# On the minimum transport required to passively suppress runaway electrons in
SPARC disruptions

R.A. Tinguely1 111Author to whom correspondence should be addressed:
<EMAIL_ADDRESS>I Pusztai2, VA Izzo3, K Särkimäki4, T Fülöp2, DT Garnier1, RS
Granetz1, M Hoppe5, C Paz-Soldan6, A Sundström2, and R Sweeney1 1 Plasma
Science and Fusion Center, Massachusetts Institute of Technology, Cambridge,
MA, USA
2 Department of Physics, Chalmers University of Technology, SE-41296 Göteborg,
Sweden
3 Fiat Lux, San Diego, CA 92101, USA
4 Max Planck Institute for Plasmaphysics, 85748 Garching, Germany
5 Ecole Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC),
CH-1015 Lausanne, Switzerland
6 Department of Applied Physics and Applied Mathematics, Columbia University,
NY, USA

###### Abstract

In [V.A. Izzo _et al_ 2022 _Nucl. Fusion_ 62 096029], state-of-the-art
modeling of thermal and current quench (CQ) MHD coupled with a self-consistent
evolution of runaway electron (RE) generation and transport showed that a non-
axisymmetric ($n=1$) in-vessel coil could passively prevent RE beam formation
during disruptions in SPARC, a compact high-field tokamak projected to achieve
a fusion gain $Q>2$ in DT plasmas. However, such suppression requires finite
transport of REs within magnetic islands and re-healed flux surfaces;
conservatively assuming _zero_ transport in these regions leads to an upper
bound of RE current ${\sim}1\,\mathrm{MA}$ compared to
${\sim}8.7\,\mathrm{MA}$ of pre-disruption plasma current. Further
investigation finds that core-localized electrons, within $r/a<0.3$ and with
kinetic energies ${\sim}0.2{-}15\,\mathrm{MeV}$, contribute most to the RE
plateau formation. Yet only a relatively small amount of transport, i.e. a
diffusion coefficient ${\sim}18\,\mathrm{m^{2}/s}$, is needed in the core to
fully mitigate these REs. Properly accounting for (i) the CQ electric field’s
effect on RE transport in islands and (ii) the contribution of significant RE
currents to disruption MHD may help achieve this.

Keywords: Runaway electrons, passive mitigation, transport, disruptions, SPARC

##  1 Introduction

In [1, 2], a novel method was proposed for _passive_ mitigation of
relativistic “runaway electrons” (REs) generated during tokamak plasma
disruptions: First, an in-vessel, non-axisymmetric coil would be passively
energized through mutual coupling to the plasma current during the
disruption’s current quench (CQ); then, the resulting magnetic field
perturbation would enhance stochasticity and transport such that the RE loss
rate would dominate the growth rate, thus preventing RE beam formation.

In [3], such a “Runaway Electron Mitigation Coil” (REMC) was proposed for the
SPARC tokamak [4], a high-field ($B_{0}=12.2\,\mathrm{T}$), compact
($R_{0}=1.85\,\mathrm{m}$, $a=0.57\,\mathrm{m}$) device currently under
construction in Devens, Massachusetts, USA. The present REMC design has a
predominantly $n=1$ structure and is located on the outboard wall; a similar
coil is planned for the DIII-D tokamak, but on the inboard wall [5]. Several
aspects of the SPARC “Primary Reference Discharge” (PRD) make the RE problem
challenging: a large plasma current ($I_{\mathrm{p}}=8.7\,\mathrm{MA}$) can
lead to dangerous exponential RE growth; high core temperatures $T_{e0}\approx
20\,\mathrm{keV}$ can cause enhanced primary and hot-tail generation; DT fuel
provides a seed of non-thermal electrons through tritium beta decay; and high
energy gammas from activated materials could accelerate electrons via Compton
scattering.

However, in [6], modeling of the PRD’s worst-case-scenario CQ
(${\sim}3\,\mathrm{ms}$) showed complete prevention of RE beam formation with
the REMC – and ${\sim}5{-}6\,\mathrm{MA}$ of RE current without it. The
modeling workflow included four steps: First, the mutual couplings of all
toroidally conducting structures were simulated in COMSOL [7] to evaluate the
REMC’s vacuum electromagnetic fields during the worst-case CQ. Second, these
magnetic fields were applied at the boundary of a nonlinear, 3D NIMROD [8]
simulation to assess the plasma response and total fields. Third, the
stochastic magnetic fields were input into the orbit-following code ASCOT5 to
calculate the advective and diffusive transport [9] of energetic electrons.
Finally, these transport coefficients – $A,D$ as functions of energy, pitch,
and radius – were supplied to the hybrid fluid-kinetic code DREAM [10] for
self-consistent evolution of the RE population. Importantly, in both NIMROD
and DREAM, the REMC vacuum fields and transport coefficients, respectively,
were evolved as functions of $I_{\mathrm{p}}$ and not time explicitly.

More recently, in [11], both the thermal quench (TQ) and CQ were modeled for
the SPARC PRD and REMC; the results of this study – which bound the maximum
expected RE current – are summarized in Section 2. Section 3 further explores
these bounds in RE phase space, as well as the minimum transport needed to
fully prevent RE beam formation. Finally, results and opportunities for future
modeling are discussed in Section 4.

##  2 REMC efficacy during the thermal and current quenches

The same workflow presented in [6] and summarized in Section 1 was used in
[11] to assess the SPARC REMC’s efficacy for a full PRD mitigated disruption,
i.e. including both the TQ and CQ. Here, the TQ (${\sim}1\,\mathrm{ms}$ in
duration) was induced by neon radiation, as in a scenario where massive gas
injection was employed. The main results are captured in Figs. 1 and 2. The
pre-disruption safety factor ($q$) profile is shown at $t=0$ in Fig. 1(a) with
$q(0)\sim 1$ and $q=2$ around a normalized poloidal flux value of
$\psi_{\mathrm{N}}\approx 0.75$. During the TQ, i.e. the first
${\sim}1\,\mathrm{ms}$ of the simulation, the plasma current $I_{\mathrm{p}}$
decreases slightly, with the $I_{\mathrm{p}}$-spike denoting the start of the
CQ. Around that time, the magnetic perturbation amplitudes $\delta B/B$ first
peak (see Fig. 1(b)), and strong nonlinear coupling among odd and even
toroidal harmonics is observed. Poincaré plots of magnetic field lines, in
Fig. 1(a), also show high stochasticity during this period.

(a)

(b)

Figure 1:  (a, upper) Poincaré plots of (mostly) stochastic magnetic field
lines from NIMROD within the simulation boundary (dashed) and SPARC first wall
(solid). (a, lower) The safety factor $q$-profile evolution vs normalized
poloidal flux ($\psi_{\mathrm{N}}$) and time, with the plasma current
($I_{\mathrm{p}}$) time-evolution overlaid. (b) Amplitudes of $n=1{-}10$ modes
in units of $\delta B/B=\sqrt{W_{\mathrm{mag}}(n)/W_{\mathrm{mag}}(n=0)}$ with
$W_{\mathrm{mag}}$ the magnetic energy Fourier component. Subplot (a) is
reproduced from Figure 5 in [11].

However, from $t\approx 1{-}1.5\,\mathrm{ms}$, $q(0)$ increases from 1 to 2,
and beyond $t>1.5\,\mathrm{ms}$, the REMC is no longer resonant with the
plasma core (refer to Figure 1 in [6] for more details). Although the
predominantly odd externally applied fields continue to grow as a the coil
current continues to increase, these are now largely non-resonant fields that
do not perturb the flux surfaces, and the nonlinearly excited resonant field
components, both odd and even, decay away. Thus, small islands start to
reform, re-healing as closed flux surfaces by $t\approx 1.8\,\mathrm{ms}$.
Note that the contribution from REs to the MHD are _not_ included in these
NIMROD simulations, but the back-reaction is expected to be small for low RE
currents early on. This will be discussed further in Section 4.

Figure 2 shows the self-consistent evolution of Ohmic and RE currents from
DREAM, including the advective and diffusive transport calculated by ASCOT5 in
DREAM’s fluid transport model [12]. Note that the time bases of the DREAM and
NIMROD simulations are not exactly the same; instead, the DREAM simulation is
initialized with profiles close to the time of NIMROD’s
$I_{\mathrm{p}}$-spike. Importantly, the TQ in DREAM is only modeled for the
final ${\sim}0.1\,\mathrm{ms}$ of NIMROD’s ${\sim}1\,\mathrm{ms}$ TQ because
DREAM requires a monotonic variation of the plasma current from which to map
transport coefficients. These transport coefficients evolve with the plasma
current until the final $I_{\mathrm{p}}$-value of the NIMROD simulation; then,
they are held constant in time (see the vertical dashed line in the upper part
of Fig. 2).222 To enforce transport ambipolarity in DREAM, any change of the
electron density on a given flux surface due to transport is compensated by a
change in the bulk electron density of similar size but opposite sign.

Figure 2: Time-traces of Ohmic (solid), RE (dot-dashed), and total (dashed)
currents from DREAM; thick/thin RE currents indicate no/transport within re-
healed flux surfaces. Times denoted above correspond to the NIMROD simulation
(see Fig. 1). Transport coefficients are fixed in time after the vertical
dashed line (upper), and surface re-healing begins at the vertical dotted line
(lower). Reproduced from Figure 6 in [11].

Two scenarios for the RE current are depicted in Fig. 2: In the first, the
transport coefficients are applied as calculated throughout the entire plasma
domain, i.e. even inside the re-healed flux surfaces, and the RE current
remains negligibly low (${\sim}1\,\mathrm{\mu A}$). However, in the second
case, transport inside islands and re-healed flux surfaces is set to zero,
which is perhaps overly conservative. (Explicitly, $A=D=0$ wherever
$D<1000\,\mathrm{m^{2}/s}$.) The result is a RE plateau with current
${\sim}1\,\mathrm{MA}$. While this value is an improvement upon the
${\sim}5{-}6\,\mathrm{MA}$ of RE current expected with _no_ REMC [3, 6], it is
likely the pessimistic upper bound on the true value.

Here, it is important to note that the initial hot-tail seed can be sensitive
to the TQ cooling time prescribed in DREAM. In the CQ-only modeling effort of
[6], a ${\sim}0.5\,\mathrm{ms}$ TQ from $20\,\mathrm{keV}$ to $4\,\mathrm{eV}$
led to a ${\sim}50\,\mathrm{kA}$ seed which was then dissipated by the REMC,
with similar $\delta B/B\sim 10^{-2}$ as that in Fig. 1(b). Further reducing
the TQ time to $0.1\,\mathrm{ms}$ resulted in a much higher transient RE beam
current of ${\sim}1\,\mathrm{MA}$, which was still suppressed by the REMC.
These results are consistent with all test-particle REs losing confinement
during the TQ in NIMROD for the scenario modeled in this paper [11]. The
inclusion of TQ transport in DREAM, as a function of non-monotonic
$I_{\mathrm{p}}$ variation, is left for future work.

##  3 Investigating transport inside re-healed flux surfaces

This section explores further the ten-order-of-magnitude difference in the
predicted RE current when transport is/not accounted for within NIMROD’s
islands and re-healed flux surfaces. Radial profiles of the diffusion
coefficient ($D$) are shown in Fig. 3(a), also as a function of normalized
electron momentum, at the last NIMROD simulation time; the values shown are
taken at a representative electron pitch $p_{\parallel}/p=0.8$. There are a
few important notes here: (i) the diffusion coefficients span five orders of
magnitude from the plasma core to edge; (ii) though not shown, the advection
coefficients are of similar magnitude ($A[{\rm m/s}]\sim D[{\rm m^{2}/s}]$);
and (iii) both transport coefficients are relatively insensitive to the
electron pitch in the relevant range $p_{\parallel}/p\in[0.8,\,1]$ (see Figure
3 in [6]).

(a) ASCOT5 results.

(b) DREAM results.

Figure 3: (a) Diffusion coefficients, $\log_{10}(D\,\mathrm{[m^{2}/s]})$, from
ASCOT5 vs normalized minor radius ($r/a$) and electron momentum normalized to
the rest mass ($p/m_{e}c$) at the time indicated by the vertical dashed lines
in Fig. 2(upper) and subplot (b). (b) Time-traces of the RE current from DREAM
when diffusion coefficients _less_ than the noted value are set to _zero_ ,
i.e. $D=0$ _within_ the similarly styled contours in (a). Note the various
linear/logarithmic scales. The legend for curves in subplot (b) applies to
contours in subplot (a).

In Fig. 3(a), general trends are seen of rapidly decreasing transport with
decreasing radius and relative insensitivity to electron energy. However,
there is a clear feature of “very low” transport ($D<30\,\mathrm{m^{2}/s}$)
for electrons localized in the core ($r/a\sim 0.05{-}0.2$) and with energies
${<}50\,\mathrm{MeV}$ ($p/m_{e}c<100$). Figure 3(b) shows the time-evolution
of RE current when the transport coefficients are zeroed in different regions
of the phase space in Fig. 3(a).333Note that the diffusivity is used for
discrimination of the phase space regions, while the advection coefficients
(not shown here) are also filtered in the same regions. The “base case” is
$D=0$ wherever $D<1000\,\mathrm{m^{2}/s}$, which effectively includes the
entire core, $r/a<0.3$, and leads to the previously seen
${\sim}1\,\mathrm{MA}$ RE beam. Yet reducing this threshold to
$D<10\,\mathrm{m^{2}/s}$ leads to negligible RE current. Thus, it is primarily
the electron population within $D\sim 10{-}18\,\mathrm{m^{2}/s}$, i.e.
localized in $r/a\sim 0.05{-}0.2$ and with kinetic energies
${\sim}0.2{-}15\,\mathrm{MeV}$ ($p/m_{e}c\sim 1{-}30$), which contributes most
to RE plateau formation.

This problem can be looked at from another angle: What is the minimum
transport needed to fully suppress RE plateau formation? More specifically,
within the region of phase space where $D<1000\,\mathrm{m^{2}/s}$ in Fig. 3(a)
(that mostly coincides with the re-healed flux surface region), which constant
value of $D$ is sufficient to yield negligible RE current? As seen in Fig. 4,
full RE beam prevention is only achieved somewhere in the range
$D=10{-}18\,\mathrm{m^{2}/s}$. Therefore, compared to the highly diffusive
edge region ($D\approx 10^{3}{-}10^{5}\,\mathrm{m^{2}/s}$), a relatively small
amount of core transport is needed. Importantly, note that the advection
coefficient $A[{\rm m/s}]$ is set to the same value as $D[{\rm m^{2}/s}]$ in
these phase space regions, but almost identical results are found when setting
$A=0$, as diffusion dominates in the narrow radial region of re-healed flux
surfaces (as long as $A[{\rm m/s}]\sim D[{\rm m^{2}/s}]$).

Figure 4: Time-traces of the RE current from DREAM when the diffusion
coefficient is set to the listed value in regions of phase space with
$D<1000\,\mathrm{m^{2}/s}$ in Fig. 3(a). The time indicated by the vertical
dashed line is the same as in Figs. 2 and 3(b).

##  4 Discussion and summary

From the previous sections, it is clear that zeroing the transport in the core
($r/a<0.3$) in DREAM is too conservative and pessimistic, resulting in a
${\sim}1\,\mathrm{MA}$ RE beam. Even so, it is important to note that this
current is 5-6 times less than that expected for an unmitigated RE beam, i.e.
no REMC, so even this conservative base case could be considered successful.
ASCOT5 simulations evaluate diffusion coefficients spanning $D\approx
1{-}1000\,\mathrm{m^{2}/s}$ in the core, but encouragingly only $D\sim
18\,\mathrm{m^{2}/s}$ is needed in that region to completely suppress a RE
beam. Perhaps this is one reason why many tokamaks struggle to generate RE
plateaus via “natural” disruptions (as in Alcator C-Mod [13, 14], for
example), and instead resort to special “recipes,” although lower plasma
current and thus lower avalanching certainly also play a role.

However, it is not yet known whether this level of transport is achievable in
SPARC . In [11], it was noted that the degree of field line stochastization
predicted by NIMROD could be affected by several approximations, most notably
the presence of a close, ideal wall which tends to _limit_ MHD mode growth.
This approximation will be explored further in future resistive-wall studies,
and perhaps this minimum $D$-value will even decrease.

We can also approach this from another direction: In what ways can the REMC
design be modified to achieve $D>18\,\mathrm{m^{2}/s}$ throughout the plasma?
Perhaps most straightforward, the coil could be moved closer to the plasma and
farther from the VV. This would (i) improve the plasma-coil mutual coupling
and reduce the coil’s self-inductance, thereby increasing the induced coil
current, (ii) decrease image currents in the conducting wall, and (iii)
enhance the magnetic perturbation amplitude $\delta B/B$ in the core. Perhaps
a design metric could be the expected diffusion coefficient computed from
vacuum fields à la [15], $D\propto(\delta B/B)^{2}$. The coil resistance could
also be lowered by changing the coil cross-section, length, and material
(resistivity). That said, many other factors constrain the design, like
available space, forces and stresses, heating, and more.

As discussed in [6], both advection and diffusion tend to increase with RE
energy, but there is a roll-over when the energetic electron drift orbits
effectively average over large regions of stochasticity (see Figure 3 in [6]
or [16, 17] and others for further details). However, for REs within healed
flux surfaces, perhaps large orbits could lead to “excursions” into stochastic
fields, thus enhancing transport. For example, KORC simulations in [16] found
that REs with Larmor radii similar to island widths could escape them. In
addition, the same electric field accelerating REs causes them to drift
radially [18], and this was not accounted for in these ASCOT5 simulations, but
will be pursued in the future.444See [17] for simulations of passing and
trapped REs during an ITER CQ, including the effects of collisions and the
electric field.

Perhaps most importantly, the effect of the RE population itself on the
magnetic field and MHD has not yet been fully assessed. Figure 5 shows the
time-evolution of the $q$-profile, its minimum value, and the internal
inductance ($\ell_{\mathrm{i}}$) in DREAM for the base case. Although slightly
later in time than in Fig. 1(a), the central safety factor $q(0)$ also
surpasses $q=2$; however, unlike the NIMROD results, the increasing RE current
then reduces $q(0)<2$ at $t\approx 5\,\mathrm{ms}$ and $q(0)<1$ at $t\approx
5.5\,\mathrm{ms}$. Thus, in theory, the REMC should regain resonance in the
core beyond $t>5\,\mathrm{ms}$, thereby enhancing transport and reducing the
RE current, but this was not captured in the current workflow. A destructive
kink instability might also be expected, as seen in experiment [19, 20], for
such low $q(0)$ and high $\ell_{\mathrm{i}}$.

(a)

(b)

Figure 5: DREAM results for the base case in Fig. 3(b) with $D=0$ below
$D<1000\,\mathrm{m^{2}/s}$: (a) evolution of the safety factor $q$-profile vs
minor radius for five times, and (b) time evolution of the minimum $q$-value
(dot-dashed) and internal inductance $\ell_{\mathrm{i}}$ (solid).

Even then it is not clear what overall effect this self-regulation would have
on the RE beam which already has a current ${\sim}1\,\mathrm{MA}$ by $t\approx
5\,\mathrm{ms}$ (for the base case with no island transport). Luckily, the
Ohmic current has almost completely decayed by then, and the relatively long
L/R time (${>}10\,\mathrm{ms}$) of the REMC will maintain the coil current and
its perturbative effect. Furthermore, any additional transport within the re-
healed flux surfaces will help lower this quasi-stationary RE current. A fluid
RE model that could capture this effect has been incorporated into the JOREK
[21, 22] and M3D-C1 [23] MHD codes; a similar model is being implemented in
NIMROD [24] and benchmarked against the existing codes. Its application to the
SPARC REMC will be pursued in future work.

## Acknowledgments

Supported by Commonwealth Fusion Systems, Swedish Research Council (Dnr.
2018-03911), US DOE Award Numbers DE-FC02-04ER54698 and DE-FG02-95ER54309.
This work has been carried out within the framework of the EUROfusion
Consortium, funded by the European Union via the Euratom Research and Training
Programme (Grant Agreement No 101052200 – EUROfusion). Views and opinions
expressed are however those of the author(s) only and do not necessarily
reflect those of the European Union or the European Commission. Neither the
European Union nor the European Commission can be held responsible for them.

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# Exciton-polaritons in GaAs-based slab waveguide photonic crystals

C. E. Whittaker  Department of Physics and Astronomy, University of Sheffield,
Sheffield S3 7RH, United Kingdom  T. Isoniemi  Department of Physics and
Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom  S.
Lovett  Department of Physics and Astronomy, University of Sheffield,
Sheffield S3 7RH, United Kingdom  P. M. Walker  Department of Physics and
Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom  S.
Kolodny  ITMO University, St. Petersburg 197101, Russia  V. Kozin  ITMO
University, St. Petersburg 197101, Russia  I. V. Iorsh  ITMO University, St.
Petersburg 197101, Russia  I. Farrer  Department of Electronic and Electrical
Engineering, University of Sheffield, S3 7HQ United Kingdom  D. A. Ritchie
Cavendish Laboratory, University of Cambridge, CB3 0HE Cambridge, UK  M. S.
Skolnick  Department of Physics and Astronomy, University of Sheffield,
Sheffield S3 7RH, United Kingdom  D. N. Krizhanovskii
<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Sheffield, Sheffield S3 7RH, United Kingdom  ITMO University,
St. Petersburg 197101, Russia

###### Abstract

We report the observation of band gaps for low loss exciton-polaritons
propagating outside the light cone in GaAs-based planar waveguides patterned
into two-dimensional photonic crystals. By etching square lattice arrays of
shallow holes into the uppermost layer of our structure, we open gaps on the
order of 10 meV in the photonic mode dispersion, whose size and light-matter
composition can be tuned by proximity to the strongly coupled exciton
resonance. We demonstrate gaps ranging from almost fully photonic to highly
excitonic. Opening a gap in the exciton-dominated part of the polariton
spectrum is a promising first step towards the realization of quantum-Hall-
like states arising from topologically nontrivial hybridization of excitons
and photons.

††preprint: AIP/123-QED

The hybridization of photons and quantum well excitons leads to the formation
of exciton-polaritons, quasiparticles combining high-speed propagation with
large nonlinearity and susceptibility to magnetic fields. These favourable
properties arising from their mixed light-matter nature have made polaritons
highly attractive candidates for novel semiconductor-based optical devices
incorporating nonlinearity Sanvitto and Kéna-Cohen (2016) and robustness
against disorder induced by topology Amo and Bloch (2016). In order to direct
and manipulate the flow of polaritons accordingly, appropriate tailoring of
the potential landscape is required Schneider  _et al._ (2016). In Bragg
microcavities, the most mature platform for polariton research to date,
lithographic patterning has been used to strongly modify the photonic mode
dispersion resulting in gapped polariton band structures enabling coherent
devices Wertz  _et al._ (2012); Nguyen  _et al._ (2013); Sturm  _et al._
(2014); Marsault  _et al._ (2015) and topological lasers St-Jean  _et al._
(2017); Klembt  _et al._ (2018).

An alternative geometry to conventional microcavities for the study of
polaritons is slab waveguides (WGs), in which a guided electromagnetic mode
confined by total internal reflection strongly couples to quantum well
excitons Walker  _et al._ (2013). This configuration not only has relative
ease of fabrication, but also far greater suitability for integration into on-
chip circuits owing to the large in-plane propagation velocities. Coherent
Jamadi  _et al._ (2018); Suárez-Forero  _et al._ (2020) and continuum Walker
_et al._ (2019); Paola  _et al._ (2020) light sources have already been
demonstrated using this “horizontal” geometry, and the thin layer structure
facilitates enhancement of nonlinearities using dipolar polaritons Rosenberg
_et al._ (2018); Suárez-Forero  _et al._ (2021).

Similar to microcavities the potential landscape can be engineered to emulate
novel physical systems, but WGs can achieve this without the need to etch
several micrometers of material Bajoni  _et al._ (2009). A full quantum theory
of strong light-matter coupling in photonic crystal (PhC) slabs with embedded
quantum wells was given by Gerace and Andreani over a decade ago Gerace and
Andreani (2007). Experimentally, PhCs and bound states in the continuum have
been demonstrated using monolayer semiconductors (MS) placed on top of
periodically patterned dielectric WGs Zhang  _et al._ (2018); Chen  _et al._
(2020); Kravtsov  _et al._ (2020). PhC were also made using pillars of hybrid
organic-inorganic perovskites embedded in a homogeneous dielectric Dang  _et
al._ (2020). In the area of topologically non-trivial polaritons, propagating
edge states protected by breaking of a pseudo-time-reversal symmetry were
demonstrated using MS on a PhC WG Liu  _et al._ (2020). Of particular interest
is the potential to realize topological polariton states protected by true
time-reversal symmetry breaking Karzig  _et al._ (2015) by combining
polarization splitting, photonic crystal bandgap and exciton Zeeman splitting
due to external magnetic field. So far similar states have only demonstrated
in microcavities in the lasing regime owing to a very small bandgap of 0.1 meV
Klembt  _et al._ (2018).

While modulated bandstructure is one key ingredient for future applications,
it is equally important to achieve long polariton lifetimes and propagation
distances. In MS and organics the lifetimes are short, which may occur due to
inhomogeneous broadening of the exciton linewidth up to $\sim$10 meV. The
longest reported polariton lifetimes are achieved in GaAs-based structures Sun
_et al._ (2017) where, owing to the high quality quantum wells, polariton
linewidths can be 10-100 times smaller. Furthermore, a crucial feature of
previous works is that the studied states are near zero in-plane momentum
where coupling to freely propagating waves in the surroundings inevitable
leads to high photonic losses. For example, in Dang et. al. Dang  _et al._
(2020) even the simulated linewidth for passive materials is of order 16 meV
suggesting only a 40 fs photon lifetime. To achieve long polariton lifetime it
is critical that states are produced at high wavenumbers where total internal
reflection (TIR) prevents radiative loss. For GaAs-based WGs lifetimes
(propagation lengths) on the order of 10 ps (500 ${\upmu}$m) can be expected
Walker  _et al._ (2019, 2013). Until now, however, strong bandstructure
modulation was not demonstrated in GaAs-based polariton WGs, and polariton PhC
states protected by TIR have not so far been demonstrated in any material
system. Achieving the former is challenging since it relies on etching through
a significant fraction of the core. However, etching through or too near the
quantum wells leads to high losses associated with surface recombination
effects. While previous works Bajoni  _et al._ (2009) have demonstrated the
persistence of strong coupling in patterned GaAs-based WGs, the modulated
region was spatially separated from the WG core leading to only a weak
perturbation and no opening of a gap.

In this work, we implement two-dimensional square lattice PhCs in GaAs-based
WGs in the strong coupling regime. We achieve PhC band gaps near the Brillouin
zone (BZ) edge where the states are protected by TIR, and show low loss
propagation over the $\sim$ 50 ${\upmu}$m wide PhC for states outside the gap.
The band gap, with width $\sim$10 meV, (roughly an order of magnitude larger
than in microcavity lattices Amo and Bloch (2016); Schneider  _et al._
(2016)), is the signature of the polariton bandstructure being strongly
modulated by the periodic lattice. Finally, we demonstrate one of the useful
features of polariton-based platforms by exploiting the excitonic content of
the states to tune the gap width using an external magnetic field.

The sample, illustrated in Fig. 1(a), is a planar WG into which holes are
etched to form a square lattice PhC. The GaAs WG core layer has total
thickness 295 nm and contains three 10 nm wide In0.06Ga0.94As quantum wells
(QWs) separated by 10 nm GaAs barriers and placed at the peak of the
transverse-electric (TE) polarised field in order to maximize the Rabi
coupling. The upper quantum well is 137 nm below the surface, which allows
etch depths up to 87 nm ($\sim$30% of the core thickness) while still leaving
50 nm GaAs cap to protect the wells. The core is separated from the GaAs
substrate by a 750 nm thick AlGaAs cladding layer with 90% average Al
composition. An extended discussion of the design of the layer structure is
given in the supplementary material.

Figure 1: (a) Schematic of photonic crystal (PhC) structure in a slab
waveguide. (b) Scanning electron microscope (SEM) image of holes etched into
the surface of the waveguide showing the centre-to-centre distance $a$. (c)
Bandstructure of the photon modes of a PhC with $a=126$ nm, $d=47$ nm, $h=64$
nm. The horizontal axis follows the triangular path $\Gamma$-X-M-$\Gamma$
indicated on the inset schematic of the first BZ (top left). Vertical lines
indicate the X and M points. Horizontal line indicates the GaAs bandgap.
Central inset is a zoom of the region around the X point. (d) Calculated
polariton dispersion relation in the vicinity of the X point
($k_{z}/(2\pi/a)=-0.5$) for the PhC in (c). The thick lines correspond to the
states which are visible in Fig. 2. (e) Optical microscope image of a PhC with
grating couplers allowing light to be coupled in and out along the $z$
direction and with etched trenches. The inset shows an angled SEM image of a
cleaved etched grating. Figure 2: (a) Angle-resolved photoluminescence spectra
corresponding to PhCs with different periodicities. The devices have holes
with diameter d = 47 nm and etch depth h = 64 nm. (b) Measured and calculated
positions of gaps (upper and lower band edges) against PhC period for devices
with different hole profiles (different diameters d and etch depths h). The
calculated positions are given by taking the gaps from photonic simulations
and assuming a Rabi splitting of 9 meV with the exciton resonance. The error
bars on the measured data points represent the standard deviation from
measurements of different devices with the same hole profiles. (c) Exciton
fraction at the center of the gap vs. periodicity for hole profiles
corresponding to the solid curves in panel (b).

The samples were grown by MBE and then patterned with a soft mask using
electron beam lithography to form PhCs with a square lattice geometry, as
shown in Figs. 1(a) and (b). The patterns were etched down into the planar
structure using inductively coupled plasma etching with a chlorine/argon
chemistry. Several samples were fabricated, with lattice constants $a$ between
120-130 nm, hole etch depths $h$ between 55-64 nm and hole diameters $d$
between 47-66 nm. The ranges for fabrication were selected by calculating the
PhC bandstructure using Lumerical® FDTD solutions finite difference time
domain software (see Supplementary Material for further details). Fig. 1(c)
shows an example of the simulated bandstructure for photons (without coupling
to the exciton) across the whole Brillouin zone (BZ). The polarisation of
states in slab photonic crystals may be classified as TE-like or TM-like, with
electric field mostly in or out of the WG plane respectively Joannopoulos  _et
al._ (2008). Only the TE-like modes couple strongly to the excitons in our
QWs. The bands for both polarisations are shown in Fig. 1(c). No narrow
linewidth states could be identified above the light cone (in the shaded
region) or above the GaAs band gap (1.519 eV) since such states are either not
guided or strongly absorbed respectively. In Fig. 1(c) the modes get broader
with increasing energy above the band gap and we choose to plot the dispersion
of the modes, which have linewidth $<100$ meV. There is a $\sim$ 10 meV gap in
the TE-like states around the X point, which we will examine in more detail
momentarily. From X towards M the bands increase in frequency so that the band
gap does not span all momentum states but exists only for modes propagating
near X. A larger gap could be engineered by using a different lattice geometry
such as honeycomb Karzig  _et al._ (2015) but even this small gap serves to
illustrate that polariton bandstructure can be modulated by many times the
linewidth, and could support edge states where the bulk 2D lattice provides
other desirable properties such as topological protection Karzig  _et al._
(2015).

The bandstructure was calculated for a range of $a$, $d$ and $h$ and close-to-
optimal values were selected for fabrication such that the gap in the TE-like
modes at the X point is near the exciton energy and so that the gap width was
maximized without etching too close to the QWs. In experiment we will observe
polariton rather than pure photon bands. Fig. 1(d) shows an example of the
calculated polariton dispersion resulting from strong coupling between the TE-
like mode and QW excitons, focussing on the region around the X point. The
polariton band gap can be seen clearly. To enable us to experimentally study
gap formation within the PhCs, grating couplers were fabricated on either side
of the PhC in the $z$ direction (Fig. 1(c)). These fold the waveguide modes
into the radiative region allowing light to be coupled in and out. On the
other two sides of the PhC, trenches were etched (through the active region)
for prospective studies of edge states, although we note that this is an
entirely optional design feature. The side length of the PhCs corresponds to
$\sim$400 periods.

For optical measurements, the sample was held at low temperatures (<8 K) and
excited by a cw laser at 637 nm focused to a $\sim$3 ${\upmu}$m spot on the
PhC surface. This nonresonant excitation incoherently populates the polariton
modes of the PhC WGs (which lie below the light cone) by multiple relaxation
processes. The polaritons may then propagate out of the PhC to the grating
where they are scattered out and recorded using a spectrometer. Spatial
filtering optics were used to collect the emission from selected regions on
the sample.

In Fig. 2(a) we show typical examples of the emission collected from gratings
adjacent to the PhCs, measured for a single set of devices and corresponding
to a cut through the X point, along the $\Gamma$-X-$\Gamma$ direction in
momentum space. The heavy attenuation of PL intensity within particular energy
windows directly results from the presence of band gaps in the PhC slabs.
Polaritons at energies in the gaps cannot propagate out of the PhC to be
detected at the grating. We stress that these gaps arise from the periodic
potential created by the PhCs, and are qualitatively very different to the
resonances induced by back-reflection of guided modes between pairs of
gratings in ref. Suárez-Forero  _et al._ , 2020. We see that as the PhC period
$a$ is made successively smaller, the gap (shown by the double-sided vertical
arrow) moves upwards in energy, becoming smaller as it approaches the exciton
resonance (dashed white line). The reduction in the gap size arises from
reduction of the photonic fraction as the gap approaches the exciton
resonance, confirming that the strong coupling is retained in the PhC region.
This can also be seen from the anticrossing behaviour of the band gap edges in
Fig. 2(b), which also includes results from devices with different hole
profiles. This is in contrast to our simulations of the purely photonic
structure (without strong coupling to an exciton) in which the gap size varies
little across this range of periods. Since the normal mode (Rabi) splitting
between the photonic and excitonic resonances is known, we can calculate the
exciton fraction of each gap using

$|X|^{2}=\frac{1}{2}\left[1+\frac{E_{G}^{ph}-E_{X}}{\sqrt{(E_{G}^{ph}-E_{X})^{2}}+\hbar\Omega^{2}}\right],$
(1)

where $E_{G}^{ph}$ is the central gap energy of the bare photonic gap,
calculated using $E_{G}^{ph}=\Omega^{2}/4(E_{X}-E_{G}^{pol})+E_{G}^{pol}$
where $E_{G}^{pol}$ is the central gap energy of the lower polariton branch.
In the data shown in Fig. 2, the gap passes from almost fully photonic
($|X|^{2}$ = 1%) to predominantly excitonic ($|X|^{2}$ = 61%), as can be seen
from Fig. 2(c). In other devices we have measured gaps with exciton fractions
as high as 73%. We note that emission from states within $\sim$1.5 meV below
the exciton line is heavily attenuated due to absorption which, along with the
broadened polariton linewidths, places an upper bound on the exciton fraction
of gaps which can be observed in our current samples. This may be improved
somewhat by reducing the exciton inhomogeneous linewidth and/or increasing the
Rabi splitting to increase $|X|^{2}$ for polariton states further from the
highly absorbing part of the exciton tail. We also note that within the gaps
some weak emission is visible, implying that the reflectivity is below unity,
which arises from the finite extent of the PhCs. As we show in Fig. 3, the
extinction within the gap depends strongly on the length of photonic crystal
(i.e. number of lattice periods) between the excitation spot and the edge of
the PhC. Increasing the number of periods strongly increases the attenuation
for energies in the gap. This is clear evidence that the observed gaps are due
to reflection of propagating polaritons from the periodic PhC structure. We
further note that outside the band gap there is little observable dependence
on excitation spot distance. This demonstrates that the decay length of freely
propagating polaritons in the photonic crystal (but outside the gap) is much
larger than the 35 ${\upmu}$m length over which the spot was moved. This is
consistent with the $\sim$500 ${\upmu}$m lengths in unpatterned polariton
waveguides. Walker  _et al._ (2019, 2013)

Figure 3: Measured angle-resolved PL spectra when the excitation spot (Exc.)
is at the near (a) and far (b) edges of the PhC (a=126 nm) with respect to the
grating from which light is collected (Col.). (c) PL spectra measured for
different excitation positions (i.e. number of PhC periods between the
excitation and detection spots).

In contrast to our system, the primary gaps studied in periodic potentials in
Bragg microcavities are typically formed in the photonic part of the spectrum
and thus have far smaller exciton fractions, which severely limits the ability
to tune the gap position using external fields. If the exciton content is
large, however, one may employ diverse methods to tune the energy of the gap
including temperature, optical excitation and electric and magnetic fields
Rahimi-Iman (2020). In order to demonstrate the feasibility of tuning gaps in
our devices, and also to show unambiguously that they are polaritonic in
nature, we placed our sample in a magnetic field in Faraday geometry. The
results are summarized in Fig. 4. The predominant effect is the diamagnetic
(blue)shift of the exciton resonance, which reduces the exciton fraction of
the gaps and hence increases their size. For the PhC with $a$ = 125 nm, the
gap size can be increased from 4.0 meV at B = 0 T to 5.4 meV at B = +9 T
(Figs.4(a)-(c)). In the case of the PhC with $a$ = 124 nm, the full gap is not
visible at B = 0 T, since the upper gap energy lies in the heavily attenuated
region below the exciton resonance. At higher magnetic fields however the gap
becomes visible, reaching a size of 2.4 meV at B = +9 T (Figs.4(d)-(f)). We
thus demonstrate that the polariton band structure can be tuned by varying the
exciton content using magnetic field. It should be noted that to obtain the
larger gap sizes requires reducing the exciton content (see Fig. 2(c)), which
will also reduce the polariton nonlinear interaction strength. The tuning can
be used as a lever to strike an optimal balance between these properties.

For finite magnetic fields there is also a Zeeman splitting of the exciton,
which reaches values exceeding 0.5 meV at the highest fields. Thus, the basic
ingredients we have presented, namely, gaps in the excitonic part of the
spectrum and an exciton Zeeman splitting are those described in ref. Karzig
_et al._ , 2015 as the criteria for quantum Hall type (chiral) edge states at
the boundaries of the system. However, for true topological protection one
needs the Zeeman splitting to exceed the size of the gap; this will require an
enhancement of the exciton $g$ factor which in our system could be achieved by
varying the In composition or width of the quantum wells Traynor, Harley, and
Warburton (1995). Alternatively one may think about employing semimagnetic
quantum wells in a Te-based system Mirek  _et al._ (2017) or using transition
metal dichalcogenides where spin-dependent strong coupling to a photonic mode
can create a giant effective Zeeman splitting exceeding 10 meV 111A. I.
Tartakovskii, private communication (unpublished). We note that even without a
global band gap the system may support topological edge states with well
defined momentum close to the X point. Losses due to scattering into bulk
modes can be minimised by maximising the gap size using other lattice
geometries such as hexagonal Karzig  _et al._ (2015).

Figure 4: (a)-(c) Magnetic field dependence of spectra for PhC with $a$ = 125
nm. The angle-resolved spectra measured at 0 T and 9 T are shown in (a) and
(c) respectively, and contour plots of the angle-integrated spectrum are shown
in (b). (d)-(f) Same as (a)-(c) for PhC with $a$ = 124 nm. The colour scale is
the same as that of Fig. 3.

In summary, we have presented a platform to study strongly modulated exciton-
polariton band structures using patterned slab waveguides in the strong
coupling regime. We observe low loss propagating states outside the light cone
and band gaps of order 10 meV. We have demonstrated that the gaps can be
controlled both through the photonic component (varying the period of the
crystal) or the excitonic component (external magnetic field). For future
studies we envisage the patterning of photonic crystals with different lattice
geometries featuring exotic dispersion relations Jacqmin  _et al._ (2014);
Whittaker  _et al._ (2018), as well as interfacing waveguides with other
excitonic materials such as atomically thin semiconductors Kravtsov  _et al._
(2020) and organic polymers Jayaprakash  _et al._ (2020). Our system could
thus offer a flexible and promising alternative to microcavity-based polariton
lattices for the study of topological states and implementation of
optoelectronic devices.

## Supplementary Material

Further details of the PhC structure and simulations used in the design
process are given in the associated supplementary material file.

###### Acknowledgements.

The work was supported by UK EPSRC Grants EP/N031776/1 and EP/R04385X/1 and by
the Russian Science Foundation (Project No. 19-72-20120).

## The data availability statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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