitic scattering suppression technique proposed in our previous works [11,12].

2. High-contrast gratings for free-space radiation

Before focusing on the integrated HCGs for surface plasmon polaritons, let us revisit the conventional high-contrast gratings for the free-space radiation.

For both the SPPs and the free-space waves (namely, TE-polarized plane waves propagating in air), we design HCGs intended to work as broadband mirrors in a 200-nm-wide spectral range centered at the telecom wavelength $\lambda = 1550~\mathrm{nm}$. We consider two incidence geometries: normal incidence and oblique incidence at an angle $\theta = 45^{\circ}$. The studied conventional HCGs [see the inset to Fig. 1(a)] consist of rectangular dielectric ridges suspended in air ($n = 1$). For the sake of simplicity, in the present work we limit our consideration to the structures with the fill factor (the ratio of the ridge width $w$ to the grating period $d$) equal to $0.5$. However, let us note that tuning of the fill factor enables slightly improving the grating efficiency as compared to the grating examples presented below.

 

Fig. 1. (a) Reflectance $R_0$ of a suspended silicon grating (inset) vs. the normalized parameters $\lambda /d$ and $h/d$ in the case of normal incidence of a TE-polarized plane wave. Fill factor of the grating is fixed at $w/d = 0.5$. Black dotted lines correspond to the examples of gratings, the reflectance and transmittance spectra of which are shown in (c). (b) Phase of the reflected wave. (c) Reflectance $R_0$ (solid lines) and transmittance $T_0$ (dashed lines) spectra of three examples of HCGs. Example 1 (upper panel): grating period $d = 992~\mathrm{nm}$, grating height $h = 146~\mathrm{nm}$; example 2 (middle panel): $d = 1128~\mathrm{nm}$, $h = 349~\mathrm{nm}$; example 3 (lower panel): $d = 921~\mathrm{nm}$, $h = 874~\mathrm{nm}$.

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Since we consider the conventional HCGs as “model” structures, let us study gratings made of a dispersionless material with the refractive index equal to the refractive index of silicon at the $1550~\mathrm{nm}$ wavelength: $n_{\mathrm{gr}} = n_{\mathrm{Si}} = 3.48$ [13]. In this case, a change in the incident wavelength $\lambda$ is equivalent to a corresponding scaling of the grating period $d$ and grating height $h$. In this regard, it is convenient to calculate the grating reflectance and transmittance vs. the normalized parameters $\lambda /d$ and $h/d$.

Figure 1(a) shows the reflectance spectrum of the conventional HCG (inset) for the case of normal incidence of a TE-polarized plane wave calculated using an in-house implementation of the rigorous coupled-wave analysis technique (RCWA; also known as the Fourier modal method) [14]. RCWA is an established numerical technique for solving Maxwell’s equations in the problems of diffraction by periodic structures. In this technique, the field over and under the grating is represented as a superposition of plane waves corresponding to the reflected and transmitted diffraction orders (the Rayleigh expansion), whereas the general representation of the field in the grating is found by solving an eigenvalue problem. The solution of the diffraction problem is then reduced to solving a system of linear equations. This approach enables finding the diffraction efficiencies (intensities) of the diffraction orders without calculating the total electromagnetic field in the structure (however, the field distributions can be calculated, if necessary). Figure 1(b) shows the phase of the reflected radiation.

In the shown $\lambda /d$ range, the grating is subwavelength ($\lambda /d > 1$), so that only the zeroth reflected and transmitted diffraction orders propagate. Moreover, in the most part of the chosen $\lambda /d$ range (at $1.075< \lambda /d < 1.92$), the grating is in the so-called “dual-mode” (or “two-mode”) regime, where it can demonstrate wideband high reflectance or high-Q resonances in the spectrum. For the sake of completeness, let us describe this regime. A binary dielectric diffraction grating such as the one shown in the inset to Fig. 1(a) can be considered as a finite-height segment of a periodic optical waveguide. Such a periodic waveguide supports a discrete (and finite) number of modes propagating in the vertical direction, which are called Bloch modes [15]. The number of the propagating Bloch modes depends on the $\lambda /d$ ratio, the fill factor, and the optical contrast between the grating materials. Neglecting the near-field interaction between the upper and lower grating interfaces, the diffraction on such a grating can be described in the following way. At the upper grating interface, the incident plane wave is coupled to several propagating Bloch modes, which bounce between the upper and lower grating interfaces, are coupled to each other at them, and are also out-coupled to the reflected and transmitted diffraction orders. Thus, the intensities of the diffraction orders are determined by the interference between these modes at the grating interfaces. A rigorous and comprehensive theoretical description of this process is given in [15,16]. In these works, it was shown that in the case of subwavelength gratings, if the incident wave is coupled only to the two Bloch modes supported by the grating, the interference between them can lead to 100% reflectance or transmittance. It is important to note that in the case of normal incidence, the incident wave is coupled only to the “even” (symmetric) modes of the periodic waveguide due to the symmetry reasons, so the grating can actually support more than two modes [the other one or two modes apart from the two symmetric ones being “odd” (antisymmetric) and, hence, symmetrically incompatible with the incident radiation]. For the sake of illustration, the electric field profiles of the two symmetric Bloch modes supported by the grating example from Fig. 1 at $\lambda /d = 1.5$ are shown on one grating period in the inset to Fig. 1(a). It was also shown in [15,16] that due to the Bloch mode interference effects, the high- and low-reflectance regions appear in a highly regular checkerboard-like pattern, which is clearly evident from Fig. 1(a).

In the case of the high-constrast gratings, the 100%-reflectance condition can be approximately fulfilled in a wide spectral range [15,16]. Indeed, Fig. 1(c) shows the reflectance and transmittance spectra of three grating examples “extracted” from Fig. 1(a) (here and in what follows, the parameters of the grating examples are given in the figure captions). Figure 1(c) demonstrates that using a high-contrast grating, it is possible to design a thin ($h/\lambda < 1$) broadband mirror: the mean reflectance values in the 1450–1650 nm spectral range amount to 97.9%, 99.2%, and 97.1% for grating examples 1, 2, and 3, respectively.

Similar high-reflectance gratings can also be designed for the case of oblique incidence. Figure 2(a) shows the dependence of the grating reflectance on the normalized wavelength $\lambda /d$ and grating height $h/d$ for the angle of incidence $\theta = 45^{\circ}$. As in the normal incidence case discussed above, in the considered $\lambda /d$ range the grating is subwavelength, and, at $\lambda /d < 3.08$, operates in the dual-mode regime. Fig. 2(b) shows the phase of the reflected wave. Fig. 2(c) shows the reflectance and transmittance spectra of three HCGs operating at $\theta = 45^{\circ}$. The corresponding mean reflectance values in this case amount to 93.7%, 97.4%, and 99.0%.

 

Fig. 2. (a) Reflectance $R_0$ of a suspended silicon grating vs. the normalized parameters $\lambda /d$ and $h/d$ in the case of oblique incidence of a TE-polarized plane wave at an angle $\theta = 45^{\circ}$. Fill factor of the grating is fixed and equals $0.5$. Black dotted lines correspond to the examples of gratings, the reflectance and transmittance spectra of which are shown in (c). (b) Phase of the reflected wave. (c) Reflectance $R_0$ (solid lines) and transmittance $T_0$ (dashed lines) spectra of three examples of HCGs. Example 1 (upper panel): grating period $d = 625~\mathrm{nm}$, grating height $h = 140~\mathrm{nm}$; example 2 (middle panel): $d = 743~\mathrm{nm}$, $h = 398~\mathrm{nm}$; example 3 (lower panel): $d = 819~\mathrm{nm}$, $h = 641~\mathrm{nm}$.

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The presented results confirm that single-layer high-contrast gratings with $w/d = 0.5$ can be used as broadband ($\Delta \lambda /\lambda >10\%$) mirrors for free-space radiation. Let us reiterate that by optimizing the grating fill factor, one can increase the mean (and minimal) reflectance in the working spectral range and/or expand the operating bandwidth. In the following section, we will extend the HCG concept to the “two-dimensional” platform of surface plasmon polaritons propagating along metal-dielectric interfaces.

3. High-contrast gratings for surface plasmon polaritons

3.1 Plasmonic HCGs without parasitic scattering suppression

The studied integrated high-contrast gratings for surface plasmon polaritons consist of periodically arranged silicon pillars located on the gold surface (Fig. 3). We consider the case when an SPP with a plane wavefront is normally or obliquely incident on the grating. At the central wavelength of 1550 nm, the effective refractive indices $n_{\mathrm{SPP}}$ (propagation constants normalized by the wavenumber $k_0 = 2\pi /\lambda$) of the SPPs supported by the Au/air and Au/Si interfaces amount to $n_{\mathrm{SPP,Air}} = \sqrt {\varepsilon _{\mathrm{Au}}/(\varepsilon _{\mathrm{Au}} + 1)} = 1.0043 + 0.0004\mathrm{i}$ and $n_{\mathrm{SPP,Si}} = \sqrt {\varepsilon _{\mathrm{Au}}\varepsilon _{\mathrm{Si}}/(\varepsilon _{\mathrm{Au}} + \varepsilon _{\mathrm{Si}})} = 3.6717 + 0.0205\mathrm{i}$, respectively. For the calculation of the effective refractive indices, the following dielectric permittivity values were used: $\varepsilon _{\mathrm{Si}} = 12.08$ and $\varepsilon _{\mathrm{Au}} = -115.11 + 11.1\mathrm{i}$ [13,17].

 

Fig. 3. Geometry of the investigated high-contrast gratings for surface plasmon polaritons consisting of periodically arranged silicon pillars on the gold surface.

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In the considered structure, the real parts of the SPP effective refractive indices are close to the refractive indices of the conventional grating considered above, therefore, one could expect that the integrated HCGs for SPPs might exhibit similar (at least qualitatively) optical properties. In the examples presented below, we investigate the plasmonic gratings in the same $\lambda /d$ and $h/d$ ranges as the conventional HCGs discussed above. For the chosen materials and central wavelength, the SPP field penetration depths into air and silicon calculated as $|k_0\mathrm{Im}(\sqrt {\varepsilon _{\mathrm{d}} - n_{\mathrm{SPP}}^{2}})|^{-1}$ [18], where $\varepsilon _{\mathrm{d}}$ is the permittivity of the dielectric, amount to $2.645~{\mu \textrm{m}}$ and $208~\mathrm{nm}$, respectively. Therefore, in the simulations we used the silicon pillar height value $p = 4~{\mu \textrm{m}}$, which exceeds both of these field penetration depths with a reasonable margin. Similarly to the conventional gratings studied in the previous section, the simulation results for the integrated HCGs presented below were obtained using an in-house parallel implementation of the aperiodic Fourier modal method (aFMM) suitable for solving the integrated optics problems [19,20]. In this method, the structure is artificially periodized in the direction perpendicular to the integrated optical waveguide (being the metal surface in the considered case of plasmonic structures). The optical interaction between the adjacent artificial periods is prevented by adding special absorbing layers (e. g., perfectly matched layers or gradient absorbers). The general field representations in the regions before and after the integrated grating and in the grating region are found by solving an eigenvalue problem in each region. Exactly as in the conventional RCWA, the solution of the diffraction problem is then reduced to solving a system of linear equations. For more details concerning the simulation of plasmonic gratings, we refer the reader to our previous work [10].

Figure 4(a) shows the reflectance of a plasmonic HCG vs. the normalized parameters $\lambda /d$ and $h/d$ calculated for the case of normal incidence of an SPP. For the calculation of Fig. 4(a), the SPP free-space wavelength was fixed at $\lambda = 1550~\mathrm{nm}$ and the grating period $d$ and ridge length $h$ were varied. It is evident from Fig. 4(a) that the optical properties of an integrated plasmonic high-contrast grating are qualitatively similar to the ones of a conventional grating [Fig. 1(a)]. However, an increase in the grating ridge length $h$ (see Fig. 3), which corresponds to the height of the conventional grating [see the inset to Fig. 1(a)], leads to a significant decrease in the mean reflectance value in the high-reflectance regions. This is due to the fact that in the case of a plasmonic grating, not only the reflected and transmitted SPPs with the intensities $R_0$ and $T_0$, respectively, are generated upon diffraction by the grating, but also the radiation scattered “out-of-plane” and propagating away from the metal surface [11,21]. In addition, some of the energy of the incident SPP is lost due to the absorption in the metal. The total losses $L = 1 - (R_0 + T_0)$ averaged over the considered $\lambda /d$ range for each $h/d$ value are shown in Fig. 4(b). According to Fig. 4(b), the average losses exceed 50% of the energy already at $h/d = 1$. The results of additional numerical simulations (not presented here) show that the dominant loss mechanism in the considered structure is the “parasitic” out-of-plane scattering of the incident SPP, whereas the absorption losses amount to 5–8% on average.

 

Fig. 4. (a) Reflectance $R_0$ of a plasmonic high-contrast grating consisting of silicon pillars vs. the normalized parameters $\lambda /d$ and $h/d$ in the case of normal SPP incidence. Fill factor of the grating is fixed and equals $0.5$. Black dotted lines correspond to the examples of gratings, the reflectance, transmittance, and losses spectra of which are shown in (c). (b) Losses $L = 1 - (R_0 + T_0)$ averaged over the considered $\lambda /d$ range vs. $h/d$. (c) Reflectance $R_0$ (solid lines), transmittance $T_0$ (dashed lines), and losses $L$ (dotted lines) spectra of three examples of plasmonic HCGs. Example 1 (upper panel): grating period $d = 1010~\mathrm{nm}$, grating ridge length $h = 121~\mathrm{nm}$; example 2 (middle panel): $d = 1168~\mathrm{nm}$, $h = 350~\mathrm{nm}$; example 3 (lower panel): $d = 929~\mathrm{nm}$, $h = 836~\mathrm{nm}$.

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Figure 4(c) shows the reflectance, transmittance and losses spectra of three integrated HCG examples. Although the parameters of the considered examples were chosen on the basis of Fig. 4(a) calculated at a single wavelength, the plots in Fig. 4(c) were rigorously calculated for the case of fixed grating parameters (grating period $d$ and grating ridge length $h$) and a varying SPP wavelength $\lambda$. Thus, the plots in Fig. 4(c) were obtained taking into account the dispersion of the dielectric permittivities of the materials of the structure and the resulting dispersion of the SPP effective refractive indices. For the chosen plasmonic gratings, the mean reflectance values amount to 86.0% (example 1), 84.3% (example 2), and 54.4% (example 3). The presented results demonstrate that by using silicon pillars on the metal surface, one can design broadband mirrors for surface plasmon polaritons with mean reflectance exceeding 85% (see example 1). However, for this example, the aspect ratio (the ratio of the pillar height $p$ to its “in-plane” length $h$) is very high and exceeds 30, which makes the structure hardly feasible. Of course, the aspect ratio can be decreased by decreasing the grating pillar height $p$. However, this leads to a significant increase in the parasitic scattering losses and, consequently, a decrease in reflectance. Figure 5 shows the dependence of the reflectance, transmittance and losses on the pillar height $p$ averaged over the operating wavelength range for the three grating examples presented in Fig. 4(c). Already at $p = 2~{\mu \textrm{m}}$, the reflectance for all the three considered examples becomes less than 60%, whereas for the third example it only slightly exceeds 30%. Similar results (not shown here for the sake of brevity) were obtained for integrated gratings designed for the case of oblique SPP incidence. Therefore, in order to design practically feasible structures with reasonable aspect ratios, one has to decrease the “parasitic” scattering losses, so that the structures with greater $h$ values could be used.

 

Fig. 5. Reflectance $R_0$ (solid lines), transmittance $T_0$ (dashed lines), and losses $L$ (dotted lines) averaged over the operating wavelength range vs. grating pillar height $p$ for the three plasmonic HCG examples presented in Fig. 4(c).

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3.2 Plasmonic HCGs with parasitic scattering suppression

The parasitic out-of-plane scattering of surface plasmon polaritons, which leads to the deterioration of the efficiency of the integrated plasmonic elements, is caused by the mismatch of the transverse SPP field profiles inside and outside the element [11,21]. If, as in the considered case, the grating pillars are high enough, the transverse field profile inside the element can be described with a reasonable accuracy using the expression for the SPP penetration depth into dielectric, which follows from the conventional dispersion relation of the SPP at the interface between metal and the grating pillar material presented above. For the considered plasmonic HCGs, the field mismatch is clearly indicated by an order-of-magnitude difference in the field penetration depth values of the SPPs supported by the Au/air and Au/Si interfaces ($2.645~{\mu \textrm{m}}$ and $208~\mathrm{nm}$, respectively). For the suppression of the parasitic scattering, several approaches have been proposed, all of which shared the same basic idea consisting in introducing additional degrees of freedom into the dispersion relation of the plasmonic mode in the element region, so that the transverse field profile of this mode could be (at least partially) matched to the transverse field profile of the incident SPP. In [21], an approach based on the utilization of anisotropic metamaterials was presented. It was shown that in this case, the transverse field profile of the SPP inside the element can be fully matched to the field profile of the incident SPP. Such an approach makes it possible to completely eliminate the parasitic scattering, however, the design and fabrication of metamaterials with required effective parameters constitute a separate nontrivial problem.

In our previous works [11,12], we proposed an alternative scattering suppression method, which does not require the use of anisotropic materials. In our method, the necessary degrees of freedom are introduced to the dispersion relation of the plasmonic mode by adding a thin dielectric layer underneath the plasmonic element [see the inset to Fig. 6(a)]. By choosing the layer thickness $t$, one can partially [in the main (upper) part of the element having the thickness $p$] match the transverse field profiles of the incident SPP and the plasmonic mode in the element. The results presented in [11,12] demonstrated that this leads to a significant (by several times) reduction of the SPP parasitic scattering. In should be noted that the thickness value $t$ providing the scattering suppression can be calculated analytically. In the present work, we adopt this approach in order to increase the efficiency of the designed plasmonic HCGs. As the material of the additional layer, we choose the silicon nitride with the dielectric permittivity $\varepsilon _{\mathrm{Si}_3\mathrm{N}_4} = 3.99$ at the central wavelength of the operating range [22]. In this case, the layer thickness providing the scattering suppression condition amounts to $t = 11~\mathrm{nm}$.

 

Fig. 6. (a) Reflectance $R_0$ of a plasmonic high-contrast grating with parasitic scattering suppression vs. the normalized parameters $\lambda /d$ and $h/d$ in the case of normal SPP incidence. Fill factor of the grating is fixed and equals $0.5$. Black dotted lines correspond to the examples of gratings, the reflectance, transmittance, and losses spectra of which are shown in (d). The inset shows the grating ridge geometry (not to scale; $p = 4~{\mu \textrm{m}}$, $t = 11~\mathrm{nm}$). The same colormap as in the previous figures is utilized. (b) Phase of the reflected wave. (c) Losses $L = 1 - (R_0 + T_0)$ averaged over the considered $\lambda /d$ range vs. $h/d$. (d) Reflectance $R_0$ (solid lines), transmittance $T_0$ (dashed lines), and losses $L$ (dotted lines) spectra of three examples of plasmonic HCGs. Example 1 (upper panel): grating period $d = 1000~\mathrm{nm}$, grating ridge length $h = 140~\mathrm{nm}$; example 2 (middle panel): $d = 1155~\mathrm{nm}$, $h = 370~\mathrm{nm}$; example 3 (lower panel): $d = 920~\mathrm{nm}$, $h = 846~\mathrm{nm}$.

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Figure 6(a) shows the reflectance of a plasmonic HCG with parasitic scattering suppression designed for the case of normal SPP incidence. Fig. 6(b) shows the phase of the reflected SPP. The total losses $L$ averaged over the considered $\lambda /d$ range for each $h/d$ value are shown in Fig. 6(c). It is evident from Figs. 6(a) and 6(c) that at relatively high $h/d$ values ($h/d>0.5$), the average reflectance in the high-reflectance regions is much higher, and the losses are much less than without the parasitic scattering suppression [see Figs. 4(a) and 4(b)]. For the most part of the considered $h/d$ range, mean losses decrease by 2.5–3 times. It is also worth noting that it follows from Figs. 6(a) and 6(b) that there exist high-reflectance regions ($R_0 > 0.8$) with the phase continuously changing by $2\pi$. This means that the studied plasmonic HCGs can be used for creating integrated focusing reflectors similar to the ones based on conventional high-contrast gratings [23,24].

Figure 6(d) shows the reflectance, transmittance and losses spectra of three integrated HCG examples with the parasitic scattering suppression. For the chosen gratings, the mean reflectance values amount to 87.5% (example 1), 88.3% (example 2), and 85.1% (example 3). The increase in efficiency is most pronounced for the third example, which has a relatively high value of $h/d = 0.92$, and exceeds 30%. Fig. 7 shows the field distributions for the third plasmonic grating examples from Figs. 4(c) and 6(d) and illustrates an increase in reflectance and a decrease in the parasitic scattering losses achieved by the utilization of two-layer grating pillars.

 

Fig. 7. Distributions of the field component $|H_x|$ at the free-space wavelength $\lambda = 1500~\mathrm{nm}$ for the third plasmonic HCG examples from Fig. 4(c) (a grating without parasitic scattering suppression) (a) and Fig. 6(d) (a grating with parasitic scattering suppression) (b). The SPP normally impinging on the gratings is incident from the left. The same colormap is used in (a) and (b). White dashed lines show the metal-dielectric interfaces and the grating pillars.

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As in the case of plasmonic gratings without the parasitic scattering suppression, the average reflectance of the grating examples presented in Fig. 6(d) decreases and the losses increase with a decrease in the grating pillar height $p$ (Fig. 8). However, as it is evident from Fig. 8, with the parasitic scattering suppression, the efficiency of the third grating example is very close to the efficiencies of the first two examples having significantly smaller $h$ values.

 

Fig. 8. Reflectance $R_0$ (solid lines), transmittance $T_0$ (dashed lines), and losses $L$ (dotted lines) averaged over the operating wavelength range vs. grating pillar height $p$ for the three plasmonic HCG examples presented in Fig. 6(d).

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Figure 9(a) shows the dependence of the reflectance of the plasmonic HCG with parasitic scattering suppression on the normalized wavelength $\lambda /d$ and grating height $h/d$ designed for the angle of incidence $\theta = 45^{\circ}$. In Fig. 9(b), the phase of the reflected SPP is shown. Fig. 9(c) shows the reflectance and transmittance spectra of three HCGs operating at $\theta = 45^{\circ}$. The corresponding mean reflectance values in this case amount to 82.3%, 85.7%, and 87.5% for the examples 1, 2, and 3, respectively.

 

Fig. 9. (a) Reflectance $R_0$ of a plasmonic high-contrast grating with parasitic scattering suppression vs. the normalized parameters $\lambda /d$ and $h/d$ in the case of oblique SPP incidence at an angle $\theta = 45^{\circ}$. Fill factor of the grating is fixed and equals $0.5$. Black dotted lines correspond to the examples of gratings, the reflectance, transmittance, and losses spectra of which are shown in (c). (b) Phase of the reflected wave. (c) Reflectance $R_0$ (solid lines), transmittance $T_0$ (dashed lines), and losses $L$ (dotted lines) spectra of three examples of plasmonic HCGs. Example 1 (upper panel): grating period $d = 651~\mathrm{nm}$, grating ridge length $h = 117~\mathrm{nm}$; example 2 (middle panel): $d = 711~\mathrm{nm}$, $h = 398~\mathrm{nm}$; example 3 (lower panel): $d = 821~\mathrm{nm}$, $h = 616~\mathrm{nm}$.

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Note that while the structures of Figs. 6 and 9 were designed for fixed angles of incidence $\theta = 0$ and $\theta = 45^{\circ}$, respectively, the designed HCGs remain functional if the angle of incidence is slightly changed. For example, Fig. 10 shows the reflectance spectra of the first examples of integrated HCGs from Figs. 6(d) and 9(c) for the initial and slightly (by $1^{\circ}$ or $5^{\circ}$) changed angles of incidence. For all the plots in Fig. 10(a), the mean reflectance does not get less than 87%. For the plots in Fig. 10(b) (except for the case of $\theta = 50^{\circ}$), the mean reflectance exceeds 80%.

 

Fig. 10. Reflectance $R_0$ at the design angle of incidence and at slightly changed angles of incidence for the first plasmonic grating examples from Fig. 6(d) (a) and Fig. 9(c) (b).

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For both third examples in Figs. 6(d) and 9(c), for which the mean reflectance over the wavelength range of operation exceeds 85%, the ratio of the grating height $p$ to the ridge length $h$ does not exceed 6.5. The structures with such aspect ratios can be fabricated using the state-of-the-art nanotechnology techniques [25–28]. It is also worth mentioning that the designed plasmonic HCGs have a subwavelength footprint ($h/\lambda < 1$) and are much more compact than the plasmonic Bragg gratings, while being competitive in terms of efficiency [29,30].

4. Conclusion

In the present work, we have proposed and numerically studied integrated high-contrast gratings for surface plasmon polaritons consisting of periodically arranged silicon pillars located on the metal surface. We have shown that the optical properties of the integrated plasmonic HCGs are close to the optical properties of the conventional HCGs operating with TE-polarized plane electromagnetic waves. In particular, we designed broadband plasmonic mirrors based on integrated HCGs, which provide mean reflectance exceeding 85% over a 200-nm-wide spectral range centered at the free-space wavelength of 1550 nm both for the cases of normal SPP incidence and of oblique incidence at an angle of 45°. In order to increase the grating efficiency and design more practical structures, we adopted a parasitic scattering suppression technique based on the utilization of two-layer grating pillars providing partial matching of the transverse SPP field profile inside and outside them. A further increase in the SPP HCG efficiency can be obtained by optimizing the fill factor of the gratings and/or “stacking” several gratings in the SPP propagation direction.

We believe that the obtained results are promising for the design of two-dimensional optical circuits for steering the SPP propagation. In the opinion of the authors, the presented results may also be extended to other integrated platforms such as Bloch surface waves [31,32] or modes of dielectric slab waveguides (sometimes referred to as semi-guided waves) [33–35]. This will be the subject of a further research.