1. Introduction

The problem of light scattering from subwavelength particles is one of the fundamental problems in electromagnetism [1], drawing much interest in various research areas, including nanoantennas, plasmon-enhanced photovoltaics, and sensors [2–4]. These applications could benefit from the use of nanoparticles that have broadband unidirectional scattering properties, where unidirectional scattering can be obtained by satisfying the first and second Kerker’s conditions [5]. The Kerker’s conditions were described in 1983 for magneto-dielectric particles that have both non-unit relative electric permittivity and magnetic permeability (${\varepsilon _r},{\mu _r} \ne 1$) [5]. However, since such particles do not exist naturally in visible wavelengths, considerable attention has been given, recently, to defining natural structures that satisfy the first and second Kerker’s conditions [6–11].

In this regard, two kinds of natural materials can be chosen: plasmonic or high-index dielectric materials. Questions remain, however, regarding which one of these two materials is better; and whether there are any differences in their mechanisms of creating the first and second Kerker’s conditions. Calculating the complete polarizability tensors of a nanoparticle is a powerful tool for answering these questions. Hence, in this paper, we investigate the polarizability tensors of plasmonic and high index dielectric structures, and discuss how to fulfill Kerker’s conditions in each of these structures.

Nanoparticles made from plasmonic materials, like gold, have attracted considerable attention, recently, because of their strong interactions with light at localized surface plasmon resonances. However, isolated plasmonic nanoparticles with simple shapes can only support electric resonances in homogeneous media. Therefore, they cannot satisfy either of the two Kerker’s conditions. Moreover, plasmonic nanostructures have limitations due to their high ohmic losses, low melting points, and high thermal conductivities, making them unsuitable candidates for certain critical applications, such as quantum optical sources and photovoltaics. In contrast, dielectric nanoparticles create displacement currents rather than actual ones [6] can overcome the problem of strong losses, and pave the way for low-loss nanophotonic devices that employ all-dielectric nanoparticles. It has also been shown that dielectric particles with highly-refractive indices can support both electric and magnetic resonances in the visible and near-infrared wavelengths [6,12,13]. Studies of magnetic resonance in all-dielectric nanoparticles, and their interactions with the electric resonance, have shown their considerable potential to be used in a wide variety of applications, such as novel low-loss nanophotonic devices [8,11].

Aside from its constituent materials, various parameters affect a nanoparticle’s scattering response, including its geometry, dimensions, and surrounding environment [14–17]. It is advocated that by carefully choosing these parameters, the interplay between the electric and magnetic resonance spectrums can be tailored, and various applications in the area of scattering can be realized. Regarding satisfying the first and second Kerker’s conditions, to achieve unidirectional scattering in the forward and backward directions, most previous studies have concentrated on nanoparticles located in homogeneous media [6,9,18,19]; while in many applications, the presence of a substrate is inevitable. Moreover, the presence of a substrate can improve the structure’s mechanical robustness, making it more favorable for practical implementations. Furthermore, the presence of a substrate strongly affects the resonance behaviors of particles, as well as the interactions between their electric and magnetic resonances. Therefore, it is important to investigate the directional conditions that occur in the presence of a substrate.

It is worth highlighting that [10] considered the effects of a substrate to satisfying the Kerker’s conditions. But, that work extracted the generalized Kerker’s conditions in terms of multipoles under normal illumination. Yet in practical applications, a particle can be illuminated by plane waves with arbitrary polarizations, and at any angle of incidence. However, to the best of our knowledge, most researchers have used structures with normal illuminations, which obviously restricts their applications in real devices. Moreover, if the Kerker’s conditions under oblique incidence are extracted in terms of the electric and magnetic moments, it is necessary to calculate the electric and magnetic moments for each incident angle, because these moments depend on the angle of incidence. But, the polarizability tensors of a particle are its characteristic parameters, which do not depend on the angle of the incident wave; and once they are calculated, they can be used as equivalent representations of the particle in any electromagnetic problems with plane wave illuminations. Hence, in this paper, the generalized Kerker’s conditions are derived for an arbitrary plane wave, and at any angle of incidence, in terms of the polarizability tensors of various nanostructures in homogeneous media and located on dielectric substrates.

The paper is organized as follows: In Section 2, the general conditions for satisfying the first and second Kerker’s conditions in the case of normal and oblique incidence, are derived, in terms of the polarizability tensors. Section 3 is then devoted to an investigation of unidirectional scattering using nanoparticles in free space; whereby, the effects of a particle’s material on its frequency response are first considered. Then, the dimensions of a nano-cylinder and nano-cone located in free space are presented, which satisfy the first and second Kerker’s conditions at two separate frequencies. Additionally, the forward-to-backward scattering ratio versus the wavelength, and the angle of incidence, are investigated. In Section 4, by modifying the base structures and placing them on substrates, the unidirectional scattering of the substrated nanoparticles is shown, under normal and oblique incidence, in terms of their polarizabilities. Finally, the study is concluded in the last section.

2. Theory

This section is aimed at deriving the general conditions for satisfying the first and second Kerker’s conditions, in terms of a particle’s polarizability tensors. First, let us consider a general case where an arbitrary nanoparticle is located at the interface, and in the upper-half space of two dielectric media that is illuminated by an arbitrarily polarized plane wave under an oblique incidence. As shown in Fig. 1, the plane wave can be written as the superposition of two orthogonally polarized plane waves, parallel (p) and perpendicular polarization (s) with respect to the plane of incidence:

The incident electric and magnetic fields of parallel polarized wave can be written as:

Here, ${\textbf {E}}_{ref}^{(p,s)}\,\,$ and ${\textbf{H}}_{ref}^{(p,s)}$ are the reflected electric and magnetic fields for p- and s-polarizations, respectively. They can be written for p-polarization as:

 

Fig. 1. Schematic representation of a substrated nanoparticle. (a) under p-polarized oblique plane wave incidence. (b) under s-polarized oblique plane wave incidence.

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and follows for the s-polarization:

Until know, the effect of plane wave illumination is modeled by a pair of p/m dipole moments, or equivalently polarizability tensors, which stands for excited nanoparticle. The excited nanoparticle then reradiates in the medium, and produces scattered fields in the upper- and lower-half spaces as:

In Eq. (9), the electric fields at the geometrical center of the substrated nanoparticle are calculated and also a general result from [22] is used. ${{\textbf E}^D}$ is the electric field of the wave that is directly scattered at the observation point, ${\textbf r}$, in the upper medium; ${{\textbf E}^R}$ is the electric field of the scattered wave that is reflected from the boundary surface, and ${{\textbf E}^T}$ is the electric field of the scattered wave that is transferred into the medium (2). Furthermore, $\bar{\bar{R}}$ and $\bar{\bar{T}}$ are the reflection and transmission tensors, respectively, which depend only on the observation point, and can be obtained from Eqs. (D.4) to (D.9) in [23]. Additionally, ${\textbf n}$, $\tilde{{\textbf n}}$, and ${{\mathop {\bf n}\limits^\approx}}$ are direction vectors of ${\textbf r}$, $\tilde{{\textbf r}}$, and ${{\mathop {\bf r}\limits^\approx}}$, respectively, where ${\textbf r} = ({x,y,z} )$, $\tilde{{\textbf r}} = ({x,y, - z} )$, ${{\mathop {\bf r}\limits^\approx}} = \left( {\sqrt {\frac{{{\varepsilon_2}}}{{{\varepsilon_1}}}} x,\sqrt {\frac{{{\varepsilon_2}}}{{{\varepsilon_1}}}} y, - \sqrt {{r^2} - \frac{{{\varepsilon_2}}}{{{\varepsilon_1}}}{\rho^2}} } \right)$, and $\rho = \sqrt {{x^2} + {y^2}}$.

Now, we can evaluate the scattered fields in any direction and study different desired cases. To consider the first (FKC) and second Kerker’s conditions (SKC), which are related to the zero backward scattering and zero forward scattering, respectively, in XZ plane of incidence ($\phi = 0$), we have $\theta = {\theta _i}$ for the FKC, and $\theta = \pi + {\theta _i}$ for the SKC, where $\theta$ is the observation point. Therefore, the direction vectors of ${\textbf r}$,$\tilde{{\mathbf r}}$, and $\mathop {\bf r}\limits^\approx $ can be written as ${\textbf n} = \sin {\theta _i}\begin{array}{c} {\hat{x} + } \end{array}\cos {\theta _i}\begin{array}{c} {\hat{z}} \end{array},$ $\tilde{{\mathbf n}} = \sin {\theta _i}\begin{array}{c} {\hat{x} - } \end{array}\cos {\theta _i}\begin{array}{c} {\hat{z}} \end{array}, {\mathop {\bf n}\limits^\approx} ={-} \sqrt {\frac{{{\varepsilon _2}}}{{{\varepsilon _1}}}} \sin {\theta _i}\begin{array}{c} {\hat{x} - } \end{array}\sqrt {1 - \frac{{{\varepsilon _2}}}{{{\varepsilon _1}}}{{\sin }^2}{\theta _i}} \begin{array}{c} {\hat{z}} \end{array}.$

To satisfying the FKC we should have ${{\textbf E}^{up}}|{_{\theta = {\theta_i}}} = 0$ while for the SKC, ${{\textbf E}^{down}}|{_{\theta = \pi + {\theta_i}}} = 0$. Therefore, the general conditions of the FKC and SKC for a substrated nanoparticle can be calculated in terms of dipole moments as:

Up to now, the first and second Kerker’s conditions have been extracted under oblique illuminations, in terms of the electric and magnetic dipole moments. In fact, the equations derived in this paper for Kerker’s conditions, are accurate enough for the structures with dominant electric and magnetic dipole moments, and higher-order multipoles are negligible. Since these dipolar moments must be calculated for each angle, it is worthy to calculate these conditions in terms of the polarizability tensors. As mentioned before, the polarizability tensors of a particle are its characteristic parameters, which do not depend on the angle of the incident wave; and once calculated, they can be used as equivalent representations of the particle in any electromagnetic problems with plane wave illuminations. Hence, substituting Eq. (4) into Eq. (10), and considering the p-polarized incident wave, the following is observed for the generalized Kerker’s condition in terms of the polarizabilities of a substrated nanoparticle:

Generally, the polarizabilities of a nanoparticle are functions of its geometry, constituent material and host medium. Therefore, with a suitable choice of nanoparticle and its surrounding environment, one can prepare the conditions for unidirectional scattering.

At this step, and before going to the next section, the similar conditions for nanoparticles located in free space, can be discussed as a special case. In such a case, the Fresnel reflection coefficients are equal to zero, and one can obtain the general first and second Kerker’s condition for nanoparticles located in homogeneous medium with characteristic impedance $\eta$ as:

3. Unidirectional scattering using nanoparticles in free space

This section deals with the study of unidirectional scattering for nanoparticles located in free space. However, before that, let us first consider the effects of a nanoparticle’s material on its frequency response. For this purpose, two cylindrical nanoparticles are considered, made from plasmonic (gold) and dielectric (silicon) materials, while the major polarizabilities of these nanoparticles are shown in Fig. 2. It should be noted that in this paper Ansys Inc. HFSS full-wave simulator was employed for all numerical calculations. As can be seen form the figure, the plasmonic cylinders only have electric polarizabilities. Therefore, they cannot satisfy either of the two Kerker’s conditions. However, it should be noted that there are other shapes of plasmonic nanoparticles, such as split-ring resonators, which can provide other polarizabilities, and also have the chance of unidirectional scattering. Moreover, plasmonic nanoparticles often suffer from inevitable optical losses, making them unsuitable candidates for low-loss nanophotonic devices. Therefore, we preferred to focus on simple shaped dielectric nanoparticles.

 

Fig. 2. The major polarizabilities of two nano-cylinders with radii and heights of 50 nm and 100 nm, respectively, made from gold and silicon materials. The permittivity of Si and Au are taken from [29] and [30], respectively. (a) A gold nano-cylinder. (b) A silicon nano-cylinder.

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Referring to Fig. 2, high-index dielectric nano-cylinder induces both electric- as well as magnetic polarizabilities. The ability to support both ED and MD resonances, in the visible range, simultaneously, offers intriguing possibilities for applications in novel nanophotonic devices, by interplaying between electric and magnetic modes [8,11,27,28]. It is worth noting that the mechanism for exciting MD resonance in nanoparticles can be attributed to the construction of an internal current loop, where the circulating current induces an MD moment that is perpendicular to the electric field in the particle [13,15]. For a silicon nanoparticle, for instance, with a radius and height of 50nm and 100nm, respectively, this circulating current is produced at a wavelength of 495nm. In fact, at this wavelength, the relative permittivity of the silicon is about 19, and the structure can make a 180° phase difference at the top and bottom of the nano-cylinder. Conversely, for a gold nano-cylinder with these dimensions, the relative permittivity is about -3, at this wavelength, and it therefore cannot produce the required phase differences inside the nanoparticle. A possible way to create a circulating current inside a gold nano-cylinder is to enlarge it, so that its height is at least 2.5 times the current value. However, as mentioned before, plasmonic nanoparticles have high losses in the visible wavelengths, and these losses will be increased with particle size.

In addition to the nanoparticle’s constituent material, different parameters affect the frequency response of the nanoparticle and the interplaying of the electric and magnetic modes. Among them, the geometrical properties are of special importance. In a previous study, we showed how different geometrical parameters (the radius and height) affect the electric and magnetic resonances of different nano-structures [17]. Based on these investigations, which determined the electromagnetic properties of different nanoparticles as a function of their radius and height, some high-index dielectric nanoparticles can now be designed that produce unidirectional scattering.

In this regard, we consider two different geometries, silicon nano-cylinder and nano-cone and study their scattering field as a function of the incident angle and wavelength. Figure 3(a) shows the forward to backward scattering spectra (F/B is defined as the ratio between scattered fields evaluated in two particular directions, i.e. $F/B = 20{\log _{10}}\left( {\frac{{|{E_{sc}^{FF, + }} |}}{{|{E_{sc}^{FF, - }} |}}} \right)$, where $E_{sc}^{FF, + }$ is the scattered field in the forward direction and $E_{sc}^{FF, - }$ is the scattered field in the backward direction [9,31,32]) for a silicon nano-cylinder with a radius and height of 50nm and 140nm, respectively, while the result for a silicon nano-cone with a radius and height of 70nm and 250nm is depicted in Fig. 3(b). As can be seen, high forward-to-backward ratios are observed for both nano-cylinder and nano-cone structures, in the case of oblique incidence, in the vicinity of 550 nm, and across a wide range of incident angles. Therefore, one could expect there to be forward unidirectional scattering in this region. To clarifying this subject, the suppression of backward scattering for incident angle of ${30^ \circ }$ is exemplified for a nano-cylinder and nano-cone. This angle of incidence in the figure is marked with a star, while the analysis of the first Kerker’s condition for the incident angle of ${30^ \circ }$ for nano-cylinder and nano-cone are given in Fig. 4. The inset of the figure shows the scattered radiation patterns of the structures at corresponding wavelengths.

 

Fig. 3. Forward to backward scattering (F/B) in dB as a function of the incidence angle and wavelength, (a) for a silicon nano-cylinder. (b) for a silicon nano-cone.

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Fig. 4. The normalized real and imaginary parts of the FKC of a silicon nanoparticle located in free space, under oblique incidence. (a) A nano-cylinder illuminated by a p-polarized plane wave in XZ plane with ${30^ \circ }$ angle of incidence from the z axis. (b) A nano-cone illuminated by a p-polarized plane wave in XZ plane with ${30^ \circ }$ angle of incidence from the z axis.

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As mentioned before, the advantage of extracting the generalized Kerker’s conditions, in terms of the polarizabilities in this paper, is that the polarizability tensors of a particle are its characteristic parameters which remain unchanged when varying the direction and polarization of incidence wave. Therefore, once calculated, they can be used as an equivalent of the particle in any electromagnetic problems.

Another interesting point for these nanoparticles can be seen under normal illumination. In this case, these nanoparticles have forward- and backward-scattering at two separate frequencies. To show that, the electric and magnetic polarizabilities of the nano-cylinder are depicted in Fig. 5. As can be observed, at wavelengths of 576nm and 494nm, the real parts of the polarizabilities match the same and opposite signs, respectively. Figure 5(b) shows the 2D scattering patterns at these wavelengths, that demonstrate zero backward and near-zero forward scattering, respectively. As mentioned in literature [33,34], for any non-invisible system, the second Kerker’s condition (zero forward scattering) is strictly forbidden by the optical theorem, resulting in the support of near-zero forward scattering only, which is also seen in Fig. 5(b).

 

Fig. 5. (a) The normalized real and imaginary parts of the FKC of a silicon nano-cylinder of radius 50 nm and height 140 nm, under normal illumination in XZ plane (red arrow in (b) label the incident direction). (b) The scattered radiation patterns of the nano-cylinder at 494 nm and 576 nm.

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4. Unidirectional scattering using substrated nanoparticles

In the previous section, the generalized Kerker’s conditions were examined for nanoparticles located in free space under normal and oblique incidence, and it was demonstrated that a nano-cylinder and a nano-cone and under normal incidence, could support both Kerker’s conditions at two different frequencies. However, these resonances are in the off-resonant range, where the scattering is weak. Since the suppression of backward scattering is a property of valuable practical interest, this section demonstrates how a modification in the above structures could generate an overlap between the ED and MD resonances, to achieve forward unidirectional scattering across a wide frequency band. Moreover, the effect of the incident angle in the scattering response of the proposed structures is investigated.

In this section, nanoparticles are considered in the presence of a dielectric substrate, to benefit from the interaction of the nanoparticle with the substrate. Although the presence of a substrate makes the structure mechanically more robust and practical, it also affects the particle’s polarizabilities. Therefore, the effects of the substrate on the polarizabilities, and the interaction between the ED and MD resonances, need to be considered. Moreover, the polarizabilities of nanoparticles can be also changed by inserting splits into planes that are perpendicular to their radiation directions, at the geometrical centers of the nanoparticles [35]. To also take into account the practical implementation considerations, this gap can be filled with a dielectric material, such as glass. The magnetic resonance would then be proportional to the gap area surrounded by the circulating currents. Then, by adjusting the gap width, an overlapping of the electric and magnetic resonances would become possible.

To better judge the effects of substrate and inserting split on the properties of silicon nanoparticles, the main component of the polarizability tensors of a single-split silicon nano-cylinder, with a radius, height, and glass gap of 50 nm, 100 nm, and 20 nm, respectively, are depicted in Fig. 6.

 

Fig. 6. The major polarizabilities of single-split silicon nano-cylinder with radius, height, and glass gap of 50 nm, 100 nm, and 20 nm, respectively, located on a semi-infinite glass substrate.

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As can be seen in the figure, in contrast to a silicon nano-cylinder in free space, which has no electromagnetic coupling, the presence of a substrate breaks the symmetry of the structure and creates an electromagnetic coupling effect that is referred to substrate-induced bianisotropy (SIB) [21,24,36]. Furthermore, the introduction of gap enhances the frequency of the MD resonance and shifts towards the electric resonance. As a result, the ED and MD polarizabilities are approximately overlapped.

The forward to backward scattering spectra (F/B) that were calculated versus wavelength and incident angle for substrated single-split silicon nano-cylinder and nano-cone with the aforementioned dimensions are shown in Fig. 7. As can be seen from the figure, across the entire frequency band, and for the case of normal incidence, and the oblique incidence with ${\theta _i}$ ranging from $- {90^ \circ }$ to ${90^ \circ }$, forward-to-backward scatterings are higher than 10 dB. Therefore, these structures can be used for wideband wide-angle unidirectional scattering.

 

Fig. 7. Forward to backward scattering (F/B) in dB as a function of the incidence angle and wavelength, (a) for single-split silicon nano-cylinder, with a radius, height, and glass gap of 50nm, 140nm, and 10nm, respectively, and located on a glass substrate. (b) for single-split silicon nano-cone, with a radius, height, and glass gap of 70nm, 250nm, and 20nm, respectively, and located on glass substrate.

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Figure 8 shows the first Kerker’s condition under oblique incidence of ${15^ \circ }$, in terms of the polarizabilities for substrated single-split nano-cylinder and nano-cone. For a single-split silicon nano-cylinder and nano-cone with the aforementioned dimensions, the polrizabilities are approximately overlapped over a wide spectral range. In the insets of Fig. 8, the scattered radiation patterns of the nanoparticles are depicted at the resonant wavelengths, where the forward-scatterings at these wavelengths are clearly shown.

 

Fig. 8. The real and imaginary parts of the FKC of single split silicon nanoparticles located on a semi-infinite glass substrate, and illuminated by a p-polarized plane wave in XZ plane with ${15^ \circ }$ angle of incidence from the z axis. (a) nano-cylinder, (b) nano-cone.

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It should be noted that the generalized Kerker’s conditions in this paper can be investigated for any arbitrary nanoparticle and plane wave illumination, and these examples are only provided to examine the accuracy of the derived equations.

5. Conclusion

In this paper, a general description of Kerker’s conditions for nanoparticles in homogeneous media and located on dielectric substrates under normal and oblique incidence are presented, while also outlining the connection to the nanoparticle’s polarizability tensors. Since the polarizability tensors of a particle are its characteristic parameters that do not depend on the direction and polarization of incidence wave, once they are obtained, they can be used as an equivalent of the particle in any electromagnetic problems. To present structures that satisfied the Kerker’s conditions, first, the electromagnetic properties of a nanoparticle are analyzed as a function of its constituent material. Then, the dimensions of nanoparticle are engineered to satisfy both the first and second Kerker’s conditions, at separate frequencies under normal illumination. It is observed that by changing the excitation wavelengths, switching could be achieved between forward and backward unidirectional scattering which creates attractive possibilities for manipulating optical pressure forces. In addition, by analyzing the scattering response of these nanoparticles in oblique incidence, it is shown that these structures have a strong forward scattering in the vicinity of specific wavelength, across a wide range of incident angles. Thereafter, to study the substrated nanoparticles, first, the effect of substrates on the nanoparticles’ polarizabilities is assessed. It is shown that the presence of a substrate affects the resonance behaviors of nanoparticles. Then, by investigating the scattering response for different angles of incidence of the proposed substrated nanoparticles, a strong asymmetrical scattering over a wide frequency range and wide angle of incidence is presented.

Appendix A

To check the importance of higher order multipoles for the cases considered in the current paper, here, we compare the contribution of strongest electric and magnetic dipole moments with the strongest electric quadrupole moment in the scattering cross-section of nanoparticles. To calculate the moments for the nanoparticles considered in this paper, the method presented in [26] can be used. According to [26], the relation for calculating moments are:

(15)$$\begin{array}{l} {\textbf p} = \int\limits_V {{\textbf r}.{\boldsymbol{\mathrm \rho }}({\textbf r})} \begin{array}{c} {dv} \end{array} = \frac{1}{{j\omega }}\int\limits_v {\textbf J} ({\textbf r})\begin{array}{c} {dv} \end{array},\\ {\textbf m} = \frac{1}{2}\int\limits_V {({{\textbf r} \times {\textbf J}({\textbf r})} )} \begin{array}{c} {dv} \end{array},\\ {q_{ij}} = \int\limits_V {{r_i}{r_j}\rho ({\textbf r})} \begin{array}{c} {dv} \end{array}. \end{array}$$

where V is the volume of the particle and r is the position vector in the Cartesian coordinate system. Subscripts i, j, k denote components of a Cartesian tensor, and ρ, J, and ${q_{ij}}$ are the induced charge and current density, and the quadrupole moment respectively.

The results are shown in Fig. 9. As can be seen from the figure, the effect of quadrupole moment is really negligible in all cases.

 

Fig. 9. scattering cross section due to the different multipole moments of different nanoparticles. (a) a silicon nano-cylinder located in free space with radius and height of 50 nm and 100 nm, respectively, (b) a silicon nano-cone located in free space with radius and height of 70 nm and 250 nm, respectively, (c) a single-split silicon nano-cylinder located on glass substrate with a radius, height, and glass gap of 50 nm, 140 nm, and 10 nm, respectively, (d) a single-split silicon nano-cone located on glass substrate with a radius, height, and glass gap of 70 nm, 250 nm, and 20 nm, respectively.

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Appendix B

In our previous papers [24,25], we proposed two different methods for calculating the induced dipole moments of nanoparticles either in free space or in the presence of substrate, far field method and induced current method. However, in this paper we have used the induced current method to calculate the dipole moments and then the polarizability tensors.

The structures have been studied in this paper are in two categories: (I) a nanoparticle located in free space, and (II) a nanoparticle located on a semi-infinite dielectric substrate. In this section, we explain step by step how to calculate the polarizabilities for these two cases:

Step1: Illuminating the nanoparticle: First, the nanoparticle is exposed to incident plane waves in HFSS. By these external electromagnetic fields, the balance between the electric charges of the nanoparticle is then affected. The exciting nanoparticle can be modeled and studied by its constituent electric and magnetic moments (p and m).

Step 2: Calculating the constituent moments of the nanoparticle: For this purpose, we use near-field responses of the particle to the plane-wave illumination. In this regard, by integration of induced currents and charges on a particle, its dipolar moments are calculated (Eq. (15) in Appendix A). These integrations are done in Ansys software.

Step 3: Calculating polarizability tensors from dipole moments: In this step, we use Eq. (4). By knowing driving fields and dipole moments, the polarizability tensors can be calculated by simple algebraic addition and subtractions. However, as previously mentioned, the driving fields differ for the case (I) and case (II). The driving fields are equal to incident fields for a nanoparticle in a homogeneous medium (case I). In the case of an individual particle near a half space substrate (case II), the scattered fields from the substrate should be considered when calculating the driving fields. Therefore, driving fields for illumination from upper and lower half spaces can be written as follows (below we consider the specific case when the particle is illuminated by a p-polarization plane wave with the ± z propagation direction):

(16)$$\begin{array}{l} {\textbf E}_0^{ - z} = {\textbf E}_{}^{inc} + {\textbf E}_{}^{ref} = \hat{x}{E_i}({{e^{j{k_1}z}} + {{\textrm{Re} }^{ - j{k_1}z}}} ),\\ {\textbf H}_0^{ - z} = {\textbf H}_{}^{inc} + {\textbf H}_{}^{ref} ={-} \hat{y}\frac{{{E_i}}}{{{\eta _1}}}({{e^{j{k_1}z}} - {{\textrm{Re} }^{ - j{k_1}z}}} ), \end{array}$$

and

(17)$$\begin{array}{l} {\textbf E}_0^{ + z} = {\textbf E}_{}^{trans} = \hat{x}{E_i}T{e^{ - j{k_1}z}},\\ {\textbf H}_0^{ + z} = {\textbf H}_{}^{trans} = \hat{y}\frac{{{E_i}}}{{{\eta _1}}}T{e^{ - j{k_1}z}}, \end{array}$$

where $R = r{e^{ - 2j{k_1}{z_0}}}$ and $T = t{e^{ - j({k_1} - {k_2}){z_0}}}$. Moreover, r and t are the Fresnel reflection and transmission coefficients, and ${\eta _i}$ and ${k_i}$ with i = 1,2 are respectively the characteristic impedance and wavenumber of the corresponding media, and ${z_0}$ is the distance between the interface and the geometrical center of the particle.

In this case, by substituting the driving fields in Eq. (4) and after simple algebraic calculations, the polarizability components could be determined as follows:

(18)$$\begin{array}{l} \alpha _{ix}^{ee} = \frac{{(1 - R)p_i^{ + z} + Tp_i^{ - z}}}{{2{E_i}T}},\\ \alpha _{iy}^{em} = \frac{{(1 + R)p_i^{ + z} - Tp_i^{ - z}}}{{2{E_i}T}}{\eta _1},\\ \alpha _{ix}^{me} = \frac{{(1 - R)m_i^{ + z} + Tm_i^{ - z}}}{{2{E_i}T}},\\ \alpha _{iy}^{mm} = \frac{{(1 + R)m_i^{ + z} - Tm_i^{ - z}}}{{2{E_i}T}}{\eta _1}, \end{array}$$

where ${\alpha ^{ee}},{\alpha ^{mm}},{\alpha ^{em}},$ and ${\alpha ^{me}}$ are electric, magnetic, magneto-electric and electromagnetic polarizabilities, respectively. The polarizabilities extracted in (18) are the transverse components of the polarizability tensors by using the z-directed illuminating waves. To calculate the other components of the polarizability tensors, the incident waves propagating along the other directions could be used as the illumination sources. It is worth mentioning that, the polarizabilities in Eq. (18) are related to the general case, when a nanoparticle located on a semi-infinite substrate (Case II). For the case of nanoparticle in free space (Case I), we have R=0 and T=1, which means that the driving fields are the same as incident fields. Therefore, the polarizabilities are simplified as follows:

(19)$$\begin{array}{l} \alpha _{ix}^{ee} = \frac{{p_i^{ + z} + p_i^{ - z}}}{{2{E_i}}},\\ \alpha _{iy}^{em} = \frac{{p_i^{ + z} - p_i^{ - z}}}{{2{E_i}}}{\eta _1},\\ \alpha _{ix}^{me} = \frac{{m_i^{ + z} + m_i^{ - z}}}{{2{E_i}}},\\ \alpha _{iy}^{mm} = \frac{{m_i^{ + z} - m_i^{ - z}}}{{2{E_i}}}{\eta _1}. \end{array}$$

Now, to show the accuracy of our method, in Fig. 10, the polarizability tensors of a special case, a silicone sphere placed in free space, are compared with analytical Mie theory. As can be seen in the figure, the results are in good agreement. It is worth highlighting that the Mie theory could only calculate the polarizability tensors of simple cases like a spherical particle located in a homogeneous environment. Therefore, for the cases in this paper (finite cylinder and cone in free space and single split finite cylinder and cone on substrate), Mie's theory cannot be applied. However, our proposed method can be used for extracting the polarizability tensors of arbitrary particles located either in a homogeneous medium, or on a dielectric substrate. To see more comparison for different structures, one can see Ref. [24–26].

 

Fig. 10. Comparison of electric and magnetic polarizabilities of a silicon sphere with a radius 65 nm placed in free space.

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Disclosures

The authors declare no conflicts of interest.

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