# All-optical Saddle Trap

Daniel Tandeitnik<EMAIL_ADDRESS>Department of Physics, Pontifical
Catholic University of Rio de Janeiro, Rio de Janeiro 22451-900, Brazil  Oscar
Kremer  Department of Electrical Engineering, Pontifical Catholic University
of Rio de Janeiro, 22451-900 Rio de Janeiro, RJ, Brazil  Felipe Almeida
Department of Physics and Astronomy, University College London, Gower Street,
WC1E 6BT London, UK  Joanna A. Zielińska  Tecnologico de Monterrey, Escuela de
Ingeniería y Ciencias, Monterrey, Nuevo León, México  Antonio Zelaquett Khoury
Instituto de Física, Universidade Federal Fluminense, Niterói, Rio de Janeiro
24210-346, Brazil  Thiago Guerreiro<EMAIL_ADDRESS>Department of Physics,
Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro 22451-900,
Brazil

###### Abstract

The superposition of frequency-shifted Laguerre-Gauss modes can produce a
rotating saddle-like intensity profile. When spinning fast enough, the optical
forces produced by this structured light saddle generate a dynamically stable
equilibrium point capable of trapping nanoparticles in a high vacuum, akin to
a Paul trap but with its unique characteristics. We analyze the stability
conditions and center-of-mass motion, dynamics and cooling of a nanoparticle
levitated in the optical saddle trap. We expect the optical saddle to find
applications in levitated optomechanics experiments requiring fast parametric
modulation and inverted squeezing potential landscapes.

##  I Introduction

Trapping charged and neutral particles in electromagnetic fields has been one
of the major tools in atomic, molecular, and nano-physics. While Earnshaw’s
theorem prevents the stable trapping of charged particles using electrostatic
forces, stable equilibrium can be achieved dynamically by time-varying
potentials [1]. These dynamic traps not only lie at the heart of trapped ion
physics [2] but can also be used to trap mesoscopic objects such as charged
dielectric nanoparticles [3]. Neutral particles, on the other hand, can be
trapped by purely electrodynamic fields, e.g. laser light, as first proposed
by Ashkin [4]. Optical tweezers can operate in different regimes depending on
the relative size of the particle compared to the laser wavelength, enabling
the trapping of single atoms all the way to nano and microparticles [5].
Moreover, levitation in vacuum tweezers offers a promising platform for
fundamental quantum physics and sensing experiments [6, 7]. Recently, hybrid
optical-electrical traps combining both optical tweezers and dynamic electric
traps have been designed and proposed [8] for fundamental experiments seeking
to delocalize the quantum state of a levitated mesoscopic object [9, 10].

Here we investigate whether the idea of a dynamic equilibrium trap can be
implemented in an all-optical setup. Considering structured light both in
space and in time, we propose a novel method for optical trapping of nano-
sized dielectric particles in the dipole regime. By superposing frequency-
shifted Laguerre-Gauss modes with a Gaussian beam, we can engineer a rotating
saddle optical potential capable of holding a Silica nanoparticle in a high-
vacuum environment, as illustrated in Figure 1. Provided the frequency shifts
of the superposition components exceed a critical value, this optical saddle
rotates at a sufficiently fast rate to develop a dynamic stable equilibrium
point at the center of the beam. This structured light trap mimics a Paul trap
but is made entirely with light and with a unique motion pattern and dynamics.

The all-optical saddle trap offers interesting possibilities for levitated
optomechanics experiments, both classically and in future quantum experiments.
A particle trapped in the rotating saddle can be cooled to a complex motional
state using optimal control methods and electric feedback [11]. Once cooled,
the trap’s rotation frequency can be tuned from several hundreds of MHz to
zero, causing an effective inverted potential upon the initially localized
particle. This not only offers novel methods for state expansion [8, 10] and
interference protocols [12, 13], but also the possibility of fast-switching
between stable and unstable potential landscapes.

Figure 1: A saddle beam goes round and round, making a nanoparticle
dynamically bound.

This paper is organized as follows. In Sec. II, we describe the optical saddle
beam as well as the associated dynamics of a levitated nanoparticle. We derive
and discuss the conditions required for achieving dynamic equilibrium and
compute the power spectrum of the particle’s motion in the laboratory, i.e.,
non-rotating frame of reference. Next, in Sec. III, we propose an experimental
setup to implement the optical saddle trap and show that feedback cooling of
the particle motion is possible by applying optimal control methods. We
conclude with a discussion of the results.

##  II Theory

###  II.1 The optical saddle beam

There are different ways of producing a saddle intensity pattern around the
focus of an optical beam. As a simple solution, we choose the superposition of
three Laguerre-Gauss (LG) beams as follows:

$\displaystyle
E_{\mathrm{s}}=\sqrt{\frac{16P}{17c\epsilon_{0}}}\left(E^{LG}_{0,0}+\frac{3}{4}e^{-i2\theta}E^{LG}_{0,2}+\frac{3}{4}e^{i2\theta}E^{LG}_{0,-2}\right),$
(1)

where $P$ is the laser power, $c$ is the speed of light in vacuum,
$\epsilon_{0}$ is the electrical permittivity of the vacuum, and $\theta$ is a
parameter that controls the orientation of the saddle on the transverse plane.
Considering the optical-axis as the $z$-axis, the normalized LG mode
$E^{LG}_{p,l}$ is defined in cylindrical coordinates as

$\displaystyle\begin{split}&E^{LG}_{p,l}(r,\phi,z)=\sqrt{\frac{2p!}{\pi(p+|l|)!}}\frac{1}{w(z)}\left(\frac{\sqrt{2}r}{w(z)}\right)^{|l|}\\\
&\times
L^{|l|}_{p}\left(\frac{2r^{2}}{w^{2}(z)}\right)\exp\left(\frac{-r^{2}}{w^{2}(z)}-ik\frac{r^{2}}{2R(z)}+il\phi+i\psi(z)\right),\end{split}$
(2)

where $w(z)$, $z_{R}$, $R(z)$, and $\psi(z)$ are the beam waist radius,
Rayleigh range, wavefront radius and Gouy phase, respectively given by

$\displaystyle w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}},$ (3a) $\displaystyle
z_{R}=\pi w_{0}^{2}/\lambda_{0},$ (3b) $\displaystyle
R(z)=z(1+(z_{R}/z)^{2}),$ (3c)
$\displaystyle\psi(z)=(|l|+2p+1)\arctan(z/z_{R}),$ (3d)

with $w_{0}$ denoting the width of the Gaussian $L^{0}_{0}$ mode at the focus,
$\lambda_{0}$ the beam’s wavelength and $L^{|l|}_{p}$ are the generalized
Laguerre polynomials. For the modes present in Eq. (1), we have
$L^{0}_{0}(x)=L^{|2|}_{0}(x)=1$.

Figure 2: Transverse absolute square of the field at the focus. The values are
normalized by the maximum value of $|E_{s}|^{2}$. The beam presents a saddle-
like profile around the origin, and the parameter $\theta$ defines its
orientation. Both axes are normalized by the distance between the origin and
the bright side peaks.

Throughout this work, we assume a linearly polarized beam. Note that the
saddle beam (1) and the LG modes (2) are normalized so that the integral of
the intensity $I=(c\epsilon_{0}/2)|E_{s}|^{2}$ over the transverse plane gives
the total power $P$.

Evaluation of the absolute square of the field results in

$\displaystyle\begin{split}|E_{s}(r,\phi,z)|^{2}&=\frac{16P}{17c\epsilon_{0}\pi
w(z)^{2}}\exp\left(\frac{-2r^{2}}{w(z)^{2}}\right)\\\
&\times\Bigg{[}1+\frac{3\sqrt{2}r^{2}}{w(z)^{2}}\cos{(2\phi+2\theta)}\cos(2\chi(z))\\\
&+\frac{9r^{4}}{2w(z)^{4}}\cos^{2}(2\phi+2\theta)\Bigg{]},\end{split}$ (4)

where $\chi(z)=\arctan(z/z_{R})$. Figure 2 shows the transverse absolute
square of the field at the focus. The obtained intensity distribution
corresponds to a saddle-like profile, whose orientation in the $xy$ plane
depends on the relative phase $\theta$ between the Laguerre-Gauss $l=\pm 2$
and the Gaussian components of the beam [see Eq. (1)]. The relationship
between the orientation of the saddle pattern and the parameter $\theta$ is
evident in Eq. (4), where the azimuthal angle $\phi$ appears only as the sum
$\phi+\theta$. Thus, as we sweep the phase $\theta$, the saddle rotates.

We can express the intensity distribution from Eq. (4) in terms of Cartesian
coordinates rotating with the saddle, i.e.
$x^{\prime}=x\cos\theta+y\sin\theta$ and $y^{\prime}=y\cos\theta-x\sin\theta$
where $x=r\cos\phi$ and $y=r\sin\phi$ are the stationary Cartesian
coordinates, obtaining:

$\displaystyle\begin{split}|E_{\mathrm{s}}(x^{\prime},y^{\prime},z)|^{2}&=\frac{16P}{17c\epsilon_{0}\pi
w(z)^{2}}\exp\left(\frac{-2(x^{\prime 2}+y^{\prime 2})}{w(z)^{2}}\right)\\\
&\times\Bigg{[}1+\frac{6\sqrt{2}}{w(z)^{2}}\cos(2\chi(z))(x^{\prime
2}-y^{\prime 2})\\\ &+\frac{18}{w(z)^{4}}(x^{\prime 2}-y^{\prime
2})^{2}\Bigg{]}.\end{split}$ (5)

Figure 3 shows the crosssections of $|E_{\mathrm{s}}|^{2}$ along the three
axes of the aforementioned rotating reference frame. The separation between
the peaks along the $x^{\prime}$-axis $\ell_{x}$ equals

$\displaystyle\ell_{x}=\frac{w_{0}}{2}\sqrt{16-3\sqrt{2}},$ (6)

and for the side peaks along the $y^{\prime}$-axis $\ell_{y}$ equals

$\displaystyle\ell_{y}=\frac{w_{0}}{2}\sqrt{16+3\sqrt{2}}.$ (7)

Figure 3: The absolute square of the saddle beam’s electrical field along the
Cartesian axes in the rotational reference frame. The intensity presents a
saddle shape around the origin for the transversal plane.

###  II.2 Dynamical model

We now model the dynamics of a dielectric particle near the beam’s focus. We
consider a particle with a radius much smaller than the beam’s wavelength, the
dipole approximation. In this case, one considers the dielectric limit where
the optical force exerted by the electric field onto the particle equals [14],

$\displaystyle\mathbf{F}_{\rm{opt}}=\frac{\real\\{\alpha\\}}{4}\nabla|E_{s}|^{2}+\frac{\sigma_{\rm{ext}}}{c}\mathbf{S},$
(8)

where $\alpha$ is the complex polarizability of the particle,
$\sigma_{\rm{ext}}$ is the particle’s extinction cross-section, and
$\mathbf{S}$ is the time-averaged Poynting vector. We omit the spin-curl force
since we are interested in uniformly linearly polarized beams. The first term
in Eq. (8) is commonly referred to as the gradient force, while the second is
known as the scattering force. In the dipole approximation, the gradient force
traps a particle near the focus of an optical tweezer. The scattering force,
on the other hand, pushes the particle along the optical axis. To achieve
stable trapping, the scattering must be smaller than the gradient force. For
the saddle beam, the scattering force is mainly provided by the Gaussian
amplitude $E^{LG}_{0,0}$ in Eq. (1) and displaces the equilibrium position of
the particle along the optical axis. We take the scattering force into account
throughout our simulations of the particle dynamics.

From the plots of $|E_{s}|^{2}$ Figure 3, one sees that the saddle intensity
in the transverse plane creates an unstable equilibrium point at the origin.
Therefore, if a dielectric particle is placed at the focus and undergoes small
perturbations, it will be pushed to one of the side peaks on the
$x^{\prime}$-axis. In analogy to electrical quadrupole and mechanical saddle
traps which can confine particles in dynamical saddle-shaped potentials [15,
16, 17], we investigate the dynamics of a dielectric nanoparticle in the
rotating optical saddle beam, i.e., by setting $\theta=\Omega t$, where
$\Omega$ is the angular velocity of rotation. As we will see, provided
$\Omega$ is large enough, the unstable origin is turned into a dynamically
stable equilibrium point.

The dynamics of a dielectric nanoparticle in the saddle becomes easier to
analyze in a frame rotating about the $z$-axis with angular velocity $\Omega$.
Without loss of generality, $|E_{s}|^{2}$ in this frame is given by Eq. (5).
The Taylor expansion of this intensity near the origin, retaining at most
quadratic terms, yields

$\displaystyle\begin{split}&|E_{s}|^{2}\approx\frac{32P}{17c\epsilon_{0}\pi
w_{0}^{2}}\\\
&\times\left(1+\frac{x^{2}}{w_{0}^{2}}(3\sqrt{2}-2)-\frac{y^{2}}{w_{0}^{2}}(3\sqrt{2}+2)-\frac{z^{2}}{z_{R}^{2}}\right).\end{split}$
(9)

At this point, we wish to derive a stability criterion for the conservative
gradient force, and for this reason, we will neglect the scattering force for
the moment. Note, however, that the effects of scattering upon the
nanoparticle’s dynamical equilibrium are discussed in Appendix A and taken
into account in the dynamical simulations shown in the next sections.

To linear order in the particle’s displacement, the gradient force
$\mathbf{F}_{g}$ given by the first term in Eq. (8) reads

$\mathbf{F}_{g}=\frac{16P\real\\{\alpha\\}}{17c\epsilon_{0}\pi
w_{0}^{2}}\begin{bmatrix}x(3\sqrt{2}-2)/w_{0}^{2}\\\
-y(3\sqrt{2}+2)/w_{0}^{2}\\\ -z/z_{R}^{2}\end{bmatrix}.$ (10)

Since we describe the dynamics in a rotating reference frame, we must account
for the centripetal and Coriolis forces that arise in such a frame. Newton’s
second law gives

$m\frac{d^{2}\mathbf{r}}{dt^{2}}\Bigg{|}_{S}=\sum_{i}\mathbf{F}_{i}(\mathbf{r},\dot{\mathbf{r}})\Bigg{|}_{S},$
(11)

where the $S$ stands for the inertial frame of the lab. To write this equation
in a rotating reference frame $S^{\prime}$ that rotates about a vector
$\mathbf{\Omega}$, one needs to rewrite the derivative of vectors as [18]

$\frac{d\mathbf{A}}{dt}\Bigg{|}_{S}=\frac{d\mathbf{A}}{dt}\Bigg{|}_{S^{\prime}}+\mathbf{\Omega}\times\mathbf{A}.$
(12)

Considering constant angular velocity, $d\mathbf{\Omega}/dt=0$ we find

$m\frac{d^{2}\mathbf{r}^{\prime}}{dt^{2}}\Bigg{|}_{S^{\prime}}=\sum_{i}\mathbf{F}_{i}(\mathbf{r}^{\prime},\dot{\mathbf{r}}^{\prime})\Bigg{|}_{S^{\prime}}-2m\mathbf{\Omega}\times\frac{d\mathbf{r}^{\prime}}{dt}\Bigg{|}_{S^{\prime}}-\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r}^{\prime}).$
(13)

where $\mathbf{\Omega}=\Omega\hat{\mathbf{z}}$. Moreover, we consider a
damping force in the lab frame of the form $-m\gamma\dot{\mathbf{r}}$, where
$m$ is the particle’s mass and $\gamma$ is the drag coefficient. In the
rotating reference frame this damping force becomes
$-m\gamma\dot{\mathbf{r}}^{\prime}-m\gamma\Omega\times\mathbf{r}^{\prime}$.

Putting it all together, Eq. (13) defines a set of three differential coupled
equations describing the motion of the particle,

$\displaystyle\ddot{x}^{\prime}$
$\displaystyle=(\Omega^{2}+\omega_{x}^{2})x^{\prime}-\gamma(\dot{x}^{\prime}-\Omega
y^{\prime})+2\Omega\dot{y}^{\prime}+\frac{F_{x}}{m},$ (14a)
$\displaystyle\ddot{y}^{\prime}$
$\displaystyle=(\Omega^{2}-\omega_{y}^{2})y^{\prime}-\gamma(\dot{y}^{\prime}+\Omega
x^{\prime})-2\Omega\dot{x}^{\prime}+\frac{F_{y}}{m},$ (14b)
$\displaystyle\ddot{z}$
$\displaystyle=-\omega_{z}^{2}z-\gamma\dot{z}+\frac{F_{z}}{m},$ (14c)

where

$\begin{bmatrix}\omega_{x}^{2}\\\ \omega_{y}^{2}\\\
\omega_{z}^{2}\end{bmatrix}=\frac{16P\real\\{\alpha\\}}{17c\epsilon_{0}\pi
w_{0}^{2}m}\begin{bmatrix}(3\sqrt{2}-2)/w_{0}^{2}\\\
(3\sqrt{2}+2)/w_{0}^{2}\\\ 1/z_{R}^{2}\end{bmatrix},$ (15)

and $F_{i}\ (i=x,y,z)$ represent additional force terms the particle might be
subject to, such as for example thermal stochastic or feedback forces. For
now, we consider $F_{i}=0$ and derive a criterion for stable trapping in the
saddle beam.

###  II.3 Stability analysis

We now consider motion along the transverse plane. Instead of solving for
$x^{\prime}$ and $y^{\prime}$ 111Note that a general solution can be found by
writing the equations as a set of four coupled first-order differential
equations and employing diagonalization. The result, however, is extremely
cumbersome and not physically illuminating. we seek a stability condition on
the rotation frequency $\Omega$. Regarding the longitudinal $z$ direction in
the presence of the scattering force, one can show that the particle follows a
simple damped harmonic motion, see Appendix A) for details.

Consider the ansatz $x^{\prime}=x^{\prime}_{0}e^{\lambda t}$ and
$y^{\prime}=y^{\prime}_{0}e^{\lambda t}$. For confinement in the transverse
plane, we require the values of $\lambda$ to have negative or null real part.
Substituting the ansatz in Eqs. (14a), (14b) and casting the result in matrix
form we find

$\begin{bmatrix}\Omega^{2}+\omega_{x}^{2}-\lambda^{2}-\gamma\lambda&\gamma\Omega+2\lambda\Omega\\\
-\gamma\Omega-2\Omega\lambda&\Omega^{2}-\omega_{y}^{2}-\lambda^{2}-\gamma\lambda\end{bmatrix}\begin{bmatrix}x^{\prime}_{0}\\\
y^{\prime}_{0}\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$ (16)

The existence of nontrivial solutions require the above matrix to be non-
invertible (vanishing determinant), which gives the characteristic equation

$\displaystyle\begin{split}\lambda^{4}&+2\gamma\lambda^{3}+\lambda^{2}\left(\gamma^{2}+2\Omega^{2}+\omega_{y}^{2}-\omega_{x}^{2}\right)\\\
&+\lambda\left(2\gamma\Omega^{2}+\gamma\omega_{y}^{2}-\gamma\omega_{x}^{2}\right)\\\
&+\Omega^{4}+\Omega^{2}(\gamma^{2}+\omega_{x}^{2}-\omega_{y}^{2})-\omega_{x}^{2}\omega_{y}^{2}=0.\end{split}$
(17)

with solutions given by

$\displaystyle\lambda_{1}^{\pm}\to\frac{1}{2}\left(-\gamma\pm\sqrt{2\sqrt{a}+b}\right),$
(18a)
$\displaystyle\lambda_{2}^{\pm}\to\frac{1}{2}\left(-\gamma\pm\sqrt{\gamma^{2}-2\left(\sqrt{a}+c\right)}\right),$
(18b)

where

$\displaystyle a$
$\displaystyle=8\Omega^{2}(-\gamma^{2}/2+\omega_{y}^{2}-\omega_{x}^{2})+(\omega_{x}^{2}+\omega_{y}^{2})^{2},$
(19a) $\displaystyle b$
$\displaystyle=\gamma^{2}-4\Omega^{2}+2\omega_{x}^{2}-2\omega_{y}^{2},$ (19b)
$\displaystyle c$ $\displaystyle=2\Omega^{2}-\omega_{x}^{2}+\omega_{y}^{2}.$
(19c)

The $\lambda_{1}^{-}$ root in Eq. (18a) is negative. On the other hand,
$\lambda_{1}^{+}$ will be negative when

$-\gamma+\sqrt{2\sqrt{a}+b}<0.$ (20)

Given $\omega_{y}>\omega_{x}$ (see Eq. (15)), this condition implies

$|\Omega|>\sqrt{\frac{-\gamma^{2}+\omega_{y}^{2}-\omega_{x}^{2}+\sqrt{(\gamma^{2}+\omega_{x}^{2}-\omega_{y}^{2})^{2}+4\omega_{x}^{2}\omega_{y}^{2}}}{2}}.$
(21)

Therefore, the first two roots are negative provided $\Omega$ is sufficiently
large.

We now turn to the remaining roots; as before $\lambda_{2}^{-}$ is negative,
while $\lambda_{2}^{+}$ will be negative when

$-\gamma+\sqrt{\gamma^{2}-2\left(\sqrt{a}+c\right)}<0.$ (22)

As it turns out, this inequality is always satisfied for
$\gamma,\omega_{x},\omega_{y}>0$ and $\omega_{y}>\omega_{x}$. Hence, the
stability criterion is given entirely by Eq. (21). Considering the dynamics in
the overdamped regime, the same criteria are obtained much more
straightforwardly; see Appendix B for more details.

###  II.4 Dynamics simulation

Figure 4: Representative trajectories of the three possible dynamics as viewed
in the laboratory reference frame. a) $\Omega/\Omega_{c}=0.1$. Trajectory
shows the last $0.4\text{\,}\mathrm{m}\mathrm{s}$ transversal positions. The
particle describes a closed trajectory following one of the rotating side
peaks of the saddle. b) $\Omega/\Omega_{c}=0.5$. The particle escapes if the
angular rotational velocity is not slow enough and well below the critical
value for stability. c) $\Omega/\Omega_{c}=1.0$. The particle displays
dynamical stability and undergoes a complex trajectory about the saddle’s
center.

We have studied the dynamics of a trapped particle in the saddle beam as well
as the validity of the stability criteria by numerical simulation. In doing
so, we have considered both the linearized model given by Eqs. (14) and the
complete form of the potential given by the gradient of Eq. (5) for the fixed
set of experimental parameters defined in Table 1. In all simulations, the
particle initializes in a position near the origin, where the trap is well
approximated by a quadratic potential landscape. All simulations consider a
stochastic external force $F_{i}=F_{\rm{th,i}}$ arising from collisions of the
particle with surrounding gas molecules. This stochastic force obeys,

$\displaystyle\langle F_{\rm{th,i}}(t)\rangle$ $\displaystyle=0,$ (23a)
$\displaystyle\langle F_{\rm{th,i}}(t)F_{\rm{th,j}}(t+\tau)\rangle$
$\displaystyle=2m\gamma k_{B}T\delta(\tau)\delta_{ij},$ (23b)

where $k_{B}$ is the Boltzmann constant and $T$ is the surrounding gas
temperature. We employed the Runge-Kutta $4^{\rm th}$-order algorithm for all
simulations, where each run simulated the dynamics for a period of
$10\text{\,}\mathrm{m}\mathrm{s}$. We refer to Appendix C for more details.

Parameter |  | Value
---|---|---
Particle composition |  | SiO2
Particle radius |  | $150\text{\,}\mathrm{n}\mathrm{m}$
Particle mass |  | $\approx$26\text{\,}\mathrm{f}\mathrm{g}$$
Initial velocity ($\approx\sqrt{k_{B}T/m}$) |  | $\approx$12.5\text{\,}\mu\mathrm{m}\mathrm{/}\mathrm{s}$$
Laser wavelength |  | $1550\text{\,}\mathrm{n}\mathrm{m}$
Tweezer power |  | $500\text{\,}\mathrm{m}\mathrm{W}$
Trapping NA |  | $0.6$
Pressure |  | $10\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$
Temperature |  | $293\text{\,}\mathrm{K}$
Table 1: Experimental parameters considered in the dynamical simulations.
Figure 5: Stability criteria, both for the linearized model (orange line) and
full optical potential (blue line). The mean value of the last
$1\text{\,}\mathrm{m}\mathrm{s}$ of the particle’s radial position is plotted
as a function of the ratio between the saddle’s rotational and critical
angular velocities $\Omega/\Omega_{c}$. The shaded region indicates the
particle is no longer trapped.

To evaluate the different regimes of motion of a particle in the rotating
saddle, we have considered different ratios of the trap’s rotating frequency
$\Omega$ to the critical angular velocity $\Omega_{c}$ given by Eq. (21).
Figure 4 qualitatively displays three regimes of motion: a) If $\Omega$ is
slow enough ($\Omega/\Omega_{c}\lesssim 0.25$), the particle is trapped by one
of the rotating side-peaks, following an approximately circular motion. As the
rotation is increased, the particle lags behind the peak due to the drag force
provided by the surrounding gas; b) increasing the rotation but keeping it
well below the critical angular velocity, the centrifugal force as felt by the
particle in the rotating frame becomes so strong that the particle is expelled
away from the beam. Going higher in angular velocity c), at
$\Omega/\Omega_{c}\gtrsim 0.92$, the particle is confined in the center and
undergoes a complex motion around this dynamically generated equilibrium
point. We note that for the parameters in Table 1 we have
$\Omega_{c}/2\pi\approx$280\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$. All
trajectories are viewed in the laboratory reference frame.

Figure 5 shows an alternative visualization of the different regimes of
motion. Here, the particle’s last radial position $\langle
r\rangle=\langle\sqrt{x^{2}+y^{2}}\rangle$ output by the simulation is
averaged over 20 runs and plotted against the $\Omega/\Omega_{c}$ ratio. We
see that for $\Omega/\Omega_{c}>1$, the particle remains trapped near the
center of the beam. On the other hand, for $\Omega/\Omega_{c}<1$, the value of
$\langle r\rangle$ blows up, indicating that the particle is unbounded (shaded
region). For sufficiently small values of $\Omega$, the particle is bound in
one of the lobes of the saddle beam, remaining at a position of the intensity
maxima given by $l_{x}/2$.

Commonly employed in levitated optomechanics experiments as a characterization
method, the power spectrum density (PSD) of the motion of a particle trapped
in the rotating saddle can be analytically computed by linearizing the Eqs. of
motion. This can be done both in the rotating and laboratory reference frames
(see Appendix D for details on the calculation) and compared to the result
obtained from numerical simulations. Figure 6 shows an example of the
laboratory frame PSD, simulated taking into account the full optical
potential, in comparison to the analytical prediction from linearized theory.
We see the overall shape of the predicted PSD as given by Eq. (65) is
consistent with the simulation, however with a shift and additional peaks due
to the nonlinear nature of the potential [20].

Figure 6: Simulated center-of-mass motion power spectra (blue line) compared
to analytical calculations in the linearized approximation (orange curve). We
see the overall form of the predicted PSD matches simulations of the motion
under the full potential a). However, as we zoom in, we notice additional
peaks and an overall shift in the PSD due to the nonlinear nature of the
potential b).

##  III Experimental proposal

We now discuss an experimental scheme for generating the rotating saddle beam
as well as the possibility of feedback cooling the center-of-mass motion of a
trapped nanoparticle.

###  III.1 Saddle-beam generation

To generate the superposition in Eq.(1) rotating with angular velocity in the
MHz range, we propose the combined use of three acoustic optical modulators
(AOM) and a variable spiral plate (VSP) – a liquid crystal element capable of
transforming an incident circularly polarized Gaussian beam into Laguerre-
Gaussian modes with $l=\pm 2$ depending on the input’s handedness [21].

Figure 7 illustrates the optical setup. The half-wave plate $\rm{HW}_{1}$, in
conjunction with the polarizing beam splitter $\rm{PBS_{1}}$, divides an
initial linearly polarized laser beam into two paths. The upper path receives
a frequency shift of $\Delta$, and $\rm{HW}_{3}$ corrects the polarization for
proper interference at the optical tweezer (OT) site. The lower path is
divided into two beams that receive distinct frequency shifts of
$\Delta\pm\delta$ and are recombined with $\rm{HW}_{2}$ and $\rm{PBS}_{2}$
into a single spatial mode. Since each component has orthogonal linear
polarizations, they have opposite handedness after passing through the
quarter-wave plate $\rm{QW}_{1}$. Therefore, the VSP transforms one of the
beams into a $l=2$ and the other into a $l=-2$ LG mode. Finally, $\rm{QW}_{2}$
reverts the circular polarization to linear, and the polarizer projects into a
common mode to interfere with the remaining Gaussian beam. The distinct
detunings received by the superposition components effectively generate a
time-dependent phase, causing the overall superposition to rotate. Tuning the
frequency shifts of each AOM by $\delta$ around a central shift $\Delta$
allows for fine control of the saddle’s rotation speed. Coupling all three
beams into a single-mode alignment fiber before inserting the VSP ensures the
spatial mode matching of the rotating saddle’s components.

Setting the transmission:reflection ratio of $\rm{HW}_{1}$ to approximately
70:30, the saddle beam can be reproduced with the proper coefficients.
Considering the AOM’s diffraction efficiency as $80\%$ and the initial beam’s
power as $1\text{\,}\mathrm{W}$, roughly $260\text{\,}\mathrm{m}\mathrm{W}$ of
power should arrive at the OT. We note that other schemes can also be
considered, such as using a single AOM with a double pass. However, in
practice, most schemes have severely low power efficiencies at the output.
Schemes using one or two SLMs to generate the LG beams were also disfavored in
favor of the VSP since the latter is a low-loss optical element.

Finally, commercially available AOMs can shift the frequency of an incoming
laser beam by tens of MHz around a central frequency $\Delta$. Thus, by
choosing a shift of $\delta=$10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$, the
saddle beam would rotate at sufficient angular velocity to have dynamical
stability.

Figure 7: Experimental setup for generating a fast rotating saddle beam. Three
AOMs are employed to give the correct relative phase to the components of the
saddle beam superposition. Gaussian beams with opposite circular polarization
generate the required $l=\pm 2$ LG modes using a VSP.

###  III.2 Feedback cooling

For most applications in high-vacuum levitated optomechanics, the ability to
control the motion of the trapped particle is required. For instance, optimal
control theory can successfully achieve the ground state of the particle’s
longitudinal center-of-mass (CoM) motion [22, 23]. We thus ask the question of
whether optimal control theory methods can be applied to cool the motion of a
levitated nanoparticle in the rotating saddle.

The issue of minimizing the energy of linear systems, as the one presented in
Eq. (14), can be effectively addressed using the optimal control policy known
as Linear Quadratic Gaussian (LQG) controller. The dynamics of a trapped
particle in the rotating reference frame can be approximated by a set of
linear differential equations, and the implementation of the feedback loop is
highly reliant on the detection of the particle’s position. In our case, the
particle’s positions are detected in the laboratory frame by means of homodyne
detection of the trap’s scattered light. One can express the detection signals
by

$\mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{m}(t),$ (24)

where we employ the state-space representation, in which $\mathbf{x}(t)$ is
the state vector for the laboratory reference frame, defined as

$\mathbf{x}(t)=\begin{bmatrix}x(t)&y(t)&z(t)&\dot{x}(t)&\dot{y}(t)&\dot{z}(t)\end{bmatrix}^{T},$
(25)

and $\mathbf{y}(t)$ is the measurement vector. Moreover, $\mathbf{C}$ is a
$3\times 6$ matrix defined by the calibration factors of the detectors and
$\mathbf{m}(t)$ is the measurement noise vector, a $3\times 1$ vector composed
by independent white Gaussian noise processes satisfying
$\langle\mathbf{m}(t)\rangle=\mathbf{0}_{3\times 1}$ and
$\langle\mathbf{m}(t)\mathbf{m}(t)^{T}\rangle=\mathbf{M}$, where $\mathbf{M}$
is a $3\times 3$ diagonal matrix.

Figure 8: Optimal control for an optical saddle trap. a) Time traces for $x$,
$y$ and $z$ coordinates of a particle levitated by an optical saddle trap and
controlled by the LQG. b) Motion of a trapped particle along the transverse
plane for different feedback gains. Numerical simulations were carried
considering SiO2 nanoparticles with a radius of
$150\text{\,}\mathrm{n}\mathrm{m}$, pressure of
$10^{-5}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, gas temperature of
$293\text{\,}\mathrm{K}$ and detection efficiencies of $1.0\%$, $1.0\%$, and
$0.1\%$ for $x$, $y$, and $z$, respectively.

Using Eq. (61), one finds the relation between $\mathbf{x}$ and the state
vector in the rotating frame $\mathbf{x}^{\prime}$,

$\mathbf{x}(t)=\mathbf{T}(\theta)\mathbf{x}^{\prime}(t),$ (26)

where $\mathbf{T}(\theta)$ is given by

$\mathbf{T}(\theta)=\begin{bmatrix}\mathbf{R}_{z}(\theta)&\mathbf{0}_{3\times
3}\\\ \dot{\mathbf{R}}_{z}(\theta)&\mathbf{R}_{z}(\theta)\end{bmatrix},$ (27)

with,

$\mathbf{R}_{z}(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta&0\\\
\sin\theta&\cos\theta&0\\\ 0&0&1\end{bmatrix}.$ (28)

From Eq. (26), we can write an expression analogous to the one in Eq. (24) for
the rotating reference frame, yielding

$\mathbf{y}^{\prime}(t)=\mathbf{C}^{\prime}\mathbf{x}^{\prime}(t)+\mathbf{m}^{\prime}(t),$
(29)

where

$\mathbf{y}^{\prime}(t)=\mathbf{R}_{z}^{T}(\theta)\mathbf{y}(t)\,\textrm{
,}\,\mathbf{C}^{\prime}=\mathbf{R}_{z}^{T}(\theta)\mathbf{C}\mathbf{T}(\theta).$
(30)

The noise vector $\mathbf{m}^{\prime}$ has zero mean and satisfies

$\langle\mathbf{m}^{\prime}(t)\mathbf{m}^{\prime}(t)^{T}\rangle=\mathbf{R}_{z}^{T}(\theta)\mathbf{M}\mathbf{R}_{z}(\theta).$
(31)

Additionally, the state-space representation also allows the rewriting of the
dynamics presented in Eq. (14), yielding

$\dot{\mathbf{x}}^{\prime}(t)=\mathbf{A}^{\prime}\mathbf{x}^{\prime}(t)+\mathbf{B}^{\prime}\mathbf{u}^{\prime}(t)+\mathbf{w}^{\prime}(t),$
(32)

with $\mathbf{w}^{\prime}(t)$ taking into account the stochastic forces acting
upon the particle, $\mathbf{B}^{\prime}\mathbf{u}(t)$ being related to the
feedback forces exerted by the controller and $\mathbf{A}^{\prime}$ the state
matrix, defined as

$\mathbf{A}^{\prime}=\begin{bmatrix}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\ \Omega^{2}+\omega_{x}^{2}&\gamma\Omega&0&-\gamma&2\Omega&0\\\
-\gamma\Omega&\Omega^{2}-\omega_{y}^{2}&0&-2\Omega&-\gamma&0\\\
0&0&-\omega_{z}^{2}&0&0&-\gamma\\\ \end{bmatrix}.$ (33)

Equations (29) and (32) form a fundamental pair in linear control theory, from
where one extracts fundamental information about the system’s dynamics,
actuators, and detection. Such information must be properly characterized to
apply LQG correctly. The optimal control law $\mathbf{u}^{\prime\ast}$ is then

$\mathbf{u}^{\prime\ast}=-\mathbf{K}_{\rm{opt}}\hat{\mathbf{x}}^{\prime}(t),$
(34)

where $\mathbf{K}_{\rm{opt}}$ is the optimal controller’s gain matrix and
$\hat{\mathbf{x}}^{\prime}$ is an estimation of the state vector returned by
the Kalman-Bucy filtering method. Physically, the control law can be
implemented using electrodes near the trap center [11]. Note that the
computation of $\mathbf{K}_{\rm{opt}}$ and $\hat{\mathbf{x}}^{\prime}$ depends
on knowledge of $\mathbf{A}^{\prime}$, $\mathbf{B}^{\prime}$,
$\mathbf{C}^{\prime}$, $\mathbf{w}^{\prime}(t)$ and $\mathbf{m}^{\prime}(t)$.
For more details regarding the estimations and optimal gain calculations, we
refer to [24].

We performed numerical simulations to examine the effectiveness of the LQG in
controlling a particle confined by the saddle beam, assuming the same particle
parameters as shown in Table 1. Figure 8 a) presents the simulation results
for the three directions in the lab fame. We observe a significant reduction
in the amplitude of the particle’s CoM amplitude for all three directions of
motion as time evolves. The motion reaches a stationary state near
$0.2\text{\,}\mathrm{m}\mathrm{s}$ after starting the application of the
control law. Figure 8b) shows the controller’s effect on the transversal plane
motion; we see the motion amplitude increase as we apply smaller feedback
gains. Finally, note that we carried out the simulations considering the
complete form of the potential, which is highly nonlinear. Even though the LQG
formalism utilizes linearized dynamics, it is still effectively capable of
appreciably damping the particle’s motion.

##  IV Conclusion

In conclusion, we proposed a rotating saddle-like structured beam for optical
levitation experiments. When rotating above a critical velocity, the saddle
beam acquires a dynamical equilibrium point capable of levitating dielectric
nanoparticles in high-vacuum. The power spectrum of the center-of-mass motion
presents unique features associated with the non-harmonic nature of the trap.
We have proposed a method to generate the rotating saddle with a controllable
rotation velocity and have shown that feedback control can be used to cool the
motion of the particle. We expect the saddle beam to find applications in
fundamental quantum physics experiments using levitated nano-objects, where
inverted and nonlinear potentials can be used to rapidly expand and interfere
with an initially localized wavepacket [13, 12].

## Funding and Competing Interests

This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001, Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq scholarship 140197/2022-2),
Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ Scholarship
No. E-26/200.252/2023, and E-26/202.330/2024), Fundação de Amparo à Pesquisa
do Estado de São Paulo (FAPESP process No. 2021/06736-5 and 2021/06823-5), the
Serrapilheira Institute (grant No. Serra – 2211-42299) and StoneLab. We
acknowledge support from EPSRC International Quantum Technology Network
LeviNet EP/W02683X/1.

All authors certify that they have no affiliations with or involvement in any
organization or entity with any financial or non-financial interest in the
subject matter or materials discussed in this manuscript.

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##  Appendix A The scattering force

To determine the stability condition for the $z$ direction the scattering
force needs to be analysed. Stable equilibrium occurs if there is a point
where the scattering and gradient forces are opposite and equal in magnitude,
and the derivative of the total force is negative. The Taylor expansion in Eq
(9) cannot be used in this analysis, since it gives a field that does not go
to zero at $z=\pm\infty$. However, since we are mainly interested in the
intensity of the beam along the $z$-axis, for $x^{\prime}=y^{\prime}=0$, Eq.
(5) yields the simple result

$|E_{s}|^{2}(z)=\frac{32P}{17\pi
c\epsilon_{0}}\frac{1}{w_{0}^{2}(1+z^{2}/z_{R}^{2})}.$ (35)

Thus, the gradient force along the $z$-axis is

$F_{g}(z)=-\frac{16P\real\\{\alpha\\}}{17\pi
c\epsilon_{0}}\frac{z}{w_{0}^{2}z_{R}^{2}(1+z^{2}/z_{R}^{2})^{2}},$ (36)

For a silica particle, the extinction cross-section is well approximated by
[14],

$\sigma_{\rm{ext}}\approx\frac{8\pi^{3}|\alpha|^{2}}{3\epsilon_{0}^{2}\lambda_{0}^{4}}.$
(37)

Therefore, the scattering force is

$F_{s}(z)=\frac{128P\pi^{2}|\alpha|^{2}}{51\lambda_{0}^{4}\epsilon_{0}^{2}c}\frac{1}{w_{0}^{2}(1+z^{2}/z_{R}^{2})}.$
(38)

Making $F_{g}(z)+F_{s}(z)=0$ leads to a quadratic equation:

$z^{2}-\frac{3\lambda_{0}^{4}\epsilon_{0}\real\\{\alpha\\}}{8\pi^{3}|\alpha|^{2}}z+z_{R}^{2}=0.$
(39)

To have real solutions, the following inequality must be satisfied:

$\left(\frac{3\lambda_{0}^{4}\epsilon_{0}\real\\{\alpha\\}}{8\pi^{3}|\alpha|^{2}}\right)^{2}-4z_{R}^{2}>0,$
(40)

where we disregarded the equality since it generates an unstable equilibrium
point. Finally, by considering the particle’s radius fixed, which fixes the
polarizability, and using the definition of the Rayleigh range, we find a
condition for $w_{0}$ to have stability along the $z$-direction:

$w_{0}<\sqrt{\frac{3\lambda_{0}^{5}\epsilon_{0}\real\\{\alpha\\}}{16\pi^{4}|\alpha|^{2}}}.$
(41)

If the above condition is satisfied, there is a stable equilibrium position
for the particle’s longitudinal motion at

$z_{\rm{eq}}=\frac{3\lambda_{0}^{4}\epsilon_{0}\real\\{\alpha\\}}{16\pi^{3}|\alpha|^{2}}-\frac{1}{2}\sqrt{\left(\frac{3\lambda_{0}^{4}\epsilon_{0}\real\\{\alpha\\}}{\pi^{3}|\alpha|^{2}}\right)^{2}-4z_{R}^{2}}.$
(42)

##  Appendix B Overdamped stability analysis

In the overdamped regime, the Eqs. of motion simplify to,

$\displaystyle 0$
$\displaystyle=(\Omega^{2}+\omega_{x}^{2})x^{\prime}-\gamma(\dot{x}^{\prime}-\Omega
y^{\prime})+2\Omega\dot{y}^{\prime}+\frac{F_{\rm{th}}}{m},$ (43a)
$\displaystyle 0$
$\displaystyle=(\Omega^{2}-\omega_{y}^{2})y^{\prime}-\gamma(\dot{y}^{\prime}+\Omega
x^{\prime})-2\Omega\dot{x}^{\prime}+\frac{F_{\rm{th}}}{m},$ (43b)
$\displaystyle 0$
$\displaystyle=-\omega_{z}^{2}z-\gamma\dot{z}+\frac{F_{\rm{th}}}{m}.$ (43c)

Similarly to Section II.3, let $x^{\prime}=x^{\prime}_{0}e^{\lambda t}$ and
$y^{\prime}=y^{\prime}_{0}e^{\lambda t}$. By substitution in equations (43a)
and (43b), and neglecting the stochastic force term, we have

$\displaystyle(\Omega^{2}+\omega_{x}^{2}-\gamma\lambda)x^{\prime}_{0}+(\gamma\Omega+2\lambda\Omega)y^{\prime}_{0}=0,$
(44a)
$\displaystyle(\Omega^{2}-\omega_{y}^{2}-\gamma\lambda)y^{\prime}_{0}-(\gamma\Omega+2\lambda\Omega)x^{\prime}_{0}=0.$
(44b)

Casting the equation in matrix form

$\begin{bmatrix}\Omega^{2}+\omega_{x}^{2}-\gamma\lambda&\gamma\Omega+2\lambda\Omega\\\
-\gamma\Omega-2\Omega\lambda&\Omega^{2}-\omega_{y}^{2}-\gamma\lambda\end{bmatrix}\begin{bmatrix}x^{\prime}_{0}\\\
y^{\prime}_{0}\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix},$ (45)

and demanding that the matrix’s determinant is equal to zero, one arrives at a
quadratic polynomial for $\lambda$:

$a\lambda^{2}+b\lambda+c=0,$ (46)

where

$\displaystyle a$ $\displaystyle=\gamma^{2}+4\Omega^{2},$ (47a) $\displaystyle
b$
$\displaystyle=2\gamma\Omega^{2}+\gamma\omega_{y}^{2}-\gamma\omega_{x}^{2},$
(47b) $\displaystyle c$
$\displaystyle=\Omega^{4}+\Omega^{2}(\gamma^{2}+\omega_{x}^{2}-\omega_{y}^{2})-\omega_{x}^{2}\omega_{y}^{2}.$
(47c)

Since the roots are $\lambda^{\pm}=-b/2a\pm\sqrt{b^{2}-4ac}/2a$ and $a,b>0$
(see Eq. (15)), then they are complex with a negative real part if
$b^{2}-4ac<0$. If $b^{2}-4ac>0$, one needs to require $-b+\sqrt{b^{2}-4ac}<0$
so both roots are negative. Taking the worst case, assuming that $b^{2}-4ac>0$
in conjunction with $a,b>0$, leads to requiring that $c>0$, i.e.,

$\Omega^{4}+\Omega^{2}(\gamma^{2}+\omega_{x}^{2}-\omega_{y}^{2})-\omega_{x}^{2}\omega_{y}^{2}>0.$
(48)

This condition is interesting because it will force both roots to be complex,
ending up in the first case.

Inequality (48) leads to demanding $|\Omega|>\Omega^{+}$ for stability, where
$\Omega^{+}$ is the real positive root of the polynomial. Moreover, note that
the polynomial necessarily has two real and two imaginary roots. Let
$\Omega=\pm\sqrt{u}$, then the roots for the quadratic equation in $u$ are

$u^{\pm}=\frac{-\gamma^{2}+\omega_{y}^{2}-\omega_{x}^{2}\pm\sqrt{(\gamma^{2}+\omega_{x}^{2}-\omega_{y}^{2})^{2}+4\omega_{x}^{2}\omega_{y}^{2}}}{2}.$
(49)

Since the argument inside the square root is positive, $u^{+}$ must give the
real roots of Eq. (48). Therefore, the stability criterion for the overdamped
case is

$|\Omega|>\sqrt{\frac{-\gamma^{2}+\omega_{y}^{2}-\omega_{x}^{2}+\sqrt{(\gamma^{2}+\omega_{x}^{2}-\omega_{y}^{2})^{2}+4\omega_{x}^{2}\omega_{y}^{2}}}{2}},$
(50)

which is identical to the criterion found for the complete dynamical model.

Figure 9: Examples of simulated position PSDs for the linearized and complete
optical potential in the rotation and laboratory reference frames. a)
linearized potential in the rotational frame; b) and c) linearized potential
in the laboratory frame; d) complete potential in the rotational frame; e) and
f) complete potential in the laboratory frame.

##  Appendix C Numerical simulation parameters

For all the simulations described in Section II.4, the particle is initially
located at

$\mathbf{r}_{0}=\begin{bmatrix}\sqrt{k_{b}T/m\omega_{x,h}^{2}}\\\
\sqrt{k_{b}T/m\omega_{y,h}^{2}}\\\
z_{eq}+\sqrt{k_{b}T/m\omega_{z,h}^{2}}\end{bmatrix},$ (51)

where $\sqrt{k_{b}T/m\omega_{i,h}^{2}}$ is the standard deviation of the
particle’s position i when it is trapped by a standard Gaussian beam optical
tweezer (OT) (considering a harmonic approximation). The $\omega_{i,h}$ are
the natural frequencies for a harmonic trap given by

$\begin{bmatrix}\omega_{x,h}^{2}\\\ \omega_{y,h}^{2}\\\
\omega_{z,h}^{2}\end{bmatrix}=\frac{2P\real\\{\alpha\\}}{c\epsilon_{0}\pi
w_{0}^{2}m}\begin{bmatrix}2/w_{0}^{2}\\\ 2/w_{0}^{2}\\\
1/z_{R}^{2}\end{bmatrix}.$ (52)

The $z_{eq}$ term is the equilibrium position along the optical axis, which is
different from the zero due to the scattering force. Most interestingly, its
value for a Gaussian beam and the proposed saddle beam is the same, given by
Eq. (42) derived in Appendix A. This arises because, at the optical axis, only
the Gaussian $E^{LG}_{0,0}$ term has a non-vanishing value for the saddle
beam. The initial velocity is given by the standard deviation of velocity for
a particle in a harmonic trap, given by $v_{0}=\sqrt{k_{B}T/m}$ for all
directions. The main motivation behind this choice of initial conditions is
the idea that one could first trap a particle using a standard Gaussian beam,
and later switching to the saddle beam.

For the beam’s overall parameters, we set its wavelength at
$1550\text{\,}\mathrm{n}\mathrm{m}$, the numerical aperture (NA) at $0.6$, and
the optical power at $500\text{\,}\mathrm{m}\mathrm{W}$. The low NA is
required to prevent the structure of the beam’s intensity from collapsing to a
non-paraxial Gaussian-like shape. The effect of high focusing was numerically
evaluating the transmission of the saddle beam by a high-NA lenses (NA $>0.8$)
using the angular spectral representation formalism [25], and we have observed
that an NA$=0.6$ is a good compromise between focusing and keeping the
structure of the saddle superposition. We considered SiO2 nanoparticles with a
$150\text{\,}\mathrm{n}\mathrm{m}$ radius. The gas pressure was set to
$10\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$. The scattering force
given by the second term of Eq. (8) was included in the simulations.

All simulations were performed using the Runge-Kutta 4th order algorithm with
a simulation step of $10\text{\,}\mathrm{n}\mathrm{s}$ and 1,000,000 steps.

##  Appendix D Position power spectral density

We now evaluate the particle’s position PSD in the rotating reference. We will
focus on the transverse dynamics given by the equations (14a), and (14b) with
$F_{i}=F_{th,i}$ obeying equations (23), since the PSD for the longitudinal
motion is well-known:

$S_{zz}(\Omega)=\frac{2\gamma_{m}k_{B}T}{m[(\Omega^{2}-\omega_{z}^{2})^{2}+\gamma^{2}\omega_{z}^{2}]}.$
(53)

We start by expressing the differential equations in the frequency domain,

$\displaystyle A_{x}\tilde{x}^{\prime}(\omega)$
$\displaystyle=B\tilde{y}^{\prime}(\omega)+\tilde{F}_{th}(\omega)/m,$ (54a)
$\displaystyle A_{y}\tilde{y}^{\prime}(\omega)$
$\displaystyle=-B\tilde{x}^{\prime}(\omega)+\tilde{F}_{th}(\omega)/m,$ (54b)

where the tilde symbolizes the variables in the frequency domain and we have,

$\displaystyle A_{x}$
$\displaystyle=-\Omega^{2}-\omega_{x}^{2}-\omega^{2}-i\omega\gamma,$ (55a)
$\displaystyle A_{y}$
$\displaystyle=-\Omega^{2}+\omega_{y}^{2}-\omega^{2}-i\omega\gamma,$ (55b)
$\displaystyle B$ $\displaystyle=\gamma\Omega-i2\omega\Omega,$ (55c)

We consider the stochastic thermal force has the same frequency spectrum in
both directions of the transverse plane. Since the above equations are linear
in $\tilde{x}^{\prime}$ and $\tilde{y}^{\prime}$, one can solve explicitly to
find

$\displaystyle\tilde{x}^{\prime}(\omega)$
$\displaystyle=\chi_{x}(\omega)\tilde{F}_{th}$ (56a)
$\displaystyle\tilde{y}^{\prime}(\omega)$
$\displaystyle=\chi_{y}(\omega)\tilde{F}_{th},$ (56b)

where $\chi_{i}(\omega)$ are the mechanical susceptibilities in the rotating
reference frame, given by

$\displaystyle\chi_{x}(\omega)$
$\displaystyle=\frac{1}{m}\frac{A_{y}+B}{B^{2}+A_{x}A_{y}},$ (57a)
$\displaystyle\chi_{y}(\omega)$
$\displaystyle=\frac{1}{m}\frac{A_{x}-B}{B^{2}+A_{x}A_{y}}.$ (57b)

Finally, the position PSDs in the rotating reference frame are

$\displaystyle S_{x^{\prime}x^{\prime}}(\omega)$ $\displaystyle=2m\gamma
k_{B}T|\chi_{x}(\omega)|^{2},$ (58a) $\displaystyle
S_{y^{\prime}y^{\prime}}(\omega)$ $\displaystyle=2m\gamma
k_{B}T|\chi_{y}(\omega)|^{2},$ (58b)

where we used

$\displaystyle S_{ii}(\omega)=|\tilde{i}(\omega)|^{2},$ (59a)
$\displaystyle|\tilde{F}_{th}|^{2}=S_{\tilde{F}_{th}\tilde{F}_{th}}(\omega)=2m\gamma
k_{B}T.$ (59b)

One can use the Fourier transform of the positions in the rotating reference
frame to evaluate the form of the PSDs in the laboratory reference frame. We
can show that,

$\displaystyle F.T.\\{f(t)\cos(\Omega t)\\}$
$\displaystyle=\frac{1}{2}(\tilde{f}(\Omega^{+})+\tilde{f}(\Omega^{-})),$
(60a) $\displaystyle F.T.\\{f(t)\sin(\Omega t)\\}$
$\displaystyle=\frac{1}{2i}(\tilde{f}(\Omega^{+})-\tilde{f}(\Omega^{-})),$
(60b)

where $F.T.\\{\cdot\\}$ is the Fourier transform and
$\Omega^{\pm}=\omega\pm\Omega$. Using this result with the coordinate
transformation

$\displaystyle x^{\prime}$ $\displaystyle=x\cos\theta+y\sin\theta,$ (61)
$\displaystyle y^{\prime}$ $\displaystyle=y\cos\theta-x\sin\theta,$ (62)

one can relate the Fourier transform of the positions from one coordinate to
the other,

$\displaystyle\begin{split}\tilde{x}(\omega)=\frac{1}{2}(\tilde{x}^{\prime}(\Omega^{+})&+\tilde{x}^{\prime}(\Omega^{-}))\\\
&-\frac{1}{2i}(\tilde{y}^{\prime}(\Omega^{+})-\tilde{y}^{\prime}(\Omega^{-})),\end{split}$
(63a)
$\displaystyle\begin{split}\tilde{y}(\omega)=\frac{1}{2i}(\tilde{x}^{\prime}(\Omega^{+})&-\tilde{x}^{\prime}(\Omega^{-}))\\\
&+\frac{1}{2}(\tilde{y}^{\prime}(\Omega^{+})+\tilde{y}^{\prime}(\Omega^{-})).\end{split}$
(63b)

Note that $\tilde{F}_{th}(\Omega^{\pm})=\tilde{F}_{th}(\omega)=\rm{costant}$
since the thermal force is a white noise. Thus, using Eq. (56),

$\displaystyle\begin{split}\tilde{x}(\omega)=\frac{\tilde{F}_{th}}{2}\Big{(}&\chi_{x}(\Omega^{+})+\chi_{x}(\Omega^{-})\\\
+&i(\chi_{y}(\Omega^{+})-\chi_{y}(\Omega^{-}))\Big{)},\end{split}$ (64a)
$\displaystyle\begin{split}\tilde{y}(\omega)=\frac{\tilde{F}_{th}}{2}\Big{(}&\chi_{y}(\Omega^{+})+\chi_{y}(\Omega^{-})\\\
+&i(\chi_{x}(\Omega^{-})-\chi_{x}(\Omega^{+}))\Big{)}.\end{split}$ (64b)

Finally, the particle’s position PSD in the laboratory reference frame is
given by

$\displaystyle\begin{split}S_{xx}=\frac{m\gamma
k_{B}T}{2}\Big{|}\Big{(}&\chi_{x}(\Omega^{+})+\chi_{x}(\Omega^{-})\\\
+&i(\chi_{y}(\Omega^{+})-\chi_{y}(\Omega^{-}))\Big{)}\Big{|}^{2},\end{split}$
(65a) $\displaystyle\begin{split}S_{yy}=\frac{m\gamma
k_{B}T}{2}\Big{|}\Big{(}&\chi_{y}(\Omega^{+})+\chi_{y}(\Omega^{-})\\\
+&i(\chi_{x}(\Omega^{-})-\chi_{x}(\Omega^{+}))\Big{)}\Big{|}^{2}.\end{split}$
(65b)

Figure 9 displays a panel with examples of simulated position PSDs against the
theoretical prediction. The plots in the first row consider the linearized
model and show a good match between theory and simulation. However, a shift in
the natural vibrational frequencies and additional peaks are observed for the
complete optical potential, as seen in the second-row plots.


# Speech and Speaker Recognition from Raw Waveform with SincNet

###### Abstract

Deep neural networks can learn complex and abstract representations, that are
progressively obtained by combining simpler ones. A recent trend in speech and
speaker recognition consists in discovering these representations starting
from raw audio samples directly. Differently from standard hand-crafted
features such as MFCCs or FBANK, the raw waveform can potentially help neural
networks discover better and more customized representations. The high-
dimensional raw inputs, however, can make training significantly more
challenging.

This paper summarizes our recent efforts to develop a neural architecture that
efficiently processes speech from audio waveforms. In particular, we propose
SincNet, a novel Convolutional Neural Network (CNN) that encourages the first
layer to discover meaningful filters by exploiting parametrized sinc
functions. In contrast to standard CNNs, which learn all the elements of each
filter, only low and high cutoff frequencies of band-pass filters are directly
learned from data. This inductive bias offers a very compact way to derive a
customized front-end, that only depends on some parameters with a clear
physical meaning.

Our experiments, conducted on both speaker and speech recognition, show that
the proposed architecture converges faster, performs better, and is more
computationally efficient than standard CNNs.

Index Terms—  ASR, CNN, SincNet, Raw samples.

##  1 Introduction

Deep learning has shown remarkable success in numerous speech tasks [1],
including speech [2, 3] and speaker recognition [4]. This paradigm exploits
the principle of compositionality to efficiently describe the world around us
and employs a hierarchy of representations that are progressively learned by
combining lower-level abstractions. To fully take advantage of deep learning,
it would thus be natural to directly feed neural networks with the lowest
possible signal representation (e.g., pixel for images or raw samples for
audio), avoiding any kind of pre-computed intermediate representations.

Nevertheless, most state-of-the-art neural networks used for speech
applications still employ hand-crafted features, such as FBANK and MFCC
coefficients. These engineered features are originally designed from
perceptual evidence and there are no guarantees that such representations are
optimal for all speech-related tasks. Standard features, for instance, smooth
the speech spectrum, possibly hindering the extraction of crucial narrow-band
speaker characteristics such as pitch and formants. To mitigate this drawback,
some recent works have proposed to directly feed the network with spectrogram
bins [5, 6, 7] or even with raw waveforms [8, 9, 10, 11, 12, 13, 14, 15, 16].
CNNs are the best candidate for processing raw speech samples, since weight
sharing, local filters, and pooling help discover robust and invariant
representations.

We believe that one of the most critical parts of current waveform-based CNNs
is the first convolutional layer. This layer not only deals with high-
dimensional inputs, but it is also more affected by vanishing gradient
problems, especially when employing very deep architectures. As we will show
in this paper, the filters learned by CNNs often take noisy and incongruous
multi-band shapes, especially when few training samples are available. These
filters certainly make some sense for the neural network, but do not appeal to
human intuition, nor appear to lead to an efficient representation of the
speech signal.

To help the CNNs discover more meaningful filters, we recently proposed a
novel convolutional architecture, called SincNet [17], that adds some
constraints on the filter shape. Compared to standard CNNs, where the filter-
bank characteristics depend on several parameters (each element of the filter
vector is directly learned), SincNet convolves the waveform with a set of
parametrized sinc functions that implement band-pass filters [17]. The low and
high cutoff frequencies of the filters are the only parameters learned from
data. This solution still offers considerable flexibility but forces the
network to focus on high-level tunable parameters that have a clear physical
meaning.

In [17] we obtained promising results on both speaker identification and
speaker verification tasks. In particular, SincNet turned out to outperform
standard CNNs fed by both raw samples and standard features. Motivated by
these encouraging results, this paper extends our recent findings to speech
recognition tasks. The speech recognition experiments conducted here are based
on the TIMIT and DIRHA [18, 19] datasets. Results confirm that the proposed
SincNet converges faster, achieves better performance, is more computationally
efficient, and leads to more interpretable filters than a standard CNN.
Interestingly, SincNet achieves a competitive performance also on the DIRHA
dataset, highlighting the effectiveness of this approach even in challenging
scenarios characterized by the presence of both noise and reverberation.

##  2 The SincNet Architecture

The first layer of a standard CNN performs a set of time-domain convolutions
between the input waveform and some Finite Impulse Response (FIR) filters
[20]. Each convolution is defined as follows:

$y[n]=x[n]*h[n]=\sum\limits_{l=0}^{L-1}x[l]\cdot h[n-l],$ (1)

where $x[n]$ is a chunk of the speech signal, $h[n]$ is the filter of length
$L$, and $y[n]$ is the filtered output. In standard CNNs, all the L elements
(taps) of each filter are learned from data. Conversely, the proposed SincNet
performs the convolution with a predefined function $g$ that depends on few
learnable parameters $\theta$ only, as highlighted in the following equation:

$y[n]=x[n]*g[n,\theta].$ (2)

A reasonable choice, inspired by standard filtering in digital signal
processing, is to define $g$ in order to employ rectangular bandpass filters.
In the frequency domain, the magnitude of a generic bandpass filter can be
written as the difference between two low-pass filters:

$G[f,f_{1},f_{2}]=rect\Big{(}\frac{f}{2f_{2}}\Big{)}-rect\Big{(}\frac{f}{2f_{1}}\Big{)},$
(3)

where $f_{1}$ and $f_{2}$ are the learned low and high cutoff frequencies, and
$rect(\cdot)$ is the rectangular function in the magnitude frequency
domain111The phase of the $rect(\cdot)$ function is considered to be linear..
After returning to the time domain (using the inverse Fourier transform [20]),
the reference function $g$ becomes:

$g[n,f_{1},f_{2}]=2f_{2}sinc(2\pi f_{2}n)-2f_{1}sinc(2\pi f_{1}n),$ (4)

where the sinc function is defined as $sinc(x)=sin(x)/x$.

The cut-off frequencies can be initialized randomly in the range
$[0,f_{s}/2]$, where $f_{s}$ represents the sampling frequency of the input
signal. As an alternative, filters can be initialized with the cutoff
frequencies of the mel-scale filter-bank, which has the advantage of directly
allocating more filters in the lower part of the spectrum, where many crucial
speech information is located. Note that the gain of each filter is not
learned at this level. This parameter is managed by the subsequent layers,
which can easily attribute more or less importance to each filter output.

An ideal bandpass filter (i.e., a filter where the passband is perfectly flat
and the attenuation in the stopband is infinite) requires an infinite number
of elements $L$. Any truncation of $g$ thus inevitably leads to an
approximation of the ideal filter, characterized by ripples in the passband
and limited attenuation in the stopband. A popular solution to mitigate this
issue is windowing [20]. Windowing is performed by multiplying the truncated
function $g$ with a window function $w$, which aims to smooth out the abrupt
discontinuities at the ends of $g$. This paper uses the popular Hamming window
that is particularly suitable to achieve high frequency selectivity [20].
However, results not reported here reveals no significant performance
difference when adopting other functions, such as Hann, Blackman, and Kaiser
windows.

All operations involved in SincNet are fully differentiable and the cutoff
frequencies of the filters can be jointly optimized with other CNN parameters
using Stochastic Gradient Descent (SGD) or other gradient-based optimization
routines. A standard CNN pipeline (pooling, normalization, activations,
dropout) can be employed after the first sinc-based convolution. Multiple
standard convolutional, fully-connected or recurrent layers [21, 22, 23] can
then be stacked together to finally perform a classification with a softmax
classifier.

Fig. 1 shows some examples of filters learned by a standard CNN and by the
proposed SincNet for a speaker identification task trained on Librispeech (the
frequency response is plotted between 0 and 4 kHz). As observed in the
figures, the standard CNN does not always learn filters with a well-defined
frequency response. In some cases, the frequency response looks noisy (see the
first CNN filter), while in others it assumes multi-band shapes (see the third
CNN filter). SincNet, instead, is specifically designed to implement
rectangular bandpass filters, leading to more meaningful CNN filters.

(a) CNN Filters

(b) SincNet Filters

Fig. 1: Filters learned by a standard CNN and by the proposed SincNet. The
first row reports the filters in the time domain, while the second one shows
their magnitude frequency response.

###  2.1 Model properties

The proposed SincNet has some remarkable properties:

  * •

Fast Convergence: SincNet forces the network to focus only on the filter
parameters with a major impact on performance. Our architecture actually
implements a natural inductive bias, utilizing knowledge about the filter
shape (similar to feature extraction methods generally deployed on this task)
while retaining flexibility to adapt to data. This prior knowledge makes
learning the filter characteristics much easier, helping SincNet to converge
significantly faster to a better solution.

  * •

Few Parameters: SincNet drastically reduces the number of parameters in the
first convolutional layer. For instance, if we consider a layer composed of
$F$ filters of length $L$, a standard CNN employs $F\cdot L$ parameters,
against the $2F$ considered by SincNet. If $F=80$ and $L=100$, we employ 8k
parameters for the CNN and only 160 for SincNet. Moreover, if we double the
filter length $L$, a standard CNN doubles its parameter count (e.g., we go
from 8k to 16k), while SincNet has an unchanged parameter count. This allows
one to derive very selective filters with many taps, without adding learnable
parameters.

  * •

Computational Efficiency: The proposed function $g$ is symmetric. This means
that we can perform convolution in a very efficient way by only considering
one side of the filter and inheriting the results for the other half. This
saves $50\%$ of the first-layer computation over a standard CNN.

  * •

Interpretability: The SincNet feature maps obtained in the first convolutional
layer are definitely more interpretable and human-readable than other
approaches. The filters, in fact, only depend on parameters with a clear
physical meaning.

Fig. 2: Cumulative frequency response of SincNet and CNN filters on speaker-
id.

##  3 Related Work

Several works have recently explored the use of low-level speech
representations to process audio and speech with CNNs. Most prior attempts
exploit magnitude spectrogram features [5, 6, 7, 24, 25, 26]. Although
spectrograms retain more information than standard hand-crafted features,
their design still requires careful tuning of some crucial hyper-parameters,
such as the duration, overlap, and typology of the frame window, as well as
the number of frequency bins. For this reason, a more recent trend is to
directly learn from raw waveforms, thus completely avoiding any feature
extraction step. This approach has shown promising in speech [8, 9, 27, 28,
10], including emotion tasks [11], speaker recognition [14], and spoofing
detection [13]. Similar to SincNet, some previous works have proposed to add
constraints on the CNN filters, for instance forcing them to work on specific
bands [24, 25].

Differently from the proposed approach, the latter works operate on
spectrogram features and still learn all the L elements of the CNN filters. An
idea related to the proposed method has been recently explored in [26], where
a set of parameterized Gaussian filters are employed. This approach operates
on the spectrogram domain, while SincNet directly considers the raw time
domain waveform. Similarly to our work, in [29] the convolutional filters are
initialized with a predefined filter shape. However, rather than focusing on
cut-off frequencies only, all the basic taps of the FIR filters are still
learned.

This work extends our previous studies on the SincNet [17]. To the best of our
knowledge, this paper is the first that shows the effectiveness of this
architecture in a speech recognition application.

##  4 Results

This section first summarizes the adopted experimental setup. We then discuss
our recent speaker recognition results achieved on TIMIT (462 spks) and
Librispeech (2484 speakers) [30] datasets and then we extend the analysis to
ASR (for more details on the speaker recognition experiments, see [17]).
Beyond adopting TIMIT, speech recognition experiments also consider the DIRHA
dataset for assessing SincNet in challenging noisy and reverberant conditions
[18]. In the spirit of reproducible research, we release the code of SincNet
on GitHub [31]222 at https://github.com/mravanelli/SincNet/..

###  4.1 SincNet Setup

The waveform of each speech sentence was split into chunks of 200 ms (with 10
ms overlap), which were fed into the SincNet architecture. The first layer
performs sinc-based convolutions as described in Sec. 2, using 80 filters of
length $L=251$ samples. The architecture then employs two standard
convolutional layers, both using 60 filters of length 5. Layer normalization
[32] was used for both the input samples and for all convolutional layers
(including the SincNet input layer). Next, three fully-connected layers
composed of 2048 neurons and normalized with batch normalization [33, 34] were
applied. All hidden layers used leaky-ReLU non-linearities. The parameters of
the sinc-layer were initialized using mel-scale cutoff frequencies, while the
rest of the network was initialized with the well-known “Glorot”
initialization scheme [35]. Frame-level speaker and phoneme classifications
were obtained by applying a softmax classifier, providing a set of posterior
probabilities over the targets. For speaker-id, a sentence-level
classification was simply derived by averaging the frame predictions and
voting for the speaker which maximizes the average posterior. Training used
the RMSprop optimizer, with a learning rate $lr=0.001$, $\alpha=0.95$,
$\epsilon=10^{-}7$, and minibatches of size 128. All the hyper-parameters of
the architecture were tuned on development data. Please, refer to the GitHub
repository for more details on the SincNet setup.

We compared SincNet with several alternative systems. First, we considered a
standard CNN fed by the raw waveform. This network is based on the same
architecture as SincNet, but replacing the sinc-based convolution with a
standard one. A comparison with popular hand-crafted features was also
performed. To this end, we computed 39 MFCCs (13
static+$\Delta$+$\Delta\Delta$) and 40 FBANKs using the Kaldi toolkit [36]. A
CNN was used for FBANK features, while a Multi-Layer Perceptron (MLP) was used
for MFCCs.

###  4.2 Speaker Recognition

Fig. 2 shows the cumulative frequency response of the filters learned by
SincNet and CNN on a speaker-id task trained with Librispeech. The cumulative
frequency response is obtained by summing up all the discovered filters and is
useful to highlight which frequency bands are covered by the learned filters.

Interestingly, there are three main peaks which clearly stand out from the
SincNet plot (see the red line in the figure). The first one corresponds to
the pitch region (the average pitch is 133 Hz for a male and 234 for a
female). The second peak (approximately located at 500 Hz) mainly captures
first formants, whose average value over the various English vowels is indeed
500 Hz. Finally, the third peak (ranging from 900 to 1400 Hz) captures some
important second formants, such as the second formant of the vowel $/a/$,
which is located on average at 1100 Hz. This filter-bank configuration
indicates that SincNet has successfully adapted its characteristics to speaker
identification. Conversely, the CNN does not exhibit such a meaningful
pattern: its filters tend to focus on the lower part of the spectrum, but
peaks tuned on first and second formants do not clearly appear.

Table 1: Performance of speaker identification (first two columns) and speaker verification (last column) on the considered corpora. | TIMIT | LibriSpeech | LibriSpeech
---|---|---|---
| CER(%) | CER(%) | EER(%)
DNN-MFCC | 0.99 | 2.02 | 0.88
CNN-FBANK | 0.86 | 1.55 | 0.60
CNN-Raw | 1.65 | 1.00 | 0.58
SincNet-Raw | 0.85 | 0.96 | 0.51

Table 1 reports the performance achieved on speaker identification and
verification tasks (Classification Error Rates - CER% for speaker-id task and
Equal Error Rate - EER% for speaker verification). The table shows that
SincNet outperforms other systems on both TIMIT (462 speakers) and Librispeech
(2484 speakers) datasets. The speaker verification system was derived from the
speaker-id neural network using the d-vector approach [37, 7], which relies on
the output of the last hidden layer and computes the cosine distance between
the enrollment and the test speaker d-vectors (see [17] for more details). Ten
utterances from impostors were randomly selected for each sentence from a
genuine speaker. Note that to assess our approach on a standard open-set
speaker verification task, all the impostors were taken from a speaker pool
different from that used for training the speaker-id DNN. The last column of
Table 1 extends our validation to speaker verification, reporting the EER(%)
achieved with Librispeech. All DNN models show promising performance, leading
to an EER lower than 1% in all cases. The table highlights that SincNet
outperforms the other models, showing a relative improvement of about 11% over
the standard CNN.

Fig. 3: Cumulative frequency responses obtained on a noisy speech recognition
task. As shown in the spectrogram, noise has been artificially added into the
band 2.0-2.5 kHz. Both the CNN and SincNet learn to avoid the noisy band, but
SincNet learns it much faster.

###  4.3 Speech Recognition

To validate the effectiveness of our model, Fig. 3 shows the cumulative
frequency response obtained on a noisy speech recognition task. In this
experiment, reported here as a showcase, we have artificially corrupted TIMIT
with a significant quantity of noise in the band between 2.0 and 2.5 kHz (see
the spectrogram) and we have analyzed how fast a CNN and the SincNet
architectures learn to avoid such a useless band. The second row of sub-
figures compares the CNN and the SincNet at an early training stage (i.e.,
after processing only one hour of speech), while the last row shows the
cumulative frequency responses at the end of training. From the figures
emerges that both CNN and SincNet have correctly learned to avoid the
corrupted band at end of training, as highlighted by the visible holes between
2.0 and 2.5 kHz in the cumulative frequency responses. SincNet, however,
learns to avoid such a noisy band much earlier. In the second row of sub-
figures, in fact, SincNet shows a valley in the cumulative spectrum even after
processing only one hour of speech, while CNN will learn it only at a later
training stage. At the end of training, the cumulative frequency responses of
SincNet and CNN look rather different. SincNet, for instance, seems to exploit
a larger bandwidth. Different from the CNN, in fact, it employs with several
filters also the part of the spectrum between 0.5-2.0 kHz.

Table 2: ASR performance obtained on the TIMIT and DIRHA. | TIMIT | DIRHA
---|---|---
| PER(%) | WER(%)
CNN-FBANK | 18.3 | 40.1
CNN-Raw | 18.1 | 40.1
SincNet-Raw | 17.2 | 37.2

Tab. 2 reports the speech recognition performance obtained by CNN and SincNet
using the TIMIT and the DIRHA dataset [18]. To ensure a more accurate
comparison between the architectures, five experiments varying the
initialization seeds were conducted for each model and corpus. Table 2 thus
reports the average speech recognition performance. Standard deviations, not
reported here, range between $0.15$ and $0.2$ for all the experiments.

For all the datasets, SincNet outperforms both a CNN trained on standard FBANK
coefficients and CNN trained or the raw waveform. The latter result confirms
the effectiveness of SincNet not only in close-talking scenarios but also in
noisy conditions characterized by the presence of both noise and
reverberation. As shown in Fig.3, SincNet is able to effectively tune its
filter-bank front-end to better address the characteristics of the noise. It
is worth noting that the relative performance gain obtained in this
challenging scenario is slightly higher than that obtained in standard close-
talking conditions (6% against 4%).

##  5 Conclusions and Future Work

This paper proposed SincNet, a neural architecture for directly processing
waveform audio. Our model, inspired by the way filtering is conducted in
digital signal processing, imposes constraints on the filter shapes through
efficient parameterization. SincNet has been extensively evaluated on
challenging speaker and speech recognition task, consistently showing some
benefits on network convergence, performance, and computational efficiency.
Moreover, analysis of the SincNet filters revealed that the learned filter-
bank is tuned to the specific task addressed by the neural network.

Inspired by the promising results obtained in this paper, we will explore the
use of SincNet for supervised and unsupervised speaker/environmental
adaptation. We believe that the proposed approach defines a general paradigm
to process time-series and can be applied in numerous other fields. Our future
effort will be thus devoted to extending to other tasks, such as emotion
recognition, speech separation, and music processing.

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