Multiple Kerker conditions in arbitrary dielectric spheres

TL;DR

This paper demonstrates how to induce multiple Kerker conditions in arbitrary dielectric spheres using focused dipolar beams, enabling control over scattering properties and generating unique optical effects.

Contribution

It introduces a method to achieve multiple Kerker conditions in arbitrary dielectric spheres with focused dipolar illumination, expanding control over light scattering.

Findings

Duality signatures can be induced on spheres of various sizes.

Topological optical vortices are generated.

Enhanced spin orbit optical mirages are observed.

Abstract

High refractive index dielectric particles present unique light scattering properties in the spectral range dominated by electric and magnetic dipolar resonances. These properties are absent in non-resonant low-index particles due to the strong overlap between dipolar and higher order multipolar contributions. Here we propose to induce duality and zero optical backscattering (first Kerker conditions) at multiple size parameter values on arbitrary dielectric spheres by illuminating with a tightly focused dipolar beam. As an illustrative example, we show that duality signatures, including topological optical vortices and enhanced spin orbit optical mirages, can be induced on arbitrary sized spheres of Polystyrene immersed in water.

Figures

Equations

$${\mathbf{E}}_{\text{inc}}$$

$${\mathbf{\Psi }}_{lm}^{\sigma }$$

$${\mathbf{M}}_{lm}$$

$${\mathbf{X}}_{lm}$$

$${\mathbf{E}}_{\text{sca}}^{\sigma }$$

$$\left(\begin{array}{c}{D}_{lm}^{+}\\ D_{lm}^{-}\end{array}\right)$$

$$⟨\Lambda ⟩$$

$$⟨\Lambda ⟩=\sigma \frac{1-T}{1+T},$$

$${\mathbf{E}}_{\text{inc}}^{D\sigma }(\mathbf{r})=E_0\left(C_{11}^{\sigma }{\mathbf{\Psi }}_{11}^{\sigma }+C_{10}^{\sigma }{\mathbf{\Psi }}_{10}^{\sigma }+C_{1-1}^{\sigma }{\mathbf{\Psi }}_{1-1}^{\sigma }\right).$$

$${\mathbf{E}}_{\text{inc}}^{D\sigma }(0)=\mathrm{i}\frac{E_0}{\sqrt{12\pi }}\left(C_{11}^{\sigma }{\mathbf{\xi }}_1+C_{10}^{\sigma }{\mathbf{\xi }}_0+C_{1-1}^{\sigma }{\mathbf{\xi }}_{-1}\right),$$

$${\mathbf{E}}_{\text{inc}}^D=E_0\mathrm{i}\sqrt{12\pi }{\mathbf{\Psi }}_{1\sigma }^{\sigma },$$

$${\mathbf{E}}_{\text{inc}}^D=E_0\mathrm{i}\sqrt{6\pi }\left({\mathbf{\Psi }}_{11}^1-{\mathbf{\Psi }}_{1-1}^{-1}\right),$$

$$C_{lm}^{\sigma }={\mathrm{i}}^l\sqrt{4\pi (2l+1)}{\delta }_{m\sigma },$$

$$g=⟨\cos \theta ⟩=\frac{\text{Re}\left\{a_1{b}_1^{\ast }\right\}}{|a_1{|}^2+|b_1{|}^2},-1/2<g\le 1/2,$$

$$T=\frac{{\sigma }_b}{{\sigma }_f}=\frac{|a_1-b_1{|}^2}{|a_1+b_1{|}^2}=\frac{1-\sigma ⟨\Lambda ⟩}{1+\sigma ⟨\Lambda ⟩}=\frac{1-2g}{1+2g},$$

$$l_z(\theta )=\frac{{\sin }^2\theta \left(1+2g\cos \theta \right)}{1+{\cos }^2\theta +4g\cos \theta },$$

Full text

High refractive index (HRI) subwavelength particles have received an increasing interest as building blocks of new all-dielectric metamaterials and optical devices (Kuznetsov et al., 2016). Their remarkable optical properties are associated to the simultaneous excitation of well-defined electric and magnetic dipolar resonant modes (Evlyukhin et al., 2010; García-Etxarri et al., 2011; Gómez-Medina et al., 2012a; Shi et al., 2013). These include strong backward-to-forward asymmetric scattering and, at the first Kerker condition (Kerker et al., 1983; Nieto-Vesperinas et al., 2011), when the duality symmetry is satisfied (Zambrana-Puyalto et al., 2013a), the emergence of zero optical backscattering (Gomez-Medina et al., 2011; Geffrin et al., 2012; Fu et al., 2013; Person et al., 2013). The scattering directionality can be tuned further by coupling dipolar electric and magnetic modes with quadrupolar responses (Alaee et al., 2015; Shamkhi et al., 2019). Under circularly polarized illumination, HRI nanoparticles can present important spin-orbit interactions (SOI) (Bliokh et al., 2015). At the first Kerker condition, these interactions lead to a backscattering topological vortex (Olmos-Trigo et al., 2019a), which in turn reveals, due to the spiraling structure of the scattered light, a diverging apparent displacement in the position of the particle (SOI optical mirages) (Olmos-Trigo et al., 2019a, 2018; Gao et al., 2018). Our main goal is to show that, under appropriate illumination, all these interesting scattering phenomena, usually associated to resonant HRI particles, can be observed in arbitrary dielectric particles, regardless their size and refactive index contrast.

Most of previous works implicitly assume plane wave (PW) illumination conditions, although, as well as the polarization, the spatial structure of the incoming light field plays a major role in the excitation of the nanoparticle resonances (Zumofen et al., 2008; Mojarad et al., 2008; Zambrana-Puyalto et al., 2012; Woźniak et al., 2015; Zambrana-Puyalto et al., 2018; Bag et al., 2018; Eismann et al., 2018; Nechayev et al., 2019; Zaza et al., 2019). On the other hand, illumination with a tightly focused beam can still only excite particle multipolar modes which are already present in the incident beam. Tailoring the multipolar expansion of the focused beam would then enable a detailed study of selected individual multipolar resonances without interference by the other modes. For instance, cylindrical beams carrying a well-defined angular momentum (AM) component along the beam axis (Allen et al., 1992), $m\hbar$, can only be coupled with multipolar modes with total AM number $l\geq|m|$ (where $l=1$ correspond to dipolar resonances, $l=2$ to quadrupolar, etc.). These beams have then been used to suppress the contribution of low order multipolar resonances (Zambrana-Puyalto et al., 2012) enabling the excitation of Kerker conditions (dual modes) of some high order multipoles. In this vein, by tuning the field patterns and polarization of tightly focused beams it is possible to selectively excite either electric or magnetic low order modes of HRI Silicon particles (Woźniak et al., 2015) or a combination of spectrally overlapping dipolar resonances leading to the transverse Kerker scattering (Bag et al., 2018). However, in contrast with HRI particles, achieving dipolar Kerker conditions for ubiquitous low refractive index (LRI) dielectric particles, (e.g. standard Polystyrene -PS- spheres) is a challenging problem (Abdelrahman et al., 2019). The strong spectral overlap between the dipolar electric and magnetic modes, and a broad range of higher order multipoles, makes these conditions to remain hidden 111Interestingly, dipolar resonances of low-index spheres can be enhanced by partial metallic dressing (Zhou et al., 2018)..

In this Letter, we show that multiple Kerker conditions can be induced in arbitrary dielectric spheres, regardless their size and refractive index contrast, by illuminating with a dipolar tightly focused beam with a well-defined total AM number, $l=1$. In analogy with HRI spheres, the scattering can be tuned from almost zero forward radiated power to perfect zero-backscattering at multiple first Kerker conditions, where the particle’s response is dual. For dipolar beams with well-defined helicity (Olmos-Trigo et al., 2019b), it is possible to induce dual scattering signatures, including topological optical vortices and enhanced optical mirages on arbitrary dielectric spheres. As a specific example, we discuss in detail multiple dipolar Kerker conditions from the scattering of Polystyrene beads immersed in water.

Let us first consider an arbitrary monochromatic light beam (with an implicit time-varying harmonic component $e^{-iwt}$) propagating through an homogeneous medium with real refractive index $n_{\rm{h}}$ and wave number $k=n_{\rm{h}}k_{0}=2\pi n_{\rm{h}}/\lambda_{0}$ (being $\lambda_{0}$ the light wavelength in vacuum). Within a helicity and AM framework (Fernandez-Corbaton et al., 2012), the beam electric field, ${\bf E}_{\text{inc}}$, can be expanded in vector spherical wavefunctions (VSWFs), $\bm{\Psi}_{lm}^{\sigma}$, with well-defined helicity, $\sigma=\pm 1$ (Olmos-Trigo et al., 2019a). In this basis,

[TABLE]

where $\bm{\Psi}_{lm}^{\sigma}$ is defined as

[TABLE]

being $\bm{M}_{lm}$ and $\bm{N}_{lm}$ are Hansen’s multipoles (Borghese et al., 2007), $\bm{X}_{lm}$ denotes the vector spherical harmonic (Jackson, 1999), $j_{l}(kr)$ are the spherical (well-defined at $r=0$) Bessel functions, $Y_{l}^{m}$ are the spherical harmonics, and ${\bf{L}}\equiv\left\{-{\rm i}{\bf r}\times\bm{\nabla}\right\}$ is the orbital angular momentum (OAM) operator. Let us recall that the multipoles $\bm{\Psi}_{lm}^{\sigma}$ can be built following the standard rules of angular momentum addition (Borghese et al., 2007; Edmonds, 1957) as simultaneous eigenvectors of the square of the total angular momentum, $\bf{J}^{2}$, and its $z$-component, $J_{z}$, with \bf{J}={\bf{L}}+\overset{\text{\scriptsize\leftrightarrow}}{{\bf S}}, being \overset{\text{\scriptsize\leftrightarrow}}{{\bf S}}\equiv-{\rm i}{\bf{I}}\times the spin angular momentum (SAM) operator and ${\bf{I}}$ the unit dyad. The multipoles $\bm{\Psi}_{lm}^{\sigma}$ are then simultaneous eigenvectors of $\bf{J}^{2}$, $J_{z}$ (Edmonds, 1957) and the helicity operator (Fernandez-Corbaton et al., 2012) $\bm{\Lambda}=(1/k)\bm{\nabla}\times$, with eigenvalues $l(l+1)$, $m$ and $\sigma$, respectively.

When a homogeneous dielectric sphere of radius $a$ and an arbitrary (absorptionless) refractive index $n_{\rm{p}}$ is centred at the origin ($r=0$), the AM numbers, $l$ and $m$, are preserved in the scattering process due to the symmetry of the system. The scattered fields outside the sphere, ${\bf E}_{\text{sca}}={\bf E}_{\text{sca}}^{+}+{\bf E}_{\text{sca}}^{-}$, can be written in terms of “outgoing” VSWFs, $\bm{\Phi}_{lm}^{\sigma^{\prime}}$ (defined as in Eq. (2) replacing $j_{l}(kr)$ by the outgoing spherical Bessel functions $h_{l}(kr)$), as

[TABLE]

where $Z{\bf H}_{\text{sca}}^{\sigma}=-{\rm i}\sigma{\bf E}_{\text{sca}}^{\sigma}$ with $Z=1/(\epsilon_{0}cn_{\rm{h}})$, and $a_{l}$ and $b_{l}$ are the electric and magnetic Mie coefficients (Bohren and Huffman, 2008).

In this framework, it can be seen that the expected value of the helicity of the scattered field is generally given by

[TABLE]

When the sphere is illuminated by a circularly polarized PW with helicity $\sigma$, or by a cylindrically symmetric beam (eigenvector of the helicity, $\bm{\Lambda}$ and $z$-component of the AM, $J_{z}$, operators with eigenvalues $m$ and $\sigma$ ), the scattered field is, in general, a combination of multipolar modes with fixed $m$. As a result, it can be shown that,

[TABLE]

is the helicity transfer function (Zambrana-Puyalto et al., 2013b). When $T$ goes to zero, the particle response is dual and the scattered helicity is the same as the helicity of the incoming beam. However, the helicity is not preserved in a multipolar scattering process except for the case $a_{l}=b_{l},\ \forall\ l$, which can only be achieved for hypothetical spheres with equal electric permittivity $\epsilon$ and magnetic permeability $\mu$, corresponding to the original first Kerker condition (Kerker et al., 1983).

To address this issue, we propose the use of dipolar beams (DB), that present a well-defined total AM, i.e., $l=1$. Under this illumination, which is solution of Maxwell’s equations, the multipolar response is by construction discarded, regardless of the $x=ka$ size parameter and the relative refractive index $n=n_{\rm{p}}/n_{\rm{h}}$. The most general DB can be written as a combination of pure dipole beams (PDB) with well-defined AM and helicity eigenvalues $(l=1,m,\sigma=\pm 1)$ (Olmos-Trigo et al., 2019b), ${\bf E}_{\text{inc}}^{D}={\bf E}_{\text{inc}}^{D+}+{\bf E}_{\text{inc}}^{D-}$, where

[TABLE]

At the focus $r=0$, it takes the simple form

[TABLE]

with $\bm{\xi}_{1}=-(\bm{\hat{x}}+{\rm i}\bm{\hat{y}})/\sqrt{2},\quad\bm{\xi}_{0}=\bm{\hat{z}},\quad\bm{\xi}_{-1}=(\bm{\hat{x}}-{\rm i}\bm{\hat{y}})/\sqrt{2}$. When a dielectric sphere is illuminated by a PDB mode, as e.g. (see top panel in Fig. 1)

[TABLE]

or by a linear combination of opposite helicities

[TABLE]

where

[TABLE]

the electric and magnetic fields at the focus are transversal with respect to the optical $z$-axis. Therefore, at the focus, these fields behave exactly the same as those induced by a circularly polarized PW of helicity $\sigma$, or by a linearly polarized PW, respectively. As a direct consequence, for small dipolar particles located at the focus, the dipolar scattered fields are strictly equivalent. Other combinations of PDBs, including $\bm{\Psi}_{11}^{\sigma}$ and $\bm{\Psi}_{10}^{\sigma}$ modes, can be used to induce focal fields with both longitudinal and transversal components leading to the so-called transversal Kerker conditions (Bag et al., 2018). In this case, the longitudinal fields arise from the interference between $\bm{\Psi}_{11}^{\sigma}$ and $\bm{\Psi}_{10}^{\sigma}$ modes since $\bm{\Psi}_{10}^{\sigma}$ modes do not carry power by themselves (Olmos-Trigo et al., 2019b). For the same reason, it is not possible to induce pure longitudinal electric and magnetic fields with a dipolar beam. According to the aforementioned, the scattering that arises from a PDB (or a linear combination of them) can be easily understood by comparison with the scattering that emerges from PW illumination.

Figure 1 (bottom panel) illustrates the behaviour of the scattering efficiency $Q$ (Bohren and Huffman, 2008), versus the size parameter $y=nka$, when typical examples of HRI and LRI lossless spheres are illuminated with both a PW (dashed blue) and a PDB (solid red line), Eqs. (10) and (11). $Q\equiv\sigma_{\rm{sca}}/(\pi a^{2})$ is proportional to the scattering cross section, $\sigma_{\rm{sca}}$, defined as the ratio between the total scattered power and the incident energy flux. Figure 1 (a) corresponds to $n=\sqrt{12}$ (similar to Si-like spheres in air -see (Fenollosa et al., 2008)-) and Fig. 1 (b) to $n=1.56/1.3=1.2$ (PS in water). As it can be inferred from the comparison between the PW and PDB responses, HRI Si-like spheres present a clear dipolar response for size parameters $y=nka\lesssim 4.5$. It is in this spectral region where it is possible to observe the dipolar Kerker condition $a_{1}=b_{1}$. For larger sizes, the overlapping with higher order multipoles masks the sequential crossing between electric and magnetic dipolar responses (dotted green and dash-dotted yellow lines), which would lead to dipolar Kerker conditions at multiple $y$ size parameters (Zambrana-Puyalto et al., 2013b). In contrast, Fig. 2 (b) illustrates that, for non-resonant LRI spheres, dipolar dual conditions cannot be observed due the strong overlap between the dipolar and higher order multipoles even at low $y$ sizes.

We shall now move to the main consequences of this DB illumination, based on the unmasking of the hidden multiple dipole Kerker conditions. To that end, it results convenient to introduce the asymmetry parameter (Gómez-Medina et al., 2012b; Pinheiro et al., 2000), defined as the average of the cosine over the scattering angle,

[TABLE]

which encodes the interference between the first electric and magnetic Mie coefficients. The maximum value, $g=1/2$, corresponds to the Kerker zero-backscattering condition $a_{1}=b_{1}$. This condition, well-known for small dipolar particles under PW illumination, can now be extended to arbitrary sized dielectric spheres illuminated by a (sectoral) dipolar beam (Olmos-Trigo et al., 2019b), regardless of the incoming helicity. A map of $g$ for a dielectric sphere as a function of the refractive index contrast $n$ and the $y$ size parameter, see Fig. 2, reveals the existence of successive first Kerker conditions even for LRI dielectrics. This phenomenon goes well beyond of previous works which were based on Kerker conditions in the limit of small particle (Gao et al., 2017).

Under illumination by a PDB of well-defined helicity, the remarkable angular dependence of the intensity and the averaged helicity are linked, and fully determined by the dipolar asymmetry parameter $g$, Eq. (13) (Olmos-Trigo et al., 2019c). Particularly, the helicity transfer function $T$ is now given by the dipolar term in Eq. (7), $l=m=\sigma$, coinciding with the ratio between backward and forward cross sections,

[TABLE]

where we have used the dipolar results for the expected value of the helicity, $\langle\Lambda\rangle=2\sigma g$ (Olmos-Trigo et al., 2019c).

Figure (3) illustrates the behaviour of both $\langle\Lambda\rangle$ and $T=\sigma_{b}/\sigma_{f}$ for a Si-like sphere in air and a PS sphere immersed in water, under PW and PDB illumination with well-defined $\sigma=+1$. Through PW illumination, the duality symmetry ($\langle\Lambda\rangle=+1$), which traduces into zero optical backscattering ($\sigma_{b}/\sigma_{f}=0$), can only be achieved in the limit of small particle for the Si case, according to Fig. (4) (a) and (b), respectively. However, under PDB, the unmasked first Kerker conditions ($g=1/2$) shown in Fig. (3) induce successively these phenomena for both examples. Notice that in the latter case the first Kerker condition is perfectly achieved, whereas under PW illumination, a slight contribution of higher multipoles may always induce a (despicable) suppression of the duality symmetry.

Interestingly, recent predictions concerning SOI phenomena in the dipolar approximation (Olmos-Trigo et al., 2019a), limited to subwavelength HRI particles, can also be extended to arbitrary spheres by PDB illumination. For a PDB with well-defined helicity, e.g. $\sigma=+1$, the (dimensionless) $z$-component of the AM, $m=\sigma$, can be written as the sum of the OAM, $l_{z}(\theta)$, and SAM, $s_{z}(\theta)$, densities $m=l_{z}+s_{z}$, which is conserved in the scattering process for axially symmetric scatterers (Moe and Happer, 1977; Schwartz and Dogariu, 2006; Bliokh et al., 2011). In this case the OAM density of the scattered photons is given by (Olmos-Trigo et al., 2019a)

[TABLE]

which approaches $l_{z}=+2$, i.e., $s_{z}=-1$, near backscattering, signaling the emergence of a topological vortex at every Kerker condition ($g=1/2$). The spiraling structure of the scattered light, associated to $l_{z}(\theta)$, is responsible of the apparent displacement in the position of the particle (SOI optical mirages) (Olmos-Trigo et al., 2019a, 2018; Gao et al., 2018) given by $\Delta/(\lambda/\pi)=l_{z}/\sin\theta$. Figure 4 illustrates the optical mirage induced from a PDB impinging on a $0.5\ \mu m$ PS sphere immersed in water in the visible spectral range. As it can be seen, the SOI optical mirage is considerably enhanced near the backscattering angles ($\theta\sim\pi$) at the successive first Kerker conditions, where the zero-backscattering is fulfilled as shown in the figure inset.

In conclusion, we have shown that dipolar beams with well-defined helicity can induce multiple Kerker conditions and duality symmetry to arbitrary dielectric spheres. Duality signatures and spin-orbit effects like perfect zero optical backscattering, emergence of topological vortex and enhanced optical mirages, predicted for subwavelength HRI spheres, could be experimentally observed in LRI particles like standard, micron-sized, Polystyrene spheres immersed in water.

This research was supported by the Spanish Ministerio de Economía y Competitividad (MINECO, MICINN) and European Regional Development Fund (ERDF) Projects FIS2015-69295-C3-3-P and FIS2017-82804-P, and by the Basque Dep. de Educación Project PI-2016-1-0041 and PhD Fellowship (PRE- 2018-2-0252).