# Inelastic scattering of transversely structured free electrons from
nanophotonic targets: Theory and computation

Austin G. Nixon  Department of Chemistry, University of Washington, Seattle,
WA 98195  Matthieu Chalifour  Department of Physics, University of Washington,
Seattle, WA 98195  Marc R. Bourgeois  Department of Chemistry, University of
Washington, Seattle, WA 98195  Michael Sanchez  Department of Physics,
University of Washington, Seattle, WA 98195  David J. Masiello
<EMAIL_ADDRESS>Department of Chemistry, University of Washington, Seattle, WA
98195

###### Abstract

Recent advancements in abilities to create and manipulate the electron’s
transverse wave function within the transmission electron microscope (TEM) and
scanning TEM (STEM) have enabled vectorially-resolved electron energy loss
(EEL) and gain (EEG) measurements of nanoscale and quantum material responses
using pre- and post-selected free electron states. This newfound capability is
prompting renewed theoretical interest in quantum mechanical treatments of
inelastic electron scattering observables and the information they contain.
Here, we present a quantum mechanical treatment of the inelastic scattering of
free electrons between pre- and post-selected transverse states with fully-
retarded electron–sample interactions for both spontaneous EEL and continuous-
wave laser-stimulated EEG measurements. General expressions for the state-
resolved energy loss and gain rates are recast in forms amenable to numerical
calculation using the method of coupled dipoles. We numerically implement our
theory within the $e$-DDA code, and use it to investigate specific examples
that highlight its versatility regarding the number, size, geometry, and
material composition of the target specimen, as well as its ability to
describe matter-wave diffraction from finite nanoscopic targets.

##  I Introduction

Leveraging recent instrumental advancements in energy monochromation and
aberration correction, inelastic scattering of free electrons has become an
effective technique to spectroscopically characterize and image atomic and
molecular [1, 2, 3], biological [4, 5, 6], solid state [7, 8, 9, 10, 11], and
nanophotonic [12, 13] systems with unprecedented spatial resolution.
Simultaneously, optical spectroscopies and microscopies based upon the
absorption, scattering, extinction, and emission of electromagnetic waves
continue to be indispensable tools used to probe the same systems, albeit with
spatial resolution limitations imposed by the optical diffraction limit.
Light-based spectromicroscopies can often be enhanced by taking advantage of
optical selection rules stemming from the intrinsic linear and spin angular
momentum degrees of freedom of the photon [14]. In addition to the
polarization degrees of freedom arising from the intrinsic spin angular
momentum, photons can also be prepared in specific orbital angular momentum
(OAM) states defined by the azimuthal phase $e^{i\ell\phi}$ [15]. Due to the
helical nature of their spiralling phase fronts, light beams characterized by
such a quantized topological charge $\ell$ are commonly referred to as optical
vortex or twisted light beams [16, 17]. Motivated in part by the infinite
dimensional Hilbert space offered by the OAM basis [18], the ability to
prepare [19], sort [20], and measure [21] these optical OAM states has driven
applications in quantum information science using photons with quantized
azimuthal and radial labels [22, 17, 23, 24, 25, 26]. Moreover, transverse
sculpting of the radiation field in general, has lead to the development of
new gauge transformations, such as the twisted light gauge [27, 28],
construction of free space optical skyrmionic beams [29], excitations of
forbidden transitions in atomic isotopes [30], and the use of vortex $\gamma$
ray photons to selectively probe high energy resonances in photonuclear
reactions [31].

Unlike photons, electrons prepared and measured in currently available TEMs,
STEMs, or ultrafast TEMs (UTEMs) are accurately described by the spinless free
particle Schrödinger equation and consequently lack intrinsic polarization
degrees of freedom [17]. Despite this limitation, linear-momentum-based
selection rules based on quantum mechanical treatments of the inelastic
scattering process have been long understood and exploited in core-loss EEL
spectroscopy [32, 33, 34, 35] and have enabled measurements of magnetic
circular dichroism [36, 37, 38, 39, 40, 41], characterization of site-specific
defects in atomic crystals [42], as well as visualization of the
electromagnetic fields of atomic-scale systems [43]. Inspired by the creation
and manipulation of optical vortex states, developments of techniques for
shaping the transverse phase profile [44] and OAM content of free electrons
via holographic masks [45, 46, 47], spiral phase plates [48], and shaped laser
pulses [49, 50] has been at the forefront of low-loss EEL spectroscopy
($\lesssim 50$ eV) [51]. Furthermore, borrowing ideas from quantum optics, the
preparation of free electron qubits carrying information in the form of
quantized energy or OAM states using laser pulses [52, 53, 54], holographic
masks, or spiral phase plates [55, 56, 57], has driven the continued
development of free electrons as holders and propagators of quantum
information. In parallel, the ability to generate phase-structured incident
electron states and sort them based upon their OAM content [58] has fueled
numerous investigations of inelastic electron scattering between states with
pre- and post-selected transverse phase profiles in the low-loss regime [59,
60, 61, 62, 63, 64, 65, 66, 67, 68].

In this paper, we expound upon a recently introduced theoretical framework
describing the fully retarded inelastic scattering of phase-shaped free
electron beams in the electron microscope [67]. Specifically, with emphasis
placed on the low-loss regime, we investigate the theory of transversely
phase-shaped EEL spectroscopy in both narrow beam and wide field limits and
laser-stimulated EEG spectroscopy in the narrow beam limit. Sec. II presents a
derivation of the energy-resolved inelastic electron scattering rate,
including both EEL and EEG processes. Transversely structured free electron
states that can be prepared within currently available TEMs and STEMs are
subsequently reviewed in Sec. III. Section IV introduces the transition
current density associated with transitions between such states. Expressions
for EEL and EEG observables are derived in Sec. V for both wide field and
focused electron beams, including those with nonuniform transverse phase-
structure, such as twisted electron beams. Numerical implementation of the
presented inelastic electron scattering theory based on the method of coupled
dipoles is described in Sec. VI, including EEL and EEG probabilities and the
EEL double differential scattering cross section (DDCS). Numerical
calculations are presented for prototypical nanophotonic systems, underscoring
particular advantages of our numerical approach, including its: (1)
flexibility regarding target size, shape, composition, and number, (2) facile
extension to accommodate arbitrary initial and final free electron transverse
states, and (3) ability to capture signatures of matter-wave diffraction and
interference arising from scattering from individual and multiple nanoscale
targets.

##  II Inelastic free electron scattering: State- and energy-resolved EEL and
EEG rates

Here we review the retarded theory of inelastic electron scattering for the
calculation of EEL and laser-stimulated EEG processes [69, 70, 71, 51, 72, 73,
74, 66, 67]. The target material is described through its bound
electromagnetic responses characterized by dielectric function $\xi(\omega)$,
while the electron probe is examined for both delocalized and localized
electron wave functions. The light-matter potential governing such
interactions takes the form

$\hat{V}=\frac{e}{2mc}\Big{(}\hat{\mathbf{A}}\cdot\hat{\mathbf{p}}+\hat{\mathbf{p}}\cdot\hat{\mathbf{A}}\Big{)}$
(1)

under minimal coupling in the generalized Coulomb gauge [75]
$\nabla\cdot\xi(\mathbf{x})\hat{\mathbf{A}}(\mathbf{x},t)=0$, where the
$\hat{\rho}\hat{\Phi}$ term makes no contribution in the absence of free
charges describing the target. The probing electron’s charge and mass are $-e$
and $m$, and $c$ is the speed of light in vacuum. $\hat{\mathbf{A}}$ and
$\hat{\mathbf{p}}$ are quantum mechanical operators for the vector potential
of the target and linear momentum of the free electron probe, respectively.

Figure 1: Scheme showing the inelastic scattering of transversely delocalized
and localized free electrons with wave functions of the separable form
$\Psi({\bf x}_{\perp})e^{ik_{\parallel}x_{3}}.$ (a) Initial electron plane
wave (red) with wave vector $\mathbf{k}=({\bf k}_{\perp},k_{\parallel})$
interacting with the vector potential (shaded green) of a target specimen.
Following the interaction, the electron is post-selected for the plane wave
state $\mathbf{k}^{\prime}=({\bf k}^{\prime}_{\perp},k^{\prime}_{\parallel})$
(blue). The momentum recoil wave vector associated with this transition is
$\mathbf{q}=\mathbf{k}-\mathbf{k}^{\prime}$. (b) A transversely localized
electron beam with pre-selected transverse wave function
$\Psi_{i}(\mathbf{x}_{\perp})$ (red) interacting with the same target and
post-selected in $\Psi_{f}(\mathbf{x}_{\perp})$ (blue).

Owing to the inherently weak nature of electron-photon coupling, the inelastic
scattering probability can be obtained using first order time-dependent
perturbation theory. The transition rate describing the scattering of an
electron from initial state $|\psi_{i}\rangle$ to final state
$|\psi_{f}\rangle$ while simultaneously exciting/de-exciting the target from
initial state $|\Phi_{\nu}\rangle$ to final state
$|\Phi_{\nu^{\prime}}\rangle$ is given by

$w_{fi}=\frac{2\pi}{\hbar}\sum_{\nu\nu^{\prime}}\Big{|}\langle\Phi_{\nu^{\prime}}|\langle\psi_{f}|\hat{V}|\psi_{i}\rangle|\Phi_{\nu}\rangle\Big{|}^{2}\delta(E_{f}-E_{i}),$
(2)

where $E_{i}=\hbar\varepsilon_{i}+\hbar\omega_{\nu}$ and
$E_{f}=\hbar\varepsilon_{f}+\hbar\omega_{\nu^{\prime}}$ are the initial and
final energies of the composite system,
$\hbar\varepsilon_{i/f}=\gamma_{i/f}mc^{2}$ represents the initial/final
energy of the probing electron, and
$\gamma_{i/f}=\big{[}1-(v_{i/f}/c)^{2}\big{]}^{-1/2}$ are the initial/final
Lorentz contraction factors. Sec. III details specific TEM and STEM electron
states, including those that are phase-shaped transversely to their
propagation direction (see Fig. 1), but for now we remain agnostic to their
identity. After some algebra, the transition matrix elements of the scattering
potential in Eq. (1) can be expressed as

$\displaystyle V_{\nu^{\prime}f\nu i}(t)$
$\displaystyle=\langle\Phi_{\nu^{\prime}}|\langle\psi_{f}|\hat{V}(t)|\psi_{i}\rangle|\Phi_{\nu}\rangle$
(3) $\displaystyle=\langle\Phi_{\nu^{\prime}}|-\frac{1}{c}\int
d\mathbf{x}\,\mathbf{A}(\mathbf{x},t)\cdot\mathbf{J}_{fi}(\mathbf{x},t)|\Phi_{\nu}\rangle,$

where
$\mathbf{J}_{fi}(\mathbf{x},t)=\mathbf{J}_{fi}(\mathbf{x})e^{-i(\varepsilon_{i}-\varepsilon_{f})t}$
is the transition current density [67] with

$\mathbf{J}_{fi}(\mathbf{x})=\frac{i\hbar
e}{2m}\big{[}\psi_{f}^{*}(\mathbf{x})\nabla\psi_{i}(\mathbf{x})-\psi_{i}(\mathbf{x})\nabla\psi_{f}^{*}(\mathbf{x})\big{]}$
(4)

defined in terms of the probe scattering states
$\psi_{i}(\mathbf{x},t)=\psi_{i}(\mathbf{x})e^{-i\varepsilon_{i}t}$ and
$\psi_{f}(\mathbf{x},t)=\psi_{f}(\mathbf{x})e^{-i\varepsilon_{f}t}$. By
continuity, i.e.,
$\nabla\cdot\mathbf{J}_{fi}(\mathbf{x},t)=-\dot{\rho}_{fi}(\mathbf{x},t)$,
$\mathbf{J}_{fi}(\mathbf{x},t)$ is connected to the transition charge density
$\rho_{fi}(\mathbf{x},t)=-e\psi_{f}^{*}(\mathbf{x},t)\psi_{i}(\mathbf{x},t)$.

EEL and EEG scattering rates each derive from Eq. (3), but involve different
vector potentials of distinct physical origin. In the case of EEL, solving the
vector Helmholtz equation on the domain of the target produces a set of
eigenmode functions $\mathbf{f}_{\nu}(\mathbf{x})$ and associated eigenmode
frequencies $\omega_{\nu}$, which serve as a basis to expand the target vector
potential [75]. In terms of this set
$\\{\mathbf{f}_{\nu}(\mathbf{x})\\}_{\nu}$,
$\mathbf{A}(\mathbf{x},t)=\sum_{\nu}[\mathbf{A}_{\nu}^{(+)}(\mathbf{x})a_{\nu}e^{-i\omega_{\nu}t}+\mathbf{A}_{\nu}^{(-)}(\mathbf{x})a_{\nu}^{\dagger}e^{i\omega_{\nu}t}]$,
with
$\mathbf{A}_{\nu}^{(+)}(\mathbf{x})=c\sqrt{2\pi\hbar/\omega_{\nu}}\mathbf{f}_{\nu}(\mathbf{x})$,
$\mathbf{A}_{\nu}^{(-)}(\mathbf{x})=c\sqrt{2\pi\hbar/\omega_{\nu}}\mathbf{f}_{\nu}^{*}(\mathbf{x})$,
and $[\mathbf{A}_{\nu}^{(+)}]^{\dagger}=\mathbf{A}_{\nu}^{(-)}$, where
$a_{\nu}^{\dagger}$ ($a_{\nu}$) are creation (annihilation) operators
responsible for inducing optical excitations (de-excitations) in the $\nu$th
target mode. In the case of EEG, the target vector potential can still be
expanded onto the $\\{\mathbf{f}_{\nu}(\mathbf{x})\\}_{\nu}$ basis, however,
it is the response vector potential induced by a stimulating laser field. In
either case, Eq. (3) becomes

$V_{\nu^{\prime}f\nu
i}(t)=-\frac{1}{c}\Big{[}\sqrt{n_{\nu}}\delta_{{\nu^{\prime}},\nu-1}\int
d\mathbf{x}\,\mathbf{A}_{\nu}^{(+)}(\mathbf{x})\cdot\mathbf{J}_{fi}(\mathbf{x})e^{-i\omega_{\nu}t}+\sqrt{n_{\nu}+1}\delta_{\nu^{\prime},\nu+1}\int
d\mathbf{x}\,\mathbf{A}_{\nu}^{(-)}(\mathbf{x})\cdot\mathbf{J}_{fi}(\mathbf{x})e^{i\omega_{\nu}t}\Big{]}e^{-i(\varepsilon_{i}-\varepsilon_{f})t},$
(5)

where $n_{\nu}$ is the occupancy of the $\nu$th target mode. When carrying out
the derivations of inelastic electron scattering processes, the first and
second terms in Eq. (5) correspond to EEG and EEL events, respectively. The
energy conserving delta function in Eq. (2), when written in the following
equivalent forms

$\delta(E_{f}-E_{i})=\frac{1}{\hbar}\int
d\omega\begin{cases}\delta(\omega-\varepsilon_{if})\delta(\omega-\omega_{\nu^{\prime}\nu})\textrm{
\ (EEL)}\\\
\delta(\omega+\varepsilon_{if})\delta(\omega-\omega_{\nu\nu^{\prime}})\textrm{
\ (EEG)},\end{cases}$ (6)

will aid in this connection to EEL and EEG.

In EEL events, the probing electron may transfer energy to and retrieve energy
from any of the target modes which must be summed over to account for all such
loss processes where the electron acts as both pump and probe. After summing
over target states $\nu,\nu^{\prime}$, the EEL scattering rate becomes

$w_{fi}^{\textrm{loss}}=\frac{2\pi}{\hbar c^{2}}\sum_{\nu}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\mathbf{A}_{\nu}^{(+)}(\mathbf{x})\mathbf{A}_{\nu}^{(-)}(\mathbf{x}^{\prime})\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\delta(E_{f}-E_{i}),$
(7)

while, in the case of EEG, the stimulating laser field is taken to populate
the specific target state $\nu$ leaving only a sum over final target states
$\nu^{\prime}$ to be performed, resulting in

$w_{fi}^{\textrm{gain}}=\frac{2\pi}{\hbar c^{2}}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\mathbf{A}_{\nu}^{(-)}(\mathbf{x})\mathbf{A}_{\nu}^{(+)}(\mathbf{x}^{\prime})\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\delta(E_{f}-E_{i}).$
(8)

Again note that the target vector potentials
$\mathbf{A}_{\nu}^{(\pm)}(\mathbf{x})$ in the EEG rate in Eq. (8) are
understood to originate in response to external laser stimulation and are not
induced by the probing electron’s transition current
$\mathbf{J}_{fi}(\mathbf{x})$.

From Eqs. (6) and (7), together with the relationship $w_{fi}=\int
d\omega\,w_{fi}(\omega)$, the frequency-resolved EEL rate
$w^{\textrm{loss}}_{fi}(\omega)$ can be derived. By expressing the target
vector potential in terms of mode functions, i.e.,
$\mathbf{A}_{\nu}^{(+)}(\mathbf{x})\mathbf{A}_{\nu}^{(-)}(\mathbf{x}^{\prime})=(2\pi\hbar
c^{2}/\omega_{\nu})\mathbf{f}_{\nu}(\mathbf{x})\mathbf{f}_{\nu}^{*}(\mathbf{x}^{\prime})$,
Eq. (7) can be written in terms of the target’s electromagnetic Green’s
tensor,
$\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)=\sum_{\nu}\mathbf{f}_{\nu}(\mathbf{x})\mathbf{f}_{\nu}^{*}(\mathbf{x}^{\prime})/(\omega^{2}-\omega_{\nu}^{2}+i0^{+})$.
More specifically, the EEL rate is formulated in terms of the imaginary part
of the Green’s dyadic,
$\textrm{Im}\big{\\{}\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\big{\\}}=-\sum_{\nu}(\pi/2\omega_{\nu})\mathbf{f}_{\nu}(\mathbf{x})\mathbf{f}_{\nu}^{*}(\mathbf{x}^{\prime})\delta(\omega-\omega_{\nu^{\prime}\nu})$.
As a result, the state- and frequency-resolved EEL transition rate then
becomes

$\begin{split}w_{fi}^{\textrm{loss}}(\omega)&=-\frac{8\pi}{\hbar}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\textrm{Im}\Big{[}\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\Big{]}\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\delta(\omega-\varepsilon_{if})\\\
&=-\frac{8\pi}{\hbar}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\textrm{Im}\Big{[}\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\Big{]}\delta(\omega-\varepsilon_{if}).\end{split}$
(9)

Note that for general phase-shaped EEL processes described by Eq. (9), the
transition current density can point arbitrarily in 3D space and is not
restricted to lie along the TEM axis.

Alternatively, for the case of laser-stimulated EEG, Eqs. (6) and (8)
determine the frequency-resolved EEG rate. The positive (negative) frequency
portion of the target’s laser-induced response field can be expressed in terms
of its induced vector potential as $\mathbf{E}_{\nu}^{(\pm)}(\mathbf{x})=(\pm
i\omega_{\nu}/c)\,\mathbf{A}_{\nu}^{(\pm)}(\mathbf{x})$. For simplicity, the
applied continuous-wave laser field is chosen to prepare the target’s $\nu$th
excited state rather than a superposition thereof. As a result, the frequency-
resolved EEG rate

$w_{fi}^{\textrm{gain}}(\omega)=2\pi\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\bigg{|}\int
d\mathbf{x}\,\mathbf{E}_{\nu}^{(+)}(\mathbf{x})\cdot\mathbf{J}_{fi}(\mathbf{x})\bigg{|}^{2}\delta(\omega+\varepsilon_{if})\delta(\omega-\omega_{\nu\nu^{\prime}})$
(10)

is proportional to the volume integral of the 3D vector transition current
density $\mathbf{J}_{fi}(\mathbf{x})$ projected onto the laser-induced
electric field $\mathbf{E}_{\nu}^{(+)}(\mathbf{x})$ of the target. As will be
shown in Section V.1, if the appropriate choices for the initial and final
electron states are made, Eqs. (9) and (10) reduce to the conventional EEL and
EEG probabilities found in the literature [70, 51, 71, 66], but, as expressed
here, are generalized to potentially describe polarized EEL and EEG
measurements where the wave function of the probing electron is phase-
structured in the plane orthogonal to its motion.

##  III Transversely phase-structured free electron states

Figure 2: Transverse free electron states. Wave functions $\Psi(x,y)$ are
visualized in the $z=0$ plane with position-dependent phase indicated by the
color of the respective intensity profiles. Black represents low wave function
density. (a) Plane wave states with $\mathbf{k}_{\perp}$ along
$\hat{\mathbf{x}}$ (top) and $\hat{\mathbf{y}}$ (bottom). (b) Bessel beam
states $\Psi_{\ell k_{\perp}}(\rho,\phi)$ with $\ell=1$, and radial wave
number $k_{\perp}$ (top) and $2k_{\perp}$ (bottom). (c) LG states $\Psi_{\ell
p}(\rho,\phi)$ with $\ell=1$, and $p=0$ (top) and $p=1$ (bottom). (d) HG
states $\Psi_{nm}(x,y)$ with $n=1$ and $m=0$ (top), and $n=0$ and $m=1$
(bottom). (e) In the $\ell=\pm 1$ OAM subspace, the first order LG states
$|L/R\rangle$ can be used as basis states spanning the Bloch sphere with
antipodal points $\big{\\{}|0\rangle,|1\rangle\\}$,
$\big{\\{}|+\rangle,|-\rangle\\},$ and $\big{\\{}|L\rangle,|R\rangle\\}$;
states with constant spatial phase are defined as $|G\rangle$ and placed at
the sphere center. Colored arrows represent transitions between $|G\rangle$
and the states at the labeled antipodal points.

This section introduces various spinless free electron states relevant to the
forthcoming discussion of phase-structured EEL and EEG measurements in TEM and
STEM instruments. The states are: (i) energy eigenstates and thereby separable
into spatial and temporal parts as
$\psi(\mathbf{x},t)=\psi(\mathbf{x})e^{-i\varepsilon t}$ and (ii) separable
within an orthogonal coordinate system $\mathbf{x}=(x_{1},x_{2},x_{3})$ into
transverse and longitudinal functions
$\psi(\mathbf{x})=\Psi(x_{1},x_{2})e^{ik_{\parallel}x_{3}}$. The electron wave
functions are translationally invariant along the TEM axis, defined as
$x_{3}\equiv z$. Transversely delocalized states are investigated first,
beginning with plane wave and vortex Bessel beam states originating as
separable solutions in the Cartesian and cylindrical coordinate systems,
respectively. Subsequently, transversely localized and nondiffracting wave
functions, including Hermite-Gauss (HG) and twisted electron Laguerre-Gauss
(LG) states, are presented.

Plane wave solutions are separable in Cartesian coordinates with well-defined
linear momentum $\mathbf{p}=\hbar\mathbf{k}$. The spatial wave function
describing a free electron in the 3D Cartesian space ${\bf x}=(x,y,z)$ is

$\psi_{\mathbf{k}}(x,y,z)=\bigg{(}\frac{1}{\sqrt{L}}\bigg{)}^{3}e^{i\mathbf{k}\cdot\mathbf{x}},$
(11)

where $\mathbf{k}=({\bf k}_{\perp},k_{\parallel})$ is the wave vector and $L$
is the box quantization length. Phase plots of two transverse plane wave
states, $\Psi_{k_{x}}(x,y)$ and $\Psi_{k_{y}}(x,y)$, with orthogonal wave
vectors $\mathbf{k}_{\perp}=k_{x}\hat{\mathbf{x}}$ and
$\mathbf{k}_{\perp}=k_{y}\hat{\mathbf{y}}$ and corresponding transverse
wavelengths $\lambda_{\perp x}=2\pi/k_{x}$ and $\lambda_{\perp y}=2\pi/k_{y}$,
and are displayed in the upper and lower panels of Fig. 2(a) for
$\lambda_{\perp x}=\lambda_{\perp y}=\lambda_{\perp}$, respectively. Electrons
can also be prepared in coherent superposition states, with one example of
such a state being
$\psi_{k_{x}k_{y}}^{\chi}(x,y,z)=L^{-3/2}\big{[}e^{ik_{x}x}+e^{ik_{y}y}e^{i\chi}\big{]}e^{ik_{\parallel}z}/\sqrt{2}$,
where the relative phase $\chi$ between the two orthogonal electron wave
vector components $k_{x}$ and $k_{y}$ can take values $0\leq\chi\leq 2\pi$
[36, 37, 38].

When expressed in the cylindrical coordinate system defined by ${\bf
x}=(\rho,\phi,z)$, the separable solutions are nondiffracting Bessel waves of
the form [76, 77, 78, 79],

$\psi_{\ell k_{\perp}}(\rho,\phi,z)\propto
J_{|\ell|}(k_{\perp}\rho)e^{i\ell\phi}e^{ik_{\parallel}z}.$ (12)

Here, $J_{|\ell|}(k_{\perp}\rho)$ are Bessel functions of the first kind,
$\ell$ is the azimuthal quantum number of the cylindrically symmetric state,
while $k_{\perp}$ is the radial wave vector component and $k_{\parallel}$ is
the longitudinal wave vector component, respectively [80]. Such states are
eigenstates of the $z$-component of the OAM operator
$\hat{L}_{z}=-i\hbar\partial/\partial{\phi}$ with eigenvalue $\ell\hbar$.
Bessel beams can be expressed in terms of a coherent superpositions of plane
waves with wave vectors that are conically-distributed in momentum space and
are additionally delocalized in the transverse plane, since they cannot be
normalized as $\int_{0}^{\infty}\rho d\mathbf{\rho}\,|\psi_{\ell
k_{\perp}}|^{2}$ diverges [80]. Specific examples of Bessel states are shown
in Fig. 2(b) for two different values of $k_{\perp}$.

Transversely localized free electron states can be constructed within the
paraxial approximation to the Schrödinger equation, where the electron
momentum along the TEM axis is much greater than its transverse momentum,
i.e., $k_{\perp}\ll|{\bf k}|$. In particular, the focused LG states well-known
in optics [15, 81, 24] arise as superpositions of Bessel beam states of
different ${k}_{\perp}$. Since they are eigenstates of the ${\hat{L}}_{z}$
operator, LG states carry a quantized azimuthal component $\ell$. In the
nondiffracting, i.e., collimated, limit defined by infinite Rayleigh range,
the LG wave functions have the form of quantized Landau states [76, 80] given
by

$\psi_{\ell
p}(\rho,\phi,z)=\frac{\big{(}\sqrt{2}\rho/w_{0}\big{)}^{|\ell|}}{w_{0}\sqrt{L}}\sqrt{\frac{2p!}{\pi(|\ell|+p)!}}\textrm{L}_{p}^{|\ell|}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}\,e^{-\frac{\rho^{2}}{w_{0}^{2}}}e^{i\ell\phi}e^{ik_{\parallel}z},$
(13)

where $\textrm{L}_{p}^{|\ell|}$ are the Laguerre polynomials, with $\ell$ and
$p$ being the azimuthal and radial quantum numbers, respectively, and $w_{0}$
is the $z$-independent beam waist. Fig. 2(c) displays two different LG modes
with finite beam waists $w_{0}$.

Similarly, the HG family of transversely localized wave functions are
solutions to the paraxial wave equation in the Cartesian coordinate system.
Since the nondiffracting LG and HG states each comprise a complete orthonormal
basis, any LG (HG) state can be synthesized from the appropriate coherent
superposition of HG (LG) states [82]. Unlike the LG transverse states, the HG
states lack a well-defined azimuthal phase, and owing to the fact that they
are not eigenstates ${\hat{L}}_{z}$, do not carry a single OAM unit of $\ell$.
In the nondiffracting limit, the HG states take the form

$\psi_{nm}(x,y,z)=\frac{2^{-\frac{n+m}{2}}}{w_{0}\sqrt{L}}\sqrt{\frac{2}{\pi
n!m!}}\,H_{n}\bigg{(}\frac{x\sqrt{2}}{w_{0}}\bigg{)}H_{m}\bigg{(}\frac{y\sqrt{2}}{w_{0}}\bigg{)}e^{-\frac{x^{2}+y^{2}}{w_{0}^{2}}}e^{ik_{\parallel}z},$
(14)

and are labeled by the indices $n$ and $m$, corresponding to the order of the
$x$\- and $y$-dependent Hermite polynomials, $H_{n}$ and $H_{m}$,
respectively. Fig. 2(d) displays the first-order $x$\- and $y$-oriented HG
modes with beam waists $w_{0}$, in contrast to the corresponding delocalized
$\Psi_{k_{x}}(x,y)$ and $\Psi_{k_{y}}(x,y)$ plane waves displayed in Fig.
2(a).

Mirroring applications of optical OAM states, free electrons with quantized
transverse degrees of freedom have recently been recognized as potential
carriers of quantum information, specifically as free electron OAM qubits.
Realization of such OAM qubits has been made possible via holographic masks
and spiral phase plates [45, 48, 46, 47] or through tailored light sources
[49, 50]. Stemming from the separability of the electron wave function
following condition (ii), the orthogonal transverse degrees of freedom can be
used as orthonormal basis states $|0\rangle$ and $|1\rangle$ on the Bloch
sphere (Fig. 2(e)). Known as the horizontal and vertical basis states,
respectively, linear combinations of $|0\rangle$ and $|1\rangle$ assemble the
remaining antipodal points $|L/R\rangle=(1/\sqrt{2})\big{[}|0\rangle\pm
i|1\rangle\big{]}$ and
$|\pm\rangle=(1/\sqrt{2})\big{[}|0\rangle\pm|1\rangle\big{]}$. One example are
the first order LG states $\Psi_{\pm 10}(\rho,\phi)=\langle
x_{1}x_{2}|L/R\rangle$ which span the truncated $\ell=\pm 1$, two dimensional
Hilbert space [82, 56, 57], and lie on the north and south poles of the Bloch
sphere. Owing to the fact that $\Psi_{\pm 10}(\rho,\phi)$ can be expressed as
linear combinations of the first order HG states [15], $\Psi_{10}(x,y)=\langle
x_{1}x_{2}|0\rangle$ and $\Psi_{01}(x,y)=\langle x_{1}x_{2}|1\rangle$ are the
wave functions associated with the vertical and horizontal basis states,
respectively. At the center of the Bloch sphere lies the Gaussian mode
$\Psi_{00}(x,y)=\Psi_{00}(\rho,\phi)=\langle x_{1}x_{2}|G\rangle=\sqrt{2/\pi
w_{0}^{2}}e^{-(x^{2}+y^{2})/w_{0}^{2}}=\sqrt{2/\pi
w_{0}^{2}}e^{-\rho^{2}/w_{0}^{2}}$. More generally, higher order electron
vortex states with topological charge of $\pm\ell$ can be used to construct
Bloch spheres for $\ell$ values other than unity [18]. Following the work of
Ref. [67], under the appropriate limits, certain superpositions of electron
plane wave states can likewise be mapped onto points on the Bloch sphere (Fig.
2(e)). In this scenario, the horizontal and vertical plane wave wave functions
$\Psi_{k_{x}}(x,y)=\langle x_{1}x_{2}|0\rangle$ and $\Psi_{k_{y}}(x,y)=\langle
x_{1}x_{2}|1\rangle$ can be used to construct the north and south antipodal
point wave functions $\Psi_{k_{x}k_{y}}^{\pm\pi/2}(x,y)$. Specifically,
$\Psi_{k_{x}k_{y}}^{\pm\pi/2}(x,y)=\langle
x_{1}x_{2}|L/R\rangle=(1/\sqrt{2})\big{[}\Psi_{k_{x}}(x,y)+\Psi_{k_{y}}(x,y)e^{i\chi}\big{]}$
with $\chi=\pm\pi/2$. At the center of the Bloch sphere is located the
electron plane wave $\Psi_{k_{\parallel}}(x,y)=\langle
x_{1}x_{2}|G\rangle=1/L$ with spatially uniform transverse phase and purely
longitudinal wave vector $\mathbf{k}=k_{\parallel}\hat{\mathbf{z}}.$ As will
be shown in the following section, transitions between OAM or linear momentum
electron states residing on the Bloch sphere produce transition current
densities with unique vector and phase profiles.

##  IV Transition current density

Figure 3: Transition current density vector fields $\mathbf{J}_{fi}({\bf x})$
in the $z=0$ plane. The arrows within each panel show the direction and
strength of the transverse component $\mathbf{J}^{\perp}_{fi}({\bf x})$, while
the underlying color map indicates the phase ($\mathbf{J}_{fi}^{\perp}({\bf
x})\cdot\hat{\bm{\eta}}$) and intensity ($|\mathbf{J}_{fi}^{\perp}({\bf
x})|$). $\hat{\bm{\eta}}$ is displayed in the bottom right hand corner of each
panel. The initial transverse wave functions for the transition current
density vector fields are labeled in the top-center of each panel, while the
final state is shown at the bottom of each column. (a) Transition current
densities involving LG $\Psi_{\ell p}(\rho,\phi)$ initial and final states.
The top panel shows the case of $\Psi_{10}(\rho,\phi)\to\Psi_{10}(\rho,\phi)$
with no transverse transition; the
$\Psi_{11}(\rho,\phi)\to\Psi_{10}(\rho,\phi)$ transition in the lower panel
involves a transition in radial, but not OAM, quantum numbers. (b) Transitions
between Bessel states $\Psi_{\ell k_{\perp}}(\rho,\phi)$. The top panel shows
the transition current density for
$\Psi_{1k_{\perp}}(\rho,\phi)\to\Psi_{1k_{\perp}}(\rho,\phi)$ with no
transverse state transition, while the lower panel considers the transition
current density for
$\Psi_{1k_{\perp}}(\rho,\phi)\to\Psi_{2k_{\perp}}(\rho,\phi)$ with an OAM
change of $\Delta\ell=1$. (c) Incoming electron plane wave states
transitioning to the pin hole state $\Psi_{k^{{}^{\prime}}_{\parallel}}(x,y)$.
The initial state is characterized by
$\mathbf{q}_{\perp}=k_{x}\hat{\mathbf{x}}$ in the top panel, while the lower
panels involve the incident superposition plane wave state
$\Psi^{\pi/2}_{k_{x}k_{y}}(x,y)$. The lower panel is a magnified view of the
middle panel in the vicinity of the origin. (d) Transition current densities
for first-order HG and LG wave functions transitioning to the Gaussian state.
(e) Transition current densities for transitions from selected HG and LG
states to the pin hole state. (f) $\hat{\bm{\mathcal{J}}}_{fi}^{\perp}$ for
transitions between free electron OAM or linear momentum states which create
transition current vector fields resembling the polarization vectors of light
on the Poincaré sphere. The antipodal points
$\big{\\{}\hat{\bm{\mathcal{J}}}_{0}^{\perp},\hat{\bm{\mathcal{J}}}_{1}^{\perp}\big{\\}}$,
$\big{\\{}\hat{\bm{\mathcal{J}}}_{+}^{\perp},\hat{\bm{\mathcal{J}}}_{-}^{\perp}\big{\\}},$
and
$\big{\\{}\hat{\bm{\mathcal{J}}}_{L}^{\perp},\hat{\bm{\mathcal{J}}}_{R}^{\perp}\big{\\}}$
are color coded to match the box colors outlining the panels in columns (c)
through (e), as well as the electron state transition arrows in Fig. 2(e).

Here the transition current density $\mathbf{J}_{fi}(\mathbf{x})$ defined in
Eq. (4) is examined for selected transitions between electron states
introduced in the previous section. This section begins with a summary of
general properties of $\mathbf{J}_{fi}(\mathbf{x})$ that arise from the
restrictions imposed on the forms of the wave functions, then highlights
several general and specific forms of the transition current density for
transitions involving delocalized and localized electron states. A note
regarding the arguments of the transition current density: recall that the
origin of time dependence in
$\mathbf{J}_{fi}(\mathbf{x},t)=\mathbf{J}_{fi}(\mathbf{x})e^{i\varepsilon_{fi}t}$
as seen in Eq. (3) is a consequence of condition (i) imposed upon the electron
wave functions introduced in Sec. III. Therefore, as a matter of convenience
and unless stated otherwise, we work with the time-independent version of the
transition current density. In addition, when discussing the transition
current density for Bessel, LG, and HG free electron states, the subscripts of
$\mathbf{J}_{fi}(\mathbf{x})$ refer explicitly to the final and initial
transverse electron states. Panels (a) and (b) of Fig. (3) present LG and
Bessel beam transitions, respectively, panel (c) involves transitions between
plane wave states, panels (d) and (e) showcase the transition current density
using focused first order HG states.

Stemming from conditions (i) and (ii) required of the electron wave functions
as stipulated in Sec. III, the transition current density given by Eq. (4) can
be re-expressed as the sum of longitudinal ($\parallel$) and transverse
($\perp$) contributions, defined relative to the TEM axis, according to

$\begin{split}\mathbf{J}_{fi}(\mathbf{x})&=\mathbf{J}_{fi}(\mathbf{x}_{\perp})e^{iq_{\parallel}z}\\\
&=\big{[}\mathbf{J}_{fi}^{\perp}(\mathbf{x}_{\perp})+J_{fi}^{\parallel}(\mathbf{x}_{\perp})\hat{\bf
z}\big{]}e^{iq_{\parallel}z}.\end{split}$ (15)

Explicitly, these components are

$\displaystyle\mathbf{J}_{fi}^{\perp}(\mathbf{x}_{\perp})$
$\displaystyle=\frac{i\hbar
e}{2mL}\big{[}\Psi_{f}^{*}(\mathbf{x}_{\perp})\nabla_{\perp}\Psi_{i}(\mathbf{x}_{\perp})-\Psi_{i}(\mathbf{x}_{\perp})\nabla_{\perp}\Psi_{f}^{*}(\mathbf{x}_{\perp})\big{]}$
(16) $\displaystyle J_{fi}^{\parallel}(\mathbf{x}_{\perp})$
$\displaystyle=-\frac{\hbar
e}{2mL}Q_{\parallel}\Psi_{f}^{*}(\mathbf{x}_{\perp})\Psi_{i}(\mathbf{x}_{\perp}),$

where $Q_{\parallel}=k_{\parallel}+k_{\parallel}^{\prime}$. It can be seen
from the perpendicular component of Eq. (16), that interchange of the initial
and final transverse states is equivalent to conjugation of the reciprocal
scattering process, i.e.,
$\mathbf{J}_{if}(\mathbf{x})=\mathbf{J}^{*}_{fi}(\mathbf{x})$. As a
consequence, when the target’s responses obey reciprocity and are described by
the electromagnetic Green’s tensor
$\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)=\tensor{\mathbf{G}}^{T}(\mathbf{x}^{\prime},\mathbf{x},\omega)$,
the EEL and EEG observables are unaffected by the interchange of the 3D
initial and final states (see Appendix). Secs. V.1 and V.2 highlight examples
of the effects of interchanging initial and final states on the transversely
phase-shaped EEL, EEG, and DDCS measurements.

###  IV.1 Plane wave states

First we consider transitions between individual incoming $\psi_{{\bf
k}_{i}}(x,y,z)$ and outgoing $\psi_{{\bf k}_{f}}(x,y,z)$ plane wave states of
the probe, defined in Eq. (11), such that $\hbar{\bf q}=\hbar({\bf k}_{i}-{\bf
k}_{f})$ is the momentum recoil associated with the transition. The transition
current density associated with this case is

$\mathbf{J}_{{{\bf k}_{f}}{{\bf
k}_{i}}}(x,y,z)=-\frac{e\hbar}{2mL^{3}}(2\mathbf{k}_{i}-\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{x}}.$
(17)

Inspection of Eq. (17) reveals that the transverse component of the transition
current density is directly proportional to the transverse recoil momentum,
i.e., $\mathbf{J}_{{{\bf k}_{f}}{{\bf
k}_{i}}}^{\perp}(x,y,z)\propto\mathbf{q}_{\perp}$. The phase and vector
structure of the transition current density for an initial plane wave state
with transverse wave vector $\mathbf{k}_{\perp}=k_{x}\hat{\mathbf{x}}$
transitioning to the outgoing plane wave state
$\psi_{k^{\prime}_{\parallel}}(x,y,z)=L^{-3/2}e^{ik^{\prime}_{\parallel}z}$ is
shown in the first panel of Fig. 3(c). This final state is referred to as the
pin hole state as it is selected by placement of a pin hole on the TEM axis in
the Fourier plane. In Fig. 3(c), it is evident that the periodicity of the
plane wave states apparent in Fig. 2(a) is inherited by the transition current
density.

When prepared in a superposition plane wave state
$\psi_{k_{x}k_{y}}^{\chi}(x,y,z)$ and post-selected in the diffraction plane
for the pin hole state, the resulting transition current density of the probe
is

$\displaystyle\mathbf{J}_{k_{\parallel}^{\prime},k_{x}k_{y}}^{\chi}(x,y,z)$
$\displaystyle=-\frac{e\hbar}{2\sqrt{2}mL^{3}}\big{[}k_{x}e^{ik_{x}x}\hat{\mathbf{x}}+k_{y}e^{i\chi}e^{ik_{y}y}\hat{\mathbf{y}}+Q_{\parallel}\big{(}e^{ik_{x}x}+e^{i\chi}e^{ik_{y}y}\big{)}\hat{\mathbf{z}}\big{]}e^{iq_{\parallel}z}.$
(18)

The phase and vector information of this transition current density are
displayed in the final two panels of Fig. 3(c) for $\chi=\pi/2$ at two
different magnification levels. Owing to the spatial dependence of the wave
functions, the $\hat{\mathbf{x}},\hat{\mathbf{y}}$-components of Eq. (18) are
functionally dependent upon $x$ and $y$. However, when working in the dipole
scattering regime, whereby $\mathbf{q}_{\perp}$ is such that
$|\mathbf{q}_{\perp}|d\ll 1$, with $d$ the transverse length scale of the
target, $\mathbf{J}_{fi}^{\perp}(\mathbf{x}_{\perp})$ becomes approximately
independent of position in the vicinity of the target. Under these
circumstances when the perpendicular components of the transition current
density become independent of position,
${\mathbf{J}}_{fi}^{\perp}(\mathbf{x}_{\perp})\to{\bm{\mathcal{J}}}_{fi}^{\perp}$.
After normalization,
$\hat{\bm{\mathcal{J}}}_{fi}^{\perp}={\bm{\mathcal{J}}}_{fi}^{\perp}/|{\bm{\mathcal{J}}}_{fi}^{\perp}|$
can thus be mapped onto the Poincaré sphere (Fig. 3(f)) [67]. In this limit,
for $\chi=\pi/2$, the transverse component of Eq. (18) becomes circularly
polarized such that it mimics the polarization of a left-handed circularly
polarized photon, evident in the final panel of Fig. 3(c) boxed in red. All
together, when $|\mathbf{q}_{\perp}|d\ll 1$, the transition current densities
given by Eqs. (17) and (18) can be tailored through appropriate pre- and post-
selection to resemble any polarization state of light on the Poincaré sphere
as shown in Fig. 3(f). Eq. (17) can be utilized to produce the horizontal and
vertical polarization vectors
$\big{\\{}\hat{\bm{\mathcal{J}}}_{0}^{\perp},\hat{\bm{\mathcal{J}}}_{1}^{\perp}\big{\\}}$,
respectively. Whereas, by varying the phase $\chi$, the transverse components
of Eq. (18) produce the remaining antipodal points
$\big{\\{}\hat{\bm{\mathcal{J}}}_{R}^{\perp},\hat{\bm{\mathcal{J}}}_{+}^{\perp},\hat{\bm{\mathcal{J}}}_{L}^{\perp},\hat{\bm{\mathcal{J}}}_{-}^{\perp}\big{\\}}$.

###  IV.2 Bessel and Laguerre-Gauss states

By virtue of the ability to prepare free electrons in defocused vortex
(Bessel) or focused vortex (LG) states, the corresponding transition current
density can exhibit both radial and azimuthal transverse vector components.
The transition current density associated with transitions between Bessel beam
states $\psi_{\ell k_{\perp}}(\rho,\phi,z)$ and
$\psi_{\ell^{\prime}k^{\prime}_{\perp}}(\rho,\phi,z)$ is [67]

$\displaystyle\mathbf{J}_{\ell^{\prime}k_{\perp}^{\prime},\ell
k_{\perp}}(\rho,\phi,z)$ $\displaystyle=\frac{i\hbar e}{2m}\Psi_{\ell
k_{\perp}}(\rho,\phi)\Psi_{\ell^{\prime}k_{\perp}^{\prime}}^{*}(\rho,\phi)\bigg{[}\bigg{\\{}\frac{k_{\perp}}{2}J_{|\ell|}(k_{\perp}\rho)^{-1}\big{[}J_{|\ell|-1}(k_{\perp}\rho)-J_{|\ell|+1}(k_{\perp}\rho)\big{]}$
(19) $\displaystyle\ \ \
-\frac{k_{\perp}^{\prime}}{2}J_{|\ell^{\prime}|}^{*}(k_{\perp}^{\prime}\rho)^{-1}\big{[}J_{|\ell^{\prime}|-1}^{*}(k_{\perp}^{\prime}\rho)-J_{|\ell^{\prime}|+1}^{*}(k_{\perp}^{\prime}\rho)\big{]}\bigg{\\}}\hat{\bm{\rho}}+\frac{i}{\rho}(\ell+\ell^{\prime})\hat{\bm{\phi}}+iQ_{\parallel}\hat{\mathbf{z}}\bigg{]}e^{iq_{\parallel}z},$

for arbitrary $\ell,k_{\perp}$ and $\ell^{\prime},k_{\perp}^{\prime}$. The
transition current density for $\ell^{\prime}=\ell=1$ and
$k_{\perp}^{\prime}=k_{\perp}$, with $\Delta\ell=0$ is displayed in the first
panel of Fig. 3(b), showcasing the expected azimuthal character inherent to
twisted electron beams. Bessel beam states can also lead to unique phase and
vector profiles as seen in the lower panel of Fig. 3(b), wherein
$\Delta\ell=1$ resulting in a transition current density with a phase-
structure similar to that of circularly polarized light, as well as to the
wave functions shown in Fig. 2(b).

Additionally, the focused LG states in Eq. (13) can also produce transition
current densities with well-defined units of OAM transferred. The transition
current density for an initial LG electron state $\psi_{\ell p}(\rho,\phi,z)$
transitioning to a final state $\psi_{\ell^{\prime}p^{\prime}}(\rho,\phi,z)$
is given by

$\displaystyle\mathbf{J}_{\ell^{\prime}p^{\prime},\ell p}(\rho,\phi,z)$
$\displaystyle=\frac{i\hbar e}{2m}\Psi_{\ell
p}(\rho,\phi)\Psi_{\ell^{\prime}p^{\prime}}^{*}(\rho,\phi)\bigg{[}\frac{1}{\rho}\bigg{\\{}\big{(}|\ell|-|\ell^{\prime}|\big{)}$
(20) $\displaystyle\ \ \
-\frac{4\rho^{2}}{w_{0}^{2}}\bigg{(}\textrm{L}_{p}^{|\ell|}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}\textrm{L}_{p^{\prime}}^{|\ell^{\prime}|*}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}\bigg{)}^{-1}\bigg{(}\textrm{L}_{p^{\prime}}^{|\ell^{\prime}|*}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}\textrm{L}_{p}^{|\ell|+1}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}-\textrm{L}_{p}^{|\ell|}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}\textrm{L}_{p^{\prime}}^{|\ell^{\prime}|+1,*}\bigg{(}\frac{2\rho^{2}}{w^{2}_{0}}\bigg{)}\bigg{)}\bigg{\\}}\hat{\bm{\rho}}$
$\displaystyle\ \ \
+\frac{i}{\rho}(\ell+\ell^{\prime})\hat{\bm{\phi}}+iQ_{\parallel}\hat{\mathbf{z}}\bigg{]}e^{iq_{\parallel}z}$

valid for arbitrary $\ell,p$ and $\ell^{\prime},p^{\prime}$. Stemming from the
commonalities in the underlying wave functions, it is unsurprising that Eq.
(19) and Eq. (20) look similar. This resemblance is most obvious when
considering events wherein no transverse transition occurs, as seen when
comparing the first panels of Fig. 3(a) and Fig. 3(b), for transitions between
LG and Bessel states, respectively. Here, shown in the top panel of Fig. 3(a),
is Eq. (20) for $p^{\prime}=p=0$ and $\ell^{\prime}=\ell=1$, which highlights
the azimuthal vector component of the LG transition current density and is
similar to the corresponding Bessel beam case displayed in the top panel of
Fig. 3(b) for $1k_{\perp}$ transitioning to $1k_{\perp}$. The bottom panel of
Fig. 3(a) shows the transition current density for $\Delta p=1$. For inelastic
scattering events where the electron does not transfer OAM to the target,
$\ell^{\prime}=\ell$ and $p^{\prime}=p$, and Eq. (20) reduces to

$\displaystyle\mathbf{J}_{\ell^{\prime}=\ell,p^{\prime}=p}(\rho,\phi,z)=-\frac{\hbar
e}{2mL}\big{|}\Psi_{\ell
p}(\rho,\phi)\big{|}^{2}\Big{[}\frac{2\ell}{\rho}\hat{\bm{\phi}}+Q_{\parallel}\hat{\mathbf{z}}\Big{]}e^{iq_{\parallel}z},$
(21)

which is the focused vortex beam form of the transition current density of a
free electron moving along $\mathbf{\hat{v}}=\mathbf{\hat{z}}$ [80, 83]. The
azimuthal component of Eq. (21) is responsible for the spiraling behavior of
the electron current typical for vortex beams, and upon spatial integration
over $(\rho,\phi)$ yields an electron current with an axial inertial OAM equal
to $\ell\hbar$ [80].

###  IV.3 Hermite-Gauss states

Transitions between transverse HG states that are naturally expressed in the
Cartesian basis provide transition current densities with any desired
$\hat{\mathbf{x}},\hat{\mathbf{y}}$-vectorial structure. Following the
approach of Ref. [67], a general form for
$\mathbf{J}_{n^{\prime}m^{\prime},nm}(x,y,z)$ involving arbitrary HG states
can be derived from Eq. (4). The transition current density for
$\psi_{nm}(x,y,z)$ transitioning to $\psi_{n^{\prime}m^{\prime}}(x,y,z)$ is

$\displaystyle\mathbf{J}_{n^{\prime}m^{\prime},nm}(x,y,z)$
$\displaystyle=\frac{i\hbar
e}{2mw_{0}L}\bigg{[}\big{\\{}\sqrt{n}\Psi_{n-1,m}(x,y)\Psi_{n^{\prime}m^{\prime}}^{*}(x,y)-\sqrt{n+1}\Psi_{n+1,m}(x,y)\Psi_{n^{\prime}m^{\prime}}^{*}(x,y)$
(22) $\displaystyle\ \ \
-\sqrt{n^{\prime}}\Psi_{nm}(x,y)\Psi_{n^{\prime}-1,m^{\prime}}^{*}(x,y)+\sqrt{n^{\prime}+1}\Psi_{nm}(x,y)\Psi_{n^{\prime}+1,m^{\prime}}^{*}(x,y)\big{\\}}\hat{\mathbf{x}}$
$\displaystyle\ \ \
+\big{\\{}\sqrt{m}\Psi_{n,m-1}(x,y)\Psi_{n^{\prime}m^{\prime}}^{*}(x,y)-\sqrt{m+1}\Psi_{n,m+1}(x,y)\Psi_{n^{\prime}m^{\prime}}^{*}(x,y)$
$\displaystyle\ \ \
-\sqrt{m^{\prime}}\Psi_{nm}(x,y)\Psi_{n^{\prime},m^{\prime}-1}^{*}(x,y)+\sqrt{m^{\prime}+1}\Psi_{nm}(x,y)\Psi_{n^{\prime},m^{\prime}+1}^{*}(x,y)\big{\\}}\hat{\mathbf{y}}$
$\displaystyle\ \ \
+iw_{0}Q_{\parallel}\Psi_{nm}(x,y)\Psi_{n^{\prime}m^{\prime}}^{*}(x,y)\hat{\mathbf{z}}\bigg{]}e^{iq_{\parallel}z},$

for arbitrary $n,m,n^{\prime},m^{\prime}$. As seen in the form of the
perpendicular component of Eq. (22), careful choice regarding the pre- and
post-selection of the initial/final HG transverse electron states can be
intuited to yield a transition current density with non-trivial
$\hat{\mathbf{x}},\hat{\mathbf{y}}$-vectorial behavior in the plane orthogonal
to the direction of propagation. Working in the $\ell=\pm 1$ OAM Hilbert
space, $\mathbf{J}_{n^{\prime}m^{\prime},nm}(x,y,z)$ is displayed in the
columns of Fig. 3(d) for first order electron wave functions
$\psi_{10}(x,y,z)$ and $\psi_{01}(x,y,z)$, and their linear combination
$\psi_{10,01}^{\chi=\pi/2}(x,y,z)=(1/\sqrt{2L})\big{[}\psi_{10}(x,y,z)+\psi_{01}(x,y,z)e^{i\pi/2}\big{]}$,
respectively, all transitioning to the final Gaussian wave function
$\psi_{00}(x,y,z)=(1/\sqrt{L})\Psi_{00}(x,y)e^{ik_{\parallel}^{\prime}z}$. It
is apparent that the various transverse state transitions create distinct
transition current densities, with similar or dissimilar phase and vector
field textures depending upon the characteristics of the underlying wave
functions. In the zero width limit whereby $w_{0}\rightarrow 0$, for the first
order transitions discussed above, the transverse components of the transition
current density become spatially independent and $\mathbf{J}^{\perp}_{fi}({\bf
x})\to{\bm{\mathcal{J}}}^{\perp}_{fi}$. Under these constraints, the
transition current density unit vector ${\bm{\mathcal{J}}}^{\perp}_{fi}$ can
thus imitate the polarization vector $\hat{\bm{\epsilon}}$ of free space
light, as illustrated on the electron analog of the Poincaré sphere presented
in Fig. 3(f). When no transverse transition of the probe occurs, i.e.,
$n^{\prime}=n$ and $m^{\prime}=m$ in Eq. (22), the transition current density
reduces to

$\mathbf{J}_{n^{\prime}=n,m^{\prime}=m}(x,y,z)=-\frac{e\hbar}{2mL}Q_{\parallel}\big{|}\Psi_{nm}(x,y)\big{|}^{2}e^{iq_{\parallel}z}\hat{\mathbf{z}},$
(23)

which is oriented parallel to the TEM axis regardless of the values for $n$
and $m$. In the zero width limit, Eq. (23) reduces to the classical current of
a point electron source [51] when $n=m=0$. Overall, the purely longitudinal
behavior of Eq. (23) differs from the vortex electron beam case in Eq. (21)
wherein the current density has a $\hat{\bm{\phi}}$-component perpendicular to
the electron’s direction of motion.

Alternatively, when the $nm$ HG state transitions to the forward directed pin
hole state rather than a transversely focused Gaussian state, the transition
current density becomes

$\displaystyle\mathbf{J}_{k_{\parallel}^{\prime},nm}(x,y,z)$
$\displaystyle=\frac{i\hbar
e}{2mw_{0}L^{2}}\bigg{[}\big{\\{}\sqrt{n}\Psi_{n-1,m}(x,y)-\sqrt{n+1}\Psi_{n+1,m}(x,y)\big{\\}}\hat{\mathbf{x}}$
(24) $\displaystyle\ \ \
+\big{\\{}\sqrt{m}\Psi_{n,m-1}(x,y)-\sqrt{m+1}\Psi_{n,m+1}(x,y)\big{\\}}\hat{\mathbf{y}}$
$\displaystyle\ \ \
+iw_{0}Q_{\parallel}\Psi_{n,m}(x,y)\hat{\mathbf{z}}\bigg{]}e^{iq_{\parallel}z}.$

Eq. (24) is displayed in Fig. 3(e) for the same initial HG states as those
used in Fig. 3(d), transitioning instead to the pin hole state
$\psi_{k^{\prime}_{\parallel}}(x,y,z)$. For the focused superposition state
$\psi_{10,01}^{\chi}(x,y,z)=(1/\sqrt{2L})\big{[}\psi_{10}(x,y,z)+\psi_{01}(x,y,z)e^{i\chi}\big{]}$
transitioning to the pin hole state $\psi_{k^{\prime}_{\parallel}}(x,y,z)$,
the transition current density explicitly has the form

$\displaystyle\mathbf{J}_{k_{\parallel}^{\prime},10,01}^{\chi}(x,y,z)$
$\displaystyle=\frac{i\hbar
e}{2\sqrt{2}mw_{0}L^{2}}\bigg{[}\big{\\{}\Psi_{00}(x,y)-\sqrt{2}\Psi_{20}(x,y)-\Psi_{11}(x,y)e^{i\chi}\big{\\}}\hat{\mathbf{x}}$
(25) $\displaystyle\ \ \
+\big{\\{}\big{(}\Psi_{00}(x,y)-\sqrt{2}\Psi_{02}(x,y)\big{)}e^{i\chi}-\Psi_{11}(x,y)\big{\\}}\hat{\mathbf{y}}$
$\displaystyle\ \ \
+iw_{0}Q_{\parallel}\big{\\{}\Psi_{10}(x,y)+\Psi_{01}(x,y)e^{i\chi}\big{\\}}\hat{\mathbf{z}}\bigg{]}e^{iq_{\parallel}z}.$

The difference between Eqs. (18) and (25) originates from the choice of the
initial electron wave function being either a superposition of linear momentum
or OAM states, respectively. Imposing the same conditions placed upon the wave
functions in Eq. (22) to obtain a transition current density whose transverse
vectorial components are spatially independent, Eqs. (24) and (25) can be used
to construct transition current densities on the surface of the Poincaré
sphere as presented in Fig. 3(f). The former equation can be manipulated to
produce the horizontal and vertical linear polarization profiles
$\big{\\{}\hat{\bm{\mathcal{J}}}_{0}^{\perp},\hat{\bm{\mathcal{J}}}_{1}^{\perp}\big{\\}}$,
whereas the latter can be used to construct the diagonal
$\hat{\bm{\mathcal{J}}}_{L/R}^{\perp}$ and circularly polarized
$\hat{\bm{\mathcal{J}}}_{\pm}^{\perp}$ profiles depending upon the value of
$\chi$. Therefore, transitions between specific states on the Bloch sphere
(Fig. 2(e)) produce transition current densities which mimic the polarization
structure of free space light and can thus be mapped onto the Poincaré sphere
(Fig. 3(f)).

##  V State- and energy-resolved Observables

Building from Secs. II, III, and IV, in this section the EEL, EEG, and DDCS
observables between phase-shaped states of the electron probe are derived.
Measurements of EEL and EEG processes are discussed first, including the
narrow beam width limit common in the low-loss electron scattering regime,
before moving on to presentation of the DDCS. A comparison of the EEL and EEG
scattering processes, and their relation to the properties of the transition
current density under interchange of initial/final electron states is briefly
presented. We illustrate that the transversely phase-shaped EEL, EEG, and DDCS
observables, reduce to the familiar forms found in the literature under the
appropriate limits.

###  V.1 Electron energy loss and gain probabilities in the narrow beam width
limit

In considering low-loss EEL and EEG scattering events in the narrow beam width
limit appropriate to the STEM it is customary to work within the nonrecoil
approximation, where the change in the energy of the electron is dictated
entirely by its momentum change along the axis of propagation [51]. In this
limit, the forward recoil momentum $\hbar q_{\parallel}\hat{\bf z}$ is small
compared to the electron’s initial momentum $\hbar{\bf k}_{i}$ so that its
change in energy can be approximated by
$\hbar\varepsilon_{fi}=\sqrt{(mc^{2})^{2}+(\hbar
c\mathbf{k}_{i})^{2}}-\sqrt{(mc^{2})^{2}+(\hbar
c(\mathbf{k}_{i}-\mathbf{q}))^{2}}\approx-\hbar\mathbf{v}_{i}\cdot\mathbf{q}$,
where $\hbar\mathbf{k}_{i}/m=\gamma_{i}\mathbf{v}_{i}$ and
$\hbar\mathbf{k}_{i}=\mathbf{p}_{i}$ for relativistic matter waves. Upon
insertion of $\varepsilon_{fi}=-\mathbf{v}_{i}\cdot\mathbf{q}$ into the
trailing delta function in Eq. (9), the state- and frequency-resolved EEL rate
becomes

$w_{fi}^{\textrm{loss}}(\omega)=-\frac{8\pi}{\hbar v_{i}}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\textrm{Im}\Big{[}\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\Big{]}\delta(q_{\parallel}-\omega/v_{i}),$
(26)

where $\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ is the
target’s electromagnetic Green’s tensor introduced in Sec. II. The state- and
energy-resolved EEL probability
$P_{fi}^{\textrm{loss}}(\omega)=(L/v_{i})w_{fi}^{\textrm{loss}}\,(\omega)$ is
obtained by integrating Eq. (26) over the time it takes the probe electron to
traverse the path length $L$ as it interacts with the target specimen. From
$P_{fi}^{\textrm{loss}}(\omega)$, the state- and energy-resolved EEL
probability is determined by summing over all possible final electron states
with
$\sum_{k_{\parallel}^{f}}\to+(L/2\pi)\int^{\infty}_{-\infty}dq_{\parallel}$
and dividing by $\hbar$, resulting in the EEL probability per unit energy

$\Gamma_{fi}^{\textrm{loss}}(\omega)=-\frac{4}{\hbar^{2}}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\textrm{Im}\Big{[}\bigg{(}\frac{L}{v_{i}}\bigg{)}\mathbf{J}_{fi}^{*}(\mathbf{x};q_{\parallel}=\frac{\omega}{v_{i}})\cdot\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\bigg{(}\frac{L}{v_{i}}\bigg{)}\mathbf{J}_{fi}(\mathbf{x}^{\prime};q_{\parallel}=\frac{\omega}{v_{i}})\Big{]}.$
(27)

Here
$\mathbf{J}_{fi}(\mathbf{x};q_{\parallel}={\omega}/{v_{i}})=\mathbf{J}_{fi}(\mathbf{x})|_{q_{\parallel}={\omega}/{v_{i}}}=\mathbf{J}_{fi}(\mathbf{x}_{\perp})e^{iq_{\parallel}z}|_{q_{\parallel}={\omega}/{v_{i}}}=\mathbf{J}_{fi}(\mathbf{x}_{\perp})e^{i(\omega/v_{i})z}$
makes explicit the locking of the longitudinal recoil wave number
$q_{\parallel}$ to $\omega/v_{i}$ imposed by the nonrecoil approximation in
Eqs. (26) and (27). Due to the frequent appearance of the $L/v_{i}$ factor
here and in the following equations, we define a new transition current as
$\mathbf{J}_{fi}(\mathbf{x},\omega=q_{\parallel}{v_{i}})\equiv(L/v_{i})\mathbf{J}_{fi}(\mathbf{x};q_{\parallel}={\omega}/{v_{i}})$
with dimensions of charge flux per unit frequency. For clarity, we also
abandon the general notation $\mathbf{x}=(\mathbf{x}_{\perp},z)$ in favor of
$\mathbf{x}=(\mathbf{R},z)$ as is common in the low-loss STEM EEL and EEG
literature.

Using the current $\mathbf{J}_{fi}(\mathbf{x},\omega=q_{\parallel}{v_{i}})$,
the EEL probability can also be cast in terms of the target’s transition
electric field $\mathbf{E}_{fi}(\mathbf{x},\omega=q_{\parallel}{v_{i}})=-4\pi
i\omega\int
d\mathbf{x}^{\prime}\,\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime},\omega=q_{\parallel}{v_{i}})$
resolved in frequency. $\mathbf{E}_{fi}$ represents the electric field
produced by the target in response to stimulation by the probing electron.
Expressed in terms of this field, the EEL probability per unit energy becomes

$\Gamma_{fi}^{\textrm{loss}}(\omega)=-\frac{1}{\pi\hbar^{2}\omega}\textrm{Re}\bigg{[}\int
d\mathbf{x}\,\mathbf{J}_{fi}^{*}(\mathbf{x},\omega=q_{\parallel}{v_{i}})\cdot\mathbf{E}_{fi}(\mathbf{x},\omega=q_{\parallel}{v_{i}})\bigg{]},$
(28)

which recovers the classical relationship [84, 51] between
$\Gamma_{fi}^{\textrm{loss}}(\omega)$ and the work performed by the electron
against its own induced field. When the electron beam waist $w_{0}$ is
negligible compared to the length scale over which the response field changes,
$\mathbf{E}_{fi}$ can be taken as constant over the spatial domain where the
current density is appreciable and approximated by its value
$\mathbf{E}_{fi}(\mathbf{R},z,\omega=q_{\parallel}{v_{i}})\approx\mathbf{E}_{fi}(\mathbf{R}_{0},z,\omega=q_{\parallel}{v_{i}})$
at the impact parameter ${\bf R}_{0}$. In this narrow beam width limit, Eq.
(28) becomes

$\displaystyle\Gamma_{fi}^{\textrm{loss}}(\mathbf{R}_{0},\omega)$
$\displaystyle=-\frac{1}{\pi\hbar^{2}\omega}\textrm{Re}\bigg{[}\int
dz\bigg{(}\int
d\mathbf{R}\,\mathbf{J}_{fi}^{*}(\mathbf{R},z,\omega=q_{\parallel}{v_{i}})e^{i\omega
z/v_{i}}\bigg{)}e^{-i\omega
z/v_{i}}\cdot\mathbf{E}_{fi}(\mathbf{R}_{0},z,\omega=q_{\parallel}{v_{i}})\bigg{]}$
(29)
$\displaystyle=-\frac{1}{\pi\hbar^{2}\omega}\textrm{Re}\bigg{[}\bm{\mathcal{J}}^{*}_{fi}\cdot\int
dz\,\mathbf{E}_{fi}(\mathbf{R}_{0},z,\omega=q_{\parallel}{v_{i}})e^{-i\omega
z/v_{i}}\bigg{]},$

where the transition current $\bm{\mathcal{J}}_{fi}=\int
d\mathbf{R}\,\mathbf{J}_{fi}(\mathbf{R},z,\omega=q_{\parallel}{v_{i}})e^{-i\omega
z/v_{i}}=(-{e}/{mv_{i}})\big{[}\langle\Psi_{f}|\hat{\mathbf{p}}_{\perp}|\Psi_{i}\rangle+(\hbar
Q_{\parallel}/2)\langle\Psi_{f}|\Psi_{i}\rangle\hat{\mathbf{z}}\big{]}$ is
$z$\- and $\omega$-independent. It has the transverse and longitudinal
components
$\bm{\mathcal{J}}_{fi}^{\perp}=-(e/mv_{i})\langle\Psi_{f}|\hat{\mathbf{p}}_{\perp}|\Psi_{i}\rangle$
and $\bm{\mathcal{J}}_{fi}^{\parallel}=-(\hbar
eQ_{\parallel}/2mv_{i})\langle\Psi_{f}|\Psi_{i}\rangle\hat{\mathbf{z}}$,
respectively, which allow the EEL probability in Eq. (29) to be separated into
the perpendicular and parallel contributions

$\displaystyle\Gamma_{fi\perp}^{\textrm{loss}}(\mathbf{R}_{0},\omega)$
$\displaystyle=-\frac{1}{\pi\hbar^{2}\omega}\textrm{Re}\bigg{[}\bm{\mathcal{J}}_{fi}^{\perp*}\cdot\int
dz\,\mathbf{E}_{fi}(\mathbf{R}_{0},z,\omega=q_{\parallel}{v_{i}})e^{-i\omega
z/v_{i}}\bigg{]}$ (30)
$\displaystyle\Gamma_{fi\parallel}^{\textrm{loss}}(\mathbf{R}_{0},\omega)$
$\displaystyle=-\frac{1}{\pi\hbar^{2}\omega}\textrm{Re}\bigg{[}\bm{\mathcal{J}}_{fi}^{\parallel*}\cdot\int
dz\,\mathbf{E}_{fi}(\mathbf{R}_{0},z,\omega=q_{\parallel}{v_{i}})e^{-i\omega
z/v_{i}}\bigg{]}.$

Based on the forms of $\bm{\mathcal{J}}_{fi}^{\perp}$ and
$\bm{\mathcal{J}}_{fi}^{\parallel}$ above, it is evident that
$\bm{\mathcal{J}}_{fi}^{\perp}={\bf 0}$ and
$\bm{\mathcal{J}}_{fi}^{\parallel}=-(\hbar eQ_{\parallel}/2mv_{i})\hat{\bf z}$
in the event of no transverse transition (i.e., $\Psi_{i}=\Psi_{f}$ and
$\Gamma_{fi\perp}^{\textrm{loss}}=0$), while
$\bm{\mathcal{J}}_{fi}^{\perp}\neq{\bf 0}$ and
$\bm{\mathcal{J}}_{fi}^{\parallel}={\bf 0}$ when a transition occurs in the
probe’s transverse wave function (i.e., $\Psi_{i}\neq\Psi_{f}$ and
$\Gamma_{fi\parallel}^{\textrm{loss}}=0$). In the former case,
$\Gamma_{fi\parallel}^{\textrm{loss}}(\mathbf{R}_{0},\omega)$ in Eq. (30)
reduces to the the well known classical form for the EEL probability [69, 85,
51, 86] in the zero width limit $w_{0}\to 0.$ Specifically,
$\Gamma_{fi\parallel}^{\textrm{loss}}(\mathbf{R}_{0},\omega)\to\Gamma_{\textrm{cl}}(\mathbf{R}_{0},\omega)=-(2e/\hbar)^{2}\int
dzdz^{\prime}\,\textrm{Im}[\hat{\bf
z}\cdot\tensor{\mathbf{G}}(\mathbf{R}_{0},z;\mathbf{R}_{0},z^{\prime},\omega)\cdot\hat{\bf
z}\,e^{-i\omega(z-z^{\prime})/v_{i}}]$, which is the EEL probability per unit
energy for a uniformly moving classical electron with current density
$\mathbf{J}_{\textrm{cl}}(\mathbf{x},\omega)=-e\delta(\mathbf{R}-\mathbf{R}_{0})e^{i\omega
z/v_{i}}\,\hat{\mathbf{z}}$ at impact parameter ${\bf R}_{0}$.

To construct the laser-stimulated phase-shaped EEG observable in the narrow
beam width limit, we again introduce
$\varepsilon_{fi}=-\mathbf{v}_{i}\cdot\mathbf{q}$ into the trailing delta
function in the frequency-resolved EEG rate presented in Eq. (10). As a
result, the state- and frequency-resolved EEG rate and scattering probability
is

$w_{fi}^{\textrm{gain}}(\omega)=\frac{2\pi}{v_{i}}\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\bigg{|}\int
d\mathbf{x}\,\mathbf{E}_{\nu}^{(+)}(\mathbf{x})\cdot\mathbf{J}_{fi}(\mathbf{x})\bigg{|}^{2}\delta(q_{\parallel}+\omega/v_{i})\delta(\omega-\omega_{\nu\nu^{\prime}})$
(31)

and
$P_{fi}^{\textrm{gain}}(\omega)=\big{(}L/v_{i}\big{)}w_{fi}^{\textrm{gain}}\,(\omega)$.
In parallel to loss, the EEG probability per unit energy is determined from
$P_{fi}^{\textrm{gain}}(\omega)$ after integrating over final states
$\sum_{k^{f}_{\parallel}}\to+(L/2\pi)\int_{-\infty}^{\infty}dq_{\parallel}$
and dividing by $\hbar$, resulting in

$\Gamma_{fi}^{\textrm{gain}}(\omega)=\frac{1}{\hbar}\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\bigg{|}\int
d\mathbf{x}\,\mathbf{E}_{\nu}^{(+)}(\mathbf{x})\cdot\mathbf{J}_{fi}(\mathbf{x},\omega=-q_{\parallel}v_{i})\bigg{|}^{2}\delta(\omega-\omega_{\nu\nu^{\prime}}).$
(32)

As in the case of loss, if the target’s induced electric field varies little
over the spatial domain of the probe’s transition current density, then
$\mathbf{E}_{\nu}^{(+)}(\mathbf{R},z)\approx\mathbf{E}_{\nu}^{(+)}(\mathbf{R}_{0},z)$
to lowest order and the state- and energy-resolved EEG probability takes the
form

$\displaystyle\Gamma_{fi}^{\textrm{gain}}(\mathbf{R}_{0},\omega)$
$\displaystyle=\frac{1}{\hbar}\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\bigg{|}\int
dz\,\mathbf{E}_{\nu}^{(+)}(\mathbf{R}_{0},z)e^{-i\omega
z/v_{i}}\cdot\bigg{[}\int
d\mathbf{R}\,\mathbf{J}_{fi}(\mathbf{R},z,\omega=-q_{\parallel}v_{i})e^{i\omega
z/v_{i}}\bigg{]}\bigg{|}^{2}\delta(\omega-\omega_{\nu\nu^{\prime}})$ (33)
$\displaystyle=\frac{1}{\hbar}\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\bigg{|}\int
dz\,e^{-i\omega
z/v_{i}}\mathbf{E}_{\nu}^{(+)}(\mathbf{R}_{0},z)\cdot\bm{\mathcal{J}}_{fi}\bigg{|}^{2}\delta(\omega-\omega_{\nu\nu^{\prime}}),$

with $z$\- and $\omega$-independent transition current $\bm{\mathcal{J}}_{fi}$
defined below Eq. (29). As in the case of EEL, the EEG probability can also be
broken into perpendicular and parallel components

$\displaystyle\Gamma_{{fi}\perp}^{\textrm{gain}}(\mathbf{R}_{0},\omega)$
$\displaystyle=\frac{1}{\hbar}\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\big{|}{\mathcal{J}}_{fi}^{\perp}\big{|}^{2}\bigg{|}\int
dz\,e^{-i\omega
z/v_{i}}\mathbf{E}_{\nu}^{(+)}(\mathbf{R}_{0},z)\cdot\hat{\bm{\mathcal{J}}}_{fi}^{\perp}\bigg{|}^{2}\delta(\omega-\omega_{\nu\nu^{\prime}})$
(34)
$\displaystyle\Gamma_{fi\parallel}^{\textrm{gain}}(\mathbf{R}_{0},\omega)$
$\displaystyle=\frac{1}{\hbar}\bigg{(}\frac{1}{\hbar\omega_{\nu}}\bigg{)}^{2}\big{|}{\mathcal{J}}_{fi}^{\parallel}\big{|}^{2}\bigg{|}\int
dz\,e^{-i\omega
z/v_{i}}\mathbf{E}_{\nu}^{(+)}(\mathbf{R}_{0},z)\cdot\hat{\bm{\mathcal{J}}}_{fi}^{\parallel}\bigg{|}^{2}\delta(\omega-\omega_{\nu\nu^{\prime}}),$

where
${\bm{\mathcal{J}}}_{fi}=\hat{\bm{\mathcal{J}}}_{fi}|{\bm{\mathcal{J}}}_{fi}|=\hat{\bm{\mathcal{J}}}_{fi}{{\mathcal{J}}}_{fi}$.
Similarly, when the transverse wave functions $\Psi_{i}=\Psi_{f}$,
$\Gamma_{fi}^{\textrm{gain}}(\mathbf{R}_{0},\omega)=\Gamma_{fi\parallel}^{\textrm{gain}}(\mathbf{R}_{0},\omega)$
and recovers the conventional EEG probability [70, 51, 71, 72, 66] in the
narrow beam limit $w_{0}\to 0$. Through combination of optical polarization
and pre- and post-selection of the probe’s transverse phase profile to define
its polarization, cross-polarized measurements in the STEM can be leveraged to
directly interrogate optically-excited target mode symmetries in 3D with
nanoscale spatial resolution [66].

The above expressions for EEL and EEG probabilities involve interrogation of
the target’s induced response field by the transition current densities
$\mathbf{J}_{fi}(\mathbf{x},\omega=+q_{\parallel}v_{i})$ and
$\mathbf{J}_{fi}(\mathbf{x},\omega=-q_{\parallel}v_{i})$ of the probe. It is
natural to consider the relationship between these currents upon interchanging
initial and final probe states in both the transverse and axial directions.
The reciprocal behavior of pre- and post-selection of the probe’s transverse
states, i.e.,
$\mathbf{J}_{fi}^{\perp}(\mathbf{R})=\mathbf{J}_{if}^{\perp*}(\mathbf{R})$,
were discussed previously in Sec. IV. Additionally, along the TEM axis, the
longitudinal component
$J_{fi}^{\parallel}(\mathbf{R})=J_{if}^{\parallel*}(\mathbf{R})$ expresses the
relationship between anti-Stokes (EEG) and Stokes (EEL) scattering processes
at the level of the transition current density. Taken together,

$\displaystyle\mathbf{J}_{fi}(\mathbf{x},\omega=+q_{\parallel}{v_{i}})$
$\displaystyle=\Big{(}\frac{L}{v_{i}}\Big{)}\big{[}\mathbf{J}_{if}^{\perp*}(\mathbf{R})+J_{if}^{\parallel*}(\mathbf{R})\hat{\bf
z}\big{]}e^{i(\omega/v_{i})z}$ (35)
$\displaystyle=\Big{(}\frac{L}{v_{i}}\Big{)}\Big{(}\big{[}\mathbf{J}_{if}^{\perp}(\mathbf{R})+J_{if}^{\parallel}(\mathbf{R})\hat{\bf
z}\big{]}e^{-i(\omega/v_{i})z}\Big{)}^{*}$
$\displaystyle=\mathbf{J}_{if}^{*}(\mathbf{x},\omega=-q_{\parallel}{v_{i}}).$

These symmetries of the transition current density under interchange of
initial and final states, and their effect on the observables, is detailed
further in the Appendix.

###  V.2 Double differential inelastic scattering cross section in the wide
field limit

When dealing with plane wave electron states, the scattering cross section is
a common observable of interest. It is attained from the EEL transition rate
$w^{\textrm{loss}}_{fi}(\omega)$ in Eq. (9) by first summing over the electron
final states $\sum_{{\bf k}_{f}}\to({L}/{2\pi})^{3}\int d{\bf k}_{f}$ and
subsequently dividing by the incoming plane wave particle flux $\hbar
k_{i}/mL^{3}$. The total frequency-resolved scattering cross section is given
by

$\sigma(\omega)=\frac{mL^{3}}{\hbar k_{i}}\Big{(}\frac{L}{2\pi}\Big{)}^{3}\int
d\Omega_{f}\,k_{f}^{2}\,dk_{f}\,w_{fi}(\omega),$ (36)

and by integration over frequency, the angle-resolved scattering cross section
is

$\begin{split}\frac{\partial\sigma}{\partial\Omega_{f}}&=\frac{mL^{3}}{\hbar
k_{i}}\Big{(}\frac{L}{2\pi}\Big{)}^{3}\int
d\omega\,k_{f}^{2}\,dk_{f}\,w_{fi}(\omega)\\\ &=-\frac{mL^{3}}{\hbar
k_{i}}\Big{(}\frac{L}{2\pi}\Big{)}^{3}\int
d\omega\,dE_{if}\,\bigg{(}\frac{\gamma_{f}m}{\hbar^{2}k_{f}}\bigg{)}k_{f}^{2}\,w_{fi}(\omega),\end{split}$
(37)

where $dE_{if}=-(\hbar^{2}/m)k_{f}dk_{f}$ and
$d\Omega_{f}=\sin\theta_{f}d\theta_{f}d\phi_{f}$. Lastly, noting that
${\partial\sigma}/\partial\Omega_{f}=\int
dE_{if}{\partial^{2}\sigma}/({\partial E_{if}\partial\Omega_{f}})$, the double
differential scattering cross section (DDCS) is defined as

$\displaystyle\frac{\partial^{2}\sigma}{\partial E_{if}\partial\Omega_{f}}$
$\displaystyle=-\frac{m^{2}L^{3}}{\hbar^{3}}\Big{(}\frac{L}{2\pi}\Big{)}^{3}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\int
d\omega\,w_{fi}(\omega)$ (38)
$\displaystyle=\Big{(}\frac{mL^{3}}{\pi\hbar^{2}}\Big{)}^{2}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\int
d\omega\,d\mathbf{x}d\mathbf{x}^{\prime}\,\textrm{Im}\Big{[}\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\Big{]}\delta(\omega-\varepsilon_{if}),$

which reduces to the DDCS common for an isolated dipolar target Eq. (40) in
core-loss EEL scattering [37, 87, 67] when the electrostatic limit where
$c\to\infty$ is taken. Further analytic progress is possible in the case of a
single target dipole located at position $\mathbf{x}_{d}$ and characterized by
frequency-dependent polarizability tensor $\tensor{\bm{\alpha}}(\omega)$. In
this case, the induced Green’s function reduces to
$\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)=-({4\pi\omega^{2}})\tensor{\mathbf{G}}^{0}(\mathbf{x},\mathbf{x}_{d},\omega)\cdot\tensor{\bm{\alpha}}(\omega)\cdot\tensor{\mathbf{G}}^{0}(\mathbf{x}_{d},\mathbf{x}^{\prime},\omega)$,
where $\tensor{\bf G}^{0}({\bf x},{\bf
x}^{\prime},\omega)=-1/({4\pi\omega^{2}})[(\omega/c)^{2}\tensor{\bf
I}+\nabla\nabla]{e^{i({\omega}/{c})|\mathbf{x}-\mathbf{x^{\prime}}|}}/{|\mathbf{x}-\mathbf{x}^{\prime}|}$
is the vacuum dipole Green’s function, and the DDCS can be expressed
analytically. Specifically, with a single incoming plane wave scattering to a
single outgoing plane wave as described by the transition current density
given by Eq. (17), the fully retarded DDCS becomes

$\frac{\partial^{2}\sigma}{\partial
E_{if}\partial\Omega_{f}}=\frac{e^{2}}{\hbar^{2}\pi}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\frac{1}{\varepsilon_{if}^{2}}\,\textrm{Im}\bigg{\\{}\mathbf{Q}\cdot\bigg{[}\bigg{(}\frac{\varepsilon_{if}}{c}\bigg{)}^{2}\tensor{\mathbf{I}}-\mathbf{q}\mathbf{q}\bigg{]}\cdot\frac{\tensor{\bm{\alpha}}(\varepsilon_{if})}{\big{|}({\varepsilon_{if}}/{c})^{2}-q^{2}\big{|}^{2}}\cdot\bigg{[}\bigg{(}\frac{\varepsilon_{if}}{c}\bigg{)}^{2}\tensor{\mathbf{I}}-\mathbf{q}\mathbf{q}\bigg{]}\cdot\mathbf{Q}\bigg{\\}},$
(39)

where $\mathbf{Q}=2\mathbf{k}_{i}-\mathbf{q}$. In the quasistatic limit
($c\to\infty$), Eq. (39), reduces to the more familiar form [88]

$\frac{\partial^{2}\sigma}{\partial
E_{if}\partial\Omega}=\frac{e^{2}}{\hbar^{2}\pi}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\frac{1}{\varepsilon_{if}^{2}}\,\textrm{Im}\bigg{[}\mathbf{Q}\cdot\mathbf{q}\mathbf{q}\cdot\frac{\tensor{\bm{\alpha}}(\varepsilon_{if})}{q^{4}}\cdot\mathbf{q}\mathbf{q}\cdot\mathbf{Q}\bigg{]}.$
(40)

##  VI Numerical Implementation

This section details the numerical implementation of the inelastic scattering
observables presented above in Secs. V.1 and V.2 for transversely phase-
structured free electrons in both the focused beam and wide field limits. Our
implementation generalizes the electron-driven discrete dipole approximation
($e$-DDA) [89, 90], built on top of the DDCSAT [91] framework, which
previously utilized the vacuum electric field ${\bf E}^{0}_{fi}$ of a point
electron as the source instead of an optical plane wave field. Other fully
retarded and quasistatic numerical methods for simulating extended
nanophotonic targets have been formulated to model low-loss electron beam
interactions, such as the finite-difference time-domain [92, 93], metal
nanoparticle boundry element (MNPBEM) [94], and finite element [95, 96]
methods. In addition, transversely structured electron beams have been
implemented in MNPBEM [60, 63, 65, 68], albeit in the quasistatic limit only.
Our treatment of the inelastic scattering of transversely structured electron
beams in $e$-DDA is distinguished by its incorporation of fully retarded
electron-sample interactions in both focused beam and wide field limits.

$e$-DDA/DDA originate from the method of coupled dipoles [97], whereby the
target is discretized into a finite collection of point electric dipoles

${\bf
p}_{k}(\omega)=\sum_{l=1}^{N}\Big{[}\tensor{\bm{\alpha}}(\omega)^{-1}-(-4\pi\omega^{2})\tensor{\mathbf{G}}^{0}(\omega)\Big{]}^{-1}_{kl}\cdot\mathbf{E}_{fi}^{0}({\bf
x}_{l},\omega)$ (41)

of polarizability $\tensor{\bm{\alpha}}(\omega)$, each driven by the vacuum
transition field $\mathbf{E}_{fi}^{0}({\bf x}_{i},\omega)$ at frequency
$\omega$ and mutually interacting via their fully-retarded electric dipole
fields $-4\pi\omega^{2}\sum_{l}\tensor{\mathbf{G}}^{0}_{kl}(\omega)\cdot{\bf
p}_{l}(\omega)$ until reaching self-consistency at that same frequency. Here
$\tensor{\bf G}^{0}_{kl}(\omega)$ are the $kl$-matrix elements of the vacuum
dipole Green’s function $\tensor{\bf G}^{0}({\bf x},{\bf x}^{\prime},\omega).$

Upon inversion of Eq. (41), all EEL observables described above may be
calculated from the resulting ${\bf p}_{k}(\omega)$ together with the applied
field ${\bf E}^{0}_{fi}({\bf x}_{k},\omega)$ (Eq. (46)) evaluated at each
dipole. Specifically, the focused beam EEL probability and the wide field DDCS
expressions in Eq. (27) and Eq. (38), respectively, can be adapted to a form
that is compatible with the $e$-DDA code via the EEL rate per unit frequency

$\displaystyle w_{fi}^{\textrm{loss}}(\omega)$
$\displaystyle=-\frac{8\pi}{\hbar}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\textrm{Im}\Big{[}\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\Big{]}\delta(\omega-\varepsilon_{if})$
(42)
$\displaystyle=\frac{2}{\hbar}\Big{(}\frac{v_{i}}{L}\Big{)}^{2}\,\textrm{Im}\Big{[}\sum_{k}\mathbf{E}^{0*}_{fi}(\mathbf{x}_{k},\omega)\cdot\mathbf{p}_{k}(\omega)\Big{]}\delta(\omega-\varepsilon_{if}).$

Here the target’s induced Green’s function $\tensor{\bf
G}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ is expanded in terms of the
polarizabilities of the $N$ dipoles representing the target. That is

$\tensor{\bf
G}(\mathbf{x},\mathbf{x}^{\prime},\omega)=-{4\pi\omega^{2}}\sum_{jj^{\prime}}\tensor{\bf
G}^{0}(\mathbf{x},\mathbf{x}_{j},\omega)\cdot\big{[}\tensor{\bm{\alpha}}^{-1}(\omega)-(-4\pi\omega^{2})\tensor{\bf
G}^{0}(\omega)\big{]}^{-1}_{jj^{\prime}}\cdot\tensor{\bf
G}^{0}(\mathbf{x}_{j^{\prime}},\mathbf{x}^{\prime},\omega),$ (43)

which can be derived from the response field of a polarizable body described
in terms of its induced Green’s function $\tensor{\bf
G}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ driven by the external current
density $\mathbf{J}^{0}(\mathbf{x})$ or by the free propagation (via
$\tensor{\bf G}^{0}(\mathbf{x},\mathbf{x}^{\prime},\omega)$) of the target’s
induced current density $\mathbf{J}(\mathbf{x})=\sum_{j}(-i\omega){\bf
p}_{j}(\omega)\delta({\bf x}-{\bf x}_{j})$ as ${\bf E}({\bf x},\omega)=-4\pi
i\omega\int d{\bf x}^{\prime}\,\tensor{\bf
G}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}^{0}(\mathbf{x}^{\prime})=-4\pi
i\omega\int d{\bf x}^{\prime}\,\tensor{\bf
G}^{0}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}(\mathbf{x}^{\prime})$.

###  VI.1 Numerical evaluation of state- and energy-resolved EEL and EEG
probabilities

From Eq. (42), the state- and energy-resolved EEL probability can be obtained
following the same procedure in Sec. V.1. Specifically, the transversely
phase-shaped EEL probability becomes

$\Gamma_{fi}^{\textrm{loss}}(\omega)=\frac{1}{\pi\hbar^{2}}\textrm{Im}\Big{[}\sum_{k}\mathbf{E}^{0*}_{fi}(\mathbf{x}_{k},\omega=q_{\parallel}v_{i})\cdot\mathbf{p}_{k}(\omega=q_{\parallel}v_{i})\Big{]},$
(44)

when working within the nonrecoil approximation introduced in Sec. V, as
appropriate to focused beams prepared in the STEM configuration. The vacuum
transition electric field $\mathbf{E}^{0}_{fi}$ appearing within Eq. (41) and
Eq. (44) can, in principle, be any of the fields sourced by the transversely
focused transition current densities described in Sec. IV. However, due to
their complexity, only those transitions involving OAM transfers
$\Delta\ell=1$ depicted as colored arrows in Fig. 2(e), and more specifically
the resulting electric fields sourced by the transition current densities seen
in Fig. 3(c) and 3(d), have been implemented within the $e$-DDA code. Explicit
forms for these fields are provided in the Appendix. Phase-shaped EEG
probability spectra of nanophotonic targets under continuous-wave laser
stimulation can also be evaluated numerically using the $e$-DDA as detailed
previously [66]. Briefly, the EEG probability in Eq. (33) is numerically
integrated by quadrature using the optically-induced response field of the
target $\mathbf{E}_{\nu}^{(+)}$ calculated using DDCSAT and the
$\hat{\mathbf{J}}_{fi}$ defined by selection of a specific pair of incoming
and outgoing free electron states.

Figure 4: Phase-shaped EEL and EEG point spectra and spectrum images of a
silver plasmonic sphere of 25 nm radius evaluated using $e$-DDA. (a) Phase-
shaped EEL (solid) and EEG (dashed) point spectra, evaluated in the narrow
beam limit, at impact parameter $\mathbf{R}_{0}=(1/\sqrt{2})(40\textrm{
nm},40\textrm{ nm})$. The optical polarization is
$\hat{\bm{\epsilon}}=\hat{\mathbf{x}}$ in the laser-stimulated EEG
calculations. Gray and magenta colors correspond to
$\hat{\bm{\mathcal{J}}}_{fi}$ along $\hat{\mathbf{z}}$ and $\hat{\mathbf{x}}$,
respectively. (b) Phase-shaped EEL spectrum images at the dipole LSP resonance
energy of 3.4 eV. Each column corresponds to the labeled
$\hat{\bm{\mathcal{J}}}_{fi}$. (c) Phase-shaped EEG spectrum images at 3.4 eV.
The optical excitation polarization state $\hat{\bm{\epsilon}}$ varies between
rows, while each column corresponds to the labeled
$\hat{\bm{\mathcal{J}}}_{fi}$. For
$\hat{\bm{\epsilon}}=\\{\hat{\bf{x}},\hat{\bf{L}}\\}$, the optical axis is
chosen to be along $\hat{\bf{z}}$, while for
$\hat{\bm{\epsilon}}=\bf{\hat{z}}$ it is chosen to be along $\hat{\bf{x}}$.
Panels with
$\hat{\bm{\mathcal{J}}}_{fi}=\\{\hat{\mathbf{x}},\,\hat{\mathbf{y}},\hat{\mathbf{L}}\\}$
and with $\hat{\bm{\epsilon}}=\\{\hat{\mathbf{x}},\hat{\mathbf{L}}\\}$ share a
common scale, while panels with $\hat{\bm{\mathcal{J}}}_{fi}=\bf{\hat{z}}$ are
separately scaled. The incident velocity of the probing electron is ${\bf
v}_{i}=0.7c\,\hat{\bf z}$ in all cases.

Figure 4 presents a comparison of normalized focused beam EEL and EEG spectra
for a 25 nm radius silver sphere calculated using $e$-DDA with dielectric data
taken from Ref. [98]. Probing electrons have a speed of 0.7$c$. Panel (a)
shows phase-structured EEL (solid traces) and EEG (dashed traces) spectra at
impact parameter $\mathbf{R}_{0}=(1/\sqrt{2})(40\textrm{ nm},40\textrm{ nm})$.
The optical excitation polarization $\hat{\bm{\epsilon}}$ in the EEG
calculations is along $\hat{\mathbf{x}}$. Gray and magenta colors correspond
to pre- and post-selection of HG transitions
$\Psi_{00}(x,y)\rightarrow\Psi_{00}(x,y)$ with
$\hat{\bm{\mathcal{J}}}_{fi}^{\parallel}=\hat{\mathbf{z}}$, and
$\Psi_{10}(x,y)\rightarrow\Psi_{00}(x,y)$ with
$\hat{\bm{\mathcal{J}}}^{\perp}_{fi}=\hat{\mathbf{x}}$, respectively. While
the $e$-DDA calculations can capture coupling to the higher-order multipoles,
as seen in Fig. 4(a), the dipolar localized surface plasmon (LSP) is clearly
evident near 3.4 eV. Here it is apparent that both transitions with
$\hat{\bm{\mathcal{J}}}_{fi}$ oriented parallel and perpendicular to the
electron trajectory couple to the quadrupolar LSP mode near 3.6 eV, albeit
with different strengths. The EEG spectra, meanwhile, are dominated by the
optically bright dipolar response of the sphere, whereas the higher-order,
dark modes are inaccessible by the stimulating optical pump field.

EEL spectrum images obtained by plotting
$\Gamma_{fi}^{\textrm{loss}}(\mathbf{R}_{0},\omega)$ as a function of the
impact parameter ${\bf R}_{0}$ at the dipole LSP energy 3.4 eV are shown in
Fig. 4(b). In the conventional EEL case, where
$\hat{\bm{\mathcal{J}}}_{fi}^{\perp}={\bf 0}$ resulting in
$\hat{\bm{\mathcal{J}}}_{fi}=\hat{\bm{\mathcal{J}}}_{fi}^{\parallel}=\hat{\mathbf{z}}$
(gray, left), the spectrum image exhibits the expected circular symmetry.
Meanwhile, when $\hat{\bm{\mathcal{J}}}_{fi}^{\perp}\neq{\bf 0}$, producing a
$\hat{\bm{\mathcal{J}}}_{fi}^{\perp}$ oriented along $\hat{\mathbf{x}}$
($\hat{\mathbf{y}}$), and outlined in magenta (green), the spectrum image
shows slight elongation along the direction of the transition current density.
As required by symmetry, the circularly-polarized
$\hat{\bm{\mathcal{J}}}_{fi}^{\perp}=\hat{\mathbf{L}}$ (red) couples with
radial symmetry to the spherical target. In Fig. 4(b), the
$\hat{\bm{\mathcal{J}}}^{\parallel}_{fi}=\hat{\mathbf{z}}$ plot is normalized
to a maximum of $10^{-2}$ eV-1, while the
$\hat{\bm{\mathcal{J}}}_{fi}^{\perp}\in\\{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{L}}\\}$
cases share a common normalization factor of $10^{-8}$ eV-1 for a beam waist
of 1 nm. The small perpendicular to parallel ratio of signals represents a
hurdle to phase-shaped EEL spectroscopy measurements regarding limits of
detection, although measurements of this type have been achieved previously
[61]. Phase-shaped EEG spectrum images at the same dipole LSP energy 3.4 eV
are presented in Fig. 4(c). The optical excitation polarization state
$\hat{\bm{\epsilon}}$ varies between rows, while each column corresponds to
the labeled component of $\hat{\bm{\mathcal{J}}}_{fi}$. For
$\hat{\bm{\epsilon}}=\\{\hat{\bf{x}},\hat{\bf{L}}\\}$, the optical axis is
chosen to be along $\hat{\bf{z}}$. There
$\hat{\bm{\mathcal{J}}}_{fi}^{\perp}=\mathbf{\hat{x},\,\hat{y},\,\textrm{and
}\hat{L}}$ share a common scale factor, while
$\hat{\bm{\mathcal{J}}}_{fi}=\bf{\hat{z}}$ is separately scaled. The ratio of
the transverse to longitudinal EEG probabilities is
$\Gamma_{{fi}\perp}^{\textrm{gain}}/\Gamma_{{fi}\parallel}^{\textrm{gain}}\approx
10^{-5}$, for a 200 keV electron beam with a waist 1 nm. These findings are
consistent with earlier theoretical [66] and experimental [99] studies. For
the $\hat{\bm{\epsilon}}=\bf{\hat{z}}$ case, the optical axis is chosen to be
along $\hat{\bf{x}}$ and the ratio of transverse to longitudinal EEG signals
remains similarly small ($\sim 10^{-5}$), but all signals are smaller by $\sim
10^{-3}$ in this excitation geometry. Comparison of Figs. 4(b) and 4(c)
highlights the differing identities of the excitation sources and roles played
by $\mathbf{J}_{fi}$ in EEL and laser-stimulated EEG processes. When
considering EEL events, the STEM electron acts as both a spatially dependent
pump and probe sourced by the transition current density
$\mathbf{J}_{fi}(\mathbf{R}_{0})$ at impact parameter $\mathbf{R}_{0}$. In
stark contrast, as alluded to in Sec. II, the pump and the probe are decoupled
for laser-stimulated EEG processes. Specifically, as seen in Eq. (10), the
pump is the optical source exciting the target’s induced field ${\bf
E}^{(+)}_{\nu}$, which is then probed by the electron’s transition current
density $\mathbf{J}_{fi}(\mathbf{R}_{0})$ at $\mathbf{R}_{0}$.

###  VI.2 Numerical evaluation of the inelastic double differential cross
section

From Eqs. (38) and (42), the wide field inelastic DDCS can be expressed as

$\displaystyle\frac{\partial^{2}\sigma}{\partial E_{if}\partial\Omega_{f}}$
$\displaystyle=-\frac{m^{2}L^{6}}{(2\pi)^{3}\hbar^{3}}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\int
d\omega\,w_{fi}^{\textrm{loss}}(\omega)$ (45)
$\displaystyle=-\frac{1}{4\pi^{3}}\frac{k_{f}}{k_{i}}\bigg{(}\frac{v_{i}mL^{2}}{\hbar^{2}}\bigg{)}^{2}\textrm{Im}\Big{[}\sum_{j}\mathbf{E}^{0*}_{fi}(\mathbf{x}_{j},\omega=\varepsilon_{if})\cdot\mathbf{p}_{j}(\omega=\varepsilon_{if})\Big{]}.$

We have implemented within $e$-DDA the plane wave transition fields defined in
Appendix Eqs. (49) and (51), sourced by the transition currents Eqs. (17) and
(18), respectively, for the cases of scattering from either a single or a
coherent superposition of two incident plane waves, to a single outgoing plane
wave state.

Figure 5: DDCSs for plasmonic nanorod monomer and dimer systems evaluated
using $e$-DDA. (a) Numerically calculated DDCS (logarithmic scale) for a
silver nanorod with Cartesian dimensions of 30 nm $\times$ 60 nm $\times$ 15
nm. The incident electron is directed along $\hat{\mathbf{z}}$ with 200 keV
kinetic energy and the collected scattering direction is varied from
$\theta_{f}=0$ $\mu$rad (along $\hat{\bf z}$) to $\theta_{f}=200$ $\mu$rad,
with $\phi_{f}=\pi/2$ fixed such that
$\mathbf{q}_{\perp}=q_{\perp}\hat{\mathbf{y}}$. The quantity
$d/\lambda_{\perp}$ (upper horizontal axis) is the rod length ($60$ nm) along
the momentum recoil direction divided by the transverse recoil wavelength
$\lambda_{\perp}=2\pi/q_{\perp}$. (b) Lineout of the DDCS in panel (a) at 2.5
eV (green). The gray trace is the DDCS of an anisotropic point dipole
evaluated using the analytic expression in Eq. (39). Vertical gray dashed
lines indicate single particle diffraction minima. Insets show schematic
depictions of the scattering and target geometries. (c) Same as (a), but for a
homodimer target composed of two copies of the nanostructure considered in (a)
and (b) arranged tip-to-tip along the $y$-axis with an $s=10$ nm gap. (d)
Lineouts from (c) at 2.30 eV (red) and 2.65 eV (blue). Vertical gray dashed
lines indicate single slit diffraction minima, while solid gray lines denote
double slit interference minima at 2.30 eV.

The DDCSs for inelastic scattering of wide field plane wave electron states
from plasmonic nanorod monomers and dimers are presented in Fig. 5. Probing
electrons have an initial kinetic energy of 200 keV and wave vector directed
along the TEM axis, i.e., $\hat{\mathbf{k}}_{i}=\hat{\mathbf{z}}$, while the
outgoing wave vectors $\mathbf{k}_{f}$ possess a nonzero $y$-component such
that the transverse momentum recoil $\hbar\mathbf{q}_{\perp}$ is along
$\hat{\mathbf{y}}$. Working in the low loss energy regime in order to observe
the plasmonic modes of interest, Fig. 5(a) shows the calculated DDCS
(logarithmic scale) for an anisotropic silver rod with Cartesian dimensions 30
nm $\times$ 60 nm $\times$ 15 nm (width, length, height) as a function of
scattering angle $\theta_{f}$ and loss energy. The nanorod is orientated with
its longest dimension $d_{y}$ along the direction of collected transverse
recoils ($\phi_{f}=\pi/2$), leading to the lowest energy long-axis dipole mode
near 2.50 eV dominating the DDCS. Since the transversely polarized components
of the transition field $\mathbf{E}^{0}_{fi}$ in Eq. (49) are proportional to
$\mathbf{q}_{\perp}$, transversely oriented LSP modes of the nanorod do not
contribute to the DDCS at $\theta_{f}=0$ $\mu$rad. The apparent features at
$\theta_{f}=0$ $\mu$rad and energies above 3.50 eV in Fig. 5(a) arise from
longitudinal multipoles oriented along $\hat{\mathbf{z}}$.

Figure 5(b) shows a lineout from panel (a) at the long-axis dipole LSP energy
marked by the dashed green line. The gray trace in Fig. 5(b) is calculated
using the analytic form of the DDCS given in Eq. (39) for a single point
dipole representing the target. The anisotropy of the nanorod response is
captured by detuning the shorter axis dipole LSP energies above 2.50 eV in the
dipole’s effective polarizability. As a consequence,
$\partial^{2}\sigma/\partial
E_{if}\partial\Omega_{f}\propto|\mathbf{E}^{0}_{fi}\cdot\hat{\mathbf{y}}|^{2}$,
which ensures $\partial^{2}\sigma/\partial
E_{if}\partial\Omega_{f}|_{\theta_{f}=0}=0$. At moderate opening angles
($\gtrsim 5$ $\mu$rad) the DDCS decreases as the opening angle increases,
which starting from Eq. (49) and $Q_{\parallel}\gg|{\bf Q_{\perp}}|=|{\bf
q_{\perp}}|$ can be shown by ${\bf E}^{0}_{fi}\cdot\hat{\mathbf{y}}\propto
Q_{\parallel}q_{\parallel}{\bf q}_{\perp}\cdot\hat{{\bf y}}/\big{(}{\bf
q}_{\perp}^{2}+q_{\parallel}^{2}-\omega^{2}/c^{2}\big{)}=Q_{\parallel}q_{\parallel}|{\bf
k}_{f}|\sin\theta_{f}/\big{[}|{\bf
k}_{f}|^{2}\sin^{2}\theta_{f}+\omega^{2}(1/v_{i}^{2}-1/c^{2})\big{]}$. For a
200 keV electron with loss energy $\hbar\epsilon_{if}=2.5$ eV, $|{\bf
k}_{f}|\sin\theta_{f}\gtrsim\omega/(\gamma_{i}v_{i})$ at opening angles
$>5\,\mu\textrm{rad}$, leading to $\partial^{2}\sigma/\partial
E_{if}d\Omega_{f}\,{\propto}\,1/\sin^{2}\theta_{f}$. This effect exists
independent of the target geometry and represents the decreasing probability
of low loss events with moderate transverse recoil. Separately the lineouts in
$\theta_{f}$ on the order of 5 $\mu$rad range are primarily dictated by the
growing in of the transverse LSP mode, as at smaller angles
$\mathbf{E}^{0}_{fi}\cdot\hat{\mathbf{y}}\propto{\bf
q}\cdot\hat{\mathbf{y}}=|{\bf k}_{f}|\sin\theta_{f}$ and the lineout is taken
at the ${\bf\hat{y}}$ oriented dipole mode energy. Unlike in the narrow beam
limit, the DDCS observable has equal magnitude contributions from transverse
and longitudinal recoils, which is a well known experimental and theoretical
result of EEL DDCS on an anisotropic target [100, 87]. In addition to tracking
the angular scattering behavior predicted by the point dipole model at small
scattering angles ($\lesssim 25$ $\mu$rad), the lineout shown in green
exhibits a progression of diffraction maxima/minima with increasing
$\theta_{f}$ arising from the finite extent of the target. The single particle
diffraction minima, indicated by vertical gray dashed lines, are located
nearby angles $\theta_{m}$ corresponding to single slit diffraction minima
predicted using $d_{y}\sin\theta_{m}=m\lambda_{e}$, where integers $m$ index
the diffraction minima, and $\lambda_{e}$ is the de Broglie wavelength of the
electron.

Figures 5(c) and 5(d) consider a nanorod dimer consisting of a pair of the
silver rods from Figs. 5(a) and 5(b) displaced along $\hat{\mathbf{y}}$ such
that there is an $s=10$ nm gap between the rod tips. The dimer’s DDCS is
presented in Fig. 5(c), where the bonding (red) and antibonding (blue)
hybridized long-axis dipole LSP modes are visible at energies slightly below
and above 2.5 eV, respectively. Lineouts from Fig. 5(c) at loss energies
corresponding to the bonding and antibonding dimer modes are shown in Fig.
5(d), again displaying multiple diffraction minima/maxima. In analogy to the
double slit experiment, each nanorod is a source of single-particle
diffraction in addition to matter-wave interference arising from the
$d_{\perp}+s=70$ nm center-to-center displacement of the the nanorods. For
example, each DDCS minimum observed at the bonding mode energy (2.30 eV)
occurs at an angle corresponding to either one of the single-particle
diffraction minima from Fig. 5(b) marked by vertical gray dashed lines, or at
angles $\theta_{n}$ satisfying the double slit interference condition
$n\lambda_{\perp}=d_{y}+s$, which are indicated by vertical gray solid lines.
The condition for constructive interference at the antibonding resonance
energy is $(n+1/2)\lambda_{\perp}=d_{y}+s$.

Figure 6: Numerical evaluation of the circular dichroic response of a chiral
nanorod structure probed using wide field EEL spectroscopy within $e$-DDA. (a)
Scheme showing the gold nanorod BK structure and probing geometries.
Superposition plane wave electron states are incident from the left, while an
aperture (black) post-selects the outgoing electron state along the TEM
($\hat{\mathbf{z}}$) axis. (b) Optical extinction spectra for incident plane
wave wave vectors along $\hat{\mathbf{z}}$ with circular polarization states
$\hat{\bm{\epsilon}}=\hat{\mathbf{R}}$ (blue) and $\hat{\mathbf{L}}$ (red).
The optical circular dichroism CD${}_{\textrm{opt}}$ is shown in black and
shares its secondary $y$-axis with panel (c). (c) DDCS spectra of the same BK
structure for superposition plane wave electron states with $\chi=+\pi/2$
(red) and $\chi=-\pi/2$ (blue). The CD${}_{\textrm{EEL}}$ spectrum is shown in
black. (d) Angle $\theta_{i}$-resolved CD${}_{\textrm{EEL}}$ spectra. The
dashed gray line in (d) indicates the fixed angle of the lineouts in (c). The
incident electron kinetic energy in (c,d) is 200 keV.

An investigation of inelastic scattering involving superposition plane wave
electron states interacting with a chiral nanophotonic target is presented in
Fig. 6. Specifically, the target is the well-understood Born-Kuhn structure
(BK) [101], composed of two gold nanorods arranged in an L-shape with a
relative displacement $\zeta$ between the rods centered along
$\hat{\mathbf{z}}$, as demonstrated in Fig. 6(a). Here, the long and short
axes of each nanorod are 100 nm and 30 nm, respectively, $\zeta=40$ nm, and
tabulated gold dielectric data are taken from the literature [98]. Fig. 6(b)
shows the optical extinction spectra of the BK target for incident wave
vectors along $\hat{\mathbf{z}}$ and polarizations
$\hat{\bm{\epsilon}}=\hat{\mathbf{L}}$ (red) and $\hat{\mathbf{R}}$ (blue). As
is well-understood (see Appendix) [102, 103] both incident field helicities
couple, albeit with unequal strengths, to the bonding and antibonding LSP
modes near 2.0 eV involving the long-axis dipoles. The optical circular
dichroism
$\textrm{CD}_{\textrm{opt}}=2(\sigma^{\hat{\mathbf{R}}}_{\textrm{ext}}-\sigma^{\hat{\mathbf{L}}}_{\textrm{ext}})/(\sigma^{\hat{\mathbf{R}}}_{\textrm{ext}}+\sigma^{\hat{\mathbf{L}}}_{\textrm{ext}})$
spectrum is shown in black, which exhibits the BK system’s chiral response and
serves as a point of comparison against the electron-based observables.

As depicted in Fig. 6(a), free electron plane wave superposition states
$\psi_{k_{x}k_{y}}^{\chi}(x,y,z)$ are incident from the left, while an
aperture situated in the diffraction plane post-selects the outgoing free
electrons along the TEM axis with $\mathbf{k}^{f}_{\perp}={\bf 0}$, i.e,
$\psi_{k^{\prime}_{\parallel}}(x,y,z)$. It has been shown previously that the
response of targets as measured by wide field EEL spectroscopy with pre- and
post-selection of such states with $k_{x}=k_{y}$ and $\chi=\pm\pi/2$ in many
ways mimic the optical response under circularly polarized optical excitation
[37, 38, 67]. Indeed, Fig. 6(c) presents DDCS spectra of the BK target for
$\psi_{k_{x}k_{y}}^{\chi}(x,y,z)\to\psi_{k_{\parallel}^{\prime}}(x,y,z)$,
producing the transition current density given by Eq. (18), for the scenarios
of $\chi=\pm\pi/2$. The circular dichroism CD${}_{\textrm{EEL}}$ (black) is
again defined as twice the difference in the $\chi=\pm\pi/2$ DDCS spectra
normalized by the sum, and strongly resembles the CD${}_{\textrm{opt}}$
spectrum in Fig. 6(b). Panel (c) presents the CD${}_{\textrm{EEL}}$ spectra as
a function of convergence angle $\theta_{i}$. Like the nanorod dimer presented
in Fig. 5, the BK system exhibits both single slit diffraction and double slit
interference effects (Fig. 6(d)), though the superposition state excitation
and BK target geometry conspire to produce more complicated beating patterns
than those observed in Fig. 5(c). The vertical dashed gray line in Fig. 6(d)
indicates the convergence angle $\theta_{i}$ of the spectra presented in panel
(c).

##  VII Conclusions

Development of capabilities to prepare, parse, and measure transversely phase-
structured electron states in the electron microscope has opened the door to
state- and energy-resolved inelastic scattering observables, adding to the
rapidly evolving toolkit used to interrogate nanoscale systems in the low-loss
regime. Here we present general expressions for transversely phase-shaped EEL
and continuous-wave laser-stimulated EEG spectroscopies for transversely
localized vortex and non-vortex electron states, and the DDCS for wide field
electron plane waves under minimal assumptions regarding the magnitudes of
electron velocity and energy exchange. By exploiting a quantum mechanical
treatment that accounts explicitly for the transverse degrees of freedom of
the probing electron wave functions, we have showcased the ability to retrieve
information about the optical near-field and electromagnetic response of
nanophotonic targets using inelastic scattering of phase-shaped free electrons
following energy-momentum post-selection. A numerical procedure for evaluating
derived observables is presented that allows for flexibility regarding
particle number, size, geometry, and material composition. Example
calculations for several prototypical plasmonic monomer and dimer systems are
investigated to highlight the utility of our approach to analyze mode
symmetries, local response field characteristics, chiral responses, and
matter-wave diffraction phenomena. The general procedures outlined for
constructing wide field plane wave and nondiffracting twisted and non-twisted
electron beams with distinct transverse polarization and topological textures,
in addition to the state- and energy-resolved observables, can be readily
applied to many areas of atomic, molecular, and materials physics. In
particular, we have drawn attention to an application of free electron qubits,
whereby transitions between different OAM qubit states produce transition
current densities with unique polarization and vector profiles, including
analogs of optical polarization states and other more general forms of
structured light. The theoretical framework presented can be extended to
describe beam coherence in electron holography [104, 105] via a density matrix
formalism [106], and utilized to explore the role of pure and mixed electron
states in inelastic scattering and the concurrent transfer of quantum
information in the form of quantized energy and OAM [107, 54, 56, 57].
Furthermore, the use of transversely phase-structured free electron states
realizable in TEMs, STEMs, and ultrafast TEMs can lead to additional
manifestations of unique electromagnetic fields [27, 28] and electron paraxial
skyrmionic beams [29], all of which could play vital roles in the
investigation of novel optically forbidden atomic, molecular, material, and
topological excitations [30, 108, 109, 110, 111].

###### Acknowledgements.

All work was supported by the U.S. Department of Energy (DOE), Office of
Science, Office of Basic Energy Sciences (BES), Materials Sciences and
Engineering Division under Award No. DOE BES DE-SC0022921.

##  VIII Appendix

###  VIII.1 Vacuum Transition Vector Potentials and Electric Fields

The transition vector potentials and electric fields associated with each of
the transition current densities in Sec. IV are presented below. Specifically,
the vacuum electric field

$\begin{split}{\bf E}^{0}_{fi}({\bf
x},\omega)&=\frac{ic}{\omega}\bigg{[}\Big{(}\frac{\omega}{c}\Big{)}^{2}{\tensor{\bf
I}}+{\nabla\nabla}\bigg{]}\cdot{\bf A}^{0}_{fi}({\bf x},\omega)\\\
&=\frac{i}{\omega}\int
d\mathbf{x}^{\prime}\,\bigg{[}\Big{(}\frac{\omega}{c}\Big{)}^{2}{\tensor{\bf
I}}+\nabla\nabla\bigg{]}\frac{e^{i({\omega}/{c})|\mathbf{x}-\mathbf{x^{\prime}}|}}{|\mathbf{x}-\mathbf{x}^{\prime}|}\,\Big{(}\frac{L}{v_{i}}\Big{)}\mathbf{J}_{fi}(\mathbf{x}^{\prime})\\\
&=-4\pi i\omega\int
d\mathbf{x}^{\prime}\,\tensor{\mathbf{G}}^{0}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\Big{(}\frac{L}{v_{i}}\Big{)}\mathbf{J}_{fi}(\mathbf{x}^{\prime})\end{split}$
(46)

plays an important role in the observable EEL and EEG processes and depends
upon the vector potential

$\mathbf{A}^{0}_{fi}(\mathbf{x},\omega)=\frac{1}{c}\int
d\mathbf{x}^{\prime}\,\frac{e^{i({\omega}/{c})|\mathbf{x}-\mathbf{x^{\prime}}|}}{|\mathbf{x}-\mathbf{x}^{\prime}|}\,\Big{(}\frac{L}{v_{i}}\Big{)}\mathbf{J}_{fi}(\mathbf{x}^{\prime})$
(47)

presented here in the Lorenz gauge. Beginning with the transition current
density in Eq. (17) associated with the transition between single plane wave
states $\psi_{{\bf k}_{i}}(x,y,z)$ and $\psi_{{\bf k}_{f}}(x,y,z)$ where
${{\bf k}_{i}}$ and ${{\bf k}_{f}}$ are arbitrary wave vectors, the vacuum
vector potential

$\mathbf{A}^{0}_{{{\bf k}_{f}}{{\bf k}_{i}}}(\mathbf{x},\omega)=\frac{2\pi
e\hbar}{mcv_{i}L^{2}}\frac{e^{i\mathbf{q}\cdot\mathbf{x}}}{\big{(}{\omega}/{c}\big{)}^{2}-q^{2}}{\bf
Q}$ (48)

and electric field

$\mathbf{E}^{0}_{{{\bf k}_{f}}{{\bf k}_{i}}}(\mathbf{x},\omega)=\frac{2i\pi
e\gamma_{i}}{k_{i}L^{2}\omega}\frac{e^{i\mathbf{q}\cdot\mathbf{x}}}{({\omega}/{c})^{2}-q^{2}}\Big{[}\Big{(}\frac{\omega}{c}\Big{)}^{2}\tensor{\bf{I}}-\mathbf{q}\mathbf{q}\Big{]}\cdot\mathbf{Q}$
(49)

are readily obtained upon using the integral identity $\int
d\mathbf{x}^{\prime}{e^{i(\omega/c)|\mathbf{x}-\mathbf{x}^{\prime}|}}e^{\mp
i\mathbf{q}\cdot\mathbf{x}^{\prime}}/{|\mathbf{x}-\mathbf{x}}^{\prime}|=-4\pi
e^{\mp i\mathbf{q}\cdot\mathbf{x}}/[(\omega/c)^{2}-q^{2}]$.

The vacuum transition vector potential sourced by the superposition plane wave
state transition current density in Eq. (18) takes the form

$\displaystyle\mathbf{A}_{k_{\parallel}^{\prime},k_{x}k_{y}}^{\chi
0}(\mathbf{x},\omega)$ $\displaystyle=\frac{2\pi
e\hbar}{\sqrt{2}mcv_{i}L^{2}}e^{iq_{\parallel}z}\bigg{[}\frac{k_{x}e^{ik_{x}x}}{({\omega}/{c}\big{)}^{2}-k_{x}^{2}-q_{\parallel}^{2}}\hat{\mathbf{x}}+\frac{k_{y}e^{i\chi}e^{ik_{y}y}}{({\omega}/{c})^{2}-k_{y}^{2}-q_{\parallel}^{2}}\hat{\mathbf{y}}$
(50) $\displaystyle\ \ \
+Q_{\parallel}\bigg{(}\frac{e^{ik_{x}x}}{({\omega}/{c})^{2}-k_{x}^{2}-q_{\parallel}^{2}}+\frac{e^{i\chi}e^{ik_{y}y}}{({\omega}/{c})^{2}-k_{y}^{2}-q_{\parallel}^{2}}\bigg{)}\hat{\mathbf{z}}\bigg{]}$

for the superposition state
$\psi_{k_{x}k_{y}}^{\chi}(x,y,z)=(1/\sqrt{2L})\big{[}\Psi_{k_{x}}(x,y)+\Psi_{k_{y}}(x,y)e^{i\chi}\big{]}e^{ik_{\parallel}z}$
introduced in Sec. III, transitioning to the final pin hole state
$\psi_{k_{\parallel}^{\prime}}(x,y,z)=L^{-3/2}e^{ik^{\prime}_{\parallel}z}$
oriented along the TEM ($\hat{\bf z}$) axis. Associated with this vector
potential is the superposition plane wave state transition electric field

$\displaystyle{\bf\mathbf{E}}^{\chi
0}_{k_{\parallel}^{\prime},k_{x}k_{y}}(\mathbf{x},\omega)=\frac{-2\pi
ie\hbar}{\sqrt{2}m\omega L^{2}v_{i}}\bigg{[}$
$\displaystyle\frac{k_{x}^{2}+Q_{\parallel}q_{\parallel}-(\frac{\omega}{c})^{2}}{k_{x}^{2}+q_{\parallel}^{2}-(\frac{\omega}{c})^{2}}k_{x}e^{ik_{x}x}{\bf\hat{x}}+\frac{k_{y}^{2}+Q_{\parallel}q_{\parallel}-(\frac{\omega}{c})^{2}}{k_{y}^{2}+q_{\parallel}^{2}-(\frac{\omega}{c})^{2}}k_{y}e^{ik_{y}y}e^{i\chi}{\bf\hat{y}}$
(51)
$\displaystyle+\bigg{(}\frac{k_{x}^{2}q_{\parallel}+Q_{\parallel}q_{\parallel}^{2}-Q_{\parallel}(\frac{\omega}{c})^{2}}{k_{x}^{2}+q_{\parallel}^{2}-(\frac{\omega}{c})^{2}}e^{ik_{x}x}+\frac{k_{y}^{2}q_{\parallel}+Q_{\parallel}q_{\parallel}^{2}-Q_{\parallel}(\frac{\omega}{c})^{2}}{k_{y}^{2}+q_{\parallel}^{2}-(\frac{\omega}{c})^{2}}e^{ik_{y}y}e^{i\chi}\bigg{)}{\bf\hat{z}}\bigg{]}e^{iq_{\parallel}z},$

where ${\bf Q}={\bf k}_{i}+{\bf k}_{f}$ and ${\bf q}={\bf k}_{i}-{\bf k}_{f}$
are the same total and recoil wave vectors defined previously.

In the case of focused beams, more specifically in the small beam-width limit,
we present only the vacuum transition fields involving one unit of OAM
exchange between initial and final electron scattering states. The narrow beam
limit, whereby the transverse electron wave function reduces to the transverse
delta function [51], is adopted in all expressions. The transition vector
potential and electric field associated with the HG transition current density
in Eq. (22) describing the scattering from $\psi_{10}(x,y,z)$ to
$\psi_{00}(x,y,z)$ is

$\mathbf{A}^{0}_{00,10}(\mathbf{x},\omega)=-\frac{2ie\hbar}{mv_{i}cw_{0}}\Big{[}\textrm{K}_{0}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\hat{\mathbf{x}}+i\frac{q_{\parallel}Q_{\parallel}w_{0}^{2}}{4\gamma_{i}}\frac{\Delta
x}{\Delta R_{0}}\textrm{K}_{1}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\hat{\mathbf{z}}\Big{]}e^{iq_{\parallel}z},$ (52)

and

$\displaystyle\mathbf{E}^{0}_{00,10}(\mathbf{x},\omega)$
$\displaystyle=\frac{2\hbar e}{mw_{0}\omega
v_{i}}\Big{[}\frac{\omega^{2}}{c^{2}}\tensor{\bf
I}+\nabla\nabla\Big{]}\cdot\Big{[}\textrm{K}_{0}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\hat{\mathbf{x}}+i\frac{q_{\parallel}Q_{\parallel}w_{0}^{2}}{4\gamma_{i}}\frac{\Delta
x}{\Delta R_{0}}\textrm{K}_{1}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\hat{\mathbf{z}}\Big{]}e^{iq_{\parallel}z}$ (53)
$\displaystyle=\frac{\hbar e}{2mw_{0}\omega
v_{i}}\Big{\\{}\Big{(}\Big{[}\frac{4\omega^{2}}{c^{2}}+\frac{q_{\parallel}^{2}}{\gamma_{i}^{2}}(4+q_{\parallel}Q_{\parallel}w_{0}^{2})\frac{\Delta
x^{2}}{\Delta R_{0}^{2}}\Big{]}\textrm{K}_{0}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}+\frac{q_{\parallel}}{\gamma_{i}}(4+{q_{\parallel}Q_{\parallel}w_{0}^{2}})\frac{(\Delta
x^{2}-\Delta y^{2})}{\Delta
R_{0}^{3}}\textrm{K}_{1}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\Big{)}\hat{\mathbf{x}}$
$\displaystyle+\frac{q_{\parallel}^{2}}{\gamma_{i}^{2}}(4+{q_{\parallel}Q_{\parallel}w_{0}^{2}})\frac{\Delta
x\Delta y}{\Delta R_{0}^{2}}\textrm{K}_{2}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\hat{\mathbf{y}}+\frac{i}{\gamma_{i}}\frac{\Delta
x}{\Delta
R_{0}}\Big{[}{q_{\parallel}Q_{\parallel}w_{0}^{2}}\frac{\omega^{2}}{c^{2}}-q_{\parallel}^{2}(1+{q_{\parallel}Q_{\parallel}w_{0}^{2}})\Big{]}\textrm{K}_{1}\bigg{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\bigg{)}\hat{\mathbf{z}}\Big{\\}}e^{iq_{\parallel}z},$

where $\Delta R_{0}=|\mathbf{R}-\mathbf{R}_{0}|$ and $\Delta x=|x-x_{0}|$ and
$\Delta y=|y-y_{0}|$. By symmetry, scattering from HG $\psi_{01}(x,y,z)$ to
$\psi_{00}(x,y,z)$ can be obtained from Eqs. (52) and by (53) interchanging
$\hat{\bf x}$ with $\hat{\bf y},$ resulting in $\mathbf{A}^{0}_{00,01}({\bf
x},\omega)$ and $\mathbf{E}^{0}_{00,01}({\bf x},\omega)$. If the transverse
state does not change in the scattering process, as is the case in the
conventional EEL signal, the vacuum vector potential and electric field
associated with the current density in Eq. (23) where $n^{\prime}=n=0$ and
$m^{\prime}=m=0$ become

$\mathbf{A}^{0}_{00,00}(\mathbf{x},\omega)=\Big{(}\frac{\hbar
e}{mv_{i}c}\Big{)}Q_{\parallel}\textrm{K}_{0}\Big{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma_{i}}\Big{)}\hat{\mathbf{z}}\,e^{iq_{\parallel}z}$ (54)

and

$\mathbf{E}^{0}_{00,00}(\mathbf{x},\omega)=\frac{\hbar e}{\omega
mv_{i}}Q_{\parallel}\Bigg{[}\frac{q_{\parallel}^{2}}{\gamma_{i}\Delta
R_{0}}\textrm{K}_{1}\Big{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma}\Big{)}\Big{(}\Delta x\hat{\mathbf{x}}+\Delta
y\hat{\mathbf{y}}\Big{)}+i\Big{(}\frac{\omega^{2}}{c^{2}}-q_{\parallel}^{2}\Big{)}\textrm{K}_{0}\Big{(}\frac{q_{\parallel}\Delta
R_{0}}{\gamma}\Big{)}\hat{\mathbf{z}}\Bigg{]}e^{iq_{\parallel}z}.$ (55)

The latter is the well known classical field of a uniformly moving point
electron [69, 51] in the nonrecoil approximation where
$q_{\parallel}=\omega/v_{i}$ and $Q_{\parallel}=2k^{i}_{\parallel}$.

Transitions between LG states $\psi_{\ell p}(\rho,\phi,z)$ and
$\psi_{\ell^{\prime}p^{\prime}}(\rho,\phi,z)$ involving one unit of OAM can be
derived from the above HG transitions by linear combination. Specifically,

$\mathbf{A}^{0}_{\ell^{\prime}=0,p^{\prime}=0,\ell=\pm
1,p=0}(\rho,\phi,z,\omega)=\frac{1}{\sqrt{2}}\big{[}\mathbf{A}^{0}_{00,10}({\bf
x},\omega)\pm i\mathbf{A}^{0}_{00,01}({\bf x},\omega)\big{]}$ (56)

and

$\mathbf{E}^{0}_{\ell^{\prime}=0,p^{\prime}=0,\ell=\pm
1,p=0}(\rho,\phi,z,\omega)=\frac{1}{\sqrt{2}}\big{[}\mathbf{E}^{0}_{00,10}({\bf
x},\omega)\pm i\mathbf{E}^{0}_{00,01}({\bf x},\omega)\big{]}.$ (57)

All electric fields introduced in the Appendix are coded within the $e$-DDA
and can be used to calculate the presented EEL observables in both wide field
and focused beam limits.

###  VIII.2 Observables Under Interchange of Initial and Final States

The EEL rate $w_{fi}$ is proportional to $\int
d\mathbf{x}\,d\mathbf{x}^{\prime}\mathbf{J}^{*}_{fi}(\mathbf{x})\cdot\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime}).$
By interchanging the 3D coordinates ${\bf x}$ and ${\bf x}^{\prime}$, and
invoking reciprocity, i.e.,
$G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)=G_{\beta\alpha}(\mathbf{x}^{\prime},\mathbf{x},\omega)$,

$\displaystyle\int d\mathbf{x}\,d\mathbf{x}^{\prime}\big{[}{\bf
J}_{fi}(\mathbf{R})e^{iq_{\parallel}z}\big{]}^{*}_{\alpha}G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)\big{[}{\bf
J}_{fi}(\mathbf{R}^{\prime})e^{iq_{\parallel}z^{\prime}}\big{]}_{\beta}$
$\displaystyle=\int d\mathbf{x}\,d\mathbf{x}^{\prime}\big{[}{\bf
J}_{fi}(\mathbf{R})\big{]}^{*}_{\alpha}G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)\big{[}{\bf
J}_{fi}(\mathbf{R}^{\prime})\big{]}_{\beta}e^{-iq_{\parallel}(z-z^{\prime})}$
(58) $\displaystyle=\int d\mathbf{x}\,d\mathbf{x}^{\prime}\big{[}{\bf
J}_{if}(\mathbf{R})\big{]}_{\alpha}G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)\big{[}{\bf
J}_{if}(\mathbf{R}^{\prime})\big{]}^{*}_{\beta}e^{-iq_{\parallel}(z-z^{\prime})}$
$\displaystyle=\int d\mathbf{x}\,d\mathbf{x}^{\prime}\big{[}{\bf
J}_{if}(\mathbf{R}^{\prime})\big{]}^{*}_{\beta}G_{\beta\alpha}(\mathbf{x}^{\prime},\mathbf{x},\omega)\big{[}{\bf
J}_{if}(\mathbf{R})\big{]}_{\alpha}e^{-iq_{\parallel}(z-z^{\prime})}$
$\displaystyle=\int d\mathbf{x}\,d\mathbf{x}^{\prime}\big{[}{\bf
J}_{if}(\mathbf{R})\big{]}^{*}_{\alpha}G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)\big{[}{\bf
J}_{if}(\mathbf{R}^{\prime})\big{]}_{\beta}e^{-iq_{\parallel}(z^{\prime}-z)}$
$\displaystyle=\int d\mathbf{x}\,d\mathbf{x}^{\prime}\big{[}{\bf
J}_{if}(\mathbf{R})e^{i(-q_{\parallel})z}\big{]}^{*}_{\alpha}G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)\big{[}{\bf
J}_{if}(\mathbf{R}^{\prime})e^{i(-q_{\parallel})z^{\prime}}\big{]}_{\beta},$

where $\alpha,\beta=x,y,z$ and Einstein summation notation has been used. Eq.
(58) can be equivalently expressed as
$w_{fi}(q_{\parallel})=w_{if}(-q_{\parallel})$ provided
$G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)=G_{\beta\alpha}(\mathbf{x}^{\prime},\mathbf{x},\omega)$.
Said differently, interchanging the initial and final transverse states
together with changing the sign of the recoil momentum wave vector
$q_{\parallel}$ leaves the EEL observable invariant in a reciprocal medium. In
the case of an isolated dipolar target at position $\mathbf{x}_{d}$
characterized by frequency-dependent polarizability
$\tensor{\bm{\alpha}}(\omega)$,
$\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ satisfies the
reciprocity condition when the polarizability tensor is complex symmetric,
i.e., when $\tensor{\bm{\alpha}}(\omega)=\tensor{\bm{\alpha}}^{T}(\omega)$.
Leveraging again the identity for the electromagnetic Green’s tensor of a
single isolated dipole in free space, the $\alpha\beta$-component of
$\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ is

$\displaystyle G_{\alpha\beta}(\mathbf{x},\mathbf{x}^{\prime},\omega)$
$\displaystyle=-{4\pi\omega^{2}}\widetilde{G}^{0}_{\alpha\gamma}(\mathbf{x},\mathbf{x}_{d},\omega)\alpha_{\gamma\gamma^{\prime}}(\omega)\widetilde{G}^{0}_{\gamma^{\prime}\beta}(\mathbf{x}_{d},\mathbf{x}^{\prime},\omega)$
(59)
$\displaystyle=-{4\pi\omega^{2}}\widetilde{G}^{0}_{\beta\gamma^{\prime}}(\mathbf{x}^{\prime},\mathbf{x}_{d},\omega)\alpha_{\gamma^{\prime}\gamma}(\omega)\widetilde{G}^{0}_{\gamma\alpha}(\mathbf{x}_{d},\mathbf{x},\omega).$

Evidently,
$\tensor{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)=\tensor{\mathbf{G}}^{T}(\mathbf{x}^{\prime},\mathbf{x},\omega)$
is guaranteed provided that the polarizability tensor satisfies
$\tensor{\bm{\alpha}}(\omega)=\tensor{\bm{\alpha}}^{T}(\omega)$, i.e., is a
(complex) symmetric matrix.

As a corollary, dipolar targets with polarizability tensors that are not
(complex) symmetric will exhibit distinct EEL spectra under
$\Psi_{i}\leftrightarrow\Psi_{f}$ interchange. In particular, consider the BK
structure from Fig. 6. Following previous treatments [101], each rod is
modeled as a harmonic oscillator with coordinate oriented along the long axis
of each constituent rod. The two rod coordinates are linearly coupled with
strength $g$. With rods oriented along $\hat{\mathbf{x}}$ and
$\hat{\mathbf{y}}$, dimer center of mass in the $z=0$ plane, and a relative
separation of $\zeta$ between the $z$ coordinates of the two rods, the
effective dipole polarizability characterizing the BK electromagnetic response
can be expressed as

$\tensor{\bm{\alpha}}_{\pm}(\omega)=\begin{bmatrix}\frac{e^{2}/m}{Z_{\pm}(\omega)}&g\frac{e^{2}/m}{Z^{2}_{\pm}(\omega)}e^{i\phi}e^{-iq_{\parallel}\zeta}&0\\\
g\frac{e^{2}/m}{Z^{2}_{\pm}(\omega)}e^{i\phi}e^{iq_{\parallel}\zeta}&\frac{e^{2}/m}{Z_{\pm}(\omega)}&0\\\
0&0&\alpha_{zz}(\omega)\\\ \end{bmatrix}.$ (60)

In this expression
$Z_{\pm}(\omega)=\big{[}(\omega^{2}-\omega^{2}_{\pm})^{2}+(\omega\gamma_{D})^{2}\big{]}^{{1}/{2}}$,
where $\gamma_{D}$ is the damping rate of the rods, and $\omega_{\pm}$ and
$\phi$ denote the eigenfrequencies and phases of the BK dimer’s dipolar hybrid
modes, respectively. Although this polarizability satisfies
$\tensor{\bm{\alpha}}(\omega)=\tensor{\bm{\alpha}}^{T}(\omega)$ in the
particular case $\zeta=0$, $\tensor{\bm{\alpha}}(\omega)$ is not complex
symmetric for any $\zeta\neq 0$, which is the source of the dichroic response
observed in Fig. 6.

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