# Versatile Super-Sensitive Metrology Using
Induced Coherence

Nathaniel R. Miller  University of Dayton, Department of Physics, Dayton, OH,
45469, United States  Louisiana State University, Department of Physics and
Astronomy, Baton Rouge, LA, 70803, United States  Sven Ramelow  Faculty of
Physics, Humboldt-University Berlin, Berlin 12489, Germany  William N. Plick
University of Dayton, Department of Physics, Dayton, OH, 45469, United States

###### Abstract

We theoretically analyze the phase sensitivity of the Induced-Coherence
(Mandel-Type) Interferometer, including the case where the sensitivity is
“boosted” into the bright input regime with coherent-light seeding. We find
scaling which reaches below the shot noise limit, even when seeding the
spatial mode which does not interact with the sample – or when seeding the
undetected mode. It is a hybrid of a linear and a non-linear (Yurke-Type)
interferometer, and aside from the super-sensitivity, is distinguished from
other systems by “preferring” an imbalance in the gains of the two non-
linearities (with the second gain being optimal at _low_ values), and non-
monotonic behavior of the sensitivity as a function of the gain of the second
non-linearity. Furthermore, the setup allows use of subtracted intensity
measurements, instead of direct (additive) or homodyne measurements – a
significant practical advantage. Bright, super-sensitive phase estimation of
an object with different light fields for interaction and detection is
possible, with various potential applications, especially in cases where the
sample may be sensitive to light, or is most interesting in frequency domains
outside of what is easily detected, or when desiring bright-light phase
estimation with sensitive/delicate detectors. We use an analysis in terms of
general squeezing and discover that super-sensitivity occurs only in this case
– that is, the effect is not present with the spontaneous-parametric-down-
conversion approximation, which many previous analyses and experiments have
focused on.

##  1 Introduction

The ability to make precision measurements is paramount in science. It is also
usually desirable to affect the system under study as little as possible.

Classical interferometry, as typified by the Mach-Zehnder interferometer [1,
2], splits light into two modes – one of which then interacts with the object
to be measured – before they are recombined. The resulting interference
pattern then provides phase information about the sample. The Mach-Zehnder
interferometer (MZI) and its numerous variants typically rely on classical
light to make measurements. However, this implies that the minimal detectable
phase shift is never below the shot noise (or “standard quantum”) limit of
$\Delta\phi_{\mathrm{min}}^{2}=1/\langle N_{\mathrm{coh}}\rangle$, where
$\Delta\phi_{\mathrm{min}}$ is the minimum detectable phase shift and
$N_{\mathrm{coh}}$ is the average number of photons in the classical
(coherent) field. Also, these systems most often use optical-frequency light
as this regime is where the best detectors and optics are available. Detection
in other domains is problematic – especially in the terahertz.

The Mach-Zehnder Interferometer, along with two other types (including the one
we study here), are presented in Figure 1.

Figure 1: Top: The Standard MZI; Middle: The Yurke or SU(1,1) Interferometer
(seeded), Bottom: Our boosted Mandel-type setup, where two modes (operated on
by $\hat{a}_{\mathrm{in}}$ and $\hat{b}_{\mathrm{in}}$, for the signal and
first idler, respectively) are inputs into the first non-linearity (NL1 with
squeezing parameter $r_{1}$) which is pumped with a laser which is also phase-
locked to the pump of the second non-linearity (NL2 with squeezing parameter
$r_{2}$). The inputs to the second non-linearity are the shared signal and
second idler modes (operated on by $\hat{c}_{\mathrm{in}}$). Between NL1 and
NL2 the signal mode interacts with an optical phase ($\varphi$), and after NL2
the mode is discarded. The idler modes are then mixed on a 50:50 beam-
splitter, erasing the “which-nonlinearity” information before detection at
$D_{b}$ and $D_{c}$. (Note that we use the opposite convention for
signal/idler from Ref.[29].) Though any of the three input modes could be
boosted (seeded) with coherent light we examine, in the most detail, coherent-
light-injection into the first idler mode ($|\beta\rangle$), as this is light
which will not interact with the phase-imprinting sample, and which
significantly improves sensitivity – as we will see.

Mitigating the limitations of shot noise is a vigorously-pursued avenue of
research starting with the landmark paper of Carlton Caves [3] where it was
shown that so-called “squeezed vacuum” light could reduce the minimum
detectable phase shift below the shot noise limit (SNL) in a MZI, when
injected into the “empty” port of the device – the reason being that, unlike
“normal” vacuum, squeezed vacuum carries phase information. This approach has
even recently been applied to GEOS – a gravitational-wave detector [4].

Measuring a phase with higher precision than is possible with classical light
has also been studied and experimentally explored in the frame-work of photon-
number entangled states. The most prominent example of such states are the so-
called “N00N States” [5]. These can be written in the number basis as
$|N\rangle_{a}|0\rangle_{b}+|0\rangle_{a}|N\rangle_{b}$ (hence the name),
where the subscripts on the kets indicate interferometric path after the
initial beam splitter in an MZI . In the ideal, but nonphysical, situation of
zero loss these states can realize a Heisenberg scaling for the phase-
sensitivity – that is a scaling that is proportional to $N$, where as
classically only the $\sqrt{N}$ of the SQL is achievable. However, in the
presence of loss, this scaling advantage quickly vanishes, as the absorption
(detection) of a single photon gives complete path information and collapses
the state. Another disadvantage of this scheme is that the efficient
experimental generation of N00N states with more than 10 photons ($N=10$) is
still an extraordinary challenge, far away from being relevant for practical
phase-sensing tasks.

It is even possible to perform metrology with _thermal_ light in an MZI.
However the scheme does indeed require a quantum resource – photon
subtraction, which is a highly-non-classical operation. For further details
see Ref.[6].

Another avenue of research involves replacing the beam splitters of the MZI
with nonlinear media ($\beta$-Barium Borate, etc.) These “non-linear” or
“entangled” interferometers (see Refs.[7], and [8] and references therein) can
push the sensitivity limit down to $\Delta\phi_{\mathrm{min}}^{2}=1/\langle
N_{\mathrm{nl}}\rangle(\langle N_{\mathrm{nl}}\rangle+2)$, where
$N_{\mathrm{nl}}$ is the average number of photons produced by the non-
linearities. Such a setup is shown in the middle diagram in Figure 1. This
shows a large improvement in scaling over the Michelson interferometer and was
first discovered by Yurke, McCall, and Klauder. [9]. Hereafter we refer to
this as a “Yurke-Type” interferometer. However, photon production by a two-
mode squeezing non-linearity goes as
$\langle\hat{N}_{\mathrm{nl}}\rangle=2\sinh^{2}(r)$, therefore since photon
numbers are extremely low relative to other sources (such as a laser) the
power of such setups (without further modification) is limited.

The physical mechanism for these phase-sensitivity improvements is illustrated
in Figure 2, where we show various states in quadrature space. Quadrature
space is defined in terms of the $p$ and $q$ quadratures of the EM field as
described in quantum optics. Light fields are defined both by the position in
this space and by the variance. These diagrams can be thought of as a
horizontal slice through the Wigner distribution.

Figure 2: Quadrature diagram in which several quantum-optical states are
living. Vacuum states (grey) have no phase information (they are rotationally-
symmetric and do not change as phase evolves). Coherent (or displaced vacuum –
red) contain phase information proportionate to their displacement from the
origin. That is, we can distinguish coherent states when the angle between
them is $\Delta\phi_{C}$, which increases as intensity increases and states
are displaced further. These are the states that traditional MZIs use.
Squeezed states (blue) change their orientation as the phase evolves due to
the asymmetry introduced by the squeezing. These are the states that are used
in the Caves setup and in the original (unseeded) Yurke interferometer.
Coherently-seeded non-linear interferometers provide the greatest potential
advantage as they utilize squeezed coherent states (purple), which can be
distinguished when the angle between them is $\Delta\phi_{SC}$. This angle is
smaller since the states are displaced in the same way as the coherent states,
but they are narrower in the phase direction.

The sensitivity can be pushed farther by seeding the modes of the original
nonlinear media with coherent light [10], which has been further theoretically
investigated [11, 12, 13, 14, 15, 16, 17] and implemented experimentally [18,
19, 20, 21, 22, 23, 24] including several variations – accounting for loss and
other experimental imperfections. Unlike the MZI the Yurke-type
interferometers need not produce light modes of the same frequency. Indeed it
can be stretched as far as to have one mode be optical light, and the other an
atomic spin wave [24]. The result is a new minimal detectable phase shift of
$\Delta\phi_{\mathrm{min}}^{2}=1/\langle N_{\mathrm{nl}}\rangle\langle
N_{\mathrm{coh}}\rangle(\langle N_{\mathrm{nl}}\rangle+2)$, where
$N_{\mathrm{coh}}$ is now light from the seeded inputs.

Several other interesting variants have recently been proposed, such as the
so-called “Pumped-up” SU(1,1) interferometry where the pump beam is also
utilized during measurement [25], and a scheme where an SU(2) interferometer
is nested inside of an SU(1,1) [26]. Another recent paper has studied the use
of optical parametric amplifiers placed into the individual arms of an MZI
[27]. All three of these schemes have yielded promising initial results.

We propose a new variation on this successful setup based on the phenomenon of
“induced coherence”, an effect first discovered by Zou, Wang, and Mandel [28]
(hereafter a “Mandel-Type” interferometer) and brought to recent prominence
with the imaging experiment of Lemos et al. which created phase images of an
object in a light field which had never interacted with the object [29, 30].
In these experiments, one mode (say, the signal) of the first non-linearity is
seeded into the same mode of the second non-linearity, which _induces
coherence_ between the idler modes. Phase information of an object in the
shared signal mode is imprinted on the idler mode which is then mixed with the
same mode from the first non-linearity on a standard beam splitter. This
experiment can be thought of a type of quantum eraser (Ref.[31] and references
therein) where _welcher-weg_ (which-path, or perhaps more accurately “which
non-linearity”) information is erased by the alignment of the signal modes of
the two non-linearities in conjunction with the beam splitter which mixes the
idler modes. In other words, after the second non-linearity information could
in-principle be obtained by seeing which idler mode contained light, but after
the beam splitter this information is erased. Then the light field that passes
through the sample is discarded. See the bottom diagram in Figure 1. This
constitutes a type of hybrid interferometer described by neither the SU(2)
formalism (standard MZI and variants), nor the SU(1,1) formalism (Yurke-Type
interferometers).

Such setups can even be used to perform tomographic bio-imaging [32]. Another
major potential application of this technique is spectroscopy, since samples
may be probed in frequency domains which are different from where detection is
performed [33, 34, 35, 36]. This has also been done with frequency combs and
induced coherence [37, 38]. The Mandel-Type interferometer has also been used
to perform optical coherence tomography [39] and microscopy [40]. An
equivalent device has also been implemented and studied in a microwave super-
conducting cavity with application to continuous-variable quantum computing
[41, 42]. It is possible that these systems could also be modified in the way
we describe below to realize the beneficial properties we present. In short,
there is the potential for the results we describe to be applied to a wide
variety of systems for use in several different applications.

We study the injection of a coherent seed into the Mandel-Type device,
allowing the sensitivity to be “boosted” into the bright super-sensitive
regime. Also, the current work represents the first time such an
interferometer has been studied specifically for phase sensitivity in
metrology.

From the perspective of optics, this device would work as a Mandel-type
interferometer with light fields displaced coherently in quadrature space,
causing a more-noticeable interaction with the phase to be probed. One could
think of how this device operates on the input optical fields in order to get
a physical intuition. From this perspective, the first interaction (NL1 in
Fig.1) would classically be described as a relatively simple parametric
amplifier – leading to an amplification of the seed mode (here either the
vacuum or a coherent state such as from a phase-locked laser), and a conjugate
copy of the seed in the second output. However, light is not a wave (or not
_entirely_ one) so this approximation has to break down when the mean number
of photons in the seed is sufficiently small and/or the gain is sufficiently
large. Then the more-general description in terms of a Bogoliubov
transformations leads to the by-now, well-known phenomenon of squeezed
coherent output states with quadrature-entanglement between the two output
modes.

In our device one of those outputs is used to measure the unknown phase and
then used as a seed in the second non-linear amplification step with a
potentially different gain (but pumped coherently with respect to the first
interaction). The output (idler) which is conjugate to the sensing field is
afterwards interfered with the seeded idler from the first interaction. This
is akin to a familiar homodyne measurement with a local oscillator (LO) – the
crucial difference being that the LO here is not a coherent state, but still
relatively strong (to boost the sensitivity), and phase locked with the idler
output of the second interaction, allowing measurement of the unknown phase.

What we find is that the device does indeed exhibit bright super-sensitivity.
However, unlike the case of the standard Yurke setup, the second non-linearity
does not need to be pumped as strongly as the first. In fact, after a certain
pumping value, no further increase in sensitivity is obtained when considering
direct intensity measurements. This will be of practical importance when
designing experiments and real-world measuring devices.

It is also the case that intensity-subtraction may be used instead of direct
(total intensity) detection, or homodyne. Intensity subtraction is a much-more
stable and straightforward procedure since noise which is shared between the
modes (such as a vibration of the device in total) is cancelled out.

Furthermore these effects are present even when the seeded mode is _not_ the
mode that passes through the phase, meaning that bright light may be used to
perform measurements on a sample (and also form images) without having that
bright light shine through the sample. This has potential application to
light-sensitive systems which are of strong interest in chemistry and biology.

Additionally, the interacting mode need not be in the same frequency domain as
the detecting mode, so the sample may be interrogated with one frequency and
detection performed in another.

Conversely the undetected mode may be seeded, yielding a similar increase in
sensitivity without exposing the detectors to bright light.

In Section II we theoretically describe the device in question, in Section III
we analyze the phase sensitivity in a number of regimes, and in Section IV we
compare this device to a several others and comment on why the first-quantized
description is inadequate. In Section V we summarize and conclude.

##  2 Theoretical Description

In 2014 Lemos et al. [29] showed that the Induced Coherence experiment
originally done by Zou et al. [28] could be used to measure phase images in an
undetected beam.

Here, we extend the study of these systems by “boosting” – seeding the non-
linearities with coherent light and calculating the minimum detectable phase
shift as a function of the available parameters. Previous work has shown that
boosting nonlinear medias with coherent light in interferometric setups can
drastically increase the phase sensitivity [10]. A diagram of the experimental
setup is shown in the bottom diagram of Figure 1, which can be modeled by
representing the output operators in terms of the input operators under the
transformations

$\displaystyle\mathbf{S_{1}^{\prime}}=\begin{bmatrix}\mu_{1}&0&0&-\nu_{1}\\\
0&\mu_{1}&-\nu_{1}^{*}&0\\\ 0&-\nu_{1}&\mu_{1}&0\\\
-\nu_{1}^{*}&0&0&\mu_{1}\\\
\end{bmatrix}\,\mathbf{\Phi^{\prime}}=\begin{bmatrix}e^{i\,\phi}&0\\\
0&e^{-i\,\phi}\\\ \end{bmatrix},\hskip 85.35826pt$
$\displaystyle\mathbf{S_{2}}=\begin{bmatrix}\mu_{2}&0&0&0&0&-\nu_{2}\\\
0&\mu_{2}&0&0&-\nu_{2}^{*}&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\
0&-\nu_{2}&0&0&\mu_{2}&0\\\ -\nu_{2}^{*}&0&0&0&0&\mu_{2}\\\
\end{bmatrix},\hskip 85.35826pt\quad$
$\displaystyle\mathbf{A_{\mathrm{in}}}=\begin{bmatrix}\hat{a}_{\mathrm{in}}\\\
\hat{a}_{\mathrm{in}}^{\dagger}\\\ \hat{b}_{\mathrm{in}}\\\
\hat{b}_{\mathrm{in}}^{\dagger}\\\ \hat{c}_{\mathrm{in}}\\\
\hat{c}_{\mathrm{in}}^{\dagger}\\\
\end{bmatrix},\mathbf{BS^{\prime}}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&i&0\\\
0&1&0&-i\\\ i&0&1&0\\\ 0&-i&0&1\\\ \end{bmatrix}.\hskip 113.81102pt$

And $\mathbf{\Phi}=\mathbf{\Phi^{\prime}}\oplus\mathbf{I_{4}}$,
$\mathbf{S_{1}}=\mathbf{S_{1}^{\prime}}\oplus\mathbf{I_{2}}$,
$\mathbf{BS}=\mathbf{I_{2}}\oplus\mathbf{BS^{\prime}}$. The $\mathbf{I}$’s are
the identity matrices of the dimension indicated by the subscript, and also
$\mu_{i}=\cosh{r_{i}}$, $\nu_{i}=e^{i\psi_{i}}\sinh{r_{i}}$, with $r_{i}$
representing the squeezing parameter of the $i$-th non-linearity, and
$\psi_{i}$ the phase. $\mathbf{BS}$ represents the beam splitter,
$\mathbf{S_{1}}$ the first squeezer (non-linearity), $\mathbf{S_{2}}$ the
second, and $\mathbf{\Phi}$ the phase to be probed. $\mathbf{A_{in}}$ is a
vector composed of all the operators on which the transformation matrices act
– the input operators. The final (output) operators can be found by applying
the transformation

$\displaystyle\mathbf{A_{out}}=\mathbf{BS\cdot S_{2}\cdot\Phi_{1}\cdot
S_{1}\cdot A_{in}}.$ (1)

Then any detection operator at output can be written in terms of input
operators working on the initial states, which we take to be coherent states
(eigenstates of the annihilation operator, displaced vacuum which becomes
vacuum in the limit of zero displacement). The reason to propagate the
operators “backwards”, rather than the state “forwards” through the device, is
because in this case state propagation is much more cumbersome – even more so
than what is employed here. Furthermore, the action of the output operators in
terms of those output states would themselves not have a simple form. It is
much more efficient to employ the analysis we use here.

The output detector we study in the most detail is $D_{b}$. That is because
the field going to this detector mixes with another mode before detection
relative to the probe phase, and when it is boosted by the injection of
coherent light optimal results are obtained in most cases. Due to the
symmetric mixing action of the beam splitter, detection at detector $D_{c}$
will be similar, up to some relative phase.

We note that injecting light into the other two ports is possible – and
beneficial in some situations, as we will discuss.

##  3 Analysis of Phase Sensitivity

We want to find the minimum detectable phase shift of the probe phase ($\phi$)
which is given by

$\displaystyle\Delta\phi_{\mathrm{min}}^{2}=\frac{\Delta\hat{O}^{2}}{|\partial_{\mathrm{\phi}}\langle\hat{O}\rangle|^{2}},$
(2)

as found in Ref.[43], for example. Where $\hat{O}$ is some general detection
operator and the variance squared is given by

$\displaystyle\Delta\hat{O}^{2}=\langle\hat{O}^{2}\rangle-\langle\hat{O}\rangle^{2}.$
(3)

In order to perform such calculations we need to find the first and second
moments of the final detection operators. These calculations are straight-
forward but extremely lengthy, therefore we created a program in the symbolic
calculation software MathematicaTM to perform them, using the NCAlgebra
package [44].

The program (a simple, commented version is available in the supplementary
materials or by request) proceeds by performing the relevant transformations.

We will consider intensity detection at each output:
$\hat{I}_{A}=\hat{a}_{\mathrm{out}}^{\dagger}\hat{a}_{\mathrm{out}}$,
$\hat{I}_{B}=\hat{b}_{\mathrm{out}}^{\dagger}\hat{b}_{\mathrm{out}}$, and
$\hat{I}_{C}=\hat{c}_{\mathrm{out}}^{\dagger}\hat{c}_{\mathrm{out}}$, as well
as the difference (subtraction) intensity operators defined as
$\hat{S}_{ij}=\hat{I}_{i}-\hat{I}_{j}$, where subscripts index the modes.

Qualitatively, the calculation can be described as follows: the first step is
find the the output operators in terms of the input operators. That is, in the
form
$\hat{a}_{\mathrm{out}}=f_{\hat{a}}(\hat{a},\hat{a}^{\dagger},\hat{b},\hat{b}^{\dagger},\hat{c},\hat{c}^{\dagger})$,
etc., where the operators in the parenthesis are the creation/annihilation
operators of those modes at input. These are obtained by multiplication of all
of the matrices representing the action of the device as in Eq.(1).

The moments of these operators, needed for the calculation of the minimum
phase shift, work on the state at input, which for the case of coherent input
into mode $b$ can be written as $|0,\beta,0\rangle$. These are then
eigenstates of the operator moments and the action of these operators is
therefore straightforward, i.e. $\hat{a}|\alpha\rangle=\alpha|\alpha\rangle$,
with $\hat{a}|0\rangle=0$ as a special case.

However what remains is to order the operators such that all daggered
operators are on the left and all un-daggered operators are on the right.
Since each operator is in-principle a function of six other operators, and
since the second moments involve multiplying four such operators; this stage
of the calculation is highly non-trivial. Thus the Mathematica program
analytically performs all the commutations using
$[\hat{a},\hat{a}^{\dagger}]=1$ etc., acts them on the states, and then
simplifies the output equation as much as possible. We show an example of one
of these calculations in a Mathematica notebook in a separate supplementary
material (or by request).

Strictly speaking, in order to achieve maximum sensitivity, there should be as
much light seeding each of the detected modes as possible. However, it is more
reasonable to consider a finite “light budget”. In this case it’s most
advantageous to inject into mode $B$ since it does not pass though the sample
and has as significant effect on sensitivity as mode $C$. Mode $A$ is less
effective (for direct intensity measurement, for intensity subtraction they
are equal – see later in this section) and must pass through the sample.
Although if it is desirable to have light shine through the sample and _not_
reach the detectors (if the detectors are very fragile for example), then the
sensitivity can be boosted in this way. A novel feature of the setup we
propose here.

The other key parameters are the squeezing values, and the sensitivity as a
function of them can behave in several different ways depending on the values
of the other parameters and which detector is monitored. We will focus on the
case of greatest interest – when only mode $B$ is boosted with a coherent
seed, phases are set to zero, and intensity measurements are also made in mode
$B$.

Figure 3 shows the minimum detectable phase shift as a function of the two
squeezing parameters.

Figure 3: The minimum detectable phase shift squared, as a function of each
squeezing value for four values of the coherent input to mode B and the other
squeezing set to one. In (a) $\beta_{\mathrm{red}}=1000$,
$\beta_{\mathrm{blue}}=1008$, $\beta_{\mathrm{brown}}=1016$, and
$\beta_{\mathrm{purple}}=1024$. In (b) $\beta_{\mathrm{red}}=1000$,
$\beta_{\mathrm{blue}}=1400$, $\beta_{\mathrm{brown}}=1800$, and
$\beta_{\mathrm{purple}}=2200$. All other parameters set to zero. The
detection operator is the direct intensity at mode B.

There are a couple features worth noting.

Firstly, the dependence of $\Delta\phi_{\mathrm{min}}^{2}$ on $r_{2}$ has a
U-shaped valley, with some optimal value of the squeezing. This is a behavior
totally unlike the SU(1,1) interferometer – where an unbalancing favoring
_higher_ gain is found [14, 15]. This is likely a significant practical
advantage as higher gains are more difficult to achieve, especially when
exacting alignment of different optical modes is required. However, it is fair
to note that the scheme in Refs.[14, 15] is very advantageous in the case of
loss. We do not study loss here, though it is an avenue we are pursuing.

Secondly, the squeezing in the first crystal has an outsize influence on the
minimum detectable phase shift compared to the squeezing in the second
crystal. We conjecture that the second non-linearity is only needed to induce
the coherence between the optical modes, further squeezing of the vacuum does
not aid in sensitivity since that mode never again interacts (or becomes
coherent with) the measured modes, and likely merely adds noise. This effect
is a significant practical advantage for experimental implementation, as one
only needs to try and “push” the squeezing in the first non-linearity.

Related to this second feature, while this manuscript was under review we
became aware of some work on a similar system by Kolobov, Giese, Lemieux,
Fickler, and Boyd [45]. In that paper the authors theoretically study a
Mandel-type interferometer with a variable-transmittance beam splitter
inserted between the two non-linearities (where we have our optical phase),
with different pump intensities for them as well. They look at the signal-to-
noise, and visibility, of the output as a function of the gain in the non-
linearities and the transmittance of the aforementioned beam splitter, paying
special attention to the qualitative and quantitative difference between
various regimes of gain. They also find that the visibility of the system may
be optimized by proper choice of the transmittance of the beam splitter. This
is likely related or linked to the effect we observe with respect to the
_sensitivity_ as a function of $r_{2}$ – seen from a different perspective. We
quote their explanation of the effect here (from the last paragraph of section
3 of Ref.[45]):

“…due to seeding crystal A, crystal B produces more photons then the first
one. Hence, both arms become unevenly populated and the visibility drops for
increasing gain. But if the transmittance is sufficiently small, the seeding
effect is suppressed (since most photons from crystal A are not transmitted)
and both arms have roughly the same intensity. This explains why the
visibility follows the coherence for a small transmittance and diminishes for
a large one.” Where their crystal A is our NL1, and crystal B is NL2. We
believe this presents a nice complementary explanation to ours. We again note,
that at least for the sensitivity, this effect can be removed by using an
intensity-subtraction detection scheme.

Also related to this second feature there is a minor, but somewhat unusual,
effect if the third mode is seeded with coherent light (the unseeded mode of
the second non-linearity, mode C in Fig. 1). One might think that this light
would only harm the sensitivity since it does not pass through the sample and
is not correlated with light from the first crystal. However, there is a minor
initial _improvement_ in the sensitivity, which is quickly ruined if the
intensity of the coherent seed is increased much further. This effect is very
small compared to the effect of seeding the other modes (a factor of
improvement of about 2-3, compared to orders of magnitude for the other
inputs), but it is there. Admittedly the reason for this is mysterious to us.
We plan to investigate further in ongoing work.

Next we examine intensity-subtraction detection, written as
$\hat{S}_{ij}=\hat{I}_{i}-\hat{I}_{j}$. This is a commonly-used technique in
interferometry as intensity noise common to both beams is canceled, and better
fringe visibility is obtained. A downside of previous SU(1,1) interferometry
schemes is that “direct detection” (total intensity measurement) is needed –
or failing that homodyne detection. In our hybrid device we have the
advantageous detection setup of an SU(2) interferometer, with the scaling
likely competitive with SU(1,1).

Though the equations generated are very large, it is useful to present a
simple case, the minimum detectable phase-shift squared for coherent light
injection into mode $B$ and intensity difference subtraction between modes $B$
and $C$ with the probe phase (and all other phases) set to zero:

$\displaystyle\Delta\phi_{\mathrm{min}}^{2}=$
$\displaystyle\frac{1}{4(1+\beta^{2})^{2}}\left[(1+\beta^{2})\coth^{2}r_{1}\right.$
$\displaystyle\left.+\coth^{2}r_{2}(\beta^{2}\operatorname{csch}^{2}r_{1}+\operatorname{sech}^{2}r_{1})\right.$
$\displaystyle\left.+(1+\beta^{2})(\tanh^{2}r_{1}-2)\right].$ (4)

Similar scaling (converging in most reasonable limits) is obtained for the
phase sensitivity when coherent light is injected into mode $A$. This means
that if the sample is robust and the detectors are sensitive, sub-shot-noise-
limit sensitivity at high brightness is possible without shining bright light
onto the detectors. Note that for subtracted measurements the “U-shape” of the
sensitivity as a function of $r_{2}$ vanishes, as $\coth^{2}(r_{2})\rightarrow
1$ as the gain gets large. This is likely because the second crystal produces
un-correlated noise which is canceled out by the subtraction. Also, in the
high gain limit the function further simplifies to $e^{-2r}/4(1+\beta^{2})$.
Note that these scalings are most-often _not_ the optimal that can be
obtained, as more judicious choices of phase can perform better. In our
Comparisons section we make further note of this.

In order to get a more-intuitive understanding of the action of the device on
the optical fields we can examine in more detail modes B and C before and
after the final beam splitter in the case when mode B is seeded with a
coherent state.

Consider a single-mode state that has been both quadrature-squeezed and
displaced (see Fig. 2). Depending on the the relative phases of the
displacement and the quadrature-squeezing operations, this state may also be
_photon-number_ (amplitude) squeezed – with maximum number-squeezing occurring
when the state is quadrature-squeezed along the same direction it is displaced
in quadrature space (this would be the opposite case of what is displayed in
purple in the figure). Such states have more-sharply-defined photon numbers
(field amplitudes), but less-certain phases.

However we have the case of a _two_ -mode-squeezed state, therefore the
squeezing occurs not in single-mode quadrature space but in superposition-mode
quadrature space – where the quadrature operators are defined as the sum of
the individual mode operators.

In our case after the first non-linearity the field is displaced only in one
mode but squeezed in superposition between both. After the second non-
linearity there is also mutual squeezing between modes A and C, so at that
point squeezing is shared both between C and A, and B and A, but _not_ between
C and B.

Since both non-linearities are pumped with phase-locked coherent beams, the
phase difference in between them ($\phi$) determines to what degree the two
squeezings shared by mode A are out of phase in quadrature space. Though modes
B and C are not mutually squeezed before the beam splitter they are connected
via mode A, which carries information about the value of $\phi$ in the form of
how it is squeezed (including both magnitude and direction of squeezing)
between both: A and B, and A and C.

Then, after the final beam splitter, due to the fact that is no longer
possible, even in principle, to determine which non-linearity any photon came
from, this mutual squeezing is transferred to a squeezing between modes B and
C (with some residual squeezing left over in the other combinations, as
controlled by the relative phases of all the beams). Now that the squeezing is
between _these_ two modes – one of which is also displaced – the state after
the beam splitter can be number-squeezed to an extent controlled by the phase
$\phi$. Therefore photon-number/intensity/amplitude-difference detection
between these modes can reveal the phase in a super-sensitive manner as we
have shown quantitatively.

##  4 Comparisons

In this section we wish to compare our setup with both the “traditional”
boosted SU(1,1) non-linear interferometer, and the induced coherence setup in
the spontaneous-parametric-down-conversion approximation.

To begin with the first point, it is instructive to examine the log plots of
many different “fair comparison” setups. We take: our boosted Mandel-type
setup (both intensity detection in one mode and intensity difference), the
boosted Yurke-type setup – and a standard coherent-light-seeded MZI with the
extra light that _would be_ needed to create the squeezing added to it. We
compare the phase sensitivity of these three devices in a log plot in Figure
4.

Figure 4: The minimum detectable phase-shift squared of several “fair
comparison” interferometric setups and detection schemes as a function of gain
(of the first crystal for Mandel-type and of _both_ crystals for Yurke-type).
Here we display the boosted Mandel-type setup with intensity detection at mode
$B$ (green), intensity difference detection between modes $B$ and $C$ (brown),
the boosted Yurke-type setup (black), and a standard coherent-light-seeded MZI
with the extra light needed to create the aforementioned squeezings added to
the initial input (red). The later is equivalent to the shot-noise limit. All
other parameters are numerically optimized at each point. Note that this is
not quite a true fair comparison as for the Yurke-type the gains of both non-
linearities are set to the same value whereas in our scheme the gain of the
second crystal is set to its (usually much lower) optimal value, giving the
SU(1,1) an artificial advantage in this plot. The circular points (upper set)
represent injected coherent light of about the same intensity as would be
needed for a high-gain non-linearity, and the square points (lower set)
represent a much-brighter coherent input. Note that the Heisenberg limit is
not plotted as at “fair comparison intensities” it would be far below any
others and not appear on the plot. At lab-available intensities for systems in
which it could be realized, it would be far above. Coherent-light boosted
schemes represent an intermediate regime where direct comparison to the HL is
not a useful metric.

Though our setup does not win against the boosted Yurke-type, this is not
quite a true fair comparison as for the Yurke-type the gains of both non-
linearities are set to the same value, whereas in our scheme the gain of the
second crystal is set to its (usually much lower) optimal value. We plan to
investigate a “more fair” comparison in later work. The most important feature
is that for all but very-low gains the boosted Mandel-type reaches orders of
magnitude below the shot-noise (standard-quantum) limit. This is in addition
to the other advantages present in the proposed device.

It should be noted that at very low gains the phase sensitivity quickly
decays. This is because at low gains the modes are only weakly-mixed – and
since interferometry is based on mode-mixing, the power of the device in these
regimes is limited in this case.

Next we examine a similar setup but instead of general squeezing we take the
“first quantized” approach and propagate an initial spontaneous-parametric-
down-conversion (SPDC) state vector through the device up until before the
final beam splitter (pre-beam-splitter), where it is

$\displaystyle|\psi_{p}\rangle=\frac{1}{\sqrt{2+\left|\beta\right|^{2}}}\hat{a}^{\dagger}_{p}\left(e^{i\phi}\hat{b}^{\dagger}_{p}+\hat{c}^{\dagger}_{p}\right)\left|0,\beta,0\right\rangle.$
(5)

Since displacement and beam-splitter operators work in different
representations/pictures it is necessary to then propagate detection operators
back through the final beam splitter to work on the state above, with the
standard transformations
$\hat{b}^{\dagger}_{f}\rightarrow\frac{1}{\sqrt{2}}(\hat{b}^{\dagger}_{p}-i\hat{c}^{\dagger}_{p})$,
etc.

The full expression for the minimum detectable phase shift is large and so we
do not report it here, but in the limit of large $|\beta|$ we find:
$\Delta\phi_{\mathrm{min}}^{2}\rightarrow 19/4$. Thus, when taking the SPDC
approximation the advantages of the induced-coherence interferometer are
totally obscured. Since most mathematical treatments of the device make this
assumption this is likely why the metrological power of this setup has gone
mostly unnoticed.

##  5 Conclusion

To conclude, we have theoretically investigated the induced-coherence
interferometer when coherent seeds are added to all three input ports. The
arrangement we study in the most detail is when the seed is injected into the
initial arm which does _not_ pass through the phase-inducing sample.

This scheme presents several practical advantages, which – strikingly – are
available simultaneously with a single setup: super-sensitivity nearly on par
with a boosted SU(1,1) interferometer, use of phase-stable intensity-
subtraction measurements instead of additive (direct detection) or homodyning,
the ability to use different frequencies for phase acquisition and detection
(up into the infrared/terahertz regime), the favoring of an imbalance in the
gains with _low_ gain in the second crystal, and the power to boost the
sensitivity with a bright coherent seed which does _not interact_ with the
sample to be studied. This last feature should in fact also be available to
the boosted SU(1,1) setups – however to the best of our knowledge this fact
has never been pointed out or pursued theoretically or experimentally.
Furthermore, the same sensitivity scaling is observed when the mode with the
phase is seeded with coherent light, even though this light does not reach the
detectors. This last effect is not possible with other non-linear
interferometers, and as far as we know is unique to the system we study here.

We conjecture that this scheme will be highly-advantageous in many real-world
physical systems, especially those where high-quality phase estimation is
desired for samples/detectors which are sensitive to bright lights (vulnerable
to bleaching) and/or are interesting at wavelengths where good optical
elements do not exist or are prohibitively expensive.

##  6 Acknowledgments

This paper is dedicated to the memory of Joe Haus, a gregarious and kind man,
and an excellent scientist. He was an example to us all. It is also dedicated
to the memory of Jon Dowling adviser to some of the authors. He always looked
after his students with kindness and care, and made doing science exciting.

N. Miller would like to thank the U. Dayton Dean’s Fellowship for funding, W.
Plick thanks the U. Dayton SEED Grant, and S. Ramelow thanks the Deutsche
Forschungsgemeinschaft (DFG) within the Emmy-Noether-Programm (RA 2842/1-1).
Discussions with Z.Y. Ou were very useful.

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# Non-destructive mapping of stress, strain and stiffness of thin elastically
deformed materials

Guo-Yang Li  Harvard Medical School and Wellman Center for Photomedicine,
Massachusetts General Hospital, Boston, MA 02139, USA.  Artur L. Gower
Department of Mechanical Engineering, University of Sheffield, Sheffield,
United Kingdom.  Michel Destrade  School of Mathematical and Statistical
Sciences, NUI Galway, Galway, Ireland.  Key Laboratory of Soft Machines and
Smart Devices of Zhejiang Province, Department of Engineering Mechanics,
Zhejiang University, Hangzhou 310027, PR China.  Corresponding authors:
<EMAIL_ADDRESS>(M.D<EMAIL_ADDRESS>(S.H.Y.).  Seok-Hyun
Yun  Harvard Medical School and Wellman Center for Photomedicine,
Massachusetts General Hospital, Boston, MA 02139, USA.  Corresponding authors:
<EMAIL_ADDRESS>(M.D<EMAIL_ADDRESS>(S.H.Y.).

###### Abstract

Knowing the stress within a soft material is of fundamental interest to basic
research and practical applications, such as soft matter devices, biomaterial
engineering, and medical sciences. However, it is challenging to measure
stress fields in situ in a non-invasive way. It becomes even more difficult if
the mechanical properties of the material are unknown or altered by the
stress. Here we present a robust non-destructive technique capable of
measuring in situ stress and strain in elastically deformed thin films without
the need to know their material properties. The technique is based on
measuring elastic wave speeds, and then using a universal dispersion curve we
derived for Lamb wave to predict the local stress and strain. Using optical
coherence tomography, we experimentally verified the method for a rubber
sheet, a cling film, and the leather skin of a musical instrument.

keywords: Mechanical stress $|$ Lamb wave $|$ Acoustoelasticity $|$ Optical
coherence elastography $|$ Soft matter

## Introduction

Soft thin films hanging in the air or confined in fluids are ubiquitous in our
daily lives as well as in natural and engineering systems. Examples include
cling film packaging food, the eardrum and the diaphragm in our body, and
various elastic sheets, membranes, vesicles and bands holding structures
together. They are typically under external and internal stress, and it is
often desirable to know the stress level to be able to understand the
environment they are exposed to or interacting with, or to monitor the changes
in and health of the materials. However, in situ non-invasive measurements of
the stress are challenging. This is even more challenging if the mechanical
properties of the material are unknown and, furthermore, if the original
configuration of the material is unknown, which precludes straightforward
measurement of strain [1, 2].

Various techniques have been devised to measure in-plane stresses [3]. The
choice of technique depends on the material type (solid/liquid/type of
molecules) and also on the length scale. Essentially all techniques so far
rely on the knowledge of the elastic moduli of the material or some specific
expected behaviour of the material, which limits their application to known or
specific materials and structures. For example, in the Langmuir–Blodgett
trough [4], a workhorse of membrane biophysics, surface tension is estimated
by measuring the amount of force required to insert a Wilhelmy plate into a
given membrane. However, this force depends on the nature of the surface
tension and its accuracy has been questioned for solid-like membranes [5].
Conventional ultrasound methods also require the elastic moduli and acousto-
elastic parameters of the materials to predict the stress [6, 12].

Here we describe a technique that allows the stress field to be determined in
soft thin films even without a priori knowledge of the material properties or
applied strain. The technique uses Lamb elastic waves propagating in the film
[7, 8] followed by a simple algorithm to determine the stress from measured
wave speeds. In this work, we use optical coherence tomography (OCT) to
visualise the elastic waves and measure their propagation speeds in an audible
frequency range. This range (1-20 kHz) is well suited for soft materials with
thickness ranging from sub-micron to a few hundreds of micrometers.

## Results

### Theoretical foundation

Figure 1 illustrates the general principle for a film under an arbitrary
stress field T which deformed it elastically. The local wave speed is affected
not only by the stiffness of the material but also by the direction and
magnitude of the local stress (Fig. 1b) [9]. The two principal stresses
$\sigma_{1}$ and $\sigma_{2}$ and stretch ratios $\lambda_{1}$ and
$\lambda_{2}$, at a location $(x,y)$ are related to the in-plane stress
(Cauchy stress) $\bm{\sigma}$ and strain (Green-Lagrange strain) $\mathbf{E}$
at the location (Fig. 1c). Mathematically, $\sigma_{1}$ and $\sigma_{2}$ are
obtained by diagonalising the stress tensor.

Figure 1: The principle of measuring stress and strain via acousto-elastic
mapping. a, Thin films elastically deformed by the stress field $\mathbf{T}$
from an (unknown) undeformed configuration. A representative dashed square is
deformed into a rectangle by the principal stresses $\sigma_{1}$ and
$\sigma_{2}$ at ($x$, $y$). b, Wave profiles in the stressed membrane showing
anisotropic wave speeds. The wave speeds along the principal directions,
$v_{1}$ and $v_{2}$, are used to recover the principal stresses and stretches
at ($x$, $y$). c, The principal stresses and stretches measured at different
locations give the stress $\bm{\sigma}$ and strain $\mathbf{E}$ map of the
membrane.

To develop our algorithm, we consider a deformed element (Fig. 2a) with local
coordinates $(x_{1},x_{2},x_{3})$, where $-h\leq x_{3}\leq h$, $2h$ being the
thickness. The film is subject to the in-plane stresses $\sigma_{1}$ and
$\sigma_{2}$ along the $x_{1}$ and $x_{2}$ axes, respectively, and the out-of-
plane stress $\sigma_{3}$ along the $x_{3}$ axis ($\sigma_{3}\simeq 0$ in thin
films). An infinitesimal elastic wave polarised in the $x_{i}-x_{3}$ plane and
propagating along the $x_{i}$ axis ($i$ = 1 or 2) is described with the
mechanical displacement field
$\mathbf{u}=\mathbf{u}_{0}\exp{(-s\,k\,x_{3})}\exp{[ik(x_{i}-v_{i}\,t)]}$,
where $\mathbf{u}_{0}$ is the amplitude vector, $k$ is the wavenumber, $s$ is
the attenuation factor, $t$ is time, and $v_{i}$ is the speed. The governing
wave equation is [10] $\mathcal{A}^{0}_{pjqk}\partial^{2}u_{k}/\partial
x_{p}\partial x_{q}-\partial\hat{\bar{p}}/\partial
x_{j}=\rho\partial^{2}u_{j}/\partial t^{2}$. Here $\mathcal{A}^{0}_{pjqk}$ is
the Eulerian elasticity tensor, which contains the effect of the stress via
the strain energy, and the second term $\hat{\bar{p}}$ denotes the increment
of the Lagrange multiplier $\bar{p}$, due to the constraint of
incompressibility, and $\rho$ is the mass density. Eliminating $\hat{\bar{p}}$
with incompressibility ($\lambda_{1}\lambda_{2}\lambda_{3}=1$,
$\nabla\cdot\mathbf{u}=0$), the wave equation is then reduced to [11]:
$\gamma_{i}\,s^{4}-(2\beta_{i}-\rho v_{i}^{2})\,s^{2}+\alpha_{i}-\rho
v_{i}^{2}=0$, where $\alpha_{i}=\mathcal{A}^{0}_{i3i3}$,
$2\beta_{i}=\mathcal{A}^{0}_{iiii}+\mathcal{A}^{0}_{3333}-2\mathcal{A}^{0}_{ii33}-2\mathcal{A}^{0}_{3ii3}$,
and $\gamma_{i}=\mathcal{A}^{0}_{3i3i}$. The stress-free boundary condition
($\sigma_{3}=0$ at $x_{3}=\pm h$) of the thin film structure gives the
complete dispersion equation of the Lamb waves (see Methods).

Figure 2: Acousto-elastic effect for Lamb waves. a, A thin structure under
tension. b, Modal shapes of the fundamental anti-symmetric (A0) and symmetric
(S0) Lamb waves. c, Global curves for the A0 and S0 modes. d, Representative
dispersion curves of the A0 and S0 modes in an incompressible material with
shear modulus $\mu=1.0$ kPa, Landau constant of third-order elasticity
$A=-3.2$ kPa and mass density $\rho=1$ g/cm3, subject to two different strains
of 5% (full lines) and 10% (dashed lines) that are applied to the wave
progagation directions. The $\alpha$, $\gamma$, and $v$ are equal to
$\alpha_{i}$, $\gamma_{i}$, and $v_{i}$, depending on direction of
propagation. The circles indicate three asymptotic values of the dispersion
curves, and arrows indicate their direction of change with tension, see
Methods for more details.

The instantaneous elastic moduli, $\alpha_{i}$ and $\gamma_{i}$ in elastically
deformed materials are directly related to the principal stresses and
stretches through the identities (see Methods)

$\sigma_{i}=\alpha_{i}-\gamma_{i};\quad\lambda_{i}^{2}/\lambda_{3}^{2}=\alpha_{i}/\gamma_{i}.$
(1)

For isotropic materials, with up to third-order elasticity [13], we have that
$2\beta_{i}=\alpha_{i}+\gamma_{i}$. This identity simplifies the dispersion
equation to:

$4\,\eta\,\tanh{(\eta kh)}/\tanh{(kh)}=(1+\eta^{2})^{2}$ (2)

for the anti-symmetric (A) modes, and $4\,\eta\,\tanh{(kh})/\tanh{(\eta
kh)}=(1+\eta^{2})^{2}$ for the symmetric (S) modes, where
$\eta\equiv\sqrt{|\alpha_{i}-\rho v_{i}^{2}|/\gamma_{i}}$. The fundamental A0
mode is a flexural bending wave, and the fundamental S0 mode has a
dilatational, out-of-plane displacement ($u_{3}$) profile (Fig. 2b).

Our algorithm to determine the in-plane stress and strain is as follows.
First, note that $\eta$ is uniquely related to $kh$ via the dispersion
equation, so that $\eta=\eta(kh)$. This relationship for the two modes is
displayed in Fig. 2c. Second, we experimentally measure $k$ or $v_{i}$ at
certain frequencies $f$. Note that $v_{i}=2\pi f/k$. Then, $\eta(kh)$ is
determined using the dispersion relationship. When $\eta$ is measured at two
different $f$’s, we can determine $\alpha_{i}$ and $\gamma_{i}$ from the two
values of $\eta$. Although two frequency data are sufficient in principle,
measurements over multiple frequencies, followed by a least square fit, lead
to a more accurate predict of $\alpha_{i}$ and $\gamma_{i}$. Finally, the
principal stress, $\sigma_{i}$, is determined from the first equation in Eq.
(1). The principal strain, $E_{i}=0.5(\lambda_{i}^{2}-1)$, is determined from
the second equation in Eq. (1). With the stress and strain, the Young’s
modulus (for small strain) can be determined as
$(\sigma_{i}-0.5\sigma_{j})/E_{i}$.

For insight, let us consider an intrinsically isotropic nonlinear material
under uni-axial tension. Figure 2d shows the dispersion curves of the wave
speeds normalized to the bulk shear wave speed, $v_{0}=\sqrt{\mu/\rho}$ of the
material in the undeformed state, where $\mu$ is the shear modulus (unknown a
priori in experiments), are plotted as a function of normalized frequency,
$fh/v_{0}$ for two different extension values of 5% (full lines) and 10%
(dashed lines), respectively. For small deformation ($\lambda_{i}$ close to
1), the A0 and S0 wave speeds along the $x_{1}$ axis in the limit of low
frequency are $v_{A0}(0)=\sqrt{\sigma_{1}/\rho}$ and
$v_{S0}(0)=2\sqrt{\mu/\rho}-\frac{1}{6}\sqrt{(\sigma_{1}+\sigma_{2})^{2}/(\rho\mu)}$
(see Methods). In this case ($\mu\gg\sigma_{i}$), the A0 wave speed has much
higher sensitivity than the S0 wave to the principal stress along the
propagation direction. A more rigorous sensitivity analysis (Supplementary
note 1) supports this conclusion for larger deformations. In our experiments,
we exclusively used A0 waves.

Although our algorithm does not require the knowledge of $\mu$ to measure
stress, the knowledge of thickness $2h$ makes the algorithm more robust in
determining the stretch and is needed to measure the elastic modulus of the
material at the deformed state. The thickness information may be obtained by
imaging. For most elastomers and biological tissues, $\rho\approx$ 0.9 - 1.1
g/cm3 and $\mu$ is in a range of 1 kPa to 1 GPa. Then, $v_{0}\approx$ 1 to
1000 m/s. Measurement over an audible range, $f$ = 1 - 20 kHz, allows us to
measure samples with $h$ $\sim$ 1 to 500 $\mu$m.

Figure 3: Imaging and analysing elastic waves in a stretched rubber sheet. a,
Photograph of our rubber sheet. b, OCT cross-sectional image. c-d, Wave
profiles measured by OCT when the film is (c) stress-free and (d) subject to a
uniaxial stress $\sigma_{1}\approx 200$ kPa. The wavefronts becomes elliptical
when stretched. e, Dispersion relations of the $A_{0}$ mode obtained at
different levels of stress. Each curve is an average over six measurements
(for clarity the error bar is not shown here). f, Comparison between the
measured and applied stress values. For reference, the dashed lines show where
the measured stresses are equal to the applied stresses. g, Comparison between
the measured and applied stretch values. Dashed lines, $1$ and $-1/2$ power
laws of the applied stretch for reference. h, The stress-strain curve from the
measured data. Dashed line, stress-strain curve measured by standard tensile
test.

### Experimental validation

To validate the method, we devised an experimental setup (Supplementary Fig.
S3) based on a home-built, swept-source OCT system [14]. The first sample used
was a rubber sheet (Fig. 3a). We applied uniaxial tension to the sample using
weights (see Methods). The thickness of the film measured by OCT (Fig. 3b)
decreased from 500 to 430 $\mu$m (Supplementary Fig. S4). Figures 3c,d show
the wave profiles in the unstressed and fully stressed states (0 and 6 20-g
weights, respectively) at $f$ = 6 kHz (see Supplementary Movies 1 and 2). In
the unstressed state the wave propagates at the same speed in all directions,
and creates a circular profile with $v_{1}=v_{2}$. The applied stress, on the
other hand, induces anisotropy for the speed, which creates an elliptical
profile, with the speed $v_{1}$ along the tensile $x_{1}$ axis being larger
than the speed $v_{2}$ along the compressive $x_{2}$ axis. Figure 3e shows the
dispersion curves of the A0 mode at different stress levels from 0 to 200 kPa.
We also measured the dispersion of the A0 wave along the $x_{2}$ axis
(Supplementary Fig. S7).

The measured wave speeds $v_{1}$ and $v_{2}$ fit very well into the dispersion
relation written in terms of $\eta$ (Supplementary Fig. S6), from which we
determined $\alpha_{1}$ and $\gamma_{1}$, and $\alpha_{2}$ and $\gamma_{2}$.
Using Eq. (1), we then obtained the stress and strain along the $x_{1}$ and
$x_{2}$ axis. Figure 3f shows the measured and actual values of the two
principal stresses with a good agreement, with errors around 5%. Figure 3g
shows the measured and actual stretch ratios again in a good agreement with
errors less than 3%. The measured stress-strain curve (Fig. 3h) gives a value
of $\mu\approx 177.8$ kPa, which agrees well with that obtained by an
independent standard tensile test (Supplementary Fig. S8).

Next, we tested the technique for a stretched plastic wrap, a.k.a. _cling
film_ (Fig. 4a) made of polyethylene with a thickness of $\sim 12$ $\mu$m.
Using a similar experimental setup, we applied a uniaxial stress to the cling
film and measured the dispersion relations at the unstressed and stressed
states (Fig. 4b). For the ultra-thin structure, the asymptotic speed in the
low frequency limit provides $\sigma_{i}$ directly. We measured the low-
frequency wave speed $v_{1}(0)$ in the stretched condition to be $\sim 30.3$
m/s. Using $\rho$ = 930 kg/m3 we obtain $\sigma_{1}\approx 0.84$ MPa. By
Taylor expansion of Eq. (2), we find $\rho
v_{1}^{2}=\sigma_{1}+(-\sigma_{1}/3+4E/9)(kh)^{2}+\mathcal{O}((kh)^{4})$,
where $E$ is the Young’s modulus in the stretched condition. By curve fitting
the measured dispersion curve (Fig. 4b), we find $E\approx 1069$ MPa. This is
slightly lower than the Young’s modulus of $\sim 1170$ MPa in the unstressed
condition. Experiments performed at different stretching force showed good
agreements between the measured and applied stress values (Fig. 4c).

Figure 4: Measuring mechanical stresses in a cling film. a, Photograph of the
cling film. b, Dispersion curves for a cling film in slack and tense states,
respectively. Markers, experiment. Dashed curves, theory. The Young’s modulus
of the cling film fitted from the unstressed dispersion curve is $\sim 1170$
MPa. c, Comparison between the applied and measured stress. Dashed line,
$45^{\circ}$ line for reference.

Finally we applied our technique to measure the stress in the drumhead of a
musical instrument called the _bodhrán drum_ , a traditional Irish drum made
with goat skin. If the skin is too taut, the pitch is higher than expected, so
players are advised to sprinkle and spread some water on the skin from the
inside just before performance. Conversely, if the skin is too loose, bodhrán
players rub their palm on the outside of the skin to make it dry and tighten
the skin to correct the pitch.

The thickness of the skin was measured to be $360\pm 30$ $\mu$m. We performed
_in situ_ measurements on the drumhead (Fig. 5a) at normal (dry) and hydrated
conditions of the skin. In the dry condition, the fundamental resonance
frequency of the instrument was 84 Hz, and it was decreased to 36 Hz after
hydration (Supplementary Fig. S9). The goat skin is intrinsically anisotropic,
but our experiments revealed almost circular wave profiles (Supplementary
Movies 3 and 4). This is well explained by the large radial stress in the
drum, which stretches collagen fibres along the stress field [15]. The applied
strain is transversely isotropic (equi-biaxial in the radial/circumferential
directions) and thus, for all intends and purposes, acoustic wave propagation
is isotropic in the drum plane. Figure 5b shows the dispersion relations of
the skin in the dry and damp states, obtained at a region in the drum head. We
determined from the experimental data that the amount of radial stretch in the
dry skin is 0.28% and that humidification relaxed it to 0.20% (Fig. 5c). The
corresponding stress is changed from 3.79 MPa (dry) to 1.31 MPa (damp). Noting
that the strain is small, we find the Young’s modulus of the skin to be $\sim$
680 MPa for dry skin and $\sim$ 330 MPa for the moisturized skin.

Figure 5: Measuring mechanical stresses in a drum head. a, Photograph of the
experimental setup. The size of the _bodhrán_ drumhead is 16 inch. A real-time
OCT image is displayed in the monitor. b, Dispersion relations measured in the
dry and damp states. Markers, experiment. Dashed curves, theory. c, Stresses
and strains measured in the dry and damp states.

The result reveals how much the humidification changed the stiffness and
consequently the tension of the skin. Hydration alters the resonance
frequencies or pitch of the instrument, which is proportional to
$\sqrt{\sigma_{1}/\rho}$. The 65% reduction in tension (and an increase of
density) predicts $\sim$ 46% decrease in resonance frequency. This is
comparable to the actual 57% decrease of the resonance frequency. The
discrepancy is probably due to spatially nonuniform hydration across the large
drum head (18-inch diameter).

## Discussion

The technique we present is nearly model-free in the sense that it is
independent of and does not require the material’s mechanical properties.
Although our method assumes the material is incompressible, extending the
method to accommodate compressibility adds a relative small error in the order
of $\mu/\lambda$, where $\lambda$ is the first Lamé constant (Supplementary
Note S2 and Fig. S2). The plastic cling film we tested is actually a
compressible material with an initial Poisson’s ratio of $\sim 0.32$ and had
significant plastic deformation. The technique is primarily for elastic
materials, but it can be applied to weakly viscoelastic materials [16].
However, highly viscoelastic materials with frequency-dependent mechanical
properties call for more involved curve fitting with parameters including the
viscosity [17].

Although we have validated the technique for relatively uniform stress, it is
easily applicable to more complex structures with spatially varying stress and
strain fields by measuring local velocity profiles, as illustrated in Fig. 1c.
The spatial resolution of the mapping is approximately in the order of one
wavelength. When $v$ = 10 m/s, for example, the resolution is $\sim$ 5 mm for
$f$ = 2-10 kHz. It is relatively straightforward to extend our method to thin
structures in contact with fluids or gel-like matters either one side or both
sides, such as dura mater of the brain after craniotomy [18] and the cornea
[14] as well as blood vessel walls [19]. Our work is expected to pave the way
to practical applications.

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## Methods

#### Theoretical model

We consider an elastic wave polarised in the $x_{i}-x_{3}$ plane, propagating
along the $x_{i}$ axis ($i=1$ or $2$), with attenuation along the $x_{3}$
axis: $\mathbf{u}=\mathbf{u}_{0}\exp{(-skx_{3})}\exp{[ik(x_{i}-v_{i}t)]}$.
Inserting $\mathbf{u}$ into $\mathcal{A}^{0}_{pjqk}\partial^{2}u_{k}/\partial
x_{p}\partial x_{q}-\partial\hat{\bar{p}}/\partial
x_{j}=\rho\partial^{2}u_{j}/\partial t^{2}$ and eliminating $\hat{\bar{p}}$
with the incompressibility $\nabla\cdot\mathbf{u}=0$, we get the secular
equation

$\gamma_{i}s^{4}-(2\beta_{i}-\rho v_{i}^{2})s^{2}+\alpha_{i}-\rho
v_{i}^{2}=0,$ (3)

where $\alpha_{i}=\mathcal{A}^{0}_{i3i3}$,
$2\beta_{i}=\mathcal{A}^{0}_{iiii}+\mathcal{A}^{0}_{3333}-2\mathcal{A}^{0}_{ii33}-2\mathcal{A}^{0}_{3ii3}$,
and $\gamma_{i}=\mathcal{A}^{0}_{3i3i}$. With the stress-free boundary
condition at $x_{3}=\pm h$, we arrive at the dispersion equation for the Lamb
wave [11]

$\left(\frac{\tanh s_{1}k_{1}h}{\tanh s_{2}k_{1}h}\right)^{\pm
1}=\dfrac{s_{2}(s_{1}^{2}+1)^{2}}{s_{1}(s_{2}^{2}+1))^{2}},$ (4)

where the exponent is $+1$ for symmetric modes and $-1$ for anti-symmetric
modes, and $s_{1}^{2}$, $s_{2}^{2}$ are the roots of Eq. (3).

To capture the acousto-elastic effect induced by a moderate strain, we
consider the strain energy of isotropic incompressible third-order elasticity
[12, 13]. The elastic moduli $\mathcal{A}^{0}_{ijml}$ are given by Equation
(9) in Li et al [12], which result in $2\beta_{i}=(\alpha_{i}+\gamma_{i})$.
Substituting into Eq. (3) we get $s_{1}^{2}=(\alpha_{i}-\rho
v_{i}^{2})/\gamma_{i}$ and $s_{2}^{2}=1$, and the dispersion equation Eq. (4)
becomes Eq. (2).

The basic idea of our acousto-elastic imaging technique is to deduce the
stress and strain with $\alpha_{i}$ and $\gamma_{i}$ from the exact formulas

$\sigma_{i}=\alpha_{i}-\gamma_{i},\qquad\
\lambda_{i}^{2}/\lambda_{3}^{2}=\alpha_{i}/\gamma_{i}.$ (5)

To show these formulas, it suffices to recall that, in general [10],

$\mathcal{A}^{0}_{jkjk}=\frac{\sigma_{j}-\sigma_{k}}{\lambda_{j}^{2}-\lambda_{k}^{2}}\lambda_{j}^{2},\quad(\text{for}\;\;j\neq
k\text{ and }\lambda_{j}\neq\lambda_{k}).$ (6)

Making use of Eq. (6), we get
$\alpha_{i}-\gamma_{i}=\mathcal{A}^{0}_{i3i3}-\mathcal{A}^{0}_{3i3i}=\sigma_{i}-\sigma_{3}$,
and
$\alpha_{i}/\gamma_{i}=\mathcal{A}^{0}_{i3i3}/\mathcal{A}^{0}_{3i3i}=\lambda_{i}^{2}/\lambda_{3}^{2}$,
which leads to Eq. (5) by taking $\sigma_{3}=0$ (thin membrane).

Expanding (4) in the low frequency limit of $f=0$ (or $kh=0$), we get $\rho
v_{A0}^{2}=\alpha_{1}-\gamma_{1}$ and $\rho v_{S0}^{2}=\alpha_{1}+3\gamma_{1}$
for the A0 and S0 modes, respectively. On the other hand, in the high
frequency limit of $f\to\infty$ (or $kh\to\infty$), we get $\rho
v_{R}^{2}=\alpha_{1}-\eta_{0}^{2}\gamma_{1}$ for both A0 and S0 modes, where
$\eta_{0}=0.2956$ is the real root of the cubic $x^{3}+x^{2}+3x-1=0$ (Rayleigh
surface wave limit). The three limits are shown in Fig. 2d.

As a simple example, suppose an initially isotropic material is subject to a
small stress. In the limit of small deformation ($\lambda_{i}\approx 1)$,
using Taylor expansion the three limits can be explicitly expressed as
functions of the principal stresses

$\begin{pmatrix}\rho\,v_{A0}^{2}(f=0)\\\ \rho\,v_{S0}^{2}(f=0)\\\
\rho\,v_{R}^{2}(f=\infty)\end{pmatrix}=\begin{pmatrix}0\\\ 4\mu\\\
0.91\mu\end{pmatrix}+\begin{pmatrix}1&0&-1\\\ -2/3&-2/3&4/3\\\
0.62&-0.15&-0.47\end{pmatrix}\begin{pmatrix}\sigma_{1}\\\ \sigma_{2}\\\
\sigma_{3}\end{pmatrix}.$ (7)

For thin structures, $\sigma_{3}\approx 0$. If two of these three limiting
wave speeds are measured, the in-plane stresses ($\sigma_{i}$) can be
calculated from this equation with $\sigma_{3}=0$. In practice, the
$kh\to\infty$ limit cannot be measured accurately because it corresponds to an
extremely thick slab. Similarly it is difficult to express the $S_{0}$ mode
for ultra-thin films; in that case, the first equation for the $A_{0}$ mode
nonetheless gives access to $\sigma_{1}-\sigma_{3}$ directly.

#### Experimental setup

Our experimental setup (Supplementary Fig. S3) is based on a home-built,
swept-source optical coherence tomography (OCT) system [14]. This system
offers an A-line rate of 43.2 kHz, axial resolution of $\sim 15$ $\mu$m (in
the air) and transverse resolution of $\sim 30$ $\mu$m, using a polygon swept
laser with a tuning range of 80 nm and a centre wavelength of 1,280 nm. The
optical beam is scanned using a two-axis galvanometer scanner. To excite Lamb
waves in the film we used a contact probe driven by a vibrating PZT
piezoelectric transducer (Thorlabs, PA4CEW). The plastic probe was 3D-printed
with a spherical tip of $\leavevmode\nobreak\ 2$ mm in diameter. A small force
($\sim 20$ mN) was applied to the probe to keep it in contact with the sample.
The optical beam scan was synchronized with the probe vibration to operate in
an M-B scan mode (see Supplementary Note 3 and Fig. S10). Their scanner axes
were aligned to the principal transverse axes ($x_{i}$). The frequency of the
vibration was step-tuned from 2 to 20 kHz with an interval of 2 kHz. At each
frequency, the amplitudes and phases of the vibrations were acquired at 96
transverse locations. The vertical displacement near the probe contact point
at the sample was in the order of 100 nm in the frequency range. To measure
this small vibration, we used the phase change in the interference signal of
the OCT [14].

To measure the wavenumber $k$ and thus the wave speed $v_{i}=2\pi f/k$ for a
given frequency, the surface displacement was Fourier-transformed from the
spatial domain to the wavenumber domain. The wavenumber was obtained by
identifying the peak that corresponds to the A0 mode (Supplementary Fig. S5).
The standard deviation error in the wavenumber measurement is estimated to be
about $0.1\%$ (Supplementary Note 4).

For the experiments of the rubber sheet and cling film, the sample was clamped
along its two short edges and one clamp was pulled horizontally by a cord
connected to 1 to 6 weights ($m=$20 g each) to apply a uniaxial tension
$\sigma_{1}$ with varying magnitudes. The Cauchy stress applied to the film is
$\sigma_{1}=\lambda_{1}\,Nm\,g/(2\,h_{0}\,W_{0})$, where $W_{0}$ and $h_{0}$
are the initial width and half-thickness of the sample, respectively, $g=9.8$
m/s2 is the acceleration of gravity and $N$ is the number of the weights. When
changing the stress state, the results obtained during loading and unloading
were averaged to minimise the effect of hysteresis.

For each stress state, the current thickness $2h$ of the rubber film was
measured from the OCT image. The uniaxial stretch ratio was then determined by
$\lambda_{1}=\lambda_{3}^{-2}=(h_{0}/h)^{2}$. The measured stretch ratio
agreed well with that obtained by the deformation of the grids drew on the
surface of the rubber film.

#### Materials

The rubber sheet has mass density $\rho\simeq$ 1,070 kg/m3 and refractive
index $n\simeq 1.4$. The initial dimension was $2h_{0}\simeq 0.5$ mm,
$W_{0}=16$ mm, and $H_{0}=40$ mm. The lateral size was large enough to avoid
wave reflections at the edges. The rubber sample was prepared from
Ecoflex${}^{\text{TM}}$ 5 material (Smooth-On Inc) by mixing the Ecoflex 1A
and 1B at 1:1 ratio by weight. The mixture was poured into a mold and cured at
room temperature overnight. Then the material was post-cured in an oven at
80∘C for 2 hours. For mechanical testing, we cut out a small piece (
$0.5\times 5\times 18$ mm3) and performed a tensile test with a uniaxial
tensile testing machine (eXpert 4000 Micro Tester, Admet, Norwood, USA).
Figure 3e shows the resulting stress-stretch curve. Applying a linear fitting
to the initial stage of the curve (stretch ratio < 1.07) we find that the
initial shear modulus (one third of the Young modulus) is approximately 180
kPa.

We used a common home-use _cling film_ (plastic wrap) made of $100\%$
polyethylene with $\rho\simeq 930$ kg/m3. The typical thickness of cling films
ranges from 8 to 13 $\mu$m. Here we used Brillouin microscopy [20] to measure
the thickness of our film to be $11.7\pm 0.3\mu$m. This is close to the axial
resolution of the OCT system, and so we could not track $h$ with the
deformation.

The _bodhrán_ instrument was purchased from Hobgoblin Music, MN, USA. The OCT
measurement was performed on the intact instrument. On separate direct
measurements after removing the skin from the frame, we found $h_{0}=360\pm
30$ $\mu$m and $\rho=831\pm 65$ kg/m3 in the dry condition. After hydration,
the density is expected to increase to $\sim$ 1000 kg/m3. To characterise the
fundamental resonance frequencies of the instrument in the dry and damp
conditions, the centre of the drumhead was beaten every 10 seconds while
recording the sound with a cellphone 10 cm away from the drumhead, using the
Google Science Journal App.

## Acknowledgements

This study was supported by grants P41-EB015903, R01-EB027653, DP1-EB024242
from the National Institutes of Health (USA) for G.Y.L and S.H.Y, and by the
111 Project for International Collaboration No. B21034 (Chinese Government, PR
China), a grant from the Seagull Program (Zhejiang Province, PR China) for M.D
and a grant from the European Commission - Horizon 2020 / H2020 - Shift2Rail
for A.L.G. The authors thank Drs Amira Eltony and Xu Feng for help with the
measurements, and Pasquale Ciarletta, Niall Colgan and Giuseppe Zurlo for
valuable feedback.

## Author contributions statement

G.Y.L., A.G., and M.D. designed the study. G.Y.L., A.G., and M.D. developed
the theoretical model. G.Y.L. conducted the experiments. G.Y.L. and S.H.Y.
analyzed the results. All authors wrote and reviewed the manuscript.