# Ergodicity in some families of Nevanlinna Functions

Tao Chen, Yunping Jiang and Linda Keen  Tao Chen, Department of Mathematics,
Engineering and Computer Science, Laguardia Community College, CUNY, 31-10
Thomson Ave. Long Island City, NY 11101 and CUNY Graduate Center, New York, NY
10016<EMAIL_ADDRESS>Yunping Jiang, Department of Mathematics, Queens
College of CUNY, Flushing, NY 11367 and Department of Mathematics and the CUNY
Graduate Center, New York, NY 10016<EMAIL_ADDRESS>Linda Keen,
Department of Mathematics, the CUNY Graduate School, New York, NY 10016
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study Nevanlinna functions $f$ that are transcendental meromorphic
functions having $N$ asymptotic values and no critical values. In [KK] it was
proved that if the orbits of all the asymptotic values have accumulation sets
that are compact and on which $f$ is a repeller, then $f$ acts ergodically on
its Julia set. In this paper, we prove that if some, but not all of the
asymptotic values have this property, while the others are prepoles, the same
holds true. This is the first paper to consider this mixed case.

###### 2010 Mathematics Subject Classification:

Primary: 37F10, 30F05; Secondary: 30D05, 37A30

This research is partially supported by gifts from the Simons Foundation
(#523341 and #942077) and PSC-CUNY awards. It was also supported by the
National Science Foundation under Grant No. 1440140, while the third author
was in residence at the Mathematical Sciences Research Institute in Berkeley,
California, during the spring semester 2022

##  1\. Introduction

An early result of McMullen [M] says that if $f$ is a rational map of degree
greater than one, and if $P(f)$ is its post-singular set, one of two things
holds: either $f$’s Julia set is equal to the whole Riemann sphere and the
action of $f$ is ergodic or, it is not the whole sphere and the spherical
distance $d(f^{n}(z),P(f))\rightarrow 0$ for almost every $z$ in $J(f)$ as
$n\to\infty$, that is, the $\omega$-limit set $\omega(z)$ is a subset of
$P(f)$ that varies with $z$. Bock [B2] proved a similar result holds for
meromorphic functions. This begs the question: what are the conditions on a
meromorphic function such that its Julia set is the whole sphere and the
meromorphic function acts on the sphere is ergodic or not? In the realm of
entire functions, Lyubich [Lyu] proved the exponential function $e^{z}$, whose
Julia set is the sphere, is not ergodic and Bock [B1] proved that if the set
of singular values of an entire function is finite, and all of these are pre-
periodic but not periodic, then the map is ergodic. In the realm of
meromorphic functions, Bock, [B2], (see also [RVS, Theorem 3.3] for a proof of
Bock’s theorem), proved that if the “radial Julia set”, a subset of the Julia
set, has positive measure, the action is ergodic. Other earlier results dealt
with the particularly simple example of meromorphic functions with two
asymptotic values and no critical values. There are partial results on the
ergodicity question for this family: Let $\lambda,\mu\in\mathbb{C}$, and

$f=\frac{\lambda e^{z}-\mu e^{-\mu}}{e^{z}-e^{-z}},$

where $\lambda,\mu$ are $f$’s two asymptotic values. Keen and Kotus [KK] have
shown that if the accumulation sets of both $\lambda$ and $\mu$ are compact,
and $f$ is a repeller on this set, then the Julia set is
$\widehat{\mathbb{C}}$ and $f$ is ergodic. By way of contrast, Skorulski [S1,
S2] has shown that if there exist natural numbers $p$ and $q$ such that
$f^{p}(\lambda)=f^{q}(\mu)=\infty$, then the Julia set is
$\widehat{\mathbb{C}}$ and $f$ is non-ergodic. (See also [CJK].)

Weiyuan Qiu asked one of the authors what happens in the remaining case where
one asymptotic value lands on a repelling cycle, and the other is a prepole.
In answering his question, we were able to prove a more general result for the
full family of functions with finitely many asymptotic values and no critical
values, so-called ”Nevanlinna functions”. Our main theorem is

######  Main Theorem.

If $f$ is a Nevanlinna function with $N$ asymptotic values of which $0<K<N$
are prepoles, and if the $\omega$-limit sets of the remaining $N-K$ are
compact repellers, then the Julia set is $\widehat{\mathbb{C}}$ and $f$ is
ergodic.

######  Remark 1.1.

Our proof of this theorem implies that for these Nevanlinna functions, the
measure of the radial Julia set is positive.

The case $K=0$ was analyzed in [KK]. For the case $K=N$, we have the following
conjecture which we are still working on and will report on in a future paper.

######  Conjecture 1.

When $K=N$, the action of $f$ on its Julia set $\widehat{\mathbb{C}}$ is not
ergodic.

The proof of our theorem depends on generalizations of some lemmas in [KK].
After an introductory section in which we give the basic definitions and
properties of Nevanlinna functions, we state and prove these lemmas and apply
them to the proof of the theorem.

Acknowledgement: We would like to thank Professor Janina Kotus for her helpful
comments and suggestions and for pointing out Skorulski’s papers [S1, S2] to
us.

##  2\. Preliminaries

In this section, we recall some of the basic theory of transcendental
meromorphic functions which we will need. Such a function,
$f:{\mathbb{C}}\rightarrow\widehat{{\mathbb{C}}}$ is holomorphic except at the
set of poles, $\\{f^{-1}(\infty)\\},$ and is a local homeomorphism everywhere
except at the set $S_{f}$ of singular points. In this paper, we will be
interested in those functions for which $\\#S_{f}$ is finite and will assume
this throughout. For such functions, the singular values are of two types:
Let $v$ be a singular value and let $V$ be a neighborhood of $v$. Then

  * •

If, for some component $U$ of $f^{-1}(V)$, there is a $u\in U$ such that
$f^{\prime}(u)=0$, then $u$ is a critical point and $v=f(u)\in V$ is the
corresponding critical value, or

  * •

If, for some component $U$ of $f^{-1}(V)$, $f:U\rightarrow V\setminus\\{v\\}$
is a universal covering map then $v$ is a logarithmic asymptotic value. We
will drop the descriptor logarithmic and call such values asymptotic values.

At regular, or non-singular points, meromorphic functions are local
homeomorphisms. The Koebe distortion theorems give estimates on the behavior
of these functions at regular points. Many proofs exist in the standard
literature on conformal mapping. (See e.g. [A], Theorem 5.3.) Since we will
use them repeatedly below we state them here without proof.

######  Theorem 1 (Koebe Distortion Theorem).

Let $f:D(z_{0},r)\to\mathbb{C}$ be a univalent function, then for any
$\eta<1$,

  1. (1)

$\displaystyle|f^{\prime}(z_{0})|\frac{\eta
r}{(1+\eta)^{2}}\leq|f(z)-f(z_{0})|\leq|f^{\prime}(z_{0})|\frac{\eta
r}{(1-\eta)^{2}}$, $z\in D(z_{0},\eta r)$,

  2. (2)

If $\displaystyle T(\eta)=\frac{(1+\eta)^{4}}{(1-\eta)^{4}}$,
$\displaystyle\frac{|f^{\prime}(z)|}{|f^{\prime}(w)|}\leq T(\eta)$, for any
$z,w\in D(z_{0},\eta r)$.

######  Theorem 2 (Koebe 1/4 Theorem).

Let $f:D(z_{0},r)\to\mathbb{C}$ be a univalent function, then

$D\Big{(}f(z_{0}),\frac{r|f^{\prime}(0)|}{4}\Big{)}\subset f(D(z_{0},r)).$

Meromorphic functions with finitely many critical points and finitely many
asymptotic values can be characterized by their Schwarzian derivatives.

######  Definition 1.

If $f(z)$ is a meromorphic function, its Schwarzian derivative is

$S(f)=(\frac{f^{\prime\prime}}{f^{\prime}})^{\prime}-\frac{1}{2}(\frac{f^{\prime\prime}}{f^{\prime}})^{2}.$

The Schwarzian differential operator satisfies the condition

$S(f\circ g)=S(f)g^{\prime 2}+S(g)$

from which it is easy to deduce that if $f$ is a Möbius transformation,
$S(f)=0$, so that $f\circ g$ and $g$ have the same Schwarzian derivative.

In [N], Chap. XI, §3, Nevanlinna, using a technique he calls rational
approximation, shows how to, given a finite set of points in the plane, and
finite or infinite branching data for these points, construct a meromorphic
function whose topological covering properties are determined by this data.
The function is defined up to Möbius transformations. He proves

######  Theorem 3.

The Schwarzian derivative of a meromorphic function with finitely many
critical points and finitely many asymptotic values is a rational function. If
there are no critical points, it is a polynomial. Conversely, if a meromorphic
function has a rational Schwarzian it has finitely many critical points and
finitely many asymptotic values.

In the literature, meromorphic functions with polynomial Schwarzian are called
Nevanlinna functions. These are the focus of this paper.

In order to prove our results, we will need estimates on the asymptotic
behavior of the poles and residues of Nevanlinna functions. These are well
known and there is an extensive literature, see e.g. [H], Chap. 5 or [L],
Chap. 4, [C] for details. Here, we state those properties of the functions
that we will need and sketch their derivation. We begin by recalling the
connection between Nevanlinna functions and the second order differential
equation,

(1) $w^{\prime\prime}+P(z)w=0,$

where $P(z)$ is a polynomial of degree $m$. The solutions of (1) are
holomorphic and form a two dimensional linear space. It is easy to verify that
if $w_{1},w_{2}$ are linearly independent solutions of (1) then
$f=w_{1}/w_{2}$ is meromorphic, has $N=m+2$ asymptotic values and satisfies
$S(f)=2P$. The following is a summary of the properties of solutions of (1)
and $S(f)=2P$.

Using the Liouville transformation

$W(Z)=P(z)^{1/4}w(z),\,Z=\int^{z}P(s)^{1/2}ds,$

we obtain a new equation in $W$ of the form

(2) $W^{\prime\prime}(Z)+(1-F(Z)W(Z)=0,\,\mbox{ where
}F(Z)=\frac{1}{4}\frac{P^{\prime\prime}(z)}{P(z)^{2}}-\frac{5}{16}\frac{P^{\prime}(z)^{2}}{P(z)^{3}}.$

Let ${\mathcal{S}}$ be the sector of the $Z$ plane defined by $|Z|>R$ and
$|\arg Z|<\pi-\delta$ for some large $R$ and some $\delta$ in $(0,\pi)$;
there, $F(Z)=\rm{O}(1/Z^{2})$ and there are linearly independent solutions
with the asymptotic expressions

(3) $W_{1}(Z)=e^{iZ}(1+\rm{O}(1/|Z|))\mbox{ and
}W_{2}(Z)=e^{-iZ}(1+\rm{O}(1/|Z|)).$

From these we obtain two linearly independent “principal solutions”

$w_{i}(z)=P(z)^{-1/4}W_{i}(Z),\,\,i=1,2,$

of the original second order equation, equation (1), defined in a sector $S$
of the $z$ plane satisfying,

$S=\\{z\,|\,|z|>R,\,\,|\arg z-\theta_{0}|<\frac{2\pi}{N}-\delta^{\prime}\\}$

where $R$ is a large constant, $\delta^{\prime}$ is a small constant depending
on $\delta$, $a$ is the leading coefficient of $P(z)$ and $\theta_{0}$ is a
solution of $\arg a+(2\pi/N)\theta_{0}=0\mod 2\pi$. The rays
$z=te^{i\theta_{0}},t>0$ are called a Julia rays. Note that there are $N$
Julia rays, equally spaced in the plane. Each sector contains the Julia ray
$te^{i\theta_{0}}$ and is contained between the rays
$te^{i(\theta_{0}+2\pi/N)}$ and $te^{i(\theta_{0}-2\pi/N)}$ , $t>0$. See
figure 1.

Figure 1. Principal solutions in sectors of the $z$-plane

The following lemma can be proved using standard techniques (see e.g. [L],
Lemma 4.3.6). We omit the proof.

######  Lemma 4.

The function $Z=Z(z)$ satisfies

$Z(z)=\frac{2a^{\frac{1}{2}}}{N}z^{\frac{N}{2}}(1+\rm{o}(1))\,\text{ as
}z\to\infty\text{ in }S$

and, for each Julia ray $te^{i\theta_{k}}$, $k=1,\ldots,N$, $Z$ is univalent
on a sector of the $z$-plane, $S_{k}=\\{z\,|\,|z|>R_{1},\,|\arg
z-\theta_{k}|<\frac{2\pi}{N}-\delta^{\prime\prime}\\}$ where $R_{1}>R$ and
$\delta^{\prime\prime}>\delta^{\prime}$. Moreover, $Z$ maps $S_{k}$ onto a
region containing the sector ${\mathcal{T}}=\\{Z\,|\,|Z|>R_{2},|\arg
Z|<\pi-\sigma\\}$ contained in ${\mathcal{S}}$, where $R_{2}$ is large and
$\sigma>N\delta/2$.

The asymptotic expressions in equation (3) show that each of the rays $Z=\pm
it+\rm{O}(1/t),t>R$ are asymptotic paths for the respective (distinct)
asymptotic values of $W_{i}$. Their images under the inverse branches of $Z$
are thus asymptotic paths in the asymptotic tracts for the principal solutions
$w_{i}$. Therefore the function $f=w_{1}/w_{2}$ has $N$ asymptotic values and,
moreover, the asymptotic values in adjacent tracts are distinct.

Equation (3) also shows that the $w_{i}$ have no zeros in the sectors where
they are defined but that, for any $A,B\in\mathbb{C}^{*}$, the equation
$Aw_{1}-Bw_{2}=0$ has infinitely many zeros. We next show that these zeroes
accumulate along the Julia rays and are thus contained in a subsector of $S$
of width $2\delta^{\prime}$ containing the Julia ray. This will imply that the
same is true for the poles of

$f=\frac{Aw_{1}-Bw_{2}}{Cw_{1}-Dw_{2}},A,B,C,D\in\mathbb{C}^{*}$

and the zeros of $f(z)-z_{0}$ for any $z_{0}\in{\mathbb{C}}$.

######  Proposition 5.

If $A,B$ are non-zero constants and $w=Aw_{1}-Bw_{2}$ then $w=0$ has
infinitely many solutions $s_{j}$. Label them so that
$\cdots\leq|s_{j}|\leq|s_{j+1}|\leq\cdots$. Then, along each Julia ray
$te^{i\theta_{k}}$, $k=1,\ldots,N$, $|s_{j}|\sim\text{O}(|j|^{2/N})$ and
$\lim_{j\to\infty}|\arg s_{j}-\theta_{k}|=0$,

###### Proof.

Set $G(z)=\frac{1}{2i}\log\frac{W_{1}}{W_{2}}$. By lemma 4, if $z\in S$ is
sufficiently large, there is a constant $c_{0}$ such that

$G(z)=Z(z)+\text{o}(1)\sim\frac{2c_{0}^{\frac{1}{2}}}{N}z^{\frac{N}{2}}(1+\text{o}(1)).$

Furthermore, the zeroes $s_{j}$ of $w$, $j\in{\mathbb{Z}}$, satisfy

$2iG(s_{j})=\log(\frac{B}{A})+2j\pi i.$

If $z\in S$, so that $|\arg Z|<\pi-\sigma$, there is a constant $c_{1}$ such
that for each zero $s_{j}$ of $w$, $|s_{j}|\sim c_{1}|j\pi|^{2/N}$, $\arg
G(s_{j})\sim\rm{o}(1)$. Hence as $j\to\infty$, the real part dominates so that
$\arg s_{j}\sim\theta_{0}$; that is, there is a constant $c_{2}$ such that
$s_{j}\sim c_{2}|j|^{2/N}$ and, as $j\to\infty$, $|\arg s_{j}-\theta_{k}|<\mu$
for some small $\mu$; moreover, $\mu\to 0$ as $R\to\infty$.

∎

Write the function $f$ as:

(4)
$f(z)=\frac{aw_{1}+bw_{2}}{cw_{1}+dw_{2}}=\frac{aW_{1}+bW_{2}}{cW_{1}+dW_{2}}=\frac{ag(Z)+b}{cg(Z)+d}$

where $ad-bc=1$ and $g(Z)=W_{1}/W_{2}\sim e^{2iZ}$ so that $g^{\prime}(Z)\sim
2ig(Z)$.

######  Proposition 6.

Let $f$ be as in equation (4). Denote the poles of $f$ and their respective
residues by $s_{j}$ and $r_{j}$, and assume the poles are labelled so that
$\cdots\leq|s_{j}|\leq|s_{j+1}|\leq\cdots$. Then

$r_{j}=\frac{1}{2i}\big{(}\frac{a}{c}-\frac{b}{d}\big{)}P(s_{j})^{-\frac{1}{2}}\sim
c_{2}\cdot s_{j}^{-\frac{N-2}{2}},$

for some constant $c_{2}$.

The relation between the residues and the poles is thus

(5) $|r_{j}|\sim c_{2}|s_{j}|^{-\frac{N-2}{2}}\sim c_{3}|j|^{-\frac{N-2}{N}}.$

######  Proposition 7.

As above, let $f$ be a solution of $S(f)=2P$ where $P$ is a polynomial of
degree $N-2$. For any $z_{0}\in\mathbb{C}$, denote its preimages by $p_{j}$
and label them so that $\cdots\leq|p_{j}|\leq|p_{j+1}|\leq\cdots$. Then there
exists a constant $c>0$ such that $|f^{\prime}(p_{j})|\sim c|j|^{(N-2)/N}$.

###### Proof.

Let

$g(z)=\frac{1}{f(z)-z_{0}};$

then $S(g)=S(f)=2P$. If $p_{j}$ are solutions of $f(z)=z_{0}$, they are poles
of $g(z)$ so that proposition 6 implies that $|\text{res}(g,p_{j})|\sim
c_{3}|j|^{-(N-2)/N}$. Furthermore, since $g(z)=1/(f(z)-z_{0})$, a simple
computation shows that

$|f^{\prime}(p_{j})|=\frac{1}{|\text{res}(g,p_{j})|}\sim c|j|^{\frac{N-2}{N}}$

∎

The zeros of $f$ are the poles of $1/f$ and vice versa. Both are determined by
the zeros of the function $w$ in the above proposition. Since for any
$z_{0}\in{\mathbb{C}}$, $f$ and $f-z_{0}$ have the same Schwarzian, it follows
from the above propositions that the zeros of $f-z_{0}$ have the same
asymptotic behavior as those of $f$. In particular, they grow at the same
rate.

##  3\. The Main Theorem

Let ${\mathcal{F}}_{N}$ be the set of Nevanlinna functions with $N$ asymptotic
values. For $f\in{\mathcal{F}}_{N}$ and $i=1,\ldots,N$ denote the asymptotic
values by $\lambda_{i}$ and the corresponding asymptotic tracts by $T_{i}$.
Assume there is an integer $K$, $1\leq K<N$, and integers $p_{i}\geq 0$,
$i=1,\ldots,K$, such that

$f^{p_{i}}(\lambda_{i})=\infty,\,\,i=1,\ldots,K.$

If $\lambda_{i}=\infty$, $p_{i}=0$. This can happen for at most $N/2$
asymptotic values and the asymptotic tracts of these infinite asymptotic
values must be separated by the asymptotic tract of a finite asymptotic value.
Also assume that for each $i=K+1,\cdots,N$, the accumulation set,
$\omega(\lambda_{i})$, of the orbit of $\lambda_{i}$, is a compact repeller;
that is, there exists a $\kappa>1$ such that for each
$z\in\omega(\lambda_{i})$, there exists an $n=n(z)$, such that
$|(f^{n})^{\prime}(z)|>\kappa$. Note that this implies that these asymptotic
values are finite.

Define

$I=I(f)=\\{z\in\mathbb{C}\ |\ f^{n}(z)\to\infty\\},$

and

$L=L(f)=\\{z\in\mathbb{C}\ |\
\omega(z)=\cup_{i=K+1}^{N}\omega(\lambda_{i})\\}.$

The proof of the main theorem depends on the following theorems:

######  Theorem 8.

The set $I$ is of measure zero.

######  Theorem 9.

The set $L$ is of measure zero.

Versions of these theorems are proved in [KK] under the assumption that all of
the asymptotic values accumulate on a compact repeller.

###  3.1. The measure of the set $I$

For each $1\leq i\leq K$, denote the orbit of the prepole asymptotic value
$\lambda_{i}$ by

$Orb(\lambda_{i})=\\{\lambda_{i},f(\lambda_{i}),\cdots,f^{p_{i}-1}(\lambda_{i}),\infty\\}.$

If $\lambda_{i}=\infty$ for some $i$, $Orb(\lambda_{i})=\\{\infty\\}$.

Let $S=\\{1,2,\cdots,K\\}$. Since there are $2^{K}-1$ distinct non-empty
subsets of $S$, label them $S_{l}$, $l=1,\ldots,2^{K}-1$ and denote the
collection $\Sigma$. For any $S_{l}$, define

$Orb_{l}=\cup_{i\in S_{l}}Orb(\lambda_{i}).$

For $S_{l}\in\Sigma$, where $l=1,\ldots,2^{K}-1$, define

$I_{l}=I_{l}(f)=\\{z\in\mathbb{C}\ |\ \omega(z)=Orb_{l}\\}.$

######  Theorem 10.

Each of the sets $I_{l}$ is of measure zero.

The proof of this theorem depends on the next two lemmas.

Fix $R>>0$, and let ${\mathcal{A}}_{R}=\\{z\in{\mathbb{C}},|z|>R\\}$; For each
$1\leq i\leq K$, and $j\in{\mathbb{Z}}$, let $b_{ij}=f^{-1}(\lambda_{i})$.

Because $\lambda_{i}$ is prepole of order $p_{i}$, one component of
$f^{-p_{i}}({\mathcal{A}}_{R})$ is the topological disk $D_{i}$ punctured at
$\lambda_{i}$. Therefore the set of components of $f^{-1}(D_{i})$ consists of
the asymptotic tract $T_{i}$ of $\lambda_{i}$ and the topological disks
$V_{ij}$ punctured at $b_{ij}$. For each $S_{l}\in\Sigma$ and $z\in\cup_{i\in
S_{l}}(\cup_{j}V_{ij}\cup T_{i})$, define the map
$\sigma_{l}(z)=f^{p_{i}+1}(z)$.

######  Lemma 11.

If $z\in\cup_{i=1}^{K}T_{i}$, then

$|\sigma_{l}^{\prime}(z)|>\frac{|\log|\sigma_{l}(z)|-\log
R|}{4\pi}\cdot\frac{|\sigma_{l}(z)|}{|z|}$

###### Proof.

Since $f^{p_{i}}:D_{i}\to A_{R}$ is conformal and $f:T_{i}\to D_{i}$ is a
universal covering, it follows that $\sigma_{l}:T_{i}\to A_{R}$ is also a
universal covering. The rest of proof given below follows along the lines of
the corresponding proof in [Lyu].

Consider $H_{R}=\log{\mathcal{A}}_{R}$, the right half plane with real part
greater than $R$ and let ${\mathcal{U}}_{l}=\log(\cup_{i\in S_{l}}T_{i})$.
Then ${\mathcal{U}}_{l}\subset H_{R}$ and consists of infinitely many disjoint
simply connected components $U_{im}$, $i\in S_{l},m\in{\mathbb{Z}}$; moreover,
there is an $\epsilon_{im}>0$, depending on $R$ such that each $U_{im}$ is
fully contained inside a strip of height $2\pi-\epsilon_{im}$. Because there
are at most $K$ sets $U_{im}$, the sum of their heights is less than $(2\pi
K/N-\epsilon_{R})$ where $\epsilon_{R}=\sum\epsilon_{im}$ depends on $R$ and
$S_{l}$.

For each $U_{im}$ there is a conformal map $F_{im}$ to $H_{R}$ such that the
following diagram commutes:

$\textstyle{U_{im}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\scriptstyle{F_{im}}$$\textstyle{H_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\textstyle{T_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{l}}$$\textstyle{{\mathcal{A}}_{R}}$.

For each point $z_{0}\in\cup T_{i}$, denote the lifts of $z_{0}$ and
$\sigma_{l}(z_{0})$ by $w_{0}\in U_{im}$ and $w_{1}\in H_{R}$ respectively.
Note that $\Re w_{1}=\log|\sigma_{l}(z)|$. Consider $D=D(w_{1},\Re w_{1}-\log
R)$ and its preimage under $F_{im}$. By the $1/4$ Koebe theorem, its preimage
contains a disk of radius

$\frac{\Re w_{1}-\log R}{4|F^{\prime}_{im}(w)|}.$

As the width of each strip is less than $2\pi$,

$|F^{\prime}_{im}(w)|\geq\frac{\Re w_{1}-\log R}{4\pi}.$

The lemma now follows from the chain rule.

∎

Theorem 8 follows directly since $I\subset\cup_{S_{l}\in\Sigma}I_{l}$.

The next lemma is the analog of lemma 11 in the case that $S_{l}\in\Sigma$ and
$z\in\cup_{i\in S_{l}}(\cup_{j}V_{i{j}})$. Fix $i\in S_{l}$ and, suppressing
the index $i$ for readability, denote the zeros of $f(z)-\lambda_{i}=0$ by
$b_{j}$. Note that by proposition 7, $|f^{\prime}(b_{j})|\sim c|j|^{(N-2)/N}$.

######  Lemma 12.

There exists a neighborhood $V_{j}^{\prime}$ and of $b_{j}$ and a constant
$b>0$ such that $V^{\prime}_{j}\subset\overline{V^{\prime}_{j}}\subset V_{j}$,

$V^{\prime}_{j}\subset D(b_{j},\frac{b}{|j|^{\frac{N-2}{N}}R}),$

and for $z\in U_{j}$ and for some constant $B>0$,

$|\sigma_{l}^{\prime}(z)|>BR|j|^{\frac{N-2}{N}}$

###### Proof.

For each $\lambda_{i}\in S_{l}$, denote the pole $f^{p_{i}-1}(\lambda_{i})$ by
$s_{i}$. Then expanding $f$ at $s_{i}$,

$f(z)=\frac{r_{i}}{z-s_{i}}(1+\phi_{i}(z))$

where $r_{i}$ is the residue of $f$ at $s_{i}$, and $\phi_{i}$ is analytic at
$s_{i}$. Consider the annular region $\mathcal{A}_{2R}\subset\mathcal{A}_{R}$,
and denote by $g$ the branch of $f^{-1}$ such that $g(\mathcal{A}_{R})$ is a
punctured neighborhood of $s_{i}$. Set $h=g(1/z):D(0,1/R)\to\mathbb{C}$, so
that $0$ is a removable singularity and let
$U=h(D(0,1/2R))=g(\mathcal{A}_{2R})\cup\\{\infty\\}$. Then $h$ is conformal
and $h^{\prime}(0)=r_{i}$. The Koebe distortion theorem applied to $h$ proves
that for any $z\in D(0,1/2R)$,

$|h(z)-h(0)|\leq|h^{\prime}(0)|\frac{\frac{1}{2}\cdot\frac{1}{R}}{(1-\frac{1}{2})^{2}}=\frac{2r_{i}}{R}.$

The Koebe $1/4$ theorem applied to $h$ on $D(0,1/2R)$ proves that
$D(s_{i},r_{i}/8R)\subset U$. Combining these gives

$D(s_{i},\frac{|r_{i}|}{8R})\subset U\subset D(s_{i},\frac{2|r_{i}|}{R}).$

Therefore for any $z\in U$,

(6)
$|f^{\prime}(z)|=|-\frac{r_{i}(1+\phi_{i}(z))}{(z-s_{i})^{2}}+\frac{r_{i}\phi^{\prime}_{i}(z)}{z-s_{i}}|\geq|\frac{r_{i}}{z-s_{i}}|\geq\frac{R}{2}.$

Since $f$ has no critical points, the disk $D(s_{i},4|r_{i}|/R)$ at the pole
$s_{i}$ is mapped univalently by the respective branches of $f^{-p_{i}}$ onto
neighborhoods of the points $b_{j}=f^{-p_{i}}(s_{i})$. Let $V_{j}^{\prime}$ be
the component of $f^{-p_{j}}(U)=f^{-(p_{i}+1)}(\mathcal{A}_{2R})$ at $s_{j}$.
It is obvious that $\overline{V_{j}^{\prime}}\subset V_{j}$, a component of
$f^{-(p_{i}+1)}(\mathcal{A}_{R})$. Since

$U_{i}\subset D(s_{i},\frac{2|r_{i}|}{R})\subset D(s_{i},\frac{4|r_{i}|}{R}),$

the Koebe distortion theorem implies that for any $z,w\in\widetilde{V}_{ij}$,

(7) $\frac{(f^{p_{i}})^{\prime}(z)}{(f^{p_{i}})^{\prime}(w)}\leq
T(\frac{1}{2}).$

By Proposition $7$, for some $c_{1}>0$, $|f^{\prime}(b_{j})|\sim
c_{1}j^{(N-2)/N}$. Since $f$ is univalent on the orbit of $\lambda_{i}$, there
exists $c_{2}>0$ such that $|(f^{p_{i}})^{\prime}(b_{j})|\sim
c_{2}j^{(N-2)/N}$ and thus

$|(f^{p_{i}})^{\prime}(z)|>c_{2}T^{-1}(\frac{1}{2})j^{\frac{N-2}{N}},\text{
for any }z\in V^{\prime}_{j}.$

Since $U\subset D(s_{i},2|r_{i}|/R)$, this implies

$V^{\prime}_{j}\subset
D(b_{{j}},\frac{2T(\frac{1}{2})|r_{i}|}{c_{2}|j|^{\frac{N-2}{N}}R}).$

This, combined with equation (6) also implies that for all $z\in
V^{\prime}_{j}$,

$\sigma_{l}^{\prime}(z)\geq\frac{c_{2}R|j|^{\frac{N-2}{N}}}{2T(1/2)}$

and thus completes the proof. ∎

###  3.2. Proof of Theorem 10

Let $E=\\{z,\sigma^{n}_{l}(z)\to\infty,n\to\infty\\}$; then

$I_{l}(f)=\cup_{n=0}^{\infty}f^{-n}(E).$

To prove theorem 10, it suffices to show that the measure of the set $E$ is
zero. We assume not and obtain a contradiction. Let $z_{0}$ be a Lebesgue
density point of the set $E$, and let $z_{n}=\sigma_{l}^{n}(z_{0})$. As
$z_{n}\to\infty$, without loss of generality, we may assume that for each $n$,
$|z_{n+1}|\geq|z_{n}|\geq R$. Set ${\mathcal{A}}_{r,s}=\\{z,\,|\,r<|z|<s\\}$.

Since by hypothesis, $K<N$, if follows that
$\cup_{K+1}^{N}T_{i}\not=\emptyset$ but
$(\cup_{K+1}^{N}T_{i})\cap\sigma_{l}^{-1}({\mathcal{A}}_{R})=\emptyset$.
Therefore, for any $s>r>R$, there is a $\tau>0$ such that

$\frac{m({\mathcal{A}}_{r,s}\cap\sigma_{l}^{-1}({\mathcal{A}}_{r,s}))}{m(A_{r,s})}<1-\tau.$

Note that if $K=N$, the asymptotic tracts fill up
$\sigma_{l}^{-1}({\mathcal{A}}_{R})$. The proof of non-ergodicity for $K=N=2$
in [CJK] uses this fact and lends support to conjecture 1.

The proof of theorem 10, is in two parts depending on the orbit of $z_{0}$.
Part 1: Assume that for all $n$, $z_{n}\in\cup_{i=1}^{K}T_{i}$. This part of
the proof depends on lemma 11 and uses the notation in that lemma.

As in the lemma, for $z_{n}\in T_{i}$, set $w_{n}=\log z_{n}\in U_{im}$ and
$r_{n}=\Re w_{n}$. Then $F_{im}^{-1}:H_{R}\to U_{im}$ is the inverse branch
such that $F_{im}^{-1}(w_{n})=w_{n-1}$. The function $F_{im}^{-1}$ is
univalent in the disk $D(w_{n},r_{n}-\log R)$. By lemma 11, it follows that

$|(F_{im}^{-1})^{\prime}(w_{n})|\leq\frac{4\pi}{r_{n}-\log R}.$

Note that it may be that $z_{n-1}\in T_{j}$, $i\neq j$, $i,j\in S_{l}$, and
similarly, $w_{n-1}$ may be in a different $U_{jm}$. For the sake of
readability, we will ignore these indices and write $U$ for whichever $U_{im}$
is meant and write $F^{-1}$ for whichever inverse branch is meant.

Next, consider the disk $D(w_{n},r_{n}/4)$. First note that since $U$ does not
intersect the any of the preimages, $f^{-1}(T_{i})$, $i=K+1,\ldots,N$, there
exists a $\tau^{\prime}>0$, such that

$\frac{m(D(w_{n},\frac{r_{n}}{4})\cap
U)}{m(D(w_{n},\frac{r_{n}}{4})}<1-\tau^{\prime}.$

Moreover, for $w\in D(w_{n},r_{n}/4)$, the Koebe distortion theorem implies
that

$|F^{-1}(w)-F^{-1}(w_{n})|\leq\frac{4\pi}{r_{n}-\log
R}\cdot\frac{\eta(r_{n}-\log R)}{(1-\eta)^{2}},$

where

$\eta=\frac{r_{n}}{4(r_{n}-\log R))}<\frac{1}{2}.$

Therefore,

$F^{-1}(D(w_{n},\frac{r_{n}}{4}))\subset D(w_{n-1},d)\text{ where }d=8\pi.$

For each $1\leq k\leq n-1$, $F^{-1}$ is univalent in the disk $D(w_{k},2d)$
and $(F^{-1})^{\prime}(w_{k})\leq 1/8$. For each $k$, $1\leq k\leq n-1$, the
Koebe $1/4$ theorem implies that

$F^{-1}(D(w_{k},d))\subset D(w_{k-1},\frac{d}{2}).$

Next, iterate $F^{-1}$ and set $B_{n}=F^{-n}(D(w_{n},r_{n}/4))\subset
D(w_{0},2^{-m+1}d)$. Now since the iterated function $F^{-n}$ is univalent on
$D(w_{n},\Re w_{n}-\log R)$, apply Koebe distortion again to get

$D(w_{0},t\rho_{n})\subset B_{n}\subset D(w_{0},\rho_{n})$

where $t$ is independent of $n$, and $\rho_{n}$ is the radius of the smallest
disk centered at $w_{0}$ containing $B_{n}$. It follows that $\rho_{n}\leq
2^{-n+1}d$ which, in turn, implies that $\rho_{n}\to 0$ as $n\to\infty$.

Part (ii) of the Koebe Distortion Theorem applied to $F^{-n}$ implies there
exists a $\tau^{\prime\prime}$ such that

$\frac{m(B_{n}\cap E)}{m(E)}\leq 1-T^{-2}(\frac{1}{2})\tau^{\prime\prime}$

for all $n$. In other words, the Lebesgue density of the point $w_{0}$ is less
than $1$, which contradicts the assumption that $w_{0}$ is a density point.

Part 2: Now consider a subsequence $z_{n_{k}}\in\cup_{i\in
S_{l}}(\cup_{j}V_{j})$. Let $W=\sigma_{l}^{-1}(\mathcal{A}_{2R})$ and let
$V_{j}$ be the component of $W$ such that $z_{n_{k}}\in V^{\prime}_{j}\subset
V_{j}$ for some $n_{k}$. There is at least one asymptotic tract $T_{k}$ such
that $W\cap T_{k}=\emptyset$; thus there exists a
$\tau^{\prime\prime\prime}>0$ such that

$\frac{m(\mathcal{A}_{2R}\cap
W)}{m(\mathcal{A}_{2R})}<1-\tau^{\prime\prime\prime}.$

By lemma 12,

$V^{\prime}_{j}\subset D(b_{j},\frac{b}{R|j|^{\frac{N-2}{N}}}),$

and for any $z\in V^{\prime}_{j}$,

$\sigma_{l}^{\prime}(z)\geq BR|j|^{\frac{N-2}{N}}\text{ and
}\frac{(f^{p_{i}})^{\prime}(z)}{(f^{p_{i}})^{\prime}(w)}\leq T(\frac{1}{2}).$

Therefore,

$\frac{m(V^{\prime}_{j}\cap\sigma_{l}^{-1}(W))}{m(V^{\prime}_{j})}<1-T(\frac{1}{2})^{-2}\tau^{\prime\prime\prime}.$

Without loss of generality, assume that $|z_{n+1}|\geq|z_{n}|>>R$ for all $n$.
Then the above inequality, together with lemmas 11 and 12, show
$|\sigma_{l}^{\prime}(z_{n})|>M>1$ for all $n$.

Let $B_{n_{k}}=\sigma_{l}^{-n_{k}}(V^{\prime}_{j})$. Then

$B_{n_{k}}\subset D(z_{0},M^{-n_{k}}\frac{b}{R|j|^{\frac{N-2}{N}}}).$

Since $\sigma_{l}^{-n_{k}}$ is univalent on $V_{j}\supset V^{\prime}_{j}$,
this implies

$D(z_{0},t\rho_{n_{k}})\subset B_{n_{j}-1}\subset D(z_{0},\rho_{k})$

where $t$ is independent of $n_{j}$, and $\rho_{n_{k}}$ is the radius of the
smallest disk centered at $z_{0}$ containing $B_{n_{j}-1}$. It follows that
$\rho_{n_{k}}\to 0$ as $n_{k}\to\infty$.

Applying part (ii) of the Koebe Distortion Theorem,

$\frac{m(B_{n_{k}}\cap E)}{m(E)}\leq
1-T^{-4}(\frac{1}{2})\tau^{\prime\prime\prime}$

for all $n_{k}$ which implies that the Lebesgue density of the point $z_{0}$
is less than $1$. This contradicts the assumption that $z_{0}$ is a density
point and completes the proof of theorem 10.

###  3.3. Proof of Theorem 9

###### Proof.

To prove that $m(L)=0$, note first that by assumption
$\Omega=\cup_{i=K+1}^{N}\omega(\lambda_{i})$ is a finite union of compact
repellers, so is again a compact repeller. This implies that the orbits of the
non-prepole asymptotic values do not accumulate on $\Omega$, but actually land
on it. The proof in [KK] assumes $K=0$. Although this proof is similar to that
proof, here we modify it to take account of the prepole asymptotic values.

Let ${\mathcal{K}}_{\epsilon}=\\{z\,|\,dist(z,\Omega)<\epsilon\\}$. We claim
there is an $\epsilon>0$ and an integer $M>0$ such that if $y=\lambda_{i}$,
$i=K+1,\ldots,N$, $n>M$ and $f^{n}(y)\in{\mathcal{K}}_{\epsilon/2}$, then
$f^{n}(y)\in\Omega$.

If $\Omega$ is finite, the claim is obviously true, so assume it is not. By
the compactness assumption, there are no prepoles in $\Omega$ and there there
are constants $\kappa>1$ and $\epsilon>0$ such that
$|(f^{n})^{\prime}(w)|\geq\kappa$ for some $m$ and all
$w\in{\mathcal{K}}_{\epsilon}$, and thus for all
$w\in\overline{{\mathcal{K}}_{\epsilon/2}}$. By the forward invariance of
$\Omega$ and this expansion property,
$\overline{{\mathcal{K}}_{\epsilon/2}}\subset
f^{n}(\overline{{\mathcal{K}}_{\epsilon/2})}$. Let $g$ be the inverse branch
of $f^{n}$ reversing this inclusion. Then set

$A_{0}=\overline{{\mathcal{K}}_{\frac{\epsilon}{2}}}-g(\overline{{\mathcal{K}}_{\frac{\epsilon}{2}}})\text{
and }A_{n+1}=g^{n}(A_{0}),n\to\infty.$

These disjoint annuli are nested and, since the inverse branches are
univalent, have the same moduli. Therefore, if for some $n$,
$f^{n}(y)\in\overline{\mathcal{K}}_{\epsilon}\setminus\Omega$, by compactness,
there are subsequences of its iterates that converge both to points in
$\overline{A_{0}}$ and to points in $\Omega$. This is a contradiction because
these sets are disjoint and the claim is proved.

Choose $\epsilon$ as above and set

${\mathcal{L}}=\cap_{n\geq 0}f^{-n}({\mathcal{K}}_{\frac{\epsilon}{2}}).$

A point $z\in{\mathcal{L}}\setminus\Omega$ if its full forward orbit belongs
to ${\mathcal{K}}_{\epsilon/2}$. We will show
$m({\mathcal{L}}\setminus\Omega)=0$. Since
$L\subset\cup_{n=0}^{\infty}f^{-n}({\mathcal{L}})$ and $\Omega$ is countable,
this will imply $m(L)=0$.

Suppose $m({\mathcal{L}}\setminus\Omega)>0$ and let $z_{0}$ be a density point
of ${\mathcal{L}}\setminus\Omega$. Since $\Omega$ is compact, a subsequence
$z_{k}=f^{n_{k}}(z_{0})$ converges to a point
$y_{0}\in\overline{{\mathcal{K}}_{\epsilon/2}}\subset{\mathcal{K}}_{\epsilon}$.
Denote the respective inverse branches by $g_{k}$. Set
$D_{k}=D(z_{k},\epsilon/4)$; then $D_{k}\subset{\mathcal{K}}_{\epsilon}$ and
$g_{k}$ is univalent on $D_{k}$. Applying Koebe distortion we obtain

$\frac{m(g_{k}(D_{k})\cap{\mathcal{L}})}{m(g_{k}(D_{k}))}\rightarrow
1\,\,\text{ and }\frac{m(D_{k}\cap
f^{n_{k}}({\mathcal{L}}))}{m(g_{k}(D_{k}))}\rightarrow 1.$

Finally, let $U$ be an open set with compact closure contained in
${\mathbb{C}}\setminus\overline{{\mathcal{K}}}$. Since the Julia set is the
whole sphere, there is an integer $M$ such that
$f^{M}(D_{k})\supset\overline{U}$ so that $m(f^{M+n_{k}}({\mathcal{L}}\cap
U))>0$. For all $k\in{\mathbb{N}}$, however,
$f^{k}({\mathcal{L}})\subset{\mathcal{K}}_{\epsilon/2}$ so that
$f^{M+n_{k}}({\mathcal{L}}\cap U)=\emptyset$. This contradiction shows
$m(L)=0$. ∎

###  3.4. Proof of the main theorem

######  Theorem 13.

If $f$ is a Nevanlinna function with $1\leq K<N$ prepole111 If infinity is an
asymptotic value, we consider it a “prepole of order 0”. asymptotic values and
$N-K$ asymptotic values that accumulate on a compact repeller, then $f$ acts
ergodically on its Julia set.

###### Proof.

Let $A$ be an $f$-invariant subset of the Julia set with positive measure. We
will show that $A=\widehat{\mathbb{C}}$ up to a set of measure zero. Let
$z_{0}$ be a Lebesgue density point of $A$ and denote its orbit by
$z_{n}=f^{n}(z)$, $n=0,1,\ldots.$ We proved above that the measures of each of
the sets $I$, $I_{l}$ and $L$ is zero. Since these three sets together contain
all points whose orbits accumulate on $\cup_{i=1}^{N}\omega(\lambda_{i})$ we
assume that $z_{0}$ is not among them.

By the above, the density point $z_{0}$ of $A$ has an accumulation point
$y\in{\mathbb{C}}\setminus\cup_{i=1}^{N}\omega(\lambda_{i})$. Recall that
$\Omega=\cup_{i=K+1}^{N}\omega(\lambda_{i})$. Hence there is an $\epsilon>0$
so that $2\epsilon=dist(y,\Omega)>0$. Thus, there is a subsequence
$\\{n_{j}\\}$ in ${\mathbb{N}}$ such that $z_{n_{j}}\to y$ as $j\to\infty$ and
$dist(z_{n_{i}},\Omega)\geq\epsilon$.

Let $B_{j}=B(z_{n_{j}},\epsilon)$ and $V_{j}=B(z_{n_{j}},\epsilon/2)$. Let
$g_{j}$ be the inverse branch of $f^{-n_{j}}$ that sends $z_{n_{j}}$ to $z$;
it is a univalent function on $B_{j}$. Let $U_{j}=g_{j}(V_{j})$. All of the
inverse branches of $f$ are contracting with respect to the hyperbolic metric
on ${\mathbb{C}}\setminus\Omega$; this implies that $g_{j}^{\prime}\to 0$ on
$V_{j}$ as $j$ goes to $\infty$, which in turn implies that the diameter of
$U_{j}$ tends to $0$. Since $g_{j}$ is univalent on $B_{j}$, the Koebe
distortion theorem shows that that $U_{j}$ is almost a disk. Since $z$ is a
density point of $A$,

$\lim_{j\to\infty}\frac{m(A\cap U_{j})}{m(U_{j})}=1.$

Applying Koebe distortion again, since $A$ is an invariant subset, we get

$\lim_{j\to\infty}\frac{m(A\cap V_{j})}{m(V_{j})}=1.$

This and the fact that $V_{j}$ approaches $B_{y}=B(y,\epsilon/2)$ as $j$ goes
to $\infty$ together imply that $B_{y}\subset A$ up to a set measure zero.
Since $A$ is an invariant subset of the Julia set $J$, $f^{n}(B_{y})\subset
A\subset J$. Since $A$ is in the Julia set, $B_{y}$ is also in the Julia set
and $f^{n}(B_{0})$ approaches ${\mathbb{C}}$ as $n$ goes to $\infty$.
Therefore, $A=\widehat{\mathbb{C}}$ up to a zero-measure set and the proof of
Theorem 13 is complete. ∎

######  Remark 3.1.

The assumption that the $\omega$-limit sets of the non pre-polar asymptotic
values are compact repellers says the Julia set is the whole sphere and gives
us the expansion we need to prove our theorem. We could replace this by
assuming that the Julia set is the sphere and that the orbits of the non-
prepolar orbits are bounded. Then the main theorem of [GKS] implies the
expansion we need exists.

######  Remark 3.2.

Another application of the results in [GKS] and [RVS, Theorem 1.1] to the
Nevanlinna functions $f$ of our main theorem is that $f$ supports no invariant
line field.

######  Remark 3.3.

Finally, the results in [KU] applied to the Nevanlinna functions of our main
theorem prove that $f$ has a $\sigma$-finite ergodic conservative
$f$-invariant measure absolutely continuous with respect to the Lebesgue
measure.

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$\sigma_{m}:G_{\mathbb{Q}}^{m}\to G_{\mathbb{Q}}^{m}$ induced by
$\sigma_{m+3}$ is an isomorphism, and there exists a unique isomorphism of
fields $\tau:\mathbb{Q}_{(m)}\to\mathbb{Q}_{(m)}$ such that $\sigma_{m}$ is
determined by $\sigma_{m}(g)=\tau g\tau^{-1}$ for all $g\in G_{K}^{m}$.

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# Convolutional Neural Networks for Mode On-Demand High Finesse Optical
Resonator Design

Denis V. Karpov<EMAIL_ADDRESS>Optoelectronics Research Centre,
University of Southampton, Southampton SO17 1BJ, United Kingdom  Sergei
Kurdiumov  Optoelectronics Research Centre, University of Southampton,
Southampton SO17 1BJ, United Kingdom  Peter Horak  Optoelectronics Research
Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom

###### Abstract

We demonstrate the use of machine learning through convolutional neural
networks to solve inverse design problems of optical resonator engineering.
The neural network finds a harmonic modulation of a spherical mirror to
generate a resonator mode with a given target topology (”mode on-demand”). The
procedure allows us to optimize the shape of mirrors to achieve a
significantly enhanced coupling strength and cooperativity between a resonator
photon and a quantum emitter located at the center of the resonator. In a
second example, a double-peak mode is designed which would enhance the
interaction between two quantum emitters, e.g., for quantum information
processing.

##  I Introduction

Machine learning methods are widely used for various applications of modern
optics, especially for inverse design problems [1, 2, 3]. Methods of forward
design, where based on a given optical device geometry and boundary conditions
researchers can find the optical field and its properties, are highly
developed and various numerical, semi-analytical, and analytical methods are
well known. On the other hand, not often is it possible to devise geometries
to generate a given target optical field using forward design methods. In this
case deep machine learning methods and in particular neural networks can play
the role of universal predictors and ultra-precise interpolators for various
types of engineering. For example, optical power beam splitters and multiport
devices for arbitrary transmission matrices have been designed with these
methods [4, 5].

The same approach can be applied to the design of optical resonators for
cavity quantum electrodynamics and quantum technology. For example, for
various applications in these areas strong coupling of a quantum emitter to
photons in a resonator is required and simultaneously a long photon lifetime
in this resonator is critically important to reach the so-called good cavity
limit [6] where the coherent cavity coupling exceeds the decay rates. At the
same time there often are geometrical restrictions, e.g., to allow for access
from the side for loading, state initialisation, optical cooling [7], or
trapping fields [8]. Other applications require strong photon coupling to two
quantum emitters for 2-qubit quantum gates [9, 10], such as for a controlled-
not (CNOT) operation between two qubits stored in trapped ions [11]. In this
case the performance could be significantly improved with an optimized cavity
mode that exhibits two peaks at the positions of the two ions.

There is thus a need for a universal method to design and engineer cavities
used in quantum information applications [12] which should combine strong
photon-particle coupling and low cavity losses by achieving a target field
topology (e.g., one or several field maxima in the center and low mode
divergence) while also taking into account constraints on geometry, materials,
and fabrication. Traditionally, only cavities with spherical mirrors
supporting Laguerre-Gaussian modes have been considered for these
applications, but they provide only a limited design space that may not
fulfill the requirements of specific applications. Allowing for non-spherical
cavity mirrors provides a much wider range of design options and can generate
cavity eigenmodes that have superior properties compared to standard Laguerre-
Gaussian modes [13, 14]. However, the large design space then makes inverse
design problems very difficult to solve, which we overcome by using a
convolutional neural network (CNN) approach here.

For most optical device inverse designs, e.g., for beam splitters, the
relevant quantity to be optimized is the electromagnetic field [4, 5].
Optimizing an optical resonator is more complicated since a resonator supports
many modes, each one of which could be a potential solution. Furthermore, it
is generally not only the electric field of the mode that is of interest but
also the mode decay rate must be taken into account; for example, a mode with
high field intensity can have very high losses and thus be useless for real
applications. In this paper we demonstrate an approach based on a coordinate-
dependent cooperativity $C$, a measure that includes the ratio of the coherent
coupling strength of the quantum emitter to the cavity photon (i.e., the
single-photon field strength) over the incoherent cavity loss rate. Our CNN
finds cavity geometries that optimize the cooperativity $C$ and that can be
achieved realistically by various modern fabrication methods such as a fiber-
optic microcavities [15, 16, 17, 18], ion beam etched dielectric resonators
[19], and micro-assembled structures[20].

This paper is organized as follows. First, in Sec. II, we describe our optical
resonator model and introduce the parameters for optimization. In Sec. III we
present a general overview of the optimization algorithm which consists of
data generation, CNN training and verification. In Sec. IV more details of the
neural network topology and computational resources are discussed. The CNN
predictions of optimized cavity mirror geometries and their verification is
presented in Sec. V. Finally we summarize our results and conclude in Sec. VI.

##  II Optical resonator coupling to a quantum emitter

We consider a Fabry-Perot optical cavity consisting of two mirrors whose
geometry we wish to optimize for applications in quantum science and
technology. A schematic of this is shown in Fig. 1.

Figure 1: Schematic cavity representation. The shape of the mirrors should be
optimized to generate a target cavity field mode $\Psi$ while keeping the
losses $\kappa$ small.

For a given cavity geometry, the cavity modes can be calculated using standard
methods, such as Gaussian beam resonator theory [21], ray transfer matrix
approaches [22, 23], or finite element methods. The electric field of a chosen
mode is then normalized to a single photon field $E(\textbf{r})$, i.e., the
total energy contained in the electromagnetic field inside the cavity is
normalized to a single photon energy $\hbar\omega$, where $\omega$ is the
angular frequency. The interaction energy between a dipole emitter at position
r and a cavity photon is given by $\hbar g(\textbf{r})=\mu E(\textbf{r})$
where $g$ is called the coupling constant and $\mu$ is the electric dipole of
the emitter. Exploiting the relationship between the dipole moment and the
spontaneous decay rate of the emitter,
$\Gamma=\frac{\omega^{3}\mu^{2}}{3\pi\varepsilon_{0}\hbar c^{3}}$, we can
write the coupling constant as [24]

$g(\textbf{r})=\sqrt{\frac{3\lambda^{2}c\Gamma}{4\pi
V_{\Psi}}}\Psi(\textbf{r}),\
\Psi(\textbf{r})=\frac{E(\textbf{r})}{|E(\textbf{r}_{m})|}$ (1)

where $\lambda$ is the wavelength corresponding to angular frequency $\omega$,
$c$ is the vacuum speed of light, $\Psi$ is the dimensionless cavity field
normalized to its maximum at position $\textbf{r}_{m}$ and the mode volume is
defined by

$V_{\Psi}=\int_{V_{cavity}}|\Psi(\textbf{r})|^{2}dV$ (2)

where the integral is over the geometric volume of the cavity.

The other important quantity of optical cavities for quantum applications is
the cavity decay rate $\kappa$, i.e., the rate at which a photon is lost from
the cavity mode. We distinguish two contributions to the cavity loss rate: the
loss per photon round trip by transmission through the mirrors or by
absorption within the mirror $D_{mir}$ (these are typically of the order of
$10^{-5}$ \- $10^{-3}$), and the loss per round trip through so-called
clipping losses $D_{clip}$ where the mode on the mirror is larger than the
mirror diameter, i.e., a fraction of the photon field misses the mirror
completely on reflection. The cavity decay rate is then

$\kappa=(D_{clip}+D_{mir})\frac{c}{2L}$ (3)

where $L$ is the length of the cavity. An alternative way to express the loss
rate is via the cavity finesse $F$ defined as

$F=\frac{2\pi}{D_{clip}+D_{mir}}=\frac{c\pi}{L}\frac{1}{\kappa}$ (4)

To achieve strong particle-cavity coupling, the coherent coupling rate
$g(\textbf{r})$ between the particle at position r and the cavity must be
larger than the strengths of any incoherent processes, i.e., energy exchange
between the particle and the cavity must occur on time scales before the
photon leaks out of the cavity or is incoherently scattered by the atom.
Therefore, the cooperativity parameter defined as

$C(\textbf{r})=\frac{g(\textbf{r})^{2}}{\kappa\Gamma}=\frac{3\lambda^{2}c}{4\pi\kappa
V_{\psi}}|\Psi(\textbf{r})|^{2}$ (5)

must be larger than one. The cooperativity therefore not only depends on the
single-photon electric field strength at the position of the atom, but it also
includes the round trip losses $D_{clip}$ and $D_{mir}$.

Our approach is to modify the shape of the cavity mirrors based on a spherical
“reference” shape. It is therefore useful to compare the cooperitivity
achieved with our designs to the maximum cooperativity obtained at the center
of this reference cavity. For two identical spherical mirrors with radius of
curvature $R$ [25] the fundamental cavity mode is given by a Gaussian beam and
the maximum cooperativity becomes

$C_{0}=\frac{6\lambda
F}{\sqrt{2RL-L^{2}}}=\frac{6\lambda}{\sqrt{2RL-L^{2}}}\frac{2\pi}{D_{mir}}$
(6)

where we assumed no clipping losses for this reference cavity mode.

For quantum optics and quantum technology applications, the cooperativity $C$
is the quantity that we wish to optimize by a convolutional neural network
approach in the following sections. Our CNN will thus be trained on a set of
functions $C(\textbf{r})$ or equivalently, once a reference spherical mirror
shape is chosen, on the relative enhancement $C(\textbf{r})/C_{0}$, which
constitutes a fundamental difference to previous CNN-based optimization of
optical devices which only take into account the electric field itself, e.g.,
in the case of integrated optical beamsplitters [4, 5].

##  III Algorithm

Figure 2: Overview of the optimization algorithm including data generation
(a-b), machine learning (c), inverse design prediction (d) and verification
(e). (b) shows modes with various topologies which exist within the design
space. (d) represents a sample target cooperativity distribution (i.e., a peak
of high intensity and low loss) as input argument for the trained CNN. (e)
compares the target mode from (d) with the mode calculated explicitly from the
CNN-generated mirror geometry.

We consider cavities with cylindrical rotational symmetry consisting of two
identical mirrors with a spherical profile perturbed by a harmonic modulation
given by

$Z(r)=\frac{r^{2}}{2R}+A\cos(r/P).$ (7)

By limiting the ranges of the period $P$ and the amplitude of the perturbation
$A$, we can choose parameters which are achievable by modern fabrication
tools, see Sec. V.3. However, we note that our approach would work for any
parameterization of the mirror surface. Eq. (7) is valid within the paraxial
approximation, i.e., we approximated the spherical profile by a parabolic one.

To generate our data training set we need a method to find the cavity
eigenmodes for a given mirror profile $Z(r)$. This mode solver will be treated
as a black box function below. Any numerical or semi-analytical method can be
used as mode solver; for the numerical calculations in this paper we employ a
mode mixing method (MMM) [26, 27]. The advantages of the MMM, compared to,
e.g., finite elements or finite differences methods, are high computation
speed, low RAM and CPU consumption and the absence of sharp corner
singularities [28, 29]. Depending on the complexity of the geometry one
simulation on a single CPU core takes no more than 20 seconds, which allowed
us to generate our CNN training data of 85000 samples on 240 CPU cores on an
HPC cluster (2.0 GHz Intel Skylake processors) in about 2 hours of computation
time. Choosing a computationally effective method for the mode solver is
critical for our approach.

Once the mode solver has calculated the modes $\Psi_{i}$ and their finesse
$F_{i}$ for a given $Z(r)$ we proceed by selecting one mode based on the
target topology, e.g., a mode with a single peak, two peaks, etc. The
spatially dependent cooperativity $C(\textbf{r})$, Eq. (5), of this selected
mode is then added to the training database. We emphasize that this selection
process is very important to generate a well balanced training set; different
training sets are required for different target mode topologies.

An overview of our algorithm is presented in Fig. 2. The first step is the
selection of a perturbation pattern imposed on the spherical mirror within
some range of parameters, Fig. 2(a). In our case we choose harmonic
perturbations with a flat distribution of periods $P$ in the range 5 $\mu m$
to 30 $\mu m$ and amplitudes $A$ in the range 0.1 to 0.6 $\mu m$. The second
step is to use these geometries as arguments for the mode solver, calculate
the modes, and select the most appropriate mode for the training data set as
outlined above, Fig. 2(b). The third step, Fig. 2(c), is to train the CNN on
the generated data set. This is described in more detail below in Sec. IV. The
fourth step is testing the trained CNN on data which were not used in the
training (more details below) and usage of the trained CNN as a predictor for
inverse mirror design: a target cooperativity distribution is given to the CNN
and it predicts the required mirror modulation parameters $P$ and $A$, Fig.
2(d). At the final, fifth step we put this predicted mirror geometry into the
mode solver and verify the result of the mode field, Fig. 2(e), as discussed
in more detail in Sec. V.

##  IV Convolutional Neural Network

###  IV.1 Introduction to neural network approach

Deep learning methods have recently been demonstrated as a powerful tool for
solving physical and in particular optical problems, for example, for deeply
subwavelength optical imaging [30, 31, 32], analysis of scatterometry data
[33, 34], enhanced resolution of SEM images [35], MRI image analysis [36, 37],
inverse design of optical components [4, 5] and many other image processing
applications, revolutionising their future development. Deep learning methods
are machine learning methods employing multilayer (3 and more layers) neural
networks where subsequent layers extract finer level features from the raw
input data. Deep neural networks automatically find the correct mathematical
transformations to convert the input data to outputs results without requiring
prior knowledge of the linear or nonlinear correlations. Convolutional neural
networks (CNN) [38], long short-term memory (LSTM) [39], variational
autoencoders (VAE) [40], and generative adversarial networks (GAN) are
examples of deep learning algorithms used for physical tasks [41].

###  IV.2 Specific network topology

In our paper we present the application of a deep convolutional neural network
for predicting the required parameters of the cavity (period and amplitude of
the sinusoidal mirror profile modulation) from a target cooperativity profile
$C(\textbf{r})$. The forward design mode solver outputs a set of modes for one
geometry. Depending on the application we then select the mode of highest
cooperativity, highest finesse, two-peaked cooperativity etc. from the full
set of modes corresponding to each geometry. Each mode is represented as a
field map in a longitudinal and radial cross-section of the cavity of 500
$\mu$m length in z-direction and 400 $\mu$m in diameter. Each field map
consists of $200\times 200$ points. Exploiting the fact that all modes are
radially and center-symmetric, we feed the neural network with only a quarter
of the field maps ($100\times 100$) in order to save computer memory.

The neural network we use in our studies consists of four convolutional layers
with 3x3 kernels with max-pool layers inserted between the first three
convolutional layers [42]. The topology of the neural network is shown in Fig.
2(c). All convolutional layers are activated by rectified linear unit (ReLU)
functions. The final convolutional layer is connected to three fully connected
layers with 512, 32 and two neurons, respectively. The hidden fully connected
layers have ReLU activation functions, and the output layer is activated by
sigmoid functions [43]. These fully-connected layers are needed for linking
the feature representations from the convolutional layers with the output
labels. This is a typical structure of a convolutional neural network which
allows for automatic feature extraction for solving the inverse problem of
finding the design of a device from the generated field map. The network was
trained with the Adam stochastic optimization method with a learning rate of
0.001. The mean square error (MSE) was used as the loss function.

The neural network takes the field map as an input, outputting the period and
amplitude both normalized to their maximum values in the whole dataset (i.e.,
between 0 and 1). 80% of the whole dataset was used for training, 10% for
validation and the last 10% for testing.

###  IV.3 Technical details

We use a self-made mode solver based on the MMM [26, 27] to generate the
training data. The data contains 85000 numerical simulations, where each
sample is a full set of modes with spatial field distribution and finesse. We
use the python programming language with libraries numpy for computation and
mpi4py for distribution of simulations over an HPC cluster.

The CNN training software was also written in the python language. The open
source machine learning framework Tensorflow was used for the deep neural
networks and training took place on an Nvidia GPU supplied with CUDA.

##  V Results and discussion

###  V.1 Mode prediction and verification

(a)
(b)

Figure 3: Correlation coefficient between the true parameters of the harmonic
mirror modulations (red dots) and the values predicted by the trained network
(blue dots) for (a) period $P$ and (b) amplitude $A$. The correlation
coefficients are 0.998826 and 0.996890 for period and amplitude, respectively,
after training with 90% (10% of that was used for validation) and for a test
set of 10% out of $N=85000$ data sets.

For the examples discussed here we choose a reference spherical cavity of
length $L=500~{}\mu$m, mirror radius of curvature $R=400~{}\mu$m, and mirror
diameter $200~{}\mu$m. Harmonic modulations of the mirror profiles are chosen
with random period $P$ and amplitude $A$ as discussed in Sec. III.

The results of the neural network training are shown in Fig. 3. This plot
compares true parameters (red line) with CNN predictions (blue dots). This
curve is based on a 10% subset of the forward design data set of 85000 samples
which were separated from the training set and have not been used in training
or validation (made after every epoch by CNN). From the figure we see that the
correlation coefficient between the true and retrieved values of both
amplitude and period are over 99%, which demonstrates a high quality of the
retrieval process.

Figure 4 presents the training convergence curves (learning curves) where
training MSE and validation MSE versus training epoch show a common, smooth
convergence. This confirms that the neural network does not overfit in the
process of training.

Figure 4: Learning curves for CNN training: mean square error of the training
set (red dots) and of the validation set (blue dots) during the learning
procedure within 150 epochs.

Next we use the trained CNN to predict the mirror modulation parameters for a
target mode profile. This target mode in Fig. 5 was chosen to provide a
desired cooperativity enhancement at the cavity center and a specific mode
shape. Then we use the predicted mirror profile in our mode solver to confirm
the generated mode field. Figure 5(a) compares the target mode (red dashed
curve) and the predicted mode (black solid) in a radial cross section. The
blue curve corresponds to the fundamental Gaussian mode of the spherical
cavity of the same size, confirming that we achieved an enhancement of
cooperativity by a factor of three. Figure 5(b) shows the predicted mode
profile in cylindrical coordinates.

(a)

(b)

Figure 5: Demonstration of CNN inverse design. (a) shows the radial cross
section of the target mode (red dashed curve), the numerically verified
predicted mode (black solid), and the fundamental Gaussian mode of the
spherical cavity of the same size (blue curve). $n$ is the mode number. (b)
shows the predicted mode in cylindrical coordinates.

In a second example, Fig. 6, we target a mode comprising two maxima. We define
the target as the sum of two Gaussian profiles shifted apart along the z-axis.
The cross section of the target and the predicted modes along the z-axis are
shown in Fig. 6(a). The shape of the predicted mode is slightly different from
the target mode, which respects the fact that not every mode topology can be
supported by our family of mirror shapes, i.e., a double-peak Gaussian mode is
not necessarily an eigenmode of a spherical mirror with any single harmonic
modulation. However, in this case the CNN finds the closest mode to the target
mode and Fig. 6(b) shows that the main feature of the target mode of two peaks
on the z-axis is well reproduced.

(a)

(b)

Figure 6: Demonstration of CNN inverse design. (a) shows the axial cross
section of the target mode (red dashed curve) and the numerically verified
predicted mode (black solid). (b) shows the predicted mode in cylindrical
coordinates confirming the two-peak structure.

The capability of the CNN to predict modes with a desired topology depends on
the training set composition, especially on the balance between desired modes
and other modes contained in the set. We should emphasize that the training
sets for the one-peak mode (Fig. 5) and the two-peak mode (Fig. 6) are very
different. For every mirror geometry our mode solver generates a set of modes;
the exact number depends on the desired precision and in our numerical
simulations was set to 30 modes. However, only one mode of this set is
included in the training data and all other modes are ignored. An essential
part of our method is therefore to define and computationally implement this
selection rule. Depending on the selection rule we create different training
sets for different target mode topologies.

###  V.2 Errors and limitations of the design method

The essential limitation of our method is given by the nature of the neural
network learning process. The neural network cannot predict modes which are
very different from the modes in the training set because the CNN is
effectively an interpolator. If we set a target mode which cannot exist within
the chosen parameter range of the training set, for example a target mode
which can only exist for a different type of mirror geometry, then the CNN
predicts the closest mode to the target one. In Fig. 7 we present such a case:
here the target mode is a Gaussian shape that decays to zero both in the
radial and the longitudinal direction. Such a mode is clearly unphysical as
the light must always be bouncing between the cavity mirrors and thus cannot
decay to zero in the longitudinal direction. Thus, the CNN finds a mode that
matches the target mode accurately in the radial direction, but the mode
deviates from the target in the axial direction. This is typical for this kind
of methods when the target field topology is not within the topologies in the
training set.

(a)

(b)

Figure 7: Attempted inverse design of a target mode (red dashed curve) that is
not within the topologies of the training set. The numerically verified mode
of the CNN prediction (black solid curve) matches in the radial direction (a)
but deviates in the longitudinal direction (b).

###  V.3 Fabrication considerations

The mirror geometries investigated here require the fabrication of a harmonic
modulation with scales of tens of micrometers in the radial direction and
hundreds of nanometers in the axial direction on a spherical mirror profile.
These scales are defined by the ranges of parameters which were used for the
generation of the training data. Since a CNN is an interpolator but not an
extrapolator, any predictions of mirror geometries beyond the training range
of parameters will be inaccurate. Mirror profiles with our specifications can
be created by a number of fabrication techniques, such as mirror shaping by
laser machining in silica glass [44, 45, 46] or by focused ion beam milling
[19]. Pulsed CO2 lasers can be used for thermal evaporation of surface
material [25] generating surfaces with extremely low roughness. Also modern
micro-machining tools can provide sufficient precision [20, 48, 49, 50].

##  VI Conclusions

We have demonstrated an approach based on convolutional neural networks for
the design of high-finesse optical resonators which support a desired
electrical mode field and, crucially, simultaneously limit the mode decay
rate. This allows us to design target fields with, for example, one or two
field maxima on the cavity longitudinal axis where significant enhancement of
the cooperativity between quantum emitters coupled to cavity photons is
achieved. Such cavity designs are crucial for various quantum optics and
quantum technology applications. The proposed “mode on-demand technique” can
be extended to many types of mode topologies by implementing appropriate mode
selection rules for the training set generation.

Numerically the most demanding part of our approach is the generation of the
training data set. We used an efficient mode mixing method that works in the
paraxial limit of cavity optics. However, other methods can be chosen for
improved accuracy or implementation of the algorithm for other inverse optical
design problems. This flexibility of our CNN formalism will make it useful for
many practical applications in the future.

The data supporting this study are openly available from the University of
Southampton repository [51].

##  VII Acknowledgments

We acknowledge financial support by the UK Quantum Technology Program under
the EPSRC Hub in Quantum Computing and Simulation (EP/T001062/1). The
University of Southampton supercomputer Iridis 5 was used for numerical
simulations.

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