# Optical mirages from spinless beams

Jorge Olmos-Trigo<EMAIL_ADDRESS>Donostia International Physics
Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain.
Centro de Física de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-
San Sebastián, Spain.  Diego R. Abujetas  Physics Department, Fribourg
University, Chemin de Musée 3, 1700 Fribourg Switzerland  Cristina Sanz-
Fernández  Multiverse Computing, Paseo de Miramón 170, E-20014 San Sebastián,
Spain  Aitzol García-Etxarri  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.
Antonio García-Martín  Instituto de Micro y Nanotecnología IMN-CNM, CSIC, CEI
UAM+CSIC, Isaac Newton 8, E-28760 Tres Cantos, Madrid, Spain.

###### Abstract

Spin-orbit interactions of light are ubiquitous in multiple branches of
nanophotonics, including optical wave localization. In that framework, it is
widely accepted that circularly polarized beams lead to spin-dependent
apparent shifts of dipolar targets, commonly referred to as optical mirages.
In contrast, these optical mirages vanish when the illumination comes from a
spinless beam such as a linearly polarized wave. Here we show that optical
localization errors emerge for particles sustaining electric and magnetic
dipolar response under the illumination of spinless beams. As an example, we
calculate the optical mirage for the scattering by a high refractive index
nanosphere under the illumination of a linearly polarized plane wave carrying
null spin, orbital, and total angular momentum. Our results point to an
overlooked interference between the electric and magnetic dipoles rather than
the spin-orbit interactions of light as the origin for the tilted position of
the nanosphere.

Light carries, together with energy and linear momentum, angular momentum
Allen  _et al._ (1992); He  _et al._ (1995). Traditionally, this last
dynamical property of light has been separated into spin and orbital
contributions Moe and Happer (1977), whose exchange after scattering gives
rise to the so-called spin-orbit interactions (SOI) of light Bliokh  _et al._
(2015).

Among all the striking effects of SOI Onoda  _et al._ (2004); Hosten and Kwiat
(2008); Yin  _et al._ (2013), perhaps the most interesting is the appearance
of spin-dependent apparent displacements of the target under the illumination
of beams with angular momentum different from zero. The first fundamental
explanation of these spin-dependent apparent shifts dates back to 1932, when
Charles G. Darwin made a beautiful analogy between electrons and photons by
pointing to the density current and Poynting vector, respectively, as the
responsible magnitudes of the tilted position of the particle Darwin (1932).
Some decades later, the very same idea was recovered by considering an
electric dipole illuminated by a circularly polarized plane wave (Arnoldus
_et al._ , 2008). In that work, it was demonstrated that the spin-dependent
apparent shift just depended on the scattering angle being its maximum
absolute value always subwavelength. That apparent shift was generally
predicted, beyond the dipolar regime, by purely using a geometrical argument,

$\bm{\Delta}=\lim_{kr\rightarrow\infty}r\frac{{\bf{S}}_{\perp}}{|{\bf{S}_{\rm{r}}}|},$
(1)

where $\bf{S}_{\perp}$ denotes the Poynting vector components
$\\{{\bf{S}}_{\varphi},{\bf{S}}_{\theta}\\}$ perpendicular to the scattered
contribution ${\bf{S}_{\rm{r}}}$. From now on, the apparent shift, described
by Eq. (1), will be referred to as an _optical mirage_. For microspheres under
the illumination of a circularly-polarized plane wave, spin-dependent optical
mirages reaching tens of wavelengths in magnitude were found Haefner  _et al._
(2009). Later on, several groups demonstrated that for high refractive index
(HRI) nanospheres under circularly-polarized plane wave illumination, this
spin-dependent optical mirage is considerably enhanced Gao  _et al._ (2018);
Olmos-Trigo  _et al._ (2019a, b); Shi  _et al._ (2019); Sun  _et al._ (2022).
Beyond the picture of circularly polarized plane waves, optical mirages
comparable to the incident wavelength were also found under the illumination
of elliptically polarized beams Araneda  _et al._ (2019).

All previous results rely on the existence of spin angular momentum in the
incident radiation for the optical mirage to be observed Haefner  _et al._
(2009); Gao  _et al._ (2018); Olmos-Trigo  _et al._ (2019a, b); Shi  _et al._
(2019); Sun  _et al._ (2022); Araneda  _et al._ (2019). Moreover, the SOI of
light after scattering has been identified as the underlying effect for
understanding the physics of optical mirages. Here, we demonstrate that
linearly polarized waves (spinless) beams can also give rise to significant
optical mirages with magnitudes well beyond the incident wavelength. In
particular, we show that optical mirages arise naturally for HRI nanoparticles
sustaining electric and magnetic dipolar responses simultaneously. We herewith
evidence that a non-radial structure of the scattered Poynting vector,
proportional to an interference of the radiation arising from the electric and
magnetic dipoles, lies in the origin of optical mirages. Moreover, we show
that we can emulate the role of the spin angular momentum for circularly
polarized beams by adequately tuning the optical response under the
illumination of spinless beams.

Next, we lay out the theoretical framework that we will use to show the
emergence of enhanced optical mirages under spinless beams. To that end, let
us consider a HRI nanoparticle of permittivity $\epsilon_{{p}}$ embedded in an
otherwise homogeneous medium with constant and real permittivity
$\epsilon_{{h}}$. The induced electric and magnetic dipoles are given by
Jackson (1999)

$\displaystyle{\bf
p}=\epsilon_{0}\epsilon_{h}\overleftrightarrow{{\alpha}}_{\rm{E}}{\bf
E}_{\rm{inc}}({\bf r}_{0}),$ $\displaystyle{\bf
m}=\overleftrightarrow{{\alpha}}_{\rm{M}}{\bf H}_{\rm{inc}}({\bf r}_{0}),$ (2)

where ${\bf E}_{\rm{inc}}({\bf r})$ and ${\bf H}_{\rm{inc}}({\bf r})$ are the
incoming electric and magnetic fields, respectively, $\epsilon_{0}$ is the
permittivity of vacuum, and $\overleftrightarrow{{\alpha}}_{\rm{E}}$ and
$\overleftrightarrow{{\alpha}}_{\rm{M}}$ denote the dyadic electric and
magnetic polarizabilities of the particle, respectively. Without loss of
generality and by assuming time-harmonic dependence ($e^{-{\rm i}\omega t}$),
we can write the scattered electromagnetic fields as Albella  _et al._ (2013);
Garcia-Etxarri (2017)

$\displaystyle{\bf E}_{\rm{scat}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{\epsilon_{0}\epsilon_{h}}{\tensor{\bm{\mathbb{G}}}}_{\rm{E}}({\bf
r}-{\bf r}_{0}){\bf p}+{\rm i}Zk^{2}{\tensor{\bm{\mathbb{G}}}}_{\rm{M}}({\bf
r}-{\bf r}_{0}){\bf m},$ (3) $\displaystyle{\bf H}_{\rm{scat}}({\bf r})$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}k^{2}\epsilon_{0}\epsilon_{h}}{Z}{\tensor{\bm{\mathbb{G}}}}_{\rm{M}}({\bf
r}-{\bf r}_{0}){\bf p}+k^{2}{\tensor{\bm{\mathbb{G}}}}_{\rm{E}}({\bf r}-{\bf
r}_{0}){\bf m}.$ (4)

Here ${\bf r}=\\{r,\theta,\varphi\\}$ denotes the spacial coordinates, ${\bf
r}_{0}$ denotes the center of the particle, $k=2\pi/\lambda$, where $\lambda$
is the radiation wavelength, and $Z=\sqrt{\mu_{0}/\epsilon_{0}\epsilon_{h}}$
is the impedance of the homogenous medium. Moreover,
${\tensor{\bm{\mathbb{G}}}}_{\rm{E}}({\bf r}-{\bf r}_{0})$ and
${\tensor{\bm{\mathbb{G}}}}_{\rm{M}}({\bf r}-{\bf r}_{0})$ refer to the
electric and magnetic dyadic Green’s functions, respectively Novotny and Hecht
(2012), which satisfy the following relations,

$\begin{split}{\tensor{\bm{\mathbb{G}}}}_{\rm{E}}({\bf r})\cdot{\bf
v}&=g(r)\Bigg{\\{}\left(1+\frac{{\rm i}}{kr}-\frac{1}{{(kr)}^{2}}\right){\bf
v}\\\ &+\left(-1-\frac{3{\rm
i}}{kr}+\frac{3}{{(kr)}^{2}}\right)\left(\hat{{\bf u}}_{\rm{r}}\cdot{\bf
v}\right)\hat{{\bf u}}_{\rm{r}}\Bigg{\\}},\\\
{\tensor{\bm{\mathbb{G}}}}_{\rm{M}}({\bf r})\cdot{\bf v}&=g(r)\left(\hat{{\bf
u}}_{\rm{r}}\times{\bf v}\right)\left({\rm i}-\frac{1}{kr}\right).\end{split}$
(5)

Here $\hat{{\bf u}}_{\rm{r}}$ denotes the radial unit vector, ${\bf v}$ is an
arbitrary vector, and $g(r)=e^{{\rm i}kr}/4\pi r$ is the scalar green
function, with $r=|{\bf r}|$. Notice that for the sake of simplicity, we have
settled ${\bf r}_{0}=0$, whereas the spatial dependence of the electromagnetic
fields (${\bf r}$) will be hereafter assumed.

The scattered electromagnetic fields by HRI nanoparticles present several
peculiar properties arising from the interference between the electric and
magnetic dipolar radiation. Most of these properties are encoded in the time-
averaged scattered Poynting vector,

${\bf{S}}=\frac{1}{2}\Re\\{{\bf E}_{\rm{scat}}\times{\bf
H}^{*}_{\rm{scat}}\\}={\bf{S}}_{\rm{E}}+{\bf{S}}_{\rm{M}}+{\bf{S}}_{\rm{EM}},$
(6)

where $\rm{E}$ (electric), $\rm{M}$ (magnetic), and $\rm{EM}$ account for the
interference between the electric and magnetic dipoles. As we will shortly
infer, this interference term plays a key role in the emergence of optical
mirages under the illumination of spinless beams. In this vein, the
interference term has also been shown to play a key role in controlling the
directionality of light García-Cámara  _et al._ (2013); Olmos-Trigo  _et al._
(2020a) and optical forces Nieto-Vesperinas  _et al._ (2010); Gómez-Medina
_et al._ (2012); Xu  _et al._ (2020), among others. To obtain a closed-
analytical expression of the optical mirage and according to Eq. (1), we need
to compute the scattered Poynting vector at shorter distances with respect to
the far-field. Now, by making use of Eqs (3)-(6), we arrive, after some
algebraic manipulation to

${\bf{\tilde{S}}}_{\rm{E}}=\left[|{{\bf
p}}|^{2}-\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
p}\right)\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
p}^{*}\right)\right]\hat{\mathbf{u}}_{\rm{r}}-\frac{2}{kr}\text{Im}\left\\{\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
p}\right){\bf p}^{*}\right\\},$ (7) ${\bf{\tilde{S}}}_{\rm{M}}=\left[|{\bf
m}|^{2}-\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
m}\right)\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
m}^{*}\right)\right]\hat{\mathbf{u}}_{\rm{r}}-\frac{2}{kr}\text{Im}\left\\{\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
m}\right){\bf m}^{*}\right\\},$ (8)
$\displaystyle\frac{{\bf{\tilde{S}}}_{\rm{EM}}}{2}$
$\displaystyle=\left(\hat{\mathbf{u}}_{\rm{r}}\cdot\text{Re}\left\\{{\bf
p}\times{\bf m}^{*}\right\\}\right)\hat{\mathbf{u}}_{\rm{r}}$
$\displaystyle+\frac{1}{kr}\text{Im}\left\\{\left(\hat{\mathbf{u}}_{\rm{r}}\times{\bf
p}\right)\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf
m}^{*}\right)+\left(\hat{\mathbf{u}}_{\rm{r}}\times{\bf
m}^{*}\right)\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf p}\right)\right\\}.$ (9)

Here
${\bf{{S}}}_{\rm{E}}=(G_{0}c/\epsilon_{h}\epsilon_{0}){\bf{\tilde{S}}}_{\rm{E}}$,
${\bf{{S}}}_{\rm{M}}=ZG_{0}{\bf{\tilde{S}}}_{\rm{M}}$, and
${\bf{{S}}}_{\rm{EM}}=ZcG_{0}{\bf{\tilde{S}}}_{\rm{EM}}$, where
$G_{0}=k^{4}/32\pi^{2}r^{2}$, and $c$ is the speed of light. Hitherto, we have
just assumed a target that can be described by ${\bf p}$ and ${\bf m}$. As a
result, Eqs. (7)-(9) are valid for any incoming polarization and any shape of
the HRI nanoparticle. Thus, they are general expressions of the Poynting
vector in the electric and magnetic dipolar regime.

Let us now turn our attention to external excitation. Hereafter, we consider a
linearly polarized (spinless) plane wave propagating in the $z$-direction,

$\displaystyle{\bf E}_{\rm{inc}}=E_{0}e^{{\rm i}kz}\hat{\mathbf{u}}_{\rm{x}},$
$\displaystyle Z{\bf H}_{\rm{inc}}=-E_{0}e^{{\rm
i}kz}\hat{\mathbf{u}}_{\rm{y}},$ (10)

where $E_{0}$ is the incoming electric field amplitude. From Eq. (10), we can
infer that a linearly polarized plane wave carries null spin, $s_{z}$, and
orbital angular momentum, $\ell_{z}$, namely,

$\displaystyle s_{z}=\frac{-{\rm i}\left\\{{{\bf E}^{*}_{\rm{inc}}}\times{\bf
E}_{\rm{inc}}\right\\}\cdot\bm{\hat{e}}_{z}}{\left|{\bf
E}_{\rm{inc}}\right|^{2}}=0,$ $\displaystyle\ell_{z}=\frac{-{\rm
i}\left\\{{{\bf E}^{*}_{\rm{inc}}}\cdot\partial_{\varphi}{\bf
E}_{\rm{inc}}\right\\}}{\left|{\bf E}_{\rm{inc}}\right|^{2}}=0.$ (11)

Consequently, for a pure dipole, the SOI of the light after scattering is, as
expected, identically zero Schwartz and Dogariu (2006); Bliokh  _et al._
(2011). As a result, we can notice that the emergence of optical mirages is
hidden in this SOI of light framework.

At this point and assuming Eq. (10) as the incoming illumination, let us
consider a HRI nanosphere sustaining an electric and magnetic dipolar
response, namely,

$\displaystyle{\bf
p}=E_{0}\epsilon_{h}\epsilon_{0}\alpha_{\rm{E}}\hat{\mathbf{u}}_{\rm{x}},$ and
$\displaystyle{\bf m}=-(E_{0}/Z)\alpha_{\rm{M}}\hat{\mathbf{u}}_{\rm{y}}$ (12)

with $\alpha_{\rm{E}}={\rm i}6\pi/k^{3}a_{1}$ and $\alpha_{\rm{M}}={\rm
i}6\pi/k^{3}b_{1}$, $a_{1}$ and $b_{1}$ being the first electric and magnetic
Mie coefficients, respectively Mie (1908). Now, by inserting Eq. (12) into
Eqs. (7)-(9), we arrive, after some cumbersome algebra to

$\lim_{kr\rightarrow\infty}kr\>\frac{{\bf{S}}_{\perp}}{S_{0}}=2G_{0}\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}\sin\theta\left(\cos
2\varphi\hat{\mathbf{u}}_{{\theta}}-\sin
2\varphi\cos\theta\hat{\mathbf{u}}_{{\varphi}}\right)$ (13)

with $S_{0}=c\epsilon_{h}\epsilon_{0}E^{2}_{0}$ being the Poynting vector of a
plane wave.

This is the key result of this Letter: a non-radial structure of the scattered
Poynting vector appears under the illumination of a linearly polarized plane
wave. In line with Eq. (1), spinless optical mirages are then expected for
particles sustaining simultaneously electric and magnetic responses, i.e.,
$\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}\neq 0$. It is essential to notice
that $\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}$ does not contribute to the
extinction, absorption, scattering, and radar cross-sections nor the asymmetry
parameter. Moreover, we cannot unravel
$\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}$ under the illumination of a
well-defined helicity beam, as occurs with the imaginary Poynting vector in
the context of optical forces Xu and Nieto-Vesperinas (2019).

Figure 1: Optical mirage when considering a linearly polarized plane wave
impinging on a HRI nanosphere ($m=3.5$) sustaining simultaneously an electric
and magnetic dipolar response (b). In the latter, the observer perceives non-
radial scattered Poynting vectors $\\{{\bf{S}}_{1},{\bf{S}}_{2}\\}$ that lead
to optical mirages $\\{\bm{\Delta}_{1},\bm{\Delta}_{2}\\}$.

At this point, let us briefly discuss why optical mirages do not appear in the
scattering of a pure electric dipole under the illumination of a spinless
beam. In absence of a magnetic response, i.e., $|{\bf m}|=0$, it is noticeable
that Eq. (8) and Eq. (9) vanishes, yielding a radial scattered Poynting vector
since $\text{Im}\left\\{\left(\hat{\mathbf{u}}_{\rm{r}}\cdot{\bf p}\right){\bf
p}^{*}\right\\}=0$. As a result, ${\bf{S}}_{\perp}=0$. According to Eq. (1),
no apparent shift can emerge for a linearly polarized plane wave impinging on
an electric dipolar sphere. The abovementioned agrees with Refs. Haefner  _et
al._ (2009); Gao  _et al._ (2018); Olmos-Trigo  _et al._ (2019a, b); Shi  _et
al._ (2019); Sun  _et al._ (2022). Note that similar reasoning can be applied
to a pure magnetic dipole. In Ref. Araneda  _et al._ (2019), it is also
discussed that there are no apparent shifts under the illumination of a
linearly polarized Gaussian beam. This physical picture remains valid
regardless of the choice of the numerical aperture, making $\bm{\Delta}=0$ a
general result for electric (or magnetic) dipolar objects.

Figure 2: a) $\eta$-parameter and b) $g$-parameter vs the scattering phase-
shifts. The dashed circle represents a fixed scattering cross-section. The
path from $A$ to $B$, which corresponds to a slight change in the Mie
coefficients, leads to $\eta\Longrightarrow-\eta$, $g\longrightarrow g$, and
$\gamma\longrightarrow\gamma$. Figure 3: Optical mirage for a Silicon-like
sphere ($m=3.5$) at different $x=ka$ size parameters vs the azimuth and
scattering angles, $\varphi$ and $\theta$, respectively. a) $x=0.85$ (dipolar
magnetic resonance) b) $x=1.15$ (dipolar electric resonance) c) $x=0.783$
($K_{1-}$) and d) $x=0.785$ ($K_{1+}$). The first Kerker condition ($K_{1}$)
emerges at $x=0.784$ for a $m=3.5$, according to Ref. García-Etxarri  _et al._
(2011).

The electromagnetic fields scattered by electric and magnetic dipoles present
a very different structure. Contrary to the purely electric (or magnetic)
case, when excited with a linearly polarized beam, the scattered Poynting
vector pirouettes from the origin of coordinates, according to Eq. (13).
Consequently, and as sketched in Fig. 1, the scattered Poynting vector makes
an angle with the line of sight, determining a spinless optical mirage.
Mathematically, we can compute this optical mirage by inserting Eq. (13) into
Eq. (1), yielding

$\frac{\bm{\Delta}}{\lambda}=\left[\frac{\eta\sin\theta\left(\cos
2\varphi\hat{\mathbf{u}}_{{\theta}}-\sin
2\varphi\cos\theta\hat{\mathbf{u}}_{{\varphi}}\right)}{1-\sin^{2}\theta\gamma+2g\cos\theta}\right]\frac{1}{\pi}.$
(14)

Here,

$\displaystyle\eta$
$\displaystyle=\frac{\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}}{|\alpha_{\rm{E}}|^{2}+|\alpha_{\rm{M}}|^{2}},$
$\displaystyle
g=\frac{\Re\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}}{|\alpha_{\rm{E}}|^{2}+|\alpha_{\rm{M}}|^{2}},$
(15)

and

$\gamma=\frac{|\alpha_{\rm{E}}|^{2}\cos^{2}\varphi+|\alpha_{\rm{M}}|^{2}\sin^{2}\varphi}{|\alpha_{\rm{E}}|^{2}+|\alpha_{\rm{M}}|^{2}}.$
(16)

From Eq. (14), two interesting limiting cases can be identified. First, in the
strict forward and backward directions, i.e., $\theta=0$ and $\theta=\pi$,
respectively, the optical mirage goes to zero, regardless of the optical
response. Secondly, whenever $\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}=0$,
the optical mirage vanishes regardless of $\theta$ and $\varphi$. To get more
insight into the physics behind the interference terms between the electric
and magnetic dipoles, it is convenient to represent $\eta$ and $g$ in the
scattering phase-shift notation Olmos-Trigo  _et al._ (2020b). That is, by re-
writing the electric and magnetic polarizabilities as
$\alpha_{\rm{E}}=-(6\pi/k^{3})\sin\alpha_{1}e^{-{\rm i}\alpha_{1}}$ and
$\alpha_{\rm{M}}=-(6\pi/k^{3})\sin\beta_{1}e^{-{\rm i}\beta_{1}}$, we arrive
from Eq. (15) to

$\eta=\frac{\Im\\{a_{1}b^{*}_{1}\\}}{|a_{1}|^{2}+|b_{1}|^{2}}=\frac{\sin\alpha_{1}\sin\beta_{1}\sin\left(\beta_{1}-\alpha_{1}\right)}{\sin^{2}\alpha_{1}+\sin^{2}\beta_{1}},$
(17)
$g=\frac{\Re\\{a_{1}b^{*}_{1}\\}}{|a_{1}|^{2}+|b_{1}|^{2}}=\frac{\sin\alpha_{1}\sin\beta_{1}\cos\left(\beta_{1}-\alpha_{1}\right)}{\sin^{2}\alpha_{1}+\sin^{2}\beta_{1}}.$
(18)

From Eq. (17), we can infer that besides a Rayleigh sphere, the pure magnetic
(or electric) scattering regimes, given by the so-called electric (or
magnetic) anapoles Miroshnichenko  _et al._ (2015); Wei  _et al._ (2016);
Parker  _et al._ (2020); Sanz-Fernández  _et al._ (2021); Coe  _et al._
(2022), i.e., $\sin\alpha_{1}=0$ ($\sin\beta_{1}=0$), give rise to
$\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}=0$. As a result, according to Eq.
(13), the streamlines of the scattered Poynting point radially. Notice that
the previous phenomenon is also reached when the sphere meets the first Kerker
condition Kerker  _et al._ (1983); Nieto-Vesperinas  _et al._ (2011); Olmos-
Trigo  _et al._ (2020c), namely,
$\alpha_{1}=\beta_{1}\Longrightarrow\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}=0$.
In short, the first Kerker condition leads to the conservation of the state of
polarization Fernandez-Corbaton  _et al._ (2013) and the absence of
backscattered light for spherical particles Person  _et al._ (2013); Fu  _et
al._ (2013); Geffrin  _et al._ (2012). It is essential to highlight that
within a slight change of the electric and magnetic scattering phase-shifts in
the vicinity of the first Kerker condition,
$\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}$ is sign-flipped. Mathematically,
the sign-flipping of $\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}$ can be
inferred from Eq. (17) while is visually confirmed in Fig 2a by the transition
from point $A$ to point $B$. Notice that the optical mirage is also sign-
flipped since it is proportional to
$\Im\\{\alpha_{\rm{E}}\alpha^{*}_{\rm{M}}\\}$. This change of sign is achieved
since $\gamma$ and $g$ are symmetrical to the first Kerker condition, as can
be inferred from Fig. 2b. Hence, we can conclude that by tuning the optical
response, we can mimic, for spinless beams, the spin angular momentum under
the illumination of a left ($s_{z}=+1$) or right ($s_{z}=-1$) circularly
polarized plane wave.

At this stage, it is convenient to discuss the absolute value of the optical
mirage for a particular HRI sphere under the illumination of a linearly
polarized plane wave. To that end, we consider the Silicon-like nanosphere of
Ref. García-Etxarri  _et al._ (2011), which has been previously introduced in
Fig. 1. First, in Fig. 3a and Fig. 3b, we show the modulus of the optical
mirage at the magnetic and electric dipolar resonances. As it can be seen, the
maximum value of the optical mirage is not subwavelength, reaching
$\Delta/\lambda\sim 1$ for several combinations of $\theta$ and $\varphi$.
Furthermore, it is important to note that at the electric (or magnetic)
resonance, the magnetic (or electric) response is not zero. We can therefore
state that the key element to observe an optical mirage is the interference
between the fields scattered by the electric and magnetic dipoles. Remarkably,
in the vicinity of the first Kerker condition and near backscattering, this
maximum is considerably enhanced, as depicted in Fig. 3c and Fig. 3d. As a
matter of fact, the optical mirage reaches tens of wavelengths in magnitude,
$\Delta/\lambda\sim 12$. It is worth mentioning that this absolute value of
the spinless optical mirage surpasses the one predicted in Ref. Haefner  _et
al._ (2009) and measured in Ref. Araneda  _et al._ (2019), showing the
importance of our finding.

In conclusion, we have demonstrated that spinless optical mirages, reaching
tens of wavelength in magnitude, can emerge in the optical response of HRI
nanoparticles under the illumination of a linearly polarized plane wave
carrying no angular momentum. SOI of light, which played a crucial role in
previous works regarding apparent displacements, does not play any role here
as the incident beam is spinless. In contrast, the spinless optical mirage is
proportional to an overlooked interference between the electric and magnetic
contributions, which only survives under the illumination of spinless beams.
Interestingly, we have also shown that we can mimic the role of the spin by
correctly tuning the optical response in the vicinity of the first Kerker
condition. Our findings can find potential applications in dielectric-based
photonic applications, such as in optical sensing, optical imaging, and in the
manipulation of all-dielectric nanoparticles.

Acknowledgements J.O.T, A.G.E, A.G.M acknowledge support from Project No.
PID2019-109905GA-C22 of the Spanish Ministerio de Ciencia, Innovación y
Universidades (MICIU). J.O.T. acknowledges support from a Juan de la Cierva
fellowship No. FJC2021-047090-I of Ministerio de Ciencia Innovación y
Universidades (MICIU). A.G.E. received funding from the IKUR Strategy under
the collaboration agreement between Ikerbasque Foundation and DIPC on behalf
of the Department of Education of the Basque Government, as well 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)
Departamento de Universidades e Investigación del Gobierno Vasco, and the
Centros Severo Ochoa AEI/CEX2018-000867-S from the Spanish Ministerio de
Ciencia e Innovación. D.R.A acknowledges financial support from the Swiss
National Science Foundation through project 197146.

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