1. Introduction

Mie scattering off a spherical object is one form of light-matter interactions [1], thus being not only of theoretical but also of practical concerns. A scattering cross section and an angle-dependent field intensity is well-documented [2]. The far-field behaviors of the total (incident plus scattered) field are largely interesting for traditional purposes. Meanwhile, an un-engineered, isotropic, homogeneous, and dielectric solid sphere can be considered as a single centrosymmetric molecule [3].

Among a myriad of reports made so far on the Mie scattering off such a simplest sphere [4], we find in a recent review [5] and the references therein that the Poynting vector exhibits fantastic near-field behaviors around a dielectric sphere, if the size parameter as large as several tens is chosen to lie at one of the resonance peaks. Recently, Mie scattering is receiving increased attention as regards metasurfaces and metamaterials [6–10]. Mie scattering has also been investigated from the viewpoint of generic electromagnetic waves [11,12].

Notwithstanding, we still find a less-explored aspect, namely, the electromagnetic (EM) chirality in the near field around a single sphere [13,14]. Of course, the EM chirality, being alternatively called ‘optical chirality’ [2,15,16], is non-vanishing even for an achiral incident plane wave if either of the embedding and embedded media is chiral (optically active). This study marks a starting point in our planned scheme of examining what we call chirality-contaminated systems. The EM chirality under this study has been partly investigated from the viewpoint of circular dichroism (CD) as seen from recent papers [17–19] and the references therein.

Let us categorize such chirality-contaminated systems into two groups. Firstly, we consider an extrinsic-chirality (incident-wave-chirality) system [20–24], where a circularly polarized (CP) incident wave propagates through an achiral embedding medium with an achiral embedded sphere. Notice that CP wave consists of two mutually orthogonal linearly polarized waves with an in-quadrature phase difference. This extrinsic system will be fully taken into consideration in the current study. This CP incident wave does carry an EM chirality. As a wave configuration complementary a CP wave, we will take an incident wave to be a linear combination of two mutually independent linearly polarized waves, where there is no phase difference between the two. It is worth remarking that this composite incident wave does not carry an EM chirality. The character of an incident wave turns out to play a crucial role in determining symmetry properties of an EM chirality [25,26].

Secondly, we plan in the future to investigate two kinds of intrinsic-chirality (material-chirality) system, [2–4,15,17,18,20,27]: (i) a linearly polarized incident wave propagating through a chiral embedding medium (say, a chiral solution), thus getting scattered off an achiral embedded object; and (ii) another intrinsic system: a linearly polarized incident wave propagating through an achiral embedding medium while getting scattered off a chiral embedded object (chiral scatterer).

Figure 1 presents basics of the Mie scattering, where Fig. 1(a) defines coordinate systems. For a linearly polarized incident wave, Figs. 1(b) and 1(c) depict wave configurations respectively on the longitudinal and transverse planes. As a simple example of all-dielectric dual systems [6,9,10,13,17,18,28], this two-dielectric system turns out to admit effective (or artificial) magnetism in the form of loop or ring currents, thus leading to a possibility of ‘dielectric magnetic resonance (DMR)’ [5,7,14,29]. In addition, several theoretical limits can be identified on the Mie-scattering formulas for such two-dielectric systems, for instance, the optical forces and torques on an embedded sphere [1,12,30]. The results from our study is certainly related to the enantio-separation [1,18,21,25,26,31].

 

Fig. 1. (a) Coordinate systems. (b) and (c) incident-wave configurations for a denser embedded sphere respectively on the longitudinal $zx$-plane and on the transverse $xy$-plane. On (b) and (c) are indicated key incident-wave properties: the propagation direction and the directions of the electric and magnetic fields. (d) A toroidal nature of the electric field on the $z\rho$-plane with varying azimuthal angle. (e) and (e) Distributions of the electric-field intensity ${|{\mathbf E} |^2}$. We set $m = \sqrt 2$ and $\alpha = 2$ on (e) and (f).

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Let $\{{{\mathbf E},{\mathbf H}} \}$ denote the electric and magnetic fields, precise dimensionless definitions being given in Supplement 1. Furthermore, the red-boundary pointers with ${{\mathbf k}^{inc}}$ shown on Figs. 1(b) and 1(c) indicate the propagation direction of an incident plane waves. In addition, the black-boundary and blue-boundary arrows signify the directions respectively of the electric- and magnetic-field polarizations. Therefore, the incident wave is achiral in this case. Figure 1(d) shows the total (a.k.a. local [25]) electric field consisting of both incident and scattered fields at a certain longitudinal location as the azimuthal angle is varied over one cycle. As a result, a toroidal feature is obtained for the loop current. On the planes depicted respectively on Figs. 1(b) and 1(c), Figs. 1(e) and 1(f) display an electric-field intensity ${|{\mathbf E} |^2}$, where the maximum intensity turns out to be about three and half in terms of the unit incident intensity, thus being the intensity localization ratio.

The EM chirality $\mathcal{C} \equiv {\mathop{\mathrm {Im}}\nolimits} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ for the total field constitutes our key result, which takes either positive or negative values in the respective quarter spheres depending on the azimuthal $\varphi$-angular range. Although both media are assumed achiral (non-chiral), the total EM wave should exhibit an EM chirality, which manifests itself very strongly in the vicinity of a spherical object. Notwithstanding, a net EM chirality vanishes when integrated over the whole three-dimensional space. Hence, we are dealing in this study with a local generation of an EM chirality due to a Mie scattering with neither geometrical nor material chirality [3,15–17,21,25]. As a sort of a complex conjugate to an EM chirality $\mathcal{C}$, we will also examine a reactive chirality $\mathcal{K} \equiv \textrm{Re} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$, which is alternatively called a ‘magneto-electric’ energy density [1,2,12].

This study is structured as follows. Section 2 deals with fundamental analysis. Section 3 presents both EM chirality and reactive chirality as our main topic. Section 4 solves for a circularly polarized incident wave. Section 5 shows how an EM chirality is altered when an incident wave consists of two linearly polarized EM waves. Section 6 and 7 provide discussions and conclusion. This text is supported by one Supplement and two Visualizations that are called out in appropriate places.

2. Fundamentals and key bilinear parameters from Mie solutions

Let us introduce our notations by starting with bold-face symbols reserved for vectors. The origin of coordinates lies on the center of a spherical object. We employ the usual coordinate systems shown on Fig. 1(a). Let ${\mathbf r} = r\hat{{\mathbf r}}$ be the position vector, whereas both time and frequency $\{{t,\omega } \}$ are taken to be positive. Let $\{{\varepsilon ,\mu } \}$ be respectively the electric permittivity and the magnetic permeability of a dielectric, which are piecewise constant. The wave number ${k^{inc}}$ of an incident plane wave through an embedding medium is given by ${k^{inc}} \equiv {\omega / {{c^{inc}}}}$ where the light speed defined by ${c^{inc}} \equiv {({{\varepsilon^{inc}}{\mu^{inc}}} )^{ - {1 / 2}}}$. To save notations, spatial coordinates are redefined by the substitution ${k^{inc}}{\mathbf r} \to {\mathbf r}$. Accordingly, we make a substitution ${k^{inc}}\nabla \to \nabla$ with $\nabla$ denoting a spatial gradient. With $\textrm{i} \times \textrm{i} \equiv{-} 1$, the time-oscillatory factor for field variables is made dimensionless into $\exp ({ - \textrm{i}t} )$ by way of a substitution $t\omega \to t$. The reference electric field is $\sqrt {{{{\mu ^{inc}}} / {{\varepsilon ^{inc}}}}}$ times the reference magnetic field. Since ${k^{inc}}$ is $2\pi$ times the inverse wavelength of an incident wave, our near field is henceforth characterized by the dimensionless range on the order of unity, being $0.3 < r < 3$ for instance. Our formulation here is suitable for analyzing a monochromatic wave of a single fixed frequency.

Let $\{{{{\mathbf E}^\upsilon },{{\mathbf H}^\upsilon }} \}$ be the field vectors as solutions to the Mie scattering, thus henceforth being called the ‘Mie solution’ [2,3,6,9–15,17,18,21,22,25–27]. Let $\upsilon$ denote superscript: $\upsilon = inc$ for an incident wave, $\upsilon = sca$ for its scattered wave, and $\upsilon = obj$ for the wave within a spherical object. As depicted by Fig. 1(a), 1(b), and 1(c), the incident plane wave is given by the following.

Field variables $\{{{{\mathbf E}^\upsilon },{{\mathbf H}^\upsilon }} \}$ are explicitly written down in Supplement 1. There is a pair of input parameters. Firstly, $m = \sqrt {{{{\varepsilon ^{obj}}} / {{\varepsilon ^{inc}}}}}$ is the refractive-index ratio of a spherical object to the embedding medium. We take either $m = \sqrt 2$ or $m = 2$ [5,9]. Secondly, $\alpha \equiv {k^{inc}}a$ is the size parameter with a as a sphere’s radius. Hence, $\alpha$ is the circumference of a spherical object normalized by the wavelength of the EM wave in the embedding medium. We set $\alpha = 2$ for most cases.

The total field (without any superscript for simplicity) outside a sphere consists of an incident wave and a scattered wave such that ${\mathbf E} = {{\mathbf E}^{inc}} + {{\mathbf E}^{sca}}$ and ${\mathbf H} = {{\mathbf H}^{inc}} + {{\mathbf H}^{sca}}$ if $r \ge \alpha$. In comparison, a field inside a sphere is taken to be the total field such that ${\mathbf E} = {{\mathbf E}^{obj}}$ and ${\mathbf H} = {{\mathbf H}^{obj}}$ if $r \le \alpha$ [11]. Together with discussions on multipolar expansions, numerical tests are carried out in Supplement 1 to make sure that we are working on correct formulas.

For all these parameters, we have neglected the factor of half that come from time averaging for time-oscillatory fields. Mie solutions involve an infinite sum $\sum\nolimits_{n = 1}^\infty {({ \cdot{\cdot} \cdot } )}$, which is replaced by a finite sum $\sum\nolimits_{n = 1}^N {({ \cdot{\cdot} \cdot } )}$ with n as a multipolar index. For the numerical evaluations in this manuscript, we set a sufficiently large value of $N = 15$. The effects of smaller N are discussed in Supplement 1.

Both panels are plotted on a longitudinal $zx$-plane. (c) Levels of ${\mathop{\mathrm {Im}}\nolimits} ({{H_\varphi }} )$ on the meridional $z\rho$-plane at $\varphi = {\textstyle{1 \over 8}}\pi$. All panels are responses to a linearly polarized incident wave. A spherical object is indicated by the circle at $r = \alpha = 2$. We set $m = \sqrt 2$.

Figure 2(a) features the streamlines formed by the pair $\{{\textrm{Re} ({{E_z}} ),Re ({{E_x}} )} \}$ on the longitudinal $zx$-plane. This electric field is temporally instantaneous since they undergo temporal oscillations. Figure 2(a) is almost identical to Supplementary Fig. 2 of [29] in the sense that circulating electric currents are induced within a sphere (indicated by a solid circle). Other vortex-like streamlines can also be identified in the embedding medium as indicated by several dotted ellipses, especially, in the downstream. Therefore, our two-dielectric system exhibit optical magnetism associated with such circulating currents [7,9,10,14,17,18]. A key to such artificial magnetism is spherically curved surfaces that are conducive to circulatory currents.

 

Fig. 2. (a) Streamlines of the real part $Re ({\mathbf E} )$. (b) is a zoomed-in view of (a) around a sphere.

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Figure 2(b) is a zoomed-in view of Fig. 2(a) around a sphere, where a single vortex within a sphere is now clearly visible on the right half of a sphere. The $zx$-plane on Fig. 2(b) is identical to the meridional $z\rho$-plane at a special azimuthal angle $\varphi = 0$. Furthermore, consider a generic pair $\{{{E_z}({\rho ,\varphi ,z} ),{E_\rho }({\rho ,\varphi ,z} )} \}$ on the meridional $z\rho$-plane. We can then prove that ${E_z},{E_\rho } \propto \cos \varphi$ as discussed in more details in Supplement 1. Consequently, all streamlines formed by $\{{{E_z},{E_\rho }} \}$ on any meridional $z\rho$-plane for any $\varphi$ look like Fig. 2(b).

Both of $\{{{E_z},{E_\rho }} \}$ are related through the Maxwell equation $\nabla \times {\mathbf E} = \textrm{i}{\mathbf H}$ to $\textrm{Re} {({\nabla \times {\mathbf E}} )_\varphi } ={-} {\mathop{\mathrm {Im}}\nolimits} ({{H_\varphi }} )$. Hence, an electric-current loop characterized by ${{\partial {E_\rho }} / {\partial z}} - {{\partial {E_z}} / {\partial \rho }}$ on the $z\rho$-plane can be alternatively examined by ${H_\varphi }$. Resultantly, ${H_\varphi } \propto \cos \varphi$. Those electric-current loops as shown on Fig. 2(b) form a toroidal configuration as illustrated on Fig. 1(d), as the azimuthal $\varphi$-angle is varied over the whole cycle. We recognize hence that the azimuthal $\varphi$-angle serves as a parameter that determines the polarization ellipticity as viewed into the $z$-direction [7,14].

Figure 2(c) presents ${\mathop{\mathrm {Im}}\nolimits} ({{H_\varphi }} )$ at a special azimuthal angle $\varphi = {\textstyle{1 \over 8}}\pi$ in correspondence to the pink-colored circle shown on Fig. 1(d). Therefore, the center of the electric-current loop shown on Fig. 2(b) corresponds to a maximum optical magnetism. Because of the common factor $\cos \varphi$, this effective optical magnetism undergoes sinusoidal variations over a toroid extending over $0 \le \varphi \le 2\pi$. Of course, the nodes of this effective optical magnetism are located at two points of $\varphi = {\textstyle{1 \over 2}}\pi ,{\textstyle{3 \over 2}}\pi$.

Notwithstanding, the time-oscillatory average of the electric loop current should vanish when integrated over both temporal and azimuthal cycles. Over a $\varphi$-cycle, ${\mathop{\mathrm {Im}}\nolimits} ({{H_\varphi }} )$ serves once as a source and another time as a drain for the flow made by $\{{{E_z},{E_\rho }} \}$. In terms of symmetry in the azimuthal direction, it is shown that ${\mathop{\mathrm {Im}}\nolimits} [{{H_\varphi }({2\pi - \varphi } )} ]= {\mathop{\mathrm {Im}}\nolimits} [{{H_\varphi }(\varphi )} ]$ [25]. Besides, ${H_\varphi }(\varphi )$ is a local pseudo-vector or an axial vector for the local vorticity $\nabla \times {\mathbf E}$. The imaginary part ${\mathop{\mathrm {Im}}\nolimits} ({\mathbf E} )$ is largely akin to its real part so that it is plotted in Supplement 1 along with those for the magnetic field [9,10,17,26].

Once field variables are evaluated by the Mie solution, we can go on to evaluate all bilinear (‘quadratic’ being included) parameters such as the intensities $\{{{{|{\mathbf E} |}^2},{{|{\mathbf H} |}^2}} \}$ and the Poynting vector ${\mathbf P} \equiv \textrm{Re} ({{{\mathbf E}^\ast } \times {\mathbf H}} )$ [5,11,12,17]. As illustrated on Fig. 1(b), an incident wave is propagating from left to right along the horizontal direction. Therefore, transversality conditions are met such that ${{\mathbf E}^{inc}}\cdot \hat{{\mathbf z}} = 0$ and ${{\mathbf H}^{inc}}\cdot \hat{{\mathbf z}} = 0$ from Eq. (1). Moreover, the Poynting vector ${{\mathbf P}^{inc}}$ of the incident wave is directed along the wave-propagation vector, i.e., ${{\mathbf k}^{inc}} \propto \hat{{\mathbf z}}$. This plane wave is free of both EM chirality and reactive chirality, namely, ${\mathcal{C}^{inc}} = 0$ and ${\mathcal{K}^{inc}} = 0$ [1].

Figures 1(e) and 1(f) show the electric-field intensity ${|{\mathbf E} |^2}$ respectively on the longitudinal and transverse planes. We find a strong intensity localization on Fig. 1(d). Here, the backward half of a sphere exhibits a rather small intensity, whereas the forward (downstream) half carries an intensity peak, especially near the shadow side of the spherical surface [5,7,14]. In addition, Fig. 1(e) displays a series of fan-like curves emanating along the backward axis and extending in the forward direction [13,22,26]. Other relevant parameters of the magnetic-field intensity, an energy sum, and an energy difference are analogous to those on Fig. 1(e) and 1(f) [5,12] as discussed in Supplement 1. It is well-established that the conservation of the total energy ${|{\mathbf E} |^2} + {|{\mathbf H} |^2}$ leads for a time-oscillatory EM field to the zero divergence $\nabla \cdot {\mathbf P} = 0$ in the Poynting vector [12,17,20].

Figures 3 shows the Poynting-vector streamlines on the longitudinal and transverse planes. Figures 3(a) and 3(b) are drawn with the data of $m = \sqrt 2$ and $\alpha = 2$ as for Figs. 1(e) and 1(f) as well as for Fig. 2. Meanwhile, Figs. 3(c) and 3(d) are drawn with the data of $m = 2$ and $\alpha = 6.09$, namely, with a higher-index sphere and about three times larger size. We have verified in Supplement 1 that this choice of $m = 2$ and $\alpha = 6.09$ corresponds to significantly strong (albeit at a shallow maximum) activity inside a sphere as discussed by [5,12], and [29]. See panels (b), (c), and (d) of Fig. S1b of Supplement 1.

 

Fig. 3. Spatial distributions of the Poynting vector ${\mathbf P}$ for a linearly polarized incident wave. (a) and (c) Drawn on the longitudinal plane located at $y = 0$ passing through the center of a sphere. (b) and (d) Drawn on the transverse plane located at $z = 0$ passing through the center of a sphere. (a) and (b) obtained with $m = \sqrt 2$ and $\alpha = 2$. (c) and (d) obtained with $m = 2$ and $\alpha = 6.09$. The dotted circles indicate spherical surfaces.

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Both Figs. 3(a) and 3(c) drawn at $y = 0$ shows that the Poynting vector is largely directed from left to right along the wave-propagation direction. Figure 3(a) shows clearly that the Poynting-vector streamlines are bent toward a higher-index spherical object according to the notion of a ‘light bending’ onto a higher-index medium (HIM) [5,17,24]. In addition, this bending of energy flows leads naturally to the hotspots of the electric-field intensity as displayed on Fig. 1(f). See Fig. 1(b) of [24], where a trapezoidal scatterer steers light onto the center of a HIM. There is however no recirculation in the longitudinal direction on Fig. 3(a) largely due to a smaller size parameter.

In comparison, we can confirm on Fig. 3(c) the light bending to a lesser extent. Instead, we encounter many vortices on Fig. 3(c). On the shadow side of a sphere on Fig. 3(c), we find jet-like streamlines as indicated by the right-parenthesis symbol ‘}’ in red color as noticed by [5,29].

In the meantime, Fig. 3(b) displays both azimuthally rotational flows and radial flows (both inward and outward depending on the radial positions) [2,6,11,12,24]. The streamlines as shown on Fig. 3(b) confirms the ‘light bending’ across the spherical surface onto a higher-index medium (HIM), which is in this case directed onto the center of a sphere from the circumference. The spherical center is in fact a saddle line, where the energy flow is altered from the centripetal direction into the downstream direction [5]. Besides, Fig. 3(d) is akin to Fig. 3(b). Furthermore, Fig. 2(a), 3(a), and 3(c) carry only longitudinal mirror planes, whereas Figs. 3(b) and 3(d) are endowed with both horizontal and vertical mirror planes.

3. Electromagnetic chirality and symmetry

Figure 4 displays the EM chirality on the longitudinal and transverse planes for a linearly polarized incident wave. See Fig. 3 for the analogous setting for preparing all panels. Figure 4(a) displays that fan-like zones of larger magnitudes are swept downstream into the forward-scattering direction. Minor peaks of alternating signs are visible on Fig. 4(b) as the radial distance is increased, thus reflecting the effect of the spherical Bessel functions.

 

Fig. 4. Spatial distributions of the electromagnetic (EM) chirality $\mathcal{C} \equiv {\mathop{\mathrm {Im}}\nolimits} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ for a linearly polarized electric field. All panels are prepared as on Fig. 3.

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Both real and imaginary parts of the electric and magnetic fields undergo temporal oscillations as illustrated on Fig. 2, thus leading to zero values of time averages. It is thus remarkable that the time-averaged EM chirality displayed on either Fig. 4(a) or 4(c) possesses certain non-zero values, although their respective constituent field variables are spatially oscillatory.

In comparison to Figs. 4(a) and 4(b), the EM chirality shown on Figs. 4(c) and 4(d) is obtained with a higher refractive-index ratio $m = 2$ and a larger size parameter $\alpha = 6.09$ as with Figs. 3(c) and 3(d). Take note that all color bars on Fig. 4 are differently scaled. Increased activity with higher m has already been confirmed on Figs. 3(c) and 3(d). In addition, Figs. 4(a) and 4(b) show that $|\mathcal{C} |$ is larger outside a sphere than inside, whereas large values of $|\mathcal{C} |$ are confined on Figs. 4(c) and 4(d) to the inside of a sphere due to the light bending onto a higher-index medium (HIM) as displayed on Fig. 3(a) for the Poynting vector [17,18,24].

Figures 4(c) and 4(d) display again that the strong activities of the EM chirality are largely confined to the inside of a sphere as can be identified by the four pairs of the EM-chirality hotspots on Fig. 4(c). Instead of the photonic jet identified on Fig. 3(c), we observe a stronger activity of the EM chirality on the upstream (backscattering) side of a sphere as indicated on Fig. 4(c) by the left-parenthesis symbol ‘{‘ in black color.

Consider a field variable ${\mathbf f} \in \{{{\mathbf E},{\mathbf H}} \}$, for which a spin angular momentum (SAM) density is defined by ${{\mathbf S}_f} \equiv {\mathop{\mathrm {Im}}\nolimits} ({{{\mathbf f}^\ast } \times {\mathbf f}} )$. This SAM density is in turn linked to a degree of circular polarization. Both constituents lead to an average SAM density ${{\mathbf S}_{avg}} \equiv {\textstyle{1 \over 2}}({{{\mathbf S}_E} + {{\mathbf S}_H}} )$ in conformance to the electric-magnetic duality [1–3,11,12,15,17,18,20,25,26,30,31]. Moreover, we obtain ${\mathbf S}_E^{inc} = {\mathbf S}_H^{inc} = {\mathbf 0}$ for the incident wave given in Eq. (1).

When an electric SAM density is normalized by the corresponding field intensity, we obtain ${{\mathbf \eta }_E} \equiv {|{\mathbf E} |^{ - 2}}{{\mathbf S}_E}$, which is a vector degree of circular polarization (DoCP). If a certain component is chosen from ${{\mathbf \eta }_E}$, we obtain a scalar electric DoCP in that direction. For instance, the longitudinal component ${\eta _{E,z}}$ is of special interest, whereby the RCP (right circular polarization) and LCP (left circular polarization) are specified respectively by ${\eta _{E,z}} ={\pm} 1$ [12]. See Supplement 1 for plots of the SAM densities along with DoCPs.

In the meantime, the reactive chirality $\mathcal{K} \equiv \textrm{Re} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ is linked to the SAM densities in the following way [12].

Here, the equality $\nabla \cdot ({{{\mathbf S}_E} + {{\mathbf S}_H}} )= 0$ or $\nabla \cdot {{\mathbf S}_{avg}} = 0$ stands for the chirality conservation for time-oscillatory fields. Resultantly, ${{\mathbf S}_{avg}}$ is nothing but the chirality flux employed in the conservation of the EM chirality $\mathcal{C} \equiv {\mathop{\mathrm {Im}}\nolimits} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ [1–3,16,17,19,31]. The EM chirality is interchangeably employed with ‘helicity’ in many other reports [7]. Sometimes, the sign of the EM chirality is taken as the helicity since $\mathcal{C}$ is a pseudo-scalar. It is worth stressing that the constituent fluxes are not conserved in general, namely, $\nabla \cdot {{\mathbf S}_E} ={-} \nabla \cdot {{\mathbf S}_H} \ne 0$. Both ${{\mathbf S}_E}$ and ${{\mathbf S}_H}$ are pseudo-vectors (viz., an axial vector) [26], since both curls $\left\{ {{\textstyle{1 \over 2}}\nabla \times {{\mathbf S}_E},{\textstyle{1 \over 2}}\nabla \times {{\mathbf S}_H}} \right\}$ constitute the spin parts decomposed from the Poynting vector [1,11,12,15,25]. See our summary in Supplement 1.

In this aspect, the EM chirality undergoes jumps across the spherical surface as displayed on Fig. 4. These jumps are one manifestation of the ‘spin-orbit interaction’, which vanishes in a homogeneous medium. However, these jumps become important at material interfaces, vertices, and corners, namely in the near field of a scatterer [18,21,25].

Meanwhile, the scalar parameter of $\mathcal{K}$ is much simpler that the vector parameters of $\{{{{\mathbf S}_E},{{\mathbf S}_H}} \}$ as seen from Eq. (2). The reactive chirality $\mathcal{K}$ discussed in Eq. (2) carries properties largely analogous to those of Fig. 4. In this regard, we provide comparative studies in Supplement 1 for pairs of $\{{\mathcal{C},\mathcal{K}} \}$ with three combinations of $m = \left\{ {\sqrt 2 ,2} \right\}$ and $\alpha = \{{2,6.09} \}$.

Let us revisit Figs. 1(b) and 1(c) for a linearly polarized incident wave of Eq. (1). The symmetry properties of various bilinear parameters are different with respect to mirror planes, across which either a mirror symmetry (MiS) or a mirror anti-symmetry (MAS) can be determined.

According to the figures examined so far, we can easily identify types of symmetry: a MiS for ${|{\mathbf E} |^2}$ on Fig. 1(e) and ${\mathbf P}$ on Figs. 3(a) and 3(c) with the mirror plane at $x = 0$; a MiS for ${|{\mathbf E} |^2}$ on Fig. 1(f) and ${\mathbf P}$ on Figs. 3(b) and 3(d) with the mirror planes at both $x = 0$ and $y = 0$. Figure 2 offers its distinct symmetry properties for $\textrm{Re} ({\mathbf E} )$: a MAS on Fig. 2(a) with the mirror plane at $x = 0$; a MiS for ${\mathop{\mathrm {Im}}\nolimits} ({{H_\varphi }} )$ on Fig. 2(c) on the transverse $xy$- or $\rho \varphi$-plane with the mirror planes at $\rho = 0$.

Figure 4 offers yet another set of symmetry properties for the EM chirality $\mathcal{C}$: a MAS on Fig. 4(a) and 4(c) with the mirror plane at $x = 0$; a MiS on Fig. 4(b) and 4(d) with the mirror plane at $y = 0$; a MAS on Fig. 4(b) and 4(d) with the mirror plane at $x = 0$ [3,17,20,25,26].

Since the EM chirality involves either of $\{{\cos ({2\varphi } ),\sin ({2\varphi } )} \}$, the azimuthal period of $\pi$ is satisfied as seen from all panels drawn on the transverse planes. Therefore, the notion of ‘quarter spaces’ arises [16,25], such that these quarter spaces are extended respectively over the azimuthal zones: ${\textstyle{1 \over 2}}\pi \ell < \varphi < {\textstyle{1 \over 2}}\pi ({\ell + 1} )$ with $\ell = 0,1,2,3$. Consequently, we can locate the mirror planes located at $\varphi = 0,\pi$ or $y = 0$ and $\varphi = {\textstyle{1 \over 2}}\pi ,{\textstyle{3 \over 2}}\pi$ or $x = 0$. Resultantly, Fig. 4 displays 4-fold patterns in the azimuthal $\varphi$-direction, depending on a MiS and/or a MAS.

4. Circularly polarized incident wave

Consider an incident wave of circular polarization (CP) as follows [1,13,15–18,20,22,26].

There is an in-quadrature (${\pm} {90^o}$ degrees) phase difference between two orthogonal constituents, which are linearly polarized and of an equal magnitude. In addition, the vector $\hat{{\mathbf x}} \pm \textrm{i}\hat{{\mathbf y}}$ common to both field variables in Eq. (3) is nothing but $({\hat{{\mathbf \rho }} \pm \textrm{i}\hat{{\mathbf \varphi }}} )\exp ({ \pm \textrm{i}\varphi } )$ in terms of the cylindrical coordinates. The combined propagation factor $\exp ({ \pm \textrm{i}\varphi + \textrm{i}z - \textrm{i}t} )$ refers thus to a helically progressive wave. The relevant Mie solution can be easily constructed as further explained by Supplement 1. Thanks to ${[{\exp ({ \pm \textrm{i}\varphi } )} ]^\ast }\exp ({ \pm \textrm{i}\varphi } )= 1$, all bilinear parameters for a CP incident wave get independent of the azimuthal $\varphi$-coordinate, namely, rotationally symmetric on the transverse $xy$- or $\rho \varphi$-plane.

We thus obtain ${{\mathbf P}^{inc}} = \hat{{\mathbf z}}$, ${{\mathbf R}^{inc}} = {\mathbf 0}$, and ${\mathcal{K}^{inc}} = 0$ for this CP incident wave. Most conspicuously, we have a nonzero incident-wave chirality ${\mathcal{C}^{inc}} ={\pm} 1$, where the upper and lower signs refer respectively to an RCP (right circular polarization) wave and an LCP (left circular polarization) wave. Equally interesting is the fact that $\nabla \cdot {\mathbf S}_E^{inc} = \nabla \cdot {\mathbf S}_H^{inc} = 0$, thereby being consistent with ${\mathcal{K}^{inc}} = 0$. Therefore, this CP incident wave does not allow for inter-modal communication between the electric and magnetic SAM densities.

Figure 5 displays $\{{\mathcal{C},\mathcal{K}} \}$ of the total field for a CP incident wave. For concreteness, we have chosen the upper signs in Eq. (3) so that ${\mathcal{C}^{inc}} ={+} 1$. On both Figs. 5(b) and 5(d), we can confirm the rotational symmetry about the longitudinal $z$-axis. As seen from Figs. 5(a) and 5(b), $\mathcal{C} > 0$ over all spaces, thus preserving the sign of the incident-wave chirality. More importantly, there are both sub-chiral zones with $0 < \mathcal{C} < 1$ and super-chiral zones with $\mathrm{C\ > }1$ for a prescribed ${\mathcal{C}^{inc}} = 1$. That is, our achiral sphere responds to a CP incident field in either stronger or weaker way [16–19,25]. Besides, Fig. 5(b) shows that the sub-chiral zone with $0 < \mathcal{C} < 1$ is largely dominant around the outside of a spherical object.

 

Fig. 5. Spatial distributions of the EM chirality $\mathcal{C} \equiv {\mathop{\mathrm {Im}}\nolimits} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ and the reactive chirality $\mathcal{K} \equiv \textrm{Re} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ for a circularly polarized incident wave. We set $m = \sqrt 2$ and $\alpha = 2$.

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Figures 5(c) and 5(d) show that the reactive chirality takes either sign over the whole space although ${\mathcal{K}^{inc}} = 0$ for a CP incident wave. Especially, Fig. 5(c) shows that the reactive chirality is negative in the downstream portion within a spherical object. In comparison, Fig. 5(d) shows that the reactive chirality is negative over a near-field annulus outside a spherical object.

Besides, Figs. 5(a) and 5(c) exhibit fan-like patterns expanding out into the downstream direction. All panels of Fig. 5 show that the super-chiral hotspots (of maximal magnitudes) occur along the longitudinal axis [16–19]. Besides, $|\mathcal{C} |$ happens to be greater than $|\mathcal{K} |$ approximately by one order of magnitude. See Supplement 1, where $|\mathcal{K} |$ could become greater than $|\mathcal{C} |$ approximately by one order of magnitude in case with a linearly polarized incident wave.

In the far field, the net reactive chirality should vanish when it is integrated over the whole space due to ${\mathcal{K}^{inc}} = 0$. To confirm this conjecture, we made integrals respectively of $\{{\mathcal{C},\mathcal{K}} \}$ over the forward half-spaces with $0 \le \theta \le {\textstyle{1 \over 2}}\pi$ and the backward half-space with ${\textstyle{1 \over 2}}\pi \le \theta \le \pi$. Suppose that $f \in \{{\mathcal{C},\mathcal{K}} \}$, both being polar-angle dependent. Let us define the integrals $I_f^ \Rightarrow \equiv \int_0^{{\pi / 2}} {f(\theta )\sin \theta d\theta }$ and $I_f^ \Leftarrow \equiv \int_{{\pi / 2}}^\pi {f(\theta )\sin \theta d\theta }$ respectively over the forward (by the superscript ‘$\Rightarrow$’) and backward (by the superscript ‘$\Leftarrow$’) hemispheres. In this way, we come up with four parameters $\{{I_{\mathcal{C}}^ \Rightarrow ,I_{\mathcal{C}}^ \Leftarrow ,I_{\mathcal{K}}^ \Rightarrow ,I_{\mathcal{K}}^ \Leftarrow } \}$. Here, both normalization constants happen to be unity, i.e., $\int_0^{{\pi / 2}} {\sin \theta d\theta } = 1$ and $\int_{{\pi / 2}}^\pi {\sin \theta d\theta }$.

Because of rotational symmetry as seen on Figs. 5(b) and 5(d), any value of $\varphi$ can be specified. Based on Figs. 5(a) and 5(c), we can thus draw Fig. 5(e) to show the radial distributions of the four integrals, thereby confirming asymptotic far-field fall-offs [26]. In other words, the upper pair on Fig. 5(e) shows that both of $\{{I_{\mathcal{C}}^ \Rightarrow ,I_{\mathcal{C}}^ \Leftarrow } \}$ are varied around ${\mathcal{C}^{inc}} = 1$, whereas the lower pair shows that both of $\{{I_{\mathcal{K}}^ \Rightarrow ,I_{\mathcal{K}}^ \Leftarrow } \}$ are varied around ${\mathcal{K}^{inc}} = 0$. The integrated EM chirality and reactive chirality undergo more violent variations as the refractive-index ratio is increased as shown in Supplement 1. This is another manifestation of the ‘light bending’ onto higher-index media (HIM) shown by the Poynting-vector streamlines on Figs. 3(a) and 3(c) [5,17,18,25].

Let us then focus on the vertical dotted line on Fig. 5(e), which marks the surface of a spherical object. We find conspicuous discontinuities in both of $\{{I_{\mathcal{C}}^ \Rightarrow ,I_{\mathcal{C}}^ \Leftarrow } \}$ across a spherical object. The discontinuities in both of $\{{I_{\mathcal{K}}^ \Rightarrow ,I_{\mathcal{K}}^ \Leftarrow } \}$ are less visible, but we could see them in Supplement 1 evaluated for refractive-index ratios not only of $m = \sqrt 2$ (closer to that of silicon dioxide or Teflon) but also of $m = 2$ (for titanium dioxide) [5,9,18]. There is a net $\mathcal{C}$ when integrated further respect to the radial distance as seen from the upper pair. Such a net $\mathcal{C}$ is greater inside than outside a spherical object. Across a spherical surface, $I_{\mathcal{C}}^ \Rightarrow > I_{\mathcal{C}}^ \Leftarrow$ (EM-chirality excess), whereas $|{I_{\mathcal{K}}^ \Rightarrow } |\approx |{I_{\mathcal{K}}^ \Leftarrow } |$ because of conservation of the reactive chirality $I_{\mathcal{K}}^ \Rightarrow + I_{\mathcal{K}}^ \Leftarrow \approx 0$ [26].

5. Linearly polarized incident wave of two orthogonal components

Consider an incident wave of other polarization states, while keeping the plane-wave nature of $\exp ({\textrm{i}z} )$. We then make of the identity $\cos ({\varphi - {\varphi_0}} )\hat{{\mathbf \rho }} - \sin ({\varphi - {\varphi_0}} )\hat{{\mathbf \varphi }} = \cos {\varphi _0}\hat{{\mathbf x}} + \sin {\varphi _0}\hat{{\mathbf y}}$ to construct the following incident wave of two-component linear polarization states with equal phase between them [7].

Here, the constant azimuthal shift ${\varphi _0} \in {\mathbb R}$ determines the relative strengths of the two orthogonal constituents for linear polarizations. However, this plane wave does not indicate elliptical polarizations, which are conventionally denoted by ${{\mathbf E}^{inc}} = ({\cos {\varphi_0}\hat{{\mathbf x}} \pm \textrm{i}\sin {\varphi_0}\hat{{\mathbf y}}} )\exp ({\textrm{i}z} )$. Thanks to $\hat{{\mathbf x}} = \cos \varphi \hat{{\mathbf \rho }} - \sin \varphi \hat{{\mathbf \varphi }}$, the solutions to the corresponding Mie solution can be readily obtained by making a substitution $\varphi \to \varphi - {\varphi _0}$, say, in ${{\mathbf E}^{gen}}({r,\theta ,\varphi } )$. Various bilinear parameters can thus be readily evaluated.

Figure 6 shows the EM chirality $\mathcal{C}$ of the total wave for the incident wave given in Eq. (4). There are several sub-panels on Fig. 6(a) obtained with $m = \sqrt 2$, whereas there is a single panel on Fig. 6(b) obtained with $m = 2$ (in the lower-left corner). On Fig. 6(a), we made variations in the azimuthal shift such that ${\varphi _0}$ is increased by ${\textstyle{1 \over 8}}\pi$ from one sub-panel to the next (in the clockwise direction). Here, the mirror planes of a MiS and/or a MAS are/is always aligned with the fixed laboratory coordinates (namely, the respective constituent field vectors). The progression among the sub-panels on Fig. 6(a) is made into a Visualization 1 in steps of ${5^0}$ between $0 \le {\varphi _0} \le {180^0}$.

 

Fig. 6. Spatial distributions of the EM chirality $\mathcal{C} \equiv {\mathop{\mathrm {Im}}\nolimits} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ for an incident wave of in-phase two-component linear polarizations. All plots are made on the transverse plane that passes through the sphere’s center. The black- and blue-boundary arrows indicate the directions respectively of the electric and magnetic fields of an incident wave. Each panel is assigned a distinct shift ${\varphi _0}$ on (a), and the patterns repeat themselves with a period of ${\varphi _0} = \pi$. We set $\alpha = 2$, and (a)$m = \sqrt 2$ and (b) $m = 2$.

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We find on Fig. 6(a) a transition in the symmetry behavior with increasing ${\varphi _0}$. Consider the first sub-panel Fig. 6(a1) with ${\varphi _0} = 0$, which is identical to Fig. 4(b). Here, a mirror anti-symmetry (MAS) holds true across the central vertical mirror plane, whereas a mirror symmetry (MiS) is established across the central horizontal plane. Let us go over to the next sub-panel of Fig. 6(a2) with ${\varphi _0} = {\textstyle{1 \over 8}}\pi$ (the second from the left on the top row). Now, there is only a MiS across the central horizontal mirror plane marked by a solid red line. On the third sub-panel of Fig. 6(a3), a MiS holds true across both central planes so on.

Let us focus on the first sub-panel on Fig. 6(a1) and Fig. 6(b), both being obtained with ${\varphi _0} = 0$, namely, for the $x$-directed electric field. Notice that the color on Fig. 6(b) indicates a range ten times larger in comparison to the color bar lying to the left of Fig. 6(a8) that applies to all sub-panels on Fig. 6(a). This larger excursion in $\mathcal{C}$ with $m = 2$ on Fig. 6(b) is another example of the ‘light bending’ onto higher-index media (HIM) discussed in connection with Figs. 3, 4, and 5(e).

In other words, the EM chirality is stronger inside a sphere with $m = 2$ on Fig. 6(b), whereas it is stronger outside of a sphere with $m = \sqrt 2$ on Fig. 6(a1) [5]. Consequently, we guess that the maximum in $|\mathcal{C} |$ is caged (or confined) from the outside of a sphere into the inside as m is increased. Our detailed plot in Supplement 1 shows that this transition from Fig. 6(a1) to Fig. 6(b) takes place around at $m = 2 + \left( {2 - \sqrt 2 } \right){\textstyle{9 \over {10}}} \approx 1.94$. The Visualization 2 shows $\mathcal{C}$ as the refractive-index ratio m is varied over $\sqrt 2 \le m \le 2.15$ in equal steps so that EM-chirality enhancement with increasing refractive-index ratio can be illustrated [5,10,17,18,29]. In addition, the resulting EM chirality on Fig. 6 carries the interference effects between $\{{{\mathbf E},{\mathbf H}} \}$ of the two constituent polarizations as specified by Eq. (4).

6. Discussions

In search for another Mie solution, a naïve approach is to make a substitution $\varphi \to \pi - \varphi$ in all pertinent formulas, say, in ${{\mathbf E}^{gen}}({r,\theta ,\varphi } )$. These apparent solutions to the Mie scattering turn out not to satisfy the governing Maxwell equations. Here, a relevant issue is more complicated since an incident wave of radial and/or azimuthal polarizations are sometimes necessary [7,10,15,18,21,32]. As a relevant problem, a two-wave system involving an evanescent incident wave has been examined by [11].

A racemic mixture of equal amounts of left-handed and right-handed EM-chiral properties could be separated into their enantiomerically pure (enantiopure) constituents. This enantiomeric separation involves EM torques due to associated asymmetry [25,30]. This enantioselectivity stems essentially from the interaction between a spherical object and an incident wave which progresses in a certain wave-propagation direction [17,21,23,26,31]. In other words, a spherical object and a Cartesian rectilinear wave vector leads to an enantio-selective EM-chirality distribution.

Our suggested enantio-separation would take advantage of the quarter spaces as shown in Fig. 4(b). This spatially compartmentalized enantio-separation (a.k.a. chiral sorting) is distinct from the conventional chromatographic separation based on fluid flows [21]. Each of such quarter spaces shown on Fig. 4(b) could thus serve as a ‘EM-chiral stationary space (EM-CSS)’ when following the idea of the ‘chiral stationary phase (CSP)’ [23]. For our method to be realizable, we need not only to hold a spherical particle fixed in a laboratory space but also to illuminate the sphere at normal incidence [17,20].

We find several measurement techniques: degrees of circular polarization (DoCPs) by a near-field polarimetry [3], EM-chirality based on the SAM densities in Eq. (2) [19], SHG (second-harmonic generation) microscopy [25], to name a few. As a relevant issue, suppose that a nano-object, being much smaller than the spherical object under consideration, is immersed in an embedding medium. This object is further supposed to be both electrically and magnetically polarizable [2,3,14,17,26]. Besides, it is assumed absorbing. In this case, the force involves both of $\{{{\mathbf P},{\mathbf R}} \}$, whereas the torque involves both of $\{{{{\mathbf S}_E},{{\mathbf S}_H}} \}$ to a good approximation [11].

The phase shift ${\varphi _0}$ employed for Eq. (4) is akin to a twist angle of ordered arrays of dielectric spheres [5]. In this respect, we could form one- or two-dimensional arrays while taking ${\varphi _0}$ as a degree of freedom, say, for information encoding or chirality spectroscopy [17]. We may take the inter-particle spacing much larger than the sphere’s diameter such that inter-particle interactions are neglected [22]. In addition, we find on Fig. 6 that both mirror planes of a MiS and MAS are available only on four panels for ${\varphi _0} = {\textstyle{1 \over 4}}\pi \ell$ with $\ell = 0,1,2,3, \cdot{\cdot} \cdot$.

The nature of incident plane wave will be rendered most general when the circular polarization in Eq. (3) and the two-component in-phase wave in Eq. (4) are combined together such that ${{\mathbf E}^{inc}} = ({{e^{\textrm{i}\beta }}\cos {\varphi_0}\hat{{\mathbf x}} + {e^{ - \textrm{i}\beta }}\sin {\varphi_0}\hat{{\mathbf y}}} )\exp ({\textrm{i}z} )$ with $\beta$ being real [10,26].

It is worth stating again that all-dielctric systems carry both electric and magnetic modes on approximately equal footing [10], in comparison to plasmonic metallic spheres [16]. Regarding Fig. 5(e), a mean (macroscopic spatially averaged) EM chirality is conserved for a body of achiral structures [2,9,15,17–19,25], whereas a vanishing net value is found for a reactive chirality. The toroidal feature as illustrated on Fig. 1(d), 2(b) and 2(c) is suggestive of ways of fabricating nano-objects to generate artificial magnetism.

There are other issues that are not covered in this study. They include the scattering off a spherical object by structured illuminations such as a Gaussian beam and a Bessel beam [1,10,13,15,18]. A candidate list goes on to include the scatterings off core-shell particles, cylinders, and non-spherical objects [3]. In this short study, we have refrained from covering wider topics such as the nonlinear magneto-chiral properties [25], Kerker conditions [12,14,17,44], optical binding, media with losses [5,9,17], complementary structures of spherical cavities or voids [14], geometrically chiral structures in the context of metamaterials and metasurfaces [2,6,13,15,17–19,21], etc.

There are other bilinear parameters such as the reactive chirality $\mathcal{K} \equiv \textrm{Re} ({{\mathbf E}\cdot {{\mathbf H}^\ast }} )$ discussed in Eq. (2). In this aspect, we discussed the dual-asymmetric (DA) energy difference ${|{\mathbf E} |^2} - {|{\mathbf H} |^2}$ and the reactive Poynting vector (magneto-electric angular momentum exchange) $|{\mathbf R} |\equiv |{{\mathop{\mathrm {Im}}\nolimits} ({{{\mathbf E}^\ast } \times {\mathbf H}} )} |$ in Supplement 1 in accordance with the formulas handled by [1] and [12]. The corresponding results exhibited on Figs. S5c, S6b, and S6c (of Supplement 1) carry characters analogous in some sense to those shown on Figs. 1, 4, and 5. Since such additional parameters play important roles in optical manipulations, we plan to make deeper investigations in separate papers when we examine spherical voids as well.

7. Conclusion

We have examined the simplest two-dielectric Mie scatterings off a spherical object with three varied forms of incident plane waves. We have thus shown how the electromagnetic chirality of an incident wave is generated or altered due to light scattering. We found the light bending onto higher-index media convenient in illustrating the EM-chirality enhancement particularly in the near field of a spherical object. Electric-field circulations are proved to play a crucial role in arousing effective magnetism within such all-dielectric photonic systems. Besides, the symmetry properties of the EM chirality have been examined with respect to possible applications. We have focused on the EM chirality and its reactive counterpart in the problems of Mie scattering which, to our knowledge, have not been investigated before.’