In this paper, we propose a novel, frequency- and angularly- broadband approach to achieve absorption rate enhancement in high-index dielectric nanostructures through the engineering of non-radiative anapole modes. We employ multipolar decomposition of numerically computed current distributions and analyze the far-field scattering power of multipole moments. By leveraging the destructive interference of electric dipole and toroidal dipole moments, we design non-radiating anapole modes and demonstrate significantly enhanced absorbed power in silicon and germanium nanostructures. We demonstrate wide wavelength tunability of the anapole-driven peak absorption enhancement for nano-disks and square nano-pixel geometries, which can be conveniently fabricated with current lithography. Finally, by combining nano-disks and nano-pixels of different sizes into functional surface units, we design nanostructured arrays with enhanced bandwidth and absorption rates that can be useful for the engineering of broadband semiconductor photodetectors driven by controllable anapole responses.
© 2016 Optical Society of America
Frequency-domain non-radiating sources (NRs) are current distributions that do not radiate electromagnetic fields outside their volume (i.e., geometrical support) at one or more desired frequencies. Although perfectly non-radiating sources cannot be realized in practice due to losses and dispersion effects, one can always achieve essentially non-radiating ones when the radiated energy is smaller than a threshold parameter e(ω) that characterizes the NRs . When induced by an external wave, non-radiating current configurations produce strongly confined near-fields that are exponentially localized inside the source region [1,2], providing opportunities to enhance light-matter interaction. In this context, induced NRs give rise to non-scattering potentials that strongly reduce the intensity of the scattered field everywhere outside their geometrical support for a particular incident plane wave. It can be rigorously proved that, for any NRs and for an arbitrary incident wave it is always possible to construct a non-scattering potential that generates zero-radiated power outside the source region [1–4].
One powerful approach to engineer the current distributions that produce essentially non-radiative modes in externally-driven dielectric nanostructures is based on the multipolar decomposition of induced currents that include the toroidal multipole moments in Cartesian coordinates [5–7]. In particular, this type of multipolar expansion [5,6] clearly identifies the possible mechanisms that lead to the far-field scattered power by nanostructures [8–14]. Toroidal moments were first proposed by Zel’dovich to explain parity symmetry violation in weak interactions , and extensively studied by Dubovik [16–19]. They form a family of multipole moments with distinctive parity-time symmetry properties other than the more familiar electric and magnetic multipole moments [5,6], and can be pictured as radiating fields generated by a solenoid bent into a torus [17–19]. Current sources on a torus, known as poloidal currents, have been shown to result in no radiation outside the volume of the torus [18–21]. Toroidal moments are well-understood in condensed matter physics where they underpin the rich physics of the magneto-electric effect and of ferrotoroids, which are materials characterized by magnetic vortices associated to the curl of the induced magnetization [18–20,22] resulting in intriguing non-reciprocal responses . Recent works [8–14,24,25] have demonstrated that significant toroidal responses can be achieved at optical frequency in purely dielectric nanostructures with large enough refractive index when excited under an external plane wave illumination [8–10,13].
The excitation of toroidal modes in nanostructures and metamaterials is highly relevant to the engineering of a type of essentially non-radiating source, known as anapole, that results from the destructive interference of the toroidal and electric dipole moments in the far-field zone. In fact, toroidal dipole and electric dipole modes have been shown to support identical far-field radiation patterns but with an opposite phase [5,6,9]. As a result, if they do not overlap in frequency with additional electromagnetic multipoles, they destructively interfere in the far-field [5,8] at all angles, corresponding to the excitation of non-radiative anapole modes.
The engineering of essentially non-radiating anapole modes in high-index dielectric nanostructures using rigorous multipolar expansion methods is a promising approach to increase light-matter interaction and hence enhance absorbed power. Compared to conventional approaches [26,27], the excitation of pure anapole modes in nanostructures provide angularly and frequency broadband responses with minimal scattering and enhanced near-fields that are very compelling for the engineering of active optical devices such as photodetectors, especially when combined to the widespread silicon and germanium material platforms that have a large refractive index [28,29].
In this paper, by numerically performing multipolar decomposition of induced currents in silicon (Si) and germanium (Ge) nanostructures with realistic dispersion data , we design anapole-driven multi-spectral absorption rate enhancement in a single dielectric nano-pixels and broadband arrays, which can be conveniently fabricated with current lithography. Following this approach, we show efficient excitation of anapole modes under plane wave excitation at oblique and large angles, and demonstrate the robustness of enhanced absorption with respect to the perturbation introduced by low-index coating layers that correspond to transparent contacts. Finally, by combining Si and Ge nano-disks and nano-pixels of different sizes into functional surface units, we design nanostructured arrays with enhanced absorption rates over a broad bandwidth, which can be useful for the engineering of broadband semiconductor photodetectors driven by tunable anapole responses.
2. Multipolar decomposition and anapole modes excitation
In this section, we discuss representative examples of current distributions induced in high-index nanostructures by an external plane wave and show how the far-field scattering power can be decomposed into individual multipole moments and their interactions. In particular, we discuss the excitation of essentially non-radiating anapole modes not only in high-index cylindrical nano-disks , but also in square nano-pixels. The multipole moments are numerically computed from the internal electric field distributions at each frequency [5,6,31] using a three-dimensional finite-difference time-domain method .
In Fig. 1 (a) we show the geometry of a Si nano-disk characterized by a diameter D and height H. A linearly polarized plane wave is used to excite this structure. Considering as an example a structure with D = 350nm and H = 60nm, in Figs. 1(b) and 1(c) we show the electric and magnetic field distributions normalized with respect to the incident one (i.e., field enhancement distributions) for a resonant anapole mode excited at 710nm. The arrows in the field plots indicate the field directions for the electric and magnetic fields. We notice that the electric field distribution in Fig. 1(b) is typical for the anapole mode, where almost all of the field amplitude circulates in the plane of the disk and it is confined inside it. The magnetic field profile shown in Fig. 1(c) is also strongly confined in the volume of the square nano-pixel, where it forms a vortex in the middle as a result of the rotation of the induced magnetic field, giving rise to a toroidal dipole moment in the plane of the nano-disk [5,6,19,20]. Based on the computed field distributions inside the structure, we can immediately obtain the induced current distribution at each frequency using the relation :Table 1 , where p, m, T, Qαβ, Mαβ indicate the electric dipole, magnetic dipole, toroidal dipole, electric quadrupole, and magnetic quadrupole moments, respectively. The indices, α and β indicate the Cartesian axes x, y, and, z. Based on the knowledge of the radiating multipole moments we can calculate the corresponding far-field scattered power contributions for each multipole moment and their interactions .
In Fig. 1(d) we plot the computed contributions to the far-field scattered power of each radiating multipole moment in the representative nano-disk geometry with D = 350nm and H = 60nm. We observe in Fig. 1(d) that all five multipole moments contribute to the far-field scattered power at shorter wavelengths, while only the electric dipole moment dominates at longer wavelengths. Notice however that for this particle geometry around 700nm the electric dipole and toroidal dipole moments are the only dominating multipole contributions and have almost identical values. Since they have opposite phases they will destructively interfere and cancel each other in the far-field region, giving rise to an anapole mode. On the contrary, around 500nm only partial cancellation between the electric and toroidal dipole can occur due to the perturbation of the additional multipole moments, which prevent the almost complete radiationless anapole condition to be satisfied. In particular, at short wavelengths the magnetic quadrupole moment is responsible for a non-zero overall scattered power near 500nm, as can be appreciated in Fig. 1(e). The total scattering efficiencies normalized to the nano-disk’s geometrical cross-section are shown in Fig. 1(e) and confirm the essentially non-radiating nature of the identified minimum at 700nm.
It is also worth noticing that, although the anapole mode is not completely radiationless in practice, this scattering cancellation effect is angularly broadband as demonstrated by the pronounced minimum in the angularly-integrated scattering cross-section shown in Fig. 1(e) around 700nm. It is also important to realize that this effect is physically distinct from the traditional scattering cancellation mechanism where the electric and magnetic dipole moments interfere destructively and cancel one another exactly only in the backward direction [33–36]. The anapole scattering cancellation is also different from Kerker’s null-scattering condition , which requires a a magnetic response of the material (μr ≠ 1). In contrast, the isotropic anapole cancellation occurs in purely dielectric media due to the destructive interference of the induced electric and toroidal dipoles that share radiation pattern with opposite phase and identical angular distribution [9,13].
From Fig. 1, we can also appreciate that the toroidal dipole moment is induced by the circulation of the magnetic field inside nanostructures, i.e., the formation of magnetic field vortex lies in the YZ-plane [Fig. 1 (c)].
We remark that compared to the work published in , where anapole excitation on nano-disk structures was demonstrated using the discrete dipole approximation (DDA) method, our simulation results are obtained using the finite-difference time-domain (FDTD) method which guarantees a better accuracy in dealing with high-index nanostructures. Moreover, the results in Fig. 1 also provide the additional information of the far-field scattered power reconstructed by interfering the first five multipoles. These results are in excellent agreement with the scattering efficiency of the nano-disk, as we show in Figs. 1(d) and 1(e).
Therefore, for a given nanostructure, it is the aspect ratio H/D rather than its geometrical shape that enables the formation of magnetic field vortices and the accompanying essentially non-radiating modes. Indeed, while the study of anapoles in Si nano-disks was pioneered in , we found that, for high-index dielectric nanostructures, we can excite anapoles in a variety of different geometries. As an example, we will discuss here a dielectric nano-disk with a square cross section, referred to as a square nano-pixel, and we will investigate its anapole modes.
Figure 2(a) shows the geometry of the investigated square nano-pixel. The normally incident plane wave has its electric field oriented along one of the sides of the square nano-pixel. In this particular example, we consider a square nano-pixel with the same height H and side length D as in the case of the nano-disk shown in Fig. 1. The multipolar decomposition in Fig. 2(b) demonstrates clearly the possibility of exciting high-quality anapole modes also in the square nano-pixel geometry. This follows from the fact that the far-field scattering power of both the electric and the toroidal dipole moments have a similar intensity around 750nm [Fig. 2(b)], resulting in the almost complete destructive interference that characterizes the anapole mode. The internal electric field distributions corresponding to this anapole conditions are shown in Figs. 2(c) and 2(d).
In the next section, we will discuss how to leverage the excitation of almost complete anapole modes in dielectric nanostructures with realistic dispersion data in order to achieve absorption rate enhancement, and we will address the important role played by the relevant geometric parameters on the performance of Si and Ge nanostructures with anapole-induced absorption rate enhancement.
3. Effect of Geometry, incidence angle, dielectric coating and material dispersion
We will now systematically study the effect of the geometry of the nanostructures considering Si and Ge nano-disks and square nano-pixels, and we will extend our analysis to include the relevant case of oblique incidence and low-index coating of the nanostructures with a transparent contact material. In what follows we define the absorption rate enhancement by comparing the power absorbed by a single nano-disk or nano-pixel to that absorbed by a thin-film with the same thickness and excited volume. We then define the absorption rate enhancement as:31]:
3.1 Effect of geometrical aspect ratio
Let us now focus on how the size in the transverse (H) and lateral (D) directions affects the anapole mode and anapole-induced absorption rate enhancement conditions. We will first consider Si square nano-pixels and its multipolar decomposition analysis. Then we extend our findings to Si and Ge nano-disks as well as square geometry nano-pixels.
In Fig. 3 we compare square nano-pixels with increasing side length for two representative cases and for a fixed value of the height H = 100nm. Each row in Fig. 3 is associated with a different side length D, with values of 500nm, and 700nm, respectively. In Fig. 3(a) and 3(c) we notice that the far-field scattering power at the shorter wavelengths is contributed by several multipole moments. Therefore, good quality anapole modes can only be excited at the intermediate wavelengths just before the electric dipole modes dominate the scattering spectra. As D increases, all multipole moments redshift and separate in wavelength, and the wavelength at which the anapole modes occur (near 1100nm and 1350nm) also increases. However, we can notice in 3(a) and 3(c) that the power scattered by the magnetic quadrupole moments contributes to the far-field, making the anapole cancellation incomplete. We plot our results for the scattering efficiency spectra in Fig. 3(b) and 3(d), which consistently show a non-zero scattering efficiency at the anapole condition. The results of the absorption rate enhancement, plotted in Fig. 3(b) and 3(d), clearly demonstrate that when the anapole condition is met the absorption rates are maximized. Notice also that additional wavelengths with incomplete anapole responses exist and result in local minima of the scattering efficiency, which are responsible for the observed secondary peaks in the absorption rate enhancement spectra. We found that these additional effects can be quite significant. In particular, in Fig. 3(d), the incomplete anapole response excited between 750nm and 1000nm drives an absorption rate enhancement of approximately a factor of 2. This provides interesting opportunities for the engineering of anapole-induced multi-spectral absorption rate enhancement.
We established before that by increasing D we can spectrally separate the anapole condition from other multipoles. We will show now that by increasing H we can approach them closer to each other. In Fig. 4 we summarize results that show the effect of increasing the square nano-pixel height H for two representative structures with fixed side length D = 400nm. Each row in Fig. 4 is associated with a different height H, with values of 80nm and 120nm, respectively. Similar to the situation where we increased D, we show a redshift of the anapole mode condition along with the other multipole moments when increasing the height of a Si square nano-pixel. We notice that as the height of the square nano-pixel increases, the magnetic dipole moment redshifts more than the anapole mode [Figs. 4(a) and 4(c)], and its long-wavelength tail overlaps partially with the anapole condition [Fig. 4(d)]. We also observed in our analysis that when the height of the square nano-pixel becomes too large compared to its side length, we lose the anapole mode condition due to the significant overlap with the magnetic dipole contribution around 1000nm in Fig. 4(d). As a result, the absorption enhancement is significantly reduced [Fig. 4(d)].
From our numerical study on the effect of the nanostructure’s geometry on the anapole mode formation we can conclude that when the aspect ratio H/D is roughly 0.2, the separation between the anapole and other multipole moments is large enough to result in significant anapole-driven absorption rate enhancement in Si square nano-pixels. Deviations from this condition show that incomplete anapole mode formation leads to progressively more radiated power away from the nanostructure and strongly reduced absorption rates.
We will now show that the behavior discussed so far in relation to the Si nano-disks and square nano-pixels is very general and can be achieved in other high-index dielectric nanostructures. In particular, we will consider here Ge as an example, since it is widely used as the absorbing material for photodetectors at telecommunication wavelengths [28,29] and has a large refractive index . As it can be appreciated already within the analytical Mie theory for spheres , the effect of increasing the refractive index generally leads to further spectral separation of the different electromagnetic multipole moments. This favors the design of the scattering cancellation condition for anapoles in Ge nanostructures compared to Si. The results in Figs. 5(a) and 5(b) that show the multipolar decompositions of a Ge nano-disk with D = 650nm and H = 100nm, and of a square nano-pixel with D = 500nm, H = 100nm. These data demonstrate that the anapole condition is met for wavelengths larger than 1000nm in Ge, with only a minimum background contribution coming from the additional multipoles [Figs. 5(a) and 5(b)].
In Figs. 5(c) and 5(d) we compare the calculated anapole-driven absorption rate enhancement values in Si and Ge nano-disks (solid curves) and square nano-pixels (dashed curves) with the same size. It is clear from the comparison in of Figs. 5(c) and 5(d) that larger values of absorption enhancement and a broader tunability can be achieved in Ge nanostructures due to the smaller perturbation of the anapole condition from the additional multipole moments. The results in Figs. 5(c) and 5(d) also demonstrate that the square nano-pixel geometry features a larger absorption rate enhancement compared to the case of nano-disks of equal volume. We attribute this effect to the enhanced internal field localization induced by the sharp edges that is attainable at the square corners of the nano-pixel structure, as shown in Figs. 2(c) and 2(d).
We can now summarize our results on the spectral position of the peak absorption rate enhancement due in Si and Ge nanostructures as a function of their geometrical parameters D and H, which are in Figs. 6 and 7.
In Fig. 6(a), we plot the wavelength shift of the anapole-induced absorption rate enhancement peak when increasing the diameter of Si nano-disks for the three representative choices of the height reported in the label. We found that the peak wavelength of absorption rate enhancement follows an almost linear trend by increasing the diameter. This is expected since the anapole condition relies on the excitation of dipolar and toroidal electric modes that shift linearly when increasing the width of the structure. In Fig. 6(b), we plot the shift of the absorption rate enhancement peak when increasing the height H of the nano-disk, for the three representative values of diameters reported in the label. In this case the trend is sub-linear and it gradually saturates with the height of the nano-disk. This behavior is also consistent with the fact that under normally incident plane wave excitation a toroidal moment can be excited with decreasing efficiency when increasing the height of the nanostructure, due to a greater magnetic dipole contribution . Figures 6(c) and 6(d) illustrate the case of Si square nano-pixels with the same D and H values. Since in this condition the volume for a square nano-pixel is larger than the one of a nano-disk with identical values of D and H, the wavelengths at which the anapole-induced absorption rate enhancement occurs are expected to be larger, as observed in our simulations. However, we notice that the general trend for the wavelength shift for both geometries is qualitatively very similar.
In Fig. 7 we show the results for the same geometries considered in Fig. 6 but replacing Si with Ge as the absorbing material. The qualitative behavior resulting from varying D or H in Ge nano-disks and square nano-pixels is identical to the previously discussed Si case. However, the wavelengths at which the anapole condition is met occur at larger wavelengths compared to the Si case due to the larger refractive index of Ge that red-shifts all the resonances. The results of our systematic analysis summarized in Figs. 6 and 7 show that it is possible to largely tune the wavelength of the anapole-driven absorption rate enhancement in Si and Ge nano-disks and nano-pixels by controlling their geometrical parameters.
3.2 Effects of incidence angle and dielectric coating
In realistic device applications it is important to consider the sensitivity of the anapole-driven enhanced absorption on incident angles and thin dielectric coatings that represent transparent contacts in photodetectors. We address both these issues in Fig. 8. In particular, in Figs. 8(a) and 8(b) we show the effect of the incidence angle on the wavelength of peak absorption enhancement and on the scattering efficiency of a Si nano-disk and a square nano-pixel, respectively. For this study we choose representative structures with D = 350nm, and H = 60nm. We also consider for each incidence angle the results corresponding to the two orthogonal states of linear polarization s and p.
In general we observed that, for both polarizations, the wavelength for the anapole-induced absorption peak blue-shifts when the nano-disk’s cross-sectional area is decreased due to the angular variation, as expected for resonant dielectric systems. However, under p-polarized plane wave excitation, the resonant anapole condition depends more strongly on the incident angle in both nano-disk and square nano-pixel geometries. This behavior reflects the less ideal quality of anapole modes excited in both the nano-disk and the square nano-pixel under p-polarized plane wave excitation. In fact, the s-polarized plane wave excitation in these geometries results in a stronger magnetic dipole contribution that degrades the spectral purity of the anapole mode. This is confirmed by multipolar decomposition analysis and visually illustrated by the corresponding angular dependence of the scattering efficiency also shown in Figs. 8(a) and 8(b). Consistently, as the incident angle increases, the values of the scattering efficiency also increase at the wavelength of the anapole mode, demonstrating the less ideal anapole response under p polarization, as it is significantly perturbed by the presence of additional electromagnetic multipoles. However, our results demonstrate that a good quality (i.e., small far-field scattering) anapole mode condition can still be obtained in Si nano-disks under oblique excitation for values of incident angles as large as approximately 20°, as measured from the normal to the disk’s surface. Figures 8(a) and 8(b) also indicate the difference in anapole responses to s and p polarization is more pronounced in the case of square nano-pixels. Similar behaviors were also obtained for the case of Ge nano-disks and square nano-pixels.
We also investigated the effect of dielectric coating on the anapole-induced absorption rate enhancement spectra. We present representative cases where a low-index (n = 1.8, as for ITO transparent contacts) coating materials is considered on both sides of a Si nano-disk [Fig. 8(c)] and Si nano-pixel [Fig. 8(d)], both with D = 350nm and H = 60nm (see insets of Figs. 8 (c) and 8(d)). Coating layers of different thicknesses, as indicated in the legends of Figs. 8(c) and 8(d), have also been studied. Our results demonstrate that the presence of the dielectric coating introduces a small red-shift in the peak of the absorption rate enhancement in both cases, but the enhancement values are only negligibly affected. This analysis demonstrates that the absorption enhancement due to the resonant excitation of non-radiating anapole modes is a robust effect in high-index dielectic materials and can be engineered at non-normal incidence even in coated structures.
4. Absorption-enhancement engineering using arrays of Ge nanostructures
In this section, we discuss the possibility of enabling broadband absorption enhancement by anapole excitation in nanostructured arrays of Ge nano-disks and square nano-pixels with different geometrical parameters. In this case, we study absorbing dimer elements formed by nano-disk or square nano-pixel of different side lengths, as specific example of nanostructured arrays.
We start by showing in Fig. 9(a) the absorption rate enhancement that is obtained in isolated Ge nano-disks with two different values of diameter and with the same height H = 100nm. In Fig. 9(b) we combine these two nano-disks separated by a variable distance d into a functional unit (see inset of Fig. 9(b)) to achieve broadband absorption enhancement atop a dielectric surface. Figure 9(b) clearly demonstrates that the absorption rate enhancements of this unit base, normalized with respect to the total volume of absorbing material, can be optimized to achieve a larger bandwidth with respect to the individual nano-disk components. Figure 9(b) summarizes the absorption rate enhancement values for five different separations d = 1000nm, 3000nm, 4000nm, 5000nm, and 6000nm, under normal incidence. Notice that when the two Ge nano-disks are sufficiently separated two distinct peaks appear in the enhancement spectrum corresponding to the two well-defined anapole conditions of the isolated nano-disks. It is also important to realize that since we normalize the enhancement values with respect to the total volume of the structures, which is larger for the combined units, the maximum absorption rate enhancement shown in Fig. 9(b) is lower than the peak value of the individual (uncoupled) structures [Fig. 9(a)]. We have found that when d = 5000nm, the largest overall absorption rate enhancement is achieved for the two particular Ge nanodisks used. Moreover, when the separation d between the disks deviates from this condition, electromagnetic coupling between the components leads to a reduction of the peak absorption values.
In Figs. 9(c) and 9(d), we consider the case of the Ge square nano-pixel geometry. Similar to the case of nano-disks, we start by designing square nano-pixels that feature optimal absorption rate enhancement peaks at different wavelengths [Fig. 9(c)] and we engineer their absorption rate enhancement over a broader spectrum by electromagnetically coupling two nano-pixels in a functional unit [Fig. 9(d)]. Comparing Figs. 9(b) and 9(d), we can notice that the separation distances that provide the largest peak absorption enhancement for nano-disks and square nano-pixels are both around 5000nm. However, for the nanodisk case, the intensity of the two resonant peaks due to individual nanodisk’s contributions are more equalized than in the case of square nano-pixels. This behavior reflects the shape-dependent scattering efficiency of the two geometries that strongly affects the coupling strength between the two resonant units, leading to spectrally different electromagnetic responses. In the case of Ge square nano-pixels, when d = 1000nm, strong electromagnetic coupling between the nano-pixels broadens the spectrum of the contribution for D = 550nm nano-pixel [Fig. 9(d)], but only at the expense of the enhancement values. This behavior is a manifestation of the intrinsic tradeoff between spectral bandwidth and resonance intensity in coupled nanostructures .
Based on the multipolar decomposition method, we have demonstrated wide and controllable wavelength tunability of anapole-driven absorption enhancement in nano-disks and square nano-pixels made of Si and Ge absorbing materials with realistic dispersion data. In addition, we have systematically addressed the effects of width D and height H of the nanostructures on the absorption enhancement spectra and we identified an optimal aspect ratio H/D ~0.2 for the excitation of anapole modes with optimal absorption enhancement. We also found that anapole modes with absorption rate enhancement can be excited not only at normal incidence but also under plane wave excitation at oblique incidence. We studied the effects of the angle of incidence and incident wave polarization and found that anapole modes are polarization sensitive and that their spectral purity, perturbed by the onset of additional electromagnetic multipoles, significantly deteriorates beyond 20° incidence especially for the p-polarization. We also studied the effect of dielectric coating and showed that anapole-induced absorption enhancement can be achieved even when coating both sides of the nanostructures with a low-index dielectric layer. Finally, by combining nano-disks and nano-pixels of different sizes into functional surface units, we design dimers as a specific example of nanostructured arrays with enhanced absorption bandwidth that can be useful for the engineering of broadband semiconductor photodetectors driven by the controllable anapole responses in high-index dielectrics.
Army Research Laboratory (ARL) [W911NF-12-2-0023]; National Science Foundation (NSF) [EAGER-ECCS 1541678]
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