#  Quantum enhanced non-interferometric quantitative phase imaging

Giuseppe Ortolano  Quantum metrology and nano technologies division, INRiM,
Strada delle Cacce 91, 10153 Torino, Italy  DISAT, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy  Alberto Paniate  Quantum
metrology and nano technologies division, INRiM, Strada delle Cacce 91, 10153
Torino, Italy  DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24,
10129 Torino, Italy  Pauline Boucher  Carmine Napoli  Quantum metrology and
nano technologies division, INRiM, Strada delle Cacce 91, 10153 Torino, Italy
Sarika Soman  Silvania F. Pereira  Imaging Physics Dept. Optics Research
Group, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg
1, 2628CJ Delft, The Netherlands  Ivano Ruo Berchera  Quantum metrology and
nano technologies division, INRiM, Strada delle Cacce 91, 10153 Torino, Italy
Marco Genovese  Quantum metrology and nano technologies division, INRiM,
Strada delle Cacce 91, 10153 Torino, Italy

###### Abstract

Quantum entanglement and squeezing have significantly improved phase
estimation and imaging in interferometric settings beyond the classical
limits. However, for a wide class of non- interferometric phase
imaging/retrieval methods vastly used in the classical domain e.g.,
ptychography and diffractive imaging, a demonstration of quantum advantage is
still missing. Here, we fill this gap by exploiting entanglement to enhance
imaging of a pure phase object in a non-interferometric setting, only
measuring the phase effect on the free-propagating field. This method, based
on the so-called “transport of intensity equation”, is quantitative since it
provides the absolute value of the phase without prior knowledge of the object
and operates in wide-field mode, so it does not need time-consuming raster
scanning. Moreover, it does not require spatial and temporal coherence of the
incident light. Besides a general improvement of the image quality at a fixed
number of photons irradiated through the object, resulting in better
discrimination of small details, we demonstrate a clear reduction of the
uncertainty in the quantitative phase estimation.

Although we provide an experimental demonstration of a specific scheme in the
visible spectrum, this research also paves the way for applications at
different wavelengths, e.g., X-ray imaging, where reducing the photon dose is
of utmost importance.

##  I Introduction

Quantum imaging Berchera and Degiovanni (2019); Moreau et al. (2019); Genovese
(2016) and sensing Degen et al. (2017); Pirandola et al. (2018); Petrini et
al. (2020) have provided genuine and valuable advantages in many measurement
applications ranging from fundamental physics Aasi et al. (2013); Pradyumna et
al. (2020) to biology Taylor and Bowen (2016); Casacio et al. (2021); Petrini
et al. (2022) from microscopy Schwartz and Oron (2012); Gatto Monticone et al.
(2014); Samantaray et al. (2017); Tenne et al. (2019)to optical sensors Lawrie
et al. (2019); Lee et al. (2021).

In particular, given the importance of optical phase measurement, appearing in
all the science fields, a considerable effort has been made to exploit quantum
entanglement or squeezing for this task. Quantum phase estimation through
first-order interference involving the mixing of two optical modes in a linear
Polino et al. (2020); Demkowicz-Dobrzanski et al. (2015) or non-linear
Chekhova and Ou (2016); Kalashnikov et al. (2016); Töpfer et al. (2022)
interaction is well understood. The ultimate uncertainty bound with quantum
optical states is known to scale with the number of probing particles $N$ as
$N^{-1}$, the so-called ’Heisenberg scaling’. In contrast, for the classical
probing state, it is limited to $N^{-\frac{1}{2}}$, referred to as the
standard quantum limit (SQL) or shot noise limit. Although the quantum
advantage would, in principle, be disruptive for $N\gg 1$ in a realistic
scenario, the gain over the SQL is rather, in the form of a constant depending
on the optical losses Demkowicz-Dobrzanski et al. (2015). Proofs of principle
of quantum-enhanced linear interferometry with the so-called entangled NOON
states have been achieved, for example, in phase contrast Ono et al. (2013)
and polarization scanning microscopy Israel et al. (2014), usually limited to
the case of N=2. However, the generation and preservation of NOON states
involving a higher number of particles are extremely demanding, so their
usability in a real-world application is questionable. More practical is the
use of squeezed states Caves (1981); Xiao et al. (1987); Aasi et al. (2013);
Schnabel (2017); Gatto et al. (2022). Non-linear interferometry involving a
parametric amplifier instead of a beam splitter for mixing the light mode is
promising for some applications, especially because the detection can be done
at a wavelength different from the probing one Kalashnikov et al. (2016);
Töpfer et al. (2022) and the quantum advantage is independent of the detection
noise Manceau et al. (2017). Moreover, with some remarkable exceptions
Camphausen et al. (2021); Frascella et al. (2019), these interferometric
schemes do not provide spatial resolution or require raster scanning for
extended samples.

Other phase imaging methods born in the quantum domain exploit second-order
intensity correlation (or two-photon coincidence) among signal and idler beams
of SPDC to retrieve the phase information. In contrast, the first-order
intensity measurement of either the signal or the idler arm does not show
interference Gatti et al. (2004). These techniques include ghost imaging and
diffraction Strekalov et al. (1995); Valencia et al. (2005); Zhang et al.
(2005); Shapiro and Boyd (2012); Losero et al. (2019) quantum holography Lemos
et al. (2014); Vinu et al. (2020); Devaux et al. (2019); Defienne et al.
(2021), quantum Fourier ptychography Aidukas et al. (2019) and phase
reconfigurable contrast microscopy Hodgson et al. (2023). In general, the
signal-to-noise ratio (SNR) is smaller compared to direct first-order
measurement if considering the same number of impinging photons on the object.
However, some advantages can be found in some cases at few photon illumination
levels, for example, the rejection of independent external noise Meda et al.
(2017); Erkmen and Shapiro (2009); Brida et al. (2011); Morris et al. (2015),
robustness through turbulence and scattering Dixon et al. (2011); Bina et al.
(2013).

Here we present a quantitative non-interferometric quantum-enhanced phase
imaging (NIQPI) scheme exploiting quantum correlations that do not belong to
any of the techniques mentioned above since it does not involve either
interference or second-order intensity correlations. In fact, only first-order
intensity in both branches is measured, so the full-field phase retrieval is
obtained in real-time by single-shot measurement. We will demonstrate, both
theoretically and experimentally that the method can provide a clear advantage
compared to the corresponding classical direct imaging thanks to the quantum
correlation.

The NIQPI protocol exploits the scheme depicted in Fig. 1. We consider two
quantum correlated beams produced by the spontaneous down-conversion process
(SPDC), usually dubbed as signal beam ($s$) and idler ($i$) beam, with
intensity patterns that are perfectly identical in the far-field, point-by-
point. Even the shot noise fluctuation is, in principle, perfectly reproduced
in the two beams, which is impossible in the classical domain. The far field
of SPDC is imaged at the sensors of a highly efficient and low-noise CCD
camera. Only the signal beam probes the object, while the idler one is used as
the reference for the noise. When the object is placed close to the far field
but not exactly there, it produces an intensity perturbation on the signal
photons propagation that is registered at the CCD camera. In particular, by
measuring the signal intensity pattern $I(\bm{x},\pm dz)$ at the detection
plane for two different ’defocused’ object positions along the $z$-axis,
namely, $+dz$ and $-dz$, it is possible to reconstruct the phase profile
$\phi(\bm{x},z=0)$, by solving the so-called transport of intensity equation
(TIE) Teague (1983):

$-k\frac{\partial}{\partial
z}I(\bm{x},z)=\nabla_{\bm{x}}\cdot\left[I(\bm{x},0)\nabla\phi(\bm{x},0)\right]$
(1)

where the derivative is approximated by the finite difference of two
measurements out of focus, $\frac{\partial}{\partial
z}I(\bm{x},z)\approx[I(\bm{x},dz)-I(\bm{x},-dz)]/(2dz)$ and $I(\bm{x},z=0)$ is
the far field intensity of the source. TIE is experimentally easy and
computationally efficient as compared to the conventional phase retrieval
techniques and, under suitable hypotheses, the method leads to a unique and
quantitative wide-field image of the phase profile Teague (1983); Paganin and
Nugent (1998); Zuo et al. (2020). However, the reconstruction obtained in the
signal arm can be strongly affected by the detection noise and by the shot
noise if low illumination is used (see $M\&M$ for a detailed discussion). On
the one hand, a faithful reconstruction through Eq. (1) requires a small
defocus distance $|dz|$ in order to well approximate the derivative on its
right-hand side. But, on the other hand, if $|dz|$ is too small, the effect of
the phase gradient on the measured intensity becomes negligible and can be
covered entirely by the shot noise. Here, we show that the noise pattern
acquired on the idler beam can be used to reduce the effect of the shot noise
in the signal beam, enhancing the overall phase image reconstruction and
reducing the uncertainty on the quantitative spatially resolved phase
retrieval Ortolano et al. (2019). NIQPI can work with partially coherent light
and has some advantages compared to interferometric schemes: it can be
directly applied to wide-field transmission microscopy settings and it is
intrinsically more stable than an interferometric setup Zuo et al. (2020).
Moreover, since NIQPI is based on free propagation effect, it can be realized
without using lenses and optical components, thus being particularly suitable
in extreme UV or X-Ray imaging, where optical components are not efficient but
where SPDC sources are available and quantum enhanced detection has been
already demonstrated Borodin et al. (2016); Sofer et al. (2019).

Figure 1: Scheme of the NIQPR. Two correlated beams labeled signal ($s$) and
idler ($i$) are generated by the spontaneous parametric down conversion (SPDC)
pumped by a CW laser @405nm and propagate through an imaging system composed
of two lenses ($L_{1}$ is the far field lens with focal length $F=1$ cm and
$L_{2}$ is the imaging lens with focal length of 3 cm) and a test object. An
interference filter (IF) is used to select a bandwidth of 40 nm around the
degenerate wavelength (@810nm) and to block the pump. $L_{2}$ images the far
field plane on the camera chip around the focal plane with a magnification
factor of about 8. The object is placed near to the far field of the source,
and only the probe beam interacts with it. Phase information can be retrieved
from intensity measurements taken at some out of focus ($\pm dz$) planes.
Figure 2: Sample. Pure phase objects used in the experiment are sketched.
Figure 3: Experimental Reconstruction of the $``\pi$” sample as a function of
the defocusing distance. First row presents the phase reconstruction when 100
intensity patterns are used. Second and third rows show the single frame
reconstructions for the classical and the quantum case, respectively. The size
of each image is $80\times 80$ $\text{pix}^{2}$

##  II Results

In our experiment, the number of photons per pixel per frame is about
$n\approx 10^{3}$, so that for the purpose of this work we can substitute the
continuous quantity $I(\bm{x})$ appearing in Eq. (1) with the number of
photons detected by the pixel at the coordinate $\bm{x}$. Actually, before the
TIE algorithm, we apply an averaging filter of size $d=4$ to the intensity
image, that consists in replacing the count in each pixel by the average count
of a neighborhood of size $4\times 4\;\text{pix}^{2}$ around it, so that the
final image conserves the same number of pixels. However, the averaging filter
does not have any influence on the classical reconstructions, neither positive
nor negative, while it improves the quantum reconstruction (see discussion in
M$\&$M and related Fig. 7). From now on we will refer to $I(\bm{x})$ with that
meaning, namely after the application of such averaging filter.

It is essential to point out that the SPDC source operates in the regime of
very low photon number per spatio-temporal mode. In this limit, the photon
statistics follows a Poisson distribution (see M$\&$M Sec.IV for details). So,
aside from the negligible contribution of electronic readout noise, the
measurement on the single beam is shot noise limited.

We image pure phase objects reported in Fig. 2 with 66 $\pm$ 3 nm thickness
estimated by profilometer (DektakXT, Bruker). It corresponds to a phase shift
of 0.230 $\pm$ 0.012 rad @ 810 nm, the central degeneracy frequency of the
SPDC photons. The samples have been realized by etching structures on a UV-
fused Silica glass window using buffered oxide etch.

Fig. 3 shows the experimental reconstructions of the $``\pi$”-shaped phase
sample of Fig. 2 as a function of the defocussing distance $dz$. Each pixel of
the phase image corresponds to a transverse resolution of about $5\mu$m in the
object plane. As a reference, the first row of Fig. 3 shows the phase
retrieved averaging 100 shots, so the shot noise effect is estimated to be
negligible compared to the other sources of disturbance. However, even in this
case, the reconstruction at small $dz$ is not perfect because of the well-
known sampling error due to the discretization of the image, while at large
defocussing the finite approximation of the derivative in $z$ fails,
essentially producing blurred images. These two opposite trends determine a
defocussing distance for which the reconstruction is optimal. The second row
of Fig. 3 shows the reconstructions obtained by single frame intensities
$I_{s}(\bm{x_{s}},\pm dz)$ measured at the CCD camera in the signal arm. In
this case, the shot noise dominates and yields a drop in the reconstruction
quality for all values of $dz$. How the noise on the intensity propagates to
the phase reconstruction through the TIE is discussed in M&M. In particular,
the region of smaller $dz$ is the most affected since the intensity variation
produced by the phase gradient is still small and is almost completely hidden
in the shot noise.

In order to take advantage of the quantum correlations here, we propose to
replace into the Eq. (1) the single beam intensity with the following one
Moreau et al. (2017); Losero et al. (2018); Ruo-Berchera et al. (2020):

$I_{s-i}(\bm{x},z)=I_{s}(\bm{x_{s}},z)-k_{opt}\delta I_{i}(\bm{x_{i}},0).$ (2)

where $\delta X\equiv\langle X\rangle-X$ represents the quantum fluctuation of
the operator $X$, and $\langle\cdot\rangle$ is the quantum expectation value.
In fact, the second term in Eq. (2) is meant to compensate for the quantum
fluctuation of the signal pattern exploiting the local correlation between
probe and reference beams. The factor $k_{opt}$ is a parameter chosen to
minimize the residual fluctuation $\langle\delta^{2}I_{s-i}\rangle$ and can be
evaluated experimentally by calibration of the system since it is related to
the detection efficiency. A phenomenological model describing noise reduction
is discussed in M$\&$M. It turns out that the fluctuation of the quantity in
Eq. (2) is reduced with respect to the shot noise according to the following
expression:

$\displaystyle\langle\delta^{2}I_{s-i}\rangle$ $\displaystyle=$
$\displaystyle\left[1-\left(1-\alpha\right)^{2}\eta^{2}\right]\langle
I(0)\rangle,$ (3)

where $0<\eta<1$ is the heralding efficiency, namely the probability of
detecting an idler photon in the pixel in $\bm{x}_{i}$ conditioned to the
detection of the correlated signal photon in the pixel in $\bm{x}_{s}$ (see
M$\&$M section). The parameter $\alpha$ is the average fraction of photons
that deviate from the original path due to the phase object and depends on the
average phase gradient. It can be experimentally evaluated as the spatial
average of the quantity
$\alpha(\bm{x})\equiv\langle|I(\bm{x}_{s},0)-I(\bm{x}_{s},dz)|\rangle/\langle
I(\bm{x}_{s},0)\rangle$. Eq. (3) states that the intensity fluctuation is
reduced below the shot noise by a factor that depends on the efficiency in
detecting quantum correlation and that it is effective if the object is weakly
affecting the intensity distribution, namely when $\alpha\ll 1$. In our
experiment, following the absolute calibration method reported in Meda et al.
(2015); Avella et al. (2016); Samantaray et al. (2017), we estimate
$\eta=0.57$ for the particular case of averaging filter size $d=4$. The value
of $\alpha$ for the faint object considered is very small, for example we
estimated $\alpha=7\cdot 10^{-3}$ for $dz=0.1$ mm.

The third row of Fig. 3 reports the reconstructions when the shot noise has
been reduced using quantum correlations between probe and reference, according
to Eq. (3). A general improvement of the reconstruction can be appreciated. As
expected, the noise reduction is more evident at smaller $dz$ leading to an
improvement in the reconstruction of higher spatial frequency.

Figure 4: Pearson correlation between reconstructed and reference images. The
light-blue and yellow curves are the result of a Fourier optics based
simulation. The line-width is the confidence interval of one standard
deviation after an average over 100 reconstructions for each $dz$. The
experimental points are represented as purple and yellow dots with uncertainty
bar also corresponding to one standard deviation. The red curve corresponds to
the reconstruction obtained by summing of 100 intensity patterns, where the
shot noise becomes negligible (in this case, quantum and classical correlation
overlaps).

A quantitative analysis of the quality of the reconstructions and of the
quantum advantage can be performed by evaluating the Pearson correlation
coefficient between the reference phase image and the reconstructed one. The
Pearson coefficient is defined as,

$\mathcal{C}=\frac{\sum_{\bm{x}}(\phi_{r}(\bm{x})-\bar{\phi_{r}})(\phi(\bm{x})-\bar{\phi})}{\sqrt{\text{Var}[\phi_{r}]\text{Var}[\phi]}}$
(4)

where $\bar{\phi}$ and Var$[\phi]$ denote the spatial mean and variance of the
phase image $\phi$, and provides a simple and commonly used figure of merit to
quantify the similarity between the two images. Fig. 4, shows the Pearson
coefficient as a function of the defocusing. Each curve has a correspondence
with each image strip in Fig. 3. The red curve corresponds to the
reconstruction using 100 frames, where shot noise is negligible (corresponding
to the first strip in Fig. 3). The lower curves present the performance of
single frame experimental reconstructions, both quantum and classical,
obtained from a simulation. Experimental points are well in agreement with
these simulations. As expected, according to this figure of merit, an optimal
reconstruction is reached for the intermediate value of defocusing. The
quantum advantage is confirmed in terms of correlation with the reference
image.

Figure 5: Phase estimation. A The estimated value of the phase step (average
of the rectangular selected region) is plotted at different defocusing
distances. Experimental points for the classical (yellow dot) and the quantum
(purple dot) phase retrieval are compared with the simulations (reported for
one standard deviation confidence band). For comparison, we also report the
nominal value, estimated by the profilometer in reflection, of the phase step
difference between the etched/non-etched areas. B The uncertainty in the phase
estimation for quantum and classical cases demonstrate the quantum advantage.

Besides the correct reconstruction of the complex phase profile assessed by
the correlation coefficient, in many cases, it is of utmost importance to
achieve a quantitative estimation of the phase. Fig. 5A reports the phase
value estimated as a function of $dz$, where, for the analysis, we have
selected the region indicated in the red rectangle in the insets. The results
indicate that the phase step is reconstructed without bias compared to the
nominal value (red horizontal line) up to $dz=100\ \mu$m for both the
classical and the quantum case. The experimental points and their error bars
agree with the confidence bandwidths provided by the simulations. However, the
uncertainty on the estimated value is smaller for the quantum case. The
quantum advantage, reported in Fig. 5B, is relatively constant in the range
considered up to a 40$\%$. The estimated phase value drops down for higher
defocusing distances because of the blurring of the image evident from the
first row in Fig. 3. However, it is clear that in this region, the method does
not provide useful reconstructions simply because the approximation of the
derivative in Eq. (1) is no longer valid.

We have also tested a different object, the pattern of regular squares
represented in Fig. 2. In Fig. 6A we report two examples of reconstructions,
at $dz=50\ \mu$m and $dz=100\ \mu$m, respectively. In Fig. 6B, the Pearson
coefficient is reported alongside the simulations. The quantum advantage is
comparable to the one obtained for the $``\pi"$, showing its robustness and
independence from the particular spatial shape of the sample. Although the
quantitative analysis of the Pearson coefficient confirms a similar quantum
advantage as the one reported in Fig. 4, by looking at the images, it appears
that the quantum advantage in the localization of dots could be even larger,
indicating the possibility of significant advantages for specific tasks
related to the recognition of finer spatial details.

In summary, these results demonstrate, for the first time, a significant
advantage of quantum phase imaging, that can be further extended in the future
with various potentially significant applications.

Figure 6: Single frame reconstruction of the squares pattern. A Examples of
classical and quantum reconstructions of the sample with squares in Fig.2
(right-hand side) for two different defocussing distances. B Pearson
correlation coefficient between the reconstructed phase image by a single
intensity frame and the reference image as a function of the defocussing.

##  III Conclusions

Here, we have demonstrated a genuine quantum enhancement in non-
interferometric quantitative phase imaging, showing that the spatially multi-
mode quantum correlations can be used to reduce the detrimental effect of
quantum noise in phase reconstruction. The present NIQPI scheme exports the
classical methods known as the transport of intensity equation to the quantum
regime, which provides real-time wide-field phase imaging and the quantitative
local estimation of the phase. The last aspect is fundamental for many
applications, providing reliable information on the object’s internal
parameters related to the phase.

We point out that, compared to the imaging of an amplitude object Brida et al.
(2010); Samantaray et al. (2017); Ruo-Berchera et al. (2020); Losero et al.
(2018), the propagation of the shot noise of the intensity measurement to the
retrieved phase in the NIQPI is not as trivial. On the one side, the noise
reduction allows reaching smaller defocussing distances for a better
approximation of the derivative in the TIE, thus providing a more faithful
reconstruction of the phase details. On the other side, artifacts due to the
noise appear at low spatial frequencies (see discussion in M$\&$M and Fig. 3)
and are known to affect mainly the reconstruction of slow phase curvature,
which produces weaker signal intensity signals Paganin et al. (2004). In this
work, in order to obtain a quantitative validation of the protocol, we studied
binary phase objects with sharp borders. However, it is expected that for an
object with smoother phase changes, e.g., biological samples, the quantum
advantage can be even more significant.

##  IV Materials and Methods

###  IV.1 Phase retrieval by TIE

A non-interferometric method Teague (1983) to retrieve the phase of an object
consists of probing the object with a beam and measuring the intensity
$I(\bm{x},z=0)$ at the object plane of coordinate $\bm{x}$ and its derivative
along the propagation axis $z$. The derivative is computed by a finite
difference of two measurements out-of-focus of a distance $dz$,
$\frac{\partial}{\partial z}I(\bm{x},z)\approx\Delta I(\bm{x},dz)/(2dz)$ with
$\Delta I(\bm{x},dz)=I(\bm{x},dz)-I(\bm{x},-dz)$. Under paraxial
approximation, the phase is retrieved using the TIE reported in Eq. (1).

Using energy conservation considerations, this equation has been proven valid
even with partially coherent sources Paganin and Nugent (1998). This feature
makes the TIE approach perfectly suited for being used with light from SPDC,
where transverse and longitudinal coherence lengths can be much smaller than
the object size and the whole illuminating beam. This is not a secondary
aspect since it is exactly due to the multimode nature of the emission that
correlation shows a local character and shot noise can be removed pixel-by-
pixel in the image. The solution of the Eq. (1) is unique provided that the
on-focus intensity $I(\bm{x},0)$ and the intensity derivative along $z$ are
known and the phase is continuous.
Following the analysis in Paganin et al. (2004), we assume that the intensity
is varying sufficiently slowly that the effects of phase curvature dominate
the intensity derivative, so that the right side of Eq. (1) can be safely
approximated as $I_{0}\nabla^{2}\phi(\bm{x},0)$. Then, we consider for a
moment that the only contribution to the finite difference $\Delta
I(\bm{x},\delta z)$ is the noise fluctuation on the intensity measurement,
$\sigma(\bm{x})$ . In this case, substituting the latter in Eq. (1), one has
that the phase artifacts in the reconstruction due to the noise are:

$-k\frac{\sigma(\bm{x})}{\sqrt{2}I_{0}\delta
z}=\nabla_{\bm{x}}^{2}\phi_{noise}(\bm{x}).$ (5)

The noise is assumed independent in the two planes $+\delta z$ and $-\delta
z$, so it has been combined in quadrature. The Eq. (5) can be solved by taking
the Fourier transform on both sides, leading to

$k\frac{\tilde{\sigma}(\bm{q})}{4\pi^{2}\sqrt{2}I_{0}\delta
z|\bm{q}|^{2}}=\tilde{\phi}_{noise}(\bm{q})$ (6)

where the tilde indicate the Fourier transform and $\bm{q}$ is the spatial
frequency. The damping factor $|\bm{q}|^{2}$ of the higher frequencies at the
denominator of Eq. (6) and the fact that the quantum noise (shot noise) has a
flat white spectrum $\sigma_{SN}(\bm{q})=\sigma_{SN}$, indicate that the
effect of shot noise is to generate artifacts especially at lower frequencies,
which are not intrinsically suppressed by the phase retrieval algorithm. This
noise at low-frequencies is evident in the single frame images reported in
Fig. 3. Moreover, in the direct propagation problem, higher frequencies of the
phase object generate a stronger effect on the intensity. Thus, based on these
remarks, the regions with rapid changes in the phase (higher frequency) are
better reconstructed than the ones characterized by slow curvature.

### Experimental details: Source, Sample, Detection

Source: In the experiment, we use SPDC in the low gain regime in which a
photon of the pump beam (p) (CW laser @405nm), thanks to the interaction with
a bulk beta-barium borate non-linear crystal as long as 15 mm, have a small
probability of converting in a couple of photons, usually called signal (s)
and idler (i), subject to conservation of energy,
$\omega_{p}=\omega_{s}+\omega_{i}$, and of momentum,
$\textbf{k}_{p}=\textbf{k}_{s}+\textbf{k}_{i}$. Thus, under the plane wave
pump approximation, signal and idler photons are perfectly correlated in
frequency and direction $\bm{q}_{s}=-\bm{q}_{i}$ (assuming $\bm{q}_{p}=0$),
although their individual spectrum is broadband both in space and frequency.
In the far field, obtained at the focal plane of a thin lens in a $f-f$
configuration, a transverse mode $\bm{q}$ is mapped in a single transverse
position $\bm{x}$ according to the transformation
$(2cf/\omega)\bm{q}\rightarrow\bm{x}$, so that momentum correlation translate
in a position correlation, $\bm{x}_{s}=-\bm{x}_{i}$ (for degenerate frequency
$\omega_{s}\approx\omega_{i}$). Signal and idler photons generate two
symmetrical intensity noise patterns, and pairs of symmetric pixels of a
camera will detect the same number of photons in the ideal lossless scenario
in the same time window. Thus, quantum fluctuation affecting the object plane
in the signal beam can be measured independently on the idler beam. The
coherence time of the SPDC sources is typically of hundreds of fs and the
spatial coherence in the far field is proportional to the inverse of the pump
transverse size. The number of photons per spatial-temporal mode is very low,
$\sim 10^{-8}$, in general, the time bandwidth of the detector is orders of
magnitude smaller than the inverse of the coherence time. Although the single
SPDC mode is thermal, in the limit above, the detected multi-thermal photon
statistics are indistinguishable from a Poisson distribution Meda et al.
(2017).

For a Gaussian distributed pump with angular full-width-half-maximum (FWHM) of
$\Delta q$ the spatial cross-correlation is also Gaussian with FWHM of $\Delta
x=2\sqrt{2\log 2}\sigma=(2cf/\omega_{p})\Delta q$: if a signal photon is
detected in the position $\bm{x}_{s}$ the twin idler photon will be detected
according to that Gaussian probability centered in $\bm{x}_{i}=-\bm{x}_{s}$.
In the experiment we have estimated $\Delta x\approx 5\mu$m

Test sample: The structures are etched on to a fused Silica glass window
(WG41010-A, Thorlabs) with an anti-reflection coating on one side. The window
is coated with positive PMMA resist and the design is exposed using electron
beams. The exposed structures are developed using a MIBK-IPA solution. After
development, the window is submerged in a buffered oxide etch for 30 seconds
to etch the structures into the window. The etch depth is determined by the
submergence time. The unexposed resist is then removed using acetone solution.

Detection: We measure the SPDC emission and the effect of the phase object by
imaging the far field of the source at the sensor of a CCD camera operated in
the conventional linear mode. Each pixel delivers a count proportional to the
number of incident photons. The proportionality coefficient is the product of
the (electronic gain) that has been carefully calibrated and the quantum
efficiency of the camera, nominally above 95% @810 nm. The electronic readout
noise is 4$e^{-}/(\text{pix}\cdot\text{frame})$. The number of photons
detected per pixel per frame is $10^{3}$, where the integration time of the
camera is set to $100$ ms, meaning that the shot noise dominates compared to
the electronic noise.

Because of the finite cross-correlation area defined in the previous section
of the M$\&$M, in order to collect most of the correlated photons, two
symmetrically placed detectors (or pixels) must have areas larger than the
cross-coherence area. Pixel size is 13 $\mu$m and a binning of $3\times 3$ is
performed to set the resolution to 5 $\mu$m at the object plane, which matches
the measured cross-coherence area. Actually, the heralding efficiency $\eta$,
i.e. the probability of detecting an idler photon conditioned to the prior
detection of the twin signal photon, depends on the pixel size $L$ and
possible misalignment $\bm{\Delta}$ of the two pixels compared to the optimal
positions, according to this expression:

$\eta(L,\bm{\Delta})=\eta_{0}L^{-2}\int_{L\times L}d\bm{x}_{s}\int_{L\times
L}d\bm{x}_{i}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(\bm{x}_{i}+\bm{x}_{s}+\bm{\Delta})^{2}}{2\sigma^{2}}}$
(7)

where, $\eta_{0}$ is the single photon detection efficiency. As the the pixel
size $L$ increases with respect to the coherence area $\Delta x$, we have that
$\eta\longmapsto\eta_{0}$.

As a consequence of that, in Eq. (3), the noise reduction depends on the pixel
size used for the measurement. This trade-off between the quantum advantage
and the spatial resolution of the intensity measurement has been reported and
analyzed in the context of sub-shot-noise imaging of amplitude objects
Samantaray et al. (2017); Ruo-Berchera et al. (2020). However, in the present
imaging of pure phase objects, the resolution issue has less impact. In fact,
as it is described in the first section of this M$\&$M, the solution of the
TIE tends by itself to suppress the higher frequency component of the
intensity perturbation. Thus, to some extent, a reduction of resolution in the
intensity measurement does not affect the phase reconstruction. In the
experiment, in order to increase the heralding efficiency, and thus the
quantum enhancement, we use an averaging filter to the intensity image that
substitutes the count in each pixel by the average of a neighborhood of size
$d\times d\;\text{pix}^{2}$ around it. The quantum correlations are then
enhanced because the effective integration area is larger, while the number of
pixels in the final image is unvaried. In Fig. 7, we report the quality of the
phase reconstruction, evaluated in terms of the Pearson correlation with the
reference image in Fig. 2, as a function of the averaging size. On the one
hand, the quantum reconstruction is enhanced as expected when the effective
resolution in the intensity measurement decreases ($d$ increases). On the
other hand, the classical reconstruction is unaffected, confirming that
classically we do not have any negative issue related to the poorer resolution
in the intensity pattern. In summary, moderate use of the averaging filter to
enhance the quantum effects is perfectly legitimate in this context.

Figure 7: Pearson correlation as a function of averaging filter size. The
purple (yellow) dots represent the values corresponding to the quantum
(classical) experimental reconstructions. The quantum (classical) confidence
bands at one standard deviation are also shown in turquoise (yellow).

### Model for the noise reduction

According to the scheme in Fig. 1, the signal beam of SPDC probes the object,
while the idler beam is used as a reference for the noise. When the object is
inserted with a defocusing distance $dz$, the photons in the signal beam are
deflected, creating local depletion or accumulation of photons at the
detection plane, and the perturbed intensity can be written as:

$I_{s}(\bm{x},z)=I_{s}(\bm{x},0)-\Delta I_{-}(\bm{x})+\Delta I_{+}(\bm{x}),$
(8)

where $I_{s}(\bm{x},0)$ is the unperturbed pattern and $\bm{x}$ indicates the
position of a pixel. The quantity $\Delta I_{-}(\bm{x})$ $(\Delta
I_{+}(\bm{x})$) represents the photons that are deflected out from (into) the
position $\bm{x}$. From now on, to simplify the notation, the spatial average
of the quantities is simply indicated by dropping the spatial dependence on
$\bm{x}$. Since the total number of photons is conserved, the spatial average
of the number of photons per pixel is unchanged, i.e. $I_{s}(z)=I_{s}(0)$ and
thus $\Delta I_{-}=\Delta I_{+}$. The loss of photons can be described as the
action of a beam splitter of transmittance $1-\alpha$ (average value) so that,
the quantum expectation value for the $\Delta I_{-}$ is simply $\langle\Delta
I_{-}\rangle=\alpha\;\langle I_{s}(0)\rangle=\langle\Delta I_{+}\rangle$ Meda
et al. (2017). In this work, we are interested in small perturbations that can
be hidden or strongly affected by the quantum noise, so we will assume
$\alpha\ll 1$.

In order to reduce spatial intensity fluctuation we replace in the TIE the
quantity in Eq. (8) with the one in Eq. (2) involving the idler measurement.

The optimal factor $k_{opt}$ appearing there is chosen to minimize the
residual fluctuation, by imposing $\frac{\partial}{\partial
k}\langle\delta^{2}I_{s-i}(\bm{x},z)\rangle=0$. We obtain,

$\displaystyle k_{opt}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{\langle\delta I_{s}(\bm{x}_{s},z)\delta
I_{i}(\bm{x}_{i},0)\rangle}{\langle\delta^{2}I_{i}(\bm{x}_{i},0)\rangle},$ (9)
$\displaystyle\langle\delta^{2}I_{s-i}(\bm{x},z)\rangle$ $\displaystyle=$
$\displaystyle\langle\delta^{2}I_{s}(\bm{x}_{s},z)\rangle-\frac{\langle\delta
I_{s}(\bm{x}_{s},z)\delta
I_{i}(\bm{x}_{i},0)\rangle^{2}}{\langle\delta^{2}I_{i}(\bm{x}_{i},0)\rangle}.$

According to the Poisson distribution of the detected photon, we can replace
the variance of the intensities appearing in Eq. (9) with the respective
quantum mean values. In particular, by performing the spatial averaging, one
gets $\langle\delta^{2}I_{i}(0)\rangle=\langle
I_{i}(0)\rangle=\langle\delta^{2}I_{s}(z)\rangle=\langle I_{i}(z)\rangle$. For
the calculation of the covariance in Eq. (9), note that $I_{s}(\bm{x}_{s},z)$
and $I_{i}(\bm{x}_{i},0)$ are correlated only for the fraction of photons that
are not lost, namely not deviated from the path due to phase effect on the
propagation along $z$. Thus, after spatial averaging Meda et al. (2017):

$\displaystyle\langle\delta I_{s}(z)\delta I_{i}(0)\rangle$ $\displaystyle=$
$\displaystyle(1-\alpha)\langle\delta I_{s}(0)\delta I_{i}(0)\rangle$ (10)
$\displaystyle=$ $\displaystyle\eta(1-\alpha)\langle I_{s}(0)\rangle.$ (11)

The last equality is justified again using the Poisson hypothesis, and
introducing the heralding efficiency $\eta$ that spoils the otherwise perfect
signal-idler correlation. By using Eq. (11), and the Poisson hypothesis above,
we can rewrite the Eq.s (9) as,

$\displaystyle k_{opt}$ $\displaystyle=$ $\displaystyle(1-\alpha)\,\eta$ (12)
$\displaystyle\langle\delta^{2}I_{s-i}\rangle$ $\displaystyle=$
$\displaystyle\left[1-\left(1-\alpha\right)^{2}\eta^{2}\right]\langle
I_{s}(0)\rangle$ (13)

##  V Acknowledgments

This project 20FUN02 POLight has received funding from the EMPIR programme co-
financed by the Participating States and from the European Union’s Horizon
2020 research and innovation programme. S.S. would like to acknowledge Dr.
Iman E. Zadeh for his supervision for the sample fabrication.

### Author contributions

GO and IRB devised the scheme of NIQPI. PB participated in the realization of
the setup and preliminary measurement with GO, while AP and CN performed the
final experimental acquisitions, which were realised in the laboratories of
the research sector directed by MG. The samples have been prepared and
characterized by SS and SP. Simulations and data analysis have been carried
out by GO and AP with the help of IRB. IRB and MG supervised the whole
project. IRB wrote the paper with the contribution of all authors.

### Competing Interests

The authors declare no competing interest.

### Data availability

All data needed to evaluate the conclusions are reported in the paper. Further
data, for reproducibility of the results, will be available in a public
repository linked to the published paper.

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# Waveguide-integrated and portable optomechanical magnetometer

Fernando Gotardo1,2  Benjamin J. Carey1,2  Hamish Greenall1,2  Glen I.
Harris1,2  Erick Romero1,2  Douglas Bulla4  Elizabeth M. Bridge5  James S.
Bennett3  Scott Foster4  and Warwick P. Bowen1,2  1School of Mathematics and
Physics, The University of Queensland, St Lucia, Queensland 4067, Australia.
2ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland
4067, Australia.
3Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4072,
Australia.
4Australian Government Department of Defence Science and Technology,
Edinburgh,
South Australia 5111, Australia.
5Quantum Brilliance, Acton, Australian Capital Territory 2601, Australia
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Optomechanical magnetometers enable highly sensitive magnetic field sensing.
However, all such magnetometers to date have been optically excited and read-
out either via free space or a tapered optical fiber. This limits their
scalability and integrability, and ultimately their range of applications.
Here, we present an optomechanical magnetometer that is excited and read-out
via a suspended optical waveguide fabricated on the same silicon chip as the
magnetometer. Moreover, we demonstrate that thermomechanical noise limited
sensitivity is possible using portable electronics and laser. The magnetometer
employs a silica microdisk resonator selectively sputtered with a
magnetostrictive film of galfenol (FeGa) which induces a resonant frequency
shift in response to an external magnetic field. Experimental results reveal
the retention of high quality-factor optical whispering gallery mode
resonances whilst also demonstrating high sensitivity and dynamic range in
ambient conditions. The use of off-the-shelf portable electronics without
compromising sensor performance demonstrates promise for applications.

##  1 Introduction

In recent years, optomechanical sensors have emerged as a powerful new type of
sensor for stimuli ranging from temperature [1] to pressure [2], forces [3, 4]
and acceleration [5]. Such sensors leverage both optical and mechanical
resonances to enable high sensitivity and high spatial resolution.
Optomechanical magnetometers are one example [6, 7, 8, 9], that are attractive
due to the crucial role that high-sensitivity magnetometers play in
applications ranging from fundamental research to medical diagnostics [10,
11], mineral exploration and surveying [12, 13], magnetic anomaly detection
[14, 15], and navigation [16, 17, 18]. Owing to their photonic nature, they
are light-weight, small-sized, low power [16, 7, 19], and can be exceedingly
resilient to detriments such as electrical interference and radiation.

At their current state of development, optomechanical magnetometers have
achieved tens-of-micron spatial resolution with sensitivities ranging from
several n$\mathrm{T}/\sqrt{\mathrm{Hz}}$ down to tens of
p$\mathrm{T}/\sqrt{\mathrm{Hz}}$ [20, 7, 16, 8]. The demonstrated sensitivity
is competitive with SQUID and diamond magnetometers of similar size but
without the need for cryogenics or high-powered optical and RF components [16,
21, 22], with theoretical models suggesting that sensitivities in the low, or
even sub-, femtotesla may be possible in future [23].

To-date, optomechanical magnetometers have used free-space or tapered-fiber
coupling for optical excitation and readout [22, 24, 9]. This prevents them
from being fully integrated on a silicon chip. Furthermore, those
demonstrations used the magnetostrictive material Terfenol-D to convert
magnetic fields into a measurable strain [6, 7]. Which is difficult to
reproducibly deposit and sensitive to corrosion and oxidation [25]. Works with
this material have also relied on high performance laser and electronic
systems [19]. Together, this introduces significant challenges for
applications outside of laboratory environments. The work reported here seeks
to address these challenges.

We develop an optomechanical magnetometer that is efficiently coupled to an
on-chip suspended waveguide, by employing galfenol (Fe82Ga18) for the first
time to convert magnetic fields to a measurable mechanical signal. This
provides low-cost sputter-coated thin-films with improved resilience to
corrosion and oxidation [25] and good magnetostriction ($\sim$400 ppm) at
lower saturation fields[26, 25]. We also use portable electronic and laser
systems to control and read-out the magnetometer, showing that they allow
performance that is limited by fundamental thermomechanical noise rather than
laser or electronic noise. Together, this represents progress towards robust,
portable and high performance mangetometers that could be employed in diverse
research and industrial settings.

##  2 Design and Simulation

###  2.1 Device Design and Functionality

The device design concept is depicted in Figure 1. It is based around a 100
$\upmu$m-diameter silica microdisk cavity on a silicon chip. This microdisk is
capable of supporting optical whispering galley modes (WGMs) throughout the
visible and near-infrared spectrum as well as megahertz frequency mechanical
resonances. A 1.5 $\upmu$m wide silica waveguide is fabricated from the same
layer of silica as the microdisk. Both microdisk and waveguide are undercut so
that the optical modes are confined to the silica, rather than leaking into
the higher-refractive index silicon substrate. The microdisk is suspended from
a central silicon pedestal. The waveguide is suspended using thin silica
tethers that are patterned along its length to reduce warping of the waveguide
that can be caused by the intrinsic stress present in the as-fabricated SiO2
films. Buckling of the waveguide could lead to severe bending losses, and out-
of-plane buckling can lead to inconsistent coupling between the waveguide and
optical cavity. The tethers are sub-wavelength (240 nm width) in order to
minimise optical scattering of light propagating within the waveguide. As
silica is lower refractive index than many other waveguide material (e.g.,
silicon) the guided wavelength is longer allowing for minimal scattering. The
waveguide is broadened (inverse-tapered) at input and output to efficiently
mode-match light into and out-of tapered optical fibers.

Figure 1: Design of the integrated SiO2 magnetometer. Here, laser light is
coupled to the trumpet waveguide via a tapered fiber. The waveguide is narrow
at the centre to optimize evanescent field coupling to the disk cavity. The
galfenol is sputtered on top of the cavity. The tethers support the waveguide,
preventing buckling.

The microdisk is coated with galfenol in a disk of diameter sufficiently
smaller than the disk diameter so as not to introduce optical absorption.
Galfenol is chosen because of its high magnetostriction, low fragility, and
low volatility [25]. When a magnetic field is applied, the expansion of the
galfenol induces strain in the microdisk, changing the optical path length and
hence the optical resonance frequency. The magnetostrictive response is
amplified at frequencies close to mechanical resonances of the microdisk,
leading to enhanced modulation of the intracavity optical field.

The light is coupled into the disk evanescently from the suspended waveguides
which are designed to support single mode propagation around 1550 nm. The
coupling of light from an optical fiber into the on-chip waveguide despite the
geometric mismatch (SMF-28 optical fiber has core and cladding diameters of 10
$\upmu$m compared to the 300 nm thickness of the waveguide) is facilitated by
mode-matched taper-drawn fibers (similar to those presented in [27]) and
trumpeted waveguide (4 $\upmu$m down to 1.5 $\upmu$m over a length of 30
$\upmu$m). This allows adiabatic coupling of light to and from the waveguides.
Further, the 4 $\upmu$m wide flared section acts as a semi-rigid anchor point
for the fiber, and its size reduces the requirement for extremely precise
positioning. This allows greater tolerance to imperfections in important
integration processes, such as bonding the fiber tip in place.

####  2.1.1 Finite-element Simulations

Finite Element Method (FEM) simulations performed with Ansys-Lumerical
software for the optical performance of the device are presented in Figure 2
a). Here the coupling from fiber to waveguide is studied by monitoring the
optical mode cross-section ($yz$) along the propagation direction ($x$). The
cross-sections shown in (i-iii) correspond to cross-sections of only fiber
(i), fiber & waveguide (ii), and only waveguide (iii). For simplicity, both
the fiber taper and the waveguide were considered uniform at the coupling
region. The taper was chosen to have a 1 $\upmu$m diameter with the optimum
waveguide width of 4 $\upmu$m then found using recursive simulations. From the
simulated cross-sections, we see that the optical mode migrates from the fiber
into the waveguide. We obtain a fiber-to-waveguide coupling efficiency of 60%
by taking the ratio of the optical power contained within the fiber at point
(i) and within the waveguide at point (iii).

Within the same simulation the waveguide-to-disk coupling and disk resonances
were also studied. Here the optical excitation frequency was swept and the
optical intensity across the geometry was recorded at each frequency. The
transmission efficiency across the device could then be calculated by
comparing the integrated intensity over the cross-sections of the input and
output of the waveguide (i.e., $T=I_{\text{out}}/I_{\text{in}}$, as measured
at points (iii) & (vi)). Fig. 2(a)(vi) shows the expected periodic
transmission dips when the frequency of the light matches WGMs of the
microdisk. The transmission is predicted to drop by as much as 70%, indicating
that efficient coupling of light into WGMs should be possible. The
correspondence of the dips with WGMs is confirmed in Fig. 2 a)(v), which shows
confinement of the light to the periphery of the disk when driven on resonance
(at a wavelength of 1551 nm in this case). To further corroborate the
confinement of the WGMs we performed an axisymmetric eigenmode-solver FEM
simulation in COMSOL Multiphysics (Figure 2 a) (iv)). This confirmed that the
WGM is contained within the outer 5 $\upmu$m of the disk, as expected.

The Free Spectral Range (FSR) of the optical resonances and corresponding
coupling were calculated from the simulated transmission in Fig. 2(a)(vi). We
find a simulated FSR of approximately 7 nm. This compares well to the expected
FSR given the circumference of the microdisk of:

$\Delta\lambda\approx\frac{\lambda^{2}}{n(\lambda)L}\approx 7.6\;\mathrm{nm},$
(1)

where $n(\lambda)$ is the effective refractive index of the cavity mode (taken
from Lumerical simulations to be 1.01) and $L$ is the length of the cavity
i.e., $L=$ 100$\pi\;\mathrm{\upmu m}$.

Figure 2: FEM simulations of the integrated optomechanical magnetometer.
a) The optical properties of the device. The cross-sectional optical mode
profiles (i-iii) at their corresponding green rectangles demonstrate the
evolution of the optical modes from fiber to SiO2 trumpet waveguide and (iv)
the cross-sectional optical mode inside the microdisk. (v) Depicts the optical
mode at the planar cross-section (yellow rectangle) of the device, and (vi)
shows the optical transmission spectrum of the system. (b) Mechanical
simulation revealing the resonance frequencies and their flexual mode-shapes

As evidenced by the results in Figure 2 a) (iv & v), the optical field of the
Whispering Gallery Mode (WGM) extends negligibly into the centre of the SiO2
disk. Hence, the addition of the optically absorbing magnetostrictive layer to
the disk’s centre should not significantly affect the quality of the optical
modes contained therein [20].

Using COMSOL Multiphysics, we performed further simulations to assess the
mechanical properties of the microdisk. As shown in Figure 2 b), we found the
mechanical eigenmodes by using fully three-dimensional geometry of the
released devices. This was necessitated because of the inclusion of stress-
release slots (discussed in 3) that break the axial symmetry of the mechanical
modes. The physical properties of the galfenol were taken from the datasheet
supplied by TdVib LLC. The lowest frequency mechanical flexural mode and two
lowest frequency crown modes are shown in Figure 2 b), with mechanical
frequencies of 3.12, 3.21, and 3.26 MHz, respectively.

##  3 Device Fabrication

The fabrication process used to produce the devices is outlined in Fig. 3 a)
(i). SiO2 (300 nm) on Si substrate wafers (500 $\upmu$m, 4") were diced into
square 15$\times$15 mm chips, large enough to fit more than 100 devices per
chip. Electron beam lithography was used to define patterns for galfenol
deposition and markers for subsequent lithography steps in the following way.
Two layers of PMMA resist were spin-coated (PMMA 950k and 495k at 2500 RPM)
onto of the SiO2/Si substrate, then patterned with an Electron-beam Pattern
Generator (EBPG) (Raith EBPG 5150) with 100 kV accelerating voltage and a 1200
$\upmu$C/cm2 dosage. Post exposure, the chips were developed in methyl
isobutyl ketone (MIBK) and rinsed with Isopropyl Alcohol (IPA).

To produce the markers, 5 nm of Ti and 50 nm of Au were e-beam evaporator
deposited (Temescal FC-2000) follow by a lift-process via submersion into
acetone and IPA. The galfenol films were then sputtered by magnetron DC
sputtering in an argon atmosphere (150 W, 2 mTorr) with a (3 inches dia.)
galfenol target. A seeding layer (5 nm Ta, plus a 16 nm Cu) and capping layer
(5 nm Ta) were used to promote adhesion and inhibit corrosion respectively.
Afterwards, the lift-off process was repeated, resulting in a 300 nm thick, 60
$\upmu$m diameter circular thin-film of galfenol on top of the SiO2 layer.

Figure 3: Representation of the fabrication process. a) PMMA e-beam resist
deposition. b) EBPG exposure and development. c). Galfenol sputter deposition.
d) lift-off. e) ARP e-beam resist deposition. f) Second EBPG exposure and
development. g) RIE of SiO2. h) Resist removal. i) Released devices after
undercutting by XeF2 etching of the Si layer.
SEM images of the final devices depicting j) trumpet wave guide and k) the
optomechanical cavity with galfenol layer in its centre and coupling
waveguide.

With markers produced and galfenol deposited, the waveguide and the disk
cavity structures were then defined. For this, 20 nm thick Ti prime adhesion
helper was spin-coated (4000 rpm) and baked (150°C, 15 min) follow by a layer
of ARP 6200.09 (CSAR-62, All Resist) 350 nm thick (1250 rpm spin-coat, 180°C
for 5 min bake). The chip was then patterned with the Raith EBPG 5150 (100 kV,
260 $\upmu$C/cm2). Proximity effect correction was performed using GenISys
Beamer software to ensure precision and reproducibility in the EBPG process.
Post exposure, the patterns were developed with All Resist AR600-546 and
rinsed with o-xylene and IPA.

RIE was used remove the unwanted SiO2 using an Oxford Instruments PlasmaPro
80. Here 25 sccm CHF3 and 45 sccm Ar at 200 W RF power for 12 min
anisotropically etched all the way through the SiO2 layer exposing the silicon
substrate. 50 sccm O2 at 50 W was then used to remove any residual resist.
Finally, the SiO2 structures were undercut by etching of the supporting
silicon with Xenon Difluoride (XeF2) gas (SPTS XACTIX e2 Series XeF2 etcher).
Here, 10 pulses of XeF2 gas at a pressure of 2 torr provides an isotropic etch
rate of about 1.4 $\upmu$m per pulse with a selectivity of >1000:1. This
removed the Si beneath the silica waveguide and WGMs of the microdisk whilst
leaving both the silica layer and galfenol unmarred.

SEM (Jeol 7800) imaging of the devices was performed in order to assess their
structural integrity. Fig. 3 j) & k) shows SEMs of the trumpet-shaped
waveguide ends for fiber coupling and the waveguide near the resonator,
supported by tethers to the main body of the wafer. It is apparent that the
waveguide shows no signs of buckling or collapse after the release process. It
can also be observed that the undercut beneath the silica layer is
approximately 18 $\upmu$m. This undercut extends under the disk, leaving
behind a silicon (Si) pedestal which is obscured by the galfenol coating
above. Measurements on a device with no galfenol revealed a pedestal width of
15 $\upmu$m (measured with a Zeta 300 3D optical profiler).

Stress release slots in the resonator were found to be necessary to prevent
buckling of the disk due to the inherent strain within both the SiO2 layer and
the galfenol film. However, as discussed in section 2, these slots are
expected to have negligible effect on the optical modes because they are
outside of the region containing appreciable intensity. The mechanical
simulation of Fig. 2 b) fully accounts for the effect of the slots on the
mechanical eigen-frequencies.

A critical parameter for consideration during fabrication of the device is the
distance between the disk and the waveguide at the coupling region ($d$). As
the light is coupled evanescently the coupling efficiency ($\kappa$) follows
the relation $\kappa\propto e^{-d}$ [28]. Devices with a range of waveguide-
microdisk coupling distances were fabricated in order to produce resonators
with optimum coupling strengths. The devices with near-critical coupling were
further investigated.

##  4 Device Performance

The experimental setup used to assess the performance of the integrated
magnetometers is depicted in Fig. 4a). Here, a continuously tuneable laser
(EXFO T100S-HP) supplied light to the resonator via tapered optical fibers
with a house-built test rig featuring two 3-axis translation stages for
precise positioning of the fibers. The transmitted light was then guided to a
Thorlabs (PDA100-CS) photodector (PD) and the photocurrent was analyzed with a
spectrum analyzer (Siglent SSA 3021x). A function generator (Rigol DG-1022)
was used to directly drive a home-wound coil (8 turns, 10 mm dia.) held
approximately 1 mm above the chip, producing a field of $\sim$50 $\upmu$TPP at
the surface of the chip (drive voltage of 10 VPP).

Figure 4: Experimental performance of the magnetometers. a) Schematic
demonstrating the experimental setup with corresponding photographs of the
laser system b), and devices on test-rig c) used for the measurements. d)
Optical transmission spectra of the devices with accompanying high-resolution
spectra around one of the WGM resonances (inset of e) which was used for the
sensing investigations. e) Power spectrum of the transmitted light with and
without externally applied magnetic field.

The emission wavelength of the laser was swept and the voltage output from the
PD (and hence power via the known responsivity and transimpedance gain of the
PD) was recorded to characterise the optical mode spectrum of fabricated
devices. The optical transmission spectrum of a typical device is presented in
Fig. 4 d), showing many dips in the transmission dips each associated to one
WGM. The observed FSR of $\approx$ 7 nm is in good agreement with the FSR as
determined from the FEM seen in Section 2 and Fig. 2a)(vi). On this device
(with a designed waveguide-microdisk separation of 550 nm) we find that the
WGM at a wavelength of 1551 nm (enclosed by the dashed box in Fig. 4 d)) is
close to critically coupled, with a transmission dip of $\sim 95\%$.

Because the 1551 nm WMG mode is close to critically coupled, we select it to
perform magnetic field measurements. A high-resolution sweep across the mode
is shown in the inset of Fig. 4e). From this the optical $Q$ of the cavity is
estimated to be:

$Q_{opt}\approx\frac{\lambda_{0}}{\Delta\lambda_{FWHM}}\approx 10^{5}.$ (2)

For many applications, it is desirable to use a low cost, low power, and
compact laser source, together with compact electronic systems, rather than
the high performance EXFO fiber-laser and associated electronics used in this
work to date. Here, we test whether it is possible to do this without
sacrificing performance. A commercially available Distributed Feedback (DFB)
laser (Eblana EP1550) with a portable laser driver (Koheron CTL101) was used
to couple light onto and off the chip (Figure 4 b)) were used for all
subsequent measurements.

Tuning the DFB laser to the side of the 1551 nm WGM allows shifts in the
resonance frequency to be directly observed as changes in the optical
intensity. This allows optical detection of mechanical vibrations, and hence
magnetic field, without the need for interferometric detection [6]. Analysing
the resulting photocurrent on a spectrum analyser reveals the mechanical mode
spectrum shown in Fig. 4b). Three mechanical modes are observed at frequencies
of at 3.55, 3.58, and 3.64 MHz. We attribute the discrepancy between the
measured and simulated mechanical frequencies to the inherent stress of the
galfenol film ($\sigma$=500 MPa) adding a stiffening effect to the mechanical
resonances.

The noise-floor of the measurement consists of two components. At frequencies
far away from the mechanical resonance frequencies it is dominated by laser
noise. This is evidenced by an increase in noise when the laser tuned to the
side of the WGM compared to when it is at a frequency far away from the mode.
At frequencies close to the mechanical resonance frequencies, it is dominated
by the thermomechanical noise. We can therefore conclude that the compact
electronic systems used introduce no degradation in performance and, close to
the mechanical resonances, neither does the optical noise of the DFB laser.

To determine the magnetic field sensitivity of the device, we apply a magnetic
field at the frequency of the most prevalent mechanical mode (3.55 MHz). This
induces a sharp peak in the Power Spectrum Density (PSD) (Figure 4e) orange-
trace), evidencing that magnetic fields can be detected. With this particular
applied field (BAC = 50 $\upmu$T) we measure a Signal-to-Noise Ratio (SNR) of
17.5 dB. The magnetic sensitivity of the device at 3.55 MHz could then be
calculated using:

$S=B_{AC}\left({10^{\frac{SNR}{10}}\times\mathrm{RBW}}\right)^{-1/2}$ (3)

where RBW is the resolution bandwidth of the spectrum analyser [6]. This
yielded a sensitivity of 2 $\upmu$$\mathrm{T}/\sqrt{\mathrm{Hz}}$, which is
comparatively less sensitive than previously demonstrated optomechanical
magnetometers that present sub n$\mathrm{T}/\sqrt{\mathrm{Hz}}$ sensitivities
[16].
This reduced sensitivity can be attributed to geometric design of the device.
With these devices the galfenol lies in part, above the pedestal, where the
silicon greatly suppresses both mechanical motion and imbued strain. Further,
the mechanical eigenmodes have very little motion where the galfenol resides,
thus do not experience the maximum possible driving force from the
magnetostriction. These effects provide a reduction of the force exerted onto
the optical eigenmodes from magnetostrictive stress. Thus, the sensitivity
could be considerably improved through the use of device geometry optimized
for deformation of the optical path from the magnetostrictive strain of the
galfenol layer.
Despite the modest sensitivity this work achieves thermomechanically limited
sensing with suspended waveguide coupling and a galfenol thin-film atop the
optomechanical resonator whilst utilising portable electronics and DFB laser.

##  5 Conclusion

Optomechanical magnetometers promise to enable a range of research and
industrial applications. Many of these will require fully integrated
magnetometers operating with compact lasers and electronics. In this work we
make progress towards this goal, demonstrating an optomechanical magnetometer
that is integrated on a silicon chip with a suspended optical waveguide,
utilises galfenol as a magnetostrictive material to provide improved
resilience to corrosion and oxidation, and achieves thermomechanical noise-
limited performance using a DFB laser and compact electronic systems. Funding
The Commonwealth of Australia (represented by the Defence Science and
Technology Group) supports this research through a Defence Science
Partnerships agreement. This work was financially supported by the Australian
Research Council (ARC) Centre of excellence for Engineered Quantum systems
(EQUS): Grant No. CE170100009, and by Orica Australia Pty Ltd.

Acknowledgments The Authors acknowledge the highly valuable advice and support
provided by Rodney Appleby. The authors also acknowledge the University of
Queensland’s Centre for Microscopy and Micro-analysis (CMM) and the Queensland
node of the Australian National Fabrication Facility (ANFF). The equipment and
staff expertise of the CMM and ANFF enabled the fabrication of the devices.

Disclosures The authors declare no conflicts of interest.

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  *[FEM]: Finite Element Method
  *[FSR]: Free Spectral Range
  *[WGM]: Whispering Gallery Mode
  *[EBPG]: Electron-beam Pattern Generator
  *[IPA]: Isopropyl Alcohol
  *[RIE]: Reactive-ion Etching
  *[SEM]: Scanning Electron Microscope
  *[Si]: silicon
  *[PD]: photodector
  *[DFB]: Distributed Feedback
  *[PSD]: Power Spectrum Density
  *[SNR]: Signal-to-Noise Ratio