# Demonstration of Parametric Instability suppression through optical feedback

Vladimir Bossilkov, Jian Liu, Carl Blair, Chunnong Zhao and Li Ju
OzGrav
University of Western Australia
Crawley, Western Australia 6009, Australia
<EMAIL_ADDRESS>affiliations: AEI & LUH

###### Abstract

We demonstrate the suppression of parametric instability using through optical
actuation in an electro-optical feedback loop, stabilising the high order
optical mode content in an 80 metre long Fabry-Perot cavity. The loop
suppression of the high order mode is achieved by injecting a high order mode
with the same frequency and opposite phase. Frequency matching is achieved by
measuring the beat note signal between the fundamental and high order mode in
transmission of the cavity and applying that signal to an electro-optical
modulator to create the required frequency sideband. Spatial mode matching of
the sideband to the high order mode is accomplished through the inherent mode
overlap between the input injected beam and the high order mode of the cavity.
The paper presents the theoretical analysis and experimental demonstration of
parametric instability suppression, for an instability which would normally
ring up with a parametric gain of approximately 2.5.

##  1 Introduction

Advanced gravitational wave detectors, including Advanced LIGO(Scientific
Collaboration, 2015) (aLIGO), Advanced Virgo(Acernese et al., 2015) and
KAGRA(Aso et al., 2013), all use long Fabry-Perot optical cavities which are
designed to contain intra-cavity optical power as high as 800 kW. In this
paper we explore the suppression of a process called three-mode parametric
instability(Braginsky et al., 2001) (PI), similar to Brillouin Scattering,
which scatters a cavity’s fundamental mode into unwanted high order modes, in
the process exciting mechanical modes in the test masses of the cavity.
Without mitigation many mechanical modes could become unstable due to
interactions with many optical modes.

The high order mode beats with the fundamental mode, exerting a radiation
pressure force that in turn can further drive the mechanical mode. PI can
occur if all of the following conditions are met: the beat frequency between
the scattered high order optical mode and the fundamental mode is close to the
test mass acoustic mode frequency; there is sufficient spatial overlap between
the high order optical mode and the acoustic mode shape; and the system has
very low loss. Under these conditions, when operating at a high power, the
radiation pressure force exerted on the test mass drives the test mass
acoustic mode. If the energy imparted to the mechanical mode surpasses the
energy dissipated by the mode it will become unstable.

This phenomenon was first observed in Gingin, Western Australia (Zhao et al.,
2015) and soon after in the aLIGO Livingston Observatory (Evans et al., 2015).
PI observed during commissioning of aLIGO and had previously limited the
operating intra-cavity power(Blair et al., 2015) and was being mitigated by
thermal tuning(Degallaix et al., 2007) and electrostatic feedback(Blair et
al., 2017) control. Recently aLIGO implemented acoustic mode dampers(Gras et
al., 2015), which reduce the quality factor of most unstable mechanical modes
by about 10 times, largely mitigating the issue for those detectors.

A number of other PI control methods have been proposed in the past(Gras et
al., 2009; Fan et al., 2010; Miller et al., 2011; Degallaix et al., 2007;
Blair et al., 2017). Zhang et al. (2010) first proposed injecting a high order
optical mode which is matched in frequency and mode shape to a high order mode
created by PI, while its phase could be tuned to suppress PI. This concept was
demonstrated by Fan et al. (2010), who used optical injection to suppress a
high order mode in a Gingin cavity, but that cavity did not have self
sustained PI occurring.

We implement a simpler version of the optical injection scheme, and Sec. 2
derives an expression for the change in Parametric Gain due to its
implementation. In Sec. 3 we present our experimental setup for this control
scheme. Sec. 4 discusses the specific aspects of the performance of our
system, and highlights its effectiveness. Lastly, Sec. 5 considers the
implementation of this scheme to Gravitational Wave detectors, such as the
aLIGO observatories.

##  2 Theoretical Derivation

We adopt the equation of motion from Danilishin et al. (2014), and consider
the feedback of the high order mode as an additional input to the high order
mode. Essentially we consider the high order mode to be pumped by feedback
amplitude, ${a_{1}}^{\text{FB}}$:

$\displaystyle\dot{a_{0}}=$
$\displaystyle-\frac{\gamma_{0}}{2}a_{0}+\textbf{i}~{}G_{\text{01}}b_{\text{m}}a_{1}+\sqrt{{\gamma_{0}}^{\text{in}}}~{}(A_{p_{0}}+{a_{0}}^{\text{in}}),$
(1) $\displaystyle\dot{a_{1}}=$
$\displaystyle-\left(\frac{\gamma_{1}}{2}+\textbf{i}~{}\Delta_{\text{m}}\right)a_{1}+\textbf{i}~{}G_{\text{01}}a_{0}b_{\text{m}}^{\dagger}$
$\displaystyle+\sqrt{{\gamma_{1}}^{\text{in}}}~{}({a_{1}}^{\text{FB}}+{a_{1}}^{\text{probe}}+{a_{1}}^{\text{in}}),$
$\displaystyle\dot{b_{\text{m}}}=$
$\displaystyle-\frac{\gamma_{\text{m}}}{2}b_{\text{m}}+\textbf{i}~{}G_{\text{01}}~{}a_{0}~{}a_{1}^{\dagger}+\sqrt{\gamma_{\text{m}}}~{}b_{\text{th}},$

where: $a_{\text{0,1}}$ is the amplitude of fundamental and high order modes
in the cavity, oscillating at frequencies $\omega_{0}$ and $\omega_{1}$
respectively; $b_{m}$ is the amplitude of the mechanical mode oscillating at
frequency $\omega_{m}$; $\gamma_{\text{0,m}}$ is the full linewidth of the
fundamental optical and test mass mechanical mode; $G_{\text{01}}$ is the
optomechanical coupling strength defined by Danilishin et al. (2014);
$A_{p_{0}}$ is the pump amplitude of the fundamental mode;
${a_{\text{0,1}}}^{\text{in}}$ are the vacuum noise fluctuations for the
optical modes; and $b_{\text{th}}$ is the thermal noise on the test mass at
the mechanical mode frequency; and ${a_{1}}^{\text{probe}}$ is a probing
signal used as an input for transfer function measurements.
${\gamma_{1}}^{\text{in}}$ is the linewidth of the high order optical mode
arising from the tranissivity input test mass. We distinguish this from the
linewidth arising from the end test mass tranissivity,
${\gamma_{1}}^{\text{end}}$, where the sum
${\gamma_{1}}^{\text{in}}+{\gamma_{1}}^{\text{out}}+{\gamma_{1}}^{\text{loss}}=\gamma_{1}$
is the full linewidth of the high order optical mode, and
${\gamma_{1}}^{\text{loss}}$ component of the linewidth arising from
absorption and scattering losses in the cavity. As $\gamma_{0}$ is expected to
be approximately equal to $\gamma_{1}$, they can be defined through known test
mass parameters:

$\displaystyle{\gamma_{0,1}}^{\text{in}}=$
$\displaystyle\frac{c~{}T_{1}}{2~{}L},$ (2)
$\displaystyle{\gamma_{0,1}}^{\text{out}}=$
$\displaystyle\frac{c~{}T_{2}}{2~{}L},$
$\displaystyle{\gamma_{0,1}}^{\text{loss}}=$
$\displaystyle\frac{c~{}\text{P}_{\text{loss}}}{L},$

where $T_{1,2}$ is the power transmission of the input and end test masses,
and $\text{P}_{\text{loss}}$ is the loss due to laser absorption and
scattering per test mass.

$a_{\text{0}}^{\text{in}}$ and $b_{\text{th}}$ are assumed to be zero for this
paper, and we consider the small signal regime where we apply the
approximation that total circulating cavity power is constant and is
equivalent to the fundamental mode power
(${a_{0}}^{2}=\bar{n_{c}}\equiv(2A_{p_{0}}/\sqrt{\gamma_{0}})^{2}$). Since we
treat the fundamental mode as a constant and $b_{\text{m}}$ and $a_{1}$ are
assumed to be very small, neglect their contribution to the fundamental mode,
and thus treat $a_{0}$ as a constant throughout.

In the absence of a feedback pump for the high order mode
(${a_{1}}^{\text{FB}}$ = 0), these equations can be solved analytically in the
frequency domain, taking into account aforementioned assumptions as follows:

$\displaystyle\textbf{i}\Omega~{}a_{1}(\Omega)=$
$\displaystyle-\left(\frac{\gamma_{1}}{2}+\textbf{i}~{}\Delta_{\text{m}}\right)a_{1}(\Omega)+\textbf{i}~{}G_{\text{01}}a_{0}b_{\text{m}}^{\dagger}(\Omega)$
(3)
$\displaystyle+\sqrt{{\gamma_{1}}^{\text{in}}}({a_{1}}^{\text{probe}}+{a_{1}}^{\text{in}}),$
$\displaystyle\textbf{i}\Omega~{}b_{\text{m}}(\Omega)=$
$\displaystyle-\frac{\gamma_{\text{m}}}{2}b_{\text{m}}(\Omega)+\textbf{i}~{}G_{\text{01}}~{}a_{0}~{}a_{1}^{\dagger}(\Omega),$

with the solution to the high order optical mode within the cavity being:

$\displaystyle\frac{a_{1}(\Omega)}{{a_{1}}^{\text{probe}}(\Omega)+{a_{1}}^{\text{in}}(\Omega)}=$
$\displaystyle\frac{\sqrt{{\gamma_{1}}^{\text{in}}}(\frac{\gamma_{m}}{2}+\textbf{i}~{}\Omega)}{(\frac{\gamma_{m}}{2}+\textbf{i}~{}\Omega)(\frac{\gamma_{1}}{2}+\textbf{i}~{}\Delta_{\text{m}}+\textbf{i}~{}\Omega)-G_{\text{01}}^{2}~{}\bar{n_{c}}},$
(4)

where $\Omega$ is the observation Fourier frequency, with $\Omega=0$
corresponding to the mechanical mode frequency (in our case 357.9 kHz). The
quantity ${a_{1}}^{\text{out}}/{a_{1}}^{\text{probe}}$ is a transfer function
of the high order mode cavity gain measured at the end test mass transmission
port, and can be measured with a swept sine probe as
${a_{1}}^{\text{probe}}\gg{a_{1}}^{\text{in}}$ to measure the response across
this optical mode for an open loop measurement, as per our Fig. 1, with
${a_{1}}^{\text{FB}}=0$. Cavity gain for the high order mode,
$G_{\text{cav}}$, is:

$\displaystyle G_{\text{cav}}=$
$\displaystyle\frac{{a_{1}}^{\text{out}}(\Omega)}{{a_{1}}^{\text{probe}}(\Omega)}=\frac{\sqrt{{\gamma_{1}}^{\text{out}}}a_{1}(\Omega)}{{a_{1}}^{\text{probe}}(\Omega)}$
(5) $\displaystyle=$
$\displaystyle\frac{\sqrt{{\gamma_{1}}^{\text{in}}{\gamma_{1}}^{\text{out}}}(\frac{\gamma_{m}}{2}+\textbf{i}~{}\Omega)}{(\frac{\gamma_{m}}{2}+\textbf{i}~{}\Omega)(\frac{\gamma_{1}}{2}+\textbf{i}~{}\Delta_{\text{m}}+\textbf{i}~{}\Omega)-G_{\text{01}}^{2}~{}\bar{n_{c}}}.$

We can obtain the stability criterion of the system by solving for the poles
of transfer function in Eq. (4), which is consistent with that derived by
Danilishin et al. (2014):

$\text{R}_{0}=\frac{4G_{\text{01}}^{2}~{}\bar{n_{c}}}{\gamma_{1}\gamma_{m}}\geq
1+\left(\frac{2\Delta_{\text{m}}}{\gamma_{m}+\gamma_{1}}\right)^{2},$ (6)

where, $\text{R}_{0}$ is the parametric gain of the interaction. When
$\text{R}_{0}$ is larger than the right hand side of this inequality,
parametric instability would occur.

Figure 1: Schematic diagram of the function of an EOM injecting a sideband
into a cavity, demonstrating the generation of the high order mode for the
cavity discussed here. The QPD yaw signal is capable of sensing the beating
product between $a_{0}$ and $a_{1}$ , and is proportional to just $a_{1}$
assuming a0 is very large and unchanging. The EOM acts to covert some of the
fundamental mode (frequency, $\omega_{0}$ )to the high order mode (frequency,
$\omega_{0}$ \- 358 kHz). In reality the EOM only generates the corresponding
sidebands at the high order mode frequency, and the beams’ misalignment with
respect to the cavity is what allows the sideband to pump the real high order
mode. The band pass filter is sufficiently broad (relative to the optical
bandwidths) such that the entire feedback is controlled by a gain defined by
the Amplifier, and a phase defined by the Phase Shifter.

Fig. 1 schematically illustrates the generation of the sideband which produces
the high order mode necessary to prevent PI. The feedback loop alters the
phase such that a new high order mode at the same frequency is injected (also
at the lower sideband frequency) and can destructively interfere with the
existing high order mode. In relation to the PI feedback loop presented by
Evans et al. (2015), this process is capable of eliminating the build-up of
undesirable high order mode content and prevent further excitation of a
mechanical mode by reducing the radiation pressure applied to the test mass.

In order to manipulate this interaction, we define the feedback, discussed in
Sec. 3, as the complex quantity $G_{\text{e}}$. With the feedback loop
implemented, the high order mode is effectively pumped with a term
${a_{1}}^{\text{FB}}=G_{\text{e}}{a_{1}}^{\text{out}}=G_{\text{e}}\sqrt{{\gamma_{1}}^{\text{out}}}a_{1}$,
where $G_{\text{e}}$ describes the gain from electronics in the feedback loop.
Now in frequency domain, we rewrite the equation for $a_{1}$ from Eq. (1) to
see how it changes in comparison to the solution in Eq. (3):

$\displaystyle\textbf{i}\Omega~{}a_{1}(\Omega)=$
$\displaystyle-\left(\frac{\gamma_{1}}{2}+\textbf{i}~{}\Delta_{\text{m}}\right)a_{1}(\Omega)+\textbf{i}~{}G_{\text{01}}a_{0}(\Omega)b_{\text{m}}^{\dagger}(\Omega)$
(7)
$\displaystyle+\sqrt{{\gamma_{1}}^{\text{in}}{\gamma_{1}}^{\text{out}}}~{}G_{\text{e}}~{}a_{1}(\Omega)+\sqrt{{\gamma_{1}}^{\text{in}}}({a_{1}}^{\text{probe}}(\Omega)+{a_{1}}^{\text{in}}(\Omega))$
$\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{1}^{\prime}}{2}+\textbf{i}~{}\Delta_{\text{m}}^{\prime}\right)a_{1}(\Omega)+\textbf{i}~{}G_{\text{01}}a_{0}(\Omega)b_{\text{m}}^{\dagger}(\Omega)$
$\displaystyle+\sqrt{{\gamma_{1}}^{\text{in}}}({a_{1}}^{\text{probe}}(\Omega)+{a_{1}}^{\text{in}}(\Omega)),$

where,
$\gamma_{1}^{\prime}=\gamma_{1}-2\sqrt{{\gamma_{1}}^{\text{in}}{\gamma_{1}}^{\text{out}}}~{}\text{Re}[G_{\text{e}}]$
and
$\Delta_{\text{m}}^{\prime}=\Delta_{\text{m}}+\sqrt{{\gamma_{1}}^{\text{in}}{\gamma_{1}}^{\text{out}}}\text{Im}[G_{\text{e}}]$
are effective changes in high order mode optical linewidth and PI detuning for
a closed loop system. Using Eq. (7) gives the solution for the cavity gain:

$\displaystyle
G_{\text{cav}}=\frac{\sqrt{{\gamma_{1}}^{\text{out}}}a_{1}(\Omega)}{{a_{1}}^{\text{probe}}(\Omega)+{a_{1}}^{\text{in}}(\Omega)}=$
$\displaystyle\frac{\sqrt{{\gamma_{1}}^{\text{in}}{\gamma_{1}}^{\text{out}}}(\frac{\gamma_{m}}{2}+\textbf{i}~{}\Omega)}{(\frac{\gamma_{m}}{2}+\textbf{i}~{}\Omega)(\frac{\gamma_{1}^{\prime}}{2}+\textbf{i}~{}\Delta_{\text{m}}^{\prime}+\textbf{i}~{}\Omega)-G_{\text{01}}^{2}~{}\bar{n_{c}}}.$
(8)

The same solution can be derived through control theory. In an open loop:

$G_{\text{open}}=G_{\text{e}}G_{\text{cav}},$ (9)

and in a closed loop where
${a_{1}}^{\text{FB}}=G_{\text{e}}{a_{1}}^{\text{out}}$:

$\frac{{a_{1}}^{\text{FB}}}{{a_{1}}^{\text{probe}}}=\frac{G_{\text{open}}}{1-G_{\text{open}}},$
(10)

yields the same solution as Eq. (8).

${a_{1}}^{\text{FB}}/{a_{1}}^{\text{probe}}$ is measured in a experimentally
in Sec.4, assuming a large enough ${a_{1}}^{\text{probe}}$ to render
${a_{1}}^{\text{in}}$ negligible.

The instability criterion may once again be evaluated:

$\displaystyle\text{R}_{0}^{\prime}=\frac{4G_{\text{01}}^{2}~{}\bar{n_{c}}}{\gamma_{1}^{\prime}\gamma_{m}}=\text{R}_{0}\left(\frac{\gamma_{1}}{\gamma_{1}^{\prime}}\right)\implies\text{R}_{0}\geq\left(\frac{\gamma_{1}^{\prime}}{\gamma_{1}}\right)\left(1+\left(\frac{2\Delta_{\text{m}}^{\prime}}{\gamma_{m}+\gamma_{1}^{\prime}}\right)^{2}\right).$
(11)

Based on the change in the instability criterion arising from the feedback
term (comparing equations Eq. (6) and Eq. (11)), we derive a term for
effective PI gain reduction factor, $\eta$, for cases where
$\Delta_{\text{m}}=0$, where parametric gain would be maximised for a specific
mechanical mode. Furthermore, since the $\gamma_{1}\gg\gamma_{\text{m}}$, the
expression can be simplified to:

$\eta=\left(1-\text{Re}[\xi]\right)\left(1+\left(\frac{\text{Im}[\xi]}{1-\text{Re}[\xi]}\right)^{2}\right)=\frac{(1-\xi)(1-\xi^{\dagger})}{1-\text{Re}[\xi]},$
(12)

where
$\xi=2\sqrt{{\gamma_{1}}^{\text{in}}{\gamma_{1}}^{\text{out}}}G_{\text{e}}/\gamma_{1}$.

As a consequence of the leading $(1-\text{Re}[\xi])$ term in Eq. (12),
feedback would suppress PI where $G_{\text{e}}$ is negative, or equivalently
having a phase of [$\frac{\pi}{2}$,$\frac{3\pi}{2}$]. A larger suppression
factor is possible near those boundary conditions due to the feedback phase
significantly contributing to effective detuning (through the $\text{Im}[\xi]$
term). Electrostatic damping and acoustic mode damper strategies for PI
mitigation, function only to increase the mechanical loss factor
($\gamma_{m}$), and in comparison, an obvious advantage to using optical
feedback is that far greater suppression can be achieved since the effective
detuning, $\Delta_{\text{m}}^{\prime}$, can be controlled through tuning the
feedback phase.

##  3 Experimental Setup Overview

The Gingin facility has been used for PI experiments since 2008(Zhao et al.,
2008). Table 1 summarises the current cavity parameters that are relevant for
this experiment.

In contrast to the previous implementation(Fan et al., 2010) of optical
feedback, where a separate out-of-phase high order mode was produced, we only
generated the frequency matched sideband through phase modulation of the
carrier mode. Due to an inherent input misalignment to the cavity, some of
this sideband couples to a cavity high order mode. The method presented could
provide an alternative way to suppress parametric for gravitational wave
observatories, and should also be applicable to controlling other
optomechanical systems where high order modes are problematic.

Table 1: Relevant Gingin cavity parameters Parameters | Value
---|---
Mass of test mass, $m$ | 0.8 kg
Test mass Q-factor | $2\times 10^{6}$
Cavity length, $L$ | 74 m
ITM, ETM radii of curvature, $R_{1}$, $R_{2}$ | 40.0 m, 40.0 m
ITM, ETM mirror power transmissivity, $T_{1}$, $T_{2}$ | 200 ppm, 25 ppm
Test Mass absorption/scattering loss, $\text{P}_{\text{loss}}$ | 110 ppm
Laser wavelength, $\lambda$ | 1064 nm
Laser injection power, $\text{P}_{\rm{in}}$ | 5 W
Intra-cavity power, $\bar{n_{c}}$ | 13 kW
Mechanical Mode frequency, $\omega_{m}$ | 2$\pi$ 357.9 krad/s
Total Optical linewidth, $\gamma_{0,1}$ | 2$\pi$ 143.47 rad/s
Input Optical linewidth, ${\gamma_{0,1}}^{\text{in}}$ | 2$\pi$ 8.06 rad/s
Transmission Optical linewidth, ${\gamma_{0,1}}^{\text{out}}$ | 2$\pi$ 64.48 rad/s
Loss Optical linewidth, ${\gamma_{0,1}}^{\text{loss}}$ | 2$\pi$ 70.93 rad/s

Figure 2: Setup of the Gingin East Cavity, and its sub-systems. The cavity
locking loop employs PDH locking (Drever et al., 1983),in blue, to lock the
laser to the cavity. The Fibre Laser Amplifier is capable of outputting laser
power between 1 W and 50 W. The optical feedback loop, in yellow, is
responsible for the injection of sideband which we use to suppress PI. The
pre-mode cleaner (PMC) loop, in pink, ensures a clean Gaussian input for our
cavity, and our injected sidebands are within the bandwidth of this mode
cleaner. Most of our feedback loops use PID controllers as the servo, and are
further amplified by High Voltage Amplifiers (HVA) to provide sufficient
dynamic range. The CO2 laser is used to assist in controlling the ITM radius
of curvature to adjust cavity mode spacing. The functionality of the Optical
Feedback system is discussed throughout. The spectrum analyser in this
configuration can be used to measure the open loop transfer function in a
close loop, by measuring $B/A$ with a swept sine source signal.

The overall set-up is illustrated in Fig. 2. The key readout component, the
Quadrant Photodiode (QPD), is placed at the transmission port of the cavity.
The ‘yaw’ signal senses the beat note power arising from beating between the
fundamental, $\text{TEM}_{00}$, and first order optical mode,
$\text{TEM}_{10}$, of the cavity. The QPD yaw signal is sensitive to both the
$\text{TEM}_{10}$ and $\text{TEM}_{01}$ modes because the cavity axes are not
aligned with the QPD splitting axes, and the modes themselves are of different
frequencies due to test mass imperfections breaking degeneracy between the two
first order modes. You can see the two beat notes as peaks in Fig. 3 and 4. We
know the lower frequency beat note is the $\text{TEM}_{10}$ as we have imaged
it with a phase-camera-like set-up, and the other is inferred to be
$\text{TEM}_{01}$.

The signal on the QPD is filtered to suppress unwanted noise, amplified and
applied to an Electro-Optic Modulator (EOM), $\text{EOM}_{\text{TEM10}}$ ,
with the phase shifter acting to optimise the stability of the feedback by
controlling the phase. The lower sideband generated by the EOM will be a
frequency matched to the high order mode participating in PI.

The power loss attributable to the test masses, $\text{P}_{\text{loss}}$ is
calculated as the required loss per test mass to yield the correct measured
cavity FinesseFang et al. (2021). This loss figure (stated in Table 1) is used
in Eq. (2), to give the $\gamma_{1}$ term in the table. There should be more
scatter loss for the high order optical modes but this difference is small for
the first order optical mode (compared to the fundamental mode), and is
consequently neglected.

The end mirror of our cavity has a mechanical mode at 357.9kHz, with a
Q-factor of approximately $2\times 10^{6}$, that has a known strong opto-
mechanical interaction. The $\text{TEM}_{10}$ mode can be thermally tuned such
that the frequency difference between it and the fundamental mode is 357.9
kHz, using a $\text{CO}_{2}$ laser to alter the radius of curvature of the
ITM. The mechanical Q-factor was measured by observing the time constant of
the ring-down of the mechanical motion of the test mass after suddenly
unlocking the cavity which had been sustaining a saturated PI. This signal
could be observed in the lever arm of the angular control system. The angular
control system provides local angular control with a bandwidth of about 6 Hz,
which has little effect on the signal at 357.9 kHz.

##  4 Performance

In the absence of the feedback loop, the dominant high order mode pumping
mechanism in our cavity is the input misalignment couplingAnderson (1984) due
to added phase delays in the cavity round-trip photon time. A small fraction
of the input power is coupled to the high order mode as not all input power is
perfectly aligned to resonate with the fundamental mode. Our feedback loop
takes advantage of directly pumping the high order mode side-band, essentially
taking advantage of the same process Anderson (1984) uses to control
alignment. The difference is that we use the signal to directly alter the high
order mode content, rather than change the alignment. This injection method
has much higher coupling to the high order mode than the coupling from the
fundamental mode and we provide an estimate for this coupling fraction in this
section, and thus transfer function measurements assume that the injection
signal contribution to ${a_{1}}^{\text{probe}}$ greatly dominates any other
effect occurring within the cavity
(${a_{1}}^{\text{probe}}\gg{a_{1}}^{\text{in}}$). If this assumption does not
hold, coherent transfer function measurements would not be possible.

Fig. 3 illustrates the difference in output observed by the QPD (illustrated
in Fig. 2), with the feedback loop on (red) and off (blue), highlighting a
change in effective optical linewidth that arises from the feedback. The two
peaks represent the beat note between $\text{TEM}_{00}$ and $\text{TEM}_{10}$
modes (left peak), and the beat note between $\text{TEM}_{00}$ and
$\text{TEM}_{01}$ mode (right peak). The apparent suppression of the high
order modes is clear when the feedback loop is turned on.

Figure 3: Power Spectral Density of the beat notes generated from beating with
the fundamental mode and the two first order modes, which are observed at the
QPD at the cavity transmission port. The amplitudes of these peaks are
proportional to the radiation pressure exerted on test masses in the PI
process due to the corresponding optical modes. The feedback loop acts to
suppress this beating in the cavity. The first peak corresponds to beating
with fundamental mode ($\text{TEM}_{00}$) and the the $\text{TEM}_{10}$ mode,
and the other is beating with the $\text{TEM}_{01}$ mode. We show that the
amplitude of both optical modes is reduced by the feedback loop.

Fig. 4 shows the open loop transfer function for our feedback loop, with the
peaks in gain magnitude corresponding to the frequencies of the optical
beatings. The open loop gain of the system can be directly measured and
represents the product of several aspects of the system, as discussed in Sec.
2. This measurement was done by adding a swept sine signal from a spectrum
analyser to the signal from the QPD, using the spectrum analyser illustrated
in Fig. 2. By dividing the response of QPD signal by the sum of a swept sine
and the QPD signal we make a direct measurement of the open loop gain within a
closed loop (as illustrated by Freise (2003)).

Within the bandwidth of the beating, the phase response is very steep:
exhibiting a $180\degree$ rotation in about 1 kHz. When the frequency of these
peaks shifts due to beam position change, thermal effects from optical or
$\text{CO}_{2}$ intensity fluctuations, the frequency of these peaks could
change on the order of 1 kHz. This causes our phase margins to change and lead
to loop stability being tricky at times. This is highlighted by transfer
function measurement issues mentioned in Fig. 5.

We can attempt to model the observations in Fig. 4, by estimating gain,
$G_{\text{e}}$, and $\Delta_{\text{m}}$ from Eq. (8). $G_{\text{e}}$ describes
the product of electronic gain, photodiode sensitivity, EOM efficiency,
subsequent amplification in the Fibre Laser Amplifier and the unknown coupling
coefficient for the fundamental-mode-shaped input sideband to the high order
mode within the cavity. $G_{\text{e}}$ is considered to have a constant gain
and phase for all frequencies within the bandwidth of an optical mode. We
approximate the losses in the optical modes to be comparable
($\gamma_{1}\approxeq\gamma_{0}$), and thus we can replicate the observed gain
and phase from Fig. 4 for the right hand beat note with
$G_{\text{e}}\approxeq-11$ (i.e. with phase of $\pi$), given $R_{0}\approxeq
2.5$. Plotting said transfer function (see Fig. 5) reveals a gain and phase
feature for frequencies near the mechanical mode (i.e. less than $\gamma_{m}$
away from $\omega_{m}$), when $R_{0}>1$. For large parametric gains, this can
manifest as a large dip in the open loop gain, particularly when detuning
($\Delta_{\text{m}}$) is small. A larger loop gain could therefore be required
to overcome both instability arising from PI, as well as potentially unstable
unity gain frequencies that could arise near the mechanical mode frequency if
the PI gain is very large. We do not see this feature in the measured data in
Fig. 5 or Fig. 4, since it would lie in a bandwidth of about 2 Hz, and the
spacing of the measured data is much too sparse to see it.

Danilishin et al. (2014) evaluates a time domain equation for PI evolution.
Assuming no knowledge of the feedback phase, we evaluate a reduction of
$\text{R}_{0}$ using Eq. (11) for the aforementioned range of stable phases
and highlight an area (in purple) in Fig. 6 where the best to worst case time
evolution scenarios would lie for $|G_{\text{e}}|=11$ and an unknown, but
stable, feedback phase. The two extremes correspond to an effective reduction
of $\text{R}_{0}$ by a factor between 6.2 and 28, for cases where
$G_{\text{e}}$ has a phase of $\pi$ and $\frac{\pi}{2}$ respectively.

Figure 4: Open Loop Transfer function across the two optical modes. Frequency
range was scanned using a chirped signal being applied to the EOM. Highlighted
in red is the frequency of the mechanical mode. Unity gains are highlighted
with black circles, with the worst unity gain phase being 43$\degree$ at 358.5
kHz. In this measurement the corresponding beat note is slightly detuned from
the mechanical mode of interest. There is an overall slope in the phase curve
of $-12\degree$ per kHz, due to the band-pass filters used in the feedback
loop, that have a bandwidth broader than this measurement.
Figure 5: Open Loop Transfer Function across the optomechanically unstable
mode, illustrating the sharp feature at 357.9 kHz, that could not be seen in
the measured data (in blue). The estimated transfer function, in purple, plots
plots the closed loop scenario ($a_{1}^{\prime}/{a_{1}}^{\text{probe}}$)
frequency shifted to 357.9 kHz, with a detuning of
$\Delta_{\text{m}}=-2\pi~{}30$, $G_{\text{e}}\approx-11$. As the optical beat
note would be moving in frequency, due to beam position and thermal state of
the cavity changing during a measurement (which consists of the average of
many measurements), we observe a disparity between the measurement and this
model. Fig. 4 shows the full measurement, where the other visible beat note
more clearly agrees with theory, though it has no optomechanical interaction.
Figure 6: Demonstration illustrating damping of observed instability. Here we
allowed PI with a Gain of approximately 2.5 to begin ringing up (seen in
blue), and enabled the feedback loop (in red) using a switch. The feedback
loop was turned on at 56 seconds illustrating a ring-down comparable to that
of the free ring-down slope observed in the mechanical mode in the absence of
any optical power (cyan line), based on the mechanical Q stated in Sec. 3. The
purple lines are the range of theoretical models based on the analysis in Sec.
2, with time-domain evaluation based on Danilishin et al. (2014). Our
recording device appears to have reached a noise floor at about -55 dBV for
the error signal, which doesn’t affect the feedback loop. The amplitude drop
at 56 seconds in the theoretical model is artificial to line up with the
sudden suppression of the visibility of the beat notes just as in Fig. 3. The
cyan and purple lines are plotted to the same scale as the Error signal.

Fig. 6 demonstrates the effectiveness of the feedback loop to suppress PI as
it is being excited with a Parametric Gain of approximately 2.5. Switching on
the feedback loop imparts an instantaneous reduction on our signal equivalent
to the loop gain at 357.9 kHz, seen at 46 and 56 seconds (this is likely
slightly different from what is shown in Fig. 4, since the experiment was
conducted on a different day, likely with slightly different cavity power and
beam positions). The mechanical mode was excited to a significantly higher
amplitude the second time. We see a ringdown of this excitation, which is
comparable to the ringdown of the mechanical mode in the absence of
optomechanical interaction (free ringdown), though with a longer time constant
since the interaction is not completely removed due to finite loop gain and
likely sub-optimal feedback phase. As expected, the observation falls
somewhere between the two modelled extremes noted in Sec. 2, as we do not know
the exact feedback phase on the day of measurement.

Since we know that we are not injecting a perfectly mode matched field we
expect only a small fraction of injected fundamental mode power to couple to a
high order mode. This parameter is encompassed in $G_{\text{e}}$ for Sec 2,
but it is possible to provide an estimate for the percentage of power from
sideband which pumps the resonant high order mode. To do so we consider a
reduction in high order mode power seen in both peaks in Fig. 3 to go from the
red to the blue curve (assuming feedback phase is such that it is only
destructively interfering, that is: $G_{\text{e}}$ has a phase of $\pi$), and
compare this to the total power which EOM is putting into the sidebands.
Comparing these ratios will provide a lower limit estimate for the coupling of
our input to the cavity.

To calculate the ratio of sideband power in the cavity injection, we consider
the signal applied to the EOM when sustained suppression visible (namely,
after 65 seconds in Fig. 6), where a signal with a peak voltage of
approximately 400 mV is applied to the EOM. Based on the modulation depth of
the EOM that we are using and assuming no further losses, this should produce
45 $\mu$W of power to the modulated sidebands of our 5 W cavity input beam,
half of which would couple to the high order mode frequency (only one of the
two sidebands), giving an injection power ratio of $4.5\times 10^{-6}$.

To calculate the change in observable high order mode within the cavity we
consider the change in power for the two curves in Fig. 3. Integrating over
each curve from 356 kHz to 364.5 kHz we obtain a change in observed power in
the beat notes. Given this, its possible to calculate (assuming a constant
intra cavity power) the high order mode amplitude ($a_{1}$) to fundamental
mode amplitude ($a_{0}$) ratio of $a_{1}/a_{0}=1.2\times 10^{-4}$, and thus a
power ratio of ${a_{1}}^{2}/{a_{0}}^{2}=1.4\times 10^{-8}$.

If we assume that this lower injected sideband power is enough to account for
all change in the high order mode, we can estimate a minimum $0.3\%$ of the
injected sideband power couples to our high order modes. This coupling
estimates how much of the sideband is essentially pumped into the cavity as a
high order mode, and encapsulates the mode-matching(Anderson, 1984) of this
sideband to the high order mode.

The small coupling ratio, combined with the very small ETM transmission of
approximately 25 ppm, means that our feedback loop required very high
electronic gain ($4\times 10^{5}$) to achieve the required gain stated in Sec.
2. Consequently strong filtering was required to prevent small noise sources
from saturating our electronics.

For our experiment, we experience no PI when the feedback loop is on
constantly, due to the corresponding reduction of effective Parametric Gain
(derived in Sec. 2). This system can be extended to other optical modes which
arise from power being scattered out from the fundamental mode, through the
addition of more sidebands. Ideally, one needs a sensing array of photo-
detectors each sensitive to only a single optical beat mode, and a feedback
loop controlling the phase of the corresponding high order mode, so that one
does not need to run into the complication of affecting multiple optical modes
with a single feedback loop.

##  5 Implementation Consideration

The major advantage of optical feedback compared to mechanical feedback to
control parametric instability is reduced complexity. The reduced complexity
arises as there are fewer optical modes ($\sim$7 in the arm cavities of aLIGO)
compared to unstable mechanical modes ($\sim$20 in each of 4 test masses(Gras,
2009) of aLIGO), resulting in a manageable number of potential independent
control loops. In this paper effective control has been demonstrated with the
simple addition of a frequency modulation feedback.

Implementing such an actuation system in a gravitational wave detector would
be a simple task. The power recycling cavity in aLIGO is designed to have a
bandwidth of 350 kHz(Arain and Mueller, 2008). Since all known unstable
mechanical modes in aLIGO occur at frequencies well below that, any optical
modes that are actuated on would be resonant in the power recycling cavity and
would therefore reach the cavity arms where they need to reduce high order
mode content. However actuation relies on a fortuitous input coupling between
the injected optical mode and the $\text{TEM}_{01}$ mode. Some investigation
would be required to determine if this coupling would be adequate in a
gravitational wave detector. If this coupling was not large enough some
spatial manipulation of the feedback optical field would be required (as
previously proposed(Fan et al., 2010; Zhang et al., 2010)) increasing
complexity.

The sensing scheme proposed here has already been demonstrated in mechanical
damping experiments at aLIGO(Blair et al., 2017). In these experiments
parametric instability was controlled with mechanical feedback using an error
signal generated from the beat note between the fundamental and higher order
mode in transmission of the arm cavity as used in this paper. The only
foreseeable sensing problem would be poor signal to noise ratio particularly
in the case that the optical beat has little overlap with degrees of freedom
that can be measured by a QPD. In this case a multi-element photodiode with
better overlap with the high order modes may be required.

One potential issue with the proposed scheme arises if suppression is required
near the detection band of a gravitational wave detector. aLIGO’s 7th order
optical modes for example are close to the detection band. In this case
feedback filters would have to be more carefully designed to avoid altering
the interferometer sensing function or the injection of noise. Fig. 3 can
illustrate this issue, where some extra noise is injected at frequencies away
from the intended actuation bandwidth (contributing extra noise where the blue
curve is above the red). Whilst the most sensitive frequency of the detector
like aLIGO is around 100 Hz, frequencies as high as 5 kHz are considered
important for aLIGO(Martynov et al., 2016). To minimise the effect on
interferometer sensitivity one could limit the feedback to a specific
frequency around the mechanically excited PI.

The existing control infrastructure already in place for PI feedback through
Electro-Static Drives (ESDs)(Blair et al., 2017) at both aLIGO sites and would
suit optical feedback design with the addition of broadband frequency
modulation.

##  6 Conclusion

We have demonstrated suppression of PI using a novel optical injection scheme
capable of alleviating parametric instability (PI) in our cavity. The method
involves the feedback of the signal produced from the beating of the
fundamental and a high order optical modes sensed at the transmission port of
the cavity. An EOM at the injection port generates the necessary frequency
matched sidebands, where the lower sideband couples to the high order mode in
the cavity to cancel out the high order mode normally produced due to
scattering in cavity.

A major improvement from previous iterations of optical feedback, is that we
did not have to produce the correct high order mode shape in order to couple
to that mode. Specifically, we estimated the coupling of the generated
sideband to the high order mode, to be at least $0.3\%$, which is still much
larger than power ratio of the high order mode to the fundamental mode within
the cavity. This makes utilising this feedback system easier and more robust
against alignment issues than previous proposals.

In Fig. 6, we demonstrate a dramatic suppression of the Parametric Gain on an
unstable mode, with a gain reduction of at between 6.2 and 28 times, as
derived in Sec. 2. This demonstrates this scheme’s ability to suppress high
gain PI, which can be further improved with higher electronic feedback gain.
Furthermore we demonstrated that continual application of this feedback would
prevent the onset of instability.

In our experiment we demonstrated the suppression of two high order modes, and
it is possible to sum signals produced by any number of high order mode beat
notes. This would allow suppression of PI produced by any number of optical
modes, remedying PI for all mechanical modes affected by that set of beat
notes. aLIGO already has infrastructure capable of sensing the required
signals and implementing this scheme is already possible.

## Acknowledgements

We wish to thank the LIGO Scientific Collaboration Advanced Interferometer
Configurations and Optics Working Group for useful advice. This work is
supported by Australian Research Council Discovery Project DP160102447, and
Center of Excellence for Gravitational Wave Discovery project CE170100004.

The authors would also like tho thank Joris van Heijningen and Jue Zhang for
useful input into countless aspects of this paper. We are also thankful Steve
Key and John Moore who contribute so much to the maintenance and technical
improvements made to the Gingin site.

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# An _In Situ_ Study of the Role of Pressure on Fe Recrystallization and Grain
Growth during Thermomechanical Processing

Darren C. Pagan<EMAIL_ADDRESS>Lukas A. Kissell  Matthew L. Whitaker
Materials Science and Engineering, The Pennsylvania State University,
University Park, PA 16802  Mineral Physics Institute, Department of
Geosciences, Stony Brook University, Stony Brook, NY 11794

###### Abstract

Elevated pressures are encountered in many metal forming processes that can
alter microstructural evolution rates. Here we measure rate changes with
pressure in recrystallization and grain growth in Fe through adaptation of
synchrotron-compatible multi-anvil presses, originally designed for study of
the mantle. Recrystallization and grain growth are monitored _in situ_ using
high-energy X-ray diffraction. Principal component analysis applied to the
diffraction images is used to quantify evolution rates, with increasing
pressure significantly slowing the process.

††journal: arXiv

Elevated pressures are often encountered in many industrially relevant forming
processes including forging, rolling, and extrusion that can reach several
times the yield strength of an alloy [1] along with direct pressure
application during hot-isostatic pressing [2]. These elevated pressures will
naturally influence the thermodynamics and kinetics of microstructural
evolution in tandem with changes in temperature. Despite extensive literature
documenting significant changes in mechanical properties like yield strength
under applied pressure [3, 4], this pressure dependence of mechanical response
is rarely accounted for in modeling efforts. Even less well investigated are
pressure’s effects on microstructural evolution. As we move towards more
sustainable alloy production processes that utilize less energy and emit less
CO2, understanding and controlling pressure’s role on microstructural
evolution can be a critical component of the effort.

While the role of pressure on mechanical response has been examined for nearly
a hundred years [5] starting with Bridgman [6, 7, 8], studies of the role of
positive pressure with respect to microstructural evolution during processing
are more limited. From a thermodynamics perspective, increasing pressure will
change the free energy to drive microstructural evolution in metallic alloys
[9], while in the context of kinetics, studies have shown that substitutional
diffusion is generally slowed under pressure [10, 11]. Regarding
recrystallization, a relatively early study on pure Cu indicates a slowing of
recrystallization with application of pressure [12] and similarly on $\alpha$
brass [13], but these studies relied on _ex situ_ characterization to estimate
recrystallization rates.

Although _in situ_ diffraction measurements during heat treatment have become
relatively common over the past two decades [14, 15, 16, 17, 18, 19, 20], _in
situ_ measurements during application of temperature _and pressure_ relevant
to common thermomechanical processing are generally not performed. A major
challenge for studying microstructural evolution under pressure is that the
devices are difficult to safely construct, particularly with inclusion of _in
situ_ monitoring. Fortunately, the mineral physics community has developed
synchrotron-X-ray-compatible loading devices for studying rock deformation in
the lower crust and upper mantle which apply complex combinations of pressure,
deviatoric stress, and high temperature. Here we demonstrate that these same
facilities can be extended to study _in situ_ microstructural evolution in
conditions simulating hot metal forming processes. The capability is
demonstrated through _in situ_ X-ray diffraction measurements of slowed
recrystallization and grain growth in pure Fe with increasing pressure
applied. Extraction of transformation rates from the diffraction data is
facilitated by machine-learning dimensionality reduction.

To study the effects of pressure on recrystallization and grain growth _in
situ_ , high-pressure measurements were performed on the X-ray Powder
Diffraction beamline (XPD, 28-ID-2) at the National Synchrotron Light Source
II [21]. An image of the Rockland Research hydraulic press in the beamline
used for testing is shown in Fig. 1a, while a schematic of the X-ray
measurement geometry is given in Fig. 1b. The specimen was illuminated with a
68.1 keV X-ray beam traveling in the -$\bm{e_{z}}$ direction. For imaging
measurements, the beam was 2.2 mm $\times$ 2.2 mm (horizontal by vertical) to
fully illuminate the specimen. The beam size was reduced to 300 $\mu$m
$\times$ 100 $\mu$m for diffraction measurements. Specimens were deformed
within the hydraulic press using 8 truncated cubic anvils which are
independently controlled with three hydraulic pumps to produce varied stress
states. A primary pump applies a hydrostatic pressure while two differential
pumps can vary the stress state along the vertical direction $\bm{e_{y}}$. The
8 anvils (2 sintered diamond and 6 WC) are arranged into a Kawai-style
geometry [22, 23] shown in Fig. 2a. The sintered diamond anvils are X-ray
transparent and arranged downstream to allow diffraction from the specimen to
be transmitted.

Figure 1: a) Image of the X-ray compatible hydraulic loading press used for
the high-pressure recrystallization and grain growth studies. b) Schematic of
the detectors used for _in situ_ X-ray measurements. A scintillator collecting
the direct beam was used to measure macroscopic strain from X-ray radiographs,
while a large-panel diffraction detector further from the specimen was used to
monitor recrystallization and grain growth.

The cylindrical specimens tested were extracted from a 99.99% Fe rod (1 in
diameter $\times$ 6 in long) attained from MilliporeSigma. The specimen
dimensions were 2 mm diameter $\times$ 2 mm tall (along $\bm{e_{y}}$) to be
compatible with the sample testing environment. A cross section of the anvil
loading system and the full sample assembly is given in Fig. 2b. The sample
was heated using direct resistance heating through the WC anvils and a BN-
doped TiB2 sleeve placed around the specimen. The specimen and sleeve were
then placed into a mullite (2Al2O3 SiO2) solid pressure medium that transmits
load from the anvils to the specimen.

Figure 2: a) Side view of the Kawai-style anvil geometry used to compress the
Fe specimens. b) Top-down cross-sectional view of the Kawai-style anvil
geometry.

The testing procedure consisted of first building a sufficient dislocation
density within the specimens by deforming the specimens to 12% macroscopic
strain in uniaxial compression. After the specimens had been deformed, they
were then pressurized to target pressures $P$ of 1 GPa and 2 GPa. Pressures
were reached by applying prescribed tonnages on the hydraulic press (222 and
374 kN respectively) determined from precalibration of the system. Temperature
in the specimens was then ramped to 675 ∘C by increasing power through direct
resistive heating (128 W target) to initiate recrystallization within the
specimens. The choice of temperature was informed by previous measurements to
determine a temperature level that would drive grain growth in a time frame
commensurate with detector speed. Active pressure control was then enabled
during heating to maintain a fixed pressure on the specimens during thermal
expansion. Power was maintained to $\pm$ 3 W during heating, corresponding to
$\pm$ 16 ∘C. The true specimen pressures and temperatures were calculated
after testing from lattice strains determined from peak shifts in the
diffraction data, before and after application of pressure and temperature.
Translation from strain to pressure and temperature was performed using linear
elasticity (bulk modulus of 164.7 GPa [24]) and linear thermal expansion
(coefficient of thermal expansion of 11.5$\times$10-6 K-1 [25]) calculations
respectively. The true specimen pressures and temperatures determined from the
diffraction data are given in Table 1. For brevity, Tests 1 and 2 will be
referred to as the 1 GPa and 2 GPa tests respectively.

| Target $P$ (MPa) | Actual $P$ (MPa) | Target $T$ (∘C) | Actual $T$ (∘C)
---|---|---|---|---
Test 1 | 1000 | 963 | 675 | 675
Test 2 | 2000 | 2097 | 675 | 685
Table 1: Test matrix showing the target and actual pressures $P$ and
temperatures $T$ for the two recrystallization and grain growth tests.

The current macroscopic strain in the specimens during prestraining was
measured using X-ray absorption radiography to monitor the top and bottom of
the specimens in real time. Radiography was performed using a YAG scintillator
placed 470 mm from the specimen. The scintillator was coupled to a PointGrey
Grasshopper 3 camera (2448 $\times$ 2048 pixels with 3.45 $\mu$m pixel size)
and a variable optical lens. Magnification was chosen such that the effective
pixel size for the imaging was 1 $\mu$m, providing effective strain resolution
of approximately $5\times 10^{-4}$. Once the target temperature was reached,
diffraction data was collected continuously at a rate of 1 Hz on a PerkinElmer
XRD 1621 detector sitting 1318 mm behind the specimen. The detector has 2048
$\times$ 2048 pixels with a size of 200 $\mu$m. In total, 1920 images were
collected during the 1 GPa test and 6560 images during the 2 GPa test. After
heating, the specimen was quenched and depressurized. We note that
uncontrolled specimen deformation that occurs during depressurization
prevented preservation of the specimens’ microstructure for post-testing
microscopy.

Figure 3 shows the evolution of the diffraction patterns at the beginning and
end of specimen heating for the 1 and 2 GPa tests. Initial diffraction
patterns for the 1 and 2 GPa tests are provided in Figs. 3a and 3b
respectively, and final diffraction patterns are provided in Figs. 3c and 3d.
The azimuthal angle $\eta$ has also been labeled. Diffraction rings from the
(110), (200), and (211) sets of lattice planes are labeled in Fig. 3a, but
diffraction rings are complete only for the (110) and (200) lattice planes.
Other diffraction rings present are from the BN-doped TiB2 sample sheathing
and mullite pressure medium. The intensity scaling in the figure is chosen to
make the diffracted intensity from the Fe clearly visible. In Figs. 3a and 3b,
at the beginning of the temperature soak, the Fe diffraction rings have non-
uniform intensity around the ring, but the intensity is generally continuous
which indicates a fine-grain microstructure or grains with large amounts of
misorientation in addition to preferred orientation within the specimens.
After the temperature soak (Figs. 3c and 3d), strong isolated diffraction
peaks are observed at the positions of the Fe diffraction rings. These
diffraction peaks correspond to newly formed, large grains with low defect
content. This transition from continuous diffraction rings to isolated peaks
is representative of the recrystallization process.

Figure 3: Diffraction images at the beginning of the experiments at 675 ∘C
with pressures of (a) 1 GPa and (b) 2 GPa applied. Diffraction images after
the recrystallization tests have completed with (c) 1 GPa and (d) 2 GPa
applied. The bright, isolated diffraction spots that form during the tests are
emitted from large grains with low defect content that form during
recrystallization and grain growth. The azimuthal angle $\eta$ on the detector
has been labeled.

Quantitative analysis of this recrystallization and grain growth process
(i.e., transition from continuous rings to discrete spots in the data) is a
challenge in comparison to more common quantitative analysis of phase
transformations using Rietveld refinement [26]. For this reason, we utilize
principal component analysis (PCA) to extract quantitative measures of the
rates of the recrystallization and grain growth process at pressure. PCA is a
form of unsupervised machine learning [27] in which high-dimensional data
(here the tens of thousands of pixels of diffracted intensity around Fe rings
of interest) are projected along orthogonal ‘principal directions’ of maximum
variance of the dataset to obtain ‘scores’ ($z_{i}$). Previous efforts have
demonstrated that lower-dimensional descriptors of diffraction data directly
reflect changes in the microstructure being probed [28, 29, 30]. In this case,
the lower-dimensional descriptors reflect the formation and growth of new
grains and the corresponding texture changes (as seen in the transition from
continuous diffraction rings to distinct diffraction spots). The principal
directions are the eigenvectors of the covariance matrix of each dataset. In
most datasets, a relatively small number of projections or scores can capture
a majority of the variance contained, thereby reducing the dimensionality. The
eigenvalues of the covariance matrix reflect the total variance captured by
each principal component direction.

To minimize contributions from the other diffracting materials in the X-ray
beam path, diffracted intensity from around the 110 and 200 diffraction rings
(the two complete diffraction rings) was utilized in the PCA. A more detailed
explanation of this peak extraction procedure prior to dimensionality
reduction can be found in [28]. Here, the sets of diffraction images from the
1 GPa and 2 GPa pressure tests are treated as separate datasets with different
corresponding principal directions. For both tests, a majority of the data
variance is described by projecting the data along the first two principal
components. As such, we focus on the first two principal component scores,
$z_{1}$ and $z_{2}$.

The first two principal components capture 71.5% and 9.6% respectively of the
1 GPa data variance (81.1% total) and 52.9% and 18.4% respectively of the 2
GPa data variance (71.3%). Figure 4a shows the evolution of the first PCA
score $z_{1}$ with time $t$ from the two recrystallization tests. The scores
from the 1 GPa test are plotted in blue, while scores from the 2 GPa test are
plotted in red. In this and other portions of Fig. 4, each marker corresponds
to a score extracted from a diffraction image. For clarity, every 10th score
is shown. We note that scale and sign of the scores are arbitrary, so in this
work, scores are rescaled between 0 and 1. The time evolution of $z_{1}$ in
both tests shows a monotonic increase and then a plateau at the end of the
test. Figure 4b shows the time evolution of the second score $z_{2}$. The
second score $z_{2}$ in both tests exhibits an increase to a maximum and then
a decrease to less than half of the maximum. Figure 5 displays both data sets
in a lower-dimensional space by projecting along the first two principal
components. In Fig. 5, the marker color corresponds to time through the
recrystallization tests. In the lower-dimensional space, the diffraction data
and underlying microstructure take similar paths, starting and ending in
similar positions. However, as can be seen in all portions of the figure, the
recrystallization process takes significantly longer under increased pressure
(nearly $4\times$).

Figure 4: a) Time $t$ evolution of the first PCA score $z_{1}$ with time for
the 1 (blue) and 2 (red) GPa tests. JMAK functions have been fit to $z_{1}$
for both tests. b) Time $t$ evolution of the second PCA score $z_{2}$ with
time for the 1 (blue) and 2 (red) GPa tests.

While determining which exact features are being captured by the PCA is
challenging, the time evolution of the first PCA score $z_{1}$ appears to
reflect the relative level of completion of the recrystallization and grain
growth process. This interpretation is supported by the observation that
$z_{1}$ evolution in both tests exhibits a characteristic sigmoidal shape. In
addition, a majority of the variance in both data sets is captured solely by
the first PCA score. With this in mind, the crystallization time $t_{c}$ of
the 1 GPa test is approximately 1500 s, and is nearly 6000 s for the 2 GPa
test. This slowing of the recrystallization and grain growth process is
consistent with previous observations that pressure slows substitutional
diffusion and, in turn, grain growth [10, 11]. Interpretation of the time
evolution of the second PCA score $z_{2}$ is less clear, but it is apparent
from the difference in $z_{2}$ evolution between the two pressures that a
larger fraction of recrystallization time is spent at the maximum for 2 GPa
compared to 1 GPa.

We can further explore the features that are influencing the scores by
directly inspecting the principal components themselves. Figures 6a and 6b
show a 120∘ azimuthal region of the first and second principal components
respectively extracted from the 1 GPa test data. The shorter azimuthal range
allows example features in the components to be more clearly discerned.
Figures 6c and 6d shows example data from the start and end of the 1 GPa test
for reference. The first principal component has contributions from both the
initial and final diffraction data configurations. When projecting the data
against this principal component, the continuous ring data at the start of the
test will produce a lower score while the distinct peaks at the end of the
test will produce a higher score. This essentially means the first score
captures the degree of transformation transitioning from low to high. The
second principal component is much more dominated by the distinct diffraction
peaks that appear during recrystallization and grain growth. There even
appears to be peaks that are not present in the final configuration which
reflect grains that were either consumed or rotated out of the diffraction
condition. Projecting the data against this principal component will produce a
score that reflects the number of grains with low defect content that are
currently diffracting to produce distinct diffraction peaks along the ring.

Figure 5: Diffraction data from the recrystallization experiments at 1 (blue)
and 2 (red) GPa plotted in a lower-dimensional space as determined from PCA.
Lower-dimensional data is colored according to time $t$.

Furthermore, with the interpretation that the first PCA score reflects the
degree of completion of the microstructure evolution process, a Johnson-Mehl-
Avrami-Kolmogorov (JMAK) function [31, 32, 33, 34] was fit to the evolution of
the first PCA scores $z_{1}(t)$ to explore the rates of the transformation
process:

$z_{1}(t)=1-e^{-Kt^{n}}$ (1)

where $K$ and $n$ are fitting parameters. The fits of the JMAK function to the
evolution of $z_{1}$ for both tests are shown in Fig. 4a. Interestingly, the
fit exponents $n$ of the two tests change fairly dramatically from $n=1.8$
(approximately 2) for the 1 GPa test to $n=3.2$ (approximately 3) for the 2
GPa test. The exponent $n$ has been related to the characteristics of
nucleation site distribution and the nucleation rate [35]. Therefore, this
increase in exponent $n$ with pressure may be related to an increase in
nucleation sites at higher pressure, despite the much slower crystallization
time. This may be rationalized by the fact that pressure will increase the
local strain energy to initiate the recrystallization, leading to more
nucleation events. Interestingly, this may be connected to the evolution of
the second PCA score $z_{2}$ (Fig. 4b) where the 2 GPa test exhibited an
extended fraction of the crystallization time at peak value. If $z_{2}$ is
connected to nucleation events, then more of the crystallization time at peak
value would coincide with a larger exponent $n$ fit to the recrystallization
process. However, further studies would be required to confirm this
supposition.

Figure 6: a) First and b) second principal components extracted from the 1 GPa
test data. Example c) initial and d) final diffraction data which is projected
on to the principal components. Only 120∘ of the full 360∘ about $\eta$ of the
peak data is shown so features can be discerned.

While _in situ_ monitoring of high pressure processes is a challenge, here we
have demonstrated that a promising approach for studying metal
thermomechanical processing is to adapt synchrotron experimental capabilities
originally designed for studying mineral physics. In this work, it was
demonstrated that pressure significantly alters the rates of the
recrystallization and grain growth process. The findings here indicate that
pressures encountered during many standard thermomechanical processes will
slow grain growth due to slowed substitutional diffusion in Fe (and likely
steels) and increase the number of nucleation sites for recrystallization due
to increased free energy. Together, these effects of pressure will lead to
smaller grains and finer microstructures, partially contributing to the fine
microstructures often found during severe plastic deformation processes at
high pressure [36, 37]. More importantly, quantitative information regarding
recrystallization and grain growth rates could be extracted from the complex
transition from full diffraction rings to isolated spots using data
dimensionality reduction (principal component analysis). As we attempt to
increase the sustainability of metal alloy thermomechanical processing,
understanding and quantifying the role of pressure on microstructural
evolution rates is a means to reach target, or even new, microstructures
without significantly increasing energy usage. The approach reported here
presents a path forward for gathering the data necessary to develop more
sustainable, heat-treatment-less, alloy processing routes across a wide range
of alloy classes.

## Acknowledgments

DCP and LAK acknowledge the support of FIERF Grant #258985. This research used
resources 28-ID-2 of the National Synchrotron Light Source II, a U.S.
Department of Energy (DOE) Office of Science User Facility operated for the
DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-
SC0012704. Use of the MAXPD endstation was supported by COMPRES, the
Consortium for Materials Properties Research in Earth Sciences, under NSF
Cooperative Agreement No. EAR 16-61511 and by the Mineral Physics Institute,
Department of Geosciences, Stony Brook University.

## Conflict of Interest Statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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