Normal light incidence

To start with, we consider a single disk with the height h = 130 nm and different diameters d. The permittivity of the disk is silicon (data from experimental results^[Reference Heavens^33]) and the surrounding material is air with ε _o = 1. We perform numerical modeling with frequency-domain solver of CST Microwave Studio commercial package.

The scattering properties of a single nanoparticle are affected by the interplay of resonances, such as Kerker effect with electric and magnetic dipoles,^[Reference Kerker, Wang and Giles^2] generalized Kerker effect involving higher multipoles,^[Reference Liu, Zhang, Lei, Ma, Xie and Hu^3] or excitation of toroidal mode resulting in non-radiating anapole response.^[Reference Miroshnichenko, Evlyukhin, Yu, Bakker, Chipouline, Kuznetsov, Luk'yanchuk, Chichkov and Kivshar^4] Thus, field enhancement or power absorption in the nanoparticle is the most reliable way to identify resonances. The same holds true for nanoparticle periodic arrangement: one should define resonances in the array absorption spectrum, not in transmittance or reflectance one. Figure 1(a) shows excitations of the EDR, MDR, and higher multipole resonances for the single disk [inset in Fig. 1(a)]. In this parameter range, the toroidal mode is also excited in the proximity to the EDR,^[Reference Miroshnichenko, Evlyukhin, Yu, Bakker, Chipouline, Kuznetsov, Luk'yanchuk, Chichkov and Kivshar^4] but it is relatively weak, and we do not consider it here. From Fig. 1(a) we observe that at the normal light incidence of a plane wave, the EDR and MDR overlap for d ≈ 350 nm, and the EDR is excited at a larger wavelength for d > 350 nm in contrast to spherical nanoparticle where λ _MDR > λ _EDR. While multipole resonances appear as maxima in absorption, the total scattering cross-section shows both increased and decreased values around the resonances [Fig. 1(b)]. In particular, the total scattering is increased on the red side of the EDR and decreased on the blue side of the resonance, which is typical for the nanoparticles with such dimensions.^[Reference Miroshnichenko, Evlyukhin, Yu, Bakker, Chipouline, Kuznetsov, Luk'yanchuk, Chichkov and Kivshar^4] Distributions of electric and magnetic fields in xz- and yz-coordinate planes, respectively, confirm the excitation of the EDR and MDR at the moment of corresponding field maximum inside the nanoparticle [insets in Fig. 1(b)].

Figure 1. Absorption and scattering properties of a single silicon disk for various disk diameters under normal light incidence: (a) absorption and (b) total scattering cross-section. The black dotted lines correspond to the diameter d = 500 nm that is chosen for further analysis. The red and magenta lines mark excitation of electric and magnetic resonances, respectively, defined as maxima in absorption. The red dots indicate the wavelength of resonances for d = 500 nm. Inset in (a): schematic of the single disk under consideration. Insets in (b): E-field distribution in the EDR (λ = 1130 nm) and H-field distribution in the MDR (λ = 978 nm) for d = 500 nm. The cross-sections are different as they are in the plane of the corresponding field component (xz-plane for the EDR and yz-plane for the MDR), and the distributions are captured at the moment of corresponding field maximum inside the disk.

Disk diameter d = 500 nm is chosen for the further analysis to ensure the EDR and MDR are spectrally distinguished from each other and the lattice effect can be separately studied for each of the resonance. Now we consider a regular array of such disks with periods D _x and D _y in x- and y-directions, respectively (Fig. 2). The polarization of the incident plane wave is along the x-axis. In this case, period D _x controls the position of the MDR and electric quadrupole (EQ) resonance,^[Reference Evlyukhin, Reinhardt, Zywietz and Chichkov^31] and period D _y controls the EDR and magnetic quadrupole (MQ) resonance. When both periods are changed simultaneously, both the EDR and MDR experience redshift (Fig. 3). Similar to the total scattering cross-section of the single disk, the reflectance is increased on the red side of the EDR and decreased on the blue side of the resonance. The resulting resonance changes under normal light incidence are summarized in Table I.

Figure 2. Schematic view of the silicon disk array in the air. Nanoparticles have diameter d and arranged into the array with the periods D _x and D _y. Light incidence k is normal to the array plane, and electric field E is along the x-axis.

Figure 3. Electric and magnetic resonances in the arrays of silicon disks under normal light incidence with field components (E _x, H _y, 0). (a),(c),(e) Absorption and (b),(d),(f) reflection. In panels (a) and (b) period D _x changes, and D _y is fixed. In panels (c) and (d) period D _y changes, and D _x is fixed. In panels (e) and (f) both periods change simultaneously D _x = D _y. The red dots mark the position of the resonances of the single disk identified in Fig. 1(a). The diffraction wavelength (black lines) is defined as λ_RA = D _x for (a) and (b), λ _RA = D _y for (c) and (d), or λ _RA = D _x = D _y for (e) and (f), and the region below the diffraction is excluded from consideration.

Table I. Lattice effects (LE) under the change of periodicity and angle of light incidence. Under normal incidence, the field is (E _x, H _y, 0). “LE” (or “No LE”) denotes the resonances are affected (or not) by the period changes or angle of incidence for the nanoparticle with scalar polarizability arranged in the infinite array.

Oblique light incidence

Now let us consider an oblique light incidence on the single disk at the different angles θ [Figs. 4(a) and 4(b)]. Additional resonances are induced in both TM and TE polarizations: in the former case, field component E _z ≠ 0 induces extra modes; and in the latter case, E _x remains in the array plane, but the oblique incidence brings an additional component of propagating wavevector k _y ≠ 0. At the angles θ ≠ 0° the conditions of resonance excitation change. In TM polarization, for 10° < θ < 80°, extra mode appears at the wavelength λ ≈ 950 nm, and for θ > 45°, the resonances correspond to the nanoparticle with inverted aspect ratio and λ _MDR > λ _EDR. In TE polarization and θ > 30°, extra resonances appear in the range 800–950 nm as well as the MDR at λ ≈ 1520 nm.

Figure 4. Electric and magnetic resonances in the single silicon disk and arrays of disks with oblique light incidence. (a) and (b) Absorption in the single disk; (c) and (d) Absorption in the disk array; (e) and (f) Reflection in the array; (a), (c), and (e) are for TM polarization; (b), (d), and (f) are for TE polarization. In (c)-(f), the calculations are performed for a square array with periods D _x = D _y = 600 nm. The red dots mark the position of the resonances of the single disk identified in Fig. 1(a). The diffraction wavelength (black lines in (c)–(f)) is defined as λ _RA = D(1 + sinθ), where D = D _x for TM polarization and D = D _y for TE polarization, and the region below the diffraction is excluded from consideration (dark red color). Insets in (c) and (d) are the schematics of TM and TE light polarization with respect to the disk, respectively.

Oblique light incidence on the periodic disk array effectively changes the distance between resonant multipoles in the lattice. In this case, the lattice resonance positions shift, and the comparison of array resonances with the single disk resonances reveals the lattice effect. The spectral position of RA can be found from the equation for TM polarization:

(1)$$\left( {\displaystyle{{2\pi} \over {D_y}}n_y} \right)^2 + \left( {\displaystyle{{2\pi} \over {D_x}}n_x + k_x} \right)^2 = k^2,$$

and for TE polarization:

(2)$$\left( {\displaystyle{{2\pi} \over {D_y}}n_y + k_y} \right)^2 + \left( {\displaystyle{{2\pi} \over {D_x}}n_x} \right)^2 = k^2,$$

where n _x and n _y are the integers 0, ± 1, ± 2,…, k _x and k _y are the in-plane components of the incident wave k _x = k _y = k sinθ, and k = 2π/λ _RA assuming the surrounding medium is air (ε _o = 1). The largest RA wavelength and the only one that corresponds to λ _RA > D _x,y is the case n _y = 0, n _x = −1, and λ _RA = D _x(1 + sinθ) for TM polarization and the case n _x = 0, n _y = −1, and λ _RA = D _y(1 + sinθ) for TE polarization. These diffraction limits are marked in Figs. 4(c)–(f) as the black lines.

Importantly, TM polarization corresponds to k _x ≠ 0 which is an equivalent of D _x change: see Eq. (1). It means that the lattice interaction of the MDR and EQR is affected in TM polarization, and the EDR and MQR are not affected. This conclusion is in a good agreement with results in Figs. 4(c) and (e). In turn, TE polarization corresponds to k _y ≠ 0 and effective change of D _y, affecting lattice interaction of the EDR and MQR, but not affecting the MDR and EQR [Figs. 4(d) and (f)]. The results of resonance changes under oblique light incidence are summarized in Table I. The extra mode that appears in TM polarization at the wavelength λ ≈ 950 nm for 10° < θ < 80° is not affected by the lattice effect. For TE polarization, the additional resonances and their spectral position changes are defined not only by the lattice effect but also by the change of the excitation conditions of each nanoparticle in the array.

The considered above silicon nanoparticles array is an example of a structure out of realistic material, dispersive and lossy. Such behavior of the EDR and MDR under change of array period and angle of light incidence is observed not only for silicon nanoparticles but also for the nanoparticles with permittivity ε = 20 + 0.01i (see Figs. 6–8 in Appendix). For the disks with such higher permittivity and lower losses, the multipole resonances are better pronounced and spectrally more separated. Numerical simulations show that the MDR or EDR changes its position under oblique light incidence with TM or TE polarization, respectively, and we expect that these array properties can be observed for a wide range of materials with high refractive index.

Finally, we confirm the effects described above are valid not only for the hypothetical silicon disk array in the air but also for nanoparticles on a substrate and with a top coating. The detailed analytical and numerical analyses of silicon nanoparticle arrays on the substrate have been performed for a variety of substrate indices and nanoparticle shapes (see e.g.^[Reference Baryshnikova, Petrov, Babicheva and Belov^10, Reference van de Groep and Polman^23, Reference Babicheva, Petrov, Baryshnikova and Belov^25]). Substrates with the high refractive index can cause a reflection that significant alternates response of the structure as a whole. However, substrates with the moderate refractive index or uniform coating of nanoparticles do not change the main effects that we discuss in the present work. To illustrate this, we consider the substrate with moderate refractive index n = 1.5, which is close to the index of silicon oxides, glasses, and polymers [Fig. 5(a)]. Although the MDR is significantly weaker than for the case of the uniform environment, it clearly shifts to the larger wavelength upon an increase of the incidence angle θ. At the same time, the EDR remains on the unchanged position. The same is seen for the case of the substrate and top coating with n = 1.5 which effectively mimics a uniform environment around the nanoparticles [Fig. 5(b)]. It is often used in the experimental work^[Reference Staude, Miroshnichenko, Decker, Fofang, Liu, Gonzales, Dominguez, Luk, Neshev, Brener and Kivshar^14, Reference Arslan, Chong, Miroshnichenko, Choi, Neshev, Pertsch, Kivshar and Staude^22, Reference Decker, Staude, Falkner, Dominguez, Neshev, Brener, Pertsch and Kivshar^24] and eliminates the effect of an additional interface and undesirable reflections from it. Thus, as we show by the examples of the substrate with and without top coating, the oblique incidence of light in TM polarization mainly affects the MDR and shifts its position to the larger wavelength moving along the RA wavelength.

Figure 5. Effect of the substrate and top coating. (a) Absorption in the silicon disk array in TM polarization in the case of nanoparticles on the substrate with refractive index n = 1.5 (inset shows the basic schematics). Lines for θ = 10° and 15° are shifted by 0.1 with respect to each other for clarity. (b) The same as (a) but with an addition of a top coating of material with the same n = 1.5. The case of the substrate and top coating effectively corresponds to the uniform environment with n = 1.5. The lines for θ = 10° and 15° are interrupted at the RA wavelength. In both cases with and without top coating, the MDR shifts to larger wavelength in agreement with the case without a substrate or coating.