# On Normal forms for Levi-flat hypersurfaces with an isolated line
singularity

Arturo Fernández Pérez  Departamento de Matemática, Universidade Federal de
Minas Gerais, UFMG Av. Antônio Carlos, 6627 C.P. 702, 30123-970 - Belo
Horizonte - MG, Brazil<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of normal forms for some local real-analytic Levi-flat
hypersurfaces with an isolated line singularity. We also give sufficient
conditions for that a Levi-flat hypersurface with a complex line as
singularity to be a pullback of a real-analytic curve in $\mathbb{C}$ via a
holomorphic function.

###### Key words and phrases:

Levi-flat hypersurfaces - Holomorphic foliations

###### 2010 Mathematics Subject Classification:

Primary 32V40 - 32S65

##  1\. Introduction

Let $M\subset U\subset\mathbb{C}^{n}$ be a real-analytic hypersurface, where
$U$ is an open set and denote by $M^{*}$ the regular part, that is, near each
point $p\in M^{*}$, the variety $M$ is a manifold of real codimension one. For
each $p\in M^{*}$, there is a unique complex hyperplane $L_{p}$ contained in
the tangent space $T_{p}M^{*}$, and consequently defines a real-analytic
distribution $p\mapsto L_{p}$ of complex hyperplanes in $T_{p}M^{*}$, the so-
called Levi distribution. We say that $M$ is Levi-flat, if the Levi
distribution is integrable in sense of Frobenius. The foliation defined by
this distribution is called Levi-foliation. The local structure near regular
points is very well understood, according to E. Cartan, around each $p\in
M^{*}$ we can find local holomorphic coordinates $z_{1},\ldots,z_{n}$ such
that $M^{*}=\\{\mathcal{R}e(z_{n})=0\\}$, and consequently the leaves of Levi-
foliation are imaginary levels of $z_{n}$. The singular case was studied by
Burns-Gong [2], The authors classified singular Levi-flat hypersurfaces in
$\mathbb{C}^{n}$ with quadratic singularities and also proved the existence of
a normal form, in the case of generic (Morse) singularities. In [4], Cerveau-
Lins Neto have proved that a local real-analytic Levi-flat hypersurface $M$
with a sufficiently small singular set is given by the zeros of the real part
of a holomorphic function.

The aim of this paper is to prove the existence of some normal forms for local
real-analytic Levi-flat hypersurfaces defined by the vanishing of real part of
holomorphic functions with an isolated line singularity (for short: ILS). In
particular, we establish an analogous result like in Singularity Theory for
germs of holomorphic functions.

The main motivation for this work is a result due to Dirk Siersma, who
introduced in [14] the class of germs of holomorphic functions with an ILS.
More precisely, let
$\mathcal{O}_{n+1}:=\\{f:(\mathbb{C}^{n+1},0)\rightarrow\mathbb{C}\\}$ be the
ring of germs of holomorphic functions and let $m$ be its maximal ideal. If
$(x,y)=(x,y_{1},\ldots,y_{n})$ denote the coordinates in $\mathbb{C}^{n+1}$
and consider the line $L:=\\{y_{1}=\ldots=y_{n}=0\\}$, let
$I:=(y_{1},\ldots,y_{n})\subset\mathcal{O}_{n+1}$ be its ideal and denote by
$\mathcal{D}_{I}$ the group of local analytic isomorphisms
$\varphi:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1},0)$ for which
$\varphi(L)=L$. Then $\mathcal{D}_{I}$ acts on $I^{2}$ and for $f\in I^{2}$,
the tangent space of (the orbit of) $f$ with respect to this action is the
ideal defined by

$\tau(f):=m.\frac{\partial{f}}{\partial{x}}+I.\frac{\partial{f}}{\partial{y}}$

and the codimension of (the orbit) of $f$ is

$c(f):=\dim_{\mathbb{C}}\frac{I^{2}}{\tau(f)}.$

A line singularity is a germ $f\in I^{2}$. An ILS is a line singularity $f$
such that $c(f)<\infty$. Geometrically, $f\in I^{2}$ is an ILS if and only if
the singular locus of $f$ is $L$ and for every $x\neq 0$, the germ of (a
representative of) $f$ at $(x,0)\in L$ is equivalent to
$y^{2}_{1}+\ldots+y^{2}_{n}$. In a certain sense ILS are the first
generalization of isolated singularities. D. Siersma proved the following
result. (The topology on $\mathcal{O}_{n+1}$ is introduced as in [5, p. 145]).

######  Theorem 1.1.

A germ $f\in I^{2}$ is $D_{I}$-simple (i.e. $c(f)<\infty$ and $f$ has a
neighborhood in $I^{2}$ which intersects only a finite number of
$D_{I}$-orbits) if and only if $f$ is $D_{I}$-equivalent to one the germs in
the following table

Type | Normal form | Conditions
---|---|---
$A_{\infty}$ | $y_{1}^{2}+y_{2}^{2}+\ldots+y^{2}_{n}$ |
$D_{\infty}$ | $xy^{2}_{1}+y^{2}_{2}+\ldots+y^{2}_{n}$ |
$J_{k,\infty}$ | $x^{k}y^{2}_{1}+y_{1}^{3}+y^{2}_{2}+\ldots+y_{n}^{2}$ | $k\geq 2$
$T_{\infty,k,2}$ | $x^{2}y^{2}_{1}+y_{1}^{k}+y_{2}^{2}+\ldots+y_{n}^{2}$ | $k\geq 4$
$Z_{k,\infty}$ | $xy_{1}^{3}+x^{k+2}y_{1}^{2}+y_{2}^{2}+\ldots+y_{n}^{2}$ | $k\geq 1$
$W_{1,\infty}$ | $x^{3}y_{1}^{2}+y_{1}^{4}+y_{2}^{2}+\ldots+y_{n}^{2}$ |
$T_{\infty,q,r}$ | $xy_{1}y_{2}+y_{1}^{q}+y_{2}^{r}+y^{2}_{3}\ldots+y_{n}^{2}$ | $q\geq r\geq 3$
$Q_{k,\infty}$ | $x^{k}y_{1}^{2}+y_{1}^{3}+xy_{2}^{2}+y^{2}_{3}\ldots+y_{n}^{2}$ | $k\geq 2$
$S_{1,\infty}$ | $x^{2}y_{1}^{2}+y_{1}^{2}y_{2}+y_{3}^{2}+\ldots+y_{n}^{2}$ |

Table 1. Isolated Line singularities

The singularities in Theorem 1.1 are analogous of the $A$-$D$-$E$
singularities due to Arnold [1]. A new characterization of simple ILS have
been proved by A. Zaharia [15]. We prove the existence of normal forms for
Levi-flat hypersurfaces with an ILS.

######  Theorem 1.

Let $M=\\{F=0\\}$ be a germ of an irreducible real-analytic hypersurface on
$(\mathbb{C}^{n+1},0)$, $n\geq 3$. Suppose that

  1. (1)

$F(x,y)=\mathcal{R}e(P(x,y))+H(x,y),$ where $P(x,y)$ is one of the germs of
the Table 1.

  2. (2)

$M=\\{F=0\\}$ is Levi-flat.

  3. (3)

$H(x,0)=0$ for all $x\in(\mathbb{C},0)$ and $j_{0}^{k}(H)=0$, for $k=\deg(P)$.

Then there exists a biholomorphism
$\varphi:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1},0)$ preserving $L$
such that

$\varphi(M)=\\{\mathcal{R}e(P(x,y))=0\\}.$

This result is a Siersma’s type Theorem for singular Levi-flat hypersurfaces.
We remark that the function $H$ is of course restricted by the assumption that
$M$ is Levi flat. Now, if $\varphi(M)=\\{\mathcal{R}e(P(x,y))=0\\}$, where $P$
is a germ with an ILS at $L$ then $\textsf{Sing}(M)=L$. In other words, $M$ is
a Levi-flat hypersurface with an ILS at $L$. If $P(x,y)$ is the germ
$A_{\infty}$, we prove that Theorem 1 is true in the case $n=2$.

######  Theorem 2.

Let $M=\\{F=0\\}$ be a germ of an irreducible real-analytic Levi-flat
hypersurface on $(\mathbb{C}^{3},0)$. Suppose that $F$ is defined by

$F(x,y)=\mathcal{R}e(y_{1}^{2}+y_{2}^{2})+H(x,y),$

where $H$ is a germ of real-analytic function such that $H(x,0)=0$ and
$j_{0}^{k}(H)=0$ for $k=2$. Then there exists a biholomorphism
$\varphi:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{3},0)$ preserving $L$ such
that $\varphi(M)=\\{\mathcal{R}e(y_{1}^{2}+y_{2}^{2})=0\\}$.

The above result should be compared to [2, Theorem 1.1]. This result can be
viewed as a Morse’s Lemma for Levi-flat hypersurfaces with an ILS at $L$. The
problem of normal forms of Levi-flat hypersurfaces in $\mathbb{C}^{3}$ with an
ILS seems difficult in the other cases. To prove these results we use
techniques of holomorphic foliations developed in [4] and [6]. Another normal
forms of singular Levi-flat hypersurfaces have been obtained in [2], [7] and
[9].

This paper is organized as follows: In Section 2, we recall some definitions
and known results about Levi-flat and holomorphic foliations. Section 3 is
devoted to prove Theorem 1. In Section 4, we prove Theorem 2. Finally, in
Section 5, using holomorphic foliations, we give sufficient conditions for
that a Levi-flat hypersurface with a complex line as singularity to be a
pullback of a real-analytic curve in $\mathbb{C}$ via a holomorphic function,
(see Theorem 5.7).

##  2\. Levi-flat hypersurfaces and Foliations

In this section we works with germs at $0\in\mathbb{C}^{n+1}$ of irreducible
real-analytic hypersurfaces and of codimension one holomorphic foliations. Let
$M=\\{F=0\\}$, where $F:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{R},0)$ is a
germ of an irreducible real-analytic function, and
$M^{*}:=\\{F=0\\}\backslash\\{dF=0\\}$. Let us define the singular set of $M$
(or “set of critical points” of $M$) by

$\textsf{Sing}(M):=\\{F=0\\}\cap\\{dF=0\\}.$ (2.1)

Note that $\textsf{Sing}(M)$ contains all points $q\in M$ such that $M$ is
smooth at $q$, but the codimension of $M$ at $q$ is at least two. In general
the singular set of a real-analytic subvariety $M$ in a complex manifold is
defined as the set of points near which $M$ is not a real-analytic submanifold
(of any dimension) and “in general” has structure of a semianalytic set; see
for instance, [11]. In this paper, we work with $\textsf{Sing}(M)$ as defined
in (2.1). We recall that (in this case) the Levi distribution $L$ on $M^{*}$
is defined by

$\displaystyle L_{p}:=ker(\partial{F}(p))\subset
T_{p}M^{*}=ker(dF(p)),\,\,\,\,\text{for any}\,\,p\in M^{*}.$ (2.2)

Let us suppose that $M$ is Levi-flat, this implies that $M^{*}$ is foliated by
complex codimension one holomorphic submanifolds immersed on $M^{*}$.

Note that the Levi distribution $L$ on $M^{*}$ can be defined by the real-
analytic 1-form $\eta=i(\partial{F}-\bar{\partial}F)$, which is called the
Levi 1-form of $F$. It is well known that the integrability condition of $L$
is equivalent to equation
$(\partial{F}-\bar{\partial}F)\wedge\partial\bar{\partial}F|_{M^{*}}=0.$

Let us consider the series Taylor of $F$ at $0\in\mathbb{C}^{n+1}$,

$F(x,y)=\sum_{i,\mu,j,\nu}F_{i\mu j\nu}x^{i}y^{\mu}\bar{x}^{j}\bar{y}^{\nu}$

where $\bar{F}_{i\mu j\nu}=F_{j\nu i\mu}$; $i,j\in\mathbb{N}$,
$\mu=(\mu_{1},\ldots,\mu_{n})$, $\nu=(\nu_{1},\ldots,\nu_{n})$,
$(x,y)\in\mathbb{C}\times\mathbb{C}^{n}$, $y^{\mu}=y_{1}^{\mu_{1}}\ldots
y_{n}^{\mu_{n}}$ and
$\bar{y}^{\nu}=\bar{y}_{1}^{\nu_{1}}\ldots\bar{y}_{n}^{\nu_{n}}$. The
complexification $F_{\mathbb{C}}\in\mathcal{O}_{2n+2}$ of $F$ is defined by
the serie

$F_{\mathbb{C}}(x,y,z,w)=\sum_{i,\mu,j,\nu}F_{i\mu
j\nu}x^{i}y^{\mu}z^{j}w^{\nu},$

where $z\in\mathbb{C}$, $w=(w_{1},\ldots,w_{n})\in\mathbb{C}^{n}$ and
$w^{\nu}=w_{1}^{\nu_{1}}\ldots w_{n}^{\nu_{n}}$. Notice that
$F(x,y)=F_{\mathbb{C}}(x,y,\bar{x},\bar{y})$. The complexification
$M_{\mathbb{C}}$ of $M$ is defined as $M_{\mathbb{C}}:=\\{F_{\mathbb{C}}=0\\}$
and defines a complex subvariety in $\mathbb{C}^{2n+2}$, its regular part is
$M^{*}_{\mathbb{C}}:=M_{\mathbb{C}}\backslash\\{dF_{\mathbb{C}}=0\\}$. Now,
assume that $M$ is Levi-flat. Then the integrability condition of

$\eta=i(\partial{F}-\bar{\partial}F)|_{M^{*}}$

implies that $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$ is integrable, where

$\eta_{\mathbb{C}}:=i[(\partial_{x}F_{\mathbb{C}}+\partial_{y}F_{\mathbb{C}})-(\partial_{z}F_{\mathbb{C}}+\partial_{w}F_{\mathbb{C}})].$

Therefore $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$ defines a codimension one
holomorphic foliation $\mathcal{L}_{\mathbb{C}}$ on $M^{*}_{\mathbb{C}}$ that
will be called the complexification of $\mathcal{L}$.

Let
$W:=M_{\mathbb{C}}^{*}\backslash\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$
and denote by $L_{\zeta}$ the leaf of $\mathcal{L}_{\mathbb{C}}$ through
$\zeta$, where $\zeta\in W$. The next results will be used several times along
of the paper.

######  Lemma 2.1 (Cerveau-Lins Neto [4]).

For any $\zeta\in W$, the leaf $L_{\zeta}$ of $\mathcal{L}_{\mathbb{C}}$
through $\zeta$ is closed in $M_{\mathbb{C}}^{*}$.

######  Definition 2.2.

The algebraic dimension of $\textsf{Sing}(M)$ is the complex dimension of the
singular set of $M_{\mathbb{C}}$.

The following result will be used enunciated in the context of Levi-flat
hypersurfaces in $\mathbb{C}^{n+1}$.

######  Theorem 2.3 (Cerveau-Lins Neto [4]).

Let $M=\\{F=0\\}$ be a germ of an irreducible analytic Levi-flat hypersurface
at $0\in\mathbb{C}^{n+1}$, $n\geq{2}$, with Levi 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$. Assume that the algebraic dimension of
$\textsf{Sing}(M)\leq 2n-2$. Then there exists a unique germ at
$0\in\mathbb{C}^{n+1}$ of holomorphic codimension one foliation
$\mathcal{F}_{M}$ tangent to $M$, if one of the following conditions is
fulfilled:

  1. (1)

$n\geq 3$ and
$cod_{M_{\mathbb{C}}^{*}}(\textsf{Sing}(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
3$.

  2. (2)

$n\geq 2$,
$cod_{M_{\mathbb{C}}^{*}}(\textsf{Sing}(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
2$ and $\mathcal{L}_{\mathbb{C}}$ admits a non-constant holomorphic first
integral.

Moreover, in both cases the foliation $\mathcal{F}_{M}$ admits a non-constant
holomorphic first integral $f$ such that $M=\\{\mathcal{R}e(f)=0\\}$.

##  3\. Proof of Theorem 1

We write

$F(x,y)=\mathcal{R}e(P(x,y_{1},\ldots,y_{n}))+H(x,y_{1},\ldots,y_{n}),$

where $P(x,y_{1},\ldots,y_{n})$ is one of the polynomials of the Table 1,
$H:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{R},0)$ is a germ of real-analytic
function such that $H(x,0)=0$ for all $x\in(\mathbb{C},0)$ and
$j_{0}^{k}(H)=0$, for $k=\deg(P)$. The complexification of $F$ is given by

$F_{\mathbb{C}}(x,y,z,w)=\frac{1}{2}P(x,y)+\frac{1}{2}P(z,w)+H_{\mathbb{C}}(x,y,z,w),$

thus
$M_{\mathbb{C}}=\\{F_{\mathbb{C}}(x,y,z,w)=0\\}\subset(\mathbb{C}^{2n+2},0)$,
where $z\in\mathbb{C}$ and $w=(w_{1},\ldots,w_{n})\in\mathbb{C}^{n}$.

Since $P(x,y)$ has an ILS at $L$, we get
$\textsf{Sing}(M_{\mathbb{C}})=\\{y=w=0\\}\simeq\mathbb{C}^{2}$. In
particular, the algebraic dimension of $\textsf{Sing}(M)$ is $2$. On the other
hand, the complexification of $\eta=i(\partial{F}-\bar{\partial}F)$ is

$\eta_{\mathbb{C}}:=i[(\partial_{x}F_{\mathbb{C}}+\partial_{y}F_{\mathbb{C}})-(\partial_{z}F_{\mathbb{C}}+\partial_{w}F_{\mathbb{C}})].$

Recall that $\eta|_{M^{*}}$ and $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$
define $\mathcal{L}$ and $\mathcal{L}_{\mathbb{C}}$ respectively. Now we
compute $\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$. We can write
$dF_{\mathbb{C}}=\alpha+\beta$, with

$\alpha:=\frac{\partial{F_{\mathbb{C}}}}{\partial{x}}dx+\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{y}_{j}}dy_{j}=\frac{1}{2}\frac{\partial{P}}{\partial{x}}(x,y)dx+\frac{1}{2}\sum_{j=1}^{n}\frac{\partial{P}}{\partial{y}_{j}}(x,y)dy_{j}+\theta_{1}$

and

$\beta:=\frac{\partial{F_{\mathbb{C}}}}{\partial{z}}dz+\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}=\frac{1}{2}\frac{\partial{P}}{\partial{z}}(z,w)dz+\frac{1}{2}\sum_{j=1}^{n}\frac{\partial{P}}{\partial{w}_{j}}(z,w)dw_{j}+\theta_{2}$

where
$\theta_{1}=\frac{\partial{H_{\mathbb{C}}}}{\partial{x}}dx+\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{z}_{j}}dz_{j}$
and
$\theta_{2}=\frac{\partial{H_{\mathbb{C}}}}{\partial{z}}dz+\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}$.

Note that $\eta_{\mathbb{C}}=i(\alpha-\beta)$, and so

$\displaystyle\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}=(\eta_{\mathbb{C}}+idF_{\mathbb{C}})|_{M^{*}_{\mathbb{C}}}=2i\alpha|_{M^{*}_{\mathbb{C}}}=-2i\beta|_{M^{*}_{\mathbb{C}}}.$
(3.1)

In particular, $\alpha|_{M^{*}_{\mathbb{C}}}$ and
$\beta|_{M^{*}_{\mathbb{C}}}$ define $\mathcal{L}_{\mathbb{C}}$. Therefore
$\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$ can be split in two
parts. In fact, let $M_{1}:=\\{(x,y,z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{z}}\neq 0$ or
$\frac{\partial{F}_{\mathbb{C}}}{\partial{w_{j}}}\neq 0$ for some
$j=1,\ldots,n\\}$ and $M_{2}:=\\{(x,y,z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{x}}\neq 0$ or
$\frac{\partial{F}_{\mathbb{C}}}{\partial{z_{j}}}\neq 0$ for some
$j=1,\ldots,n\\}$, then $M_{\mathbb{C}}=M_{1}\cup M_{2}$. If we denote by
$A_{0}=\frac{\partial{H_{\mathbb{C}}}}{\partial{x}}$,
$A_{j}=\frac{\partial{H_{\mathbb{C}}}}{\partial{z}_{j}}$ for all $1\leq j\leq
n$ and by $B_{0}=\frac{\partial{H_{\mathbb{C}}}}{\partial{z}}$,
$B_{j}=\frac{\partial{H_{\mathbb{C}}}}{\partial{w}_{j}}$ for all $1\leq j\leq
n$, we obtain that
$\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=X_{1}\cup X_{2}$,
where

$X_{1}:=M_{1}\cap\\{\frac{\partial{P}}{\partial{x}}(x,y)+A_{0}=\frac{\partial{P}}{\partial{y}_{1}}(x,y)+A_{1}=\ldots=\frac{\partial{P}}{\partial{y}_{n}}(x,y)+A_{n}=0\\}$

and

$X_{2}:=M_{2}\cap\\{\frac{\partial{P}}{\partial{z}}(z,w)+B_{0}=\frac{\partial{P}}{\partial{w}_{1}}(z,w)+B_{1}=\ldots=\frac{\partial{P}}{\partial{w}_{n}}(z,w)+B_{n}=0\\}.$

Since $P$ is a polynomial with an ILS at $L=\\{y=0\\}$, we conclude that

$cod_{M^{*}_{\mathbb{C}}}\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=n.$

By hypothesis $n\geq 3$, then it follows from Theorem 2.3, part $(1)$ that
there exists a germ $f\in\mathcal{O}_{n+1}$ such that the holomorphic
foliation $\mathcal{F}$ defined by $df=0$ is tangent to $M$. Moreover
$M=\\{\mathcal{R}e(f)=0\\}$. Note that if
$M=\\{\mathcal{R}e(f)=0\\}=\\{F=0\\}$, with $F$ an irreducible germ, we must
have that $\mathcal{R}e(f)=U\cdot F$, where $U$ is a germ of real-analytic
function with $U(0)\neq 0$. Without loss of generality, we can assume that
$U(0)=1$. In particular, $\mathcal{R}e(f)=U\cdot F$ implies that $f=P+h.o.t$.
According to Theorem 1.1, there exists a biholomorphism
$\varphi:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1},0)$ preserving $L$
such that $f\circ\varphi^{-1}=P$, ($f$ is $D_{I}$-equivalent to $P$, because
$f$ is a germ with ILS at $L$). Therefore,
$\varphi(M)=\\{\mathcal{R}e(P)=0\\}$ and the proof ends.

##  4\. Proof of Theorem 2

The idea is to use Theorem 2.3, part (2). In order to prove our result in the
case $n=2$, we are going to prove that $\mathcal{L}_{\mathbb{C}}$ has a non-
constant holomorphic first integral.

We begin by a blow-up along
$C:=\\{y_{1}=y_{2}=w_{1}=w_{2}=0\\}\simeq\mathbb{C}^{2}\subset\mathbb{C}^{6}$.
Let $F(x,y_{1},y_{2})=\mathcal{R}e(y^{2}_{1}+y^{2}_{2})+H$ and $M=\\{F=0\\}$
Levi-flat. Its complexification can be written as

$F_{\mathbb{C}}(x,y_{1},y_{2},z,w_{1},w_{2})=\frac{1}{2}(y^{2}_{1}+y^{2}_{2})+\frac{1}{2}(w^{2}_{1}+w^{2}_{2})+H_{\mathbb{C}}(x,y_{1},y_{2},z,w_{1},w_{2}).$

Note that

$\textsf{Sing}(M_{\mathbb{C}})=\\{y=w=0\\}=C.$

Let $E$ be the exceptional divisor of the blow-up
$\pi:\tilde{\mathbb{C}}^{6}\rightarrow\mathbb{C}^{6}$ along $C$. Denote by
$\tilde{M}_{\mathbb{C}}:=\overline{\pi^{-1}(M_{\mathbb{C}}\setminus\\{C\\})}\subset\tilde{\mathbb{C}}^{6}$
the strict transform of $M_{\mathbb{C}}$ via $\pi$ and by
$\tilde{\mathcal{F}}:=\pi^{*}(\mathcal{L}_{\mathbb{C}})$ the foliation on
$\tilde{M}_{\mathbb{C}}$.

Now, we consider an especial situation. Suppose that $\tilde{M}_{\mathbb{C}}$
is smooth and set $\tilde{C}:=\tilde{M}_{\mathbb{C}}\cap E$. Moreover, assume
that $\tilde{C}$ is invariant by $\tilde{\mathcal{F}}$. Take
$S=\tilde{C}\setminus\textsf{Sing}\tilde{\mathcal{F}}$, then $S$ is a smooth
leaf of $\tilde{\mathcal{F}}$. Pick $p_{0}\in S$ and a transverse section
$\sum$ through $p_{0}$. Let $G\subset\operatorname{Diff}(\sum,p_{0})$ be the
holonomy group of the leaf $S$ of $\tilde{\mathcal{F}}$. Since
$\textsf{dim}(\sum)=1$, we can assume that
$G\subset\operatorname{Diff}(\sum,0)$. We state a fundamental lemma.

######  Lemma 4.1 (Fernández-Pérez [9]).

In the above situation, suppose that the following properties are verified:

  1. (1)

For any $p\in S\backslash\textsf{Sing}{(\tilde{\mathcal{F}})}$ the leaf
$L_{p}$ of $\tilde{\mathcal{F}}$ through $p$ is closed in $S$.

  2. (2)

$g^{\prime}(0)$ is a primitive root of unity, for all $g\in
G\backslash\\{id\\}$.

Then $\mathcal{L}_{\mathbb{C}}$ admits a non-constant holomorphic first
integral.

###### Proof.

Let $G^{\prime}=\\{g^{\prime}(0)/g\in G\\}$ and consider the homomorphism
$\phi:G\rightarrow G^{\prime}$ defined by $\phi(g)=g^{\prime}(0)$. We claim
that $\phi$ is injective. In fact, assume that $\phi(g)=1$ and suppose by
contradiction that $g\neq id$. In this case $g(z)=z+az^{r+1}+\ldots$, where
$a\neq 0$. According to [12], the pseudo-orbits of this transformation
accumulate at $0\in(\sum,0)$, contradicting the fact that the leaves of
$\tilde{\mathcal{F}}$ are closed and so the assertion is proved. Now, it
suffices to prove that any element $g\in G$ has finite order (cf. [13]). In
fact, $\phi(g)=g^{\prime}(0)$ is a root of unity thus $g$ has finite order
because $\phi$ is injective. Hence, all transformations of $G$ have finite
order and $G$ is linearizable.

This implies that there is a coordinate system $w$ on $(\sum,0)$ such that
$G=\langle w\rightarrow\lambda w\rangle$, where $\lambda$ is a
$d^{th}$-primitive root of unity (cf. [13]). In particular, $\psi(w)=w^{d}$ is
a first integral of $G$, that is $\psi\circ g=\psi$ for any $g\in G$.

Let $\Gamma$ be the union of the separatrices of $\mathcal{L}_{\mathbb{C}}$
through $0\in\mathbb{C}^{6}$ and $\tilde{\Gamma}$ be its strict transform
under $\pi$. The first integral $\psi$ can be extended to a first integral
$\varphi:\tilde{M}_{\mathbb{C}}\backslash\tilde{\Gamma}\rightarrow\mathbb{C}$
by setting

$\varphi(q)=\psi(\tilde{L}_{q}\cap\sum),$

where $\tilde{L}_{p}$ denotes the leaf of $\tilde{\mathcal{F}}$ through $q$.
Since $\psi$ is bounded (in a compact neighborhood of $0\in\sum$), so is
$\varphi$. It follows from Riemann extension theorem that $\varphi$ can be
extended holomorphically to $\tilde{\Gamma}$ with $\varphi(\tilde{\Gamma})=0$.
This provides the first integral of $\mathcal{L}_{\mathbb{C}}$. ∎

The rest of the proof is devoted to prove that we are indeed in the conditions
of Lemma 4.1. It is follows from Lemma 2.1 that the leaves of
$\mathcal{L}_{\mathbb{C}}$ are closed. Therefore, we need to prove that each
generator of the holonomy group $G$ of $\tilde{\mathcal{F}}$ with respect to
$S$ has finite order.

Consider for instance the chart $(U_{1},(x,t,s,z,u,v))$ of
$\tilde{\mathbb{C}}^{6}$ where

$\pi(x,t,s,z,u,v)=(x,tu,su,z,u,vu)=(x,y_{1},y_{2},z,w_{1},w_{2}).$

We have

$\tilde{M}_{\mathbb{C}}\cap U_{1}=\\{(x,t,s,z,u,v)\in
U_{1}|1+t^{2}+s^{2}+v^{2}+uH_{1}(x,t,s,z,u,v)=0\\},$

where $H_{1}=H(x,ut,us,z,u,uv)/u^{3}$ and this fact imply that

$E\cap\tilde{M}_{\mathbb{C}}\cap U_{1}=\\{(x,t,s,z,u,v)\in
U_{1}|1+t^{2}+s^{2}+v^{2}=u=0\\}.$

It is not difficult to see that these complex subvarieties are smooth. Now,
let us describe the foliation $\tilde{\mathcal{F}}$ on $U_{1}$. In fact, note
that the foliation $\mathcal{L}_{\mathbb{C}}$ is defined by
$\alpha|_{M^{*}_{\mathbb{C}}}=0$, where

$\alpha=\frac{1}{2}\frac{\partial{P}}{\partial{x}}dx+\frac{1}{2}\frac{\partial{P}}{\partial{y_{1}}}dy_{1}+\frac{1}{2}\frac{\partial{P}}{\partial{y_{2}}}dy_{2}+\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\sum^{2}_{j=1}\frac{\partial{H}_{\mathbb{C}}}{\partial{y_{j}}}dy_{j}.$

It follows that
$\alpha=y_{1}dy_{1}+y_{2}dy_{2}+\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\sum^{2}_{j=1}\frac{\partial{H}_{\mathbb{C}}}{\partial{y_{j}}}dy_{j}$,
then $\tilde{\mathcal{F}}|_{U_{1}}$ is defined by
$\tilde{\alpha}|_{\tilde{M}_{\mathbb{C}}\cap U_{1}}=0$, where

$\displaystyle\tilde{\alpha}=(t^{2}+s^{2})du+utdt+usds+u\tilde{\theta},$ (4.1)

and

$\tilde{\theta}=\frac{\pi^{*}(\frac{\partial{H_{\mathbb{C}}}}{\partial{x}}dx+\sum^{2}_{j=1}\frac{\partial{H_{\mathbb{C}}}}{\partial{y_{j}}}dy_{j})}{u^{2}}.$

Therefore, the singular set of $\tilde{\mathcal{F}}|_{U_{1}}$ is given by

$\textsf{Sing}\tilde{\mathcal{F}}|_{U_{1}}=\\{u=t+is=0\\}\cup\\{u=t-is=0\\}.$

On the other hand, note that the exceptional divisor $E$ is invariant by
$\tilde{\mathcal{F}}$ and the intersection with
$\textsf{Sing}\widetilde{\mathcal{F}}$ is

$\textsf{Sing}\tilde{\mathcal{F}}|_{U_{1}}\cap
E=\\{u=t+is=v^{2}+1=0\\}\cup\\{u=t-is=v^{2}+1=0\\}.$

In particular,
$S:=(E\cap\tilde{M}_{\mathbb{C}})\backslash\textsf{Sing}\widetilde{\mathcal{L}}_{\mathbb{C}}$
is a leaf of $\widetilde{\mathcal{F}}$. We calculate the generators of the
holonomy group $G$ of the leaf $S$. We work in the chart $U_{1}$, because of
the symmetry of the variables in the definition of the variety
$\tilde{M}_{\mathbb{C}}$.

Pick $p_{0}=(0,1,0,0,0,0)\in S\cap U_{1}$ and a transversal
$\sum=\\{(0,1,0,0,\lambda,0)|\lambda\in\mathbb{C}\\}$ parameterized by
$\lambda$ at $p_{0}$. We have that

$\textsf{Sing}\tilde{\mathcal{F}}|_{U_{1}}\cap
E=\\{u=t+is=v^{2}+1=0\\}\cup\\{u=t-is=v^{2}+1=0\\}.$

For each $j=1,2$; let $\rho_{j}$ be a $2^{td}$-primitive root of $-1$. The
fundamental group $\pi_{1}(S,p_{0})$ can be written in terms of generators as

$\pi_{1}(S,p_{0})=\langle\gamma_{j},\delta_{j}\rangle_{1\leq j\leq 2},$

where for each $j=1,2$; $\gamma_{j}$ are loops that turn around
$\\{u=t+is=v-\rho_{j}=0\\}$ and $\delta_{j}$ are loops that turns around
$\\{u=t-is=v-\rho_{j}=0\\}$. Therefore, $G=\langle f_{j},g_{j}\rangle_{1\leq
j\leq 2}$, where $f_{j}$ and $g_{j}$ correspond to $[\gamma_{j}]$ and
$[\delta_{j}]$, respectively. We get from (4.1) that
$f^{\prime}_{j}(0)=e^{-\pi i}$ and $g^{\prime}_{j}(0)=e^{-\pi i}$ for all
$1\leq j\leq 2$. The proof of the theorem is complete.

##  5\. Levi-flat hypersurfaces with a complex line as singularity

In this section, we work with the system of coordinates
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$. The canonical local models examples
of Levi-flat hypersurfaces $M$ in $\mathbb{C}^{3}$ such that
$\textsf{Sing}(M)=L=\\{z_{1}=z_{2}=0\\}$ are
$\\{\mathcal{R}e(z^{2}_{1}+z^{2}_{2})=0\\}$ and
$\\{z_{1}\bar{z}_{2}-\bar{z}_{1}z_{2}=0\\}$.

Recently, Burns and Gong [2] classified, up to local biholomorphism, all germs
of quadratic Levi-flat hypersurfaces. Namely, up to biholomorphism, there is
only five models:

Type | Normal form | Singular set
---|---|---
$Q_{0,2k}$ | $\mathcal{R}e(z_{1}^{2}+z_{2}^{2}+\ldots+z^{2}_{k})$ | $\mathbb{C}^{n-k}$
$Q_{1,1}$ | $z^{2}_{1}+2z^{2}_{1}\bar{z}_{1}+z^{2}_{1}$ | empty
$Q^{\lambda}_{1,2}$ | $z^{2}_{1}+2\lambda z^{2}_{1}\bar{z}_{1}+z^{2}_{1}$ | $\mathbb{C}^{n-1}$
$Q_{2,2}$ | $(z_{1}+\bar{z}_{1})(z_{2}+\bar{z}_{2})$ | $\mathbb{R}^{2}\times\mathbb{C}^{n-2}$
$Q_{2,4}$ | $z_{1}\bar{z}_{2}-\bar{z}_{1}z_{2}$ | $\mathbb{C}^{n-2}$

Table 2. Levi-flat quadrics

We address the problem of provide conditions to characterize singular Levi-
flat hypersurfaces with a complex line as singularity. Using the
classification due to Burns and Gong [2], it is not hard to prove the
following proposition.

######  Proposition 5.1.

Suppose that $M$ is a quadratic real-analytic Levi-flat hypersurface in
$\mathbb{C}^{n}$, $n\geq 3$ such that
$\textsf{Sing}(M)=\\{z_{1}=z_{2}=\ldots=z_{n-1}=0\\}$. Then

  1. (1)

If $n=3$, $M$ is biholomorphically equivalent to $Q_{0,2}$ or $Q_{2,4}$.

  2. (2)

If $n\geq 4$, $M$ is biholomorphically equivalent to $Q_{0,2(n-1)}$.

###### Proof.

To prove part (1), observe that only there are two models of $M$ which admits
$\textsf{Sing}(M)=\\{z_{1}=z_{2}=0\\}$ as singularity, $Q_{0,2}$ or $Q_{2,4}$.
Now to prove part (2), note that if $n\geq 4$, the real hypersurface
$\\{z_{1}\bar{z}_{2}-\bar{z}_{1}z_{2}=0\\}$ has a complex subvariety of
dimension $n-2$ as singularity. It is follows that $M$ is biholomorphically
equivalent to $Q_{0,2(n-1)}$. ∎

In order to obtain a characterization, we define the Segre varieties
associated to real-analytic hypersurfaces. Let $M$ be a real-analytic
hypersurface defined by $\\{F=0\\}$. Fix $p\in M$, the Segre variety
associated to $M$ at $p$ is the complex variety in $(\mathbb{C}^{n},p)$
defined by

$Q_{p}:=\\{z\in(\mathbb{C}^{n},p):F_{\mathbb{C}}(z,\bar{p})=0\\}.$ (5.1)

Now assume that $M$ is Levi-flat and denote by $L_{p}$ the leaf of
$\mathcal{L}$ through $p\in M^{*}$. We denote by $Q^{\prime}_{p}$ the union of
all branches of $Q_{p}$ which are contained in $M$. Observe that
$Q^{\prime}_{p}$ could be the empty set when $p\in\textsf{Sing}(M)$.
Otherwise, it is a complex variety of pure dimension $n-1$.

The following result is classical, we proved it here for completeness.

######  Proposition 5.2.

In above situation, $L_{p}$ is an irreducible component of $(Q_{p},p)$ and
$Q^{\prime}_{p}=L_{p}$.

###### Proof.

Since $p\in M^{*}$, E. Cartan’s theorem assures that there exists a
holomorphic coordinate system such that near of $p$, $M$ is given by
$\\{\mathcal{R}e(z_{n})=0\\}$ and $p$ is the origin. In this coordinates
system the foliation $\mathcal{L}$ is defined by $dz_{n}|_{M^{*}}=0$. In
particular, $L_{0}=\\{z_{n}=0\\}$ and obviously $\\{z_{n}=0\\}$ is a branch of
$Q_{0}$. Furthermore, $L_{0}$ is the unique germ of complex variety of pure
dimension $n-1$ at $0$ which is contained in $M$. Hence
$Q^{\prime}_{0}=L_{0}$. ∎

Let $p\in\textsf{Sing}(M)$, we say that $p$ is a Segre degenerate singularity
if $Q_{p}$ has dimension $n$, that is, $Q_{p}=(\mathbb{C}^{n},p)$. Otherwise,
we say that $p$ is a Segre nondegenerate singularity.

Suppose that $M$ is defined by $\\{F=0\\}$ in a neighborhood of $p$, observe
that $p$ is a degenerate singularity of $M$ if $z\longmapsto
F_{\mathbb{C}}(z,\bar{p})$ is identically zero.

######  Remark 5.3.

If $V$ is a germ of complex variety of dimension $n-1$ contained in $M$ then
for $p\in V$, we have $(V,p)\subset(Q_{p},p)$. In particular, if there exists
distinct infinitely many complex varieties of dimension $n-1$ through $p\in M$
then $p$ is a Segre degenerate singularity.

To continuation, we consider a germ at $0\in\mathbb{C}^{n}$ of a codimension
one singular holomorphic foliation $\mathcal{F}$.

######  Definition 5.4.

We say that $\mathcal{F}$ and $M$ are tangent, if the leaves of the Levi
foliation $\mathcal{L}$ on $M$ are also leaves of $\mathcal{F}$.

######  Definition 5.5.

A meromorphic (holomorphic) function $h$ is called a meromorphic (holomorphic)
first integral for $\mathcal{F}$ if its indeterminacy (zeros) set is contained
in $\textsf{Sing}(\mathcal{F})$ and its level hypersurfaces contain the leaves
of $\mathcal{F}$.

Recently, Cerveau and Lins Neto proved the following result.

######  Theorem 5.6 (Cerveau-Lins Neto [4]).

Let $\mathcal{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq{3}$, of
holomorphic codimension one foliation tangent to a germ of an irreducible real
analytic hypersurface $M$. Then $\mathcal{F}$ has a non-constant meromorphic
first integral.

In our context, we prove the following result.

######  Theorem 5.7.

Let $M$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$ of an irreducible real-
analytic Levi-flat hypersurfaces such that
$\textsf{Sing}(M)=L:=\\{z_{1}=z_{2}=\ldots=z_{n-1}=0\\}$. Suppose that:

  1. (1)

Every point in $\textsf{Sing}(M)$ is a Segre nondegenerate singularity.

  2. (2)

The Levi-foliation $\mathcal{L}$ on $M^{*}$ extends to a holomorphic foliation
$\mathcal{F}$ in some neighborhood of $M$.

Then there exists $f\in\mathcal{O}_{n}$ and a real-analytic curve
$\gamma\subset\mathbb{C}$ such that $M=f^{-1}(\gamma)$.

###### Proof.

Since the Levi-foliation $\mathcal{L}$ on $M^{*}$ extends to a holomorphic
foliation $\mathcal{F}$, we can apply directly Theorem 5.6, this means that
$\mathcal{F}$ has a non-constant meromorphic first integral $f=g/h$, where $g$
and $h$ are relatively prime. We asserts that $f$ is holomorphic. In fact, if
$f$ is purely meromorphic, we have that for all $\zeta\in\mathbb{C}$, the
complex hypersurfaces $V_{\zeta}=\\{g(z)-\zeta h(z)=0\\}$ contains leaves of
$\mathcal{F}$. In particular, $M$ contains an infinitely many of hypersurfaces
$V_{\zeta}$, because $M$ is closed and $\mathcal{F}$ is tangent to $M$. Set
$\Lambda:=\\{\zeta\in\mathbb{C}:V_{\zeta}\subset M\\}$. Note also that the
foliation $\mathcal{F}$ is singular at $L$, so that
$\mathcal{I}_{f}:=\\{h=g=0\\}$ the indeterminacy set of $f$ intersect $L$.
Therefore, we have a point $q$ at $\mathcal{I}_{f}\cap L$ which would be a
Segre degenerate singularity, because $q\in V_{\zeta}$, for all
$\zeta\in\Lambda$. It is a contradiction and the assertion is proved.

The foliation $\mathcal{F}$ is defined by $df=0$, $f\in\mathcal{O}_{n}$ and is
tangent to $M$. Without loss generality, we can assume that $f$ is an
irreducible germ in $\mathcal{O}_{n}$. According to a remark of Brunella [3,
pg. 8], there exists a real-analytic curve $\gamma\subset\mathbb{C}$ through
the origin such that $M=f^{-1}(\gamma)$. ∎

######  Remark 5.8.

In [11], J. Lebl gave conditions for the Levi-foliation on $M^{*}$ does
extended to a holomorphic foliation. One could be considered these hypothesis
and establish a theorem more refined. Note also that if $\textsf{Sing}(M)$ is
a germ of smooth complex curve, it is possible adapted the proof of Theorem
5.7. In general, the holomorphic extension problem for the Levi-foliation of a
Levi-flat real-analytic hypersurface remains open and is of independent
interest, for more details see [8].

Acknowledgments.– This work was partially supported by PRPq - Universidade
Federal de Minas Gerias UFMG 2013 and FAPEMIG APQ-00371-13. I would like to
thank Maurício Corrêa JR for his comments and suggestions, and the referee for
pointing out corrections.

## References

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  * [2] D. Burns, X. Gong: Singular Levi-flat real analytic hypersurfaces. Amer. J. Math. 121, no. 1, pp. 23-53, $(1999)$.
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# REAL-TIME ADAPTIVE ABSTRACTION AND APPROXIMATION USING VALIDITY FRAMES - AN
EXPERIENCE REPORT

Raheleh Biglari
Joachim Denil
Department of Electronics and ICT
Faculty of Applied Engineering
Cosys-lab, University of Antwerp
Flanders Make@Uantwerpen
Groenenborgerlaan 171, Antwerp, Belgium
{raheleh.biglari,
<EMAIL_ADDRESS>

## ABSTRACT

Designing a Cyber-Physical System (CPS), including modeling the control
components and services, is a challenging task. Using models and simulations
during run-time is crucial for successfully implementing advanced control and
prediction components. The complexity of designing an effective CPS system
increases due to real-time constraints. Generating accurate predictions and
making decisions using detailed models in various contexts is computationally
demanding and complex to manage within the available computational resources.
Employing approximated models and switching to the most suited model
adaptively at run-time is an effective technique. But an approximated model is
most probable not valid in all the different contexts the system will be in.
This experience report uses the Validity Frame concept to enable adaptation at
run-time. In each environment, some influencing factors are outside the
model’s control, but these properties influence the model’s behavior. By
defining Validity Frames, based on specific contexts and related models, we
present a possible perspective to address the issue of selecting the more
appropriate model in various contexts. Furthermore, we discuss the insights
and lessons obtained and determine future challenges.

Keywords: Cyber-Physical Systems (CPS), Model-Based Systems Engineering
(MBSE), Validity Frames, Adaptive Approximation, Real-time.

##  1 INTRODUCTION

In the design and development of software-intensive systems, particularly in
the avionics and automotive industries, engineers are faced with the challenge
of dealing with highly complex systems composed of various interrelated and
deeply integrated components [Lee, 2008]. These systems, known as Cyber-
Physical Systems (CPS) typically operate in harsh conditions with rapidly
fluctuating environmental conditions. In these dynamic environments, the
ability to predict the behavior and performance of the system is often not
guaranteed[Palumbo et al., 2017].
To address the challenges presented by a changing environment, adaptivity is
integrated into the design of CPSs to allow the system to respond to
environmental changes. This adaptation must typically be performed in a
decentralized manner due to the autonomous nature of many real-time CPSs,
which increases the overall complexity of designing these adaptive systems
[Kit et al., 2015].
Cyber-Physical Systems, which integrate embedded control, mechanics, and
networking, are typically developed using a model-based approach. This
approach involves utilizing one or more physics-based models for designing and
simulating the system. Since a CPS is a complicated system, its model also
tends to be complicated. Additionally, these models are employed for control
of the system at run-time. Complicated models have a heavy computational load
on the embedded devices that typically implement the control on the system. As
such, implementing these computationally demanding models is challenging,
possibly resulting in missed deadlines and increased costs. One way to balance
improved performance and cost reduction in CPS is to use adaptive abstraction
and approximation techniques [Franceschini et al., 2019b, Franceschini et al.,
2019a, Biglari et al., 2022]. Adaptive approximation specifically switches
between multiple prediction models to achieve improved computational
efficiency while maintaining an acceptable level of control performance. The
adaptive approximation technique enables the utilization of a simpler, less
detailed model with a lower computational cost.

In this experience report, we applied the idea of utilizing the Validity Frame
to ease the development of real-time Cyber-Physical Systems while using the
run-time adaptive abstraction and approximation technique. We use the
properties of interest and environment properties which are known as
influencing factors within a Validity Frame. Moreover, we exploit the
relations between different Validity Frames to choose the most appropriate
model from the library of models.

The structure of this paper is as follows: section 2 provides the related
work. Next, section 3 introduces our case study and details the adaptive
approximation method applied to the system under study. Afterward, section 4
discusses the challenges encountered, lessons gained, and insights of this
research. Finally, section 5 concludes with a summary and future research
based on the insights gained.

##  2 RELATED WORK

This section describes related work for both adaptive abstraction and
approximation and validity frames.

###  2.1 Adaptive Abstraction and Approximation

Due to the complexity of CPSs, CPS models also possess similar complexity.
Utilizing models with heavy computational requirements is challenging as it
can cause missing deadlines.

To avoid computationally expensive models, there is a possibility of
eliminating the reasoning on a set of properties within a model. This is known
as a more abstract model. As the model removes certain properties, it is
typically also less computationally expensive to simulate. An alternative way
to exclude factors from a model is the approximation of resolution which means
removing detail, while retaining the set of properties one can reason on with
the model. As such a more approximate model uses a simpler model to
approximate the results of a more complicated model, assuming the difference
between the two is insignificant for the results of the simulation [Frantz and
Ellor, 1996].

It is possible to have multiple abstracted and approximated models. To switch
between these models at run-time, we benefit from the idea of adaptation by
[Mathieu et al., 2018, Franceschini et al., 2019a, Franceschini et al., 2019b]
that dynamically switches between abstractions during simulation or execution
or simulation.

Therefore, the adaptive approximation technique allows for the use of a
simpler, less detailed model with a lower computational cost. Allowing
adaptivity at run-time is one technique for resolving the conflict between
better performance and reduced cost in CPS.

However, for this adaptation between the different models, we need to
establish that these models are substitutable in a specific situation. Figure
1 represents the conceptual foundation from [Biglari et al., 2022] for
reasoning about run-time adaptive approximation in a dynamic situation. This
conceptual foundation is based on [Barroca et al., 2014] that combines the use
of ontology and language engineering.
A simulation of a model([[.]]) produces a trace. By analyzing this trace, we
can use a function $(f())$ to calculate the system’s prediction value related
to a logical property. This prediction value is used in the decision-making
process to make a decision. To reason about an allowed approximation of a
model, we must consider the goal models, decision-making processes, and
context. The sensitivities of the logical property in relation to decision-
making in a specific context and for a specific goal must be mapped to enable
this reasoning. The same reasoning process is applied when a different, more
approximate decision-making model is used in a specific context.

Figure 1: Conceptual Foundation for Model Approximation Technique from
[Biglari et al., 2022].

We explore the impact of approximating a model on cost reduction as a
foundation for a framework that enables real-time system adaptation. This
framework allows for swapping models with more approximate versions based on
the available library of models. Utilizing the insights from the conceptual
framework, Biglari et al. proposed an architecture for adaptive approximation
in real-time systems. This architecture in figure2 is based on MAPE-K
architecture. MAPE-K is a high-level control loop for self-adaptive systems
from IBM [Kephart and Chess, 2003].

Figure 2: Real-Time Adaptive Abstraction and Approximation Architecture from
[Biglari et al., 2022].

An approximated model is most probably not valid in all the different contexts
the system will be in. Consequently, the question arises: How can we do the
adaptive approximation and abstraction at run-time, and find the most
appropriate model for a specific context?
We use the Validity Frame concept to enable the adaptive approximation at run-
time.

###  2.2 Validity Frames

The concept of frames has been present for some time, dating back to the early
1980s when the original idea of ”experimental frames” was introduced by
[Zeigler, 1984].
[Klikovits et al., 2017] addressed the structure of modeling frames, which
depends on the activity performed, and this work describes why different
activities require different frames. Extending the work of [Klikovits et al.,
2017], [Van Acker et al., 2021] formalizes the Validity Frame concept to
specify the range-of-validity of a model.

A Validity Frame ($VF$) contains all necessary data and processes for
determining the appropriate usage of the model. A $VF$ is defined as a
structure:

$VF=<S_{M},\pi_{M},\gamma_{M},map_{\pi}\rightarrow
S_{M},map_{\gamma}\rightarrow S_{M},SPEC_{exe},Val_{M},\alpha_{val}>$ (1)

Where:

$S_{M}$ is the model structure, which defines the set of model components and
their relationship, $\pi_{M}$ is the modeled properties.
$\gamma_{M}$ is the influencing factor that is captured within $VF$ and
enables reasoning about the utilization of the model in terms of what
properties the model can provide valid answers for and under which influence
factors. Properties of the environment related to the system under study that
can potentially influence a model’s behavior and/or the range of validity
concerning a particular property, even though they are outside of the model’s
control, are referred to as ”influencing factors” [Mierlo et al., 2020].

$map_{\pi}$→$S_{M}$ and $map_{\gamma}$→$S_{M}$ are explicitly mapped onto the
model’s implementation. The $SPEC_{exe}$ shows the specification of the
simulation environment, the simulator, or the specification of the embedded
platform ($SPEC_{MBD}$) on which M is executed. Model validity, $Val_{M}$, is
a crucial part of $VF$, allowing reason about how well model M reflects the
real-world counterpart $Sys$.

Last is the set of validation activities $\alpha_{val}$ includes a set of
validation conditions $v$, used to validate or invalidate the behavior of M in
known or unknown contexts.

A $VF$ is used to explicitly identify the influence factors and range-of-
validity for a model and to provide methods and processes to ensure the model
accurately represents the source system. This includes methods for calibrating
the model, experiment design for validation, validation metrics, etc.
The purpose of validity frames is to clearly express the contexts in which a
model generates valid results for a specific set of properties concerning a
real-world system.

To convey different contexts, specific properties of the environment must be
considered within our system.

##  3 ADAPTIVE ABSTRACTION AND APPROXIMATION USING VALIDITY FRAMES

In this section, we present the process we followed to implement the adaptive
approximation technique using $VF$ in our case study.

###  3.1 Case study: Lane Changing

We use the Highway Lane change system as a case study to demonstrate our idea.
A lane-changing system controls the movement of the ego (target car) in both
the longitudinal (forward/backward) and lateral (left/right) directions.
According to the environmental properties, the ego car changes or follows the
lane.
This research presents a scenario of lane changing depicted in Figure 3. We
refer to this illustration throughout the paper to demonstrate our concepts
and findings. In this example, there are three cars, ego car (Ego), middle car
(MC), and front car (FC), which have the velocity of $v_{ego}$, $v_{mc}$, and
$v_{fc}$ respectively. Moreover, $v_{fc}$ $<$ $v_{mc}$ $<$ $v_{ego}$

Figure 3: Scenario.

To predict the positions of the different cars, the simulation can use several
models with different levels of abstraction and approximation. The first uses
the kinematic equation $x(t)=1/2*a*t^{2}+v*t+x_{0}$. In this formula, $a$ is
the vehicle’s acceleration in $m/s^{2}$, $v$, the velocity of the vehicle in
$m/s$, and $x$, the vehicle’s position. This model is the original model.
However, there is a possibility of using a more simplified model. We obtain
this model by removing the acceleration term. We call this model the
approximated model, which is less detailed but has a lower computational cost.
These two models only take into account that the cars do not change lanes
during the prediction of their position.

In [Biglari and Denil, 2022], the adaptive approximation is applied to a
similar lane changing case study using these two different models. In the
study of Biglari and Denil, a controller needs to select the most appropriate
model at run-time between an original and approximated model. However, both
models reason on the same set of properties and have the same environmental
context, namely, the cars do not change lanes. Hence, selecting one valid
model within the specific context is essential during model selection. The
concept of a Validity Frame is highly beneficial in this regard.

The scenario in this paper describes an additional feature, a turning or
blinking light. A blinking light indicates the driver’s intention to change
lanes. In our running example, blinking is the influencing factor of interest,
an environmental factor that is outside of the model’s control but impacts
model selection.

We also added a much more computationally expensive model. This model
simulates the lane changing behaviour of the different cars as well not only
the ego car, and as such is valid both in the blinking and non-blinking light
context. In this model, we simulate every car and considered each of them as
an ego car, and run the simulation to perceive the behaviour of the system.

###  3.2 Adaptation using a Validity Frame

Our validity frame defines the different contexts of the system based on
_influencing factors_ in which the original model and approximated models are
valid. For applying the adaptation with frames, two challenges arise:

Challenge 1 - Properties of the environment and frames:

Influencing factors are those properties of the environment that influence a
model’s behavior and are quantified. We must identify these properties as
factors that exert a significant impact. Once the properties are properly
identified, characterised, a validity frame needs to be created. We will not
focus on the aspect of building a frame but assume that proper techniques are
available for creating the frame.

Challenge 2 - Run-time selection of models:

Having different models available, both abstracted, approximated and
combinations, requires model management techniques. Organising these models is
required. Furthermore, selecting the correct model at run-time requires proper
data-structures and algorithms.

###  3.3 Design-time Model Organisation

As a starting point, we use the Validity Frame Graph (VFG) proposed by Van
Acker in [Van Acker et al., 2021]. The validity graph encodes the relations
between different models based on their validity. It is therefore classified
as a mega-model [Favre, 2006].

Mega-modeling, as defined in [Favre, 2006], is the practice of creating a
model that illustrates the global relationships between various modeling
artifacts without focusing on their specific content. This is precisely what a
Validity Frame Graph (VFG) does, capturing the abstract relationships between
interrelated Validity Frames and their containing models without considering
their specific modeling details.

However, in our case, the data-structure is quite naive, in a sense that all
models are represented as a Vertex within the graph. Edges between the
vertices encode how properties are removed, added, or the range of properties
is changed between the models. The organisation in this data-structure results
in a fully connected graph.

Figure 4 depicts an example of $VFG$. This figure shows five interrelated
Validity Frames and corresponding models, which create a Validity Frame Graph.
It visualizes the relationships between various models using the sets of
properties and factors of influence, $\pi$, and $\gamma$. In the $VFG$,
instances that depict the same $\pi$ and $\gamma$ are connected by an
abstraction relation represented by a solid arrow. If an instance represents
only a portion of the $\pi$ and/or $\gamma$, it is connected to the
corresponding instance via a view decomposition relation represented by a
dashed arrow [Van Acker et al.,].

Figure 4: Validity Frame Graph example from [Van Acker et al.,].

Using the $VFG$ technique, we define and classify models within a Validity
Frame. Figure 5 illustrates the $VFG$ for this Highway Lane change scenario.
In Figure 5, $\pi$ is the set of properties and $\gamma$ is the set of
influencing factors. $VF_{d}$ is the head of the $VFG$, and the contained
model, $M_{d}$ which is the most detailed model. The following relations are
between the contained models of the VFs:

  * –

$M_{a}$ = $Abstract(M_{d})$

  * –

$M_{a_{1}}$ = $Approximate(M_{a})$

  * –

$M_{b}$ = $Abstract(M_{d})$

  * –

$M_{b_{1}}$ = $Approximate(M_{b})$

$M_{a}$ is the abstract version of $M_{d}$ and $M_{a1}$ is the approximated
model based on $M_{a}$. And also $M_{b}$ is the abstracted model of $M_{d}$,
and $M_{b1}$ is the approximated model based on $M_{b}$.

Figure 5: validity frames Graph for the Highway Lane change scenario.

###  3.4 Run-time Model Organisation: Decision Tree

As hinted in the previous section, a fully connected graph is not ideal for
selecting models at run-time. As such, another data-structure is needed. We
utilize a decision tree to select the most suitable models for our system in a
specific context, as depicted in figure 6.

The organisation in a decision tree also requires a significant amount of
domain knowledge. The designer needs to select in what order the properties
and influence factors are ordered in the tree or possibly combined within a
single splitting of the tree.

To select a set of models, we use Decision Tree, and its time complexity is
$O(depth)$. Using a fully connected graph needs choosing between Breadth First
Search (BFS) and Depth First Search (DFS). The time complexity of the DFS and
BFS graphs are represented in the form of $O(|V|+|E|)$, where $V$ is the
number of nodes and $E$ is the number of edges. Therefore decision tree is
computationally less expensive than BFS or DFS [Cormen et al., 2022].

Figure 6: Decision Tree of Highway Lane Change Scenario.

We select the model during _run-time_ but build the decision tree at _design
time_ to avoid increasing run-time computational cost. In Figure 6, we
traverse from the root to the leaf nodes according to the context. Afterward,
we employ the response surface graph for the model selection method suggested
by Biglari and Denil. The approximated model is a model with uncertainty and
there is no possibility of using as much uncertainty as you like. So, you can
switch to the approximated model if the uncertainty is within the bound. In
[Biglari and Denil, 2022], the response surface graph shows how to find the
bound and where you are allowed to swap between models. Algorithm 1 shows the
pseudocode of selecting from the decision tree’s leaf nodes and deciding to
use the original or approximated model. This algorithm compares
nextTrajectories, which correspond to different decisions. For example, the
ego car may decide to change lanes or not, and if so, when and where. If
simulations conclude the same decision, then we use the approximated model,
which has less computational cost.

Algorithm 1

$out1\leftarrow simulation\ result\ of\ the\ original\ model$

$out2\leftarrow simulation\ result\ of\ the\ approximated\ model$

if $out1.nextTrajectories=out2.nextTrajectories$ then

$Use\ Approximated\ model.$

else

$Use\;Original\;model.$

end if

Finally, to ensure that the model tree is optimized for real-time performance,
we follow the MAPE-K control loop in figure 2, and constraints are applied to
eliminate infeasible solutions. As this is a real-time system, not all
possible solutions are suitable. Thus, solutions that cannot meet deadlines
are pruned.

###  3.5 Experiments and Results of the Adaptation

In our running example scenario in Figure 3, the velocity of $v_{ego}$,
$v_{mc}$, and $v_{fc}$ are as follows: $v_{fc}$ ¡ $v_{mc}$ ¡ $v_{ego}$. We
show the simulation results for different contexts. Figure 7 shows the
simulation result when there is a non-blinking context.

Figure 7: Non-Blinking Context Simulation.

Accordingly, we add the influencing factor of blinking. Figure 8 shows the
simulation result in the blinking context. The optimal trajectory that the ego
car follows is depicted by the arrowed line, while the other lines represent
alternative trajectories that are deemed inappropriate.

Figure 8: Blinking Context Simulation.

The simulation results depict that in a blinking context when the ego car
knows that MC intends to change the lane, in Figure 8, the ego car keeps the
lane but in a non-blinking context, the simulation result is different, Figure
7, the ego car decides to change the lane.
Additionally, our scenario involves both an original and an approximated
model. The simulation will determine if either model leads to a differing
conclusion. By using Algorithm 1 and response surface graph, if there is no
different conclusion and the next trajectories are the same, the approximated
model is considered suitable, as it has lower computational cost. Logically,
using an approximated model instead of the original model decreases the
computational cost because these models have a simpler structure with fewer
parameters compared to more detailed models.

##  4 DISCUSSION

In section 3, we described the experiments and simulations we performed to
reduce the computational cost while using the run-time adaptive approximation
technique. Consequently, we chose the more appropriate model from our library
of models and interrelated Validity Frames for our system under study.

In this section, we discuss some of these steps, presenting lessons learned
and highlighting additional questions raised and identified challenges:

Challenge A: A library of related models’ data-structure.

We use the library of related models. Accordingly, we need to find out what
such a library of related models looks like. In our case, we used the validity
frame graph at design time and the decision tree at run-time. Multiple models
are created for a single system, $Sys$. By clearly defining the validity of
each model within its Validity Frame ($VF$), using a Validity Frame Graph
$VFG$ [Van Acker et al.,] can visualize the relationships between the models.
To overcome challenge A, we also need to solve some sub-challenges.

Encoding relations between properties.

There are three types of properties in a system: [Van Acker et al.,] system
properties, properties of interest, and Influencing factors (environment
properties).

  * •

System properties: related to the real-world entity. It is part of the
possibly infinite set of properties required to grasp all possible contexts
and facets of the considered real-world entity. In practice, one cannot fully
specify this property set, as it is impossible to capture all object’s facets
in all its possible contexts.

  * •

Properties of interest. The set of properties of interest gives rise to the
abstraction relation between model M and system Sys: a model only provides
(correct) answers with respect to these properties, while other properties are
abstracted away. More specifically, the implemented set of properties of model
M is a subset of the system properties.

  * •

Influencing factors. The set of influence factors can also influence the
abstraction relation between model M and system Sys: a model is only supposed
to provide answers concerning the modeled properties under the modeled
influence factors, while other influence factors are abstracted away. More
specifically, the implemented set of influence factors of the model is a
subset of the system properties.

Therefore, properties of interest and influencing factors are properties that
we consider. After bringing out properties in our example are distance and
blinking, the next step is finding the relation between these properties. How
to encode this relation is still the question.

Order in the property set.

There seems to be an intuitive concept on ordering in the set of models that
exist. However, the ordering can only be partial as we cannot really compare
different properties and their ranges to each other and make a conclusion if
the model has a bigger validity. As such a partially ordered set or poset
might be a better visualisation. In a poset, each element can be compared to
each other element and results in a bigger, smaller, equal or incomparable
result. However, when we create relations per property, we get a fully ordered
relation. Although these options still need more thinking, handling multiple
properties is still one of the unsolved challenges.

Challenge B: How to select a model at run-time to use in a specific situation?

We can not use all our models at run-time in all different situations
(context). Then two questions pop up.

Search Algorithm.

We use Decision Tree so the time complexity is $O(depth)$. The designer must
decide on the arrangement of properties and impact factors in the tree or
combine them into a single splitting. There are some splitting decision tree
methods to choose from.

Translation beforehand to a data-structure

CPS is a real-time system, so we can not do everything at run-time and tend to
increase the computational cost at run-time. To avoid the overhead at run-
time, we propose to do translation beforehand to a data-structure at design
time that is more feasible. For instance, making a decision tree at design
time.

Challenge C: Can we automatically convert a mega-model to a decision tree (or
related data-structure)?

Currently, converting a mega-model to a decision tree is performed manually as
we fully understand our system, but this may not be the case in other
environments and systems. As previously mentioned, the order of the decisions
and organisation of properties and influence factors within the decision tree
affects its run-time performance. The challenge is to automatically create a
decision tree that is optimal to use at run-time. Figure 6 represents our
first idea of implementing the decision tree. In this data structure, level 1
of the tree includes influencing factors. The nodes would be sorted down as
the number of branches in level 1 is less than level 2 of the tree, which is
depicted in Figure 9.

Figure 9: Decision Tree Representation.

For building the decision tree, we utilize properties of interest and
influencing factors, and there are relationships between these properties.
However, there is a need to optimise the organisation of the tree. For this,
there is a need for domain knowledge about which contexts are accessed often.
A _heat map_ is a graphical representation of data to visualize the
relationship between variables. Heat maps might help discover the feature
importance of each property in the tree. A heat map of feature importance
provides a clear view of which properties have the greatest impact and which
play a less significant role. Figure 10 represents the heatmap where the
possibility of non-braking/non-blinking context is more than other ones.

Figure 10: Heat Map.

Furthermore, it is essential to show that the resulting tree can be used in
all different contexts as the mega-model. However, this does not mean that all
models in the mega-model should be present in the run-time data structure.
This is closely related to the next challenge.

Challenge D: Which surrogate models to create?

It is possible to create a set of surrogate models starting from a single
model. Typically, the smaller the context, the less computational effort is
required to simulate the model. However, creating a surrogate model, either by
hand or computationally, requires a tremendous amount of effort. Balancing the
number of models and their operating range will be needed. As such, guidelines
and heuristics will need to be created to assist designers in making
appropriate decisions. Again, the domain knowledge about often used contexts
is important to see which surrogates to create for the model at design time.
Information from the operation phase of the system might be needed to make
optimised decisions. Digital twins are needed for such a setup.

This experience report acknowledges that validation is a crucial step in
modeling. However, for this study, we proceed with the assumption that the
models have been thoroughly validated and are functioning as intended.

In our running example, we used blinking as the influencing factor of interest
to represent the validity frame concept. There are other influencing factors
that we could add to our validity frame, for example, traffic, road and
weather conditions.

##  5 CONCLUSIONS

Real-time adaptive approximation in Cyber-Physical Systems is a method used to
reduce the computational cost of running models. However, an approximated
model is not valid in all the various contexts the system may operate in. In
this experience report, we applied the adaptive approximation technique using
the Validity Frames concept in our case study. We utilized the concept of
Validity Frames to enable adaptation at run-time in different contexts. In
this work, we used the properties of interest and environment properties known
as influencing factors within a validity frame, and the relations between
different validity frames to choose the most appropriate model from our
library of models. We discussed identified challenges, lessons learned, and
additional questions raised.

## Acknowledgments

Raheleh Biglari is funded by the BOF fund at the University of Antwerp.

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## Author Biographies

RAHELEH BIGLARI is a doctoral student at the University of Antwerp, Faculty of
Applied Engineering in Electronics and ICT. Her email address is
<EMAIL_ADDRESS>
JOACHIM DENIL is an associate professor at the University of Antwerp, Faculty
of Applied Engineering in Electronics and ICT. His email address is
<EMAIL_ADDRESS>


# Integrated waveguide-based acousto-optic modulation with near-unity
conversion efficiency

Liang Zhang  1 Chaohan Cui  1 Pao-Kang Chen  1 and Linran Fan1  1James C.
Wyant College of Optical Sciences, The University of Arizona, Tucson, Arizona
85721, USA<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Acousto-optic modulation in piezoelectric materials offers the efficient
method to bridge electrical and optical signals. It is widely used to control
optical frequencies and intensities in modern optical systems including
Q-switch lasers, ion traps, and optical tweezers. It is also critical for
emerging applications such as quantum photonics and non-reciprocal optics.
Acousto-optic devices have recently been demonstrated with promising
performance on integrated platforms. However, the conversion efficiency of
optical signals remains low in these integrated devices. This is attributed to
the significant challenge in realizing large mode overlap, long interaction
length, and high power robustness at the same time. Here, we develop acousto-
optic devices with gallium nitride on sapphire substrate. The unique
capability to confine both optical and acoustic fields in sub-wavelength
scales without suspended structures allows efficient acousto-optic
interactions over long distances under high driving power. This leads to the
near-unity optical conversion efficiency with integrated acousto-optic
modulators. With the unidirectional phase matching, we also demonstrate the
non-reciprocal propagation of optical fields with isolation ratio above 10 dB.
This work provides a robust and efficient acousto-optic platform, opening new
opportunities for optical signal processing, quantum transduction, and non-
magnetic optical isolation.

##  1 Introduction

Large-scale integration and device minimization are powerful methods to
improve system functionality and efficiency. This is witnessed by the recent
development of nonlinear optics in integrated photonic circuits [1, 2, 3, 4].
Compared with optical fields, acoustic fields have much lower propagation
speed and stronger coupling with electric fields, thus providing complementary
benefits for signal processing [5]. Therefore, tailored interactions between
optical and acoustic fields in hybrid photonic-phononic circuits attract
significant attentions recently with potential applications ranging from
quantum transduction [6, 7, 8] and comb generation [9, 10] to photonic machine
learning [11].

Acousto-optic modulation (AOM) plays a critical role in such hybrid circuits
for signal conversion among different degrees of freedom. Intensive efforts
have been devoted to the development of integrated acousto-optic modulators
based on different piezoelectric materials including lithium niobate [12, 9,
13], aluminum nitride [14, 15, 16], gallium arsenide [17], and indium
phosphire [18]. Efficient AOM requires the simultaneous confinement of optical
and acoustic fields in sub-wavelength structures [13]. While optical
confinement can be readily realized using materials with higher refractive
index for waveguides, acoustic confinement is challenging as integrated
photonic materials typically have acoustic velocities higher than substrates
[19]. The lack of acoustic confinement leads to small coupling strengths
between optical and acoustic fields due to the small mode overlapping. While
the simultaneous confinement of optical and acoustic fields can be realized in
suspended structures, the interaction length and power handling capability are
limited due to the mechanical fragility [13]. The complex fabrication process
also causes high acoustic propagation losses, which further deceases the
interaction length. As a result, high optical conversion efficiency is still
out of reach for integrated acousto-optic devices.

In this article, we overcome these challenges to realize near-unity conversion
efficiencies with integrated acousto-optic modulators. This is achieved by
developing the gallium nitride (GaN) on sapphire platform. The refractive
index of GaN is significantly larger than sapphire [20]. More importantly,
velocities of both transverse and longitudinal acoustic waves in GaN are
remarkably lower than sapphire [19]. Therefore, we can realize sub-wavelength
confinement of both optical and acoustic fields in GaN waveguides on sapphire
substrates without suspended structures. Strong acousto-optic coupling can be
realized over long interaction lengths under high driving power with minimal
propagation losses, leading to the near-unity optical conversion efficiency.

Figure 1: GaN-on-sapphire platform for acousto-optic modulation. a. Schematic
of the integrated acouto-optic modulator. Waveguides are aligned along the GaN
$[11\bar{2}0]$ direction. b. Phase matching condition for co-propagating
acoustic field (green), input TE0 (blue), and output TE1 (red) optical modes.
c. Simulated electric field profiles of input TE0 and output TE1 optical modes
along the x direction. d. Calculated acousto-optic coupling coefficient
between TE0 and TE1 optical modes as a function of waveguide width and
acoustic frequency. e. Simulated displacement profile of the fundamental
Rayleigh (R0) acoustic mode.

##  2 Results

The integrated acousto-optic modulator is schematically depicted in Fig. 1a.
Acoustic fields are launched by interdigital transducers (IDT) through the
piezoelectric effect, and focused into the waveguide. Optical fields are
launched through a separate waveguide, and transferred into the same waveguide
with acoustic fields through a directional coupler. For co-propagating
acoustic and optical fields, AOM can occur as both Stokes and anti-Stokes
processes. Input optical fields at angular frequency $\omega_{0}$ are
scattered by acoustic fields at angular frequency $\Omega$ to generate the
output optical field at $\omega_{1}=\omega_{0}-\Omega$ and
$\omega_{1}=\omega_{0}+\Omega$ in the Stokes and anti-Stokes processes
respectively. The corresponding phase matching conditions are
$\beta_{0}-q=\beta_{1}$ and $\beta_{0}+q=\beta_{1}$ with $q$, $\beta_{0}$, and
$\beta_{1}$ the acoustic, optical input, and optical output wavevectors
respectively (Fig. 1b). We can switch between Stokes and anti-Stokes processes
by interchanging the optical input and output modes [15]. Here, we use the
fundamental and first-order transverse-electric (TE0 and TE1) modes for the
input and output optical fields respectively (Fig. 1c). As the TE0 mode has a
larger wave-vector ($\beta_{0}>\beta_{1}$), the Stokes process dominates the
acousto-optic modulation in our device. The conversion between TE0 and TE1
optical modes can be mediated by different acoustic modes. We perform
numerical simulations to calculate the acousto-optic coupling coefficient
[21]. Multiple acoustic modes can mediate efficient acousto-optic coupling
between TE0 and TE1 modes as shown in Fig. 1d. We choose the fundamental
Rayleigh (R0) acoustic mode, which shows the largest coupling strength.
Moreover, with the significant out-of-plane displacement, the fundamental
Rayleigh mode can be efficiently excited by IDTs on GaN (0001) plane (Fig. 1e)
[22]. Due to the strong sub-wavelength confinement, the acousto-optic
modulation process shows significant geometric dispersion. Therefore, the
acoustic frequency can be tuned by the waveguide width. The waveguide width of
our device is designed to be 1 $\mu$m. Therefore, the phase matching condition
can be satisfied near the acoustic frequency $\Omega=2\pi\,\times\,$1 GHz.

Figure 2: Integrated acousto-optic modulators. a. Optical image of the
fabricated device. b-d. Scanning electron microscopy (SEM) images of the IDT
(b), IDT electrodes (c), and directional coupler (d). e. Acoustic transmission
of the directional coupler in the bar (blue) and cross (red) waveguides. f.
Optical transmission of the directional coupler in the bar (blue) and cross
(red) waveguides.

The device is fabricated with 1-$\mu$m thick GaN template wafers grown on
sapphire substrates using metal-organic chemical vapor deposition (Fig. 2a).
Acousto-optic devices are patterned by the electron-beam lithography (EBL)
using FOX-16 resist. After developing in TMAH, we etch the GaN layer with
reactive ion etching using Cl2/BCl3/Ar gases. IDTs are defined with EBL in
ploymethyl methacrylate (PMMA) resist, followed by Ti/Al/Au deposition and
lift-off in acetone. The total waveguide length is $L=3$ mm to ensure
efficient acousto-optic interaction. IDTs consist of a 5-nm titanium bottom
layer, 100-nm aluminium middle layer, and 10-nm gold top layer. The IDT period
is designed as 4.9 $\mu$m to match the acoustic frequency around
$\Omega=2\pi\,\times\,$1 GHz (Supplementary Section 1). The IDT aperture and
electrode width are 100 $\mu$m and 1.22 $\mu$m respectively (Fig. 2b and c).
The directional coupler consists of two parallel 800 nm wide waveguides with
400 nm gap (Fig. 2d). As acoustic and optical fields have different coupling
strengths between the two waveguides, the same directional coupler structure
can be designed to show different functions for acoustic and optical fields.
Here, we set the directional coupler length at 250 $\mu$m, in which case
acoustic fields remain in the same waveguide and optical fields are completely
transferred into the other waveguide (Supplementary Section 2).

We first use the vector network analyzer (VNA) to measure the acoustic
transmission between IDT pairs directly connected by straight GaN waveguides
with different lengths. The dependence of the acoustic transmission on the
device length allows us to estimate the propagation loss of the fundamental
Rayleigh mode, which is $\alpha_{b}=0.85$ dB/mm (Supplementary Section 1). By
extrapolating the acoustic transmission to zero device length, we can obtain
the IDT power efficiency $\eta^{2}=-44$ dB. With 50 periods, IDTs show a
center frequency of 0.99 GHz and 3-dB bandwidth of 12 MHz. To verify the
directional coupler performance, we fabricated test devices with four
input/output ports connected all with IDTs for acoustic field characterization
or all with optical couplers for optical field characterization (Fig. 2e and
f). We only observe strong acoustic signal in the bar configuration of the
directional coupler (Fig. 2e). The oscillation in the acoustic transmission
spectrum is caused by the reflection between input and output IDTs. Extinction
ratio above 10 dB between the bar and cross waveguides can be achieved. For
optical fields, we only observe strong output in the cross configuration with
average extinction ratio above 14 dB (Fig. 2f). This shows that optical fields
and acoustic fields can be efficiently combined by the directional coupler.

Figure 3: Acousto-optic modulation performance. a. Measurement setup. EDFA:
Erbium-Doped Fiber Amplifier. FPC: Fiber Polarization Controller. VNA: Vector
Network Analyszer. RF AMP: RF Amplifier. HV DC AMP: High voltage DC Amplifier.
AWG: Arbitrary Waveform Generator. OSC: Oscilloscope. PD: Photo-Detector. b.
Acoustic modulation spectrum with optical input wavelength 1551.7 nm. c.
Acoustic modulation spectrum with different optical input wavelengths. d.
Optical modulation spectrum with acoustic frequency $\Omega/2\pi=0.998$ GHz.
e. Optical signals filtered by the free-space cavity. f. Power of TE0 and TE1
modes with different RF driving amplitudes. g. Optical conversion efficiency
with different RF driving amplitudes.

The modulation response is characterized by the measurement setup illustrated
in Fig. 3a. CW light generated by a tunable semiconductor laser is amplified
by the erbium-doped fiber amplifier (EDFA) and delivered to the device with a
lensed fiber. RF signals from the vector network analyzer (VNA) is loaded into
IDTs after amplification to excite acoustic fields. The optical transmission
is monitored by a slow high-sensitivity photodetector (DC PD). The output
optical field is also measured by a fast photodetector (AC PD), whose electric
signal is sent back to VNA. The representative RF modulation spectrum is shown
in Fig. 3b. The input optical wavelength is set at $\lambda=$1551.7 nm. Strong
modulation can be clearly observed. The acoustic modulation 3-dB bandwidth is
measured as $\Delta$ = 0.57 MHz. With the modulation bandwidth and acoustic
propagation loss, we can calculate the acoustic group velocity $v_{\rm
g}=2\pi\Delta/\alpha_{b}\approx\,$4400 m/s, which agrees with the simulated
value (see Supplementary Section 1) [14]. We further measure the RF modulation
response at different optical wavelengths (Fig. 3c). The acoustic frequency
with maximum modulation efficiency shifts with the optical wavelength, showing
the change of the phase-matching condition due to dispersion. Figure 3d shows
the peak modulation response at different optical wavelengths with the fixed
acoustic frequency $\Omega/2\pi$= 0.998 GHz. The optical bandwidth of the
modulation process is measured as 3.5 nm, which agrees the theoretical value
$\delta\lambda=2.78\lambda^{2}/(\pi L\delta n_{g})\approx 3.2$ nm with $\delta
n_{\rm g}=0.22$ the group refractive index difference between TE0 and TE1
modes [15].

To measure the optical conversion efficiency, we use a free-space cavity with
1-MHz linewidth as the optical filter (Fig. 3a). A piezo-transducer is
attached to the free-space cavity mirror to modify the resonant wavelengths.
The driving voltage of the piezo-transducer is continuously swept. Therefore,
we can separate optical powers from the residual input and converted output
optical fields in the time domain (Fig. 3e). Without acoustic fields, we only
observe the TE0 input optical field. With acoustic fields, the amplitude of
the TE0 input optical field decreases. More importantly, we can observe the
emergence of the converted TE1 output optical field. The converted TE1 output
optical field only shows up on one side of the residual TE0 input optical
field, proving the single-sideband nature of acousto-optic modulation. By
increasing the RF drive amplitude, we can observe the sinusoidal oscillation
of the optical power between TE0 and TE1 modes (Fig. 3f). With driving power
$P=560$ mW, the TE0 power is close to zero with the TE1 mode reaching the
maximum amplitude. Therefore, the on-chip conversion efficiency ($\theta$)
close to unity can be achieved at driving power $P=560$ mW (Fig. 3g). This
further allows us to estimate the acousto-optic coupling coefficient
(Supplementary Section 3) [13]

$\displaystyle\frac{g}{\sqrt{\hbar\Omega}}=\frac{\pi\alpha_{b}}{4\eta\sqrt{P}(1-e^{-\alpha_{b}L/2})}\approx
255\;\mathrm{mm^{-1}W^{-1/2}}$ (1)

which is more than two orders of magnitude higher than acousto-optic devices
without simultaneous sub-wavelength acoustic and optical confinement (Table
1).

Figure 4: Nonreciprocal propagation. a. Power spectrum of the input TE0 mode
and output Stokes and anti-Stokes TE1 modes in the forward direction. b. Power
spectrum of the input TE0 mode and output Stokes and anti-Stokes TE1 modes in
the backward direction. c. Optical signals filtered by the free-space cavity
in forward and backward directions. d. Power ratio between forward and back
directions.

Due to difficulties in integrating high-quality magneto-optical materials, it
is challenging to build on-chip optical isolators and circulators. With the
direction-dependent phase-matching condition, acousto-optic modulation
provides the non-magnetic method to realize on-chip optical isolation [23, 15,
13]. We verify the non-reciprocal behavior of our acousto-optic device by
measuring the optical transmission in both forward and backward directions
(Fig. 4a and b). The RF driving signal has fixed frequency $\Omega/2\pi=0.998$
GHz and power $P=560$ mW. In the forward direction, phase-matching condition
is satisfied for the Stokes process, where the input TE0 mode at frequency
$\omega_{0}$ is converted into the output TE1 mode at frequency
$\omega_{0}-\Omega$ (Fig. 4a) [13]. No anti-Stokes signal at frequency
$\omega_{0}+\Omega$ is observed. In the backward direction, no mode conversion
should happen ideally and all optical power should remain in the TE0 mode at
frequency $\omega_{0}$. However, with the finite interaction length, we
observe the anti-Stokes process in the backward direction (Fig. 4b). The anti-
Stokes process in the backward direction has much lower efficiency due to
phase mismatch. Therefore, the input TE0 mode at frequency $\omega_{0}$ is
only partially converted into the output TE1 mode at frequency
$\omega_{0}+\Omega$ even at the maximum modulation optical wavelength
$\lambda=1551.7$ nm (Fig. 4c). If we compare the TE0 power in forward and
backward directions, we can clearly see that isolation ratio above 10 dB has
been achieved (Fig. 4d).

##  3 Discussion

To benchmark the performance of our device, a summary of recent works on
integrated acousto-optic modulators is presented in Table 1. The GaN-on-
sapphire platform in this work is the only one that is capable of confining
both optical and acoustic fields in sub-wavelength scales without using
suspended structures. Therefore, the interaction length has been significantly
improved while maintaining the high acousto-optic coupling coefficient. In
addition, the GaN-on-sapphire platform also has excellent power-handling
capability. No performance degradation is observed under high driving powers
(Supplementary Section 4). Moreover, the GaN-on-sapphire platform shows
remarkably lower acoustic propagation loss. The excellent performance in all
critical merits leads to the demonstration of the near-unity optical
conversion efficiency. The robustness of unsuspended structures also allows
the incorporation of integrated acousto-optic modulators into large-scale
photonic-phononic circuits with complex functionalities. The elimination of
supporting tethers and membranes to anchor suspended structures further
enables the flexible design of circuit patterns.

The acoustic driving efficiency can be further improved. Propagation loss
below 0.05 dB/mm has been demonstrated with GaN-on-sapphire acoustic
waveguides [19]. Therefore, the acousto-optic interaction length can be
extended by more than 10 times. This will lead to the reduction of the driving
power by more than two orders of magnitude, below 10 mW with the current IDT
design. The IDT efficiency can be further improved using uni-directional IDT
designs with more periods and better electrical impedance matching [24, 25].
The heterogeneous structure of aluminum nitride (AlN) and GaN, which is widely
used for power electronics [26], can further increase the driving efficiency
by leveraging the larger piezoelectric coefficient in AlN [27]. As a result,
we expect the driving power to achieve complete optical conversion can be
decreased to the micro-watt level.

Table 1: Integrated acousto-optic modulators

Work | Year | Platform | $\Omega/2\pi$ | $\alpha_{b}$ | $L$ | $g/\sqrt{\hbar\Omega}$ | $\theta$ | Suspended | Optical | Acoustic
---|---|---|---|---|---|---|---|---|---|---
|  |  | (GHz) | (dB/mm) | (mm) | (mm-1W-1/2) | ($\%$) |  | confined | confined
Liu [14] | 2019 | AlN | 16.4 | — | 0.5 | 0.041 | 2.5e-4 | Y | Y | Y
Kittlaus [15] | 2020 | Si/AlN | 3.11 | — | 0.96 | 4.5 | 13.5 | N | Y | N
Christopher [12] | 2020 | LiNbO3 | 0.67 | — | 0.6 | 1.66 | 0.9 | N | Y | N
Shao [9] | 2020 | LiNbO3 | 2.89 | — | 0.1 | 0.004 | 3.5 | N | N | N
Ahmed [28] | 2021 | LiNbO3 | 1.16 | 4 | 0.45 | 0.417 | 1 | Y | Y | N
Christopher [13] | 2021 | LiNbO3 | 0.44 | 11.7 | 0.25 | 377 | 18 | Y | Y | Y
Wan [29] | 2022 | LiNbO3 | 0.84 | — | 0.12 | 0.035 | 3.2e-3 | N | Y | N
This work | 2023 | GaN | 0.99 | 0.85 | 3 | 255 |  $\sim$100 | N | Y | Y

##  4 Conclusion

In conclusion, we have developed the GaN-on-sapphire platform for acousto-
optic devices. We achieve high acousto-optic coupling strength, long
interaction length, low-loss acoustic propagation, and large power handling
capability at the same time. This leads to the first demonstration of near-
unity conversion efficiency with integrated acousto-optic modulators. This
work will enable the exploration of hybrid photonic-phoninc circuits at large
scale for advanced signal processing, with important applications in microwave
photonics and quantum transduction.

Funding This material is based upon work supported by the Office of the Under
Secretary of Defense for Research and Engineering under DEPSCoR program award
number FA9550-21-1-0225 managed by Army Research Office, and NSF Grant No.
ITE-2134830.

Disclosures The Authors declare no competing financial or non-financial
interests

Data availability The data that support the findings of this study are
available from the corresponding author upon reasonable request

Supplemental document See Supplement 1 for supporting content.

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