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# Resonant helicity mixing of electromagnetic waves propagating through matter

Jon Lasa-Alonso  [  Centro de Física de Materiales, Paseo Manuel de Lardizabal
5, 20018 Donostia-San Sebastián, Spain  Donostia International Physics Center,
Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain  Jorge
Olmos-Trigo  Donostia International Physics Center, Paseo Manuel de Lardizabal
4, 20018 Donostia-San Sebastián, Spain  Chiara Devescovi  Donostia
International Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San
Sebastián, Spain  Pilar Hernández  IFIC (CSIC-UVEG), Edificio Institutos
Investigación, Apt. 22085, 46071 Valencia, Spain  Aitzol García-Etxarri
Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018
Donostia-San Sebastián, Spain  IKERBASQUE, Basque Foundation for Science,
María Díaz de Haro 3, 48013 Bilbao, Spain  Gabriel Molina-Terriza  [  Centro
de Física de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-San
Sebastián, Spain  Donostia International Physics Center, Paseo Manuel de
Lardizabal 4, 20018 Donostia-San Sebastián, Spain  IKERBASQUE, Basque
Foundation for Science, María Díaz de Haro 3, 48013 Bilbao, Spain

###### Abstract

Dual scatterers preserve the helicity of an incident field, whereas antidual
scatterers flip it completely. In this setting of linear electromagnetic
scattering theory, we provide a completely general proof on the non-existence
of passive antidual scatterers. However, we show that scatterers fulfilling
the refractive index matching condition flip the helicity of the fields very
efficiently without being in contradiction with the law of energy
conservation. Moreover, we find that this condition is paired with the
impedance matching condition in several contexts of electromagnetism and, in
particular, within Fresnel’s and Mie’s scattering problems. Finally, we show
that index-matched media induce a resonant helicity mixing on the propagating
electromagnetic waves. We reach to this conclusion by identifying that the
refractive index matching condition leads to the phenomenon of avoided level-
crossing. Our contribution not only closes a historical discussion within the
Nanophotonics community, but also opens up new possibilities in the fields of
Metamaterials and Particle Physics.

##  I Introduction

The coupling of two harmonic oscillators is one of the most studied systems in
Physics textbooks with long-stretching implications in many areas of research
Allen and Eberly (1975). Under suitable modifications, coupled harmonic
oscillators allow us to understand such distant phenomena as the strong-
coupling of light matter interactions Novotny (2010), hybridization of
plasmonic and polaritonic systems Tame  _et al._ (2013); Chikkaraddy  _et al._
(2016), atomic dark states Lukin  _et al._ (1999); Vewinger  _et al._ (2003)
or even neutrino oscillations Wolfenstein (1978); Mikheyev and Smirnov (1986,
1989). Here we explore a hitherto hidden form of this kind of coupling which
appears within the polarization of electromagnetic waves propagating through
matter.

A natural way of describing the polarization of electromagnetic fields is to
use helicity as a degree of freedom. The electromagnetic helicity, $\Lambda$,
can be understood as the circular polarization basis in the plane wave
expansion of an electromagnetic field. In an operator formalism, helicity is
defined as the projection of the spin angular momentum, $\mathbf{S}$, onto the
direction of the linear momentum, $\mathbf{P}$, i.e.
$\Lambda=\mathbf{S}\cdot\mathbf{P}/|\mathbf{P}|$, and for monochromatic
electromagnetic fields in homogeneous media it takes the particularly simple
form of $\Lambda=k^{-1}\nabla\times$ Bialynicki-Birula (1994, 1996); Messiah
(1962); Fernández-Corbaton  _et al._ (2013); Bliokh  _et al._ (2019), where
$k$ is the modulus of the wavevector. The electromagnetic helicity is
conserved in the interaction with dual samples, i.e. systems that have the
same response to electric and magnetic fields. Thus, a dual sample, when
illuminated with a beam with well-defined helicity, produces a scattered
electromagnetic field with the same helicity of the incident field (see Fig.
1a). On the other hand, an antidual sample produces a scattered
electromagnetic field only with the opposite helicity (see Fig. 1b). These
concepts have been essential to understand the anomalous scattering of light
by spherical particles Person  _et al._ (2013); Fu  _et al._ (2013) and
nanodisks Staude  _et al._ (2013a). Such scattering phenomena were first
described by Kerker and coworkers in what they are now known as the Kerker
conditions Kerker  _et al._ (1983). It has been shown that the absence of
backscattered field, in the first Kerker condition, is directly related to
dual spherical scatterers. On the other hand, the second Kerker condition,
which leads to the zero forward scattering condition, has been shown to be
related to antidual particles Zambrana-Puyalto  _et al._ (2013a). While the
properties of dual particles are well understood and have been exploited for a
wide variety of applications both experimentally and theoretically Nieto-
Vesperinas  _et al._ (2011a); García-Etxarri and Dionne (2013); Pfeiffer and
Grbic (2013); Liu and Kivshar (2018); Solomon  _et al._ (2019); Lasa-Alonso
_et al._ (2020a); Bliokh  _et al._ (2015); Allen  _et al._ (2003); Lasa-Alonso
_et al._ (2020b), antidual particles remain elusive. Even if similar
scientific efforts have been dedicated, difficulties have arisen when trying
to identify experimentally realizable antidual scatterers Mehta  _et al._
(2006); García-Cámara  _et al._ (2007); Ramachandran and Kumar (2008); García-
Cámara  _et al._ (2008); Alú and Engheta (2010); García-Cámara  _et al._
(2011); Geffrin  _et al._ (2012); Lee  _et al._ (2018); Zambrana-Puyalto  _et
al._ (2013b); Shen  _et al._ (2017); Sanz-Fernández  _et al._ (2021); Ali
(2022).

Figure 1: a) Sketch of a dual sample, i.e. one which preserves the helicity of
an incident field. b) Sketch of an antidual sample, i.e. one which completely
flips the helicity of an incident field.

In this work, we demonstrate in full generality that antidual scatterers which
are not externally pumped with energy are precluded in linear electromagnetic
scattering theory. However, we show that modes of electromagnetic helicity can
be maximally mixed under a resonant condition that respects the law of energy
conservation. Such a resonant condition appears in a symmetric manner with
respect to the duality condition, i.e. just as a dual medium occurs when
$\nabla Z=0$, where $Z$ is the local impedance, the resonant helicity mixing
condition is found for media in which $\nabla n=0$, where $n$ is the local
refractive index. This closes a long-standing controversy about the
implementation of helicity flipping scatterers, while at the same time, we
indicate where the experimental efforts should be directed instead. Finally,
we characterize the resonant helicity mixing condition with the aid of time-
independent perturbation theory and find that it can be related to the
phenomenon of avoided level-crossing. The connection with quantum two-level
systems aligns the resonant helicity mixing effect with other relevant
physical phenomena in Particle Physics.

##  II Antiduality and energy conservation

Let us first show that passive antidual scatterers are not compatible with
energy conservation in linear electromagnetic scattering theory. To that end,
let us consider the most general situation in which an arbitrary incident
field with a definite frequency $\omega$,
$[\mathbf{E}_{i}(\mathbf{r},\omega),\mathbf{H}_{i}(\mathbf{r},\omega)]$,
scatters off a sample which, in turn, emits a field at the same frequency,
$[\mathbf{E}_{s}(\mathbf{r},\omega),\mathbf{H}_{s}(\mathbf{r},\omega)]$. With
such fields one naturally computes the total fields outside the scatterer just
by taking the sum of the incident and the scattered electromagnetic fields,
i.e. $\mathbf{E}_{t}=\mathbf{E}_{i}+\mathbf{E}_{s}$ and
$\mathbf{H}_{t}=\mathbf{H}_{i}+\mathbf{H}_{s}$. With the total fields, the
absorbed, scattered, or extincted powers are computed in terms of the time-
averaged Poynting vector,
$\mathbf{S}_{t}=\frac{1}{2}\text{Re}[\mathbf{E}_{t}\times\mathbf{H}_{t}^{*}]$
Bohren and Huffman (1998); Novotny and Hetch (2006); Jackson (1999). From
Poynting’s theorem, the absorbed power is calculated through the integral of
the flux of the total Poynting vector across a surface $S$ which encloses the
sample, i.e. $W_{a}=-\int\mathbf{S}_{t}\cdot\mathbf{n}~{}dS$, where
$\mathbf{n}$ is a unitary vector perpendicular to the surface. By expanding
$\mathbf{S}_{t}$ in terms of the incident and scattered fields, one obtains
the most general form of the energy conservation law in electromagnetic
scattering theory: $W_{a}=W_{ext}-W_{s}$, where $W_{ext}$ and $W_{s}$ are the
extincted and scattered powers, respectively. The extincted and scattered
powers also depend on the flux of a vector, in a similar fashion to $W_{a}$,
but using
$\mathbf{S}_{ext}=-\frac{1}{2}\text{Re}[\mathbf{E}_{i}\times\mathbf{H}_{s}^{*}+\mathbf{E}_{s}\times\mathbf{H}_{i}^{*}]$
for $W_{ext}$ and
$\mathbf{S}_{s}=\frac{1}{2}\text{Re}[\mathbf{E}_{s}\times\mathbf{H}_{s}^{*}]$
for $W_{s}$.

Now, we will show that $\mathbf{S}_{ext}$ vanishes identically for a generic
antidual sample. Indeed, an antidual scatterer is most generally defined as
one which completely flips the helicity of the incident electromagnetic field.
This implies that given an incident illumination with helicity eigenvalue
$\lambda=\pm 1$, the scattered field emitted by an antidual sample is of
helicity $-\lambda$. Using Maxwell’s equations and the helicity operator, such
a situation can most generally be represented by the following constraints
over the incident and scattered electromagnetic fields:
$\sqrt{\varepsilon_{m}}\mathbf{E}_{i}=i\lambda\sqrt{\mu_{m}}\mathbf{H}_{i}$
and
$\sqrt{\varepsilon_{m}}\mathbf{E}_{s}=-i\lambda\sqrt{\mu_{m}}\mathbf{H}_{s}$,
where $\varepsilon_{m}$ and $\mu_{m}$ are the electric permittivity and
magnetic permeability of the (lossless) medium in which the sample is
embedded. Substituting these relations into the definition of the extinction
component of the Poynting vector, it is obtained that $\mathbf{S}_{ext}=0$ for
any antidual scatterer. This implies that, regardless of the size, form,
material constituents, or, even, the spatial shape of the incident
illumination, an antidual scatterer produces a null extincted power. Thus, the
relation among the absorbed, extincted and scattered powers enforces that, for
an antidual sample, $W_{a}=-W_{s}$. As the scattered power is positive by
definition, one is only left with two possibilities: either the scatterer is
externally pumped with energy ($W_{a}<0$) or both the absorbed and scattered
powers must vanish. As a result, we conclude that passive antidual scatterers,
i.e. those in which there is no external energy pumping, are precluded.

Let us underline the importance of this first contribution by briefly
discussing previous works which are mainly devoted to the study of antidual
dipolar spheres under plane-wave illumination Mehta  _et al._ (2006); García-
Cámara  _et al._ (2007); Ramachandran and Kumar (2008); García-Cámara  _et
al._ (2008); Alú and Engheta (2010); García-Cámara  _et al._ (2011); Geffrin
_et al._ (2012); Lee  _et al._ (2018); Zambrana-Puyalto  _et al._ (2013a, b);
Shen  _et al._ (2017); Sanz-Fernández  _et al._ (2021); Ali (2022). In those
works, it was essentially found that the zero forward scattering condition
Mehta  _et al._ (2006); García-Cámara  _et al._ (2007); Ramachandran and Kumar
(2008); García-Cámara  _et al._ (2008); Alú and Engheta (2010); García-Cámara
_et al._ (2011); Geffrin  _et al._ (2012); Lee  _et al._ (2018); Shen  _et
al._ (2017), which follows from dealing with a cylindrically symmetric
antidual particle Zambrana-Puyalto  _et al._ (2013a, b); Fernández-Corbaton
(2013), cannot be met according to the optical theorem. Indeed, the optical
theorem states that the extincted power is proportional to the scattering
amplitude in the forward direction Alú and Engheta (2010); Bohren and Huffman
(1998); Novotny and Hetch (2006); Jackson (1999) and, thus, it is
straightforward to deduce that $W_{ext}=0$ for this type of scatterers. It was
then concluded that Kerker’s original second condition could not be achieved
for passive dipolar spheres under plane-wave illumination. Now, our
contribution generalizes this key result of linear electromagnetic scattering
theory to an arbitrary antidual scatterer under general illumination
conditions. Please note that in our derivation it is possible that the
scattering amplitude has a non-vanishing value in the forward direction, which
could be the case for non-cylindrical antidual particles. Our findings finally
settle this long debate on the existence of passive antidual scatterers since
forty years ago the second Kerker condition was formalized by Kerker, Wang and
Giles until today.

Even though the construction of antidual scatterers with natural materials is
not possible, we have found that there exists a condition which is not in
contradiction with the energy conservation law and leads to scatterers that
flip the helicity very efficiently.

##  III Helicity flipping passive scatterers

We will be dealing with linear electromagnetic scatterers. Then, as a starting
point, we use the source-less time-independent Maxwell’s equations:

$\displaystyle-i\omega\mathbf{D}(\mathbf{r},\omega)$
$\displaystyle=\nabla\times\mathbf{H}(\mathbf{r},\omega),~{}~{}~{}\nabla\cdot\mathbf{D}(\mathbf{r},\omega)=0,$
$\displaystyle i\omega\mathbf{B}(\mathbf{r},\omega)$
$\displaystyle=\nabla\times\mathbf{E}(\mathbf{r},\omega),~{}~{}~{}\nabla\cdot\mathbf{B}(\mathbf{r},\omega)=0.$
(1)

Let us consider, as a simplification of the problem, that the scattering
process takes place upon an isotropic sample, i.e.
$\mathbf{D}(\mathbf{r},\omega)=\varepsilon(\mathbf{r})\mathbf{E}(\mathbf{r},\omega)$
and
$\mathbf{B}(\mathbf{r},\omega)=\mu(\mathbf{r})\mathbf{H}(\mathbf{r},\omega)$,
where $\varepsilon(\mathbf{r})$ and $\mu(\mathbf{r})$ are the local electric
permittivity and magnetic permeability, respectively. There is another
formulation of Maxwell’s equations, based on the the Riemann-Silberstein (RS)
vector Weber (1901); Silberstein (1907); Bialynicki-Birula and Bialynicka-
Birula (2013). Such a vector is defined as a linear combination of the
electric and magnetic fields, i.e.
$\mathbf{F}_{\lambda}(\mathbf{r},\omega)=2^{-1/2}[\sqrt{\varepsilon(\mathbf{r})}\mathbf{E}(\mathbf{r},\omega)+i\lambda\sqrt{\mu(\mathbf{r})}\mathbf{H}(\mathbf{r},\omega)]$,
with $\lambda=\pm 1$. This formulation is better for our purposes because it
explicitly captures the polarization degree of freedom of electromagnetic
waves. Actually, following the previous notation, the label $\lambda$ denotes
the eigenvalue of the helicity operator $\Lambda$. When studying the
properties of electromagnetic waves, it can be shown that the RS vector
represents each of the two polarization degrees of freedom or helicities. This
can be checked by employing the usual representation of the helicity operator,
$\Lambda\rightarrow k^{-1}\nabla\times$, and expressing the time-independent
Maxwell’s equations in a homogeneous medium as a function the RS vector:
$k^{-1}\nabla\times\mathbf{F}_{\lambda}=\lambda\mathbf{F}_{\lambda}$.
Additionally, as we will see, this formulation of Maxwell’s equations can be
cast in terms of a hamiltonian notation, which will prove to be very useful in
the next Section.

Let us rewrite Maxwell’s equations given by (1) in terms of the RS vector for
inhomogeneous media Bialynicki-Birula (1994, 1996):

$\displaystyle\omega\mathbf{F}_{+}=\frac{1}{\sqrt{n}}\nabla\times\left(\frac{\mathbf{F}_{+}}{\sqrt{n}}\right)+\frac{1}{n}\nabla\ln\sqrt{Z}\times\mathbf{F}_{-}$
(2)
$\displaystyle\omega\mathbf{F}_{-}=-\frac{1}{\sqrt{n}}\nabla\times\left(\frac{\mathbf{F}_{-}}{\sqrt{n}}\right)-\frac{1}{n}\nabla\ln\sqrt{Z}\times\mathbf{F}_{+}$
(3)
$\displaystyle\nabla\cdot\mathbf{F}_{+}=-\nabla\ln\sqrt{n}\cdot\mathbf{F_{+}}+\nabla\ln\sqrt{Z}\cdot\mathbf{F}_{-}$
(4)
$\displaystyle\nabla\cdot\mathbf{F}_{-}=-\nabla\ln\sqrt{n}\cdot\mathbf{F}_{-}+\nabla\ln\sqrt{Z}\cdot\mathbf{F}_{+}$
(5)

where, for convenience, we have omitted the $(\mathbf{r},\omega)$ dependencies
and we have also defined the local impedance,
$Z(\mathbf{r})=\sqrt{\mu(\mathbf{r})/\varepsilon(\mathbf{r})}$, and refractive
index, $n(\mathbf{r})=\sqrt{\varepsilon(\mathbf{r})\mu(\mathbf{r})}$. Please
note that Eqs. (2)-(5) represent the time-independent Maxwell’s equations in
an isotropic and inhomogeneous medium exactly as those given by (1). In this
form, Faraday’s and Ampère’s laws given by (2) and (3) can be written as an
eigenvalue problem:

$\displaystyle H\begin{pmatrix}\mathbf{F}_{+}\\\
\mathbf{F}_{-}\end{pmatrix}=\omega\begin{pmatrix}\mathbf{F}_{+}\\\
\mathbf{F}_{-}\end{pmatrix},~{}~{}\text{with}~{}~{}$ $\displaystyle
H=\begin{pmatrix}\frac{1}{\sqrt{n}}\nabla\times\left(\frac{\cdot}{\sqrt{n}}\right)&\frac{1}{n}\nabla\ln\sqrt{Z}\times\\\
-\frac{1}{n}\nabla\ln\sqrt{Z}\times&-\frac{1}{\sqrt{n}}\nabla\times\left(\frac{\cdot}{\sqrt{n}}\right)\end{pmatrix}.$
(6)

This particular way of expressing Faraday’s and Ampère’s laws has lead some
authors to denote the operator $H$ as the ”photon hamiltonian”, in analogy
with the time-independent Schrödinger’s equation Bialynicki-Birula (1994,
1996); Barnett (2014); Bialynicki-Birula and Bialynicka-Birula (2017);
Kiessling and Tahvildar-Zadeh (2018). In this line, note that the frequency
(or energy) of the system is completely determined by (6). However, for our
immediate purpose, (6) indicates that the role of a generic inhomogeneous
medium is to interconnect the different helicity components of the
electromagnetic field through the spatial derivatives of $Z$ and $n$.

The study of the impedance matching condition, i.e. media in which $\nabla
Z=0$, has been particularly fruitful for the communities of linear scattering
theory and metamaterials Kerker  _et al._ (1983); Person  _et al._ (2013);
Solomon  _et al._ (2019); Babicheva and Evlyukhin (2017); Pfeiffer and Grbic
(2013); Staude  _et al._ (2013b); Pfeiffer  _et al._ (2014); Decker  _et al._
(2015). Under this condition the duality symmetry is restored in macroscopic
Maxwell’s equations, leading to the conservation of helicity Fernández-
Corbaton  _et al._ (2013). Indeed, in terms of Faraday-Ampère’s laws given by
(6), it can be checked that the time-evolution of the helicity components are
decoupled under this condition. Such a matching condition was first identified
by Giles and Wild in the expressions of the Fresnel coefficients for a plane
wave impinging on a plane surface Giles and Wild (1982). In particular, they
found that when imposing the impedance matching condition, the reflection,
$r$, and transmission, $t$, coefficients are the same for both s and p
polarizations, i.e. $r_{\text{s}}=r_{\text{p}}$ and
$t_{\text{s}}=t_{\text{p}}$, independently of the incidence angle. Following
this pioneering work, Kerker, Wang and Giles one year later reported the
effects of the impedance matching condition within Mie theory, i.e. in the
problem of an electromagnetic wave scattering off a sphere Kerker  _et al._
(1983). Their findings confirmed that such a condition was very particular as
it leads to specific analytical relations between the Mie coefficients, i.e.
the amplitudes of the scattered electric/magnetic multipolar modes
($a_{\ell}$/$b_{\ell}$) and also the amplitudes of the internal
electric/magnetic multipolar modes ($d_{\ell}$/$c_{\ell}$). In particular, it
was found that the amplitude of the electric and magnetic multipolar modes,
for both internal and scattered fields, is the same under this particular
condition: $a_{\ell}=b_{\ell}\text{ and }d_{\ell}=c_{\ell},~{}\forall~{}\ell$.
Moreover, they realized that the analytical relation between the scattering
coefficients implied an absence of backscattered fields. Since then, the
impedance matching condition has most commonly been denoted as the first
Kerker condition in the literature and adopts the analytical form
$\varepsilon=\mu$ for a scatterer embedded in vacuum.

| Fresnel | Mie
---|---|---
$\nabla Z=0$ | $r_{\text{s}}/r_{\text{p}}=t_{\text{s}}/t_{\text{p}}=1$ | $a_{\ell}/b_{\ell}=d_{\ell}/c_{\ell}=1$
$\nabla n=0$ | $~{}r_{\text{s}}/r_{\text{p}}=-t_{\text{s}}/t_{\text{p}}=-1~{}$ | $~{}a_{\ell}/b_{\ell}=-d_{\ell}/c_{\ell}\sim-1~{}$
Table 1: Fresnel and Mie coefficients under the impedance and refractive index
matching conditions. $r_{\text{s}/\text{p}}$ and $t_{\text{s}/\text{p}}$ are
Fresnel reflection and transmission coefficients for s and p polarized waves.
$a_{\ell}$ and $b_{\ell}$ are the scattering Mie coefficients, whereas
$d_{\ell}$ and $c_{\ell}$ are the internal Mie coefficients. Note that the
last approximate equality only holds in the limit $s\rightarrow 1$ and cannot
be exactly fulfilled due to energy conservation.

While the role of the impedance has been analyzed in a wide variety of
problems, the refractive index has been quite generally overlooked. Taking
into account the time-evolution of the fields described by (6), the index
matching condition, i.e. media which fulfill $\nabla n=0$, would imply having
an environment in which the helicity components are mixed. At first sight,
however, one is not able to tell the difference with a common dielectric
medium ($\mu=1$ and, thus, $Z=1/n$) in which the helicity components are also
coupled. The index matching condition was first reported by Giles and Wild in
the expressions of the Fresnel coefficients Giles and Wild (1982). Under this
condition they found that the reflection coefficients are changed by a
relative sign regardless, again, of the incidence angle, i.e.
$r_{\text{s}}=-r_{\text{p}}$ and $t_{\text{s}}=t_{\text{p}}$. Interestingly,
this implies that, when scattering off an index-matched plane surface, the
reflected light always has the opposite helicity Lakhtakia (1990). An attempt
to study this condition within Mie theory was also made in Kerker’s seminal
paper but the authors were not so successful as they were with the analysis of
the impedance matching effect. In fact, they exclusively focused on the study
of the index matching condition for small spherical particles, overlooking the
implications that this condition generally had on the analytical expressions
of the Mie coefficients (see Section 4 in Ref. Kerker  _et al._ (1983)). In
what follows, we show that the refractive index matching condition does lead
to spherical scatterers which very efficiently flip the helicity of
electromagnetic waves, while still respecting the energy conservation law.

Figure 2: Helicity expectation value, $\langle\Lambda\rangle$, for a passive
sphere of size parameter $X=3$ illuminated by a circularly polarized plane
wave. $\langle\Lambda\rangle$ is given as a function of the impedance
contrast, $s=Z_{1}/Z_{2}$, and refractive index contrast, $m=n_{1}/n_{2}$. In
particular, $\langle\Lambda\rangle=1$ implies a dual scatterer and
$\langle\Lambda\rangle=-1$ an antidual one. The impedance matching condition
is fulfilled through the line $s=1$ and refractive index matching condition
through the line $m=1$.

Whenever the refractive indices of the sphere ($n_{1}$) and the surrounding
medium ($n_{2}$) are equal, we have found that the following analytical
relation is fulfilled within the Mie coefficients:
$a_{\ell}c_{\ell}=-b_{\ell}d_{\ell},~{}\forall~{}\ell$. Furthermore, and most
importantly, it can also be checked that the limit $\lim_{s\rightarrow
1}a_{\ell}=-b_{\ell},~{}\forall~{}\ell$ holds, where $s$ is the impedance
contrast (see Supporting Information). This implies that, under the index
matching condition ($n_{1}=n_{2}$), if one takes the impedances of the sphere
($Z_{1}$) and the surrounding medium ($Z_{2}$) to be increasingly similar
($s=Z_{1}/Z_{2}\rightarrow 1$), then, the helicity of the scattered field is
the opposite of the incident field. In other words, an antidual scatterer
emerges Zambrana-Puyalto  _et al._ (2013a). However, note that such a limit
can only be reached if there is no scatterer, as expected from our previous
result on antiduality and energy conservation (see Section II). To shed light
on this, we have computed the helicity expectation value,
$\langle\Lambda\rangle$, for the fields scattered by a sphere of radius $a$
when illuminated by a circularly polarized plane wave. In Fig. 2 we show the
result in terms of the size parameter $X=k_{0}a$ of the sphere, where $k_{0}$
is the wavevector modulus of the incident wave in vacuum. The helicity
expectation value is a bounded observable,
$\langle\Lambda\rangle\in\left[-1,1\right]$, and its extreme values imply a
dual, $\langle\Lambda\rangle=1$, or an antidual, $\langle\Lambda\rangle=-1$,
scatterer Lasa-Alonso  _et al._ (2022); Olmos-Trigo  _et al._ (2020a). As it
can be observed from Fig. 2, an outstanding behaviour is found when the index
matching condition is fulfilled, through the line $m=1$. Values of
$\langle\Lambda\rangle\sim-1$ are obtained in the vicinities of such a line
but never reaching the exact antiduality condition. Similar conclusions can be
drawn from helicity maps for the large and the small sphere regime (see
Supporting Information).

So, we have just shown that, even if passive antidual scatterers are precluded
by the energy conservation law, one can obtain a pretty similar behaviour for
index-matched spheres (see Fig. 2). This is based on the analytical
expressions of the Mie coefficients under the refractive index matching
condition, which had previously been overlooked in the literature. In
addition, our analysis indicates that the impedance and refractive index
matching conditions are paired in several contexts of electromagnetism.
Indeed, we have explicitly shown that these two matching conditions have
parallel implications over Fresnel and Mie coefficients (see Table 1). The
pairing of the impedance and index matching conditions may be particularly
useful in the field of Metamaterials, where one could in principle control the
effective impedance and refractive index by properly engineering the
inclusions and meta atoms. As it is known, helicity preserving and flipping
surfaces emit, respectively, in the forward and backward directions provided
that they have certain cylindrical symmetries Fernández-Corbaton (2013). On
the other hand, note that Fresnel coefficients do not depend on the angle of
incidence for index-matched surfaces. Furthermore, reflected and transmitted
powers are also independent of the helicity of the incident plane wave
whenever $\nabla n=0$. These properties could serve to construct angle- and
polarization-independent ultrathin beam-splitters.

##  IV Refractive index matching and the avoided level-crossing phenomenon

The previous analysis points in the direction that the $\nabla n=0$ condition
allows for helicity flipping media as close as energy conservation allows.
However, the exact mechanism for this helicity transfer still remains hidden.
In this last part, we characterize the phenomenon that leads to the
construction of efficient helicity flipping scatterers. To that aim, we
evaluate the eigenfrequencies appearing in (6) for one-dimensional
inhomogeneous media, i.e. systems in which $Z=Z(x)$ and $n=n(x)$. Due to the
complexity of such a general problem, we have constrained to slowly varying
environments for which time-independent perturbation theory may be applied. In
other words, we analytically derive the form of Faraday’s and Ampère’s laws
for smooth one-dimensional inhomogeneous media and compute the energies of the
propagating electromagnetic waves in terms of $Z$ and $n$.

Standard time-independent perturbation theory may only be employed if the
operator given by (6), can be expressed as a sum of two terms, i.e.
$H=H_{0}+V$, where $H_{0}$ is the ”hamiltonian” of a homogeneous medium and
$V$ is the perturbation ”potential” due to small inhomogeneities. Moreover,
depending on whether we consider the perturbation in the direction of constant
impedance or constant refractive index, $V$ has a different analytical form.
Most generally, the perturbation potential in an arbitrary direction of the
$(Z,n)$ parameter space can be constructed as a linear combination of the
perturbation potentials in each direction. It can be shown that in the
perturbative regime, the operator given by (6) breaks up into two terms:
$H_{0}=n^{-1}_{0}\bigl{(}\begin{smallmatrix}\nabla\times&0\\\
0&-\nabla\times\end{smallmatrix}\bigr{)}$ and
$V=n^{-2}_{0}\bigl{(}\begin{smallmatrix}D&A\\\ -A&-D\end{smallmatrix}\bigr{)}$
where $n_{0}$ is a real constant representing the refractive index of the
unperturbed system. The operators which define the components of the $V$
operator are $D=-[ik_{t}e(x)\hat{u}_{y}+\delta_{x}\hat{u}_{x}]\times$ and
$A=(2Z_{0})^{-1}n_{0}\partial_{x}{d}(x)\hat{u}_{x}\times$. We have defined the
functions $d(x)=Z(x)-Z_{0}$ with $d(x)/Z_{0}<<1$, where $Z_{0}$ is the
impedance of the unperturbed system. Moreover, we have set $e(x)=n(x)-n_{0}$
with $e(x)/n_{0}<<1$ and we have defined the operator
$\delta_{x}=\sqrt{e(x)}\partial_{x}[\sqrt{e(x)}~{}\cdot~{}]$. Finally, without
loss of generality, we consider that the solutions can be written as
$\mathbf{\Psi}(\mathbf{r})=\mathbf{\Psi}(x)e^{ik_{t}y}$ (see Supporting
Information). Notice also that $V$ depends on the derivatives of $e(x)$ and
$d(x)$. Thus, for the perturbation theory to be valid, both
$\partial_{x}e(x)/e(x)$ and $\partial_{x}d(x)/d(x)$ must be small compared to
$k_{t}$. This precludes the use of sharp and step-like functions.

Once we have obtained the complete perturbation potential, we need a set of
exact solutions of the unperturbed system. Such a set of solutions is obtained
by solving the system of coupled differential equations
$H_{0}\mathbf{\Psi}(x)=\omega\mathbf{\Psi}(x)$. As expected, the solutions are
circularly polarized plane waves propagating in the $XY$ plane. We find four
different types of normalized eigenstates:
$\mathbf{\Psi}_{\lambda\pm}(x)=\sqrt{\omega}\hat{u}_{\lambda}\exp(\pm
ik_{x}x)$, which are classified by their helicity, $\lambda$, and direction of
propagation, $\pm k_{x}$, with $k_{x}^{2}=\omega^{2}n_{0}^{2}-k_{t}^{2}$; and
$\hat{u}_{\lambda}$ is a normalized polarization vector. All of these
eigenvectors fulfill
$H_{0}\mathbf{\Psi}_{\lambda\pm}(x)=\omega\mathbf{\Psi}_{\lambda\pm}(x)$,
implying that they are degenerate (see Supporting Information). This means
that the exact solutions of Maxwell’s equations in a homogeneous medium share
the same energy and, thus, degenerate time-independent perturbation theory
must be employed. In our case, the approximate solutions to the perturbed
system, $\mathbf{\Psi}^{(1)}$, can be found as linear superpositions of the
unperturbed system eigenstates
($\mathbf{\Psi}^{(1)}=\sum_{j}K_{j}\mathbf{\Psi}_{j}$) by diagonalizing the
$4\times 4$ matrix constructed from the matrix elements
$\braket{\mathbf{\Psi}_{j}}{V\mathbf{\Psi}_{i}}$ where in $\mathbf{\Psi}_{k}$,
$k\in\\{++,+-,-+,--\\}$. Finally, the eigenvalues of the matrix correspond to
the first order energy corrections.

As the problem we are interested in is one dimensional, the matrix elements
are computed as
$\braket{\mathbf{\Psi}_{j}}{V\mathbf{\Psi}_{i}}=\int_{-\infty}^{\infty}dx~{}[\mathbf{\Psi}_{j}(x)]^{*}\cdot[V\mathbf{\Psi}_{i}(x)]$.
The computation of such integrals results in the matrix equation
M$\mathbf{K}=\frac{\Delta\omega}{\omega}\mathbf{K}$, where
$\mathbf{K}=(K_{1},K_{2},K_{3},K_{4})^{T}$. In addition, $\Delta\omega$
represents the first order correction to the energy such that the perturbed
energies are obtained as $\omega^{(1)}=\omega+\Delta\omega$. After a
similarity transformation, the matrix M can be written in a form M′ that
better allows us to identify the physical properties of the problem. Thus, we
are left with the following eigensystem:
M${}^{\prime}\mathbf{K}^{\prime}=\frac{\Delta\omega}{\omega}\mathbf{K}^{\prime}$,
where

$\text{M}^{\prime}=\begin{pmatrix}\mathbb{D}+|\mathbb{F}|&\mathbb{C}^{*}&0&0\\\
\mathbb{C}&-(\mathbb{D}-|\mathbb{F}|)&0&0\\\
0&0&\mathbb{D}-|\mathbb{F}|&\mathbb{C}^{*}\\\
0&0&\mathbb{C}&-(\mathbb{D}+|\mathbb{F}|)\end{pmatrix}$ (7)

and
$\mathbf{K}^{\prime}=(K^{\prime}_{1},K^{\prime}_{4},K^{\prime}_{2},K^{\prime}_{3})^{T}$.
Please note that the helicity components are reordered within the vector
$\mathbf{K}^{\prime}$. In the new basis, $K^{\prime}_{1}$ and $K^{\prime}_{2}$
still represent positive helicity components, and also, the negative helicity
components are still associated with $K^{\prime}_{3}$ and $K^{\prime}_{4}$
(see Supporting Information). Parameter $\mathbb{C}$ in (7) captures the
effect of the inhomogeneities in the impedance, i.e. $\mathbb{C}$ vanishes
whenever $\partial_{x}Z(x)=0$. On the other hand, $\mathbb{D}$ and
$\mathbb{F}$ capture the effect of the inhomogeneities in the refractive
index, i.e. $\mathbb{D}$ and $\mathbb{F}$ are zero whenever
$\partial_{x}n(x)=0$.

Bearing this in mind, we can straightforwardly analyze the effect of
inhomogeneities from the elements of the matrix in (7). First of all, it is
clear that for an arbitrary perturbation in the $(Z,n)$ parameter space, the
whole system can be split into two independent subsystems. Moreover, due to
their dimensionality, each of the subsystems can be analyzed in close analogy
to a quantum mechanical two-level system. In particular, the case of impedance
matching ($\mathbb{C}=0$) implies, in this formalism, that the levels are
decoupled and, thus, there is no mixing between the helicities, whereas the
refractive index matching condition ($\mathbb{F}=\mathbb{D}=0$) makes the two-
level systems be simultaneously degenerate and coupled. On the other hand, the
expression of the energy corrections is
$\Delta\omega=\pm\omega(\mathbb{S}\pm|\mathbb{F}|)$, where
$\mathbb{S}=\sqrt{\mathbb{D}^{2}+|\mathbb{C}|^{2}}$, and for index-matched
inhomogeneous media the perturbed energies are computed as
$\omega^{(1)}=\omega\pm\omega|\mathbb{C}|$. This is the exact mathematical
form that energies take in a resonant two-level system. In addition, whenever
$n(x)$ is a constant function, perturbed states can be written as linear
superpositions of the unperturbed states with amplitudes of the same modulus,
which is also a key characteristic of the systems undergoing quantum resonance
Cohen-Tannoudji  _et al._ (1977).

Figure 3: a) Sketch of an inhomogeneous medium with $Z(x)=Z_{0}+d(x)$ and
$n(x)=n_{0}+e(x)$, with the functions $d(x)=\alpha/[1+\exp{(x/\ell)}]$ and
$e(x)=-\beta\exp{(-x^{2}/\ell^{2})}$. b) First order energy corrections:
$Z_{0}=n_{0}=1$, $\omega=1$, $k_{t}=0.4$, $\ell=1$ and $\alpha=0.01$ (in
colors). All units are normalized to an arbitrary frequency $\omega_{0}$ and
$c=1$. The grey dashed lines are obtained with the same parameters but fixing
$\alpha=0$, which implies that $\mathbb{C}=0$ in this case. c) Mixing
parameter of the two-level systems Cohen-Tannoudji  _et al._ (1977). Whenever
$\beta=0$, the refractive index matching condition is fulfilled.

In Fig. 3 we have numerically computed a particular case of a one-dimensional
inhomogeneous system. We have considered that the perturbation is given by the
functions $d(x)=\alpha/[1+\exp{(x/\ell)}]$ and
$e(x)=-\beta\exp{(-x^{2}/\ell^{2})}$ (see Fig. 3a). The amplitude of the
perturbation in the impedance is determined by the parameter $\alpha$ and in
the refractive index by $\beta$. Note that whenever $\alpha=0$ helicity is
preserved and, on the other hand, whenever $\beta=0$ the refractive index
matching condition is fulfilled. To meet the requirements imposed by
perturbation theory we have chosen $\alpha=0.01\ll Z_{0}$ and $|\beta|\leq
0.01\ll n_{0}$. In Fig. 3b we have computed the corrections to the energy as a
function of $\beta$ (in colors). For comparison, in the same figure, we also
display the energy corrections for the case in which $\alpha=0$ (grey dashed
lines). Whenever $\alpha=0$ we have that $\mathbb{C}=0$ and, thus, the
helicity components are decoupled in this case. From the form of the matrix in
(7) it can be checked that the ascending (descending) diagonal grey lines
correspond to solutions with well-defined positive (negative) helicity.
Finally, in Fig. 3c, we have computed the mixing parameters of the two-level
systems identified in (7), which happen to be the same. Among others, this
parameter has been employed to study the resonant mixing of neutrino flavours
in matter and it is a signature of quantum resonance when it approaches unity
Mikheyev and Smirnov (1986, 1989). The reported results are completely
compatible with the phenomenon of the avoided level-crossing.

To sum up, our results indicate that index-matched media induce a resonant
helicity mixing on the electromagnetic waves that propagate through them. We
have reached to this conclusion by applying perturbation theory to Maxwell’s
equations for smooth varying environments and computing the energies of the
propagating waves in terms of the medium parameters. Strikingly and completely
unexpectedly, we have identified that the refractive index matching condition
($\nabla n=0$) leads to the phenomenon of avoided level-crossing (see Fig. 3).
A great deal of phenomena in Physics are related to the avoided crossing of
energy levels, but this is the first time that such a connection is made for
the helicity components of electromagnetic fields. In our view, there is an
effect which closely resembles the resonant helicity mixing condition: the
Mikheyev–Smirnov–Wolfenstein (MSW) effect in neutrino physics Wolfenstein
(1978); Mikheyev and Smirnov (1986). This effect, which has been regarded as
the solution to the solar neutrino problem Bahcall  _et al._ (2018), takes
place when neutrinos propagate through a medium with varying matter density,
$\rho(x)$. Similarly, there also exists a condition over the medium for which
neutrino flavours are resonantly mixed, i.e. whenever $\rho(x)=\rho_{R}$.
However, the resonant density, $\rho_{R}$, is a function of the initial
neutrino energy and, thus, only neutrinos that are released with enough energy
hit the resonance in their way out of the sun. This turns out on an almost
complete conversion of the high energy electron neutrinos generated by the sun
into other neutrino flavours. One can observe the mathematical concordance of
the analogy between neutrino flavours and electromagnetic helicity components,
and the density matching and refractive index matching conditions.

##  V Conclusions

In conclusion, we have shown that passive antidual scatterers are precluded by
the energy conservation law in linear electromagnetic scattering theory. The
result is completely general and it is independent of the size, shape,
material constituent or, even, the spatial form of the illuminating field.
Then, we have presented the condition of refractive index matching and showed
that it appears paired with the impedance matching condition in several
contexts of electromagnetism including Fresnel and Mie coefficients.
Evaluating the analytical form of Mie coefficients under the index matching
condition, something which had been previously overlooked, we have found that
this condition leads to spherical scatterers which flip the helicity of the
incoming light very efficiently. Importantly, these scatterers do not
contradict the energy conservation law and, thus, could in principle be
constructed from natural materials or metamaterials. Finally, we have
concluded that, index-matched media induce a resonant helicity mixing on the
electromagnetic waves that propagate through them. We have reached to this
conclusion by evaluating the energies of propagating waves in terms of the
medium parameters and identifying that the index matching condition leads to
the phenomenon of avoided level-crossing.

We believe that our findings will have an impact on, at least, three different
communities. Our first contribution generalizes a key result of linear
electromagnetic scattering theory to generic passive scatterers under general
illumination conditions. This closes a historical discussion on the zero
forward scattering problem Mehta  _et al._ (2006); García-Cámara  _et al._
(2007); Ramachandran and Kumar (2008); García-Cámara  _et al._ (2011); Alú and
Engheta (2010); Geffrin  _et al._ (2012); García-Cámara  _et al._ (2008); Lee
_et al._ (2018); Shen  _et al._ (2017), the second Kerker condition Alú and
Engheta (2010); Nieto-Vesperinas  _et al._ (2011b); Lee  _et al._ (2017,
2018); Olmos-Trigo  _et al._ (2020b, a); García-Cámara  _et al._ (2011); Shen
_et al._ (2017) and the possible existence of passive antidual scatterers
Zambrana-Puyalto  _et al._ (2013a, b); Fernández-Corbaton (2013); Olmos-Trigo
_et al._ (2020b, a); Ali (2022). Moreover, identifying the refractive index
matching condition as being paired to the impedance matching condition in
several contexts of electromagnetism opens up a great deal of new
possibilities in the field of Metamaterials. Exactly as impedance-matched meta
atoms are commonly employed to maximize transmissivity Pfeiffer and Grbic
(2013); Staude  _et al._ (2013b); Pfeiffer  _et al._ (2014); Decker  _et al._
(2015), index-matched particles are good candidates to build reflective
photonic devices. Also, index-matched metasurfaces could serve to construct
angle- and polarization-independent ultrathin beam-splitters. Finally, the
connection with quantum two level-systems aligns the resonant helicity mixing
condition with other phenomena such as the MSW effect. This identification
opens up a window of possible cross-pollination of ideas with Particle
Physics.

###### Acknowledgements.

J.L.A. acknowledges the warm hospitality that Iwo Bialynicki-Birula and Zofia
Bialynicka-Birula have extended to him in Warsaw where part of this work has
been developed. J.L.A., J.O.T., C. D. and A.G.E acknowledge support from
Project No. PID2019-109905GA-C22 of the Spanish Ministerio de Ciencia,
Innovación y Universidades (MICIU), IKUR Strategy under the collaboration
agreement between Ikerbasque Foundation and DIPC on behalf of the Department
of Education of the Basque Government, the Basque Government Elkartek program
(KK-2021/00082), Programa de ayudas de apoyo a los agentes de la Red Vasca de
Ciencia, Tecnología e Innovación acreditados en la categoría de Centros de
Investigación Básica y de Excelencia (Programa BERC) from the Departamento de
Universidades e Investigación del Gobierno Vasco and Centros Severo Ochoa
AEI/CEX2018-000867-S from the Spanish Ministerio de Ciencia e Innovación.
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.
G.M.T received funding from the IKUR Strategy under the collaboration
agreement between Ikerbasque Foundation and DIPC/MPC on behalf of the
Department of Education of the Basque Government. P. H. acknowledges support
from the DIPC visitor program.

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