# Spherical wedge billiard: from chaos to fractals and Talbot carpets

Tomáš Tyc<EMAIL_ADDRESS>Department of Theoretical Physics and
Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno,
Czechia  Darek Cidlinský  Department of Theoretical Physics and Astrophysics,
Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czechia

###### Abstract

We introduce the spherical wedge billiard, a dynamical system consisting of a
particle moving along a geodesic on a closed non-Euclidean surface of a
spherical wedge. We derive the analytic form of the corresponding Poincaré map
and find very complex dynamics, ranging from completely chaotic to very
regular, exhibiting fractal features. Further, we show that upon changing the
billiard parameter, the fixed points of the Poincaré map merge in complex
ways, which has origin in the spherical aberration of the billiard mapping. We
also analyze in detail the regular regime when phase space diagram is closely
related to Talbot carpets.

††preprint: APS/123-QED

In the standard billiard problem Katok ; Gutkin2012 , a particle or light ray
is confined to a certain region of a plane where it moves freely with specular
reflection at its boundary. The particle dynamics can be regular, completely
chaotic or a combination of the two Chernov ; Andreasen2009 , depending on the
shape of the boundary. Billiards play an important role in nonlinear physics
and statistical mechanics, are related to many areas of physics Douglas ;
Ponomarenko ; friedman2001 ; gao2015 ; bittner2012 ; Bunimovich2005  and have
been subject to extensive theoretical and experimental investigation; they are
invaluable in investigating relations between the classical dynamics and
properties of the corresponding quantum system ishikawa1985 ; cheon1989 ;
Bogomolny2004 .

Consider now a related situation, when the particle or light ray is confined
to a closed non-Euclidean surface where it moves freely. As there is no
boundary, the particle is not subject to reflection and it continues moving
forward along a geodesic, which is the most straight line on the surface. We
will call such a system a reflectionless geodesic billiard (RGB). In contrast
to the standard billiard problem, the motion in the RGB is governed by the
curvature of the surface rather than by the shape of the boundary. This is a
very similar situation as in geodesic lenses that are attracting increasing
attention from both the theoretical and experimental point of view
schultheiss2010 ; mitchell2014 ; chen2015 ; Quevedo-Teruel2022 ; there too, it
is the curvature of the non-Euclidean surface that yields the desired
functionality, e.g. focusing of light, rather than a refractive index profile
as in usual lenses. The simplest example of RGB is a sphere where all
trajectories are periodic and form great circles. Less trivial examples
include surfaces of regular polyhedra recently analyzed in garcia2020  or a
deformed torus wang-noneuclidean2020 .

Here we present a RGB with the shape of a spherical wedge, see Fig. 1 (a—b);
we call it spherical wedge billiard (SWB). Despite the simplicity of its
geometry, the dynamics of this system exhibit extremely complex behavior,
ranging from almost completely chaotic to highly regular behavior with fractal
features, depending on the value of the billiard parameter $\Phi$. Remarkably,
for certain parameter values the phase space diagram of SWB reflects the
structure of the set of rational numbers, closely resembling Talbot carpets
known from diffraction theory Berry2001 . A great advantage of this system is
that the trajectories can be described analytically by formulas containing
just trigonometric functions, which allows to analyze many features of the
billiard analytically or semi-analytically.

Figure 1: The spherical wedge billiard with a sample geodesic trajectory for
(a) $\Phi=0.6\,\pi$ and (b) $\Phi=0.25\,\pi$. (c) The angles $\vartheta$ and
$\omega$ labeling the straight particle trajectory segment (blue) on the disk.
Alternatively, the segment will be specified with the pair of angles
$(\xi,\omega)$.

The spherical wedge billiard consists of three parts (see Fig. 1(a)): the
spherical lune $P_{1}$ with dihedral angle $\Phi$, and two half-disks $P_{2}$
and $P_{3}$ meeting along the polar axis of the spherical lune. We can imagine
that we “unfold” $P_{2}$ and $P_{3}$ into a single disk; the particle will
then alternate between moving on the disk and on the spherical lune. Parts of
the trajectory on the disk are straight line segments while on the spherical
lune they are parts of great circles. When the particle crosses the border
between the disk and lune or back, the angle between the trajectory and the
border is preserved because the trajectory is a geodesic. The angle $\Phi$ is
the sole parameter of the billiard.

To describe the straight ray segments on the disk, we use a pair of angles
$(\vartheta,\omega)$, see Fig. 1(c)). Here $\vartheta$ runs from $-\pi$ to
$\pi$ and measures the polar angle on the disk of the point where the particle
enters the disk. The $\omega$ runs from $-\pi/2$ to $\pi/2$ and it is the
angle between the trajectory and radial line to the entry point; negative and
positive $\omega$ corresponds to the particle running clockwise and counter-
clockwise, respectively, around the center.

Figure 2: The phase space diagram in the plane $(\xi,\omega)$ showing 1000
iterations the Poincaré map $M$ for different initial conditions and $\Phi$
equal to (a) $0.129\,\pi$, (b) $0.461\,\pi$, (c) $0.73\,\pi$ and (d)
$0.98\,\pi$. Only one half of the plane $(\xi,\omega)$ is shown; the other
half looks the same due to the symmetry of the SWB. The orbits are
distinguished by the color hue.

Instead of analyzing the whole trajectory on the SWB, we construct the
corresponding Poincaré map $M$ by iterating the straight line segments on the
disk. This creates a sequence of points
$\\{(\vartheta_{k},\omega_{k});\,k=1,2,\dots\\}$ that represent the particle
trajectory. By a direct calculation we get analytic formulas for the mapping
$M$ from one point $(\vartheta_{k},\omega_{k})$ to the next one,
$(\vartheta_{k+1},\omega_{k+1})$:

$\displaystyle\vartheta_{k+1}$
$\displaystyle=\frac{\pi}{2}\operatorname{sign}[\sin(\vartheta_{k}-2\omega_{k})]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!-\arctan\left\\{\frac{\tan\omega_{k}\sin\Phi}{|\sin(\vartheta_{k}-2\omega_{k})|}-\frac{\cos\Phi}{\tan(\vartheta_{k}-2\omega_{k})}\right\\},$
(1) $\displaystyle\omega_{k+1}$
$\displaystyle=-\arcsin\\{\sin\omega_{k}\cos\Phi$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!+\operatorname{sign}[\sin(\vartheta_{k}-2\omega_{k})]\cos\omega_{k}\cos(\vartheta_{k}-2\omega_{k})\sin\Phi\\}.$
(2)

It is convenient to introduce an angle
$\xi=(\vartheta-\omega)\,\operatorname{mod}2\pi$, see Fig. 1(c)), where in
this paper we define the modulo function ($\operatorname{mod}2\pi$) such that
the result belongs to the interval $(-\pi,\pi]$. It turns out that the
variables $(\xi,\omega)$ reflect the symmetry of the problem better than
$(\vartheta,\omega)$, so we will prefer them. Liouville’s theorem in the phase
space yields an invariant measure
$\mathrm{d}\xi\,\mathrm{d}(\sin\omega)=\cos\omega\,\mathrm{d}\xi\,\mathrm{d}\omega$
of the Poincaré map $M$.

Figure 3: The phase space diagram for $\Phi=0.828\,\pi$ with several periodic
orbits marked with dots of the same color, and the corresponding trajectory
segments on the disk, shown in the same color. The dark-blue dot marks the
point $\Gamma$ discussed in the text.

The formulas (1) and (2), together with the relations between
$(\vartheta,\omega)$ and $(\xi,\omega)$, i.e.,

$\xi=(\vartheta-\omega)\,\operatorname{mod}2\pi,\quad\vartheta=(\xi+\omega)\,\operatorname{mod}2\pi,$
(3)

fully describe the behavior of the spherical wedge billiard and conceal
enormous complexity whose character varies with the value of the angle $\Phi$.
This can be visualized by plotting the phase space diagram in the plane
$(\xi,\omega)$, see Fig. 2 and supplemental Video 1. See also supplemental
Video 2 for the correspondence of the disk trajectory segments with the points
in the plane $(\xi,\omega)$.

For small $\Phi$, the behavior is mostly chaotic, with occasional islands of
quasiregular motion, see Fig. 2(a-b). For $\Phi\geq\pi/2$, two large regions
of quasiregular motion appear, centered at the points
$(\vartheta=\pm\pi/2,\omega=0)$ [one of them can be seen in Fig. 2(c-d)]. As
$\Phi$ increases further, more islands of regular motion appear [see Fig.
2(c-d)] centered around points belonging to periodic orbits, see Fig. 3. When
$\Phi$ approaches $\pi$, a very interesting feature can be observed: the plane
$(\xi,\eta)$ becomes covered by stacks of ellipses of various sizes that
seemingly fill the whole plane, see Fig. 2(d) and Fig. S1 in Supplemental
Material supplement . Still, even then there are tiny regions of irregular
behavior between these ellipses, due to fragmentation of the phase space
caused by the two poles of the billiard. For $\Phi$ just below $\pi$, the
phase space diagram strongly resembles Talbot carpets known from diffraction
theory Berry2001 , see Fig. 4. It is out of scope of this Letter to describe
all of this complexity; we therefore focus on just a few of the most
interesting features.

Figure 4: Comparison of (a) the phase space diagram for $\Phi=0.997\pi$ (here
shown for irregular trajectories starting from the North pole) with (b) a
Talbot carpet Berry2001  corresponding to a binary grating with period of $a$
and opening ratio of 1/40. $z_{\mathrm{T}}$ is the Talbot distance.

First, we investigate the situation when $\Phi$ is just slightly smaller than
$\pi$, and write it as $\Phi=\pi-\varepsilon$ with $\varepsilon\ll 1$. The the
phase space diagram (shown for $\varepsilon=0.02\,\pi$ in Fig. 2(d)) reveals
that the plane $(\vartheta,\omega)$ contains a plethora of stacks of ellipses
with similar axes ratios and the same orientations. It is possible to find
their approximate equations; the calculation is shown in Supplemental Material
supplement  for the ellipses centered near the points
$(\xi,\omega)=(0,\pi/(2n))$ with $n=2,3,4,\dots$ and it yields the formulas

$\displaystyle\xi_{2nk}$ $\displaystyle=A\cos(k\alpha_{n}+\phi_{0}),$ (4)
$\displaystyle\omega_{2nk}$
$\displaystyle=AR_{n}\sin(k\alpha_{n}+\phi_{0})+\frac{\pi}{2n}-\frac{\varepsilon}{n\sin\frac{\pi}{n}\sin\frac{\pi}{2n}}\,,$
(5)

where $R_{n}=\sqrt{\varepsilon/\\{2n\sin[\pi/(2n)]\\}}$ is the ratio of the
ellipse axes in the $\omega$ and $\xi$ directions, $\alpha_{n}=4nR_{n}$
determines the rate of the point moving along the ellipse, and $\phi_{0}$ is
an initial phase. These equations are valid only up to a certain value of the
constant $A$, so a single stack of ellipses fills only a certain region,
leaving space for other stacks of ellipses. The higher is the number $n$, the
smaller are the stacks. This is the reason why for small $\varepsilon$ the
Poincaré map resembles Talbot carpets Berry2001 , see Fig. 4: it reflects the
structure of rational numbers on the $\omega$ axis in a similar way as is
found in the fractional Talbot effect along the light propagation axis,
probably including the fractal features; see Supplemental Material supplement
for more detail.

Next we examine the properties of the map near the point
$(\xi,\omega)=(\pi/2,0)$. We will denote this point by $\Gamma$; in Fig. 3, it
is marked by a dark-blue dot. The path segment on the disk corresponding to
$\Gamma$ is a horizontal diameter line going from left to right, and on the
spherical part $P_{1}$ of the spherical wedge billiard, the path follows the
equator, $\theta=\pi/2$. Obviously, the point $\Gamma$ is a fixed point of the
map $M$. For $\Phi\in[0,\pi/2)$ this point is unstable while for
$\Phi\in[\pi/2,\pi)$ it is stable, which can be checked by a direct
calculation of the Jacobi matrix eigenvalues. Further, Eqs. (1), (2) and (3)
reveal that for $\Phi\in(0,\pi/2)$, there is a period-2 stable orbit
consisting of points $(\xi,\omega)=(\pi/2,\pm\arccos(\tan\frac{\Phi}{2}))$, so
these points are the fixed points of the mapping $M^{2}$. As $\Phi$ approaches
$\pi/2$, these two points approach each other and merge at the point $\Gamma$
when $\Phi=\pi/2$. Similarly, for $\Phi\in(\pi/2,3\pi/4)$, there is a period-4
stable orbit consisting of points $(\pi/2,\pm\arccos(-\cot\Phi))$ and
$(\pi/2\pm\arccos(\cot^{2}\Phi),0)$, which are hence fixed points of the
mapping $M^{4}$. These points merge together at the point $\Gamma$ when
$\Phi=3\pi/4$. It turns out that this is a general behavior, and in the
interval $\Phi\in(\pi/2,\pi)$ there are infinitely many such mergers of fixed
points of mappings $M^{q}$ (with different orders $q\in\mathbb{N}$). We
conjecture that an even stronger statement is valid: in any arbitrarily narrow
subinterval $I\subset(\pi/2,\pi)$ there are infinitely many such mergers.
Conversely, when we decrease the spherical wedge billiard parameter $\Phi$
somewhere within the interval $(\pi/2,\pi)$, the point $\Gamma$ becomes a
source of $q$-tuples of stable fixed points of $M^{q}$ emerging from it, and
the set of values of $\Phi$ at which these $q$-tuples emerge is dense.

Although we have not been able to prove this statement in general, there is a
strong analytical and numerical indication that it is valid, which we briefly
outline next. To examine the neighborhood of the point $\Gamma$
semianalytically for $\Phi\in(\pi/2,\pi)$, we make substitutions
$\xi=\pi/2+\tau$ and $\omega=\eta/\sqrt{-2\cot\Phi}$, and rewrite Eqs. (1 – 3)
for the variables $(\tau,\eta)$; assuming that they are small, we first keep
just terms up to the first order in these variables, which is a good
approximation since all the quadratic terms vanish and next order terms are of
the third order in $\tau$ and $\eta$. This enables to put the result into the
following matrix form,

$\begin{pmatrix}\tau_{i+1}\\\
\eta_{i+1}\end{pmatrix}=\begin{pmatrix}\cos\gamma&-\sin\gamma\\\
\sin\gamma&\cos\gamma\end{pmatrix}\begin{pmatrix}\tau_{i}\\\
\eta_{i}\end{pmatrix}$ (6)

where $\gamma(\Phi)=\arccos(-\cos\Phi-\sin\Phi)\in(0,\pi)$. We see that up to
the first order, the iteration in the $(\tau,\eta)$ plane simply corresponds
to rotation by the angle $\gamma$ around the origin. Within this
approximation, for such values of $\Phi$ that give $\gamma$ as a rational
multiple of $\pi$, i.e., $\gamma=\gamma_{pq}\equiv 2\pi p/q$ with
$p,q\in\mathbb{N}$, all points $(\tau,\eta)$ would be fixed points of the
mapping $M^{q}$. However, the cubic terms that were neglected in Eq. (6)
slightly alter the transformation, making the rotation angle $\gamma$ slightly
decreasing with the radius $r=\sqrt{\tau^{2}+\eta^{2}}$, in addition to its
dependence on $\Phi$. A closer inspection reveals that in terms of the action
of the map $M$ on the rays on the disk, this corresponds simply to spherical
aberration. This way, for given integers $p,q$ the stable fixed points of the
mapping $M^{q}$ merge at the point $\Gamma$ for
$\Phi=\Phi_{pq}\equiv\gamma^{-1}(\gamma_{pq})$, where $\gamma^{-1}$ denotes
the function inverse to the function $\gamma(\Phi)$ defined above; this is
illustrated in supplemental Video 3. It turns out that the distance of these
fixed points from the origin of the plane $(\tau,\eta)$ is almost the same, so
they lie very close to a circle centered at the origin, and are almost
uniformly spaced along it, thus forming an almost regular $q$-gon. One (in
case of odd $q$) or two (for even $q$) vertices of this polygon always lie on
the $\eta$ axis. The stable points are interlaced with hyperbolic points,
which are also fixed points of the mapping $M^{q}$; however, these points lie
on a slightly smaller circle than the stable points, whereas the relative
difference of the radii quickly becomes negligible as $n$ increases. This
behavior can be illustrated on the case $p=1,q=4$ described above
analytically.

Moreover, it turns out that the point $\Gamma$ is not the only one having this
property of being a source of infinitely many stable fixed points emerging
from it as $\Phi$ decreases. In fact, the same holds also for other fixed
points of the mapping, in particular the centers of the ellipse stacks
discussed above; in this case, however, the positions of these fixed points
themselves depend on $\Phi$, as Eqs. (4) and (5) show. We thus arrive at a
striking behavior of SWB: when changing the parameter $\Phi$, stable fixed
points become sources of $q$-tuples of stable fixed points of the “higher
order”, and this way we can continue to infinity. This is a specific form of
fractal behavior.

In summary, we have introduced the concept of reflectionless geodesic billiard
and presented its remarkable realization, the spherical wedge billiard. Having
established an analytic form of the Poincaré map that describes it, we have
analyzed the system dynamics near a particular fixed point of the map and
shown that there is a complex fixed points merging with fractal features,
related to spherical aberration. We have also described the system dynamics
for the parameter $\Phi$ approaching $\pi$ and shown that the phase space gets
covered with stacks of ellipses whose positions reflect the structure of
rational numbers; this leads to a striking similarity of SWB phase space
diagrams with Talbot carpets that are known to have fractal features.

The spherical wedge billiard is unique in several aspects: its dynamics can be
described analytically but at the same time it is very complex, ranging from
chaos to regular behavior and having fractal features. It is related to
several rather different fields of physics, namely nonlinear dynamics, optics
of geodesic lenses, diffraction and number theory (both related to Talbot
carpets), and quantum physics. Whereas the analysis of the quantum version of
SWB is out of the scope of this Letter, it is the obvious next research focus
because of the close relation of energy spectrum of a quantized system to
trajectories in its classical counterpart. Moreover, the similarity between
RGBs and geodesic lenses enables to use the same mathematical tools for
describing both systems, e.g. solving the geodesic equations, or the Helmholtz
equation, on the curved surface.

We believe that the reflectionless geodesic billiard may become an important
model system for investigating a number of features of dynamical systems, both
classical and quantum. Moreover, the recent achievements in experimental
investigation of geodesic lenses should soon enable to explore RGBs
experimentally as well.

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# Weak amenability of weighted measure algebras and their second duals

M. J. Mehdipour and A. Rejali111Corresponding author

Abstract. In this paper, we study the weak amenability of weighted measure
algebras and prove that $M(G,\omega)$ is weakly amenable if and only if $G$ is
discrete and every bounded quasi-additive function is inner. We also study the
weak amenability of $L^{1}(G,\omega)^{**}$ and $M(G,\omega)^{**}$ and show
that the weak amenability of theses Banach algebras are equivalent to
finiteness of $G$. This gives an answer to the question concerning weak
amenability of $L^{1}(G,\omega)^{**}$ and $M(G,\omega)^{**}$. 11footnotetext:
2020 Mathematics Subject Classification: 43A10, 43A20, 47B47,47B48. Keywords:
Locally compact group, Weak amenability, Weighted measure algebras, Second
dual of group algebras.

##  1 Introduction

Let $G$ be a locally compact group with an identity element $e$. Let us recall
that a continuous function $\omega:G\rightarrow[1,\infty)$ is called a
_weight_ _function_ if for every $x,y\in G$

$\omega(xy)\leq\omega(x)\;\omega(y)\quad\hbox{and}\quad\omega(e)=1.$

Let $C_{0}(G,1/\omega)$ be the space of all functions $f$ on $G$ such that
$f/\omega\in C_{0}(G)$, the space of all bounded continuous functions on $G$
that vanish at infinity. Let also $M(G,\omega)$ be the Banach space of all
complex regular Borel measures $\mu$ on $G$ for which $\omega\mu\in M(G)$, the
measure algebra of $G$. It is well-known that $M(G,\omega)$ is the dual space
of $C_{0}(G,1/\omega)$ [4, 21, 26], see [22, 24] for study of weighted
semigroup measure algebras; see also [13, 14, 17]. Note that $M(G,\omega)$ is
a Banach algebra with the norm $\|\mu\|_{\omega}:=\|\omega\mu\|$ and the
convolution product “$\ast$”defined by

$\mu\ast\nu(f)=\int_{G}\int_{G}f(xy)\;d\mu(x)d\nu(y)\quad\quad(\mu,\nu\in
M(G,\omega),f\in C_{0}(G,1/\omega)).$

Let $L^{1}(G,\omega)$ be the Banach space of all Borel measurable functions
$f$ on $G$ such that $\omega f\in L^{1}(G)$, the group algebra of $G$. Then
$L^{1}(G,\omega)$ with the convolution product “$\ast$” and the norm
$\|f\|_{1,\omega}=\|\omega f\|_{1}$ is a Banach algebras.

A Borel measurable function $p$ from $G\times G$ into ${{C}}$ is called
_quasi-additive_ if for almost every where $x,y,z\in G$

$p(xy,z)=p(x,yz)+p(y,zx).$

If there exists $h\in L^{\infty}(G,1/\omega)$ such that

$p(x,y)=h(xy)-h(yx)$

for almost every where $x,y\in G$, then $p$ is called _inner_. Let
$D(G,\omega)$ be the set of all quasi-additive functions $p$ on $G$ such that

$\sup_{x,y\in G}\frac{|p(x,y)|}{\omega^{\otimes}(x,y)}<\infty.$

We denote by $I(G,\omega)$ the set of inner quasi-additive functions. For
$\mu\in M(G,\omega)$, let $L^{\infty}(|\mu|,\omega)$ be the Banach space of
all $\omega-$bounded Borel measurable functions $p$ on $G$ such that
$\|p\|_{\omega,\mu}=\|p/\omega\|_{\mu}<\infty.$ An element

$P=(p_{\mu})_{\mu}\in\Pi\\{L^{\infty}(|\mu|,\omega):\mu\in M(G,\omega)\\}$

is called a $\omega-$_generlized function_ on $G$ if

$\sup\\{\|p_{\mu}\|_{\omega,\mu}:\mu\in M(G,\omega)\\}<\infty$

and for every $\mu,\nu\in M(G,\omega)$ with $|\mu|\ll|\nu|$ we have
$p_{\mu}=p_{\nu}$,$\;|\mu|-a.e.$. The space of all $\omega-$generlized
function on $G$ is denoted by $GL(G,1/\omega)$. It is well-known from [25]
that $GL(G,1/\omega)$ is the dual of $M(G,\omega)$ for the pairing

$\langle(p_{\mu})_{\mu},\nu\rangle=\int_{G}p_{\nu}\;d\nu.$

A function $F=(F_{\mu\otimes\nu})_{\mu,\nu\in M(G,\omega)}\in GL(G\times
G,\omega^{\otimes})$ is called a _generalized quasi-additive function_ if

$F_{(\mu\ast\nu)\otimes\eta}(xy,z)=F_{\mu\otimes(\nu\ast\eta)}(x,yz)+F_{\nu\otimes(\eta\ast\mu)}(y,zx)$

for all $\mu,\nu,\eta\in M(G,\omega)$ and $x,y,z\in G$. The set of all
generalized quasi-additive functions is denoted by $GD(G,\omega)$. If there
exists $p=(p_{\mu})\in GL(G,1/\omega)$ such that for every $\mu,\nu\in
M(G,\omega)$ and for almost every where $x,y\in G$

$F_{\mu\otimes\nu}(x,y)=p_{\mu\ast\nu}(xy)-p_{\nu\ast\mu}(yx),$

then $F$ is said to be a _generalized inner quasi-additive function_. The set
of all generalized inner quasi-additive functions is denoted by
$GI(G,\omega)$.

Let $A$ be a Banach algebra and $D:A\rightarrow A^{*}$ be a bounded linear
operator. Then $D$ is called _cyclic_ if $\langle D(a),a\rangle=0$ for all
$a\in A$. Let us recall that a bounded linear operator $D:A\rightarrow A^{*}$
is called a _derivation_ if

$D(ab)=D(a)\cdot b+a\cdot D(b)$

for all $a,b\in A$. The space of all bounded continuous derivations from $A$
into $A^{*}$ is denoted by $\cal{Z}(A,A^{*})$. If every element of
$\cal{Z}(A,A^{*})$ is cyclic, then $A$ is called _cyclically_ _weakly
amenable_ , however, $A$ is called _weakly amenable_ if every derivation
$D\in\cal{Z}(A,A^{*})$ is inner; that is, there exists $z\in A^{*}$ such that
for every $a\in A$

$D(a)=\hbox{ad}_{z}(a):=z\cdot a-a\cdot z.$

The weak amenability of group algebras have been study by several authors. For
example, Brown and Moran [2] studied the weak amenability of measure algebra
of locally compact Abelian groups and showed that if zero is the only
continuous point derivation of $M(G)$, then $G$ is discrete. Note that if $G$
is discrete, then $M(G)$ is weakly amenable, because in this case
$M(G)=\ell^{1}(G)$ is always weakly amenable [8]. One can prove that if $d$ is
a non-zero continuous point derivation of $M(G)$ at

$\varphi\in\Delta(M(G))\cup\\{0\\},$

then the map $\mu\mapsto d(\mu)\varphi$ is a continuous non-inner derivation
from $M(G)$ into $M(G)^{*}$. In other words, $M(G)$ is not weakly amenable.
These facts give rise to the conjecture that for a locally compact group $G$,
the Banach algebra $M(G)$ is weakly amenable if and only if $G$ is discrete;
or equivalently, zero is the only continuous point derivation of $M(G)$ at a
character. Dales, Ghahramani and Helemskii [3] proved this conjecture. Some
authors investigated the weak amenability of the second dual of Banach
algebras. For instance, Ghahramani, Loy and Willis [7] proved that if $G$ is a
locally compact Abelian group and $L^{1}(G)^{**}$ is weakly amenable, then $G$
is discrete. Forrest [6] investigated the weak amenability of the dual of a
topological introverted subspace $X$ of $VN(G)$. Under certain conditions, he
showed that if $A(G)^{**}$ is weakly amenable, then every Abelian subgroup of
$G$ is finite. As a consequence of this result, he improved the result of
Ghahramani, Loy and Willis. In fact, for a locally compact Abelian group $G$,
he proved that weak amenability of $L^{1}(G)^{**}$ is equivalent to the
finiteness of $G$. Lau and Loy [9] considered a left introverted subspace of
$L^{\infty}(G)$ containing $AP(G)$, say $X$, and studied weak amenability of
$X^{*}$. One can obtain the result of weak amenability of $L^{1}(G)^{**}$ from
Lau-Loy’s theorem. Finally, Dales, Lau and Strauss [5] proved that
$L^{1}(G)^{**}$ is weakly amenable if and only if there is no non-zero
continuous point derivation of $L^{1}(G)^{**}$ at the discrete augmentation
character; or equivalently, $G$ is finite.

This paper is organized as follow. In Section 2 we study the weak amenability
of $M(G,\omega)$ and show that $M(G,\omega)$ is weakly amenable if and only if
$G$ is discrete and every bounded quasi-additive function is inner. We also
prove that cyclic weak amenability and point amenability of $M(G,\omega)$ are
equivalent to weak amenability of it. Section 3 is devoted to studies of the
weak amenability of second dual of $L^{1}(G,\omega)$ and $M(G,\omega)$. We
proved that $L^{1}(G,\omega)^{**}$ is weakly amenable if and only if
$M(G,\omega)^{**}$ is weakly amenable; or equivalently, $G$ is finite. We
verify that cyclic weak amenability and point amenability of
$L^{1}(G,\omega)^{**}$ and $M(G,\omega)^{**}$ are equivalent to finiteness of
$G$.

##  2 Weighted measure algebras

Let $\omega_{i}$ be a weight function on locally compact group $G_{i}$ for
$i=1,2$. Define the weight function $\omega_{1}\otimes\omega_{2}$ on
$G_{1}\times G_{2}$ by

$\omega_{1}\otimes\omega_{2}(x_{1},x_{2})=\omega_{1}(x_{1})\omega_{2}(x_{2})$

for all $x_{1}\in G_{1}$ and $x_{2}\in G_{2}$. In the case where,
$G_{1}=G_{2}=G$ and $\omega_{1}=\omega_{2}=\omega$, we set
$\omega^{\otimes}=\omega_{1}\otimes\omega_{2}$. The following result is needed
to prove our results.

###### Proposition 2.1

Let $\omega_{i}$ be a weight function on locally compact group $G_{i}$ for
$i=1,2$. Then

$M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})=M(G_{1}\times
G_{2},\omega_{1}\otimes\omega_{2}).$

Proof. Let $\eta_{i}\in M(G_{i},\omega_{i})$, for $i=1,2$. Then for every
$f\in C_{0}(G_{1}\times G_{2})$, we have

$\langle\eta_{1}\otimes\eta_{2},f\rangle=\int_{G_{1}}\int_{G_{2}}f(x,y)\;d\eta_{1}(x)d\eta_{2}(y).$

It is easy to prove that

$\eta_{1}\otimes\eta_{2}\in C_{0}(G_{1}\times
G_{2},1/\omega_{1}\otimes\omega_{2})^{*}=M(G_{1}\times
G_{2},\omega_{1}\otimes\omega_{2}).$

Conversely, let $\eta\in M(G_{1}\times G_{2},\omega_{1}\otimes\omega_{2})$. In
view of Theorem Lusin’s theorem, there exists sequences $(f_{n})$ and
$(g_{n})$ in the unit ball $C_{c}(G_{1},1/\omega_{1})$ and
$C_{c}(G_{2},1/\omega_{2})$ with compact support, respectively, such that for
almost every where $x\in G_{1}$ and $y\in G_{2}$

$f_{n}(x)\rightarrow 1\quad\hbox{and}\quad g_{n}(y)\rightarrow 1$

as $n\rightarrow\infty$. We define the functionals $\eta_{1}$ and $\eta_{2}$
by

$\eta_{1}(f)=\lim_{n}\eta(f\otimes
g_{n})\quad\hbox{and}\quad\eta_{2}(g)=\lim_{n}\eta(f_{n}\otimes g)$

for all $f\in C_{0}(G_{1},1/\omega_{1})$ and $g\in C_{0}(G_{2},1/\omega_{2})$.
Then $\eta_{1}\in M(G_{1},\omega_{1})$, $\eta_{2}\in M(G,\omega_{2})$. In
fact,

$|\eta_{1}(f)|\leq\|\eta\|\|f\|_{\infty,1/\omega}\quad\hbox{and}\quad|\eta_{2}(g)|\leq\|\eta\|\|g\|_{\infty,1/\omega}.$

On the other hand,

$\displaystyle\eta_{1}\otimes\eta_{2}(f\otimes g)$ $\displaystyle=$
$\displaystyle\lim_{n}\eta(f\otimes g_{n})\eta(f_{n}\otimes g)$
$\displaystyle=$ $\displaystyle\lim_{n}\int_{G_{1}\times
G_{2}}f(x)f_{n}(x)g(y)g_{n}(y)\;d\eta(x,y).$

Since $\eta$ is bounded, it follows from Lebesgue dominated convergence
theorem that $\|f_{n}\otimes g_{n}\|_{\infty,1/\omega}\leq 1$ and $1\in
L^{1}(\eta)$. Furthermore, $f_{n}\otimes g_{n}(x,y)\rightarrow 1$ for every
$x\in G_{1},y\in G_{2}$. For every $f\in C_{0}(G_{1},1/\omega_{1})$ and $g\in
C_{0}(G_{2},1/\omega_{2})$

$\eta_{1}\otimes\eta_{2}(f\otimes g)=\int_{G_{1}\times G_{2}}f\otimes
g(x,y)\;d\eta(x,y)=\eta(f\otimes g).$

It follows that for every $h\in C_{0}(G_{1},1/\omega_{1})\otimes
C_{0}(G_{2},1/\omega_{2})$

$\eta_{1}\otimes\eta_{2}(h)=\eta(h)$

and so for every $h\in C_{0}(G_{1}\times G_{2},1/\omega_{1}\otimes\omega_{2})$

$\eta_{1}\otimes\eta_{2}(h)=\eta(h).$

Therefore, $\eta_{1}\otimes\eta_{2}=\eta$.$\hfill\mathchar 1027\relax$

For every $f\in L^{1}(G,\omega)$, we define the seminorm
$T_{f}:M(G,\omega)\rightarrow[0,\infty)$ by

$T_{f}(\mu)=\|f\ast\mu\|_{1,\omega}+\|\mu\ast f\|_{1,\omega}.$

The locally convex topology defined by the family of seminorms $(T_{f})_{f\in
L^{1}(G,\omega)}$ is called the _strict topology_ on $M(G,\omega)$ with
respect to $L^{1}(G,\omega)$ (or briefly strict topology).

###### Proposition 2.2

Let $G$ be a locally compact group and $\omega$ be a weight function on $G$.
If $p\in D(G,\omega)$, then there exists a unique bounded derivation
$D\in{\cal Z}(M(G,\omega),M(G,\omega)^{*})$ such that $p(x,y)=\langle
D(\delta_{x}),\delta_{y}\rangle$ for all $x,y\in G$, where $\delta_{\cdot}$ is
the Diract measure at $\cdot$.

Proof. Let $p\in D(G,\omega)$. Then $\Gamma(D_{1})=p$ for some $D_{1}\in{\cal
Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. By Proposition 2.1.6 [23], there
exists $D_{2}\in{\cal Z}(M(G,\omega),L^{\infty}(G,1/\omega))$ such that
$D_{2}$ is strict-weak∗ continuous and $D_{2}|_{L^{1}(G,\omega)}=D_{1}$. Hence
for every $f\in L^{1}(G,\omega)$,

$\displaystyle\langle D_{2}(\delta_{x}),f\rangle$ $\displaystyle=$
$\displaystyle\lim\langle D_{1}(e_{\alpha}\ast\delta_{x}),f\rangle$
$\displaystyle=$
$\displaystyle\lim\int_{G}\int_{G}p(x,y)(e_{\alpha}\ast\delta_{x})(z)f(z)\;dzdy$
$\displaystyle=$
$\displaystyle\int_{G}\int_{G}p(x,y)e_{\alpha}(zx^{-1})f(y)\;dzdy.$

On the other hand, there exists a linear functional $T_{1}:L^{1}(G\times
G,\omega^{\otimes})\rightarrow{{C}}$ such that

$\langle T_{1},f\otimes g\rangle=\langle D_{1}(f),g\rangle$

for all $f,g\in L^{1}(G,\omega)$. Since $L^{1}(G\times G,\omega^{\otimes})$ is
a closed ideal in $M(G\times G,\omega^{\otimes})$, it follows that $T_{1}$ has
a strict continuous extension, say
$T_{2}:M(G,\omega)\hat{\otimes}M(G,\omega)\rightarrow{{C}}$. Define
$D:M(G,\omega)\rightarrow M(G,\omega)^{*}$ by

$\langle D(\mu),\nu\rangle=\langle T_{2},\mu\otimes\nu\rangle$

for all $\mu,\nu\in M(G,\omega)$. If $(e_{\alpha})$ is a bounded approximate
identity of $L^{1}(G,\omega)$, then for every $x\in G$,
$e_{\alpha}\ast\delta_{x}\rightarrow\delta_{x}$ in the strict topology. So

$T_{2}(e_{\alpha}\ast\delta_{x}\otimes e_{\alpha}\ast\delta_{y})\rightarrow
T_{2}(\delta_{x}\otimes\delta_{y}).$

Therefore,

$\displaystyle\langle D(\delta_{x}),\delta_{y}\rangle$ $\displaystyle=$
$\displaystyle\lim\langle T_{2}(e_{\alpha}\ast\delta_{x}\otimes
e_{\alpha}\ast\delta_{y})$ $\displaystyle=$ $\displaystyle\lim\langle
D_{2}(e_{\alpha}\ast\delta_{x}),e_{\alpha}\ast\delta_{y}\rangle=p(x,y),$

as claimed.$\hfill\mathchar 1027\relax$

In the following, let ${\cal I}_{nn}(M(G,\omega),M(G,\omega)^{*})$ be the set
of all inner derivations from $M(G,\omega)$ into $M(G,\omega)^{*}$, and let
${\cal B}(M(G,\omega),M(G,\omega)^{*})$ be the space of bounded linear
operators from $M(G,\omega)$ into $M(G,\omega)^{*}$. Define the isometric
isomorphism $\Gamma$ from Banach space ${\cal B}(M(G,\omega),M(G,\omega)^{*})$
onto $(M(G,\omega)\hat{\otimes}M(G,\omega))^{*}$ by

$\langle\Gamma(T),\mu\otimes\nu\rangle=\langle T(\mu),\nu\rangle,$

$M(G,\omega)\hat{\otimes}M(G,\omega)$ is the projective tensor product of
$M(G,\omega)$; see Proposition 13 VI in [1].

###### Proposition 2.3

Let $G$ be a locally compact group and $\omega$ be a weight function on $G$.
Then the following statements hold.

_(i)_ The function $\Gamma:{\cal Z}(M(G,\omega),M(G,\omega)^{*})\rightarrow
GD(G,\omega)$ is an isometric isomorphism. Furthermore, $\Gamma({\cal
I}_{nn}(M(G,\omega),M(G,\omega)^{*}))=GI(G,\omega)$.

_(ii)_ If $D\in{\cal Z}(M(G,\omega),M(G,\omega)^{*})$, then for every $\mu\in
M(G,\omega)$ there exists $F=(F_{\mu\otimes\nu})_{\nu}\in GD(G,\omega)$ such
that $D(\mu)=(p_{\mu,\nu})_{\nu}$ and
$p_{\mu,\nu}(y)=\int_{G}F_{\mu\otimes\nu}(x,y)\;d\mu(x)$ for almost every
where $y\in G$.

Proof. Let $D\in{\cal Z}(M(G,\omega),M(G,\omega)^{*})$. Then $D\in{\cal
B}(M(G,\omega),M(G,\omega)^{*})$. Putting $A=B=M(G,\omega)$ in the definition
of $\Gamma$, we have

$F:=\Gamma(D)\in(M(G,\omega)\hat{\otimes}M(G,\omega))^{*}=GL(G\times
G,1/\omega^{\otimes})$

and

$\displaystyle\langle D(\mu),\nu\rangle$ $\displaystyle=$
$\displaystyle\langle F,\mu\otimes\nu\rangle$ $\displaystyle=$
$\displaystyle\int_{G}F_{\mu\otimes\nu}(x,y)\;d(\mu\otimes\nu)(x,y)$
$\displaystyle=$
$\displaystyle\int_{G}\int_{G}F_{\mu\otimes\nu}(x,y)\;d\mu(x)d\nu(y).$

On the other hand, if $P=(p_{\mu})_{\mu\in M(G,\omega)}$, then

$\displaystyle\langle\hbox{ad}_{P}(\mu),\nu\rangle$ $\displaystyle=$
$\displaystyle\langle P\cdot\mu-\mu\cdot P,\nu\rangle$ $\displaystyle=$
$\displaystyle\langle P,\mu\ast\nu\rangle-\langle P,\nu\ast\mu\rangle$
$\displaystyle=$
$\displaystyle\int_{G}p_{\mu\ast\nu}\;d(\mu\ast\nu)-\int_{G}p_{\nu\ast\mu}\;d(\nu\ast\mu)$
$\displaystyle=$
$\displaystyle\int_{G}\int_{G}(p_{\mu\ast\nu}(xy)-p_{\nu\ast\mu}(yx))\;d\mu(x)d\nu(y).$

Now, by the argument used in the proof of Theorem 2.3 in [18], it can be shown
that the statement (i) holds. For (ii), assume that $D\in{\cal
Z}(M(G,\omega),M(G,\omega)^{*})$ and $\mu\in M(G,\omega)$. Then

$D(\mu)\in M(G,\omega)^{*}=GL(G,1/\omega).$

Thus $D(\mu)=(p_{\mu,\nu})_{\nu}$ for some $(p_{\mu,\nu})_{\nu}\in
GL(G,1/\omega)$. Hence for every $\nu\in M(G,\omega)$, we have

$\langle D(\mu),\nu\rangle=\int_{G}p_{\mu,\nu}\;d\nu.$

This together with (2) s shows that

$p_{\mu,\nu}(y)=\int_{G}F_{\mu\otimes\nu}(x,y)\;d\mu(x)$

for almost every where $y\in G$.$\hfill\mathchar 1027\relax$

We are now in a position to prove the main result of this section.

###### Theorem 2.4

Let $G$ be a locally compact group and $\omega$ be a weight function on $G$.
Then the following assertions are equivalent.

_(a)_ $M(G,\omega)$ is weakly amenable.

_(b)_ For every $D\in{\cal Z}(M(G,\omega),M(G,\omega)^{*})$ there exists
$P=(p_{\mu})_{\mu\in M(G,\omega)}$ such that $\langle
D(\mu),\nu\rangle=\int_{G}\int_{G}(p_{\mu\ast\nu}(xy)-p_{\nu\ast\mu}(yx))\;d\mu(x)d\nu(y)$
for all $\mu,\nu\in M(G,\omega)$.

_(c)_ Every generalized quasi-additive function is inner.

_(d)_ $M(G)$ is weakly amenable and $D(G,\omega)=I(G,\omega)$.

_(e)_ $G$ is discrete and every non-inner quasi-additive function in
$L^{\infty}(G,1/\omega)$ is unbounded.

Proof. The implications (a)$\Rightarrow$(b)$\Rightarrow$(c)$\Rightarrow$(a)
follow from Proposition 2.3. By Theorem 1.2 in [3] the implication
(d)$\Leftrightarrow$(e) holds. Also, the implication (e)$\Rightarrow$(a)
follows from Corollary 2.5 in [18]. For (a)$\Rightarrow$(e), let $M(G,\omega)$
be weakly amenable and $\varphi$ be a character of $M(G)$. If $d$ is a
continuous point derivation at $\varphi$ on $M(G)$, then $d|_{M(G,\omega)}$ is
a continuous point derivation of $M(G,\omega)$ at $\varphi|_{M(G,\omega)}$.
Hence $d$ is zero on $M(G,\omega)$. Since $M(G,\omega)$ is dense in $M(G)$, we
have $d=0$ on $M(G)$ which is implies $G$ is discrete. Apply Theorem 2.4 in
[18] to conclude that $D(G,\omega)=I(G,\omega)$.$\hfill\mathchar 1027\relax$

From Theorem 4.8 in [19] and Theorem 2.4 and its proof, we may prove the next
result.

###### Corollary 2.5

Let $G$ be a locally compact group. Then the following assertions are
equivalent.

_(a)_ $M(G,\omega)$ is weakly amenable.

_(b)_ $M(G,\omega)$ is cyclically weakly amenable.

_(c)_ $M(G,\omega)$ is point amenable.

_(d)_ $G$ is discrete and every non-inner quasi-additive function in
$L^{\infty}(G,1/\omega)$ is unbounded.

An elementary computation shows that the functions $\omega^{\prime}$ and
$\omega^{*}$ defined by

$\omega^{\prime}(x)=\omega(x^{-1})\quad\hbox{and}\quad\omega^{*}(x)=\omega\otimes\omega^{\prime}(x,x)$

are weight functions on $G$. Combining Theorem 2.4 and the result of [18] we
have the following result.

###### Corollary 2.6

Let $\omega$ and $\omega_{0}$ be weight functions on a locally compact group
$G$. Then the following statements hold.

_(i)_ If $\omega\leq m\omega_{0}$ for some $m>0$, $M(G,\omega_{0})$ is weakly
amenable and $I(G,\omega_{0})=D(G,\omega_{0})$ , then $M(G,\omega)$ is weakly
amenable.

_(ii)_ If $\omega$ and $\omega_{0}$ are equivalent, then weak amenability of
$M(G,\omega)$ is equivalent to weak amenability of $M(G,\omega_{0})$.

_(iii)_ $M(G,\omega^{\prime})$ is weakly amenable if and only if $M(G,\omega)$
is weakly amenable.

_(iv)_ If $M(G,\omega^{*})$ is weakly amenable and
$I(G,\omega^{*})=D(G,\omega^{*})$, then $M(G,\omega)$ is weakly amenable.

_(v)_ If $G$ is Abelian, then $M(G,\omega^{*})$ is weakly amenable if and only
if $M({D},\omega^{\otimes})$ is weakly amenable, where
${D}:=\\{(x,x^{-1}):x\in G\\}$.

Let $\phi:G\rightarrow G$ be a group epimorphism and $\omega$ be a weight
function on $G$. Then the function
$\overleftarrow{\omega}:G\rightarrow[1,\infty)$ defined by
$\overleftarrow{\omega}(x)=\omega(\phi(x))$ is a weight function on $G$. For
every quasi-additive function $p$, let ${S}(p)$ be the quasi-additive function
defined by

${S}(p)(x,y)=p(\phi(x),\phi(y))\quad\quad(x,y\in G).$

Theorem 2.4 together with Proposition 4.1 and Theorem 4.6 in [18] proves the
next result.

###### Corollary 2.7

Let $\omega$ be weight function on locally compact group $G$. Then the
following statements hold.

_(i)_ If $\phi:G\rightarrow G$ is a continuous group epimorphism,
$M(G,\overleftarrow{\omega})$ is weakly amenable and
${S}(I(G,\omega))=I(G,\overleftarrow{\omega})$, then $M(G,\omega)$ is weakly
amenable.

_(ii)_ If $G$ is Abelian and $M(G,\tilde{\omega})$ is weakly amenable, then
$M(H,\omega|_{H})$ is weakly amenable, where $H$ is a subgroup of $G$.

###### Corollary 2.8

Let $\omega_{i}$ be a weight function on a locally compact group $G_{i}$, for
$i=1,2$. Then the following assertions are equivalent.

_(a_) $M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$ is weakly
amenable.

_(b_) $M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$ is cyclically
weakly amenable.

_(c_) $M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$ is point amenable.

_(d_) $M(G_{i},\omega_{i})$ is weakly amenable and $G_{i}$ is discrete, for
$i=1,2$.

Proof. Let $M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$ be point
amenable. Since $M(G_{i},\omega_{i})$ is unital, for $i=1,2$, from Proposition
2.1 we infer that then $M(G_{1}\times G_{2},\omega_{1}\otimes\omega_{2})$ is
point amenable. By Theorem 2.4, $G_{1}\times G_{2}$ is discrete. It follows
that $G_{i}$ is discrete, for $i=1,2$. Hence
$M(G_{i},\omega_{i})=\ell^{1}(G_{i},\omega_{i})$ and so

$\ell^{1}(G_{1},\omega_{1})\hat{\otimes}\ell^{1}(G_{1},\omega_{1}))=M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$

is weakly amenable. In view of Corollary 4.8 in [18],
$\ell^{1}(G_{i},\omega_{i})$ is weakly amenable. So (c) implies (d).

Let $M(G_{i},\omega_{i})$ is weakly amenable and $G_{i}$ is discrete, for
$i=1,2$. By Corollary 2.5, $M(G_{i},\omega_{i})$ is point amenable, for
$i=1,2$. It follows from Theorem 4.1 in [20] and Proposition 2.1 that

$M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})=M(G_{1}\times
G_{2},\omega_{1}\otimes\omega_{2})$

is point amenable. Again, apply Corollary 2.5 to conclude that
$M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$ is weakly amenable. That
is, (d) implies (a).$\hfill\mathchar 1027\relax$

As a consequence of Corollary 2.8, we give the next result.

###### Corollary 2.9

Let $\omega_{i}$ be a weight function on a locally compact discrete group
$G_{i}$, for $i=1,2$. Then the following assertions are equivalent.

_(a_) $\ell^{1}(G_{1},\omega_{1})\hat{\otimes}\ell^{1}(G_{2},\omega_{2})$ is
weakly amenable.

_(b_) $\ell^{1}(G_{1},\omega_{1})\hat{\otimes}\ell^{1}(G_{2},\omega_{2})$ is
cyclically weakly amenable.

_(c_) $\ell^{1}(G_{1},\omega_{1})\hat{\otimes}\ell^{1}(G_{2},\omega_{2})$ is
point amenable.

_(d_) $\ell^{1}(G_{i},\omega_{i})$ is weakly amenable and $G_{i}$ is discrete,
for $i=1,2$.

We say that $T\in M(G,\omega)^{*}$ _vanishes at infinity_ if for every
$\varepsilon>0$, there exists a compact subset $K$ of $G$, for which $|\langle
T,\mu\rangle|<\varepsilon$, where $\mu\in M(G,\omega)$ with $|\mu|(K)=0$ and
$\|\mu\|_{\omega}=1$. We denote by $M_{*}(G,\omega)$ the subspace of
$M(G,\omega)^{*}$ consisting of all functionals that vanish at infinity. In
the case where, $\omega(x)=1$ for all $x\in G$, we write

$M_{*}(G,\omega):=M_{*}(G).$

The space $M_{*}(G,\omega)$ is a norm closed subspace of $M(G,\omega)^{*}$. It
is proved that $M_{*}(G,\omega)^{*}$ with the first Arens product is a Banach
algebra [16]. For each $f\in L^{1}(G,\omega)$, we may consider $f$ as a linear
functional in $M_{*}(G)^{*}$. One can prove that $L^{1}(G,\omega)$ is a closed
ideal in $M_{*}(G,\omega)^{*}$ and $M_{*}(G,\omega)^{*}=L^{1}(G,\omega)$ if
and only if $G$ is discrete [16]; see [15] for the case $\omega=1$.

###### Corollary 2.10

Let $G$ be a locally compact group. Then the following assertions are
equivalent.

_(a)_ $M_{*}(G,\omega)^{*}$ is weakly amenable.

_(b)_ $M_{*}(G,\omega)^{*}$ is cyclically weakly amenable.

_(c)_ $M_{*}(G,\omega)^{*}$ is point amenable.

_(d)_ $G$ is discrete and every non-inner quasi-additive function in
$L^{\infty}(G,1/\omega)$ is unbounded.

Proof. Let $M_{*}(G,\omega)^{*}$ be point amenable. Since $M(G,\omega)$ is a
direct summand of $M_{*}(G,\omega)^{*}$, by Theorem 3.7 in [20], $M(G,\omega)$
is point amenable. Hence $G$ is discrete and every non-inner quasi-additive
function in $L^{\infty}(G,1/\omega)$ is unbounded. Thus (c) implies (d). It is
easy to see that if $G$ discrete, then

$M_{*}(G,\omega)*=\ell^{1}(G,\omega)=M(G,\omega).$

It follows that (d) implies (a).$\hfill\mathchar 1027\relax$

Let $L^{\infty}(G,1/\omega)$ be the space of all Borel measurable functions
$f$ on $G$ with $f/\omega\in L^{\infty}(G)$, the Lebesgue space of bounded
Borel measurable functions on $G$. Let also $L_{0}^{\infty}(G,1/\omega)$
denote the subspace of $L^{\infty}(G,1/\omega)$ consisting of all functions
$f\in L^{\infty}(G,1/\omega)$ that vanish at infinity. It is proved that
$L_{0}^{\infty}(G,1/\omega)$ is left introverted in $L^{\infty}(G,1/\omega)$.
So $L_{0}^{\infty}(G,1/\omega)^{*}$ is a Banach algebra with the first Arens
product [10]; see also [4, 11, 12, 21].

###### Corollary 2.11

Let $G$ be a locally compact group. Then the following assertions are
equivalent.

_(a)_ $L_{0}^{\infty}(G,1/\omega)^{*}$ is weakly amenable.

_(b)_ $L_{0}^{\infty}(G,1/\omega)^{*}$ is cyclically weakly amenable.

_(c)_ $L_{0}^{\infty}(G,1/\omega)^{*}$ is point amenable.

_(d)_ $G$ is discrete and every non-inner quasi-additive function in
$L^{\infty}(G,1/\omega)$ is unbounded.

###### Corollary 2.12

Let $\omega_{i}$ be a weight function on a locally compact group $G_{i}$, for
$i=1,2$. Then the following assertions are equivalent.

_(a)_ $M_{*}(G_{1},\omega_{1})^{*}$ and $M_{*}(G_{2},\omega_{2})^{*}$ are
weakly amenable.

_(b)_ $L_{0}^{\infty}(G,1/\omega)^{*}$ and $L_{0}^{\infty}(G,1/\omega)^{*}$
are weakly amenable.

_(c)_ $M_{*}(G_{1},\omega_{1})^{*}\hat{\otimes}M_{*}(G_{2},\omega_{2})^{*}$ is
weakly amenable and $G_{i}$ is discrete, for $i=1,2$.

_(d)_
$L_{0}^{\infty}(G,1/\omega)^{*}\hat{\otimes}L_{0}^{\infty}(G,1/\omega)^{*}$ is
weakly amenable and $G_{i}$ is discrete, for $i=1,2$.

Proof. Assume that $M_{*}(G_{1},\omega_{1})^{*}$ and
$M_{*}(G_{2},\omega_{2})^{*}$ are weakly amenable. By Corollary 2.10, $G_{i}$
is discrete and $M(G_{i},\omega_{i})=M_{*}(G_{i},\omega_{i})^{*}$ is weakly
amenable, for $i=1,2$. It follows from Corollary 2.8 that

$\displaystyle
M_{*}(G_{1},\omega_{1})^{*}\hat{\otimes}M_{*}(G_{2},\omega_{2})^{*}=M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$

is weakly amenable. So (a) implies (c).

Let $M_{*}(G_{1},\omega_{1})^{*}\hat{\otimes}M_{*}(G_{2},\omega_{2})^{*}$ be
weakly amenable and $G_{i}$ is discrete, for $i=1,2$. This implies that
$M(G_{1},\omega_{1})\hat{\otimes}M(G_{2},\omega_{2})$ is weakly amenable. Thus
$M_{*}(G_{i},\omega_{i})^{*}=M(G_{i},\omega_{i})$ is weakly amenable. Hence
(c) implies (a). Similarly, (b) and (d) are equivalent.$\hfill\mathchar
1027\relax$

Let $\hbox{LUC}(G,1/\omega)$ be the space of all continuous function $f$ on
$G$ such that $f/\omega$ is a left uniformly continuous functions on $G$; for
study of this space see [27]. Let $WAP({A})$ be the space of all weakly almost
periodic functionals on Banach algebra ${A}$, that is, $f\in{A}^{*}$ such that
the map $a\mapsto af$ from ${A}$ into ${A}^{*}$ is weakly compact, where
$\langle af,b\rangle=\langle f,ba\rangle$ for all $b\in{A}$.

###### Corollary 2.13

Let $\hbox{WAP}(L^{1}(G,\omega))^{*}$ or $\hbox{LUC}(G,\omega)^{*}$ be 0-point
amenable. Then $G$ is discrete.

Let ${A}$ be one of the Banach algebras $M(G,\omega)$, $M_{*}(G,\omega)^{*}$,
$L_{0}^{\infty}(G,1/\omega)^{*}$, $\hbox{WAP}(L^{1}(G,\omega))^{*}$ or
$\hbox{LUC}(G,\omega)^{*}$.

###### Proposition 2.14

Let $G$ be a locally compact group. If ${A}$ is cyclically amenable, then
every element of $CD(G,\omega)$ is inner.

Proof. Let $M(G,\omega)$ be cyclically amenable. Since $L^{1}(G,\omega)$ is a
direct summand of $M(G,\omega)$, by Theorem 3.7 in [20], the Banach algebra
$L^{1}(G,\omega)$ is cyclically amenable. It follows from Theorem 5.6 in[18]
that every element of $CD(G,\omega)$ is inner. For the other cases, we only
need to recall that

${A}=M(G,\omega)\oplus{B}$

for some closed subspace ${B}$ of ${A}$.$\hfill\mathchar 1027\relax$

##  3 The second dual of Banach algebras

The main result of this section is the following which solves an open problem
posed in [9].

###### Theorem 3.1

Let $G$ be a locally compact group. Then the following assertion are
equivalent.

_(a)_ $L^{1}(G,\omega)^{**}$ is weakly amenable.

_(b)_ $L^{1}(G,\omega)^{**}$ is cyclically weakly amenable.

_(c)_ $L^{1}(G,\omega)^{**}$ is point amenable.

_(d)_ $G$ is finite.

Proof. Let $\iota:L^{1}(G,\omega)\rightarrow L^{1}(G)$ be the inclusion map.
Since $L^{1}(G,\omega)$ is dense in $L^{1}(G)$, $\iota$ is a continuous
homomorphism with dense range. So $\iota^{**}:L^{1}(G,\omega)^{**}\rightarrow
L^{1}(G)^{**}$ is epimorphism. Hence if $L^{1}(G,\omega)^{**}$ is point
amenable, then by Theorem 2.1 in [20] the Banach algebra $L^{1}(G)^{**}$ is
point amenable. It follows that every continuous point derivation of
$L^{1}(G)^{**}$ at the discrete augmentation character is zero. From Theorem
11.17 in [5] infer that $G$ is finite. So (c)$\Rightarrow$(d). The
implications (a)$\Rightarrow$(b)$\Rightarrow$(c) follows from Theorem 4.1 in
[19]. $\hfill\mathchar 1027\relax$

###### Corollary 3.2

Let $G$ be a locally compact group. Then the following assertion are
equivalent.

_(a)_ $M(G,\omega)^{**}$ is weakly amenable.

_(b)_ $M(G,\omega)^{**}$ is cyclic amenable.

_(c)_ $M(G,\omega)^{**}$ is point amenable.

_(d)_ $G$ is finite.

Proof. Let $M(G,\omega)^{**}$ is point amenable. By Proposition 5.2 in [20],
the Banach algebra $M(G,\omega)$ is point amenable. In view of Corollary 2.5,
$G$ is discrete. Hence $L^{1}(G,\omega)^{**}$ is weakly amenable. Now, apply
Theorem 3.1.$\hfill\mathchar 1027\relax$

Let us recall that if there exists a compact invariant neighborhood of $e$ in
$G$, then $G$ is called an $[IN]-$_group._ The following result is an
improvement of Theorem 3.4 in [9].

###### Theorem 3.3

Let $G$ be a connected locally compact group. If either $G_{d}$ is amenable or
$G$ is an $[IN]-$group, then the following assertions are equivalent.

_(a)_ $L^{1}(G,\omega)^{**}$ is weakly amenable.

_(b)_ $M(G,\omega)$ is weakly amenable.

_(c)_ $G=\\{e\\}$.

Proof. Let $L^{1}(G,\omega)^{**}$ be weakly amenable. Since

$L^{1}(G,\omega)^{**}=M(G,\omega)\oplus C_{0}(G,\omega)^{\perp}$

and $C_{0}(G,\omega)^{\perp}$ is an ideal in $L^{1}(G,\omega)^{**}$ , we have
$M(G,\omega)$ is weakly amenable. So (a)$\Rightarrow$(b). Let’s show that
(b)$\Rightarrow$(c). To this end, let $M(G,\omega)$ be weakly amenable. It
follows from Theorem 2.4 that $G$ discrete and $M(G)$ is weakly amenable. If
$G_{d}$ is amenable, then from Theorem 3.3 in [9] we infer that $G=\\{e\\}$.
If $G$ is an $[IN]-$group, then by Theorem 3.4 in [9], $G$ is compact. Since
$G$ is also discrete, it follows that $G$ is finite. Hence $G_{d}$ is
amenable. Thus $G=\\{e\\}$. So (b)$\Rightarrow$(c). The implication
(c)$\Rightarrow$(a) is clear. $\hfill\mathchar 1027\relax$

## References

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Mohammad Javad Mehdipour
Department of Mathematics,
Shiraz University of Technology,
Shiraz 71555-313, Iran
e-mail<EMAIL_ADDRESS>
Ali Rejali
Department of Pure Mathematics,
Faculty of Mathematics and Statistics,
University of Isfahan,
Isfahan 81746-73441, Iran
e-mail<EMAIL_ADDRESS>


# Constraining continuous topology optimizations to discrete solutions for
photonic applications

Conner Ballew  Gregory Roberts  Tianzhe Zheng  Andrei Faraon∗ Kavli
Nanoscience Institute and Thomas J. Watson Sr. Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
<EMAIL_ADDRESS>
###### Abstract

Photonic topology optimization is a technique used to find the electric
permittivity distribution of a device that optimizes an electromagnetic
figure-of-merit. Two common techniques are used: continuous density-based
optimizations that optimize a grey-scale permittivity defined over a grid, and
discrete level-set optimizations that optimize the shape of the material
boundary of a device. More recently, continuous optimizations have been used
to find an initial seed for a concluding level-set optimization since level-
set techniques tend to benefit from a well-performing initial structure.
However, continuous optimizations are not guaranteed to yield sufficient
initial seeds for subsequent level-set optimizations, particularly for high-
contrast structures, since they are not guaranteed to converge to solutions
that resemble only two discrete materials. In this work, we present a method
for constraining a continuous optimization such that it converges to a
discrete solution. This is done by inserting a constrained sub-optimization at
each iteration of an overall gradient-based optimization. This technique can
be used purely on its own to optimize a device, or it can be used to provide a
nearly discrete starting point for a level-set optimization.

††journal: osajournal

##  1 Introduction

Photonic inverse design is the process of choosing a figure-of-merit (FoM) and
finding an optimal photonic device that maximizes or minimizes the FoM. A
subset of photonic inverse design is topology optimization, which entails
altering the shape of the photonic structure (i.e. its topology) to achieve
the desired performance [1]. In this case the device is characterized by its
electric permittivity $\epsilon(x,y,z)$ in a 3D design region.

A useful technique for inverse-design is gradient descent. This is the process
of finding a local minimum of a function $f(x)$ by following the gradient
$\nabla f$. The functions being minimized are generally non-linear, thus a
computed gradient is only valid in a locally linear region of $f$. Gradient
descent is therefore an iterative optimization, comprised of computing $\nabla
f$, stepping $f$ in the direction of $-\alpha\nabla f$ where $\alpha$ is a
sufficiently small step size, and concluding the optimization when $\nabla f$
is sufficiently close to 0. A major enabling technique for gradient-based
optimization is the adjoint method for efficiently computing a gradient. The
technique has been applied to many fields (see [2] for a recent overview of
the topic), including photonic design [3].

The gradient-descent procedure becomes increasingly complex as the dimensions
of the parameter space increase and as constraints are imposed. For optimizing
the topology of photonic devices, the parameter space includes the
permittivity of every voxel in the design region, which can be arbitrarily
large. For many photonic devices, the desired solution is constrained due to
the limitations of available fabrication techniques that make only certain
solutions physically realizable. Some common requirements of current
fabrication techniques include:

  * •

Minimum feature size: the size of the smallest uniform material piece. This is
typically dictated by lithography, material etching, and material deposition
capabilities.

  * •

Connectivity: one or more materials are topologically connected.

  * •

Binary: the device is made of only two materials.

There are many topology optimization algorithms and many reviews summarizing
them [4]. They can be characterized as density-based optimization or shape
optimization: density-based optimizations feature a density variable $\rho$
that is evaluated across a grid of points, thus describing the device in an
element-wise way. Density-based optimization can be further classified as
discrete, where $\rho=0$ or $\rho=1$, or continuous, where $0\leq\rho\leq 1$.
Shape-based optimizations instead describe the boundary of the device
$\Omega$. One common technique for shape-based optimizations involves
describing the boundary $\Omega$ as the level-set contour of a higher-
dimensional function, called the level-set method. In the context of photonic
inverse design, two methods have dominated the literature: continuous density-
based optimizations, and shape optimizations using the level-set method.

Of the fabrication constraints listed above, the binary constraint is a
substantial difference between the continuous density-based and the discrete
level-set approaches to photonic optimization, with the former inherently
describing a non-binary device and the latter inherently describing a
binarized device. While the level-set method has the advantage of describing
binarized devices, a known limitation is its ability to easily nucleate holes,
bridge gaps, and generally converge to shapes that are substantially different
from the starting topology [4, 5]. A technique that is now employed in
photonic optimization is to begin with a density-based optimization, then
switch to a level-set method to conclude the optimization [6]. Given the
limitations of the level-set method, it is important that the density-based
optimization converge to a sufficiently good (high-performing and nearly
binary) starting point for the level-set method. This represents a challenge
for devices that feature materials with large refractive index contrast, which
will be shown in Section 3 of this work.

The goal of binarizing a continuous density-based optimization is not new, and
is not limited to photonic inverse-design. Commonly used methods for this task
include projection filters, which encourage the device to become more binary
but fall short of explicitly requiring the device become binary [7]. More
advanced methods employing gradually strengthened sigmoidal or Heaviside
filters have been used to aid convergence to a binary solution [8, 9].
However, these do not guarantee that the device becomes binary, and the final
thresholding operation (in which all voxels are rounded to 0 or 1) can thus
incur a severe performance penalty that requires substantial level-set re-
optimization to overcome.

We present a method here that reformulates the gradient-based continuous
density optimization in a manner that strictly enforces that the design become
fully binary, thus preventing the option of settling at a non-binary solution.
We have found this technique to be particularly useful in optimizing high
index contrast devices. The method uses the gradient information at each
iteration to step the permittivity in a direction that maximizes the
performance of the device while constraining the permittivity step to binarize
the device by some amount that is greater than zero. We refer to this
procedure as "sub-optimization", since it occurs at every iteration of the
overall photonic optimization. An important quality of this sub-optimization
is that much of the computational burden of finding the ideal constrained
permittivity is eased by instead solving the Lagrange dual-problem of the
original sub-optimization, which converts the original N-dimensional linear
optimization (where $N$ is the number of voxels in the design region) into a
1-dimensional non-linear optimization which can be easily solved
computationally.

The remainder of the paper is composed as follows: first, we formulate the
sub-optimization and derive the solution of the Lagrange dual problem; then,
we provide an example of an optimized spectral demultiplexer. The examples are
intended to illustrate the practical usage of the method, compare its
performance to a traditional steepest-ascent optimization, integrate the
technique with a subsequent level-set optimization, and describe a technique
to control the overall fraction of material in the final device.

##  2 Lagrange dual problem

###  2.1 Setup

Consider a discretized 3D domain of interest $\mathcal{P}$. The domain
contains a region of dielectric permittivity
$\epsilon_{i},i\in\mathcal{D}\subseteq\mathcal{P}$, where $\mathcal{D}$ is
called design region or device. The objective function
$f(\textbf{E},\textbf{H},\vec{\epsilon})$ depends on the electric and magnetic
fields output from the device, and the permittivity of the design region
$\vec{\epsilon}(x)$ with
$\epsilon_{i}\in[\epsilon_{min},\epsilon_{max}]\>\forall i\in\mathcal{D}$. The
general goal of the inverse-design optimization is to find the optimal
permittivity of the design region that maximizes
$f(\textbf{E},\textbf{H},\vec{\epsilon})$, and here we wish to constrain the
solutions to only permittivity distributions consisting of two materials.
$\epsilon_{min}$ and $\epsilon_{max}$ are the lower and upper bounds of the
permittivity, respectively, representing the two materials the final device is
comprised of. The electric field $\mathbf{E}$ and magnetic field $\mathbf{H}$
inside the domain can be solved for using Maxwell’s equation solvers such as
the finite-difference-time-domain (FDTD). The adjoint method computes the
gradient $\partial f/{\partial\vec{\epsilon}}$ at all points within the design
region.

The quantities involved in 3D simulations, such as the device permittivity
$\vec{\epsilon}$ and FoM gradient $\partial f/{\partial\vec{\epsilon}}$, are
often computed and stored as 3D matrices. Here, we find it convenient to
flatten these into N-dimensional column vectors, where $N$ is the total number
of voxels in the device region. For the remainder of this derivation, all
multi-dimensional quantities are considered N-dimensional vectors.

We define a metric to quantify how binary the device is. In this case, we use
a simple absolute value function to define the binarization of a single pixel,
whose mean across all pixels quantifies the binarization of the device. The
absolute value function is scaled and shifted such that it evaluates to 1 at
$\epsilon=\epsilon_{min}$ and $\epsilon=\epsilon_{max}$, 0 at the permittivity
midpoint, $\epsilon_{mid}$.

$\displaystyle
B=\frac{1}{N}\sum_{i}\left|\frac{\epsilon_{i}-\epsilon_{mid}}{\epsilon_{max}-\epsilon_{mid}}\right|,i\in\mathcal{D}$
(1) $\displaystyle\epsilon_{mid}=\frac{\epsilon_{min}+\epsilon_{max}}{2}$ (2)

If we define $\vec{b}=dB/d\vec{\epsilon}$, then the approximate change in
binarization of the device given a permittivity step $\vec{\Delta\epsilon}$ is
$\vec{b}\cdot\vec{\Delta\epsilon}$. This term can thus be constrained to
predictably control the change in device binarization.

Direct gradient ascent would shift the device permittivity in the direction of
$\partial f/{\partial\vec{\epsilon}}$. However, the gradient may not point in
a direction that increases the binarization of the device, so we reformulate
the optimization to improve the objective function as much as possible while
enforcing that the binarization increases by some amount $\beta$. This
corresponds to solving the following constrained optimization problem, which
we refer to as the sub-optimization problem from now on since it is solved at
every iteration of the main photonic optimization.

→Δϵ→∂f∂ϵ

⋅→Δϵ →b ⋅→Δϵ≥β Δϵ_i≤u_i ∀i Δϵ_i≥l_i ∀i

The vectors $\vec{u}$ and $\vec{l}$ control the maximum step size that the
permittivity can undergo in a single iteration. They are also used to enforce
the maximum and minimum permittivity constraints (i.e.
$\epsilon_{i}\in[\epsilon_{min},\epsilon_{max}]\>\forall i$) since they can be
set to 0 for a permittivity voxel at $\epsilon_{min}$ or $\epsilon_{max}$.

We convert (2.1) to a canonical form.

→x→c

⋅→x β\- →b ⋅→x≤0 x_i - u_i≤0 ∀i l_i - x_i≤0 ∀i

Since $\vec{x}\in\mathbb{R}^{N}$, there are $2N+1$ inequality constraints in
(2.1) in addition to the $N$ dimensions of the non-constrained optimization
problem. For large $N$, this problem is challenging to solve numerically.
Fortunately, this multi-dimensional linear optimization can be reduced to a
single-dimensional non-linear optimization, regardless of the value of $N$.
This is done by instead solving the Lagrange dual problem, which in this case
will have the same optimal point as (2.1) since the property of strong duality
holds for linear optimizations [10].

###  2.2 Solution

To obtain the dual problem, we start with the Lagrangian. This is defined as:

${{}}\begin{split}L(\vec{x},\vec{\lambda_{1}},\vec{\lambda_{2}},\nu)&=\vec{c}\cdot\vec{x}+\vec{\lambda_{1}}\cdot(\vec{x}-\vec{u})+\vec{\lambda_{2}}\cdot(\vec{l}-\vec{x})+\nu(\beta-\vec{b}\cdot\vec{x})\\\
&=\vec{x}\cdot(\vec{c}+\vec{\lambda_{1}}-\vec{\lambda_{2}}-\nu\vec{b})-\vec{\lambda_{1}}\cdot\vec{u}+\vec{\lambda_{2}}\cdot\vec{l}+\nu\beta\end{split}$
(3)

The dual function is defined as the infimum of the Lagrangian over $\vec{x}$:

${{}}\begin{split}g(\vec{\lambda_{1}},\vec{\lambda_{2}},\nu)=\inf_{\vec{x}}L(\vec{x},\vec{\lambda_{1}},\vec{\lambda_{2}},\nu)\end{split}$
(4)

Since (3) is linear, the infimum is $-\infty$ if any slope is non-zero.

${{}}g(\vec{\lambda_{1}},\vec{\lambda_{2}},\nu)=\left\\{\begin{array}[]{ll}-\infty&\quad\vec{c}+\vec{\lambda_{1}}-\vec{\lambda_{2}}-\nu\vec{b}\neq\vec{0}\\\
-\vec{\lambda_{1}}\cdot\vec{u}+\vec{\lambda_{2}}\cdot\vec{l}+\nu\beta&\quad\vec{c}+\vec{\lambda_{1}}-\vec{\lambda_{2}}-\nu\vec{b}=\vec{0}\end{array}\right.$
(5)

The optimization we wish to solve is called the Lagrange dual problem:

g(→λ_1, →λ_2, ν) →λ_1≥0 →λ_2≥0 ν≥0

Incorporating (5) into Eq. 2.2, we get:

-→λ_1 ⋅→u + →λ_2 ⋅→l + νβ →λ_1≥0 →λ_2≥0 ν≥0 →c + →λ_1 - →λ_2 - ν→b= 0

The fourth constraint of (2.2) can be used to write the objective function and
the second constraint in terms of only $\nu$ and $\lambda_{1}$.

-→λ_1 ⋅→u + (→c + →λ_1 - ν→b) ⋅→l + νβ →λ_1≥0 →λ_1≥ν→b \- →c ν≥0

For any $\nu$, the optimal elements of $\vec{\lambda_{1}}$ are as small as
possible since $\vec{u}\geq 0$ and $\vec{l}\leq 0$. The solution to
$\vec{\lambda_{1}}$ is obtained by satisfying the constraints while minimizing
$\vec{\lambda_{1}}$. The solution for the i-th element of $\vec{\lambda_{1}}$
is thus:

${{}}\lambda_{1,i}=\left\\{\begin{array}[]{ll}0&\quad\nu b_{i}-c_{i}\leq 0\\\
\nu b_{i}-c_{i}&\quad\nu b_{i}-c_{i}>0\end{array}=(\nu b_{i}-c_{i})H(\nu
b_{i}-c_{i})\right.$ (6)

where $H(x)$ is the Heaviside step function. $\vec{\lambda_{2}}$ is solved
using the fourth constraint of (2.2).

${{}}\vec{\lambda_{2}}=\vec{c}+\vec{\lambda_{1}}-\nu\vec{b}=\vec{c}+(\nu\vec{b}-\vec{c})H(\nu\vec{b}-\vec{c})^{T}-\nu\vec{b}$
(7)

Here, $H(\vec{x})$ represents the Heaviside function applied element-wise to
$\vec{x}$. Since the optimal $\vec{\lambda_{1}}$ and $\vec{\lambda_{2}}$ are
dependent on $\nu$, the dual problem is a function of only $\nu\in\mathbb{R}$.
The optimization can be reduced to a single dimension as follows.

νν→b ⋅→l + (→u-→l) ⋅(ν→b \- →c)H(ν→b \- →c)^T - νβ ν≥0

This problem can be solved numerically using open-source functions. We denote
the optimal dual variables as
$(\nu^{\star},\vec{\lambda_{1}^{\star}},\vec{\lambda_{2}^{\star})}$. The
optimal solutions of the dual problem can be mapped back to original variable
$x$ using the Karush–Kuhn–Tucker (KKT) conditions, which are a set of
conditions that hold under certain regularity conditions. Fortunately,
linearity of the constraints is a sufficient condition for the KKT conditions
to hold. We use the KKT condition of complementary slackness to translate
$(\nu^{\star},\vec{\lambda_{1}^{\star}},\vec{\lambda_{2}^{\star}})$ to the
optimal $\vec{x}$ that we call $\vec{x^{\star}}$.

$\displaystyle{{}}\vec{\lambda_{1}^{\star}}(\vec{x^{\star}}-\vec{u})^{T}=\vec{0}$
(8)
$\displaystyle\vec{\lambda_{2}^{\star}}(\vec{l}-\vec{x^{\star}})^{T}=\vec{0}$
(9)

The solution is

${{}}x_{i}^{\star}=\left\\{\begin{array}[]{ll}u_{i}&\quad\lambda_{1,i}\neq
0\\\ l_{i}&\quad\lambda_{1,i}=0\end{array}\right.$ (10)

##  3 Example

This section provides example optimizations to illustrate the described
techniques usage and elaborate on several key points. We study device
performance relative to a basic implementation of regular gradient ascent in
which the permittivity is stepped in the exact direction of the gradient. We
study this for three different cases of material-to-void refractive index
contrast: a low index contrast of 1.5:1, a medium index contrast of 2.5:1, and
a high index contrast of 3.0:1. Next, we pair this optimization technique with
a level-set optimization that thresholds the continous permittivity to a
binary solution and re-optimizes the device with a level-set optimization to
recover the lost performance from the threshold operation. Finally, we
demonstrate how this technique can be used to control the overall ratio of
material:void in the final device simply by shifting the binarization function
in Eq. 1.

The example optimization we use here is a spectral demultiplexer, or "color
splitter". Fig. 1a illustrates the purpose of the device, which focuses three
equally-sized frequency bands of a normally incident TE plane-wave to three
distinct points in the focal plane. The FoM function is defined as the
intensity at the desired point in the focal plane, which allows us to use a
simple dipole source for the adjoint source. When quantifying the performance
of the device we use the power transmission through apertures centered at the
dipole source. These apertures are drawn in Fig. 1b as red, green, and blue
horizontal lines in the focal plane. The gradient of every FoM (there is one
FoM for each simulated frequency) are combined using a weighted average to
obtain a single gradient vector, which is input as $\partial
f/{\partial\vec{\epsilon}}$ in Eq. 2.1. This weighting procedure is described
in detail in [11].

The device is optimized over a 42% fractional bandwidth, which is comparable
to fractional bandwidth of visible light. $\lambda_{0}$ and $f_{0}$ denote the
center wavelength and frequency of the full bandwidth of the device. In Fig.
1b the intensity evaluated at the various frequencies are overlaid using their
equivalent hue in the visible regime. The quantitative performance of the
device is shown in Fig. 1c, which plots the power transmission into three
monitors of size $1.56\lambda_{0}$, centered at the relevant focal points.
Fabrication tolerance limits are not strictly enforced in this design,
although the discretization of the geometry on a $\lambda_{0}/30$ grid and the
discretization of the FDTD simulation on a $\lambda_{0}/10n_{max}$ grid
preclude arbitrarily small features in the design. After all optimizations, we
simulated the final device on a finer grid to ensure that the FDTD results had
properly converged. We noticed very little change to device behavior and
efficiency after halving the simulation grid size in all directions.

Figure 1: Example device. (a) A schematic of the device. The device is a
spectral demultiplexer that focuses three equally spaced frequency bins to
three distinct points marked in the focal plane. The input is a normally
incident TE-polarized planewave. (b) The intensity $|E|^{2}$ of each frequency
overlaid. Each frequency is drawn in its equivalent hue in the visible regime.
(c) The power transmission of the output fields through the apertures. Each
trace is drawn in the same color as the corresponding aperture drawn in (b).

In the optimization problem described by Eq. 2.1, the quantity $\beta$
describes the amount that the device must binarize after stepping the
permittivity. In our implementation we set this quantity to be a factor of the
max possible binarization, denoted $\beta_{max}$, during the current
iteration. $\beta_{max}$ is not constant during the optimization, since points
on the device eventually reach a material boundary. The maximum and minimum
allowable step size for each permittivity point in the device is contained in
the vectors $\vec{u}$ and $\vec{l}$, respectively, which can be used to find
$\beta_{max}$. For this example, we use $\beta=0.2\beta_{max}$. The maximum
and minimum permittivity step sizes are chosen to be 0.02. The algorithm used
in this example is summarized in Algorithm 1.

initialize device permittivity;

while  _$B <B_{end}$_ do

compute gradient using adjoint method, averaging multiple FoM gradients as
necessary for multi-functional performance;

compute $\vec{u}$ and $\vec{l}$ based on the current permittivity and the
maximum allowable step size;

compute $\beta_{max}$ based on the values of $\vec{u}$ and $\vec{l}$, and
choose $\beta_{0}$ such that $\beta=\beta_{0}\beta_{max}$;

solve the optimization problem Eq. 2.2 using a numerical solver;

use Eq. 6 to solve for $\lambda_{1}^{\star}$, then use Eq. 10 to solve for
$x^{\star}$;

update the permittivity of the device by $x^{\star}$;

end while

Algorithm 1

We compare our method with a basic implementation of gradient ascent in order
to demonstrate the differences in convergence properties and final device
performance. In the basic implementation of gradient ascent, we step the
device permittivity in the exact direction of the electromagnetic FoM gradient
(i.e. a steepest ascent optimization). The permittivity update is computed
with $\alpha(\partial f/{\partial\vec{\epsilon}})$, where $\alpha$ controls
the specific step size. To ensure both stability and acceptable convergence,
we dynamically alter $\alpha$ such that the maximum step in the absolute value
of permittivity is 0.1. The reason for comparing these two methods is to
observe their different behaviors, particularly with respect to converging to
binary devices. We do not compare the speed of convergence. However,
techniques for improving convergence speed (e.g. a line search for computing
an optimal step size) apply to both methods.

The results of the traditional gradient ascent, which we call direct gradient
ascent, and the technique that this paper describes, which we call modified
gradient ascent here, are shown in Fig. 2 for different value of material:void
refractive index contrast. The plotted average FoM is the transmission through
the desired aperture averaged over all frequencies. In the low contrast case
(1.5:1) in Fig. 2a, the optimizations perform equally well in terms of FoM. In
the medium contrast case (2.5:1) in Fig. 2b, the direct gradient ascent
optimization converges to a final FoM value of 84% while the binarization
stagnates at 54%. This implies that optimization has found a local maximum of
the FoM where the device is not binary. This type of solution is forbidden in
the modified gradient ascent approach, which after 500 iterations is at a
point where the FoM is 80% for both the FoM and binarization (the fact that
FoM and binarization percentage are nearly identical is a coincidence). In
fact, the FoM reaches a maximum at iteration 300, but continues to search for
a solution that is binary in order to satisfy the binarization constraint. It
is necessary that the maximum FoM of a binarized device is less-than-or-equal
to the maximum FoM of a greyscale device, since the former is a subset of the
latter (it is possible that a binarized device exists at a better local
optimum than a seperately optimized greyscale device). The difference between
the two methods becomes yet more apparent in the high index contrast case
(3.0:1) in Fig. 2c. In this case, the direct gradient ascent optimization
converges to a 48% binary device after 1000 iterations while the modified
gradient ascent optimization reaches 80% binarization after 1000 iterations.

Figure 2: FoM and binarization convergence curves for different index
contrasts. The FoM (averaged across all functionalities) and binarization are
drawn as red and blue traces, respectively. The modified gradient ascent
(G.A.) and direct gradient ascent methods are drawn as solid and dashed lines
respectively. The final index profile of each device is shown as a greyscale
colormap. In each subset, the device optimized with the modified gradient
ascent procedure is the top device while the device optimized with the direct
gradient ascent procedure is the bottom device. The index contrasts are: (a)
1.5:1, (b) 2.5:1, (c) 3.0:1.

While the modified gradient ascent approach described above will eventually
converge to a 100% binary device, this is not strictly required. Instead, the
optimization can be concluded before the device is 100% binary, thresholded to
the nearest material boundary, and then used as an initial seed to a level-set
optimization. Level-set optimizations tend to be dependent on initial seed
[4], and we study the dependence of the converged performance of a level-set
optimization with respect to the binarization percentage of an initial seed
obtained with the modified gradient ascent approach. For this study, we use an
implementation of level-set optimization based on signed distance functions
[12]. Fig. 3a shows the converged average FoM of a 2.5:1 medium index contrast
device as a function of the binarization percentage of the initial seed and
Fig. 3b shows the convergence curves for the individual level-set
optimizations. The results suggest that the level-set optimization does indeed
depend on initial seed, and that in general the more binarized starting seeds
yield a better level-set optimized device. Beyond 40% binarization, the
performance monotonically increased with starting binarization, however the
diminishing gains suggest that terminating the density-based optimization
around 80% will yield nearly optimal performance while minimizing the time
spent in the density-based phase of the optimization.

Figure 3: Combining the continuous density-based optimization with a level-set
optimization. (a) The level-set optimization uses an initial seed given by the
results of the continuous density-based optimization at different starting
binarization values. The average FoM is plotted as a function of this starting
binarization. The red curve is a quadratic fit. (b) The convergence curves of
each level-set optimization.

The definition of the binarization function can be shifted with respect to
permittivity in order to converge to devices with different overall
material:void ratio. Eq. 1 uses the mid-point of the two material
permittivities as the center of the absolute value function. We can introduce
a shift term, $\delta\epsilon$:

$B=\frac{1}{N}\sum_{i}\left|\frac{\epsilon_{i}+\epsilon_{mid}-\delta\epsilon}{\epsilon_{max}-\epsilon_{mid}}\right|,i\in\mathcal{D}$
(11)

Intuitively the binarization constraint in Eq. 2.1 can be thought of as
defining whether a permittivity point should increase or decrease in order to
binarize the device more, depending on its current value of permittivity. In
Eq. 2.1 (with no shift about the midpoint), points that are above the midpoint
are encouraged to increase further since this is the direction of
binarization, and similarly points below the midpoint are encouraged to
decrease further. Thus, shifting the cusp away from the midpoint affects how
the permittivity is encouraged to become more binary. In general, shifting the
cusp of $B$ to the left (positive $\delta\epsilon$) yields a device with more
material, and vice versa. This can be a useful feature for enabling shorter
maskless lithography write times or ensuring mechanical robustness without
incurring a substantial decrease in device performance.

This technique is demonstrated in a low index contrast (1.5:1) device. Fig. 4a
shows the final material:void fraction as a function of $\delta\epsilon$,
which is varied from -0.5 to +0.5 about the mid-point permittivity of 1.625.
The resulting material distribution is also shown for each optimized device.
Fig. 4b shows the evolution of the material:void fraction over the course of
the optimization for the different $\delta\epsilon$ values. The largest change
in material:void fraction tends to occur in the early stages of optimization
in this example. Fig. 4c shows the evolution of the average FoM for different
values of $\delta\epsilon$. The devices with the worst final performance are
the ones with the most extreme $\delta\epsilon$, with the device consisting of
primarily material being substantially worse than the rest. The best
performing device was optimized with $\delta\epsilon=-0.25$.

Figure 4: Controlling the overall fraction of material. (a) The material-to-
void ratio for different shifts in the definition of binarization. To the
right of the plots is a picture of the resulting optimized device, where white
is material and black is void. From top to bottom the devices correspond to
$\delta\epsilon=-0.5$, $\delta\epsilon=-0.25$, $\delta\epsilon=0$,
$\delta\epsilon=0.25$, $\delta\epsilon=0.5$. (b) The evolution of the
material-to-void ratio during the optimization. Each traces shows the raw data
and a 5-iteration rolling average. (c) The evolution of the FoM during the
optimization.

##  4 Conclusion

We have presented a sub-optimization technique that is constrained to converge
to fully binary solutions. The technique involves solving a sub-optimization
at each iteration that maximizes the change in device performance while
forcing the step in dielectric permittivity to increase the overall
binarization of the device by a certain amount. This technique is
computationally infeasible to solve directly with a numerical solver, but the
optimization problem is reduced to a one-dimensional non-linear optimization
problem by instead solving the Lagrange dual problem.

The usefulness of the technique becomes apparent when optimizing devices with
a large refractive index contrast between the two material boundaries, where
traditional gradient-based techniques tend to stagnate at non-binary
solutions. It is shown that our level-set optimization depends on the initial
seed, as has been discussed in literature over the years, and that beginning
the level-set optimization with more binary seeds leads to better final device
performance. Lastly, we showed an example where the material-to-void fraction
of the optimized device can be controlled by simply shifting the definition of
binarization.

The specific sub-optimization presented here was intended to constrain the
device only to binary solutions. However, there were no assumptions made on
the binarization function $B$ other than the existence of a derivative with
respect to $\epsilon$. Thus, this function can describe other properties of
the device and the optimization procedure can be used to constrain other
device properties in future works.

Acknowledgments This work was supported by the Defense Advanced Research
Projects Agency EXTREME program (HR00111720035), the Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, CA, USA, under a
contract with the National Aeronautics and Space Administration (APRA
399131.02.06.04.75 and PDRDF 107614-19AW0079), the National Institutes of
Health (NIH) brain initiative program grant NIH 1R21EY029460-01, and Caltech
Rothenberg Innovation Initiative.

The authors thank Oscar P. Bruno for helpful discussion during the course of
this work.

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