# Spheres of maximum electromagnetic chirality

Jorge Olmos-Trigo<EMAIL_ADDRESS>Centro de Física de Materiales, Paseo
Manuel de Lardizabal 5, 20018 Donostia-San Sebastian, Spain. Manuel Nieto-
Vesperinas Instituto de Ciencia de Materiales de Madrid, Consejo Superior de
Investigaciones Científicas. Campus de Cantoblanco, Madrid 28049, Spain.
Gabriel Molina-Terriza Centro de Física de Materiales, Paseo Manuel de
Lardizabal 5, 20018 Donostia-San Sebastian, Spain. Donostia International
Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian,
Spain. IKERBASQUE, Basque Foundation for Science, María Díaz de Haro 3, 48013
Bilbao, Spain.

###### Abstract

The search for objects that yield maximum electromagnetic chirality in their
emitted wavefield has garnered significant attention in recent years. However,
achieving such maximum chirality is challenging, as it typically requires
complex chiral metamaterials. Here we demonstrate that chiral spheres can
yield maximum chirality in their emitted wavefield. Specifically, we
analytically find the spectral trajectories at which chiral spheres become
optically transparent to a given helicity of the incident field, while for its
opposite helicity, they behave as dual objects, i.e., on scattering, they
preserve helicity. Since chiral spheres behave as dual objects at the first
Kerker condition of zero backscattering, we significantly simplify this
condition in terms of a Riccati-Bessel function. Importantly, all our results
are exact and applicable regardless of the multipolar order, refractive index
contrast, optical size, and intrinsic chirality of the chiral sphere. Thus,
our exact findings can serve as building blocks for designing novel
metasurfaces or metamaterials with maximum electromagnetic chirality
properties.

Chiral objects cannot be superimposed with their mirror images by translations
or rotations Petitjean (2003); Hentschel  _et al._ (2017); Chen  _et al._
(2022). Many crucial organic molecules, such as Glucose, DNA, and most
biological amino-acids, are chiral. Nowadays, it is recognized that chirality
plays a vital role in the chemical interactions of chiral drugs in the human
body regarding their curative potency and toxicity Chen  _et al._ (1982). For
instance, the thalidomide tragedy demonstrated that any drug that manifests in
enantiomers (mirror pairs of chiral molecules) must be considered as two
different drugs whose efficacy and side effects must be treated and determined
separately Lenz and Knapp (1962). Consequently, detecting and characterizing
enantiomers has been of utmost relevance in the biomedical and pharmaceutical
industries McBride (1961); Nugent  _et al._ (1993); Kim  _et al._ (2022).
Besides its geometry, another feasible way to infer the chirality of an object
is based on its interactions with chiral light.

In electromagnetism, the most usual magnitude to unravel the chirality of a
sample is the handedness of circularly-polarized (CPL) fields Tang and Cohen
(2010); Sanz-Hernández  _et al._ (2020), commonly referred to as helicity
Calkin (1965). In addition to the potential applications of helicity in the
context of optical forces Cameron  _et al._ (2014); Canaguier-Durand  _et al._
(2013); Ding  _et al._ (2014); Tkachenko and Brasselet (2014); Nieto-
Vesperinas (2015a); Hayat  _et al._ (2015); Liu  _et al._ (2018); Li  _et al._
(2019); Nieto-Vesperinas and Xu (2021); Genet (2022); Sifat  _et al._ (2022);
Wu  _et al._ (2022), spin-orbit interactions of light Marrucci  _et al._
(2006); Bliokh  _et al._ (2015); Olmos-Trigo  _et al._ (2019a, b); Forbes and
Andrews (2019); Guo  _et al._ (2019); Müllner  _et al._ (2022); Gautier  _et
al._ (2022), or its fundamental connection to Kerker conditions Zambrana-
Puyalto  _et al._ (2013a); Olmos-Trigo  _et al._ (2020a); Olmos-Trigo and
Zambrana-Puyalto (2022), helicity can also be used to probe chiral and achiral
objects Yoo and Park (2015); Nieto-Vesperinas (2017); Poulikakos  _et al._
(2019); Crimin  _et al._ (2019); Olmos-Trigo  _et al._ (2023). Now, an object
is electromagnetically achiral if all components of the scattered field under
the illumination with CPL electromagnetic waves of one sign can also be
mimicked with incident CPL waves of the opposite sign. Otherwise, the object
is electromagnetically chiral Hanifeh and Capolino (2020).

However, chiroptical responses are typically weak; thus, the intrinsic
chirality of samples is often not detected Tang and Cohen (2010), and
researchers have recently turned their attention to the search of objects of
maximum electromagnetic chirality (MECh) Fernandez-Corbaton  _et al._ (2016).
In short, all chiral samples that are transparent to all fields of one
helicity are objects of MECh Fernandez-Corbaton  _et al._ (2016).
Additionally, if such object is reciprocal, the previous argument is
bidirectional: all objects of MECh are optically transparent to a given
helicity of the field. Moreover, reciprocal objects of MECh preserve the
opposite helicity of the incident field upon interaction. That is, such
objects have electromagnetic duality as a result of the fact that they are
reciprocal Fernandez-Corbaton  _et al._ (2016).

However, to date, objects yielding MECh have only been unveiled in complex
chiral metamaterials. For instance, in a double-turn helix Garcia-Santiago
_et al._ (2022), on metasurfaces made of pairs of dielectric bars Gorkunov
_et al._ (2020), in dielectric metamaterials consisting of a random colloid of
meta-atoms Asadchy  _et al._ (2020), or in a photonic crystal with a chiral
array of perforating holes Semnani  _et al._ (2020), among others Höflich  _et
al._ (2019); Gorkunov  _et al._ (2021); Kühner  _et al._ (2022); Lim  _et al._
(2022); Zhao  _et al._ (2022); Shi  _et al._ (2022). In this regard, we should
note that numerical methods are involved in all previous examples devoted to
the search for objects yielding MECh Garcia-Santiago  _et al._ (2022);
Gorkunov  _et al._ (2020); Asadchy  _et al._ (2020); Semnani  _et al._ (2020);
Höflich  _et al._ (2019); Gorkunov  _et al._ (2021); Kühner  _et al._ (2022);
Lim  _et al._ (2022); Zhao  _et al._ (2022); Shi  _et al._ (2022). This is
because the constitutive relations of chiral media have only been analytically
solved for the case of spherical objects by Craig Bohren in 1974 Bohren
(1974). However, whether spherical chiral particles may behave as objects of
MECh remains unexplored Shang  _et al._ (2013); Hu  _et al._ (2022).

In this work, we demonstrate that chiral spheres can behave as objects of
MECh. Based on energy conservation, we first derive the equation that such a
sphere satisfies. Shortly after, we solve this equation by analytically
finding the spectral trajectories at which chiral spheres are optically
transparent to a given helicity of the incident field. In contrast, for the
opposite helicity of the incident field, they behave as dual objects, i.e., on
scattering, they preserve helicity as a consequence of reciprocity for MECh
objects 111Note that chiral spheres are by construction reciprocal, and thus,
meet the Onsager relations. As a consequence of reciprocity, if a chiral
sphere is transparent to a given helicity of the incident field, then the
opposite helicity of the incident field is necessarily preserved upon
interaction.. Moreover, since electromagnetic duality is a characteristic of
spheres also when they meet the first Kerker condition (K1) of zero
backscattering Nieto-Vesperinas (2015b), we disclose the analytical
requirements of K1 for chiral spheres in terms of a Riccati-Bessel function.
This offers a straightforward, systematic procedure to establish the K1
condition. Note that the so-called generalized Kerker effects Liu and Kivshar
(2018); Alaee  _et al._ (2015); Bag  _et al._ (2018); Shamkhi  _et al._ (2019)
do not generally correspond to dual particles and, thus, do not emerge for
spheres of MECh.

We first address magnetodielectric chiral spheres of high refractive index
(HRI), which, as far as we know, have not yet been synthesized; subsequently,
we present results for so far existing dielectric chiral spheres of lower
refractive index.

Our findings are exact and remain valid regardless of the multipolar order,
refractive index, optical size, and intrinsic chirality. Spheres yielding MECh
can be used as building blocks of novel maximum electromagnetic chirality
nanoantennas, metasurfaces, and metamaterials. Therefore, our predictions
suggest experimental observations.

In what follows, we will derive the general conditions that must be imposed on
the scattering coefficients for a sphere to become an object of MECh. They can
be summarized with a simple formula:

$\displaystyle\boxed{\;a_{\ell}=b_{\ell}=i\sigma
c_{\ell}\;\forall\ell\quad\text{and}\quad c_{\ell}\neq 0.}$ (1)

The implication of Eq. (1) is that for the incident helicity $\sigma=\pm 1$,
we get $\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=0$, (cf. Appendix A),
while for the helicity $-\sigma$,
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=2ic_{\ell}$ 222Note that the
theoretical framework is introduced in Appendix. A while the explicit form of
the $a_{\ell}$, $b_{\ell}$, annd $c_{\ell}$ chiral Mie coefficients are given
in B. . Note that $\\{\tilde{a}_{\ell,\sigma},\tilde{b}_{\ell,\sigma}\\}$ are
given by Eq. (13). As a result, the scattering cross-section for any incident
field will be zero for helicity $\sigma$, while it will be non-zero for an
incident field carrying the opposite helicity.

To start the derivation, let us use the GLMT in order to compute the
extinction and scattering cross-sections, $Q^{\sigma}_{\rm{ext}}$ and
$Q^{\sigma}_{\rm{sca}}$, respectively. We generalize the work by Gouesbet et
al., devoted to the GLMT for achiral spheres Gouesbet  _et al._ (1985), to
obtain

$\displaystyle k^{2}Q^{\sigma}_{\rm{ext}}$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\ell=\infty}|C^{\sigma}_{\ell
m}|^{2}\Re\\{\tilde{a}_{\ell,\sigma}+\tilde{b}_{\ell,\sigma}\\},$ (2)
$\displaystyle k^{2}Q^{\sigma}_{\rm{sca}}$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\ell=\infty}|C^{\sigma}_{\ell
m}|^{2}\left(|\tilde{a}_{\ell,\sigma}|^{2}+|\tilde{b}_{\ell,\sigma}|^{2}\right).$
(3)

Moreover, let us also address the absorption cross-section,
$Q^{\sigma}_{\rm{abs}}=Q^{\sigma}_{\rm{ext}}-Q^{\sigma}_{\rm{sca}}$, which we
will use later on.

The derivation of Eq. 1 proceeds in the following way:

  1. 1)

We will show that objects of MECh must be lossless.

  2. 2)

Then, we shall show that an antidual sphere must be completely transparent,
i.e., the extinction cross-section is identical to zero.

  3. 3)

We will apply 1) and 2) to a chiral sphere and see that the conditions for
MECh are those of Eq. (1).

Figure 1: Scattering features of a sphere of maximum electromagnetic chirality
under the illumination of a well-defined helicity field. Red and green colors
correspond to left and right circularly polarized fields, respectively. a)
Antiduality condition (helicity is sign-flipped after scattering). b) Optical
transparency (no scattering). c) Duality condition, i.e., preservation of
helicity after scattering, which is also a characteristic of the first Kerker
condition, which leads to zero backscattering for cylindrically symmetry
objects. The left panel highlights that optical transparency is reached for an
antidual chiral sphere if and only if it is also lossless. The right panel
highlights that duality is reached for reciprocal spheres of MECh upon
flipping the helicity content of the incident field.

To start with, we observe that objects of MECh are optically transparent to a
given helicity of the incident field Fernandez-Corbaton  _et al._ (2016).
Therefore, such objects cannot absorb photons with that helicity. In this
regard, Kirchhoff’s thermal radiation law states that the emissivity of a body
must be equal to its absorptivity Kirchhoff (1860); Planck (1914); Greffet and
Nieto-Vesperinas (1998). In other words, a lossy generic object must emit.
Therefore, to avoid emission, the object must necessarily be without losses.

Now, we turn to the derivation that an antidual sphere must be completely
transparent. By definition, such sphere must not scatter light with the same
helicity as the incident field Zambrana-Puyalto  _et al._ (2013b). Let us
impose optical transparency in the same helicity channel as the incident field
($\sigma=\sigma^{\prime}$), namely, $|{\bf E}^{\sigma\sigma}_{\rm{sca}}({\bf
r})|=0$ (see Appendix A). Then from Eqs. (11)-(12), we straightforwardly note
that

$\displaystyle\tilde{a}_{\ell,\sigma}=-\tilde{b}_{\ell,\sigma}\;\forall\ell$
$\displaystyle\Longleftrightarrow a_{\ell}+b_{\ell}=2i\sigma
c_{\ell}\;\forall\ell$ for $\displaystyle\;\;\sigma^{\prime}=\sigma.$ (4)

From Eq. (2), we note that Eq. (4) is a sufficient condition for total
transparency to a given incident helicity since then the extinction cross
section goes to zero. That is, imposing partial transparency in the same
helicity channel as the incident field ($\sigma=\sigma^{\prime}$) naturally
leads to $Q^{\sigma}_{\rm{ext}}=0$. Note that in this derivation, we have
employed the formulas for a chiral sphere to apply them to the next part of
the derivation directly. Although complete transparency of an antidual sphere
is a general result Olmos-Trigo  _et al._ (2020b, c); Lasa-Alonso  _et al._
(2023), when applied to chiral spheres, if they are total transparent for a
given incident helicity, they may be non-transparent for the opposite incident
helicity.

It is crucial to notice that the non-absorbing version of the conservation
energy ($Q^{\sigma}_{\rm{ext}}=Q^{\sigma}_{\rm{sca}}$), together with Eq. (4),
also imposes $Q^{\sigma}_{\rm{sca}}=0$. Now, according to Eq. (3), the only
possible solution is then given by
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=0$, which leads to 333Note
that this argument is bidirectional. That is, if
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=0$, then
$Q^{\sigma}_{\rm{sca}}=0$.

$\displaystyle\boxed{\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=0\;\forall\ell\Longleftrightarrow
a_{\ell}=b_{\ell}=i\sigma c_{\ell}\;\forall\ell.}$ (5)

This ends our derivation, as these equations lead directly to the requirements
we anticipated in Eq. (1). Therefore, while for incident helicity $\sigma$,
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=0$, for the opposite helicity
of the incident field, we obtain
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=2ic_{\ell}$. As expected and
due to the fact that chiral spheres are reciprocal objects Onsager (1936),
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}=2ic_{\ell}$ preserves the
helicity content of the incident field.

Equations (1),(4)-(5) are important results of the present Letter. In this
connection, notice that the conclusions of Ref. Fernandez-Corbaton  _et al._
(2016) particularized, on the one hand, to magnetic scatterers, described by
$\epsilon$ and $\mu$, and, on the other hand, to magneto-electric dipolar
particles, are consistent with our results 444An important insight that we get
within the current approach is that we have not imposed transparency to the
opposite helicity of the incident field, $\sigma^{\prime}=-\sigma$. In other
words, we have not imposed $|{\bf E}^{\sigma-\sigma}_{\rm{sca}}({\bf r})|=0$
to arrive to total transparency. In contrast, we have just required
transparency in the same helicity channel of the incident field
($\sigma=\sigma^{\prime}$), finding that the latter is sufficient for total
transparency to a given helicity of the incident field as extinction
vanishes..

To get a visual insight into these effects, Fig. 1 depicts the scattering
features of a sphere of MECh under the illumination of an incident field with
well-defined helicity.

At this stage, we establish the solutions to Eq. (5). That is, we will find
the spectral trajectories at which spheres can yield MECh, in terms of a
Riccati-Bessel function. Moreover, we will also unravel the first Kerker
condition of zero backscattering,
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}$, in terms of a Riccati-
Bessel function.

Spheres of MECh satisfy Eq. (5). However, it is challenging to find analytical
solutions to Eq. (5) when several multipolar orders contribute to the optical
response of a chiral sphere. Then, let us start this section by imposing Eq.
(5) for a fixed multipolar order $\ell$. Now, by inspecting Eqs. (14a)-(14c)
and Eqs. (15a)-(15d), we notice that

$a_{\ell}=b_{\ell}=i\sigma
c_{\ell}\Longleftrightarrow\left\\{\begin{array}[]{lr}A_{\ell}(R)=\sigma
B_{\ell}(R).\\\ A_{\ell}(L)=-\sigma B_{\ell}(L).\end{array}\right\\}$ (6)

Equation (6) represents an important simplification to find the analytical
condition ruling spheres of MECh. Note that $W_{\ell}(J)$ and $V_{\ell}(J)$
for $J=L,R$ do not play any role in finding a sphere of MECh. Moreover, it is
vital to notice that $A_{\ell}(R)=\sigma B_{\ell}(R)$ and $A_{\ell}(L)=-\sigma
B_{\ell}(L)$ must be satisfied simultaneously and independently. In other
words, such spheres depend on the helicity of the incident field. With this
information in mind, we can now simplify Eq. (6) to

$\boxed{a_{\ell}=b_{\ell}=i\sigma
c_{\ell}\Longleftrightarrow\begin{array}[]{lr}\psi_{\ell}(m_{R}x)\psi^{\prime}_{\ell}(x)=-\sigma\psi_{\ell}(x)\psi^{\prime}_{\ell}(m_{R}x).\\\
\psi_{\ell}(m_{L}x)\psi^{\prime}_{\ell}(x)=\sigma\psi_{\ell}(x)\psi^{\prime}_{\ell}(m_{L}x).\end{array}}$
(7)

Equation (7) is one of the primary key results of this Letter, as it can be
used as a general road map for finding the spectral trajectories at which
chiral spheres can behave as objects of MECh. It is essential to emphasize
that Eq. (7) is exact and, thus, remains valid regardless of the multipolar
order $\ell$, refractive index contrast $m$, and optical size $x$ of the
chiral sphere. Moreover, it is also crucial to note that we have not imposed
any restriction on the intrinsic chirality of the sphere, $\chi$. Hence, Eq.
(7) could, in principle, be satisfied for an infinity of values of the set
$\\{\sigma,\ell,m,x,\chi\\}$, underscoring the generality of our analytical
derivation.

At this point, we should also highlight another important aspect included in
Eq. (7): The Kerker condition, K1, given by
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}\;\forall\ell$ Kerker  _et
al._ (1983); Nieto-Vesperinas  _et al._ (2011); Person  _et al._ (2013);
Carretero  _et al._ (2019). Notice that at K1, the electromagnetic helicity is
preserved since $|{\bf E}^{\sigma-\sigma}_{\rm{sca}}({\bf r})|=0$ (see
Appendix A). We also note from Eq. (7) that
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}\;\forall\ell$ is a necessary
yet insufficient condition for chiral spheres to be objects of MECh. Now,
taking into account the effective relations of Mie coefficients, which are
given by Eq. (13), we infer that
$\tilde{a}_{\ell,\sigma}=\tilde{b}_{\ell,\sigma}\;\Longleftrightarrow{a}_{\ell}={b}_{\ell}\;$.
That is, the chiral coefficient $c_{\ell}$ and the incident helicity of the
field $\sigma$ do not play any role in the emergence of K1 in the scattering
by chiral spheres. Taking this fact into account, we can now manipulate Eq.
(7) to obtain after some algebra

$\boxed{a_{\ell}=b_{\ell}\Longleftrightarrow\psi_{\ell}(m_{R}x)\psi^{\prime}_{\ell}(m_{L}x)+\psi_{\ell}(m_{L}x)\psi^{\prime}_{\ell}(m_{R}x)=0.}$
(8)

Equation (8) gives the spectral trajectories at which K1 is satisfied. Again,
we remark that no restrictions have been imposed on Bohren´s exact solution
Bohren (1974). Hence, Eq. (8) is exact and remains valid for finding the first
Kerker condition of zero backscattering, given by $a_{\ell}=b_{\ell}$, as a
function of $\\{\ell,m,x,\chi\\}$.

Hitherto, we have discussed spheres of MECh from the point of view of energy
conservation. Later on, we found the exact analytical expression of such
spheres in terms of a Riccati-Bessel function, inferred from Eq. (7).
Moreover, from Eq. (8), we derived the first Kerker condition of zero
backscattering in terms of the same Riccati-Bessel function. However, up to
now, we have not explicitly shown the solutions to Eqs. (7)-(8).

Figure 2 shows surfaces at which K1, $a_{1}=b_{1}$, holds versus the optical
size $x$, refractive index contrast $m$, and intrinsic chirality $\chi$ of a
HRI chiral sphere. Note that there are several sheets that emerge from Eq. (8)
for $l=1$ as the Riccati-Bessel functions $\psi_{\ell}$ and
$\psi_{\ell}^{{}^{\prime}}$ oscillate with $x$ and $m$, while their variation
with $\chi$ is very slow within its range of values.

In addition, we show the spectral trajectories of MECh for incident helicity
$\sigma=+1$ (blue curves) and $\sigma=-1$ (green curves). In this regard, it
is essential to notice that two experiments are embedded in Fig. 2: one with
$\sigma=1$ (green curves vanish) and the other with $\sigma=-1$ (blue curves
vanish).

We should acknowledge that, as mentioned before and to the best of our
knowledge, chiral HRI spheres have not yet been synthesized. To encourage
experiments at the current state-of-the-art, Fig. 3 shows that spheres may
exhibit MECh at lower refractive index contrasts ($1.5<m<2$) for larger
optical sizes ($6<x<8$). This range of parameters is of utmost current
interest because previous experimental studies on chiral microspheres with low
refractive indices have been reported in the literature Cipparrone  _et al._
(2011); Donato  _et al._ (2014); Shi  _et al._ (2020); Venkatakrishnarao  _et
al._ (2017).

Last but not least, let us briefly discuss the common features of Figs. 2-3.
First, we can note that a sphere of MECh surfs the first Kerker condition of
zero backscattering for both helicities of the incident field. That is, the
blue and green lines sail the red surfaces in Figs. 2-3. In this regard, we
note that these trajectories of MECh predominantly vanish upon flipping the
helicity of the incident field from $\sigma\Longleftrightarrow-\sigma$. In
other words, the blue and green lines do not generally intersect. However,
this is not a general feature (see the purple circles).

Figure 2: Chiral spheres of high refractive index: First Kerker condition, Eq.
(8) , given by $a_{1}=b_{1}$ (red surfaces), and spheres of MECh, given by
$a_{1}=b_{1}={\rm i}\sigma c_{1}$, (i.e., only $\ell=1$ contributes), as a
function of the optical size of the chiral sphere $x$, refractive index
contrast $m$, and intrinsic chirality $\chi$. The helicities of the incident
field are given by $\sigma=1$ (blue curves) and $\sigma=-1$ (green curves),
respectively. Purple circles denote hybrid anapoles given by
$a_{1}=b_{1}=c_{1}=0$.  Figure 3: Chiral spheres of low refractive index:
First Kerker condition, Eq. (8), given by $a_{1}=b_{1}$ (red surfaces), and
spheres of MECh, given by $a_{1}=b_{1}={\rm i}\sigma c_{1}$, (i.e., only
$\ell=1$ contributes) as a function of the optical size of the chiral sphere
$x$, refractive index contrast $m$, and intrinsic chirality $\chi$. The
helicities of the incident field are given by $\sigma=1$ (blue curves) and
$\sigma=-1$ (green curves), respectively. Purple circles denote hybrid
anapoles given by $a_{1}=b_{1}=c_{1}=0$.

In the following, we provide a physical explanation of these points. First, we
can infer that the crossing of spheres of MECh for $\sigma=+1$ and $\sigma=-1$
arises whenever $a_{\ell}=b_{\ell}=c_{\ell}=0$. In fact, the purple circle
represents a chiral sphere that behaves as a non-radiating source for both
helicities of the incident field. In other words, a chiral sphere satisfying
$a_{\ell}=b_{\ell}=c_{\ell}=0$ meets the hybrid anapole condition Luk’yanchuk
_et al._ (2017); Sanz-Fernández  _et al._ (2021); Coe  _et al._ (2022). In
short, an hybrid anapole emerges when the electric and magnetic scattering
coefficients simultaneously vanish regardless of the incident helicity. In
Refs Luk’yanchuk  _et al._ (2017); Sanz-Fernández  _et al._ (2021); Coe  _et
al._ (2022), all illuminated samples are achiral ($\chi=0$) and satisfy the
hybrid anapole condition $a_{\ell}=b_{\ell}=0$. In our case, we have a very
different scenario: from the purple circles in Figs. 2-3, we can infer that
$\chi\neq 0$. That is, even if the sphere is chiral, it yields a scattered
field that is identical to zero for both helicities. This fact arises since
$c_{\ell}=0$. In short, we can conclude that $\chi=0\rightarrow c_{\ell}=0$
while $c_{\ell}=0$ does not generally imply $\chi=0$.

In conclusion, we have demonstrated that chiral spheres may be objects
yielding maximum electromagnetic chirality. That is, we have analytically
found spectral trajectories in which chiral spheres are optically transparent
to a given helicity of the incident field, while for its opposite helicity,
they behave as dual objects as a result of reciprocity.

In addition, we have unveiled the first Kerker condition of zero
backscattering in terms of a Riccati-Bessel function. We recall that this
condition preserves the incident helicity upon interaction, yielding zero
backscattered light for cylindrically symmetry targets such as chiral spheres
Zambrana-Puyalto  _et al._ (2013a). In this connection, note that the so-
called generalized Kerker effects Liu and Kivshar (2018); Alaee  _et al._
(2015); Bag  _et al._ (2018); Shamkhi  _et al._ (2019), which are other
important cases of directional scattering for higher multipolar orders, do not
generally correspond to dual particles, and thus, have not been addressed
here.

Our results are exact and remain valid regardless of multipole order,
refractive index, optical size, and intrinsic chirality of the sphere.
Therefore, our predictions can be used in experiments aimed to improve
dichroism characterization and the optical manipulation and sorting of
present-day low-refractive index chiral spheres.

Additionally, our findings are of use to develop building blocks for novel
maximum electromagnetic chirality nanoantennas, metasurfaces, or
metamaterials, encouraging experiments, and inspiring the fabrication of high
refractive index chiral magnetoelectric spheres.

JO-T wishes to acknowledge his former thesis supervisor, Prof. Juan José Sáenz
(Mole), for sharing his passion for science and helping him to become an
independent physicist. He also thanks Jon Lasa-Alonso for initial
conversations and Adrian Juan-Delgado for valuable insights. He also would
like to thank Dr. Cristina Sanz-Fernández for her helpful quantum comments.
J.O-T. acknowledges support from the Juan de la Cierva fellowship No.
FJC2021-047090-I of MCIN/AEI/10.13039/501100011033 and NextGenerationEU/PRTR.
Research financial aid from this ministry is also thanked by MN-V.

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##  Appendix A Theoretical framework

Next, we lay out the framework to find that chiral spheres can be objects of
MECh. To that end, let us consider an incident field with well-defined
helicity $\sigma=\pm 1$ Olmos-Trigo  _et al._ (2019c),

$\displaystyle{\bf E}^{\sigma}_{\text{inc}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{+\ell}C_{\ell
m}^{\sigma}\bm{\Psi}_{\ell m}^{\sigma}({\bf r}).$ (9)

Here $C_{\ell m}^{\sigma}$ denotes the incident coefficients characterizing
the nature of the wave, $k$ is the radiation wavevector, and

$\displaystyle\bm{\Psi}_{\ell m}^{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[{\bm{N}}_{\ell m}+\sigma{\bm{M}}_{\ell
m}\right],$ (10a) $\displaystyle{\bm{M}}_{\ell m}$ $\displaystyle\equiv$
$\displaystyle j_{\ell}(kr)\bm{X}_{\ell m},\quad{\bm{N}}_{\ell
m}\equiv\frac{1}{k}\bm{\nabla}\times{\bm{M}}_{\ell m},$ (10b)
$\displaystyle\bm{X}_{\ell m}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{\ell(\ell+1)}}{\bf{L}}Y_{\ell
m}(\theta,\varphi).$ (10c)

Here $\bm{M}_{\ell m}$ and $\bm{N}_{\ell m}$ are Hansen’s multipoles Jackson
(1999), $j_{\ell}(kr)$ are the spherical Bessel functions, $Y_{\ell
m}(\theta,\varphi)$ are spherical harmonics, $\theta$ and $\varphi$ being the
polar and azimuthal angles, and ${\bf{L}}=\left\\{-{\rm i}{\bf
r}\times\bm{\nabla}\right\\}$ represents the total angular momentum operator.

Let us expand the electric field scattered by a chiral sphere. The Generalized
Lorentz-Mie theory (GLMT) gives the exact solution to Maxwell’s equations for
spherical particles (both chiral and achiral) in a homogeneous medium under
general illumination conditions Gouesbet  _et al._ (2011). Hence, we can
employ the GLMT for the particular case in which an incident field with well-
defined helicity illuminates a chiral sphere. In the same basis of multipoles
of well-defined helicity, the electric field scattered by a chiral sphere
reads

$\displaystyle{\bf E}^{\sigma}_{\rm{sca}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\sigma^{\prime}=\pm 1}{\bf
E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf r}),$ (11) $\displaystyle{\bf
E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{+\ell}D^{\sigma\sigma^{\prime}}_{\ell
m}\bm{\Phi}_{\ell m}^{\sigma^{\prime}}({\bf r}).$ (12)

Here $D^{\sigma\sigma^{\prime}}_{\ell m}=C^{\sigma}_{\ell
m}\left(\tilde{a}_{\ell,\sigma}+\sigma\sigma^{\prime}\tilde{b}_{\ell,\sigma}\right)/{2}$,
$\sigma^{\prime}=\pm 1$, denotes the decomposition of the scattered field into
left - and right-handed CPL waves, and $\bm{\Phi}_{\ell m}^{\sigma}$ is
defined like in Eq. (10a) with the substitution of $h_{\ell}$ in place of
$j_{\ell}$, $j_{\ell}$ and $h_{\ell}$ being the spherical Hankel functions of
first and second kind, respectively Jackson (1999). Moreover,
$\tilde{a}_{\ell,\sigma}=a_{\ell}+i\sigma d_{\ell}$ and
$\tilde{b}_{\ell,\sigma}=b_{\ell}-i\sigma c_{\ell}$ denote the effective Mie
coefficients Ali  _et al._ (2020b), $a_{\ell}$, $b_{\ell}$, $d_{\ell}$, and
$c_{\ell}$ being the Mie coefficients of the scattered field by a chiral
sphere Bohren and Huffman (2008).

Chiral spheres are, by their very nature, reciprocal objects. Hence, chiral
spheres fulfill the Onsager-like relations, namely, $d_{\ell}=-c_{\ell}$
Onsager (1936). Following this fact, we can simplify the previous effective
relations to Ali  _et al._ (2020b)

$\displaystyle\tilde{a}_{\ell,\sigma}=a_{\ell}-i\sigma c_{\ell},$
$\displaystyle\tilde{b}_{\ell,\sigma}=b_{\ell}-i\sigma c_{\ell}.$ (13)

We shall see in the next section that the relationships given by Eq. (13) have
significant consequences concerning MECh spheres.

##  Appendix B Chiral Mie coefficients

In 1974 Craig Bohren analytically solved the constitutive relations of chiral
and reciprocal media for the case of a sphere Bohren (1974). The exact results
that he derived are fully captured by the scattering coefficients $a_{\ell}$,
$b_{\ell}$, and $c_{\ell}$, which according to Ref. Bohren and Huffman (2008)
read:

$\displaystyle a_{\ell}$ $\displaystyle=$
$\displaystyle\Delta^{-1}_{\ell}\left[V_{\ell}(R)A_{\ell}(L)+V_{\ell}(L)A_{\ell}(R)\right],$
(14a) $\displaystyle b_{\ell}$ $\displaystyle=$
$\displaystyle\Delta^{-1}_{\ell}\left[W_{\ell}(L)B_{\ell}(R)+W_{\ell}(R)B_{\ell}(L)\right],$
(14b) $\displaystyle c_{\ell}$ $\displaystyle=$ $\displaystyle
i\Delta^{-1}_{\ell}\left[W_{\ell}(R)A_{\ell}(L)-W_{\ell}(L)A_{\ell}(R)\right],$
(14c)

where $\Delta_{\ell}=W_{\ell}(L)V_{\ell}(R)+V_{\ell}(L)W_{\ell}(R)$, with

$\displaystyle W_{\ell}(J)$ $\displaystyle=$ $\displaystyle
m\psi_{\ell}(m_{J}x)\xi^{{}^{\prime}}_{\ell}(x)-\xi_{\ell}(x)\psi^{{}^{\prime}}_{\ell}(m_{J}x),$
(15a) $\displaystyle V_{\ell}(J)$ $\displaystyle=$
$\displaystyle\psi_{\ell}(m_{J}x)\xi^{{}^{\prime}}_{\ell}(x)-m\xi_{\ell}(x)\psi^{{}^{\prime}}_{\ell}(m_{J}x),$
(15b) $\displaystyle A_{\ell}(J)$ $\displaystyle=$ $\displaystyle
m\psi_{\ell}(m_{J}x)\psi^{{}^{\prime}}_{\ell}(x)-\psi_{\ell}(x)\psi^{{}^{\prime}}_{\ell}(m_{J}x),$
(15c) $\displaystyle B_{\ell}(J)$ $\displaystyle=$
$\displaystyle\psi_{\ell}(m_{J}x)\psi^{{}^{\prime}}_{\ell}(x)-m\psi_{\ell}(x)\psi^{{}^{\prime}}_{\ell}(m_{J}x).$
(15d)

Here $J=L,R$, with $m_{L}=m-\chi$ and $m_{R}=m+\chi$, $m$ being the refractive
index contrast particle/surrounding medium, $\chi$ is a small parameter
modeling the intrinsic chirality of the sphere, $x=ka$ is the size parameter
of the particle, $a$ being its radius. $\psi_{\ell}(z)=zj_{\ell}(z)$ and
$\xi_{\ell}(z)=zh_{\ell}(z)$ are Riccati-Bessel functions.