## Abstract

Starting from a general description of light scattering by a nanoparticle in homogeneous surroundings and situated near a substrate, we outline the connection to multipole expansion of scattered light and derive conditions and limits on achievable half-space scattering asymmetry, including the possibility of unidirectional scattering along the propagation direction of the incident light (i.e., generalized Kerker conditions). As a way of realizing strongly asymmetric scattering, we perform a parametric study of the optical properties of disk-shaped gap-surface plasmon (GSP) resonators, consisting of a glass spacer sandwiched between two gold disks, with numerical calculations that corroborate the conditions derived from the multipole expansion. Finally, we present proof-of-principle experiments of asymmetric scattering by GSP-resonators on a glass substrate.

© 2015 Optical Society of America

## 1. Introduction

Scattering of light by small particles constitutes one of the most studied branches of light-matter interaction [1] and is subject to a host of daily life observations, such as the blue sky and reddish sunset, while it underlies other optical phenomena, like diffraction and refraction [2]. Light scattering is, in general, dependent on size, shape and composition of the particle, but physical understanding and qualitative predictions can often be gained from studying the archetypical case of scattering by a spherical particle, also known as the Mie theory [3]. For example, it follows from Mie theory that subwavelength-sized homogeneous particles scatter light as an electric dipole, while larger particles may feature both electric and magnetic resonances. These general findings of light scattering by particles are, of course, closely related to multipole theory for which similar scattering characteristics can be derived [2].

In recent decades, in connection with the development and prevalence of nanofabrication techniques, scattering of light by small particles has gained renewed interest, particularly focusing on nanoscale engineering of the strength and direction of scattered light. Here, metallic particles have attracted most attention due to the strong light-matter interaction occurring at the localized surface plasmon (LSP) resonances, thus leading to optical scattering cross sections that typically are an order of magnitude larger than the geometrical counterparts [4, 5]. As prominent examples of applications utilizing light control by metallic nanoparticles, we mention optical antennas [6, 7], sensors [8, 9], plasmon-enhanced photovoltaics [10, 11], cloaking [12], and two- and three-dimensional metamaterials [13–16]. Among those mentioned applications, the first three may also benefit from particle configurations featuring unidirectional scattering, which is typically achieved in two ways. The first approach, also known as the detuned electric dipoles approach [9, 17], uses two or more nanoparticles with electric dipolar responses that are both spectrally (with respect to their LSP resonances) and spatially separated, hereby ensuring at the design wavelength constructive interference in the desired direction while other directions show reduced or suppressed scattering. One particular example of this design approach is the Yagi-Uda antenna [18]. In the second approach of creating unidirectional scattering, one can utilize the fact that light scattered by electric and magnetic dipoles (EDs and MDs) show different parity with respect to scattering in two opposite directions. As a result, nanoparticles featuring spectrally overlapping electric and magnetic dipolar resonances of same strength (at the design wavelength) will display unidirectional scattering by suppressing backward scattering relative to the direction of the incoming wave – a prerequisite known as the first Kerker condition [19, 20]. This kind of unidirectional scattering can be reached by utilizing high-dielectric nanoparticles [21–24] or coupled metallic nanoparticles, like core-shell particles [25, 26] and dimer nanoantennas [27–29]. The latter configuration, typically consisting of two coupled nanorods, can be viewed within the hybridization picture where the electric dipolar resonances of the individual nanorods couple, thus creating electric and magnetic resonances with in-phase and out-of-phase current distributions, respectively [30]. Note that the magnetic resonance has been frequently utilized in optical metamaterials to create negative permeability [31, 32]. Also, it should be noted that in the retardation-based regime the magnetic resonance is related to the lowest order standing-wave gap-surface plasmon (GSP) mode originating from the GSP propagating in the gap between two metal parts and being efficiently reflected at the structure terminations [33, 34]. For this reason, metal-insulator-metal configurations (featuring GSP resonances) are also referred to as GSP-resonators.

In this work, we begin with a general discussion of light scattering by nanoparticles while emphasizing the connection to the associated multipole expansion valid for homogeneous surroundings and in the presence of a substrate. From the multipole expansion of scattered light, we derive conditions for maximum half-space scattering asymmetry and unidirectional scattering along the direction of the incident wave (i.e., generalized Kerker conditions). For both homogeneous and substrated surroundings, we present numerical parametric studies of disk-shaped GSP-resonators, particularly demonstrating the possibility of almost complete suppression of backward scattering that nicely correlates with the derived conditions. Finally, we experimentally probe the asymmetric scattering present in the forward and backward half-spaces of disk GSP-resonators on glass substrates. In regard to related work, we would like to point out that scattering asymmetry arising from the interference between modes of different multipole orders is directly related to the study of Fano resonances in nanoscale configurations [21, 35, 36], likewise a previous numerical study of GSP-like resonators in homogeneous surroundings has documented the influence of electric quadrupole (EQ) moments in light scattering [29]. Here, we specifically derive the Kerker conditions and present limits on half-space scattering asymmetry from general multipole theory, with the influence of a substrate being analyzed.

## 2. Multipole expansion of scattered light

When light interacts with a nanoparticle, being either of metallic or dielectric character, the scattered light is generated by the induced polarization current **J*** _{p}*, which for time-harmonic fields [convention: exp(−

*iωt*)] is given by

**J**

*(*

_{p}**r**) = −

*iω*

**P**(

**r**) = −

*iωε*

_{0}(

*ε*−

_{p}*ε*)

_{d}**E**(

**r**). Here,

*ω*is the angular frequency,

**r**is the spatial coordinate vector,

**P**is the induced polarization,

*ε*

_{0}= 8.854·10

^{−12}F/m is the vacuum permittivity,

*ε*

_{p}_{(}

_{d}_{)}is the relative permittivity of the particle (dielectric surrounding), and

**E**is the electric field inside the particle. Once the current source is defined, we can calculate the scattered electric field anywhere in space by the use of the Green’s dyadic of the reference system (i.e., without the particle)

*μ*

_{0}= 4π · 10

^{−7}H/m is the vacuum permeability and V

*is the volume of the nanoparticle. In order to proceed, we now make two assumptions: first, we limit the discussion to the far-field zone (*

_{p}*λ*≪ |

**r**−

**r**′|) in which light locally can be treated as plane waves; second, we assume the nanoparticle to be small compared to the wavelength) [(V

*)]*

_{p}^{1}

^{/}^{3}≪

*λ*. The first assumption allow us to replace the full Green’s dyadic

*Ĝ*with the simpler far-field part

*Ĝ*

^{FF}that for homogeneous and substrate environments can be written as [37]

*ĝ*and

**N**do not depend on the nanoparticle position

**r**

*′*, and ${k}_{d}={k}_{0}\sqrt{{\epsilon}_{d}}$ is the wave number in the surrounding medium. The second assumption motivate us to Taylor expand the exponential function in Eq. (2)) around the center of mass r

_{0}of the nanoparticle, which results in the following expression for the scattered far-field

**r**=

**r**

*′*−

**r**

_{0}. Equation (3)) can also be written in the more familiar multipole expansion, which for a truncation of the sum at

*n*= 2 reads

*c*is the speed of light in vacuum,

**p**(

**m**) is the electric (magnetic) dipole moment,

*Ô*$(\widehat{M})$ is the electric (magnetic) quadrupole tensor, and

*Ô*is the tensor of the electric octupole moment. It should be noted that Eq. (4)) can be obtained from Eq. (3)) by using the identities $\int}_{{\mathrm{V}}_{p}}(\mathbf{N}\cdot \mathrm{\Delta}\mathbf{r}){\mathbf{J}}_{p}{\mathrm{d}}^{3}{\mathbf{r}}^{\prime}=\omega /(6i)\widehat{Q}\mathbf{N}-\mathbf{N}\times \mathbf{m$ and $\frac{1}{2}{{\displaystyle {\int}_{{\mathrm{V}}_{p}}\left(\mathbf{N}\cdot \mathrm{\Delta}\mathbf{r}\right)}}^{2}{\mathbf{J}}_{p}{\mathrm{d}}^{3}{\mathbf{r}}^{\prime}=\omega /(6i)\widehat{O}(\mathbf{NN})-\frac{1}{2}(\mathbf{N}\times \widehat{M}\mathbf{N})$ [37], with the multipole moments up to quadrupole order defined as

Here, *Î* is the 3 *×* 3 identity matrix, while expressions like Δ**rJ*** _{p}* and

**J**

*Δ*

_{p}**r**correspond to the outer product between the two vector quantities. It is worth noting that except from the electric dipole moment all multipole moments depend on the origin of the multipole expansion

**r**

_{0}, hereby emphasizing that multipoles are not uniquely defined, nor is the pace with which the series converges. In this work, we conventionally choose the center of mass of the nanoparticle as the origin of the multipole expansion, as this choice typically only requires the calculation of the lowest order multipoles for satisfactory description of the scattering process. Moreover, we choose to work with traceless and symmetric quadrupole tensors [see Eqs. (7)) and (8))] which, in principle, also leads to the existence of toroidal multipole moments [38]. These moments, however, are typically vanishingly small and, hence, ignored in the following discussion.

## 3. Unidirectional scattering from nanoparticles in homogeneous surroundings

In this section we derive the conditions for unidirectional scattering in the forward and backward directions and half-spaces (relative to the direction of the incident wave) for nanoparticles in homogeneous surroundings. As a way to realize strong scattering asymmetry, we perform a parametric numerical study of disk-shaped GSP-resonators, hereby verifying the delicate interplay between different multipole moments and the effect on scattering asymmetry.

#### 3.1. The generalized Kerker conditions and limits on half-space scattering asymmetry

Scattering of a nanoparticle in homogeneous surroundings is interesting not only from the purely theoretical viewpoint but also from the viewpoint of practical applications, as it is related to the scattering by a dilute suspension of particles in a liquid or particles placed on a substrate covered with index-matching oil. In either case, the far-field Green’s dyadic reads

*r*= |

**r**| and

**N**=

**n**=

**r**/

*r*is the unit vector in the direction of observation. By substituting Eq. (9)) into Eq. (4)) and using the identity (

*Î*−

**nn**)

**a**=

**n**

*×*(

**a**

*×*

**n**), we can write the scattered far-field (for multipole moments up to quadrupole order) as

Regarding asymmetric scattering, it is important to notice how the contribution from the MD and EQ changes sign when scattered light is viewed in two opposite directions, i.e., along *±***n**. It transpires that strongly asymmetric scattering may occur in properly designed configurations due to either constructive or destructive interference between multipoles (of the same strength) depending on the direction of observation. In order to carefully examine the conditions for asymmetric scattering, we will now consider an illustrative example in which a *x*-polarized plane wave propagates along the +*z*-direction and interacts with a highly symmetric nanoparticle that features mirror symmetry in all three orthogonal planes defined by the coordinate system and intersecting **r**_{0}. An example of such a configuration is shown in Fig. 1(a), where the symmetry of the nanoparticle leads to an induced polarization current of the form **J*** _{p}* = (

*J*, 0, 0)

_{x}*, thus resulting in the only non-zero multipole components being*

^{T}*p*,

_{x}*m*,

_{y}*Q*=

_{xz}*Q*, and

_{zx}*M*=

_{yz}*M*. At the same time, we are interested in the forward and backward scattered electric field [corresponding to

_{zy}**n**= (0, 0,

*±*1)

*] which only contains the*

^{T}*x*-component given by

Here, it is readily seen that the contribution from MD and EQ changes sign when evaluating forward and backward scattered light, respectively. Moreover, complete suppression of backward scattering (i.e., ${E}_{sc,x}^{\text{FF},-}=0$) can be achieved when

which is the well-known Kerker condition generalized to take into account quadrupole moments. As also pointed out elsewhere [20, 24], a similar complete suppression of forward scattering cannot be achieved for passive particles since the terms in square brackets in Eq. (11)) all feature positive imaginary parts due to causality. The suppression of forward scattering is mostly pronounced for off-resonant conditions (meaning weak scattering), where the imaginary parts are small and the real parts satisfy the equationThe above equation is known as the second Kerker condition, now also generalized to the quadrupole order. Nevertheless, since optimum suppression of forward scattering occurs away from resonances, and suppression of backward scattering is typically of more practical interest, we focus in the next subsection (containing a numerical study of disk-shaped GSP-resonators) on the first Kerker condition [Eq. (13))].

When studying unidirectional or, more broadly, asymmetric scattering it is also of general interest to know the asymmetry of light scattered into the forward and backward half-spaces. Such quantities were studied by Kerker for magnetic spheres [19] and, hence, inspire us to derive similar expression for the multipole expansion of the scattered light in the highly symmetric case discussed above. By using the fact that the scattering Poyting’s vector in the far-field can be expressed as
${\mathbf{S}}_{sc}^{\text{FF}}=1/(2{\eta}_{d}){\left|{\mathbf{E}}_{sc}^{\text{FF}}\right|}^{2}\mathbf{n}$, where *η _{d}* is the wave impedance in the surroundings, and performing the necessary integrations, we obtain the following expressions for the power flowing into the forward and backward half-spaces

^{*}denotes complex conjugate. It should be noted that the first four terms in the square bracket correspond to light scattered by the individual multipoles and, hence, scatter light symmetrically into the two half-spaces. The latter three terms, on the other hand, represent the asymmetric scattering (i.e., sign difference in the two half-spaces) that occurs due to interference between the multipoles. It is important to notice that maximum scattering asymmetry into the two half-spaces does not necessary coincide with the Kerker conditions, nor is it, in general, possible to reach complete suppression of scattering into one of the half-spaces. Regarding suppression of scattering into the backward half-space, we find by analyzing Eq. (15)) that the highest scattering asymmetry ${P}_{sc}^{+}/{P}_{sc}^{-}$ is ≃ 157.4 when

*m*=

_{y}*αv*,

_{d}p_{x}*Q*=

_{xz}*β*6

*i/k*,

_{d}p_{x}*M*=

_{yz}*γi*2

*v*, and the proportionality constants take on the values

_{d}/k_{d}p_{x}*α*= 1 and $\beta =\gamma =\sqrt{8/3}-1$. Note that the real values of

*α*,

*β*and

*γ*indicate that the four terms in Eq. (10)) should be in-phase for optimum scattering asymmetry. Also, note that the first Kerker condition [Eq. (13))] corresponds to 1 +

*γ*=

*α*+

*β*, meaning that it is also satisfied at the condition for maximum scattering asymmetry in the two half-spaces. Moreover, as a figure of merit for possible scattering asymmetry in GSP-like resonators, it is instructive to consider the case of negligible magnetic quadrupole moment (i.e.,

*γ*= 0), which leads to maximum scattering asymmetry of ${P}_{sc}^{+}/{P}_{sc}^{-}\simeq 17.6$for $\alpha =\sqrt{12/17}$ and $\beta =5\sqrt{51}$. Interestingly, these parameters do not coincide with the first Kerker condition. Finally, it ought to be mentioned that when only dipole moments contribute to scattering (i.e.,

*β*=

*γ*= 0) the scattering asymmetry reaches ${P}_{sc}^{+}/{P}_{sc}^{-}=7$ when

*α*= 1.

#### 3.2. Unidirectional scattering by gap-plasmon resonators

In this subsection we will exemplify the suppression of backward scattering by using disk-shaped GSP-resonators, as depicted in Fig. 1(a). These resonators consist of a glass disk of diameter *d* and thickness *t _{s}* sandwiched between two equally sized gold disks of thickness

*t*. The surrounding medium is assumed to be air. It should be noted that all numerical calculations are performed using the commercially available finite element software Comsol Multiphysics, ver. 5.1, with the gold permittivity described by interpolated experimental values [39] and the constant refractive index of glass assumed to be 1.45. In order to limit the number of free parameters in the geometry, we now fix the disk diameter and metal thickness to

*d*= 140 nm and

*t*= 30 nm, respectively, and study the scattering and absorption cross sections as a function of wavelength and spacer thickness for a

*x*-polarized incident plane wave propagating along the

*z*-direction [Figs. 1(b) and 1(c)]. As expected, the spectra feature two peaks that spectrally separate when the spacer thickness is decreased due to the stronger interaction between the gold disks. Moreover, the long-wavelength resonance is spectrally narrower and becomes progressively suppressed in the scattering cross section as the spacer thickness decreases. Referring to the mode profiles of the two resonances [Figs. 1(d) and 1(e)], it is clear that the short-wavelength peak corresponds to the ED resonance, as the lack of any appreciable concentration of the magnetic field in the spacer region and the behavior of the electric near-field are consequences of in-phase polarization currents in the two nanodisks. In contrast, the long-wavelength peak, thus corresponding to the magnetic resonance, features out-of-phase polarization currents in the two gold nanodisks which is signified by strong enhancement of the magnetic field in the spacer and an electric near-field that connects the two gold disks.

Having clarified the role of the two resonances, it is instructive to consider the multipole expansion of the scattered light. Figures 1(f)–1(h) present the decomposition of the scattered light for three different spacer thicknesses, which is obtained by evaluating the first four terms in Eq. (15)) separately. Firstly, one notices that the magnetic quadrupole contribution is vanishing small in all cases and, hence, will be neglected in the remaining discussion. Secondly, the contribution from ED is practically independent of the spacer thickness, while MD and EQ contributions are increasing for larger spacer thicknesses, with MD even exceeding ED in a small wavelength range for *t _{s}* = 90 nm. The increase in MD and EQ for increasing spacer thickness is somewhat expected since their moments are proportional to the gap area surrounded by circulating currents [Fig. 1(e)]. The rate of increase, however, is different for MD and EQ as seen in Figs. 1(f)–1(h), where the MD-EQ ratio to scattering decreases from ~8.7 to ~3.4 when spacer thickness increases from 30 nm to 90 nm. Note that the EQ contribution can not, in general, be disregarded when optimizing GSP-resonator configurations for unidirectional scattering.

We now consider the scattering into the forward and backward half-spaces with respect to the direction of the incident plane wave. The division of the total scattering cross section into the two half-spaces are depicted in Figs. 2(a) and 2(b) as a function of wavelength and spacer thickness. It is evident that strong scattering asymmetry exists, with forward scattering following the increase in MD and EQ for increasing spacer thicknesses and displaying a maximum at the magnetic resonance. The level of backward scattering, on the other hand, is only weakly influenced by the spacer thickness, though it is characterized by a minimum at the long-wavelength side of the magnetic resonance [Fig. 2(b)]. In order to better understand the markedly different scattering in forward and backward half-space, Figs. 2(c) and 2(d) display the symmetric and asymmetric scattering contribution, respectively, as calculated from Eq. (15)). It is clear that the asymmetric contribution, corresponding to the interference between the multipoles, can for certain (*t _{s}, λ*)-values become close in value to the symmetric part of the scattering, hence leading to significant suppression in backward space and approximately a doubling of light scattering in the forward space [with respect to the symmetric scattering in Fig. 2(c)]. The full-wave numerically calculated half-space scattering asymmetry, as obtained by dividing Fig. 2(a) with Fig. 2(b), is shown in Fig. 2(e), together with the derived conditions for maximum scattering asymmetry:
${m}_{y}=\sqrt{12/17}{v}_{d}{p}_{x}$and
${Q}_{xz}=30i/(\sqrt{51}{k}_{d}){p}_{x}$. It is seen that the maximum scattering ratio of ≃ 15.1 is reached at (

*t*) = (90, 700) nm but, otherwise, the ratio (for the proper wavelength) remains high and above ~ 14 for

_{s}, λ*t*> 90 nm. This insensitivity to spacer thickness follows naturally from the derived conditions, as it is evident that the four solid lines are close to each other for

_{s}*t*> 90 nm, but do not intersect each other simultaneously. Overall, for a large range of (

_{s}*t*)-parameters we approach the condition for maximum scattering asymmetry, but the condition is never perfectly satisfied. This is also the reason why the maximum scattering asymmetry never reaches the ultimate value of 17.6, as derived from Eq. (15)).

_{s}, λAs a final numerical experiment, we demonstrate the possibility to satisfy the first Kerker condition [i.e., Eq. (13))] using disk GSP-resonators. In evaluating the scattering asymmetry in the forward and backward direction, we calculate the directivity
$D=10{\mathrm{log}}_{10}\left({S}_{sc,z}^{\text{FF},+}/{S}_{sc,z}^{\text{FF},-}\right)$, where
${S}_{sc,z}^{\text{FF},\pm}$ is the *z*-component of the scattered Poynting vector evaluated at *z* = *±*∞, respectively. The numerically obtained directivity is displayed in Fig. 3(a) together with the real and imaginary parts of Eq. (13)). It is evident that one experiences the largest asymmetry at the long-wavelength side of the magnetic resonance (when the multipole moments are in-phase), with the maximum directivity of ~ 38 dB at (*t _{s}, λ*) = (65, 720) nm closely coinciding with the ful-fillment of the Kerker condition, i.e., the intersection of the blue and green curves. We ascribe the small discrepancy between the intersection of the two curves and maximum directivity to the finite discretization of the (

*t*)-space, but also the fact, that the scattered light is not fully equivalent to the interaction of ED, MD and EQ multipoles, will slightly change the condition. Moreover, numerical noise might also play a role when ${S}_{sc,z}^{\text{FF},-}$ approaches zero. In any case, however, it is clear that scattering from GSP-resonators in homogeneous environments can with high accuracy be described by ED, MD and EQ moments. As a final comment, Fig. 3(b) shows the normalized radiation pattern at maximum directivity, clearly demonstrating perfect suppression of backward scattering, while the scattering asymmetry into the two half-spaces is ${P}_{sc}^{+}/{P}_{sc}^{-}~9$ with a normalized scattering cross section of ~ 4. This shows that suppression of backward scattering can be achieved for configurations strongly interacting with the incident light.

_{s}, λ## 4. Unidirectional scattering from particles near substrates

In the following, we will discuss the important practical configuration of nanoparticles situated near a substrate. In the first subsection, we present the Kerker conditions and discuss half-space scattering asymmetry when taking into account the effect of the substrate, while the second subsection includes a parametric study of GSP-resonators on a glass substrate.

#### 4.1. The generalized Kerker conditions and limits on half-space scattering asymmetry

When considering an arbitrarily-shaped nanoparticle on top of a substrate, we assume the substrate to be of semi-infinite extend (*z* < 0) with the interface at *z* = 0 and described by the permittivity *ε _{s}*. It should be noted that the light generated by the induced polarization current in the nanoparticle will either directly propagate away from the interface, reflect at the interface before propagating away, or transmit into the substrate. For this reason, the electric far-field in the upper and lower half-spaces may be written as
${\mathbf{E}}_{sc}^{\text{FF},+}={\mathbf{E}}_{sc}^{0,\text{FF}}+{\mathbf{E}}_{sc}^{r,\text{FF}}$ and
${\mathbf{E}}_{sc}^{\text{FF},-}={\mathbf{E}}_{sc}^{t,\text{FF}}$, respectively, where
${\mathbf{E}}_{sc}^{0,\text{FF}}$ [as defined in Eq. (10))] represents the part of the light in the upper medium that directly propagates away from the interface, and
${\mathbf{E}}_{sc}^{r,\text{FF}}$ and
${\mathbf{E}}_{sc}^{t,\text{FF}}$ are the reflected and transmitted fields. Due to the increased mathematical complexity in describing the latter two field components, we have moved the explicit expressions to the Appendix, focusing here on the main results and physical consequences of the material interface. Moreover, in relation to Kerker conditions we

*conventionally*define, in contrast to the homogeneous case studied above, the incident wave to be propagating in the −

*z*-direction, which means that suppression of backward scattering corresponds to minimizing ${\mathbf{E}}_{sc}^{\text{FF},+}$. In fact, if we again assume

*p*,

_{x}*m*,

_{y}*Q*=

_{xz}*Q*, and

_{zx}*M*=

_{yz}*M*to be the only non-zero multipole moments, the condition for complete suppression of backward scattering occurs when

_{zy}*p*-polarized light, and

*z*

_{0}is the

*z*-component of the center of the multipole expansion. In comparison with the expression for homogeneous surroundings [Eq. (13))], it should be noted that the negative sign on the right hand side of the equation occurs due to the different choices of coordinate system with respect to forward/backward scattering, while the fraction $\mathrm{\Theta}=(1+{r}_{p}{e}^{i2{k}_{d}{z}_{0}})/(1-{r}_{p}{e}^{i2{k}_{d}{z}_{0}})$ is a result of interference between direct and reflected light. It is interesting to note that in most cases |Θ| > 1 (i.e.,

*ε*), meaning that unidirectional scattering occurs for smaller MD and EQ contributions than in homogeneous surroundings. Moreover, Θ is, in general, complex, hereby underlining that so too are the proportionality constants

_{s}> ε_{d}*α*and

*β*that relate the MD and EQ contributions to the ED.

Considering the suppression of forward scattered light, it corresponds to evaluating
${\mathbf{E}}_{sc}^{\text{FF},-}$ along the −*z*-direction and requiring the field to be zero, which results in the second Kerker condition

At first sight, it may seem that, unlike the case of homogenous surroundings, it is possible to completely suppress forward scattering. However, it should be noted that reverting back to the coordinate system of Fig. 1(a) implies a sign change on *m _{y}* and

*Q*, hence reproducing the result of Eq. (14)). Moreover, it should be emphasized that the multipole moments considered in Eqs. (16)) and (17)) [and defined in Eqs. (5))–(8))] are

_{xz}*effective*moments that incorporate the interaction with the substrate. Consequently, typical causality constraints, like Im {

*p*} > 0, are no longer obeyed and, for this reason, it is difficult to make any general conclusions on the level of scattering suppression.

_{x}In the remainder of this section, we will focus on asymmetric half-space scattering. The explicit expressions of power scattered into the upper and lower half-spaces can be found in the Appendix, but generally speaking the scattered power can be represented as
${P}_{sc}^{\pm}={P}_{d}^{\pm}+{P}_{i}^{\pm}$, where
${P}_{d}^{\pm}$ is the sum of direct contributions from the individual multipoles, while
${P}_{i}^{\pm}$ represents the sum of interference terms between the different multipoles and can take on both positive and negative values. The two terms may be seen as the equivalents of the symmetric and asymmetric terms for homogeneous environments [Eq. (15))], though the presence of the substrate entails
${P}_{d}^{+}\ne {P}_{d}^{-}$ and
${P}_{i}^{+}\ne {P}_{i}^{-}$. In this work, we want to maximize the scattering asymmetry
${P}_{sc}^{-}/{P}_{sc}^{+}$, which corresponds to finding the proper relation between the multipoles so practically all light is scattered in the forward direction. It is important to realize that the proportionality constants *α*, *β*, and *γ* for maximum scattering asymmetry do depend on both the relative refractive index
$N=\sqrt{{\epsilon}_{s}/{\epsilon}_{d}}$ and the ratio between the height of the multipole expansion and the wavelength in the upper medium, i.e, *z*_{0}*/λ _{d}*. Here, we limit the discussion to the case relevant for GSP-like resonators in which the contribution from MQ can be neglected (

*γ*= 0). The theoretical maximum achievable scattering asymmetry and associated proportionality constants are displayed in Figs. 4(a) and 4(b)–4(e), respectively, as a function of

*N*and

*z*

_{0}

*/λ*. It is clear that a subwave-length particle (

_{d}*z*

_{0}≪

*λ*) situated on a substrate shows an increasing urge to scatter light into the substrate as the refractive index contrast increases, with half-space scattering asymmetry of the order ~ 10

_{d}^{3}and ~10

^{5}for the realistic case of nanoparticles in air and placed atop a glass (

*N*≃ 1.5) and silicon substrate (

*N*≃ 3.5), respectively. The strong scattering asymmetry is attributed to the near-field interaction with the substrate, which diminishes as the center of mass (i.e., center of multipole expansion) of the nanoparticle moves away from the interface. In fact, when

*z*

_{0}

*/λ*> 0.25 the ratio ${P}_{sc}^{-}/{P}_{sc}^{+}$decreases with increasing

_{d}*N*, which we ascribe to the negligible near-field interaction and the increasing magnitude of the interface reflection coefficient, thus preventing nanoparticles from efficiently scatter light into the substrate. In achieving the maximum scattering asymmetry for a certain (

*z*

_{0}

*/λ*) combination, it is worth noting that the associated multipole proportionality constants (

_{d}, N*α*and

*β*) also display dependencies on those parameters [Figs. 4(b)–4(e)], particularly featuring an oscillatory behavior with respect to

*z*

_{0}

*/λ*. We note that this dependence follows naturally from the fact that the electric field in the upper medium represents the interference of direct and reflected light.

_{d}In completing the theoretical study of maximum achievable half-space scattering asymmetry, we also discuss the relevant situation of pure ED and MD scattering (*β* = *γ* = 0) near a substrate [Figs. 4(f)–4(h)]. It is seen that the behavior of the scattering asymmetry and proportionality constant as a function of *z*_{0}*/λ _{d}* and

*N*is qualitatively equivalent to the previous case of both ED, MD, and EQ scattering. The level of scattering asymmetry, however, is reduced, with nanoparticles on top of glass and silicon substrates demonstrating maximum asymmetry of the order ~ 10

^{2}and ~ 10

^{3}, respectively.

#### 4.2. Unidirectional scattering from GSP-resonator situated on a glass substrate

In order to better judge on the influence of a glass substrate on the optical properties of GSP-resonators, we continue the numerical study with disk-shaped resonators, as depicted in Fig. 5(a). In the following calculations we fix the disk diameter to *d* = 130 nm and the gold thickness to *t* = 30 nm, while the spacer thickness and wavelength are the parameters to be varied. The scattering and absorption cross sections are displayed in Figs. 5(b) and 5(c), and it is seen that the presence of the glass substrate does not fundamentally alter the optical properties. Similar to the case of homogeneous surroundings (Fig. 1), the scattering and absorption spectra feature two optically-different resonances that separate as the spacer thickness is decreased. Moreover, by analyzing the electric and magnetic near-fields [Figs. 5(d) and 5(e)] it is evident that the short- and long-wavelength resonances correspond to in-phase and out-of-phase induced polarization currents, hereby resulting in an electric and magnetic response, respectively. This conclusion is further substantiated by an illustrative multipole decomposition of the scattered light into ED, MD, and EQ contributions [Figs. 5(f)–5(h)]. It is clear that the short-wavelength resonance is purely of ED character, while the long-wavelength counterpart displays a dominant MD response, though the EQ contribution becomes progressively more important as the spacer thickness increases. Interestingly, and unlike the case of homogeneous surroundings, the interaction of the polarization current with the substrate results in an anti-resonant ED response at the magnetic resonance, thus ensuring a scattered field of purely MD and EQ origin.

Having clarified the fundamental optical properties of GSP-resonators in air and situated on a glass substrate, we proceed with the decomposition of scattered light into direct and interference contributions evaluated in both half-spaces (Fig. 6). Regarding the direct contribution [Figs. 6(a) and 6(d)], corresponding to the sum of power scattered by the ED, MD, and EQ multipoles, it is apparent that the presence of the resonator at the material interface entails a dominant scattering in the forward (i.e., substrate) direction, particularly pronounced at the ED resonance. In contrast, the contribution to scattering from interference between the multipoles [Figs. 6(b) and 6(e)] is close to be, although not exactly, of same magnitude but opposite sign. As the interference contribution changes sign across the magnetic resonance, it is evident that the forward-to-backward scattering asymmetry must be maximized at the long-wavelength side of the resonance. This is also seen in the total scattering cross sections of the two half-spaces [Figs. 6(c) and 6(f)], where backward scattering features a clear minimum that approaches zero on the long-wavelength side of the magnetic resonance, thus giving rise to strongly asymmetric scattering. The exact level of half-space scattering asymmetry, as found by dividing Fig. 6(c) with Fig. 6(f), is shown in Fig. 7(a) together with the appropriate conditions for maximum asymmetry (solid lines). We emphasize that the theoretical conditions for maximum scattering asymmetry result from the calculated multipoles [using Eqs. (5))–(7))] and the proportionality constants in Figs. 4(b)–4(e) when *N* = 1.45 and the center of the multipole expansion is related to the spacer thickness by *z*_{0} = *t* + *t _{s}*/2. It is seen that ultimate scattering asymmetry, as indicated by a perfect overlap of all four solid lines in one or several (

*t*) points, is not achieved in the studied GSP-resonators, though a contraction of three of the lines (blue, green, magenta) at maximum scattering asymmetry for

_{s}, λ*t*> 25 nm does demonstrate ED, MD, and EQ multipole moments that in unison create strongly asymmetric scattering. Since the fraction of scattering from EQ relative to the ED and MD contributions diminishes as the spacer thickness decreases [see Figs. 5(f)–5(h)], the region in Fig. 7(a) for

_{s}*t*< 25 nm approximates scattering from a GSP-resonator featuring ED and MD responses only. This explains somewhat the scattering asymmetry of the order 10

_{s}^{2}– 10

^{3}, although the numerical values actually in several cases exceed the theoretically predicted maximum. For example, the largest scattering asymmetry in numerical calculations amounts to ~ 600 for (

*t*) = (17.5 nm, 858 nm), with the theoretical limit for the pure ED-MD configuration being ~ 100. We ascribe this discrepancy to the crucial fact that disk GSP-resonators (for any spacer thickness) are always situated on the substrate, while the scattering from the associated collection of point multipoles assumes a distance to the interface of

_{s}, λ*z*

_{0}. For this reason, a part of the near-field of disk GSP-resonators will always interact with the interface, which is not accounted for in the multipole description of scattering.

Finally, we would like to show that the first Kerker condition, representing complete suppression of back-scattered light, can be satisfied for GSP-resonators situated on a glass substrate. Figure 7(b) displays the directivity together with the real and imaginary part of Eq. (16)). It is clearly seen that the maximum directivity of ~ 30 dB at *t _{s}* = 15 − 17.5 nm closely coincides with the intersection of the blue and green solid lines. As a final remark, Fig. 7(c) shows the radiation pattern for (

*t*) = (17.5 nm, 858 nm), which confirms (almost) complete suppression of scattered light in the backward direction and half-space. It should be noted that the kink observed in the radiation pattern at angles ~ 226

_{s}, λ*°*and 314

*°*signifies, cf. the Appendix, the transition between the allowed and forbidden zones. As a final remark, we note that the normalized scattering cross section at the first Kerker condition is only ≃ 0.8, so one might consider for practical applications to increase the spacer thickness at the expense of slightly decreased scattering asymmetry.

## 5. Experiment: probing the scattering asymmetry in GSP-resonators

From the above theoretical and numerical discussion of light scattering by GSP-resonators, it is clear that light scattered in the forward and backward half-spaces shows noticeably different behavior with respect to both magnitude and spectral response, thus allowing for a strongly asymmetric response. Incited by this property, we have as a proof-of-principle experiment fabricated (using standard electron beam lithography) arrays of disk-shaped GSP-resonators on a glass substrate with disks and spacer thicknesses (determined by atomic force microscopy measurements) of 30 nm and disk diameters being 120 nm and 130 nm (measured with scanning electron microscopy). The disk and spacer materials are gold and silicon dioxide, respectively. As a way of probing the scattering response of GSP-resonators, we have performed dark-field spectroscopy on the individual resonators, as schematically depicted in Fig. 8(a). During forward scattering measurements, the incident light (that interacts with the sample) originates from the lower dark-field condenser lens with incidence angles varying between 53*°* – 66*°* from the surface normal (red lines), and the light scattered into the upper medium is collected by a *×*50 (NA=0.75) objective. For backward scattering measurements, the incident light emanates from the upper objective with incidence angles between 68*°* – 77*°* from the surface normal (blue lines), while the light scattered into the upper medium is collected in the same way as for forward scattering measurements. It should be noted that the spectra from the individual resonators are obtained by utilizing a 150 *µ*m pinhole in the image plane. In the same way, we remove background noise from the spectra by recording dark-field spectra from the glass-air interface next to the resonators. In order to remove the spectral dependencies of the halogen lamp and optical setup, the dark-field spectra in Fig. 8(b) are all normalized by scattering from a 700 nm large gold disk. It should be noted that it has not been possible to faithfully record dark-field spectra for wavelengths larger than 800 nm due to scattering intensity being reduced to the noise level of background scattering.

We would like to point out that several complicating factors do not allow us to directly compare the obtained scattering spectra with the numerical counterparts [see Figs. 6(c) and 6(f) for *t _{s}* = 30 nm]. First of all, the wavelength-dependence of the reference particle is imprinted on the scattering spectra. Secondly, the excitation condition with a cone of inclined light is very much different from a normal incident plane wave, which is expected to change the relative strength of the two resonances and smear out details in the scattering spectra. Finally, it is well known that planar fabrication of GSP-like structures entails inclined side walls, thus making the two gold disks of different size [31], while evaporated gold typically features significant higher losses than tabular values from thin gold films [42]. In addition, Ohmic losses are enhanced by the presence of 3 nm adhesive titanium layers. For this reason, experimental scattering spectra are expected to feature less pronounced resonances with decreased (increased) spectral separation (linewidth), hereby making the electric and magnetic resonances harder to discern. With all the above reservations, we emphasize that the key objective in these proof-of-principle experiments is to observe scattering asymmetry related to the interference of different multipole moments.

Returning to the scattering spectra in Fig. 8(b), it is evident that light is resonantly scattered with an increase and redshift of the peak value for increasing disk diameter. Unlike numerical calculations, we only observe one distinct peak in the spectra and the level of scattering is roughly the same in the two half-spaces. The presence of both ED and MD (and probably weak EQ) multipole moments, however, is revealed by the slope of decay in scattering intensity on the long-wavelength side of the resonance. Here, it is seen that the constructive and destructive interference between the multipoles for forward and backward directions, respectively, leads to a faster decrease in light scattering as a function of wavelength for backward scattered light compared to forward scattering, thus given rise to noticeable scattering asymmetry for *λ* > 750 nm [see inset in Fig. 8(b)].

## 6. Conclusion

In summary, we have presented a general description of light scattering by nanoparticles in homogeneous surroundings and situated near substrates, while also outlining the connection to multipole expansion of scattered light. The multipole expansion allowed us to derive conditions and limits on achievable half-space scattering asymmetry for multipole contributions up to quadrupole order, including the possibility of unidirectional scattering along the propagation direction of the incident light (i.e., the so-called generalized Kerker conditions). In order to realize strongly asymmetric scattering by suppressing backward scattering (defined by the direction of the incident wave), we performed a parametric study of light scattering from disk-shaped GSP-resonators, featuring both ED, MD, and EQ contributions, which enabled us to recognize configurations satisfying the first Kerker condition or demonstrating half-space scattering asymmetry larger than 10 and 100 for homogeneous and substrated environments, respectively. Finally, proof-of-principle experiments were conductor on gold disk GSP-resonators situated on a glass substrate. Despite the fact that several factors complicated a clear experimental verification of strong scattering asymmetry, the recorded dark-field spectra did display features of interference between light scattered from ED and MD moments. Overall, we believe that the derived formulas directly connecting scattering asymmetry with multipole moments of nanoparticles, together with the thorough study of the optical properties of GSP-resonators, shed light on key aspects of light engineering at the nanoscale. As such, this work can find usage in establishing a connection between required scattering characteristics (for a certain application) and guidance towards appropriate nanoparticle composition, shape, and size or ensembles thereof.

## Appendix: Light scattering from nanoparticles near substrates

As stated in the main manuscript, light scattered from a nanoparticle on (or close to) a substrate can be split into three terms corresponding to direct propagation of light in the upper medium, propagation of light in the upper medium after reflection at the interface, and transmission of light into the lower medium (i.e., substrate). The corresponding far-field Green’s dyadics take the form [37]

**n**= (

*n*,

_{x}*n*,

_{y}*n*)

_{z}*= (*

^{T}*x*,

*y*,

*z*)

*/*

^{T}*r*,

**ñ**= (

*x*,

*y*, −

*z*)

*/*

^{T}*r*, $\tilde{\tilde{\mathbf{n}}}={(x,y,-\sqrt{{r}^{2}-{N}^{2}{\rho}^{2}})}^{T}/r$, $\rho =\sqrt{{x}^{2}+{y}^{2}}$ and $N=\sqrt{{\epsilon}_{s}/{\epsilon}_{d}}$. The tensors $\widehat{R}$ and $\widehat{T}$ can be written as [43]

*n*=

_{ρ}*ρ/r*, and

*r*,

_{p}*r*,

_{s}*t*, and

_{p}*t*are the Fresnel reflection and transmission coefficients for p- ans s-polarized light. The Fresnel coefficients, expressed with respect to the polar angle

_{s}*θ*measured from the

*z*-axis, are given by

It is important to notice that the transmission coefficients are written with respect to the angle of transmission *θ _{t}*, which in our configuration corresponds to the lower half-space defined by

*θ*[

_{t}∈*π/*2;

*π*].

Having defined the necessary Green’s dyadics for the description of light scattering by nanoparticles on substrates, we can write the expressions for the scattered far-field in the upper (+) and lower (−) half-spaces

*p*,

_{x}*m*,

_{y}*Q*=

_{xz}*Q*, and

_{zx}*M*=

_{yz}*M*. The associated scattered far-field along

_{zy}**n**= (0, 0,

*±*1)

*only contains a*

^{T}*x*-component given by

From the above equations, one can easily derive the generalized Kerker conditions, as presented in Eqs. (16)) and (17)).

Having defined general expressions for the electric far-field in Eqs. (25)) and (26)), it is also possible, though rather tedious, to calculate the far-field Poynting vector and obtain expressions for the power scattered in the two half-spaces. The interaction of the light with the extended interface, however, do not allow us to derive explicit expressions for the scattered power, as obtained for homogeneous surroundings [Eq. (15))]. That said, since we consider a multipole expansion up to quadrupole order the scattered power can, in general, be written as

*p*,

_{x}*m*,

_{y}*Q*=

_{xz}*Q*, and

_{zx}*M*=

_{yz}*M*being the only non-zero multipole components, we can set up semi-analytical expressions for the ten different terms in Eq. (29)) that can be easily solved numerically. For example, if we consider light scattered in the upper medium the different terms constituting the total power can be written as

_{zy}*D*represents the power contained in the direct propagation of light (i.e., like in a homogeneous medium), while the integrals related to the constants

*C*and

_{r}*C*represent power contained in the reflected light and interference between the direct and reflected light, respectively. It should be noted that the functions

_{i}*f*,

_{s}*f*, and

_{p}*g*depend on the order of the multipole in consideration or interference between multipoles. Table 1 lists the constants and functions for the different ten terms making up the total power in the upper medium.

If we consider scattered light that is transmitted into the lower medium with *ε _{s} > ε_{d}*, it is appropriate to mention light propagation in the allowed and forbidden zones. The allowed zone corresponds the the angular range

*θ*∈ [

_{t}*θ*;

_{m}*π*] where

*θ*=

_{m}*π*− sin

^{−1}(1

*/N*), and light transmitted into this range does not depend on the height of the center of the nanoparticle above the substrate (i.e., independent of

*z*

_{0}). In contrast, power transmitted into the forbidden zone, corresponding to

*θ*∈ [

_{t}*π/*2;

*θ*), shows an exponential dependence on

_{m}*z*

_{0}with practically all light absent when

*z*

_{0}~

*λ*. The power transmitted into the lower half-space by the ten different multipole terms can in a compact way be written as

The remaining constants and functions of Eq. (31)) can be found in Table 2.

## Acknowledgments

We acknowledge financial support for this work from the Danish Council for Independent Research (the FNU project, contract no. 12-124690), the European Research Council (the PLAQ-NAP project, Grant 341054), and the University of Southern Denmark ( SDU 2020 funding).

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