# Fabrication of Nanostructured GaAs/AlGaAs Waveguide for Low-Density
Polariton Condensation from a Bound State in the Continuum

F. Riminucci  Molecular Foundry, Lawrence Berkeley National Laboratory, One
Cyclotron Road, Berkeley, California 94720, USA  Dipartimento di Matematica e
Fisica, ‘Ennio de Giorgi’, Università del Salento, Lecce, Italy  V. Ardizzone
CNR Nanotec, Institute of Nanotechnology, via Monteroni, 73100, Lecce  L.
Francaviglia  Molecular Foundry, Lawrence Berkeley National Laboratory, One
Cyclotron Road, Berkeley, California 94720, USA  M. Lorenzon  Molecular
Foundry, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley,
California 94720, USA  C. Stavrakas  Molecular Foundry, Lawrence Berkeley
National Laboratory, One Cyclotron Road, Berkeley, California 94720, USA  S.
Dhuey  Molecular Foundry, Lawrence Berkeley National Laboratory, One Cyclotron
Road, Berkeley, California 94720, USA  A. Schwartzberg  Molecular Foundry,
Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley,
California 94720, USA  S. Zanotti  Dipartimento di Fisica, Università di
Pavia, via Bassi 6, Pavia (IT)  D. Gerace  Dipartimento di Fisica, Università
di Pavia, via Bassi 6, Pavia (IT)  K. Baldwin  PRISM, Princeton Institute for
the Science and Technology of Materials, Princeton University, Princeton, New
Jersey 08540, USA  L. N. Pfeiffer  PRISM, Princeton Institute for the Science
and Technology of Materials, Princeton University, Princeton, New Jersey
08540, USA  G. Gigli  Dipartimento di Matematica e Fisica, ‘Ennio de Giorgi’,
Università del Salento, Lecce, Italy  CNR Nanotec, Institute of
Nanotechnology, via Monteroni, 73100, Lecce  D. F. Ogletree  Molecular
Foundry, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley,
California 94720, USA  A. Weber-Bargioni  Molecular Foundry, Lawrence Berkeley
National Laboratory, One Cyclotron Road, Berkeley, California 94720, USA  S.
Cabrini  Molecular Foundry, Lawrence Berkeley National Laboratory, One
Cyclotron Road, Berkeley, California 94720, USA  D. Sanvitto
<EMAIL_ADDRESS>CNR Nanotec, Institute of Nanotechnology, via
Monteroni, 73100, Lecce

###### Abstract

Exciton-polaritons are hybrid light-matter states that arise from strong
coupling between an exciton resonance and a photonic cavity mode. As bosonic
excitations, they can undergo a phase transition to a condensed state that can
emit coherent light without a population inversion. This aspect makes them
good candidates for thresholdless lasers, yet short exciton-polariton lifetime
has made it difficult to achieve condensation at very low power densities. In
this sense, long-lived symmetry-protected states are excellent candidates to
overcome the limitations that arise from the finite mirror reflectivity of
monolithic microcavities. In this work we use a photonic symmetry protected
bound state in the continuum coupled to an excitonic resonance to achieve
state-of-the-art polariton condensation threshold in GaAs/AlGaAs waveguide.
Most important, we show the influence of fabrication control and how surface
passivation via atomic layer deposition provides a way to reduce exciton
quenching at the grating sidewalls.

##  I Introduction

Efficient low-threshold lasing is a long-sought goal for various applications
spanning from integrated circuits to biological sensing [1, 2]. However, as of
today the emission of coherent and monochromatic light has relied on
population inversion, which requires to overcome a threshold excitation power
to achieve lasing. A different and new approach exploits hybrid light-matter
particles, known as exciton-polaritons[3, 4], which arise in a semiconductor
when the energy exchange rate between a photon and an exciton is higher than
their losses. These particles possess properties inherited from their photonic
component, such as a small effective mass, and from their excitonic component
such as high nonlinearities[5, 6] that can be further enhanced via dipolar
interactions[7, 8]. All these properties make them interesting candidates for
devices[9] such as optical transistors[10, 11], fast optical switches[12, 13],
and electrically-injected light sources [14].
Because of their bosonic nature, they can undergo a phase transition to a
coherent state, known as Bose-Einstein condensate[15, 16] (BEC), without the
need for population inversion, making thresholdless lasing theoretically
possible[17]. A necessary condition for this phase transition to happen is
that the rate of scattering processes[18] that populate the BEC state must
exceed the loss rates[9]. This has been one of the main limitations in the
historical achievement of exciton-polariton condensates, as polaritons tend to
accumulate in the so-called ”bottleneck”[19, 20]. Such limitation was
overcome[15] by growing highly reflective mirrors with layers exceeding 40
pairs in both sides of a microcavity as well as using many stacks of QWs. In
the last few years a new approach has enabled the achievement of polariton
states in horizontal platforms[21, 22, 23, 5, 24, 25], making the fabrication
easier and less time consuming. A similar waveguide configuration with
multiple QWs was used to investigate the enhancement of dipolar polariton
interactions[26, 27, 28, 8]. More recently, the same heterostructure led to
the achievement of a Bose-Einstein condensate from a BIC[29] while also
proving the topological charge possessed by the condensate.
In this work we show the achievement of a low-density condensate in a
GaAs/AlGaAs waveguide by tuning the parameters of a low-loss shallow grating
and then we study the effect of the reduction of excitonic losses on the
threshold. In order to reduce the radiative losses we make use of a
topologically protected bound state in the continuum [30, 31, 32] (BIC) which
is introduced in the polariton dispersion through the realization of a 1D
etched grating[33]. Since the BIC has an antisymmetric profile, it cannot
couple to outgoing plane waves, resulting in a perfectly dark state with zero
radiative linewidth [34]. The exciton-polariton dispersion can be modified by
changing the grating periodicity, which defines the energy at which the
photonic modes cross. Such crossing can then be adjusted to bring the BIC
closer to or further from the exciton, changing its excitonic fraction. This
will ultimately change the threshold since the excitonic fraction has also an
effect on the polariton relaxation process towards the BIC. This control is
something that has no counterparts in classical vertical microcavities and can
be tuned with extreme precision on several different gratings on the same
chip. The excitonic fractions were estimated throughout this work by using the
coupled oscillators model described in the supplementary materials.
A horizontal configuration and such long-lived state can drastically ease the
fabrication with respect to vertical microcavities in order to achieve
thermalization of polaritons into a single quantum state. Here we demonstrate
the achievement of low-threshold condensate thanks to simple, yet extremely
effective sample processing. We will study three different shallow etched 1D
gratings: 40nm, 90nm, and 130nm deep grooves. Finally, we show the reduction
of the exciton-polariton linewidth and condensate threshold by passivating the
trap states that form on the groove sidewalls during the etching process.

##  II Results

The sample is composed of a GaAs substrate on which a waveguide slab is grown
via molecular beam epitaxy. 500nm of Al0.8Ga0.2As are firstly grown onto the
substrate, forming the waveguide cladding. A set of 12 GaAs quantum wells
(QWs) and 13 Al0.4Ga0.6As barriers 20nm thick were grown on top of the
cladding, forming the waveguide core. Lastly, 10nm of GaAs were grown on top
of the last barrier to act as a cap layer. The optical mode is confined in the
waveguide core through total internal reflection, and here it exchanges energy
with the QW excitons leading to propagating polaritons. A shallow etched 1D
grating (Fig.1a) can be used to couple propagating and counterpropagating
photonic modes via the slab refractive index modulation, hence we can easily
introduce new properties by engineering the photonic dispersion. In our system
we can exploit such coupling between photonic modes to introduce a symmetry
protected bound state in the continuum which appears in the so-called dark
state. If we consider E${}_{\text{Bright}}$ and E${}_{\text{Dark}}$ as the
photonic modes originated from the grating coupling and separated by an energy
gap, the energy exchange with the QWs exciton can lead to the following system
Hamiltonian

$H=\left(\begin{matrix}\text{E}_{\text{Bright}}&\frac{\Omega}{2}&0&0\\\
\frac{\Omega}{2}&X&0&0\\\ 0&0&\text{E}_{\text{Dark}}&\frac{\Omega}{2}\\\
0&0&\frac{\Omega}{2}&X\end{matrix}\right)$ (1)

The photonic bright and dark states are then hybridized with the exciton,
hence the diagonalization leads to exciton-polariton eigenstates with a BIC in
the dark lower state (Fig.1b)[35]. Since the gap depends on the coupling
strength between the photonic modes, it also depends on the grating’s
fabrication parameters, such as the etching depth and the filling factor,
defined as $\text{FF}=1-\frac{\text{width}}{\text{pitch}}$ (Fig.1a). We used
Stanford Stratified Structure Solver (S4) [36] to simulate the structure in
Fig.1a, from which we obtained the energy dispersion shown in Fig.1b. The
exciton was introduced in the simulations with a Lorentzian resonance in the
GaAs dielectric constant, with a Rabi splitting $\Omega=13.9$meV [8]. The
bound state in the continuum appears as a saddle point of the dark state as
shown in [29], which creates a direct path for polaritons to thermalize from
the exciton reservoir.
The fabricated gratings are 50$\mu$m wide, 300$\mu$m long with a filling
factor 70$\%$ \- 75$\%$. The periodicity was set around 240nm to have the
photonic mode coupling close to the exciton resonance. We realized the grooves
by writing an electron beam sensitive resist, and subsequent pattern transfer
into the heterostructure via ICP-Chlorine etching[37, 38, 39, 40]. The etched
grating penetrated into the waveguide core, and three different depths were
chosen, resulting in different degrees of photonic coupling.

Figure 1: (a) Sketch of the final processed waveguide sample with a 90nm deep
grating, along with a conformal passivating layer of Al2O3. (b) Simulations of
the exciton-polariton dispersion with bright and dark states coupled to the
exciton. The BIC appears here as a dark state at the maximum energy point of
the dispersion, corresponding to vertical emission (k∥ = 0) in the far field.

The sample was cooled down to 4K in a cryostat and polaritons were created by
nonresonantly pumping with a laser onto the grating (See Methods). Light was
outcoupled by the grating itself and the energy dispersion imaged on a CCD
camera. Accumulation of polaritons in the long-lived state triggers the onset
of a coherent emission by bosonic stimulation. The polariton energy dispersion
at and above threshold for a grating with filling factor FF = 75$\%$ is shown
in Fig.2a and Fig.2b.

Figure 2: (a,b) Exciton-polariton energy dispersion and coherent light
emission at (a) and above (b) the excitation threshold. The condensate in (b)
is blueshifted as a result of polariton-polariton interactions. (c) Emission
intensity from the condensate as a function of the pumping power.

The power at which the condensation occurs was found to be strongly dependent
on the grating parameters. Fig.3 shows the threshold intensity in a 90nm deep
grating as a function of the periodicity. We chose gratings with periodicities
equal to 240nm, 242nm, and 244nm which correspond to excitonic fractions
51.2$\%$, 22.6$\%$, and 12.1$\%$, respectively. We observed a lower threshold
for the 240nm pitch with respect to the 242nm and 244nm cases, ascribable to
the higher exciton fraction. In our system, the grating periodicity moves the
BIC closer to or further from the exciton, making the polariton thermalization
process more or less efficient for the two cases.

Figure 3: Emission intensity from the condensate for periodicities 240nm
(blue), 242nm (orange), and 244nm (green). The excitonic fraction $|X|^{2}$ is
modified by the grating pitch, favoring a polariton thermalization towards the
BIC and reducing the power threshold in the 240nm case.

In order to reduce the threshold, different etching depths $d$ were studied
with filling factors close to 70$\%$: 40nm, 90nm, 130nm and the measured
dispersions are reported in Fig.S6d-f. The etching is performed in the
waveguide core, which has consequences on the polariton modes quality. The
trap states in the electronic band gap created during the sample’s fabrication
lead to nonradiative exciton surface recombination at the QWs sidewalls.
Surface recombination can be minimized through oxide removal and surface
passivation, achievable through etching at sufficiently low ion energy to
remain in the ion-assisted chemical etching regime[41, 42, 43]. For this
reason, the sample was immersed in a hydrochloric acid bath to remove the
surface oxide, and then 8nm of Al2O3 were deposited via atomic layer
deposition[44, 45, 46, 47] (ALD) at 300°C [48] (see Supplementary Material).
The observed emission intensities in the three cases along with the gap size
and excitonic fraction are shown in Fig.4a,b. The lower threshold observed in
deeper gratings is caused by a larger gap. In fact, the BIC state is more
protected from the lossy bright state when the gap is larger. In other words,
the smaller the gap, the closer the BIC is to the bright state, the higher the
losses induced by leakage of the polariton population into the lossy state.

Figure 4: (a) Emission intensity as a function of the pump power. The
threshold is minimum for a grating 90nm deep, while it is higher for a 40nm
deep grating due to smaller gap size, and for a 130nm deep grating due to
surface damage. (b) Energy gap and excitonic fraction as a function of the
etching depth. The filling factor is approximately the same in all the three
cases.

Concurrently with the etching depth, the surface damage increases as
demonstrated by the 130nm deep grating in which the condensate was achieved
only from one structure with a high filling factor, and only after post-
processing, as a consequence of a more pronounced damage (see Fig.S11). The
damage induced by the etching was found to act mainly on the excitonic
component, as shown in Fig.5 and it is discussed in the supplementary material
where cathodoluminescence hyperspectral maps elucidate the effect of surface
damage on the emission[49, 50, 51].

---

---

Figure 5: (a,b) Exciton-polariton energy dispersion pre- and post-processing,
respectively. (c) Emission intensity at the threshold before and after
aluminum oxide deposition for a 90nm deep grating, 240nm pitch. (d,e) Exciton-
polariton dispersions corresponding to the case (a) and (b). Colors represent
the HWHM fit of the linewidths.

Lastly we observed the effect of post-processing on gratings characterized by
240nm pitch and 90nm deep grooves before and after passivation. The polariton
dispersion for this grating is shown in Fig.5a and Fig.5b, where the threshold
reached an energy per pulse corresponding to a few
$\frac{\mu\text{J}}{\text{cm}^{2}}$ after ALD as displayed in Fig.5c. This
measurement was performed on several gratings, and all of them showed the same
characteristic reduction (See Fig.S10). The longer polariton lifetime, which
we associate to lower nonradiative exciton recombination at the groove
sidewalls, can be observed in terms of linewidth reduction after Al2O3
deposition (Fig.5e). This result, even though limited by the spectrometer
resolution and inhomogeneous broadening, clearly highlights the reduction of
exciton inhomogeneous broadening which ultimately corresponds to a longer
exciton-polariton lifetime. A more detailed discussion about the BIC lifetime
is presented in [29] where a lower limit is set by means of polariton
propagation under the grating. While the effect on the excitonic component is
clear, no significant shift in the polariton dispersion was observed after the
aluminum oxide deposition, a sign that the lower Al2O3 refractive index (n =
1.6 for thin film) compared to the photonic mode effective refractive index (n
= 3.359) does not alter the designed structure behaviour.

##  III Conclusions

The main limitation in achieving thresholdless exciton-polariton lasing is
hindered by non-radiative and radiative losses that are inevitably present in
microcavities. In this work we have realized a low-density exciton-polariton
condensate in a horizontal cavity that makes use of the long lifetime of a
quasi-bound state in the continuum in which lifetime is increased by several
orders of magnitude due to the special nature of this dark state. However,
surface damage during processing introduces nonradiative channels through
which excitons can decay. By performing surface ALD post-processing we greatly
reduce such an effect, increasing the polariton lifetime at the BIC. Our
method substantially lowers the power needed to achieve a polariton
condensate. This system is not only a good candidate for low threshold lasers,
but also for those experiments that have been affected by the simultaneous
presence of horizontal and vertical polariton lasing as recently reported in
[22]. In conclusion we assessed the role of grating parameters and fabrication
in the achievement of state-of-the-art polariton BEC arising from a bound
state in the continuum coupled to an excitonic resonance. The extremely long
lifetime of the polariton BIC facilitates the thermalization of the bosonic
gas with the onset for condensation at a low record polariton density despite
the presence of lower energy states. Finally we also observed the reduction of
the threshold and improvement in the polariton lifetime after hydrochloric
acid and passivation of the sidewalls via atomic layer deposition, reducing
the density of nonradiative recombination centers at the QW sidewalls. In the
case of deep etching, the role of ALD was crucial, given it was able to
restore the possibility to observe the condensate from one 130nm deep grating,
while it reduced the threshold in all the 90nm deep gratings. The linewidth
narrowing observed after post-processing is related to the reduction of non-
radiative recombination, improving the overall quality factor of the sample
leading to an estimated polariton lifetime of a few hundreds of picoseconds in
the BIC[29]. Since BEC formation from a BIC has the great advantage of
reaching the phase transition at extremely low densities, it can be used to
obtain ultra-low threshold lasers. The techniques and ideas investigated and
developed here will not only finally boost the realization of polariton-based
thresholdless lasers, but could also be a way to realize new horizontal
electrically pumped polariton light sources.

##  IV Materials and Methods

Optical Measurements. The sample was kept at 4K for all the experiments. All
measurements were performed using a 150fs, 80MHz pump laser with its central
energy tuned at 1.588meV (780nm), and a spot size of 15$\mu$m FWHM. The
emission from the sample was sent to a SpectraPro-300i spectrometer with a
1200lines/mm grating in order to reconstruct the reciprocal space. The optical
setup allowed the simultaneous acquisition of Fourier space and real space
images, which made it possible to keep the sample in focus at all times. To
remove spurious effects, the laser tail was cut at around 800nm by a short
pass filter, combined with a spatial filter which removes any emission outside
the selected grating region.

Acknowledgments We are grateful to Ronen Rapaport for inspiring discussions
and for sharing information about the sample design. The authors acknowledge
the project PRIN Interacting Photons in Polariton Circuits INPhoPOL [Ministry
of University and Scientific Research (MIUR), Grant No. 2017P9FJBS_001]. Work
at the Molecular Foundry was supported by the Office of Science, Office of
Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. We acknowledge the project FISR—C. N. R. Tecnopolo di
nanotecnologia e fotonica per la medicina di precisione—CUP B83B17000010001
and ”Progetto Tecnopolo per la Medicina di precisione, Deliberazione della
Giunta Regionale Grant No. 2117. This research is funded in part by the Gordon
and Betty Moore Foundations EPiQS Initiative, Grant No. GBMF9615 to L. N.
Pfeiffer, and by the National Science Foundation MRSEC Grant No. DMR 1420541.
L. Francaviglia acknowledges funding from the Swiss National Science
Foundation (SNSF) via Early PostDoc Mobility Grant No. P2ELP2_184398. Triennal
Programm 2021 -2023. Italian Ministry of Research (MIUR) through the FISR 2020
– COVID, project “Sensore elettro-ottico a guida d’onda basato
sull’interazione luce-materia” (WaveSense), FISR2020IP_04324.

Disclosures The authors declare no conflicts of interest.

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Supplementary Material: Fabrication of Nanostructured GaAs/AlGaAs Waveguide
for Low-Density Polariton Condensation from a Bound State in the Continuum

##  V Fabrication and post-processing

The heterostructure was grown via molecular beam epitaxy on a GaAs substrate.
The waveguide cladding is made of Al0.8Ga0.2As 500nm thick, while the core is
composed of 13 barriers of Al0.4Ga0.2As barrier (20nm thick), 12 GaAs QWs
(20nm thick), and a final cap layer made of GaAs (10nm thick). We then
fabricated a 1D grating by etching the waveguide, inducing a coupling between
propagating and counter-propagating modes. The grating period was tuned to
have the coupled photonic states at energies close to the exciton.
An electron beam sensitive resist (ZEP520a diluted 50$\%$ in anisole) was spun
onto the heterostructure and baked at 180°C for 5 minutes, leading to 160nm
thick final resist layer. The resist was exposed with a Vistec VB300 Electron
Beam Lithography System, with doses that range from 140$\mu$C/cm2 up to
280$\mu$C/cm2. The electron beam dose and design allow to achieve a wide
variety of filling factors (see Fig.6) that range from 30$\%$ to 75$\%$,
necessary to tune the BIC and band gap. Each grating has a dimension of
300$\mu$m x 50$\mu$m. The resist was then developed in amyl acetate, and the
heterostructure was etched using Oxford PlasmaLab 150 Inductively Coupled
Etcher. A recipe with Cl2:BCl3:N2=11:3:11[40] was found to be ideal for low
selectivity between GaAs and Al0.4Ga0.6As, while guaranteeing vertical
sidewalls and good selectivity with the softmask ($\simeq$ 2.73). In the past,
the damage induced by energetic ions below 200eV was proved to be caused by an
ion-assisted chemical etching which can be recovered through passivation[42].
For this reason, the RF and IPC power were also tuned to reduce the ion energy
down to 138eV in order to limit the etching induced damage in the GaAs QWs.

Figure 6: Scanning electron microscope images of gratings with filling factors
a) 70$\%$ and b) 55$\%$ with darker regions corresponding to the etched
grooves.

The GaAs oxide at the surface was removed via hydrochloric acid and was
prevented to be formed again by passivating the surface with 8nm of Al2O3
conformal layer right after the acid step. The sample was placed in a Oxford
FlexAl-Plasma Enhanced Atomic Layer Deposition and kept for 10 minutes at a
temperature of 300°C before starting the deposition[48]. The alumina layer was
conformally grown all over the sample to reduce the nonradiative recombination
rate at the GaAs grating sidewalls[44, 47, 46]. The SEM image in Fig.7 shows
the top view of the grating with the thin film at the etched grooves. The
results shown in Fig.5d and Fig.5e of the main text show the linewidth fit of
the exciton-polariton dispersions before and after the passivation (Fig.5a and
Fig.5b, respectively). The reduction of the exciton linewidth led to a quality
improvement of the sample after the passivation with the consequent reduction
of the minimum polariton density needed to achieve condensation.

Figure 7: SEM image of the 8nm thick aluminum oxide grown via atomic layer
deposition. The layer is conformal with the sample surface, hence it covers
the grooves sidewalls, bottom and top surface alike.

##  VI Simulations

Simulations of the grating were performed using Stanford Stratified Structure
Solver (S4). The strong coupling is introduced by means of a modulation of the
GaAs dielectric function[4] as a function of the energy E

$\epsilon(E)=\epsilon_{\text{GaAs}}+\epsilon_{\text{GaAs}}\frac{\omega_{LT}}{X-E-i\gamma}$
(2)

where $\epsilon_{\text{GaAs}}=3.54$ is the GaAs dielectric constant at 10K,
$X=1527.5$ meV and $\gamma=0.01$ meV are the exciton resonance and its losses,
respectively. $\omega_{LT}$ is the longitudinal transverse splitting which, in
close proximity to the resonance frequency, is defined as

$\omega_{LT}=\frac{\Omega^{2}}{2X}$ (3)

where $\Omega=13.9$ meV is the Rabi splitting calculated from the dispersion
fitting in [8, 27]. The barrier dielectric constant was set to
$\epsilon_{\text{Al}_{0.4}\text{Ga}_{0.6}\text{As}}=3.3$. While the bound
state in the continuum is introduced through the coupling of pure photonic
modes, the presence of the exciton does not induce a finite linewidth as the
state is symmetry protected. Nevertheless, its lifetime is still limited by
the intrinsic losses inside the waveguide. Fig. 8.a shows the photonic modes
without the resonance, which is then introduced in Fig. 8.b leading to the 4
polariton branches: two above and two below the exciton energy. For the rest
of the discussion we will focus on the lower energy branches.

(a)

(b)

Figure 8: (a) Energy dispersion of the coupled photonic modes. (b) Upper and
lower exciton-polariton dispersion branches that arise from the coupling
between the two photonic modes in (a) with the exciton.

As the filling factor is reduced from 75$\%$ to 50$\%$ (Fig.9a,b.), the gap is
reduced with a blueshifted dark state. This was also confirmed experimentally
as reported in Fig.13a. An interesting aspect is the switch of the BIC from
the lower state to the upper one (Fig.9c.), which was also predicted from
previous theoretical calculations[29]. We used S4 to obtain the exciton-
polariton dispersion with its geometrical parameter dependence. However, in
order to fit our experiments we relied on a coupled oscillator model,
described in the next section, which also provides us with the corresponding
states excitonic fractions.

(a)

(b)

(c)

Figure 9: Simulations of the sample for three different filling factors: (a)
75$\%$, (b) 50$\%$, and (c) 25$\%$.

##  VII Coupled oscillators model

As reported by Ardizzone et al.[29], this system can be described as a 2-steps
perturbation method. Using this method, they have developed a theory that
describes the polariton states along both kx and ky directions. The first step
is to introduce a coupling between the transverse electric propagating
($\hat{\omega}_{+1}$) and counter-propagating ($\hat{\omega}_{-1}$) modes.
This is done by introducing a refractive index modulation in our waveguide,
while the uncoupled modes were obtained from the fitted dispersion in [8].
Here we show the result of the first step where the photonic modes are coupled
with a strength U, limiting our considerations to ky=0. Such term can be
changed experimentally by varying both filling factors and the depth of the
grooves. The Hamiltonian can then be written as

$H=\left(\begin{matrix}\hat{\omega}_{+1}&\text{U}\\\
\text{U}&\hat{\omega}_{-1}\end{matrix}\right)+i\left(\begin{matrix}\gamma_{r}&-\gamma_{r}\\\
-\gamma_{r}&\gamma_{r}\end{matrix}\right)$ (4)

where $\gamma_{r}$ represents the radiative losses and $\hat{\omega}_{\pm 1}$
are the photonic dispersion characteristics obtained through the fitting of
the uncoupled modes[8]. The eigenvalues for the pure photonic coupled modes
are

$E_{\text{Bright}}=\frac{\hat{\omega}_{+1}+\hat{\omega}_{-1}}{2}+i\gamma_{r}+\frac{1}{2}\sqrt{(\hat{\omega}_{+1}-\hat{\omega}_{-1})^{2}+4(U+i\gamma_{r})^{2}}$
(5)
$E_{\text{Dark}}=\frac{\hat{\omega}_{+1}+\hat{\omega}_{-1}}{2}+i\gamma_{r}-\frac{1}{2}\sqrt{(\hat{\omega}_{+1}-\hat{\omega}_{-1})^{2}+4(U+i\gamma_{r})^{2}}$
(6)

The second step consists in the coupling of the resulting states with the
exciton resonance which eigenvalues can be found by solving the 4x4
Hamiltonian

$H=\left(\begin{matrix}\text{E}_{\text{Bright}}&\frac{\Omega}{2}&0&0\\\
\frac{\Omega}{2}&X&0&0\\\ 0&0&\text{E}_{\text{Dark}}&\frac{\Omega}{2}\\\
0&0&\frac{\Omega}{2}&X\end{matrix}\right)$ (7)

The matrix diagonalization leads to the E${}_{\text{dark}}^{\text{pol}}$ and
E${}_{\text{bright}}^{\text{pol}}$ which describe our polaritonic system

$E_{\text{Bright},\pm}^{\text{pol}}=\frac{E_{\text{Bright}}+X}{2}\pm\frac{1}{2}\sqrt{(E_{\text{Bright}}-X)^{2}+\Omega^{2}}$
(8)
$E_{\text{Dark},\pm}^{\text{pol}}=\frac{E_{\text{Dark}}+X}{2}\pm\frac{1}{2}\sqrt{(E_{\text{Dark}}-X)^{2}+\Omega^{2}}$
(9)

The two separate solutions identified by $\pm$ for both bright and dark states
correspond to the lower and upper bright and dark states, respectively. These
states are also shown in Fig. 8.b as upper and lower polariton branches. From
now on, we will consider only the lower branches, hence the states below the
exciton energy $E_{\text{Dark},-}^{\text{pol}}$ and
$E_{\text{Bright},-}^{\text{pol}}$. This model will also be used to determine
the excitonic fractions for the rest of the work. This simple model, even
though limited by the lack of explicit definition of the grating coupling
strength U, it can qualitatively describe our system. One of the things
mentioned and shown in Fig.4b is the experimental trend of the energy gap and
excitonic fraction as a function of the etching depth. Here we show that
increasing the coupling U between the pure photonic modes can lead to a
similar behaviour illustrated in Fig.4b where gratings with the same filling
factors were compared when the etching depth was varied. The dispersions
obtained from the model for three different coupling factors are shown along
with the fitted dispersions for the different etching depths considered.
Results are shown in Fig.11.

Figure 10: Energy gap opened in the polaritonic dispersion along with the
excitonic fraction of the BIC obtained through the coupled oscillators model
as a function of the grating coupling strength U. Figure 11: (a-c) Theoretical
polariton dispersions extracted from the two coupled oscillators model for
different coupling coefficients U. (d-f) Fit of the measured dispersions for
the etching depths $d$ considered in this work. The colors represent the
excitonic fraction.

##  VIII Control of the polariton dispersion via processing

As mentioned in the main text, the BIC energy, its excitonic fraction, and the
gap are parameters that can be tuned by changing the grating periodicity which
is a peculiar advantage of our horizontal configuration. In Fig.12 we show
such control over the dispersions for pitch equal to 240 nm (a) and 241 nm
(b). Along with the tuning of the polariton dispersion, the grating pitch
modifies the BIC energy position and excitonic fraction (c).

Figure 12: (a,b) Dispersions for a pitch equal to 240 nm and 241 nm. (c) BIC
energy position (red) and excitonic fraction (blue) as a function of the
grating periodicity.

The energy gap is another controllable parameter which is modified by the
coupling between propagating and counter-propagating mode. While it is almost
constant for a 40 nm deep grating, its size is changed with the filling factor
in deeper gratings(Fig.13a), allowing a further control over the BIC energy
position and its excitonic fraction (Fig.13b). Fig.13 shows the energy gap and
excitonic fraction obtained through a coupled oscillator model fitting of the
polariton dispersions.

(a)

(b)

Figure 13: (a,b) Energy gap and excitonic fraction as a function of the
filling factor for the 40nm and 90nm deep gratings.

The data in Fig.14 show the situation for two different filling factors where
the gap closes for values close to 50$\%$. No lasing was observed for such
gratings as a consequence of higher losses due to a BIC closer to the bright
dispersion branch.

Figure 14: Polariton energy dispersion for (a) FF = 75$\%$ and (b) FF =
50$\%$. Passing from (a) to (b), the bright exciton-polariton state remains
mostly unaltered while the dark exciton-polariton state blueshifts. The gap
closes towards FF$\simeq$ 50$\%$ in which case no lasing was observed.

##  IX Condensation after Atomic Layer Deposition

The ALD processing was proved to be effective in reducing the condensate
density, hence the lasing emission from the BIC, by improving the sample’s
quality. In Fig.15 we show the emission intensity as a function of the
incident laser pumping power before and after aluminum oxide deposition. This
was done on different gratings characterized by a pitch = 240nm, depth = 90nm,
and filling factors around 70$\%$ \- 75$\%$. The results show a reduction in
the threshold value of our lasing from BIC that we associate to a lower
nonradiative recombination rate of the excitons at the etched sidewalls.

Figure 15: Lasing intensity of 4 pre- and post-processed gratings.

It is worth noting that in the case of 130nm deep grooves no lasing threshold
was observed prior passivation, and the processing was able to restore the
sample quality to achieve lasing from a high filling factor structure. Even
though not all the structures recovered enough quality to achieve lasing, this
result highlights the importance of this post-processing step as displayed in
Fig.16.
Despite being a long lifetime state, the lifetime of the photonic bound state
in the continuum coupled to a matter state is reduced by the intrinsic
excitonic losses, giving rise to the so-called quasi-BIC. As a matter of fact,
from theoretical calculations[29] its lifetime only depends on the BIC
excitonic fraction and nonradiative losses. For this reason the state acquires
an intrinsic finite lifetime, which can be increased by post-processing the
sample. In this sense, the presence of the aluminum oxide layer reduces the
density of defects where excitons can decay nonradiatively at the grooves.
This step allows to reduce the lasing threshold and also to increase the
exciton-polariton lifetime, as demonstrated in the main text. Even though the
lifetime of the quasi-BIC is finite, its linewidth always appears to be zero
since it cannot couple to the far field.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 16: Far field of the polariton emission for different excitation powers
for the 130nm deep grating before passivation (a,b,c) and after ALD
passivation (d,e,f)

##  X Cathodoluminescence

Hyperspectral cathodoluminescence (CL) microscopy at 15K was used to
investigate the role of etching depth on the excitonic losses. This technique
uses electrons from a scanning electron microscope (SEM) to locally excite the
sample, which is kept at 15K during the measurements. The emission is
collected by a parabolic mirror, which sends the exciton light towards a
spectrometer, allowing the reconstruction of a hyperspectral image. As
electrons pass through the layers of the heterostructure, they undergo
inelastic scattering events that excite a broadband of energy states. Carriers
then relax and contribute to the emission from the QWs. The energy of the
primary beam was tuned to 3keV to ensure high spatial resolution and surface
sensitivity. The electron acceleration energy controls the size of the
interaction volume of primary electrons in the sample: the lower the energy,
the smaller the interaction volume. The interaction volume of a 3kV electron
beam reaches depths of few tens of nm from the top surface of GaAs. The
electron beam penetration and interaction with the heterostructure was
simulated using CASINO Monte Carlo simulations[50, 51] (Fig.17b). At the same
time, an electron beam probing a small volume also provides a better lateral
spatial resolution. We used this technique to measure the emission from the
40nm and 130nm deep gratings after ALD processing. In both cases, only the
first QW is populated directly through electron interaction, while the second
one can be populated through carriers photogenerated in the AlGaAs barrier.
Other QWs can be indirectly populated with the exciton emission from the first
two. This process of exciton passage among QWs is the same that is responsible
for the achievement of strong coupling under optical pumping. When excitons
are created between the grooves, they diffuse through the QWs to the sidewalls
where they may decay in a non-radiative way on etching-induced surface
states[41, 42, 43]. Because some of the excitations can effectively change the
QW in which they are located, the non-radiative recombination rate will be
more important if the etching is deeper.

(a)

(b)

Figure 17: (a) Sketch of the cathodoluminescence interaction with the
heterostructure. A low-voltage (3kV) electron beam interacts within a small
volume of the sample (dark green) generating excitons and free carriers that
diffuse (lighter green). The light emitted is collected by a parabolic mirror
and sent to a spectrometer. Part of the generated carriers recombine at
surface defects induced by the etching. The two beams shown here represent two
possible different excitation positions that occur while the beam scans the
sample surface: the beam incident on the plateau probes a larger portion of
the surface defects compared to a beam in the groove. As a consequence, a
larger part of carriers recombine radiatively in the second case. (b) Monte
Carlo simulation of the deposited energy distribution for the 3kV electron
beam penetrating the heterostructure.

The surface damage is a limitation of the current waveguide approach which can
be addressed via surface treatment. A deeper groove generates a wider surface
area through etching, thus increasing the number of non-radiative
recombination channels, that is, the system losses in the excitonic component.
Therefore, the depth of the grooves can play an important role in the
achievement of lasing. In Fig18 we show two hyperspectral maps of the CL
emission from a sample with 40nm-deep grooves (b) and 130nm-deep grooves (c).
The intensity of each pixel depends on the probability that excitons excited
in that localized position recombine through radiative channels while
diffusing away from the interaction volume. If excitons generated at a certain
position find non-radiative recombination channels along their random path,
the overall intensity associated with that pixel decreases. In both maps in
Fig18 we observe that excitation inside the grooves always correspond to a
higher CL signal compared to the emission collected when the electron beam
hits the plateaus (flat area between two grooves). Finite difference time
domain simulations (not shown) discard that this modulation of the CL
intensity is due to optical interference effects of etched patterned on the
emitted light. We rather hypothesize that the etched sidewalls of the plateaus
contain a large number of non-radiative states that quench the exciton
emission. We observe that the CL intensity modulation depends on the grooves
depth. We thus calculated the ratio of CL intensity of the pixels in the
grooves over the pixels on the plateaus. The post-processed sample shows that
for the 40nm deep grooves, such ratio is around 4.94 (Fig18.b), meaning that,
on average, the light coming from excitation in the grooves is almost five
times more intense than the light collected when exciting on the plateaus. For
the 130nm case, this ratio reaches values up to 13.22 (Fig18.c). We suggest
that a deeper etching exposes a larger surface area to the presence of non-
radiative defects, thus increasing the probability that diffusing charge
carriers recombine at the surface in a non-radiative way. Indeed, the
intensity from bottom grooves is comparable to the intensity from unetched
areas.

(a)

(b)

(c)

Figure 18: (a) CL emission spectrum under a 3kV electron beam. The gray area
shows the energy range considered in the hyperspectral maps (b,c) which
corresponds to the exciton energy position. (b,c) Hyperspectral maps of 40nm
(b) and 130nm (c) deep grating after ALD processing.


Gaussian Process, Motion control, Learning based feedforward

§ INTRODUCTION

§ MAGNETICALLY LEVITATED PLANAR MOTOR SYSTEM

§ GAUSSIAN-PROCESS BASED LEARNING OF  DISTURBANCES

§.§ Disturbance characteristics and experiment design

This section describes the modelling of static behaviour of residual dynamics using the GP framework. The main focus of this work is directed towards the suppression of disturbances acting on the $z$ axis, which exhibits the largest deviations from the desired position in the investigated planar motor due to effects of residual dynamics, see <cit.>. Apart from cancelling the known non-linear effects associated with the rigid-body dynamics, the commutation algorithm, presented in Section <ref>, further facilitated the control design by decoupling the system and reducing the number of inputs from $40$, i.e. the number of active coils, to $6$, i.e. the rigid-body of force and torque components. For similar reasons, it is desirable to view the static effect of residual dynamics acting on the $z$ axis as load disturbances acting on the output $\hat{F}_z$ of the RB decoupled feedback controller.
Using the Gaussian Process modelling framework, the disturbance force model is described by

\begin{equation}
\begin{aligned}
{F}_{z}^{\mathrm{dist}}  \left( \boldsymbol{w} \left( i \right) \right)
\end{aligned}
\label{eq:gpGenerateSystem},
\end{equation}

where $i$ is the time index and ${e}$ is the noise source, modelled as i.i.d. white Gaussian noise, ${e} (i) \sim \mathcal{N} \left( 0, \sigma_e^2 \right) $ <cit.>, while $y \in \mathbb{R}$ is the measured output. The disturbance force ${F}_{z}^{\mathrm{dist}}$ is assumed to be dependent on the input vector $\boldsymbol{w} \in \mathbb{R}^{n_u}$ and will be modelled by a Gaussian Process <cit.>,

\begin{equation}
\begin{aligned}
(\boldsymbol{w}) &\sim \mathcal{G}\mathcal{P}
\big(
k \left( \boldsymbol{w}(i), \boldsymbol{w}(j) \right)
\big),
\end{aligned}
\label{ch6:eq:gp_def2}
\end{equation}

i.e. its covariance is fully characterized by the kernel function $k$. The Gaussian Process framework exploits the assumption of Gaussian distribution of both the function to be estimated and the involved noise process, in order to arrive at an analytically computed predictor, which is based on the posterior predictive distribution of ${F}_{z}^{\mathrm{dist}}$ conditioned on  a training set of input-output observations. The mean of the predictive distribution is also the maximum a posteriori estimate

\begin{equation}
\bar{F}_{z}^{\mathrm{dist}}
\left(\boldsymbol{w}^* \right)
\boldsymbol{k}
\left(\boldsymbol{w}^*, W_N \right)
\left(
\mathcal{K} \left( W_N, W_N \right)
\sigma _e ^2 I_N \right)^{-1} \boldsymbol{y},
\end{equation}

where $\boldsymbol{w}^* \in \mathbb{R}^{n_u}$ is a test input, and $W_N$ with $\boldsymbol{y}$ contain the training input and output data, respectively, at the $N$ training points. For the vector $\boldsymbol{k}
\left(\boldsymbol{w}^*, W_N \right) \in \mathbb{R}^{1 \times N}$, its $i$th element is equal to $k \left( \boldsymbol{w}^*, \boldsymbol{w} \left(i\right) \right)$, and in a similar fashion, for the matrix $\mathcal{K} \left( W_N, W_N \right) \in \mathbb{R}^{N \times N}$ the element on the  $i$th row and $j$th column is equal to $k \left( \boldsymbol{w} \left(i\right), \boldsymbol{w} \left( j \right) \right)$.

Based on the discussion so far, three main questions have to be answered towards the modelling of the disturbance via the Gaussian Process framework. First, in spite of the model postulated in (<ref>), the static effects of the residual dynamics cannot be directly measured. Consequently, we need a methodology to infer the non-measurable values of these static effects from measurable data. Secondly, the relevant inputs have to be defined. Thirdly, the kernel function has to be chosen.

In order to answer the aforementioned questions, in ch6:fig:fzGrid, the total force command on $z$ axis on a selected grid of positions in the $x^\mathcal{M}_\mathcal{T}-y^\mathcal{M}_\mathcal{T}$ plane is shown, after the dynamic behaviour has settled. The two plots correspond to two independent measurements. At steady state, due to the integral action in the feedback controller, the resulting total control effort in each DOF is equal to the feedforward effort, minus the constant disturbances that act on the same DOF, i.e.

\begin{equation}
\begin{aligned}
\hat{F}_{z} = \hat{F}_{z}^{\mathrm{FF}} - {F}_{z}^{\mathrm{dist}}
\end{aligned}
\label{eq:Force_effort},
\end{equation}

with $\ \hat{F}_{z}^{\mathrm{FF}}$ denoting the feedforward effort. By making use of (<ref>), it becomes apparent that the measurements in ch6:fig:fzGrid can be directly used to estimate the magnitude of the disturbance.

Two different measurements of $\hat{F}_{ z}$ on the $\hat{x}^{\mathcal{M}}_{\mathcal{T}}-\hat{y}^{\mathcal{M}}_{\mathcal{T}}$ plane.

ch6:fig:fzGrid can be used further to draw conclusions, which are important for inferring which signals are affecting the static behaviour of residual dynamics, which are viewed as load disturbances. As a starting point, due to the position-dependent characteristics of the planar motor behaviour, it is natural to assume that the disturbances exhibit position-dependent characteristics as well. Secondly, it is asserted that the load disturbance exhibits a sinusoidal behaviour over the  $x^\mathcal{M}_\mathcal{T}-y^\mathcal{M}_\mathcal{T}$ plane. This is verified by computing the 2D Fourier transform of the measured $\hat{F}_{z}$ with respect to the latter coordinates, shown in ch6:fig:fft2DSpatial, where it becomes evident that a sinusoidal behaviour with a spatial frequency of approximately $2.5-3$ cm is dominant. This value is related to the expected magnet pitch, see <cit.>.
Thirdly, the differences in the two measurements, presented in ch6:fig:fzGrid, reveal that the residual dynamics cannot be solely described as a function of $\hat{x}^{\mathcal{M}}_{\mathcal{T}},\hat{y}^{\mathcal{M}}_{\mathcal{T}}$. The deviation per grid point is on average equal to $0.46 \mathrm{N}$, which is approximately equal to $0.5 \%$ of the total mass feedforward control force $\hat{F}_{z}^{\mathrm{FF}}$. Equation (<ref>) reveals a plausible source of residual dynamics; the vector $\hat{\boldsymbol{W}}_{\mathcal{M}}$ depends on the relative position between the coil, metrology and translator frame. Consequently, any unforeseen, neglected or inaccurately modelled dynamics will depend, to some extent, on the relative distance between the aforementioned frames. Consequently, at least six input variables are required in order to relate the coordinate frames to each other.

Spatial Fourier transform with respect to the $\hat{x}^{\mathcal{M}}_{\mathcal{T}},\hat{y}^{\mathcal{M}}_{\mathcal{T}}$ coordinates.

The attenuation capabilities of the resulting feedforward control depend on how well the input space has been explored. The variables $\hat{\boldsymbol{q}}_{\mathcal{T}}^{\, \mathcal{M}}$ are actively controlled, therefore it is straightforward to define the excitation region with respect to them. The experiments have been performed on a $\hat{x}^{\mathcal{M}}_{\mathcal{T}}-\hat{y}^{\mathcal{M}}_{\mathcal{T}}$ grid, with $\hat{x}^{\mathcal{M}}_{\mathcal{T}}, \hat{y}^{\mathcal{M}}_{\mathcal{T}} \in \left[ 0.01, 0.1 \right]$ m. The position difference between two consecutive measurement points is approximately equal to $2$ mm. On the other hand, the $\hat{z}^{\mathcal{M}}_{\mathcal{T}}$ as well as the rotation of the translator with respect to the metrology frame are kept constant during operation. Regarding the $\boldsymbol{q}^{\mathcal{C}}_{\mathcal{M}}$ coordinates, the mounting of the metrology frame on passively controlled air mounts, as  explained in Section <ref>, means that the relative displacement between the coil and the metrology frame cannot be actively controlled. Hence, the input space exploration is limited to the variability of the experiments, which is affected by environmental or other, unforeseen physical phenomena. For this reason, the experiment on the aforementioned $\hat{x}^{\mathcal{M}}_{\mathcal{T}}, \hat{y}^{\mathcal{M}}_{\mathcal{T}}$ grid has been repeated six times and the corresponding $\hat{\boldsymbol{q}}^{\mathcal{C}}_{\mathcal{M}}$ have been estimated. In total, the input vector is defined as

\begin{equation}
\begin{aligned}
\boldsymbol{w} ( i)
\begin{bmatrix}
\hat{x}^{\mathcal{M}}_{\mathcal{T}}( i)
\hat{y}^{\mathcal{M}}_{\mathcal{T}}( i)
\hat{x}_{\mathcal{M}}^{\mathcal{C}}( i)
\hat{y}_{\mathcal{M}}^{\mathcal{C}}( i)
\hat{z}_{\mathcal{M}}^{\mathcal{C}}( i)
\hat{\chi}_{\mathcal{M}}^{\mathcal{C}}( i)
\hat{\psi}_{\mathcal{M}}^{\mathcal{C}}( i)
\hat{\zeta}_{\mathcal{M}}^{\mathcal{C}}( i)
\end{bmatrix}^{\top},
\end{aligned}
\label{ch6:eq:inputSignals}
\end{equation}

with $\boldsymbol{w} ( i) \in \mathbb{R}^8$ and $i = \left\{ 1, \ldots, N_p \right\}$, where $N_p$ denotes the number of data points. Moreover, ${w_\nu} ( i)$ denotes the $\nu^\text{th}$ element of the vector. Based on these inputs, an appropriate kernel selection is sought next.

§.§ Kernel design and validation

In the Gaussian Process framework, the kernel structure constitutes an important aspect, since it specifies the hypothesis space for the resulting estimator. As such, it is beneficial to incorporate any
observations to specify the structure of the expected function class. To this end, the aforementioned load disturbance characteristics can be inscribed in the kernel structure, thus facilitating the fitting of an appropriate GP model to the experimental data. Based on the experimental observations in Section <ref>, the kernel is defined as the product of a periodic kernel <cit.>, which captures the coil array characteristics, a Radial Basis Function (RBF) kernel <cit.>, which captures inconsistencies in the magnetic force distribution and a linear kernel, which explores linear trends with respect to the  $\hat{x}^{\mathcal{M}}_{\mathcal{T}}$-
$\hat{y}^{\mathcal{M}}_{\mathcal{T}}$ frame, which would
cause $\hat{z}^{\mathcal{M}}_{\mathcal{T}}$ to vary as a function of $\hat{x}^{\mathcal{M}}_{\mathcal{T}}$,
$\hat{y}^{\mathcal{M}}_{\mathcal{T}}$ . The resulting kernel is mathematically described by

\begin{equation}
\begin{aligned}
\left(
\boldsymbol{w} \left( i \right),
\boldsymbol{w} \left( j \right)
\right)
\sigma_{1}^2 \exp
\bigl(
\overbrace{
\sum\limits_{\nu = 1}^{8}
\frac{
\left(
{w_\nu} \left( i \right)
{w_\nu} \left( j \right)
\right)^2
2 {\lambda}_{\nu, \mathrm{rbf}}^2
}}^{\text{RBF kernel}}
\overbrace{
\sum\limits_{\nu = 1}^{8}
\frac{
\sin^2
\left(
\frac{
\pi
\left(
{w_\nu} \left( i \right)
{w_\nu} \left( j \right)
\right)
\right)
2 \lambda_{\mathrm{sin}}^2
}}^{\text{Periodic kernel}}
\bigl)
\\
%%% LINEAR
& \underbrace{\sum\limits_{\nu = 1}^{2}
{\sigma}_{\nu, 2}^2
\left(
{w_\nu} \left( i \right)
{c}_{\nu, \mathrm{lin}}
\right)
\left(
{w_\nu} \left( j \right)
{c}_{\nu, \mathrm{lin}}
\right)}_{\text{Linear kernel}},
\end{aligned}
\label{ch6:eq:kernelFun}
\end{equation}

where $\sigma_{\nu, 2}, {c}_{\nu, \mathrm{lin}}$ is the $\nu^{\text{th}}$ element of the vector $\boldsymbol{\sigma}_{2}, \boldsymbol{c}_{\mathrm{lin}}$ respectively. In total, there are 16 hyperparameters to be optimized, including the noise variance $\sigma_e$. Additionally, variations of the kernel structure, given by  (<ref>) have been investigated. More specifically, the kernel structure in (<ref>) without the sinusoidal term, i.e. linear plus RBF, has been investigated, as well as a standalone RBF kernel. These two kernel structures are comprised of $14$ and $10$ hyperparameters, respectively. For all the aforementioned kernels, the hyperparameters have been computed by maximizing the marginal likelihood with respect to the hyperparameters <cit.>, using the collected training set.

For the validation of the results, 5 data sets with sufficient variations in $\boldsymbol{q}^{\mathcal{C}}_{\mathcal{M}}$ are used for training purposes, where in total 3600 training points have been used. The predicted output $\hat{\boldsymbol{y}}$ is compared to the measured output $\boldsymbol{y}$ on a fresh data set using the Best Fit Ratio (BFR)

\begin{equation}
\begin{aligned}
\text{BFR} &= 100 \%
\cdot
\max
\left(
\frac{
\left| \left|
\boldsymbol{y} - \hat{\boldsymbol{y}}
\right| \right|_2
\left| \left|
\boldsymbol{y} - \bar{{y}}
\right| \right|_2}
\right),
\end{aligned}
\label{ch6:eq:bfr}
\end{equation}

where $\bar{y}$ denotes the sample mean of $\boldsymbol{y}$. The BFR criterion delivers an estimate between $0\%$ for no match and $100\%$  for the perfect match between the estimated and the measured output. The corresponding BFR results are presented in Table <ref>.

Validation of the estimated GP compensator on the validation data set

Linear + RBF $\times$ Sinusoidal

Linear + RBF

BFR $\%$

A few conclusions can be drawn from Table <ref>. First of all, it is evident that the employed kernels manage to efficiently capture the major characteristics of the disturbances that act on the motor prototype. Among the three kernels, the kernel containing the sinusoidal term leads to the highest accuracy, thus verifying the physics-based intuition for selecting such a kernel. However, the remaining two kernels also manage to predict the steady state behaviour of residual dynamics in a satisfactory manner, which highlights the flexibility of the RBF kernel in capturing non-linear trends. Finally, it is observed that any linear trend can be captured by the RBF kernel alone, thus explaining the similarity in results for the Linear+RBF and RBF kernels. In total, the results show that the GP framework is a powerful tool for modelling highly complex, position-dependent static effects of residual dynamics, which can severely limit the positioning accuracy of planar motors.

§ EXPERIMENTAL RESULTS

In this section, the GP-based predictor is used for compensation of  static disturbances, which are induced by residual dynamics that the feedback controller must compensate. Real-time implementation of the feedforward augmentation on the experimental prototype requires additional consideration. First of all, for $\hat{x}^{\mathcal{M}}_{\mathcal{T}}$, $\hat{y}^{\mathcal{M}}_{\mathcal{T}}$, the corresponding reference signals are used in the predictor instead of their measurement-based estimates. This selection is justified by the fact that for these two controlled variables, the error in this high-precision motor is relatively small compared to the sensitivity of the GP with respect to these variables. Moreover, introduction of additional feedback control loops is avoided, which could otherwise endanger stability. Additionally, the total control effort, shown in ch6:fig:fzGrid, contains the feedforward control effort, i.e. the gravity compensation term. Therefore, in order to only compensate for the disturbance, the gravity compensation term is subtracted from the predicted disturbance force.

In order to evaluate the attenuation capabilities of the proposed feedforward controller  in the NAPAS prototype, its performance is tested under both diagonal as well as straight motion profiles with respect to the $\hat{x}^{\mathcal{M}}_{\mathcal{T}}$-$\hat{y}^{\mathcal{M}}_{\mathcal{T}}$ plane.
For experimental validation of the designed feedforward augmentation, a low bandwidth feedback controller ($\approx$ 9.5 Hz) is applied to obtain a better visualization of the effects of the designed feedforward augmentation on the position tracking performance. The selected plane is defined as $\hat{x}^{\mathcal{M}}_{\mathcal{T}}$,$\hat{y}^{\mathcal{M}}_{\mathcal{T}} \in [0.015, 0.055] \mathrm{m}$. The employed 4th order reference trajectory is generated based on the method by <cit.>. The designed reference trajectory is illustrated in ch6:fig:MlStaticExpResults.

For the real-time implementation of the GP-based feedforward augmentation, the computational complexity of the predictor has to be taken into consideration. For demonstration purposes, the linear plus RBF kernel is implemented in the experimental prototype, which, in practice, offers a good balance between prediction accuracy and required real-time calculation time. To further reduce the computation time, the initial predictor, which is formulated using the $3600$ training points, is further approximated using a subset of $200$ points. To accomplish this approximation, the Subset of Regressors (SR) <cit.> method is applied, for which is empirically shown <cit.> that it offers a good balance between approximation error and computation time. For the selection of the subset of training points, a heuristic approach is considered <cit.>.
First, 200 samples are randomly selected for which the corresponding kernel is constructed. Secondly, the prediction capability of the resulting predictor is validated on a fresh data set by computing the BFR value. The heuristic approach is repeated 1000 times, after which the predictor with the highest BFR is kept. Using this approach, the approximated predictor achieved a BFR value of $65.37\%$, compared to the original BFR value of  $77.60\%$.

The experimental results are shown in Figure <ref>. The trajectory consists of a motion on the $+{y}^{\mathcal{M}}_{\mathcal{T}}$ direction, followed by a diagonal motion on the $+{x}^{\mathcal{M}}_{\mathcal{T}}, -{y}^{\mathcal{M}}_{\mathcal{T}}$ direction, another $+{y}^{\mathcal{M}}_{\mathcal{T}}$ motion and finally the magnet plate returns to the starting point. Moreover, the error on the $\hat{z}^{\mathcal{M}}_{\mathcal{T}}$ axis is plotted for two cases, namely when the GP-based feedforward augmentation is active and inactive. Due to the coupling between the various DOFs, it is observed that the motion in the ${x}^{\mathcal{M}}_{\mathcal{T}}$ and/or ${y}^{\mathcal{M}}_{\mathcal{T}}$ axes leads to higher error in $\hat{z}^{\mathcal{M}}_{\mathcal{T}}$, too, compared to steady-state intervals. As a consequence, attenuating the disturbances in the former intervals is more imperative, especially in the constant velocity part, for which scanning takes place in the lithography process. Indeed, it can be directly observed in Figure <ref> that the GP-based feedforward augmentation leads to significant improvement both in position tracking performance during steady-state, as well as position tracking performance during motion.

Finally, in order to quantify the resulting improvement with the GP-based feedforward augmentation, the $\ell_2$ norm, scaled by the square root of the number of data points, and the $\ell_\infty$ norms of the $\hat{z}^{\mathcal{M}}_{\mathcal{T}}$  error are computed. In Table <ref> the constant velocity results are presented, while in Table <ref> the results for the whole trajectory are presented. Based on the results shown in both tables, it is asserted that the improvement is approximately $50 \%$, thus highlighting the effectiveness of the proposed approach towards achieving higher positioning accuracy.

Experimental results of the feedforward augmentation.

No compensation

Kernel-based FF

Relative reduction

$\ell _2 / \sqrt{N}$

$6.4 \cdot 10^{-6}$

$3.39 \cdot 10^{-6}$

$47.03 \%$

$\ell _\infty$

$2.19  \cdot 10^{-5}$

$9.97 \cdot 10^{-6}$

$54.47 \%$

From top to bottom: Reference trajectory in terms of position $[m]$ (thoughtProvRed), velocity $[\frac{m}{s}]$ (thoughtProvDark) and scaled acceleration $[\frac{m}{s^2}]$ by a factor of 10 (thoughtProvGreen) for $\hat{x}^{\mathcal{M}}_{\mathcal{T}}$.
Reference position $[m]$ (thoughtProvRed), velocity $[\frac{m}{s}]$ (thoughtProvDark) and scaled acceleration $[\frac{m}{s^2}]$ by a factor of 10 (thoughtProvGreen) for $\hat{y}^{\mathcal{M}}_{\mathcal{T}}$. Error in $z$-direction $[m]$ with (thoughtProvGreen) and without (thoughtProvRed) kernel-based feedforward compensation for static disturbance rejection. The yellow areas represent the time intervals for which the velocity is constant.

Experimental results of the feedforward augmentation during constant velocity.

No compensation

Kernel-based FF

Relative reduction

$\ell _2 / \sqrt{N}$

$1.02 \cdot 10^{-5}$

$4.38 \cdot 10^{-6}$

$\ell _\infty$

$2.19  \cdot 10^{-5}$

$9.97 \cdot 10^{-6}$

§ CONCLUSIONS

This paper presents a kernel based modelling approach that is able to successfully capture the steady state behaviour of residual dynamics for moving-magnet planar actuator systems as a function of the generalized coordinates associated to the coil - metrology and the translator - metrology reference frames. Additionally, capturing the static behaviour of residual dynamics using the GP framework allows for augmentation of the feedforward controller, such that unforeseen static effects of residual dynamics are timely attenuated for. Consequently, position tracking error in $z_\mathcal{T}^\mathcal{M}$ direction is reduced by more than 50 $\%$ during application.

Finally, in order to further improve position tracking performance of a planar motor stage, the kernel based modelling strategy can be extended, such that the full dynamic residual model is considered for feedforward compensation.