Highlights

1\. Momentum diffusion in an ordinary gas owing to Unruh radiation associated
with the collisions is estimated.

2\. The effect is sufficient to scramble the motion of the molecules on
nanosecond timescales at standard pressure and temperature.

3\. Effects such as this illustrate that quantum field theory implies that
physical systems are open, not isolated, unless extreme measures are taken to
modify the quantum vacuum.

4\. It follows that the evolution of the observable universe is not unitary
and thermodynamic irreversibility is a fact not an approximation.

# Irreversible behaviour of a gas owing to Unruh radiation

Andrew M. Steane  Department of Atomic and Laser Physics, Clarendon
Laboratory, Parks Road, Oxford OX1 3PU, England.

###### Abstract

When gas molecules collide, they accelerate, and therefore encounter the
Fulling-Davies-Unruh and Moore-DeWitt effects. The size of these effects is
sufficient to randomize the motion of the gas molecules after about 1
nanosecond at standard temperature and pressure. Such observations show that
quantum field theory modifies what is required to isolate a physical system
sufficiently for its behaviour to be unitary. In practice the requirements are
never satisfied exactly. Therefore the evolution of the observable universe is
non-unitary and thermodynamically irreversible.

###### keywords:

quantum chaos , Unruh effect , irreversibility , quantum field theory ,
quantum foundations

In the first part of the argument we consider the exponential growth of the
separation of trajectories in a sequence of hard sphere collisions, and of the
out-of-time-order correlator in a quantum treatment of a gas. In the second
part we consider momentum diffusion owing to effects associated with the
quantum vacuum in the case of bodies undergoing otherwise elastic collisions.
In the final part we combine the results so as to estimate the randomizing
effect of its own Unruh radiation, and Moore-DeWitt radiation, on a gas which
is otherwise isolated. Implications for irreversibility and determinism in
general are discussed.

Consider a collection of rigid spheres in motion and colliding with one
another in elastic collisions, as described by classical physics. This is a
simple model of a gas so we will refer to these spheres as ‘atoms’. Suppose
that the direction of motion of some given atom is uncertain by
$\Delta\theta_{0}$. If the distance travelled by an atom between collisions is
$\lambda$, then the position at which two atoms hit one another is imprecise
by approximately $\lambda\Delta\theta_{0}$. If the atomic radius is $r$ then
the collision point subtends an angle at either atom’s centre that is
uncertain by approximately $\lambda\Delta\theta_{0}/r$. The collision causes
the direction of motion after the collision to become uncertain by an amount
of order twice this (see figure 1), so the direction of motion after the
collision is uncertain by $\Delta\theta_{1}\simeq
2(\lambda/r)\Delta\theta_{0}$. By the same argument, the direction of motion
after two collisions is uncertain by $\Delta\theta_{2}\simeq
2(\lambda/r)\Delta\theta_{1}=4(\lambda/r)^{2}\Delta\theta_{0}.$ Further
collisions continue to amplify the uncertainty. After $n$ collisions, and
assuming the angles remain small compared to one radian, the uncertainty is of
order

$\displaystyle\Delta\theta_{n}\simeq(2\lambda/r)^{n}\Delta\theta_{0}.$ (1)

The exponential growth here is an example of a Lyapunov exponent. Our simple
calculation matches quite well the results of more sophisticated treatments in
the literature.[18] For the quantum treatment one may study the out-of-time-
order correlator which is closely related to the classical Lyapunov
exponent.[17, 13] Zhang obtains values consistent with the above for a gas at
high temperature (far from degeneracy).[28]. On this basis we claim that, at
high temperatures, the number of collisions required to randomize the motion
of a gas is the number required to obtain $\Delta\theta_{n}\sim 1$:

$\displaystyle n\simeq-\frac{\ln\Delta\theta_{0}}{\ln(2\lambda/r)}.$ (2)

This concludes the first part of the argument.

$\theta$ Figure 1: The final direction of motion when one sphere collides with
another changes by $2\delta\theta$ when the angle $\theta$ changes by
$\delta\theta$.

Now let’s consider Unruh radiation [25, 11, 4]. The Fulling-Davies-Unruh
effect can be presented in various ways [4, 26, 9, 3, 24, 23]. (A related
phenomenon is the Moore-DeWitt effect, also called (by a misnomer) the
dynamical Casimir effect [20, 6, 21, 5, 7].) At the heart of it is Unruh’s
observation that, in the coordinate system of the Rindler frame (constantly
accelerating frame in flat spacetime), the vacuum state of quantum field
theory takes the form of a thermal state with temperature $T_{\rm U}={\hbar
a}/{(2\pi k_{\rm B}c)}$ where $a$ is the proper acceleration. There is not yet
a consensus on the detail of the physical implications [10, 9, 12, 2], but it
will be sufficient to our purpose to take the following broadly standard point
of view. We consider an inertial reference frame, with the electromagnetic
field initially in its vacuum state. In this frame we suppose there exists a
system A which provides a force on system B, causing B to accelerate. B may be
any system having electromagnetic interactions, such as a charged body, a
detector, an atom, a lump of fused silica. In this situation, we claim, the
net force on B will fluctuate. If B has internal structure, then it will
undergo internal excitations, and subsequently emit photons, in a stochastic
way. The energy is provided by system A; the complete process involves the
system’s response to correlations in the vacuum state of the electromagnetic
field.[24, 23] If the proper acceleration is constant on average and goes on
for long enough, then the resulting fluctuation is the same as if B were
bathed in thermal radiation at the Unruh temperature in its instantaneous rest
frame. If B has no internal structure but is an accelerating charged particle,
the QED treatment of radiation reaction equally leads to a fluctuating force
[8, 9, 16, 15].

In view of the above, we make the following claim. A physical entity
accelerating through the vacuum will undergo internal excitations similar to
those it would undergo if it were moving inertially and subject to radiation
such that the density matrix of the electromagnetic field is diagonal in the
Fock basis, with a mean excitation per mode

$\displaystyle\bar{n}(\omega)=\left[{\exp({2\pi c\omega/a})-1}\right]^{-1}.$
(3)

A major source of imprecision in the following calculation is the fact that
the motion under consideration will only involve acceleration for short
periods of time, so the Unruh result does not apply exactly. (It is also true
that the acceleration during a particle collision is not constant, but we
expect a brief period of constant acceleration to be a reasonable
approximation for rough calculations.) Acceleration for finite periods is
discussed in [19, 4, 1]. The approximation that the Unruh temperature applies
to the majority of the elapsed proper time is good when the product of proper
acceleration and proper time is of order $c$, but the motion we shall consider
is far from that limit. To further support the main argument of the paper, we
shall also present a rough estimate from the Moore-DeWitt effect, after the
Unruh-effect estimate.

Momentum diffusion owing to scattering of thermal radiation is described by
Joos [14]. Adopting that model, we find that an accelerating body experiences
momentum diffusion such that after a time $\tau$ the momentum variance owing
to Unruh radiation at frequency $ck$ is $\Delta
p_{k}^{2}=\hbar^{2}k^{2}\Gamma_{k}\tau$, where

$\displaystyle\Gamma_{k}=\int\frac{{\rm d}\sigma}{{\rm d}\Omega}{\rm
d}\Omega\frac{\bar{n}c}{V}2(1-\cos\theta)\,\simeq\frac{\bar{n}c}{V}\sigma$ (4)

where $\sigma$ is the cross-section for scattering of radiation by the body,
$\bar{n}$ is given by Eqn (3) with $\omega=ck$ and $V$ is the volume of space
containing the electromagnetic field; it will go to infinity when we integrate
over $k$.

It will emerge that the wavelengths concerned are in the radio wave part of
the spectrum, where a suitable approximation for atoms and molecules is to use
the polarizability and hence scattering cross-section of a conducting sphere:
$\sigma\simeq(10\pi/3)r^{2}(kr)^{4}.$ We thus find for the total momentum
diffusion

$\displaystyle\Delta p^{2}=\int\Delta p_{k}^{2}\frac{V\rm
d^{3}k}{(2\pi)^{3}}\,\simeq\,\frac{5\hbar^{2}}{3\pi
r^{2}}\frac{c}{v}\int\frac{x^{8}}{\exp(\alpha x)-1}\rm dx$ (5)

where $x=kr$ and $\alpha=2\pi c^{2}/ar$. For two molecules of size $r$ the
acceleration during an elastic collision at initial speed $v$ can be estimated
as $a\simeq v/\tau\simeq v^{2}/r$. For a gas at temperature $T$, $\langle
v^{2}\rangle=3k_{\rm B}T/m$ so we find $a\simeq 3k_{\rm B}T/mr$ and
$\alpha\simeq(2/3)\pi mc^{2}/k_{\rm B}T$. We shall be considering a gas far
from the relativistic limit, so $\alpha\gg 1$. In this situation the integrand
in (5) is sharply peaked at $\alpha x\simeq 8$ and therefore the $-1$ in the
denominator can be neglected and the integral is well approximated by
$8!/\alpha^{9}$.

Owing to the momentum diffusion associated with a single collision, the
direction of motion of either molecule is liable to vary by the standard
deviation $\Delta\theta_{0}=\Delta p/p$. Hence we find

$\displaystyle\Delta\theta_{0}\simeq\frac{\hbar}{rmv}\sqrt{\frac{8!5c}{3\pi
v}}\left(\frac{3k_{\rm B}T}{2\pi mc^{2}}\right)^{9/2}.$ (6)

We now apply the above to nitrogen gas at standard temperature and pressure.
We find $rm\bar{v}\simeq 103\hbar$, $\alpha=2\times 10^{12}$, so $\alpha kr=8$
when $k=0.007\,{\rm m}^{-1}$ which corresponds to a wavelength around 850 m.
Eqn (6) gives $\Delta\theta_{0}\sim 3\times 10^{-53}$ and then (2) gives
$n\simeq 34$. It takes about a nanosecond for any one N2 molecule to undergo
34 collisions at STP. Therefore we find that nitrogen gas is randomized by the
effects of Unruh radiation (amplified by collisions) after about one
nanosecond at STP.

Now let’s consider the Moore-DeWitt effect. We will adopt the results of
Dalvit and Neto,[5] with appropriate adjustments. Dalvit and Neto (DN)
consider a mirror of imperfect reflectivity oscillating in the direction
normal to its surface, coupled to a scalar quantum field. They use a coupling
parameter $\Omega$ which is defined such that the amplitude reflectivity is
$R(\omega)=-i\Omega/(\omega+i\Omega)$. We will crudely transfer this model to
our situation by treating a single collision as an oscillation of a mechanical
oscillator through one half-period, with frequency $\omega_{0}=v/r$. The
colliding atoms constitute the ‘mirror’ and we estimate the coupling parameter
by adopting $k^{2}\sigma\sim|R(\omega)|^{2}$ using the scattering cross-
section $\sigma$ as above. This gives
$\Omega\simeq\sqrt{10\pi/3}(kr)^{3}\omega$. DN find momentum diffusion at the
rate $\Gamma m\omega_{0}\hbar/2$ where $\Gamma$ is given in their eqn (9) as

$\displaystyle\Gamma=\frac{\hbar\Omega\omega_{0}}{2\pi
mc^{2}}\zeta(\omega_{0}/\Omega)\simeq\frac{\hbar\Omega^{2}}{2\pi
mc^{2}}{\ln(\omega_{0}/\Omega)},$ (7)

where $\zeta(u)=\ln(1+u^{2})/2u-\arctan(u)/u^{2}-1/u$ and the second
expression is for the case $\Omega\ll\omega_{0}$. Using $\Delta p^{2}=\Gamma
m\omega_{0}\hbar\tau/2$ we obtain

$\displaystyle\Delta\theta_{0}=\frac{\Delta
p}{p}\simeq\frac{\hbar\Omega}{2mvc}\sqrt{\ln(\omega_{0}/\Omega)}$ (8)

where we used $\omega_{0}\tau=\pi$. Our expression for $\Omega$ gives it as
proportional to $\omega^{4}$. We must use the value $\omega\sim\omega_{0}$ in
this expression because that is the order of magnitude of the range of
frequencies sampled by the (one half-period) ‘oscillation’. This gives
$kr=v/c$ and we obtain

$\displaystyle\Delta\theta_{0}\simeq\frac{\hbar}{rmv}\sqrt{(5\pi/6)\ln(\omega_{0}/\Omega)}\left(\frac{v}{c}\right)^{4}.$
(9)

Ignoring for a moment the $\sqrt{\ln}$ term in (9) (which has only a weak
dependence on $v/c$), (6) and (9) both have the form
$\Delta\theta_{0}\propto(\hbar/rmv)(v/c)^{n}$ but with different powers $n$.
Even if we have estimated the coupling $\Omega$ poorly, the result for
$\ln\Delta\theta_{0}$ will not change by much. We find that the Moore-DeWitt
effect dominates the Unruh effect in the collisions under discussion, and the
value of $\ln(\Delta\theta_{0})$ is predicted to be about twice as large as
the one predicted from the Unruh effect, so 16 collisions are sufficient to
randomize the motion at STP. It is not necessary for either calculation to be
very accurate in order that they are sufficient to support the main argument
of this paper.

There are many larger physical effects which will influence the motion of gas
molecules more quickly than this. We have drawn attention to the role of
quantum field theory (QFT) because of the implications for determinism and
reversibility in physics more generally. The Fulling-Davies-Unruh effect,
Hawking radiation and the Moore-DeWitt effect all call into question the
degree to which QFT predicts deterministic, unitary evolution. One can observe
from the role of Schrödinger’s equation in QFT that it predicts globally
unitary behaviour of an entirely isolated system. However QFT modifies our
understanding of what is required to achieve such isolation. It is not
sufficient merely to put other systems far away. One must also ‘switch off’
the vacuum effects by banishing black holes and surrounding the system in
question with perfectly reflecting walls.

Both the Unruh and Hawking calculation (which are related) can be formulated
by expressing the vacuum state in the entangled form

$\displaystyle\left|0\right>$ $\displaystyle=$
$\displaystyle\prod_{i}C_{i}\sum_{n_{i}=0}^{\infty}e^{-\pi
n_{i}\omega_{i}c/a}\left|n_{i},R\right>\otimes\left|n_{i},L\right>$ (10)

where $C_{i}$ are constant coefficients and $R,L$ refer to spacetime regions
which may be spacelike-separated and on different sides of a horizon. In order
to describe physical effects in one region one traces over the variables
associated with the other. Taking the trace corresponds to throwing away
information and this is what leads to non-unitary evolution. The evolution of
the density matrix remains deterministic, but the realisation of physical
outcomes is not entirely deterministic because the physical system adopts one
state or another from a pointer basis—the prediction for a Schrödinger cat,
for example, gives probabilities of one half but in any given trial the cat is
either alive or dead. (A more careful statement would be: a rational observer
wishing to plan for their future is informed by quantum theory that one or the
other outcome, not evidence of a superposition of both, will be observable,
and the theory furnishes the probabilities and no more.)

It is well known that open quantum systems can exhibit non-unitary behaviour
which can be modelled by calculating their entanglement with other systems and
then tracing over the latter.[22] A lesson from effects associated with the
QFT vacuum is that all physical systems are open in this sense, and therefore
liable to have non-unitary dynamics, unless measures are taken to modify the
QFT vacuum by the use of mirrors of extremely high reflectivity. In short, QFT
does not predict unitary behaviour of any part of the universe, nor even of
the whole observable universe, except in certain idealized circumstances which
never hold exactly. This same conclusion was affirmed by Unruh and Wald in
quite general terms.[27] They discussed it more fully in the context of black
holes; we have shown that such considerations affect everyday phenomena.

A related issue is what kind of physical system warrants comparison with an
ordinary household cat, when one is considering issues in quantum interference
and measurement raised by the Schrödinger’s cat thought-experiment. Among the
significant features of a cat is the chaotic motion of its water molecules,
such that they exhibit the kind of sensitivity to their own Unruh radiation
which we have discussed. No physical system exhibiting empirical evidence of
overall quantum superposition has ever been even remotely comparable to a cat
on this measure.

It would be of interest to calculate the QFT effects during a collision more
carefully than was done here. One must account, in particular, for the very
small duration of the periods of acceleration, where motion never approaches
the speed of light, in the Unruh effect. One might also wish to set up the
Moore-DeWitt calculation afresh. Such efforts may considerably modify the
estimate of $\Delta\theta_{0}$. However if it does not change
$\log(\Delta\theta_{0})$ by very much then we would still have the prediction
of randomization of a gas on nanosecond timescales.

I thank a thoughtful referee for drawing my attention to some further work and
for the insight that it would be valuable to flesh out the Moore-DeWitt part
of the argument more fully.

## References

  * [1] G Barton and A Calogeracos.  Acceleration-induced radiative excitation of ground-state atoms.  Journal of Physics A: Mathematical and Theoretical, 41(16):164030, 2008.
  * [2] S. De Bièvre and M. Merkli.  The Unruh effect revisited.  Class. Quant. Grav., 23:6525–6541, 2006.
  * [3] Timothy H. Boyer.  Thermal effects of acceleration through random classical radiation.  Phys. Rev. D, 21:2137–2148, Apr 1980.
  * [4] Luís C. B. Crispino, Atsushi Higuchi, and George E. A. Matsas.  The Unruh effect and its applications.  Rev. Mod. Phys., 80:787–838, Jul 2008.
  * [5] Diego A. R. Dalvit and Paulo A. Maia Neto.  Decoherence via the dynamical casimir effect.  Phys. Rev. Lett., 84:798–801, Jan 2000.
  * [6] Bryce S. DeWitt.  Quantum field theory in curved spacetime.  Physics Reports, 19:295–357, 1975.
  * [7] V. V. Dodonov.  Dynamical casimir effect: some theoretical aspects.  Journal of Physics: Conference Series, 161:012027, 2009.
  * [8] G. W. Ford and R. F. O’Connell.  Radiation reaction in electrodynamics and the elimination of runaway solutions.  Physics Letters A, 157:217–220, 1991.
  * [9] S. A. Fulling.  Review of some recent work on acceleration radiation.  J. Mod. Opt., 52:2207–2213, 2005.
  * [10] S. A. Fulling and W. G. Unruh.  Comment on “boundary conditions in the Unruh problem”.  Phys. Rev. D, 70:048701, Aug 2004.
  * [11] Stephen A. Fulling.  Nonuniqueness of canonical field quantization in Riemannian space-time.  Phys. Rev. D, 7:2850–2862, May 1973.
  * [12] P G Grove.  On an inertial observer’s interpretation of the detection of radiation by linearly accelerated particle detectors.  Classical and Quantum Gravity, 3(5):801, sep 1986.
  * [13] Maldacena J, Shenker S H, and Stanford D.  A bound on chaos.  J. High Energy Phys., 2016:106, 2016.
  * [14] E. Joos and H.D. Zeh.  The emergence of classical properties through interaction with the environment.  Z. Phys. B, 59:223–243, 1985.
  * [15] Sanved Kolekar.  Directional dependence of the Unruh effect for spatially extended detectors.  Phys. Rev. D, 101:025002, Jan 2020.
  * [16] Sanved Kolekar and T. Padmanabhan.  Drift, drag and brownian motion in the Davies-Unruh bath.  Phys. Rev. D, 86:104057, Nov 2012.
  * [17] A. Larkin and Y. N. Ovchinnikov.  Quasiclassical method in the theory of superconductivity.  Sov. Phys. JETP, 28:1200, 1969.
  * [18] A. Latz, H. van Beijeren, and J. R. Dorfman.  Lyapunov spectrum and the conjugate pairing rule for a thermostatted random lorentz gas: Kinetic theory.  Phys. Rev. Lett., 78:207–210, Jan 1997.
  * [19] Pierre Martinetti and Carlo Rovelli.  Diamond’s temperature: Unruh effect for bounded trajectories and thermal time hypothesis.  Class. Quantum Grav., 20:4919–4931, 2003.
  * [20] G. T. Moore.  Quantum theory of the electromagnetic field in a variable-length one‐dimensional cavity.  Journal of Mathematical Physics, 11:2679–2691, 1970.
  * [21] Gheorghe Sorin Paraoanu and Göran Johansson.  Listening to the quantum vacuum: a perspective on the dynamical Casimir effect.  Europhysics News, 51/4:18–20, 2020.
  * [22] Angel Rivas and Susana F. Huelga.  Open Quantum Systems.  Springer, Berlin, Heidelberg, 2012.
  * [23] Marlan O. Scully, Anatoly Svidzinsky, and William Unruh.  Entanglement in Unruh, Hawking, and Cherenkov radiation from a quantum optical perspective.  Phys. Rev. Res., 4:033010, Jul 2022.
  * [24] Anatoly Svidzinsky, Arash Azizi, Jonathan S. Ben-Benjamin, Marlan O. Scully, and William Unruh.  Unruh and Cherenkov radiation from a negative frequency perspective.  Phys. Rev. Lett., 126:063603, Feb 2021.
  * [25] W. G. Unruh.  Notes on black-hole evaporation.  Phys. Rev. D, 14:870–892, Aug 1976.
  * [26] William G. Unruh and Robert M. Wald.  What happens when an accelerating observer detects a Rindler particle.  Phys. Rev. D, 29:1047–1056, Mar 1984.
  * [27] William G. Unruh and Robert M. Wald.  Information loss.  Rep. Prog. Phys., page 092002, 2017.
  * [28] Pengfei Zhang.  Quantum chaos for the unitary Fermi gas from the generalized Boltzmann equations.  J. Phys. B: At. Mol. Opt. Phys., 52:135301, 2019.


# Probing the Polarization of Low-Energy Excitations in 2D Materials from
Atomic Crystals to Nanophotonic Arrays using Momentum-Resolved Electron Energy
Loss Spectroscopy

Andrew W. Rossi  Department of Chemistry, University of Washington, Seattle,
WA, 98195  Marc R. Bourgeois  Department of Chemistry, University of
Washington, Seattle, WA, 98195  Caleb Walton  Department of Chemistry,
University of Washington, Seattle, WA, 98195  David J. Masiello
<EMAIL_ADDRESS>Department of Chemistry, University of Washington, Seattle,
WA, 98195

###### Abstract

Spectroscopies utilizing free electron beams as probes offer detailed
information on the reciprocal-space excitations of 2D materials such as
graphene and transition metal dichalcogenide monolayers. Yet, despite the
attention paid to such quantum materials, less consideration has been given to
the electron-beam characterization of 2D periodic nanostructures such as
photonic crystals, metasurfaces, and plasmon arrays, which can exhibit the
same lattice and excitation symmetries as their atomic analogs albeit at
drastically different length, momentum, and energy scales. Due to their lack
of covalent bonding and influence of retarded electromagnetic interactions,
important physical distinctions arise that complicate interpretation of
scattering signals. Here we present a fully-retarded theoretical framework for
describing the inelastic scattering of wide field electron beams from 2D
materials and apply it to investigate the complementarity in sample excitation
information gained in the measurement of a honeycomb plasmon array versus
angle-resolved optical spectroscopy in comparison to single monolayer
graphene.

###### keywords:

momentum-resolved electron energy loss spectroscopy ($q$-EELS), high
resolution electron energy loss spectroscopy (HREELS), 2D materials, graphene,
plasmon arrays

These authors contributed equally These authors contributed equally

Historically, techniques employing low energy electrons, such as low energy
electron diffraction (LEED), scanning tunneling microscopy (STM), high
resolution electron energy-loss spectroscopy (HREELS), and photoemission
spectroscopy (PES), have played a critical role in understanding a range of
material properties, from local atomic structure 1, 2, 3, 4 to dispersion
relations of materials hosting collective excitations, such as phonons 5, 6
and plasmons 7, 8, 9, 10. The isolation of graphene 11 and other 2D atomic
crystals 12, 13, 14 has furthered the need for state-of-the-art
characterization techniques where the excitonic 15, 16, 17, phononic 18, 19,
20, 21, 22, 23, and plasmonic 24, 25, 26, 27 excitations in these materials
underlie important applications in optoelectronic devices 25, 28, 29, 16 and
quantum information technologies 30. Due to its atom scale spatial resolution
and ability to measure broadband spectral responses, electron energy-loss
spectroscopy (EELS) performed inside a scanning transmission electron
microscope (STEM) has played an important role in characterizing such 2D
materials at their native response scales 31, 32, 33, 34, 27. However, the
high spatial resolution offered by STEM-EELS relies on momentum space
integration, limiting its use as a probe of reciprocal-space excitations. By
increasing the incident beam width, parallel beam momentum-resolved EELS
($q$-EELS) in a STEM or wide field $q$-EELS in a TEM overcome this challenge,
offering sufficient momentum resolution to characterize the dispersive
responses of graphene 23, 24 and other 2D materials 23, 15, 34. Similarly,
HREELS, adapted from its original uses in surface science, has demonstrated
the ability to retrieve detailed excitation information of 2D materials with
even higher momentum and energy resolution 20, 21, 22, 35.

Like 2D quantum materials with atomic scale periodicities, 2D periodic
nanophotonic structures, such as photonic crystals, metasurfaces, and plasmon
arrays, have been under intense study owing to their facilely tunable optical
properties and role in the formation of new hybrid light-matter states via
interaction with excitonic media in both weak and strong coupling regimes.
Actively tunable array lasers 36, 37, 38, 39 and exciton polariton structure,
dynamics 40, 41, 42, 43, 44, 45, and room temperature condensation 46, 47
represent examples that exploit the energy-momentum dispersion arising from
the discrete space translational symmetry of the lattice 48, 49 to create
coherent optical states or simulate complex quantum many-body phenomena. Due
to their ability to support directional and vector-polarized lasing emission,
bound states in the continuum (BIC) realized in periodic nanophotonic
structures have also been in the spotlight 50, 51, 52, 53, 54, 55, 56.
Measurements of these phenomena necessitate the use of characterization tools
with simultaneously high energy, momentum, and polarization resolution. While
optical-based probes have been successful in the characterization of such
materials, the use of $q$-EELS has been limited to the study of photonic
density of states in the vicinity of plasmonic thin film architectures 57 and
crystal defects 58, despite its ability to probe sample excitations with high
energy and momentum resolution that are inaccessible to light. Like STEM-EELS,
a powerful tool for mapping single nanoparticle excitations 59, 60, 61, 62,
momentum-resolved EEL measurements using wide field electron beams have strong
potential in the characterization of periodically structured nanophotonic
materials 63, as exemplified by the routine use of $q$-EELS and HREELS to
interrogate the dispersive features of 2D atomic crystals.

Here we present a common theoretical framework for the interpretation of wide
field $q$-EELS signals in the measurement of the broadband reciprocal-space
excitations in 2D crystalline materials spanning from atomic to nanophotonic
dimension and expose the unexplored ability of $q$-EELS to probe the 3D
polarization-resolved responses of plasmonic arrays. Reflection scattering
processes typical of HREELS are also equally treated from the same formalism.
While well understood as a probe of responses intrinsic to solid state 2D
materials with Ångstrom-scale bond lengths, the $\gtrsim 100$ nm periodicity
scale of 2D nanophotonic materials such as photonic crystals, metasurfaces,
and plasmon arrays present a distinctly new physical regime where
electromagnetic retardation effects prevail and companion wide field inelastic
electron scattering observables must be interpreted accordingly. Through the
lens of expanded selection rules appropriate to fully retarded light-matter
interactions, we elucidate the conditions under which $q$-EELS can be adapted
for measuring the broadband excitations of nanophotonic 2D materials with
emphasis placed upon the evolution of the 3D selection rules for scattering
processes both within and beyond the first Brillouin zone (BZ), the excitation
of optically dark transitions on and off of the light cone, and role of
electron speed and collection geometry (i.e., transmission versus reflection).
Throughout, contrast and comparison between the IR vibrational excitations of
graphene and the optical-frequency excitations of a plasmonic honeycomb
lattice as encoded in the inelastic electron scattering signal are made to
illustrate the utility of $q$-EELS in probing 2D materials excitations
spanning broadly across disparate spatial, momentum, and energy scales.

Figure 1: Probing reciprocal-space excitations in 2D material systems with
$q$-EELS. a) Scheme of $q$-EELS measurement under forward scattering
conditions. Momentum transfers between initial $\mathbf{k}_{i}$ (blue) and
final $\mathbf{k}_{f}$ (red) free electron states are dictated by the
scattering angle $\theta_{f}$ and recoil momentum $\hbar{\bf q}$ (yellow) with
projection ${\bf q}_{\parallel}=({\bf q}\cdot\hat{\bf x})\hat{\bf x}$ oriented
along the $\Gamma-\textrm{M}$ direction of the sample. b) Scheme of $q$-EELS
measurement under reflection scattering conditions, otherwise sharing the same
kinematical parameters as in transmission. The inset defines the $xz$
scattering plane relevant to both transmission and reflection geometries. The
detection angle is $\theta_{f}$ in transmission and
$\Delta\theta_{f}=\theta_{f}-\theta_{\textrm{spec}}$ in reflection, where
$\theta_{\textrm{spec}}$ is the specular angle. Lab (primed) and sample
(unprimed) coordinates are distinguished, and the $xz$-scattering plane is
always perpendicular to the 2D sample $xy$-plane. Beginning with coincident
lab and sample axes as in (a), where only the sample axes are explicitly
labeled, and collecting momentum recoils within the
$x^{\prime}z^{\prime}$-plane, an arbitrary planar scattering configuration can
be reached by (i) rotating the sample by angle $\phi$ about the
$z^{\prime}$-axis, followed by (ii) rotation by angle $\theta$ about the lab
$y^{\prime}$-axis. c) Graphene (upper) and a plasmonic array (lower) are
examples of 2D periodic honeycomb lattices structured on the atomic and
photonic length scales, respectively. $a$ is the magnitude of the primitive
vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ spanning the real space array.
In either case, point $\textrm{M}_{0}$ is located at
$\mathbf{q}_{\parallel}=(2\pi/\sqrt{3}a,0)$.

The schemes in Figs. 1a,b show a pair of generic wide field inelastic electron
scattering events whereby an incoming free electron plane wave with wave
vector $\mathbf{k}_{i}$ (blue) scatters to an outgoing plane wave with wave
vector $\mathbf{k}_{f}$ (red) via interaction with a 2D periodic material
sample. The linear momentum
$\hbar\mathbf{q}=\hbar(\mathbf{k}_{i}-\mathbf{k}_{f})$ lost by the probing
electron is transferred into excited sample states, inducing the transition
$|0\rangle\to|n\rangle$. Experimental conditions, such as detection of
transmitted (Fig. 1a) versus reflected (Fig. 1b) electrons as well as the
specific incident and collection angles involved, fix $\mathbf{q}$ (yellow),
while the projection $\mathbf{q}_{\parallel}$ of $\mathbf{q}$ onto the $xy$
sample plane can be manipulated by rotating the sample. Unless stated
otherwise, $\phi=0^{\circ}$ such that transverse recoils ${\bf q}\cdot\hat{\bf
x}$ in the $xz$ scattering plane are connected with reciprocal space
excitations in the 2D-periodic samples with Bloch vectors
$(\mathbf{K}\cdot\hat{\bf x})\hat{\bf x}.$

As representative 2D-periodic samples that support reciprocal space
excitations in the low-loss regime, we consider graphene and a plasmonic
honeycomb array (Fig. 1c), which support phonons and lattice plasmon
polaritons (LPPs), respectively. The 2D atomic (nanoparticle) positions
$\mathbf{x}_{\mathbf{n}\kappa}=\mathbf{x}_{\mathbf{n}}+{\bf
r}_{\kappa}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+{\bf r}_{\kappa}$ in the
graphene (plasmonic array) case are situated on a honeycomb lattice, which is
a hexagonal Bravais lattice described by primitive lattice vectors
$\mathbf{a}_{1}$ and $\mathbf{a}_{2}$, with
$|\mathbf{a}_{1}|=|\mathbf{a}_{2}|=a$ and $s=2$ sites per unit cell labeled by
$\kappa$. These sites are colored red and blue in Figs. 1a,b. While both
honeycomb arrays share common real space and reciprocal space symmetry, the
characteristic length scale is $a=1.42$ Å for graphene, and $a=460$ nm for the
plasmonic array.

The Hamiltonian involving either class of 2D material can be partitioned as
$\hat{H}=\hat{{H}}_{0}+\hat{{H}}_{\textrm{int}}$, where
$\hat{{H}}_{0}=\hat{{H}}_{s}+\hat{{H}}_{\textrm{el}}$ is the sum of sample
$\hat{{H}}_{s}$ and free electron $\hat{{H}}_{\textrm{el}}$ components, and
$\hat{{H}}_{\textrm{int}}$ is the interaction Hamiltonian. Beginning from
minimal coupling and working in the generalized Coulomb gauge 64, the
interaction Hamiltonian is
$\hat{{H}}_{\textrm{int}}=(e/2mc)[\hat{\mathbf{A}}\cdot\hat{\mathbf{p}}+\hat{\mathbf{p}}\cdot\hat{\mathbf{A}}]$,
where $m$ and $-e$ are the electron mass and charge, and $\hat{\mathbf{A}}$
and $\hat{\mathbf{p}}$ are the quantum mechanical operators associated with
the vector potential of the target and the linear momentum of the free
electron probe, respectively. The wide field double differential cross section
(DDCS) with fully-retarded probe-sample interactions becomes 65, 66

$\frac{\partial^{2}\sigma}{\partial
E_{if}\partial\Omega_{f}}=\Big{(}\frac{mL^{3}}{\pi\hbar^{2}}\Big{)}^{2}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\int
d\omega\,d\mathbf{x}d\mathbf{x}^{\prime}\,\textrm{Im}\Big{[}\mathbf{J}_{fi}^{*}(\mathbf{x})\cdot\bar{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot\mathbf{J}_{fi}(\mathbf{x}^{\prime})\Big{]}\delta(\omega-\varepsilon_{if}),$
(1)

where $L$ is the box quantization length,
$E_{if}=\hbar\varepsilon_{if}=\hbar\varepsilon_{i}-\hbar\varepsilon_{f}$ is
the loss energy, and $\hbar\varepsilon_{i/f}=\gamma_{i/f}mc^{2}$ with
initial/final Lorentz contraction factors
$\gamma_{i/f}=\big{[}1-(v_{i/f}/c)^{2}\big{]}^{-1/2}$. The transition current
density produced as the probing electron transitions between incoming
$\psi_{i}({\bf x})=L^{-3/2}e^{i{\bf k}_{i}\cdot{\bf x}}$ and outgoing
$\psi_{f}({\bf x})=L^{-3/2}e^{i{\bf k}_{f}\cdot{\bf x}}$ plane wave states is
$\mathbf{J}_{fi}(\mathbf{x})=-(e\hbar/2mL^{3})(2\mathbf{k}_{i}-\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{x}}$,
while $\bar{\mathbf{G}}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ is the dyadic
Green’s tensor characterizing the electromagnetic responses of the sample.
Leveraging the plane wave form of $\mathbf{J}_{fi}(\mathbf{x})$, Eq. (1) can
be recast as

$\begin{split}\frac{\partial^{2}\sigma}{\partial
E_{if}\partial\Omega_{f}}&=-\frac{1}{\pi}\Big{(}\frac{mv_{i}L^{2}}{2\pi\hbar^{2}}\Big{)}^{2}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\sum_{\kappa\kappa^{\prime}}\textrm{Im}\Big{[}\mathbf{E}_{fi,{\bf
q}\kappa}^{0*}(\varepsilon_{if})\cdot\bar{\boldsymbol{\Pi}}_{\mathbf{q}\kappa\kappa^{\prime}}(\varepsilon_{if})\cdot\mathbf{E}^{0}_{fi,{\bf
q}\kappa^{\prime}}(\varepsilon_{if})\Big{]}\\\
&=-\frac{1}{\pi}\Big{(}\frac{mv_{i}L^{2}}{2\pi\hbar^{2}}\Big{)}^{2}\bigg{(}\frac{k_{f}}{k_{i}}\bigg{)}\sum_{\kappa}\textrm{Im}\Big{[}\mathbf{E}_{fi,{\bf
q}\kappa}^{0*}(\varepsilon_{if})\cdot\mathbf{p}_{n0,{\bf
q}\kappa}(\varepsilon_{if})\Big{]}\\\ \end{split}$ (2)

where

$\mathbf{E}^{0}_{fi,\mathbf{q}\kappa}=\frac{2\pi
ie\gamma_{i}}{k_{i}L^{2}\omega}\frac{({\omega}/{c})^{2}\bar{\bf{I}}-\mathbf{q}\mathbf{q}}{({\omega}/{c})^{2}-q^{2}}\cdot(2\mathbf{k}_{i}-\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{r}_{\kappa}}$
(3)

is the Fourier coefficient of the probe’s vacuum transition field
$\mathbf{E}^{0}_{fi}(\mathbf{x},\omega)=-4\pi i\omega\int d{\bf
x}^{\prime}\,\bar{\mathbf{G}}_{0}(\mathbf{x},\mathbf{x}^{\prime},\omega)\cdot(L/v_{i})\mathbf{J}_{fi}(\mathbf{x}^{\prime})$
with free space Green’s dyadic
$\bar{\mathbf{G}}_{0}(\mathbf{x},\mathbf{x}^{\prime},\omega)=(-1/4\pi\omega^{2})[(\omega/c)^{2}\bar{\bf
I}+\nabla\nabla]{e^{i({\omega}/{c})|\mathbf{x}-\mathbf{x^{\prime}}|}}/{|\mathbf{x}-\mathbf{x}^{\prime}|}$.
The Fourier coefficient is related to the position- and frequency-dependent
field, evaluated at ${\bf x}=(\mathbf{x}_{\mathbf{n}\kappa},z=0)$ and
$\omega=\varepsilon_{if}$, as $\mathbf{E}^{0}_{fi}({\bf x}_{{\bf
n}\kappa},\varepsilon_{if})\equiv\mathbf{E}^{0}_{fi,{\bf
n}\kappa}(\varepsilon_{if})=\mathbf{E}^{0}_{fi,{\bf q}\kappa}e^{i{\bf
q}\cdot{\bf x}_{\bf n}}$. Intrinsic excitations of the sample are encoded
within the tensor-valued $\mathbf{q}$-dependent response function
$\bar{\boldsymbol{\Pi}}_{\mathbf{q}\kappa\kappa^{\prime}}(\varepsilon_{if}),$
while the subset that are excited by the electron probe are characterized by
the induced moment ${\bf p}_{n0,{\bf
q}\kappa}(\varepsilon_{if})=\sum_{\kappa^{\prime}}\bar{\boldsymbol{\Pi}}_{\mathbf{q}\kappa\kappa^{\prime}}(\varepsilon_{if})\cdot\mathbf{E}^{0}_{fi,{\bf
q}\kappa^{\prime}}(\varepsilon_{if})$.

Despite apparent physical distinctions, the discrete translation symmetry
shared by each structure necessitates periodic energy-momentum dispersion of
both phonon and LPP quasiparticles excitations. In either class of 2D
material, Bloch wave excitations with in-plane momentum $\hbar\mathbf{K}$ can
be described by the equations of motion (SI)

$-\omega^{2}{\mathbf{u}}_{\mathbf{K}\kappa}-i\eta\omega{\mathbf{u}}_{\mathbf{K}\kappa}+\sum_{\kappa^{\prime}}\bar{\boldsymbol{\cal
D}}_{\kappa\kappa^{\prime}}(\mathbf{K},\omega)\cdot{\mathbf{u}}_{\mathbf{K}\kappa^{\prime}}=Z_{\kappa}eM_{\kappa}^{-1}\mathbf{E}_{\mathbf{K}\kappa}^{0},$
(4)

where the displacement of the coordinate at lattice position
$\mathbf{x}_{\mathbf{n}\kappa}$ is
$\mathbf{u}_{\mathbf{n}\kappa}=\mathbf{u}_{\mathbf{K}\kappa}e^{i\mathbf{K}\cdot\mathbf{x}_{\mathbf{n}}}e^{-i\omega
t}$. Electron beams serve as one example of a probe of reciprocal space
lattice excitations where the in plane recoil momentum $\hbar{\bf
q}_{\parallel}$ is the Bloch momentum $\hbar{\bf K}$. However, propagating
free space light can also drive lattice responses at specific ${\bf K}$ points
corresponding to the projection ${\bf k}_{\parallel}$ of its wavevector ${\bf
k}$ onto the sample plane. In either case, the induced moments are ${\bf
p}_{n0,\mathbf{K}}=[Z_{1}e{\bf u}_{\mathbf{K}1},\dots,Z_{s}e{\bf
u}_{\mathbf{K}s}]=[(-\omega^{2}-i\eta\omega)\bar{\bar{\mathbf{I}}}+\bar{\bar{\boldsymbol{\cal
D}}}(\mathbf{K},\omega)]^{-1}[(Z_{1}e)^{2}\mathbf{E}_{{\bf
K}1}^{0}/M_{1},\ldots,(Z_{s}e)^{2}\mathbf{E}_{{\bf
K}s}^{0}/M_{s}]=\bar{\bar{\boldsymbol{\Pi}}}_{\bf
K}(\omega)\mathbf{E}^{0}_{\mathbf{K}}$, where the $3s\times 1$ vector
$\mathbf{E}^{0}_{\mathbf{K}}=[\mathbf{E}_{{\bf
K}1}^{0},\ldots,\mathbf{E}_{{\bf K}s}^{0}]$ represents the specific probe
under consideration evaluated at each of the $s$ sublattice sites. Double bars
have been introduced to distinguish $3s\times 3s$ matrices from $3\times 3$
matrices, which have a single bar, e.g.,
$\big{[}\bar{\bar{\boldsymbol{\Pi}}}_{\mathbf{K}}(\omega)\big{]}_{\kappa\kappa^{\prime}}=\bar{\boldsymbol{\Pi}}_{\mathbf{K}\kappa\kappa^{\prime}}(\omega)$,
while the absence of a $\kappa$ subscript indicates that a vector is $3s\times
1$.

Figure 2: Energy-momentum dispersion of graphene phonons probed along the
$\Gamma-\textrm{M}$ direction. a) and b) display the $q$-EELS DDCS in the
transmission and reflection geometries, respectively. Each spectrum at fixed
${\bf q}_{\parallel}$ is independently normalized to the maximum DDCS value at
that ${\bf q}_{\parallel}$ point as in Ref.23. Gray traces indicate the
dispersion of the graphene phonon eigenenergies. c) and d) show the normalized
magnitude of the in-plane $\mathbf{E}^{0}_{fi}\cdot\hat{\mathbf{x}}$ (red) and
out-of-plane $\mathbf{E}^{0}_{fi}\cdot\hat{\mathbf{z}}$ (blue) components of
the electron’s vacuum transition field at $E_{if}=100$ meV versus
$\mathbf{q}_{\parallel}$ along $\Gamma_{0}-\textrm{M}_{1}$, computed in the
fully-retarded (solid) and quasistatic (dotted) limits in transmission (c) and
reflection (d) geometries. Note the scale difference between $\theta_{f}$ and
$\Delta\theta_{f}$ in (c) and (d). The inset in c) shows the dispersion of
$\mathbf{E}^{0}_{fi}$ as well as that of free space plane wave light ${\bf
E}^{0}=E_{0}\hat{\boldsymbol{\epsilon}}({\bf k})$ in the immediate vicinity of
the $\Gamma_{0}$ point, where the dashed lines represent the in-plane
$\mathbf{E}^{0}\cdot\hat{\mathbf{x}}$ (red) and out-of-plane
$\mathbf{E}^{0}\cdot\hat{\mathbf{z}}$ (blue) components of the incident
optical field at $\hbar\omega=100$ meV versus in-plane wave vector ${\bf
k}_{\parallel}=(\mathbf{k}\cdot\hat{\mathbf{x}})\hat{\bf x}$. In a) and c),
the incident electron wave vector $\mathbf{k}_{i}$ is oriented normal to the
sample plane ($\theta_{i}=0^{\circ}$) with initial velocity $v_{i}=0.3c$. In
b) and d), $\theta_{i}=80^{\circ}$ with respect to the surface normal in the
$xz$ scattering plane and $v_{i}=0.01c$. All calculations include an empirical
damping rate of $\eta=20$ $\textrm{meV}/\hbar$ at zero temperature and a
vacuum background.

Focusing first on atomic crystal phonons where the resonant IR wavelengths
together with subnanometer bond lengths justify neglect of the finite speed of
light, the generalized dynamical matrix $\bar{\boldsymbol{\cal
D}}_{\kappa\kappa^{\prime}}(\mathbf{K},\omega)$ reduces to the conventional
dynamical matrix
$\bar{\mathbf{D}}_{\kappa\kappa^{\prime}}(\mathbf{K})=\bar{\mathbf{F}}_{\kappa\kappa^{\prime}}(\mathbf{K})+\bar{\mathbf{C}}_{\kappa\kappa^{\prime}}(\mathbf{K})$
67, which accounts for the sum of covalent
$\bar{\mathbf{F}}_{\kappa\kappa^{\prime}}(\mathbf{K})$ and ionic
$\bar{\mathbf{C}}_{\kappa\kappa^{\prime}}(\mathbf{K})$ interactions (SI). We
compute all electron-induced graphene excitations within the harmonic crystal
approximation 67, 68 based upon Eq. (4) using an empirically-parameterized
model 69, 70 for
$\bar{\mathbf{F}}_{\kappa\kappa^{\prime}}(\mathbf{q}_{\parallel})$ and
$\mathbf{q}$-dependent effective charges $Z_{\kappa}({\bf q}_{\parallel})$
along $\Gamma-\textrm{M}$ from Ref. 23 (SI). Within the sample and scattering
planes lie the longitudinal acoustic (LA) and optical (LO) phonons, while the
transverse acoustic (TA) and optical (TO) phonons are polarized within the
sample plane but are oriented perpendicular to the scattering plane.
Oscillations of the $z$-directed acoustic (ZA) and optical (ZO) phonons are
directed normal to the sample plane. Figs. 2a,b display the calculated energy-
momentum dispersion $\hbar\omega_{\lambda}({\bf K}^{\prime})$ of the phonon
eigenmodes (gray traces), where $\lambda=1-6$ denotes the band indices, as
well as the phononic responses induced under wide field (plane wave) electron
excitation as encoded in the $q$-EELS DDCS collected in the transmission
(panel a) and reflection (panel b) geometries, where momentum conservation
requires $\mathbf{K}=\mathbf{q}_{\parallel}=({\bf q}\cdot\hat{\bf x})\hat{\bf
x}=\mathbf{K}^{\prime}+\mathbf{G}$, with $\mathbf{G}$ a reciprocal lattice
vector. At each $\mathbf{q}_{\parallel}$ point, the spectrum is independently
normalized to the maximum DDCS value, as in Ref. 23. For unnormalized $q$-EELS
DDCS spectra, see Fig. S1. In Fig. 2a, the incident plane wave wave vector
$\mathbf{k}_{i}$ of the electron probe is normal to the graphene surface
($\theta_{i}=0^{\circ}$) and its initial speed is $v_{i}=0.3c$, which are
parameters typical of $q$-EELS experiments performed in the transmission
geometry 23, 15. Reflection measurements, on the other hand, generally involve
momentum transfers with a larger out-of-plane component (Figs. 1a,b) such that
the DDCS is largest for low incident electron kinetic energies and large
angles of incidence $\theta_{i}$ 20, 21, 22. Thus, for the reflection
calculations in Fig. 2b, we choose $\theta_{i}=80^{\circ}$ from the surface
normal and $v_{i}=0.01c$, typical of HREELS experiments.

It is evident from Eq. (2) that the selection rule for electron beam
excitation of a specific phonon mode ${\boldsymbol{\xi}}_{\lambda}$ at ${\bf
q}_{\parallel}$ of energy $\hbar\varepsilon_{if}$ requires overlap of the
electron’s vacuum field $\mathbf{E}^{0*}_{fi,\mathbf{q}}$ in Eq. (3) with the
induced phonon moment ${\bf p}_{n0}$ dominated by
${\boldsymbol{\xi}}_{\lambda}$ at ${\bf q}_{\parallel}$ and
$\hbar\varepsilon_{if}$. However, in the quasistatic ($c\to\infty$) and dipole
($a\ll 2\pi c/\varepsilon_{if}$) limits relevant to graphene, the electron’s
transition field $\mathbf{E}^{0}_{fi,\mathbf{q}}(\mathbf{x})\propto q^{-2}{\bf
q}$, and the general induced phonon moments become the induced dipole moments,
i.e., ${\bf p}_{n0}\to{\bf d}_{n0}$, so that the selection rule
$\mathbf{E}^{0*}_{fi,\mathbf{q}}\cdot{\bf p}_{n0,\mathbf{q}}$ reduces to
$\sum_{\kappa}{\bf q}\cdot{\bf d}_{n0,\mathbf{q}\kappa}=\sum_{\kappa}{\bf
q}_{\parallel}\cdot{\bf
d}_{n0,\mathbf{q}\kappa}^{\parallel}+{q}_{z}{d}_{n0,\mathbf{q}\kappa}^{z}$
when projected onto the sample coordinates. Based upon the dipole limit of the
scattering form factor of a single Coulombically bound target electron,
$\langle n|e^{i{\bf q}\cdot{\bf x}}|0\rangle\approx\langle n|1+i{\bf
q}\cdot{\bf x}+\cdots|0\rangle\propto{\bf q}\cdot{\bf d}_{n0}$ 71, this result
is expected as electromagnetic retardation effects are negligible for
graphene. The $\sum_{\kappa}{\bf q}\cdot{\bf d}_{n0,\mathbf{q}\kappa}$
selection rule is also identical to that invoked to rationalize HREELS and
$q$-EELS phonon measurements 72, 73.

While the $\sum_{\kappa}{\bf q}\cdot{\bf d}_{n0,\mathbf{q}\kappa}$ selection
rule explains the absence of the TO and TA bands in the spectra presented in
Fig. 2 on grounds of symmetry, the degree to which accessible bands contribute
to the DDCS spectra inside (and outside) the first BZ can be more clearly
understood by considering the top line of Eq. (2). In particular, the DDCS is
proportional to
$|\mathbf{E}_{fi,\mathbf{q}}^{0}|^{2}\,\textrm{Im}\Big{[}\hat{\mathbf{E}}_{fi,\mathbf{q}}^{0*}\cdot\bar{\bar{\boldsymbol{\Pi}}}_{\mathbf{q}}\cdot\hat{\mathbf{E}}^{0}_{fi,\mathbf{q}}\Big{]}$,
showing that the characteristics of the transition field largely dictate how
strongly each tensor component of $\bar{\bar{\boldsymbol{\Pi}}}_{\mathbf{q}}$
contributes to the DDCS. Although the eigenmode dispersion is periodic in
reciprocal space, the momentum dependence of the effective charge
$Z(\mathbf{q})$ causes
$\bar{\bar{\boldsymbol{\Pi}}}_{\mathbf{K}^{\prime}+\mathbf{G}}\neq\bar{\bar{\boldsymbol{\Pi}}}_{\mathbf{K}^{\prime}}$.
The dispersions in ${\bf q}_{\parallel}$ of the electron’s field
$\mathbf{E}^{0}_{fi}$ (solid red/blue traces) as well as that of its limiting
quasistatic form $\mathbf{E}^{0}_{fi}\propto\bf q$ (dotted red/blue traces)
are displayed in Figs. 2c,d for both transmission and reflection geometries,
clearly showing the reduction of the former to the latter in the quasistatic
limit appropriate to graphene. Notably, in transmission, the probe is strongly
$x$-polarized except in the immediate vicinity of $\Gamma_{0}$, explaining the
preferential excitation of both LA and LO phonons at both small and large
momenta, as recently observed in $q$-EELS measurements of individual
freestanding graphene monolayers 23.

In contrast, the probe’s polarization in the reflection geometry (Fig. 2d)
evolves from predominantly $z$-polarized in the first BZ to predominantly
$x$-polarized in the second BZ. This crossover in polarization content of the
electron’s field elucidates the computed DDCS in Fig. 2b, where primary
excitation of ZA and ZO phonons at low momenta 20, 21 give way to the
additional excitation of both LA and LO phonons outside of the first BZ. Taken
together, these results contrast the different information contained within
$q$-EELS measurements of graphene performed in transmission or reflection
geometries at IR energies in the first two BZs. For comparison, optical
excitation via propagating free space light is strongly limited in the IR by
the light cone boundary, where ${\bf K}=|\mathbf{k}|\hat{\mathbf{x}}\sim
10^{-3}$ nm-1 (see dashed red/blue traces in the inset of Fig. 2c).

Figure 3: Energy-momentum dispersion of LPPs along the $\Gamma-\textrm{M}$
direction in a honeycomb plasmonic array composed of 100 nm wide $\times$ 60
nm high silver nanodisks separated by 460 nm. The localized surface plasmon
energies of each isolated nanodisk are 3.13 eV and 3.36 eV in the $x$ and $z$
directions, respectively. a) Normalized optical extinction cross section under
$p$-polarized excitation as a function of incident wave vector ${\bf
k}_{\parallel}$ in the first BZ. b) Normalized $q$-EELS DDCS calculated in the
transmission geometry versus recoil wave vector ${\bf q}_{\parallel}$ in the
first BZ. The electron’s incident wave vector ${\bf k}_{i}$ is oriented normal
to the sample surface with an initial velocity of $v_{i}=0.7c$. c) Induced LPP
polarizations extracted at the boxed points in the optical extinction and
$q$-EELS DDCS spectra. d) Polarization dispersion of the electron’s vacuum
transition field $\mathbf{E}^{0}_{fi}$ (solid red/blue) and of the free space
plane wave optical field $\mathbf{E}^{0}$ (dashed red/blue) contrasted against
that of the electron’s vacuum field $\mathbf{E}^{0}_{fi}\propto{\bf q}$ in the
quasistatic limit (dotted red/blue) along the $\Gamma_{0}-\textrm{M}_{2}$
direction, all computed at 3 eV. e) $q$-EELS DDCS under same conditions as b)
along $\Gamma_{2}-\textrm{M}_{2}$. The white traces in (a), (b), and (e) are
the photonic dispersion of the empty honeycomb lattice. f) Normalized
momentum-resolved extinction (blue traces) and retarded/quasistatic DDCS
spectra (black/gray-dashed traces) at $\Gamma_{0}$ and M0 points from (a) and
(b). All calculations use a vacuum background.

Sharing the same hexagonal lattice structure as graphene but with a lattice
constant of $a=460$ nm, Fig. 3 considers a 2D plasmonic honeycomb array
composed of 100 nm diameter $\times$ 60 nm high silver nanodisks. In contrast
to graphene, the large lattice constant and optical excitation frequencies of
the array renders the entire BZ accessible to optical excitation 38. Displayed
in Fig. 3a is the angle-resolved optical extinction cross section
$\sigma_{\textrm{ext}}({\bf
k},\omega)=(4\pi\omega/c)\textrm{Im}\big{[}\sum_{\kappa}\mathbf{E}^{0*}_{\mathbf{k}\kappa}\cdot\mathbf{p}_{n0,\mathbf{k}\kappa}\big{]},$
where $\mathbf{E}^{0}_{\mathbf{k}\kappa}=E_{0}\hat{\boldsymbol{\epsilon}}({\bf
k})e^{i{\bf k}\cdot{\bf r}_{\kappa}}$ is the Fourier coefficient of the
stimulating optical plane wave field with wave vector $\mathbf{k}$, while the
fully retarded $q$-EELS DDCS (Eq. (2)) is shown in Fig. 3b, both within the
first BZ. The inelastic electron scattering spectra are computed in the
forward scattering geometry with ${\bf k}_{i}$ oriented along the $z$-axis
($\theta_{i}=0^{\circ}$) and $v_{i}=0.7c$. Both optical and electron beam LPP
spectra are calculated from Eq. (4) using the method of coupled dipoles where
the multipolar plasmon moments ${\bf p}_{n0}$ in each silver nanodisk are
approximated by individual point electric dipole moments ${\bf d}_{n0}$ 74, 75
with anisotropic polarizability evaluated using tabulated dielectric data for
silver 76 (SI). Unlike for graphene, site-to-site coupling in nanoparticle
arrays is mediated by long-range, causal interactions between nanophotonic
antenna modes localized to each nanoscale scattering element. As a
consequence, the generalized dynamical matrix $\bar{\boldsymbol{\cal
D}}_{\kappa\kappa^{\prime}}({\bf K},\omega)$ for LPPs encodes the individual
nanoantenna resonances as well as the retarded dipole-dipole interactions
between sites (SI). The white traces in Figs. 3a,b denote the empty lattice
photon dispersion in the reduced zone scheme 77.

The honeycomb lattice LPPs are photonic analogs of the acoustic (in-phase) and
optical (out-of-phase) transverse (TA, TO), longitudinal (LA, LO), and
$z$-directed (ZA, ZO) phonon modes of graphene. Under $p$-polarization,
optical extinction (Fig. 3a) reveals the in-plane ($x$-polarized) in-phase LPP
at $\Gamma_{0}$, while the electron DDCS (Fig. 3b) probes the complementary
out-of-plane ($z$-polarized) in-phase LPP at the same reciprocal space point,
as seen by the color-coded induced moment plots in Fig. 3c. The induced
dipoles exhibit the expected in-phase and out-of-phase polarizations between
sites within the unit cell at the high-symmetry points shown in Figs. 3a,b.
The complementary behavior exhibited by photon and electron probes in Figs.
3a,b can again be understood by examining the dispersion of the electron’s
vacuum transition field $\mathbf{E}^{0}_{fi}$ (solid blue/red traces) versus
that of plane wave light ${\bf E}^{0}$ (solid black/gray traces) shown in Fig.
3d. In moving from the first to second BZ, the $p$-polarized optical field
${\bf E}^{0}$ switches from dominantly $x$-polarized to dominantly
$z$-polarized at the light cone edge near M1, while $\mathbf{E}^{0}_{fi}$ from
Eq. (3) is predominantly polarized in the $z$ direction in the first BZ, but
evolves to be largely $x$-polarized at larger ${\bf
K}=\mathbf{q}_{\parallel}$. Indeed the computed optical extinction spectrum in
Fig. 3a shows clear signatures of the $x$-polarized in-phase and out-of-phase
LPPs with increasing ${\bf k}_{\parallel}$ in the first BZ. Near the light
cone edge, the $z$-polarized in-phase LPP becomes evident in
$\sigma_{\textrm{ext}}({\bf k},\omega)$, however, the $z$-polarized out-of-
phase LPP is not apparent (Fig. S2a). The evolution of
$\sigma_{\textrm{ext}}({\bf k},\omega)$ under $s$\- and $p$-polarized
excitation as $\mathbf{k}_{\parallel}$ traverses the full boundary of the
irreducible BZ is shown in Fig. S3a,b. Oppositely, but reflecting the
crossover behavior in $\mathbf{E}^{0}_{fi}$ polarization at larger ${\bf
q}_{\parallel}$ values in Fig. 3d, the electron probe excites primarily the
$z$-polarized out-of-phase LPP at M0 as well as both $x$-polarized out-of-
phase and in-phase LPPs at M0/M2 and $\Gamma_{2}$, respectively, in the third
BZ (Fig. 3e), a region inaccessible to free space optical excitation. The
$\mathbf{q}_{\parallel}$ value where $x$\- and $z$-polarized
$\mathbf{E}^{0}_{fi}$ cross over can be tuned across multiple BZs in the
transmission geometry by varying the primary kinetic energy of the probing
electron (Fig. S4). In the reflection geometry typical of HREELS, and in
contrast to optical excitation, the electron’s transition field is strongly
$z$-polarized across many BZs, which limits the induced sample polarizations
to be predominantly $z$-polarized (Fig. S5).

In contrast to the near indistinguishability of quasistatic and fully-retarded
transition fields under the conditions presented in Figs. 2c,d for graphene,
Fig. 3d displays noticeable differences in $\mathbf{E}_{fi}^{0}$ arising from
the finite speed of light. The consequences of these differences are evident
in Fig. 3f, which displays normalized momentum-resolved optical (blue traces)
and electron beam spectra at $\Gamma_{0}$ and M0, where both fully retarded
(black traces) and quasistatic (dashed gray traces) descriptions of the
electron’s vacuum transition field are adopted. While the normalized DDCS
spectra are indistinguishable at the $\Gamma_{0}$ point, where the quasistatic
and fully-retarded fields in Fig. 3d coincide, it is evident from the M0 point
spectra that a quasistatic description of the electron’s field fails to fully
capture the electron’s ability to excite the $x$-polarized out-of-phase LPP in
the first BZ at $\sim 3.05$ eV. As compared to the quasistatic case, a fully
retarded description of the electron’s field involves a larger ratio of $x$\-
to $z$-polarized $\mathbf{E}^{0}_{fi}$ components, leading to stronger
excitation of the $x$-polarized out-of-phase LPP.

While of considerable utility in the investigation of 2D atomic materials,
spectroscopies derived from the inelastic scattering of free electron beams
have found less consideration as probes of 2D periodic nanostructures. Here,
we have presented a fully-retarded theoretical approach to describe the low
energy inelastic scattering of wide field electron beams from 2D periodic
materials spanning from atomic-scale quantum materials to nanophotonic array
structures. In both forward and reflection geometries typical of $q$-EELS and
HREELS measurements, we have investigated the reciprocal-space excitations
native to a plasmonic honeycomb lattice under both inelastic electron
scattering and angle-resolved optical extinction spectroscopies, and compared
these observables to the momentum-resolved phonon excitation spectrum of
monolayer graphene. We find important differences arise in the selection rules
between 2D atomic and nanophotonic materials under optical- and electron-based
probes, leading to complementary behavior in their polarization-dependent
responses both inside and outside of the BZ. These differences, together with
a breakdown of the quasistatic approximation, thus require consideration of
fully retarded light-matter interactions to properly interpret inelastic
electron scattering signals from periodic nanophotonic materials.

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.

Additional details related to the employed graphene phonon model, two-
dimensional lattice plasmon polariton response matrix, additional graphene and
lattice plasmon dispersion diagrams, and the incident electron speed
dependence of the transition field.

## References

  * Hamers 1989 Hamers, R. Atomic-resolution surface spectroscopy with the scanning tunneling microscope. _Annu. Rev. Phys. Chem._ 1989, _40_ , 531–559
  * Saldin and De Andres 1990 Saldin, D.; De Andres, P. Holographic LEED. _Phys. Rev. Lett._ 1990, _64_ , 1270
  * Bauer 1994 Bauer, E. Low energy electron microscopy. _Rep. Prog. Phys._ 1994, _57_ , 895
  * Wolkow 1992 Wolkow, R. A. Direct observation of an increase in buckled dimers on Si(001) at low temperature. _Phys. Rev. Lett._ 1992, _68_ , 2636
  * Siebentritt et al. 1997 Siebentritt, S.; Pues, R.; Rieder, K.-H.; Shikin, A. M. Surface phonon dispersion in graphite and in a lanthanum graphite intercalation compound. _Phys. Rev. B_ 1997, _55_ , 7927
  * Aizawa et al. 1990 Aizawa, T.; Souda, R.; Otani, S.; Ishizawa, Y.; Oshima, C. Anomalous bond of monolayer graphite on transition-metal carbide surfaces. _Phys. Rev. Lett._ 1990, _64_ , 768
  * Moresco et al. 1999 Moresco, F.; Rocca, M.; Hildebrandt, T.; Henzler, M. Plasmon confinement in ultrathin continuous Ag films. _Phys. Rev. Lett._ 1999, _83_ , 2238
  * Petek and Ogawa 1997 Petek, H.; Ogawa, S. Femtosecond time-resolved two-photon photoemission studies of electron dynamics in metals. _Prog. Surf. Sci._ 1997, _56_ , 239–310
  * Diaconescu et al. 2007 Diaconescu, B.; Pohl, K.; Vattuone, L.; Savio, L.; Hofmann, P.; Silkin, V. M.; Pitarke, J. M.; Chulkov, E. V.; Echenique, P. M.; Farías, D.; Rocca, M. Low-energy acoustic plasmons at metal surfaces. _Nature_ 2007, _448_ , 57–59
  * Bostwick et al. 2007 Bostwick, A.; Ohta, T.; Seyller, T.; Horn, K.; Rotenberg, E. Quasiparticle dynamics in graphene. _Nat. Phys._ 2007, _3_ , 36–40
  * Novoselov et al. 2004 Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.-e.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric field effect in atomically thin carbon films. _Science_ 2004, _306_ , 666–669
  * Novoselov et al. 2005 Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T.; Khotkevich, V.; Morozov, S.; Geim, A. K. Two-dimensional atomic crystals. _Proc. Natl. Acad. Sci. U.S.A._ 2005, _102_ , 10451–10453
  * Naguib et al. 2011 Naguib, M.; Kurtoglu, M.; Presser, V.; Lu, J.; Niu, J.; Heon, M.; Hultman, L.; Gogotsi, Y.; Barsoum, M. W. Two-dimensional nanocrystals produced by exfoliation of Ti3AlC2. _Adv. Mater._ 2011, _23_ , 4248–4253
  * Liu et al. 2014 Liu, H.; Neal, A. T.; Zhu, Z.; Luo, Z.; Xu, X.; Tománek, D.; Ye, P. D. Phosphorene: an unexplored 2D semiconductor with a high hole mobility. _ACS Nano_ 2014, _8_ , 4033–4041
  * Hong et al. 2020 Hong, J.; Senga, R.; Pichler, T.; Suenaga, K. Probing exciton dispersions of freestanding monolayer WSe2 by momentum-resolved electron energy-loss spectroscopy. _Phys. Rev. Lett._ 2020, _124_ , 087401
  * Mak and Shan 2016 Mak, K. F.; Shan, J. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. _Nat. Photonics_ 2016, _10_ , 216–226
  * Ross et al. 2013 Ross, J. S.; Wu, S.; Yu, H.; Ghimire, N. J.; Jones, A. M.; Aivazian, G.; Yan, J.; Mandrus, D. G.; Xiao, D.; Yao, W.; Xu, X. Electrical control of neutral and charged excitons in a monolayer semiconductor. _Nat. Commun._ 2013, _4_ , 1474
  * Maultzsch et al. 2004 Maultzsch, J.; Reich, S.; Thomsen, C.; Requardt, H.; Ordejón, P. Phonon dispersion in graphite. _Phys. Rev. Lett._ 2004, _92_ , 075501
  * Mohr et al. 2007 Mohr, M.; Maultzsch, J.; Dobardžić, E.; Reich, S.; Milošević, I.; Damnjanović, M.; Bosak, A.; Krisch, M.; Thomsen, C. Phonon dispersion of graphite by inelastic x-ray scattering. _Phys. Rev. B_ 2007, _76_ , 035439
  * Politano et al. 2015 Politano, A.; de Juan, F.; Chiarello, G.; Fertig, H. A. Emergence of an out-of-plane optical phonon (ZO) Kohn anomaly in quasifreestanding epitaxial graphene. _Phy. Rev. Lett._ 2015, _115_ , 075504
  * Politano 2017 Politano, A. Spectroscopic investigations of phonons in epitaxial graphene. _Crit. Rev. Solid State_ 2017, _42_ , 99–128
  * Li et al. 2023 Li, J.; Li, J.; Tang, J.; Tao, Z.; Xue, S.; Liu, J.; Peng, H.; Chen, X.-Q.; Guo, J.; Zhu, X. Direct observation of topological phonons in graphene. _Phys. Rev. Lett._ 2023, _131_ , 116602
  * Senga et al. 2019 Senga, R.; Suenaga, K.; Barone, P.; Morishita, S.; Mauri, F.; Pichler, T. Position and momentum mapping of vibrations in graphene nanostructures. _Nature_ 2019, _573_ , 247–250
  * Kramberger et al. 2008 Kramberger, C.; Hambach, R.; Giorgetti, C.; Rümmeli, M. H.; Knupfer, M.; Fink, J.; Büchner, B.; Reining, L.; Einarsson, E.; Maruyama, S.; Sottile, F.; Hannewald, K.; Olevano, V.; Marinopoulos, A. G.; Pichler, T. Linear plasmon dispersion in single-wall carbon nanotubes and the collective excitation spectrum of graphene. _Phys. Rev. Lett._ 2008, _100_ , 196803
  * Chen et al. 2012 Chen, J.; Badioli, M.; Alonso-González, P.; Thongrattanasiri, S.; Huth, F.; Osmond, J.; Spasenović, M.; Centeno, A.; Pesquera, A.; Godignon, P.; Zurutuza Elorza, A.; Camara, N.; de Abajo, F. J. G.; Hillenbrand, R.; Koppens, F. H. L. Optical nano-imaging of gate-tunable graphene plasmons. _Nature_ 2012, _487_ , 77–81
  * Ni et al. 2018 Ni, G. X.; McLeod, A. S.; Sun, Z.; Wang, L.; Xiong, L.; Post, K. W.; Sunku, S. S.; Jiang, B.-Y.; Hone, J.; Dean, C. R.; Fogler, M. M.; Basov, D. N. Fundamental limits to graphene plasmonics. _Nature_ 2018, _557_ , 530–533
  * Zhou et al. 2012 Zhou, W.; Lee, J.; Nanda, J.; Pantelides, S. T.; Pennycook, S. J.; Idrobo, J.-C. Atomically localized plasmon enhancement in monolayer graphene. _Nat. Nanotechnol._ 2012, _7_ , 161–165
  * Liu et al. 2011 Liu, M.; Yin, X.; Ulin-Avila, E.; Geng, B.; Zentgraf, T.; Ju, L.; Wang, F.; Zhang, X. A graphene-based broadband optical modulator. _Nature_ 2011, _474_ , 64–67
  * Bonaccorso et al. 2010 Bonaccorso, F.; Sun, Z.; Hasan, T.; Ferrari, A. Graphene photonics and optoelectronics. _Nat. Photonics_ 2010, _4_ , 611–622
  * Liu and Hersam 2019 Liu, X.; Hersam, M. C. 2D materials for quantum information science. _Nat. Rev. Mater._ 2019, _4_ , 669–684
  * Krivanek et al. 2014 Krivanek, O. L.; Lovejoy, T. C.; Dellby, N.; Aoki, T.; Carpenter, R. W.; Rez, P.; Soignard, E.; Zhu, J.; Batson, P. E.; Lagos, M. J.; Egerton, R. F.; Crozier, P. A. Vibrational spectroscopy in the electron microscope. _Nature_ 2014, _514_ , 209–212
  * Dwyer et al. 2016 Dwyer, C.; Aoki, T.; Rez, P.; Chang, S.; Lovejoy, T.; Krivanek, O. Electron-beam mapping of vibrational modes with nanometer spatial resolution. _Phys. Rev. Lett._ 2016, _117_ , 256101
  * Lagos et al. 2017 Lagos, M. J.; Trügler, A.; Hohenester, U.; Batson, P. E. Mapping vibrational surface and bulk modes in a single nanocube. _Nature_ 2017, _543_ , 529–532
  * Hage et al. 2018 Hage, F. S.; Nicholls, R. J.; Yates, J. R.; McCulloch, D. G.; Lovejoy, T. C.; Dellby, N.; Krivanek, O. L.; Refson, K.; Ramasse, Q. M. Nanoscale momentum-resolved vibrational spectroscopy. _Sci. Adv._ 2018, _4_ , eaar7495
  * Politano 2021 Politano, A. On the fate of high-resolution electron energy loss spectroscopy (HREELS), a versatile probe to detect surface excitations: Will the Phoenix rise again? _Phys. Chem. Chem. Phys._ 2021, _23_ , 26061–26069
  * Yang et al. 2015 Yang, A.; Hoang, T. B.; Dridi, M.; Deeb, C.; Mikkelsen, M. H.; Schatz, G. C.; Odom, T. W. Real-time tunable lasing from plasmonic nanocavity arrays. _Nat. Commun._ 2015, _6_ , 6939
  * Guo et al. 2019 Guo, R.; Nečada, M.; Hakala, T. K.; Väkeväinen, A. I.; Törmä, P. Lasing at K Points of a Honeycomb Plasmonic Lattice. _Phys. Rev. Lett._ 2019, _122_ , 013901
  * Juarez et al. 2022 Juarez, X. G.; Li, R.; Guan, J.; Reese, T.; Schaller, R. D.; Odom, T. W. M-point lasing in hexagonal and honeycomb plasmonic lattices. _ACS Photonics_ 2022, _9_ , 52–58
  * Hirose et al. 2014 Hirose, K.; Liang, Y.; Kurosaka, Y.; Watanabe, A.; Sugiyama, T.; Noda, S. Watt-class high-power, high-beam-quality photonic-crystal lasers. _Nat. Photonics_ 2014, _8_ , 406–411
  * Liu et al. 2020 Liu, W.; Ji, Z.; Wang, Y.; Modi, G.; Hwang, M.; Zheng, B.; Sorger, V. J.; Pan, A.; Agarwal, R. Generation of helical topological exciton-polaritons. _Science_ 2020, _370_ , 600–604
  * Bourgeois et al. 2022 Bourgeois, M. R.; Beutler, E. K.; Khorasani, S.; Panek, N.; Masiello, D. J. Nanometer-scale spatial and spectral mapping of exciton polaritons in structured plasmonic cavities. _Phys. Rev. Lett._ 2022, _128_ , 197401
  * Chen et al. 2020 Chen, Y.; Miao, S.; Wang, T.; Zhong, D.; Saxena, A.; Chow, C.; Whitehead, J.; Gerace, D.; Xu, X.; Shi, S.-F.; Majumdar, A. Metasurface Integrated Monolayer Exciton Polariton. _Nano Lett._ 2020, _20_ , 5292–5300
  * Väkeväinen et al. 2014 Väkeväinen, A. I.; Moerland, R. J.; Rekola, H. T.; Eskelinen, A.-P.; Martikainen, J.-P.; Kim, D.-H.; Törmä, P. Plasmonic surface lattice resonances at the strong coupling regime. _Nano Lett._ 2014, _14_ , 1721–1727
  * Shi et al. 2014 Shi, L.; Hakala, T.; Rekola, H.; Martikainen, J.-P.; Moerland, R.; Törmä, P. Spatial coherence properties of organic molecules coupled to plasmonic surface lattice resonances in the weak and strong coupling regimes. _Phys. Rev. Lett._ 2014, _112_ , 153002
  * Baranov et al. 2018 Baranov, D. G.; Wersall, M.; Cuadra, J.; Antosiewicz, T. J.; Shegai, T. Novel nanostructures and materials for strong light-matter interactions. _ACS Photonics_ 2018, _5_ , 24–42
  * Hakala et al. 2018 Hakala, T. K.; J., M. A.; Väkeväinen, A. I.; Guo, R.; Martikainen, J.-P.; Daskalakis, K. S.; Rekola, H. T.; Julku, A.; Törmä, P. Bose-Einstein Condensation in a Plasmonic Lattice. _Nat. Phys._ 2018, _14_ , 739–744
  * Berghuis et al. 2023 Berghuis, A. M.; Castellanos, G. W.; Murai, S.; Pura, J. L.; Abujetas, D. R.; van Heijst, E.; Ramezani, M.; Sánchez-Gil, J. A.; Rivas, J. G. Room Temperature Exciton–Polariton Condensation in Silicon Metasurfaces Emerging from Bound States in the Continuum. _Nano Lett._ 2023, _23_ , 5603–5609
  * Cherqui et al. 2019 Cherqui, C.; Bourgeois, M. R.; Wang, D.; Schatz, G. C. Plasmonic surface lattice resonances: theory and computation. _Acc. Chem. Res._ 2019, _52_ , 2548–2558
  * Baur et al. 2018 Baur, S.; Sanders, S.; Manjavacas, A. Hybridization of Lattice Resonances. _ACS Nano_ 2018, _12_ , 1618–1629
  * Zhen et al. 2014 Zhen, B.; Hsu, C. W.; Lu, L.; Stone, A. D.; Soljačić, M. Topological nature of optical bound states in the continuum. _Phys. Rev. Lett._ 2014, _113_ , 257401
  * Azzam et al. 2018 Azzam, S. I.; Shalaev, V. M.; Boltasseva, A.; Kildishev, A. V. Formation of bound states in the continuum in hybrid plasmonic-photonic systems. _Phys. Rev. Lett._ 2018, _121_ , 253901
  * Liang et al. 2020 Liang, Y.; Koshelev, K.; Zhang, F.; Lin, H.; Lin, S.; Wu, J.; Jia, B.; Kivshar, Y. Bound states in the continuum in anisotropic plasmonic metasurfaces. _Nano Lett._ 2020, _20_ , 6351–6356
  * Guan et al. 2020 Guan, J.; Sagar, L. K.; Li, R.; Wang, D.; Bappi, G.; Wang, W.; Watkins, N.; Bourgeois, M. R.; Levina, L.; Fan, F.; Hoogland, S.; Voznyy, O.; Martins de Pina, J.; Schaller, R. D.; Schatz, G. C.; Sargent, E. H.; Odom, T. W. Quantum dot-plasmon lasing with controlled polarization patterns. _ACS Nano_ 2020, _14_ , 3426–3433
  * Liang et al. 2021 Liang, Y.; Lin, H.; Lin, S.; Wu, J.; Li, W.; Meng, F.; Yang, Y.; Huang, X.; Jia, B.; Kivshar, Y. Hybrid anisotropic plasmonic metasurfaces with multiple resonances of focused light beams. _Nano Lett._ 2021, _21_ , 8917–8923
  * Ha et al. 2018 Ha, S. T.; Fu, Y. H.; Emani, N. K.; Pan, Z.; Bakker, R. M.; Paniagua-Domínguez, R.; Kuznetsov, A. I. Directional lasing in resonant semiconductor nanoantenna arrays. _Nat. Nanotechnol._ 2018, _13_ , 1042–1047
  * Mohamed et al. 2022 Mohamed, S.; Wang, J.; Rekola, H.; Heikkinen, J.; Asamoah, B.; Shi, L.; Hakala, T. K. Controlling topology and polarization state of lasing photonic bound states in continuum. _Laser Photonics Rev._ 2022, _16_ , 2100574
  * Shekhar et al. 2017 Shekhar, P.; Malac, M.; Gaind, V.; Dalili, N.; Meldrum, A.; Jacob, Z. Momentum-resolved electron energy loss spectroscopy for mapping the photonic density of states. _ACS Photonics_ 2017, _4_ , 1009–1014
  * Saito et al. 2019 Saito, H.; Lourenço-Martins, H.; Bonnet, N.; Li, X.; Lovejoy, T. C.; Dellby, N.; Stéphan, O.; Kociak, M.; Tizei, L. H. G. Emergence of point defect states in a plasmonic crystal. _Phys. Rev. B_ 2019, _100_ , 245402
  * Batson 1982 Batson, P. A new surface plasmon resonance in clusters of small aluminum spheres. _Ultramicroscopy_ 1982, _9_ , 277–282
  * Nelayah et al. 2007 Nelayah, J.; Kociak, M.; Stéphan, O.; García de Abajo, F. J.; Tencé, M.; Henrard, L.; Taverna, D.; Pastoriza-Santos, I.; Liz-Marzán, L. M.; Colliex, C. Mapping surface plasmons on a single metallic nanoparticle. _Nature Physics_ 2007, _3_ , 348–353
  * García de Abajo 2010 García de Abajo, F. J. Optical excitations in electron microscopy. _Rev. Mod. Phys._ 2010, _82_ , 209
  * Cherqui et al. 2016 Cherqui, C.; Thakkar, N.; Li, G.; Camden, J. P.; Masiello, D. J. Characterizing localized surface plasmons using electron energy-loss spectroscopy. _Annu. Rev. Phys. Chem._ 2016, _67_ , 331–357
  * Polman et al. 2019 Polman, A.; Kociak, M.; García de Abajo, F. J. Electron-beam spectroscopy for nanophotonics. _Nat. Mater._ 2019, _18_ , 1158–1171
  * Glauber and Lewenstein 1991 Glauber, R. J.; Lewenstein, M. Quantum optics of dielectric media. _Phys. Rev. A_ 1991, _43_ , 467
  * Bourgeois et al. 2023 Bourgeois, M. R.; Nixon, A. G.; Chalifour, M.; Masiello, D. J. Optical polarization analogs in inelastic free-electron scattering. _Sci. Adv._ 2023, _9_ , eadj6038
  * Nixon et al. 2023 Nixon, A. G.; Chalifour, M.; Bourgeois, M. R.; Sanchez, M.; Masiello, D. J. Inelastic scattering of transversely structured free electrons from nanophotonic targets: Theory and computation. 2023; https://arxiv.org/abs/2312.08513
  * Dove 1993 Dove, M. T. _Introduction to Lattice Dynamics_ ; Cambridge University Press, 1993
  * Hohenester et al. 2018 Hohenester, U.; Trügler, A.; Batson, P. E.; Lagos, M. J. Inelastic vibrational bulk and surface losses of swift electrons in ionic nanostructures. _Phys. Rev. B_ 2018, _97_ , 165418
  * Bosak et al. 2007 Bosak, A.; Krisch, M.; Mohr, M.; Maultzsch, J.; Thomsen, C. Elasticity of single-crystalline graphite: Inelastic x-ray scattering study. _Phys. Rev. B_ 2007, _75_ , 153408
  * Michel and Verberck 2008 Michel, K.; Verberck, B. Theory of the evolution of phonon spectra and elastic constants from graphene to graphite. _Phys. Rev. B_ 2008, _78_ , 085424
  * Sakurai 1985 Sakurai, J. J. _Modern Quantum Mechanics_ ; Benjamin/Cummings Publishing Company, 1985
  * de Juan et al. 2015 de Juan, F.; Politano, A.; Chiarello, G.; Fertig, H. A. Symmetries and selection rules in the measurement of the phonon spectrum of graphene and related materials. _Carbon_ 2015, _85_ , 225–232
  * Nicholls et al. 2019 Nicholls, R.; Hage, F.; McCulloch, D.; Ramasse, Q.; Refson, K.; Yates, J. Theory of momentum-resolved phonon spectroscopy in the electron microscope. _Phys. Rev. B_ 2019, _99_ , 094105
  * Bourgeois et al. 2021 Bourgeois, M. R.; Rossi, A. W.; Chalifour, M.; Cherqui, C.; Masiello, D. J. Lattice Kerker effect with plasmonic oligomers. _J. Phys. Chem. C_ 2021, _125_ , 18817–18826
  * Bourgeois et al. 2022 Bourgeois, M. R.; Rossi, A. W.; Khorasani, S.; Masiello, D. J. Optical control over thermal distributions in topologically trivial and non-trivial plasmon lattices. _ACS Photonics_ 2022, _9_ , 3656–3667
  * Johnson and Christy 1972 Johnson, P. B.; Christy, R. W. Optical constants of the noble metals. _Phys. Rev. B_ 1972, _6_ , 4370–4379
  * Guo et al. 2017 Guo, R.; Hakala, T. K.; Törmä, P. Geometry dependence of surface lattice resonances in plasmonic nanoparticle arrays. _Phys. Rev. B_ 2017, _95_ , 155423