1. Introduction

The photon-counting in superconducting nanowire single-photon detectors (SNSPDs) is based on the hot-spot formation in a meandered pattern of absorbing niobium-nitride (NbN) nanowires due to near-infrared light illumination [1]. In order to achieve high SNSPD responsivity, nanowire widths of ~100 nm or less are required, while to maximize the resistive barrier across an NbN stripe pattern, wire feature sizes of 100 nm are ideal [2]. The typical geometrical size parameters applied in commercially available devices are a pitch of p = 100-200 nm, a fill factor of f = 50%, and an NbN thickness of t = 4 nm [3].

Detection efficiency of SNSPDs is determined by optical and electronic efficiencies according to the relation: DE = AP_R, where A refers to the absorptance of the detector and P_R is the probability of measurable electronic signal caused by an absorbed photon. This implies that the ideal SNSPD detector has to be optimized both optically and electronically. Detection efficiency of SNSPDs is limited optically by losses accompanying reflection from and transmission through the structure, as well as by absorptance in other materials in the device, therefore optimization of these photodetectors’ detection efficiency requires the maximization of NbN patterns’ absorptance.

The absorptance inherently depends on the E-field oscillation orientation with respect to the NbN stripes during polarized light illumination. Two specific orientations are referred usually as P-orientation, when the plane of incidence is parallel to the wires, and S-orientation, when the plane of incidence is perpendicular to the wires [4].

Comparison of results from previous literature suggests that the device composition determines whether p- or s-polarization is the most advantageous. Higher absorptance was always detected during illumination by perpendicularly incident light, when the E-field oscillation was parallel to the NbN wires embedded into dielectric media [5]. This polarization preference can be explained by the enhanced penetration promoted by the parallelism of the E-field oscillation direction to the wires, which can be used to maximize the intrinsic loss [6].

The tilting of SNSPD devices is capable of resulting in considerable enhancement. It was shown that the absorption of s-polarized near-infrared light having a wavelength of 775 nm approaches A = 100%, when NbN patterns in S-orientation are illuminated from the substrate side at the angle of Total Internal Reflection (TIR) [7].

The integration of an optical cavity and an Anti-Reflection-Coating (ARC) can be used to overcome some of the optical losses, and results in detection efficiency of DE ~50% in case of illumination by near-infrared, λ = 1550 nm light [8]. Our previous polar and azimuthal angle dependent studies have proven that in optical-cavity-integrated detector designs the largest absorptance is reachable in case of p-polarized light illumination of NbN structures in P-orientation through all polar angles [9, 10].

The novel idea in detector designs is the improvement of absorptance via integrated noble metal structures [11, 12]. Recent investigations have shown that integration of appropriately designed noble metal nano-antenna-arrays into SNSPDs results in A = 96% absorptance in NbN wires, when polarized light is incident perpendicularly and the E-field oscillation is perpendicular to the bi-grating consisting of grating-like noble-metal and a meandered NbN pattern. The conclusion of design considerations was that the optimum integrated structure parameters resulting in strong near-field concentration around 100 nm wide NbN segments are p = 200 nm pitch, no-gap between the metal nano-antenna and the NbN wires, and l_OC = 220 nm length of a cavity filled with a dielectric material such as hydrogen silsesquioxane (HSQ) [11].

The purpose of our present work is to determine the optimum conditions for polarized light illumination that maximize the absorption in NbN patterns integrated with nano-cavity-array. Previously, such integrated systems have been investigated theoretically only for normally incident light. We extend the method presented in Refs [11, 12], by varying both the polar and azimuthal directions during illumination by p- and s-polarized light.

We study two different detector designs, a 200-nm-pitch and a 600-nm-pitch design, each of these is proposed for specific applications. The 200-nm-pitch design maximizes the absorbed optical power in the active NbN wires, but requires high pattern density and thus is more difficult to fabricate. This pitch leads to a longer meandered nanowire with concurrently longer reset time or smaller device area. The 600-nm-pitch design reduces the pattern density, thus simplifying fabrication. The reduced density can then be used to either expand the device active area correspondingly, or to achieve a faster reset time, to a point, as reset times apparently cannot be scaled without limit, or latching will occur [13–15].

Based on our previously developed three-dimensional numerical method described in Ref [9], we observe different optical phenomena occurring under specific illumination conditions, which relative importance in absorptance enhancement depends on the device design: (1) Brewster-like phenomenon [4]; (2) total internal reflection [7]; (3) non-resonant intensity enhancement at boundaries, edges and corners of gold segments that concentrate the E-field around NbN stripes [16, 17]; (4) overlapping of the resonant nano-plasmonic modes antinodes with the absorbing NbN stripes in appropriately designed nano-cavities [18–24].

This paper is organized as follows: in Section 2, the investigated device designs are introduced followed by detailed description of the applied three-dimensional numerical procedure and the complementary Transfer Matrix Method (TMM). In Section 3, the phenomena governing the optical response are identified and the near-field distribution accompanying these phenomena is analyzed. Finally, in Section 4, we conclude that simultaneous optimization of the polar and azimuthal illumination angles of appropriately designed structures can maximize SNSPD devices absorptance.

2. Theoretical method

2.1. Model systems

Theoretical investigation of two different nano-cavity-array integrated SNSPD device designs was performed numerically and by TMM. The first investigated device design presented in Fig. 1(a) is an optical system consisting of subwavelength NbN pattern with geometrical parameters according to conventional SNSPDs but integrated with an array of (HSQ)-filled nano-cavities and vertical and horizontal closing gold segments in a 200 nm periodic grating-like pattern.

 

Fig. 1 (a, b) Schematic drawings of the investigated device designs: NbN patterns with p = 200 nm and p = 600 nm periodicity, t = 4 nm thickness and f = 50% and f = 16.7% fill factor, covered by t = 2 nm NbNO_x layer, are arrayed below hydrogen silsesquioxane-filled nano-cavities having l_OC = 220 nm length, and surrounded by vertical and horizontal gold segments (a) in 200-nm-pitch design and (b) in 600-nm-pitch design. (c, d) These integrated NCAI-SNSPD structures are illuminated by polarized λ = 1550 nm light from sapphire substrate side, the φ polar angle is measured from the surface normal, while the azimuthal orientation is specified by the γ angle between the plane of light incidence and the NbN pattern. (c) P-polarized light illumination of periodic patterns in S-orientation; (d) s-polarized light illumination of periodic patterns in P-orientation.

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The second investigated device design presented in Fig. 1(b) is a three-times larger, 600 nm periodic pattern of the NbN stripes having the same geometry and integrated with dielectric-filled nano-cavity-array closed by analogous horizontal gold reflector but separated by wider vertical gold segments. These integrated patterns might be considered as arrays of closed nano-cavities [18], while the entire gold-pattern acts as a nano-cavity-grating [19–24].

2.1.1. The 200-nm-pitch design

The p = 200 nm periodic NbN pattern with f = 50% filling factor, consisting of t = 4 nm thick NbN stripes, and t = 2 nm thick NbNO_x dielectric cover-layer shown in Fig. 1(a) is the conventional structure in SNSPDs devices. This NbN pattern was combined with a nano-cavity-array closed by metal segments similar to the nano-antenna-array presented in [11], and using gold as in [12], since gold is chemically more stable than silver. Vertical gold segments with w = 100 nm width and l = 226 nm length were arrayed parallel to the NbN stripes in a grating-like pattern with the same periodicity.

HSQ was used as a dielectric material in the nano-cavities because it is an ideal electron-beam resist, and additionally has negligible absorptance in the near-IR. HSQ layers with l_OC = 220 nm length were included into the device-designs, since previous studies revealed the largest E-field intensity enhancement due to standing wave development inside 100 nm wide HSQ-filled nano-cavities with this length [11]. A continuous gold film with t = 60 nm thickness was used as the enclosing layer of the nano-cavities.

2.1.2. The 600-nm-pitch design

SNSPDs with longer wire lengths have a longer reset time because of their larger kinetic inductance, but increasing the nanowire periodicity across an area can reduce the wire length and thus reduce the reset time [13–15]. Therefore we evaluated a device design with a 600-nm-pitch that provides appropriate absorptance simultaneously with reduced inductance. This approach helps to scale devices to larger areas without sacrificing device speed.

Numerical calculations were performed using the same w = 100 nm width and t = 4 nm thick NbN wires covered by t = 2 nm NbNO_x layer as in 200-nm-pitch design, but arrayed in a p = 600 nm periodic pattern, and integrated with larger periodic nano-cavity-array, as presented in Fig. 1(b). Vertical gold segments with w = 500 nm width were arranged in between the NbN wires with the same periodicity. A continuous gold film with t = 60 nm thickness was the closing layer of the arrays of l_OC = 220 nm long nano-cavities filled with HSQ resist, analogously with the 200-nm-pitch design.

2.2. Transfer Matrix Method for complementary polar angle dependent study

We performed first TMM calculations on different unpatterned multilayers composing the vertical segments in the above described optical systems based on Ref [25], in order to determine the dominant optical phenomena and to identify the corresponding characteristic extrema on the optical response, as described in our previous works [9, 10]. The complex refractive indices of the materials in investigated NCAI-SNSPDs are summarized in Table 1 .

Table 1. Real (n₁) and imaginary (n₂) parts of the refractive indices of materials in NCAI-SNSPD designs.

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TMM calculations were then realized also for horizontally patterned multilayers, approximating them by composites made of following vertical stacks: (A) is a film stack of NbN-NbNO_x bi-layers, HSQ, and gold cover-layer; (B) is a vertical gold segment. All of layers in (A)-(B) stacks have thicknesses according to those in NCAI-SNSPD designs. The absorptance, reflectance and transmittance in the integrated optical systems composed of horizontally alternating segments were then determined based on responses of unpatterned films multiplied by their fill-factor, as in references [3, 9, 10].

The optical responses of different composing multilayers, and the TMM approximation of optical responses of 200-nm-pitch and 600-nm-pitch designs are presented in Fig. 2 .

 

Fig. 2 (a, b) Absorptance, (c, d) reflectance and (e, f) transmittance in case of (a, c, e) p-polarized and (b, d, f) s-polarized light illumination, determined by TMM for all layers composing the NCAI-SNSPD systems, and for 200-nm-pitch and 600-nm-pitch NCAI-SNSPDs computed by weighting the optical responses of the composing stacks according to their fill-factors.

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2.3. Finite Element Method for dual-angle dependent computations

The three-dimensional numerical computation method used for simulating off-axis illumination with polarized light at arbitrary azimuthal angle is described in our previous paper [9]. Similar three-dimensional finite element method (FEM) based models consisting of two- and one-unit-cells of the 200-nm-pitch and 600-nm-pitch integrated patterns according to Figs. 1(a, b) were prepared by using the Radio Frequency module of Comsol Multiphysics software package (COMSOL AB).

The NCAI-SNSPD structures are illuminated by 1550 nm monochromatic plane wave from sapphire substrate side by applying a port-boundary-pair, and Floquet periodic boundary condition is applied at the vertical sides of the models. The absorptance is computed by dividing the resistive heating in the absorbing NbN and Au segments by the total incoming power.

The threefold novelty of our present approach is 1) numerical investigation method of integrated optical systems in case of 2) off-axis illumination and 3) in conical-mounting. The strength of present numerical computations is that it is possible to provide an overview about the dual-angle dependent absorptance in the entire polar-azimuthal angle region, i.e. also above TIR, where measurements would require special methods for light in-coupling. One limitation of these models is that an infinite structure is supposed to be illuminated by a plane wave, i.e. the edge effects of meanders are neglected, as in references [11, 12]. Indeed, the short perpendicular connecting segments may have small optical effect taking into account their small fraction in comparison to the long parallel wires in the meandered NbN pattern.

In present theoretical work, φ indicates the polar angle measured from the surface normal of the substrate and γ refers to the azimuthal angle between the plane of polarized light incidence and the nanowires’ axes, as sown in Figs. 1(c, d). The parametric sweep method was used to realize angle dependent computations with different angular resolutions. First we have performed dual-angle dependent studies, when the polar- and azimuthal-angles were varied in φ = [0°, 85°] and γ = [0°, 90°] angle intervals with Δφ = Δγ = 5° resolution. This large-step sweep was used to select the φ - γ angles, which are capable of resulting in high absorption (Figs. 3 , 4 ).

 

Fig. 3 Dual-angle dependent absorptance of NbN stripes (a) in 200-nm-pitch design and (b) in 600-nm-pitch design, illuminated by p-polarized light from sapphire substrate side.

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Fig. 4 Dual-angle dependent absorptance of NbN stripes (a) in 200-nm-pitch design and (b) in 600-nm-pitch design, illuminated by s-polarized light from sapphire substrate side.

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Specific angular intervals as p-polarized illumination of the integrated patterns in S-orientation shown in Fig. 1(c), and s-polarized illumination of the integrated patterns in P-orientation shown in Fig. 1(d), were then investigated with a smaller-step sweep, i.e. with Δφ = 1° polar angle resolution. An important difference between the two configurations is that in former case the intensity modulation originated from illumination by oblique incident light and the polar angle dependent E-field projection are perpendicular to the periodic pattern, while in latter case the intensity modulation is along but the polar angle independent E-field is perpendicular to the periodic pattern.

Finally we performed higher resolution numerical computations by tuning the polar angle with Δφ = 0.05° resolution in intervals, where extrema are observed, i.e. in φ = [25°, 36°] interval in p-to-S configuration and in φ = [20°, 55°] interval in s-to-P configuration of 200-nm and 600-nm pitch designs (Fig. 5 ).

 

Fig. 5 The comparison of the optical responses in case of p-polarized illumination in S-orientation ($\gamma =90°$, closed symbols) and s-polarized illumination in P-orientation ($\gamma =0°$, open symbols) (a) of 200-nm-pitch design, and (b) in 600-nm-pitch design.

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The power ratio of the applied λ = 1550 nm wavelength illuminating homogeneous infrared light beams ensured that the normalized E-field was comparable in all investigated optical systems consisting of two unit cells of 200-nm-pitch and one unit cell of 600-nm-pitch integrated patterns (Figs. 6 , 7 ).

 

Fig. 6 (a-d) The normalized (E)-field distribution in case of p-polarized light illumination of NCAI-structures in S-orientation at planes shown in Figs. 7(e, f), at polar angles corresponding to (a) global maximum, and (b) local minimum on the absorptance of 200-nm-pitch design; (c) global minimum, and (d) local maximum on the absorptance of 600-nm-pitch design. The video recordings in (Media 1 and Media 2) about the time evolution of the (E)-field cross-sections taken in vertical plane presented by (e) model scheme indicate (f) efficient back-reflection at 27.4° polar angle (Media 1); while (g) a backward propagating Brewster-wave that promotes (E)-field concentration below each third cavity oscillating in-phase can be seen below the “boundary” at 29° polar angle (Media 2). (Notice that the apparent wavelength is doubled on the presented${[(E_{\text{x}}^2+E_{\text{y}}^2+E_{\text{z}}^2)]}^{1/2}$ quantity).

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Fig. 7 The normalized (E)-field distribution in case of s-polarized light illumination of NCAI-structures in P-orientation at (a) perpendicular incidence, and (b) at TIR in 200-nm-pitch design; (c) at perpendicular incidence, and (d) at TIR in 600-nm-pitch design. The (e-f) schematic drawings indicate unit cells in the planes where the near-field cross-sections were taken, and (g) model scheme shows the relationship between the schematic drawings 7(f) and 6(e).

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The comparison of the optical responses determined by TMM and FEM is presented in Supplemental information (Fig. 8 ).

 

Fig. 8 FEM optical responses (lines with symbols) in case of (a, b) p-polarized illumination of NCAI-SNSPDs in S-orientation ($\gamma =90°$, lines with closed symbols) and (c, d) s-polarized illumination of NCAI-SNSPDs in P-orientation ($\gamma =0°$, open symbols) in (a, c) 200-nm-pitch design, and in (b, d) 600-nm-pitch design compared to optical responses determined by TMM.

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3. Results and discussion

The computation results prove that a number of nano-optical effects are at play simultaneously in these integrated systems, some of them dominate at specific illumination conditions, thus resulting in a considerable local absorptance enhancement.

3.1. Phenomena governing the optical response

The phenomena governing the optical response and especially the absorptance in NbN wires are complex. In present theoretical work special emphasis was laid on looking for reflectance minima caused by Brewster- and plasmonic-phenomena occurring in case of p-polarized illumination, and for TIR phenomenon arising in case of s-polarized illumination, since these phenomena might be accompanied by absorptance maxima beneficial in SNSPD detection efficiency optimization.

The complementary TMM computations have shown that fingerprints of nano-photonical phenomena in different multilayers are observable with a dominance, which strongly depends on the composition of the integrated systems (Fig. 2).

(1) The rigorous coupled wave analyses (RCWA) of wire-grid polarizers predicts the appearance of equivalent Brewster-angle manifesting itself in a reflection minimum in case of p-polarized light illumination of periodic structures in both S- and P-orientation [4]. This equivalent Brewster-angle can be tuned by the geometrical parameters and by the orientation of periodic patterns. Based on complementary TMM calculations, a Brewster-phenomenon related reflectance minimum may appear at ~30° polar angle in optical systems containing NbN-NbNO_x-HSQ multilayers on sapphire, with film thicknesses according to those in present integrated patterns (Fig. 2(c), cyan curve).

(2) The complementary TMM calculations confirmed the predictions based on former RCWA studies about that TIR phenomenon results in zero absorptance at 34.85° in case of p-polarized illumination of NbN layers on sapphire (Fig. 2(a), light magenta curve), and in maximal absorptance in case of illumination by s-polarized light (Fig. 2(b), light magenta curve) [7, 10]. Similarly to our previous results about single-cavity-integrated structures, the presence of lossy Au and NbN layers causes attenuated TIR, when the nano-cavity-array integrated systems are illuminated by polarized light (Figs. 2(c, d)) [9, 10].

(3) The opposite sign and ratio of the bounding materials’ dielectric parameters fulfill the conditions of surface plasmon polaritons (SPP) excitation at λ = 1550 nm wavelength at every dielectric-metal interface in the studied integrated systems, except the NbN-NbNO_x bounding media. TMM computations predict propagating plasmonic modes related local maxima on the gold absorptance at 35.1° polar angle, when p-polarized light illuminates the integrated systems consisting of 60 nm thick horizontal gold reflectors (Fig. 2(a), purple, pink and wine curves). The transmittance indicates a cut-off above TIR in case of s-polarization, while in case of p-polarization a finite transmitted signal appears in all multilayers consisting of 60 nm thick Au layer and in the integrated systems due to SPP excitation (Figs. 2(e)-to-2(f), green, dark gray and olive curves).

(4) Although the geometrical size parameters are strongly subwavelength in our detector designs, the plasmonic modes are capable of resulting in localization and transportation of the EM-field, which is a widely applied principle also at telecom wavelength [16, 17]. The film stacks (B)-(A)-(B) compose a Metal-Insulator-Metal (MIM) pattern, where each HSQ layer acts as an embedded 100 nm wide dielectric cavity, array of stacks (B) composed of 100 or 500 nm wide vertical gold segments and the 60 nm thick horizontal gold reflectors close these nano-cavities, thus finally the entire complex integrated pattern might be considered as an MIM nano-cavity-array. Dominantly nano-plasmonic waveguide modes may exhibit cavity resonances in MIM patterns of similar geometry, which will be taken into account, when their role in absorptance maximization is analyzed in NCAI-SNSPDs [16–24]. Further FEM computations confirmed that the l_OC = 220 nm long HSQ layer acts as a perfect quarter-wavelength cavity for the nano-plasmonic modes supported by each MIM unit in the integrated patterns [11].

3.2. Dual-angle dependent optical response of NCAI-SNSPDs

Figures 3 and 4 indicates the most important result of present computations, namely different optimum direction exists for illumination each of the integrated patterns, since there are extrema on the dual-angle dependent absorptance of the NbN stripes, which fundamentally depend also on the device composition.

In case of p-polarized light illumination of nano-cavity-array integrated systems the absorptance inside the NbN stripes is larger, when they are in S-orientation, i.e. the plane of incidence is rotated to $\gamma =90°$ (Fig. 3, “S-orientation”).

Figure 3(a) presents that in 200-nm-pitch design ~95% NbN absorptance is observable in case of perpendicular incidence (φ = 0°). The absorptance is almost perfect in a wide polar angle interval, when the integrated structure is in S-orientation, and exhibits a slow variation with the polar angle at any azimuthal angle. More rapid decrease is observable, when the azimuthal angle is decreased at fixed polar angle.

Figure 3(b) presents that in 600-nm-pitch design the main characteristic of the dual-angle dependent NbN absorptance is enhancement by increasing either of the polar or azimuthal angles, but there is a wide angular interval, where the simultaneous φ−γ variation causes significant local changes. According to this, the absorptance exhibits pronounced polar angle dependence also in S-orientation, which is accompanied by appearance of local extrema, that we describe later in Section 3.3.

In case of s-polarized light illumination of nano-cavity-array integrated optical systems the NbN absorptance reaches the largest values through all polar angles, when the wires are in P-orientation, i.e. the plane of incidence is rotated to $\gamma =0°$ (Fig. 4, “P-orientation”).

Figure 4(a) presents that in 200-nm-pitch design the largest absorptance is observed in case of perpendicular incidence (φ = 0°), and then the absorptance monotonously decreases by increasing either the polar or the azimuthal angles, but with larger gradient along the azimuthal axes.

Figure 4(b) presents that in 600-nm-pitch design the absorptance of the NbN pattern exhibits a global maximum at one specific polar angle through a wide azimuthal angular interval neighboring the P-orientation. Comparison of Figs. 3(b) and 4(b) shows that the dual-angle variation causes a modulation in similar φ−γ angular interval, where sudden changes are observable in case of p-polarized illumination. The absorptance modification is less significant and is restricted into narrower angular interval in case of s-polarized illumination.

Comparison of these nano-cavity-array integrated systems with single-cavity-integrated structure presented in our previous papers [9, 10] proves that in presence of vertical gold segments the E-field oscillation perpendicular to the vertical Au segments is preferred (i.e. p-to-S and s-to-P configurations) rather than E-field oscillation parallel to the NbN stripes. This indicates that the vertical gold segments play crucial role in absorptance enhancement.

The explanation of the preferential p-polarized illumination in S-orientation and s-polarized illumination in P-orientation is that two nano-optical effects contribute to E-field enhancement in NCAI-SNSPDs in these configurations with commensurate efficiency [12]. Both NCAI-SNSPD structures might be considered as two intertwined gratings made of gold and NbN, where the non-resonant concentration of the E-field expelled from better metal, which is gold, results in enhanced absorptance in NbN segments. In addition to this, cavity-resonant nano-plasmonic modes might be excited with high efficiency inside the quarter-wavelength nano-cavities, when the E-field oscillates perpendicularly to the vertical gold-segments. As a result, resonant absorptance phenomena might occur [12, 18–24], which are described in Sections 3.3. and 3.4.

If p- and s-polarized light illumination related phenomena would only play significant role in the appearance of extrema, than the illumination angle dependence would be analogous in the two systems. The present results prove that the pitch has fundamental impact on the absorptance of these integrated systems (Figs. 3(a)-to-3(b) and Figs. 4(a)-to-4(b)).

It is described in the literature that excitation of coupled modes is capable of resulting in resonant absorption, when the nano-cavity-array made of noble-metal has strongly subwavelength period [19, 20, 22, 23]. These coupled nano-cavity resonances explain the large and illumination angle independent absorptance in 200-nm-pitch design.

In stand-alone MIM nano-cavities the coupling of propagating to cavity-resonant plasmonic modes manifests itself in pronounced reflection dips at specific polar angles [18]. The stronger polar angle dependence in the 600-nm-pitch design is related to this phenomenon, and indicates that the entire in-coupling efficiency strongly depends on the inter-cavity distance in arrays. The larger separation of the nano-cavities in 600-nm-pitch design by wider Au segments enable smaller NbN absorption in this optical system, in addition to the larger gold absorptance.

3.3. Polar angle dependent optical responses of NCAI-SNSPDs illuminated at optimum azimuthal orientation

Figure 5 presents the results of higher resolution computations on polarized illumination, showing that the main advantage of the 200-nm-pitch and 600-nm-pitch nano-cavity-array integrated NbN patterns in comparison to the single-cavity-integrated design is the fundamental perturbation of the reflectance due to subwavelength vertical gold segments’ inclusion [9, 10]. This manifests itself in reflection suppression and in appearance of pronounced reflectance minima. However, the optical response exhibits strong dependence on the periodicity-to-wavelength ratio, as well as on the type of polarization.

Figure 5(a) presents the comparison of polar angle dependent FEM optical responses computed, when 200 nm periodic NCAI-structures are illuminated by p-polarized light in S-orientation (closed symbols, p-to-S configuration), and when the same integrated structure is illuminated by s-polarized light in P-orientation (open symbols, s-to-P configuration).

The detailed inspection of p-to-S configuration (Fig. 5(a), closed symbols) indicates that the peculiarity of 200-nm-pitch design is the almost polar angle independent perfect absorptance. The high resolution computations have shown that the NbN absorptance increases from 94.6% to 95.7%, when the polar angle is tuned from 0° to 34.35°, than a small local minimum appears at 35.1°. There is a minimum-maximum pair on the absorptance of the gold segments at these orientations, while on the transmittance a small local maximum appears at 35.05°. The overall reflectance is suppressed through the entire polar angle interval, and shows small local minimum-maximum pair at 34.95° and 35.2° polar angles.

The NbN absorptance decreases monotonously and more rapidly in 200-nm-pitch design, when the integrated structures are illuminated by oblique incident s-polarized light in P-orientation (Fig. 5(a), open symbols). The high resolution computations did not provide evidence of any local extrema neither on gold absorptance, nor on transmittance. The reflectance monotonously increases through the entire polar angle interval in s-to-P configuration of 200-nm-pitch design.

Figure 5(b) presents the comparison of polar angle dependent FEM optical responses computed, when 600 nm periodic NCAI-structures are illuminated by p-polarized light in S-orientation and by s-polarized light in P-orientation.

The calculations on 600-nm-pitch integrated structures illuminated by p-polarized light in S-orientation (Fig. 5(b), closed symbols, p-to-S configuration) prove the advantage of the nano-cavity-arrays’ integration. The 46.5% NbN absorptance observed at perpendicular incidence is commensurate with the absorptance in 200-nm-pitch single-cavity-integrated structure [9, 10] in contempt of the small fill-factor. A well defined local NbN absorptance maximum of 37.2% is observed at 29°, which is surrounded by global and local minima at 27.4° and 30.95° polar angles. Finally a global NbN absorptance maximum of 69% appears at 77° tilting. The gold absorptance shows much smaller modulation with a global maximum, which is observable at 27.9° polar angle slightly back-shifted with respect to the NbN absorptance maximum. Surprisingly, the Au absorptance indicates only a weak propagating plasmonic modes related enhancement at 35.1°, while the transmittance is slightly enhanced again at 35.05°, similarly to 200-nm-pitch design. The reflectance shows complement modulation characteristics at polar angles corresponding to the extrema on NbN absorptance.

The high resolution calculations on 600-nm-pitch integrated structures illuminated by s-polarized light in P-orientation (Fig. 5(b), open symbols, s-to-P configuration) show a well defined global maximum of 52.3% at 33.55° polar angle, which maximum overrides the absorptance observable at perpendicular incidence, proving that tilting is advantageous. The gold absorptance and the transmittance do not indicate any extrema, while the reflectance shows complementary modulation characteristics.

The polar angle values corresponding to the shallow optical response modulations in case of p-polarized illumination of 200-nm-pitch design in S-orientation are in accordance with the TMM computations and indicate that the origin of these perturbations are TIR and propagating surface plasmon modes excitation phenomena (Fig. 5(a), closed symbols, Figs. 2(a, c, e)). The local maximum is in accordance with our previous computations proving that in case of p-polarized illumination the TIR related minimum is transformed into a maximum in presence of quarter-wavelength cavity (Fig. 2(a)) [10]. The small transmittance maximum is caused by the small EM-intensity out-coupling via propagating plasmonic modes into the near-field above the horizontal 60 nm thick Au segment, in accordance with the TMM computations (Fig. 5(a), closed green symbols and Fig. 2(e) dark grey curve).

Our present observations help to understand, how the strongly subwavelength periodicity of the 200-nm-pitch design causes the lack of optical far-field diffracted orders. The literature about subwavelength gratings describe that coupled modes in nano-cavity-arrays may be insensitive to excitation conditions in special geometries, which results in large bandwidth in polar angle.

If the grating is deep enough so that resonance modes might be excited, a narrow band may be absorbed over the whole range of polar angles [19, 20, 22, 23]. The observed almost perfect absorptance is in accordance with this, and indicates the effect of resonant nano-plasmonic modes excitation on subwavelength array of quarter-wavelength MIM cavities with large, almost polar angle independent strength. The near-field evidence of the cavity resonant modes will be presented in Section 3.4.

According to our present results, excitation of localized nano-plasmonic modes in the subwavelength cavities exhibits stronger polar angle dependence in case of s-polarized illumination of 200-nm-pitch NCAI-structures in P-orientation, even though the E-field projection, which is perpendicular to the vertical Au segments, is polar angle independent. We plan to investigate in more details the reason of the more pronounced tilting-related effects in case of s-polarized illumination of the 200-nm-pitch design.

The polar angle values corresponding to strong optical response modulation in case of p-polarized illumination of 600-nm-pitch design in S-orientation indicate relation to Brewster-phenomenon (Fig. 5(b), closed symbols and Fig. 2(c), cyan curve). Complementary high-resolution FEM computations have shown that these extrema exist only when the integrated pattern is in S-orientation, proving that the intensity modulation caused by oblique incidence has to be perpendicular to the stripes.

Additionally, there are no similar extrema at the Brewster-angle in 200-nm-pitch design, indicating that the p-polarized illumination of NCAI-structures in S-orientation is a required, but not a sufficient condition. Previous studies also indicated that the periodicity has a key role in the appearance of such extrema on NbN absorptance, and the ~600 nm periodicity region seems to be special [12].

To inspect the phenomena observed in case of 600-nm-pitch design, we made the following considerations: the nano-optical cavities surrounded by gold-segments enhance the EM-field intensity around the NbN stripes most effectively, when the oblique incident light illumination results in collective resonance oscillation on the highest number of them. The condition is to maintain such L light-path-length difference between the beams entering subsequent series of k-cavities, which equals to a multiple of the light wavelength in sapphire medium. The primary condition of collective oscillation on nano-cavity-arrays regarding the $\varphi$ polar angle is:

The characteristic spatial frequency corresponding to specular reflection appears in the diffracted optical signal always, and indicates a polar angle dependent “photonic modulation” on the EM-field amplitude:

When this photonic modulation is perpendicular to the pattern, this signal results in novel Fourier spectral line corresponding to an amplitude modulated “adaptive grating” with periodicity:

The global NbN absorptance minimum and the nearby Au absorptance maximum appear at $\varphi =27.4°$ and 27.9° polar angles in 600-nm-pitch design, where the Rayleigh-condition is fulfilled for the periodicity corresponding to this adaptive grating [26]:

The grating equation for $\lambda /n_{\text{sapphire}}$ periodic adaptive grating predicts zero order diffraction at φ‘ = −φ polar angle. According to this, on the near-field cross-sections at 27.4° polar angle wave-fronts tilted at φ‘ angle are observable with considerable intensity, indicating that this intensity is missing from absorption, as it is presented in Figs. 6(c, f) and in (Media 1) in Section 3.4.

The peculiarity of the local maximum observed at $\varphi =29°$in 600-nm-pitch design is that the adaptive grating periodicity fulfills the condition:

The polar angle value, where the global maximum appears on NbN absorptance in case of s-polarized illumination of 600-nm-pitch design in P-orientation (Fig. 5(b), open symbols) corresponds to TIR phenomenon shown in Figs. 2(b, d, f). This observation indicates that TIR phenomenon is capable of resulting in maximal absorptance in integrated systems, similarly to previous results about s-polarized illumination of bare NbN patterns in the literature [7]. The weak polar angle dependence in case of s-polarized illumination of the 600-nm-pitch design in P-orientation is in accordance with the E-field projection independence on tilting.

Based on these observations one can conclude that p-polarized illumination of 200-nm-pitch integrated structures in S-orientation is optically the most efficient in the entire polar angle interval, while in 600-nm-pitch design larger absorptance might be reached in φ = [0°-44.55°] interval, when the integrated pattern is illuminated in P-orientation by s-polarized beams (Fig. 5, closed-to-open symbols). Although there is a global maximum on the NbN absorptance in case of p-polarized illumination of 600-nm-pitch NCAI-SNSPD in S-orientation, it is difficult to reach such large polar angles as 77° experimentally. The comparison of Figs. 5(a) and 5(b) indicates that the absorptance in gold grating with 600 nm periodicity is larger than in 200-nm-pitch design according to the larger 63.3% fraction of gold in vertical segments. This shows that it is more advantageous to use gold-grating with smaller pitch in order to avoid this loss, and to have a local-field enhancement around each NbN stripes of the meandered pattern.

The extrema on FEM optical signals reveal to the key role of the quarter wavelength nano-cavity-array, which was inspected further based on the near-field pictures.

3.4. Near field-distribution in NCAI-SNSPDs

Figures 6 and 7 present near-field distribution at vertical planes indicated on the schematic drawings in Figs. 6(e) and 7(e,g). The E-field images taken across three cavities according to Fig. 6(e) indicate simultaneously the incoming and the back-reflected waves directly below the cavities, while in the area below “boundary”, corresponding to the boundary pair in our FEM models, the reflected E-field is shown separately. Detailed investigation of the near-field distribution and its time evolution shows that characteristic intensity changes occur on the nano-scale around the absorbing NbN segments, when the angle of incidence is varied.

The normalized E-field presented in Figs. 6(a-d) shows asymmetrical field concentration at the NbN-Au-sapphire edges in case of p-polarized illumination of integrated structures in S-orientation, while the E-field distribution is symmetrical in case of s-polarized illumination of integrated structures in P-orientation, as shown in Figs. 7(a-d). Interestingly, significant penetration is observable into the NbNO_x layer at the NbNO_x-Au boundary at all maxima.

In case of p-polarized illumination the E-field antinodes overlap with NbN segments at any polar angle in 200-nm-pitch design indicating nano-cavity resonances, and the highest near-field intensity is observable at the local maximum in Fig. 6(a). When the condition of propagating plasmonic modes excitation is fulfilled at φ = 35.1°, the E-field enhancement above the gold-air interface indicates out-coupling from the nano-cavity-array in Fig. 6(b).

In 600-nm-pitch design illuminated by p-polarized light in S-orientation the detailed investigation of the normalized E-field shows more strongly polar angle dependent distribution manifesting itself in a pronounced asymmetry across the cavities.

The normalized E-field is enhanced below the gold segments rather than around NbN stripes at 27.4° corresponding to the global NbN absorptance minimum, shown in Fig. 6(c). The time evolution of the E-field shows intense specular reflection at the global minimum, i.e. wave fronts tilted at φ‘ = −27.4° appear below “boundary”, see Fig. 6(f) (Media 1).

At 29° polar angle, which corresponds to the local NbN absorptance maximum, the antinodes of the E-field result in relatively intense E-field concentration around the NbN-stripes at the local absorptance maximum, shown in Fig. 6(d). On the time evolution of the E-field a synchronous oscillation is observable in each third nano-cavity. In comparison to the global minimum, instead of reflected waves, a plane-wave propagating backward with wave-fronts perpendicular to sapphire interface is identifiable below “boundary”, see Fig. 6(g) (Media 2). As the ~900 nm wavelength of this plane wave is larger than the light wavelength in sapphire, these results reveal to the existence of a Brewster-wave [27, 28]. This surface wave plays crucial role in E-field concentration below the nano-cavities at the polar angle corresponding to the local maximum.

Comparing Figs. 6 and 7 it can be concluded that the main difference in E-field distribution in case of s-polarized light illumination of the integrated patterns in P-orientation in comparison to the case of p-polarized illumination of analogous patterns in S-orientation is that there is no asymmetry horizontally along the cavities in latter case, in accordance with the parallelism of the intensity modulation to the stripes at γ = 0° azimuthal orientation.

The E-field antinodes properly overlap with NbN segments at any polar angle in 200-nm-pitch design, and the highest intensity is observable at perpendicular incidence shown in Fig. 7(a). The local E-field intensity monotonously decreases with the polar angle and there is no additional enhancement at the orientation corresponding to TIR presented in Fig. 7(b).

In case of 600-nm-pitch design the E-field concentration in quarter-wavelength cavities is maximal in s-to-P configuration at the polar angle corresponding to TIR phenomenon, as it can be seen by comparing Figs. 7(c) and 7(d).

These near-field observations prove that the quarter-wavelength MIM cavities ensure the E-field concentration at their entrance due to the $\pi$ phase-shift maintained by standing plasmonic waves, and indicate that additional role of vertical gold-segments is to promote the E-field penetration into NbN stripes at their boundaries [11, 12, 16].

The near-field phenomena observed in 200-nm-pitch design confirm that the strong absorptance is due to polar angle independently large E-field concentration around the lossy thin NbN stripes lying at the entrance of quarter-wavelength nano-cavities in both of p-to-S and s-to-P configurations [18, 19, 22].

In case of 600-nm-pitch design the near-field study indicates that polar angle dependent nano-optical phenomena might play crucial role, when the cavities are weakly coupled. It is proven that a backward propagating Brewster-wave exists at the local maximum observed during p-polarized light illumination, which promotes the E-field concentration at the entrance of the nano-cavities. According to our knowledge this is the first demonstration of the possibility to attain absorptance enhancement in lossy thin metal layers between two dielectric media having different refractive indices via Brewster-wave excitation. The enhancement in E-field intensity at the tilting corresponding to TIR in case of s-polarized illumination indicates the pronounced role of local phase phenomena in larger periodic integrated patterns (Figs. 7(b)-to-(d)). The highest local intensities are reached at the maxima in 600-nm-pitch design, so the larger absorptance attainable in 200-nm-pitch design is obviously due to the larger fill-factor (Figs. 6(a,b)-to-(c,d) and 7(a,b)-to-(c,d)).

4. Conclusions

In present work we have shown that the method of synchronous polar-azimuthal orientation is applicable to enhance the near-field intensity around the NbN stripes and to maximize their absorptance inside different integrated structures consisting of subwavelength nano-cavity-arrays. The presented dual-angle dependent results indicate that the E-field redistribution on intertwined NbN-Au bi-grating and the coupling of the propagating to cavity-resonant nano-plasmonic modes can be used in SNSPDs absorptance maximization, since the propagating EM-field can be transported directly to the NbN stripes under specific illumination conditions in specific device designs.

The non-resonant E-field concentration and the excitation of collective resonances on subwavelength 200 nm periodic arrays of quarter-wavelength MIM cavities results in almost illumination angle independent perfect absorptance. The 200-nm-pitch design, or perhaps a future improvement to it, may be useful in quantum-information-processing applications, where high detection efficiency is required, but device reset time is not important.

The E-field concentration in quarter-wavelength nano-cavities arrayed in larger 600 nm periodic pattern exhibits pronounced illumination angle dependence caused by weaker coupling between the cavities. The propagating waves appearing at the Brewster-angle in case of p-polarized illumination and the phase shift at TIR during s-polarized illumination is capable of resulting in absorptance enhancement in this design. In the latter case it is possible to obtain an absorptance larger than 50%. With the 600-nm-pitch design it is possible to attain reduced wire length and thus reduced reset time or expanded device area, while maintaining a significantly improved absorptance at the optimum polar angle. This design would be of more general use in situations, where coupling efficiency, speed, and overall detection efficiency are all equally important variables.

We conclude that it is possible to attain considerably higher absorptance in nano-cavity-integrated SNSPDs by optimizing both the polar and azimuthal angles of the incident polarized light. The developed method is general, namely efficiency of all detecting devices might be maximized by selecting appropriate polar-azimuthal illumination directions.

Supplemental information

Comparing the FEM and TMM results it can be concluded that at perpendicular incidence FEM computations predict ~2 and ~3-times larger total absorptance than TMM in 200-nm and 600-nm-pitch designs, as it is presented in Fig. 8. The course of FEM optical responses is significantly different in case of p-polarized illumination of NCAI-SNSPDs in S-orientation in comparison to TMM signals, shown in Figs. 8(a, b). Although the local extrema determined by FEM and TMM are in agreement in 200-nm-pitch design, only FEM predicts the pronounced local perturbations of the optical signals in 600-nm-pitch design. The optical responses determined by FEM in case of s-polarized illumination of NCAI-SNSPDs in P-orientation and the TMM signals are similar, but the absorptances computed by TMM are significantly smaller through the entire polar angle interval, as it is presented in Figs. 8(c, d). Only FEM predicts local maximum in 600-nm-pitch design at TIR.