# Identifying regions of minimal back-scattering by a relativistically-moving
sphere

Mitchell R. Whittam<EMAIL_ADDRESS>Institut für Theoretische
Festkörperphysik, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
Aristeidis G. Lamprianidis  Institut für Theoretische Festkörperphysik,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany  Yannick
Augenstein  Institut für Theoretische Festkörperphysik, Karlsruhe Institute of
Technology, 76131 Karlsruhe, Germany  Carsten Rockstuhl  Institut für
Theoretische Festkörperphysik, Karlsruhe Institute of Technology, 76131
Karlsruhe, Germany  Institute of Nanotechnology, Karlsruhe Institute of
Technology, 76021 Karlsruhe, Germany

###### Abstract

The far-field back-scattering amplitude of an electric field from a
relativistically-moving sphere is analyzed. Contrary to prior research, we do
so by expressing the fields in the helicity basis, and we highlight here its
advantages when compared to the commonly-considered parity basis. With the
purpose of exploring specific scattering phenomena considering relativistic
effects, we identify conditions that minimize the back-scattered field,
leading to a relativistic formulation of the first Kerker condition. The
requirements to be satisfied by the sphere are expressed in terms of Mie
angles, which constitute an effective parametrization of any possible optical
response a sphere might have. We are able to identify multiple combinations of
Mie angles up to octupolar order via gradient-based optimization that satisfy
our newly formulated relativistic Kerker condition, yielding minima for the
back-scattered energy as low as $0.016\text{\,}\mathrm{\char 37\relax}$ of the
average scattered energy. Our results can be extended to involve multiple
particles forming a metasurface, potentially having direct implications on the
design of light sails as considered by the Breakthrough Starshot Initiative.

††preprint: APS/123-QED

##  I Introduction

The scattering of light by a sphere is a canonical problem in optics and
electrodynamics and has been investigated for many years, particularly for
stationary spheres [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The scattering of
light by spheres is best described using Mie theory, which involves expressing
the incident and scattered electromagnetic fields in terms of vector spherical
harmonics (VSHs). The amplitude coefficients that weight these VSHs are
collected in a vector and are mutually linked by a matrix-vector product.
Moreover, all optical properties of the object are captured by the
corresponding matrix, called the transition or T-matrix. For an arbitrary
object, the T-matrix can be dense, but is diagonal for a sphere, and the
diagonal entries are the Mie coefficients [12, 13].

Controlling an object’s geometrical and material properties provides a unique
way of tailoring the scattered field on demand, and many intriguing aspects
have been explored, an example being the so-called Kerker condition [14, 15,
16, 17, 18]. The first Kerker condition contains the necessary composition of
multipolar excitations such that the object exhibits zero back-scattering. A
second Kerker condition implies a vanishing scattering in the forward
direction, but this is considered less often, since optical gain is necessary
for its observation [19].

While initially formulated for objects that can be safely described in dipolar
approximation, it was soon realized that similar effects are encountered while
capitalizing on higher-order multipole moments. This coined the notion of
generalized Kerker conditions [20], and Kerker effects have been explored in a
large variety of settings. These studies are motivated by high-impact
applications related to nanoantennas, chiral molecules, and metamaterials, to
name a few [20, 21, 22, 23, 24, 25].

This paper provides a further yet, to our knowledge, unexplored perspective on
the Kerker effect. It considers the Kerker effect in the relativistic regime.
The basic setting of our exploration is that of a relativistically moving
sphere illuminated with a monochromatic Gaussian beam characterized by an
incident angle $\Theta_{\mathrm{i}}$ relative to the direction of motion of
the sphere. Of course, unlike the case of a stationary sphere, one cannot
assume that there exists a combination of multipole excitations that yield
zero back-scattering with the inclusion of motion. However, one can aim to
minimize the back-scattering, which depends on the multipolar contribution to
the scattering response, for a given speed and incident electric field angle.
This leads to an approximate Kerker condition in the case of a
relativistically moving sphere.

Our work has clear implications for future technology developments. For
example, within the Breakthrough Starshot Initiative [26, 27], micro-gram
satellites equipped with light sails, potentially made from metasurfaces
consisting of a tailored arrangement of scatterers, are to be accelerated with
an Earth-based laser system up to a significant fraction of the speed of
light. Using these satellites, neighbouring galaxies shall be explored. The
design of such systems has many facets, and among them is the accurate
description of the optical response from scattering objects in the form of
metasurfaces. The formulation of the light scattering by an isolated object
under relativistic conditions, as a pursuit in this contribution, is an
important prerequisite to study such more advanced devices.

The structure of the paper is as follows. In Section II, the physical setup is
outlined, and all necessary coordinate systems are defined. Moreover, the
field of the considered incident beam is transformed from the lab frame to the
reference frame of the sphere based on the transformation of each constitutive
plane wave of its angular spectrum representation. Afterwards, the scattered
field is obtained by solving an ordinary Mie problem in the rest frame of the
sphere. We rely on a parametrization of its response using Mie angles [28].
These Mie angles constitute a minimalist model to express all possible
responses from a sphere, which allows a generic analysis of the back-
scattering response. To conclude Section II, the scattered field will be
transformed back to the lab frame, in which the back-scattering is observed.

In Section III, the back-scattering amplitude is visualized with respect to
some given Mie angles for a sphere with a fixed velocity and a field with a
fixed incident angle. We implement all calculations using the Julia
programming language [29] and implement a gradient-based optimization scheme
by leveraging automatic differentiation within the JuMP modelling framework
[30], much in the spirit of recent works on differentiable physics solvers
[31, 32]. Using this scheme, we design spheres that provide minimum values for
the back-scattering and identify the corresponding combinations of Mie angles.
We find multiple suitable combinations, and minimize the back-scattered energy
to a negligible $0.016\text{\,}\mathrm{\char 37\relax}$ of the average
scattered energy. In Section IV, we conclude our findings.

##  II Description of the scattering scenario

Before delving into the mathematical description of the scattered field, it is
first necessary to specify the geometry and constraints of the system. We
consider a spherical particle moving at a relativistic velocity
$\mathbf{v}=v\hat{\mathbf{z}}$ within a pervading incident electric field
$\mathbf{E}_{\mathrm{i}}(\mathbf{r},t)$ with incident angle
$\Theta_{\mathrm{i}}$ as observed by an external lab frame $S$. Although an
accompanying magnetic field will always exist, to avoid repetition, we omit
explicit reference to this. Two further frames are considered, namely the
beam’s frame $S^{\parallel}$, which is the frame where the direction of motion
of the beam moves parallel to its corresponding $z^{\parallel}$-axis, and the
boosted frame $S^{\prime}$, which represents the inertial reference frame of
the sphere (see Fig. 1). Accordingly, the corresponding quantities are denoted
without a prime in $S$ and with a prime in $S^{\prime}$, while all quantities
in $S^{\parallel}$ are denoted with a $\parallel$ superscript.

A further quantity of interest is the polar angle $\Theta_{\mathrm{i}}$
between $\hat{\mathbf{k}}_{\mathrm{i}}$ and $\mathbf{v}$, i.e., the angle
between the beam’s propagation direction and the axis of movement of the
scatterer (see Fig. 1). Given the symmetry of the system, we set the azimuthal
angle of the incident field $\Phi_{\mathrm{i}}$ to be zero. Moreover, the
direction of back-scattering is given by $\hat{\mathbf{k}}_{\text{BS}}$, the
opposite direction to $\hat{\mathbf{k}}_{\mathrm{i}}$.

To determine the scattered field in $S$, we implement the ‘frame-hopping
method’ (FHM) as described in Garner  _et al._ [33]. For reference, this
process is outlined below:

  1. 1.

Lorentz-boost the incident electric field from $S$ to $S^{\prime}$.

  2. 2.

Solve the scattering problem in $S^{\prime}$.

  3. 3.

Inverse Lorentz-boost the scattered field from $S^{\prime}$ back to $S$.

The reason for computing the scattered field in $S^{\prime}$ and not $S$ is a
matter of mathematical simplicity. In $S^{\prime}$, the scattering calculation
is analogous to a stationary system, thus avoiding any superfluous variable
transformations.

Figure 1: Pictorial representation of the sphere in a) the beam’s frame
$S^{\parallel}$ and the rotated frame $S$ containing an external observer
(represented by the eye) and b) the sphere’s inertial reference frame
$S^{\prime}$. In $S$, the sphere is seen to be moving with velocity
$v\hat{\mathbf{z}}$. The wave vector $\hat{\mathbf{k}}_{\mathrm{i}}$ is
incident on the sphere with angle $\Theta_{\mathrm{i}}$. In a), the direction
of back-scattering $\hat{\mathbf{k}}_{\mathrm{BS}}$ (denoted by the red arrow)
is in the opposite direction to $\hat{\mathbf{k}}_{\mathrm{i}}$ and it is this
direction that shall be considered when formulating a relativistic Kerker
condition. In b), the sphere is stationary ($v=0$) while the Lorentz-boosted
wave vector of the incident field is given by
$\hat{\mathbf{k}}^{\prime}_{\mathrm{i}}$ and is incident with angle
$\Theta^{\prime}_{\mathrm{i}}\neq\Theta_{\mathrm{i}}$ when
$\Theta_{\mathrm{i}}\notin\\{0,\pi\\}$, and
$\Theta^{\prime}_{\mathrm{i}}=\Theta_{\mathrm{i}}$ when
$\Theta_{\mathrm{i}}\in\\{0,\pi\\}$. Moreover, the direction of back-
scattering $\hat{\mathbf{k}}_{\mathrm{BS}}^{\prime}$ in this frame is in
general not opposite to $\hat{\mathbf{k}}_{\mathrm{i}}$ or
$\hat{\mathbf{k}}^{\prime}_{\mathrm{i}}$.

###  II.1 Lorentz-boosting the incident field into the scatterer’s reference
frame

First, we need to consider the incident field in the beam’s reference frame
$S^{\parallel}$. As an incident field, we consider a single monochromatic
Gaussian beam of well-defined helicity (_i.e._ , handedness) expanded in terms
of circularly polarized plane waves, which are eigenstates of the
electromagnetic wave equation. We use the following ket in abstract Dirac
notation to denote such plane waves as eigenstates of free space characterized
by helicity $\lambda^{\parallel}=\pm 1$, temporal frequency
$\omega^{\parallel}$, and direction of propagation
$\hat{\mathbf{k}}^{\parallel}$:

$\ket{\lambda^{\parallel}\ \hat{\mathbf{k}}^{\parallel}\
\omega^{\parallel}}\doteq\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{k}}^{\parallel})\exp\\{{\mathrm{i}\omega^{\parallel}[(\hat{\mathbf{k}}^{\parallel}\cdot\mathbf{r}/c)-t}]\\}\mbox{\quad,\quad}$
(1)

where the symbol $\doteq$ refers to the spatiotemporal representation of the
plane wave eigenstate. The quantity $c$ is the speed of light in vacuum, and
the polarization unit vector
$\hat{\mathbf{e}}_{\lambda^{\parallel}}(\hat{\mathbf{k}}^{\parallel})$ is
given by

$\displaystyle\hat{\mathbf{e}}_{\lambda^{\parallel}}(\hat{\mathbf{k}}^{\parallel})$
$\displaystyle=\frac{-\lambda^{\parallel}\hat{\theta}(\hat{\mathbf{k}}^{\parallel})-\mathrm{i}\hat{\phi}(\hat{\mathbf{k}}^{\parallel})}{\sqrt{2}}\mbox{\quad,\quad}$
(2)

where $\lambda^{\parallel}=\pm 1$ corresponds to left/right circularly-
polarized waves. The quantities $\hat{\theta}$ and $\hat{\phi}$ are,
respectively, the polar and azimuthal spherical unit vectors perpendicular to
the direction of propagation $\hat{\mathbf{k}}^{\parallel}$ that is
characterized by the polar and azimuthal angles of propagation
$\theta^{\parallel}$, $\phi^{\parallel}$ of each constituent plane wave.

Denoting quantities that belong to the incident field with the subscript
‘$\mathrm{i}$’, we represent a general electric field in terms of its angular
spectrum, i.e., as a plane wave expansion:

$\displaystyle\ket{\mathbf{E}^{\parallel}_{\mathrm{i}}}$
$\displaystyle=\sum_{\lambda}\int_{0}^{2\pi}\mathrm{d}\phi^{\parallel}\int^{\pi}_{0}\mathrm{d}\theta^{\parallel}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\parallel}$
$\displaystyle\hskip 2.0pt\mathcal{G}^{\parallel}_{\lambda^{\parallel},\hskip
1.13809pt\mathrm{i}}(\omega^{\parallel},\theta^{\parallel},\phi^{\parallel})\ket{\lambda^{\parallel}\
\hat{\mathbf{k}}^{\parallel}\
\omega^{\parallel}}+\mathrm{c.c.}\mbox{\quad,\quad}$ (3)

where the amplitudes for a monochromatic Gaussian beam focused at the origin
of $S$ of waist $w_{0}$ , frequency $\omega_{\mathrm{i}}$ and helicity
$\lambda_{\mathrm{i}}$ propagating along the $+z$-axis are given by:

$\displaystyle\mathcal{G}^{\parallel}_{\lambda^{\parallel},\hskip
1.13809pt\mathrm{i}}(\omega^{\parallel},\theta^{\parallel},\phi^{\parallel})$
$\displaystyle=E_{0}\sin(2\theta^{\parallel})$
$\displaystyle\cdot\exp[-\frac{\omega_{\mathrm{i}}^{2}w_{0}^{2}\sin^{2}(\theta^{\parallel})}{4c^{2}}]$
$\displaystyle\cdot\delta_{\lambda^{\parallel}\lambda_{\mathrm{i}}}\delta(\omega^{\parallel}-\omega_{\mathrm{i}})H(\pi/2-\theta^{\parallel})\mbox{\quad,\quad}$
(4)

where $E_{0}$ is a constant, and $H(\pi/2-\theta^{\parallel})$ is the
Heaviside step function which eliminates all counter-propagating waves.
Moreover, we consider the waist $w_{0}$ to be very large, such that
$\ket{\mathbf{E}^{\parallel}_{\mathrm{i}}}$ approximates to a plane wave but
is nonetheless still finite in space. The reason for doing this is that, for a
regular plane wave infinitely extended in space, the interaction of the
incident wave with the moving sphere would be incessant and, therefore, the
scattered power flux would have a cylindrical symmetry with respect to the
axis of movement of the scatterer. That is, the scattered power flux would be
translationally invariant with respect to this axis and would only vary
azimuthally. On the other hand, an excitation of finite spatial extent ensures
that the interaction of light with the scatterer is localized in space (around
the origin of $S$), therefore, yielding a spherical-like scattering of waves
emanating from the region where the interaction takes place.

To consider an electric field of arbitrary angle of incidence
$\Theta_{\mathrm{i}}$, we apply a rotation operator
$\hat{\mathbf{R}}_{y}(\Theta_{\mathrm{i}})$ about the $y$-axis to Eqn. (II.1)
to transit from a representation of the beam with respect to the
$S^{\parallel}$ to one with respect to $S$ such that

$\displaystyle\ket{\mathbf{E}_{\mathrm{i}}}=\hat{\mathbf{R}}_{y}(\Theta_{\mathrm{i}})\ket{\mathbf{E}^{\parallel}_{\mathrm{i}}}\mbox{\quad,\quad}$
(5)

where

$\displaystyle\hat{\mathbf{R}}_{y}(\Theta_{\mathrm{i}})\ket{\lambda^{\parallel}\
\hat{\mathbf{k}}^{\parallel}\ \omega^{\parallel}}$
$\displaystyle=\sum_{\lambda}\int_{0}^{2\pi}\mathrm{d}\phi\int^{\pi}_{0}\mathrm{d}\theta\int^{\infty}_{0^{+}}\mathrm{d}\omega$
$\displaystyle\hskip
2.0pt\mathcal{R}_{\lambda^{\parallel},\lambda}(\omega^{\parallel},\theta^{\parallel},\phi^{\parallel},\omega,\theta,\phi;\Theta_{\mathrm{i}})$
$\displaystyle\ket{\lambda\ \hat{\mathbf{k}}\ \omega}\mbox{\quad.\quad}$ (6)

The transformation coefficients are given by

$\displaystyle\mathcal{R}_{\lambda^{\parallel},\lambda}(\omega^{\parallel},\theta^{\parallel},\phi^{\parallel},\omega,\theta,\phi;\Theta_{\mathrm{i}})$
$\displaystyle=\mathcal{P}(\theta^{\parallel},\phi^{\parallel},\Theta_{\mathrm{i}})$
$\displaystyle\cdot\delta_{\lambda\lambda^{\parallel}}$
$\displaystyle\cdot\delta(\omega-\omega^{\parallel})$
$\displaystyle\cdot\delta\left[\theta-\mathrm{arccos}(\hat{k}_{z})\right]$
$\displaystyle\cdot\delta\left[\phi-\mathrm{atan2}(\hat{k}_{y},\hat{k}_{x})\right]\mbox{\quad,\quad}$
(7)

where

$\displaystyle\begin{pmatrix}[1.3]\hat{k}_{x}\\\ \hat{k}_{y}\\\
\hat{k}_{z}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}[1.3]\cos{\Theta_{\mathrm{i}}}&0&\sin{\Theta_{\mathrm{i}}}\\\
0&1&0\\\
-\sin{\Theta_{\mathrm{i}}}&0&\cos{\Theta_{\mathrm{i}}}\end{pmatrix}\begin{pmatrix}[1.3]\hat{k}^{\parallel}_{x}\\\
\hat{k}^{\parallel}_{y}\\\ \hat{k}^{\parallel}_{z}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}[1.3]\sin\theta^{\parallel}\cos\phi^{\parallel}\cos{\Theta_{\mathrm{i}}}+\cos\theta^{\parallel}\sin{\Theta_{\mathrm{i}}}\\\
\sin\theta^{\parallel}\sin\phi^{\parallel}\\\
-\sin\theta^{\parallel}\cos\phi^{\parallel}\sin{\Theta_{\mathrm{i}}}+\cos\theta^{\parallel}\cos{\Theta_{\mathrm{i}}}\end{pmatrix}\mbox{\quad,\quad}$
(8)

and $\mathcal{P}(\theta^{\parallel},\phi^{\parallel},\Theta_{\mathrm{i}})$ is
a prefactor corresponding to the acquired phase due to the rotation:

$\displaystyle\mathcal{P}(\theta^{\parallel},\phi^{\parallel},\Theta_{\mathrm{i}})=\mathrm{exp}[\mathrm{i}p(\theta^{\parallel},\phi^{\parallel},\Theta_{\mathrm{i}})]\mbox{\quad,\quad}$
(9)

with

$\displaystyle p(\theta^{\parallel},\phi^{\parallel},\Theta_{\mathrm{i}})$
$\displaystyle=\mathrm{atan2}\Big{[}-\lambda\sin(\Theta_{\mathrm{i}})\sin(\phi^{\parallel}),$
$\displaystyle\hskip 11.66573pt\hskip
2.84544pt\cos(\theta^{\parallel})\sin(\Theta_{\mathrm{i}})\cos(\phi^{\parallel})$
$\displaystyle\hskip
1.42271pt+\sin(\theta^{\parallel})\cos(\Theta_{\mathrm{i}})\Big{]}\mbox{\quad.\quad}$
(10)

After doing this, one can follow the first step of the frame-hopping method
and compute the Lorentz boost of the incident electric field from $S$ to
$S^{\prime}$. In App. A, we calculate the Lorentz boost of plane waves. We
denote with $\hat{\mathbf{LB}}_{z}(\beta)$ the operator that boosts fields
along the $z$-axis with speed $v=\beta c$, where $0\leq\beta<1$ from $S$ to
$S^{\prime}$. This operator acts on the eigenstates of monochromatic plane
waves with well-defined helicity in the following way:

$\displaystyle\hat{\mathbf{LB}}_{z}(\beta)\ket{\lambda\ \hat{\mathbf{k}}\
\omega}$
$\displaystyle=\sum_{\lambda^{\prime}}\int_{0}^{2\pi}\mathrm{d}\phi^{\prime}\int^{\pi}_{0}\mathrm{d}\theta^{\prime}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}$
$\displaystyle\hskip
14.22636pt\mathcal{L}_{\lambda\lambda^{\prime}}(\omega,\theta,\phi,\omega^{\prime},\theta^{\prime},\phi^{\prime};\beta)$
$\displaystyle\hskip 11.38092pt\ket{\lambda^{\prime}\
\hat{\mathbf{k}}^{\prime}\ \omega^{\prime}}\mbox{\quad,\quad}$ (11)

with the transformation coefficients given by

$\displaystyle\mathcal{L}_{\lambda\lambda^{\prime}}(\omega,\theta,\phi,\omega^{\prime},\theta^{\prime},\phi^{\prime};\beta)$
$\displaystyle=\mathcal{C}\left(\beta,\theta\right)$
$\displaystyle\cdot\delta_{\lambda^{\prime}\lambda}$
$\displaystyle\cdot\delta\left(\omega^{\prime}-\mathcal{C}\left(\beta,\theta\right)\omega\right)$
$\displaystyle\cdot\delta\left[\theta^{\prime}-\mathrm{arccos}\left(\frac{\cos\theta-\beta}{1-\beta\cos\theta}\right)\right]$
$\displaystyle\cdot\delta\left(\phi^{\prime}-\phi\right)\mbox{\quad,\quad}$
(12)

where $\gamma=1/\sqrt{1-\beta^{2}}$,
$\cos\theta=\hat{\mathbf{k}}\cdot\hat{\mathbf{z}}$ and

$\displaystyle\mathcal{C}\left(\beta,\theta\right)=\gamma\left[1-\beta\cos\theta\right]\mbox{\quad,\quad}$
(13)

which is derived in App. A. We see from Eqn. (II.1) that

$\displaystyle\theta^{\prime}=\mathrm{arccos}\left(\frac{\cos\theta-\beta}{1-\beta\cos\theta}\right)\mbox{\quad,\quad}$
(14)

and

$\displaystyle\omega^{\prime}=\mathcal{C}\left(\beta,\theta\right)\omega$ (15)

correspond to the Lorentz boost of $\theta$ and $\omega$, respectively. Since
the motion occurs solely along the $z$-axis, the azimuthal angle $\phi$
remains unchanged under the Lorentz boost, that is,

$\displaystyle\phi^{\prime}=\phi\mbox{\quad.\quad}$ (16)

Figure 2: a) The Doppler shift $\Theta^{\prime}_{\mathrm{i}}$ of
$\Theta_{\mathrm{i}}$ as a function of $\Theta_{\mathrm{i}}$ and $\beta$
determined using Eqn. (14). This demonstrates that the direction of the
incident wave as perceived in $S^{\prime}$ is different to that in $S$. b) The
normalized Doppler-shifted incident frequency
$\omega^{\prime}_{\mathrm{i}}/\gamma\omega_{\mathrm{i}}$ as a function of
$\Theta_{\mathrm{i}}$ and $\beta$, where $\omega^{\prime}_{\mathrm{i}}$ is the
Doppler-shifted incident frequency determined using Eqn. (15). When
$\Theta_{\mathrm{i}}=0$ and $\beta\to 1$, the object is moving away from the
field source, thus causing the incident frequency in $S^{\prime}$ to decrease
(redshift). When $\Theta_{\mathrm{i}}=\pi$ and $\beta\to 1$, the object is
moving towards the field source, thus causing the incident frequency in
$S^{\prime}$ to increase (blueshift). When $\Theta_{\mathrm{i}}=\pi/2$,
$\omega^{\prime}_{\mathrm{i}}/\gamma\omega_{\mathrm{i}}=1$ for all $\beta$.
This is due to $\hat{\mathbf{k}}_{\mathrm{i}}$ and $\mathbf{v}$ being
perpendicular to each other. The factor $1/\gamma$ is necessary to eliminate
the over-exaggeration of $\omega^{\prime}_{\mathrm{i}}$ when $\beta\to 1$ and
$\Theta_{\mathrm{i}}\to\pi$. Without this factor, the other frequency values
would appear too suppressed.

The Lorentz boost $\theta^{\prime}$ of $\theta$ given by Eqn. (14) explains
the perceived change in direction of the beam in $S^{\prime}$ compared to $S$
as shown by $\hat{\mathbf{k}}_{\mathrm{i}}$ and
$\hat{\mathbf{k}}_{\mathrm{i}}^{\prime}$ in Fig. 1. In Fig. 2 a), this is
visualized for the Lorentz boost $\Theta^{\prime}_{\mathrm{i}}$ of the polar
angle of the incident field with respect to the incident angle
$\Theta_{\mathrm{i}}$ as seen in $S$ and speed ratio $\beta$. Moreover, the
Doppler shift $\omega^{\prime}$ of $\omega$ is displayed in Fig. 2 b) with the
same functional dependency. Note that, for an incident angle of
$\Theta_{\mathrm{i}}=0$ and a speed ratio $\beta\to 1$, the Doppler-shifted
frequency becomes zero. This corresponds to the sphere moving away from the
external observer in $S$ at a speed tending to that of light, thus exhibiting
a complete redshift. In other words, the incident wave is perceived by the
sphere to be so stretched out that the frequency disappears in its reference
frame. Conversely, when $\Theta_{\mathrm{i}}=\pi$ and $\beta\to 1$, the wave
is seen to be infinitely blueshifted in $S^{\prime}$, corresponding to a
completely compressed wave with infinite frequency. This corresponds to the
sphere moving towards the source of the incident field.

Note that the same expression for the scaling factor
$\mathcal{C}\left(\beta,\theta\right)$ is given by Eqn. (27) in De Cupis  _et
al._ [34]. Importantly, we observe that for Eqn. (II.1) to be non-zero, the
helicity of the field must remain invariant upon the Lorentz boost
transformation due to the $\delta_{\lambda^{\prime}\lambda}$ term. This
invariance demonstrates the power of expressing the fields in the helicity
instead of the parity basis, that is, specifically making use of circularly-
polarized plane waves instead of TE/TM plane waves.

Generally speaking, the change in direction of the beam upon boosting is given
by the following transformation of the wavevectors:

$\displaystyle\hat{\mathbf{k}}^{\prime}=\frac{\hat{\mathbf{k}}+\left[(\gamma-1)\cos\theta-\gamma\beta\right]\hat{\mathbf{z}}}{\mathcal{C}\left(\beta,\theta\right)}\mbox{\quad.\quad}$
(17)

Finally, putting all the above together, and after some straightforward
algebra, we can get the following relation between the amplitudes of the
initially considered and non-rotated incident beam in $S^{\parallel}$ and the
rotated one in $S^{\prime}$:

$\displaystyle\ket{\mathbf{E}_{\mathrm{i}}^{\prime}}$
$\displaystyle=\hat{\mathbf{LB}}_{z}(\beta)\hat{\mathbf{R}}_{y}(\Theta_{\mathrm{i}})\ket{\mathbf{E}_{\mathrm{i}}^{\parallel}}$
$\displaystyle=\sum_{\lambda^{\prime}}\int_{0}^{2\pi}\mathrm{d}\phi^{\prime}\int^{\pi}_{0}\mathrm{d}\theta^{\prime}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}$
$\displaystyle\hskip 14.22636pt\mathcal{G}^{\prime}_{\lambda^{\prime},\hskip
1.13809pt\mathrm{i}}(\omega^{\prime},\theta^{\prime},\phi^{\prime})\ket{\lambda_{\mathrm{i}}^{\prime}\
\hat{\mathbf{k}}_{\mathrm{i}}^{\prime}\ \omega_{\mathrm{i}}^{\prime}}$
$\displaystyle+\hskip 2.27626pt\mathrm{c.c.}\mbox{\quad,\quad}$ (18)

where $\omega^{\prime}_{\mathrm{i}}$ and
$\hat{\mathbf{k}}^{\prime}_{\mathrm{i}}$ are determined by Eqns. (15) and
(17), respectively and

$\displaystyle\mathcal{G}^{\prime}_{\lambda^{\prime},\hskip
1.13809pt\mathrm{i}}(\omega^{\prime},\theta^{\prime},\phi^{\prime})$
$\displaystyle=\mathcal{J}(\theta^{\prime},\phi^{\prime},\Theta_{\mathrm{i}})$
$\displaystyle\cdot\mathcal{G}^{\parallel}_{\lambda^{\prime},\hskip
1.13809pt\mathrm{i}}\Bigg{\\{}\frac{\omega^{\prime}}{\mathcal{C}\left[\beta,\theta^{\parallel}(\theta^{\prime},\phi^{\prime})\right]},$
$\displaystyle\theta^{\parallel}(\theta^{\prime},\phi^{\prime}),\phi^{\parallel}(\theta^{\prime},\phi^{\prime})\Bigg{\\}}\mbox{\quad,\quad}$
(19)

where $\theta^{\parallel}(\theta^{\prime},\phi^{\prime})$ and
$\phi^{\parallel}(\theta^{\prime},\phi^{\prime})$ express $\theta^{\parallel}$
and $\phi^{\parallel}$ as viewed from $S^{\prime}$:

$\displaystyle\theta^{\parallel}(\theta^{\prime},\phi^{\prime})$
$\displaystyle=\arccos\\{\frac{1}{{\gamma(1+\beta\cos\theta^{\prime})}}$
$\displaystyle\cdot\Big{[}\sin\theta^{\prime}\cos\phi^{\prime}\sin\Theta$
$\displaystyle+\gamma(\cos\theta^{\prime}+\beta)\cos\Theta\Big{]}\Bigg{\\}},$
(20) $\displaystyle\phi^{\parallel}(\theta^{\prime},\phi^{\prime})$
$\displaystyle=\mathrm{atan2}[\sin\theta^{\prime}\sin\phi^{\prime},$
$\displaystyle\hskip 11.38092pt\sin\theta^{\prime}\cos\phi^{\prime}\cos\Theta$
$\displaystyle-\gamma(\cos\theta^{\prime}+\beta)\sin\Theta]\mbox{\quad,\quad}$
(21)

with the Jacobian

$\displaystyle\mathcal{J}(\theta^{\prime},\phi^{\prime},\Theta_{\mathrm{i}})$
$\displaystyle=\begin{vmatrix}[1.8]\frac{\partial\theta^{\parallel}}{\partial\theta^{\prime}}&\frac{\partial\theta^{\parallel}}{\partial\phi^{\prime}}\\\
\frac{\partial\phi^{\parallel}}{\partial\theta^{\prime}}&\frac{\partial\phi^{\parallel}}{\partial\phi^{\prime}}\end{vmatrix}$
(22)

that converts $\mathrm{d}\phi^{\parallel}\mathrm{d}\theta^{\parallel}$ to
$\mathrm{d}\phi^{\prime}\mathrm{d}\theta^{\prime}$.

Recall that we do this since, for simplicity, we wish to carry out the
scattering calculation in $S^{\prime}$, that is, where it is equivalent to
that in the stationary case. For a given incident field described by Eqn.
(II.1), we can use Eqn. (II.1) to calculate the amplitudes that are needed to
describe the incident field in $S^{\prime}$ in terms of the plane wave
representation given by Eqn. (II.1).

###  II.2 Solving the scattering problem in the sphere’s reference frame

The approach taken to calculate the amplitude of the scattered field begins by
expressing the incident field $\ket{\mathbf{E}_{\mathrm{i}}^{\prime}}$ as a
series of spherical waves with respect to the coordinates describing
$S^{\prime}$ [12]:

$\ket{\mathbf{E}_{\mathrm{i}}^{\prime}}=\sum_{\lambda^{\prime}\ell^{\prime}m^{\prime}}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}\mathcal{A^{\prime}}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})\ket{\omega^{\prime}\hskip
2.0pt\lambda^{\prime}\hskip 2.0pt\ell^{\prime}\hskip
2.0ptm^{\prime}}^{(1)}+\mathrm{c.c.}\mbox{\quad,\quad}$ (23)

where $\ket{\omega^{\prime}\hskip 2.0pt\lambda^{\prime}\hskip
2.0pt\ell^{\prime}\hskip 2.0ptm^{\prime}}^{(1)}$ signifies a regular VSH
attached to $S^{\prime}$ with frequency $\omega^{\prime}$, helicity
$\lambda^{\prime}$, multipolar index $\ell^{\prime}$ ($\ell^{\prime}=1$
corresponds to dipoles, $\ell^{\prime}=2$ corresponds to quadrupoles etc.),
and angular momentum along the $z$-axis
$m^{\prime}=-\ell^{\prime},-(\ell^{\prime}-1),\hskip 2.84526pt...\hskip
2.84526pt,\ell^{\prime}$. The ${(1)}$ superscript denotes that the VSH
corresponds to a first-order spherical Bessel function
$j_{\ell^{\prime}}(k^{\prime}r^{\prime})$, and the coefficients of the
expansion are given by

$\displaystyle\mathcal{A^{\prime}}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}\phi^{\prime}\int^{\pi}_{0}\mathrm{d}\theta^{\prime}\hskip
2.0pt\mathcal{G}^{\prime}_{\lambda^{\prime},\hskip
1.13809pt\mathrm{i}}(\omega^{\prime},\theta^{\prime},\phi^{\prime})$
$\displaystyle\cdot\mathcal{S}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime},\hat{\mathbf{k}}^{\prime})\mbox{\quad,\quad}$
(24)

where the transformation coefficients between the plane waves and the
spherical waves (under which transformation the helicity and frequency of the
waves remain unchanged) are given by:

$\displaystyle\mathcal{S}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime},\hat{\mathbf{k}}^{\prime})$
$\displaystyle=\hskip 1.0pt4\pi\mathrm{i}^{\ell^{\prime}+2m^{\prime}+1}\hskip
1.0pt\Omega_{\ell^{\prime}m^{\prime}}$
$\displaystyle\cdot\tau^{(\lambda^{\prime})}_{\ell^{\prime}m^{\prime}}[\mathrm{\theta}^{\prime}(\hat{\mathbf{k}}^{\prime})]\hskip
1.0pt\mathrm{e}^{-\mathrm{i}m^{\prime}\phi^{\prime}(\hat{\mathbf{k}}^{\prime})}\mbox{\quad,\quad}$
(25)

where $\Omega_{\ell^{\prime}m^{\prime}}$ is a normalization constant and
$\tau^{(\lambda^{\prime})}_{\ell^{\prime}m^{\prime}}[\mathrm{\theta}^{\prime}(\hat{\mathbf{k}}^{\prime})]$
is a function which we define in App. B. The expression given in Eqn. (II.2)
is derived by applying Eqn. (2) to Eqn. (12) in Lamprianidis and
Miroshnichenko [35].

For the case of monochromatic excitation (in $S$), like the one we consider
here, we have that

$\displaystyle\mathcal{G}^{\parallel}_{\lambda^{\parallel},\hskip
1.13809pt\mathrm{i}}(\omega^{\parallel},\theta^{\parallel},\phi^{\parallel})=\mathcal{G}^{\parallel,0}_{\lambda^{\parallel},\hskip
1.13809pt\mathrm{i}}(\theta^{\parallel},\phi^{\parallel})\delta(\omega^{\parallel}-\omega_{\mathrm{i}})\mbox{\quad.\quad}$
(26)

Using this expression, we get the following simplified expression for the
incident spherical amplitudes in $S^{\prime}$:

$\displaystyle\mathcal{A^{\prime}}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})$
$\displaystyle=-4\pi i^{\ell^{\prime}+2m^{\prime}+1}\hskip
1.0pt\Omega_{\ell^{\prime}m^{\prime}}\tau^{(\lambda^{\prime})}_{\ell^{\prime}m^{\prime}}(\theta^{\prime}_{0})$
$\displaystyle\cdot\delta\left(\omega^{\prime}\in\left[\frac{\omega_{\mathrm{i}}}{\gamma(1+\beta)},\frac{\omega_{\mathrm{i}}}{\gamma(1-\beta)}\right]\right)$
$\displaystyle\cdot\frac{(1+\beta\cos{\theta^{\prime}_{0}})}{\beta\sin{\theta^{\prime}_{0}}}$
$\displaystyle\cdot\int_{0}^{2\pi}\mathrm{d}\phi^{\prime}\hskip
1.0pt\mathrm{e}^{-\mathrm{i}m^{\prime}\phi^{\prime}}$
$\displaystyle\cdot\mathcal{P}\left[\theta^{\parallel}(\theta^{\prime}_{0},\phi^{\prime}),\phi^{\parallel}(\theta^{\prime}_{0},\phi^{\prime}),\Theta_{\mathrm{i}}\right]$
$\displaystyle\cdot\mathcal{J}(\theta^{\prime}_{0},\phi^{\prime},\Theta_{\mathrm{i}})\mathcal{G}^{\parallel,0}_{\lambda^{\prime},\hskip
1.13809pt\mathrm{i}}[\theta^{\parallel}(\theta^{\prime}_{0},\phi^{\prime}),\phi^{\parallel}]\mbox{\quad,\quad}$
(27)

where:

$\displaystyle\theta^{\prime}_{0}=\arccos{\left(\frac{\gamma\omega_{\mathrm{i}}-\beta^{2}\gamma\omega_{\mathrm{i}}-\omega^{\prime}}{\beta\omega^{\prime}}\right)}\mbox{\quad.\quad}$
(28)

Next, in conjunction with Step 2 of the FHM, we need to express the scattered
field $\ket{\mathbf{E}_{\mathrm{s}}^{\prime}}$ in a series of radiating VSHs
in $S^{\prime}$, denoted as $\ket{\omega^{\prime}\hskip
2.0pt\lambda^{\prime}\hskip 2.0pt\ell^{\prime}\hskip 2.0ptm^{\prime}}^{(3)}$.
Analogous to Eqn. (23), this can be written as

$\ket{\mathbf{E}_{\mathrm{s}}^{\prime}}=\sum_{\lambda^{\prime}\ell^{\prime}m^{\prime}}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}\mathcal{B}^{\prime}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})\ket{\omega^{\prime}\hskip
2.0pt\lambda^{\prime}\hskip 2.0pt\ell^{\prime}\hskip
2.0ptm^{\prime}}^{(3)}+\mathrm{c.c.}\mbox{\quad,\quad}$ (29)

where the ${(3)}$ superscript denotes that the VSHs correspond to a third-
order spherical Bessel (Hankel) function
$h_{\ell^{\prime}}(k^{\prime}r^{\prime})$. Specific expressions of the
radiating and regular VSHs are given in App. B. Moreover, the ‘s’ subscript
denotes quantities which correspond to the scattered field.

Finally, the scattering coefficients
$\mathcal{B^{\prime}}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})$
can be related to the incident coefficients
$\mathcal{A^{\prime}}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})$
by way of the T-matrix formalism [12]:

$\mathbf{B}^{\prime}=\mathbf{T}^{\text{H}}\mathbf{A}^{\prime}\mbox{\quad,\quad}$
(30)

where $\mathbf{A}^{\prime}$ and $\mathbf{B}^{\prime}$ are vectors containing
the incident and scattering coefficients, respectively, and
$\mathbf{T}^{\text{H}}$ is the corresponding T-matrix expressed in the
helicity basis (see App. C). The T-matrix fully describes the scattering
response of the individual scatterer in the stationary case, which we can
safely use in the rest frame of the sphere. Let us note that the time-
invariance of the stationary system implies a matrix that is diagonal with
respect to frequency $\omega^{\prime}$, whereas duality-symmetry implies a
diagonal matrix with respect to helicity $\lambda^{\prime}$, and the spherical
symmetry of the scatterer implies a diagonal matrix with respect to the
multipolar indices $\ell^{\prime},m^{\prime}$.

Specifically, for a spherical scatterer, we can write

$\displaystyle\mathcal{B}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})=\sum_{\lambda_{0}}\mathrm{T}_{\lambda^{\prime}\lambda_{0},\ell^{\prime}}(\omega^{\prime})\mathcal{A}_{\lambda_{0}\ell^{\prime}m^{\prime}}(\omega^{\prime})\mbox{\quad,\quad}$
(31)

where $\lambda_{0}=\pm 1$ is a dummy index representing helicity and the term
$\mathrm{T}_{\lambda^{\prime}\lambda_{0},\ell^{\prime}}$ is defined at the end
of App. C. Moreover, in this work, we will make the assumption that the
T-matrix of the scatterer is non-dispersive, _i.e._ , invariant with respect
to frequency. This assumption is logical as long as we are exciting with a
monochromatic beam with a narrow angular spectrum, _i.e._ , a large waist. One
must consider this, since the plane-wave components of the beam all Doppler-
shift differently depending on their polar angles of propagation. However, a
small angular width in $S$ minimizes this difference, thus allowing us to
assume a non-dispersive T-matrix in $S^{\prime}$. As we will see, this
assumption significantly simplifies the final equations used for numerical
computation.

Finally, we require an expression for the electric field in the far-field
region of $S^{\prime}$. For this, we need to use the following asymptotic
expression for the radiating spherical waves:

$\displaystyle\lim_{\omega^{\prime}r^{\prime}/c\rightarrow\infty}\ket{\omega^{\prime}\hskip
2.0pt\lambda^{\prime}\hskip 2.0ptm^{\prime}\hskip 2.0pt\ell^{\prime}}^{(3)}$
$\displaystyle\equiv(-\mathrm{i})^{\ell^{\prime}}\hskip
3.0pt\mathbf{f}_{\lambda^{\prime},\ell^{\prime}m^{\prime}}(\hat{\mathbf{r}}^{\prime})\hskip
3.0pt$
$\displaystyle\cdot\frac{\mathrm{e}^{\mathrm{i}\omega^{\prime}(r^{\prime}/c-t^{\prime})}}{\omega^{\prime}r^{\prime}/c}\mbox{\quad,\quad}$
(32)

which, from Eqn. (29), readily gives the following expression for the electric
field in the far-field region of $S^{\prime}$:

$\displaystyle\mathbf{E}^{{}^{\prime}\mathrm{ff}}_{\mathrm{s}}(\mathbf{r}^{\prime},t^{\prime})$
$\displaystyle=\sum_{\lambda^{\prime}\ell^{\prime}m^{\prime}}\int^{+\infty}_{0^{+}}\mathrm{d}\omega^{\prime}\mathcal{B^{\prime}}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})(-\mathrm{i})^{\ell^{\prime}}$
$\displaystyle\cdot\mathbf{f}_{\lambda^{\prime},\ell^{\prime}m^{\prime}}(\hat{\mathbf{r}}^{\prime})\hskip
3.0pt\frac{\mathrm{e}^{\mathrm{i}\omega^{\prime}(r^{\prime}/c-t^{\prime})}}{\omega^{\prime}r^{\prime}/c}+\mathrm{c.c.}\mbox{\quad,\quad}$
(33)

where
$\mathbf{f}_{\lambda^{\prime},\ell^{\prime}m^{\prime}}(\hat{\mathbf{r}}^{\prime})$
is a vector function defined in App. B.

As shown in Garner  _et al._ [36], the angular density of the total radiation
energy flux in a given direction in $S^{\prime}$ specified by
$\theta^{\prime}$ and $\phi^{\prime}$, which we denote as
$U^{\prime}(\theta^{\prime},\phi^{\prime})$, is calculated by integrating the
amplitude of the electric field
$\mathbf{E}^{\prime\mathrm{ff}}_{\mathrm{s}}(\mathbf{r}^{\prime},t^{\prime})$
in the far-field limit. As a result, we have

$\displaystyle
U^{\prime}(\theta^{\prime},\phi^{\prime})=\lim_{r^{\prime}\to\infty}\int^{\infty}_{-\infty}(r^{\prime})^{2}\frac{|\mathbf{E}^{\prime\mathrm{ff}}_{\mathrm{s}}(\mathbf{r}^{\prime},t^{\prime})|^{2}}{\eta_{0}}\mathrm{d}t^{\prime}\mbox{\quad,\quad}$
(34)

where $\eta_{0}$ is the impedance of free space. An expanded, and numerically-
efficient form of Eqn. (34) is given in App. D.

At this point, the second step of the FHM is complete.

###  II.3 Solution to the scattering problem in the lab frame

To investigate the back-scattering, we analyse the directivity
$D(\theta,\phi)$ of the sphere. This is defined as [37]

$\displaystyle
D(\theta,\phi)=\frac{U(\theta,\phi)}{W_{\mathrm{tot}}/4\pi}\mbox{\quad,\quad}$
(35)

where
$U(\theta,\phi)=\sum_{\lambda_{\mathrm{s}}}U_{\lambda_{\mathrm{s}}}(\theta,\phi)$
is the angular density of the total radiation energy in a given direction in
$S$ specified by $\theta$ and $\phi$, $U_{\lambda_{\mathrm{s}}}$ is the
component of $U(\theta,\phi)$ corresponding to the scattered helicity
$\lambda_{\mathrm{s}}=\pm 1$, and $W_{\mathrm{tot}}$ is the total scattered
energy.

Considering the directivity of the sphere allows us to obtain a physically
meaningful and intuitive formulation from which the behaviour of the back-
scattering can be interpreted. Qualitatively speaking, the directivity is the
ratio of the total angular energy $U(\theta,\phi)$ to the average scattered
energy $W_{\mathrm{tot}}/4\pi$ by an analogous isotropic scatterer.
Consequently, a directivity $>$1 means that the contribution of back-scattered
energy outweighs that of the average scattered energy. Conversely, a
directivity $<$1 implies that the back-scattered energy is lower than the
average energy scattered by the sphere.

We are now in a position to carry out the final step of the FHM, that is,
transforming the directivity from $S^{\prime}$ back to $S$. The power of the
FHM really becomes apparent here, since the angular energy $U(\theta,\phi)$ in
$S$ can easily be related to quantities in $S^{\prime}$. More specifically, we
have

$\displaystyle W_{\mathrm{tot}}$
$\displaystyle=\int^{2\pi}_{0}\int^{\pi}_{0}U(\theta,\phi)\sin{\theta}\mathrm{d}\theta\mathrm{d}\phi\mbox{\quad,\quad}$
(36)

where, as we see from Eqn. (21) in Garner  _et al._ [36],

$\displaystyle
U(\theta,\phi)=[\gamma\left(1+\beta{\cos\theta^{\prime}}\right)]^{3}\hskip
2.84544ptU^{\prime}(\theta^{\prime},\phi^{\prime})\mbox{\quad,\quad}$ (37)

where $\theta^{\prime}$ and $\phi^{\prime}$ can be transformed using Eqns.
(14) and (16) to obtain an expression for $U(\theta,\phi)$ in $S$. For back-
scattering, we have $\theta=\Theta_{\mathrm{BS}}$ and
$\phi=\Phi_{\mathrm{BS}}$, where

$\displaystyle\Theta_{\mathrm{BS}}=\pi-\Theta_{\mathrm{i}}\quad\text{and}\quad\Phi_{\mathrm{BS}}=\pi\mbox{\quad,\quad}$
(38)

respectively.

The third step of the FHM is now complete, and the back-scattered directivity
$D_{\mathrm{BS}}$ of the sphere in $S$ can be calculated by substituting Eqns.
(36) and (37) into Eqn. (35) such that

$\displaystyle
D_{\mathrm{BS}}=D(\Theta_{\mathrm{BS}},\Phi_{\mathrm{BS}})\mbox{\quad.\quad}$
(39)

##  III Relativistic Kerker condition

###  III.1 Visualizing the directivity

The final theoretical result of our work has been formulated in Eqn. (39),
which expresses the contribution of the back-scattered energy compared to the
average scattered energy for a given scenario. While Eqn. (35) is more
general, for the sake of this discussion, we choose to investigate a possible
suppression of the back-scattering (that is, the first Kerker condition) [38,
39, 40].

As an example, we consider a lossless dielectric sphere and parametrize the
multipolar response using what are known as Mie angles [41, 42] (cf. App. E).
Each Mie angle is bounded between $-\nicefrac{{\pi}}{{2}}$ and
$\nicefrac{{\pi}}{{2}}$, and we consider them to be non-dispersive. A value of
zero corresponds to a resonance of the respective Mie coefficient.

Since the objective function describing $D_{\mathrm{BS}}$ does not have a
clear analytical solution, we implement numerical routines to identify
properties for the sphere which minimize the back-scattering. More
specifically, we wish to seek the optimized combination of Mie angles such
that the back-scattering is minimized. Moreover, we carry out the optimization
when $\beta$ and $\Theta_{\mathrm{i}}$ are fixed values; as an example we
consider $\beta=0.2$ and $\Theta_{\mathrm{i}}=\frac{\pi}{4}$.

Figure 3: Figures a) and b) represent the $\lambda_{\mathrm{s}}=+1$ and
$\lambda_{\mathrm{s}}=-1$ components of $\mathrm{log}_{10}D_{\mathrm{BS}}$,
respectively when $\theta_{\mathrm{ED}}=\theta_{\mathrm{MD}}=\frac{\pi}{3}$.
Note in b) the dark diagonal line, which means that the back-scattering
vanishes when $\theta_{\mathrm{EQ}}=\theta_{\mathrm{MQ}}$. This corresponds to
the case where the sphere is a dual scatterer. Since the incident helicity is
given by $\lambda_{\mathrm{i}}=+1$, these vanishing components are to be
expected. Figure c) shows the total directivity
$\mathrm{log}_{10}D_{\mathrm{BS}}$ when combining Figs. a) and b), while Fig.
d) shows for comparison $\mathrm{log}_{10}D_{\mathrm{BS}}$ when
$\theta_{\mathrm{ED}}=\frac{\pi}{9}$ and
$\theta_{\mathrm{MD}}=-\frac{\pi}{4}$. In all cases we have that $\beta=0.2$
and $\Theta_{\mathrm{i}}=\frac{\pi}{4}$.

Before doing this, it makes sense to visualize how $D_{\mathrm{BS}}$ varies
with respect to some chosen Mie angles. For this purpose, we sweep across the
possible electric quadrupole ($\theta_{\text{EQ}}$) and magnetic quadrupole
($\theta_{\text{MQ}}$) angles while fixing the dipole angles
$\theta_{\text{ED}}$ and $\theta_{\text{MD}}$.

We visualize the $\lambda_{\mathrm{s}}=+1$ and $\lambda_{\mathrm{s}}=-1$
components of $\mathrm{log}_{10}D_{\mathrm{BS}}$ with
$\theta_{\mathrm{ED}}=\theta_{\mathrm{MD}}=\pi/3$ in Fig. 3 a) and Fig. 3 b),
respectively, followed by the corresponding total directivity in Fig. 3 c). In
all cases, the helicity of the incident field is $\lambda_{\mathrm{i}}=1$. One
observes in Fig. 3 b) that the diagonal representing
$\theta_{\mathrm{EQ}}=\theta_{\mathrm{MQ}}$, that is, when the sphere is a
dual scatterer, displays values of below $-30$, implying that the back-
scattering is zero at these points. This is expected since the incoming
helicity is given by $\lambda_{\mathrm{i}}=+1$ and the scattered helicity in
the dual case must remain the same, so we are only left with non-zero values
when $\lambda_{\mathrm{s}}=+1$. For comparison, Fig. 3 d) shows the total
directivity where $\theta_{\mathrm{ED}}=\pi/9$ and
$\theta_{\mathrm{MD}}=-\pi/4$.

###  III.2 Numerically minimizing the back-scattering

The fact that Fig. 3 b) provides a physically known result is a justification
of the numerical implementation and allows us to proceed in minimizing the
directivity. A further verification is the fact that the directivity is
independent of the incident helicity $\lambda_{\mathrm{i}}$. This is expected
since the system is both rotationally and mirror-symmetric about the $x$-axis.
From the mirror symmetry, $\lambda_{\mathrm{i}}$ would flip sign [43].
However, when rotated to the same position, $\lambda_{\mathrm{i}}$ would
preserve its sign. These situations both describe the same physical scenario,
that is, a system where the sphere moves in the opposite direction with
$\mathbf{v}\to-v\hat{\mathbf{z}}$ and
$\Theta_{\mathrm{i}}\to\pi+\Theta_{\mathrm{i}}$. Therefore, the scattered
energy remains unchanged.

To minimize the back-scattering with respect to the Mie angles, we implement
the directivity calculation using the Julia programming language [29] and
leverage the automatic differentiation capabilities included in the modelling
toolkit JuMP [30] for gradient-based optimization. This enables us to
efficiently take derivatives of $D_{\mathrm{BS}}$ with respect to Mie angles
up to arbitrary order. We then formulate our optimization problem as the
minimization of $D_{\mathrm{BS}}$ using the interior-point optimizer IPOPT
[44].

Using this method, we find higher order combinations of Mie angles which yield
minima below our defined cut-off point of $D_{\mathrm{C}}=10^{-3}$, that is,
the value below which we consider the back-scattering to be negligible. We
consider this value appropriate, since it corresponds to a back-scattered
energy which contributes a mere $0.1\text{\,}\mathrm{\char 37\relax}$ to the
average scattered energy. For a single optimization run, we randomly
initialize a set of Mie angles between
$\quantity(-\nicefrac{{\pi}}{{2}},\nicefrac{{\pi}}{{2}})$ and minimize
$D_{\mathrm{BS}}$ with respect to these angles. Owing to quasi-analytical
gradients, the optimization quickly converges to high-quality minima one order
of magnitude lower than $D_{\mathrm{C}}$. Finding a single set of Mie angles
up to octupolar order takes less than a second on average (measured over 100
optimization runs on Intel Xeon Platinum 8368 CPU @
$2.4\text{\,}\mathrm{GHz}$). A selection of possible combinations up to
octupolar order is given in Table 1.

Table 1: Examples of optimized Mie angles for minimum back-scattering up to octupolar order. $\theta_{\text{ED}}$ | $\theta_{\text{MD}}$ | $\theta_{\text{EQ}}$ | $\theta_{\text{MQ}}$ | $\theta_{\text{EO}}$ | $\theta_{\text{MO}}$ | $D_{\mathrm{BS}}\times 10^{-4}$
---|---|---|---|---|---|---
$-0.18$ | $1.21$ | $1.38$ | $1.23$ | $1.54$ | $1.55$ | $1.57$
$-1.39$ | $-1.32$ | $-1.12$ | $-1.44$ | $-1.51$ | $-1.57$ | $1.95$
$-1.32$ | $-1.32$ | $-1.21$ | $-1.21$ | $-1.53$ | $-1.53$ | $2.02$

These minimized values of $D_{\mathrm{BS}}$ are all smaller than
$D_{\mathrm{T}}=10^{-3}$ and thus satisfy our cut-off criterion, providing
evidence of the existence of the first Kerker condition in the relativistic
regime. The most pronounced minimum located ($D_{\mathrm{BS}}=1.57\times
10^{-4}$) describes a case where the back-scattered energy contributes a
negligible $0.016\text{\,}\mathrm{\char 37\relax}$ to the average scattered
energy. It must be emphasized that there exist many combinations of Mie angles
that fulfil this condition, making those in Table 1 just few of many.

##  IV Conclusion

The main goal of this paper was to demonstrate the utility of expressing
incident and scattered fields in the helicity basis for the case of a sphere
moving at a relativistic speed. In doing this, we were able to transform fewer
variables (namely the helicity) as opposed to the same problem in the parity
basis, thus greatly simplifying calculations. Moreover, we obtained an
expression for the back-scattering amplitude of the scattered field observed
from an external lab frame in the form of the directivity of the sphere.
Finally, the directivity was minimized with respect to Mie angles, providing
evidence for the existence of the first Kerker condition in the relativistic
regime.

Opportunities for future work are plentiful. Since our implementation can be
differentiated with respect to all continuous parameters of the considered
system, it is, in principle, possible to optimize not only for the Mie angles
but also for parameters beyond the scope of this work, such as incident angle
and velocity. Furthermore, the current work could be extended to analyze a
composition of particles describing a metasurface as opposed to just a single
particle. In this case, a cluster T-matrix as described in Mishchenko  _et
al._ [12] would need to be implemented. The motivation for considering a
surface of particles links to the future applications of light sails proposed
by the Breakthrough Starshot Initiative.

Of course, for this to be done, the current model would have to be refined to
consider the opposite case of maximum back-scattering, resulting in the ideal
case of maximum momentum transfer to the sail [45].

## Data availability

The code used to produce the results in this manuscript can be accessed via
the following link: https://github.com/tfp-photonics/Jorkle.jl.

###### Acknowledgements.

M.R.W., A.G.L., and C.R. acknowledge support from the Max Planck School of
Photonics, which is supported by BMBF, Max Planck Society, and the Fraunhofer
Society. M.R.W. and A.G.L. also acknowledge support from the Karlsruhe School
of Optics and Photonics (KSOP). Y.A. and C.R. acknowledge support by the
German Research Foundation within the Excellence Cluster 3D Matter Made to
Order (EXC 2082/1 under project number - 390761711) and by the Carl Zeiss
Foundation. The optimizations in Section III were carried out on the HoreKa
supercomputer funded by the Ministry of Science, Research and the Arts Baden-
Württemberg and by the Federal Ministry of Education and Research.

##  Appendix A Lorentz boost of helical plane waves

As stated in Garner  _et al._ [33], the Lorentz boost of the electric field
$\mathbf{E}(\mathbf{r},t)$ is given by

$\displaystyle\mathbf{E}^{\prime}(\mathbf{r}^{\prime},t^{\prime})$
$\displaystyle=\gamma\left[\mathbf{E}(\mathbf{r},t)+v\hat{\mathbf{v}}\times\mathbf{B}(\mathbf{r},t)\right]$
$\displaystyle+(1-\gamma)\left[\hat{\mathbf{v}}\cdot\mathbf{E}(\mathbf{r},t)\right]\hat{\mathbf{v}}\mbox{\quad,\quad}$
(40)

where $\mathbf{B}(\mathbf{r},t)$ is the corresponding magnetic field and
$\gamma=1/\sqrt{1-\beta^{2}}$ with $\beta=v/c$ being the ratio of the boosting
speed to the speed of light and boosting takes place along the direction
$\hat{\mathbf{v}}$. For boosting along $+\hat{\mathbf{z}}$, the coordinates of
the primed (boosted) and unprimed coordinate systems are related with the
following formulas:

$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle x$ (41)
$\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle y$ (42)
$\displaystyle z^{\prime}$ $\displaystyle=$ $\displaystyle\gamma(z-\beta ct)$
(43) $\displaystyle t^{\prime}$ $\displaystyle=$ $\displaystyle\gamma(t-\beta
z/c)\mbox{\quad.\quad}$ (44)

Here, we want to consider the boost of a monochromatic plane wave of well-
defined helicity $\lambda$. Specifically, its electric field is given by
$\mathbf{E}(\mathbf{r},t)=\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{k}})\exp\\{{\mathrm{i}\omega[(\hat{\mathbf{k}}\cdot\mathbf{r}/c)-t}]\\}$
(see main text). By making use of the above coordinate transformations we can
get the following transformation of the exponent:
$\\{{\mathrm{i}\omega[(\hat{\mathbf{k}}\cdot\mathbf{r}/c)-t}]\\}=\\{{\mathrm{i}\omega^{\prime}[(\hat{\mathbf{k}}^{\prime}\cdot\mathbf{r}^{\prime}/c)-t^{\prime}}]\\}$,
with $\omega^{\prime},\hat{\mathbf{k}}^{\prime}$ being given by Eqs. (15,17)
of the main text. Thus, we have transformed the scalar part of the fields,
which gave us the transformed frequency and direction of propagation of the
boosted plane wave, $\omega^{\prime},\hat{\mathbf{k}}^{\prime}$, respectively.

Next, we need to transform the polarization vector. For this we need to take
into account that our considered plane wave, being an eigenstate of the
helicity operator $\frac{\nabla\times}{k}$ with well-defined helicity
$\lambda$ [46], has the following property:

$\displaystyle\nabla\times\mathbf{E}(\mathbf{r},t)$ $\displaystyle=\lambda
k\mathbf{E}(\mathbf{r},t)\mbox{\quad,\quad}$ (45)

and, therefore, we get the following for its corresponding magnetic field from
Maxwell’s equations:

$\displaystyle\mathbf{B}(\mathbf{r},t)$
$\displaystyle=\frac{\lambda}{\mathrm{i}c}\mathbf{E}(\mathbf{r},t)\mbox{\quad.\quad}$
(46)

Therefore, by substituting the right-hand-side of Eqn. (46) into Eqn. (A), we
find that

$\displaystyle\mathbf{E}^{\prime}(\mathbf{r}^{\prime},t^{\prime})$
$\displaystyle=\gamma\left[\mathbf{E}(\mathbf{r},t)+\frac{\lambda
v}{\mathrm{i}c}\hat{\mathbf{z}}\times\mathbf{E}(\mathbf{r},t)\right]$
$\displaystyle+(1-\gamma)\left[\hat{\mathbf{z}}\cdot\mathbf{E}(\mathbf{r},t)\right]\hat{\mathbf{z}}\mbox{\quad.\quad}$
(47)

What remains then, is to project the boosted helical polarization vector
$\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{k}})$ onto the boosted polarization
basis $\hat{\mathbf{e}}_{\lambda^{\prime}}(\hat{\mathbf{k}}^{\prime})$. That
is to say, we need to find the coefficients
$\mathcal{E}_{\lambda\lambda^{\prime}}(\beta,\hat{\mathbf{k}})$ in the
expansion bellow:

$\displaystyle\gamma\left[\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{k}})+\frac{\lambda
v}{\mathrm{i}c}\hat{\mathbf{z}}\times\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{k}})\right]$
$\displaystyle+(1-\gamma)\left[\hat{\mathbf{z}}\cdot\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{k}})\right]\hat{\mathbf{z}}$
$\displaystyle=\sum_{\lambda^{\prime}}\mathcal{E}_{\lambda\lambda^{\prime}}(\beta,\hat{\mathbf{k}})\hat{\mathbf{e}}_{\lambda^{\prime}}(\hat{\mathbf{k}}^{\prime})\mbox{\quad.\quad}$
(48)

By making use of the following orthogonality relation [43]:

$\hat{\mathbf{e}}_{\lambda^{\prime}}(\mathbf{k}^{\prime})\cdot\hat{\mathbf{e}}_{-\lambda^{\prime}_{0}}(\mathbf{k}^{\prime})=-\delta_{\lambda^{\prime}\lambda^{\prime}_{0}}\mbox{\quad,\quad}$
(49)

we readily get after some algebra that
$\mathcal{E}_{\lambda\lambda^{\prime}}(\beta,\hat{\mathbf{k}})=\delta_{\lambda\lambda^{\prime}}\mathcal{C}_{\lambda}(\beta,\theta)$,
with $\theta$ being the polar angle of the propagation direction
$\hat{\mathbf{k}}$ and $\mathcal{C}_{\lambda}(\beta,\theta)$ being given by:

$\displaystyle\mathcal{C}_{\lambda}(\beta,\theta)$
$\displaystyle=\gamma\left[1-\beta\cos\theta\right]\mbox{\quad.\quad}$ (50)

The same expression calculated using the parity basis is given by Eqn. (27) in
De Cupis  _et al._ [34]. Note that the helicity $\lambda$ of massless
particles (and hence, elecromagnetic fields) is invariant under Lorentz boosts
[47]. Finally, summing up all the above, we get the Lorentz boost
transformation given by Eq. (II.1) in the main text.

##  Appendix B Vector spherical harmonics of well-defined helicity

We begin by using the following definition of the spherical harmonics:

$\displaystyle\mathrm{Y}_{\ell}^{m}(\theta,\phi)$ $\displaystyle\triangleq$
$\displaystyle\Omega_{\ell
m}P_{\ell}^{m}(\mathrm{cos}\theta)\mathrm{e}^{\mathrm{i}m\phi}\mbox{\quad,\quad}$
(51)

where $P_{\ell}^{m}(\mathrm{cos}\theta)$ is the associated Legendre function
of the 1st kind, with $\Omega_{\ell
m}\triangleq\mathrm{i}^{m}\sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi\ell(\ell+1)(\ell+m)!}}$
being the corresponding normalization factor.

Then, the VSHs of well-defined parity, $\mathbf{M}_{\ell mk}^{(j)}$ and
$\mathbf{N}_{\ell mk}^{(j)}$, are defined as follows [48]:

$\displaystyle\mathbf{M}_{\ell mk}^{(j)}\left(\mathbf{r}\right)$
$\displaystyle\triangleq\nabla\times\left[\mathbf{r}z^{(j)}_{\mathrm{M},\ell}(kr)\mathrm{Y}_{\ell}^{m}(\theta,\phi)\right]$
$\displaystyle=\mathrm{i}z_{\mathrm{M},\ell}^{(j)}(kr)\mathbf{m}_{\ell
m}(\hat{\mathbf{r}}),$ (52) $\displaystyle\mathbf{N}_{\ell
mk}^{(j)}\left(\mathbf{r}\right)$
$\displaystyle\triangleq\frac{1}{k}\nabla\times\mathbf{M}_{\ell
mk}^{(j)}\left(\mathbf{r}\right)$
$\displaystyle=\hat{\mathbf{r}}\frac{\ell(\ell+1)}{\mathrm{k_{0}r}}z_{\mathrm{M},\ell}^{(j)}(kr)\mathrm{Y}_{\ell}^{m}(\theta,\phi)$
$\displaystyle+z_{\mathrm{N},\ell}^{(j)}(kr)\mathbf{n}_{\ell
m}(\hat{\mathbf{r}})\mbox{\quad,\quad}$ (53)

where

$\displaystyle\mathbf{m}_{\ell m}(\hat{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\Omega_{\ell m}\left[\hat{\theta}\tau_{\ell
m}(\theta)+\mathrm{i}\hat{\phi}\tau_{\ell
m}^{\prime}(\theta)\right]\mathrm{e}^{\mathrm{i}m\phi},$ (54)
$\displaystyle\mathbf{n}_{\ell m}(\hat{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\Omega_{\ell m}\left[\hat{\theta}\tau_{\ell
m}^{\prime}(\theta)+\mathrm{i}\hat{\phi}\tau_{\ell
m}(\theta)\right]\mathrm{e}^{\mathrm{i}m\phi}\mbox{\quad.\quad}$ (55)

The index $\ell$ stands for the angular momentum quantum number that takes the
values 1,2,$\dots$ and corresponds to dipoles, quadrupoles, etc., and the
index $m$ stands for the angular momentum along the $z$-axis which takes the
values $-\ell,...,-2,-1,0,1,2,...,\ell$. The superscript $j$ refers to the
corresponding Bessel ($j=1$) and Hankel ($j=3$) functions,
$z_{\mathrm{M},\ell}^{(j)}(kr)$, of the first kind. The functions
$z_{\mathrm{N},\ell}^{(j)}(kr)\triangleq\frac{1}{kr}\frac{\mathrm{d}}{\mathrm{d}(kr)}[krz_{\mathrm{M},\ell}^{(j)}(kr)]$
are the corresponding Riccati functions and $\tau_{\ell m}(\theta)\triangleq
m\frac{P_{\ell}^{m}(\mathrm{cos}\theta)}{\mathrm{sin}\theta}$ and $\tau_{\ell
m}^{\prime}(\theta)\triangleq\frac{\mathrm{d}P_{\ell}^{m}(\mathrm{cos}\theta)}{\mathrm{d}\theta}$
are the generalized Legendre functions.

The VSHs $\mathbf{\Lambda}^{(j)}_{\lambda,\ell mk}(\mathbf{r})$ of well-
defined helicity $\lambda=\pm 1$ are defined with respect to the VSHs of well-
defined parity according to the formula:

$\displaystyle\mathbf{\Lambda}^{(j)}_{\lambda,\ell mk}(\mathbf{r})$
$\displaystyle=\frac{\mathbf{M}^{(j)}_{\ell
mk}(\mathbf{r})+\lambda\mathbf{N}^{(j)}_{\ell mk}(\mathbf{r})}{\sqrt{2}}$ (56)
$\displaystyle=\frac{\lambda}{\sqrt{2}}\frac{\ell(\ell+1)}{kr}z_{\mathrm{M},\ell}^{(j)}(kr)\mathrm{Y}_{\ell}^{m}(\theta,\phi)\hskip
3.0pt\hat{\mathbf{r}}$ $\displaystyle+\sum_{\lambda^{\prime}=\pm
1}\Big{[}\frac{\mathrm{i}z^{(j)}_{\mathrm{M},\ell}(kr)+\lambda\lambda^{\prime}z^{(j)}_{\mathrm{N},\ell}(kr)}{2}$
$\displaystyle\cdot\hskip 5.69046pt\mathbf{f}_{\lambda^{\prime},\ell
m}(\hat{\mathbf{r}})\Big{]}\mbox{\quad,\quad}$ (57)

where we have defined:

$\displaystyle\mathbf{f}_{\lambda,\ell m}(\hat{\mathbf{r}})$
$\displaystyle=\frac{\mathbf{m}_{\ell
m}(\hat{\mathbf{r}})+\lambda\mathbf{n}_{\ell m}(\hat{\mathbf{r}})}{\sqrt{2}}$
$\displaystyle=\Omega_{\ell m}\tau^{(\lambda)}_{\ell
m}(\theta)\mathrm{e}^{\mathrm{i}m\phi}\hskip
3.0pt\hat{\mathbf{e}}_{\lambda}(\hat{\mathbf{r}})$ (58)

and

$\displaystyle\tau^{(\lambda)}_{\ell m}(\theta)$ $\displaystyle=$
$\displaystyle-\tau^{\prime}_{\ell m}(\theta)-\lambda\tau_{\ell
m}(\theta)\mbox{\quad,\quad}$ (59)

which has the property $\Omega_{-\ell m}\tau^{(\lambda)}_{-\ell
m}(\theta)=\Omega_{\ell m}\tau^{(-\lambda)}_{\ell
m}(\theta)=(-1)^{\ell+m+1}\Omega_{\ell m}\tau^{(\lambda)}_{\ell
m}(\pi-\theta)$.

One can show that the functions $\mathbf{\Lambda}^{(j)}_{\lambda,\ell mk}$
have the property [46]:

$\displaystyle\frac{\nabla\times}{k}\mathbf{\Lambda}^{(j)}_{\lambda,\ell mk}$
$\displaystyle=$ $\displaystyle\lambda\mathbf{\Lambda}^{(j)}_{\lambda,\ell
mk}\mbox{\quad,\quad}$ (60)

that is, $\mathbf{\Lambda}^{(j)}_{\lambda,\ell mk}$ is an eigenstate of the
helicity operator $\frac{\nabla\times}{k}$ with eigenvalue $\lambda$. For the
functions $\mathbf{f}_{\lambda,\ell m}$, there exists the orthogonality
property:

$\displaystyle\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{\pi}$
$\displaystyle\mathrm{sin}\theta\mathrm{d}\theta$
$\displaystyle\mathbf{f}_{\lambda,\ell
m}(\hat{\mathbf{k}})\cdot\left[\mathbf{f}_{\lambda^{\prime},\ell^{\prime}m^{\prime}}(\hat{\mathbf{k}})\right]^{*}$
$\displaystyle=\delta_{\lambda\lambda^{\prime}}\delta_{\ell^{\prime}m^{\prime}}\delta_{\ell^{\prime}m^{\prime}}\mbox{\quad.\quad}$
(61)

Moreover, if we employ the large argument property of the Hankel functions:

$\displaystyle
z_{\alpha,\ell}^{(3)}(x)\xrightarrow{x>>1}\begin{cases}\frac{\mathrm{e}^{\mathrm{i}x}}{x}(-\mathrm{i})^{\ell}&\text{for}\quad\alpha=\mathrm{N}\\\
\frac{\mathrm{e}^{\mathrm{i}x}}{x}(-\mathrm{i})^{\ell+1}&\text{for}\quad\alpha=\mathrm{M}\mbox{\quad,\quad}\end{cases}$
(62)

and also reject the $\mathrm{O}(\nicefrac{{1}}{{\mathrm{r_{0}^{2}}}})$ radial
term as negligible, for the radiating helical VSHs we can get the following
asymptotic form in the far field:

$\displaystyle\left[\mathbf{\Lambda}_{\lambda,\ell
mk}^{(3)}\left(\mathbf{r}\right)\right]^{\mathrm{ff}}$ $\displaystyle=$
$\displaystyle(-\mathrm{i})^{\ell}\hskip 3.0pt\mathbf{f}_{\lambda,\ell
m}(\hat{\mathbf{r}})\hskip
3.0pt\frac{\mathrm{e}^{\mathrm{i}kr}}{kr}\mbox{\quad.\quad}$ (63)

##  Appendix C T-matrix in the helicity basis for a sphere

In the parity basis, the T-matrix is given by

$\mathbf{T}=\begin{pmatrix}\mathbf{T}_{\text{NN}}&\mathbf{T}_{\text{MN}}\\\
\mathbf{T}_{\text{NM}}&\mathbf{T}_{\text{MM}}\end{pmatrix}\mbox{\quad,\quad}$
(64)

where each element of Eqn. (64) is a diagonal
$\ell_{\text{max}}(2+\ell_{\text{max}})\times\ell_{\text{max}}(2+\ell_{\text{max}})$
matrix and $\ell_{\text{max}}$ is the maximum multipolar excitation order of
the sphere.

This can be transformed to the T-matrix $\mathbf{T}^{\text{H}}$ in the
helicity basis by using [49]

$\mathbf{T}^{\text{H}}=\mathbf{P}^{-1}\mathbf{T}\mathbf{P}\mbox{\quad,\quad}$
(65)

where, as can be seen from Eqn. (2), $\mathbf{P}$ is given by

$\mathbf{P}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\mbox{\quad.\quad}$ (66)

In the case of a dielectric sphere, we have that
$\mathbf{T}_{\text{MN}}=\mathbf{T}_{\text{NM}}=0$, so Eqn. (65) reduces to

$\displaystyle\mathbf{T}^{\text{H}}$
$\displaystyle=\begin{pmatrix}\mathbf{T}_{++}&\mathbf{T}_{+-}\\\
\mathbf{T}_{-+}&\mathbf{T}_{--}\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{bmatrix}(\mathbf{T}_{\text{NN}}+\mathbf{T}_{\text{MM}})&(\mathbf{T}_{\text{NN}}-\mathbf{T}_{\text{MM}})\\\
(\mathbf{T}_{\text{NN}}-\mathbf{T}_{\text{MM}})&(\mathbf{T}_{\text{NN}}+\mathbf{T}_{\text{MM}})\end{bmatrix}\mbox{\quad,\quad}$
(67)

where

$\mathbf{T}_{\text{NN}}=\begin{pmatrix}a_{1}&\dots&0\\\
\vdots&\ddots&\vdots\\\
0&\dots&a_{\ell_{\text{max}}}\end{pmatrix}\mbox{\quad,\quad}$ (68)
$\mathbf{T}_{\text{MM}}=\begin{pmatrix}b_{1}&\dots&0\\\
\vdots&\ddots&\vdots\\\
0&\dots&b_{\ell_{\text{max}}}\end{pmatrix}\mbox{\quad,\quad}$ (69)

and the values $a_{\ell}$ and $b_{\ell}$ are respectively the electric and
magnetic Mie coefficients defined in App. E.

The components of the T-matrix $\mathbf{T}^{\mathrm{H}}$ are given by
$\mathrm{T}_{\lambda_{\mathrm{s}},\lambda_{\mathrm{i}},\ell}$, and relate to
the entries in $\mathbf{T}^{\mathrm{H}}$ corresponding to the $\ell$’th
multipolar order, along with an incident helicity $\lambda_{\mathrm{i}}$ and
scattered helicity $\lambda_{\mathrm{s}}$. That is,

$\displaystyle\mathrm{T}_{\lambda_{\mathrm{s}},\lambda_{\mathrm{i}},\ell}$
$\displaystyle=a_{\ell}+\lambda_{\mathrm{i}}\lambda_{\mathrm{s}}b_{\ell}.$
(70)

##  Appendix D Computation-friendly expansion of Eqn. (34)

By substituting Eqn. (II.2) into the Eqn. (34), and using the orthogonality
relation given by Eqn. (B), we can express the angular energy density
$U^{\prime}(\theta^{\prime},\phi^{\prime})$ in the following form that is
suitable for efficient numerical evaluation:

$\displaystyle U^{\prime}(\theta^{\prime},\phi^{\prime})$
$\displaystyle=\sum_{\lambda^{\prime}}\frac{4\pi
c^{2}}{\eta_{0}}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}\frac{1}{(\omega^{\prime})^{2}}$
$\displaystyle\cdot\left|\sum_{\ell^{\prime}m^{\prime}}\mathcal{B}_{\lambda^{\prime}\ell^{\prime}m^{\prime}}(\omega^{\prime})(-\mathrm{i})^{\ell^{\prime}}\Omega_{\ell^{\prime}m^{\prime}}\tau^{(\lambda^{\prime})}_{\ell^{\prime}m^{\prime}}(\theta^{\prime})\mathrm{e}^{\mathrm{i}\ell^{\prime}\phi^{\prime}}\right|^{2}$
$\displaystyle=\sum_{\lambda^{\prime}}\sum_{\ell^{\prime}m^{\prime}}\sum_{\bar{\ell}^{\prime}\bar{m^{\prime}}}Q_{\lambda^{\prime}\ell^{\prime}m^{\prime}}^{\bar{\ell}^{\prime}\bar{m^{\prime}}}(\theta^{\prime},\phi^{\prime})$
$\displaystyle\cdot\sum_{\lambda_{0}\bar{\lambda}_{0}}J_{\lambda_{0}\ell^{\prime}m^{\prime}}^{\bar{\lambda}_{0}\bar{\ell}^{\prime}\bar{m^{\prime}}}\mathrm{T}_{\lambda^{\prime}\lambda_{0},\ell^{\prime}}\mathrm{T}_{\lambda^{\prime}\bar{\lambda}_{0},\bar{\ell}^{\prime}}^{*}\mbox{\quad,\quad}$
(71)

where the T-matrix elements
$\mathrm{T}_{\lambda^{\prime}\lambda_{0},\ell^{\prime}}$ are defined in App.
C. For the latter equation, we have assumed a non-dispersive T-matrix and have
also defined the integral:

$\displaystyle
J_{\lambda_{0}\ell^{\prime}m^{\prime}}^{\bar{\lambda}_{0}\bar{\ell}^{\prime}\bar{m^{\prime}}}$
$\displaystyle=$
$\displaystyle\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}\frac{\mathcal{A}_{\lambda_{0}\ell^{\prime}m^{\prime}}(\omega^{\prime})\mathcal{A}^{*}_{\bar{\lambda}_{0}\bar{\ell^{\prime}}\bar{m^{\prime}}}(\omega^{\prime})}{(\omega^{\prime})^{2}}\mbox{\quad,\quad}$
(72)

and the quantity:

$\displaystyle
Q_{\lambda^{\prime}\ell^{\prime}m^{\prime}}^{\bar{\ell}^{\prime}\bar{m^{\prime}}}(\theta^{\prime},\phi^{\prime})$
$\displaystyle=\frac{4\pi
c^{2}}{\eta_{0}}(-\mathrm{i})^{\ell^{\prime}-\bar{\ell}^{\prime}}\Omega_{\ell^{\prime}m^{\prime}}\Omega^{*}_{\bar{\ell}^{\prime}\bar{m^{\prime}}}$
$\displaystyle\cdot\tau^{(\lambda^{\prime})}_{\ell^{\prime}m^{\prime}}(\theta^{\prime})\tau^{(\lambda^{\prime})*}_{\bar{\ell}^{\prime}\bar{m^{\prime}}}(\theta^{\prime})\mathrm{e}^{\mathrm{i}(m^{\prime}-\bar{m^{\prime}})\phi^{\prime}}\mbox{\quad,\quad}$
(73)

where ∗ denotes the complex conjugate.

Furthermore, we can define express the total scattered energy
$W_{\mathrm{tot}}$ given by Eqn. (36) as follows:

$\displaystyle W_{\mathrm{tot}}$
$\displaystyle=\int_{0}^{\pi}\mathrm{d}\theta\int_{0}^{2\pi}\mathrm{d}\phi\sin\theta
U(\theta,\phi)$
$\displaystyle=\sum_{\lambda^{\prime}}\int^{\infty}_{0^{+}}\mathrm{d}\omega^{\prime}\frac{1}{(\omega^{\prime})^{2}}\sum_{\ell^{\prime}m^{\prime};\mathrm{min}\\{|m^{\prime}|,1\\}\leq\ell\leq\ell^{\prime}}$
$\displaystyle\hskip
14.22636pt\real\left\\{\mathcal{B}_{\omega^{\prime}\lambda^{\prime}\ell^{\prime}m^{\prime}}\mathcal{B}^{*}_{\omega^{\prime}\lambda^{\prime}\ell
m^{\prime}}I_{\lambda^{\prime}\ell^{\prime}m^{\prime}}^{\ell}\right\\}$
$\displaystyle=\sum_{\lambda^{\prime}}\sum_{\ell^{\prime}m^{\prime};\mathrm{min}\\{|m^{\prime}|,1\\}\leq\ell\leq\ell^{\prime}}\sum_{\lambda_{0}\bar{\lambda}_{0}}$
$\displaystyle\real\left\\{I_{\lambda^{\prime}\ell^{\prime}m^{\prime}}^{\ell}J_{\lambda_{0}m^{\prime}\ell^{\prime}}^{\bar{\lambda}_{0}m^{\prime}\ell}\mathrm{T}_{\lambda^{\prime}\lambda_{0},\ell^{\prime}}\mathrm{T}_{\lambda^{\prime}\bar{\lambda}_{0},\ell}^{*}\right\\}$
(74)

with

$\displaystyle I_{\lambda^{\prime}\ell^{\prime}m^{\prime}}^{\ell}$
$\displaystyle=2^{4-\delta_{\ell\ell^{\prime}}}\frac{\pi^{2}c^{2}}{\eta_{0}}(-\mathrm{i})^{\ell^{\prime}-\ell}\Omega_{\ell^{\prime}m^{\prime}}\Omega_{\ell
m^{\prime}}^{*}$
$\displaystyle\int_{0}^{\pi}\mathrm{d}\theta\sin\theta\left\\{\gamma\left[1+\beta\,\theta^{\prime}(\beta,\theta)\right]\right\\}^{3}$
$\displaystyle\times\tau^{(\lambda^{\prime})}_{\ell^{\prime}m^{\prime}}\left[\theta^{\prime}(\beta,\theta)\right]\tau^{(\lambda^{\prime})*}_{\ell
m^{\prime}}\left[\theta^{\prime}(\beta,\theta)\right]\mbox{\quad,\quad}$ (75)

where the expression for $\theta^{\prime}(\beta,\theta)$ is given by Eqn.
(14).

Figure 4: The absolute value of the dipole expansion coefficients
$\mathcal{A}_{10}$ ($\ell^{\prime}=1,m^{\prime}=0$) given by Eqn. (II.2) as a
function of the normalized frequency
$(\omega_{\mathrm{i}}-\omega^{\prime})/\omega_{\mathrm{i}}$ when a):
$\beta=0.1$ (blue line), $\beta=0.2$ (orange line), $\beta=0.5$ (yellow line)
and $\beta=0.9$ (purple line). In all cases,
$\Theta_{\mathrm{i}}=\frac{\pi}{4}$, and $\lambda_{\mathrm{i}}=+1$ which,
since helicity is conserved under Lorentz boosts, means that
$\lambda^{\prime}=\lambda_{\mathrm{i}}=+1$. Note that the width of the peak
increases with speed. This is due to the increasing effect of the Doppler
shift $\omega^{\prime}$ of $\omega_{\mathrm{i}}$. Since the incident beam is
monochromatic, $|\mathcal{A}_{10}|$ tends to a delta-distribution-like peak as
the speed decreases. As the speed increases, the plane wave components of the
incident beam all Doppler shift differently due to their differing angular
orientations. This leads to non-zero values for $|\mathcal{A}_{10}|$ when
$\omega_{\mathrm{i}}\neq\omega^{\prime}$.

Note the importance of writing Eqn. (D) as nested integrals instead of a
standard triple integral. Computationally speaking, we are able to determine
Eqn. (72) with a very low tolerance, while using a higher tolerance for the
other integrals, thus significantly reducing computation time. The reason for
this is because the expansion coefficients $\mathcal{A}_{\lambda^{\prime}\ell
m}(\omega^{\prime})$ (and hence the integrand in Eqn. (72)) form Gaussian-like
peaks centered about $\omega^{\prime}$ which tend to a delta distribution as
the speed of the sphere decreases (see Fig. 4). If the tolerance is too high,
the numerical integration could miss this peak entirely, thus ignoring vital
non-zero values.

Moreover, by separating the integrals Eqn. (D) we are able to obtain
$100\times 100$ grids for the directivity (like those used to generate Fig. 3)
in as little as $\sim$$40\text{\,}\mathrm{s}$. The reason for this is that, as
the numerically-demanding integral
$J_{\lambda_{0}\ell^{\prime}m^{\prime}}^{\bar{\lambda}_{0}\bar{\ell}^{\prime}\bar{m^{\prime}}}$
is independent of the Mie angles, we only need to compute it once for a given
incident angle $\Theta_{\mathrm{i}}$ and speed parameter $\beta$. If the
integrals were combined,
$J_{\lambda_{0}\ell^{\prime}m^{\prime}}^{\bar{\lambda}_{0}\bar{\ell}^{\prime}\bar{m^{\prime}}}$
would be computed for each combination of Mie angles, thus significanly
increasing computation time.

##  Appendix E Mie angles for a lossless sphere

The electric and magnetic Mie coefficients $a_{\ell}$ and $b_{\ell}$,
respectively for each multipolar order $\ell$ can be represented using Mie
angles $\theta_{\text{E}\ell}$ and $\theta_{\text{M}\ell}$. In the lossless
case, one can write [42, 28]

$a_{\ell}=-\mathrm{i}\sin{\alpha_{\ell}}\exp(-\mathrm{i}\alpha_{\ell})$ (76)

and

$b_{\ell}=-\mathrm{i}\sin{\beta_{\ell}}\exp(-\mathrm{i}\beta_{\ell})\mbox{\quad,\quad}$
(77)

where

$\alpha_{\ell}=\frac{\pi}{2}-\theta_{\text{E}\ell},\quad-\frac{\pi}{2}\leq\theta_{\text{E}\ell}\leq\frac{\pi}{2}\mbox{\quad,\quad}$
(78)

and

$\beta_{\ell}=\frac{\pi}{2}-\theta_{\text{M}\ell},\quad-\frac{\pi}{2}\leq\theta_{\text{M}\ell}\leq\frac{\pi}{2}\mbox{\quad.\quad}$
(79)

Note that the convention used in Rahimzadegan  _et al._ [28] omits the use of
the minus sign in Eqns. (76) and (77). In our case, the minus sign is required
for energy conservation.

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