††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work††thanks: These authors contributed
equally to this work

# Modulation of Brillouin optomechanical interactions via acoustoelectric
phonon-electron coupling

Nils T. Otterstrom  Microsystems Engineering, Science, and Applications,
Sandia National Laboratories, Albuquerque, New Mexico, USA  Matthew J. Storey
Microsystems Engineering, Science, and Applications, Sandia National
Laboratories, Albuquerque, New Mexico, USA  Ryan O. Behunin  Department of
Applied Physics and Materials Science, Northern Arizona University, Flagstaff,
Arizona 86011, USA  Center for Materials Interfaces in Research and
Applications, Northern Arizona University, Flagstaff, Arizona 86011, USA  Lisa
Hackett  Microsystems Engineering, Science, and Applications, Sandia National
Laboratories, Albuquerque, New Mexico, USA  Peter T. Rakich  Department of
Applied Physics, Yale University, New Haven, Connecticut 06520, USA  Matt
Eichenfield<EMAIL_ADDRESS>Microsystems Engineering, Science, and
Applications, Sandia National Laboratories, Albuquerque, New Mexico, USA

###### Abstract

Optomechanical Brillouin nonlinearities—arising from the coupling between
traveling photons and phonons—have become the basis for a range of powerful
optical signal processing and sensing technologies. The dynamics of such
interactions are largely set and limited by the host material’s elastic,
optical, and photo-elastic properties, which are generally considered
intrinsic and static. Here we show for the first time that it is feasible to
dynamically reconfigure the Brillouin nonlinear susceptibility in transparent
semiconductors through acoustoelectric phonon-electron coupling.
Acoustoelectric interactions permit a wide range of tunability of the phonon
dissipation rate and velocity, perhaps the most influential parameters in the
Brillouin nonlinear susceptibility. We develop a Hamiltonian-based analysis
that yields self-consistent dynamical equations and noise coupling, allowing
us to explore the physics of such acoustoelectrically enhanced Brillouin (AEB)
interactions and show that they give rise to a dramatic enhancement of the
performance of Brillouin-based photonic technologies. Moreover, we show that
these AEB effects can drive systems into new regimes of fully-coherent
scattering that resemble the dynamics of optical parametric processes,
dramatically different than the incoherent traditional Brillouin limit. We
propose and computationally explore a particular semiconductor heterostructure
in which the acoustoelectric interaction arises from a piezoelectric phonon-
electron coupling. We find that this system provides the necessary
piezoelectric and carrier response ($k^{2}\approx 6\%$), favorable
semiconductor materials properties, and large optomechanical confinement and
coupling ($|g_{0}|\approx 8000$ (rad/s)$\sqrt{\text{m}}$) sufficient to
demonstrate these new AEB enhanced optomechanical interactions.

##  I Introduction

Optomechanical Brillouin interactions are unique amongst nonlinear optical
processes in that they are enabled by phonons and consequently depend
sensitively on the elastic properties of a material [1, 2, 3]. In optical
waveguides, the supported phonon modes determine the types of allowed
stimulated Brillouin scattering (SBS) processes, including the traditional
intra-modal backward (Fig. 1a), intra-modal forward, or inter-modal forward
SBS (Fig. 1b), as well as the characteristic SBS frequencies, nonlinear
coupling strengths, and bandwidths. Leveraging these distinctive properties,
SBS has become the basis for a range of flexible and powerful optical signal
processing technologies such as amplifiers [4, 5, 6], lasers [7, 8, 9, 10, 11,
12, 12], filters [13, 14, 15, 16, 17], nonreciprocal devices [18, 19, 20, 21],
and optical delay lines [22, 23, 24]. Despite the wide application space for
SBS devices, however, the dynamics and resulting utility of these interactions
have historically been limited by the host material’s intrinsic elastic and
optical properties [25]. In particular, two fundamental material properties
that determine the Brillouin susceptibility are the phononic dissipation rate
and velocity. Although these properties and the resulting Brillouin
susceptibility can be significantly enhanced by patterning materials into
waveguides [26], the underlying material properties are generally considered
intrinsic and static, and in a given material they ultimately limit the
strength of these scattering processes and the performance of devices
utilizing them.

Figure 1: Essential physical system for acoustoelectrically (AE) enhanced
stimulated Brillouin scattering (SBS). $B(z,t)$, $A_{\rm s}(z,t)$ and $A_{\rm
p}(z,t)$ represent the phonon, Stokes, and pump fields, respectively. (a)
Backward intra-modal and (b) forward inter-modal SBS. (c) Elastic wave
creating an acoustoelectric current through the acoustoelectric effect. (d) AE
gain applied with an external DC field. (e) Composite system permitting AE-
enhanced stimulated Brillouin scattering. (f) Bulk model prediction of the
relative change of (left) the SBS gain spectrum with AE amplification as
compared to the peak ABS gain without AE amplification, and (right) the SBS
dispersion compared to peak SBS gain rate ($v_{\rm g,s}G_{\rm B}P_{\rm p}$)
without AE effects. For clarity, the curves with drift fields of
(1.6,1.2,0.8,0.4,0.0) kV/cm are respectively displaced vertically by
(20,15,10,5,0).

In this context, the ability to electrically control phononic loss, or even
induce phononic gain, could enable dynamic reconfigurability of the Brillouin
susceptibility, suppressing or enhancing the nonlinear response by orders of
magnitude on demand. Coulomb drag interactions between electrons and phonons
in semiconductors provide a powerful mechanism to deterministically modify the
phonon velocity and dissipation [27]. This Coulomb drag effect is achieved in
practice by applying external, quasi-static electric fields to the
semiconductor charge carriers to produce a drift current; the drifting
carriers then interact with and become spatially polarized by the generated
charge distribution of the phonons via the electric fields they induce, as
depicted in Fig. 1c-d. The resulting coupling between the two moving charge
distributions ultimately allows non-reciprocal amplification, attenuation, and
velocity modification of the phonons. Although the acoustoelectric effect can
be produced by different mechanisms in many materials [28, 29], the strongest
interactions were predicted to exist [27] and then observed in piezoelectric
semiconductors [30]. Going beyond the effects intrinsic to single materials,
it was proposed that heterostructures that combine the strongest piezoelectric
materials with high-mobility semiconductors could produce exceptionally large
acoustoelectric effects [31], and this ultimately allowed demonstrations of
devices such as radio frequency acoustic amplifiers [32, 33, 34], convolvers
[35, 36], and correlators [37, 38, 39]. Recent advances in semiconductor
epitaxy, heterogeneous integration, and nanofabrication have led to a new
class of acoustoelectric heterostructures, enabling ultra-compact acoustic
amplifiers [40, 41, 42, 43, 44], circulators [41, 45], and switches [46], with
performance that greatly supercedes that which was previously possible.

Here, for the first time, we describe how Brillouin interactions can be
modified by the acoustoelectric effect using externally applied electric
fields, enabling new degrees of control over the nonlinear optical
susceptibility. We show that straight-forward modifications of our recently
developed heterostructure platform for acoustoelectric radio-frequency signal
processing devices [40, 41] provide a powerful platform for demonstrating and
using these effects, simultaneously allowing large optomechanical confinement
and coupling ($|g_{0}|\approx 8000$ (rad/s)$\sqrt{\text{m}}$) and
acoustoelectric coupling ($k^{2}\approx 6\%$). We derive a general Hamiltonian
framework that describes the coupled acoustoelectric and optomechanical
interactions and show that acoustoelectric modifications to both the real and
imaginary parts of the Brillouin susceptibility lead to novel effects and
dynamics that would not be expected with intrinsic material properties. Using
the proposed heterostructure as an example system for concrete physical
predictions, we show how modification of the phonon dissipation rates through
application of quasistatic electric fields in the semiconductor can
drastically improve the performance of archetypal Brillouin photonic devices
such as Brillouin amplifiers, lasers, nonreciprocal devices, and delay lines.
Moreover, we show that these acoustoelectrically enhanced Brillouin
interactions allow Brillouin scattering processes to go from a regime where
the phonon coherence lengths ($\sim 100$ $\upmu$m) are significantly less than
those of the photons ($\sim$ cm)—behaving effectively as incoherent
scatterers—to one in which the phonon coherence lengths can achieve parity
with and even exceed those of the photons, enabling fully coherent scattering
processes that resemble the dynamics of optical parametric amplification [47].
We then discuss future prospects for achieving these effects in other
materials platforms, novel devices, and applications, as well as their
application to cavity optomechanical systems.

##  II Proposed Candidate Physical System

To enable reconfigurable Brillouin optomechanics through acoustoelectric
control of phonon dissipation, we seek a traveling-wave optomechanical
physical system that (1) guides and tightly confines optical and elastic
waves, (2) supports non-zero Brillouin coupling, and (3) exhibits an
acoustoelectric coupling between phonons and electrons—as depicted in Fig. 1e.
This combination of system properties allows us to modulate the Brillouin
susceptibility using an applied electric field.

Figure 2: (a) Illustration of the InGaAsP on Y-cut lithium niobate on silicon
material system for acoustoelectrically enhanced Brillouin devices (not to
scale). (b) The key geometric design parameters are shown as well as (c) the
defined material orientation parameters (not to scale). (d) Phase-matched
phonon wavelength as a function of waveguide geometry parameters for backward
intra-modal Brillouin scattering (blue circle highlights the chosen waveguide
dimensions). (e) Simulated elastic and optical mode shapes with profiles of
each strain component. For each normalized strain profile, the relative
magnitude between the strain components is shown in parenthesis. (f) The
calculated optomechanical coupling coefficient (black) and piezoelectric
coupling coefficient (red) for each phase-matched elastic mode simulated with
the chosen waveguide dimensions. The elastic mode with the highest
optomechanical and piezoelectric coupling is highlighted (black box).

Building on our recent work in acoustoelectric heterostructures [40, 41], we
propose the following physically realizable system to explore and harness
these new acoustoelectric Brillouin dynamics. The system seen in Fig. 2a-c
consists of a hybrid photonic-phononic waveguide patterned out of epitaxially
grown In0.712Ga0.288As0.625P0.375, which is bonded to a Y-cut lithium niobate
thin film on a silicon substrate (InGaAsP-LN-Si). Here, the InGaAsP thin film
is used as the optical guiding layer in addition to the semiconductor layer
that provides the necessary free carriers for the acoustoelectric effect.
Because a small carrier density ($\sim 10^{16}$ $\rm cm^{-3}$) is sufficient
to produce a strong acoustoelectric gain, these free carriers have negligible
to minimal impact on the linear optical losses [41, 48]. For the elastic modes
and quaternary composition considered in this work, the longitudinal and shear
acoustic phase velocity in InGaAsP is smaller than in lithium niobate,
allowing for a high degree of acoustic confinement. The higher thermal
conductivity of the silicon substrate improves the heat dissipation of the
system, which reduces the thermal effects of the Joule heating in the InGaAsP
and allows for continuous operation of the acoustoelectrically enhanced
Brillouin devices (see Supplementary Section B). The optomechanical coupling,
$g_{0}$, is a function of the overlap between the co-localized optical and
elastic modes (see Supplementary Section A), while the acoustoelectric gain is
governed by the interaction of the electric potential of the elastic wave in
the piezoelectric and the free carriers in the semiconductor. Given these
considerations, the InGaAsP-LN-Si heterostructure provides the essential
acoustoelectric and optomechanical ingredients for acoustoelectrically
reconfigurable Brillouin optomechanics.

To determine the accessible experimental performance of such a waveguide
structure, we carry out finite element simulations of the optical modes, the
electric and strain fields of the elastic modes, along with $g_{0}$ and the
relevant piezoelectric coupling $k^{2}$ (see Supplementary Section A), for a
variety of Brillouin optomechanical interactions and device geometries. Figure
2b-c highlights key geometric parameters to consider when designing an
acoustoelectrically enhanced Brillouin device (see Supplementary Section A for
more details). The width ($w$) and thickness ($t$) of the InGaAsP waveguide
not only determines the confinement and effective index of the optical modes,
but it also determines the lateral and vertical confinement of the elastic
mode in the InGaAsP-LN-Si system. The phonon wavelength ($\lambda$m) sets the
operating phonon frequency and elastic mode shape given a particular waveguide
cross section. Finally, since this physical system includes anisotropic
materials, the propagation angle ($\beta$) in the device plane and the
acoustic polarization play a critical role in determining the acoustoelectric
and optomechanical coupling.

With these considerations, we simulate the relevant optical modes, determine
the phonon wavelength through the phase-matching condition, simulate the
elastic mode with the appropriate periodic boundary condition, and then
calculate the optomechanical overlap integrals and piezoelectric coupling
($k^{2}$) (see Fig. 2d-f and Supplementary Section A for more details). Figure
2e highlights a particular mode triplet (required by the 3-wave Brillouin
process) that demonstrates excellent Brillouin optomechanical and
acoustoelectric properties, with a distributed optomechanical coupling rate of
$|g_{0}|=7943$ (rad/s)$\rm\sqrt{m}$ and electromechanical coupling
$k^{2}=5.87$% (for details on these calculations, see Supplementary Section
A). The elastic mode that mediates this backward Brillouin interaction at
telecom wavelengths is a Rayleigh-like waveguide mode with a center frequency
of 8.78 GHz and an acoustic velocity of 3072 $\rm m/s$.

Figure 3: Acoustoelectric phonon amplification as a function of drift field,
highlighting the accessible regimes of dynamics.

Equipped with the optomechanical and acoustoelectric couplings, we now have
the necessary parameters to examine the accessible dynamics for this
particular mode triplet (see Eq. 1). For acoustoelectric gain, a key parameter
is the electric field at which the free carrier drift velocity,
$v_{\text{d}}$, equals the phonon phase velocity, $v_{\text{m}}$, which we
call the equal-velocity point ($v_{\text{d}}=v_{\text{m}}$). At electric
fields lower (higher) than this point, the acoustoelectric material causes
phononic loss (gain). Using the normal mode theory developed by Kino and
Reeder [49], the acoustoelectric phonon gain as a function of applied drift
field near this equal-velocity point is plotted in Fig. 3. Here we assume an
electron mobility of 2000 cm2/Vs and an intrinsic phonon spatial decay rate of
3100 dB/cm, corresponding to an acoustic quality factor of 250. This loss rate
defines the distinct, acoustoelectric-enabled regimes of operation that will
be explored in Section III. With these conditions, the drift field needed to
reach the equal velocity point is 150 V/cm. Increasing the drift field
further, we enter the acoustoelectrically enhanced Brillouin (AEB) regime in
which the phonon loss is reduced by the net acoustoelectric gain. As the
strength of the drift field increases, we approach the acoustoelectric
coherent Brillouin limit, defined as the point at which the acoustoelectric
gain matches the intrinsic phonon losses; as we will see in Section III.3,
this yields parametric-like three-wave dynamics. If, by contrast, we apply a
field below the equal-velocity point or with a negative polarity, additional
phonon loss is introduced (lossy Brillouin regime). In this case, the
Brillouin coupling can either be dramatically enhanced or reduced, allowing
for dynamical switching of the optomechanical response.

##  III Acoustoelectric Brillouin Dynamics

We present a general analysis of the effects of acoustoelectric phonon gain on
the nonlinear susceptibility and spatio-temporal dynamics of optomechanical
Brillouin interactions using the InGaAsP-LN-Si physical system as a concrete
example for physical predictions (see Section II). In this system, the
Hamiltonian governing a stimulated (Stokes) optomechanical Brillouin response
can be expressed as [25],

$\displaystyle H_{\rm B}^{\rm int}=\hbar\int dz\big{(}g_{0}A^{\dagger}_{\rm
p}(z,t)A_{\rm s}(z,t)B(z,t)\big{)}e^{i(q_{\rm m}-\Delta k_{\rm s})z}+h.c.$ (1)

where $A_{\rm p}(z,t)$, $A_{\rm s}(z,t)$, and $B(z,t)$ are the envelope
operators for the pump, Stokes, and phonon fields, respectively, $g_{0}$ is
the distributed optomechanical coupling of the waveguide system, and $q_{\rm
m}-\Delta k_{\rm s}$ represents the phase-mismatch between the wavevectors of
the phonon ($q_{\rm m}$) and the optical beat pattern ($\Delta k_{\rm
s}=k_{\rm p}-k_{\rm s}$) [25].

When phase-matching is satisfied (i.e., $q_{\rm m}-\Delta k_{s}=0$), the
Heisenberg equations of motion yield

$\displaystyle\frac{\partial\bar{B}}{\partial t}$
$\displaystyle=-i(\Omega_{\rm m}-\Omega)\bar{B}-\frac{\Gamma}{2}\bar{B}+v_{\rm
g,b}\frac{\partial\bar{B}}{\partial z}-ig_{0}^{*}\bar{A}_{\rm
s}^{\dagger}\bar{A}_{\rm p}+\eta$ (2) $\displaystyle\frac{\partial\bar{A}_{\rm
p}}{\partial t}$ $\displaystyle=-\frac{\gamma_{\rm p}}{2}\bar{A}_{\rm
p}+v_{\rm g,p}\frac{\partial\bar{A}_{\rm p}}{\partial z}-ig_{0}\bar{A}_{\rm
s}\bar{B}+\xi_{\rm p}$ $\displaystyle\frac{\partial\bar{A}_{\rm s}}{\partial
t}$ $\displaystyle=-\frac{\gamma_{\rm s}}{2}\bar{A}_{\rm s}-v_{\rm
g,s}\frac{\partial\bar{A}_{\rm s}}{\partial z}-ig^{*}_{0}\bar{A}_{\rm
p}\bar{B}^{\dagger}+\xi_{\rm s}$

where $\bar{B}(z,t)=B(z,t)\exp(i\Omega t)$, $\bar{A}_{\rm p}(z,t)=A_{\rm
p}(z,t)\exp(i\omega_{\rm p}t)$, and $\bar{A_{\rm s}}(z,t)=A_{\rm
p}(z,t)\exp(i\omega_{\rm s}t)$ are the slowly varying phonon, pump, and Stokes
envelopes in the rotating frame, with the condition $\Omega=\omega_{\rm
p}-\omega_{\rm s}$; $\Omega_{\rm m}$ is the natural mechanical frequency of
the phonon mode, while $v_{\rm g,b}$, $v_{\rm g,p}$, and $v_{\rm g,s}$
represent the group velocities for the phonon, pump, and Stokes fields. We
also note that Eq. 1 includes the effects of dissipation—denoted by $\Gamma$,
$\gamma_{\rm p}$, and $\gamma_{\rm s}$ for the phonon, pump, and Stokes
fields, respectively—and therefore requires thermal and vacuum noise terms
$\eta$, $\xi_{\rm p}$, and $\xi_{\rm s}$, according to the fluctuation-
dissipation theorem.

We next examine the impact of the acoustoelectric phonon gain or loss on these
dynamics. For this purpose we develop a Hamiltonian formulation for the
coupled dynamics of the electric potential, drift current, and elastic waves.
This treatment provides (1) a general description for the acoustoelectric
coupling for waveguides of arbitrary cross-sectional geometry, (2) yields the
acoustoelectric gain, loss and dispersion in terms of coupling parameters (see
$\kappa_{\omega,\ell}$ below) quantified by (overlap) integrals of products of
the piezoelectric coupling and elastic and electric potential modes over the
waveguide cross section, and (3) naturally lends itself to noise analysis
leveraging the fluctuation-dissipation theorem [50]. These acoustoelectric
dynamics can be captured by the Hamiltonian given by

$\displaystyle H_{\rm AE}=\hbar\sum_{\ell}\int d\omega$ $\displaystyle\int
dz\bigg{[}\Phi_{\omega\ell}^{\dagger}(z)\hat{\omega}\Phi_{\omega\ell}(z)$ (3)
$\displaystyle+(\kappa_{\omega,\ell}\Phi_{\omega
j}(z)B^{\dagger}(z)+H.c.)\bigg{]},$

where $\Phi_{\omega\ell}$ and $B$ are the envelope operators for the electric
potential and phonon fields, respectively, $j$ labels bulk and surface modes
of the potential and $\kappa_{\omega,\ell}$, quantifies the acoustoelectric
coupling defined in Supplementary Section C. Within this coupled-envelope
framework, the oscillation frequency of the mode $\hat{\omega}=\omega+v_{\rm
d}q_{\rm m}-i{\bf v}_{\rm d}\cdot\nabla$ is operator valued, capturing the
effects of temporal oscillations, current drift (given by ${\bf v}_{\rm
d}=\mu{\bf E}_{\rm DC}$, where ${\bf E}_{\rm DC}$ is the applied drift field)
and slowly-varying spatial dynamics of the potential amplitude. Here, $\omega$
denotes the normal mode frequency of the potential in the absence of a drift
current of speed $v_{\rm d}$ and $q_{\rm m}$ is the carrier wavevector of the
phonon and potential within the slowly-varying envelope approximation.

When an applied electric field causes free carriers to drift at a velocity
greater than the acoustic wave, the acoustoelectric effect produces a spatial
phonon gain given by $\alpha_{\rm AE}=-\sum_{\ell}2\pi|\kappa_{\omega-v_{\rm
d}q_{\rm m},\ell}|^{2}/v_{\rm g,b}$. Noting that $|\kappa_{\omega-v_{\rm
d}q_{\rm m},\ell}|^{2}\propto(v_{\rm m}-v_{\rm d})$ where $v_{\rm m}$ is the
phonon phase velocity, we recover the essential features of acoustoelectric
gain. Namely, acoustoelectric amplification (excess attenuation) occurs when
the drift velocity exceeds (falls below) the phonon phase velocity. In the
case of a Brillouin-active optomechanical system, the effects of
acoustoelectric gain and the accompanying velocity shift as illustrated by
Fig. 1f can be described by a modified complex dissipation
$\Gamma\rightarrow\tilde{\Gamma}$ within the coupled envelope equations (Eq.
2), where $\tilde{\Gamma}=2i\Delta\Omega_{\rm AE}+\Gamma-G_{\rm AE}$. Here
$\Delta\Omega_{\rm AE}$ denotes the acoustoelectric shift in the resonance
frequency and $G_{\rm AE}$ quantifies the phonon gain as a time rate such that
$G_{\rm AE}=v_{\rm g,b}\alpha_{\rm AE}$ (see Supplementary information for
more details).

To elucidate the impact of acoustoelectric gain on the nonlinear Brillouin
susceptibility, we will semi-classically treat this phenomena in the
undepleted pump limit, for the moment ignore the noise terms (for noise
analysis see Supplementary Section D), and move to the Fourier domain
($f[\omega]=\int_{-\infty}^{\infty}f(t)\exp(i\omega t)dt$), which yields

$\displaystyle\left(-i\omega+\frac{\gamma_{\rm s}}{2}\right)\bar{A}_{\rm
s}[z,\omega]+v_{\rm g,s}\frac{\partial\bar{A}_{\rm s}[z,\omega]}{\partial
z}=-ig_{0}^{*}\bar{A}_{\rm p}\bar{B}^{\dagger}[z,\omega]$ (4) $\displaystyle
i(\Omega-\Omega_{\rm m}-\Delta\Omega_{\rm
AE}-\omega)\bar{B}[z,\omega]+\frac{\Gamma-G_{\rm AE}}{2}\bar{B}[z,\omega]+$
$\displaystyle v_{\rm g,b}\frac{\partial\bar{B}[z,\omega]}{\partial
z}=-ig_{0}^{*}\bar{A}_{\rm p}\bar{A}^{\dagger}_{\rm s}[z,\omega].$

These spatial equations in the frequency domain are the basis for a range of
new acoustoelectric Brillouin device physics that we explore below.

###  III.1 Acoustoelectrically enhanced Brillouin (AEB) limit

We now focus on the particular case in which the spatial coherence length of
the phonon field, despite gain from the acoustoelectric effect, is
significantly shorter than that of the optical fields [51]. We will explore
the case of near equal coherence lengths in Section III.3. In this
acoustoelectrically enhanced Brillouin (AEB) limit, the spatial dynamics of
the phonon field can be adiabatically eliminated (i.e., $v_{\rm
g,b}\frac{\partial\bar{B}[z,\omega]}{\partial z}\approx 0$) [51], yielding

$\displaystyle\bar{B}[z,\omega]=-ig_{0}^{*}\chi^{\rm AE}_{\rm
B}[\omega]\bar{A}_{\rm p}\bar{A}_{\rm s}^{\dagger}[z,\omega].$ (5)

Where the acoustoelectrically modified phonon susceptibility is given by
$\chi^{\rm AE}_{\rm B}[\omega]=(i(\Omega-\Omega_{\rm m}-\Delta\Omega_{\rm
AE}-\omega)+(\Gamma-G_{\rm AE})/2)^{-1}$. Through the stimulated Brillouin
process, this modification translates into an acoustoelectric Brillouin
nonlinear optical susceptibility $\chi_{\rm s}^{\rm AEB}[\omega]$ in the
dynamics of the Stokes wave such that

$\displaystyle-i\omega\bar{A}_{\rm s}[\omega,z]$ $\displaystyle+\frac{\alpha
v_{\rm g,s}}{2}\bar{A}_{\rm s}[\omega,z]$ (6) $\displaystyle+v_{\rm
g,s}\frac{\partial\bar{A}_{\rm s}[z,\omega]}{\partial z}=\chi_{\rm s}^{\rm
AEB}[\omega]\bar{A}_{\rm s}[\omega,z],$

where $\alpha$ is the spatial optical decay rate and $\chi_{\rm s}^{\rm
AEB}[\omega]=|g_{0}|^{2}|\bar{A}_{\rm p}|^{2}\chi_{\rm B}^{\rm AE}[\omega]$.

Thus, through this acoustoelectrically enhanced stimulated Brillouin process,
a Stokes field with initial conditions $\bar{A}_{\rm s}[0,\omega]$ is
amplified as

$\displaystyle\bar{A}_{\rm s}[z,\omega]=\bar{A}_{\rm
s}[0,\omega]\exp\bigg{[}\bigg{(}\frac{-i\omega}{v_{\rm
g,s}}-\frac{\alpha}{2}+\frac{G_{\rm B}P\Gamma\chi_{\rm B}^{\rm
AE*}[\omega]}{4}\bigg{)}z\bigg{]},$ (7)

where the Brillouin gain coefficient power product $G_{\rm B}P$ is given by
$G_{\rm B}P=4|g_{0}|^{2}|\bar{A}_{\rm p}|^{2}/(\Gamma v_{\rm s})$. Inspecting
the resonance condition (i.e., $\omega=0,\Omega=\Omega_{\rm
m}+\Delta\Omega_{\rm AE}$), reveals a gain (loss) factor that is exponentially
increased (decreased) by the acoustoelectric modified phonon dissipation,
specifically

$\displaystyle\frac{|\bar{A}_{\rm s}[z,0]|^{2}}{|\bar{A}_{\rm
s}[0,0]|^{2}}=\exp\bigg{[}\bigg{(}G_{\rm B}P\frac{\Gamma}{\Gamma-G_{\rm
AE}}-\alpha\bigg{)}z\bigg{]},$ (8)

revealing that the ability to modify the nonlinear susceptibility through
acoustoelectric coupling enables in situ reconfigurability of the optical SBS
gain.

In addition to these gain dynamics, we explore thermal-mechanical Brillouin
noise under the influence of acoustoelectric gain and find that the SBS noise
factor from these thermal fluctuations scales as

$\displaystyle F\approx 1+n_{\rm th}\left(\frac{\Gamma}{\Gamma-G_{\rm
AE}}\right),$ (9)

where $n_{\rm th}$ is the thermal occupation of the phonon mode given by the
Bose-Einstein distribution (for a detailed derivation, see Supplementary
Section D). When the intrinsic acoustoelectric noise is small relative to
thermomechanical noise (see Supplementary Section D for more details), these
results suggest that near quantum limited amplification may be possible in the
limit of low temperatures (i.e., when $k_{\rm B}T\ll\hbar\Omega_{\rm m}$).

Figure 4: Performance enhancement from acoustoelectric coupling for archetypal
Brillouin devices in the acoustoelectrically enhanced Brillouin (AEB) limit
using the selected mode triplet of the InGaAsP-LN-Si heterostructure waveguide
as an example (see Fig. 2). In addition to the simulated optomechanical and
electromechanical coupling and acoustic quality factor of 250 (see
Supplementary Section A for more details), for (a) and (c) we assume a pump
power of 50 mW and a device length of 1 cm. (a) Gain improvement in a backward
SBS amplifier as a function of bias field strength. (b) Potential improvement
in pump-limited phonon linewidth for an acoustoelectrically enhanced Brillouin
optomechanical oscillator operating in the phonon linewidth narrowing regime.
Here the phonon linewidth is not limited by Schawlow-Townes narrowing. The
phonon linewidth is normalized by the pump linewidth. (c) Nonreciprocal
scattering efficiency for Brillouin-based AE-enhanced isolator as a function
of bias field strength (neglecting optical loss). The particular
implementation of this enhanced nonreciprocal scattering process is shown in
insets i-ii, and requires optomechanical waveguides that are coupled
phononically. Counter-propagating pump and signal waves in the drive waveguide
(light gray) set up an acousto-optic grating in the modulator waveguide (dark
gray) that can be enhanced by the acoustoelectric effect. (i) When light at
the pump frequency is injected into the modulator waveguide in the forward
direction, it is not phase-matched to and passes unaffected by the acousto-
optic grating. (ii) By contrast, in the backward direction, the light can
experience complete acousto-optic scattering providing effective optical
isolation. (d) Phonon delay enhancement and conversion length reduction due to
the acoustoelectric effect. As diagrammed in the inset, pump (blue) and write
(red) fields set up an acoustoelectrically enhanced acoustic-grating, with a
conversion efficiency of 1 (i.e., one phonon from one photon) over the so-
called conversion length (black). At the same time, the acoustoelectric effect
increases the effective delay (green) for the phononic memory. After the
delay, the write information can be recovered by the pump pulse, which upon
acousto-optic scattering, creates a back-propagating read pulse (red).

###  III.2 Performance of acoustoelectric Brillouin-based devices

Acoustoelectric phonon gain directly modifies the nonlinear optical
susceptibility through an enhancement in the effective Brillouin gain
coefficient, which is inversely proportional to the phonon dissipation rate.
These enhanced dynamics have important consequences for Brillouin-based
devices such as amplifiers, lasers, nonreciprocal devices, optomechanical
delay among others—especially in chip-scale systems where the accessible
levels of Brillouin gain have been historically limited.

####  III.2.1 Brillouin amplifiers

The performance of Brillouin-based amplifiers—used in a range of signal
processing [52], filtering [13, 5], and optical nonreciprocity applications
[19, 6]—can be radically improved as acoustoelectric phonon gain dramatically
reduces the effective phonon lifetime. The effective exponential gain
enhancement is highlighted by the power dynamics of a small-signal Stokes
wave, given by

$\displaystyle P_{\rm s}[z]=P_{\rm s}[0]\exp\bigg{[}\big{(}G_{\rm
B}P\frac{\Gamma}{\Gamma-G_{\rm AE}}-\alpha\big{)}z\bigg{]}.$ (10)

Figure 4a plots AEB-enhanced SBS gain as a function of applied drift field for
the InGaAsP-LN-Si physical system under consideration. Without acoustoelectric
gain, this Brillouin-optomechanical system might struggle to yield even net
amplification (Brillouin gain compensating for propagation loss), but with it,
50 dB or more of dynamically reconfigurable amplification may be possible on-
chip. Moreover, this degree of AEB amplification is accompanied by an
increasingly narrow gain bandwidth (from $\sim 35$ MHz to $\sim 300$ kHz),
enabling selective narrowband operations in RF-photonic filtering
applications.

####  III.2.2 Brillouin-based laser oscillators

The ability to dynamically reconfigure phonon dissipation rates and Brillouin
gain by orders of magnitude with DC electric fields also can be leveraged in
Brillouin laser oscillator applications. For one, acoustoelectric phonon gain
directly reduces the pump power required to reach self-oscillation.
Specifically, the threshold in this case is given by

$\displaystyle P^{\rm AEB}_{\rm th}=\frac{\Gamma-G_{\rm AE}}{\Gamma}P_{\rm
th}.$ (11)

Thus, provided the acoustoelectric interaction can produce net phonon
amplification, acoustoelectric gain enables near arbitrary control of the
self-oscillation threshold. For example, a drift field of 1.62 kV/cm applied
to the InGaAsP would reduce the threshold condition by approximately
$30\times$.

Second, acoustoelectric gain allows one to precisely shape the linewidth
narrowing dynamics of Brillouin lasers. The well-known ability of a Brillouin
laser to produce ultra-low noise self-oscillation (even in cases where the
pump is spectrally much broader) is contingent on a sufficient asymmetry
between the optical and acoustic temporal dissipation rates. Specifically,
systems in which the optical dissipation rate exceeds that of the acoustic
(and vice versa) can produce acoustic (optical) linewidth narrowing. Using
external DC fields to change the phonon dissipation rate through the
acoustoelectric effect permits flexible in situ control of the linewidth
narrowing dynamics.

Figure 5: (a) Exponential growth rate enhancement ($\kappa/(G_{\rm B}P)$) as a
function of unmodified coupling strength ($G_{\rm B}PL_{\rm c}$ at the
coherent Brillouin limit, where $L_{\rm c}$ is the intrinsic phonon coherence
length $L_{\rm c}=v_{\rm g,b}/\Gamma$ ). The potential improvement by
operating in this lossless regime is accentuated in the case of low intrinsic
Brillouin coupling. The insets illustrate the acoustoelectric phonon
amplification as a function of drift field, highlighting the coherent
Brillouin limit (left) and the exponential growth rate enhancement by entering
the coherent Brillouin regime (right). (b) Stokes and phonon number densities
as a function of device length, normalized independently for illustration.
Disparate parametric-like dynamics emerge in the case of co- and counter-
propagating Stokes and phonon fields, as described by Eq. 19,21.

For oscillator applications, it may be desirable to achieve high degrees of
spectral compression of the microwave-frequency phonon field. In the case of a
linewidth limited by pump noise, this ‘noise eating’ process can be quantified
by the ratio of phonon linewidth ($\Delta\nu_{\rm B}$) to the input pump
linewidth ($\Delta\nu_{\rm p}$), given by

$\displaystyle\frac{\Delta\nu_{\rm B}}{\Delta\nu_{\rm
p}}=\frac{1}{(1+\frac{\gamma}{\Gamma-G_{\rm AE}})^{2}},$ (12)

in analogy with the optical linewidth narrowing predicted by [53]. Figure 4b
plots the pump-limited phonon linewidth as a function of acoustoelectric gain,
revealing phonon linewidth narrowing beyond what is possible without the
acoustoelectric effect. Looking forward, this material stack may be compatible
with quantum well systems that yield optical gain in addition to
acoustoelectric phonon gain. In principle, this could allow complete control
over the photonic and phononic dissipation rates, enabling dynamics ranging
from the significant optical linewidth narrowing to the acoustic linewidth
narrowing regime described above.

####  III.2.3 Brillouin-based acousto-optic isolators

Acoustoelectrically enhanced Brillouin processes may be the key to significant
performance improvements in chip-scale non-reciprocal technologies based on
traveling-wave optomechanics. Some of the most promising broadband
technologies for on-chip isolators rely on inter-modal waveguide acousto-
optics, in which a moving acoustic Bragg grating produces unidirectional
optical mode conversion due to a nonreciprocal phase-mismatch [18, 20].
However, it has been challenging to achieve the scattering efficiencies
necessary to produce large contrast optical nonreciprocity with low insertion
loss over broad bandwidths [54, 21, 55].

With recourse to acoustoelectric phonon gain, the scattering efficiency
$\eta_{\rm ef}$ (in the case of lossless optical fields) can be improved as
[21]

$\displaystyle\eta_{\rm ef}=\tanh^{2}\bigg{(}\frac{G_{\rm
B}PL\Gamma}{4(\Gamma-G_{\rm AE})}\bigg{)}.$ (13)

As a result, acoustoelectrically enhanced transduction may bring unity
efficiencies within reach in the InGaAsP-LN-Si system—even with intrinsic
scattering efficiencies of $\sim 10^{-3}$—as plotted in Fig. 4c. Moreover, the
acoustoelectric effect yields additional suppression of unwanted backward
propagating phonons on account of the attenuation experienced by
contradirectionally propagating phonons, further enhancing the overall
nonreciprocity [41]. As a result, the AEB interactions in this exemplar system
or others may play a key role in enabling the first generation of practical
acousto-optic on-chip isolators.

####  III.2.4 Optomechanical delay

Control of the phonon dissipation through acoustoelectric phonon gain may also
be used to enhance the performance of Brillouin-based memory and
optomechanical delay [23]. The most apparent improvement is the ability to
extend the phonon lifetime ($\tau\propto 1/\Gamma$), and hence the delay time
by

$\displaystyle\tau^{\rm AEB}=\tau\Big{(}\frac{\Gamma}{\Gamma-G_{\rm
AE}}\Big{)}.$ (14)

A less obvious, but equally useful benefit of increasing the phonon lifetime
is that one can achieve unity photon-phonon conversion lengths (in the case of
optomechanical memory for instance) over much shorter distances. The
conversion length scales as

$\displaystyle L^{\rm AEB}=L\Big{(}\frac{\Gamma-G_{\rm AE}}{\Gamma}\Big{)},$
(15)

hence simultaneously permitting substantially longer delays in a much smaller
geometry, as shown in Fig. 4d. For the system under consideration, tunable
chip light storage of more than 0.1 $\upmu$s may be possible in the AEB limit
(corresponding to $\sim 20$ m of fiber optic path length), with potential for
orders of magnitude larger improvements with the acoustoelectrically induced
coherent Brillouin limit outlined below.

###  III.3 Acoustoelectrically induced coherent Brillouin (ACB) limit

We now explore what we term the acoustoelectrically induced coherent Brillouin
(ACB) limit, in which the mean-free path of the phonons is on par with that of
the photons. In this limit, we can no longer treat the phonon field as local
in space, in sharp contrast with the standard assumption for SBS processes
[51]. Specifically, we consider an interaction region in which both the
optical and acoustic waves can be considered lossless, with the latter enabled
by acoustoelectric gain that compensates for any phonon propagation loss.
Under these conditions, the spatial equations of motion (i.e., steady state in
time) become

$\displaystyle v_{\rm g,s}\frac{\partial\bar{A}_{\rm s}[z,\omega]}{\partial
z}$ $\displaystyle=-ig_{0}^{*}\bar{A}_{\rm p}\bar{B}^{\dagger}[z,\omega]$ (16)
$\displaystyle v_{\rm g,b}\frac{\partial\bar{B}[z,\omega]}{\partial z}$
$\displaystyle=-ig_{0}^{*}\bar{A}_{\rm p}\bar{A}^{\dagger}_{\rm s}[z,\omega].$

We note that, as a vector quantity, the phonon velocity $v_{\rm g,b}$ plays a
singularly important role in the resulting coupled dynamics. Equation 16 can
be decoupled through differentiation, yielding

$\displaystyle\frac{\partial^{2}\bar{A}_{\rm s}[z]}{\partial z^{2}}$
$\displaystyle=\frac{|g_{0}|^{2}|\bar{A}_{\rm p}|^{2}}{v_{\rm g,s}v_{\rm
g,b}}\bar{A}_{\rm s}[z]$ (17)
$\displaystyle\frac{\partial^{2}\bar{B}[z]}{\partial z^{2}}$
$\displaystyle=\frac{|g_{0}|^{2}|\bar{A}_{\rm p}|^{2}}{v_{\rm g,s}v_{\rm
g,b}}\bar{B}[z].$

In the case of a phonon field that co-propagates with the Stokes field (e.g.,
forward intra- or inter-modal SBS), the solution for $\bar{A}_{\rm s}[z]$ and
$\bar{B}[z]$ take the form

$\displaystyle\bar{A}_{\rm s}[z]$ $\displaystyle=A\sinh[\kappa
z]+B\cosh[\kappa z]$ (18) $\displaystyle\bar{B}[z]$
$\displaystyle=C\sinh[\kappa z]+D\cosh[\kappa z]$

where $\kappa^{2}=(|g_{0}|^{2}|\bar{A}_{\rm p}|^{2}/(v_{\rm g,s}v_{\rm
g,b}))$, and $A$, $B$, $C$, and $D$ are determined by the initial conditions.

For instance, an input Stokes wave of the form ($\bar{A}_{\rm
s}[0]=\bar{B}[0]=0$) yields the solutions

$\displaystyle\bar{A}_{\rm s}[z]$ $\displaystyle=\bar{A}_{\rm
s}[0]\cosh(\kappa z)$ (19) $\displaystyle\bar{B}[z]$
$\displaystyle=-\frac{ig_{0}^{*}\bar{A}_{\rm p}}{|g_{0}||\bar{A}_{\rm
p}|}\sqrt{\frac{v_{\rm g,s}}{v_{\rm g,b}}}\bar{A}^{\dagger}_{\rm
s}[0]\sinh(\kappa z).$

We see that the dynamics in the ACB limit resemble those of optical parametric
(e.g. $\chi^{(2)}$) processes [51]—distinct from those of traditional
stimulated Brillouin scattering. In particular, we note there is a
qualitatively different gain-like behavior. The nontrivial acoustoelectric
gain enhancement can be characterized by the ratio of the standard gain per
unit length in the coherent ($\kappa$) to traditional ($G_{\rm B}P$) limits,
given by

$\displaystyle\frac{\kappa}{G_{\rm
B}P}=\frac{\Gamma}{4|g_{0}|}\sqrt{\frac{\hbar\omega_{\rm p}v_{\rm g,p}v_{\rm
g,s}}{Pv_{\rm g,b}}},$ (20)

where $\Gamma$ is the phonon dissipation rate in the absence of
acoustoelectric gain. We note that the ACB regime is particularly advantageous
(from a perspective of accessible optical gain) in the case of relatively low
optical pump powers and optomechanical coupling rates, as shown in Fig. 5a.

The optomechanical dynamics of this acoustoelectric induced coherent Brillouin
limit diverge even more dramatically from those of traditional SBS if we
consider a counter-propagating phonon field—such as is required for backward
stimulated Brillouin scattering. Switching the sign of the phonon group
velocity yields an imaginary coupling $\kappa=i|\kappa|$, and the solution
becomes Rabi-like as

$\displaystyle\bar{A}_{\rm s}[z]$ $\displaystyle=\bar{A}_{\rm
s}[0]\cos(|\kappa|z)$ (21) $\displaystyle\bar{B}[z]$
$\displaystyle=\frac{g_{0}^{*}\bar{A}_{\rm p}}{|g_{0}||\bar{A}_{\rm
p}|}\sqrt{\frac{v_{\rm g,s}}{v_{\rm g,b}}}\bar{A}^{\dagger}_{\rm
s}[0]\sin(|\kappa|z).$

Thus, in this case, acoustoelectric gain can lead to strong-coupling-like
optomechanical dynamics—a dramatic departure from the parametric gain behavior
of the co-propagating case (see Fig. 5b).

##  IV Discussion and outlook

In this paper, we have proposed the concept for and explored the dynamics of
acoustoelectrically enhanced Brillouin (AEB) interactions for the first time.
Through a Hamiltonian-based formalism, we have determined that the
acoustoelectric modification of the Brillouin susceptibility drastically
enhances the performance of useful Brillouin photonic devices and opens the
door to new regimes of traveling-wave optomechanical dynamics. Moreover, we
have proposed an experimentally realizable chip-based system that provides the
necessary optomechanical and acoustoelectric degrees of freedom to demonstrate
the full range of this new device physics. This work lays the foundation for a
new class of powerful optomechanics-based classical and quantum signal
processing applications, synergistically combining the unique properties of
photons, phonons, and electrons.

Beyond the material system and device geometry proposed in this work,
acoustoelectrically enhanced Brillouin physics may be accessible in a number
of promising integrated phononic/photonic platforms. Such physics requires
high carrier mobility, appreciable acoustoelectric response, and sufficient
traveling-wave optomechanical guidance and coupling. Example systems with
these properties may include piezoelectric semiconductors such as GaN [56],
GaAs [57, 58, 59], and GaP [60, 61], or semiconductor-piezoelectric hybrids
(like the example presented in this work), such as $\rm GaAs-LiNbO_{3}$ [62,
63] or co-integrated AlN in SOI-based silicon photonics [64]. In addition,
systems that utilize forms of non-piezoelectric-based acoustoelectric
coupling, for instance deformation potential coupling [28, 65], may enable AEB
physics in an even broader class of semiconductors. We anticipate the
emergence of a range of other material systems and device geometries that will
be able to leverage the new concepts proposed in this work.

By modifying the nonlinear Brillouin susceptibility through acoustoelectric
gain and nonreciprocity, we are able to drastically transform the performance
of important Brillouin signal processing applications—long valued for their
ability to amplify, modulate, or filter light over narrow bandwidths [52]. At
the core of these modified interactions is the ability to enhance the
Brillouin amplification process by orders of magnitude, enabling exceptionally
narrow bandwidths and state-of-the-art gain in systems that would otherwise
yield little or no net amplification. Moreover, our noise analysis suggests
that when thermomechanical noise is dominant (see Supplementary Section D),
this approach may enable high levels of near quantum-limited amplification at
cryogenic temperatures. Building on this work, inducing optical gain via
manipulation of electrical carriers may enable a new generation of flexible
and narrow linewidth Brillouin laser systems with reconfigurable dynamics and
improved noise performance.

The impact of acoustoelectrics on optomechanics extends beyond traditional
Brillouin photonic devices. Whereas we have focused here on traveling-wave
optomechanical interactions, these same concepts are applicable to cavity
optomechanical systems [66]. The ability to dynamically modify the phonon
decay rate and cooperativity in cavity-optomechanical systems should lead to
enhanced coherent photon-phonon interactions in devices such as optomechanical
amplifiers [67, 68, 69, 70], oscillators [71, 72, 73], filters [74], and
quantum memories [75]. Finally, applying these concepts to acousto-optic
devices may allow externally applied electric fields to modify local phonon
dissipation rates and populations, enabling faster amplitude modulation in
systems that would otherwise be limited by intrinsic phonon lifetimes.

## Funding Information

This material is based upon work supported by the Laboratory Directed Research
and Development program at Sandia National Laboratories. Sandia National
Laboratories is a multi-program laboratory managed and operated by National
Technology and Engineering Solutions of Sandia, LLC., a wholly owned
subsidiary of Honeywell International, Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-
NA-0003525. This paper describes objective technical results and analysis. Any
subjective views or opinions that might be expressed in the paper do not
necessarily represent the views of the U.S. Department of Energy or the United
States Government. R. Behunin acknowledges support through the Center for
Materials Interfaces in Research and Applications (¡MIRA!) Exploratory
Research Program.

## Author contributions

*N.T.O., M.J.S., and R.O.B. contributed equally to this work.

M.E. and N.T.O. jointly developed concept and proposed physical system with
input from M.J.S., L.H., and P.T.R.. M.J.S. examined system properties and
performed numerical simulations and analysis of the optical, elastic,
optomechanical, electromechanical, and acoustoelectric properties of the
proposed system with assistance from N.T.O., M.E., and P.T.R.. N.T.O. and
R.O.B. developed analysis of acoustoelectric-modified Brillouin optomechanical
dynamics and noise. R.O.B. developed comprehensive Hamiltonian formalism as
the basis for the acoustoelectric-modified optomechanical interactions. M.J.S.
generated 3D illustrative diagrams. L.H. analyzed prospects of cw operation.
All authors contributed to the writing and composition of the manuscript.

##  Supplementary Information A Physical System Design

Phase-matched Brillouin scattering is a three-wave mixing process involving
two optical modes (pump and Stokes) and one elastic mode. This supplementary
section covers the design procedure for computing the optical and elastic mode
triplet for acoustoelectric (AE) Brillouin interactions in a given physical
system. The primary physical system focused on in this supplement is an
epitaxially grown In0.712Ga0.288As0.625P0.375 (referred to as InGaAsP in the
manuscript for convenience) bonded to a Y-cut lithium niobate thin film on a
silicon substrate (InGaAsP-LN-Si). The simulated cross-sectional 2D optical
domain is shown in Fig. 6a where the optical guiding layer is the patterned
InGaAsP waveguide, the lower cladding is the lithium niobate, and the top and
side cladding is air. Since the FEA software has a solver for time-harmonic
optical field distributions, a cross-sectional 2D Mode analysis is performed
to compute the propagation constant and propagating mode shapes for a given
optical frequency. Due to the high index contrast of the InGaAsP waveguide,
the optical fields are confined such that no perfectly matched layers (PMLs)
are needed at the boundaries of the simulation domain. Additionally, the
silicon is assumed to be far enough away from the optical fields as to not
cause optical loss, making absorption losses in that material unnecessary to
include (an assumption that is easily verified for the specific modes we find
as solutions). The simulated 3D acoustic domain is shown in Fig. 6b and
includes the addition of PMLs to absorb any elastic energy at the edges of the
simulation domain. For certain polarizations and phase-matched phonon
wavelengths of the elastic modes, there is low contrast between the phonon
velocities in InGaAsP and lithium niobate, which results in minimal guiding of
the elastic mode in the waveguide. The amount of elastic energy absorbed by
the PMLs help us characterize whether the simulation boundary has been made
large enough and whether a given elastic mode is guided. For backward intra-
modal and forward inter-modal SBS processes, the elastic modes in this
physical system have periodic variations in their strain along the direction
of propagation, which invalidates the cross-sectional 2D plain strain
approximation. Therefore, analysis of these elastic modes require a 3D
simulation domain with floquet periodic boundary conditions in the direction
of the defined phonon wavelength. The elastic mode is guided within the
InGaAsP/lithium niobate heterostructure and the silicon substrate assists in
heat dissipation for continuous wave (CW) operation (see Supplementary Section
B).

Figure 6: Simulation domains (not to scale) for the epitaxial InGaAsP bonded
to a Y-cut lithium niobate thin film on silicon substrate (InGaAsP-LN-Si). The
materials and key design parameters are shown in the (a) optical and (b)
acoustic simulation domains as well as the (c) defined material orientations.

The three key geometric parameters to consider when designing an AE Brillouin
device are shown in Fig. 6b. The width ($w$) and thickness ($t$) of the
InGaAsP waveguide not only determines the confinement and effective index of
the optical modes, but it also determines the lateral and vertical confinement
of the elastic mode in the InGaAsP/lithium niobate heterostructure. The phonon
wavelength ($\lambda_{\text{m}}$) sets the operating phonon frequency and
elastic mode shape, as determined by the width, thickness, and mode-dependent
dispersion. Since this physical system includes anisotropic materials, Fig. 6c
describes all of the material orientations. The lithium niobate thin film has
its +Y-axis normal to the device plane and the +X-axis is parallel to the
[110] direction in the cubic InGaAsP and silicon crystals. The lithium niobate
thin film has a thickness of 5 $\mu$m and the +Y-axis was chosen to be normal
to the device plane because these LN-Si wafers are commercially available and
have high piezoelectric coupling to both the Rayleigh and Shear Horizontal
surface acoustic waves. In the plane of the device, the propagation angle
($\beta$) not only determines the optical axis of lithium niobate, but also
the piezoelectric coupling strength of the elastic mode. The optomechanical
coupling strength is a function of the overlap between the optical mode and
elastic mode, while the acoustoelectric coupling strength is a function of the
overlap between the elastic mode, piezoelectric potential, and free carriers
in the InGaAsP. Therefore, careful consideration must be made when designing
the optical and elastic modes to ensure good overlap between all domains,
resulting in strong optomechanical and acoustoelectric coupling.

The general outline for the design procedure is detailed below. First, the
type of stimulated Brillouin scattering (SBS) is specified and the appropriate
acoustic and optical phase matching conditions are applied. The two types of
SBS considered here are intra-modal backward scattering (i.e. two
contradirectionally propagating optical modes with the same spatial mode
profile) and inter-modal forward scattering (i.e. two co-directionally
propagating optical modes with different spatial mode profiles). For each case
of SBS, the following design parameters must be considered to cover the full
design space: optical mode pair, propagation angle, InGaAsP waveguide width,
and InGaAsP waveguide thickness. The pair of optical modes used for the pump
and Stokes photons will determine the phonon wavelength needed for phase
matching, which together with the mode-dependent acoustic dispersion
determines the elastic mode frequency. The propagation angle chosen is largely
influenced by the optical and acoustic properties of the lithium niobate thin
film. Longitudinal strains have high piezoelectric coupling parallel to the
Z-axis, whereas shearing strains have a high piezoelectric coupling parallel
to the X-axis. This corresponds to either a 0° or 90° propagation angle,
respectively. These two propagation angles also correspond to the
extraordinary and ordinary optical axes in lithium niobate. Since the optimal
waveguide width strongly depends on the chosen elastic mode, the range of
simulated waveguide widths must be carefully considered. Finally, the
simulated InGaAsP waveguide thickness ranges from 100 nm to 300 nm to remain a
single mode optical waveguide in the vertical direction and to reduce the
electric field required to achieve a given phonon gain (see Eq. 88).

After the range of design parameters is set, the simulations and calculations
are carried out in the following order: optical domain simulation, acoustic
domain simulation, optomechanical coupling calculation, and acoustoelectric
coupling calculation. Using COMSOL Multiphysics, a commercial finite element
method (FEM) software, the 2D optical domain simulation is used to identify
the effective indices of the pump and Stokes photons, which are used to
compute the phonon wavelength required for phase matching. Additionally, the
normalized electric fields for the pump and Stokes photons are extracted from
the simulation to use in the optomechanical coupling rate calculations. Then,
the 3D acoustic domain is simulated for the free and grounded eigenfrequencies
of the elastic modes. The normalized displacement and strain profiles for the
elastic modes are extracted from the simulation to use in the optomechanical
coupling rate calculations. Using all of the extracted values from simulation,
the following optomechanical coupling rates are calculated: photoelastic
coupling in the InGaAsP waveguide, photoelastic coupling in the surrounding
lithium niobate, radiation pressure coupling in the waveguide, and total
optomechanical coupling rate. Finally, the sets of elastic modes from the free
and grounded eigenfrequency simulations are correlated to determine the
piezoelectric coupling coefficient of each elastic mode. Using the full
treatment of the normal mode theory for an AE amplifier, the acoustoelectric
gain is calculated as a function of applied drift field for each elastic mode.

###  A.1 InGaAsP Material

Before any of the simulations are performed, we calculate the interpolated
material parameters for the quaternary InGaAsP compound. For the InGaAsP-LN-Si
physical system, the operating optical wavelength is 1.55 $\mu$m. Some of the
figures and interpolation schemes consider this operating optical wavelength
in terms of its photon energy $E=\hbar\omega$. The corresponding photon energy
at a 1.55 $\mu$m wavelength is approximately 0.8 eV. The InGaAsP considered
for this physical system will be grown on a (100) InP wafer—a typical
substrate orientation for ternary and quaternary epitaxial growth—and bonded
to a Y-cut lithium niobate on silicon wafer. Based on previous work in InGaAsP
optical devices grown on a (100) InP wafer [76, 77, 78, 79], the [110]
direction of the InGaAsP will be oriented parallel to the [110] direction of
the silicon substrate [80] and the +X-axis of the lithium niobate thin film.
The following subsections will outline the proposed fabrication procedure,
quaternary composition, and interpolation schemes used to determine the
InGaAsP material parameters for use in this physical system.

####  A.1.1 Acoustoelectric Fabrication Modified for InGaAsP Waveguides

Based on our previous work in designing and fabricating high performance
acoustoelectric devices [40, 41, 63, 81, 46], we can leverage our experience
in epitaxial growth and nanofabrication to make acoustoelectrically enhanced
Brillouin devices. Based on the fabrication process flow from our previous
work, the modified process is shown in Fig. 7. The only modification in the
epitaxial growth stack of the InGaAsP/InP wafer (Fig. 7a) from our previous
work is during the growth of the waveguide layer with a free carrier
concentration of 1 x 1016 cm-3. Here, the addition of phosphorus alters the
growth of a lattice-matched ternary In1-xGaxAs to a lattice-matched quaternary
In1-xGaxAsyP1-y. All the other InP and InGaAs layers in the growth process
(etch stop layers, ohmic contact layers, capping layer) remain unchanged from
the fabrication process of previous work. The fabrication steps in Fig. 7b-e
make use of two different wet etches, one selective to InP and one selective
to InGaAs [82], which will be the same wet etch chemistries that were used in
the fabrication process of our previous acoustoelectric work. If the InGaAsP
waveguide layer (Fig. 7f) is patterned through wet etching, there are several
chemistries available that etch both InGaAs and InGaAsP alloys [83, 84, 85].
Due to the isotropic nature of wet etching, the undercut makes resolving small
features difficult. For the waveguide dimensions needed for Brillouin devices,
anisotropic dry etching is typically required. There are several chemistries
available for etching InGaAsP using reactive ion etching (RIE) including BCl3
[86], Cl2/H2 [87, 88], and CH4/H2 [89]. The remaining fabrication steps (Fig.
7g-h) to pattern the metal transducers and DC electrical contacts remain
unchanged from our previous work. Overall, only slight modifications are
needed in our established acoustoelectric device fabrication process to
incorporate phosphorus into the InGaAsP waveguide layer and pattern the
Brillouin devices.

Figure 7: Fabrication process flow from our previous work in acoustoelectric
devices modified to include the InGaAsP waveguide layer for
acoustoelectrically enhanced Brillouin devices (not to scale).

####  A.1.2 Quaternary Composition for InP Lattice Matching

The quaternary compound of indium, gallium, arsenic, and phosphorus can be
expressed in terms of two composition parameters, $x$ and $y$
(In1-xGaxAsyP1-y). For lattice matching with InP, the following expression
relates the two composition parameters [90]

$\displaystyle x\approx\frac{0.1894\,y}{0.4184-0.013\,y}\,\,\,\,\left(0\leq
y\leq 1\right).$ (22)

Using this relationship between the two composition parameters, the
interpolation schemes for all of the material parameter calculations can be
expressed in terms of just one composition parameter ($y$). In order to
operate at an optical wavelength of 1.55 $\mu$m, the direct band gap energy
needs to be large enough to be optically transparent and the optical
wavelength needs to be far enough away from the absorption edge in order to
minimize loss from absorption. The direct gap energy (in eV) at 300 K is given
as

$\displaystyle E_{\text{0}}=1.35-0.72\,y+0.12\,y^{2}.$ (23)

For the case of the ternary compound InGaAs ($y=1$), the band gap energy is
0.75 eV, which is too small to be optically transparent at the desired optical
wavelength. The band gap energy can be increased through the introduction of
phosphorus into the compound. For the physical system in this manuscript, the
chosen composition parameter is $y=0.625$, giving the full description of the
quaternary composition as In0.712Ga0.288As0.625P0.375. At this composition,
the band gap energy is 0.95 eV, which is now optically transparent at the
operating optical wavelength and 0.15 eV away from the absorption edge.
Reference [91] examines the interpolated InGaAsP absorption coefficient as a
function of free carrier concentration. Given their operating optical
wavelength, the absorption coefficient is proportional to the free carrier
concentration when operating 0.15 eV away from the absorption edge. At our
free carrier concentration of $N$ = 1 x 1016 cm-3 (for optimal acoustoelectric
interaction), they see an absorption coefficient of approximately 0.02 cm-1.
From this, we can estimate a loss of 0.1 dB/cm in the InGaAsP from optical
absorption.

####  A.1.3 Refractive Index

The treatment for calculating the index of refraction as a function of photon
energy and composition parameter can be found in the appendix of Ref. [92].
Assuming an operating temperature of 300 K, the index of refraction can be
expressed as

$\displaystyle
n^{2}_{\text{r}}=A\left(y\right)\left[f\left(z\right)+\frac{1}{2}\left(\frac{E_{\text{0}}}{E_{\text{0}}+\Delta_{0}}\right)^{3/2}f\left(z_{0}\right)\right]+B\left(y\right),$
(24)

where the split-off valence band gap (in eV) at 300 K is

$\displaystyle\Delta_{0}=0.118+0.225\,y.$ (25)

The normalized energy terms ($z$ and $z_{0}$) are a function of the operating
photon energy, $E$, and the related terms can be expressed as

$\displaystyle
f\left(z^{\prime}\right)=\left(2-\sqrt{1+z^{\prime}}-\sqrt{1-z^{\prime}}\right)/z^{\prime\,2},$
(26)

$\displaystyle z=E/E_{\text{0}},$ (27)

$\displaystyle z_{0}=E/\left(E_{\text{0}}+\Delta_{0}\right).$ (28)

Finally, the fitting parameters $A$ and $B$ are a function of the composition
parameter

$\displaystyle A\left(y\right)=8.616-3.886\,y,$ (29)

$\displaystyle B\left(y\right)=6.621+3.461\,y.$ (30)

For the composition parameter and photon energy used in this physical system,
the calculated refractive index is $n_{\text{r}}=3.4013$.

####  A.1.4 Dielectric Constant

The interpolation for both the static and high frequency dielectric constants
as a function of composition parameter are given in Ref. [90]. For the
frequency range of elastic modes simulated, the static dielectric constant
will be used, which is given as

$\displaystyle\epsilon_{\text{s}}=12.4+1.5\,y.$ (31)

For the composition parameter used in this physical system, the calculated
dielectric constant is $\epsilon_{\text{s}}=13.3375$.

####  A.1.5 Density

The density varies almost linearly between InP and InGaAs depending on the
composition parameter, which is given in Ref. [90] (in units of kg/m3)

$\displaystyle\rho=4787+712\,y.$ (32)

For the composition parameter used in this physical system, the calculated
density is $\rho=5232$ kg/m3.

####  A.1.6 Elasticity and Compliance Matrix

InGaAsP has a zincblende crystal structure, which means the elasticity matrix
has three independent coefficients ($C_{11}$, $C_{12}$, $C_{44}$) and the
matrix takes the form

$\displaystyle\left[C\right]=\begin{bmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\\
C_{12}&C_{11}&C_{12}&0&0&0\\\ C_{12}&C_{12}&C_{11}&0&0&0\\\
0&0&0&C_{44}&0&0\\\ 0&0&0&0&C_{44}&0\\\ 0&0&0&0&0&C_{44}\end{bmatrix}.$ (33)

The interpolation of the elasticity coefficients as a function of the
composition parameter is given in Chapter 3 of Ref. [93]. These coefficients
(in GPa) are given as

$\displaystyle C_{11}=101.1-1.2\,y,$ (34)

$\displaystyle C_{12}=56.1-6.8\,y,$ (35)

$\displaystyle C_{44}=45.6+3.3\,y.$ (36)

For the composition parameter used in this physical system, the calculated
elasticity coefficients are $C_{11}$ = 100.35 GPa, $C_{12}$ = 51.85 GPa, and
$C_{44}$ = 47.6625 GPa. Similarly, the three independent compliance matrix
coefficients can be calculated using the interpolated elasticity matrix
coefficients from Eq. 34-36 with the following relations

$\displaystyle
S_{11}=\frac{C_{11}+C_{12}}{\left(C_{11}-C_{12}\right)\left(C_{11}+2\,C_{12}\right)},$
(37)

$\displaystyle
S_{12}=\frac{-C_{12}}{\left(C_{11}-C_{12}\right)\left(C_{11}+2\,C_{12}\right)},$
(38)

$\displaystyle S_{44}=\frac{1}{C_{44}}.$ (39)

Using the interpolated elasticity matrix coefficients above, the calculated
compliance matrix coefficients are $S_{11}$ = 1.538 x 10-11 Pa-1, $S_{12}$ =
-5.239 x 10-12 Pa-1, and $S_{44}$ = 2.098 x 10-11 Pa-1.

####  A.1.7 Photoelastic Coupling Constants

Since there is no complete set of interpolated photoelastic constants for
InGaAsP in the literature, the following subsection will detail the
methodology used for estimating the full photoelastic coupling matrix. Using
the base set of binaries {InAs, InP, GaAs, GaP}, Chapter 9 in Ref. [93] uses
interpolation methods to calculate the linear piezobirefringence (PB)
coefficient for the [100] and [111] stress directions. Using these PB
coefficients, the method for estimating the photoelastic coupling constants as
a function of the composition parameter is outlined. Since InGaAsP has a
zincblende crystal structure, there are three independent photoelastic
coupling constants ($p_{11}$, $p_{12}$, $p_{44}$) and the matrix takes the
form

$\displaystyle\left[p\right]=\begin{bmatrix}p_{11}&p_{12}&p_{12}&0&0&0\\\
p_{12}&p_{11}&p_{12}&0&0&0\\\ p_{12}&p_{12}&p_{11}&0&0&0\\\
0&0&0&p_{44}&0&0\\\ 0&0&0&0&p_{44}&0\\\ 0&0&0&0&0&p_{44}\end{bmatrix}.$ (40)

Chapter 9 in Ref. [93] also details the procedure to compute the relationship
between photoelastic coupling constants and the interpolated linear PB
coefficients ($\alpha_{\text{pe}}$). The interpolated results are based on
fitting the experimental data of the four binary compounds {InAs, GaAs, InP,
GaP}, which were taken using uniaxial stress measurements to extract the
linear PB coefficients

$\displaystyle\alpha_{\text{pe}}=\frac{\Delta\epsilon_{ij}}{X}=-\sum_{mn}\epsilon_{ii}\epsilon_{jj}p_{ijkl}S_{klmn}.$
(41)

Here, $\epsilon_{ii,jj}$ is the component of the dielectric constant tensor in
the absence of the applied stress, $p_{ijkl}$ is the photoelastic tensor
component, $S_{klmn}$ is the component of elastic compliance tensor, $X$ is
the directional applied stress, and $\Delta\epsilon_{ij}$ is the difference
between the real part of the dielectric constant parallel and perpendicular to
the applied stress direction. Since all of the higher order band gaps for the
four binary compounds are far away from the first direct gap, the linear PB
coefficient can be modeled with dispersive effects from only the first direct
band gap and include the influence of the higher order gaps as a constant
background effect. This model for the linear PB coefficient is given as

Figure 8: Dispersion of the photoelastic constants as a function of the
optical wavelength for (a) GaAs [94] and (b) InP [95]. (c) A similar
dispersion trend is shown for the photoelastic constants of InGaAsP as a
function of composition parameter. The green points in (b) and (c) correspond
to the same photoelastic constant values on the InP and InGaAsP dispersion
curves.

$\displaystyle\alpha_{\text{pe}}=C^{*}\left\\{-g\left(z\right)+\frac{4E_{0}}{\Delta_{0}}\left[f\left(z\right)-\left(\frac{E_{\text{0}}}{E_{\text{0}}+\Delta_{0}}\right)^{3/2}f\left(z_{0}\right)\right]\right\\}+D^{*},$
(42)

where $C^{*}$ and $D^{*}$ are parameters used to fit to the experimental data
and $g\left(z\right)$ is the normalized function

$\displaystyle
g\left(z^{\prime}\right)=\left(2-\frac{1}{\sqrt{1+z^{\prime}}}-\frac{1}{\sqrt{1-z^{\prime}}}\right)/z^{\prime\,2}.$
(43)

The fit parameters $C^{*}$ and $D^{*}$ for InGaAsP can be interpolated as a
function of the fit parameters for each binary compound shown in Table 9.1
(Chapter 9) of Ref. [93]. The interpolation scheme for these fit parameters is
described in Ref. [96] and given as

$\displaystyle\phi_{\text{InGaAsP}}\left(x,y\right)=\left(1-x\right)y\,\psi_{\text{InAs}}+\left(1-x\right)\left(1-y\right)\,\psi_{\text{InP}}+xy\,\psi_{\text{GaAs}}+x\left(1-y\right)\,\psi_{\text{GaP}},$
(44)

where $\phi$ is the quaternary parameter and each $\psi$ is the corresponding
binary parameter. The interpolated linear PB coefficients correspond to a
stress applied parallel to the [100] direction in InGaAsP
($\alpha_{\text{pe}}^{\text{[100]}}$) and the stress applied parallel to the
[111] direction ($\alpha_{\text{pe}}^{\text{[111]}}$). Using Eq. 41, each
linear PB coefficient can be expressed in terms of the photoelastic constants,
depending on the orientation of the applied stress. For a stress applied in
the [100] direction, the linear PB coefficient can be expressed as

$\displaystyle\alpha_{\text{pe}}^{\text{[100]}}=-n_{\text{r}}^{4}\left(p_{11}-p_{12}\right)\left(S_{11}-S_{12}\right),$
(45)

and for a stress applied in the [111] direction, the linear PB coefficient can
be expressed as

$\displaystyle\alpha_{\text{pe}}^{\text{[111]}}=-n_{\text{r}}^{4}p_{44}S_{44}.$
(46)

Using the previously calculated index of refraction and compliance matrix
coefficients, the relationships between photoelastic constants can be
expressed as

$\displaystyle
p_{11}-p_{12}=\frac{-n_{\text{r}}^{4}\,\alpha_{\text{pe}}^{\text{[100]}}}{\left(S_{11}-S_{12}\right)},$
(47)

$\displaystyle
p_{44}=\frac{-n_{\text{r}}^{4}\,\alpha_{\text{pe}}^{\text{[111]}}}{S_{44}}.$
(48)

For the composition parameter and photon energy used in this physical system,
the interpolated relationships between the photoelastic constants are
$p_{11}-p_{12}=0.115$ and $p_{44}=0.027$.

It is not possible with PB measurements to determine the magnitude and sign of
$p_{11}$ and $p_{12}$ [94, 95]. In order to estimate the $p_{11}$ and $p_{12}$
photoelastic constants for simulating Brillouin optomechanics in InGaAsP, the
measured dispersion relations of the GaAs and InP binary compounds will be
used to estimate the dispersive trends of the InGaAsP interpolation scheme. As
the photon energy approaches the direct band gap energy, the photoelastic
constants experience large dispersion [94, 95, 97]. While most dispersion
measurements of the binary compounds are plotted as a function of the
operating optical wavelength, the interpolated InGaAsP dispersion will be
plotted as a function of the composition parameter. The operating optical
wavelength will be held constant at 1.55 $\mu$m and the composition parameter
will be swept from $y$ = 0 (InP) to $y$ = 0.625 (InGaAsP). Since the
photoelastic constants of InP at 1.55 $\mu$m are known from Ref. [95], they
can be used as a starting point for the InGaAsP dispersion curve. The first
step is to estimate the sign of the $p_{11}$ and $p_{12}$ constants in
InGaAsP. The binary compounds of InGaAsP, all with different band gap
energies, have similar signs and magnitudes of their photoelastic constants at
photon energies below the band gap energy. Reference [98] has the photoelastic
constants of InP, which are $p_{11}=-0.150$, $p_{12}=-0.115$, $p_{44}=-0.056$.
Reference [99] has the photoelastic constants of GaAs, which are
$p_{11}=-0.165$, $p_{12}=-0.140$, $p_{44}=-0.072$. Finally, Ref. [100] has the
photoelastic constants of GaP, which are $p_{11}=-0.151$, $p_{12}=-0.082$,
$p_{44}=-0.074$. From the signs of the binary compounds’ photoelastic
constants, the signs of $p_{11}$ and $p_{12}$ in InGaAsP are estimated to be
negative for photon energies below the band gap.

The next step is to determine the magnitude of the dispersive photoelastic
constants in InGaAsP as a function of the composition parameter. At the
starting point ($y$ = 0), the band gap of InP is 1.35 eV which is much larger
than the photon energy of 0.8 eV. At the desired composition ($y$ = 0.625),
the band gap reduces to 0.95 eV, which is much closer to the photon energy.
With regard to the dispersion trends, lowering the band gap has a similar
effect to increasing the photon energy (reducing the operating optical
wavelength). To illustrate these similarities, the dispersion trends of GaAs
and InP as a function of operating optical wavelength are plotted along side
the dispersion of InGaAsP as a function of composition parameter in Fig. 8.
All three dispersion curves show a similar trend as the photon energy
approaches the band gap. The values of $p_{44}$ and ($p_{11}-p_{12}$) become
less negative until crossing into positive values. The values of $p_{11}$ and
$p_{12}$ become more negative, with the magnitude of $p_{12}$ eventually
becoming larger than the magnitude of $p_{11}$. The green points in Fig. 8
correspond to the starting InP photoelastic constants for the InGaAsP
interpolation scheme. These starting InP values at an operating optical
wavelength of 1.55 $\mu$m are calculated from the fit lines in Ref. [95]
($p_{11}=-0.127$, $p_{12}=-0.109$, $p_{44}=-0.064$). For the InGaAsP
dispersion curve in Fig. 8c, the $p_{44}$ value was explicitly calculated from
Eq. 48. Conservatively, the value of $p_{11}$ is estimated to be the same as
the starting InP value. Then, the value of $p_{12}$ can be estimated using the
calculated difference ($p_{11}-p_{12}$) from Eq. 47. Using this methodology,
the estimated InGaAsP photoelastic constants used in our simulations are
$p_{11}=-0.127$, $p_{12}=-0.242$, and $p_{44}=0.027$.

####  A.1.8 Carrier Transport

From Chapter 10 in Ref. [93], the mobility of InGaAs at room temperature is
similar in magnitude to the mobility of InGaAsP at the composition parameter
used in this physical system. The mobility, $\mu$, of InGaAs thin films made
at Sandia for use in AE delayline devices [41] is 2000 cm2/Vs, which is lower
than the calculated values for bulk InGaAs as a result of surface effects
limiting the mobility. Since these same surface effects will be present in the
fabricated InGaAsP thin film, its reasonable to assume a similar limit to the
mobility for the physical system in this work ($\mu$ = 2000 cm2/Vs). The
optimal carrier concentration for AE interactions with MHz and GHz elastic
modes typically falls within the range of 1015 to 1017 cm-3 [41]. The carrier
concentration must also be low enough so significant optical losses are not
introduced [48]. Therefore, the simulations and calculations for this physical
system will use a carrier concentration of $N$ = 1 x 1016 cm-3.

The effective mass and plasma frequency of the InGaAsP are necessary to
compute the Brillouin nonlinear susceptibility. The effective mass of the
electrons in the conduction band edge is given by a linear interpolation as a
function of the InGaAsP composition parameter [90]

$\displaystyle m^{*}=\left(0.080-0.039\,y\right)\,m_{\text{e}}.$ (49)

For the composition parameter used in this physical system, the effective mass
is $m^{*}=0.055625\,\,m_{\text{e}}$, where $m_{\text{e}}$ is the mass of an
electron. Finally, the plasma frequency in a semiconductor material is given
as

$\displaystyle\omega_{\text{e}}^{2}=\frac{Ne^{2}}{m^{*}\epsilon_{0}},$ (50)

where $e$ is the charge of an electron and $\epsilon_{0}$ is the permittivity
of free space. Given the carrier concentration and effective mass of this
physical system, the plasma frequency is $\omega_{\text{e}}/2\pi=3.8069$ THz.

Table 1: Summary of the calculated InGaAsP parameters for the physical system used in this paper. InGaAsP Parameter |  Calculated Value
---|---
$\lambda_{\text{optical}}$ |  1.55 $\mu$m
$E$ |  0.7999 eV
$y$ |  0.625
$x$ |  0.288
$C_{11}$ |  100.35 GPa
$C_{12}$ |  51.85 GPa
$C_{44}$ |  47.6625 GPa
$S_{11}$ |  1.538 x 10-11 Pa-1
$S_{12}$ |  -5.239 x 10-12 Pa-1
$S_{44}$ |  2.098 x 10-11 Pa-1
$\rho$ |  5232 kg/m3
$E_{0}$ |  0.9469 eV
$\Delta_{0}$ |  0.2586 eV
$\epsilon_{\text{s}}$ |  13.3375
$A\left(y\right)$ |  6.1873
$B\left(y\right)$ |  8.7841
$n_{\text{r}}$ |  3.4013
$C^{*}$ [100] |  -1.327 x 10-11 cm2/dyn
$C^{*}$ [111] |  -0.799 x 10-11 cm2/dyn
$D^{*}$ [100] |  1.932 x 10-11 cm2/dyn
$D^{*}$ [111] |  2.315 x 10-11 cm2/dyn
$\alpha_{\text{pe}}^{\text{[100]}}$ |  -3.186 x 10-11 cm2/dyn
$\alpha_{\text{pe}}^{\text{[111]}}$ |  -0.764 x 10-11 cm2/dyn
$p_{11}-p_{12}$ |  0.115
$p_{11}$ |  -0.127
$p_{12}$ |  -0.242
$p_{44}$ |  0.027
$\mu$ |  2000 cm2/Vs
$N$ |  1 x 1016 cm-3
$m^{*}$ |  0.055625 $m_{\text{e}}$
$\omega_{\text{e}}$ |  $2\pi\cdot 3.8069$ THz.

###  A.2 Optical Simulations

Given a physical system and its material parameters, the first step in
designing an acoustoelectric Brillouin device is to examine the phase-matching
criteria between the optical mode pair and the phonon wavelength. This section
of the supplement will cover both backward intra-modal and forward inter-modal
Brillouin scattering processes. In the case of a backward intra-modal Stokes
process, a higher energy pump photon creates a forward scattered phonon and
backward scattered lower energy Stokes photon in the same optical mode.
Similarly, in the case of a forward inter-modal Stokes process, a higher
energy pump photon from one optical mode creates a forward scattered phonon
and lower energy Stokes photon in a second optical mode. Due to momentum
conservation, the phonon wave vector can be expressed as the difference
between the pump and Stokes photon wave vectors

$\displaystyle
q_{\text{m}}=k_{\text{p}}\left(\omega_{\text{p}}\right)-k_{\text{s}}\left(\omega_{\text{s}}\right),$
(51)

where $k_{\text{p}}$ is the pump photon wave vector, $k_{\text{s}}$ is the
Stokes photon wave vector, $\omega_{\text{p}}$ is the pump photon frequency,
and $\omega_{\text{s}}$ is the Stokes photon frequency. The required energy
conservation can be expressed as

$\displaystyle\Omega_{\text{m}}=\omega_{\text{p}}-\omega_{\text{s}},$ (52)

Figure 9: Illustrations of (a) backward intra-modal and (b) forward inter-
modal dispersion diagrams showing the phase-matched pump photon, Stokes
photon, and phonon.

where $\Omega_{\text{m}}$ is the phonon frequency. Illustrations of the intra-
modal and inter-modal dispersion diagrams with phase-matched phonons are shown
in Fig. 9. As the dispersion diagrams show, the phase-matched phonon wave
vector is much larger for backward intra-modal scattering than forward inter-
modal scattering. In general, this means backward scattering Brillouin
processes require small phonon wavelengths while forward scattering requires
much larger phonon wavelengths. The phase-matched phonon wavelength can be
calculated using Eq. 51 along with expressions for the photon and phonon wave
vectors. The optical wave vectors ($k_{\text{p,s}}$) can be expressed as

$\displaystyle
k_{\text{p,s}}=n_{\text{eff}}^{\text{p,s}}\,\frac{2\pi}{\lambda_{\text{p,s}}},$
(53)

where $n_{\text{eff}}^{\text{p}}$ is the pump optical mode’s effective index,
$n_{\text{eff}}^{\text{s}}$ is the Stokes optical mode’s effective index,
$\lambda_{\text{p}}$ is the pump photon wavelength, and $\lambda_{\text{s}}$
is the Stokes photon wavelength. Similarly, the wave vector of the phonon
($q_{\text{m}}$) is given as

$\displaystyle q_{\text{m}}=\frac{2\pi}{\lambda_{\text{m}}},$ (54)

where $\lambda_{\text{m}}$ is the phonon wavelength. In the case of backward
intra-modal Brillouin scattering, the Stokes photon wave vector is
approximately equal to the pump photon wave vector, but opposite in sign
($k_{\text{s}}\approx-k_{\text{p}}$). Using Eq. 51, the phonon wave vector can
be expressed in terms of the pump wave vector

$\displaystyle q_{\text{m}}=2\,k_{\text{p}}\left(\omega_{\text{p}}\right).$
(55)

The phonon wavelength can then be solved for in terms of the optical
wavelength and effective mode index

$\displaystyle\lambda_{\text{m}}=\frac{\lambda_{\text{p}}}{2\,n_{\text{eff}}^{\text{p}}}.$
(56)

The phase-matched phonon wavelength for backward intra-modal scattering is
smaller than half of the optical wavelength and strongly depends on the
optical mode’s effective index. Using the TE0 optical mode in the InGaAsP-LN-
Si physical system, the phase-matched phonon wavelength for backward intra-
modal Brillouin scattering as a function of waveguide geometry is shown in
Fig. 10b. The phonon wavelength, which has an average value around 300 nm, is
almost independent of both the waveguide width and thickness. This phonon
wavelength is the same order of magnitude as the waveguide dimensions, which
leads to elastic modes with the strain almost entirely confined in the InGaAsP
waveguide. These elastic waveguide modes will have strain profiles
characteristic of either the traditional Rayleigh or Shear Horizontal (SH)
surface acoustic wave (SAW) modes, which means these waveguide modes will have
the same favorable propagation angle as the SAW mode it resembles most.

Figure 10: Optical simulation results for both backward intra-modal and
forward inter-modal phase matching conditions. (a) Example TE-like and TM-like
optical mode shapes for the InGaAsP-LN-Si physical system. Surface plots of
the phase-matched phonon wavelength for (b) backward intra-modal and (c-f)
forward inter-modal Brillouin scattering as a function of the waveguide width
and thickness.

This phase-matching analysis can then be extended to forward inter-modal
Brillouin scattering. Since the phonon frequency is much smaller than the
photon frequencies ($\Omega_{\text{m}}\,$$<<$$\,\omega_{\text{p,s}}$), both
the pump and Stokes photons can be approximated as having the same optical
wavelength when solving for the phase-matched phonon wavelength. Using this
assumption, the required phonon wavelength for forward inter-modal Brillouin
scattering is

$\displaystyle\lambda_{\text{m}}=\frac{\lambda_{\text{p}}}{n_{\text{eff}}^{\text{p}}-n_{\text{eff}}^{\text{s}}}.$
(57)

Due to the relatively small difference between the pump and Stokes effective
mode indices, the phonon wavelength for forward inter-modal Brillouin
scattering is typically larger than the optical wavelength. For different
combinations of TE-like and TM-like optical modes (mode shapes shown in Fig.
10a), the phase-matched phonon wavelengths for forward inter-modal Brillouin
scattering range from 2 $\mu$m to 20 $\mu$m. This wavelength range is
consistent with traditional SAW devices in both bulk and thin film lithium
niobate material systems. Therefore, the desired elastic modes for forward
inter-modal Brillouin scattering will be traditional SAW modes with high
piezoelectric coupling coefficients ($k^{2}$) for a strong AE interaction. The
two SAW modes with the highest $k^{2}$ in Y-cut lithium niobate are the
Rayleigh [101] and Shear Horizontal [81] elastic modes. Additionally, these
SAW modes will be horizontally guided and laterally confined by the density
mismatch and sound speed difference between the InGaAsP and lithium niobate.
As a result of each SAW mode’s strain profile, the Rayleigh SAW will have the
strongest optomechanical coupling to optical modes of the same parity, while
the SH SAW will have the strongest optomechanical coupling to optical modes of
different parity. Therefore, the TE0/TE1 and TE0/TM0 optical mode pairs (Fig.
10c,e) are considered for the SH SAW elastic mode and the TE0/TE2 and TE0/TM1
optical mode pairs (Fig. 10d,f) are considered for the Rayleigh SAW elastic
mode.

###  A.3 Acoustic Simulations

The next stage in the design process is to consider the strain profiles and
piezoelectric coupling coefficient, $k^{2}$, for each elastic mode given the
phase-matched phonon wavelength. The overlap between the strain profiles and
optical fields determines the strength of the optomechanical coupling, and a
high $k^{2}$ is an indicator of a strong AE interaction. In the presence of a
perfect conductor, the piezoelectric coupling coefficient of a surface
acoustic wave can be calculated by the shift in acoustic velocity as the
perfect conductor is brought from infinitely far away to the surface of the
piezoelectric material [102, 103]. In FEA simulation, this can be achieved
through two different eigenfrequency simulations. One will have an
electrically free lithium niobate surface and the other will have a grounded
lithium niobate surface. The piezoelectric coupling coefficient can be
computed as

$\displaystyle
k^{2}=\frac{\left|v_{\text{free}}^{\,2}-v_{\text{ground}}^{\,2}\right|}{v_{\text{free}}^{\,2}}=\frac{\left|f_{\text{free}}^{\,2}-f_{\text{ground}}^{\,2}\right|}{f_{\text{free}}^{\,2}},$
(58)

Here, $v_{\text{free}}$ is the acoustic velocity with a free piezoelectric
surface and $v_{\text{ground}}$ is the acoustic velocity with a grounded
piezoelectric surface. Since the phonon wavelength is the same for both
simulations, the piezoelectric coupling coefficient can be expressed in terms
of the acoustic eigenfrequency for the free piezoelectric surface
($f_{\text{free}}$) and grounded piezoelectric surface ($f_{\text{ground}}$).

Figure 11: Acoustic mode simulation results for both inter-modal and intra-
modal SBS processes. Example periodic mode shape and cross-sectional strain
profiles are shown for (a) Rayleigh SAW mode, (b) SH SAW mode, (c) waveguide
mode 1, (d) waveguide mode 2—elastic mode highlighted in the main manuscript,
and (e) waveguide mode 3. For each normalized strain profile, the relative
magnitudes between the strain components are shown in parenthesis.

In the case of an unguided Rayleigh SAW mode in Y-cut lithium niobate, the
$k^{2}$ is highest in the 0° propagation direction with a value around 5 %.
Similarly, the unguided SH SAW mode has its highest $k^{2}$ around 18 % in the
90° propagation direction. The definition of the calculated $k^{2}$, however,
must be adjusted with the inclusion of the InGaAsP waveguide into the acoustic
simulation. In order to consider the interaction strength between the elastic
mode and free carriers in the InGaAsP, the grounded eigenfrequency is
simulated with the perfect conductor placed only at the interface between the
InGaAsP waveguide and lithium niobate surface. This means the Rayleigh and SH
SAW modes used for forward inter-modal Brillouin scattering will only have a
fraction of the $k^{2}$ found in unguided lithium niobate propagation. As the
phonon wavelength approaches the waveguide width from the large wavelength
side, a larger fraction of the elastic mode is guided within the waveguide,
resulting in a piezoelectric coupling coefficient approaching the unguided
$k^{2}$ limit. For backward intra-modal Brillouin scattering, where the phase-
matched phonon wavelength is smaller than the InGaAsP waveguide width, almost
all of the elastic energy is confined in the waveguide. For these elastic
waveguide modes, the piezoelectric coupling strength depends on the
displacement profile within the waveguide.

The inclusion of the InGaAsP waveguide on top of the lithium niobate guides
the Rayleigh and SH SAW modes within the InGaAsP, with the amount of guiding
dependent on the relative size difference between the waveguide width and
phonon wavelength. Example elastic mode shapes and strain profiles for guided
Rayleigh and SH SAW modes are shown in Fig. 11a-b. Additionally, as more of
the elastic energy is guided within the InGaAsP waveguide, there is a larger
overlap between the optical fields and elastic strains. Therefore, it is
advantageous in the design process for inter-modal Brillouin devices to reduce
the required phonon wavelength to as close to the waveguide dimensions as
possible in order to achieve higher acoustoelectric and optomechanical
coupling strength. Operating near the optical mode cutoff, however, comes at
the expense of additional optical losses. This trade-off between optical
losses and coupling strength will need to be carefully considered when
designing acoustoelectric Brillouin devices.

For backward intra-modal Brillouin scattering, example elastic mode shapes and
strain profiles for three elastic waveguide modes are shown in Fig. 11c-e. For
waveguide mode 1, the displacement is primarily at the edges of the waveguide,
which has poor overlap with the TE0 optical fields. Since the
$\varepsilon_{\text{zz}}$ and $\varepsilon_{\text{xz}}$ strain components are
similar to the SH SAW mode, the propagation angle with the highest $k^{2}$ is
90°. Waveguide mode 2—the elastic mode highlighted in the main manuscript—has
displacement throughout the entire width of the waveguide and therefore a much
larger overlap with the optical fields. Waveguide mode 2 and the Rayleigh SAW
mode have similar strain profiles and the same dominant strain components
($\varepsilon_{\text{yy}}$ and $\varepsilon_{\text{zz}}$), which means
waveguide mode 2 will have the highest $k^{2}$ at a 0° propagation angle.
Finally, waveguide mode 3 has a mix of shear and longitudinal strains with
several nodes and anti-nodes along the width of the waveguide. As a result,
there is no clear preference in propagation direction and waveguide mode 3 has
similar $k^{2}$ values in both 0° and 90° propagation directions.

###  A.4 Optomechanical Coupling Calculation

After simulating the optical modes, acoustic modes, and piezoelectric coupling
coefficient, the next step in the design process is to compute the
optomechanical coupling coefficient ($g_{0}$). The two main contributions to
optomechanical coupling are from the photoelastic ($g_{\text{pe}}$) and
radiation pressure ($g_{\text{rp}}$) effects, where
$g_{0}=g_{\text{pe}}+g_{\text{rp}}$. This section of the supplement will
outline the theory for photoelastic and radiation pressure optomechanical
coupling for Stokes scattering processes and then implement this theory for
the InGaAsP-LN-Si physical system.

####  A.4.1 Photoelastic Coupling

The photoelastic coupling coefficient (in units of (rad/s)$\sqrt{\text{m}}$)
is given as [25]

$g_{\text{pe}}=\left|C_{\text{m}}\right|\left|C_{\text{p}}\right|\left|C_{\text{s}}\right|\,\xi\iint_{A}\left(\mathrm{D}_{\text{p}}^{i}\right)^{*}\mathrm{D}_{\text{s}}^{j}\,p^{ijkl}\varepsilon^{kl}\,dA.$
(59)

where $\mathrm{D}_{\text{p}}$ and $\mathrm{D}_{\text{s}}$ are the electric
displacement fields of the pump and Stokes photons extracted from simulation,
$p$ is the fourth-rank photoelastic tensor, $\varepsilon$ is the second-rank
strain tensor extracted from simulation, and $\xi$ is the frequency constant
defined as

$\xi=\frac{1}{\epsilon_{0}}\sqrt{\frac{\omega_{\text{p}}}{2}}\sqrt{\frac{\omega_{\text{s}}}{2}}\sqrt{\frac{\hbar\Omega_{\text{m}}}{2}}.$
(60)

The tensor product is integrated over the cross-sectional area perpendicular
to the wave propagation and the elastic strain and electric displacement
fields need to be normalized as described in Ref. [25]. A scaling factor can
be implemented for the phonon displacement ($\left|C_{\text{m}}\right|$), pump
photon electric displacement field ($\left|C_{\text{p}}\right|$), and Stokes
photon electric displacement field ($\left|C_{\text{s}}\right|$)

$\left|C_{\text{m}}\right|^{2}=\frac{1}{\Omega_{\text{m}}^{2}\iint_{A}\rho\left(\left|\mathrm{u}_{\text{x}}\right|^{2}+\left|\mathrm{u}_{\text{y}}\right|^{2}+\left|\mathrm{u}_{\text{z}}\right|^{2}\right)\,dA},$
(61)

$\left|C_{\text{p}}\right|^{2}=\frac{1}{\frac{1}{\epsilon_{0}}\iint_{A}\frac{1}{\epsilon_{\text{r}}}\left(\left|\mathrm{D}_{\text{x}}^{\text{p}}\right|^{2}+\left|\mathrm{D}_{\text{y}}^{\text{p}}\right|^{2}+\left|\mathrm{D}_{\text{z}}^{\text{p}}\right|^{2}\right)\,dA},$
(62)

$\left|C_{\text{s}}\right|^{2}=\frac{1}{\frac{1}{\epsilon_{0}}\iint_{A}\frac{1}{\epsilon_{\text{r}}}\left(\left|\mathrm{D}_{\text{x}}^{\text{s}}\right|^{2}+\left|\mathrm{D}_{\text{y}}^{\text{s}}\right|^{2}+\left|\mathrm{D}_{\text{z}}^{\text{s}}\right|^{2}\right)\,dA},$
(63)

where $\mathrm{u}$ is the elastic displacement field, $\epsilon_{\text{r}}$ is
the relative permittivity, and $\mathrm{D}^{\text{p,s}}$ is the electric
displacement field (pump or Stokes). Since the elastic strain is the symmetric
gradient of the displacement, the $\left|C_{\text{m}}\right|$ scaling factor
is used for the strain tensor in Eq. 59. To compute this coupling coefficient,
the simulation data is extracted and converted into matrices. The tensor
product within the integration is then computed through matrix multiplication
[104] and integrated.

A compacted Voigt notation is used to reduce the second and fourth-rank
tensors so that matrix multiplication can be used within the integration. The
compacted notation is 1 = (xx), 2 = (yy), 3 = (zz), 4 = (yz) = (zy), 5 = (xz)
= (zx), 6 = (xy) = (yx). This reduces the photoelastic coupling coefficient to

$g_{\text{pe}}=\left|C_{\text{m}}\right|\left|C_{\text{p}}\right|\left|C_{\text{s}}\right|\,\xi\iint_{A}\left[\mathrm{D}\right]\left[p\right]\left[\varepsilon\right]\,dA,$
(64)

where the electric displacement field product ($\left[\mathrm{D}\right]$) is a
[1x6] matrix, the photoelastic constants ($\left[p\right]$) are a [6x6]
matrix, and the strain ($\left[\varepsilon\right]$) is a [6x1] matrix. InGaAsP
has a cubic crystal structure with 3 independent photoelastic constants and
the coupling matrix takes the form

$\left[p\right]=\begin{bmatrix}p_{11}&p_{12}&p_{12}&0&0&0\\\
p_{12}&p_{11}&p_{12}&0&0&0\\\ p_{12}&p_{12}&p_{11}&0&0&0\\\
0&0&0&p_{44}&0&0\\\ 0&0&0&0&p_{44}&0\\\ 0&0&0&0&0&p_{44}\end{bmatrix}.$ (65)

The strain matrix is given as

$\left[\varepsilon\right]=\begin{bmatrix}\varepsilon_{\text{xx}}\\\
\varepsilon_{\text{yy}}\\\ \varepsilon_{\text{zz}}\\\
2\,\varepsilon_{\text{yz}}\\\ 2\,\varepsilon_{\text{xz}}\\\
2\,\varepsilon_{\text{xy}}\end{bmatrix},$ (66)

and the electric displacement field product matrix is given as

$\left[\mathrm{D}\right]=\begin{bmatrix}\left\\{\left(\mathrm{D}_{\text{x}}^{\text{p}}\right)^{*}\mathrm{D}_{\text{x}}^{\text{s}}\right\\}&\left\\{\left(\mathrm{D}_{\text{y}}^{\text{p}}\right)^{*}\mathrm{D}_{\text{y}}^{\text{s}}\right\\}&\left\\{\left(\mathrm{D}_{\text{z}}^{\text{p}}\right)^{*}\mathrm{D}_{\text{z}}^{\text{s}}\right\\}&2\left\\{\left(\mathrm{D}_{\text{y}}^{\text{p}}\right)^{*}\mathrm{D}_{\text{z}}^{\text{s}}\right\\}&2\left\\{\left(\mathrm{D}_{\text{x}}^{\text{p}}\right)^{*}\mathrm{D}_{\text{z}}^{\text{s}}\right\\}&2\left\\{\left(\mathrm{D}_{\text{x}}^{\text{p}}\right)^{*}\mathrm{D}_{\text{y}}^{\text{s}}\right\\}\end{bmatrix}.$
(67)

To include the contribution of lithium niobate in the photoelastic coupling
coefficient, the treatment above is modified to use the photoelastic coupling
matrix of a trigonal crystal structure when integrating over regions of
lithium niobate

$\left[p\right]=\begin{bmatrix}p_{11}&p_{12}&p_{13}&p_{14}&0&0\\\
p_{12}&p_{11}&p_{13}&-p_{14}&0&0\\\ p_{31}&p_{31}&p_{33}&0&0&0\\\
p_{41}&-p_{41}&0&p_{44}&0&0\\\ 0&0&0&0&p_{44}&p_{41}\\\
0&0&0&0&p_{14}&\frac{1}{2}\left(p_{11}-p_{12}\right)\end{bmatrix}.$ (68)

Here, there are eight independent photoelastic constants ($p_{11}$, $p_{33}$,
$p_{44}$, $p_{12}$, $p_{13}$, $p_{31}$, $p_{14}$, $p_{41}$) and these values
can be found in Ref. [105, 106]. Since the optical modes are guided within the
InGaAsP, there is little overlap between the elastic mode and optical fields
in the lithium niobate, which means the contribution to the overall
photoelastic coupling is very small.

####  A.4.2 Radiation Pressure Coupling

The radiation pressure coupling coefficient (in units of
(rad/s)$\sqrt{\text{m}}$) is given as [25]

$g_{\text{rp}}=\left|C_{\text{m}}\right|\left|C_{\text{p}}\right|\left|C_{\text{s}}\right|\,\xi\iint_{A}\epsilon_{0}\left[\left(\mathbf{E}_{\text{p}}^{\parallel*}\cdot\mathbf{E}_{\text{s}}^{\parallel}\right)\nabla\epsilon-\left(\mathbf{D}_{\text{p}}^{\perp*}\cdot\mathbf{D}_{\text{s}}^{\perp}\right)\nabla\epsilon^{-1}\right]\cdot\mathbf{u}\,\,dA,$
(69)

where $\mathbf{E}_{\text{p,s}}^{\parallel}$ is the component of the electric
field parallel to the defined normal and $\mathbf{D}_{\text{p,s}}^{\perp}$ is
the component of the electric displacement field perpendicular to the defined
normal. Since the physical system presented in this paper only has discrete
permittivities, there exists permittivity discontinuities at the material
interfaces. Since the gradient of the permittivity is zero within the
materials, this integral collapses to a line integral around the boundaries of
the waveguide. At the waveguide boundaries, the permittivity is a step
function between the permittivity of the waveguide ($\epsilon_{1}$) and
surrounding material ($\epsilon_{2}$). Since the gradient of a unit step
function is a delta function, the radiation pressure coupling coefficient can
be written as [107, 108]

$g_{\text{rp}}=\left|C_{\text{m}}\right|\left|C_{\text{p}}\right|\left|C_{\text{s}}\right|\,\xi\int_{L}\epsilon_{0}\left[\left(\mathbf{E}_{\text{p}}^{\parallel*}\cdot\mathbf{E}_{\text{s}}^{\parallel}\right)\Delta\epsilon-\left(\mathbf{D}_{\text{p}}^{\perp*}\cdot\mathbf{D}_{\text{s}}^{\perp}\right)\Delta\epsilon^{-1}\right]\hat{n}\cdot\mathbf{u}\,\,dL,$
(70)

where $\Delta\epsilon=\epsilon_{1}-\epsilon_{2}$,
$\Delta\epsilon^{-1}=\epsilon_{1}^{-1}-\epsilon_{2}^{-1}$, and $\hat{n}$ is
the normal of the line integral path.

Figure 12: Calculated optomechanical coupling coefficient as a function of the
simulated elastic modes for various waveguide geometries. The optomechanical
coupling for key elastic modes is plotted for (a) backward intra-modal and
(b-e) forward inter-modal Brillouin scattering. For each elastic mode, the
black arrow indicates the trend in optomechanical coupling as the waveguide
width is increased.

The optomechanical coupling for key simulated elastic modes is plotted in Fig.
12 at a fixed waveguide thickness as the waveguide width is swept from 0.4
$\mu$m to 1.75 $\mu$m (black arrows). For the case of backward intra-modal
Brillouin scattering, the optomechanical coupling trends for three elastic
waveguide modes are shown in Fig. 12a. Both waveguide mode 1 and 2 have an
optimal waveguide width which maximizes the optomechanical coupling
coefficient. This is a result of two competing trends found in this physical
system. Since all the elastic modes examined in this supplement have a slower
phase velocity in InGaAsP compared to lithium niobate, a larger fraction of
the strain is guided within the InGaAsP as the waveguide width is increased,
increasing the overlap with the optical fields. On the other hand, increasing
the waveguide width reduces the magnitude of the optical fields at the
boundaries of the InGaAsP waveguide, reducing the overlap with the strain.
When directly comparing the first two waveguide modes, waveguide mode 2 will
have a larger optomechanical coupling coefficient compared to waveguide mode
1, which is a result of the different strain profiles between the two elastic
modes. Waveguide mode 1 has a majority of its strain around the edges of the
InGaAsP waveguide, which has a lower overlap with the optical fields. For
waveguide mode 3, there is only a reduction in optomechanical coupling as the
waveguide width is increased. Since waveguide mode 3 has several displacement
nodes along the waveguide width, there exists a minimum waveguide width where
this mode is allowed. Overall, waveguide mode 2 has the highest optomechanical
coupling, which is why this simulated elastic mode is used throughout the main
manuscript.

For the cases of forward inter-modal Brillouin scattering shown in Fig. 12b-e,
the trends in optomechanical coupling as a function of waveguide width are
influenced by additional factors. For each simulated optical mode, a different
set of waveguide cutoff dimensions exists, creating different cutoff waveguide
widths for each optical mode pair. In addition, the phase-matched phonon
wavelength varies as a function of waveguide width, which will change the
confinement of the elastic mode and overlap with the optical fields. For the
optical mode pairs shown in Fig. 12b,c,e, the dominant factor in the
optomechanical coupling trends for the Rayleigh and SH SAW modes is the change
in phase-matched phonon wavelength. As the waveguide width is increased, the
phase-matched phonon wavelength also increases, reducing the confinement and
overlap with the optical fields. In the case of the TE0 & TM0 optical mode
pair (Fig. 12d), the phase-matched phonon wavelength does not change
significantly with waveguide width and the trend in optomechanical coupling is
similar to those found in backward intra-modal Brillouin scattering.

###  A.5 Phonon Dissipation

This subsection of the supplement outlines the estimated phonon dissipation of
the elastic mode used in the main manuscript. Given the simulated optimal
waveguide geometry for the elastic waveguide mode 2 ($w$ = 1.1 $\mu$m and $t$
= 0.1 $\mu$m), the frequency of the simulated elastic mode is
$\Omega_{\text{m}}$/$2\pi$ = 8.78 GHz and the phase velocity is $v_{\text{m}}$
= 3072 m/s. Since the simulated elastic waveguide mode 2 in the InGaAsP-LN-Si
physical system has similar waveguide dimensions and frequency to elastic
modes reported in silicon with mechanical quality factors ($Q_{\text{m}}$)
ranging from 250-1800 [109, 52], a conservative approximation for the quality
factor in our physical system is $Q_{\text{m}}$ = 250. The phonon dissipation
rate, $\Gamma_{\text{m}}$, can be expressed as [109]

$\displaystyle\Gamma_{\text{m}}=\frac{\Omega_{\text{m}}}{Q_{\text{m}}}.$ (71)

For the simulated elastic mode used in this paper, the phonon dissipation rate
is estimated to be $\Gamma_{\text{m}}$/$2\pi$ = 35.1 MHz. From Chapter 8 in
Ref. [51], the loss coefficient of the elastic mode is given as

$\displaystyle\alpha_{\text{m}}=\frac{\Gamma_{\text{m}}}{v_{\text{m}}}.$ (72)

For the simulated elastic mode used in this paper, the loss coefficient is
estimated to be $\alpha_{\text{m}}$ = 71800 m-1. This loss coefficient can be
converted into units of [dB/cm] through the following conversion

$\displaystyle\alpha_{\text{m}}\left[\frac{dB}{cm}\right]=\frac{1\,m}{100\,cm}\left(\frac{10}{\text{ln}\left(10\right)}\right)\alpha_{\text{m}}\left[\frac{1}{m}\right].$
(73)

This results in an estimated loss coefficient of $\alpha_{\text{m}}$ = 3100
dB/cm.

###  A.6 Acoustoelectric Gain Calculation

The two predominant acoustoelectric theories for an amplifier consisting of a
semiconductor thin film on a piezoelectric substrate are the normal mode
theory [49] and simple theory [110]. Later, the simple theory for AE gain was
expanded to include the effects of loss in the elastic mode [40]. Each of
these theories come with its own set of limitations and assumptions. This
subsection of the supplement will outline each theory, and compare the results
for backward intra-modal Brillouin scattering using the simulated elastic mode
in the main manuscript and an example elastic mode for forward inter-modal
Brillouin scattering.

The first AE gain theory examined is based on the normal mode theory as
described by Kino and Reeder [49], modified to include a dielectric gap
material between the amplifier layer and the piezoelectric substrate. In the
presence of a semiconductor with a finite film thickness, the elastic wave’s
propagation constant can be expressed as

$\displaystyle q=q_{\text{m}}+q_{\text{AE}}+i\frac{\alpha_{\text{AE}}}{2},$
(74)

where $q_{\text{m}}$ is the elastic wave’s unperturbed propagation constant,
$\alpha_{\text{AE}}$ is the acoustoelectric gain, and $q_{\text{AE}}$ is the
acoustoelectric phase delay. The acoustoelectric contributions to the gain and
phase delay are given as

$\displaystyle\alpha_{\text{AE}}=\frac{\left(v_{\text{d}}/v_{\text{m}}-1\right)\omega_{\text{c}}\epsilon_{\text{s}}Z^{\prime}_{\text{m}}\left(q_{\text{m}}h\right)q_{\text{m}}\text{tanh}\left(q_{\text{m}}t\right)}{\left(v_{\text{d}}/v_{\text{m}}-1\right)^{2}+\left(R\omega_{\text{c}}/\Omega_{\text{m}}+D\right)^{2}},$
(75)

$\displaystyle
q_{\text{AE}}=\frac{1}{2}\frac{\left(R\omega_{\text{c}}/\Omega_{\text{m}}+D\right)\omega_{\text{c}}\epsilon_{\text{s}}Z^{\prime}_{\text{m}}\left(q_{\text{m}}h\right)q_{\text{m}}\text{tanh}\left(q_{\text{m}}t\right)}{\left(v_{\text{d}}/v_{\text{m}}-1\right)^{2}+\left(R\omega_{\text{c}}/\Omega_{\text{m}}+D\right)^{2}}.$
(76)

Here, $\Omega_{\text{m}}$ is the phonon frequency, $h$ is the gap height
between the lithium niobate and InGaAsP waveguide, $t$ is the InGaAsP
waveguide thickness, $\epsilon_{\text{s}}$ is the InGaAsP permittivity,
$v_{\text{m}}$ is the elastic wave phase velocity, and $v_{\text{d}}$ is the
free carrier drift velocity. The dielectric relaxation frequency
($\omega_{\text{c}}$) and diffusion frequency ($\omega_{\text{D}}$) can be
expressed as

$\displaystyle\omega_{\text{c}}=\frac{e\mu N}{\epsilon_{\text{s}}},$ (77)

$\displaystyle\omega_{\text{D}}=\frac{v_{\text{m}}^{2}e}{\mu\,k_{\text{B}}T},$
(78)

where $e$ is the charge of an electron, $\mu$ is the free carrier mobility,
$N$ is the free carrier concentration, $k_{\text{B}}$ is the Boltzmann
constant, and $T$ is the temperature. The space charge potential factor ($M$)
above the piezoelectric surface is given as

$\displaystyle
M\left(q_{\text{m}}h\right)=\frac{\epsilon_{\text{g}}+\epsilon_{\text{p}}\text{tanh}\left(q_{\text{m}}h\right)}{\left(\epsilon_{\text{g}}+\epsilon_{p}\right)\left(1+\text{tanh}\left(q_{\text{m}}h\right)\right)},$
(79)

where $\epsilon_{p}$ and $\epsilon_{g}$ are the permittivities of the
piezoelectric (lithium niobate) and gap dielectric (InP), respectively. The
interaction impedance ($Z_{\text{m}}$) of the elastic wave above the
piezoelectric surface is a measure of the overlap between the electric
potential and free carriers and is given as

$\displaystyle
Z_{\text{m}}\left(q_{\text{m}}h\right)=Z_{\text{m}}\left(0\right)\mathrm{e}^{-2q_{\text{m}}h}.$
(80)

The strength of the interaction impedance decays exponentially as the height
of the dielectric gap is increased, with a maximum occurring at the
piezoelectric surface

$\displaystyle
Z_{\text{m}}\left(0\right)=\frac{k^{2}}{\Omega_{\text{m}}\left(\epsilon_{\text{g}}+\epsilon_{\text{p}}\right)},$
(81)

where $k^{2}$ is the piezoelectric coupling coefficient of the elastic mode.
The introduction of the dielectric above the semiconductor thin film will
perturb the fields as a result of the top dielectric permittivity
($\epsilon_{d}$). The perturbed values for the space charge potential factor
and interaction impedance are given as

$\displaystyle
M^{\prime}\left(q_{\text{m}}h\right)=\frac{M\left(q_{\text{m}}h\right)}{1+\left(\epsilon_{d}/\epsilon_{g}-1\right)M\left(q_{\text{m}}h\right)},$
(82)

$\displaystyle
Z_{\text{m}}^{\prime}\left(q_{\text{m}}h\right)=\frac{Z_{\text{m}}\left(q_{\text{m}}h\right)}{\left[1+\left(\epsilon_{d}/\epsilon_{g}-1\right)M\left(q_{\text{m}}h\right)\right]^{2}}.$
(83)

Considering semiconductor films of a finite thickness, the space-charge
reduction factor can be expressed as

$\displaystyle
R=\left(\epsilon_{s}/\epsilon_{g}\right)M^{\prime}\left(q_{\text{m}}h\right)\text{tanh}\left(q_{\text{m}}t\right).$
(84)

The diffusion term, which comes from carrier diffusion in the semiconductor,
can be expressed as

$\displaystyle
D=\sqrt{\frac{\omega_{\text{c}}}{\omega_{\text{D}}}}\,\frac{\text{tanh}\left(q_{\text{m}}t\right)}{\text{tanh}\left(\gamma_{\text{d}}t\right)},$
(85)

where $\gamma_{\text{d}}$ is related to the Debye length
($\lambda_{\text{d}}$) in the semiconductor

$\displaystyle\gamma_{\text{d}}\cong\frac{\sqrt{\omega_{\text{c}}\omega_{\text{D}}}}{v_{\text{m}}}=\frac{1}{\lambda_{\text{d}}}.$
(86)

Finally, the conversion of this AE gain coefficient to units of (dB/cm) is
given as

$\displaystyle\alpha_{\text{AE}}\left[\frac{dB}{cm}\right]=\frac{1\,m}{100\,cm}\left(\frac{10}{\text{ln}\left(10\right)}\right)\alpha_{\text{AE}}\left[\frac{1}{m}\right].$
(87)

The expression for the AE gain from the simple theory is given as [110]

$\displaystyle\alpha_{\text{AE}}=q_{\text{m}}\left[k^{2}\frac{\gamma_{v}\Omega_{\text{m}}\tau}{1+\left(\gamma_{v}\Omega_{\text{m}}\tau\right)^{2}}\right],$
(88)

where the time constant ($\tau$) and non-dimensionalized velocity
($\gamma_{v}$) are given as

$\displaystyle\tau=\frac{\epsilon_{\text{p}}+\epsilon_{\text{d}}}{e\mu
Nq_{\text{m}}t},$ (89)

$\displaystyle\gamma_{v}=1-\frac{v_{\text{d}}}{v_{\text{m}}}.$ (90)

The simple theory has a couple of limitations when compared to the normal mode
theory. First, the simple theory does not include any carrier diffusion
effects. Second, there is no gap height defined in the simple theory, which
means the semiconductor thin film is assumed to be in direct contact with the
piezoelectric substrate. The normal mode theory can be made to match the
assumptions found in the simple theory by setting the temperature to zero ($T$
= 0) for the diffusion frequency in Eq. 78 and by setting the gap height to
zero ($h$ = 0) in all of the expressions.

The inclusion of loss from the elastic mode into the expression for AE gain is
done by representing the free ($v_{\text{f}}=v_{\text{fr}}+i\,v_{\text{fi}}$)
and metalized ($v_{\text{l}}=v_{\text{lr}}+i\,v_{\text{li}}$) phase velocities
as complex values. The real component of the free velocity is simply the phase
velocity of the given elastic mode ($v_{\text{fr}}=v_{\text{m}}$). The real
component of the metalized velocity can be determined from the definition of
the piezoelectric coupling coefficient. The imaginary part of the free and
metalized velocities can be computed from the complex propagation constant of
the elastic mode ($q_{\text{f,l}}=q_{\text{m}}-i\,\alpha_{\text{f,l}}/2$) and
is given as

$\displaystyle
v_{\text{fi,li}}=\frac{\Omega_{\text{m}}\alpha_{\text{f,l}}}{2\,q_{\text{m}}^{2}}.$
(91)

The full expression for the AE gain from the simple theory with the inclusion
of losses from the elastic mode is given as [40]

$\displaystyle\alpha_{\text{AE}}=q_{\text{m}}\left[k^{2}\frac{\gamma_{v}\Omega_{\text{m}}\tau}{1+\left(\gamma_{v}\Omega_{\text{m}}\tau\right)^{2}}+\frac{2\,v_{\text{li}}}{v_{\text{fr}}}\frac{\left(\gamma_{v}\Omega_{\text{m}}\tau\right)^{2}}{1+\left(\gamma_{v}\Omega_{\text{m}}\tau\right)^{2}}+\frac{2\,v_{\text{fi}}}{v_{\text{fr}}}\frac{1}{1+\left(\gamma_{v}\Omega_{\text{m}}\tau\right)^{2}}\right].$
(92)

This AE gain can again be converted into units of (dB/cm) using Eq. 87.

Figure 13: Calculated AE gain curves as a function of applied drift field. The
normal mode theory and simple theory are plotted for (a) elastic waveguide
mode 2 (backward intra-modal)—elastic mode used in the main manuscript—and (b)
SH SAW mode (forward inter-modal).

A comparison of these AE theories for an example backward intra-modal and
forward inter-modal Brillouin scattering process is shown in Fig. 13. The four
different AE theories include: the full treatment of the normal mode theory
with no assumptions, the simple theory, the normal mode theory with the same
assumptions as the simple theory, and the simple theory including the losses
from the elastic mode. The elastic mode from the main manuscript (waveguide
mode 2) is chosen as the example for backward intra-modal Brillouin scattering
(Fig. 13a). The full set of parameters for the AE calculations are
$\Omega_{\text{m}}$/2$\pi$ = 8.78 GHz, $\lambda_{\text{m}}$ = 0.35 $\mu$m,
$v_{\text{m}}$ = 3072 m/s, $\alpha_{\text{f,l}}$ = 71800 m-1, $k^{2}$ = 5.87
%, $t$ = 0.1 $\mu$m, $h$ = 5 nm, $\epsilon_{\text{p}}$ = 29.2$\epsilon_{0}$,
$\epsilon_{\text{s}}$ = 13.3$\epsilon_{0}$, $\epsilon_{\text{g}}$ =
12.5$\epsilon_{0}$, $\epsilon_{\text{d}}$ = $\epsilon_{0}$, $T$ = 293.15 K,
$\mu$ = 2000 cm2/Vs, $N$ = 1 x 1016 cm-3, and $Q_{\text{m}}$ = 250. For
waveguide mode 2, all four AE theories show significant differences. The large
difference between the full treatment of the normal mode theory and simple
theory can be attributed to significant carrier diffusion effects when
operating at this small phonon wavelength. The two conditions for minimal
diffusion effects are
$\Omega_{\text{m}}^{2}<<\omega_{\text{c}}\omega_{\text{D}}$ and
$R\omega_{\text{c}}/\Omega_{\text{m}}>>D$. Both conditions relate the phonon
frequency to constants that are a function of the semiconductor properties and
elastic mode’s phase velocity. Assuming the elastic mode has minimal
dispersion, then the amount of carrier diffusion is proportional to the phonon
frequency. Most reported AE amplifiers operate at phonon frequencies in the
range of MHz up to 1 GHz and the carrier diffusion is minimal in this
frequency range. Backward intra-modal Brillouin scattering processes, however,
have phase-matched elastic modes at frequencies at least an order of magnitude
larger, which no longer satisfy the two conditions for negligible diffusion
effects. Additionally, the phase-matched phonon wavelength is now the same
order of magnitude as the semiconductor film thickness. As a result, the
phonon wave vector-thickness product ($q_{\text{m}}t$) is too large for the
small angle approximation in Eq. 75. The full evaluation of the hyperbolic
tangent in the normal mode theory (with assumptions) accounts for the
discrepancy with the simple theory. Both theories have similar magnitudes in
AE gain, but the difference in shape with respect to the applied drift field
is a result of the invalid small angle approximation for this $q_{\text{m}}t$
product. Finally, the addition of losses from the elastic mode to the simple
theory shifts the AE gain curve, but does not significantly change the shape
of the curve. With the conservative estimate of 250 for the mechanical quality
factor, the additional loss terms added to Eq. 92 have a relative magnitude of
6.8 %. Since the carrier diffusion effects are significant for backward intra-
modal Brillouin scattering in the InGaAsP-LN-Si physical system, the full
treatment of the normal mode theory was chosen as the AE theory to use in the
main manuscript and supplement to illustrate the different regimes of AE
Brillouin dynamics. If the elastic mode of an acoustoelectrically enhanced
Brillouin device exhibits these losses in experiment, the full treatment of
the normal mode theory would need to be expanded to include the effects of
losses in the elastic mode for the most accurate AE theoretical model.
Additionally, the acoustoelectric coupling defined using the Hamiltonian
formulation for the acoustoelectric dynamics (see Supplementary Section C) is
most similar to the full treatment of the normal mode theory minus the
diffusion effects. Both the normal mode theory and Hamiltonian formulation
take into consideration the cross-sectional overlap between the elastic wave
and free carriers including the gap dielectric between the piezoelectric
surface and semiconductor thin film.

For forward inter-modal Brillouin scattering, the phase-matched phonon
wavelength is similar to those found in traditional AE delayline amplifiers
using the Rayleigh and SH SAW modes. As an example, the AE gain calculations
for the best simulated SH SAW mode are plotted in Fig. 13b. The full set of
parameters for the AE calculations are $\Omega_{\text{m}}$/2$\pi$ = 1.24 GHz,
$\lambda_{\text{m}}$ = 3.45 $\mu$m, $v_{\text{m}}$ = 4289 m/s,
$\alpha_{\text{f,l}}$ = 1800 m-1, $k^{2}$ = 8.84 %, $t$ = 0.2 $\mu$m, $h$ = 5
nm, $\epsilon_{\text{p}}$ = 43.6$\epsilon_{0}$, $\epsilon_{\text{s}}$ =
13.3$\epsilon_{0}$, $\epsilon_{\text{g}}$ = 12.5$\epsilon_{0}$,
$\epsilon_{\text{d}}$ = $\epsilon_{0}$, $T$ = 293.15 K, $\mu$ = 2000 cm2/Vs,
$N$ = 1 x 1016 cm-3, and $Q_{\text{m}}$ = 1000. With a lower phonon frequency
and longer phonon wavelength, the two conditions for minimal carrier diffusion
effects in the normal mode theory are valid. The only significant difference
between the full treatment of the normal mode theory and the simple theory is
the introduction of the 5 nm InP gap dielectric between the InGaAsP and
lithium niobate. The longer phonon wavelength also makes the small angle
approximation for the $q_{\text{m}}t$ product valid. In this case, the normal
mode theory with the additional assumptions is in good agreement with the
simple theory. With a estimated higher mechanical quality factor at these
lower phonon frequencies, the additional loss terms added to the simple theory
only have a relative magnitude of 1.1 %. This amount of loss has a minimal
effect on the AE gain curve, which means all four AE theories have good
agreement with each other.

###  A.7 Acoustoelectric Noise Figure Calculation

We next estimate the levels of intrinsic acoustic noise relative to thermal
fluctuations in the acoustoelectric gain medium for our analysis of the
acoustoelectric enhanced optical Brillouin noise figure in Supplementary
Section D. To do this, we analyze the predicted noise figure in the absence of
acoustic losses. For the InGaAsP-LN-Si physical system, the internal noise
figure resulting from the noise source in the semiconductor coupling to the
elastic wave is given as [111]

$\displaystyle
F_{\text{n}}=1+\frac{t\,e^{2}D_{\text{c}}\,N\left|q_{\text{d}}\right|^{2}Z_{\text{m}}\left(q_{\text{m}}h\right)\mathrm{e}^{-2q_{\text{m}}h}}{k_{\text{B}}T\alpha_{\text{AE}}}\left(\frac{\mathrm{e}^{\alpha_{\text{AE}}l}-1}{\mathrm{e}^{\alpha_{\text{AE}}l}}\right).$
(93)

Here, $D_{\text{c}}$ is a diffusion coefficient, $q_{\text{d}}$ is the
perturbed carrier wave propagation constant, and the interaction impedance,
$Z_{\text{m}}$, is the same as in Eq. 80-81. The diffusion coefficient can be
expressed as $D_{\text{c}}=D_{\text{TH}}+D_{\text{TR}}$, where the thermal
diffusion constant ($D_{\text{TH}}$) and trapping effects ($D_{\text{TR}}$)
are given as

Figure 14: Calculated internal noise figure as a function of acoustoelectric
gain for a 1 cm long device, in the absence of acoustic loss. The
acoustoelectric gain values for the elastic waveguide mode 2 (backward intra-
modal)—elastic mode used in the main manuscript—and the SH SAW mode (forward
inter-modal) are taken from Fig. 13 using the full treatment of the normal
mode theory. The range of plotted acoustoelectric gain values corresponds to
the enhanced Brillouin regime illustrated in Fig. 3.

$\displaystyle D_{\text{TH}}=\frac{\mu\,k_{\text{B}}T}{e},$ (94)

$\displaystyle
D_{\text{TR}}=\frac{f_{\text{TR}}\left(1-f_{\text{TR}}\right)v_{\text{d}}^{2}\,\tau_{\text{TR}}}{1+\left(\Omega_{\text{m}}\tau_{\text{TR}}\right)}.$
(95)

The factor $f_{\text{TR}}$ represents the fraction of untrapped carriers and
$\tau_{\text{TR}}$ is the carrier trapping relaxation time, which can be
estimated as $\tau_{\text{TR}}$ = 1/$\Omega_{\text{m}}$. For the case where
$\omega_{\text{c}}d/v_{\text{d}}>>1$ and $q_{\text{d}}h<<1$, the carrier wave
propagation constant can be approximated as [111]

Table 2: Table summarizing the mode triplets for the acoustoelectrically enhanced backward intra-modal Brillouin dynamics in the InGaAsP-LN-Si physical system. The AE gain value is calculated using the normal mode theory and evaluated at a drift field of 2 kV/cm. For each elastic mode, the rows with the best simulated results are highlighted in gray. Optical Pair |  Acoustic Mode |  $\beta$ |  $t$ ($\mu$m) |  $w$ ($\mu$m) |  $\lambda_{\text{m}}$ ($\mu$m) |  $\Omega_{\text{m}}/2\pi$ (GHz) |  $v_{\text{m}}$ (m/s) |  $\left|g_{\text{pe}}\right|$ ((rad/s)$\sqrt{\text{m}}$) |  $\left|g_{\text{rp}}\right|$ ((rad/s)$\sqrt{\text{m}}$) |  $\left|g_{0}\right|$ ((rad/s)$\sqrt{\text{m}}$) |  $k^{2}$ (%) |  $\alpha_{\text{AE}}$ (dB/cm)
---|---|---|---|---|---|---|---|---|---|---|---|---
TE0 / TE0 |  WG 1 |  0° |  0.1 |  0.9 |  0.356 |  7.597 |  2704 |  2228 |  -358 |  1843 |  0.65 |  413.7
TE0 / TE0 |  WG 1 |  0° |  0.15 |  0.75 |  0.328 |  7.346 |  2410 |  2352 |  -707 |  1626 |  0.23 |  150.0
TE0 / TE0 |  WG 1 |  0° |  0.2 |  0.75 |  0.302 |  7.558 |  2283 |  2545 |  -643 |  1876 |  0.04 |  27.2
TE0 / TE0 |  WG 1 |  0° |  0.3 |  0.65 |  0.281 |  7.915 |  2224 |  2927 |  -611 |  2304 |  0.02 |  14.1
TE0 / TE0 |  WG 1 |  90° |  0.1 |  0.95 |  0.347 |  7.896 |  2740 |  1586 |  -236 |  1350 |  4.22 |  2093.7
TE0 / TE0 |  WG 1 |  90° |  0.15 |  0.75 |  0.324 |  7.463 |  2418 |  2122 |  -626 |  1496 |  1.29 |  649.5
TE0 / TE0 |  WG 1 |  90° |  0.2 |  0.75 |  0.301 |  7.604 |  2289 |  2362 |  -601 |  1761 |  0.69 |  357.1
TE0 / TE0 |  WG 1 |  90° |  0.3 |  0.65 |  0.280 |  7.947 |  2225 |  2835 |  -592 |  2244 |  0.19 |  100.9
TE0 / TE0 |  WG 2 |  0° |  0.1 |  1.1 |  0.350 |  8.777 |  3072 |  7294 |  848 |  7943 |  5.87 |  3725.9
TE0 / TE0 |  WG 2 |  0° |  0.15 |  0.75 |  0.328 |  8.401 |  2755 |  7861 |  1461 |  9281 |  1.63 |  1054.2
TE0 / TE0 |  WG 2 |  0° |  0.2 |  0.7 |  0.305 |  8.579 |  2617 |  5783 |  1158 |  6906 |  0.21 |  141.0
TE0 / TE0 |  WG 2 |  0° |  0.3 |  0.6 |  0.285 |  9.079 |  2588 |  3200 |  307 |  3502 |  0.01 |  6.9
TE0 / TE0 |  WG 2 |  90° |  0.1 |  1.15 |  0.343 |  9.067 |  3110 |  6605 |  967 |  7572 |  0.62 |  305.7
TE0 / TE0 |  WG 2 |  90° |  0.15 |  0.8 |  0.321 |  8.545 |  2743 |  7518 |  1540 |  9059 |  0.03 |  15.0
TE0 / TE0 |  WG 2 |  90° |  0.2 |  0.75 |  0.301 |  8.672 |  2610 |  5504 |  1173 |  6676 |  0.01 |  5.1
TE0 / TE0 |  WG 2 |  90° |  0.3 |  0.6 |  0.285 |  9.082 |  2588 |  3043 |  383 |  3426 |  0.06 |  31.4
TE0 / TE0 |  WG 3 |  0° |  0.15 |  0.8 |  0.325 |  10.453 |  3397 |  2186 |  161 |  2345 |  0.82 |  524.2
TE0 / TE0 |  WG 3 |  0° |  0.2 |  0.75 |  0.302 |  10.870 |  3283 |  2345 |  182 |  2527 |  0.66 |  437.8
TE0 / TE0 |  WG 3 |  90° |  0.15 |  0.8 |  0.321 |  10.611 |  3406 |  1978 |  113 |  2091 |  0.41 |  202.1
TE0 / TE0 |  WG 3 |  90° |  0.2 |  0.7 |  0.304 |  11.205 |  3406 |  2086 |  250 |  2332 |  0.00 |  0.0

$\displaystyle
q_{\text{d}}\approx-i\,\frac{\Omega_{\text{m}}\left(\epsilon_{\text{g}}+\epsilon_{\text{p}}\right)}{e\mu
Nt}.$ (96)

Finally, the internal noise figure can be expressed in decibels through the
conversion
$F_{\text{n}}\left[\mathrm{dB}\right]=10\cdot\mathrm{log}_{10}\left(F_{\text{n}}\right)$.

Using the previous two example elastic modes (Fig. 13), the internal noise
figure is plotted in Fig. 14 as a function of the acoustoelectric gain using
the full treatment of the normal mode theory for a 1 cm long device in the
absence of acoustic loss. For both elastic modes, the fraction of untrapped
carriers is estimated to be $f_{\text{TR}}=0.95$. For amorphous InSb, the
fraction of untrapped carriers is approximately 0.8, so for our nearly single
crystalline epitaxial films we expect $f_{\text{TR}}$ to be substantially
closer to 1, with 0.95 being a conservative estimate. This range of
acoustoelectric gain corresponds to operating in the enhanced Brillouin regime
(shown in Fig. 3). The noise figure is initially large when entering the
enhanced Brillouin regime from the lossy Brillouin regime
($\alpha_{\text{AE}}>0$). As the acoustoelectric gain is increased, the noise
figure decreases to a minimum value before increasing again as the diffusion
coefficient begins to dominate at high carrier drift velocities. In the case
of the elastic waveguide mode 2, the elastic mode used in the main manuscript,
the noise figure does not reach its minimum value in the enhanced Brillouin
regime. This means that the internal noise figure will continue to decrease
for larger acoustoelectric gains up until the coherent Brillouin limit is
reached.

###  A.8 Mode Triplet Simulation Results

For both backward intra-modal and forward inter-modal Brillouin scattering, an
iterative design process is implemented using all the previously described
metrics in this section of the supplement. First, the waveguide geometry is
set and simulated to determine the optical modes, which dictates the phase-
matched phonon wavelength. The elastic modes are then simulated with
electrically free and grounded boundary conditions at the InGaAsP and lithium
niobate interface. The piezoelectric coupling coefficient is then calculated
using the difference in eigenfrequencies for each elastic mode. Finally, the
optomechanical coupling coefficient is computed using the optical and elastic
field profiles and the piezoelectric coupling coefficient is used to compute
the AE gain using the normal mode theory. This design process is repeated for
different waveguide geometries until a set of modes are identified with both
high AE gain and optomechanical coupling. Tables 2 and 3 show the simulation
results with the rows containing the best performance for each elastic mode
highlighted in gray.

Table 3: Table summarizing the mode triplets for the acoustoelectrically enhanced forward inter-modal Brillouin dynamics in the InGaAsP-LN-Si physical system. The AE gain value is calculated using the normal mode theory and evaluated at a drift field of 2 kV/cm. For each elastic mode, the rows with the best simulated results are highlighted in gray. Optical Pair |  Acoustic Mode |  $\beta$ |  $t$ ($\mu$m) |  $w$ ($\mu$m) |  $\lambda_{\text{m}}$ ($\mu$m) |  $\Omega_{\text{m}}/2\pi$ (GHz) |  $v_{\text{m}}$ (m/s) |  $\left|g_{\text{pe}}\right|$ ((rad/s)$\sqrt{\text{m}}$) |  $\left|g_{\text{rp}}\right|$ ((rad/s)$\sqrt{\text{m}}$) |  $\left|g_{0}\right|$ ((rad/s)$\sqrt{\text{m}}$) |  $k^{2}$ (%) |  $\alpha_{\text{AE}}$ (dB/cm)
---|---|---|---|---|---|---|---|---|---|---|---|---
TE0 / TE2 |  Rayleigh |  0° |  0.15 |  1.55 |  4.066 |  0.865 |  3516 |  15.7 |  15.9 |  31.5 |  2.21 |  299.7
TE0 / TE2 |  Rayleigh |  0° |  0.2 |  1.3 |  2.902 |  1.195 |  3469 |  21.3 |  12.4 |  33.6 |  3.08 |  415.0
TE0 / TE2 |  Rayleigh |  0° |  0.3 |  1.1 |  2.102 |  1.619 |  3402 |  17.5 |  81.1 |  97.7 |  4.91 |  630.3
TE0 / TE2 |  Rayleigh |  90° |  0.15 |  1.65 |  4.696 |  0.802 |  3766 |  13.5 |  13.8 |  27.3 |  0.16 |  22.4
TE0 / TE2 |  Rayleigh |  90° |  0.2 |  1.35 |  3.192 |  1.166 |  3723 |  27.8 |  17.4 |  45.3 |  0.22 |  30.9
TE0 / TE2 |  Rayleigh |  90° |  0.3 |  1.15 |  2.332 |  1.570 |  3661 |  23.1 |  63.3 |  86.4 |  2.05 |  259.3
TE0 / TM1 |  Rayleigh |  0° |  0.3 |  0.8 |  2.303 |  1.495 |  3444 |  -19.2 |  133.3 |  114.3 |  3.67 |  434.4
TE0 / TM1 |  Rayleigh |  90° |  0.3 |  0.8 |  2.412 |  1.536 |  3706 |  -15.7 |  129.3 |  113.6 |  1.40 |  172.6
TE0 / TE1 |  SH SAW |  0° |  0.15 |  1.05 |  5.180 |  0.702 |  3636 |  42.4 |  49.4 |  91.6 |  0.00 |  0.0
TE0 / TE1 |  SH SAW |  0° |  0.2 |  0.85 |  3.170 |  1.129 |  3577 |  49.9 |  75.6 |  123.7 |  0.01 |  1.3
TE0 / TE1 |  SH/Ray |  0° |  0.3 |  0.7 |  2.314 |  1.418 |  3281 |  374.6 |  1049.3 |  1419.8 |  1.43 |  169.3
TE0 / TE1 |  SH SAW |  90° |  0.15 |  1.1 |  5.848 |  0.773 |  4519 |  36.2 |  44.7 |  80.9 |  2.91 |  342.6
TE0 / TE1 |  SH SAW |  90° |  0.2 |  1.0 |  4.759 |  0.935 |  4449 |  57.7 |  68.9 |  126.6 |  2.81 |  298.0
TE0 / TE1 |  SH/Ray |  90° |  0.3 |  0.75 |  2.689 |  1.373 |  3691 |  113.2 |  396.2 |  509.4 |  1.31 |  149.5
TE0 / TM0 |  SH SAW |  0° |  0.2 |  1.75 |  3.123 |  1.123 |  3507 |  34.8 |  20.7 |  58.9 |  0.12 |  15.3
TE0 / TM0 |  SH SAW |  0° |  0.3 |  1.5 |  4.673 |  0.772 |  3607 |  16.3 |  4.5 |  22.5 |  0.02 |  1.3
TE0 / TM0 |  SH SAW |  90° |  0.2 |  1.75 |  3.453 |  1.242 |  4289 |  46.0 |  7.7 |  50.9 |  8.84 |  1166.1
TE0 / TM0 |  SH SAW |  90° |  0.3 |  1.5 |  4.972 |  0.889 |  4418 |  47.4 |  9.3 |  55.6 |  7.56 |  541.6

##  Supplementary Information B Feasibility of Continuous Operation for
Acoustoelectric Brillouin Devices

Acoustoelectric devices for radiofrequency signal processing applications have
historically had difficulty operating continuously with large gain, primarily
due to effects related to thermal dissipation of the current required to drive
the acoustoelectric current. [33, 34, 40, 41]. Joule heating arises in
acoustoelectric devices when a voltage is applied to the semiconductor. If the
thermal conductivity out of the device is not sufficient, this resistive
heating can cause variations in the device response as temperature increases,
and ultimately can lead to thermal runaway and even irreversible damage. The
thermal behavior depends on the semiconductor resistance and geometry, the
bias field required to achieve the necessary acoustic gain, and the material
thermal properties of the semiconductor and all the other layers, including
the substrate. However, in contrast to radiofrequency acoustic wave amplifiers
that have been previously experimentally demonstrated, acoustoelectric-
Brillouin devices have different metrics and geometries that make continuous
operation significantly less challenging.