# The discrete logarithmic Minkowski problem for the electrostatic
${\mathfrak{p}}$-capacity

Minhyun Kim  Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld,
Germany<EMAIL_ADDRESS>and  Taehun Lee  School of Mathematics,
Korea Institute for Advanced Study, Seoul 02455, Korea<EMAIL_ADDRESS>
###### Abstract.

The Minkowski problem for electrostatic capacity characterizes measures
generated by electrostatic capacity, which is a well-known variant of the
Minkowski problem. This problem has been generalized to $L_{p}$ Minkowski
problem for ${\mathfrak{p}}$-capacity. In particular, the logarithmic case
$p=0$ relates to cone-volumes and therefore has a geometric significance. In
this paper we solve the discrete logarithmic Minkowski problem for
$1<{\mathfrak{p}}<n$ in the case where the support of the given measure is in
general position.

###### Key words and phrases:

logarithmic Minkowski problem, ${\mathfrak{p}}$-capacity, polytope, convex
body

###### 2020 Mathematics Subject Classification:

52A20, 31B15, 52B11

##  1\. Introduction

A cornerstone of the Brunn–Minkowski theory is the Minkowski problem which
asks if a given measure on the unit sphere $\mathbb{S}^{n-1}$ arises as the
surface area measure of a convex body. This problem was completely solved by
Minkowski himself [21], Aleksandrov [1, 2], and Fenchel–Jessen [11].
Precisely, if a measure $\mu$ on $\mathbb{S}^{n-1}$ is not concentrated on any
closed hemisphere of $\mathbb{S}^{n-1}$, then $\mu$ is the surface area
measure of a convex body if and only if the centroid of $\mu$ is the origin,
i.e., $\int_{\mathbb{S}^{n-1}}\xi\,\mathrm{d}\mu(\xi)=0$. The regularity of
the solution has been studied by Nirenberg [22], Cheng–Yau [8], Pogorelov
[23], and Caffarelli [5].

An important variant of the Minkowski problem initiated by Jerison [14] is the
Minkowski problem for the electrostatic capacity. In a similar way, it asks if
a given measure on the unit sphere $\mathbb{S}^{n-1}$ arises as the
electrostatic capacitary measure. The electrostatic capacitary measure is
defined by the differential of the capacity as the surface area measure is
defined by the differential of the volume. Jerison found the necessary and
sufficient conditions for the existence of a convex body which are
surprisingly identical to the corresponding conditions in the classical
Minkowski problem (for volume).

A lot of research has been inspired by the work of Jerison. Namely, $L_{p}$
Minkowski problem for the electrostatic ${\mathfrak{p}}$-capacity, which will
be described below, has been investigated intensively. The aim of this paper
is to consider the logarithmic case $p=0$ for discrete measures which relates
to the cone-volumes111For a polytope that has the origin in its interior, the
cone-volume of a face of the polytope is the volume of the convex hull of the
face and the origin..

The result for the discrete Minkowski problem can be understood as prescribing
the areas of faces of a polytope. From this perspective, a natural, important
variant among other Minkowski type problems is the problem that prescribes the
cone-volumes of a polytope instead of the surface areas. The discrete
Minkowski problem for the cone-volumes can be stated in the following way:

Discrete logarithmic Minkowski problem. Let
$u_{1},\cdots,u_{N}\in\mathbb{S}^{n-1}$ be unit vectors with a set of numbers
$\gamma_{1},\cdots,\gamma_{N}>0$. Find necessary and sufficient conditions on
the set of unit vectors and the numbers so that there exists an $N$-faced
polytope whose outer unit normals are $u_{1},\cdots,u_{N}$ and the
corresponding cone-volumes are $\gamma_{1},\cdots,\gamma_{N}$.

Böröczky, Lutwak, Yang, and Zhang [4] solved the problem when the unit vectors
are even, i.e., $\\{u_{1},\cdots,u_{N}\\}=\\{-u_{1},\cdots,-u_{N}\\}$. In
[28], Zhu proved the problem in the case that the unit vectors are in general
position (see Definition 2.1) and are not concentrated on any closed
hemisphere. Later, the discrete logarithmic Minkowski problem was solved under
a more general assumption that contains the two results above as special cases
[3].

It is worth noting that the logarithmic Minkowski problem for general measures
was solved by Böröczky et al. [4] in the case of even measures. For non-even
measures, Chen, Li, and Zhu [6] proved the existence of the solution under the
same assumption as in [4]. We also note that there is no known conjecture
characterizing cone-volume measures.

Both the classical Minkowski problem and the logarithmic Minkowski problem are
incorporated in $L_{p}$ Minkowski problem. The $L_{p}$ Minkowski problem was
initiated by Lutwak in [17] for $p\geq 1$ and has been intensively studied in,
e.g., [7, 9, 12, 13, 15, 16, 18, 19, 20, 25, 29].

To describe the $L_{p}$ Minkowski problem, we recall the notion of the $L_{p}$
surface area. The surface area measure $S(K,\cdot)$ of $K$ appears in the
celebrated Aleksandrov variational formula: for convex bodies $K$ and $L$,

$\displaystyle\frac{\mathrm{d}V(K+tL)}{\mathrm{d}t}\Big{|}_{t=0+}=\int_{\mathbb{S}^{n-1}}h_{L}(u)\,\mathrm{d}S(K,u),$

where $V$ is the $n$-dimensional volume. For an index $p\in{\mathbb{R}}$, the
$L_{p}$ surface area measure $S_{p}(K,\cdot)$ is then defined by

$\displaystyle S_{p}(K,\omega)=\int_{\omega}h_{K}(u)^{1-p}\,\mathrm{d}S(K,u)$

for any Borel set $\omega\subset\mathbb{S}^{n-1}$.

$L_{p}$ Minkowski problem. Let $p\in{\mathbb{R}}$ and $\mu$ be a finite Borel
measure on the unit sphere $\mathbb{S}^{n-1}$. Find necessary and sufficient
conditions on $\mu$ so that $\mu$ is the $L_{p}$ surface area measure
$S_{p}(K,\cdot)$ of a convex body $K$.

The $L_{1}$ Minkowski problem is the classical Minkowski problem, and the
$L_{0}$ Minkowski problem is the logarithmic Minkowski problem. Another
important special case is the $L_{-n}$ Minkowski problem, also known as the
centro-affine Minkowski problem.

Along with the $L_{p}$ Minkowski problem for volume, there is a parallel
$L_{p}$ Minkowski-type problem. This problem is concerned with the
electrostatic ${\mathfrak{p}}$-capacity of convex bodies, and is known as the
$L_{p}$ Minkowski problem for ${\mathfrak{p}}$-capacity. To describe the
problem, we recall the definition of the electrostatic
${\mathfrak{p}}$-capacity and its variational formula.

Let $1<{\mathfrak{p}}<n$. For a compact set $K$ in ${\mathbb{R}}^{n}$, the
electrostatic ${\mathfrak{p}}$-capacity is defined as

$\displaystyle C_{\mathfrak{p}}(K)=\inf\left\\{\int_{{\mathbb{R}}^{n}}|\nabla
u|^{\mathfrak{p}}\,\mathrm{d}x:u\in
C^{\infty}_{c}({\mathbb{R}}^{n}),u\geq\chi_{K}\right\\},$

where $\chi_{K}$ is the characteristic function of $K$ and
$C^{\infty}_{c}({\mathbb{R}}^{n})$ denotes the set of smooth functions with
compact supports. The case ${\mathfrak{p}}=2$ is the classical electrostatic
(or Newtonian) capacity of $K$.

As in the volume case, the differential of the electrostatic
${\mathfrak{p}}$-capacity produces a geometric measure. Indeed, the
electrostatic ${\mathfrak{p}}$-capacitary measure
$\mu_{{\mathfrak{p}}}(K,\cdot)$ of $K$ appears in the Hadamard variational
formula for ${\mathfrak{p}}$-capacity [10]: for convex bodies $K$ and $L$,

$\displaystyle\frac{\mathrm{d}C_{\mathfrak{p}}(K+tL)}{\mathrm{d}t}\bigg{|}_{t=0+}=({\mathfrak{p}}-1)\int_{\mathbb{S}^{n-1}}h_{L}(u)\,\mathrm{d}\mu_{{\mathfrak{p}}}(K,u).$

In particular, the formula implies the Poincaré ${\mathfrak{p}}$-capacity
formula

$\displaystyle
C_{\mathfrak{p}}(K)=\frac{{\mathfrak{p}}-1}{n-{\mathfrak{p}}}\int_{\mathbb{S}^{n-1}}h_{K}(u)\,\mathrm{d}\mu_{\mathfrak{p}}(K,u).$

In a similar way as in the construction of the $L_{p}$ surface area measure
from the surface area measure, we introduce the $L_{p}$ electrostatic
${\mathfrak{p}}$-capacitary measure.

######  Definition 1.1.

Let $p\in{\mathbb{R}}$ and $1<{\mathfrak{p}}<n$. Suppose that $K$ is a convex
body in ${\mathbb{R}}^{n}$ with the origin in its interior, and the measure
$\mu_{p,{\mathfrak{p}}}(K,\cdot)$ is defined by

$\displaystyle\mu_{p,{\mathfrak{p}}}(K,\omega)=\int_{\omega}h_{K}(u)^{1-p}\,\mathrm{d}\mu_{\mathfrak{p}}(K,u)$

for any Borel set $\omega\subset\mathbb{S}^{n-1}$. The measure
$\mu_{p,{\mathfrak{p}}}(K,\cdot)$ is called the $L_{p}$ electrostatic
${\mathfrak{p}}$-capacitary measure.

Note that the $L_{p}$ electrostatic ${\mathfrak{p}}$-capacitary measure
appears in the $L_{p}$ variational formula of the electrostatic
${\mathfrak{p}}$-capacity [30]. Precisely, for convex bodies $K$ and $L$ with
the origin in their interiors, it follows that for $p\geq 1$

$\displaystyle\frac{\mathrm{d}C_{\mathfrak{p}}(K+_{p}t\cdot_{p}L)}{\mathrm{d}t}\bigg{|}_{t=0+}=\frac{{\mathfrak{p}}-1}{p}\int_{\mathbb{S}^{n-1}}h_{L}(u)^{p}\,\mathrm{d}\mu_{p,{\mathfrak{p}}}(K,u).$

We now state the following $L_{p}$ Minkowski problem for the electrostatic
${\mathfrak{p}}$-capacity.

$L_{p}$ Minkowski problem for ${\mathfrak{p}}$-capacity. Let
$p\in{\mathbb{R}}$, $1<{\mathfrak{p}}<n$, and $\mu$ be a finite Borel measure
on the unit sphere $\mathbb{S}^{n-1}$. Find necessary and sufficient
conditions on $\mu$ so that $\mu$ is the $L_{p}$ electrostatic
${\mathfrak{p}}$-capacitary measure of a convex body $K$.

As mentioned above, Jerison [14] initiated the Minkowski type problem for the
electrostatic capacity and solved the problem for the classical case $p=1$ and
${\mathfrak{p}}=2$. The necessary and sufficient conditions for the existence
of a convex body are the centroid of $\mu$ on $\mathbb{S}^{n-1}$ is the
origin, which is identical to the corresponding conditions in the classical
Minkowski problem (for volume). We point out that a solution for general
measures was obtained by the existence of a solution for discrete measures and
an approximation argument.

Recently, many authors extended Jerison’s work to other ranges of
$(p,{\mathfrak{p}})$. For general measures, the $L_{p}$ Minkowski problems for
${\mathfrak{p}}$-capacity were studied by Colesanti, Nyström, Salani, Xiao,
and Zhang [10] for $p=1$ and $1<{\mathfrak{p}}<n$; by Zou and Xiong [30] for
$p>1$ and $1<{\mathfrak{p}}<n$. We note that the existence results in these
paper are also obtained from the existence results to the discrete case. For
$0<p<1$ and $1<{\mathfrak{p}}<2$, the problem for discrete measures was solved
by Xiong, Xiong, and Xu [27]. As indicated in [30], however, the logarithmic
case $p=0$ with $1<{\mathfrak{p}}<n$ is still open although it is an important
case.

In this paper we consider the case $p=0$ for discrete measures. More
precisely, we solve the logarithmic Minkowski problem for
${\mathfrak{p}}$-capacity in the case of discrete measures whose support is in
general position.

######  Theorem 1.2.

Let $n\geq 2$, $1<\mathfrak{p}<n$, and $\mu$ be a discrete measure on the unit
sphere $\mathbb{S}^{n-1}$ whose support is not concentrated on a closed
hemisphere. Then $\mu$ is the measure $\mu_{0,{\mathfrak{p}}}(P,\cdot)$ of a
polytope $P$ whose outer unit normals are in general position if and only if
the support of $\mu$ is in general position. In that case, the polytope $P$
contains the origin in its interior, and its unit normals are exactly the
support of $\mu$.

The importance of the discrete $L_{p}$ Minkowski problem for
${\mathfrak{p}}$-capacity follows from that the problem for general measures
or for measures with density can be obtained by an approximation argument with
discrete measures. Indeed, by first solving the polytopal case, an
approximation argument provides a solution to the problem for general measures
when $p\geq 1$ and $1<{\mathfrak{p}}<n$ [14, 10, 30]. See also [6] for an
approximation argument in the logarithmic ($p=0$) Minkowski problem.

The paper organized as follows. In Section 2 we provide some basic notations
and facts on convex bodies. In Section 3 we study an extreme problem whose
minimizer solves the logarithmic Minkowski problem for
${\mathfrak{p}}$-capacity. Section 4 is devoted to obtaining a uniform
diameter control for a family of polytopes with the same
${\mathfrak{p}}$-capacity. Finally, Theorem 1.2 is proved in Section 5.

##  2\. Preliminaries

Throughout the paper we assume that $n\geq 2$ and $1<{\mathfrak{p}}<n$. In the
Euclidean space ${\mathbb{R}}^{n}$ a convex body is a compact convex set with
non-empty interior. We denote by $\mathcal{K}^{n}$ the set of convex bodies in
${\mathbb{R}}^{n}$ and by $\mathcal{K}^{n}_{0}$ the set of convex bodies with
the origin $o$ in their interior.

We write $x\cdot y$ for the standard inner product of $x,y\in{\mathbb{R}}^{n}$
and $|x|=(x\cdot x)^{1/2}$ for the standard norm of $x\in{\mathbb{R}}^{n}$.
Let $\mathbb{S}^{n-1}$ be the boundary of the unit ball $B_{1}$ in
${\mathbb{R}}^{n}$. For a set $A$ in ${\mathbb{R}}^{n}$, we write
$\mathrm{Int}(A)$ for the interior of $A$. We denote the volume, area, and
diameter of $A$ by $V(A)$, $\mathrm{Area}(A)$, and $\mathrm{diam}(A)$,
respectively.

The support function $h_{K}$ of $K\in\mathcal{K}^{n}$ is defined by

$\displaystyle h_{K}(x)=\max\\{x\cdot y:y\in K\\}.$

For $K,L\in\mathcal{K}^{n}$, the Hausdorff metric is defined by

$\displaystyle\delta(K,L)=\max_{u\in\mathbb{S}^{n-1}}|h_{K}(u)-h_{L}(u)|.$

We define the support hyperplane $H(K,u)$ for $K\in\mathcal{K}^{n}$ and
$u\in\mathbb{S}^{n-1}$ as

$\displaystyle H(K,u)=\\{x\in{\mathbb{R}}^{n}:x\cdot u=h_{K}(u)\\},$

and then the support set $F(K,u)$ is defined by

$\displaystyle F(K,u)=K\cap H(K,u).$

A polytope in ${\mathbb{R}}^{n}$ is the convex hull of a finite set of points
in ${\mathbb{R}}^{n}$ with positive $n$-dimensional volume. If the convex hull
of a subset of these points lies entirely on the boundary of the polytope and
has positive $(n-1)$-dimensional volume, then it is called a facet of the
polytope.

We write a discrete measure $\mu$ of the form

$\displaystyle\mu=\sum_{i=1}^{N}\gamma_{i}\delta_{u_{i}},$

where $\gamma_{1},\cdots,\gamma_{N}$ are positive constants and $\delta_{u}$
denotes the Dirac delta measure on $\mathbb{S}^{n-1}$ defined by

$\displaystyle\delta_{u}(A)=\begin{cases}1\quad\text{if }u\in A,\\\
0\quad\text{otherwise}.\end{cases}$

We need some definitions on the support of measures:

######  Definition 2.1.

A finite subset $U$ (with no less than $n$ elements) of $\mathbb{S}^{n-1}$ is
said to be in general position if any $n$ elements of $U$ are linearly
independent.

######  Definition 2.2.

A subset $U$ of $\mathbb{S}^{n-1}$ is said to be concentrated on a closed
hemisphere if there exists $u\in\mathbb{S}^{n-1}$ such that
$U\subset\\{v\in\mathbb{S}^{n-1}:u\cdot v\geq 0\\}$.

Suppose that the unit vectors $u_{1},\cdots,u_{N}\in\mathbb{S}^{n-1}$ are not
concentrated on any closed hemisphere of $\mathbb{S}^{n-1}$. We denote by
$\mathcal{P}(u_{1},\cdots,u_{N})$ the set of polytopes of the form
$\cap_{i=1}^{N}\\{x\in\mathbb{R}^{n}:x\cdot u_{i}\leq a_{i}\\}$ for
$a_{1},\dots,a_{N}\in{\mathbb{R}}$ and by
$\mathcal{P}_{N}(u_{1},\cdots,u_{N})$ the subset of
$\mathcal{P}(u_{1},\cdots,u_{N})$ whose element has exactly $N$ facets. Note
that any $P\in\mathcal{P}(u_{1},\cdots,u_{N})$ has at most $N$ facets, and the
outer normals of $P$ are a subset of $\\{u_{1},\cdots,u_{N}\\}$.

Note that for convex bodies $K,L\in\mathcal{K}^{n}$, the (first) mixed
${\mathfrak{p}}$-capacity $C_{\mathfrak{p}}(K,L)$ is defined as

$C_{\mathfrak{p}}(K,L)=\frac{1}{n-{\mathfrak{p}}}\frac{\mathrm{d}C_{\mathfrak{p}}(K+tL)}{\mathrm{d}t}\bigg{|}_{t=0+}=\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\int_{\omega}h_{L}(u)\,\mathrm{d}\mu_{\mathfrak{p}}(K,u)$

and by taking $K=L$, we obtain a measure corresponding to the cone-volume
measure, namely for a Borel set $\omega\subset\mathbb{S}^{n-1}$,

$\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\int_{\omega}h_{K}(u)\,\mathrm{d}\mu_{\mathfrak{p}}(K,u).$

Finally, we recall the Blaschke selection theorem (see [24, Theorem 1.8.7]).

######  Theorem 2.3 (Blaschke selection theorem).

Every bounded sequence of convex bodies has a subsequence that converges to a
convex body.

##  3\. An extreme problem

Let us fix $N\geq n+1$ and let
$\\{u_{1},\dots,u_{N}\\}\subset\mathbb{S}^{n-1}$ be in general position and
not concentrated on a closed hemisphere. Let $\gamma_{1},\dots,\gamma_{N}$ be
$N$ positive real numbers. For a polytope $P\in\mathcal{P}(u_{1},\dots,u_{N})$
and the discrete measure $\mu=\sum_{i=1}^{N}\gamma_{i}\delta_{u_{i}}$, we
define a functional $\Phi_{\mu,P}:\mathrm{Int}(P)\to\mathbb{R}$ by

$\Phi_{\mu,P}(\xi)=\sum_{i=1}^{N}\gamma_{i}\log(h_{P}(u_{i})-\xi\cdot u_{i}).$

Note that an extreme problem for the functional $\Phi_{\mu,P}$ with volume
constraint has been widely used to establish logarithmic Minkowski problem
(see, for instance, [4, 28, 3, 6]). However, in this paper we consider the
extreme problem

(3.1)
$\inf\left\\{\max_{\xi\in\mathrm{Int}(Q)}\Phi_{\mu,Q}(\xi):Q\in\mathcal{P}_{N}(u_{1},\dots,u_{N})~{}\text{and}~{}C_{\mathfrak{p}}(Q)=\sum_{i=1}^{N}\gamma_{i}\right\\}$

with $\mathfrak{p}$-capacity constraint to prove Theorem 1.2.

Within this section, we prove that any minimizer of the problem (3.1) solves
the discrete logarithmic Minkowski problem for $\mathfrak{p}$-capacity. Before
we prove this, let us recall known results about the functional $\Phi_{\mu,P}$
(see [28, Section 3]).

######  Lemma 3.1.

Suppose that $\\{u_{1},\dots,u_{N}\\}\subset\mathbb{S}^{n-1}$ is in general
position and not concentrated on a closed hemisphere. Let
$\gamma_{1},\dots,\gamma_{N}$ be $N$ positive real numbers and define
$\mu=\sum_{i=1}^{N}\gamma_{i}\delta_{u_{i}}$. If
$P\in\mathcal{P}(u_{1},\dots,u_{N})$, then there exists a unique point
$\xi(P)\in\mathrm{Int}(P)$ such that

$\Phi_{\mu,P}(\xi(P))=\max_{\xi\in\mathrm{Int}(P)}\Phi_{\mu,P}(\xi).$

Moreover, if a sequence of polytopes $P_{m}\in\mathcal{P}(u_{1},\dots,u_{N})$
converges to $P$ with respect to the Hausdorff metric, then

$\lim_{i\to\infty}\xi(P_{i})=\xi(P)$

and

$\lim_{i\to\infty}\Phi_{\mu,P_{i}}(\xi(P_{i}))=\Phi_{\mu,P}(\xi(P)).$

Let us now prove that a minimizer of the extreme problem (3.1) solves the
discrete logarithmic Minkowski problem for $\mathfrak{p}$-capacity.

######  Theorem 3.2.

Let $u_{1},\dots,u_{N}$ and $\gamma,\dots,\gamma_{N}$ be given as in Lemma
3.1. If $P\in\mathcal{P}_{N}(u_{1},\dots,u_{N})$ satisfies $\xi(P)=o$,
$C_{\mathfrak{p}}(P)=\sum_{i=1}^{N}\gamma_{i}$, and

(3.2)
$\Phi_{\mu,P}(o)=\inf\left\\{\max_{\xi\in\mathrm{Int}(Q)}\Phi_{\mu,Q}(\xi):Q\in\mathcal{P}_{N}(u_{1},\dots,u_{N})~{}\text{and}~{}C_{\mathfrak{p}}(Q)=\sum_{i=1}^{N}\gamma_{i}\right\\},$

then $\mu_{0,\mathfrak{p}}(P,\cdot)=\mu$.

###### Proof.

Since $\xi(\lambda P)=\lambda\xi(P)$, we may assume that
$C_{\mathfrak{p}}(P)=\sum_{i=1}^{N}\gamma_{i}=1$. Let
$\delta_{1},\dots,\delta_{N}\in\mathbb{R}$ and define

$P_{t}:=\bigcap_{i=1}^{N}\\{x\in\mathbb{R}^{n}:x\cdot u_{i}\leq
h_{P}(u_{i})+t\delta_{i}\\}.$

Since $P\in\mathcal{P}_{N}(u_{1},\dots,u_{N})$, we choose $|t|$ sufficiently
small so that $P_{t}$ is the polytope with exactly $N$ facets. Let

$\alpha(t)=C_{\mathfrak{p}}(P_{t})^{-1/(n-\mathfrak{p})}$

and define $\tilde{P}_{t}=\alpha(t)P_{t}$. Then,
$\tilde{P}\in\mathcal{P}_{N}(u_{1},\dots,u_{N})$,
$C_{\mathfrak{p}}(\tilde{P}_{t})=1$, and $\tilde{P}_{t}\to P$ with respect to
the Hausdorff metric as $t\to 0$. Since $P$ is an optimizer of the extreme
problem (3.2), we will compute the first variation of the functional.

We first have from [10, Theorem 5.2] and

(3.3)
$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}h_{P_{t}}(u_{i})=\delta_{i}$

that

(3.4)
$\begin{split}\alpha^{\prime}(0)&=-\frac{1}{n-\mathfrak{p}}\left.\frac{\mathrm{d}C_{\mathfrak{p}}(P_{t})}{\mathrm{d}t}\right|_{t=0}\\\
&=-\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\int_{\mathbb{S}^{n-1}}\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}h_{P_{t}}(\xi)\,\mathrm{d}\mu_{\mathfrak{p}}(P,\xi)\\\
&=-\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\sum_{i=1}^{N}\delta_{i}\mu_{\mathfrak{p}}(P,\\{u_{i}\\}).\end{split}$

We set $\xi(t):=\xi(\tilde{P}_{t})$ and define

$\Phi(t):=\Phi_{\mu,\tilde{P}_{t}}(\xi(t))=\max_{\xi\in\mathrm{Int}(\tilde{P}_{t})}\Phi_{\mu,\tilde{P}_{t}}(\xi)=\sum_{i=1}^{N}\gamma_{i}\log(\alpha(t)h_{P_{t}}(u_{i})-\xi(t)\cdot
u_{i}).$

Since the maximum is attained in an interior point $\xi(t)$, we obtain

(3.5)
$0=\sum_{i=1}^{N}\gamma_{i}\frac{u_{i,j}}{\alpha(t)h_{P_{t}}(u_{i})-\xi(t)\cdot
u_{i}}$

for each $j=1,\dots,n$, where $u_{i,j}$ is the $j$-th element of the vector
$u_{i}$. In particular, when $t=0$ we have

(3.6) $\sum_{i=1}^{N}\gamma_{i}\frac{u_{i}}{h_{P}(u_{i})}=o$

as vectors.

Let us next show that $\Phi$ is differentiable at zero. We define avector
valued function $F$ by

$F(t,\xi)=\sum_{i=1}^{N}\gamma_{i}\frac{u_{i}}{\alpha(t)h_{P_{t}}(u_{i})-\xi\cdot
u_{i}}.$

Then, the Jacobian matrix of $F$ at $(0,0)$ is given by

$\frac{\partial
F}{\partial\xi}(0,0)=\sum_{i=1}^{N}\frac{\gamma_{i}}{h_{P_{t}}^{2}(u_{i})}u_{i}u_{i}^{T}.$

Let $x\in\mathbb{R}^{n}$ be a nonzero vector, then by the assumption that
$\\{u_{1},\dots,u_{N}\\}$ is in general position, there exists a vector
$u_{i_{0}}\in\\{u_{1},\dots,u_{N}\\}$ such that $u_{i_{0}}\cdot x\neq 0$.
Thus, we have

$x^{T}\left(\sum_{i=1}^{N}\frac{\gamma_{i}}{h_{P_{t}}^{2}(u_{i})}u_{i}u_{i}^{T}\right)x=\sum_{i=1}^{N}\frac{\gamma_{i}}{h_{P_{t}}^{2}(u_{i})}(x\cdot
u_{i})^{2}\geq\frac{\gamma_{i_{0}}}{h_{P_{t}}^{2}(u_{i_{0}})}(x\cdot
u_{i_{0}})^{2}>0,$

which indicates that the Jacobian matrix $\frac{\partial F}{\partial\xi}(0,0)$
is positive definite. Therefore, by $\xi(0)=o$, (3.5), and the implicit
function theorem, $\xi(t)$ is differentiable at zero and so is $\Phi(t)$.

From the assumption that $\tilde{P}_{0}=P$ is a minimizer of (3.2), we have
$\Phi(0)\leq\Phi(t)$ for sufficiently small $|t|$, which implies that
$\Phi^{\prime}(0)=0$. Thus, by using (3.3), (3.4), and (3.6), we obtain

$\begin{split}0=\Phi^{\prime}(0)&=\sum_{i=1}^{N}\gamma_{i}\left(\frac{\alpha(0)}{h_{P}(u_{i})}\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}h_{P_{t}}(u_{i})+\alpha^{\prime}(0)-\frac{\xi^{\prime}(0)\cdot
u_{i}}{h_{P}(u_{i})}\right)\\\
&=\sum_{i=1}^{N}\gamma_{i}\left(\frac{\delta_{i}}{h_{P}(u_{i})}-\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\sum_{j=1}^{N}\delta_{j}\mu_{\mathfrak{p}}(P,\\{u_{j}\\})\right)\\\
&=\sum_{i=1}^{N}\delta_{i}\left(\frac{\gamma_{i}}{h_{P}(u_{i})}-\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\left(\sum_{j=1}^{N}\gamma_{j}\right)\mu_{\mathfrak{p}}(P,\\{u_{i}\\})\right).\end{split}$

Since we assumed that $\sum_{j=1}^{N}\gamma_{j}=1$, we arrive at

$\sum_{i=1}^{N}\delta_{i}\left(\frac{\gamma_{i}}{h_{P}(u_{i})}-\frac{\mathfrak{p}-1}{n-\mathfrak{p}}\mu_{\mathfrak{p}}(P,\\{u_{i}\\})\right)=0.$

Recalling that we have chosen $\delta_{i}$ arbitrary, we conclude that

$\frac{\mathfrak{p}-1}{n-\mathfrak{p}}h_{P}(u_{i})\mu_{\mathfrak{p}}(P,\\{u_{i}\\})=\gamma_{i}$

for all $i=1,\dots,n$. Therefore, $\lambda P$ satisfies
$\mu_{0,\mathfrak{p}}(\lambda P,\cdot)=\mu$, where
$\lambda=(\frac{{\mathfrak{p}}-1}{n-{\mathfrak{p}}})^{-1/(n-{\mathfrak{p}})}$.
∎

##  4\. Compactness

In the previous section, we proved that any minimizer of the extreme problem
(3.1) solves the discrete logarithmic Minkowski problem for
$\mathfrak{p}$-capacity. In order to prove Theorem 1.2, we need to show that
there exists a minimizer of the extreme problem (3.1). To this end, we provide
a compactness result for this problem in this section. That is to say, we
prove that a sequence of polytopes from $\mathcal{P}(u_{1},\dots,u_{N})$
having a bounded $\mathfrak{p}$-capacity also has a bounded diameter.

######  Theorem 4.1.

Suppose that $\\{u_{1},\dots,u_{N}\\}\subset\mathbb{S}^{n-1}$ is in general
position and not concentrated on a closed hemisphere. If
$\\{P_{m}\\}\subset\mathcal{P}(u_{1},\dots,u_{N})$ is a sequence of polytopes
satisfying $o\in P_{m}$ and $C_{\mathfrak{p}}(P_{m})=1$, then $\\{P_{m}\\}$ is
bounded.

A statement similar to Theorem 4.1 with the assumption
$C_{\mathfrak{p}}(P_{m})=1$ replaced by $V(P_{m})=1$ was first proved by Zhu
in [28]. The idea of proof of Theorem 4.1 is to combine Zhu’s result and the
isoperimetric inequality for $\mathfrak{p}$-capacity. Let us first recall the
result in [28].

######  Theorem 4.2.

[28, Theorem 4.3] Let $\\{u_{1},\dots,u_{N}\\}\subset\mathbb{S}^{n-1}$ be
given as in Theorem 4.1. If $\\{P_{m}\\}\subset\mathcal{P}(u_{1},\dots,u_{N})$
is a sequence of polytopes satisfying $o\in P_{m}$ and $V(P_{m})=1$, then
$\\{P_{m}\\}$ is bounded.

We need the isoperimetric inequality for $\mathfrak{p}$-capacity for the proof
of Theorem 4.1.

######  Theorem 4.3.

[26, Theorem 2.1] Let $\mathfrak{p}\in(1,n)$ and $P$ be a convex body. Then,

$\left(\frac{\mathrm{Area}(P)}{\mathrm{Area}(B_{1})}\right)^{\frac{1}{n-1}}\leq\left(\frac{\mathfrak{p}(n-1)}{n(\mathfrak{p}-1)}\right)^{\frac{\mathfrak{p}-1}{n-\mathfrak{p}}}\left(\frac{C_{\mathfrak{p}}(P)}{C_{\mathfrak{p}}(B_{1})}\right)^{\frac{1}{n-\mathfrak{p}}},$

where $B_{1}$ is the unit ball in $\mathbb{R}^{n}$.

We are now in a position to prove Theorem 4.1 by using Theorem 4.2 and Theorem
4.3.

###### Proof of Theorem 4.1.

Suppose that $\\{P_{m}\\}$ is unbounded. Then, by Theorem 4.2, a sequence of
volumes $\\{V(P_{m})\\}$ is unbounded. By the isoperimetric inequality (for
volume), there exists a dimensional constant $c_{1}=c_{1}(n)>0$ such that

$V(P_{m})^{\frac{1}{n}}\leq c_{1}\mathrm{Area}(P_{m})^{\frac{1}{n-1}}$

holds for every $m$. Moreover, by Theorem 4.3, we have

$\mathrm{Area}(P_{m})^{\frac{1}{n-1}}\leq
c_{2}C_{\mathfrak{p}}(P_{m})^{\frac{1}{n-\mathfrak{p}}}$

for some $c_{2}=c_{2}(n,\mathfrak{p})>0$. Thus, we conclude that
$\\{C_{\mathfrak{p}}(P_{m})\\}$ is unbounded, which leads us to a
contradiction. Therefore, $\\{P_{m}\\}$ is bounded. ∎

##  5\. Discrete logarithmic Minkowski problem for $\mathfrak{p}$-capacity

In this section, we prove the main result, Theorem 1.2, which follows from the
existence of a minimizer of the extreme problem (3.2).

######  Theorem 5.1.

Suppose that $\\{u_{1},\dots,u_{N}\\}\subset\mathbb{S}^{n-1}$ is in general
position and not concentrated on a closed hemisphere. Let
$\gamma_{1},\dots,\gamma_{N}$ be $N$ positive real numbers and define
$\mu=\sum_{i=1}^{N}\gamma_{i}\delta_{u_{i}}$. Then, there exists a polytope
$P\in\mathcal{P}_{N}(u_{1},\dots,u_{N})$ such that $\xi(P)=o$,
$C_{\mathfrak{p}}(P)=\sum_{i=1}^{N}\gamma_{i}$, and

$\Phi_{\mu,P}(o)=\inf\left\\{\max_{\xi\in\mathrm{Int}(Q)}\Phi_{\mu,Q}(\xi):Q\in\mathcal{P}_{N}(u_{1},\dots,u_{N})~{}\text{and}~{}C_{\mathfrak{p}}(Q)=\sum_{i=1}^{N}\gamma_{i}\right\\}.$

In particular, the measure $\mu$ is $L_{0}$ electrostatic
${\mathfrak{p}}$-capacitary measure, i.e.,
$\mu_{0,\mathfrak{p}}(P,\cdot)=\mu$.

###### Proof.

It is enough to prove the theorem for the case $\sum_{i=1}^{N}\gamma_{i}=1$.
Let us take a minimizing sequence
$\\{P_{m}\\}\subset\mathcal{P}_{N}(u_{1},\dots,u_{N})$ for

(5.1)
$\inf\left\\{\max_{\xi\in\mathrm{Int}(Q)}\Phi_{\mu,Q}(\xi):Q\in\mathcal{P}_{N}(u_{1},\dots,u_{N})~{}\text{and}~{}C_{\mathfrak{p}}(Q)=1\right\\}.$

Moreover, we may assume that $\xi(P_{m})=o$. Indeed, otherwise we may take
$P_{m}-\xi(P_{m})$ instead of $P_{m}$, which also minimizes (5.1) since
$\Phi_{\mu,P_{m}}(\xi(P_{m}))=\Phi_{\mu,P_{m}-\xi(P_{m})}(\xi(P_{m}-\xi(P_{m})))$.

We know from Theorem 4.1 that $\\{P_{m}\\}$ is bounded. Thus, by Blaschke
selection theorem, Lemma 3.1, and continuity of $C_{\mathfrak{p}}$ on
$\mathcal{K}^{n}$, there exists a subsequence of $\\{P_{m}\\}$ that converges
to a convex set $P$ satisfying $\xi(P)=o$, $C_{\mathfrak{p}}(P)=1$, and

$\Phi_{\mu,P}(o)=\inf\left\\{\max_{\xi\in\mathrm{Int}(Q)}\Phi_{\mu,Q}(\xi):Q\in\mathcal{P}_{N}(u_{1},\dots,u_{N})~{}\text{and}~{}C_{\mathfrak{p}}(Q)=1\right\\}.$

Furthermore, it follows from the fact
$P_{m}\in\mathcal{P}_{N}(u_{1},\dots,u_{N})$ that
$\mathrm{supp}(P)\subset\\{u_{1},\dots,u_{N}\\}$.

We now prove that $P$ has a positive $n$-dimensional volume. Otherwise
$P\subset L$ for some linear subspace $L$ in $\mathbb{R}^{n}$ with
$\mathrm{dim}(L)=n-m$, $0<m\leq n$. Then, there exist $m+1$ vectors
$u_{i_{1}},\dots,u_{i_{m+1}}\in\\{u_{1},\dots,u_{N}\\}$ such that
$u_{i_{j}}\in L^{\perp}$ for $j=1,\dots,m+1$. Since
$\mathrm{dim}(L^{\perp})=m$, any set of $n$ vectors containing
$u_{i_{1}},\dots,u_{i_{m+1}}$ is linearly dependent, which contradicts to the
assumption that $\\{u_{1},\dots,u_{N}\\}$ is in general position. Therefore,
we have $\mathrm{Int}(P)\neq\emptyset$.

To show that $P\in\mathcal{P}_{N}(u_{1},\dots,u_{N})$, we need to show that
$P$ has exactly $N$ facets. We first argue by contradiction that $F(P,u_{i})$
are facets for all $i=1,\dots,N$. Assume that there is an index
$i_{0}\in\\{1,\dots,N\\}$ such that $F(P,u_{i_{0}})$ is not a facet. For
$t>0$, we define the polytope $P_{t}$ by

$P_{t}=P\cap\\{x\in\mathbb{R}^{n}:x\cdot u_{i_{0}}\leq h_{P}(u_{i_{0}})-t\\}$

and let $\lambda=C_{\mathfrak{p}}(P_{t})^{-1/(n-\mathfrak{p})}$. Then, we have
$C_{\mathfrak{p}}(\lambda P_{t})=1$ and $\lambda P_{t}\to P$ as $t\searrow 0$.
Moreover, it follows from Lemma 3.1 that
$\xi(P_{t})\to\xi(P)=o\in\mathrm{Int}(P)$ as $t\searrow 0$. We now choose
$t>0$ sufficiently small so that $P_{t}$ has exactly one more facet than that
of $P$ and $h_{P}(u_{i})>\xi(P_{t})\cdot u_{i}+t$ for all $i=1,\dots,N$.

We show that $\Phi_{\mu,\lambda P_{t}}(\xi(\lambda P_{t}))<\Phi_{\mu,P}(o)$,
which contradicts the fact that $\Phi_{\mu,P}(o)$ is a minimum. Let
$d_{0}=\mathrm{diam}(P)$, then $d_{0}>h_{P}(u_{i_{0}})-\xi(P_{t})\cdot
u_{i_{0}}>0$. Since $h_{P_{t}}(u_{i})=h_{P}(u_{i})$ for all $i\neq i_{0}$, we
have

$\begin{split}&\prod_{i=1}^{N}(h_{\lambda P_{t}}(u_{i})-\xi(\lambda
P_{t})\cdot u_{i})^{\gamma_{i}}\\\
&=\lambda\prod_{i=1}^{N}(h_{P_{t}}(u_{i})-\xi(P_{t})\cdot
u_{i})^{\gamma_{i}}\\\
&=\lambda\left(\prod_{i=1}^{N}(h_{P}(u_{i})-\xi(P_{t})\cdot
u_{i})^{\gamma_{i}}\right)\left(\frac{h_{P}(u_{i_{0}})-\xi(P_{t})\cdot
u_{i_{0}}-t}{h_{P}(u_{i_{0}})-\xi(P_{t})\cdot
u_{i_{0}}}\right)^{\gamma_{i_{0}}}\\\
&\leq\left(\prod_{i=1}^{N}(h_{P}(u_{i})-\xi(P_{t})\cdot
u_{i})^{\gamma_{i}}\right)\frac{(1-t/d_{0})^{\gamma_{i_{0}}}}{C_{\mathfrak{p}}(P_{t})^{1/(n-\mathfrak{p})}}.\end{split}$

If we define a function

$g(t)=C_{\mathfrak{p}}(P_{t})^{\frac{1}{(n-\mathfrak{p})\gamma_{i_{0}}}}+\frac{1}{d_{0}}t-1,$

then $g(0)=0$ and

$g^{\prime}(t)=\frac{1}{(n-\mathfrak{p})\gamma_{i_{0}}}C_{\mathfrak{p}}(P_{t})^{\frac{1}{(n-\mathfrak{p})\gamma_{i_{0}}}-1}\frac{\mathrm{d}}{\mathrm{d}t}C_{\mathfrak{p}}(P_{t})+\frac{1}{d_{0}}.$

By the variational formula for $\mathfrak{p}$-capacity [10, Theorem 5.2], we
have

$\left.\frac{\mathrm{d}C_{\mathfrak{p}}(P_{t})}{\mathrm{d}t}\right|_{t=0}=(\mathfrak{p}-1)\sum_{i=1}^{N}\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}h_{P_{t}}(u_{i})\mu_{\mathfrak{p}}(P,\\{u_{i}\\}).$

Since $\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{i=0}h_{P_{t}}(u_{i})=0$
for $i\neq i_{0}$,
$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{i=0}h_{P_{t}}(u_{i_{0}})=-1$,
and $\mu_{\mathfrak{p}}(P,\\{u_{i_{0}}\\})=0$, we obtain that

$\left.\frac{\mathrm{d}C_{\mathfrak{p}}(P_{t})}{\mathrm{d}t}\right|_{t=0}=0.$

Here we have used the fact that $\mu_{\mathfrak{p}}(P,\cdot)$ is absolutely
continuous with respect to $S(P,\cdot)$. Thus, $g(t)>0$ for sufficiently small
$t>0$, which yields that

$\frac{(1-t/d_{0})^{\gamma_{i_{0}}}}{C_{\mathfrak{p}}(P_{t})^{1/(n-\mathfrak{p})}}<1.$

Therefore, we arrive at

$\prod_{i=1}^{N}(h_{\lambda P_{t}}(u_{i})-\xi(\lambda P_{t})\cdot
u_{i})^{\gamma_{i}}<\prod_{i=1}^{N}(h_{P}(u_{i})-\xi(P_{t})\cdot
u_{i})^{\gamma_{i}},$

which implies $\Phi_{\mu,\lambda P_{t}}(\xi(\lambda
P_{t}))<\Phi_{\mu,P}(\xi(P_{t}))\leq\Phi_{\mu,P}(o)$. We now conclude that $P$
has exactly $N$ facets. ∎

## Acknowledgement

We want to thank Kyeongsu Choi for his interest in our work and valuable
comments. Minhyun Kim gratefully acknowledges financial support by the German
Research Foundation (GRK 2235 - 282638148). Taehun Lee was supported by a KIAS
Individual Grant (MG079501) at Korea Institute for Advanced Study

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# High pressure control of optical nonlinearity in the polar Weyl semimetal
TaAs

Chen Li  Department of Physics, California Institute of Technology, Pasadena,
CA 91125, USA  Institute for Quantum Information and Matter, California
Institute of Technology, Pasadena, CA 91125, USA  Xiang Li  Department of
Physics, California Institute of Technology, Pasadena, CA 91125, USA  T.
Deshpande  Department of Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Xinwei Li  Department of Physics, California
Institute of Technology, Pasadena, CA 91125, USA  Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena, CA
91125, USA  N. Nair  Department of Physics, University of California,
Berkeley, CA 94720, USA  Materials Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  J. G. Analytis  Department of
Physics, University of California, Berkeley, CA 94720, USA  Materials Sciences
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  D.
M. Silevitch  Department of Physics, California Institute of Technology,
Pasadena, CA 91125, USA  T. F. Rosenbaum  Department of Physics, California
Institute of Technology, Pasadena, CA 91125, USA  D. Hsieh<EMAIL_ADDRESS>Department of Physics, California Institute of Technology, Pasadena, CA 91125,
USA  Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA

###### Abstract

The transition metal monopnictide family of Weyl semimetals recently has been
shown to exhibit anomalously strong second-order optical nonlinearity, which
is theoretically attributed to a highly asymmetric polarization distribution
induced by their polar structure. We experimentally test this hypothesis by
measuring optical second harmonic generation (SHG) from TaAs across a
pressure-tuned polar to non-polar structural phase transition. Despite the
high pressure structure remaining non-centrosymmetric, the SHG yield is
reduced by more than 60% by 20 GPa as compared to the ambient pressure value.
By examining the pressure dependence of distinct groups of SHG susceptibility
tensor elements, we find that the yield is primarily controlled by a single
element that governs the response along the polar axis. Our results confirm a
connection between the polar axis and the giant optical nonlinearity of Weyl
semimetals and demonstrate pressure as a means to tune this effect $in$
$situ$.

The Weyl semimetal is a gapless three-dimensional phase of matter in which
spin-polarized valence and conduction bands touch at isolated points —Weyl
nodes —in the Brillouin zone Armitage  _et al._ (2018); Wan  _et al._ (2011);
Xu  _et al._ (2015); Lv  _et al._ (2015); Yang  _et al._ (2015). Weyl nodes
serve as sinks and sources of Berry curvature in momentum space and can thus
only be removed by pairwise annihilation, rendering them topologically stable.
The presence of Weyl nodes, which is only allowed in the absence of time-
reversal or inversion symmetry, has been shown to endow materials with exotic
DC transport properties including the chiral anomaly Zyuzin and Burkov (2012);
Zhang  _et al._ (2016); Jia  _et al._ (2016); Parameswaran  _et al._ (2014);
Huang  _et al._ (2015), anomalous Hall Yang  _et al._ (2011); Burkov (2014);
Sun  _et al._ (2016) and surface Fermi arc mediated cyclotron motion Moll  _et
al._ (2016); Koshino (2016); Li  _et al._ (2017a).

The transition metal monopnictide family of inversion broken Weyl semimetals
has garnered additional attention for its band structure geometry induced
second-order nonlinear AC response Orenstein  _et al._ (2021). Headlined by
TaAs, observations include strong light helicity-dependent injection Ma  _et
al._ (2017); Sirica  _et al._ (2019); Gao  _et al._ (2020) and helicity-
independent shift photocurrents Sirica  _et al._ (2019); Osterhoudt  _et al._
(2019); Ma  _et al._ (2019), as well as a record-setting shift current-induced
optical second harmonic generation (SHG) efficiency in the near-infrared
spectral range Wu  _et al._ (2017); Patankar  _et al._ (2018). This SHG
response has been attributed theoretically to a large skewness in the
polarization distribution originating from the polar structure of these
materials Patankar  _et al._ (2018); Li  _et al._ (2018). Experimentally
testing this hypothesis would not only advance our fundamental understanding
of the nonlinear optical properties of inversion broken Weyl semimetals, but
also illuminate new pathways to control them.

Figure 1: (a) Schematic of the diamond anvil cell (DAC) based SHG-RA setup.
Incident and reflected SHG light are shown in red and blue, respectively. The
angle between the polarization of the incident beam and the [1,1,-1] axis of
TaAs is defined as $\varphi$. (b) Ambient pressure SHG-RA data from GaAs (001)
obtained in a parallel polarization geometry through type Ia and type IIac
diamonds (blue lines), compared to data taken on the bare sample outside the
DAC (orange lines). The slight asymmetries in the SHG-RA patterns are due to
GaAs surface imperfections. (c) Microscope image showing a TaAs (112) crystal
sealed inside the DAC and the locations of ruby spheres for checking pressure
homogeneity. (d) Ambient pressure SHG-RA data from bare TaAs (112) obtained in
parallel and crossed polarization geometries. Figure 2: (a) Unit cell of
ambient pressure tetragonal ($I4_{1}md$) TaAs projected onto the (112)
surface. (b) Projection of high-pressure hexagonal TaAs ($P\overline{6}m2$)
onto the same surface. (c) SHG-RA patterns at select pressures measured upon
compression and (d) decompression after reaching a maximum pressure of 19.2
GPa. Pressure was tuned $in$ $situ$, allowing all patterns to be acquired
under identical alignment conditions. Horizontal and vertical dashed lines lie
along the [1,1,-1] and [1,-1,0] axes respectively. The loss of horizontal
mirror symmetry in the 14.9 GPa data can be seen from the presence of an
intensity dip at $\varphi\approx 100^{\circ}$ contrasted with hump at
$\varphi\approx 260^{\circ}$ (red arrows).

Powder x-ray diffraction measurements on TaAs Zhou  _et al._ (2016), supported
by density functional theory calculations Guo  _et al._ (2018); Gupta  _et
al._ (2018); Zhang  _et al._ (2017); Lu  _et al._ (2016), have shown that
hydrostatic pressurization above a critical value $P_{c}$ = 14 GPa induces a
phase transition from an inversion broken polar Weyl semimetal to an inversion
broken non-polar Weyl semimetal. In this Letter, we present a pressure-
dependent SHG rotational anisotropy (RA) study of single crystalline TaAs. We
observe a structural symmetry change above $P_{c}$ consistent with powder
x-ray diffraction data, which is accompanied by a sharp decrease in overall
SHG efficiency —approaching 60% at 20 GPa —despite the crystal symmetry
remaining inversion broken. By tracking the pressure dependence of different
components of the SHG susceptibility tensor, we show that this effect is
dominated by the response along the polar axis, in line with a previous
theoretical proposal. Moreover, we find evidence of reversibility upon slow
decompression, providing a pathway to $in$ $situ$ tunable optical nonlinearity
in Weyl semimetals.

Large hydrostatic pressures were applied using a 3-pin Merrill–Bassett diamond
anvil cell (DAC) Moggach  _et al._ (2008). Continuous $in$ $situ$ pressure
tuning was implemented using a helium gas driven membrane and monitored via
ruby fluorescence [Fig. 1(a)]. Use of low defect density type IIac diamond
(Almax easyLab) was essential for sufficiently reducing the SHG background to
allow isolation of reflected SHG light from the sample. A comparative study
conducted on a GaAs test sample shows that its intrinsic SHG-RA pattern is
hardly changed through a type IIac diamond, whereas it is completely obscured
through a conventional type Ia diamond [Fig. 1(b)]. Single crystals of TaAs
were grown by vapor transport Nair  _et al._ (2020). As-grown (112) surfaces
were identified for SHG measurements and polished from the backside to a
thickness of 20 $\mu$m. Crystals were loaded inside a pre-indented MP35N
gasket pressed between two diamonds with 600 $\mu$m culets [Fig. 1(c)]. A
mixture of methanol-ethanol (4:1) was used as the pressure medium. The sample
was maintained at room temperature for all measurements. Light from a
regeneratively amplified Ti:sapphire laser (1.5 eV photon energy, 100 kHz
repetition rate) was focused at normal incidence onto a 30 $\mu$m spot (FWHM)
on TaAs (112) with a fluence of 0.03 mJ/cm2. Retro-reflected SHG light was
reflected from a dichroic beamsplitter at a 45∘ angle of incidence onto a
photomultiplier tube. The difference in reflectance for $s$\- versus
$p$-polarized light from the dichroic beamsplitter was less than 2 %, below
our measurement noise level. Rotational anisotropy patterns were acquired by
co-rotating a pair of input and output linear polarizers Torchinsky  _et al._
(2014); Harter  _et al._ (2015). Data collected in both parallel and crossed
polarization geometries under ambient pressure agree closely with previously
reported results [Fig. 1(d)].

Under ambient pressure, TaAs crystallizes in a body-centered tetragonal
structure. Its $I4_{1}md$ space group (point group 4$mm$) includes a polar
axis along the $z$ direction and two mirror planes ($M_{x}$, $M_{y}$) and two
glide planes ($M_{xy}$, $M_{-xy}$) that contain the $z$ axis. On the (112)
surface, which is spanned by the [1,1,-1] and [1,-1,0] axes [Fig. 2(a)], the
$z$ axis and $M_{xy}$ are projected onto [1,1,-1]. This results in SHG-RA
patterns that are symmetric under reflection across the horizontally oriented
[1,1,-1] axis [Fig. 1(d)]. The high pressure phase is characterized by a non-
polar hexagonal structure (space group $P\overline{6}m2$, point group
$\overline{6}m2$) that breaks $M_{xy}$ symmetry Zhou  _et al._ (2016).
Therefore, the tetragonal-to-hexagonal phase transition can in principle be
detected through the loss of horizontal reflection symmetry in the SHG-RA
patterns [Fig. 2(b)].

Figure 3: Pressure dependence of the $\varphi$-integrated SHG intensity from
TaAs (112) measured upon compression for both parallel (blue squares) and
crossed (orange triangles) polarization geometries. Each curve is normalized
to its ambient pressure value. Error bars represent the intensity variation
measured across different sample locations. Gray curve is a guide to the eye.

Figure 2(c) shows SHG-RA data on TaAs (112) in a crossed polarization
geometry, acquired upon compression at an average rate of 1 GPa per hour. The
intensities at different pressures are normalized to unity in order to better
visualize the symmetry changes. At $P$ = 0.9 GPa, the sealing pressure of the
DAC, the pattern is nearly indistinguishable from ambient [Fig. 1(d)]. Upon
increasing pressure to 4.7 GPa, 9.1 GPa and 11 GPA, the intensity at 90∘ and
270∘ increases relative to the four major lobes, but the patterns continue to
exhibit reflection symmetry about the horizontal axis. Therefore, these
changes predominantly should arise from tetragonal symmetry-preserving lattice
distortions. In contrast, at 14.9 GPa and 19.2 GPa the patterns become six-
lobed and exhibit a clear loss of horizontal reflection symmetry, indicating a
transition into the hexagonal structure. This is consistent with a previous
high-pressure powder x-ray diffraction study Zhou  _et al._ (2016) that
assigned $P_{c}$ = 14.4 GPa and found phase coexistence over a finite pressure
range around $P_{c}$, which cannot be ruled out in our SHG data. However,
whereas the x-ray diffraction study showed that the hexagonal phase persists
upon decompression below $P_{c}$, we find that the tetragonal SHG-RA pattern
is restored upon decompression, although the intensity does not recover its
ambient pressure value. This demonstrates that the transition is hysteretic
but possibly reversible [Fig. 2(d)]. This discrepancy may be due to a
combination of our use of slow decompression over 8 hours, our lower peak
pressure (19.2 GPa as opposed to 53 GPa) and differences in the behavior of
single crystal versus powder samples.

A measure of the overall SHG conversion efficiency can be obtained by
integrating the SHG-RA intensity over $\varphi$ from $0$ to $2\pi$. Figure 3
shows the pressure dependence of the $\varphi$-integrated intensity in both
parallel and cross polarization channels. The intensity remains essentially
constant at low pressures and then starts to decrease above approximately 13
GPa, dropping to nearly 40% of the ambient pressure value by 19.2 GPa. The
fact that the intensity decrease occurs over a wide pressure window suggests a
gradual transfer of population from the tetragonal to the hexagonal structure,
consistent with powder x-ray diffraction measurements Zhou  _et al._ (2016).
Upon scanning the laser spot across different regions of the crystal, we
observe a slight variation in intensity, which is captured by the error bars
in Fig. 3. However, the overall trend versus pressure is the same. These
results indicate that the large second-order optical nonlinearity of the
tetragonal phase should be primarily attributed to the presence of a polar
axis rather than the absence of an inversion center, which is retained in the
hexagonal phase.

To directly evaluate the contribution of the polar axis to the SHG intensity,
we investigate the pressure dependence of the individual electric-dipole SHG
susceptibility elements $\chi_{ijk}$, which govern the relationship between
the incident electric field at frequency $\omega$ and the induced polarization
at 2$\omega$ via $P_{i}^{2\omega}=\chi_{ijk}E_{j}^{\omega}E_{k}^{\omega}$.
Under normal incidence and cross-polarized geometry, the most general form of
the electric-dipole SHG-RA intensity is given by
$I_{\perp}(\varphi)\propto|A_{1}\cos^{3}(\varphi)+A_{2}\cos^{2}(\varphi)\sin(\varphi)+A_{3}\cos(\varphi)\sin^{2}(\varphi)+A_{4}\sin^{3}(\varphi)|^{2}$
where $A_{1}\rightarrow A_{4}$ are complex numbers representing different
$\chi_{ijk}$ combinations. In the case where light is normally incident on the
(112) face of TaAs in the tetragonal $4mm$ phase, $A_{1}$ = $A_{3}$ = 0 by
symmetry, $A_{2}=\gamma(2a^{2}\chi_{zxx}+c^{2}\chi_{zzz}-2c^{2}\chi_{xzx})$
and $A_{4}=\gamma(2a^{2}+c^{2})\chi_{zxx}$, where
$\gamma=1/\left(\frac{2a^{3}}{c}+ac\sqrt{2+\frac{c^{2}}{a^{2}}}\right)$ and
$a$ and $c$ are the TaAs lattice constants Li  _et al._ (2022). Notably, the
SHG response along the polar axis $\chi_{zzz}$ is contained in $A_{2}$, which
was previously shown to be an order of magnitude larger than both $\chi_{zxx}$
and $\chi_{xzx}$ Wu  _et al._ (2017). In the hexagonal $\overline{6}m2$ phase,
$A_{1}\rightarrow A_{4}$ all become symmetry allowed.

Figure 4: (a) Representative SHG-RA data acquired in crossed polarization
geometry below and above $P_{c}$ (green circles). Fits to the expression for
$I_{\perp}(\varphi)$ given in the main text are overlaid as black lines. A
decomposition of the fits into its $A_{1}\rightarrow A_{4}$ components is
shown to the right. Filled versus empty lobes represent opposite signs of the
associated trigonometric function. The fitted amplitude of the various terms
is shown below, which are normalized to the maximum value of $A_{2}$ for
$P<P_{c}$. (b) Pressure dependence of the fitted amplitudes of $A_{2}$ (blue)
and $A_{3}$ (orange).

The SHG-RA pattern at each measured pressure was fitted to the general
expression for $I_{\perp}(\varphi)$. Figure 4(a) shows representative examples
below and above $P_{c}$, which demonstrates a high quality of fit, alongside
the fitted amplitudes of $A_{1}\rightarrow A_{4}$. As expected, for $P<P_{c}$
the fit is dominated by $A_{2}$ with an order of magnitude smaller
contribution from $A_{4}$. We also note a small contribution from $A_{1}$ that
is present even in data collected outside the DAC Li  _et al._ (2022),
possibly pointing to a slight departure of the ambient structure from $4mm$
symmetry. For $P>P_{c}$, all four terms become necessary in order to fit the
SHG-RA patterns, with the growth of $A_{1}$ and $A_{3}$ being responsible for
the loss of reflection symmetry about the horizontal axis. Moreover, the fit
is no longer dominated by $A_{2}$, resulting in SHG-RA patterns with reduced
anisotropy. The pressure dependence of $|A_{2}|$ and $|A_{3}|$ are plotted in
Fig. 4(b), which can be used to approximately track the order parameters
associated with the polar tetragonal phase and non-polar hexagonal phase,
respectively. We find that while neither $|A_{2}|$ nor $|A_{3}|$ exhibit
strong pressure dependence below $P_{c}$, they undergo clear downward and
upward trends respectively above $P_{c}$, with $A_{2}$ falling to less than 50
% of its ambient pressure value by 19.2 GPa. This reveals that the observed
drop in overall SHG intensity above $P_{c}$ (Fig. 3) is predominantly due to a
drastic reduction of $\chi_{zzz}$ induced by the loss of a polar axis in the
hexagonal structure.

These observations are qualitatively consistent with a recent theoretical
proposal that relates $\chi_{zzz}$ to the third cumulant of the polarization
distribution ($C_{3}$) Patankar  _et al._ (2018). In contrast to the first
moment ($C_{1}$), which describes the average macroscopic polarization,
$C_{3}$ characterizes the intrinsic asymmetry of the polarization
distribution, independent of the electronic center-of-mass. Although $C_{3}$
is generally allowed whenever inversion symmetry is broken, a simplified two-
band tight-binding model of TaAs showed that it is greatly enhanced by the
polar structure. Therefore, the observed decrease in $\chi_{zzz}$ above
$P_{c}$ in our experiments can possibly be attributed to a drop in $C_{3}$
induced by the loss of a polar axis. Verification of this hypothesis will
require detailed calculations of $C_{3}$ for realistic models of TaAs under
pressure. By extension, our work opens a route to manipulate the large second-
order nonlinearity of transition metal monopnictide Weyl semimetals by tuning
their polar order parameter with hydrostatic pressure. More generally, our
demonstration of high pressure SHG-RA from single crystalline TaAs provides
the means to search for symmetry breaking in the vicinity of high pressure
superconducting phases in non-centrosymmetric materials like TaP Li  _et al._
(2017b) or Cd2Re2O7 Harter  _et al._ (2017); Malavi  _et al._ (2016); Yamaura
_et al._ (2017), which may be key to understanding their topological
properties.

###### Acknowledgements.

We thank Darius Torchinsky, George Rossman, Jennifer Jackson and Liang Wu for
helpful discussions. High pressure optical second harmonic generation
measurements, as well as construction of the instrument, were supported by the
U.S. Department of Energy under Grant No. DE SC0010533. T.F.R. acknowledges
support from AFOSR grant No. FA9550-20-1-0263. X.L. acknowledges support from
an IQIM postdoctoral fellowship. Work by N.N. and J.G.A. was supported by the
National Science Foundation under Grant No. 1905397.

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Supplemental Material
“High pressure control of optical nonlinearity in the polar Weyl semimetal
TaAs”

##  I I. SHG-RA data in parallel polarization geometry

Figure 5: SHG-RA patterns at select pressure in parallel polarization geometry
measured upon compression.

The most general form of the electric-dipole SHG-RA intensity in parallel
polarized geometry under normal incidence is given by
$I_{\parallel}(\varphi)\propto|B_{1}\cos^{3}(\varphi)+B_{2}\cos^{2}(\varphi)\sin(\varphi)+B_{3}\cos(\varphi)\sin^{2}(\varphi)+B_{4}\sin^{3}(\varphi)|^{2}$,
similar to the case for crossed polarized geometry described in the main text.
From the (112) face of TaAs in the tetragonal $4mm$ phase, $B_{2}$ = $B_{4}$ =
0 by symmetry,
$B_{1}=\gamma(2a^{2}\chi_{zxx}+4a^{2}\chi_{xzx}+c^{2}\chi_{zzz})$ and
$B_{3}=\gamma(4a^{2}+2c^{2})\chi_{xzx}+(2a^{2}+c^{2})\chi_{zxx}$, where
$\gamma=1/\left(\frac{2a^{3}}{c}+ac\sqrt{2+\frac{c^{2}}{a^{2}}}\right)$ and
$a$ and $c$ are the TaAs lattice constants.

Figure S1 shows SHG-RA patterns at various pressures measured in parallel
polarization geometry. For all pressures, the patterns are dominated by the
$\chi_{zzz}$ containing $B_{1}$ term, which was previously shown to be an
order of magnitude larger than both $\chi_{zxx}$ and $\chi_{xzx}$ Wu  _et al._
(2017). Nonetheless, careful examination of the patterns shows that the nodes
at 90∘ and 270∘ are lifted at high pressure, evidencing appearance of a
$B_{4}$ contribution that is only allowed in the hexagonal phase.

##  II II: Departure from 4$mm$ point group at ambient pressure

Figure 6: Typical cross-polarized SHG-RA pattern from TaAs (112) acquired at
ambient pressure outside the DAC (left). A zoom-in (middle) shows a lifting of
the nodes at 0∘ and 180∘, indicating a finite $A_{1}$ term. Fits to the
general form for $I_{\perp}(\varphi)$ are overlaid as black lines. A
decomposition of the fits into its $A_{1}\rightarrow A_{4}$ components is
shown to the right. Filled versus empty lobes represent opposite signs of the
associated trigonometric function. The fitted amplitude of the various terms
is shown below, which are normalized to $A_{2}$ for $P<P_{c}$.

For an ideal tetragonal crystal with point group 4$mm$, the $A_{1}$ term in
$I_{\perp}(\varphi)$ is forbidden by symmetry (see main text) and the pattern
should exhibit nodes at 0∘, 90∘, 180∘ and 270∘. Yet we observe a small but
finite $A_{1}$ component in our ambient pressure SHG-RA patterns taken outside
the DAC 6, manifested as a lifting of the nodes at 0∘ and 180∘. Our simulation
results show that this is not due to misalignment, implying that the crystals
may exhibit a slight departure from 4$mm$ symmetry.

##  III III: Pressure dependence of $A_{1}$ and $A_{4}$

The SHG-RA patterns measured in crossed polarized geometry were fit to
$I_{\perp}(\varphi)\propto|A_{1}\cos^{3}(\varphi)+A_{2}\cos^{2}(\varphi)\sin(\varphi)+A_{3}\cos(\varphi)\sin^{2}(\varphi)+A_{4}\sin^{3}(\varphi)|^{2}$.
The fitted amplitudes of $A_{2}$ and $A_{3}$ are shown in the main text.
Figure S3 shows the pressure dependence of the fitted amplitudes of $A_{1}$
and $A_{4}$. Subtle kinks appear in $|A_{1}|$ and $|A_{4}|$ around the
critical pressure $P_{c}$. However, as expected, $|A_{4}|$ does not change as
drastically as $|A_{2}|$ because $A_{4}$ does not contain $\chi_{zzz}$. Also,
as expected, $|A_{1}|$ does not exhibit a clear a kink at $P_{c}$ as $|A_{3}|$
because $A_{1}$ is present in both the low and high pressure phases.

Figure 7: Pressure dependence of the fitted amplitudes of $A_{1}$ (green) and
$A_{4}$ (purple) normalized to the maximum value of $A_{2}$ for $P<P_{c}$

## References

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# Numerical Approximation Methods for Antenna Radiation Patterns for Motus
Wildlife Tracking Systems

Erik Carlson, Douglas Gobeille
Department of Physics
University of Rhode Island
Kingston, RI
{erikvcarlson<EMAIL_ADDRESS>
Robert Deluca
Physics Systems LLC
Xenia, OH
<EMAIL_ADDRESS>
Pam Loring
Division of Migratory Birds
US Fish and Wildlife Service
Charlestown, RI
<EMAIL_ADDRESS>
###### Abstract

As plans for offshore wind energy development increase in the US, the
developing methods to monitor migratory birds and bats offshore is an
important area of research. To contribute to this research, current guidance
recommends the deployment of Motus Wildlife Tracking stations by each
developer for pre- and post- construction monitoring. To understand the
characteristics of each of the stations, calibration techniques are
recommended for each deployment. Despite this, previous attempts to calibrate
these stations has failed to provide sufficient detail to allow for high-
resolution tracking techniques. In this paper, we introduce an affordable and
robust methodology to calibrate stations in the off-shore environment and
develop processes to turn this raw calibration data into a numerical
approximation of the antennas radiation pattern.

_K_ eywords Motus Wildlife Tracking System $\cdot$ RF Calibration $\cdot$ Off-
Shore Wind

##  1 Introduction

With the increase in off-shore wind (OSW) lease areas in the last several
years, monitoring offshore bird and bat species has become an issue of upmost
importance. For small aerofauna ($<150$ g), technologies traditionally
utilized for monitoring individual movements utilizing GPS tracking are
limited due to the nature of their weight. Following the work of Loring
(2016), digital VHF transmitters (RF-Tags) were identified as a safe
alternative for monitoring small-bodied species including the federally
threatened Piping Plover (Charadrius melodus) and the federally endangered
Roseate Tern (Sterna dougallii). This work is conducted in coordination with
the Motus Wildlife Tracking System (Motus) an international automated radio

While these techniques were successful at tracking the movements of these
aerofauna on a regional level, more precise tracking required for monitoring
these aerofauna relative to OSW lease areas are required to directly monitor
site specific movements in the offshore areas. To accomplish this, the
intersection of near-simultaneous detections from $>3$ stations was purposed
(Bai, 2018; Janaswamy et al., 2018). While the underlining theory behind this
methodology is correct, the utilization of theoretical radiation patterns
fails to take into account the nuances of each stations deployed environment.
For example, as the station ages and dielectric material builds up on the
antenna, the radiation pattern of each deployed antenna will change
dramatically from the theoretical pattern. Additionally, the theoretical
pattern does not account for site-specific factors including antenna blinds
spots, reflections and other interference which must be quantified
numerically.

This paper’s goal, in support of upcoming federal guidance documents, is to
outline the methodology and analysis of calibration data of OSW and land-based
Motus Stations in an effort to quantify and standardize methods to produce
numerical approximation the radiation pattern of yagi and omni-direction
antennas.

##  2 Methods

The primary method for producing the models for our radiation patterns
revolves around co-locating GPS locations with RF-Tag detections. Although
various types of calibration data collection have been used in previous
efforts, there is a need to standardized key elements including in the GPS and
tags temporal resolution, power regulation and final data analysis in order to
apply the method consistently and collect the amount and quality of data
needed to estimate radiation patterns.

###  2.1 Pre-Calibration Equipment

To test our calibration methods, three stations were deployed and tested
throughout this investigation. Each station was 20ft tall and was deployed
with four horizontally mounted circularly polarized yagi-uda antennas and one
vertically mounted omni-direction antenna, all operating at 434 MHz. One
station was located at a local residence for initial testing and to represent
a deployed land-based station. Two stations were placed at Block Island, Rhode
Island at South East Lighthouse (SEL) and Black Rock Beach (BRR) to represent
shore-based monitoring systems. Finally, one station was placed at the Block
Island Wind Farm (BIWF) Turbine 1 to represent a deployed station in an off-
shore environment. The locations of each of these stations is located in Table
1

Station Name | Latitude | Longitude
---|---|---
South East Lighthose | 41.1534 | -71.5521
Black Rock Beach | 41.1479 | -71.5901
BIWF Turbine 2 | 41.1418 | -71.5302
Table 1:  Station Locations

A standard Cell Track Tech (CTT) PowerTag operating at 434 MHz was utilized
for all of our calibration efforts with a one-second ping rate to correspond
with the ping rate of our GPS. These tags were also outfitted with larger
batteries and activators to conserve battery life in between calibrations.
Utilizing a PowerTag over the CTT LifeTag ensured a constant ping strength
over the course of the calibration cycle.

To provide GPS location, we utilized a Garmin Edge 130. This device provided
high frequency (1 Hz) GPS locations and relied on barometric pressure readings
to provide accurate elevation information. It was also light enough to be used
in all successful calibration methods. Finally, this device was not cost
prohibitive for station operators to purchase and utilize in the field for
their own calibration studies.

###  2.2 Traditional Calibration Methods

Traditional calibrations methods, such as those used in Janaswamy et al.
(2018), attached a test tag to a partially frozen bird carcass that was
suspended from a kite and flown from the back of a boat. During Janaswamy’s
calibration surveys, the boat was approximated to travel at a constant speed
and transmitter at an approximate height of $30$ m. This approximation
introduced large errors into the calibration data. Additionally, a boat hook
with a test tag was attached to the hull of the boat and carried throughout
their calibration process to approximate the radiation pattern at an assumed
$3$ m ASL (above sea-level).

To continue the theme of inexpensive, minimal effort calibration efforts, we
built on and improved the techniques used in Janaswamy et al. (2018). First,
we utilized a non-conducting pole which lowered the interference of the tag’s
signal. Secondly, we adhered a barometric GPS to our kite which allowed for
precise measurement of the tag’s altitude and position.

###  2.3 Small Unmanned Aerial System Operations

Throughout this effort EC conducted Unmanned Aerial System (UAS) operations
utilizing a DJI Mavic Air 2 with a co-located GPS and RF-Tag. Flying this UAS
around each of the stations provided detailed calibration information between
$0$ m AGL (above ground-level) to $122$ m AGL. These flights are a far
superior method for collecting detailed altitude information throughout the
study as they allow for fine control of the position and altitude of the UAS.
This fine control allows the user to fly the UAS precisely into areas where
previous calibration data was not collected, thus optimizing the time and cost
of each calibration survey.

###  2.4 Aviation Calibration

To gather high ($>300$ m AGL) altitude information about each station, small
aviation aircraft were utilize to conduct flights over the stations in the
study. These flights, conducted by with a Cessna 182, provided similar
calibration information to UAS operations. However, due to the stall speed of
the Cessna 182, high resolution calibration was not possible. This was abated
by utilizing the winds aloft to lower the overall ground speed of the
aircraft.

##  3 Results and Analysis

Five Calibration surveys were conducted over the course of 2021, their
specific properties111Raw Calibration data is located at:
https://tinyurl.com/BIWFCalData and tracks are located in Table 2 and Figure
3.

Date | Calibration Type | GPS Points Collected
---|---|---
Apr 17 | UAS | 4,253
Aug 30 | Traditional | 18,715
Aug 31 | Traditional | 10,864
Sep 18/19 222Calibration started on September 18 and continued past 0:00 UTC into September 19 | Aviation | 13,543
Sep 26 | Aviation | 12,277
Sep 27 | Aviation | 14,862
Table 2:  Calibration Event Properties

Upon the completion of the calibration runs, test tag data was downloaded from
the Motus database for each of the land-based stations and was manually
collected via USB drive from the station deployed in the Block Island Wind
Farm. The tag data was first filtered to exclude all tag data from tags not
used specifically for calibration. We then identified simultaneous detections
of the GPS and tag, allowing us to assign a tag signal strength to a given
location using the timestamps of each tag detection and GPS ping. Knowing the
position of each station allowed us to transition our GPS detections to a
relative position to each antenna. We present the results of this co-location
for each of the three stations in Figure 3. To continue our analysis of any
given station, we utilized k-nearest neighbor regression to estimate a signal
strength for arbitrary points around each station. To do this we generated
100,000 random points within a 20 km by 30 km by 1.5 km volume assigning
values for the RSSI utilizing scikit-learn’s K-neighbors regressor with 5
neighbors (Pedregosa et al., 2011). We present the results of this analysis in
Figure 4.

As we can see from the results, most of our top down images have defined
structure along the positive y-axis of each calibration run. This is
particularly prominent in the South East Lighthouse approximation which shows
the a clearly defined elliptical structure stemming from the origin and
protruding to approximately 15km away. This structure aligns with theoretical
approximates of the primary beam of a standard yagi-antenna beam pattern.

Although not obvious in the images below, each station presented some
elliptical structure in line with what would be the putative primary beam of
each antenna when the scaling is reduced to only include the strongest
detections. We stipulate that the South East Lighthouse Station best presented
this structure due to the targeted UAS calibration flights directly within the
primary beam while other surveys over or under flew the primary radiation
pattern resulting in an overall lesser number of strong detections.

No station was able to produce any type of structure in the edge-on plane.
This is due to the fact that our vertical calibration plane remained poorly
sampled despite our attempts to utilize various calibration methods. We
present our data density in the vertical plane in Figure 1.

Figure 1: Detected Points from South East Lighthouse in the Vertical Plane

To further examine our data, we examine how well each station model reproduced
the calibration data for its own station as well as the calibration data for
the other stations. The results of which are presented in Table 3.

Station 1 | Station 2 | Mean Error (dBm) | Standard Deviation (dBm)
---|---|---|---
Black Rock | BIWF Turbine | 5 | 3
Black Rock | Black Rock | 3 | 2
Black Rock | South East Lighthouse | 7 | 6
South East Lighthouse | South East Lighthouse | 6 | 5
BIWF Turbine | South East Lighthouse | 7 | 2
BIWF Turbine | BIWF Turbine | 8 | 8
Table 3:  Data Reproduction Errors

Of all the stations tested the Black Rock Station Model had the best results
when reproducing the calibration data for its own station. This is most likely
due to the absence of any large conducting materials around the station as
well as clear lines of sight to the horizon. Both the South East Lighthouse
and the BIWF station had large reflectors around each station causing
increased error.

Finally, we see that no two station’s models reproduce each other. This is
expected given the unique nature of each station’s unique environment and
demonstrates the need for numerical methods for calibrating Motus stations.

##  4 Conclusion

Through this paper, we have introduced the methodology to generate numerical
approximations of radiation patterns for Motus Wildlife Tracking Systems. We
have demonstrated that this method can approximate the received signal
strength of a tag to within an average of 8 dbm. Given the nature of decibel-
milliwatts, this represents a small error for detections close to the station
and a much larger error farther away from the station. To quantify this error
into meters, more calibration data is needed. This methodology, in conjunction
with a the analytical approach in Janaswamy et al. (2018) are being used to
approximate the effectiveness of Motus tracking stations in Off-Shore Wind
Farm application Adams et al. (in preparation). We have also identified the
time intensive nature of high-resolution calibration surveys. Future work will
aim to reduce the overall survey effort required to conduct these surveys
while producing more robust altitude information. This work will be used in
the generation of a tracking algorithm utilizing the intersections of
tessellated radiation patterns produced by this approach.

## Acknowledgments and Disclaimers

This study was funded by the New York State Energy Research and Development
Authority under Agreement No. 143771. NYSERDA has not reviewed the information
contained herein, and the opinions expressed in this report do not necessarily
reflect those of NYSERDA or the State of New York. Any use of trade, firm, or
product names is for descriptive purposes only and does not imply endorsement
by the US Government. The findings and conclusions in this article are those
of the author(s) and do not necessarily represent the views of the US Fish and
Wildlife Service. EC was also supported by the NASA Rhode Island Space Grant
Consortium during this work. We thank Kate Williams, Evan Adams, and Andrew
Gilbert at the Biodiversity Research Institute for their insights. For pilot
work on Block Island, we thank Block Island Wind Farm (Orsted), Scott Comings
(The Nature Conservancy), Lisa Nolan (Southeast Lighthouse Foundation), and
Brett Still (University of Rhode Island). For calibration surveys, we thank
Suzanne Paton (U.S. Fish and Wildlife Service Coastal Program), Tom Halavik,
Nathan Fueller, and Rhode Island Civil Air Patrol.

## References

  * Adams et al. (in preparation) Evan Adams, Andrew Gilbert, Kate Williams, Erik Carlson, Douglas Gobeille, Stuart MacKenzie, and Pamela Loring.  Study design guidance for tracking aerofauna offshore with the motus network.  in preparation.
  * Bai (2018) Hua Bai.  Estimation of Parameters in Avian Movement Models.  _arXiv e-prints_ , art. arXiv:1810.02416, October 2018.
  * Janaswamy et al. (2018) Ramakrishna Janaswamy, Pamela Loring, and James McLaren.  A State Space Technique for Wildlife Position Estimation Using Non-Simultaneous Signal Strength Measurements.  _arXiv e-prints_ , art. arXiv:1805.11171, May 2018.
  * Loring (2016) Pamela H. Loring.  _Evaluating Digital VHF Technology to Monitor Shorebird and Seabird Use of Offshore Wind Energy Areas in the Western North Atlantic_.  PhD thesis, Doctoral Dissertations, 2016.
  * Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay.  Scikit-learn: Machine learning in Python.  _Journal of Machine Learning Research_ , 12:2825–2830, 2011\.

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(a) Boat Calibration Track

(b) UAS Calibration Track

(c) Aircraft Calibration Track

Figure 2:

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(a) South East Lighthouse Detections

(b) Black Rock Detections

(c) Turbine Detections

Figure 3:

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(a) South East Lighthouse Top-Down

(b) South East Lighthouse Side-On

(c) Black Rock Station Top-Down

(d) Black Rock Station Side-On

(e) Turbine Station Top-Down

(f) Turbine Station Side-On

Figure 4: K-Means Results Colorized by dBm with Antenna Beam Pointed towards
the Positive Y-Axis