Magnetic dipole radiation tailored by substrates: numerical investigation

Nanoparticles of high refractive index materials can possess strong magnetic polarizabilities and give rise to artificial magnetism in the optical spectral range. While the response of individual dielectric or metal spherical particles can be described analytically via multipole decomposition in the Mie series, the influence of substrates, in many cases present in experimental observations, requires different approaches. Here, the comprehensive numerical studies of the influence of a substrate on the spectral response of high- index dielectric nanoparticles were performed. In particular, glass, perfect electric conductor, gold, and hyperbolic metamaterial substrates were investigated. Optical properties of nanoparticles were characterized via scattering cross-section spectra, electric field profiles, and induced electric and magnetic moments. The presence of substrates was shown to introduce significant impact on particle's magnetic resonances and resonant scattering cross-sections. Variation of substrate material provides an additional degree of freedom in tailoring properties of emission of magnetic multipoles, important in many applications.

types of substrates. In particular, glass, gold and hyperbolic metamaterial substrates were considered. Optical properties of nanoparticles were characterized via scattering cross-section spectra, electric field profiles, and 49 induced electric and magnetic moments. 50 2 Theoretical and numerical frameworks 51 Fig. 1: Dielectric nanoparticle on a substrate illuminated with a plane wave. The regions inside and outside the green box depicts total (incident and scattered) electric field (TF) and scattered electric field (SF), respectively.
Optical response of individual spherical particles embedded in homogeneous host materials can be described 52 analytically via the Mie series decomposition [18]. Generally, any shape with boundaries belonging to "coordinate 53 surfaces" sets at coordinate systems where Helmholtz equation is separable, i.e., has analytic solution for 54 scattering problems. However, the separation of variables method breaks down once substrates are introduced. 55 As the first order approximation, the image theory could be employed [19,20,21], while more complex treatments 56 can account for retardation effects and higher multi-pole interactions [22,23,24]. Considerations of anisotropic 57 substrates lead to major complexity even in the image theory (point charge images should be replaced by 58 certain distributions) [25] and make fully analytical approaches to be of limited applicability. Moreover, in 59 the case of high-index dielectric particles, the resonances are overlapping in frequency, making even the image 60 theory to be inapplicable. At the same time, this spectral overlap can be employed for super-directive antenna 61 applications [16,26].

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One of the commonly used techniques for numerical analysis of scattering processes is the so-called "total-63 field scattered-field"(TFSF) approach [27]. The key advantage of this method is the separation of relatively 64 weak scattered field (SF) from predominating high amplitude total field (TF) which contains both incident and 65 scattered fields, in a distinct simulation domain. The TFSF method also allows to subtract the electric field, 66 reflected backwards by the substrate in the SF domain, enabling the calculation of a scattering cross-section.

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The geometrical arrangement of the considered scenario is represented in Fig. 1: a small dielectric spherical 68 particle (r = 70 nm, = 20) is placed on the substrate. The centre of the particle coincides with the coordinate 69 origin. The system is illuminated with a short pulse (in time) with broadband spectrum covering the spectral 70 range from 400 to 750 nm. FDTD method allows analyzing spectral responses via single simulation by adopting 71 the Fourier decomposition method with subsequent normalization. The key parameters characterizing the system 72 are the electric and magnetic dipole moments defined as

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where 0 is the vacuum permittivity, V sphere 1.44e −3 µm 3 is the volume of the particle, (r, ω) is the frequency-76 dependent particle permittivity, ω, is the frequency of the incident light, and E(r, ω) is the electric field in 77 frequency domain. Generally, these dipolar moments in the asymmetric system considered here depend on the 78 illumination polarization, direction of incidence and spatial shape of the beam. In the following, a normally 79 incident y-polarized plane wave was considered. To avoid the interplay between geometrical parameters and 80 chromatic dispersion of the particle's material, the latter was neglected. To verify the accuracy of the approach, optical properties of the particle in free space were evaluated 84 numerically and compared to analytic description. Scattering cross-section spectra ( Fig. 2 is observed for a MD resonance (Fig. 2e), the drop-shaped distribution for an ED resonance (Fig. 2d), and the 90 double resonant sectors for a MQ resonance (Fig. 2c).

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Knowing the electric field distribution in the entire space allows to calculate the moments defined in Eq. 1.

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These results are summarized in Fig. 2 analytical Mie theory approach reported in [12] and differ by no more than 5%, verifying the accuracy of the 100 employed numerical method.  Fig. 2: Optical properties of a dielectric ( particle = 20) nanoparticle of 70 nm radius in free space: (a) scattering cross-section spectra, (b) the summary of the resonant wavelengths, the scattering cross-section normalised to the geometric cross-section, and electric p y and magnetic moments m z , (c-e) the spatial distribution of the electric field amplitudes at resonant wavelengths (normalised to the incident field).

Dielectric substrate
In many cases, dielectric particles are placed on glass substrates as the result of their fabrication and for 103 optical characterisation [17]. This type of substrates is expected to introduce the smallest distortions in the 104 optical properties of a particle, compared to other types of substrates, e.g., metallic ones. We considered a 105 glass substrate with = 3.1 corresponding to the family of flint glasses. Comparing to the nanoparticles in free 106 space, the glass substrate influence is in significant suppression of the high-order multipoles (Fig. 3), with both 107 electric and, especially, magnetic dipolar resonances being less affected. These conclusions are confirmed by the 108 field amplitude distributions at the resonant wavelengths. Minor electric field amplitude amplification for ED 109 and MD resonance and nearly two times reduction for MQ resonance can be seen compared to the particle in 110 free-space. The electric moment of the particle is slightly increased while the magnetic one is slightly reduced.

PEC substrate
In order to test the applicability of the image theory for high-index dielectric particles, a perfect electric

Gold substrate
The illumination of a scatterer placed on a gold film gives rise to excitation of surface plasmon polaritons [28].

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The excitation of additional propagating surface wave substantially changes the optical properties of dielectric 128 spheres (Fig. 5). It can be seen that the electric dipole resonance is split in two. This is the result of strong 129 coupling by anti-crossing between two coupled dipoles (SPP mode and ED resonance of the particle). Moreover, 130 the electric dipole mode excites the SPP with much higher efficiency than the other particle resonances. Electric 131 field amplitudes experience minor amplification comparing to the case of free space, except for the MQ resonance.

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The electric moment experiences nearly ten fold enhancement due to the presence of the substrate, whilst the 133 magnetic moments retain the values close to those in free space.  an emitter (or scatterer) placed in the near-field proximity to a metamaterial [30,31]. Hereafter, we compare 142 the layered realization of hyperbolic metamaterial [32] with its homogeneous counterpart described via effective 143 medium theory [33] neglecting the effects of spatial dispersion [34].

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The considered nanostructured substrate consists of dielectric layers with d = 3.1 and silver metal layers 145 with m (the Drude model for silver for considered [29]). The layers have the same subwavelength thickness (30 146 nm) and can be described using a diagonalised effective permittivity tensor with components substrate, but they are suppressed in metal-dielectric slab (cf. Fig. 4 in Ref. [30]). This is the result of the  i) j) Fig. 7: Optical properties of a dielectric nanoparticle ( particle = 20) of 70 nm radius on (a,c,e,g,i) a metaldielectric multilayered substrate ( dielectric = 3.1, see the inset in (a) for silver ) and (b,d,f,h,j) an effective homogeneous medium with the effective permittivity as in Fig. 6: (a,b) scattering cross-section spectra, (c,d) the summary of the resonant wavelengths, the scattering cross-section normalised to the geometric cross-section, and electric p y and magnetic moments m z , (e-j) the spatial distribution of the electric field amplitudes at resonant wavelengths (normalised to the incident field). 8 In this paper, we performed comprehensive numerical studies of the substrate influence on the optical 168 properties of high-index dielectric nanoparticles. Different types of substrates such as flint glass, perfect 169 electric conductor, gold, and hyperbolic metamaterial were investigated. Retardation effects were shown to 170 play significant role in the particle-substrate interactions making the "classical" tools such as image theory to

Acknowledgements
This work was supported, in part, by the government of the Russian Federation (Grant 074-U01 and 11.G34.31.0020) and EPSRC (UK). AZ acknowledges support from the Royal Society and the Wolfson Foundation. Authors acknowledge discussions with Prof. Yuri S. Kivshar and Dr. Andrey Miroshnichenko.