# Self-similarity and recurrence in stability spectra of near-extreme Stokes
waves

B. Deconinck1  S. A. Dyachenko2  A. Semenova1<EMAIL_ADDRESS>1Department of
Applied Mathematics, University of Washington 2Department of Mathematics, SUNY
Buffalo.

###### Abstract

We consider steady surface waves in an infinitely deep two–dimensional ideal
fluid with potential flow, focusing on high-amplitude waves near the steepest
wave with a 120$\degree$ corner at the crest. The stability of these solutions
with respect to coperiodic and subharmonic perturbations is studied, using new
matrix-free numerical methods. We provide evidence for a plethora of
conjectures on the nature of the instabilities as the steepest wave is
approached, especially with regards to the self-similar recurrence of the
stability spectrum near the origin of the spectral plane.

###### keywords:

Water waves, Stokes waves, Benjamin-Feir instability, High-frequency
instability, superharmonic instability

##  1 Introduction

We consider the stability of spatially periodic waves that propagate with
constant velocity without change of form, in potential flow of an ideal
(incompressible and inviscid) two-dimensional fluid of infinite depth. The
study of such wave profiles has been the subject of much previous work, going
back to Stokes (1847) (also published in Stokes (1880a)). Stokes’ work was
followed by numerical computation of these waves by Michell (1893), and
existence was proved in the works of Nekrasov (1921) and Levi-Civita (1925),
see also Toland (1996); Hur (2006) for existence of the global branch of wave
profiles. The numerical study of these waves and the nature of their
singularities was continued by Grant (1973), Schwartz (1974), Williams (1981),
Williams (1985), Tanveer (1993), Cowley et al. (1999), Baker & Xie (2011) and
others.

In the context of water waves, such waves are usually referred to as Stokes
waves. It was suggested by Stokes (1880b) that there exists a progressive wave
of maximum height, and that the angle at the crest of this limiting wave
should be $2\pi/3$. Rigorous proofs of these Stokes conjectures came much
later. Toland (1978) showed global existence of the limiting Stokes wave, but
did not prove that the angle at the crest is $2\pi/3$. This result was proved
by Amick et al. (1982) and Plotnikov (1982) (Reported in English in Plotnikov
(2002)) independently. We refer to the Stokes wave of greatest height as the
extreme wave and to waves of near-maximal amplitude as near-extreme waves.
Even with the original Stokes conjectures resolved, the study of the graph of
the wave profiles remains active, with a number of open problems, as detailed
by Dyachenko et al. (2023), see also below. The works by Longuet-Higgins & Fox
(1977, 1978) and by Longuet-Higgins (2008) study the near-extreme waves using
both asymptotic and numerical methods. The review by Haziot et al. (2022)
discusses many currently active research directions.

The investigation of the stability of Stokes waves was begun in the works of
Benjamin (1967), Benjamin & Feir (1967), Lighthill (1965) and Whitham (1967).
Except for the influential experimental work by Benjamin & Feir (1967), the
focus of these works was on the dynamics of small disturbances of small-
amplitude periodic Stokes waves. They unveiled the presence of the
modulational or Benjamin-Feir instability with respect to long-wave
disturbances in water of sufficient depth, $kh>1.363\ldots$. Here $h$ is the
depth of the water and $k=2\pi/L$, with $L$ the period of the Stokes wave. The
first rigorous results on the Benjamin-Feir instability were established by
Bridges & Mielke (1995), followed up very recently by Nguyen & Strauss (2023)
and Hur & Yang (2023). The numerical results of Deconinck & Oliveras (2011)
reveal the presence of a figure-8 curve in the complex plane of the spectrum
of the linear operator governing the linear evolution of the Stokes wave
disturbances. Approximations to this figure-8 are obtained by Creedon &
Deconinck (2023) and by Berti et al. (2022), where the existence of the
figure-8 was proven rigorously. Berti et al. (2023) also examined the critical
case $kh=1.363\ldots$.

Deconinck & Oliveras (2011) also brought to the fore the presence of the so-
called high-frequency instabilities, existing for narrow ranges of the
disturbance quasi-periods. These instabilities were further studied by Creedon
et al. (2022) and by Hur & Yang (2023), where their existence was proven
rigorously.

The instabilities mentioned above play a role in our study of the dynamics of
large-amplitude Stokes waves, but we illustrate other instability mechanisms,
not present for small-amplitude waves. Understandably, the study of large-
amplitude Stokes waves, which cannot be thought of as perturbations of flat
water, is harder, both from a computational and an analytical point of view.
Nonetheless, some groundbreaking examinations have been done, for instance by
Tanaka (1983), Longuet-Higgins & Tanaka (1997) and for near-extreme waves by
Korotkevich et al. (2023). These authors all consider perturbations of the
Stokes waves with respect to co-periodic (or superharmonic) disturbances,
i.e., the Stokes waves and the disturbance have the same minimal period. Their
results are recapped in detail below, as they are instrumental to our own
investigations. The results in this manuscript follow those of Deconinck et
al. (2023), as we present a computational study of the instabilities of
periodic Stokes waves, under the influence of disturbances parallel to the
propagation direction of the wave. It should be emphasized that all figures
presented below are quantitatively correct unless they are described as
“schematic” in the caption. Similarly, all floating-point numbers given are
approximate, of course, but all digits provided are believed to be correct.

##  2 One-dimensional waves in water of infinite depth

The equations of motion governing the dynamics of the one-dimensional free
surface of a two-dimensional irrotational, inviscid fluid (see left panel of
Fig. 1) are the Euler equations:

$\displaystyle\Phi_{xx}+\Phi_{yy}=0,$ $\displaystyle
x\in\mathbb{R},~{}y\in(-\infty,\eta(x,t)),$ (1)
$\displaystyle\eta_{t}=\left.\left(-\Phi_{x}\eta_{x}+\Phi_{y}\right)\right|_{y=\eta(x,t)},$
$\displaystyle x\in\mathbb{R},~{}y=\eta(x,t),$ (2)
$\displaystyle\left.\left(\Phi_{t}+\frac{1}{2}\left(\Phi_{x}^{2}+\Phi_{y}^{2}\right)\right)\right|_{y=\eta(x,t)}+g\eta=0,$
$\displaystyle x\in\mathbb{R},~{}y=\eta(x,t),$ (3)
$\displaystyle\lim_{y\rightarrow-\infty}\Phi_{y}=0,$ $\displaystyle
x\in\mathbb{R}.$ (4)

Here $y=\eta(x,t)$ is the equation of the free surface and $\Phi(x,y,t)$ is
the velocity potential (i.e., the velocity in the fluid is
$v=(\Phi_{x},\Phi_{y})$), subscripts denote partial derivatives, $x$ and $y$
are the horizontal and vertical coordinate respectively, $t$ denotes time, and
$g$ is the acceleration of gravity. We ignore the effects of surface tension.
Although the Stokes waves are $2\pi$-periodic, it is important to pose the
problem above on $x\in\mathbb{R}$, since the perturbations we consider are not
necessarily periodic. The first equation expresses the divergence-free
property of the flow under the free surface determined by $\eta(x,t)$. The
second and third equation are nonlinear boundary conditions determining the
free surface: the kinematic condition (2) expresses that the free surface
changes in the direction of the normal derivative to the surface (particles on
the surface remain on the surface), whereas the dynamic condition (3) states
the continuity of pressure across the surface. Atmospheric pressure has been
equated to zero, without loss of generality.

Since the location of the surface $y=\eta(x,t)$ is the main focus of the water
wave problem, different reformulations have been developed that eliminate the
velocity potential in the bulk of the fluid as an unknown. Zakharov (1968)
shows how the problem (1)–(4) can be recast in terms of only the surface
variables $\eta(x,t)$ and $\varphi(x,t)=\Phi(x,\eta(x,t),t)$, and the dynamics
of $\eta(x,t)$ and $\varphi(x,t)$ is governed by an infinite-dimensional
Hamiltonian system with $\eta(x,t)$ and $\varphi(x,t)$ as canonical variables.
The Hamiltonian is the total energy of the system (with potential energy
renormalized to account for infinite depth), which depends on the velocity
potential $\Phi(x,y,t)$ in the bulk of the fluid.

To avoid the dependence on the bulk, Zakharov’s formulation uses the
Dirichlet-to-Neumann operator (DNO), producing the normal derivative of the
velocity potential at the free surface (the right-hand side of (2)) from the
values of $\varphi(x,t)$. For small-amplitude waves, the DNO is conveniently
expressed as a series, as done by Craig & Sulem (1993). For large-amplitude
waves, such an expansion is not readily available, and the DNO has to be
approximated numerically. To avoid doing so, we use conformal variables, see
Fig. 1: for a $2\pi$-periodic wave, a time-dependent conformal transformation
maps the half plane in the $w=u+iv$ plane
($(u,v)\in[-\pi,\pi]\times(-\infty,0]$) into the area
$(x,y)\in[-\pi,\pi]\times(-\infty,\eta]$ in the physical $z=x+iy$ plane
occupied by the fluid. The horizontal line $v=0$ is mapped into the fluid
surface $y=\eta(x,t)$. The implicit equations of motion in conformal variables
are constructed in the works of Ovsyannikov (1973), see also Tanveer (1991),
Zakharov et al. (1996), Dyachenko (2001). We use this implicit formulation to
study the stability of Stokes waves.

Figure 1: A schematic of the conformal map from the lower half plane (right)
to the physical domain (left).

From these works, the conformal map $z(w,t)=x(w,t)+iy(w,t)$ is a complex-
analytic function in $\mathbb{C}^{-}$ that approaches the identity map
$z(w,t)\to w$ as $w\to-i\infty$, the image of a point at infinite depth. In
the conformal variables, the Hamiltonian has the form

$\displaystyle\mathcal{H}=\frac{1}{2}\int_{-\pi}^{\pi}\psi\hat{k}\psi\,du+\frac{g}{2}\int_{-\pi}^{\pi}y^{2}x_{u}\,du,$
(5)

where $\psi(u,t)=\varphi(x,t)$ and the operator
$\hat{k}=-\hat{H}\partial_{u}$. Here $\hat{H}$ is the periodic Hilbert
transform defined by the principal-value integral

$\displaystyle\hat{H}f(u)=\frac{1}{2\pi}\fint^{\pi}_{-\pi}f(u^{\prime})\cot{\frac{u^{\prime}-u}{2}}du^{\prime}.$
(6)

Equivalently, the Hilbert transform can be defined by its action on Fourier
harmonics, $\hat{H}e^{iku}=i\,\mbox{sign}(k)\,e^{iku}$.The equations of motion
are derived by taking variational derivatives of the action
$\mathcal{S}=\int\mathcal{L}\,dt$ with respect to $x$, $y$ and $\psi$. The
Lagrangian has the form

$\displaystyle\mathcal{L}\\!=\\!\int_{-\pi}^{\pi}\\!\\!\\!\psi\left(y_{t}x_{u}-y_{u}x_{t}\right)du\\!-\\!\mathcal{H}\\!+\\!\int_{-\pi}^{\pi}\\!\\!\\!f(u)\left(y-\hat{H}\left[x-u\right]\right)du,$
(7)

where the Lagrange multiplier $f(u)$ is chosen to enforce the relation
$x(u,t)=u-\hat{H}\left[y(u,t)\right]$. We refer to the work of Dyachenko et
al. (1996) for the complete derivation of the equations of motion in conformal
variables:

$\displaystyle y_{t}x_{u}-y_{u}x_{t}$ $\displaystyle=-\hat{H}\psi_{u},$ (8)
$\displaystyle
x_{t}\psi_{u}-x_{u}\psi_{t}-\hat{H}\left[y_{t}\psi_{u}-y_{u}\psi_{t}\right]$
$\displaystyle=g\left(x_{u}y-\frac{1}{2}\hat{H}\partial_{u}y^{2}\right).$ (9)

###  2.1 Traveling Waves

Using the conformal variables formulation (8-9), the Stokes waves are obtained
by looking for a solution $y=y(u-ct)$, $\psi=\psi(u-ct)$, corresponding to
stationary solutions in a frame of reference moving with constant speed $c$ in
physical variables, see Dyachenko et al. (2014). This gives rise to the so-
called Babenko (1987) equation:

$\displaystyle\left(c^{2}\hat{k}-g\right)y-\frac{g}{2}\left(\hat{k}y^{2}+2y\hat{k}y\right)=0.$
(10)

Since we are interested in the stability of near-extreme Stokes waves, the
accurate numerical solution of (10) for near-limiting values of the speed is
required. Details of such computations for the Babenko equation (10) are given
by Dyachenko et al. (2016). In what follows, the ratio of the crest-to-trough
height $H$ to wavelength $L$ is used as the definition of wave steepness
$s=H/L$. The limiting Stokes wave has the steepness
$s_{lim}=0.1410634839\ldots$ and speed $c_{lim}=1.0922850485\ldots$ as
computed by Dyachenko et al. (2023).

It is known that the speed $c$ and the Hamiltonian ${\cal H}$ oscillate as a
function of the wave steepness $s$, for values near the limiting value
$s_{lim}$, see Longuet-Higgins & Fox (1978). It is believed there is an
infinite number of such oscillations. They are presented schematically in Fig.
2. The other details of this figure constitute some of the main results of
this paper, discussed below.

Figure 2:  A schematic of the oscillations of the Hamiltonian ${\cal H}$
(green) and the velocity $c$ (red) relative to their limit values as steepness
$s$ increases to its limiting 120$\degree$ Stokes wave value, using a
logarithmic scale for the steepness $s$ relative to its limit value $s_{lim}$.
In the green region, the waves are unstable with respect to the first
Benjamin-Feir branch and the first localized instability branch, see Section
3. In the yellow region, they are unstable with respect to the second
branches, and so on. The steepnesses $s_{c,n}$ and $s_{H,n}$ correspond to the
steepness values where the velocity $c$ and the Hamiltonian ${\cal H}$ have
extreme values.

###  2.2 Linearization about a Stokes wave

In a reference frame traveling with the velocity of a Stokes wave, the Babenko
equation describes a stationary solution of the equations (8)–(9). The linear
stability of these Stokes waves is determined by the eigenvalue spectrum of
the linearization in this traveling frame. To obtain the linearization, we
transform (8-9) to a moving frame, using

$\displaystyle y(u,t)$ $\displaystyle\to y(u-ct,t),\quad\mbox{and}\quad
x(u,t)\to u-\hat{H}y(u-ct,t),$ (11) $\displaystyle\psi(u,t)$
$\displaystyle\to\phi(u-ct,t)-c\hat{H}y(u-ct,t),$ (12)

as in Dyachenko & Semenova (2023a). The nonlinear system (8–9) in the moving
frame becomes

$\displaystyle x_{u}y_{t}-y_{u}x_{t}$
$\displaystyle=-\hat{H}\left(\phi_{u}-c\right),$ (13) $\displaystyle
x_{u}\phi_{t}-x_{t}\phi_{u}-\hat{H}\left[y_{u}\phi_{t}-y_{t}\phi_{u}\right]+2cx_{t}$
$\displaystyle=c^{2}\hat{k}y-g\left(x_{u}y-\hat{H}\left[yy_{u}\right]\right).$
(14)

The linearization about a Stokes wave is found by substituting

$\displaystyle y(u,t)$ $\displaystyle\to y(u)+\delta
y(u,t)+\ldots\quad\mbox{and}\quad\phi(u,t)\to 0+\delta\phi(u,t)+\ldots.$ (15)

Here $y(u)$ corresponds to the Stokes wave and $\delta y(u,t)$,
$\delta\phi(u,t)$ are small perturbations. Retaining only linear terms in
$\delta y$ and $\delta\phi$ leads to the following evolution equations for the
perturbations $\delta y$ and $\delta\phi$:

$\displaystyle\hat{\Omega}_{21}^{\dagger}\delta\phi_{t}-2c\hat{H}\delta y_{t}$
$\displaystyle=\hat{S}_{1}\delta y,$ (16)
$\displaystyle\hat{\Omega}_{21}\delta y_{t}$
$\displaystyle=\hat{k}\delta\phi,$ (17)

where the operator $\hat{\Omega}_{21}$ is defined via
$\hat{\Omega}_{21}f=x_{u}f+y_{u}\hat{H}f$. The operator
$\hat{\Omega}^{\dagger}_{21}$ is its adjoint:
$\hat{\Omega}^{\dagger}_{21}f=x_{u}f-\hat{H}\left[y_{u}f\right]$ and

$\displaystyle\hat{S}_{1}(y)\delta y=\left(c^{2}\hat{k}-g\right)\delta
y-g\left[y\hat{k}\delta y+\delta y\hat{k}y+\hat{k}(y\delta y)\right].$ (18)

This is rewritten in matrix form as

$\displaystyle\hat{L}\partial_{t}\left[\begin{array}[]{c}\delta\phi\\\ \delta
y\end{array}\right]=\hat{M}\left[\begin{array}[]{c}\delta\phi\\\ \delta
y\end{array}\right],$ (23)

with

$\displaystyle\hat{L}=\begin{bmatrix}0&\hat{\Omega}_{21}\\\
\hat{\Omega}_{21}^{\dagger}&-2c\hat{H}\end{bmatrix},\,\,\hat{M}=\begin{bmatrix}\hat{k}&0\\\
0&\hat{S}_{1}\end{bmatrix}.$ (24)

We examine the effect of quasi-periodic perturbations $\delta y$, $\delta\phi$
using the Fourier-Floquet-Hill (FFH) approach described in Deconinck & Kutz
(2006) and Deconinck & Oliveras (2011). The time dependence for $\delta y$ and
$\delta\phi$ is found using separation of variables. Moreover, in order to
consider quasiperiodic perturbations we use a Floquet-Bloch decomposition in
space,

$\displaystyle\left[\begin{array}[]{c}\delta y(u,t)\\\
\delta\phi(u,t)\end{array}\right]=e^{i\lambda t+i\mu
u}\left[\begin{array}[]{c}\delta y(u)\\\ \delta\phi(u)\end{array}\right],$
(29)

where $\mu\in(-1/2,1/2]$ is the Floquet parameter and
$\lambda(\mu)\in\mathbb{C}$ is the eigenvalue. The resulting $\mu$-dependent
spectral problem is solved using a Krylov-based method and the shift-and-
invert technique, see Dyachenko & Semenova (2023b). Details on Krylov methods
are presented by Stewart (2002). We refer to the spectrum obtained this way as
the stability spectrum of the Stokes wave. Note that the Floquet parameter is
defined modulo 1, thus $\mu=1/2$ is equivalent to $\mu=-1/2$, see Deconinck &
Kutz (2006).

##  3 Instability

###  3.1 The oscillating velocity and Hamiltonian

Both the velocity $c$ and the Hamiltonian ${\cal H}$ depend on the Stokes
wave. As the steepness $s$ of the Stokes wave increases and approaches its
limiting value $s_{lim}$, both quantities are not monotone, as observed by
Longuet-Higgins (1975). In fact, Longuet-Higgins & Fox (1977, 1978) produce an
asymptotic result that implies the presence of an infinite number of
oscillations for both quantities. These oscillations were studied more by
Maklakov (2002), Dyachenko et al. (2016) and Lushnikov et al. (2017), and very
recently by Silantyev (2019). To our knowledge, no proof of an infinite number
of oscillations in velocity $c$ and Hamiltonian ${\cal H}$ exists.

We denote the steepness of a wave at the $n$th turning point of the speed by
$s_{c,n}$, $n=0,1,2,\ldots$, with $s_{c,0}=0$. Similarly, the $n$th extremizer
of the Hamiltonian is denoted by $s_{H,n}$. These critical values of the
velocity and the Hamiltonian are important for changes in the stability
spectrum, as shown below. For the Hamiltonian, the importance of these values
is known, due to the works of Tanaka (1983, 1985); Saffman (1985); Longuet-
Higgins & Tanaka (1997), for instance. It appears that these extremizing
values interlace, so that $s_{c,n}<s_{H,n+1}<s_{c,n+1}$, $n=0,1,\ldots$.

In the recent work of Dyachenko & Semenova (2023b), a conjecture is made about
an infinite number of secondary bifurcations associated with the Floquet
multiplier $\mu=1/2$, corresponding to perturbations whose period is twice
that of the Stokes wave. It is unclear how these bifurcations are related to
those in the works of Chen & Saffman (1980), Longuet-Higgins (1985) and
Zufiria (1987), since those works do not introduce a Floquet parameter.
However, their importance to the stability results presented here is
demonstrated below. We denote the steepness associated with the $n$th
secondary bifurcation point by $s_{n}^{1/2}$, $n=1,2,\ldots$. Further, we
observe that $s_{c,n}<s_{n}^{1/2}<s_{H,n}$. These values are included in the
schematic of Fig. 2.

It is convenient to break up the range of steepness $s$ from $s=0$ to
$s=s_{lim}$ in intervals from one extremizer of the wave speed to the next.
For example, the first interval starts at the primary bifurcation $s=0$ and
ends at the first maximizer of the wave speed $s=s_{c,1}$; the second interval
starts at $s=s_{c,1}$ and ends at the first minimizer of the wave speed
$s_{c,2}$, and so on. The length of each interval shrinks as the extreme wave
is approached, and following Longuet-Higgins & Fox (1978), we use a
logarithmic scaling as illustrated in Fig. 2. Note that, because of the
observed interlacing of the extremizers of the wave speed, those of the
Hamiltonian, and the secondary bifurcations, each interval contains one
extremizer of the Hamiltonian and one secondary bifurcation point.

###  3.2 A cyclus of changes in the spectrum

As the steepness increases and each interval is traversed, an instability
emerges from $\lambda=0$ in the spectral plane, giving rise to a sequence of
$\lambda(\mu)$-curves with changing topology, see Fig. 3. These changes for
$s\in[s_{c,0}=0,s_{c,1})$ are described below.

  1. 1.

Initially, at $s_{c,0}$, a figure-8 emerges from the origin, expanding in size
as steepness increases, Panels (a) and (b).

  2. 2.

At an isolated value of the steepness $s=s_{h,1}$, both tangents of the
figure-8 at the origin become vertical, resulting in an hourglass shape, Panel
(c).

  3. 3.

Next, the lobes of the figure-8 detach from the origin, forming two disjoint
isles qualitatively reminiscent of the high-frequency instabilities of
Deconinck & Oliveras (2011). The band of Floquet values parameterizing the
isles shrinks away from $\mu=0$ as the steepness increases, Panels (d)-(h).

  4. 4.

At $s=s_{1}^{1/2}$, eigenvalues with Floquet parameter $\mu=1/2$ bifurcate
away from the origin onto the real line, creating an oval of eigenvalues with
center at the origin, parameterized by Floquet values centered about
$\mu=1/2$, see Panels (f)-(h).

  5. 5.

As the steepness increases, the oval around the origin deforms to a bean
shape, eventually re-absorbing the remnants of the figure-8, Panel (j). More
detail on the changes in these remnants and their re-absorption is presented
in Figs. 5 and 8.

  6. 6.

At $s=s_{H,1}$, the bean shape pinches to form a figure-$\infty$. The double
point of the figure-$\infty$ is at the origin and has a Floquet parameter
$\mu=0$. Thus it corresponds to perturbations with the same period as the
Stokes wave, see Korotkevich et al. (2023); Dyachenko & Semenova (2023a). In
Panel (k) this co-periodic (or superharmonic) eigenvalue is marked in green.
The unstable eigenvalue with $\mu=1/2$ is marked in red and gives rise to the
most unstable mode for the wave with steepness $s=s_{H,1}$.

  7. 7.

As $s$ increases beyond $s_{H,1}$, the figure-$\infty$ splits off from the
origin into a pair of symmetric lobes, one moving to the right, the other to
the left, Panel (l). Further interesting changes in the shape of these lobes
are observed as the steepness increases and they move away from the origin,
see Deconinck et al. (2023) and Fig. 4. Importantly, we observe that the most
unstable mode for this range of steepness $s$ is either co-periodic with the
Stokes wave ($\mu=0$, green dot in Fig. 4) or has twice its period, i.e., it
is subharmonic with $\mu=1/2$ (red dot in Fig. 4). Figure 4 illustrates two
interchanges between these modes. We conjecture that such interchanges recur
an infinite number of times as $s\rightarrow s_{lim}$. Note that the
difference between $s_{lim}=0.1410634839$ and the steepness in the final panel
of Fig. 4 is about $0.000425$ or $0.3\%$.

Below we focus on what happens near the origin of the spectral plane as the
steepness continues to increase, ever getting closer to its extreme value.

Figure 3:  Spectra in the vicinity of the origin for increasing steepness $s$.
A detailed description is found in the main text. Here (a) $s=0.0449032652$,
(b) $s=0.1042102092$, (c) $s=0.1090618215$, (d) $s=0.1122542820$, (e)
$s=0.1214481620$, (f) $s=0.1289100582$, (g) $s=0.1292029131$, (h)
$s=0.1307253066$, (j) $s=0.1364173038$, (k) $s=0.1366036552$, (l)
$s=0.1368557681$. Note that Panel (h) is repeated, with changing scales.

Figure 4: Spectrum component lobe with highest real part, for increasing
steepness. A detailed description is found in the main text. Here (a)
$s=0.1394245282$, (b) $s=0.1394647831$, (c) $s=0.1394802926$, (d)
$s=0.1394894509$, (e) $s=0.1394970022$, (f) $s=0.1395148411$, (g)
$s=0.1395380437$, (h) $s=0.1395744737$, (i) $s=0.1405658442$, (j)
$s=0.1405850778$, (k) $s=0.1405964046$, (l) $s=0.1406007221$, (m)
$s=0.1406050801$, (n) $s=0.1406080087$, (o) $s=0.1406169126$, (p)
$s=0.1406384552$.

Figure 5: Changes in the Benjamin-Feir remnant as it approaches the oval at
the origin, with steepness (from left to right panels): $s=0.1307253066$,
$s=0.1323979204$, and $s=0.1329490573$.

###  3.3 The Benjamin-Feir Instability

We observe that a figure-8 shape in the stability spectrum emerges from the
origin when the steepness $s=s_{c,n}$, an extremizer of the velocity. The
first three extrema of the velocity appear at the following values:

$\displaystyle s_{c,0}$ $\displaystyle=0,$ (30) $\displaystyle s_{c,1}$
$\displaystyle=0.138753,$ (31) $\displaystyle s_{c,2}$
$\displaystyle=0.140920.$ (32)

For small-amplitude waves (i.e., waves with steepness $s$ near $s_{c,0}=0$),
the figure-8 corresponds to the well-studied classical Benjamin-Feir, or the
modulational instability. In what follows we refer to instabilities manifested
through a figure-8 in the spectral plane as Benjamin-Feir instabilities. We
refer to the Benjamin-Feir instability branches starting at steepness
$s_{c,n}$ as the $(n+1)$-th Benjamin-Feir branch, denoted BFI, BFII, BFIII,
and so on. Below, we show that eigenvalues on the figure-8 near the origin
(for BFII and BFII) give rise to modulational instabilities, as they do for
small-amplitude waves, see Benjamin (1967); Whitham (1967).

All the Benjamin-Feir branches that we compute experience the sequence of
changes for increasing steepness described above: they grow in size, their
tangents at the origin become vertical followed by pinching off of the
figure-8, resulting in the formation of isole on the positive and negative
imaginary axis. For each branch BFI, BFII and BFIII, we determine the figure-8
that gives rise to the eigenvalue with the largest real part, i.e., the
maximal growth rate, see Fig. 6. Table 1 displays these values of steepness,
the corresponding eigenvalue with maximal real part and its Floquet exponent,
for BFI, BFII and BFIII. These computations illustrate that the widest
figure-8 (green curves in Fig. 6) settles down to a universal shape as the
extreme wave is approached, since the second and third panel appear
indistinguishable). The values in Table 1 confirm this visual inspection.
Further, for BFI, BFII and BFIII, we compute the hourglass shapes resulting
from the figure-8’s with vertical tangents at the origin, see Fig. 6. We
overlay these shapes in Fig. 7, plotting the real part of the spectrum as a
function of the Floquet parameter. This figure illustrates convergence to a
universal hourglass shape as the steepest wave is approached. We conjecture
that an infinite number of Benjamin-Feir instability branches exist as the
steepest wave is approached and that all of them experience a universal
sequence of transitions.

Finally, using the points marked by triangles in Fig. 6, we examine the
eigenfunctions of (23). The eigenfunctions related to BFI, BFII and BFIII are
visibly different, while their spectra in Figs. 6 and 7 are, to the eye,
indistinguishable. A second observation is that these eigenfunctions are
indeed modulational in nature: Fig. 7 displays $10~{}(=1/\mu)$ Stokes wave
periods of the eigenfunctions. Although their effect is increasingly localized
at the crest in each wave period, there is a more global modulational effect
when many periods are considered. Thus even the BFII and BFIII instabilities,
for $\mu$ close to zero, deserve the modulational instability moniker.

Figure 6:  The figure-8 component of the eigenvalue spectrum, showing the Benjamin-Feir instability branches BFI, BFII and BFIII for Stokes waves of steepness $s=0.1045109822$ (left, green) and $s=0.1092129256$ (left, red); $s=0.1398401087$ (center, green) and $s=0.1401021466$ (center, red); $s=0.1409908317$ (right, green) and $s=0.1410079370$ (right, red). The green curves are associated with the maximal instability growth on the corresponding Benjamin-Feir branch, with the black points marking the eigenvalues with largest real part, given in Table 1. The red hourglass curves correspond to the steepness when the figure-8 tangents at the origin become vertical, leading to the figure-8 detaching from the origin in the spectral plane. The points marked by gold triangles close to the origin have Floquet exponent $\mu=1/10$.  | $s$ | $\lambda$ | $\mu$
---|---|---|---
BFI | 0.1045109822 | 0.0235702 + 0.2029603i | 0.46376
BFII | 0.1398401087 | 0.0288896 + 0.1746076i | 0.45743
BFIII | 0.1409908317 | 0.0288299 + 0.1747554i | 0.45774
Table 1: The Benjamin-Feir instability parameters for the figure-8 with the
largest growth rate, first three branches.

Figure 7: (Left Panel) The growth rate Re $\lambda$ as a function of the
Floquet exponent $\mu$ for BFI, BFII and BFIII are shown in purple, dashed
green and dashed red, respectively. The curves are associated with hourglass
cases of the BFI, BFII and BFIII figure-8’s, the red curves in Fig. 6). (Right
Panel) The unstable eigenfunctions associated with the triangular marker in
Fig. 6, left, center and right panel, are colored purple, red and green
respectively. The envelope for all three is well described by $\cos(\mu x/L)$
with $\mu=1/10$ and $L=2\pi$. Strong localization of the peaks of the
eigenfunctions at the wave crests of the Stokes waves is observed as the wave
steepness increases.

###  3.4 The localized instability

The $n$th oval emerges from the origin of the spectral plane as the steepness
increases past $s^{1/2}_{n}$. We observe that $s^{1/2}_{n}>s_{h,n}$, the value
of the steepness for which the $n$-th Benjamin-Feir figure-8 separates from
the origin. Thus prior (i.e., for $s<s^{1/2}_{n}$) to these ovals emerging
from the origin, the spectrum near the origin is confined to the imaginary
axis. The primary, secondary, and tertiary ovals form at the steepnesses

$\displaystyle s_{1}^{1/2}$ $\displaystyle=0.128903,$ (33) $\displaystyle
s_{2}^{1/2}$ $\displaystyle=0.140487,$ (34) $\displaystyle s_{3}^{1/2}$
$\displaystyle=0.141032049,$ (35)

which correspond to steepnesses at which $4\pi$-periodic Stokes waves
bifurcate from the primary, $2\pi$-periodic wave branch, see Dyachenko &
Semenova (2023b).

Figure 8: (Left) Bean-shaped stage of the localized instability for
$s=0.1365552495<s_{H,1}$ (green), $s=0.1365917123<s_{H,1}$ (gold) and
$s=0.1366066477>s_{H,1}$ (blue). (Right) Zoom-in on the remnant of the primary
Benjamin-Feir isole as it is absorbed into the first localized branch at
$s=0.1365546598$ (red), $s=0.1365552495$ (green) and $s=0.1365558392$ (pink).

The changing topology of the primary oval for $s>s^{1/2}_{1}$ is shown in
Figs. 3 and 4. More detail is presented in Fig. 8. The oval develops for
$s_{1}^{1/2}\leq s<s_{H,1}$. The oval stage is followed by a symmetric bean
shape with a narrowing neck as steepness approaches $s_{H,1}$. The maximal
growth rate associated with the localized instability quickly overtakes the
maximal growth rate associated with the Benjamin-Feir isole higher on the
imaginary axis, see Deconinck et al. (2023). Shortly before the steepness $s$
reaches $s_{H,1}$, the first extremizer of the Hamiltonian, the remnant of the
Benjamin-Feir instability isola merges with the localized instability branch
bean, as shown in the right panel of Fig. 8.

We observe the recurrence of the process described above two more times, for
the secondary and tertiary ovals that form at $s=s^{1/2}_{2}$ and
$s=s^{1/2}_{3}$, respectively. This leads to the conjecture of an infinite
number of such recurrences, the $n$-th one born at $s=s^{1/2}_{n}$, leading to
the formation of the oval, gradually deforming to a bean shape, which pinches
off at $s=s_{H,n}$, after which the resulting lobes move away from the origin
along the real axis, ever decreasing in diameter. For $s>s_{H,n}$, the lobes
are parameterized by the full range of Floquet exponents $\mu\in[-1/2,1/2)$.
Further, for $s\in(s_{H,n},s_{c,n+1})$ there is no component of the spectrum
other than the imaginary axis.

Figure 9:  (Left) The figure-$\infty$ component of the instability spectrum
appears at the first extremum of the Hamiltonian at $s_{H,1}=0.1366035$.
(Right) The second figure-$\infty$ component of the instability spectrum
occurs for the wave with steepness $s_{H,2}=0.1407965$ at the second extremum
of the Hamiltonian. The difference between the real and imaginary parts of the
two curves as a function of the Floquet exponent $\mu$ is less than $10^{-3}$.

Figure 10: (Left) For the wave with $s_{H,1}=0.1366035$ in the left panel of
Fig. 9, the perturbation associated with $\mu=0.01$ (green), with the
eigenvalue $\lambda=0.06106383+0.03263364i$ and its complex conjugate
$\bar{\lambda}$. The perturbation is given by $\delta
p=\mathrm{Re}\left[e^{i\mu u}\delta y\right]$. The red curve shows the same
for $\mu=0.2$, with eigenvalue $\lambda=0.1600750+0.04326491i$ and its complex
conjugate $\bar{\lambda}$. Only the interval $-3\pi<x<3\pi$ is shown from the
$2\pi/\mu$-periodic function. (Right) Polar plot, $e^{i\mu u}\delta y(u)$,
where real and imaginary parts are plotted along the horizontal and vertical
axes respectively.

Figure 11: (Left) For the wave with $s_{H,2}=0.1407965$ in the right panel of
Fig. 9, the perturbation associated with $\mu=0.01$ (gold) with the eigenvalue
$\lambda=0.06075090+0.03248890i$, and its complex conjugate $\bar{\lambda}$.
The perturbation is given by $\delta p=\mathrm{Re}\left[e^{i\mu u}\delta
y\right]$. The purple curve shows the same for $\mu=0.2$. Only the interval
$-3\pi<x<3\pi$ is shown from the $2\pi/\mu$-periodic function. (Right) Polar
plot, $e^{i\mu u}\delta y(u)$, where real and imaginary parts are plotted
along the horizontal and vertical axes respectively.

The first two figure-$\infty$’s are shows in Fig. 9. Like the figure-$8$’s,
they settle down to a universal shape as $s\rightarrow s_{lim}$. The
difference between the real and imaginary parts of these first two
figure-$\infty$’s as a function of the Floquet exponent never exceeds
$10^{-3}$.

Some eigenfunctions associated with the figure-$\infty$ are displayed in Figs.
10 and 11. For Floquet exponents close to zero (green and gold graphs), the
modulational effect of the perturbation is clear from the polar plots. For
other Floquet exponents (e.g. red and blue), the perturbation does not have a
distinct modulational character. As for other high-amplitude Stokes waves, the
localization of the eigenfunction (and thus the perturbation) near the crest
of the waves is increasingly pronounced as $s\rightarrow s_{lim}$.

When the oval forms at $s=s_{n}^{1/2}$, its eigenvalue with largest real part
is real and has Floquet exponent $\mu=1/2$, leading to eigenfunctions that
have double the period of the Stokes wave. After pinch off, $s>s_{H,n}$, the
left-most eigenvalue of the right lobe has $\mu=0$ (co-periodic
eigenfunctions). As for the primary lobe, we conjecture that the most unstable
mode on the right lobe is either the $\mu=0$ or the $\mu=1/2$ mode, which
interchange an infinite number of times as $s\rightarrow s_{lim}$, see
Deconinck et al. (2023). As remarked above, the profile of the eigenfunction
is strongly localized at the crests of the Stokes wave. Modes with $\mu$ close
to 0 have an envelope containing roughly $1/\mu$ periods of the Stokes wave
and could be called modulational. However, in contrast to the Benjamin-Feir
instabilities, for these instabilities the $\mu=0$ mode itself is unstable.
For this reason, we refer to the instabilities emerging from the
figure-$\infty$’s as localized instabilities.

Figure 12:  The primary BF instability emerges at steepness $s=0$ for small-
amplitude waves. This is marked with a green diamond. The green curve shows
the maximal growth rate associated with this BF instability. At the secondary
period-doubling bifurcation, $s_{1}^{1/2}=0.128903$, marked with a green
triangle, the first branch of the localized instability appears (maximal
growth rate in black solid and dashed). The near-vertical appearance of this
localized instability branch is a consequence of the rapid changes in the
spectrum of this branch for steep waves. The inset $s\in[0.12878,0.12908]$,
$\gamma\in[0.005,0.018]$ shows a zoom-in of the intersection of the maximal
growth rate of the localized branch with the primary BF branch. The remnant of
the BF instability merges with the localized branch at the edge of the gold
region I. In this region, the BF branch is no longer distinguishable from the
localized branch. The secondary branch of BF (maximal growth rate plotted in
blue) emerges at the first maximizer of the speed at $s_{c,1}=0.138753$ (blue
diamond) and follows the same sequence of steps merging with the localized
branch at the edge of the golden rectangle II. The secondary localized branch
emerges at the second period-doubling bifurcation at $s_{2}^{1/2}=0.140487$
(blue triangle). For the secondary inset
$s\in[0.1404795,0.1404955],\gamma\in[0.005,0.018]$. The tertiary BF emerges at
the turning point of speed, $s_{c,2}=0.140920$ (maximal growth rate plotted in
purple), the tertiary localized branch appears at $s_{3}^{1/2}=0.141032049$
(purple triangle).

###  3.5 The maximal growth rate

We track the maximum growth rate $\gamma$ (i.e., we track the eigenvalues with
the largest real part) of the Benjamin-Feir and localized instabilities
(plotted in black dotted and solid lines) as a function of the steepness of
the Stokes wave in Fig. 12. The maximum growth rate for BFI, BFII, and BFII
are presented by green, blue, and purple curves respectively. Steepnesses at
which the dominant instability switches from the Benjamin-Feir to the
corresponding localized branch are marked by circles. These switches are
presented in the corresponding insets. The steepness values where the maximal
growth rates for BFI and BFII vanish, correspond to the case where the
Benjamin-Feir remants are absorbed into the localized instabilities.

###  3.6 The high-frequency instabilities

Since the work presented here focuses on the evolution of the spectrum for
increasing steepness in the vicinity of the origin in the spectral plane, we
discuss the high-frequency instabilities only briefly. As shown by Deconinck &
Oliveras (2011) and Creedon et al. (2022), the high-frequency instabilities
emanate from purely imaginary double eigenvalues for steepness $s=0$, giving
rise to an isola of unstable eigenvalues centered on the imaginary axis, away
from the origin. As steepness is increased, these isole may collapse back on
the imaginary axis, and new ones may form, see MacKay & Saffman (1986). Unlike
the Benjamin-Feir (figure-8’s) and localized instability branches
(figure-$\infty$’s), the high-frequency isole are highly localized in the
space of Floquet exponents: indeed, Deconinck & Oliveras (2011) show that
often a range of Floquet exponents of width no more than $10^{-4}$
parameterizes an isola. This complicates their numerical detection. Figure 13
presents plots of a few high-frequency isole for near-extreme increasing
steepness, showing the collapse of one into the imaginary axis. For the top
isola plotted, $\mu\in[0.00092,0.00095]$, for the one below
$\mu\in[0.00057,0.00059]$. For the two isole on bottom,
$\mu\in[0.000251,0.000294]$ (outer), $\mu\in[0.000263,0.000282]$ (inner). This
demonstrates the isole can be captured using our method. A detailed study of
the evolution of the high-frequency instabilities as steepness increases is
kept for future work.

Figure 13: (Left) Detached figure-$\infty$ spectrum of the localized
instability for $s=0.1366171604>s_{H,1}$ (blue). (Right) Zoom-in on the high-
frequency instability for four increasing values of the steepness,
$s=0.1366066477$ (red), $s=0.1366141405$ (green), $s=0.1366171424$ (golden),
and $s=0.1366171604$ (blue, the value for the left panel), showing the
vanishing of a high-frequency isola at a value just exceeding $s=s_{H,1}$.

##  4 Conclusions

Figure 14: A schematic view of the appearance (and existence for a range of
steepnesses) of localized and Benjamin-Feir type instabilities. The first
Benjamin-Feir instability figure-8 appears at the steepness $s=s_{c,0}=0$, the
second one appears at $s=s_{c,1}$, and the third one at $s=s_{c,2}$. Localized
instabilities, manifested by an oval at the origin deforming to a
figure-$\infty$ appear at steepnesses labeled $s^{1/2}_{n}$ with $n=1,2,3,...$
that correspond to bifurcations to double-period Stokes waves. Once localized
instabilities appear, they continue to exist for all larger values of the
steepness in contrast to Benjamin-Feir type instabilities that emerge and
vanish as the steepness is increased. We conjecture that infinitely many
Benjamin-Feir type and localized instabilities appear as $s\rightarrow
s_{lim}$, the steepness of the extreme wave.

We have presented a numerical exploration of the stability spectrum of Stokes
waves near the origin of the spectral plane, focusing on the topological
changes in the stability spectrum as the wave steepness grows. The main
challenge of the study is due to the non-smooth nature of the extreme Stokes
wave, which has a 120$\degree$ corner at its crest. Thus for waves whose
steepness approaches that of this extreme wave, our Fourier-based method
requires the use of millions of Fourier modes. Indeed, for the computation of
waves with steepness near the third extremizer of the Hamiltonian (see Fig. 2)
nearly ten million Fourier modes are used. To examine the stability of these
waves, we linearize about them, resulting in a generalized operator eigenvalue
problem. The numerical approximation of this problem results in a generalized
matrix eigenvalue problem with matrices of dimension equal to the square of
twice the number of Fourier modes used for approximating the underlying Stokes
wave, since each component of the perturbation requires a comparable number of
Fourier modes to reach the same accuracy. Storing and manipulating such
matrices is prohibitive, and our investigations are only possible because of
the matrix-free approaches to the conformal mapping formulation, introduced by
Dyachenko & Semenova (2023a).

In Deconinck et al. (2023) we used this same method to investigate the largest
growth rate of perturbations of near-extreme Stokes waves, as a function of
their steepness. Among other outcomes, this led to the conclusion that long-
lived ocean swell is confined to moderate amplitudes. In this paper, we focus
instead on the behavior of the stability spectrum near the origin of the
spectral plane, as the recurring, self-similar behavior may provide an
indication of how the stability of near-extreme Stokes waves may be approached
analytically. Specifically, previous work and our numerical explorations lead
to the following conjectures.

  1. 1.

The Hamiltonian ${\cal H}$ and the velocity ${c}$ have an infinite number of
local extrema ($s_{H,n}$, $n=1,2,\ldots$ and $s_{c,n}$, $n=0,1,\ldots$
respectively) as the steepness $s$ increases to $s_{lim}$, the steepness of
the extreme wave. This conjecture is not new, and the asymptotics of Longuet-
Higgins & Fox (1977, 1978) provides a strong indication as to its validity. We
include this conjecture here because all others below depend upon it.

  2. 2.

The maximal instability growth rate approaches infinity as the steepness
increases to that of the steepest wave. Convincing evidence for this
conjecture is presented by Korotkevich et al. (2023). This would imply that
the Euler water wave problem for the evolution of the extreme Stokes wave is
ill posed. This is not a surprise as capillary effects need to be incorporated
when the curvature at the crest is too large. This is discussed in more detail
by Deconinck et al. (2023).

The conjectures below are a direct outcome of the investigations presented in
this paper. A graphical overview of which features occur at which steepness,
according to these conjectures, is presented in Fig. 14.

  1. (iii)

As the steepness $s$ increases from 0 to that of the steepest wave, there
exists an infinite number of Benjamin-Feir figure-8 instabilities. These
emanate from the origin at each extremum $s_{c,n}$, $n=0,1,\ldots$ of the
velocity. Upon formation, these figure-8’s persist for a range of steepness.
After their tangents at the origin become vertical (resulting in an hourglass
shape) at steepness $s_{h,n}$, $n=1,2,\ldots$, the figure-8 separates in two
lobes on the imaginary axis.

  2. (iv)

After the Benjamin-Feir figure-8 separates from the origin, the $n$th oval
appears at the origin, for the steepness $s^{1/2}_{n}$, $n=1,2,\dots$,
corresponding to the period-doubling bifurcation points from the primary
branch of Stokes waves.

  3. (v)

As the steepness $s$ increases from 0 to that of the extreme wave, there are
an infinite number of figure-$\infty$ instabilities. These occur at the origin
at each extremum of the Hamiltonian/energy. As the steepness is increased, the
figure-$\infty$’s detach instantaneously. In other words, the figure-$\infty$
shapes occur only at isolated values of the steepness, for which the
Hamiltonian has a local extremum.

  4. (vi)

These figure-8’s and figure-$\infty$’s alternate in their occurrence. Stated
differently, the extremizers of the Hamiltonian interlace the extremizers of
the velocity.

  5. (vii)

For Stokes waves with amplitude greater than that of the first maximizer of
the Hamiltonian, the most unstable mode is either superharmonic (co-periodic)
or subharmonic with twice the period of the Stokes wave (anti-periodic).
Further, there exists an infinite number of interchanges between which of
these two modes is dominant. Throughout these interchanges, no other mode is
the most unstable. These observations were already discussed by Deconinck et
al. (2023).

In the context of the full water wave problem, the present work reveals two
primary mechanisms for the breaking of ocean waves: (i) for Stokes waves with
$s<s_{2}=0.12894$ (the steepness at which the dominant instability switches
from Benjamin-Feir to the localized branch, the abscissa of the green circle
in Fig. 12), when the unstable envelope of multiple periods of a train of
Stokes waves enters the nonlinear stage, a complicated pattern of waves is
observed on the free surface. The pattern tends to self focus leading to the
formation of a large unsteady wave whose crest forms a plunging breaker
Clamond et al. (2006); Onorato et al. (2013); (ii) for Stokes waves with
$s>s_{2}$, the dynamics of the wave is dominated by the localized instability
at the wave crest, see Dyachenko & Newell (2016); Baker & Xie (2011); Duncan
(2001). The localized instability immediately leads to wave breaking of either
every other wave crest in the train (if $\mu=0$ is dominant), or every other
crest (if $\mu=1/2$ is dominant) as discussed by Deconinck et al. (2023). More
work is needed to understand the fully nonlinear stage of the many different
instabilities computed.

A complete understanding of the stability of Stokes waves with respect to
bounded perturbations (see Haragus & Kapitula (2008); Kapitula & Promislow
(2013)) requires further study of the spectrum of the operators associated
with the linearization of the Euler equations governing the dynamics of these
perturbed waves. Nonetheless, our study provides numerical evidence that the
(quasi-) periodic eigenfunctions that we examine, are fundamental to this
problem.

Acknowledgements. The authors wish to thank Eleanor Byrnes, Diane M.
Henderson, and Pavel M. Lushnikov for helpful discussions. Also, the authors
thank Frigo & Johnson (2005), the developers of FFTW and the whole GNU project
for developing and supporting this important and free software. S.D. thanks
the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, for
support and hospitality during the programme “Dispersive hydrodynamics” where
work on this paper was partially undertaken. Partial support for A. S. is
provided by a PIMS-Simons postdoctoral fellowship.

Declaration of Interests. The authors report no conflict of interest.

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# 10 ps timing with highly irradiated 3D trench silicon pixel sensors

A. Lampis,11footnotetext: Corresponding author. F. Borgato  D. Brundu  A.
Cardini  G.M. Cossu  G.-F. Dalla Betta  M. Garau  L. La Delfa  A. Lai  A. Loi
M. Obertino  G. Simi  S. Vecchi

###### Abstract

In this paper the results of a beam test characterization campaign of 3D
trench silicon pixel sensors are presented. A time resolution in the order of
10 ps was measured both for non-irradiated and irradiated sensors up to a
fluence of $2.5\cdot 10^{16}\,1\,MeV\,n_{eq}\,cm^{-2}$. This feature and a
detection efficiency close to $99\%$ make this sensors one of the best
candidates for 4D tracking detectors in High-Energy-Physics experiments.

##  1 Introduction

High-Energy-Physics experiments are demanding higher and higher event rates to
increase statistics that allows to access to new physics processes and to
decrease statistical uncertainties. Upgraded colliders are fulfilling this
demand running at increased luminosity and bringing new challenges to tracking
detectors. The main problem that these detectors have to cope is a higher
radiation damage up to $10^{17}\,1\,MeV\,N_{eq}\,cm^{-2}$. The increase in
luminosity will also bring higher pile-up and the event reconstruction will
become more difficult. The LHCb collaboration is evaluating the possibility to
build a new vertex detector with time information at the hit level to recover
tracking and vertexing capabilities, a 4D tracker [1]. Looking to the LHCb
Upgrade Phase II specifications the ideal sensor technology must have, at the
same time, a spatial resolution in the order of 10 $\mu$m, a time resolution
less than 50 ps and a radiation hardness from $10^{16}$ to
$10^{17}\,1\,MeV\,n_{eq}\,cm^{-2}$. Recent results from the INFN funded
project TimeSPOT have shown the excellent time resolution of 3D silicon
sensors optimized for timing performances [2][3]. In the next sections new
beam test characterizations of these innovative 3D silicon sensors are
presented, with a focus on the comparison between non-irradiated and
irradiated sensors at very high fluences.

##  2 3D sensors

3D sensors are silicon pixel sensors with the electrodes directly built into
the sensor thickness. This feature allows to develop sensors with a very short
inter-electrode distance with respect to standard planar sensors, Figure 1,
without reducing the active material in which the crossing particles can
create e-h pairs. This allows to exploit short inter-electrodes distances
without loosing in signal amplitude.

Figure 1: Comparison of the charge collection process between a planar sensor
(left) and a 3D sensor (right).

One of the main features of these sensors are the extremely fast signals,
given by the shorter path travelled by the charge carriers before being
collected by the electrodes. Moreover, higher electric fields and thus charge
carriers mobility saturation are obtained with a lower bias voltages. Another
important advantage of sensors with short inter-electrode distance is a high
radiation hardness, that for a columnar 3D silicon sensor was tested up to
$1\cdot 10^{17}\,1\,MeV\,n_{eq}\,cm^{-2}$ [4]. These two characteristics lead
3D sensors to be one of the most appealing sensors for tracking detectors for
HL-LHC experiments and beyond.

##  3 The TimeSPOT sensor

The TimeSPOT project, by means of a very complete simulation work, has
designed new pixel structures based on the trench design [2][3]. This
configuration is based on trench electrodes that recreate a planar-like
geometry. This electrode shape leads to a weighting field [5], uniform by
design, and to a high and uniform electric field that give access to velocity
saturation regime in almost the entire active area of the sensor [6]. TCAD
simulations allow to compare electric field and weighting field maps of a
standard 5 columns 3D geometry and the trench geometry, that is shown in
Figure 2. The trench design, despite the other classic 3D geometry, presents
more uniform electric and weighting field. As a result of this, Figure 3 shows
that the simulated charge collection curves, obtained for Minimum Ionizing
Particles (MIP) that uniformly crosses the pixels sensitive area, provide
shorter and much more uniform charge collection time for the trench design.

Figure 2: TCAD simulations of weighting field and electric field for a 3D
detector with a 5 columns electrodes geometry (left) and for the TimeSPOT
3D-trench geometry (right). -150 V bias voltage simulation. Figure 3:
Simulations of charge collection curves for MIPs uniform incidence for the
five columns geometry (left) and for the trench geometry (right).

The TimeSPOT pixel, shown in Figure 4, has a dimension of $55\times 55\,\mu$m2
with an active thickness of 150 $\mu$m. In each pixel a 40 $\mu$m long n++
trench, the readout electrode, is placed between two p++ trenches used for the
bias. The readout electrode is only 135 $\mu$m deep and under the pixel there
is a support wafer of 350 $\mu$m thickness.

Figure 4: The TimeSPOT 3D trench silicon pixel sensor.

Two batches of sensors were produced by FBK (Fondazione Bruno Kessler, Italy)
in 2019 and 2021 using the Deep Reactive Ion Etching Technique (DRIE) Bosch
process. Several test structure were fabricated: single pixels, double pixels,
pixel-strips

##  4 Tested devices and front end amplifier board

The sensors characterized in this paper, reported in Figure 5, are single
pixels and triple-strips structures in which 30 pixels are read by the same
amplifier channel. The tested devices are both non-irradiated and irradiated
sensors up to a fluence of $2.5\cdot 10^{16}\,1\,MeV\,n_{eq}\,cm^{-2}$. To
amplify the fast sensors currents a custom-made front-end electronics board is
used. It is composed of two stages of transimpedance amplifier (TIA), made
with low noise and high bandwidth silicon germanium BJTs and features a very
low jitter, for MIP equivalent charge deposition, of about 6 ps [7].

Figure 5: Pictures of some of the 3D trench silicon pixel test structures used
in this work. For each structure the active area is outlined in red. (A)
Single pixel sensor; (B) triple strip sensor: 30 pixels located in three
adjacent rows.

##  5 Beam test setup

The sensors are tested in the H8 line of SPS with a 180 GeV/c $\pi^{+}$ beam.
The experimental setup, shown in Figure 6, consists of 2 MCP-PMTs aligned to
the beam providing, when used together, a time resolution of about 4 ps. The
Device Under Test (DUT) is placed at the center of a black RF shielded box,
used to reduce electronics pick up on the DUT and also to operate the sensors
in the dark.

Figure 6: The setup used for the measurements described in this work. The
sensors mounted on their FEE boards inside the RF shielded and light tight box
and the two MCP-PMTs downstream.

The DUT mount allows to tilt the board in order to test sensors with various
tilting angles with respect to the normal incidence. A trigger sensor is
mounted upstream on two piezoelectric stages allowing to fine align two
pixels. The alignment of two small 3D sensors allows to perform unbiased
characterizations without using any amplitude selection on the DUT signals and
also to perform efficiency measurements. With this setup it is also possible
to test irradiated sensors at low temperature, from -40° to -20° C placing a
polystirene box, filled with dry ice, inside the black box. A temperature
sensor in thermal contact with the amplifier board is used to check the DUT
temperature during the measurements.

The sensors signals are recorded via $8$ GHz analog bandwidth, $20$ GSa/s 4
channels oscilloscope, the Rhode&Schwartz RTP084. A typical event consists of
four signals, two from the MCP-PMTs and two from the 3D sensors; the signals
are then analyzed offline in order to measure their amplitude and Time of
Arrival (ToA).

##  6 Analysis and results

In this section the analysis of the recorded signals and the results are
presented. The analysis consists of measuring the DUT signal amplitude,
measured as the peak of the waveform subtracted by the baseline, calculated
event by event, and the signal ToA that is evaluated with three different
methods. A leading edge algorithm gets the time in which the signal exceeds 15
mV, a linear interpolation is done around the threshold. The Spline method, a
CFD based algorithm, gets the time at the 20% of the signal amplitude; the
signal is interpolated with splines in order to reduce the jitter due to the
oscilloscope sampling frequency. The last method is the Reference method,
known in literature as the Amplitude Rise time Compensated discriminator (ARC)
[8]: in this case a delayed copy of the signal is subtracted to itself, then
the ToA is given as the time in which this new signal reaches the 50% of its
amplitude. The benefit of the Reference method is to reduce the time jitter
due to rise time fluctuations.

###  6.1 Amplitude

The amplitude measurements allow to evaluate and compare the overall charge
collection performances of the non-irradiated and irradiated sensors. Figure 7
shows the amplitude distributions for a non-irradiated 3D trench single pixel
operated at different bias voltages for normal beam incidence. A good charge
collection performance even at very low bias voltages (-7 V) is measured. The
measurements of the irradiated pixel at the fluence of $2.5\cdot
10^{16}\,1\,MeV\,n_{eq}\,cm^{-2}$ confirm the high radiation hardness of 3D
devices, in fact increasing the bias voltage higher than -100 V the
performances of the non-irradiated pixel are fully recovered.

Figure 7: Amplitude distributions for non-irradiated (left) and irradiated at
a fluence of $2.5\cdot 10^{16}\,1\,MeV\,N_{eq}\,cm^{-2}$ (right) 3D trench
single pixels for different bias voltages.

###  6.2 Time Resolution

The time resolution is estimated measuring multiple times the DUT ToA with
respect to the absolute time measured with the time reference detector and
measuring its fluctuation. Figure 8 shows the distribution of a non-irradiated
3D trench single pixel with respect to the time of the MCP-PMTs. To estimate
the time resolution a two gaussians fit is performed on the distribution.

Figure 8: (left) Time distribution of a non-irradiated 3D trench single pixel
at -100 V. (right) Time resolution as a function of the applied bias voltage
for the same pixel.

The two gaussians fit function allows to take into account the small
asymmetric tale of late signals. Then the time resolution is quoted as the
standard deviation of the distribution, calculated as

$\sigma_{t}^{2}=f_{1}(\sigma_{1}^{2}+\mu_{1}^{2})+(1-f_{1})\cdot(\sigma_{2}^{2}+\mu_{2}^{2})-\mu^{2},$
(6.1)

where $f_{1}$ is the fraction of the Gaussian core and $\mu$ is

$\mu=f_{1}\mu_{1}+(1-f_{1})\cdot\mu_{2}.$ (6.2)

This leads to a time resolution of 11 ps at a bias voltage of -100 V using the
Reference method. Figure 8 (right) shows no evidence of a strong dependence of
the time resolution to the bias voltage down to -20 V for all the proposed
methods. Excellent timing performances, below 30 ps, are obtained even with
the leading edge algorithm, a simple discriminator implementable in ASICs,
with low area impact. The trench pixel irradiated at $2.5\cdot
10^{16}\,1\,MeV\,n_{eq}\,cm^{-2}$ shows very similar performances to the non-
irradiated pixel featuring a time resolution of 10 ps at a bias voltage of
-150 V. Also in this case, no significant time resolution degradation emerges
for biases up to -20 V, Figure 9 (left).

Figure 9: (Left) Single pixel time resolution as a function of the bias
voltage for (red) irradiated sensor and (black) non-irradiated. (Right) Triple
strip detection efficiency for a non-irradiated (red) and irradiated (black)
as a function of the tilt angle with respect to the normal incidence.

###  6.3 Efficiency

The trenches electrodes of the TimeSPOT pixels are not active volume so if a
particle channels inside a trench it will not be detected, decreasing the
pixel detection efficiency. A typical solution to recover 3D sensor
geometrical efficiency is to operate them with a tilted angle with respect to
the normal incidence. Dedicated measurements of 3D trench pixel detection
efficiency are presented in this paper. In order to measure the detection
efficiency a 3D trench single pixel is aligned to the DUT, that in this case
is a 3D trench triple-strip structure. This configuration allows to estimate
the efficiency as $\eta=N_{ts}/N_{tr}$, where $N_{ts}$ is the number of events
detected by the triple-strip sensor and $N_{tr}$ is the number of trigger
(number of particles that geometrically cross the triple strip sensor). The
results are shown in Figure 9 (right), for both the non-irradiated and the
irradiated at $2.5\cdot 10^{16}\,1\,MeV\,n_{eq}\,cm^{-2}$ triple strips
sensors. The efficiency increases as the tilting angle increases reaching 99%
detection efficiency for tilt angles higher than $10^{\circ}$. Also the
irradiated sensor recovers the full efficiency, in this case measured at a
bias voltage of -130 V.

##  7 Conclusion

New particle beam characterizations of 3D trench silicon pixel sensors are
shown in this paper. 3D-trench pixel sensors have shown a 10 ps time
resolution for MIP detection with a 99% detection efficiency when operated at
tilt angles higher than $10^{\circ}$ with respect to normal incidence. The
same results are obtained for 3D trench sensors irradiated at $2.5\cdot
10^{16}\,1\,MeV\,N_{eq}\,cm^{-2}$, indicating that the radiation hardness
limit of this technology it is not reached yet at this fluence. The results
presented in this paper are the first beam test timing characterization of
highly irradiated 3D silicon pixel sensors and they prove that 3D trench
sensors are one of the best candidate for 4D tracking detectors at very high
fluences in high-energy-physics colliders.

## Acknowledgments

This work was supported by the Fifth Scientific Commission (CSN5) of the
Italian National Institute for Nuclear Physics (INFN), within the Project
TIMESPOT and by the ATTRACT-EU initiative, INSTANT project. The authors wish
to thank Heinrich Schindler that, as LHCb beam test coordinator, has given
logistic support for the setup installation at the SPS H8 line.

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