# ONE-PARAMETER FAMILIES OF CONFORMAL MAPPINGS OF THE HALF-PLANE ONTO
POLYGONAL DOMAINS WITH SEVERAL SLITS

A. Posadskii  Lebedev Physical Institute, Moscow, 119991; Saint Petersburg
University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia
<EMAIL_ADDRESS>and  S. Nasyrov  Kazan Federal University, Kazan,
420008; Saint Petersburg University, 7/9 Universitetskaya nab., St.
Petersburg, 199034 Russia<EMAIL_ADDRESS>
###### Abstract.

Among various methods of finding accessory parameters in the Schwarz-
Christoffel integrals, Kufarev’s method, based on the Loewner differential
equation, plays an important role. It is used for describing one-parameter
families of functions that conformally map a canonical domain onto a polygon
with a slit the endpoint of which moves along a polygonal line starting from a
boundary point. We present a modification of Kufarev’s method for the case of
several slits, the lengths of which have depend of each other in a certain
way. We justify the method and find a system of ODEs describing the dynamics
of accessory parameters. We also present the results of numerical calculations
which confirm the efficiency of our method.

Keywords: Schwarz-Christoffel integral, accessory parameters, Loewner
equation, parametric method, Kufarev method, Cauchy problem, ODE system.

Mathematics Subject Classification: 30C30, 30-08.

## Introduction

As it is known, the integral formula giving a conformal mapping of a canonical
domain onto a given polygonal domain was independently derived by E.
Christoffel [6] and H. Schwarz [27]. The Schwarz-Christoffel integral contains
unknown (accessory) parameters, and in general case the problem to find them
is very difficult (see, e.g. [9]).

Many approaches have developed to solve this problem, including analytical
methods suggested by M. A. Lavrentiev [20, 21], A. Weinstein [31, 32], S.
Bergman [2, 3] and others. With the development of computing power, numerical
methods are being actively developed; some of them are presented in the works
of V. V. Sobolev [15, 28], L. N. Trefesen [30], R. T. Davis [7] and others.
Later these methods were implemented in software packages (see, e.g., the
Driscoll MATLAB toolbox [8]). We also note a new interesting method, suggested
by S. I. Bezrodnykh [4]; it is based on analytic continuation of the
Lauricella function.

The famous Loewner parametric method [11, 1] has influenced the process of
solving the accessory parameters problem. The main idea of the method is to
consider families of univalent functions depending on a real parameter and
mapping some canonical domain (such as the unit disk or the upper half-plane)
onto a family of domains with a growing slit. If the trajectory of the slit is
a polygonal line and the endpoint of the slit moves along one of its line
segment, then the family satisfies some differential equation. Using this
approach, P. P. Kufarev (see, e.g. [14], [1, Ch.VI]) reduced the problem of
finding the accessory parameters to integrating several Cauchy problems for
systems of ordinary differential equations. In this method, the process of
finding the accessory parameters consists of several steps. At every step, one
of such systems is solved and its initial conditions are determined from the
results obtained on the previous step. The values obtained at the last step,
are desirable ones. Some numerical results based on the Kufarev’s method can
be found in [5] and [13].

Later, V. Ya. Gutlyansky and A. O. Zaidan [12] proposed a modification of
Kufarev’s method. They consider conformal mappings in the upper half-plane
$\mathbb{H}^{+}$ and cut the needed polygon $P$ out of another polygon
$P^{\prime}$, not a canonical domain. Geometrically, this means that at first
a rectilinear slit grows from a boundary point of a polygon $P^{\prime}$,
then, at the second step, the slit changes its direction and grows from the
endpoint of the slit obtained at the first step and so on. At the last step,
the endpoint of the slit reaches the boundary of $P^{\prime}$ (see Fig. 1.).
The whole trajectory $L$ of the endpoint of the slit subdivides $P^{\prime}$
into two parts, every of which is a polygonal domain; they are kernels of the
family of domains with growing slit. If one of the kernels is the needed
polygonal domain $P$ and the normalization of functions, mapping conformally
$\mathbb{H}^{+}$ onto domains with slits along subarcs of $L$, is fixed
appropriately, then, according to the Caratheodory kernel convergence theorem
(see, e.g., [11, ch. II], the function corresponding to the limiting (final)
values of the accessory parameters maps the upper half-plane onto $P$. Thus,
we cut out, like scissors, the polygonal domain $P$ with vertices $A_{1}$,
$A_{2},\ldots,A_{n}$ from the original polygon $P^{\prime}$ with vertices
$A^{\prime}_{1}$, $A^{\prime}_{2},\ldots,A^{\prime}_{n^{\prime}}$. On Fig. 1
we illustrate the situation in a case where; here $P^{\prime}$ and $P$ have
common vertices $A^{\prime}_{j}=A_{j}$,
$j\in\left\\{1,\ldots,l,k+3,\ldots,n\right\\}$).

Figure 1. Polygon $P^{\prime}$ with a growing slit along a part of the
boundary of the polygon $P$.

Kufarev’s method showed its efficiency and there arises an interest in its
generalization. Thus, using the parametric method, it is possible to find
approximate conformal mappings of canonical domains onto polycircular arc
domains (see, for example, [1]). Recently, I. A. Kolesnikov [16, 17] proposed
an improvement of this approach. Also he used the parametric method to find
accessory parameters for conformal mappings of polygonal domains with a
countable set of vertices and the translation symmetry [18]. Moreover I. A.
Kolesnikov [19] developed a more general method that allows to carry out
parallel translation of the sides of a polygon. L. Yu. Nizamieva [24]
suggested an approximate method for finding accessory parameters in the
Schwarz-Christoffel integrals, based on Kufarev’s method and the apparatus of
Hilbert’s boundary value problems. In addition, she develop this approach for
solving some boundary value problems with a free boundary [25]. A
generalization for polygonal domains on Riemann surfaces was given by N. N.
Nakipov and S. R. Nasyrov [23].

In this paper, we suggest a modification of the results obtained by V. Ya.
Gutlyansky and A. O. Zaidan [12] for the case where several slits in an
initial polygon grow simultaneously. The advantage of the proposed approach is
that it becomes possible to reduce significantly the number of successive
steps, in comparison to [12]. When solving the problem, we can perform
calculations in fewer steps. For example, for a convex polygon we can make it
in one step.

Now we will describe one of the possible variants. Let we have a polygonal
domain $P^{\prime}$ with vertices $A^{\prime}_{1}$,
$A^{\prime}_{2},\ldots,A^{\prime}_{n^{\prime}}$ and we want to cut out of
$P^{\prime}$ a convex polygonal domain $P$ with vertices $A_{1}$,
$A_{2},\ldots,A_{n}$ from $P^{\prime}$. Assume that we obtain $P$ from
$P^{\prime}$ drawing the slit along the polygonal line $A_{l+1}A_{l+2}\ldots
A_{k+2}$ (Fig. 2). Then, for every $j$, $l+1\leq j\leq k+1$, we draw the ray
$L_{j}$ originated at $A_{j}$ and passing through the next vertex $A_{j+1}$.
Let the ray $L_{j}$ intersect the boundary of the polygon $P^{\prime}$ at some
point $B_{j}$ ($B_{k+1}=A_{k+2}$). Then we simultaneously release slits from
the points $B_{j}$, $l+1\leq j\leq k+1$, which endpoints move along the rays
$L_{j}$ to the points $A_{j}$. The growth velocities of slits can be chosen so
that the endpoints of the $j$th slits reach simultaneously the points $A_{j}$,
$l+1\leq j\leq k+1$. As a result, we see that, unlike the classic Kufarev’s
method, we reduced the number of steps to one. The only thing is that for
carrying out such an algorithm it is necessary to solve a more complicated
system of ordinary differential equations. However, the system does not
essentially differ from the systems of ODEs used in the case of one slit.
Thus, we see that the developed method can be used for a more simple and fast
search for the accessory parameters.

Figure 2. Cutting a convex polygonal domain $P$ out of $P^{\prime}$ in one
stage. On the figure, $A^{\prime}_{i^{\prime}}=A_{i}$ for
$i^{\prime}=i\in\left\\{1,\ldots,l,k+3,\ldots,n\right\\}$.

To justify the method, it is important to choose an appropriate parameter with
respect to which the family of mappings is differentiable. We consider the
Schwarz-Christoffel integral of a sufficiently arbitrary form giving a
conformal mapping onto a polygonal domain with fixed angles and investigate
the dependence between the lengths of the sides of polygon and the accessory
parameters. We show that the accessory parameters depends smoothly on the
lengths of the sides (Actually, this fact was established by A. Weinstein
[31], [32]). In the case of a fixed polygonal domain with growing slits, such
that the lengths are connected with each other smoothly, this fact allows us
to take, as a suitable parameter, the length of one of these slits.

##  1\. One-parameter families of conformal mappings onto polygons with
several slits

Let $D_{n}$ be a simply connected domain in $\mathbb{C}$, whose boundary is a
(not necessarily bounded) $n$-gon, $n\geq 3$, with vertices
$A_{1},\ldots,A_{n}$ and interior angles $\pi\alpha_{k}$,
$\left|\alpha_{k}\right|\leq 2$, $1\leq k\leq n$ (the value of $\alpha_{k}$ is
negative if $A_{k}=\infty$). Then the conformal mapping of the upper half-
plane $\mathbb{H}^{+}$ onto $D_{n}$ exists and can be represented by the
Schwarz-Christoffel formula:

(1) $f\left(z\right)=c\int_{0}^{z}\prod_{k=1}^{n}\left(\zeta-
a_{k}\right)^{\alpha_{k}-1}d\zeta+c_{1},$

where $a_{1},\ldots,a_{n}$ are the preimages of the vertices
$A_{1},\ldots,A_{n}$, $c\neq 0$ and $c_{1}$ are some complex numbers. The
numbers $a_{k}$, $1\leq k\leq n$, $c$ and $c_{1}$ are not known beforehand,
they are called accessory parameters. Moreover, due to the properties of
conformal mappings, we can fix any three of the parameters $a_{k}$. Then the
remaining accessory parameters and, therefore, the conformal mapping are
defined in a unique way. Further we will assume that $a_{n-2}=0$, $a_{n-1}=1$,
$a_{n}=\infty$, and the vertices $A_{n-2}$, $A_{n-1}$, $A_{n}$ are at finite
points of the plane; this does not essentially restrict the generality. In
this case, $-\infty<a_{1}<a_{2}<\ldots<a_{n-3}<0$ and (1) takes the form

$f\left(z\right)=c\int_{0}^{z}\prod_{k=1}^{n-1}\left(\zeta-
a_{k}\right)^{\alpha_{k}-1}d\zeta+A_{n-2}.$

Let us fix $m$ different points $B_{1}$, $B_{2},\ldots,B_{m}$ on $\partial
D_{n}$, not lying on the segments $A_{n-2}A_{n-1}$ and $A_{n-1}A_{n}$, and
release from them inside $D_{n}$ disjoint rectilinear slits of growing lengths
depending on a real parameter $t\in[0,T]$. Denote by
$\Lambda_{1}(t),\ldots,\Lambda_{m}(t)$ the endpoints of the slits. The domain
$D_{n}$ with slits, corresponding to the value of the parameter $t$, will be
denoted by $D_{n}(t)$. We also assume that at $t=0$ the lengths of all slits
are equal to zero, and for $t\geq 0$ they are strictly increasing. Since the
domain $D_{n}(t)$ is also polygonal, the conformal mapping
$f\left(z,t\right):\mathbb{H}^{+}\rightarrow D_{n}(t)$, satisfying the
normalization conditions
$f\left(0,t\right)=A_{n-2},f\left(1,t\right)=A_{n-1},f\left(\infty,t\right)=A_{n}$,
can be also represented as a Schwarz-Christoffel integral.

For every $i$, $1\leq i\leq m$, the length of the $i$-th slit grows, and at
the initial moment $t=0$ the point $B_{i}$ bifurcates. Therefore, for $t>0$
there are two boundary elements (prime ends in the sense of Caratheodory) in
$D_{n}(t)$ supported at the point $B_{i}$; they lie on the different sides of
the slit. Denote by $a_{i1}(t)$ and $a_{i2}(t)$ their preimages under the
conformal mapping of the upper half-plane onto $D_{n}(t)$, and let
$\pi\alpha_{i1}$ and $\pi\alpha_{i2}$ be the angles of $D_{n}(t)$ at these
points. We numerate $a_{ij}$ so that $a_{i1}(t)<a_{i2}(t)$, $1\leq i\leq m$.
Denote by $\lambda_{i}(t)$ the preimage of the endpoint of the $i$-th slit; it
is located on the segment $\left[a_{i1}(t),a_{i2}(t)\right]$. Let also
$\sigma_{k}=\alpha_{k}-1$, $\sigma_{ij}=\alpha_{ij}-1$.

Now we will write the Schwarz-Christoffel integral mapping the upper half-
plane onto $D_{n}(t)$. Here we distinguish two cases.

1) If none of the slits leaves a vertex of the polygon $\partial D_{n}$, then
for all $i$ we have $\alpha_{i1}+\alpha_{i2}=1$, i.e.
$\sigma_{i1}+\sigma_{i2}=-1$, and

(2)
$f\left(z,t\right)=c(t)\int_{0}^{z}\prod_{l=1}^{m}\left(\zeta-\lambda_{l}(t)\right)\prod_{i=1}^{m}\prod_{j=1}^{2}\left(\zeta-
a_{ij}(t)\right)^{\sigma_{ij}}\prod_{k=1}^{n-1}\left(\zeta-
a_{k}(t)\right)^{\sigma_{k}}d\zeta+A_{n-2}.$

2) If one of the initial points of the slits coincides with a vertex of the
initial polygon, for example, $B_{l}=A_{k}$, then we omit the factor in the
formula $\left(\ref{2.3}\right)$ with $k=l$; we do the same for a larger
number of slits growing from vertices.

In what follows, for greater clarity, we will consider only case 1), unless
otherwise stated, although considering the case 2) is not difficult.

##  2\. Derivation of the Loewner equation

Information about the dynamics of the mapping (2) with a change in the
parameter $t$ is given by the Loewner equation. Moreover, the equation allows
us to find a system of ordinary differential equations to determine the
accessory parameters. For the case of mapping of the unit disk, the
corresponding equations can be found, for example, in the monograph [1].

Here we consider the case of several slits. When deriving the Loewner
equation, we will follow the scheme, proposed in [12] where the case of a
single slit was considered.

We start by proving a consequence of the Schwarz lemma.

######  Lemma 2.1.

Let $w(z)$ be a conformal mapping of $\mathbb{H}^{+}$ onto $\mathbb{H}^{+}$
with $m$ slits along disjoint analytic arcs $\Gamma_{1}$,
$\Gamma_{2},\ldots,\Gamma_{m}$, growing from the points
$\lambda_{1},\lambda_{2},\ldots,\lambda_{m}$ on $\partial H$ such that
$-\infty<\lambda_{1}<\lambda_{2}<\ldots<\lambda_{m}<0$ and $w(0)=0$, $w(1)=1$,
$w(\infty)=\infty$. Then $w(z)$ is holomorphic at $z=1$ and
$\left.\frac{dw}{dz}\right|_{z=1}\leq 1$. If
$\left.\frac{dw}{dz}\right|_{z=1}=1$, then $w\left(z,t\right)\equiv z$.

###### Proof.

There are $m$ segments
$\left[\alpha_{1},\beta_{1}\right],\left[\alpha_{2},\beta_{2}\right],\ldots,\left[\alpha_{m},\beta_{m}\right]\in\mathbb{R}^{-}$
which the conformal mapping $w\left(z\right)$ maps to the slits along the arcs
$\Gamma_{1}$, $\Gamma_{2},\ldots,\Gamma_{m}$. We have
$\alpha_{i}<\beta_{i}<\alpha_{i+1}<\beta_{i+1}$, $1\leq i\leq m-1$.

By the Riemann-Schwarz symmetry principle, $w\left(z\right)$ can be
analytically continued into the lower half-plane through $\mathbb{R}^{+}$. In
particular, $w(z)$ is holomorphic at the point $z=1$ and, using the geometric
meaning of the derivative argument, we conclude that $\mathop{\rm
Im}\nolimits\left.\frac{dw}{dz}\right|_{z=1}=0$. The function
$\phi\left(\zeta\right)=(\zeta+1)^{2}(\zeta-1)^{-2}$ provides a conformal
mapping of the unit circle $\left|\zeta\right|<1$ onto a plane with a slit
along $\mathbb{R^{-}}$.

Applying the Schwarz lemma to the holomorphic in $|\zeta|<1$ function
$\Phi\left(\zeta\right)=\left(\phi^{-1}\circ
w\circ\phi\right)\left(\zeta\right)$ we obtain the required assertion. ∎

Now we will return to consideration of the family of functions $f(z,t)$,
defined by (2). We introduce the function
$w\left(z,t\right)=f^{-1}\left(f\left(z,T\right),t\right)$, where $0\leq t\leq
T$, conformally mapping $\mathbb{H}^{+}$ onto $\mathbb{H}^{+}$ with $m$ slits
and satisfying $w(0,t)=0$, $w(1,t)=1$, $w(\infty,t)=\infty$. The points on
$\partial H$ from which the slits grow are located on $\mathbb{R^{-}}$. By the
Caratheodory kernel convergence theorem, $w(z,\,\cdot\,)$ is continuous on
$\left[0,T\right]$ for a fixed $z\in\mathbb{H}^{+}$. By virtue of Lemma 2.1
and the Weierstrass theorem on the convergence of a sequence of holomorphic
functions,

(3) $\frac{dw}{dz}\left(1,t\right)=q(t)$

is a non-negative continuous strictly increasing function of the parameter
$t$. Indeed, if $0\leq s<t\leq T$ then, by Lemma 2.1, applied to the function
$w^{-1}\left(w\left(z,s\right),t\right)$, we conclude that
$\frac{dw}{dz}\left(1,s\right)\left(\frac{dw}{dz}\left(1,t\right)\right)^{-1}<1$.

For convenience, we will reparametrize our family, taking the value $\tau=\ln
q(t)$ as a new parameter. In what follows, for simplicity of notation, we will
assume that $q(t)=\exp(t)$; such a parametrization will be called canonical.

Now consider the function $h(z,t,s)=f^{-1}(f(z,t),s)$, $0\leq s\leq t\leq T$,
conformally mapping $\mathbb{H}^{+}$ onto $\mathbb{H}^{+}$ with $m$ slits; it
keeps the points $0$, $1$, and $\infty$. According to the Schwarz formula for
the half-plane, we obtain:

(4)
$h\left(z,t,s\right)=z+\frac{z\left(z-1\right)}{\pi}\sum_{i=1}^{m}I_{i}\left(z,t,s\right),$

where

$I_{i}\left(z,t,s\right)=\int_{\alpha_{i}\left(s\right)}^{\beta_{i}\left(s\right)}\frac{\mathop{\rm
Im}\nolimits h\left(x,t,s\right)}{\left(x-z\right)\left(x-1\right)x}dx.$

Since $h\left(w\left(z,t\right),t,s\right)=w\left(z,s\right)$, this implies

(5)
$w\left(z,s\right)-w\left(z,t\right)=\frac{w\left(z,t\right)\left(w\left(z,t\right)-1\right)}{\pi}\sum_{i=1}^{m}I_{i}\left(w\left(z,t\right),t,s\right).$

Going to the limit, as $z\rightarrow 1$, from (4) and the definition of
$h(z,t,s)$ we obtain

$\exp\left(s-t\right)=\frac{dh}{dz}\left(1,t,s\right)=1+\frac{1}{\pi}\sum_{i=1}^{m}J_{i}\left(t,s\right),$

where

$J_{i}\left(t,s\right)=I_{i}\left(1,t,s\right)=\int_{\alpha_{i}\left(s\right)}^{\beta_{i}\left(s\right)}\frac{\mathop{\rm
Im}\nolimits h\left(x,t,s\right)}{\left(x-1\right)^{2}x}\,dx<0.$

Therefore, we have

(6)
$s-t\sim\exp\left(s-t\right)-1=\frac{1}{\pi}\sum_{i=1}^{m}J_{i}\left(t,s\right),\quad
s\to t.$

By the mean value theorem, we have

$I_{i}\left(w\left(z,t\right),t,s\right)=\int_{\alpha_{i}\left(s\right)}^{\beta_{i}\left(s\right)}\frac{\mathop{\rm
Im}\nolimits h\left(x,t,s\right)}{\left(x-1\right)^{2}x}dx\cdot
K_{i}=J_{i}\left(t,s\right)\cdot K_{i},$

where

$K_{i}=\mathop{\rm
Re}\nolimits\frac{x_{i,1}-1}{x_{i,1}-w\left(z,t\right)}+i\mathop{\rm
Im}\nolimits\frac{x_{i,2}-1}{x_{i,2}-w\left(z,t\right)},\quad
x_{i,1},x_{i,2}\in\left[\alpha_{i}\left(s\right),\beta_{i}\left(s\right)\right],$

therefore, from (6) we obtain

(7) $\frac{\partial w}{\partial t}=\lim_{s\rightarrow
t}\frac{w\left(z,t\right)\left(w\left(z,t\right)-1\right)}{\pi}\sum_{i=1}^{m}\left(J_{i}\left(t,s\right)\cdot
K_{i}\right)\left(\frac{1}{\pi}\sum_{k=1}^{m}J_{k}\left(t,s\right)\right)^{-1}.$

Below we will show that if the lengths of the slits smoothly depend on each
other, for every $i$, $1\leq i\leq m$, there exist the limit

$C_{i}(t)=\lim_{s\rightarrow
t}\left(J_{i}\left(t,s\right)\left(\sum_{k=1}^{m}J_{k}\left(t,s\right)\right)^{-1}\right).$

We note that $C_{i}(t)\geq 0$ as the limit of the ratio of two negative
numbers. In addition, there is a relationship between these coefficients:

$\sum_{i=1}^{m}C_{i}(t)=1.$

Then, taking into account the introduced notation, we obtain

$\frac{\partial w}{\partial
t}=w\left(z,t\right)\left(w\left(z,t\right)-1\right)\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)-1}{\lambda_{i}(t)-w\left(z,t\right)}.$

Recalling that $w\left(z,t\right)=f^{-1}\left(f\left(z,T\right),t\right)$, we
get

(8) $\frac{\partial f}{\partial t}=-\frac{\partial f}{\partial
z}z\left(z-1\right)\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)-1}{\lambda_{i}(t)-z}.$

At last, we should note that, in fact, we consider the limit if $s$ approaches
$t$ from the left; the limit from the right is studied in a similar way.

######  Remark 2.1.

If we did not use the change to the canonical parameter, we would get the
equation similar to (8) but with the additional multiplier $d(\ln q(t))/dt$ in
the right-hand side; under some natural conditions, we will show the
smoothness of the function $q(t)$ in Subsection 6.2. Therefore, the form of
(8) do not change, since we can multiply every $C_{i}(t)$ by this expressions
and denote the result of multiplication, say, by $\widetilde{C}_{i}(t)$.

######  Remark 2.2.

We can consider similar families of conformal mappings with normalization such
that $f(0,t)$, $f(1,t)$ and $f(\infty,t)$ do not depend on $t$ but are not
necessarily angular points of the corresponding polygonal domains. Then, as it
is easy to see, the form of the derived equation does not change.

##  3\. ODE system for accessory parameters

Now we will derive a system of equations for determining the accessory
parameters.

######  Theorem 3.1.

The accessory parameters satisfy on $[0,T]$ the following system of
differential equations:

$-\frac{d\lambda_{p}}{dt}=\lambda_{p}(t)\left(\lambda_{p}(t)-1\right)\left(\sum_{l=1,l\neq
p}^{m}C_{l}(t)\frac{\lambda_{l}(t)-1}{\lambda_{p}(t)-\lambda_{l}(t)}+C_{p}(t)\left(\lambda_{p}(t)-1\right)\right.$
$\left.\times\sum_{l=1,l\neq
p}^{m}\frac{1}{\lambda_{p}(t)-\lambda_{l}(t)}\right)+C_{p}(t)\left(2\lambda_{p}(t)-1\right)\left(\lambda_{p}(t)-1\right)+C_{p}(t)\lambda_{p}(t)$
(9)
$\times\left(\lambda_{p}(t)-1\right)^{2}\left(\sum_{i=1}^{m}\sum_{j=1}^{2}\frac{\sigma_{ij}}{\lambda_{p}(t)-a_{ij}(t)}+\sum_{k=1}^{n-1}\frac{\sigma_{k}}{\lambda_{p}(t)-a_{k}(t)}\right),\
1\leq p\leq m,$ (10)
$\frac{da_{l}}{dt}=-a_{l}(t)\left(a_{l}(t)-1\right)\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)-1}{a_{l}(t)-\lambda_{i}(t)},\
1\leq l\leq n-3,$ (11)
$\frac{da_{ij}}{dt}=-a_{ij}(t)\left(a_{ij}(t)-1\right)\sum_{l=1}^{m}C_{l}(t)\frac{\lambda_{l}(t)-1}{a_{ij}(t)-\lambda_{l}(t)},\
1\leq i\leq m,\ 1\leq j\leq 2,$ (12)
$\frac{1}{c(t)}\frac{dc}{dt}=-\alpha_{n}\sum_{i=1}^{m}C_{i}(t)\left(\lambda_{i}(t)-1\right).$

###### Proof.

Consider the function $\phi\left(z,t\right)=\ln\frac{\partial f}{\partial z}$.
Because of (8), it satisfies the differential equation

$-\frac{\partial\phi}{\partial t}=\frac{\partial\phi}{\partial
z}z\left(z-1\right)\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)-1}{\lambda_{i}(t)-z}$
(13)
$+\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)\left(\lambda_{i}(t)-1\right)^{2}}{\left(\lambda_{i}(t)-z\right)^{2}}-\sum_{i=1}^{m}C_{i}(t)\left(\lambda_{i}(t)-1\right).$

From (2) it follows that

$\phi\left(z,t\right)=\ln
c(t)+\sum_{l=1}^{m}\ln\left(z-\lambda_{l}(t)\right)+\sum_{i=1}^{m}\sum_{j=1}^{2}\sigma_{ij}\ln\left(z-a_{ij}(t)\right)+\sum_{k=1}^{n-1}\sigma_{k}\ln\left(z-a_{k}(t)\right).$

Therefore,

(14) $-\frac{\partial\phi}{\partial
t}=-\frac{1}{c(t)}\frac{dc}{dt}+\sum_{l=1}^{m}\frac{1}{z-\lambda_{l}(t)}\frac{d\lambda_{l}}{dt}+\sum_{i=1}^{m}\sum_{j=1}^{2}\frac{\sigma_{ij}}{z-a_{ij}(t)}\frac{da_{ij}}{dt}+\sum_{k=1}^{n-1}\frac{\sigma_{k}}{z-a_{k}(t)}\frac{da_{k}}{dt},$
(15) $\frac{\partial\phi}{\partial
z}=\sum_{l=1}^{m}\frac{1}{z-\lambda_{l}(t)}+\sum_{i=1}^{m}\sum_{j=1}^{2}\frac{\sigma_{ij}}{z-a_{ij}(t)}+\sum_{k=1}^{n-1}\frac{\sigma_{k}}{z-a_{k}(t)}.$

Substituting (14) and (15) into (13), we get

$-\frac{1}{c(t)}\frac{dc}{dt}+\sum_{l=1}^{m}\frac{1}{z-\lambda_{l}(t)}\frac{d\lambda_{l}}{dt}+\sum_{i=1}^{m}\sum_{j=1}^{2}\frac{\sigma_{ij}}{z-a_{ij}(t)}\frac{da_{ij}}{dt}+\sum_{k=1}^{n-1}\frac{\sigma_{k}}{z-a_{k}(t)}\frac{da_{k}}{dt}=$
$=\left(\sum_{l=1}^{m}\frac{1}{z-\lambda_{l}(t)}+\sum_{i=1}^{m}\sum_{j=1}^{2}\frac{\sigma_{ij}}{z-a_{ij}(t)}+\sum_{k=1}^{n-1}\frac{\sigma_{k}}{z-a_{k}(t)}\right)z\left(z-1\right)\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)-1}{\lambda_{i}(t)-z}$
$+\sum_{i=1}^{m}C_{i}(t)\frac{\lambda_{i}(t)\left(\lambda_{i}(t)-1\right)^{2}}{\left(\lambda_{i}(t)-z\right)^{2}}-\sum_{i=1}^{m}C_{i}(t)\left(\lambda_{i}(t)-1\right).$

Comparing the residues of the left- and right-hand parts of the last relation
at the points $\lambda_{j}(t),a_{ij}(t),a_{l}(t)$ and the free terms, we
obtain the system (9)–(12). ∎

We note that the converse statement is also true. If the accessory parameters
satisfy the system (9)–(12), then the family of mappings $f(z,t)$ satisfies
(8). This implies that the solution to the ODE is unique, since the family of
conformal mappings given by the Loewner equation is uniquely determined.

Integration of equations (9)–(12) gives the dependence of the accessory
parameters on the parameter $t$.

Let us now describe how to define accessory parameters for a given polygon
$P$. First, one should fix the polygon $P_{0}$ inside of which slits grow. Its
specific form is chosen from considerations of convenience or the specifics of
the problem, while all accessory parameters for $P_{0}$ must be known. Next,
the Cauchy problem is solved, in which the initial conditions (for $t=0$) are
the values of the accessory parameters for the polygon $P_{0}$. The values of
the parameters $a_{i}$, $a_{ij}$, $\lambda_{i}$ at $t=T$ give the desired
values for the polygon $P$.

If the shape of $P$ is rather complicated and it is not possible to cut it out
of $P_{0}$ in one step, the above procedure is repeated. At subsequent stages,
the initial conditions for the Cauchy problem are determined by the polygonal
domain obtained at the previous step. Since the domains remains polygonal
after cutting off, this procedure can be carried out sequentially any finite
number of times.

Note that when integrating the system of equations (9)–(12), there is a
difficulty associated with the presence of degeneracy at the initial and,
possibly, at the final moments.

As $t\to 0$, three the accessory parameters corresponding to the end of the
slit and two vertices at its base, lying on different sides of the slit merge
into one. When solving the system numerically, one can get rid of the first
problem by slightly changing the initial data, for example, by taking
$a_{ij}\left(0\right)=\lambda_{i}\left(0\right)+(-1)^{j}\epsilon_{ij}$, $1\leq
i\leq m$, $j=1$, $2$ where $\epsilon_{ij}>0$ is sufficiently small111In the
numerical examples, given below, we took $\epsilon_{ij}=10^{-15}$., and thus
removing the degeneracy. However, with a rigorous justification of this
method, the question arises about the stability of the system with respect to
changes in the initial conditions.

As $t\to T$, a degeneracy may appear, since the end of the slit
$\Lambda_{k}(t)$ approaches some point of the boundary $E_{k}$. As we
mentioned in Introduction, the polygonal trajectory of the slit subdivide the
polygonal domain $P^{\prime}$ into two subdomains, $P$ and another one,
denoted it by $P^{\prime\prime}$. According to the G. D. Suvorov convergent
theorems [29], the preimages of all the vertices lying on the boundary of
$P^{\prime\prime}$ and the preimage of the endpoint of the slit converge to
the same point on $\partial\mathbb{H}^{+}$, as $t\to T$. This fact may cause
complications in numerical calculations.

##  4\. Solution of the system with the help of power series

After we have derived the system of differential equations for determining the
accessory parameters, the question arises of how to solve it.

Consider a special case. Assume that $C_{i}(t)$ are some given real-analytic
functions of the variable $x=\sqrt{t}$, $t\in[0,T]$. Then they can be
represented by power series

$C_{p}(t)=\widetilde{C}_{p}\left(x\right)=\sum_{n=0}^{\infty}\widetilde{C}_{p,n}x^{n},\
1\leq p\leq m.$

Similarly to the case of one slit, we use the change of variable $x=\sqrt{t}$
in the system (9)–(12) and seek for its solution in the form of power series:

(16) $\lambda_{p}\left(x\right)=\sum_{n=0}^{\infty}\lambda_{p,n}x^{n},\quad
1\leq p\leq m,$ (17)
$a_{k}\left(x\right)=\sum_{m=0}^{\infty}a_{k,m}x^{m},\quad 1\leq l\leq n-3,$
(18)
$a_{ij}\left(x\right)=\lambda_{i,0}+\sum_{m=1}^{\infty}a_{ij,m}x^{m},\quad
1\leq i\leq m,\ j=1,2.$

Substituting these expansions into the system (9)–(12) and equating the
coefficients at the first power of $x$ in the left- and right-hand parts, we
obtain:

$\lambda_{p,1}=q_{p}\sum_{j=1}^{2}\frac{\sigma_{pj}}{\lambda_{p,1}-a_{pj,1}}\,,\quad
1\leq p\leq m,$ $a_{ij,1}=\frac{q_{i}}{a_{ij,1}-\lambda_{i,1}}\,,\quad 1\leq
i\leq m,\ j=1,2,$ $a_{l,1}=0,\quad 1\leq l\leq n-3,$

where
$q_{\mu}=-2\widetilde{C}_{\mu,0}\lambda_{\mu,0}\left(\lambda_{\mu,0}-1\right)^{2}$.
From these equalities it follows that

$\begin{cases}\displaystyle\lambda_{i,1}=q_{i}\sum_{j=1}^{2}\frac{\sigma_{ij}}{\lambda_{i,1}-a_{ij,1}}\,,\\\\[5.69054pt]
\displaystyle
a_{ij,1}=-\frac{q_{i}}{\lambda_{i,1}-a_{ij,1}}\,,\end{cases}\Longrightarrow\ \
\begin{cases}\displaystyle\lambda_{i,1}=\left(\alpha_{i1}-\alpha_{i2}\right)\sqrt{\frac{q_{i}}{\alpha_{i1}\alpha_{i2}}}\,,\\\\[5.69054pt]
\displaystyle
a_{i1,1}=-\sqrt{q_{i}\frac{\alpha_{i2}}{\alpha_{i1}}}\,,\\\\[8.53581pt]
\displaystyle
a_{i2,1}=\sqrt{q_{i}\frac{\alpha_{i1}}{\alpha_{i2}}}\,.\end{cases}$

Similarly, comparing the coefficients at $x^{s}$, we obtain

$a_{i1,1}\lambda_{i,s}+\left(sa_{i2,1}-a_{i1,1}\right)a_{i1,s}=\phi_{i1},$
$a_{i2,1}\lambda_{i,s}+\left(sa_{i1,1}-a_{i2,1}\right)a_{i2,s}=\phi_{i2},$
$\lambda_{i,s}+\sum_{k=1}^{n-3}\sigma_{k}a_{k,s}+\sum_{j=1}^{2}\sigma_{ij}a_{ij,s}=\phi_{i}^{*},\
1\leq i\leq m,$ $a_{k,s}=\phi_{k},\ 1\leq k\leq n-3,$

where $\phi_{k},\ \phi_{ij}$ and $\phi_{i}^{*}$ only depend on the
coefficients of the preceding orders.

Further, arguing similarly to the case of one slit (see [1] and [12]), we
obtain that the power series (16)–(18) converge and represent the only
solution that is real-analytic with respect to the variable $x=\sqrt{t}$ in
some neighborhood of the point $0$.

However, as we will show below, another case is more important for our
purposes. It is interesting to study the case where the functions $C_{i}(t)$
are chosen not arbitrarily, but in such a way that provides a given dynamics
of the slit lengths (up to a parametrization). In this regard, we will discuss
this case in the next section.

##  5\. The control of slit lengths

Now we will study how the lengths of the slits depend on the parameter $t$.
Consider the dynamics of the endpoint $\Lambda_{r}(t)$ of the $r$-th slit that
is the image of the point $\lambda_{r}(t)$ under the mapping $f(z,t)$, i.e.
$\Lambda_{r}(t)=f(\lambda_{r}(t),t)$. Differentiating this equality, with the
help of (2), we can obtain an expression for the growth rate $v_{r}(t)$ of the
length of the $r$-th slit:

(19)
$v_{r}(t)=\biggl{|}\frac{d\Lambda_{r}(t)}{dt}\biggr{|}=|c(t)|A_{r}(t)C_{r}(t),$

where

(20)
$A_{r}(t)=\prod_{i=1}^{m}\prod_{j=1}^{2}|\lambda_{r}(t)-a_{ij}(t)|^{\sigma_{ij}}\prod_{k=1}^{n-1}|\lambda_{r}(t)-a_{k}(t)|^{\sigma_{k}}\\\
\times\,|\lambda_{r}(t)|\,|\lambda_{r}(t)-1|^{2}\prod_{i=1,i\neq
r}^{m}|\lambda_{r}(t)-\lambda_{i}(t)|\,.$

We see that $v_{r}(t)$ is proportional to $C_{r}(t)$ and does not depend
explicitly on the other functions $C_{j}(t)$, $j\neq r$. This fact can be used
to control the ratio between the slit lengths.

###  5.1. The case of two slits

Consider the case $m=2$, i.e. when there are only two slits. Assume, for
simplicity, that the ratio of the velocities of their lengths is a given
positive number $\alpha$, i. e.

(21) $\frac{v_{1}(t)}{v_{2}(t)}\equiv\alpha,$

where $v_{1}(t)$ and $v_{2}(t)$ are the growth rates of the lengths of the
first and second slits. Then from (19) and (21) we obtain

$\frac{C_{1}(t)}{C_{2}(t)}=\alpha\,\frac{A_{2}(t)}{A_{1}(t)}\,.$

Moreover, $C_{1}(t)+C_{2}(t)=1$, therefore,

(22) $C_{1}(t)=\frac{\alpha A_{2}(t)}{A_{1}(t)+\alpha A_{2}(t)}\,,\quad
C_{2}(t)=\frac{A_{1}(t)}{A_{1}(t)+\alpha A_{2}(t)}\,.$

Here $A_{1}(t)$ and $A_{2}(t)$ are given by (20) where we put $m=2$. If we
substitute the obtained expressions for $C_{1}(t)$ and $C_{2}(t)$ into the
right-hand sides of system (9)–(12), then we obtain the system to determine
the accessory parameters of conformal mappings $f(z,t)$ with the needed ratio
$\alpha$ of the growth rates of the lengths of the slits.

Now we will give some examples.

######  Example 5.1.

Consider a family of conformal mappings $f(z,t)$ from the upper half-plane
onto the upper half-plane with two rectilinear slits going orthogonally
upwards from given points $x_{1}$ and $x_{2}$, $x_{1}<x_{2}<0$; for any $t$
(Fig. 3).

Figure 3. The image of the Cartesian grid under the conformal mapping of the
upper half-plane onto a half-plane with two slits.

The conformal mapping $f(z,t)$ has the form

(23)
$f\left(z,t\right)=c(t)\int_{0}^{z}\prod_{l=1}^{2}\left(\zeta-\lambda_{l}(t)\right)\prod_{i,j=1}^{2}\left(\zeta-
a_{ij}(t)\right)^{-\frac{1}{2}}d\zeta,$

where the branch of the integrand is fixed such that it is positive for large
positive real $\zeta$; moreover, $f(z,t)$ is equal to the identity mapping at
$t=0$. From (23) it follows that the growth rates of the lengths
$L_{j}=L_{j}(t)$ of the slits are

$v_{1}(t)=\frac{dL_{1}(t)}{dt}=\frac{C_{1}(t)\left|c(t)\right|\lambda_{1}(t)\left(\lambda_{1}(t)-1\right)^{2}\left(\lambda_{1}(t)-\lambda_{2}(t)\right)}{\sqrt{\left(\lambda_{1}(t)-a_{11}(t)\right)\left(a_{12}(t)-\lambda_{1}(t)\right)\left(a_{21}(t)-\lambda_{1}(t)\right)\left(a_{22}(t)-\lambda_{1}(t)\right)}}\,,$
$v_{2}(t)=\frac{dL_{2}(t)}{dt}=\frac{C_{2}(t)\left|c(t)\right|\lambda_{2}(t)\left(\lambda_{2}(t)-1\right)^{2}\left(\lambda_{1}(t)-\lambda_{2}(t)\right)}{\sqrt{\left(\lambda_{2}(t)-a_{11}(t)\right)\left(\lambda_{2}(t)-a_{12}(t)\right)\left(\lambda_{2}(t)-a_{21}(t)\right)\left(a_{22}(t)-\lambda_{2}(t)\right)}}\,.$

Taking into account the ratio of growth rates of the slit lengths (21), we get

$\frac{1}{C_{2}(t)}=\alpha\sqrt{\frac{\left(\lambda_{1}(t)-a_{11}(t)\right)\left(a_{12}(t)-\lambda_{1}(t)\right)\left(a_{21}(t)-\lambda_{1}(t)\right)\left(a_{22}(t)-\lambda_{1}(t)\right)}{\left(\lambda_{2}(t)-a_{11}(t)\right)\left(\lambda_{2}(t)-a_{12}(t)\right)\left(\lambda_{2}(t)-a_{21}(t)\right)\left(a_{22}(t)-\lambda_{2}(t)\right)}}$
$\times\,\frac{\lambda_{2}(t)}{\lambda_{1}(t)}\left(\frac{\lambda_{2}(t)-1}{\lambda_{1}(t)-1}\right)^{2}+1,\qquad
C_{1}(t)=1-C_{2}(t),$

what is equivalent to (22).

Now we will give the results of some numerical calculations. Let $x_{1}=-2$,
$x_{2}=-1$ and we need to find the conformal mapping of the upper half-plane
onto the half-plane with two slits orthogonal to the boundary of lengths
$L_{1}=1$ and $L_{2}=2$ (Fig. 3). We put $\alpha=L_{1}/L_{2}=0.5$. Solving the
Cauchy problem for the system of ODEs we obtain the numerical values of the
accessory parameters in (23). They are given in Table 1. The calculations were
carried out using the Wolfram Mathematica package. For comparison, we also
give in the table the values calculated with the help the well-known
Driscoll’s SC Toolbox package for MATLAB [8]; the values are given with $7$
digits after the decimal point. As you can see, the obtained values of the
parameters coincides up to 6 digits after the decimal point.

Table 1. The values of the accessory parameters for the conformal mapping of the upper half plane onto the half-plane with two slits. Parameter | Our method | SC Toolbox
---|---|---
$c$ | $0.5867804$ | $-$
$a_{11}$ | $-9.8974995$ | $-9.8974994$
$\lambda_{1}$ | $-8.5126732$ | $-8.5126732$
$a_{12}$ | $-7.3979258$ | $-7.3979257$
$a_{21}$ | $-6.8108252$ | $-6.8108251$
$\lambda_{2}$ | $-3.7393888$ | $-3.7393887$
$a_{22}$ | $-0.3978735$ | $-0.3978735$

To eliminate the degeneracy of the system at the initial moment, a slight
change in the initial data was made: instead of
$a_{ij}\left(0\right)=\lambda_{i}\left(0\right)$ we took
$a_{ij}\left(0\right)=\lambda_{i}\left(0\right)+(-1)^{j}\epsilon_{ij}$, $i$,
$j=1$, $2$, $\epsilon_{ij}=10^{-15}$.

######  Example 5.2.

Consider the problem of finding the accessory parameters for the conformal
mapping of the upper half-plane onto the hexagon $H$ that is the rectangle
$R=[-1,1]\times[0,1]$ with the removed corner $[-1,-0.5]\times[0.5,1]$ (Fig.
4).

As an initial polygon, we take the rectangle $R$. The initial mapping of
$\mathbb{H}^{+}$ onto $R$ is

(24)
$f\left(z,0\right)=-c\left(0\right)\int_{0}^{z}\left(\zeta\left(\zeta-1\right)\right)^{-\frac{1}{2}}\left(\zeta-
a_{2}\left(0\right)\right)^{-\frac{1}{2}}\left(\zeta-
a_{1}\left(0\right)\right)^{-\frac{1}{2}}d\zeta+1,$

where

$c\left(0\right)=\frac{2+\sqrt{2}}{K\left({1}/{\sqrt{2}}\right)}\,=1.84146496\ldots,\quad
a_{1}\left(0\right)=-(3+2\sqrt{2})=-5.82842712\ldots,$
$a_{2}\left(0\right)=-{2}(\sqrt{2}+1)=-4.82842712\ldots$

Here $K\left(k\right)$ is the complete elliptic integral of the first kind.
The branch of the integrand is fixed so that it is positive for large positive
real $\zeta$; the same applies to the mappings $f(z,t)$ given below.

Let two slits go, orthogonally to the boundary of the rectangle $R$, from the
points $-0.5+i$ and $-1+0.5i$ with the same growth rates. We note that the
preimages of the points $-0.5+i$ and $-1+0.5i$ under the mapping (24) are

$\lambda_{1}=-(\sqrt{2+\sqrt{2}}+1)(1+\sqrt{2})=-6.87509856\ldots,\quad\lambda_{2}=-(1+\sqrt[4]{2})(1+\sqrt{2})=-5.28521351\ldots.$

The Schwarz-Christoffel integral, mapping $\mathbb{H}^{+}$ onto $R$ with two
slits, described above, and normalized by the conditions
$f\left(0,t\right)=1,f\left(1,t\right)=1+i,f\left(\infty,t\right)=i$, has the
form

(25)
$f\left(z,t\right)=-c(t)\int_{0}^{z}(\zeta(\zeta-1))^{-\frac{1}{2}}\prod_{l=1}^{2}\left(\zeta-\lambda_{l}(t)\right)\prod_{i,j=1}^{2}\left(\zeta-
a_{ij}(t)\right)^{-\frac{1}{2}}\prod_{k=1}^{2}\left(\zeta-
a_{k}(t)\right)^{-\frac{1}{2}}d\zeta+1,$

where
$-\infty<a_{11}(t)<\lambda_{1}(t)<a_{12}(t)<a_{1}(t)<a_{21}(t)<\lambda_{2}(t)<a_{22}(t)<a_{2}(t)<0$.
We note that the image of $\infty$ is not an angular point but it does not
matter, according to Remark 2.2. For $t=0$, we have
$a_{11}(0)=\lambda_{1}(0)=a_{12}(0)=\lambda_{1}$,
$a_{21}(0)=\lambda_{2}(0)=a_{22}(0)=\lambda_{2}$.

At the chosen growth rates, the slits simultaneously reach the point
$0.5+0.5i$ for some $t=T$, and the family of mappings $f(z,t)$ converges to
the mapping $g\left(z\right)$ of $\mathbb{H}^{+}$ onto the needed hexagon $H$;
it has the form

$f(z,T)=-c\int_{0}^{z}(\zeta(\zeta-1)(\zeta-a))^{-\frac{1}{2}}(\zeta-\lambda)^{1/2}\prod_{i=1}^{2}(\zeta-
b_{i})^{-\frac{1}{2}}d\zeta+1,$

$-\infty<b_{1}<\lambda<b_{2}<a<0$. Moreover, $a_{11}(t)\to b_{1}$;
$\lambda_{1}(t)$, $a_{12}(t)$, $a_{1}(t)$, $\lambda_{2}(t)$ and
$a_{21}(t)\to\lambda$, $a_{22}(t)\to b_{2}$, $a_{2}(t)\to a$, as $t\to T$.
Table 2 compares the results of our method and those obtained by the SC
Toolbox package; the values are given with $7$ digits after the decimal point.
As we can see, most of the obtained values of the parameters agree up $6$
digits after the decimal point; the values of $\lambda$ agree up $4$ digits;
this is caused by the fact that the convergence of the mappings is non-uniform
near this point.

Table 2. The values of the accessory parameters for the conformal mapping of the upper half plane $\mathbb{H}^{+}$ onto the hexagon $H$. Parameter | Our method | SC Toolbox
---|---|---
$c$ | $1.90896$ | $-$
$b_{1}$ | $-8.6039921$ | $-8.6039920$
$\lambda$ | $-4.6992541$ | $-4.6992949$
$b_{2}$ | $-4.0805629$ | $-4.0805627$
$a$ | $-4.0225626$ | $-4.0225623$
Figure 4. The image of the Cartesian grid under the conformal mapping of the
half-plane $\mathbb{H}^{+}$ onto the hexagon $H$.

###  5.2. General case

In general, by setting the relationships between the velocities $v_{j}$ of the
lengths of the slits, which could be even non-stationary, we have $m-1$
equations for $m$ functions $C_{i}(t)$:

$g_{\nu}\left(v_{1}(t),v_{2}(t)\ldots,v_{m}(t),t\right)=0,\ 1\leq\nu\leq m-1.$

The condition

(26) $\sum_{i=1}^{m}C_{j}(t)=1$

closes the system of equations to determine $C_{i}(t)$, $1\leq i\leq m$.
Together with the equations (9)-(12) they form a complete system to find the
accessory parameters.

Now consider the very important case where the ratios of the rates $v_{j}$ are
constant. Let, for example,

$\frac{v_{j}(t)}{v_{m}(t)}\equiv\alpha_{j},\quad 1\leq j\leq m-1,$

where $\alpha_{j}$ are some positive constants. Taking into account (19) and
(26) we obtain

$C_{j}^{-1}(t)=\alpha_{j}^{-1}A_{j}(t)\sum_{k=1}^{m}\alpha_{k}A_{k}^{-1}(t),\quad
1\leq j\leq m,$

where we put $\alpha_{m}=1$; the values of $A_{j}(t)$ is given by (20).

Therefore we find $C_{j}(t)$ as functions depending explicitly on the
accessory parameters. Putting them in (9)-(12) we obtain a system of ODEs to
determine the accessory parameters.

##  6\. Relationship between accessory parameters and side lengths

The search of a suitable parameter with respect to which the family of
mappings will be continuously differentiable is an important and interesting
problem. In this section we consider families corresponding to deformations of
a polygon with fixed angles. As the above parameter, it is proposed to take
any varying side length (in our case, the length of one of the slits). This
choice is based on the idea of a bijective and differentiable mapping between
the prevertices and the lengths of polygon sides (with fixed angles). The
proof of this fact allows us to justify the existence of the derivative
$\frac{\partial f}{\partial t}$ of the mapping $f(z,t)$, as well as the
existence of the functions $C_{i}(t).$

###  6.1. Differentiability of polygon side lengths as a function of accessory
parameters

Consider the set of polygons $P$ satisfying the conditions:

(1) the boundary of $P$ is a closed polygonal line $A_{1}A_{2}\ldots
A_{n}A_{n+1}$, where $A_{n+1}=A_{1}$, and the value of $A_{n-2}$ is fixed;

(2) the angle of inclination $\beta$ of the segment $A_{1}A_{2}$ to the real
axis is fixed;

(3) the values of interior angles of $P$,
$\alpha_{1}\pi,\alpha_{2}\pi,\ldots\alpha_{n}\pi$ at the vertices
$A_{1}A_{2}\ldots A_{n}$ are fixed.

Then the conformal mapping of the upper half-plane onto $P$ is given by the
formula

(27) $f\left(z\right)=de^{i\beta}\int_{0}^{z}\prod_{k=1}^{n-1}\left(\zeta-
a_{k}\right)^{\alpha_{k}-1}d\zeta+A_{n-2};$

Here, as above, we assume that $a_{n-2}=0$, $a_{n-1}=1$ and the preimage of
$A_{n}$ is $a_{n}=\infty$.

The length of the $\nu$-th side of the polygon is:

(28)
$l_{\nu}=d\int_{a_{\nu}}^{a_{\nu+1}}\prod_{k=1}^{n}\left|x-a_{k}\right|^{\sigma_{k}}dx.$

We note that if we fix the values of $l_{\nu}$ with $\ 1\leq\nu\leq n-2$, then
the values $l_{n-1}$ are $l_{n}$ are uniquely determined, because of (2).
Consider the mapping $\Phi:(d,a_{1},a_{2},\ldots
a_{n-3})\mapsto(l_{1},l_{2},\ldots,l_{n-2})$ where $d>0$,
$a_{1}<a_{2}<a_{3}<\ldots<a_{n-3}<0$ where $l_{\nu}$ is given by (28). We note
that if some of $\alpha_{k}-1$ is positive, the mapping (27) may be non-
univalent and the corresponding image is a multi-sheeted Riemann surface.

######  Lemma 6.1.

The mapping $\Phi$ is smooth and non-degenerate at every point of the set

$\mathcal{A}=\\{(d,a_{1},a_{2},a_{3},\ldots,a_{n-3})\in\mathbb{R}^{n-2}\mid
d>0,a_{1}<a_{2}<a_{3}<\ldots<a_{n-3}<0\\}.$

######  Remark 6.1.

We note that the relationship between the accessory parameters and the side
lengths was considered in the work of A. Vainshtein [31]; the result obtained
was used to substantiate the continuity method he developed. Variations of the
this idea were also used by other scientists: K. Reppe [26], L. N. Trefesen
[30] and others. In particular, they form the basis for the numerical mapping
of the Schwarz-Christoffel integral [9]. Below we will give a sketch of the
proof of this fact based on the ideas given in [22, ch. 3].

Differentiating $l_{\nu}$ in (28) with respect to $a_{\mu}$ and $d$, we can
easily show that exist the mapping $\Phi$ is continuously differentiable. It
remains to prove that the Jacobian of the mapping $\Phi$ is non-degenerate.
Assume the contrary. Then at some point of $\mathcal{A}$ there exist
variations $\delta d$, $\delta a_{1},\ldots,\delta a_{n-2}$ under with the
values of all $l_{\nu}$ do not change. If the straight line containing the
segment $A_{\nu}A_{\nu+1}$ is described as
$\\{(\xi,\eta)\in\mathbb{R}^{2}:\beta_{\nu}\xi+\gamma_{\nu}\eta=\epsilon_{\nu}\\}$,
where $\beta_{\nu}$, $\gamma_{\nu}$, $\epsilon_{\nu}$ are some constants, then
the variation $\delta f$ satisfies the boundary condition of a homogeneous
Hilbert boundary value problem with piecewise-constant coefficients (see, e.g.
[10] or [22]):

$\beta_{\nu}\mathop{\rm Re}\nolimits\delta f(x)+\gamma_{\nu}\mathop{\rm
Im}\nolimits\delta f(x)=0,\quad a_{\nu}<x<a_{\nu+1},$

for all $\nu$. Studying the behavior of $\delta f$ at the points $a_{\nu}$ we
conclude that the Hilbert problem has only a trivial solution. Lemma 6.1 is
proved.

From Lemma 6.1 it follows that the accessory parameters in (2) depends
smoothly on the lengths of the slits.

###  6.2. Differentiability of the family of mappings

Now consider a family of polygons with are obtained from a fixed polygon by
drawing $m$ slits of lengths $l_{1},l_{2}\ldots,l_{m}$. We will assume that
these lengths depends smoothly on each other. For definiteness, we will assume
that $l_{j}=\phi_{j}(l_{1})$, $2\leq j\leq m$, where $\phi_{j}$ are smooth
functions. Then the accessory parameters and the corresponding conformal
mappings also depends smoothly on $l_{1}$. Therefore, we can put, as a
parameter $t$ for the family $f(z,t)$, the value of $l_{1}$ and, consequently,
consider $f(z,t)$ as a smooth family.

Then we can take the expression following from Loewner’s equation
$(\ref{2.16})$ as the definition of the functions $C_{k}(t)$, since all the
quantities in it now are well defined:

$C_{k}(t)=\frac{2\pi
i}{\lambda_{k}(t)\left(\lambda_{k}(t)-1\right)^{2}}\underset{z=\lambda_{k}(t)}{\mathrm{res}}\left(\frac{\partial
f}{\partial t}\left(\frac{\partial f}{\partial z}\right)^{-1}\right),\ 1\leq
k\leq m.$

Now we will show that, under such the smooth parametrization, the function
$q(t)$, defined by (3), is a smooth function of the parameter $t=L_{1}$. Since
$f(z,T)=f(w(z,t),t)$ and $f(z,T)$ is a Christoffel-Schwarz integral,
differentiating by $z$ we have
$f^{\prime}(z,T)=f^{\prime}(w(z,t),t)w^{\prime}(z,t)$, therefore,

$w^{\prime}(z,t)=\frac{f^{\prime}(z,T)}{f^{\prime}(w(z,t),t)}\,.$

From (2) we obtain

$f^{\prime}(w(z,t),t)\sim\Psi(w(z,t)-1)^{\sigma_{n-1}},\quad{f^{\prime}(z,T)}\sim
C(z-1)^{\sigma_{n-1}},\ z\to 1,$

where $C$ is a constant and $\Psi$ is a non-zero smooth function of $c(t)$ and
the preimages of the vertices of the polygonal domain $D_{n}(t)$. Therefore,
we can consider that $\Psi=\Psi(t)$ is a smooth function of $t$. Then

$w^{\prime}(1,t)=\lim_{z\to
1}\frac{f^{\prime}(z,T)}{f^{\prime}(w(z,t),t)}\,=\frac{C}{\Psi(t)(w^{\prime}(1,t))^{\sigma_{n-1}}},$

consequently,

$q(t)=w^{\prime}(1,t)=\left(\frac{C}{\Psi(t)}\right)^{1/\alpha_{n-1}}.$

This shows that $q(t)$ is smooth.

We also note that the parametrization by the length of one of the slits allows
us to avoid some technical difficulties. Let we want to obtain the first slit
with length $L_{1}$ (the lengths of the remaining slits are uniquely
determined). Then the system of ODEs (9)–(12) must be solved on the interval
$\left[0,L_{1}\right]$. In the case of an arbitrary parametrization, it is
necessary to look for the final value $T$ of the parameter $t$, at which the
slits have desired lengths. Here, this value is determined automatically.

Weinstein’s result helped us to prove the differentiability of a family of
mappings in the case of several slits. However, the consequences are not
limited to this: the justification is also applicable to a more general case
considered by I. A. Kolesnikov in [17].

##  7\. Conclusion

We propose a generalization of the modification of the well-known Kufarev’s
method for approximate finding the conformal mapping of the upper half-plane
onto a polygon, given by V. Ya. Gutlyanskii and A. O. Zaidan. It is based on
the Loewner parametric method and considering one-parameter families $f(z,t)$,
$0\leq t\leq T$, of Schwarz-Christoffel integrals; for a fixed $t$, the
functions $f(z,t)$ maps the upper half-plane conformally onto a polygon with a
few rectilinear slits and the lengths of the slits increases with the growth
of $t$. The integral representation of $f(z,t)$ depends on some unknown
(accessory) parameters. The differential equations that describe the dynamics
of the mappings $f(z,t)$ and the corresponding accessory parameters are
derived. The existence and uniqueness of a one-parameter family implies the
existence and uniqueness of a solution to the resulting system of ODEs. We
also investigate the problem of finding a suitable parameter $t$. We show that
if the parameter $t$ coincides with the length of some slit, and other lengths
are depends on this length smoothly, then the family $f(z,t)$ depends on it
smoothly, and the accessory parameters can be found by solving a Cauchy
problem for a system of ODEs. Numerical calculations confirms the efficiency
of the method and its good accuracy.

However, some questions remained unexplored. For example, the stability of the
ODE system with respect to the variation of initial data needs to be
justified, as well as the use of the power series method in the general case.
Also we would like to achieve a greater accuracy in the numerical solution of
the obtained system of ODEs to determine the accessory parameters in the
Schwarz-Christoffel integral.

##  8\. Acknowledgements

The work of A. Posadsky and S. Nasyrov was supported by the Ministry of
Science and Higher Education of the Russian Federation (agreement no.
075-15-2022-287).

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Imaging Scatterometer for Observing Changes to Optical Coatings During Air
Annealing

Michael Rezac1, Daniel Martinez1, Amy Gleckl1, Joshua R. Smith1,∗

1 The Nicholas and Lee Begovich Center for Gravitational-Wave Physics and
Astronomy, California State University, Fullerton, US

<EMAIL_ADDRESS>
###### Abstract

Air annealing generally reduces absorption, scattering, and mechanical loss in
amorphous coatings up to temperatures where damage occurs. Our instrument uses
an industrial oven with viewports to observe coating scatter and damage during
annealing.

##  1 Introduction

Optical interference coatings with as low as possible optical and mechanical
losses are in demand for high-precision measurement applications. The
gravitational-wave observatories LIGO and Virgo use coatings formed by
TiO2-doped Ta2O5 (high index) and SiO2 (low index) layers, in which the dopant
is added to frustrate crystallization in Ta2O5, allowing higher annealing
temperatures [1]. Post-deposition annealing of such coatings to 600∘C, in air,
has been shown to reduce their scatter [2, 3], absorption [4], and mechanical
loss [1]. High-temperature post-deposition annealing remains a promising path
toward producing coatings with even lower mechanical loss using materials such
as TiO2-doped GeO2 with SiO2.

The practical limits to the maximum annealing temperatures achievable for
doped coatings are set by crystallization or damage mechanisms such as
delamination, bubbles, and cracks. Traditionally, optics are inspected before
and after a given annealing regimen, providing only weak insight to guide the
manufacturing process. Our previous work introduced an in-situ method to
observe optical scattering from coatings while they are annealed in vacuum and
showed that scattered light from TiO2:Ta2O5, decreases during annealing in
vacuum [3]. However, air annealing is likely superior to vacuum annealing in
terms of stoichiometry and absorption losses. Here we describe a new
instrument that was developed with the goal of imaging scattered light from
the coatings while they are being annealed in air to temperatures as high as
800∘C. This instrument will provide deeper insight to coating damage and
crystallization mechanisms by measuring their onset and evolution versus
temperature.

##  2 Experimental Setup and Process

Fig. 1: Setup of the Air Annealing Scatterometer. Labeled components are
described in the text.

The setup of the Air Annealing Scatterometer (AAS) is shown in Figure 1. The
basic operation is as follows. A sample coated optic is mounted within the
oven, then illuminated by an external light source and imaged at regular
intervals (once per minute here) using a CCD camera while an annealing
temperature profile is carried out. This system was designed with the goal of
imaging coating scatter in-situ to 800∘C. The components that were required to
meet that goal and their interplay are described below.

The setup requires a programmable oven (a) capable of reaching 800∘C and of
accurately carrying out heating profiles with multiple “ramp”
(increase/decrease temperature) and “dwell” (maintain constant temperature)
segments. We chose the industrial annealing oven ST-1500C-121012 by
SentroTech, which uses Molybdenum disilicide (MoSi2) heating elements and
thick ceramic insulation to reach, for an unmodified oven, 1500∘C. We worked
with Sentrotech’s engineering team to add observation and instrument ports (b)
to both the front (door) and rear. Because of these holes through the outer
walls and insulation, the oven’s maximum temperature will be reduced to
900∘C-1100∘C. We also added an air circulation fan option to the interior of
the oven to ensure temperature consistency. The temperature is read by an
S-type thermocouple that communicates with the programmable controller. The
oven’s controller (Nanodac from Eurotherm) uses
proportional–integral–derivative control and provides an interface for
creating heating profiles with up to 30 segments using the software package,
iTools.

A coated sample optic (c) is mounted within the oven on a solid stainless
steel pedestal (d) so that it can be illuminated and imaged through a viewport
on the back of the oven. To provide useful information for gravitational-wave
detectors, the setup requires monitoring scattered light from samples
illuminated at normal incidence by a light source similar in wavelength to
that used by LIGO and Virgo (1064 nm). To avoid time-dependent speckle effects
associated with coherent light (allowing association of small changes in
scatter with physical changes in the coatings), we use a 1050 nm
superluminescent diode (SLD) (e, Thorlabs S5FC1050P). To monitor fluctuations
in the incident power, a few percent of the SLD’s output is picked off by a
beam sampler (f, Thorlabs BSF10-C) and recorded by a power meter (g, Thorlabs
PM100D). The transmitted light is measured by a second power meter after
passing the viewport on the front door of the oven. The setup could be
modified to record the reflected power should in-situ measurement of
reflectivity be desired.

A low-noise and high resolution camera is required to image the light
scattered from the coated optic and identify defects such as point scatterers,
bubbles, and crystals. The AAS uses a cooled 4096x4096-pixel astronomical CCD
camera (h, Apogee Alta F16M) with adjustable exposure times, programmable
capture, and high linearity over a large illumination range. An image of the
sample’s surface at a scattering angle of $\theta_{s}=8^{\circ}$ is cast on
the CCD chip, with 2X magnification, using a single ($f$=200 mm) lens (i) and
adjustable iris (j). The SLD beams and camera are at the same height. A
narrow-band filter (Edmunds 1050nm/50nm) is installed at the front of the lens
tube to limit blackbody radiation from the oven’s heaters and room light from
entering the camera. To further limit the effects of blackbody radiation and
to account for “hot” pixels in the camera, for each “bright” image that is
taken with the SLD illumination on, a “dark” image is also taken with the SLD
off, that can be subtracted from the bright image during analysis. Images are
recorded using the Flexible Image Transport System (FITS) format, which
incorporates metadata such as time stamp, exposure time, camera temperature,
and saturation levels.

A LabView Virtual Instrument (VI) is used to automatically control and acquire
data from the SLD, oven, camera, and power monitors. A typical experiment goes
as follows. A sample is installed and focus of the imaging optics is checked.
The desired heating profile, with typical duration of 1-2 days, is created
using iTools and loaded to the oven’s controller. The VI is configured with
the desired camera exposure time and imaging cadence. The VI is started and it
executes the following sequence (where times indicate the duration spent in
each state): i) read oven set point, heater power, and thermocouple
temperature ($<$1s); ii) turn SLD on (1.5s); iii) read incident power monitor
($<$1s); iv) bright image exposure (5s); v) transfer bright image (20s); vi)
turn SLD off (1.5s); vii) dark image exposure (5s); viii) transfer dark image
(20s); ix) wait (roughly 9s) until the total elapsed time of the entire
sequence reaches 60s, then repeat. In this way, one bright image and dark
image along with one data point each for incident laser power and oven
temperature data is collected per minute. The images and data are written to
disk on the PC running LabView and backed up for analysis. The setup is
located on a passive seismic isolation optical bench, within a laminar flow
softwall cleanroom and the room is kept closed and dark during measurement.

At elevated temperatures, thermal radiation from the heating elements provides
an alternative side illumination of the coatings that is particularly useful
for viewing bubbles and delamination, especially in the “dark” images with the
SLD off. Results making use of this will be described in a future publication.
We note that a dedicated side illumination could be added to the setup to
achieve the same outcomes throughout the duration of the experiment.

##  3 Results: crystallization of TiO2:Ta2O5

Figure 2 shows the results of annealing sample optic PL003, consisting of a
single quarter-wavelength-thick layer of TiO2:Ta2O5 coated (by Laboratoire des
Matériaux Avancés) on a superpolished ($\sigma<0.1$ nm) Corning 7979 fused
silica substrate (from Coastline Optics), to 750∘C. This simple heating
profile was chosen to achieve crystallization in the coating layer based on
previous studies [4]. The bottom left panel shows the time evolution of the
oven’s temperature and the optic’s bidirectional reflectance distribution
function (BRDF). The BRDF exhibits a first bright peak at low temperature due
to point scatterers getting very bright as the sample moves up (due to thermal
expansion of the tall stainless steel holder) and the beam sweeps over them.
At mid-temperatures the BRDF is modulated somewhat, likely also due to the
sample translating within the beam pattern. Starting around 735∘C there is a
factor of ten, and seemingly permanent, increase in BRDF which we associated
with the onset of crystallization. This increased scatter has the onset
temperature and “frosted” appearance we expect, as shown in the images along
the top row, which resemble the incident beam pattern shown in the bottom
right.

Fig. 2: Results of annealing sample PL003, a single quarter-wavelength layer
of TiO2:Ta2O5 on a superpolished fused silica substrate. Bottom, left to
right: Measured oven temperature (orange) and sample BRDF (green) of the
sample, both versus elapsed time; sample before annealing; sample after
annealing (note, the stainless steel holder changed color); intensity profile
of the incident laser beam at the location of the sample (beam diameter 4 mm).
The top row shows cropped images of the (7 mm-wide) region of the sample
illuminated by the SLD. Top, left to right: At 734.8∘C some point scatterers
are visible, but no crystallization is seen; at 736.5∘C weak diffuse
scattering from the onset of crystallization is seen; at 746.6∘C the beam
intensity profile is scattered somewhat uniformly by the crystallized coating;
at 724.8∘C on the ramp down, the same pattern is seen more brightly.

##  4 Conclusion

The AAS instrument is capable of observing coating optical scatter,
crystallization, and damage throughout in-air annealing to 750∘C, or higher.
The images and BRDF presented here, with SLD normal illumination, reveal the
onset of coating crystallization. Followup studies will be done to more
closely observe crystal formation, for example heating profiles with slower
temperature increases or longer soaks around the crystallization onset
temperature. An upcoming paper will present similar onset and evolution of
damage mechanisms such as bubbles and delamination seen also with side
illumination from the oven’s heaters. Together, these capabilities allow
deeper study of the damage mechanisms at play in coatings during high-
temperature annealing. Such studies may lead to improvements in the coating
manufacture process for low-optical-loss applications.

## References

  * [1] M. Granata _et al._ , “Amorphous optical coatings of present gravitational-wave interferometers,” Class. Quant.Grav. 37, 095004 (2020).
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  * [3] E. M. Capote, A. Gleckl, J. Guerrero, M. Rezac, R. Wright, and J. R. Smith, “In-vacuum measurements of optical scatter versus annealing temperature for amorphous Ta2O5 and TiO2:Ta2O5 thin films,” J. Opt. Soc. Am.A 38, 534–541 (2021).
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