# Contact line catch up by growing ice crystals

Rodolphe Grivet<EMAIL_ADDRESS>Laboratoire
d’Hydrodynamique (LadHyX), UMR 7646 CNRS-Ecole Polytechnique, IP Paris, 91128
Palaiseau CEDEX, France  Antoine Monier  Sorbonne Université, CNRS, UMR 7190,
Institut Jean Le Rond $\partial$’Alembert, F-75005 Paris, France  Axel Huerre
MSC, Université de Paris, CNRS (UMR 7057), 75013 Paris, France  Christophe
Josserand  Laboratoire d’Hydrodynamique (LadHyX), UMR 7646 CNRS-Ecole
Polytechnique, IP Paris, 91128 Palaiseau CEDEX, France  Thomas Séon  Sorbonne
Université, CNRS, UMR 7190, Institut Jean Le Rond $\partial$’Alembert, F-75005
Paris, France

###### Abstract

The effect of freezing on contact line motion is a scientific challenge in the
understanding of the solidification of capillary flows. In this letter, we
experimentally investigate the spreading and freezing of a water droplet on a
cold substrate. We demonstrate that solidification stops the spreading because
the ice crystals catch up with the advancing contact line. Indeed, we observe
the formation and growth of ice crystals along the substrate during the drop
spreading, and show that their velocity equals the contact line velocity when
the drop stops. Modelling the growth of the crystals, we predict the shape of
the crystal front and show that the substrate thermal properties play a major
role on the frozen drop radius.

††preprint: APS/123-QED

When freezing occurs during capillary flows, such as for droplets impacting
very cold substrates [1, 2] or trickles flowing over subzero substrates [3,
4], the modification of wetting due an ice layer formation leads to the
creation of surprising ice patterns. The interplay between contact line motion
and phase change received increasing attention over the past years [5, 6, 7].
In particular, understanding the dynamics of contact line in the presence of
solidification enables the tuning of the interaction between the ice and the
solid [8, 9] and finds numerous applications over various fields going from
metallurgy [10] to icing/de-icing in aeronautics [11] or inkjet like ice
printing [12].

An important question that arises often in these situations, is the arrest of
a moving contact line because of solidification. Despite numerous studies
looking at the arrest criteria for a moving contact line on a sub-cooled
substrate, the different potential mechanisms are still debated, lacking a
global understanding. They include dissipation increase at the contact line
due to solidification [13, 14], solidification of a critical volume [15], or
crystal growth pinning [16, 17, 18].

In this letter, we investigate experimentally the spreading and arrest of a
water drop deposited on a cooled substrate for temperatures down to $T_{\rm
s}=-25^{\circ}\mathrm{C}$. We observe that, in this temperature range, the
colder the substrate, the smaller the arrest radius. We demonstrate for the
first time that this is due to ice crystals that catch up with the advancing
contact line. Indeed, we observe the ice crystals growing in the drop at the
contact with the substrate while the droplet spreads, and show that the
crystal growth velocity along the substrate equals the contact line advancing
velocity at the time of arrest. Based on the strong heat transfer between the
ice and the substrate, we propose a model for the crystal growth dynamics that
matches our experimental measurements and allows us to predict the shape of
the crystal front. Our study discriminates therefore between the debated
mechanisms behind contact line arrest due to solidification.

In the experiments, a droplet of initial tip radius $r_{\rm 0}=3.1\pm
0.23\,\mathrm{mm}$ is brought in contact with a cold sapphire of temperature
$T_{\rm s}$, thermal diffusivity $D_{\rm s}=11.5\,\mathrm{mm^{2}.s^{-1}}$, and
thickness $5\,\mathrm{mm}$, cooled using liquid nitrogen. The drop-sapphire
contact angle was measured as $\theta\approx 61^{\circ}$. The droplet is
attached below a flat glass plate (see Figure 1 (a)) which is then moved
downward at low velocity until the drop touches the cold sapphire. As the
spreading lasts less than a second, the position and the temperature of the
glass plate can be considered constant during the experiment. To avoid frost
formation, the setup is placed in a humidity controlled box with relative
humidity $H_{\rm r}<3\%$. The droplet initial temperature $T_{\rm d}$ is equal
to the box temperature and constant in all experiments, close to
$10^{\circ}\mathrm{C}$. The spreading dynamics is recorded from two
synchronised high speed cameras at 6,000 fps. One provides a side-view of the
spreading, using a 205mm lens and the other one a top-view, using a x12
Navitar lens. The top-view pictures are recorded under cross-polarised light
to directly observe the crystal growth.

Figure 1: (a) Schematic side-view of the experiment. (b) Temporal sequence of
a droplet spreading on sapphire cooled at $T_{\rm s}=-3.2^{\circ}\,\rm C$
observed from the side. The contact line speed decreases with time until it
gets pinned by solidification as seen in the final picture.

Figure 1 (b) shows a side view recorded time sequence of a droplet spreading
on the cold substrate. After contact with the substrate, the contact line
advances at early times symmetrically and ultimately the spreading stops, with
an arrest radius $r_{\rm a}$. The spreading typically lasts few tenths of
millisecond.

Figure 2 displays the rescaled arrest radius $r_{\rm a}/r_{\rm 0}$ plotted
against the undercooling $\Delta T=T_{\rm m}-T_{\rm s}$ where $T_{\rm
m}=0^{\circ}\rm{C}$ is the melting temperature of the ice. It shows a clear
decrease of the maximum spreading radius as $\Delta T$ increases. For some
experiments, the arrest radius is not the same on the left and the right sides
of the image, an effect represented with the error-bars on the graph.
Noticeably, the arrest radius decreases by a factor 3 over the range of
temperatures explored. The inset of Figure 2 displays, in a log-log scale, the
evolution of the rescaled radius $r(t)/r_{\rm 0}$, with the time normalized by
the inertial-capillary time $\tau_{\rm c}=\sqrt{\frac{\rho\,r_{\rm
0}^{3}}{\gamma}}\approx 21\,\rm{ms}$, for three undercoolings. It confirms
that the substrate temperature has a strong effect on the arrest radius (see
horizontal dashed lines $r_{\rm a}/r_{\rm 0}$), and shows that the spreading
dynamics itself is neither affected by $\Delta T$ nor by the solidification
[16].

In the first stage of the drop spreading, inertia resists to the liquid motion
and the evolution of the droplet radius follows :

$\frac{r(t)}{r_{\rm 0}}=C\left(\frac{t}{\tau_{\rm c}}\right)^{\alpha}$ (1)

where the theoretical prediction gives $\alpha=1/2$ [19], valid for both
wetting and partially wetting substrates at short times [20].

The inset of Figure 2 shows that (i) the spreading dynamics observed on cooled
substrates can be well modelled by this law until the time of pinning, (ii)
the fitted coefficients $C$ and $\alpha$ are independent of the substrate
temperature, and (iii) the exponent $\alpha=0.47\pm 0.05$ agrees with the
theoretical prediction (see also Supplementary Materials). These results are
in agreement with former studies realised with other liquids and smaller
temperature range [16, 18].

Figure 2: Rescaled arrest radius $r_{\rm a}/r_{\rm 0}$ as a function of the
undercooling $\Delta T$. Error bars are computed based on spreading asymmetry.
The circular marker represents the isothermal case. The inset shows a log-log
temporal evolution of the rescaled spreading radius. Before pinning, all
curves superimpose and the coefficients fitted from Eq 1 are ${C}=1.15\pm
0.06$ and $\alpha=0.47\pm 0.05$. The arrest diameter $r_{\rm a}/r_{\rm 0}$, in
dotted lines ( ) is determined at the time where the curve departs from the
power-law behaviour. Figure 3: (a) Temporal evolution of the spreading at
$-13^{\circ}\mathrm{C}$. Nucleation is designated by white arrows. Local
pinning events are emphasised with small arrows at $t=3.7$ ms. (b) Final
contact line shapes for different temperatures.

To understand the mechanism responsible for the arrest of the contact line and
its dependence with the substrate undercooling, we performed direct
visualisations of the ice growth and its interaction with the spreading front.
Figure 3 shows top views of the experiment under cross-polarised light,
enabling a clear visualisation of the crystal growth, as seen on the sequence
of Figure 3 (a), and on the final shape of the deposits (Figure 3 (b)). The
temporal evolution can be divided into four phases. First, the drop spreads
without ice nucleation: the contact line is circular (t$<$0.8 ms). Then,
nucleation happens within the wetted disc (t=0.8 ms), and nuclei grow without
affecting the contact line shape nor the dynamics (t$<$3.7 ms). Third, the
contact line gets locally pinned (t=3.7 ms), either by nucleation of crystals
very close to it, as it is the case at the lower part of the drop, or by
growing crystals catching it, as seen on the right of the drop between 2.7 and
3.7 ms. Here it seems that pinning occurs when the growing crystals catch the
contact line. Finally, the contact line is entirely stopped when pinned on its
full perimeter by the crystals (t=24 ms). Then, the ice grows perpendicular to
the substrate in the droplet bulk.

The final contact line shapes of the deposits, displayed on Figure 3 (b), show
a global decrease of the frozen drop radius with lower substrate temperatures,
as seen on Figure 2. These pictures also enable to see clearly the crystal
shape when the drop stops spreading. In Figure 2, $r_{\mathrm{a}}$ is the mean
radius determined from the side views, and larger error-bars at intermediate
undercoolings are a consequence of the symmetry breaking in the contact line
shape. Other definitions for the arrest radius have also been tested, such as
the equivalent radius determined from the area of the final deposit, or the
radius of the first pinned point of the contact line, showing however no
qualitative change in the results (see Supplementary Materials).

From the time-sequences of Figure 3 (a), the crystal-growth radial velocity
$V_{\rm c}$ can be extracted for different temperatures of the substrate.
$V_{\rm c}$ is observed to be constant with time as shown in the Supplementary
Materials. In a typical experiment, around 10 crystals are observed and their
velocity is the same with small variations that can be attributed to thermal
fluctuations within the water or the substrate. Thus, for each temperature, a
single crystal-growth radial velocity $V_{\rm c}$ is measured with the
associated relative error.

Similarly, the arrest velocity $V_{\rm a}$, the velocity of the contact line
just before stopping, can be determined from the side views, taking the
derivative of Eq. (1) at the time of arrest $t_{\rm a}$, and using the fitted
values of $C$ and $\alpha$ for each experiment:

$V_{\rm a}=\frac{r_{\rm 0}C\alpha}{\tau_{\rm c}^{\alpha}}\,t_{\rm
a}^{\alpha-1}=\frac{r_{\rm 0}C\alpha}{\tau_{\rm c}}\,\left(\frac{r_{\rm
a}}{Cr_{\rm 0}}\right)^{\frac{\alpha-1}{\alpha}}.$ (2)

As illustrated in the inset of Figure 2, the transient time from spreading to
arrest is very short, allowing a clear definition for the time of arrest
$t_{\rm a}$ and its corresponding arrest velocity. The remaining uncertainty
on $t_{\rm a}$ leads to error bars smaller than the ones induced by the
asymmetry of the arrest. In Figure 4 (a), the arrest velocity $V_{\rm a}$ is
plotted as a function of the crystal growth velocity $V_{\rm c}$ for a large
range of temperatures shown with a color bar. The dashed line represents the
line $V_{\rm a}=V_{\rm c}$ and it is striking to notice that the arrest
velocity is always roughly equal to the crystal growth velocity. This confirms
our hypothesis on the mechanism responsible for the arrest of a contact line
catched up by ice crystals: the contact line has to slow down to the crystal
velocity to get caught by the crystal. Strictly speaking, this provides an
upper limit for the arrest velocity. Hence, the pinning occurs when $V_{\rm
a}\lesssim V_{\rm c}$. Consequently, we observe smaller deposits when $\Delta
T$ increases (Figure 3(a)): the contact line is caught earlier by the crystal
as the crystal velocity increases with the undercooling.

Noticeably, in this scenario, the nucleation process plays a small role on the
arrest criterion but could explain the small deviations observed, namely that
the arrest velocity always seems to be slightly smaller than the crystal one
(Figure 4 (a)). Once the velocity criterion is met, a ”lag time” would still
be necessary so that the contact line can be caught up by the crystals.
Qualitatively, we often see that the contact line is first arrested by
crystals nucleating very close to it and hence catching it quickly when the
velocity criterion is met. This lag becomes negligible at high undercoolings,
where nucleation is very dense and uniformly distributed on the surface [18,
21]. This could explain the small influence of the nucleation rate on the
arrest mechanism.

Figure 4: (a) Arrest velocity against crystal growth velocity. Colours
indicate the surface temperature. The inset shows a sketch of the model of ice
crystal growth near a substrate. (b) Arrest and crystal growth velocities as a
function of the substrate undercooling. The dashed line represents the
dentritic growth velocity and the plain one the model we developed. The inset
shows that the tip radius of the crystal parabola increases linearly with
$\Delta T$.

Thanks to the visualisations, we can now discuss the measured crystal growth
velocities. As can be seen on Figure 3 (a), part of the liquid is not
immediately solidified after contact, and subsequently gets cooled by the
substrate. We can assume that the liquid temperature close to the substrate is
the so-called contact temperature that arises when two bodies of different
temperature are suddenly in contact $T_{\rm c}=T_{\rm s}+(T_{\rm d}-T_{\rm
s})(1+e_{\rm s}/e_{\rm w})$, where $e_{\rm s}$ and $e_{\rm w}$ are the
effusivities of the substrate and water respectively, $e_{\rm
k}=\sqrt{\lambda_{\rm k}\rho_{\rm k}C_{\rm p,k}}$ [22]. In such configuration,
the temperature in the water varies on a typical diffusive lengthscale
$\delta\approx\sqrt{D_{w}\tau_{c}}\approx 10^{-5}$ m, with $D_{w}$ the heat
diffusion coefficient in water. $\delta$ is thus far larger than the
characteristic size of an ice nucleus (few nm) and of the tip radius of an ice
dendrite growing in water at supercooled temperature $T_{c}$ (few $\mu$m) for
our range of substrate temperatures [23]. This allows us to approximate the
temperature of the water surrounding the ice crystals as $T_{\rm c}$.

When ice grows in supercooled water, the ice formation occurs through a series
of nucleation and growth of crystals [24]. In the liquid bulk, the growth
velocity is constant [25] but the mechanism that sets the velocity is not
completely understood. Indeed, at large undercoolings, despite a good
agreement between the experimental data available, theoretical models hardly
catch the growth dynamics [26, 27]. More precisely, the work of Shibkov et al.
[27] on ice growing freely in an infinite supercooled bulk provides a strong
set of experimental data, and fitting a power-law over their whole
experimental range gives $V_{\rm d}=K(T_{\rm m}-T_{\rm w})^{\gamma}$, $T_{\rm
w}$ being the liquid undercooling, with $K=4.83\times 10^{-5}$ S.I and
$\gamma=2.78$ for the growth velocity, consistent with other studies [28, 17].
This experimental fit is shown with the grey dashed line on Figure 4 (b) using
$T_{\rm w}=T_{\rm c}$. In this Figure, our velocities $V_{\rm a}$ (black
diamonds) and $V_{\rm c}$ (blue dots) are represented as a function of $\Delta
T$. Our crystal growth velocities are systematically higher than the above
power-law accounting for the ice growth velocity in liquid bulk. In fact,
several differences can be pointed out between these two experimental
configurations, among which the heterogeneity of the water temperature, the
motion in the surrounding fluid (however the two effects should lead to a
decrease of the Shibkov prediction), and finally the presence of a cold
substrate in the present case.

The effect of the latter was recently explored [29, 30] [31] showing a strong
coupling between the advancing crystal and the substrate. Indeed, latent heat
due to solidification is also transferred to the substrate and in the case of
a very effusive substrate ($e_{\rm s}>e_{\rm i}$) a higher solidification rate
was clearly observed experimentally. To rationalize our experimental
observations, we propose to enrich the model originally proposed by Schremb et
al. [29], which fails to reproduce our data on the whole range of investigated
temperatures. Following their approach, we assume that the vertical growth of
the ice is well accounted by the 1D solidification problem, as illustrated on
the inset of figure 4 (a). However, here, we take into account the thermal
diffusion in all three phases (water, ice and substrate), and obtain that the
growth of the ice front obeys a square-root law in time $h=\sqrt{D_{\rm
eff}\Delta t}$ where $\Delta t$ is the time of solidification. $D_{\rm eff}$
depends on the substrate and water temperatures and is deduced from the self
similar solution of the coupled heat equations for the three phases (see Sup.
Mat.), generalising previous results [32, 33]. Using this square-root law
solution and knowing that the crystal growth radial velocity is constant, we
obtain: $h(x,t)=\sqrt{D_{\rm eff}(t-\frac{x}{V_{\rm c}})}=\sqrt{D_{\rm
eff}\frac{x_{\mathrm{tip}}-x}{V_{\rm c}}}$. Subsequently, the shape of the
crystal is a parabola and its tip radius is expressed as $R_{\rm
tip}=\frac{D_{\mathrm{eff}}}{2\,V_{\rm c}}$.

Measuring $V_{\rm c}$ and calculating the theoretical value of
$D_{\mathrm{eff}}$, we can compute $R_{\rm tip}$ for each of our experiments,
plotted in the inset of Figure 4 (b) as a function of the undercooling. By
contrast with former studies, where $R_{\rm tip}$ was found constant [29, 30]
or with dendritic growth where the tip radius decreases with the undercooling,
we obtain a clear linear dependence $R_{\rm tip}=k\Delta T$, with
$k=2.10^{\mathrm{-8}}\,\rm m/K$. Using this fitted law for $R_{\rm tip}$ in
the formula above relating the tip radius and $V_{c}$, we obtain
$V_{c}=\frac{D_{\rm eff}}{2k\Delta T}$ corresponding to the green plain curve
on Figure 4 (b) which, by construction, shows a very good agreement with our
experiments. The linear law for $R_{\rm tip}$, together with our refined
model, allows a good understanding of our results over the whole temperature
range.

In conclusion, when a drop spreads on a cold substrate, its decelerating
contact line is eventually caught up and arrested by ice crystals growing at
the substrate-water interface. The race between the spreading line and the
solidification front determines an arrest criterion, qualitatively similar to
the ones proposed in previous studies [16, 17, 18]. However, the new
experiments proposed here, on water and at large undercoolings, provide a
direct visualisation of the growing crystals, and the model reveals the role
of the substrate thermal properties on the arrest. Finally, the physical
mechanism leading to the pinning of the contact line when reached by the
crystal still needs to be elucidated. Several theories have already been
proposed [15, 13, 14] and this phenomenon, which takes place at very small
time and length scales, should be the object of future experimental
investigations.

###### Acknowledgements.

We thank the Direction Générale de l’Armement (DGA) for financial support.
R. Grivet and A. Monier contributed equally.

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# FIBER-OPTIC VECTOR ACOUSTIC RECEIVER BASED ON ADAPTIVE HOLOGRAPHIC
INTERFEROMETER

R.V. Romashko<EMAIL_ADDRESS>Institute of Automation and Control
Processes, Far Eastern Branch, Russian Academy of Science, Vladivostok 690041,
Russia  Far Eastern Federal University, Vladivostok, Russia  D.V. Storozhenko
Institute of Automation and Control Processes, Far Eastern Branch, Russian
Academy of Science, Vladivostok 690041, Russia  M.N. Bezruk  Institute of
Automation and Control Processes, Far Eastern Branch, Russian Academy of
Science, Vladivostok 690041, Russia  D.A. Bobruyko  Institute of Automation
and Control Processes, Far Eastern Branch, Russian Academy of Science,
Vladivostok 690041, Russia  Yu.N. Kulchin  Institute of Automation and Control
Processes, Far Eastern Branch, Russian Academy of Science, Vladivostok 690041,
Russia  Far Eastern Federal University, Vladivostok, Russia

###### Abstract

A mobile scalar-vector acoustic receiver is proposed, experimentally
implemented and investigated. The key components of the receiver are (i) the
six-channel fiber-optic coil-type sensor configured as to detect three
projections of acoustic intensity vector, (ii) the six-channel optical phase
demodulator based on six-channel adaptive holographic interferometer
configured with use of dynamic holograms multiplexed in a photorefractive
crystal of cadmium telluride and (iii) the signals recording ADC-based system
combined with software package for data processing. Field tests of the
developed receiver applied for obtaining scalar and vector parameters of
acoustic waves generated by a stationary and moving acoustic source in open
air and water area are carried out. Experimental results show perceptiveness
of use of the fiber-optical adaptive interferometry system for bearing of weak
acoustic sources in real conditions.

##  I Introduction

Acoustic monitoring of sea areas, observation of the underwater situation,
control of underwater vehicles, including of unmanned-type and a number of
other applied problems require highly sensitive means of registering weak
hydroacoustic signals Howe  _et al._ (2019); Venkatesan  _et al._ (2018). Such
problems are usually solved by using both classical hydroacoustic sensors
based on electrical transducers (piezoelectric, electrodynamic, capacitive,
etc.) and currently developing fiber-optic sensors Bjørnø (2017); Udd (1995).
Using interferometry principles the for fiber-optic sensors signal processing
makes it possible to detecting ultra-weak signals due to potentially high
sensitivity of optical interferometer Wagner and Spicer (1987); Abbott and et
al. (2016); Beverini and et al. (2019); A.Gordeev  _et al._ (2019). In
addition, methods of adaptive interferometry based on two-wave mixing at
dynamic hologram recorded in a photorefractive crystal (PRC) make it possible
to effectively stabilize the operating point of the optical interferometer
without any additional stabilization procedures Kamshilin and Petrov (1985);
Stepanov (1991); Shandarov and et al. (2008). Such technique allows to
successfully apply the adaptive interferometers for measuring weak signals
under the conditions of influence of external noise factors such as random
mechanical impacts, industrial noisy sound and vibrations, temperature or
pressure drift, etc. Kamshilin  _et al._ (2009); Romashko  _et al._ (2008,
2014a, 2016a); Stepanov  _et al._ (2007); Girolamo  _et al._ (2007);
Bryushinin and et al. (2020). Adaptive interferometers were being already
applied for acoustic and hydroacoustic measurements Romashko and et al.
(2015); Romashko  _et al._ (2016b); Kulchin and et al. (2020); Bashkov and et
al. (2017); Romashko  _et al._ (2014b), including use of scalar-vector methods
for detection of an acoustic signal Romashko  _et al._ (2020); Romashko and et
al. (2021). In the paper Romashko and et al. (2021), the adaptive laser
scalar-vector hydroacoustic measuring system which provides determination of
full vector of acoustic intensity is proposed. However experimental study of
this system was carried out in a narrow space (water tank) which dimensions
are comparable with acoustic wavelength, and only for one frequency of sound.
These conditions cannot exclude distortion of the obtained experimental
results caused by influence of sound reflections from the water boundaries.
Therefore, the applicability of the proposed system for acoustic bearing
requires a confirmation.

In this work, a mobile scalar-vector acoustic receiver based on six fiber-
optic coil type acoustic sensors and six-channel adaptive interferometer using
dynamic holograms multiplexed in a photorefractive crystal is developed and
investigated. The optimized construction of the fiber-optic acoustic sensors
has allowed increasing the receiver sensitivity up to 51 V/Pa and reducing the
detection threshold down to 0.5 mPa. The performance of the developed receiver
has been studied in the field tests on detection and measurement of weak
hydroacoustic signals in a marine area and acoustic signals in an open air.
The obtained results confirm the possibility of reliable bearing an acoustic
source in a real environment which is characterized by industrial vibrations,
artificial and natural acoustic noises, wind, sea waves, etc.

##  II The receiver architecture

The developed acoustic receiver consists of two main blocks: (i) the six-
channel fiber-optic sensor which provides primary registration of the acoustic
signal, and (ii) the registering system for demodulation, recording and
processing of optical signals from the sensor (Fig. 1). The fiber-optic sensor
comprises six spatially spaced coil-type sensing elements. The base of each
element is fabricated from XPS extruded polystyrene and has cylindrical shape
with the diameter of 65 mm and the height of 25 mm. A multimode optical fiber
with a core diameter of 62.5 $\mu$m, numerical aperture $NA$ = 0.22 and length
of 10 m is wound on the cylinder (30 winds). The impact of acoustic pressure
on the coil material causes its stretching deformation which, in turn, leads
to phase modulation of the light passing through the optical fiber wound on
the cylinder. The six coils are arranged on a metal frame in three pairs in
such way that each pair is oriented along one of the three orthogonal axes of
space (Fig. 1). This arrangement of the coils of the fiber-optic sensor
ensures that all three components of the acoustic pressure gradient are to be
measured Romashko and et al. (2021). The distance between coils for each pair
determines not only the overall dimension of the sensor, but also the
operating frequency range of the receiver. According to the recommendations of
Ref. Gade (1985), the acoustic signals with wavelength in the range 4d0÷10d0
can be correctly detected and processed by means of a vector receiver. The
distance between coils, d0, for the developed sensor is amounted to 25 cm.
Consequently, the wavelength range is 1÷2.5 m. It corresponds to the frequency
range of 130÷340 Hz for air and 600÷1000 Hz for aquatic environment.

The registering system includes a six-channel adaptive holographic
interferometer, a six-channel ADC and PC (Fig. 1). The adaptive interferometer
performs phase demodulation of light waves coming from the fiber-optic
sensors. The interferometer is based on a photorefractive crystal (CdTe:V) of
(100)-cut where six dynamic holograms associated with six measuring channels
are continuously recorded in the orthogonal geometry of vector two-wave mixing
Romashko and et al. (2021). The dynamic hologram operates as an adaptive beam
combiner and provides stabilization of the interferometer operating point.
This allows an interferometer to adapt itself to all low-frequency noises and
measure the weak physical quantities (acoustic signals, etc.) in out-of-
laboratory conditions which are characterized by accidental mechanical
impacts, industrial noises, temperature or pressure drift etc. It is worth
noting that the dynamic holograms are recorded in orthogonal geometry of
vectorial two-wave mixing where all signal waves from optical fibers are
directed to the photorefractive crystal orthogonally to the reference wave
Girolamo  _et al._ (2010). Such geometry provides independent operation of all
the measuring channels of the adaptive interferometer, and excludes a
crosstalk between the channels Romashko and et al. (2012); Girolamo  _et al._
(2008).

All parts of the registering system are packaged into the portable waterproof
box and supplied from the power battery with capacity of 3 A·h which provides
autonomous operation of the whole system during 3 hours. The fiber-optic
sensor is connected with the adaptive interferometer via optical patch cords
with length of 30 m. Due to high coherence of the laser used in the receiver
(Nd:YAG laser, $\lambda$ =1064 nm, optical power 1000 mW, SLM, coherence
length 300 m), the distance between the fiber-optic sensor and the
registration system can be extended up to 100 m if necessary.

Figure 1: Diagram of the mobile fiber-optic scalar-vector acoustic receiver on
the base of an adaptive holographic interferometer.

The recorded signals from the six photodetectors PD1-PD6 which measure the
intensity of light beams in six interferometer channels are fed, through a DC-
blocking filter, to the input of a multichannel amplifier with a controlled
gain. Then each signal is digitized by a 12-bit ADC with a sampling rate of 57
kHz per a channel and fed to the processing software. At the first stage of
signal processing the algorithm performs signal filtering and Fourier
transform. The second stage, the scalar and vector parameters of the acoustic
field, in particular the acoustic intensity vector, are calculated for
selected frequency components. The mathematical procedure for the calculations
is described in details in work Romashko and et al. (2021). Single measurement
and processing cycle takes time 0.2÷0.6 sec.

Here we also applied the two-stage calibration of the developed receiver which
was proposed in the paper Romashko and et al. (2021). Such calibration allows
compensating the inevitable difference in the adaptive interferometer channels
sensitivity caused by the inhomogeneity of a PRC properties over its volume,
different light wave amplitudes at the output from the fiber-optic splitter,
difference in the coils geometry, length of wound fiber in a fiber-optic
sensing elements, etc.

In contrast to the adaptive hydrophone system presented in the work Romashko
_et al._ (2014b), the size of fiber-optic sensor coils was increased by 30%,
the length of optical fiber wound on the coils was increased from 5 to 10 m.
This results in increasing the sensitivity of the receiver from (6.7 $\pm$
0.1)rad/Pa or (29 $\pm$ 0.12)V/Pa up to (11.7 $\pm$ 0.2)rad/Pa or (51 $\pm$
0.19)V/Pa. In addition, the use of a low-noise amplifier made it possible to
improve the signal-to-noise ratio (SNR) of the sensor. Such optimization of
the system has provided a reduction of the minimum detectable acoustic
pressure from 3 mPa down to 0.5 mPa.

##  III Testing the vector receiver on an open air

The developed vector acoustic receiver was tested in an air environment on an
open area. Diagram of the experiment arrangement is shown in Fig.2. The fiber-
optic sensor was fixed on a rod at the height of 2 m from the ground. The
acoustic source generating a harmonic acoustic signal at the frequency of 340
Hz and sound level 40dB (re 20 µPa) was moving along a circle trajectory with
radius of 5 m around the rod located in the center with variable angular
velocity (in the range of 5÷10 deg/s). In the experiment, the trajectory plane
was set at three different levels of height from the ground: (1) h1 = 2 m (at
the same height with the sensor), (2) h2 = 1 m (below the sensor), and (3) h3
= 3 m (above the sensor).

Full turn of the acoustic source along the circle was completed during 1 min.
The recorded signal was processed by the software algorithm that calculated
the acoustic intensity vector I inside the time interval of 0.6 sec. Thus, for
a single pass of the source, a set of 95 values of the vector was obtained.
Fig. 2b shows the temporal evolution of orthogonal components of the vector I.
As seen, the z-components of the vector I obtained for three height levels
(I1z, I2z, and I3z) were kept constant during the movement and correspond to
constant heights of the acoustic source. In contrast, the x- and y-components
(I1x and I1y) were changed in the way close to harmonic dependency which
corresponds to the trajectory of the source movement along the circle. It
worth to note that deviation I1x,y(t) from harmonic functions is related with
non-constant speed of the source movement (it accelerates in the time interval
of 45÷55 samples and slowdown in the time interval of 65÷85 samples).

Figure 2: Experimental results of testing the fiber-optic scalar-vector
receiver on an open air: (a) arrangement of the receiver position and the
acoustic source trajectories (1, 2, 3) accompanied by the bearing vectors D
(dashed lines) obtained with use of the receiver for three levels of the
source trajectory at heights h1 (black), h2 (blue) and h3 (red); (b) temporal
evolution of the selected orthogonal components of the acoustic intensity
vector. t1 = 25th sample, t2 = 42th sample, t3 = 62th sample.

Bearing to the acoustic source is determined by the unit vector inverse to the
vector of the acoustic intensity D=-I/—I—. Fig. 2a shows the scheme of the
experimental arrangement with the trajectories of the acoustic source and the
obtained bearing vectors D directed the to the source positions. In
particular, the continuations of the vectors D built at time moments t1 = 25,
t2 = 42 and t3 = 62 are traced to the corresponding trajectory of the source
motion. As seen, the angular displacement of the source in the interval
between t2 and t3 is twice greater than the displacement between t1 and 2,
which is related with increase of the source speed mentioned above. The
maximum root-mean-square (RMS) deviation of the azimuthal angle of direction
to the source did not exceed 3.5o. Thus, the fiber-optic scalar-vector
receiver can provide reliable real-time obtaining of the direction to a signal
source from the measured acoustic intensity vector at open air.

##  IV Testing the vector receiver in a seawater area

The field tests of the developed scalar-vector acoustic receiver applied for
the problem of obtaining the direction to a moving source of a hydroacoustic
signal were carried out in a sea area. The diagram of the experiment is shown
in Fig. 3. The acoustic source being submersed into sea water at the depth of
0.5 m was linearly displaced at the constant speed 0.2 m/sec along a straight
line perpendicular to the coastline from point A1 to point A2 at the distance
of 12 m. The source emits a periodic acoustic signal with the amplitude of 30
mPa at the frequency of 600 Hz and amplitude modulation at the frequency of
200 Hz. The fiber-optic sensor was located stationary at the depth of 6 m, at
the distances of 0.5 m from the bottom, 30 m from the coastline, and 4 m from
the trajectory of the source movement. The spatial orientation of the receiver
was chosen in such a way that the z-axis was perpendicular to the sea bottom,
while the x-axis makes the angle of 20o with the perpendicular to the
coastline.

Figure 3: Diagram of field test of the fiber-optic vector acoustic receiver in
seawater area.

The output signal of the receiver was continuously recorded during the full
time of the acoustic source movement ( 1 min). Fourier spectrum of the
received signal is shown in the inset of Fig.4(a). As seen it contains three
discrete frequency components – at central, sum and difference frequencies,
600 Hz, 800 Hz and 400 Hz, respectively. For the calculation of the acoustic
intensity vector, only two frequency components (600 and 800 Hz) were used,
since the rest one (400 Hz) is out of the range which is appropriate for the
correct determining of vector parameters (600÷1000 Hz) as was explained above.

Whole measurement time for each discrete frequency component was subdivided
into the intervals of 0.5 sec. Fig.4(a) shows the temporal evolution of all
orthogonal components of the acoustic intensity vector obtained at frequencies
of 600 Hz and 800 Hz. It worth to note that the behavior of the horizontal
components of the vector (Ix and the Iy) is similar for both frequencies.
Contrary, the vertical component (Iz) obtained at the frequency of 600 Hz
differ from one obtained at the frequency of 800 Hz. This can be caused by
sound reflection from the bottom and the surface in condition of shallow sea
(the depth of 6.5 m) which affects the acoustic intensity in the z-direction
differently at different frequencies.

Figure 4: The temporal evolution of the orthogonal components of the acoustic
intensity vector (a) and the directional angles together with the average
acoustic pressure (b) obtained at the frequencies 600 Hz and 800 Hz during the
acoustic source movement. The inset at the upper figure shows the Fourier-
spectrum of the received acoustic signal.

The bearing vector D being calculated in the same way as in the previous
section was used to obtain both the azimuthal angle, $\phi$, (between D and
y-axis) and the zenith angle, $\theta$, (between D and the z-axis). Fig. 4(b)
shows the temporal evolution of $\phi$ and $\theta$ accompanied by evolution
of the average acoustic pressure, ⟨p⟩, measured at both frequencies (600 and
800 Hz). As seen the level of the ⟨p⟩ defere for these frequencies by more
than 6 folds and amount in the maximum 9 mPa and 1.5 mPa, respectively, while
the background acoustic noise is at the level of 0.2 mPa. It is worth to note
that, in spite of worse SNR at 800 Hz (17.5 dB) in comparison with SNR at 600
Hz (33 dB), the angles $\phi$ and $\theta$ can still be determined by means of
the fiber-optic scalar-vector receiver with the RMS 20o and 3.5o obtained at
the side and the central frequencies, respectively. As seen, the zenith angle,
$\theta$, obtained at the frequency of 600 Hz was remained unchanged within
some fluctuations during the entire measurement period. The deviation of the
angle $\theta$ obtained at the frequency of 800 Hz in the period of 65÷90
samples is associated with the interference of the initial acoustic wave with
the wave reflected from the bottom.

Figure 5: Experimentally obtained bearing vectors (black) and their
continuations (blue) to the acoustic source moving along the linear trajectory
(red). Points 1, 2, 3 and 4 correspond to time moments of 40th, 60th, 70th and
75th samples, respectively.

Fig. 5 shows 3D view of the experimental space arrangement which contains the
stationary fiber-optic sensor and the acoustic source moving along the
trajectory A1–A2. The bearing vectors D obtained in the experiment at the
central frequency (600 Hz) are shown by black lines. Part of them (obtained at
time moments of 40th, 60th, 70th, 75th samples) are prolonged so that they
intersect with the trajectory of the source movement. The intersection points
coincide with the locations of the source at the corresponding time moments
with appropriate accuracy. It is clear that correctness of acoustic source
bearing depends on acoustic conditions in a water area. In case of shallow sea
when a reflection from water surface and bottom becomes significant, the
registering the of acoustic signals at different frequencies and following
application of averaging procedure can provide improvement of the bearing
accuracy.

##  V Conclusion

In this work, we developed the mobile fiber-optic scalar-vector acoustic
receiver based on adaptive holographic interferometer. The receiver possesses
high sensitivity to acoustic pressure at the level (11.7 $\pm$ 0.2) rad/Pa or
(51 $\pm$ 0.19) V/Pa and provides a detection of weak acoustic signals with
pressure above 0.5 mPa. Field tests of the developed receiver applied for
obtaining scalar and vector parameters of acoustic waves generated by a
stationary and moving acoustic source in open air and water area are carried
out. Experimental results show perceptiveness of use of the fiber-optical
adaptive interferometry system for bearing of weak acoustic sources in real
conditions with accuracy not worse than 3.5o. The use of the developed
acoustic receiver on the base of adaptive interferometer opens up new
possibilities for solving problems related to the determination of the energy,
interphase, coherent and probabilistic properties of acoustic field which, in
turn, can become a basis for the new effective detection algorithms and
methods both for precise bearing of weak acoustic sources and their
classification.

##  VI Acknowledgements

The research is supported by the Russian Science Foundation (grant No.
19-12-00323).

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