# Conoscopic interferometry for optimal acoustic pulse detection in ultrafast
acoustics

Martin Robin<EMAIL_ADDRESS>Ruben Guis  Mustafa Umit Arabul  Zili
Zhou  Nitesh Pandey  Gerard J. Verbiest<EMAIL_ADDRESS>Department of
Precision and Microsystems Engineering, Delft University of Technology,
Mekelweg 2, Delft, 2628CD, Netherlands  ASML Netherlands B.V., De Run 6501,
Veldhoven, 5504DR, Netherlands

###### Abstract

Conoscopic interferometry is a promising detection technique for ultrafast
acoustics. By focusing a probe beam through a birefringent crystal before
passing it through a polarizer, conoscopic interferences sculpt the spatial
profile of the beam. The use of these patterns for acoustic wave detection
revealed a higher detection sensitivity over existing techniques, such as
reflectometry and beam distortion detection. However, the physical origin of
the increased sensitivity is unknown. In this work, we present a model,
describing the sensitivity behaviour of conoscopic interferometry with respect
to the quarter-wave plate orientation and the diaphragm aperture, which is
validated experimentally. Using the model, we optimize the detection
sensitivity of conoscopic interferometry. We obtain a maximal sensitivity of
detection when placing the diaphragm edge on the dark fringes of the
conoscopic interference patterns. In the configurations studied in this work,
conoscopic interferometry can be 8x more sensitive to acoustic waves than beam
distortion detection.

###### keywords:

Picosecond ultrasonics , Acoustic waves detection , Conoscopic Interferometry
, Beam Distortion Detection , Reflectometry

††journal: Photoacoustics

##  1 Introduction

Photoacoustics uses pulsed lasers to excite high-frequency acoustic waves
ranging from hundreds of kHz to hundreds of GHz [1][2] for non-destructive
testing [3], material characterization [4], and for medical imaging and
diagnosis [5]. Usually, a nanosecond (ns) [6] to femtosecond (fs) [7] pulsed
laser – the pump - generates bulk, guided, or surface acoustic waves in a
sample of interest [8]. The detection of the same acoustic waves with a second
laser beam – the probe - enables non-contact measurements on samples with
complex geometries, in tough environmental conditions, and without
contaminating their surface [3].

The most common implementation for acoustic wave detection with lasers is
reflectometry. The strain associated with the acoustic waves changes the
refractive index of the material through the photoelastic effect [9]. Hence,
the power of the probe beam reflected from the material surface has a
component directly proportional to the elastic strain. The resulting relative
variation in laser power is usually in the range of $10^{-6}-10^{-4}$ [2] [10]
[11]. The photoelastic constants of the material at the probe laser wavelength
set the detection sensitivity.

The strong dependence of the photoacoustic signal on the photoelastic
constants limits the applicability of reflectometry and thus inspired the
development of Beam Distortion Detection (BDD) [10] [12] and Conoscopic
Interferometry (CI) [11]. In BDD, the Gaussian spatial profile of the acoustic
wave incident on the sample surface causes slight fluctuations in the
divergence angle of the reflected probe beam. This results in diameter
variations of the reflected probe beam that are proportional to the acoustic
displacement, hence variations in power density. By masking a part of the
probe beam with a diaphragm, the power measured with a photodetector becomes
proportional to the displacement of the sample surface. This technique has the
advantage of a detection sensitivity _independent_ of the properties of the
sample material: BDD does allow the detection of acoustic waves in materials
with very low photoelastic constants. In this case, Chigarev et al. reported a
clear improvement of the Signal-to-Noise Ratio (SNR) with respect to
reflectometry [10]. In general, the measured signal is a sum of the BDD and
the reflectometry signal. In materials with high photoelastic coefficients,
BDD and reflectometry signals are therefore difficult to distinguish from each
other [10] [13].

CI makes use of Conoscopic Interference Patterns (CIPs), which are well-known
for the characterization of birefringent crystals [11] [14] [15]. By focusing
the probe beam with a given polarization through a birefringent crystal and
then collimating it before passing through a polarizer, one can obtain a
succession of bright (isochromates) and dark (isogyres) fringes. The fringes
form a pattern characteristic of the birefringence properties of the crystal
and the input and output polarizations. Liu et al. [11] implemented this
phenomenon in BDD by adding a birefringent crystal (sapphire plate) between
the objective and the sample and by using a Polarizing Beam Splitter (PBS) as
a polarizer to change the spatial profile of the probe beam to a CIP. The
resulting CIP is controlled by rotating a quarter-wave plate placed between
the PBS and the objective. Liu et al. observed surprisingly high SNR for some
CIPs, with respect to BDD and reflectometry in identical configurations on two
different samples. However, to this day, an analytical model that predicts the
sensitivity of acoustic wave detection by CI is still missing. This is of
great interest in view of pushing the sensitivity higher to allow measurements
of weak acoustic signals from thick structures, reflections from interface
with low acoustic impedance mismatch, or materials with high acoustic damping.

In this paper, we present an analytical model for the CIPs and predict their
sensitivity to acoustic waves. Using this model, we identified the key
parameter to optimize the performance of CI: maxima of sensitivity is obtained
by placing the diaphragm edges on the dark fringes (isogyres) of the CIPs. In
the configurations considered in this work, we found a sensitivity up to 8
times higher than that of BDD. We experimentally validate the model on a 2.4
µm thick silicon plate (Si) coated with $\sim 30$ nm aluminum (Al) indicating
that the model correctly predicts the sensitivity of CI to acoustic waves.

Figure 1: Picosecond ultrasonics ASOPS setup with Conoscopic Interferometry
detection. HWP: Half-Wave Plate, QWP: Quarter-Wave Plate, (P)BS: (Polarizing)
Beam Splitter, LP/SP: Long Pass/Short Pass dichroic mirrors (cut-off
wavelength). The UHFLI from Zürich Instruments records and analyzes the
photodetector signal before sending it to the computer. A white LED array
illuminates the sample. We use a camera for aligning the lasers with respect
to each other and the sample.

##  2 Materials and methods

The experimental setup, shown in Figure 1, contains an ASynchronous OPtical
Sampling (ASOPS) system [16] consisting of two synchronized Erbium lasers from
Menlo Systems with a pulse duration of around 100 fs. The pump pulses locally
heat the sample, which results in an extremely short temperature increase and
to the thermomechanical generation of a longitudinal acoustic pulse [9] in a
bandwidth of a few tens of GHz ($\sim 10-100$ GHz). The probe pulses allow us
to measure the acoustic reflections arriving back at the surface of the
sample. The pump laser has a wavelength of 1560 nm, a repetition rate of 100
MHz, and an average output power of around 100 mW. The probe laser has a
wavelength of 780 nm, an average output power of $\sim 500$ µW, and a $\sim
10$ kHz lower repetition rate than the pump laser. This offset in repetition
rate allows the reconstruction of the 10 ns time window between two pump
pulses within 100 µs. The time window is thus probed with $10^{4}$ discrete
time samples and consequently offers a temporal resolution of 1 ps. In this
section, we describe the experimental setup by introducing successively the
paths of the pump beam, probe beam and the illumination of the sample as well
as the data acquisition and measurement methodologies.

###  2.1 Pump beam path

The pump beam fiber output is first collimated by a collimator and reflected
at 90° by a Short-Pass (SP) 950 nm dichroic mirror to make a common path with
the probe beam. The P-polarized component is transmitted by a polarized beam
splitter (PBS) and then crosses a Quarter-Wave Plate (QWP) before it is
focused on the sample through a sapphire plate by an objective. The near-
infrared long working distance Plan-Apochromat objective from Mitutoyo has a
magnification of 20 and a wavelength correction from visible range to 1800 nm.
A part of the pump beam reflected off the sample is redirected towards a
camera using a Beam Splitter (BS) for aligning the pump and probe beam. The
radius of the pump beam, defined by the Half Width Half Maximum (HWHM) of the
intensity, is estimated as $r_{pu}\approx 2$ mm directly after the collimator
and $a_{pu}\approx 2$ µm on a sample in focus.

###  2.2 Probe beam path

The probe beam is free-space and first passes through a Half-Wave Plate (HWP)
to make it P-polarized. This maximizes the power transmitted by the PBS.
Before crossing the PBS, the probe beam travels through two dichroic mirrors;
a Long-Pass (LP) with a cut-off wavelength of 650 nm and the SP with a cut-off
wavelength of 950 nm. We place these dichroic mirrors before the PBS to avoid
any shift in polarization of the probe beam after the PBS as this would affect
the CIPs. After crossing the PBS, we place a QWP to controllably rotate the
probe beam polarization.

The objective focuses the probe beam on the sample through a 1 mm thick C-axis
cut (0001) birefringent sapphire plate, which modifies the beam polarization
and gives it a spatial dependence. After reflection of the probe beam by the
sample, it passes again through the sapphire plate, the objective, and the
QWP. Now, the PBS acts as a polarizer and reflects the S-polarized component
only towards the detection arm of the setup.

In the detection arm, a BS splits the probe beam into two beams of equal
power. One of these beams is focused on a camera, to visualize the CIPs and to
align the pump and probe beam on the sample. The other beam is truncated by an
iris diaphragm, of which the aperture diameter can be set between 0.4 mm and 8
mm. This diaphragm is used to detect the acoustic waves in a BDD or CI
configuration. After the diaphragm, the probe beam is focused on a
photodetector to ensure a spot size smaller than the photosensitive area and
thereby avoid additional truncation of the beam. The probe beam radius (HWHM)
is estimated as $r_{pr}\approx 0.4$ mm directly at the laser output and
$a_{pr}\approx 3$ µm on a sample in focus.

###  2.3 Sample illumination

A white LED array illuminates the sample to localize the pump and probe beam
spots with respect to the sample. The white light of the LED is first
collimated to a beam by the use of a lens and a diaphragm. This allows us to
control the white beam’s diameter and power by adjusting its aperture. A lens
with a long focal length then focuses the white beam to avoid loss of power by
truncation on the aperture of the other optical components in the setup.
Before reaching the sample, the white beam is first reflected with an angle of
90° by the LP 650 nm dichroic mirror and then crosses the SP dichroic mirror,
the PBS, the QWP, the objective, and the sapphire plate. The white beam is
then reflected by the surface of the sample and crosses the sapphire plate,
the objective, the QWP, and the PBS before moving into the detection arm of
the setup. Part of the white beam is reflected by the BS and is focused on the
camera to visualize the position of the pump and probe spots with respect to
the sample.

Figure 2: Typical signal ($\Delta P_{ac}/P_{0}$ vs. time) measured in BDD for
a diaphragm aperture of $P_{D}/P_{0}$ = 0.63. Inset: zoom on the first
acoustic echo. We use the amplitude of the thermal peak and the peak-to-peak
amplitude of the first acoustic echo to quantify the measurements sensitivity.

###  2.4 Data acquisition

We detect the probe beam pulses using a Si amplified photodetector from Menlo
System (FPD510-FS-VIS) that is sensitive in a wavelength range from 400 nm to
1000 nm and has a bandwidth of 250 MHz. The photosensitive area of the
photodetector has a diameter of 0.4 mm. The signal coming from the
photodetector is processed by a lock-in amplifier (Ultra High Frequency Lock-
In amplifier from Zürich Instruments, 600 MHz bandwidth) with the Boxcar +
Periodic Waveform Analyzer function [17]. This allows the accurate
reconstruction of the individual probe pulses. By using a trigger signal from
the ASOPS system at a frequency equal to the difference in the repetition rate
between both lasers ($\sim 10$ kHz), we probe the full-time delay window from
0 to 10 ns. The measured signals correspond to a variation in the probe pulse
power induced by the response of the sample. We normalize these signals by
dividing them by the probe power incident on the photodetector when the
diaphragm is fully open.

###  2.5 Measurement methodology for BDD and CI

To investigate the influence of the diaphragm aperture in the probe beam path
on the detection sensitivity of CI, we study two experimental configurations:

  * 1.

BDD configuration: without the sapphire plate present in the setup (Figure 1),
to validate the experimental methodology in the well-known BDD case.

  * 2.

CI configuration: using a sapphire plate with a thickness $h$=1 mm and three
different orientations of the QWP’s fast axis orientation, $\theta_{1/4}$ =
0°, 25°, 45°.

We perform measurements on a 2.4 µm thick Si sample coated with $\sim 30$ nm
of Al (Atomic Force Microscopy probe, model CONTR from NanoWorld). By
measuring the full probe power $P_{0}$ with the diaphragm fully open before
each measurement, we ensure that $P_{0}$ is the same for all the measurements.
We determine the power ratio $P_{D}/P_{0}$ between the power after ($P_{D}$)
and before the diaphragm by measuring the power incident on the photodetector
after partly closing the diaphragm. Depending on the configuration, between 7
and 9 diaphragm aperture diameters are used, ranging from $P_{D}/P_{0}=0$ to
1.

The pump and probe lasers are both focused on the free surface of the Al film.
Figure 2 shows a typical measurement of the relative variation of probe power
incident on the photodetector $\Delta P_{ac}/P_{0}$ induced by the response of
the sample to the pump pulse. The thermal response starts at $\sim 0.08$ ns,
consisting in a peak due to the very fast temperature increase and then an
exponential decay due to cooling. The acoustic reflection coefficient between
the aluminium and silicon is very low (<1% of the acoustic energy), which
makes the amplitude of the acoustic reflection at the Al/Si interface very
weak. Therefore, we observe the first clear acoustic reflection (longitudinal
wave) from the backside of the sample, arriving at $\sim 0.65$ ns. The time
delay between the thermal peak and the acoustic echo (0.57 ns) corresponds to
a Si thickness of 2.4 µm, which is within the range specified by Nanoworld. In
each acquisition, we extract the amplitude of the thermal peak as well as the
peak-to-peak amplitude of the first acoustic echo to quantify the sensitivity
of detection.

###  2.6 Measurement methodology for reflectometry

For a good comparison between reflectometry and BDD/CI measurements, we pay
particular attention to the distinct contributions of reflectometry and BDD/CI
components to the experimental signals. The aluminium in the sample offers an
interband transition around 780 nm [18] [19] resulting in high photoelastic
constants at the probe wavelength and thus in a high reflectometry component.
The measured signals are therefore a sum of the reflectometry and the BDD/CI
contributions, as explained by Chigarev et al. [10].

To extract the BDD contribution, we compare the signals for BDD with similar
measurements performed only in reflectometry. For these reflectometry
measurements, we completely open the diaphragm and decrease the power at the
output of the probe laser until we have the same incident power on the
photodetector as in the corresponding BDD measurement. This emulates the loss
of power induced by the diaphragm. The measurements are normalized in the same
way as for the BDD measurements. Since reflectometry is based on the variation
of the local refractive index by the acoustic strain [9], its sensitivity is
directly proportional to the probe power incident on the photodetector.

Similar to BDD and CI measurements, we use the thermal peak amplitude and the
peak-to-peak amplitude of the first acoustic reflection inside the sample to
characterize the sensitivity of reflectometry (see Figure 2).

##  3 Theory

The analytical model we present here combines the influence of the diaphragm
aperture on the acoustic wave detection sensitivity in BDD [10] with the Jones
calculus formulation for the CIPs [11] [14] [15]. BDD is thus a particular
case of the model, where the probe beam is spatially Gaussian. The full model
can be applied to any kind of beam shape.

Without loss of generality, we assume a probe beam that is spatially uniform
and purely S-polarized. In the experimental setup, after crossing the HWP, the
PBS, the QWP, the sapphire plate and reflection by the sample, the electric
field $\vec{E}$ of the beam is as follows:

$\vec{E}(x,y)=M_{R}W_{S}(x,y)W_{1/4}P_{t}W_{W1/2}\begin{pmatrix}0\\\
1\end{pmatrix},$ (1)

wherein $\begin{pmatrix}0,1\end{pmatrix}^{T}$, represents the S-polarized beam
at the output of the laser. $W_{n}$ is the Jones calculus formulation of the
wave plates ($n=1/4,1/2$, indicating if it is a QWP or a HWP and $n=S$
indicating the sapphire plate), and $P_{t}=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}$ is the one of the PBS in transmission for the P-polarized
component. The matrix $M_{R}=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$ models
the reflection of the probe beam on the sample, which simply acts as a mirror
[11]. $W_{n}$ is defined as follows:

$W_{n}=R(\theta_{n})^{T}\begin{pmatrix}1&0\\\
0&e^{-j\delta_{n}}\end{pmatrix}R(\theta_{n}),$ (2)

wherein $\theta_{n}$ is the angle of the QWP’s or HWP’s fast axis with respect
to the x axis, $R(\theta_{n})$ the corresponding rotation matrix (and
$R(\theta_{n})^{T}$ its transpose), and $\delta_{n}$ the phase shift of the
wave plate. The phase shift $\delta_{n}$ equals $\pi$ for the HWP and $\pi/2$
for the QWP. The angle of orientation $\theta_{1/4}$ of the QWP controls the
CIPs [11]. The sapphire plate acts as a wave plate due to its birefringent
properties [11] [15] and is represented by the matrix $W_{S}$. As the probe
beam is focused through the sapphire plate, the angle $\theta_{S}$ and phase
shift $\delta_{S}$ depend on the location in the $(x,y)$ plane. Therefore,
unlike for the HWP and the QWP, $\theta_{S}$ and $\delta_{S}$ in $W_{S}(x,y)$
are position dependent and are expressed as follows [11]:

$\theta_{S}(x,y)=\tan^{-1}(y/x),$ (3)

$\delta_{S}(x,y)=\frac{2\pi}{\lambda}h(n_{e}-n_{o})\sin^{2}\left(\tan^{-1}\left(\frac{\sqrt{x^{2}+y^{2}}}{f_{obj}}\right)\right).$
(4)

Here, $\lambda=780$ nm is the wavelength of the probe laser, $h$ is the
thickness of the sapphire plate, $n_{e}=1.760$ and $n_{o}=1.768$ are the
extraordinary and the ordinary refractive indices, respectively, and
$f_{obj}=20$ mm is the focal length of the objective.

Since the pump beam is spatially Gaussian, the displacement induced by the
acoustic pulse when it reaches the sample’s surface is Gaussian as well. This
produces a slight variation in the reflected probe beam divergence angle, as
explained in [10]. The relative variation $\xi$ on the objective plane of the
reflected probe beam radius $r_{pr}^{{}^{\prime}}$ with respect to the
incident beam radius $r_{pr}$ (considering
$r_{pr}^{{}^{\prime}}=(1+\xi)r_{pr}$) is [10]:

$\xi=2\frac{2\pi z_{0}a_{pr}^{2}}{\lambda z_{r}a_{pu}^{2}}A_{0},$ (5)

where $z_{0}$ is the distance between the sample position and the probe beam
focus position, $z_{r}\approx 35$ µm is the Rayleigh length of the probe beam,
and $A_{0}$ is the displacement amplitude of the sample’s surface due to the
acoustic wave.

Before reaching back the objective, the probe beam again crosses the sapphire
plate. The variation in the reflected probe beam divergence angle caused by
the acoustic wave induces a shift in coordinate on the sapphire plate with
respect to the probe beam which was initially incident on the sample.
Furthermore, due to the reverse propagation direction of the reflected light,
the orientation of the fast axis with respect to the beam is mirrored with
respect to the beam incident on the sample. Therefore, we now use
$\theta_{S,r}=\pi-\theta_{S}$ and
$\delta_{S,r}(x,y)=\delta_{S}((1+\xi)x,(1+\xi)y)$ in $W_{S}(x,y)$ to obtain
$W_{S,r}(x,y)$. After being collimated by the objective, the probe beam
crosses the QWP with a reverse propagation direction (i.e.,
$\theta_{r,1/4}=\pi-\theta_{1/4}$ in $W_{1/4,r}$) and the PBS. The PBS now
reflects the S-polarized component of the beam, $P_{r}=\begin{pmatrix}0&0\\\
0&1\end{pmatrix}$. All combined, this results in the following expression for
the electric field $\vec{E}$ arriving at the photodetector:

$\displaystyle\vec{E}(x,y)=$ $\displaystyle
P_{r}W_{1/4,r}W_{S,r}(x,y)M_{R}\times$ (6) $\displaystyle
W_{S}(x,y)W_{1/4}P_{t}W_{1/2}\begin{pmatrix}0\\\ 1\end{pmatrix}.$

From the electric field $\vec{E}$, we obtain the probe beam intensity $I(x,y)$
incident on the photodetector: $I(x,y)=c\epsilon_{0}|\vec{E}(x,y)|^{2}/2$,
where $c=3*10^{8}m.s^{-1}$ is the speed of light, and
$\epsilon_{0}=8.85*10^{-12}F.m^{-1}$ is the vacuum permittivity. The spatial
dependence of the intensity directly gives us the CIPs induced in the probe
beam. In reality, the probe beam is spatially Gaussian at the output of the
laser. Therefore, the beam intensity as seen by the photodetector becomes in
polar coordinates $(r,\phi)$:

$I_{G}(r,\phi)=I(r,\phi)e^{-r^{2}/(r_{pr}(1+\xi))^{2}}/(\pi(r_{pr}(1+\xi))^{2}).$
(7)

Finally, we take into account the influence of the diaphragm, for which we
assume a circular aperture perfectly aligned with the center of the beam. The
relative variation of probe power $\Delta P_{ac}/P_{0}$ incident on the
photodetector is given by:

$\frac{\Delta
P_{ac}}{P_{0}}=\frac{\int_{0}^{r_{D}}\int_{0}^{2\pi}(I_{G}(r,\phi)-I_{G,0}(r,\phi))rdrd\phi}{\int_{0}^{\infty}\int_{0}^{2\pi}I_{G,0}(r,\phi)rdrd\phi},$
(8)

where $r_{D}$ is the radius of the diaphragm aperture, $\infty$ represents the
radius of the diaphragm aperture when it is completely open, and
$I_{G,0}(r,\phi)$ is the probe beam intensity incident on the photodetector
when the sample is not excited by the pump beam. Equation 8 directly gives the
relative variation in probe power induced by the acoustic waves in presence of
a CIP and a diaphragm.

By removing the Gaussian profile from Equation 7 and the acoustic wave
contribution ($\xi=0$), we find back the CIPs as presented by Liu et al. in
[11]. By assuming a non-birefringent crystal ($n_{e}=n_{o}$ and thus
$\delta_{S}=\delta_{S,r}=0$), Equation 8 reduces to the BDD signal as derived
by Chigarev et al. [10]. BDD is thus a particular case of Equation 8 when the
crystal used is not birefringent.

The theory resulting in Equation 8 highlights the main parameters influencing
the sensitivity of CI to acoustic waves:

  * 1.

The QWP orientation $\theta_{1/4}$.

  * 2.

The refractive indexes, $n_{e}$ and $n_{o}$ of the birefringent crystal.

  * 3.

The thickness $h$ of the birefringent crystal.

  * 4.

The angle of the focused probe beam w.r.t. the birefringent crystal set by the
focal length $f_{obj}$ and thus the probe beam radius on the objective
$r_{pr}$.

  * 5.

The position of the sample $z_{0}$ with respect to the probe beam focus.

  * 6.

The ratio between the probe and pump spot radii on the sample,
$(a_{pr}/a_{pu})^{2}$.

  * 7.

The diaphragm aperture $r_{D}$ with respect to the beam radius $r_{pr}$.

For the calculations below, we set $z_{0}=-0.5$ µm and $A_{0}=0.1$ nm. The
precision of the translation stage used to adjust the position of the sample
provides a resolution of 0.5 µm for $z_{0}$. Although the pump laser
characteristics and the sample material and geometry determine $A_{0}$, its
value is typically of the order of several tenths of pm [10]. Due to the
negative value of $z_{0}$, $\xi$ is thus negative in our calculations below.

Figure 3: Conoscopic interference patterns for $\theta_{1/4}$ = 0°, 25°, 45°
and for different values of $r_{pr}$ and $h$: a $h=1$ mm and $r_{pr}=0.4$ mm,
b $h=2$ mm and $r_{pr}=1.3$ mm, c $h=1$ mm and $r_{pr}=2.5$ mm, and d $h=2$ mm
and $r_{pr}=2.5$ mm. Panel b shows both the theoretical and measured
conoscopic interference patterns. All shown conoscopic interference patterns
have a physical size of 12x12 mm2. e Table with the maximum intensity value
$I_{G,0}$ ($\%$) for each calculated pattern relative to an input intensity of
$7171W/m^{2}$.

##  4 Results and discussion

The results and discussion section is organized as follows. We present in
Section 4.1 the calculated CIPs and validate them with the experiment. In
Section 4.2 we show good agreement between both the theoretical and
experimental sensitivity of CI and BDD to the acoustic waves. Finally, in
Section 4.3, we elucidate the dependence of the sensitivity of CI to the probe
beam radius $r_{pr}$, the QWP orientation $\theta_{1/4}$, and diaphragm
aperture size $P_{D}/P_{0}$, in order to optimize the sensitivity.

###  4.1 Conoscopic interference patterns

Figure 3 shows CIPs for several probe beam radii and thicknesses of the
sapphire plate. For each configuration, we show three QWP orientations
corresponding to $\theta_{1/4}$ = 0°, 25° and 45°. The patterns calculated in
the Figure 3a correspond to the configuration studied experimentally in the
Section 4.2, with $r_{pr}=0.4$ mm and $h=1$ mm. In this configuration, the
phase shift $\delta_{S}(x,y)$ induced by the sapphire plate (Equation 4) is
only -1.5° for light leaving the objective at a distance $r_{pr}$ from the
optical axis. The patterns observed for $\theta_{1/4}$ = 25°, 45° are very
close to a spatial profile of a purely Gaussian beam. For $\theta_{1/4}$ = 0°,
the pattern is different, showing bright (isochromates) and dark fringes
(isogyres). For $\theta_{1/4}=$ 25°, 45°, the polarization is elliptical and
circular, respectively, whereas the beam is purely P-polarized when
$\theta_{1/4}=$ 0°. Since the PBS reflects only the S-polarized component
towards the detection arm of the setup, and since the sapphire plate does not
induce a phase shift at the center of the probe beam, this results in an
isogyre. The influence of a weak phase shift $\delta_{S}$ between the P and
S-polarized components of the beam, is therefore only clearly visible when
$\theta_{1/4}=$ 0°. Due to this, the CIPs for $\theta_{1/4}$ = 25°, 45° also
have an intensity $1000\times$ higher than for $\theta_{1/4}$ = 0° (see Figure
3e).

To enable the experimental observation of the CIPs, we use a sapphire plate
with $h=2$ mm and a beam expander directly at the probe laser output to
increase the diameter to $r_{pr}\approx 1.3$ mm. Consequently, the maximum
value $\delta_{S}(x,y)$ at a distance $r_{pr}$ from the center of the beam in
this configuration increases to -31°, which induces a more significant
difference between the CIPs and a Gaussian profile. The calculated and
measured CIPs are presented in Figure 3b. Note that the beam expander reduces
the ratio between $a_{pr}$ and $a_{pu}$, which decreases the sensitivity of CI
and BDD (Equation 5, [10]) and therefore we do not consider this configuration
in Section 4.2. For $\theta_{1/4}$ = 0°, the pattern is similar to that in
Figure 3a, but the CIPs for $\theta_{1/4}$ = 25° and $\theta_{1/4}$ = 45° are
different. The $\theta_{1/4}$ = 25° loses its circular symmetry and both the
$\theta_{1/4}$ = 25° and $\theta_{1/4}$ = 45° CIPs contain fringes around a
central maximum in intensity. The intensity of the different patterns is also
now of the same order of magnitude ($\sim 1000W/m^{2}$, see Figure 3e). Due to
the initially spatially Gaussian profile of the probe beam, the fringes of the
CIPs have a lower intensity than their centers (see Equation 7). We observe
the same features and patterns experimentally which validates the model
presented in Section 3. We attribute the slight rotation between the patterns
obtained theoretically and experimentally to the unknown reference coordinate
for the polarization of the probe beam in the experiment.

To gain more insight into the parameters determining the CIPs, we plot them
for different combinations of $r_{pr}$ and $h$ in Figures 3c and 3d. When
$r_{pr}$ increases from $0.4$ mm to $2.5$ mm (Figures 3a to 3d), we observe
the appearance of more bright (isochromates) and dark (isogyres) fringes
around the central shape. The appearance of more fringes is due to the
increased convergence angle ($\tan^{\rm-1}(r_{pr}/f_{obj})$ in Equation 4) of
the light passing through the sapphire plate. As a consequence, $\delta_{S}$
increases resulting in stronger conoscopic interferences and thus in the
appearance of more bright and dark fringes in the pattern. For the same
reason, also more bright and dark fringes appear in the CIP when increasing
$h$ from 1 to 2 mm (Figures 3c to 3d). Also, the CIP for $\theta_{1/4}$ = 25°
in Figures 3c and 3d clearly differs from that for $\theta_{1/4}$ = 45°. This
is due to the large $r_{pr}$ which ensures significant intensity in the bright
fringes. As a result, all different CIPs in Figures 3c and 3d show similar
intensities.

###  4.2 Sensitivity to sample deformations

Figure 4: Measurements of the detection sensitivity of BDD and reflectometry
and associated fits: a thermal peak amplitude and b peak-to-peak amplitude of
the first acoustic echo. The blue data points represent the BDD measurements
and the corresponding continuous blue lines the fit to Equation 8. Similarly,
the black data points and lines show the reflectometry measurements and fit.
The light blue data points and light dashed blue lines show the BDD
measurement and fit from which the reflectometry component has been
subtracted. The grey lines show the sensitivity obtained using the model of
Chigarev et al. [10] scaled to match the amplitude of the dashed light blue
curve.

Table 1: Values of the fit parameters, $A$ (reflectometry sensitivity) and $B$
(BDD/CI sensitivity) and their associated error, for the different
experimental configurations studied. The last two colomns show the ratio (%)
between the $A$($B$) fit parameters of the first acoustic echo and the thermal
peak.

To further validate the model presented in Section 3, we now focus on the
measurements obtained in the BDD configuration and compare it to the
sensitivity profile presented by Chigarev et al. [10]. In order to identify
the BDD contribution to the total signal, we compare the measurements
presented in Figures 4a and 4b for BDD (in blue) with similar measurements
performed only in reflectometry (in black). Then, we fit the experimental data
of the BDD configuration using the following function $f_{fit}$:

$f_{fit}(P_{D}/P_{0})=A\cdot P_{D}/P_{0}+B\cdot f_{th}(P_{D}/P_{0}),$ (9)

where $f_{th}$ is the theoretical sensitivity function of BDD or CI calculated
from Equation 8 ($\Delta P_{ac}/P_{0}$) normalized to its maximum value. The
fit parameters $A$ and $B$ correspond to the amplitude of reflectometry and
BDD, respectively. As reflectometry is directly proportional to the probe
power, $A$ is simply multiplied with $P_{D}/P_{0}$. The continuous blue line
in Figures 4a and 4b shows the best fit result. The continuous black line
depicts the reflectometry part ($A\cdot P_{D}/P_{0}$) and the dashed blue line
the BDD contribution ($B\cdot f_{th}(P_{D}/P_{0})$). For comparison, we also
plot the experimental data points from which we subtracted the reflectometry
component ($A\cdot P_{D}/P_{0}$), and the calculation (gray line) from the
model of Chigarev et al. [10]. The values of the fit parameters $A$ and $B$
are listed in Table 1 and are similar to values for reflectometry on aluminium
[18] and BDD [10, 11] reported in the literature. Figures 4a and 4b thus show
that the model of Section 3 predicts the sensitivity of both the thermal peak
amplitude and the peak-to-peak amplitude of the first acoustic echo.

As the fitting procedure was validated for BDD, we now focus on the CI
configuration with $h$ = 1 mm, $r_{pr}$ = 0.4 mm. In this configuration, we
cannot measure the reflectometry contribution independently due to the
sapphire plate. Therefore, we rely on the fitting procedure to separate the
reflectometry contribution from the CI contribution. Figure 5 shows the CI
measurements before and after the subtraction of the reflectometry component.
We compare three different orientations of the QWP (yellow: $\theta_{1/4}$=0°,
purple: $\theta_{1/4}$=25°, green: $\theta_{1/4}$=45°) with the BDD
measurements (blue). Note that in contrast to the BDD sensitivity, the CI
sensitivity is not necessarily zero when the diaphragm is fully open (see
Section 4.3). The experimental results show a similar trend for QWP
orientations of $\theta_{1/4}$=25°, 45° and BDD, but a different one when
$\theta_{1/4}$=0°. The sensitivity is even negative for $\theta_{1/4}$=0° (see
Section 4.3 for explanation). Similar to the BDD case, the model thus
correctly predicts the sensitivity in CI for the thermal peak amplitude and
the peak-to-peak amplitude of the first acoustic echo.

Figure 5: Measurements of the sensitivity of detection of BDD and CI and
associated fits: a thermal peak, b first acoustic echo. The top panels show
the measured data points and fits to equation 8. The lower panels show the
measured data points and fits after subtracting the reflectometry component.
For completeness, we show the BDD measurement of Figure 4 in blue. The CI data
are shown in yellow for $\theta_{1/4}=$ 0°, purple for $\theta_{1/4}=$ 25°,
and green for $\theta_{1/4}=$ 45°.

Although the model correctly predicts the the sensitivity in BDD and CI, the
extracted fit parameters $A$ and $B$ differ between the different
measurements. We attribute this to experimental uncertainties; slightly
different alignment for each measurement, the reflection of pump power ($\sim
10\%$) on the sapphire plate and variations in experimental conditions (e.g.
room temperature). However, the ratio between the fit parameter $A$ for the
thermal peak amplitude and that of the peak-to-peak amplitude of the first
acoustic echo equals around 16% ($\pm 3\%$) except for the $\theta_{1/4}$ = 0°
configuration that has a ratio of 33 %. This also holds for the $B$ parameter
for which the ratio is 50 % ($\pm 5\%$) and 79% for the $\theta_{1/4}$ = 0°
configuration. We attribute the different ratios of the $\theta_{1/4}$ = 0°
configuration to the fact that exactly this configuration was measured several
days after the other configurations and therefore had to be re-aligned
significantly. The further constant ratio of $A$ and $B$ further support the
validity of the model.

Figure 6: Calculated CI sensitivity for $\theta_{1/4}$ = 0° (yellow), 25°
(purple), 45° (green) and for different values of $r_{pr}$ and $h$: a $h=1$ mm
and $r_{pr}=0.4$ mm, b $h=2$ mm and $r_{pr}=1.3$ mm, c $h=1$ mm and
$r_{pr}=2.5$ mm, and d $h=2$ mm and $r_{pr}=2.5$ mm. For comparison, the
sensitivity curve of BDD (blue) is shown in all panels. Panel a also shows the
CI sensitivity for $\theta_{1/4}$ = 1° (dashed yellow) to indicate the large
change in sensitivity for a small change of $\theta_{1/4}$ around 0°. The
overlapping lines in the inset of panel a indicates that the sensitivity
barely depends on $\theta_{1/4}$ between 25° and 45°. Figure 7: Calculation of
the difference ($I_{G}-I_{G,0}$) between the pattern with and without an
acoustic wave for $h=$ 1 mm, $r_{pr}=$ 0.4 mm, and a $\theta_{1/4}=$ 0°, and b
$\theta_{1/4}=$ 45°. Panels are normalized in intensity.

###  4.3 Optimizing the sensitivity

To optimize the CI sensitivity, we calculate the relative power variation
$\Delta P_{ac}/P_{0}$ for different values of $h$ and $r_{pr}$ and
$\theta_{1/4}$=0°, 25°, and 45° as a function of the diaphragm opening (see
Figure 6). The CIPs corresponding to these sensitivities are depicted in
Figure 3. In Figure 6, we observe the following features. The values for
$\Delta P_{ac}/P_{0}$ are similar to those obtained experimentally (see Figure
5) despite we do not know the exact values of $z_{0}$ and $A_{0}$ (see
Equation 5) in the experiment. In contrast to BDD, we observe a nonzero
sensitivity for CI in case of a completely opened diaphragm in several
configurations. By increasing the probe radius $r_{pr}$ and/or the sapphire
plate thickness $h$, the sensitivity changes from the one of BDD into one with
distinct maxima in sensitivity even exceeding that of BDD for $\theta_{1/4}=$
25° and 45°. For $\theta_{1/4}=$ 0°, the sensitivity changes sign and becomes
positive. For higher values of $r_{pr}$ and/or $h$, even more local maxima in
the sensitivity appear.

When comparing the calculated sensitivities in Figure 6a to the corresponding
experimental results presented in Figure 5, we find that despite the agreement
in trend, the relative amplitudes are different. The sensitivity of BDD and CI
for $\theta_{1/4}=$ 25°, 45° should in theory almost overlap, while
differences are measured experimentally. We attribute this to the variation in
alignment and experimental conditions, as discussed in Section 4.2. For
$\theta_{1/4}=$ 0°, the theoretical difference in sensitivity with the other
QWP angles is much higher than the one measured in reality. We attribute this
to the unknown values of $z_{0}$ and $A_{0}$ in the experiment, the several
days delay between the $\theta_{1/4}=$ 0° measurement and the other ones, and
also to the experimental error in QWP angle. By comparing the continuous
($\theta_{1/4}=$ 0°) and dashed ($\theta_{1/4}=$ 1°) yellow lines in Figure
6a, we find that the sensitivity strongly depends on the QWP angle, at
$\theta_{1/4}=$ 1° already almost halves the sensitivity. As an experimental
error of 1° or less in the QWP orientation is realistic, we attribute the
difference in sensitivity between experiment and calculations at
$\theta_{1/4}=$ 0° to it.

Figure 8: Calculated absolute CI sensitivity $|P_{ac}|$ for $\theta_{1/4}$ =
0° (yellow), 25° (purple), 45° (green) and for different values of $r_{pr}$
and $h$: a $h=1$ mm and $r_{pr}=0.4$ mm, b $h=2$ mm and $r_{pr}=1.3$ mm, c
$h=1$ mm and $r_{pr}=2.5$ mm, and d $h=2$ mm and $r_{pr}=2.5$ mm. The
intensities are normalized w.r.t. the maximum of the BDD sensitivity shown by
dashed blue line. The inset in panel a shows a zoom of the $\theta_{1/4}$ = 0°
case. Figure 9: Periodicity in the CI sensitivity at $h$=2 mm, $r_{pr}$=2.5
mm. a and b show the CI sensitivity for $\theta_{1/4}$=0° and
$\theta_{1/4}$=45°, respectively. The dashed black (gray) lines indicate the
minima (maxima) of sensitivity for a given diaphragm opening quantified by
$P_{D}/P_{0}$. c and d show the corresponding conoscopic interference patterns
$I_{G,0}$ and the diaphragm openings corresponding to minima (red) and maxima
(green) in CI sensitivity. e and f calculations of the difference
($I_{G}-I_{G,0}$) between the patterns with and without an acoustic wave
corresponding to the CIPs shown in panel c and d. Panels c-f are normalized in
intensity.

Let us now focus on understanding the negative sign of the sensitivity for
$\theta_{1/4}$=0°. We attribute this to the change in CIP. For
$\theta_{1/4}$=0° (see Figure 3), we see that the center of the CIP is an
isogyre. In contrast, the CIPs for $\theta_{1/4}$=25° and $\theta_{1/4}$=45°
have a maximum in intensity at the center and therefore show similar
sensitivities in Figure 6. The fast thermal expansion and acoustic pulses
cause a small change in divergence angle of the probe beam. Consequently,
$r_{pr}$ decreases by the factor $1+\xi$ (see Equation 5) as $\xi$ is
negative. In turn, this slightly shrinks the CIP. Therefore, relatively more
light will pass closer to the optical axis through the sapphire plate. This
light acquires a smaller phase shift $\delta_{S}$ than rays further away from
the optical axis. For $\theta_{1/4}$=0°, all light is P-polarized before going
through the sapphire plate. Due to the PBS, the photodetector only detects
light that has a S-polarization component which is thus less in presence of
the acoustic pulse. In contrast, the total probe power increases for
$\theta_{1/4}$=25° and $\theta_{1/4}$=45°. The light has both P- and
S-polarization components before going through the sapphire plate. The
S-polarization component is also focused on the center and experience less
phase shift $\delta_{S}$. Hence, more of this S-polarized light will arrive at
the photodetector resulting in an increase of the total measured probe power.
Considering this argument, the relative probe power $\Delta P_{ac}/P_{0}$ (see
Equation 8) will be negative for $\theta_{1/4}$=0° and positive for
$\theta_{1/4}$=25° and $\theta_{1/4}$=45°, as shown in Figure 7. The observed
negative sensitivity for $\theta_{1/4}$=0° reverses the sign of the acoustic
signal. In case where this signal has both a reflectometry and BDD/CI
component, as seen in Section 4.2, this can reduce the total sensitivity of
detection. However, this can be circumvented by changing the sign of $z_{0}$
(see Equation 5 and Ref. [10]) by moving the sample to the other side of the
probe beam focus.

The reason causing the negative sensitivity for $\theta_{1/4}$=0° also makes
CI sensitive to acoustic waves without a diaphragm (see Figure 6), i.e. the
sensitivity is not zero when the diaphragm is fully open ($P_{D}/P_{0}$ = 1).
Due to the slight variation in $r_{pr}$, the reflected probe beam experiences
a slightly different phase shift $\delta_{S}$ when propagating through the
sapphire plate. In turn, this results in a slightly different CIP (see Figure
7). Therefore, the incident intensity on the photodetector is varying, even
without the use of a diaphragm.

The reduction in probe power $P_{0}$ incident on the photodetector for
$\theta_{1/4}$=0° (see Figure 3e) also has an effect on the relative probe
power $\Delta P_{ac}/P_{0}$. As seen in Figure 6, the sensitivity for
$\theta_{1/4}$ = 0° is much larger than the one for BDD and the other values
of $\theta_{1/4}$ for small $h$ and $r_{pr}$. When $P_{0}$ is not used to
normalize $\Delta P_{ac}$, the sensitivity of BDD will exceed that of CI in
almost all studied configurations (see Figure 8). The high sensitivity of CI
is thus a direct consequence of the normalization by $P_{0}$ in the
calculation of the relative probe power $\Delta P_{ac}/P_{0}$.

In order to understand the local maxima in the sensitivity shown in Figure 6,
we compare the CIPs for $\theta_{1/4}$=0° and $\theta_{1/4}$=45° obtained with
different diaphragm apertures in Figure 9. All maxima in the sensitivity
correspond to an aperture with the edges of the diaphragm placed in the
isogyres (dark fringes). The minima exactly occur when the diaphragm edges are
on top of the bright fringes (isochromates). To find out why the sensitivity
is maximum (minimum) at the isogyres (isochromates), we show the intensity
difference $I_{G}-I_{G,0}$ between a pattern with and without acoustic wave
(see Equation 8) in Figure 9e and 9f. The acoustic waves induce a variation in
the CIP due to the slight change in divergence angle as well as a change in
$\delta_{S}$. The sign of this intensity variation is alternatingly positive
and negative. By placing the diaphragm edges on the isochromates of the CIP,
the same number of positive and negative variations are incident on the
photodetector. As we integrate this CIP over the open area of the diaphragm,
the light intensity variation partly cancels out and thus results in a minimal
sensitivity. In contrast, by placing the diaphragm edges on the dark fringes
of the CIP, more positive than negative variations of intensity are obtained,
resulting in a maximum sensitivity to acoustic waves.

Finally, we compare the sensitivity of CI with that of BDD. As Figure 6 shows,
CI is not always more sensitive than BDD. However, by choosing the right
diaphragm opening and QWP orientation, CI can be made more sensitive than BDD.
For example, the CI configuration with $r_{pr}=$ 2.5 mm, $h=$ 1 mm,
$\theta_{1/4}=$ 45°, and $P_{D}/P_{0}=\sim 0.73$, has a total sensitivity
almost twice that of BDD. The total sensitivity of CI configuration with
$r_{pr}=$ 0.4 mm, $h=$ 1 mm, $\theta_{1/4}=$ 0°, and no diaphragm, is even up
to 8 times higher than that of BDD.

##  5 Conclusion

Conoscopic Interferometry (CI) is a promising detection technique for
ultrafast acoustics that can offer an improved SNR compared to Beam Distortion
Detection (BDD) and reflectometry. We developed a model that predicts the
sensitivity of CI and BDD. Our results show that for a given probe power
incident on the photodetector, CI can be more sensitive than BDD for detecting
the surface displacement of a sample, if one carefully chooses the right
parameters. By using a 1 mm thick sapphire plate, a probe beam radius of 0.4
mm, a Quarter Wave Plate orientation of 0° and no diaphragm, CI is up to 8
times more sensitive than BDD. Moreover, we showed that the CI sensitivity is
optimal when the diaphragm aperture cuts the radially symmetric conoscopic
interference patterns in its dark fringes. We validated these observations
experimentally on a 2.4 µm thick silicon substrate coated with $30$ nm
aluminum. We foresee significant improvements of the CI detection sensitivity
by using different birefringent crystals, by beam shaping the probe beam, or
using different diaphragm geometries. Because of the enhanced sensitivity
compared to BDD and reflectometry on materials with low photoelastic
constants, optimized CI detection schemes could play a central role in the
future of ultrafast acoustics.

## Acknowledgment

M.R., R.G., G.J.V. acknowledge support from project TKI-HTSM/19.0172.

## Author contribution

M.R.: Conceptualization, Methodology, Software, Formal Analysis,
Investigation, Writing - original draft. R.G.: Conceptualization,
Investigation, Writing - review & editing. U.A.: Conceptualization, Writing -
review & editing, Supervision, Project administration. Z.Z.:
Conceptualization, Supervision. N.P.: Conceptualization, Supervision. G.J.V.:
Conceptualization, Methodology, Writing - review & editing, Supervision,
Project administration, Funding acquisition.

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## Vitae

Martin Robin received his PhD degree in the field of Laser-Ultrasonics at the
Université Polytechnique des Hauts-de-France, France, 2019, after obtaining
his M.Sc. degree in acoustics at Le Mans Université, France, 2015. His
doctorate work was about the characterization of the adhesion in film-on-
substrate using Surface Acoustic Wave, generated and detected by lasers. He
then pursued as a postdoctoral researcher at the Technische Universiteit
Delft, Netherlands, between 2020 and 2022. There, he worked on the design and
installation of a Picosecond Ultrasonic setup for imaging and Non-Destructive
Testing applications. He is now employed as a postdoctoral researcher at
Institut Lumière Matière, Université Claude Bernard Lyon 1, France, to
characterize bio-metamaterials by Surface Acoustic Waves using Laser-
Ultrasonics.

Ruben Guis obtained his master’s degree in Physics at Leiden University,
Netherlands in 2020. For his thesis project he worked towards 3D imaging at
the nanoscale by attempting to measure the subsurface information using
resonant hydrogen spins in Magnetic Resonance Force Microscopy. Started in
October 2020, he is currently doing his PhD at the Technische Universiteit
Delft, Netherlands, where he is again working on 3D imaging at the nanoscale,
this time using ultrasound to get the subsurface information, by integrating
nanoscale acoustics into atomic force microscopy.

M. Ümit Arabul received the B.Sc. degree (2011) in Electrical & Electronics
Engineering from the Middle East Technical University (METU), Ankara, Turkey.
Next, he joined the Medical Imaging Lab. in the Institute of Biomedical
Engineering, Bogaziçi University (BOUN), Istanbul and obtained the M.Sc.
degree in Biomedical Engineering (2013). He pursued his Ph.D. in the
Photoacoustics and Ultrasound Laboratory of Eindhoven (PULS/e Lab), in the
Cardiovascular Biomechanics group of the department of Biomedical Engineering
in the Eindhoven University of Technology, The Netherlands. His research
focused on photoacoustic imaging of carotid arteries and fundamental
characterization and preclinical validation of photoacoustic imaging (2018).
Currently, he is working as a Research Scientist at ASML with the focus on
semiconductor metrology.

Zili Zhou obtained his Master’s degree of Optical Engineering from Chinese
Academy of Sciences, Shanghai, China, in 2010. Then he received his PhD degree
from Applied Physics department of Eindhoven University of Technology, the
Netherlands, in 2014, where he focused on multi-photon detection with
superconducting nanowires. Since 2015, he works at Research department of ASML
on optical nanometrology projects, as a senior research scientist.

Nitesh Pandey (PhD 2011, NUI Maynooth) is a Principal Research Engineer
working at the Advanced Technology Development group at ASML. He joined ASML
in 2011 and has worked on various topics related to Optical sensors in
Lithography scanners and Overlay metrology. He is currently working in the
area of pattern fidelity control, OPC and stochastic placement error
modelling.

Gerard J. Verbiest obtained his M.Sc. degree in theoretical physics (cum
laude, 2009), and a PhD in experimental physics (2013) from Leiden University.
His doctorate work focused on the application of ultrasound at MHz frequencies
in atomic force microscopes to enable subsurface atomic force microscopy. He
then worked as postdoctoral researcher at the RWTH Aachen in Germany between
2013 and 2018 on graphene mechanics and dynamics. Since August 2018, Gerard
works as assistant professor at the Delft University of Technology. His work
focusses on nanoscale acoustics with applications in atomic force microscopes,
graphene and other two-dimensional materials, and plant physiology.