# Conjunction Data Messages for Space Collision Avoidance behave as a Poisson
Process

1st Francisco M. Caldas  NOVA LINCS
NOVA School of Science and Technology
Caparica, Portugal
<EMAIL_ADDRESS>2nd Cláudia Soares  NOVA LINCS
NOVA School fo Science and Technology
Caparica, Portugal
<EMAIL_ADDRESS>3rd Cláudia Nunes  CEMAT
Instituto Superior Técnico
Lisboa, Portugal
<EMAIL_ADDRESS>4th Marta Guimarães  Neuraspace
Lisboa, Portugal
<EMAIL_ADDRESS>
###### Abstract

Space debris is a major problem in space exploration. International bodies
continuously monitor a large database of orbiting objects and emit warnings in
the form of conjunction data messages. An important question for satellite
operators is to estimate when fresh information will arrive so that they can
react timely but sparingly with satellite maneuvers. We propose a statistical
learning model of the message arrival process, allowing us to answer two
important questions: (1) Will there be any new message in the next specified
time interval? (2) When exactly and with what uncertainty will the next
message arrive? The average prediction error for question (2) of our Bayesian
Poisson process model is smaller than the baseline in more than 4 hours in a
test set of 50k close encounter events.

###### Index Terms:

component, formatting, style, styling, insert

##  I Introduction

Since the early 1960s, the space debris population has extensively increased
[1]. It is estimated that more than 36,000 objects larger than 10 centimetres,
and millions of small er pieces, exist in Earth’s orbit [2]. Collisions with
debris give rise to more debris, leading to more collisions in a chain
reaction known as Kessler syndrome [3]. To avoid catastrophic failures,
satellite owners/operators need to be aware of the collision risk of their
assets [4]. Currently, this monitoring process is done via the global Space
Surveillance Network (SSN). To assess possible collisions, a physics simulator
uses SSN observations to propagate the evolution of the state of the objects
over time [5, 6]. Each satellite (also referred to as target) is screened
against all the objects of the catalogue in order to detect a conjunction,
i.e., a close approach. Whenever a conjunction is detected between the target
and the other object (usually called chaser), SSN propagated states become
accessible and a Conjunction Data Message (CDM) is issued, containing
information about the event, such as the Time of Closest Approach (TCA) and
the probability of collision. Until the TCA, more CDMs are issued with updated
and better information about the conjunction. Roughly in the interval between
two and one day prior to TCA, the O/Os must decide whether to perform a
collision avoidance manoeuvre, with the available information. Therefore, the
CDM issued at least two days prior to TCA is the only guaranteed information
that the O/Os have and, until new information arrives, the best knowledge
available.

Several approaches have been explored to predict the collision risk at TCA,
using statistics and machine learning [7], but only a few were developed with
the aim of predicting when the next CDM is going to be issued. Very recently,
[8] developed a recurrent neural network architecture to model all CDM
features, including the time of arrival of future CDMs. GMV is currently
developing an autonomous collision avoidance system that decides if the
current information is enough for the O/Os to decide, or if they should wait
for another CDM to have more information . However, the techniques used are
not publicly available. In this work, we propose a novel statistical learning
solution for the problem of modeling and predicting the arrival of CDMs based
on a homogeneous Poisson Process (PP) model, with Bayesian estimation. We note
that standard machine learning and statistical learning solutions are data-
hungry and cannot model our problem, suffering from a special type of data
scarcity. Further, as our application is high-stakes, we require confidence or
credibility information added to a point estimate.

###  I-A Background

The present work formalizes the problem of modelling and predicting the
arrival of a new CDM with a probabilistic generative model. The present
formulation has the advantage of providing a full description of the problem,
stating clearly the required assumptions. Our proposed model shows high
accuracy, decreasing error in more than 4 hours, when compared with the
baseline, for predicting the next event.

Figure 1: Graphical model representation of the proposed model. The boxes are
plates representing replication of the structure. The outer plate represents
the dataset of $K$ events, and the inner one the repeated prediction of
$N_{k}$ occurrences in each event.

We leverage on a data model depicted in Figure 1. Probabilistic Graphical
Models [9, 10] is a probabilistic framework that allows for highly expressive
models, while keeping computational complexity to a minimum. A few statistical
learning developments like the Latent Dirichlet Allocation for topic modelling
[11], are now the basis for deep learning-powered probabilistic models like
the Variational Autoencoder [12] and Normalizing flows [13], and as a means of
creating modular and interpretable machine learning models [14]. We leverage
on the homogeneous Poisson Point Process [15], a counting process of messages
occurring over continuous time. Thus, we assume that the inter-CDM times are
independent and identically distributed random variables with exponential
distribution with a rate $\lambda$. The homogeneity property comes from an
event rate constant in time. Phenomena like radioactive decay of atoms and
website views are well modeled by this process [16]. The homogeneity is,
nevertheless, a limitation that will be addressed in future work, as there
might be some temporal trend in the data. Not considering it, though, delivers
a fast to compute solution that can outperform a standard baseline.

###  I-B Contributions

  * •

Formalizing an arrival problem within a probabilistic graphical model
framework

  * •

Few-shot learning using a Bayesian Approach

  * •

Model testing with a real world problem, surpassing the Baseline

  * •

analysis of the limitations of the homogeneous Poisson Process

##  II Poisson Point Process

A homogeneous Poisson Process (PP) can be seen as a counting process of events
over continuous time. A process is said to be Poisson when the inter-event
times are independent and identically distributed random variables with
exponential distribution and parameter $\lambda$. If $\lambda$ is constant
over the continuous time interval, then the PP is homogeneous [15].

A Poisson Process is a specific type of Renewal Process, when the interarrival
times have an exponential distribution. A renewal process may have any
independent and identically distributed (i.i.d.) inter-event times with finite
mean.

The Poisson process related with the CDMs is defined as follows:

  1. 1.

{${X_{i}}:i\in\mathbb{N}\\}$ be a sequence of random variables, where $X_{i}$
denotes the inter-CDM time, with $X_{i}\sim$ Exponential($\lambda$);

  2. 2.

Then if we define $S_{n}=\sum_{i=1}^{n}X_{i}$, it follows that $S_{n}$ is the
time of occurrence of the $n_{th}$ CDM;

  3. 3.

Finally, $N(t)=\text{max}\\{n\in\mathbb{N}_{0}:S_{n}\leq
t\\},\text{for}~{}t\geq 0$ is the number of CDMs issued in the interval (0,t].

The counting process $N(t)$ is therefore a PP with rate $\lambda$, i.e.,

$\\{N(t):t\geq 0\\}\sim PP(\lambda)$

There are two very important characteristics in a Poisson process. For any
interval $[a,b]$ let $N([a,b])$ denote the number of CDMs at that interval.
There, from the fact that $\\{N(t):t\geq 0\\}$ is a PP, it follows that
$N([a+s,b+s])$ is Poisson distributed, with parameter equal to $\lambda(b-a)$,
for all $s,a,b\geq 0$ (stationary increments). Furthermore, given two disjoint
time intervals, $[a,b]$ and $[c,d]$, $N([a,b])$ and $N([c,d])$ are independent
random variables (independent increments).

Finally, we postulate that each event has a specific unknown arrival rate
$(\lambda_{k})$, and the determination of that parameter is the objective of
the graphical model.

##  III Problem Formulation

Our generative probabilistic model of the CDM arrival process will allow
answering some questions as: (1) what is the probability that, during the
decision phase, new information will be received (2) what is the best estimate
for the next CDM arrival time, and the uncertainty of this prediction.

Following the literature, we define the stochastic quantities, with $k$ as the
ID of the object:

  * •

$X_{n+1}^{k}$ is time (in days) between the $n^{th}$ and the $n+1^{t}h$ CDM,
where $X_{n+1}^{k}\sim Exp(\lambda_{k})$, i.e., $X_{n+1}$ has an exponential
distribution of rate $\lambda_{k}$;

  * •

$N^{k}(t)$ is the number of CDMs received in the interval (0,t], in days,
where $N^{k}(t)\sim\text{Poisson}(\lambda_{k}t)$, and $\lambda_{k}$ is the
rate of CDMs issued per day;

  * •

$S_{n}^{k}=\sum_{i=1}^{n}X_{i}^{k}$ is the time of occurrence of the $n^{th}$
CDM, which, in view of the definition of $X_{n}^{k}$ is such that:

$S_{n}^{k}\sim Gamma(n,\lambda_{k})$

.

Probability of Interest: We are interested in the probability of receiving one
or more CDMs in the time interval from the last available observation
$s_{n}^{k}$, until a constant security threshold $(t_{sec})$. In this
application, we consider 1.3 days prior to TCA, and the interval will be
$(s_{n}^{k},t_{sec}]$.

$\displaystyle P(N^{k}(t_{sec}-s_{n}^{k})\geq 1)$
$\displaystyle=1-P(N^{k}(t_{sec}-s_{n}^{k})=0)$ (1)
$\displaystyle=1-e^{-\lambda_{k}(t_{sec}-s_{n}^{k})}$

Point Prediction: Furthermore, as:

$\displaystyle S^{k}_{n+1}=S^{k}_{n}+X^{k}_{n}\qquad\qquad\qquad X^{k}_{n}\sim
Exp(\lambda_{k})$ it follows that:
$\displaystyle(S^{k}_{n+1}-S^{k}_{n}|s^{k}_{n})\sim Exp(\lambda_{k})$

and therefore we may estimate the time until the next CDM to be issued by
$\frac{1}{\hat{\lambda^{k}}}$, the estimate of the expected value of the
inter-CDM time for event $k$.

###  III-A Baseline

The conjunction events are composed of a series of real CDMs for a set of
satellites during a 4 year period. The training data set has around 50,000
independent conjunctions events, with each event having at least 2 messages
and 15 messages in average. The complete dataset and variables are described
in [17], however, in this case, the only required information are the times
between message arrivals for each event. We assume the baseline model to be
the simplest one, where we assume that the next inter-CDM time is equal to the
last one, i.e.:

$X_{n+1}^{k}=x_{n}^{k}$

for each event k. This baseline implies a Markovian property on the variable
of interest.

In addition, we will also consider another model — hereby called by Classical
— which is nothing more than estimating $\lambda_{k}$ as in the classical
frequentist estimation. Because, for most events, the number of inter-CDM
times is very small, we expect this estimator process will not be as robust
and will provide biased results. Therefore, we need other models, besides the
Baseline and the Classical PP.

##  IV Bayesian Approach

To have a statistically significant estimation of $\lambda_{k}$, and in order
to use information from the other events, we adopt the Bayesian model
presented in Figure 1. More specifically, we use the extensive amount of
events in the dataset with an empirical Bayes framework to get an informative
prior, and then we update that distribution with the specific inter-CDMs times
of each event. We end up with a posterior distribution for $\lambda_{k}$, of
which we can extract a Maximum a Posteriori (MAP) estimate,
$\hat{\lambda}_{k}$ [18]. This method is basically described in the following
steps:

The posterior $\lambda$ distribution is defined as:

  1. 1.

the prior distribution for $\lambda_{k}$ is

$\lambda_{k}\sim\text{Gamma}(\alpha,\beta)$

where $\alpha$ and $\beta$ are hyperparameters estimated using data for all
the events in the train data.

  2. 2.

The likelihood function, for each event $k$, based on the information
$(X_{1}^{k},X_{2}^{k},...,X_{n}^{k})$ is given by:

$l(\lambda_{k}|X_{1}^{k},X_{2}^{k},...,X_{n}^{k})=\lambda_{k}^{n}e^{-\lambda_{k}\sum^{n}_{i=1}X_{i}^{k}}$
(2)

where $(X_{1}^{k},X_{2}^{k},...,X_{n}^{k})$ is the information regarding the
inter-CDM times for the $k^{th}$ event

  3. 3.

Then, it follows from straightforward manipulations, that the posterior
distribution of $\lambda_{k}$ is [15]:

$\displaystyle
f(\lambda_{k}|n,T)\propto\lambda_{k}^{n}e^{-\lambda_{k}T}\lambda_{k}^{\alpha-1}e^{-\beta\lambda_{k}}$
$\displaystyle
f(\lambda_{k}|n,T)\propto\lambda_{k}^{\alpha+n-1}e^{-\lambda_{k}(\beta+T)}$
$\displaystyle\lambda_{k}|n,T\sim\text{Gamma}(\alpha+n,\beta+T^{k}).$ (3)

With $T^{k}=\sum^{n}_{i=1}X_{i}^{k}$.

Note that the posterior in (3) can be derived either from the set of inter-
event times or through the number of CDMs received in a given interval
$N^{k}(t)$.

Figure 2: Prior distribution of $\lambda$, and a posterior distribution for a
single event in the train data.

In Figure 2, we observe the initial trained prior, and an example of a
posterior distribution, for a particular event. We can see that while
informative, the prior has enough variance that with few updates there is a
shift in the distribution of the posterior rate of arrival ($\lambda_{k}$). We
can also see that the posterior is narrower, meaning that the credible
interval of the posterior is smaller.

##  V Results

Next, we present the results obtained with the Bayesian, Classical and
Baseline models. For the computation of the estimations errors, we have used
half of the data as training data and the other half as an unbiased test set.
In Table I, we show the prediction errors (according to the measures: Mean
Absolute Error (MAE), Mean Squared (MSE) and Root Mean Squared Error (RMSE))
for the three models.

TABLE I: MAE, MSE and RMSE for next CDM time prediction, in days. | MAE | MSE | RMSE
---|---|---|---
Bayesian | 0.15170 | 0.06282 | 0.25064
Classic | 0.17581 | 0.08789 | 0.29646
Baseline | 0.26075 | 0.19105 | 0.43709

These results were obtained using an unbiased test data of 50523 independent
events, that was not seen during the study and was not used to derive the
hyperparameters. The RMSE for the proposed Bayesian model is $0.25064$ days,
which corresponds to six hours. Comparing the Bayesian with the classical
estimation of the parameter, we note that we expect better accuracy, as the
number of CDMs for each event is small, and thus, the classical approach is
more sensitive to extreme values. As can be seen in Figure 3, the proposed
model outperforms the classic counterpart and the baseline.

Figure 3: Empirical CDF for each of the models. There is a perceivable
difference in the predictive capability of each model. Because the three
models use the same formula to predict the expected time of the next CDM, the
difference lies in parameter determination, that is, the rate of arrival
$\lambda_{k}$.

A more detailed look into the performance of the Bayesian model, leads us to
an interesting observation. From Figure 4, we conclude that the distribution
of the error is not Gaussian.

Figure 4: Error Distribution for the Bayesian Poisson Process. Difference in
days between the Predicted Time of Arrival and the Observed Time of Arrival.

This might indicate that our assumptions are not adapted to the data that we
are analysing. This means, in particular, that the assumption of exponential
distribution for inter-CDM times may not be adequate. Another problem that may
cause this behaviour is related with the fact that some CDMs are not issued.
In fact, there might be a probability $p>0$ that a CDM is issued but not
received (leading to a filtered Poisson Process), then the apparent observed
time of the $n+1$ CDM, is actually the time of the $n+2$ CDM.

In order to confirm the good performance of our approach, we compare the
estimated probability of receiving a CDM in a decision interval with the
empirical probability. The estimated probability is the probability obtained
using (1). To obtain the empirical probability, the events are grouped by the
estimated probability intervals as presented in Table II. By computing ratio
of events in each group that actually receive a CDM in that time interval,
thus getting an empirical probability. For example, for events which where
determined to have an estimated probability in the interval $(0.803,0.952]$
with equation (1), we analyzed how many of them actually received a CDM in the
time interval. We note that there is a small positive drift between the
estimated probability under the Poisson assumption and the empirical one,
meaning that the estimated probability is conservative when compared to
reality. We also note that, as it should be, a lower estimated probability
will indeed represent a lower empirical probability, because along the groups,
the empirical probability increases.

TABLE II: Grouping of events by estimated probability, and the empirical results for each group. The Estimated probability is not over-confident when determining if a CDM is going to be issued during the decision interval. Estimated Prob. | Empirical Prob. | Deviation
---|---|---
(0.704, 0.753] | 0.7647 | 0.0117
(0.753, 0.803] | 0.83992 | 0.03692
(0.803, 0.852] | 0.90365 | 0.05165
(0.852, 0.901] | 0.9401 | 0.0391
(0.901, 0.951] | 0.96740 | 0.0164
(0.951, 1.0] | 1.0 | 0

In Figure 5 we can see two examples of the temporal prediction for two events
in our test data, with 90% credible intervals. This figure shows that when the
number of observations used for the Bayesian estimation is not too low (15, in
top panel), the observed and estimated times are close, and more important
from a statistical point of view, the observed value is contained within the
credibility interval. In the lower panel case, the credible intervals have a
large range (spanning over almost one day), and the observations and
predictions are far apart. Therefore when the number of observations in low,
the prediction results are less meaningful.

Figure 5: Example of the predictive capability of the model for different
examples. Top panel uses 15 observations to update the prior. Bottom panel
uses 3 observations.

##  VI Conclusion and Future Work

To conclude this section, we found that this real-life problem of predicting
the time of arrival of the next CDM can be successfully modeled by a
stochastic process, and that by using a Bayesian scheme, we can overcome data
scarcity. By modeling this problem as a PP, we can estimate the arrival of
CDMs during the decision period, which can be used in practice to aid with
expert decision process, helping the operator to confidently delay the
manoeuvre decision until new information is received. The proposed model is
able to predict the time of the next CDM with accuracy exceeding both the
baseline and a simpler predictor. The error distribution of the model suggests
further improvements to the model, such as the generalization of this model as
a renewal process, or the indication that the model should be a filtered
Poisson Process. The model is also slightly conservative when it comes to the
empirical probability of receiving a CDM in a given interval, and this might
mean that the initial hypothesis of homogeneity might not completely match to
the data. While the deviation is small, it is consistent, and it might
indicate that in the last two days, there is a bias in the model to predict a
smaller number of CDMs. However, for the practical case of a high-stakes
application like space awareness, it is always safer to be under-confident
than over-confident.

## Acknowledgment

The authors of this paper would like to thank FCT/PT Space under the PhD grant
PRD/BD/153601/2021 and Neuraspace for supporting this research.

## References

  * [1] J. Radtke, C. Kebschull, and E. Stoll, “Interactions of the space debris environment with mega constellations—using the example of the oneweb constellation,” _Acta Astronautica_ , vol. 131, pp. 55–68, 2017.
  * [2] ESA, “Space debris by the numbers,” https://www.esa.int/Safety_Security/Space_Debris/Space_debris_by_the_numbers, 2023\.
  * [3] H. Krag, M. Serrano, V. Braun, P. Kuchynka, M. Catania, J. Siminski, M. Schimmerohn, X. Marc, D. Kuijper, I. Shurmer _et al._ , “A 1 cm space debris impact onto the sentinel-1a solar array,” _Acta Astronautica_ , vol. 137, pp. 434–443, 2017.
  * [4] S. Le May, S. Gehly, B. Carter, and S. Flegel, “Space debris collision probability analysis for proposed global broadband constellations,” _Acta Astronautica_ , vol. 151, pp. 445–455, 2018.
  * [5] A. Horstmann and E. Stoll, “Investigation of propagation accuracy effects within the modeling of space debris,” in _7th European Conference on Space Debris_ , 2017.
  * [6] A. K. Mashiku and M. D. Hejduk, “Recommended methods for setting mission conjunction analysis hard body radii,” 2019.
  * [7] G. Acciarini, F. Pinto, F. Letizia, J. A. Martinez-Heras, K. Merz, C. Bridges, and A. G. Baydin, “Kessler: A machine learning library for spacecraft collision avoidance,” in _8th European Conference on Space Debris_ , 2021, pp. 1–9.
  * [8] F. Pinto, G. Acciarini, S. Metz, S. Boufelja, S. Kaczmarek, K. Merz, J. A. Martinez-Heras, F. Letizia, C. Bridges, and A. G. Baydin, “Towards automated satellite conjunction management with bayesian deep learning,” in _AI for Earth Sciences Workshop at NeurIPS 2020, Vancouver, Canada_ , 2020.
  * [9] D. Koller and N. Friedman, _Probabilistic graphical models: principles and techniques_. MIT press, 2009.
  * [10] C. M. Bishop, _Pattern recognition and machine learning_. springer, 2006.
  * [11] D. M. Blei, A. Y. Ng, and M. I. Jordan, “Latent dirichlet allocation,” _the Journal of machine Learning research_ , vol. 3, pp. 993–1022, 2003.
  * [12] D. P. Kingma, M. Welling _et al._ , “An introduction to variational autoencoders,” _Foundations and Trends® in Machine Learning_ , vol. 12, no. 4, pp. 307–392, 2019.
  * [13] I. Kobyzev, S. Prince, and M. Brubaker, “Normalizing flows: An introduction and review of current methods,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2020.
  * [14] W. J. Murdoch, C. Singh, K. Kumbier, R. Abbasi-Asl, and B. Yu, “Definitions, methods, and applications in interpretable machine learning,” _Proceedings of the National Academy of Sciences_ , vol. 116, no. 44, pp. 22 071–22 080, 2019.
  * [15] V. G. Kulkarni, _Modeling and analysis of stochastic systems_. Crc Press, 2016.
  * [16] S. Ross, _Introduction to Probability Models_. Elsevier Science, 2014.
  * [17] T. Uriot, D. Izzo, L. F. Simões, R. Abay, N. Einecke, S. Rebhan, J. Martinez-Heras, F. Letizia, J. Siminski, and K. Merz, “Spacecraft collision avoidance challenge: design and results of a machine learning competition,” _Astrodynamics_ , 2021.
  * [18] D. R. Insua and F. Ruggeri, _Robust Bayesian Analysis_. Springer Verlag, New York, 2012.

  *[SSN]: Space Surveillance Network
  *[CDM]: Conjunction Data Message
  *[TCA]: Time of Closest Approach
  *[CDMs]: Conjunction Data Message
  *[O/Os]: owner/operator
  *[PP]: Poisson Process
  *[MAE]: Mean Absolute Error
  *[MSE]: Mean Squared
  *[RMSE]: Root Mean Squared Error


# Probing Gigahertz Coherent Acoustic Phonons in TiO2 Mesoporous Thin Films

E. R. Cardozo de Oliveira111These authors contributed equally to this work,
Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies,
91120 Palaiseau, France  C. Xiang∗, Université Paris-Saclay, CNRS, Centre de
Nanosciences et de Nanotechnologies, 91120 Palaiseau, France  M. Esmann
Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies,
91120 Palaiseau, France  Institute for Physics, University of Oldenburg, 26129
Oldenburg, Germany  N. Lopez Abdala  Instituto de NanoSistemas - Universidad
Nacional de San Martín-CONICET, Buenos Aires, Argentina  M. C. Fuertes
Gerencia Química, Inst. de Nanociencia y Nanotecnología, CNEA-CONICET, Buenos
Aires, Argentina  A. Bruchhausen  Centro Atómico Bariloche, Inst. de
Nanociencia y Nanotecnología, CNEA-CONICET, Rio Negro, Argentina  H. Pastoriza
Centro Atómico Bariloche, Inst. de Nanociencia y Nanotecnología, CNEA-CONICET,
Rio Negro, Argentina  B. Perrin  Sorbonne Université, CNRS, Institut des
NanoSciences de Paris, INSP, F-75005 Paris, France  G. J. A. A. Soler-
Illia222Corresponding authors<EMAIL_ADDRESS>gsoler-
<EMAIL_ADDRESS>Instituto de NanoSistemas - Universidad Nacional de San
Martín-CONICET, Buenos Aires, Argentina  N. D. Lanzillotti-Kimura∗∗,
Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies,
91120 Palaiseau, France

###### Abstract

Ultrahigh-frequency acoustic-phonon resonators usually require atomically flat
interfaces to avoid phonon scattering and dephasing, leading to expensive
fabrication processes, such as molecular beam epitaxy. In contrast, mesoporous
thin films are based on inexpensive wet chemical fabrication techniques. Here,
we report mesoporous titanium dioxide-based acoustic resonators with
resonances up to 90 GHz, and quality factors from 3 to 7. Numerical
simulations show a good agreement with the picosecond ultrasonics experiments.
We also numerically study the effect of changes in the speed of sound on the
performance of the resonator. This change could be induced by liquid
infiltration into the mesopores. Our findings constitute the first step
towards the engineering of building blocks based on mesoporous thin films for
reconfigurable optoacoustic sensors.

##  1 Introduction

Coherent acoustic phonons, with frequencies in the GHz - THz range, have
associated wavelengths between a few and hundreds of nanometers. [1, 2, 3, 4,
5, 6, 7, 8, 9] Among other applications, they are suitable for high resolution
nanoimaging, non-destructive testing, and sensing. Acoustic phonon dynamics
have been explored in systems such as plasmonic nanostructures, [10, 11, 12,
13] metasurfaces, [14] oxides, [15, 16, 17, 18] and semiconductor
heterostructures [19, 20, 21, 22] with layer thicknesses on the nanometric
scale. [23, 19, 24] To precisely tailor the nanophononic response and obtain
high quality devices, expensive and complex growth and processing techniques
are usually employed, including molecular beam epitaxy and electron beam
lithography.

Conversely, mesoporous structures rely on cheap and reproducible bottom-up
fabrication processes derived from soft chemistry, carried out in mild
conditions, [25, 26] and are able to sustain gigahertz acoustic resonances.
[27, 28, 29] Brillouin light scattering experiments have been employed to
demonstrate phononic band gaps in the 10 - 20 GHz range in periodic stacks of
alternating layers of porous silicon dioxide (SiO2) and poly- (methyl
methacrylate) (PMMA). [27, 30, 31] More recently, mesoporous thin films (MTFs)
based on SiO2 have been shown to support coherent acoustic modes between 5 and
100 GHz, with Q-factors ranging from 5 to 17. [29]. MTFs are also suitable for
photonic sensing applications due to the high surface-to-volume ratio and
tailorable mesopores. [32, 33] By liquid infiltration into the nanopores, a
modulation of the optical and elastic properties of the material could be
achieved, also enabling chemical functionalization in nanoacoustic devices.
[28, 34]

A well-established material, with numerous applications for both the dense and
mesoporous forms, is titanium dioxide (TiO2). Applications include
photocatalysis, [35, 36] implants, [37] photovoltaics, [38] energy harvesting
and storage, and sensing. [39] These materials are promising candidates for
nanoacoustics. For instance, TiO2 anatase nanoparticles presented a
particularly strong induced exciton shift when applying an acoustic strain
pulse. [40, 41]

In this work, we employ coherent phonon generation and detection techniques to
study acoustic resonators based on TiO2 MTFs, with resonances up to 90 GHz.
Our results indicate that GHz acoustic resonators based on mesoporous
structures are not limited to SiO2, and open the possibility to more complex
heterostructures formed by different materials. In addition, we theoretically
investigate the effect of changes in the elastic properties of the mesoporous
matrix on the acoustic resonances, which can be achieved via liquid
infiltration into the pores. This platform constitutes a promising building
block for developing environment-responsive nanosystems for nanoacoustic
sensing and reconfigurable optoacoustic nanodevices based on soft and
inexpensive fabrication methods.

##  2 Experiments and Simulations

###  2.1 Sample Fabrication

The TiO2 thin films are synthesized using the sol-gel method. The ordered
mesoporosity is obtained using the evaporation-induced self-assembly of
surfactants. [25] The fabrication details can be found in reference 29 for
silica MTFs, in which the same approach is employed. Ethanolic solutions of
TiCl4 are first prepared, and then the water is added; to obtain the porosity,
the surfactant Pluronic F127 is finally added to the precursor solution. The
final molar ratio of the sols is TiCl4:H2O:EtOH 1:10:40 to obtain the dense
films and TiCl4:F127:H2O:EtOH 1:0.005:10:40 to synthesize the mesoporous
materials. These sols are used immediately after preparation to deposit the
films on top of clean Si substrates by dip coating.

Figure 1:  (a) Schematics of the mesoporous TiO2 sample design. (b)
Experimental setup of the reflectometric pump-probe setup. An interferometric
measurement can also be implemented, indicated by the gray box. [42, 43] (c)
Reflectivity timetrace of the mesoporous TiO2 sample C. (d) Phononic spectrum
corresponding to the timetrace shown in (c).

The schematics of the studied structure is shown in Fig 1(a). First, a
mesostructured titania film is deposited on the substrate and submitted to a
stabilization process up to 200ºC to consolidate the oxide and stabilize the
mesostructure. [44] Then, a capping layer of dense titania is synthesized and
stabilized with the same treatment. Afterwards, the bilayer is submitted to a
final calcination step of 2 h at 350°C with a 1°C/min heating ramp; this
treatment is performed to eliminate the pores template and generate an
accessible and interconnected mesoporosity. [45] Besides, it leads to a phase
transition from amorphous titania to anatase. [46] After these steps,
homogeneous and crack-free thin films of cm dimensions are achieved.

Afterwards, a $\sim$32 nm-thick nickel cover layer is deposited by vacuum
thermal evaporation using a homemade system. The dense layer is necessary to
avoid nickel diffusion through the mesopores, and the Ni film acts as the
acousto-optical transducer for coherent phonon generation and detection. [47]

Table 1: Structural details of the layers from samples A, B, C and Control. Best fit (nominal) thickness values of the dense TiO2, nickel and mesoporous layers, and the porosity. Sample |  Mesoporous TiO2 |  Dense TiO2 | Ni
---|---|---|---
Thickness (nm) |  Porosity (%) | Thickness (nm)
A | 110 (103) | 48 | 18 (14) | 35 (32)
B | 160 (144) | 40 | 30 (26) | 31 (32)
C | 185 (191) | 44 | 32 (25) | 45 (32)
Control | – | – | 68 (60) | 29.5 (32)

Four samples are fabricated: three structures designed with different
mesoporous layer thicknesses, defined by the dip-coating withdrawal speeds of
1, 2 and 3 mm/s, and one control sample without the mesoporous thin film. The
dense layer withdrawal speed is set constant to 0.2 mm/s. The structures are
respectively labeled as A, B, C, and Control. Table 1 presents the thickness
and porosity parameters, derived from ellipsometry measurements (SOPRA GES5E
ellipsometer). [48]

###  2.2 Coherent phonon generation and detection

The coherent acoustic phonon dynamics is studied via time-domain Brillouin
scattering (TDBS) [49] in a pump-probe setup (see Fig. 1(b)). An incident
pulsed pump laser ($\lambda=$ 758 nm, 200 fs pulse duration, 80 MHz repetition
rate, 275 mW equivalent CW power) is partially absorbed by the Ni transducer
and excites acoustic phonons via photoinduced stress processes. [49] A second
ultrafast laser pulse, namely the probe (typically 5 mW power), delayed with
respect to the pump, detects the instantaneous optical reflectivity modulated
by the coherent acoustic phonons. Two different processes drive these
modulations: the photoelastic interaction, i.e., the changes in the optical
properties of the structure due to strain; and the surface displacement
induced by the presence of phonons. For the first process, the coherent
acoustic phonons induce a change in the index of refraction in the structure
proportional to the strain, which will then modulate the reflectivity of the
probe. For the latter process, surface displacement detection is performed by
implementing a Sagnac interferometer. [42, 43] Further experimental details
can be found in Ref. 29. The reflectivity setup is simpler in terms of optical
alignment. However, depending on the studied materials and layer ordering, the
modulation of the photoelastic properties is weak and the detection of
coherent acoustic phonons becomes impossible. In such cases, the
interferometric setup must be employed. [42]

Figure 1(c) depicts a typical transient reflectivity timetrace in the
interferometric configuration for a TiO2-based mesoporous sample. Coherent
oscillations are visible for time delay <120 ps. They result from longitudinal
coherent phonons modulating the optical properties of the structure. By
performing a Fourier transform, we extract the phononic response in the
frequency domain (Fig 1(d)).

###  2.3 Numerical Simulation

To investigate the acoustic resonances, we simulate the acoustic strain,
displacement, and electric fields by implementing a transfer matrix method to
solve the respective wave equations for a multilayered structure with
contrasting refractive indices and acoustic impedances. [50, 51, 52, 53] By
calculating the normalized optical and acoustic solutions we can simulate the
phonon generation spectrum by integrating the strain, the electric field
square modulus and the photoelastic constant. The phonon detection is then
obtained by an overlap integral of the generation spectrum with the
photoelastic constant, the strain, and the electric field square. [51, 52, 53]
The material parameters used in the simulations are shown in Table 2. The
surface displacement is also simulated by taking the product of the solutions
of the phonon displacement amplitude at the interface between the air and the
last layer of the structure, with the phonon generation spectrum. It is
assumed that the optical absorption and coherent phonon generation are
entirely limited to the nickel transducer layer.

Table 2: Optical and elastic properties of the studied materials for the numerical simulation. For the mesoporous TiO2, the index of refraction is derived from the Bruggemann approximation for a mixture of two dielectric media states [54]: dense TiO2 matrix and the air (pores), according to the porosity on table 1. Material |  Index of refraction |  Speed of sound (m/s) |  Density (g/cm3)
---|---|---|---
TiO2 | 2.56 | 6700 | 2.9
SiO2 | 1.5375 | 5750 | 2.2
Air | 1.00028 | 343 | 1.275E-3
Nickel | 2.218+4.893i | 4970 | 8.908
TiO2 (mesoporous) | (1.76 – 1.90) [54] |  $4355$ |  $2.9$

##  3 Results and Discussion

###  3.1 Surface displacement and photoelastic interaction

The experimental surface displacement spectra of the TiO2-based mesoporous
samples A, B and C, and the control sample without mesoporous layer, are
presented in Fig 2(a). An intense and broad peak is present between 55 GHz and
90 GHz. This peak is associated to an acoustic resonance in the nickel and
dense TiO2 bilayer. [29] At lower frequencies, low amplitude peaks are
resolved, which correspond to coherent acoustic phonons confined in the
mesoporous layer. For comparison, the results for the mesoporous SiO2 sample,
reported in reference 29, are also displayed in Fig 2(a,b), where the acoustic
modes at the mesoporous layer are better resolved. In contrast, the Control
structure, without a mesoporous layer, does not show any resonances below 40
GHz, as expected. The spectrum associated to this sample presents a peak at
84.8 GHz, corresponding to the resonance on metallic and dense layers, with a
dip at 81.2 GHz, probably originating from a destructive interference between
the surface displacement and Brillouin scattering contributions of the signal.

Figure 2:  (a) Experimental and (b) simulated spectra of the surface
displacement of TiO2-based samples A, B, C and Control, and SiO2-based sample
C (Ref. 29), for comparison. The low-frequency-shaded areas (up to 50 GHz)
indicate the first confined modes in the mesoporous layer. (c) Displacement
field of the modes at 13, 25.5 and 38 GHz on TiO2 sample B, respectively
indicated as black, red and blue circles in (b).

The simulation of the surface displacements are shown in Fig. 2(b). A good
agreement between the experimental data and simulation results is achieved.
The interference dip present in the Control sample is not reproduced in the
simulation as Brillouin scattering is not considered simultaneously with the
surface displacement effect. Furthermore, the experimental results of sample C
show a dip at $\sim$ 100 GHz, whereas a peak at the same frequency is present
in the simulation.

In order to get a finer picture of the phononic behavior of these structures,
it is worth comparing the signals -measured and calculated- corresponding to
the surface displacement (interferometry) and modulation of the index of
refraction (reflectometry, photoelastic effect). Figures 3(a) and (b) display
the phononic spectra of both modulation effects for the TiO2 (sample B) and
SiO2 structures, respectively. For both materials, the intense peak is present
in the two cases. However, the lower frequency confined modes are observed
uniquely in the interferometric dataset, in which the first five acoustic
resonances at 12.6, 23.9, 36.9, 47.7 and 57.9 GHz have respective quality
factors of 2.7, 4.7, 4.1, 3.9 and 6.9, within a 10$\%$ error.

Figure 3: Experimental results (black line) and the respective TMM simulations
(red area) on surface displacement and photoelastic interaction for (a) TiO2
sample B and (b) SiO2 sample C, from reference 29. Acoustic modes at the low-
frequency region are only present at the surface displacement spectra for both
TiO2 and SiO2.

Simulations for the surface displacement and the photoelastic interaction for
both structures are displayed as red-shadowed lines in Figs. 3(a) and (b).
They reproduce the main peaks of the experimental datasets for both cases.
Furthermore, the low-frequency modes in the photoelastic interaction
calculation, associated to the mesoporous resonances, are negligible when
compared to the interferometric simulation, in accordance with the
experimental results.

The lack of a photoelastic interaction contribution in both experiment and
simulation supports the hypothesis that the respective vibrational resonances
are confined in the mesoporous layer. Note that the mesoporous material is
transparent for the incident laser wavelength -with an associated weak
photoelastic constant-. For the surface displacement results, the acoustic
modes in the soft mesoporous layer leads the whole structure to vibrate, thus,
contributing to the interferometric detection.

###  3.2 Surface displacement dependence on TiO2 speed of sound

Elastic properties of TiO2 thin films, such as mass density ($\rho$), Young’s
modulus ($Y$) and speed of sound ($v$) depend on several factors such as
fabrication method, layer thickness, annealing temperature and crystallinity.
This dependence leads to a broad range of reported values for $\rho$ (from 2.9
to 3.9 g/cm3) and $Y$ (from 80 to 250 GPa), for TiO2 in its anatase phase.
[55, 56, 57, 58, 59, 60, 61, 62] The speed of propagation of longitudinal
acoustic modes is calculated according to:

$v=\sqrt{\frac{Y(1-\nu)}{\rho(1+\nu)(1-2\nu)}},$ (1)

where $\nu$ is the Poisson ratio. Considering a Poisson ratio of 0.27 for
TiO2, the broad range of values for Y and $\rho$ lead to inaccurate values of
longitudinal acoustic waves speed, spanning from 5500 to 11000 m/s. A precise
determination of these parameters is out of the scope of this work, however a
quantitative analysis is required for the numerical simulations and further
discussion.

We simulate the surface displacement at different TiO2 sound speeds for the
control sample, within the range of 6000 to 10000 m/s, and present the results
on Fig. 4(a), in a colorplot. The spectrum that best matches with the
experiment is simulated with v${}_{TiO_{2}}=$ 6700 m/s, indicated by a dashed
red line. The density used in the simulation is set constant at 2.9 g/cm3. The
TiO2 dense layer parameters are then used as input for the calculation of the
mesoporous samples.

The mesoporous TiO2 layer parameters are extracted according to the layer
porosity and the dense TiO2 constants. The refractive index is calculated
using the Bruggemann approximation of a complex composite by an effective
homogeneous medium for a mixture of two dielectric media states. [54] The mass
density is considered to be the same as the dense layer, [29] and the speed of
sound in the mesoporous material is varied to match with the experimental
results. The evolution of the surface displacement spectrum as a function of
the ratio between the sound velocities of the mesoporous and dense TiO2 layers
for the three samples is shown on the three panels of Fig. 4(b). The dashed
red lines indicate the ratio of the sound velocities for which the simulations
best match with the experimental results ($v_{meso}=4355$ m/s), and the
respective spectra are displayed in Fig. 2(b). They exhibit a good agreement
with the experiment after adjustments on the layers thicknesses according to
the values from Table 1, with a maximum deviation of 11% from the nominal
values for the MTFs. We also consider acoustic losses with an effective phonon
decay length of $\sim 75$ nm. [29, 63]. Considering the values $v_{meso}=4355$
m/s, $\rho=2.9$ g/cm3 and the Poisson ratio for mesoporous materials
$\nu=0.2$, we deduce that the Young’s modulus of the thin films is $\sim 49.5$
GPa. Although this method of deducing Young’s modulus is not standard, the
estimated value is in close agreement with the ones reported in the
literature, between 37 and 50 GPa. [64, 65]

In Fig. 4(b) we observe a clear increase in mode frequency for the low-
frequency modes upon rising the speed of sound in the mesoporous layer. In
contrast, the high-frequency modes are hardly affected. This implies that the
low-frequency resonances are indeed mainly confined within the MTF. It is
worth mentioning that the studied variation of the sound velocity in the
mesoporous layer can be experimentally achieved by liquid infiltration into
the nanopores, which modifies the elastic properties of the effective medium
composed of the TiO2 dense matrix and air in the pores. As chemical adsorption
and capillary condensation are reversible processes, MTFs can be employed as
active elements in optoacoustic sensing devices.

Figure 4: Colorplot of surface displacement amplitude vs frequency and (a)
dense TiO2 speed of sound for control sample, and (b) speed of sound ratio
between mesoporous and dense materials for mesoporous TiO2-based samples. The
dashed lines at V${}_{TiO2}=6700$ m/s and V${}_{meso}/$V${}_{TiO2}=0.65$
correspond to the values that best match with experimental results.

The results exhibited in this section are compatible with what has been
reported on SiO2 mesoporous systems, [29] and reinforce the feasibility of
coherent acoustic phonon generation and detection in different mesoporous
materials. Furthermore, this work reinforces the concept of mesoporous thin
films as potential environment-responsive platforms able to transduce a
physicochemical process (adsorption, capillary condensation) into optical [26,
32, 33] or nanoacoustic signals [29]. This building block represents an
important step towards the engineering of more complex structures based on
such soft materials for practical applications, e.g., sensing.

##  4 Conclusion

In this work we have demonstrated the confinement of gigahertz coherent
acoustic phonons in mesoporous titanium dioxide structures, extending the
concept of mesoporous acoustic resonators to different material systems.
Simulations on photoelastic interaction and surface displacement convey
appreciable agreement with the experimental results. Furthermore, we discussed
the effects of changes in the elastic parameters on the mesoporous resonator
performance. These changes can be induced by chemical compound infiltration.
This concept can be extended and applied to novel sensing devices based on
ultrahigh-frequency resonators, with the mesoporous layer as the active
optoacoustic element. Our findings unlock the way to a promising platform for
nanoacoustic sensing and reconfigurable optoacoustic nanodevices based on
soft, inexpensive fabrication methods.

##  5 Acknowledgment

The authors acknowledge funding by the European Research Council Starting
Grant No. 715939, Nanophennec, the ECOS-Sud Program through the project TUNA-
Phon (PA19N03), and through a public grant overseen by the ANR as part of the
“Investissements d’Avenir” program (Labex NanoSaclay Grant No.
ANR-10-LABX-0035). GJAASI acknowledges ANPCyT for projects PICT 2017-4651,
PICT-2018-04236 and PICT 2020-03130. MCF acknowledges ANPCyT for projects PICT
2015-0351 and 2017-1133.ME acknowledges funding by the University of Oldenburg
through a Carl von Ossietzky Young Researchers’ Fellowship. NDLK and GJAASI
acknowledge funding from CNRS through the project IEA RANas.

##  6 Authors contributions

GJAASI and NDLK proposed the concepts and directed the research. NLA, MCF, AB,
HP, and GJAASI fabricated and characterized the samples. BP and NDLK performed
the picosecond ultrasonics experiments. ERCO, CX, ME, NLA, and NDLK performed
the simulations and analyzed the data.

## References

  * [1] M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg.  Confinement of acoustical vibrations in a semiconductor planar phonon cavity.  Phys. Rev. Lett., 89:227402, Nov 2002.
  * [2] Alexander A. Balandin.  Nanophononics: Phonon Engineering in Nanostructures and Nanodevices.  Journal of Nanoscience and Nanotechnology, 5(7):1015–1022, July 2005.
  * [3] G. Rozas, M. F. Pascual Winter, B. Jusserand, A. Fainstein, B. Perrin, E. Semenova, and A. Lemaître.  Lifetime of THz Acoustic Nanocavity Modes.  Physical Review Letters, 102(1):015502, January 2009.
  * [4] R. P. Beardsley, A. V. Akimov, M. Henini, and A. J. Kent.  Coherent Terahertz Sound Amplification and Spectral Line Narrowing in a Stark Ladder Superlattice.  Physical Review Letters, 104(8):085501, February 2010.
  * [5] Sebastian Volz, Jose Ordonez-Miranda, Mika Prunnila Andrey Shchepetov, Jouni Ahopelto, Thomas Pezeril, Gwenaelle Vaudel, Vitaly Gusev, Pascal Ruello, Eva M.Weig, Martin Schubert, Mike Hettich, Martin Grossman, Thomas Dekorsy, Francesc Alzina, Bartlomiej Graczykowski, Emigdio Chavez-Angel, J. Sebastian Reparaz, Markus R. Wagner, Clivia M. Sotomayor-Torres, Shiyun Xiong, Sanghamitra Neogi, and Davide Donadio.  Nanophononics: State of the art and perspectives.  European Physical Journal B, 89(1):15, 2016.
  * [6] Fabricio Della Picca, Rodrigo Berte, Mohsen Rahmani, Pablo Albella, Juan M. Bujjamer, Martín Poblet, Emiliano Cortés, Stefan A. Maier, and Andrea V. Bragas.  Tailored Hypersound Generation in Single Plasmonic Nanoantennas.  Nano Letters, 16(2):1428–1434, February 2016.
  * [7] F. R. Lamberti, Q. Yao, L. Lanco, D. T. Nguyen, M. Esmann, A. Fainstein, P. Sesin, S. Anguiano, V. Villafañe, A. Bruchhausen, P. Senellart, I. Favero, and N. D. Lanzillotti-Kimura.  Optomechanical properties of GaAs/AlAs micropillar resonators operating in the 18 GHz range.  Optics Express, 25(20):24437, October 2017.
  * [8] Marta De Luca, Claudia Fasolato, Marcel A. Verheijen, Yizhen Ren, Milo Y. Swinkels, Sebastian Kölling, Erik P. A. M. Bakkers, Riccardo Rurali, Xavier Cartoixà, and Ilaria Zardo.  Phonon Engineering in Twinning Superlattice Nanowires.  Nano Letters, 19(7):4702–4711, July 2019.
  * [9] M. Esmann, F. R. Lamberti, A. Harouri, L. Lanco, I. Sagnes, I. Favero, G. Aubin, C. Gomez-Carbonell, A. Lemaître, O. Krebs, P. Senellart, and N. D. Lanzillotti-Kimura.  Brillouin scattering in hybrid optophononic Bragg micropillar resonators at 300 GHz.  Optica, 6(7):854, July 2019.
  * [10] A. Arbouet, N. Del Fatti, and F. Vallee.  Optical control of the coherent acoustic vibration of metal nanoparticles.  The Journal of Chemical Physics, 124(14):144701, April 2006.
  * [11] Kevin O’Brien, N. D. Lanzillotti-Kimura, Junsuk Rho, Haim Suchowski, Xiaobo Yin, and Xiang Zhang.  Ultrafast acousto-plasmonic control and sensing in complex nanostructures.  Nature Communications, 5(1):4042, September 2014.
  * [12] Yannick Guillet, Allaoua Abbas, Serge Ravaine, and Bertrand Audoin.  Ultrafast microscopy of the vibrational landscape of a single nanoparticle.  Applied Physics Letters, 114(9):091904, March 2019.
  * [13] Martin Poblet, Rodrigo Berté, Hilario D. Boggiano, Yi Li, Emiliano Cortés, Gustavo Grinblat, Stefan A. Maier, and Andrea V. Bragas.  Acoustic Coupling between Plasmonic Nanoantennas: Detection and Directionality of Surface Acoustic Waves.  ACS Photonics, 8(10):2846–2852, October 2021.
  * [14] Norberto D. Lanzillotti-Kimura, Kevin P. O’Brien, Junsuk Rho, Haim Suchowski, Xiaobo Yin, and Xiang Zhang.  Polarization-controlled coherent phonon generation in acoustoplasmonic metasurfaces.  Physical Review B, 97(23):235403, June 2018.
  * [15] A. Soukiassian, W. Tian, D. A. Tenne, X. X. Xi, D. G. Schlom, N. D. Lanzillotti-Kimura, A. Bruchhausen, A. Fainstein, H. P. Sun, X. Q. Pan, A. Cros, and A. Cantarero.  Acoustic Bragg mirrors and cavities made using piezoelectric oxides.  Applied Physics Letters, 90(4):042909, January 2007.
  * [16] N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, A. Soukiassian, X. X. Xi, and D. G. Schlom.  Enhancement and Inhibition of Coherent Phonon Emission of a Ni Film in a BaTiO 3 / SrTiO 3 Cavity.  Physical Review Letters, 104(18):187402, May 2010.
  * [17] Thomas Vasileiadis, Jeena Varghese, Visnja Babacic, Jordi Gomis-Bresco, Daniel Navarro Urrios, and Bartlomiej Graczykowski.  Progress and perspectives on phononic crystals.  Journal of Applied Physics, 129(16):160901, April 2021.
  * [18] Yu Cang, Yabin Jin, Bahram Djafari-Rouhani, and George Fytas.  Fundamentals, progress and perspectives on high-frequency phononic crystals.  Journal of Physics D: Applied Physics, 55(19):193002, May 2022.
  * [19] N. D. Lanzillotti-Kimura, A. Fainstein, B. Jusserand, and A. Lemaître.  Resonant Raman scattering of nanocavity-confined acoustic phonons.  Physical Review B, 79(3):035404, January 2009.
  * [20] S. Anguiano, A. E. Bruchhausen, B. Jusserand, I. Favero, F. R. Lamberti, L. Lanco, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, P. Senellart, and A. Fainstein.  Micropillar Resonators for Optomechanics in the Extremely High 19–95-GHz Frequency Range.  Physical Review Letters, 118(26):263901, June 2017.
  * [21] D. L. Chafatinos, A. S. Kuznetsov, S. Anguiano, A. E. Bruchhausen, A. A. Reynoso, K. Biermann, P. V. Santos, and A. Fainstein.  Polariton-driven phonon laser.  Nature Communications, 11(1):4552, December 2020.
  * [22] O. Ortiz, P. Priya, A. Rodriguez, A. Lemaitre, M. Esmann, and N. D. Lanzillotti-Kimura.  Topological optical and phononic interface mode by simultaneous band inversion.  Optica, 8(5):598, May 2021.
  * [23] G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura.  Coherent generation and detection of acoustic phonons in topological nanocavities.  APL Photonics, 4(3):030805, March 2019.
  * [24] N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, L. Largeau, O. Mauguin, and A. Lemaitre.  Enhanced optical generation and detection of acoustic nanowaves in microcavities.  Physical Review B, 83(20):201103, May 2011.
  * [25] G. J. A. A. Soler-Illia and P. Innocenzi.  Mesoporous Hybrid Thin Films: The Physics and Chemistry Beneath.  Chemistry - A European Journal, 12(17):4478–4494, June 2006.
  * [26] Rodrigo Martínez Gazoni, Martín G. Bellino, M. Cecilia Fuertes, Gustavo Giménez, Galo J. A. A. Soler-Illia, and María Luz Martínez Ricci.  Designed nanoparticle–mesoporous multilayer nanocomposites as tunable plasmonic–photonic architectures for electromagnetic field enhancement.  Journal of Materials Chemistry C, 5(14):3445–3455, 2017.
  * [27] N. Gomopoulos, D. Maschke, C. Y. Koh, E. L. Thomas, W. Tremel, H.-J. Butt, and G. Fytas.  One-Dimensional Hypersonic Phononic Crystals.  Nano Letters, 10(3):980–984, March 2010.
  * [28] Giulio Benetti, Marco Gandolfi, Margriet J. Van Bael, Luca Gavioli, Claudio Giannetti, Claudia Caddeo, and Francesco Banfi.  Photoacoustic sensing of trapped fluids in nanoporous thin films: Device engineering and sensing scheme.  ACS Applied Materials & Interfaces, 10(33):27947–27954, 2018.  PMID: 30039696.
  * [29] Nicolas Lopez Abdala, Martin Esmann, Maria C. Fuertes, Paula C. Angelomé, Omar Ortiz, Axel Bruchhausen, Hernan Pastoriza, Bernard Perrin, Galo J. A. A. Soler-Illia, and Norberto D. Lanzillotti-Kimura.  Mesoporous thin films for acoustic devices in the gigahertz range.  The Journal of Physical Chemistry C, 124(31):17165–17171, 2020\.
  * [30] Dirk Schneider, Faroha Liaqat, El Houssaine El Boudouti, Youssef El Hassouani, Bahram Djafari-Rouhani, Wolfgang Tremel, Hans-Jürgen Butt, and George Fytas.  Engineering the Hypersonic Phononic Band Gap of Hybrid Bragg Stacks.  Nano Letters, 12(6):3101–3108, June 2012.
  * [31] Dirk Schneider, Faroha Liaqat, El Houssaine El Boudouti, Ossama El Abouti, Wolfgang Tremel, Hans-Jürgen Butt, Bahram Djafari-Rouhani, and George Fytas.  Defect-Controlled Hypersound Propagation in Hybrid Superlattices.  Physical Review Letters, 111(16):164301, October 2013.
  * [32] Baptiste Auguié, María Cecilia Fuertes, Paula C. Angelomé, Nicolás López Abdala, Galo J. A. A. Soler Illia, and Alejandro Fainstein.  Tamm Plasmon Resonance in Mesoporous Multilayers: Toward a Sensing Application.  ACS Photonics, 1(9):775–780, September 2014.
  * [33] M. C. Fuertes, F. J. López-Alcaraz, M. C. Marchi, H. E. Troiani, V. Luca, H. Míguez, and G. J. A. A. Soler-Illia.  Photonic Crystals from Ordered Mesoporous Thin-Film Functional Building Blocks.  Advanced Functional Materials, 17(8):1247–1254, May 2007.
  * [34] Marc Thelen, Nicolas Bochud, Manuel Brinker, Claire Prada, and Patrick Huber.  Laser-excited elastic guided waves reveal the complex mechanics of nanoporous silicon.  Nature Communications, 12(1):3597, June 2021.
  * [35] S. Suárez, N. Arconada, Y. Castro, J.M. Coronado, R. Portela, A. Durán, and B. Sánchez.  Photocatalytic degradation of TCE in dry and wet air conditions with TiO2 porous thin films.  Applied Catalysis B: Environmental, 108–109:14–21, October 2011\.
  * [36] Abdulmenan M. Hussein, Luther Mahoney, Rui Peng, Harrison Kibombo, Chia-Ming Wu, Ranjit T. Koodali, and Rajesh Shende.  Mesoporous coupled ZnO/TiO 2 photocatalyst nanocomposites for hydrogen generation.  Journal of Renewable and Sustainable Energy, 5(3):033118, May 2013\.
  * [37] N. Harmankaya, J. Karlsson, A. Palmquist, M. Halvarsson, K. Igawa, M. Andersson, and P. Tengvall.  Raloxifene and alendronate containing thin mesoporous titanium oxide films improve implant fixation to bone.  Acta Biomaterialia, 9(6):7064–7073, June 2013.
  * [38] Ianina L. Violi, M. Dolores Perez, M. Cecilia Fuertes, and Galo J. A. A. Soler-Illia.  Highly Ordered, Accessible and Nanocrystalline Mesoporous TiO 2 Thin Films on Transparent Conductive Substrates.  ACS Applied Materials & Interfaces, 4(8):4320–4330, August 2012\.
  * [39] Jia Hong Pan, X.S. Zhao, and Wan In Lee.  Block copolymer-templated synthesis of highly organized mesoporous TiO2-based films and their photoelectrochemical applications.  Chemical Engineering Journal, 170(2-3):363–380, June 2011.
  * [40] Edoardo Baldini, Tania Palmieri, Adriel Dominguez, Pascal Ruello, Angel Rubio, and Majed Chergui.  Phonon-Driven Selective Modulation of Exciton Oscillator Strengths in Anatase TiO 2 Nanoparticles.  Nano Letters, 18(8):5007–5014, August 2018.
  * [41] Edoardo Baldini, Adriel Dominguez, Tania Palmieri, Oliviero Cannelli, Angel Rubio, Pascal Ruello, and Majed Chergui.  Exciton control in a room temperature bulk semiconductor with coherent strain pulses.  Science Advances, 5(11):eaax2937, November 2019.
  * [42] B Perrin, C Rossignol, B Bonello, and J.-C Jeannet.  Interferometric detection in picosecond ultrasonics.  Physica B: Condensed Matter, 263–264:571–573, March 1999.
  * [43] O. Matsuda and O. B. Wright.  Reflection and transmission of light in multilayers perturbed by picosecond strain pulse propagation.  Journal of the Optical Society of America B, 19(12):3028, December 2002.
  * [44] P. C. Angelomé, M. C. Fuertes, and G. J. A. A. Soler-Illia.  Multifunctional, Multilayer, Multiscale: Integrative Synthesis of Complex Macroporous and Mesoporous Thin Films with Spatial Separation of Porosity and Function.  Advanced Materials, 18(18):2397–2402, September 2006.
  * [45] María Cecilia Fuertes, Silvia Colodrero, Gabriel Lozano, Agustín R. González-Elipe, David Grosso, Cédric Boissière, Clément Sánchez, Galo J. de A. A. Soler-Illia, and Hernán Míguez.  Sorption Properties of Mesoporous Multilayer Thin Films.  The Journal of Physical Chemistry C, 112(9):3157–3163, March 2008\.
  * [46] Galo J. A. A. Soler-Illia, Paula C. Angelomé, M. Cecilia Fuertes, David Grosso, and Cedric Boissiere.  Critical aspects in the production of periodically ordered mesoporous titania thin films.  Nanoscale, 4(8):2549, 2012.
  * [47] A. Huynh, N. D. Lanzillotti-Kimura, B. Jusserand, B. Perrin, A. Fainstein, M. F. Pascual-Winter, E. Peronne, and A. Lemaître.  Subterahertz Phonon Dynamics in Acoustic Nanocavities.  Physical Review Letters, 97(11):115502, September 2006.
  * [48] Cédric Boissiere, David Grosso, Sophie Lepoutre, Lionel Nicole, Aline Brunet Bruneau, and Clément Sanchez.  Porosity and Mechanical Properties of Mesoporous Thin Films Assessed by Environmental Ellipsometric Porosimetry.  Langmuir, 21(26):12362–12371, December 2005.
  * [49] Pascal Ruello and Vitalyi E. Gusev.  Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action.  Ultrasonics, 56:21–35, February 2015.
  * [50] C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc.  Surface generation and detection of phonons by picosecond light pulses.  Physical Review B, 34(6):4129–4138, September 1986.
  * [51] Alejandro Fainstein and Bernard Jusserand.  Raman scattering in resonant cavities.  In Manuel Cardona and Roberto Merlin, editors, Light Scattering in Solid IX, pages 17–110. Springer Berlin Heidelberg, Berlin, Heidelberg, 2007.
  * [52] N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, and B. Jusserand.  Theory of coherent generation and detection of THz acoustic phonons using optical microcavities.  Physical Review B, 84(6):064307, August 2011.
  * [53] N. D. Lanzillotti-Kimura, A. Fainstein, A. Lemaitre, B. Jusserand, and B. Perrin.  Coherent control of sub-terahertz confined acoustic nanowaves: Theory and experiments.  Physical Review B, 84(11):115453, September 2011.
  * [54] Tsun Hang Fung, Tom Veeken, David Payne, Binesh Veettil, Albert Polman, and Malcolm Abbott.  Application and validity of the effective medium approximation to the optical properties of nano-textured silicon coated with a dielectric layer.  Optics Express, 27(26):38645, December 2019.
  * [55] Sanjay Gopal Ullattil and Pradeepan Periyat.  Sol-Gel Synthesis of Titanium Dioxide.  In Suresh C. Pillai and Sarah Hehir, editors, Sol-Gel Materials for Energy, Environment and Electronic Applications, pages 271–283. Springer International Publishing, Cham, 2017.
  * [56] Li-Lan Yang, Yi-Sheng Lai, J.S. Chen, P.H. Tsai, C.L. Chen, and C. Jason Chang.  Compositional Tailored Sol-Gel SiO 2 –TiO 2 Thin Films: Crystallization, Chemical Bonding Configuration, and Optical Properties.  Journal of Materials Research, 20(11):3141–3149, November 2005\.
  * [57] C. R. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, and K. Bange.  Young’s Modulus and Density of thin TiO 2 Films Produced by Different Methods.  MRS Proceedings, 436:251, 1996.
  * [58] M C Ferrara, L Pilloni, S Mazzarelli, and L Tapfer.  Hydrophilic and optical properties of nanostructured titania prepared by sol–gel dip coating.  Journal of Physics D: Applied Physics, 43(9):095301, March 2010\.
  * [59] A.O Olofinjana, J.M Bell, and A.K Jämting.  Evaluation of the mechanical properties of sol–gel-deposited titania films using ultra-micro-indentation method.  Wear, 241(2):174–179, July 2000.
  * [60] A Bendavid, P J Martin, and H Takikawa.  Deposition and modification of titanium dioxide thin films by filtered arc deposition.  Thin Solid Films, page 9, 2000.
  * [61] Paulo Soares, Alexandre Mikowski, Carlos M. Lepienski, Emanuel Santos, Glória A. Soares, Vitoldo Swinka Filho, and Neide K. Kuromoto.  Hardness and elastic modulus of TiO2 anodic films measured by instrumented indentation.  Journal of Biomedical Materials Research Part B: Applied Biomaterials, 84B(2):524–530, February 2008.
  * [62] L. Borgese, M. Gelfi, E. Bontempi, P. Goudeau, G. Geandier, D. Thiaudière, and L.E. Depero.  Young modulus and Poisson ratio measurements of TiO2 thin films deposited with Atomic Layer Deposition.  Surface and Coatings Technology, 206(8-9):2459–2463, January 2012\.
  * [63] O. Ortíz, M. Esmann, and N. D. Lanzillotti-Kimura.  Phonon engineering with superlattices: Generalized nanomechanical potentials.  Physical Review B, 100(8):085430, August 2019.
  * [64] Diego F. Lionello, Juan Ignacio Ramallo, Galo J. A. A. Soler-Illia, and María Cecilia Fuertes.  Mechanical properties of ordered mesoporous oxides thin films.  Journal of Sol-Gel Science and Technology, 101(1):114–139, January 2022.
  * [65] Diego F. Lionello, Paula Y. Steinberg, M. Mercedes Zalduendo, Galo J. A. A. Soler-Illia, Paula C. Angelomé, and M. Cecilia Fuertes.  Structural and Mechanical Evolution of Mesoporous Films with Thermal Treatment: The Case of Brij 58 Templated Titania.  The Journal of Physical Chemistry C, 121(40):22576–22586, October 2017.