# The mathematical foundations of the asymptotic iteration method

D. Batic<EMAIL_ADDRESS>Department of Mathematics,
Khalifa University of Science and Technology,
Main Campus, Abu Dhabi,
United Arab Emirates  M. Nowakowski<EMAIL_ADDRESS>Departamento de
Fisica,
Universidad de los Andes, Cra.1E No.18A-10, Bogota, Colombia

###### Abstract

We introduce a new approach to the the asymptotic iteration method (AIM) by
means of which we establish the standard AIM connection with the continued
fractions technique and we develop a novel termination condition in terms of
the approximants. With the help of this alternative termination condition and
certain properties of continuous fractions, we derive a closed formula for the
asymptotic function $\alpha$ of the AIM technique in terms of an infinite
series. Furthermore, we show that such a series converges pointwise to
$\alpha$ which, in turn, can be interpreted as a specific term of the minimal
solution of a certain recurrence relation. We also investigate some conditions
ensuring the existence of a minimal solution and hence, of the function
$\alpha$ itself.

##  I Introduction

Ciftci  developed the so-called asymptotic iteration method (AIM) in order to
solve certain eigenvalues problems described by second order linear ODEs of
the form

$y^{{}^{\prime\prime}}=\lambda_{0}(x)y^{{}^{\prime}}+s_{0}(x)y,\quad\lambda_{0},s_{0}\in
C^{\infty}(I),\quad I\subseteq\mathbb{R}$ (1)

with $\lambda_{0}\neq 0$ for any $x\in I$ and furthermore, $\lambda_{0}$ and
$s_{0}$ do not need to be bounded at the endpoints of the interval $I$. The
spectral parameter is usually encoded in the function $s_{0}$. The smoothness
condition on $\lambda_{0}$, and $s_{0}$ is not too restrictive because in most
applications such functions are usually represented by some rational functions
which can be always made smooth by suitably choosing their domain of
definition. Moreover, Ciftci  tacitly assumed that the general solution of (1)
is also smooth on the interval $I$ where both $\lambda_{0}$ and $s_{0}$ are
smooth. This assumption is reasonable because the solution of (1) is
discontinuous at most at those points where $\lambda_{0}$ and $s_{0}$ are not
continuous. Ciftci  differentiates equation (1) $n$-times with respect to the
independent variable to end up with

$y^{(n+2)}=\lambda_{n}(x)y^{{}^{\prime}}+s_{n}(x)y$ (2)

where the functions $\lambda_{n}$ and $s_{n}$ are determined by the following
coupled system of recurrence relations

$\displaystyle\lambda_{n}$ $\displaystyle=$
$\displaystyle\lambda_{n-1}^{{}^{\prime}}+\lambda_{0}\lambda_{n-1}+s_{n-1},\quad\forall
n=1,2,\cdots$ (3) $\displaystyle s_{n}$ $\displaystyle=$ $\displaystyle
s^{{}^{\prime}}_{n-1}+s_{0}\lambda_{n-1}.$ (4)

At this point, Ciftci  considers the ratio of the $(n+2)$-th and $(n+1)$-th
derivatives which can be written by means of (2) as

$\frac{y^{(n+2)}}{y^{(n+1)}}=\frac{\lambda_{n}}{\lambda_{n-1}}\frac{y^{{}^{\prime}}+\frac{s_{n}}{\lambda_{n}}y}{y^{{}^{\prime}}+\frac{s_{n-1}}{\lambda_{n-1}}y}$
(5)

together with the asymptotic condition in Ciftci

$\frac{s_{n}}{\lambda_{n}}=\frac{s_{n-1}}{\lambda_{n-1}}:=\alpha(x),\quad n\gg
1,$ (6)

which is equivalent to requiring that there exists an $N\in\mathbb{N}$ such
that for all $n\geq N$

$\frac{s_{n}(x)}{\lambda_{n}(x)}=\alpha(x)$ (7)

for all $x\in I$. Finally, if the condition (7) holds and we make use of (3)
and (4), we conclude that (5) admits the solution

$y^{(n+1)}(x)=C_{1}\lambda_{n-1}(x)\mbox{exp}\left(\int_{x}\left[\lambda_{0}(t)+\alpha(t)\right]~{}dt\right)$
(8)

for all $n\geq N$. Substituting (8) into
$y^{(n+1)}=\lambda_{n-1}y^{{}^{\prime}}+s_{n-1}y$ and applying the asymptotic
condition (7), we come up with the following representation for the solution
of (1), namely

$y(x)=\mbox{exp}\left(-\int_{x}\alpha(\rho)d\rho\right)\left[C_{1}\int_{x}\mbox{exp}\left(\int_{\tau}\left[\lambda_{0}(t)+\alpha(t)\right]~{}dt\right)d\tau+C_{2}\right].$
(9)

Ciftci  claims that (9) is the general solution of (1). Such a claim should be
taken with some caution. First of all, (9) would need that (7) holds for all
$n\in\mathbb{N}$ which is not the case in general. Moreover, if we replace (9)
into (1), we find that (9) is a solution of the original ODE if and only if
the unknown function $\alpha$ is a solution of the Riccati equation

$\alpha^{{}^{\prime}}=\alpha^{2}(x)+\lambda_{0}(x)\alpha(x)-s_{0}(x).$ (10)

It is straightforward to verify that a class of solutions to (1) can be
represented as in (9) with $\alpha$ solution to the Riccati equation (10) if
and only if the second order differential operator in (1) admits the
factorization

$\left(\frac{d}{dx}+\alpha(x)\right)\left(\frac{d}{dx}+\beta(x)\right)y=0,\quad\beta=-\lambda_{0}(x)-\alpha(x).$
(11)

This property is completely independent on the AIM asymptotic condition (6)
whose origin can be related to the requirement that the general solution of
(1) is real-analytic on some interval $I$. As we will see here below, this
observation is a key ingredient in order to derive new asymptotic conditions
for the AIM method. To see that, let us take any $x_{0}\in I$. Note that the
interval $I$ has been already so defined that $\lambda_{0}$ never vanishes
there. Consider the initial condition $y(x_{0})=\mathfrak{c}_{1}$. Then, we
can set up a Taylor series about $x_{0}$ for the solution of (1)

$y(x)=\mathfrak{c}_{1}+(x-x_{0})\sum_{n=0}^{\infty}f_{n}(x),\quad
f_{n}(x)=\frac{y^{(n+1)}(x_{0})}{(n+1)!}(x-x_{0})^{n}$ (12)

and

$\left|\frac{f_{n+1}(x)}{f_{n}(x)}\right|=\frac{|x-x_{0}|}{n+2}\left|\frac{y^{(n+2)}(x_{0})}{y^{(n+1)}(x_{0})}\right|=\frac{|x-x_{0}|}{n+2}\left|\frac{\lambda_{n}(x_{0})}{\lambda_{n-1}(x_{0})}\right|\left|\frac{y^{{}^{\prime}}(x_{0})+\mathfrak{c}_{1}\frac{s_{n}(x_{0})}{\lambda_{n}(x_{0})}}{y^{{}^{\prime}}(x_{0})+\mathfrak{c}_{1}\frac{s_{n-1}(x_{0})}{\lambda_{n-1}(x_{0})}}\right|,$
(13)

where in the last step we used (5) evaluated at the point $x_{0}$ together
with the initial condition $y(x_{0})=\mathfrak{c}_{1}$. If the asymptotic
condition

$\lim_{n\to\infty}\frac{s_{n}(x_{0})}{\lambda_{n}(x_{0})}=\alpha(x_{0}),\quad\alpha(x_{0})<\infty$
(14)

and the asymptotic expansion

$\left|\frac{\lambda_{n}(x_{0})}{\lambda_{n-1}(x_{0})}\right|=\mathfrak{a}_{0}+\mathfrak{a}_{1}n+\mathcal{O}\left(\frac{1}{n^{\gamma}}\right),\quad\mathfrak{a}_{i}\geq
0~{}\forall i=1,2\quad,\gamma>0$ (15)

hold, then the Taylor series (12) will converge in some open interval about
$x_{0}$ with convergence radius $\rho=1/\mathfrak{a}_{1}$. At this point a
remark is in order. Pointwise convergence of the sequence
$(s_{n}/\lambda_{n})_{n\in\mathbb{N}}$ together with the asymptotic expansion
(15) is sufficient to ensure convergence of (12) for any $x_{0}\in I$, and
therefore, the smoothness of the solution to (1) on the interval $I$. However,
if we also require that the general solution to (1) admits the representation
(9), we need to demand that $\alpha$ is at least continuous on $I$ but this is
equivalent to imposing uniform convergence for the sequence
$(s_{n}/\lambda_{n})_{n\in\mathbb{N}}$ on the interval $I$. Note that Ciftci ;
Ismail  do not discuss which kind of convergence should be imposed on (6). Let
us introduce the sequence $(\alpha_{n})_{n\in\mathbb{N}}$ with
$\alpha_{n}=s_{n}/\lambda_{n}$. If such a sequence converges at least
pointwise on $I$, then the new AIM asymptotic condition can be formulated as

$\lim_{n\to\infty}\left[\alpha_{n}(x)-\alpha_{n-1}(x)\right]=0$ (16)

or, in the case a termination behaviour occurs as

$\alpha_{N}(x)-\alpha_{N-1}(x)=0$ (17)

for some positive integer $N$. If the termination condition (17) is not
satisfied by the particular problem at hand, it is necessary to consider (16).
Ciftci ; Cho1 ; Cho2  noticed that the stability and number of iterations
needed in (16) to compute eigenvalues of certain eigenvalues problems rely in
a sensitive way on the choice of the point $x_{0}\in I$. This signals that
uniform convergence for the sequence $(\alpha_{n})_{n\in\mathbb{N}}$ will be
more the exception than the rule. In that regard, we would like to mention
that Cama  showed that in the case (1) is a generic Fuchsian differential
equation in the complex plane, the aforementioned sequence converges uniformly
to the logarithmic derivative of the holomorphic solution to (1) in compact
subsets of
$U_{i}=\\{z\in\mathbb{C}~{}|~{}|z-z_{i}|<|z-z_{j}|~{}|~{}j=1,2,\cdots,r,~{}j\neq
i\\}$ where the index $r$ counts the singular points of (1). Concerning the
optimal choice for $x_{0}$, Ismail  gives indeed a condition represented by
equation (56) therein, which is mutatis mutandis equivalent to (16) and
specifies when the AIM should work, however no algorithm or method is offered
by Ismail  leading to the optimal choice of $x_{0}$. Moreover, the connection
between AIM and continuous fractions was also studied in Matamala . Last but
not least, the power of the AIM becomes evident if we recall that the new
branch of quasinormal modes for a massless scalar field in the Schwarzschild
metric derived analytically in 34 ; 35 ; 36  can also be obtained numerically
by using the improved AIM Ciftci ; Cho1 ; Cho2 ; 37 . The paper is organised
as follows: in Section II we prove the uniqueness of the solution to (3)
subject to an initialisation condition. In Section III we show how the unknown
function $\alpha$ appearing in (10) can be represented in terms of a certain
continued fraction and we also discuss the convergence problem of the latter.
We also construct a new formula for the unknown function $\alpha$ represented
by (68) and we show that if the conditions in Theorem IV.3 are met, the
sequence with terms $\alpha_{n}=s_{n}/\lambda_{n}$ in the AIM will converge
pointwise on some interval $I$ to the function $\alpha$ which in turn can be
interpreted as the term $\psi_{-1}(x)$ of the minimal solution $\psi_{n}(x)$
of the recurrence relation (87). We also investigate several conditions
ensuring the existence of a minimal solution and we observe that they have the
effect to shrink the interval over which the optimal value for $x_{0}$ must be
chosen when we use the AIM.

##  II The coupled system of recurrence relations (3) and (4)

Let $I_{2}$ denote the $2\times 2$ identity matrix. Define

$\Phi_{n}:=\left(\begin{array}[]{c}\lambda_{n}\\\
s_{n}\end{array}\right)~{}\forall n\in\mathbb{N},\quad
T:=I_{2}\frac{d}{dx}+\mathfrak{A}(x),\quad\mathfrak{A}(x):=\left(\begin{array}[]{cc}\lambda_{0}&1\\\
s_{0}&0\end{array}\right).$ (18)

Note that at this step the differential operator $T$ is a formal linear
operator defined on $\mathbb{C}(x)\times\mathbb{C}(x)$ with $\mathbb{C}(x)$
denoting the vector space of all rational functions in the real variable $x$
where the polynomials in the numerator and denominator may have complex
coefficients. Then, the recurrence relations (3) and (4) can be written in
matrix form as

$\Phi_{n}=T\Phi_{n-1}\quad\forall n=1,2,\cdots$ (19)

with initialisation condition

$\Phi_{0}=\left(\begin{array}[]{c}\lambda_{0}\\\ s_{0}\end{array}\right).$
(20)

It is straightforward to observe that for any $n$ the system (19) has the
solution

$\Phi_{n}=T^{n}\Phi_{0}.$ (21)

Since for any $n$ the functions $\lambda_{n}$ and $s_{n}$ are real-analytic on
the same interval $I$ where $s_{0}$ and $\lambda_{0}$ are real-analytic, we
can expand the vector-valued function $\Phi_{n}$ into power series about some
$x_{0}\in I$ with radius of convergence $R$ up to the nearest pole of
$\lambda_{0}$ and $s_{0}$. Likewise, we can expand the matrix $A$ into a
matrix power series about $x_{0}$ with radius of convergence $R$. More
precisely, if we substitute

$\Phi_{n}(x)=\sum_{m=0}^{\infty}\mathfrak{C}_{m,n}(x-x_{0})^{m},\quad\mathfrak{A}(x)=\sum_{k=0}^{\infty}\mathfrak{A}_{k}(x-x_{0})^{k},\quad|x-x_{0}|<R$
(22)

with $\mathfrak{C}_{m,n}\in\mathbb{C}^{2}$ and
$\mathfrak{A}_{k}\in\mathbb{C}^{2\times 2}$ into (19) and we make use of the
Cauchy product of power series together with a trivial shift of indices, we
end up with the following recurrence relation

$\mathfrak{C}_{m,n+1}=(m+1)\mathfrak{C}_{m+1,n}+\sum_{\ell=0}^{m}\mathfrak{A}_{m-\ell}\mathfrak{C}_{\ell,n}\quad\forall~{}m,n=0,1,\cdots.$
(23)

Given $x_{0}\in I$ and for each fixed $n$ we can recursively compute the
coefficients $\mathfrak{C}_{m,n}$ for $m=1,2,\cdots$ from (23). For instance,
for $n=0$ equation (23) allows to compute all coefficients in the Taylor
expansion for $\Phi_{1}$ and so on. This shows the uniqueness of the solution
$\Phi_{n}$ to (19) for any $n$.

##  III The link between AIM and the continued fraction method

In this section, we show how the unknown function $\alpha$ appearing in (10)
can be represented in terms of a certain continued fraction. Furthermore, we
also discuss the convergence problem of the latter. To do that, we follow a
different approach to the one outlined in Section I. If we differentiate (1)
one time with respect to $x$, we get

$y^{1+2}=\lambda_{0}y^{{}^{\prime\prime}}+(\lambda_{0}^{{}^{\prime}}+s_{0})y^{{}^{\prime}}+s_{0}^{{}^{\prime}}y.$
(24)

On the other hand, we can solve (1) with respect to $y$ to get

$y=\frac{y^{{}^{\prime\prime}}}{s_{0}}-\frac{\lambda_{0}}{s_{0}}y^{{}^{\prime}}$
(25)

which replaced into (24) gives

$y^{(1+2)}=p_{1}y^{{}^{\prime\prime}}+q_{1}y^{{}^{\prime}}$ (26)

with

$p_{1}=\lambda_{0}+\frac{s_{0}^{{}^{\prime}}}{s_{0}},\quad
q_{1}=s_{0}+\lambda_{0}^{{}^{\prime}}-\lambda_{0}\frac{s_{0}^{{}^{\prime}}}{s_{0}}.$
(27)

Differentiating once more (26) and substituting there

$y^{{}^{\prime}}=\frac{y^{(1+2)}}{q_{1}}-\frac{p_{1}}{q_{1}}y^{{}^{\prime\prime}}$
(28)

yields

$y^{(2+2)}=p_{2}y^{(1+2)}+q_{2}y^{{}^{\prime\prime}}$ (29)

with

$p_{2}=p_{1}+\frac{q_{1}^{{}^{\prime}}}{q_{1}},\quad
q_{2}=q_{1}+p_{1}^{{}^{\prime}}-p_{1}\frac{q_{1}^{{}^{\prime}}}{q_{1}}y^{{}^{\prime\prime}}.$
(30)

At this point, if we proceed iteratively, we obtain for all $n\geq 0$

$y^{(n+2)}=p_{n}(x)y^{(n+1)}+q_{n}(x)y^{(n)}$ (31)

where

$\displaystyle p_{n}$ $\displaystyle=$ $\displaystyle
p_{n-1}+\frac{q^{{}^{\prime}}_{n-1}}{q_{n-1}},$ (32) $\displaystyle q_{n}$
$\displaystyle=$ $\displaystyle
q_{n-1}+p^{{}^{\prime}}_{n-1}-p_{n-1}\frac{q^{{}^{\prime}}_{n-1}}{q_{n-1}},$
(33)

with $p_{0}:=\lambda_{0}$, $q_{0}:=s_{0}$ and $p_{-1}=q_{-1}=0$. Equation (31)
is a second order linear ODE for the unknown function $y^{(n)}$ and,
therefore, for each $n\geq 0$ we can associate to it a Riccati equation of the
form

$\varphi^{{}^{\prime}}_{n}=\varphi_{n}^{2}(x)+p_{n}(x)\varphi_{n}(x)-q_{n}(x),\quad\varphi_{n}:=-\frac{y^{(n+1)}}{y^{(n)}}.$
(34)

It is straightforward to observe that in the case $n=0$ equation (34) coupled
with the conditions $p_{0}:=\lambda_{0}$ and $q_{0}:=s_{0}$ reduces to the
Riccati equation (10) and hence, we can make the identification
$\varphi_{0}=\alpha$. If we rewrite (31) as

$\frac{y^{(n+2)}}{y^{(n+1)}}=p_{n}(x)+q_{n}(x)\frac{y^{(n)}}{y^{(n+1)}}$ (35)

and we take into account that $\varphi_{n+1}=-y^{(n+2)}/y^{(n+1)}$, then (35)
becomes

$\frac{q_{n}}{\varphi_{n}}=p_{n}+\varphi_{n+1}$ (36)

from which the following recurrence relation for $\varphi_{n}$ emerges, namely

$\varphi_{n}(x)=\frac{q_{n}(x)}{p_{n}(x)+\varphi_{n+1}(x)}\quad\forall n\geq
0.$ (37)

Finally, the solution of the Riccati equation (10) can be expressed as a
continued fraction by means of (37) as follows

$\alpha(x)=\varphi_{0}(x)=\frac{q_{0}(x)}{p_{0}(x)+\varphi_{1}(x)}=\cfrac{q_{0}(x)}{p_{0}(x)+\cfrac{q_{1}(x)}{p_{1}(x)+\cfrac{q_{2}(x)}{p_{2}(x)+\dots}}}=\operatorname*{K}_{n=0}^{\infty}\left(\frac{q_{n}(x)}{p_{n}(x)}\right).$
(38)

At this point a couple or remarks are in order.

  1. 1.

If for some positive integer $N$ we have $q_{N}(x)=0$ for all $x\in I$, then
(37) induces a termination into the continued fraction (38) and by means of
(37) and (38) it follows that

$\alpha(x)=\cfrac{q_{0}(x)}{p_{0}(x)+\cfrac{q_{1}(x)}{p_{1}(x)+\cfrac{q_{2}(x)}{\ddots
p_{N-2}(x)+\cfrac{q_{N-1}(x)}{p_{N-1}(x)}}}}.$ (39)

  2. 2.

If no termination condition occurs, a careful analysis of the
convergence/divergence property of the continuous fraction (38) is mandatory.

##  IV A series representation for the function $\alpha$

In the following, the N-th approximant of (38) is defined as

$C_{N}(x)=\frac{A_{N}(x)}{B_{N}(x)}=\operatorname*{K}_{n=0}^{N}\left(\frac{q_{n}(x)}{p_{n}(x)}\right),$
(40)

where the sequence of functions $\left(A_{N}(x)\right)_{n\in\mathbb{N}}$ and
$\left(B_{N}(x)\right)_{n\in\mathbb{N}}$ are the $N$-th partial numerator and
the N-th partial denominator of the continued fraction (38), respectively. We
will always assume that $A_{N}/B_{N}$ is in reduced form, i.e. $A_{N}$ and
$B_{N}$ have no common factors. Furthermore, we say that the continued
fraction (38) converges pointwise to $\alpha$ on some interval $I$ of the real
line if

$\lim_{N\to\infty}C_{N}(x)=\alpha(x)\quad\forall~{}x\in I.$ (41)

By observing that

$\displaystyle C_{0}(x)$ $\displaystyle=$
$\displaystyle\frac{A_{0}(x)}{B_{0}(x)}=\frac{q_{0}(x)}{p_{0}(x)}=\frac{s_{0}(x)}{\lambda_{0}(x)}=\alpha_{0}(x),$
(42) $\displaystyle C_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{A_{1}(x)}{B_{1}(x)}=\frac{p_{1}(x)q_{0}(x)}{p_{0}(x)p_{1}(x)+q_{1}(x)}=\frac{s_{0}^{{}^{\prime}}(x)+s_{0}(x)\lambda_{0}(x)}{\lambda_{0}^{{}^{\prime}}(x)+\lambda_{0}^{2}(x)+s_{0}(x)}=\frac{s_{1}(x)}{\lambda_{1}(x)}=\alpha_{1}(x),$
(43) $\displaystyle\vdots$ (44)

we discover that the sequence of approximants of (38) is equivalent to the
sequence $(\alpha_{n})_{n\in\mathbb{N}}$ appearing in the AIM method. Hence,
we proved that the AIM condition (6) introduced by Ciftci  is equivalent to
require that $\alpha$ is given by the $n$-th approximant of the continued
fraction (38) for $n$ sufficiently large. On the other hand, if (38) or,
equivalently, the sequence of its approximants converges pointwise on some
interval $I$, then, the AIM asymptotic condition (16) can be replaced by the
condition

$\lim_{n\to\infty}\left[C_{n}(x)-C_{n-1}(x)\right]=0\quad\forall~{}x\in I.$
(45)

In the case a termination behaviour occurs in a problem amenable to the
treatment with the AIM method, the termination condition (17) can be replaced
by the equivalent condition

$q_{N}(x)=0\quad\forall~{}x\in I$ (46)

with $N$ some positive integer. The next theorem will allow us to reformulate
the condition (45) as a certain limit involving only the terms of the sequence
of the partial denominators of the continued fraction (38) and those belonging
to sequence $(q_{n})_{n\in\mathbb{N}}$ defined through the recurrence relation
(32) and (33).

###### Theorem IV.1

Consider the continued fraction
$\operatorname*{K}_{n=0}^{N}\left(q_{n}(x)/p_{n}(x)\right)$ whose $n$-th
approximant is given by (40). Then, the $n$-th partial numerator $A_{n}(x)$
and denominator $B_{n}(x)$ of the continued fraction satisfy the three-term
recurrence relation

$\displaystyle A_{n}(x)$ $\displaystyle=$ $\displaystyle
p_{n}(x)A_{n-1}(x)+q_{n}(x)A_{n-2}(x),\quad
A_{-1}(x)=0,~{}A_{-2}(x)=1\quad\forall x\in I,$ (47) $\displaystyle B_{n}(x)$
$\displaystyle=$ $\displaystyle p_{n}(x)B_{n-1}(x)+q_{n}(x)B_{n-2}(x),\quad
B_{-1}(x)=1,~{}B_{-2}(x)=0\quad\forall x\in I$ (48)

for all $n\geq 0$.

  * Proof.

We proceed by induction. First of all, we observe that for $n=0$ equations
(47) and (48) reproduce correctly

$\displaystyle A_{0}(x)$ $\displaystyle=$ $\displaystyle
p_{0}(x)A_{-1}(x)+q_{0}(x)A_{-2}(x)=q_{0}(x),$ (49) $\displaystyle B_{0}(x)$
$\displaystyle=$ $\displaystyle p_{0}(x)B_{-1}(x)+q_{0}(x)B_{-2}(x)=p_{0}(x).$
(50)

In the case $n=1$, we obtain as expected

$\displaystyle A_{1}(x)$ $\displaystyle=$ $\displaystyle
p_{1}(x)A_{0}(x)+q_{1}(x)A_{-1}(x)=p_{1}(x)q_{0}(x),$ (51) $\displaystyle
B_{1}(x)$ $\displaystyle=$ $\displaystyle
p_{1}(x)B_{0}(x)+q_{1}(x)B_{-1}(x)=p_{0}(x)p_{1}(x)+q_{1}(x).$ (52)

Let us assume that (47) and (48) are true for $n=m$. Then, we have

$\displaystyle A_{m}(x)$ $\displaystyle=$ $\displaystyle
p_{m}(x)A_{m-1}(x)+q_{m}(x)A_{m-2}(x),$ (53) $\displaystyle B_{m}(x)$
$\displaystyle=$ $\displaystyle p_{m}(x)B_{m-1}(x)+q_{m}(x)B_{m-2}(x).$ (54)

From

$\displaystyle C_{m+1}(x)$ $\displaystyle=$
$\displaystyle\operatorname*{K}_{n=0}^{m+1}\left(\frac{q_{n}(x)}{p_{n}(x)}\right)=\cfrac{q_{0}(x)}{p_{0}(x)+\cfrac{q_{1}(x)}{p_{1}(x)+\cfrac{q_{2}(x)}{\ddots
p_{m-1}(x)+\cfrac{q_{m}(x)}{p_{m}(x)+\frac{q_{m+1}(x)}{p_{m+1}(x)}}}}},$ (55)
$\displaystyle C_{m}(x)$ $\displaystyle=$
$\displaystyle\cfrac{q_{0}(x)}{p_{0}(x)+\cfrac{q_{1}(x)}{p_{1}(x)+\cfrac{q_{2}(x)}{\ddots
p_{m-1}(x)+\frac{q_{m}(x)}{p_{m}(x)}}}}$ (56)

we realize that $C_{m+1}(x)$ can be obtained from $C_{m}(x)$ by replacing
there $p_{m}(x)$ with $p_{m}(x)+(q_{m+1}(x)/p_{m+1}(x))$. Hence, we can write

$C_{m+1}(x)=\frac{A_{m+1}(x)}{B_{m+1}(x)}=\frac{\widehat{A}_{m}(x)}{\widehat{B}_{m}(x)},$
(57)

with

$\displaystyle\widehat{A}_{m}(x)$ $\displaystyle=$
$\displaystyle\frac{p_{m+1}(x)A_{m}(x)+q_{m+1}(x)A_{m-1}(x)}{p_{m+1}(x)},$
(58) $\displaystyle\widehat{B}_{m}(x)$ $\displaystyle=$
$\displaystyle\frac{p_{m+1}(x)B_{m}(x)+q_{m+1}(x)B_{m-1}(x)}{p_{m+1}(x)},$
(59)

where we made use of (53) and (54). Finally, by means of (58) and (59) we get

$\frac{A_{m+1}(x)}{B_{m+1}(x)}=\frac{p_{m+1}(x)A_{m}(x)+q_{m+1}(x)A_{m-1}(x)}{p_{m+1}(x)B_{m}(x)+q_{m+1}(x)B_{m-1}(x)}$
(60)

and the proof is completed. $\Box$

A formula for the $N$-th approximant can be found multiplying (47) by
$B_{n-1}(x)$, (48) by $A_{n-1}(x)$ followed by subtraction of both. This
procedure leads to

$A_{n}(x)B_{n-1}(x)-A_{n-1}(x)B_{n}(x)=-q_{n}(x)\left[A_{n-1}(x)B_{n-2}(x)-A_{n-2}(x)B_{n-1}(x)\right].$
(61)

Let $v_{n}(x)=A_{n}(x)B_{n-1}(x)-A_{n-1}(x)B_{n}(x)$. Using the initial
conditions in (47) and (48) yields $v_{-1}(x)=-1$. Furthermore, (61) can be
rewritten as

$v_{n}(x)=-q_{n}(x)v_{n-1}(x)$ (62)

and its solution is

$v_{n}(x)=(-1)^{n}\prod_{j=0}^{n}q_{j}(x)\quad\forall n\geq 0$ (63)

with $v_{-1}(x)=-1$. At this point, by means of (40) and the definition of
$v_{n}$ we observe that

$\lim_{n\to\infty}\left[C_{n}(x)-C_{n-1}(x)\right]=\lim_{n\to\infty}\frac{A_{n}(x)B_{n-1}(x)-A_{n-1}(x)B_{n}(x)}{B_{n}(x)B_{n-1}(x)}=\lim_{n\to\infty}\frac{v_{n}(x)}{B_{n}(x)B_{n-1}(x)}.$
(64)

It is straightforward to verify with the help of (62) and (63) that the AIM
condition (45) becomes

$\lim_{n\to\infty}\frac{(-1)^{n}\prod_{j=0}^{n}q_{j}(x)}{B_{n}(x)B_{n-1}(x)}=0\quad\forall
x\in I.$ (65)

The above formula clearly shows that the termination behaviour occurs when one
of the $q_{j}$ vanishes. Furthermore, the question of whether or not the limit
in (65) vanishes on the interval $I$ may be answered by constructing uniform
asymptotic representations for $B_{n}(x)$ solution of the recurrence relation
(54). If we divide (63) by $B_{n}(x)B_{n-1}(x)$ and introduce the difference
operator $\Delta x_{n}:=x_{n+1}-x_{n}$ Elady  , we get

$\Delta
C_{n-1}(x)=\frac{A_{n}(x)}{B_{n}(x)}-\frac{A_{n-1}(x)}{B_{n-1}(x)}=\frac{(-1)^{n}\prod_{j=0}^{n}q_{j}(x)}{B_{n}(x)B_{n-1}(x)}\quad\forall
n\geq 0$ (66)

and applying the antidifference operator $\Delta^{-1}$ defined in $(2.1.16)$
in Elady  yields

$C_{n-1}(x)=\sum_{k=0}^{n-1}(-1)^{k}\frac{q_{0}(x)\cdots
q_{k}(x)}{B_{k}(x)B_{k-1}(x)}\quad\forall n\geq 1.$ (67)

Combining (67) with (41) and letting $n\to\infty$ leads to the following
formal representation of the continued fraction (38) or equivalently for the
unknown function $\alpha$, namely

$\alpha(x)=\operatorname*{K}_{n=0}^{\infty}\left(\frac{q_{n}(x)}{p_{n}(x)}\right)=\sum_{n=0}^{\infty}(-1)^{n}\frac{q_{0}(x)\cdots
q_{n}(x)}{B_{n}(x)B_{n-1}(x)}.$ (68)

In order to formulate a necessary and sufficient condition for the continued
fraction (68) to converge pointwise to some function on a given interval of
the real line, we first recall that two continued fractions are said to be
equivalent (denoted by the symbol $\approx$) if both have the same sequence of
approximants.

###### Theorem IV.2

For all $n\in\mathbb{N}\cup\\{0\\}$ and for all
$x\in\mathfrak{I}\subseteq\mathbb{R}$ suppose that $p_{n}(x)>0$ and $q_{n}(x)$
does not vanish. Further assume that $\mathcal{I}=\mathfrak{I}\cap
I\neq\emptyset$ where $I$ is the interval introduced in (1). Then, the
continued fraction (68) converges pointwise on $\mathcal{I}$ to some function
$\alpha$ if and only if the series

$\sum_{n=0}^{\infty}p_{n}(x)$ (69)

diverges for every $x\in\mathcal{I}$.

  * Proof.

Fix any $x_{0}\in\mathcal{I}$. We start with the observation that the
equivalence relation

$\operatorname*{K}_{n=0}^{\infty}\left(\frac{q_{n}(x_{0})}{p_{n}(x_{0})}\right)\approx\operatorname*{K}_{n=0}^{\infty}\left(\frac{\gamma_{n-1}\gamma_{n}q_{n}(x_{0})}{\gamma_{n}p_{n}(x_{0})}\right)$
(70)

holds for any sequence of non zero complex numbers $\gamma_{-1}=1$,
$\gamma_{0}$, $\gamma_{1}\cdots$ Elady . In particular, if we choose the
sequence $(\gamma_{n})_{n\in\mathbb{N}}$ such that $\gamma_{n-1}=1/q_{n}$,
then

$\operatorname*{K}_{n=0}^{\infty}\left(\frac{q_{n}(x_{0})}{p_{n}(x_{0})}\right)\approx
q_{0}(x_{0})\operatorname*{K}_{n=0}^{\infty}\left(\frac{1}{p_{n}(x_{0})}\right).$
(71)

From (68) it follows that

$\operatorname*{K}_{n=0}^{\infty}\left(\frac{1}{p_{n}(x_{0})}\right)=\sum_{n=0}^{\infty}S_{n},\quad
S_{n}:=\frac{(-1)^{n}}{B_{n-1}(x_{0})B_{n}(x_{0})}$ (72)

with $B_{n}(x_{0})$ given by the recurrence relation (54) with $q_{n}=1$ for
all $n$, namely

$B_{n}(x_{0})=p_{n}(x_{0})B_{n-1}(x_{0})+B_{n-2}(x_{0}),\quad
B_{-2}(x_{0})=0,\quad B_{-1}(x_{0})=1.$ (73)

Note that $p_{n}(x)>$ for all $n\in\mathbb{N}\cup\\{0\\}$ and for all
$x\in\mathcal{I}$ ensures that all terms $B_{n}(x_{0})$ in the above
recurrence relation are positive. From (72) we realize that (68) converges
whenever the alternating series in (72) converges. To this purpose, we observe
that

$B_{n+1}(x_{0})=p_{n+1}(x_{0})B_{n}(x_{0})+B_{n-1}(x_{0})>B_{n-1}(x_{0}).$
(74)

This implies that $B_{n}(x_{0})B_{n+1}(x_{0})>B_{n}(x_{0})B_{n-1}(x_{0})$ and
hence,

$\frac{1}{B_{n}(x_{0})B_{n+1}(x_{0})}<\frac{1}{B_{n}(x_{0})B_{n-1}(x_{0})}$
(75)

for all $n\in\mathbb{N}\cup\\{0\\}$. Thus, the sequence $(|S_{n}|)$ is
monotonically decreasing. Therefore, the series in (72) converges provided
that the following limit diverges

$\lim_{n\to\infty}B_{n-1}(x_{0})B_{n}(x_{0}).$ (76)

Furthermore, from (73) we have

$\displaystyle B_{1}(x_{0})$ $\displaystyle=$ $\displaystyle
1+p_{0}(x_{0})p_{1}(x_{0})>1,$ (77) $\displaystyle B_{2}(x_{0})$
$\displaystyle=$ $\displaystyle
p_{0}(x_{0})+p_{2}(x_{0})+p_{0}(x_{0})p_{1}(x_{0})p_{2}(x_{0})>p_{0}(x_{0}),$
(78) $\displaystyle B_{3}(x_{0})$ $\displaystyle=$ $\displaystyle
1+p_{0}(x_{0})p_{1}(x_{0})+p_{0}(x_{0})p_{3}(x_{0})+p_{2}(x_{0})p_{3}(x_{0})+p_{0}(x_{0})p_{1}(x_{0})p_{2}(x_{0})p_{3}(x_{0})>1,$
$\displaystyle\vdots$

Hence, $B_{n}(x_{0})>\min{\\{1,p_{0}\\}}=:\mu$ and we conclude that

$B_{n-1}(x_{0})B_{n}(x_{0})>\mu^{2}\sum_{k=0}^{n}p_{k}(x_{0}).$ (80)

This shows that if $\sum_{n=0}^{\infty}p_{n}(x_{0})$ diverges, the limit in
(76) diverges as well and the continued fraction converges. Suppose now that
$\sum_{n=0}^{\infty}p_{n}(x_{0})$ converges. Then, (73) implies that

$B_{n-1}(x_{0})+B_{n}(x_{0})=B_{n-2}(x_{0})+\left[1+p_{n}(x_{0})\right]B_{n-1}(x_{0})<\left[1+p_{n}(x_{0})\right]\left[B_{n-1}(x_{0})+B_{n-2}(x_{0})\right]$
(81)

and by induction

$B_{n-1}(x_{0})+B_{n}(x_{0})<\prod_{k=0}^{n}\left[1+p_{k}(x_{0})\right]<\prod_{k=0}^{n}e^{p_{k}(x_{0})}<\mbox{exp}\left(\sum_{n=0}^{\infty}p_{n}(x_{0})\right).$
(82)

Since $\sum_{n=0}^{\infty}p_{n}(x_{0})$ converges, let $L$ denote the
corresponding limit. This implies that

$B_{n-1}(x_{0})+B_{n}(x_{0})<e^{L}$ (83)

and hence,

$B_{n-1}(x_{0})B_{n}(x_{0})<B^{2}_{n-1}(x_{0})+B_{n}^{2}(x_{0})+2B_{n-1}(x_{0})B_{n}(x_{0})=\left[B_{n-1}(x_{0})+B_{n}(x_{0})\right]^{2}<e^{2L}.$
(84)

This results violates the divergence condition for (76) and (68) must diverge
at $x_{0}$. Since the choice of $x_{0}\in\mathcal{I}$ was arbitrary, the
result is valid for any $x\in\mathcal{I}$. $\Box$

In order to probe that the function $\alpha$ can be interpreted as a certain
term of a minimal solution of a recurrence relation we derive hereafter, we
briefly recall that a solution $\psi_{n}$ of the difference equation

$\mathfrak{x}_{n+2}+\mathfrak{f}_{1}(n)\mathfrak{x}_{n+1}+\mathfrak{f}_{2}(n)\mathfrak{x}_{n}=0$
(85)

is called minimal or subdominant if

$\lim_{n\to\infty}\frac{\psi_{n}}{\mathfrak{x_{n}}}=0$ (86)

for any solution $\mathfrak{x}_{n}$ that is not a multiple of $\psi_{n}$.

###### Theorem IV.3

Let $\mathcal{I}$ be defined as Theorem IV.2. Consider the recurrence relation

$\mathfrak{x}_{n}(x)-p_{n}(x)\mathfrak{x}_{n-1}(x)-q_{n}(x)\mathfrak{x}_{n-2}(x)=0$
(87)

with $q_{n}(x)\neq 0$ for all $x\in\mathcal{I}$ and for every
$n\in\mathbb{N}\cup\\{-1,0\\}$. If for any $x\in\mathcal{I}$ (87) has a
minimal solution $\psi_{n}(x)$ with $\psi_{-1}(x)\neq 0$ for all
$x\in\mathcal{I}$, then the continued fraction (68) converges pointwise to
$\alpha$ on the interval $\mathcal{I}$.

  * Proof.

Suppose that (87) has a minimal solution $\psi_{n}(x)$ with $\psi_{-1}(x)\neq
0$ for all $x\in\mathcal{I}$. If we divide (87) by $\mathfrak{x}_{n-1}$ and
introduce $\mathfrak{y}_{n}(x):=\mathfrak{x}_{n}(x)/\mathfrak{x}_{n-1}(x)$
with $\mathfrak{x}_{n-1}(x)\neq 0$ for all $x\in\mathcal{I}$, we obtain

$\mathfrak{y}_{n}(x)-p_{n}(x)=\frac{q_{n}(x)}{\mathfrak{y}_{n-1}(x)}$ (88)

and hence,

$\mathfrak{y}_{n-1}(x)=\frac{q_{n}(x)}{-p_{n}(x)+\mathfrak{y}_{n}(x)}.$ (89)

A recursive application of the formula above gives

$\mathfrak{y}_{n-1}(x)=\cfrac{q_{n}(x)}{-p_{n}(x)+\cfrac{q_{n+1}(x)}{-p_{n+1}(x)+\cfrac{q_{n+2}(x)}{-p_{n+2}(x)+\cdots}}}.$
(90)

Taking into account that
$\mathfrak{y}_{n-1}(x):=\mathfrak{x}_{n-1}(x)/\mathfrak{x}_{n-2}(x)$, the case
$n=0$ yields

$\mathfrak{y}_{-1}(x)=\frac{\mathfrak{x}_{-1}(x)}{\mathfrak{x}_{-2}(x)}=\cfrac{q_{0}(x)}{-p_{0}(x)+\cfrac{q_{1}(x)}{-p_{1}(x)+\cfrac{q_{2}(x)}{-p_{2}(x)+\cdots}}}=\operatorname*{K}_{n=0}^{\infty}\left(\frac{q_{n}(x)}{-p_{n}(x)}\right)$
(91)

with $n$-th approximant

$\widehat{C}_{n}(x)=\frac{\widehat{A}_{n}(x)}{\widehat{B}_{n}(x)}.$ (92)

On the other hand, $\widehat{A}_{n}(x)$ and $\widehat{B}_{n}(x)$ are two
linearly independent solutions of (87) with initial conditions
$\widehat{A}_{-1}(x)=0$, $\widehat{A}_{-2}(x)=1$ and $\widehat{B}_{-1}(x)=1$,
$\widehat{B}_{-2}(x)=0$ for all $x\in\mathcal{I}$. Hence, without loss of
generality let for some $\ell(x)\neq 0$ and for all $x\in\mathcal{I}$

$\psi_{n}(x)=\widehat{A}_{n}(x)+\ell(x)\widehat{B}_{n}(x)\quad\forall
n\geq-1.$ (93)

Note that for all $x\in\mathcal{I}$

$\psi_{-1}(x)=\ell(x),\quad\psi_{-2}(x)=1.$ (94)

Furthermore, we have

$\frac{\psi_{n}(x)}{\widehat{B}_{n}(x)}=\frac{\widehat{A}_{n}(x)}{\widehat{B}_{n}(x)}+\ell(x).$
(95)

Since $\psi_{n}(x)$ is a minimal solution of (87) for each $x\in\mathcal{I}$,
it follows that

$\lim_{n\to\infty}\frac{\psi_{n}(x)}{\widehat{B}_{n}(x)}=0$ (96)

pointwise on $\mathcal{I}$. If we take $n\to\infty$ in (95), we obtain again
pointwise on $\mathcal{I}$

$\lim_{n\to\infty}\widehat{C}_{n}(x)=\lim_{n\to\infty}\frac{\widehat{A}_{n}(x)}{\widehat{B}_{n}(x)}=-\ell(x).$
(97)

On the other hand, from (91) we have

$\mathfrak{y}_{-1}(x)=\frac{\psi_{-1}(x)}{\psi_{-2}(x)}=\ell(x)$ (98)

which combined with (97) and $\psi_{-2}(x)=1$ gives

$\lim_{n\to\infty}\widehat{C}_{n}(x)=-\psi_{-1}(x).$ (99)

Taking into account the general property that if a convergent continued
fraction $\operatorname*{K}_{n=0}^{\infty}\left(\frac{a_{n}}{-b_{n}}\right)$
has limit $L$, then
$\operatorname*{K}_{n=0}^{\infty}\left(\frac{a_{n}}{b_{n}}\right)$ has also
limit $L$, it follows that

$\lim_{n\to\infty}C_{n}(x)=-\lim_{n\to\infty}\widehat{C}_{n}(x)=\psi_{-1}(x)$
(100)

and the proof is completed. $\Box$

This result tells us that if the conditions in the theorem above are met, the
sequence with terms $\alpha_{n}=s_{n}/\lambda_{n}$ in the AIM method will
converge pointwise on $\mathcal{I}$ to some function $\alpha$ which can be
interpreted as the term $\psi_{-1}(x)$ of the minimal solution $\psi_{n}(x)$
of the recurrence relation (87). We investigate now some conditions ensuring
the existence of a minimal solution. To this purpose, it is convenient to
introduce the change of variable

$\mathfrak{x}_{n}(x)=\left(-\frac{1}{2}\right)^{n-1}\prod_{j=n_{0}}^{n-2}p_{j}(x)\mathfrak{y}_{n}(x),$
(101)

which applied to (87) leads to the recurrence relation

$\mathfrak{y}_{n}(x)-2\mathfrak{y}_{n+1}(x)+\frac{4q_{n}(x)}{p_{n-1}(x)p_{n}(x)}\mathfrak{y}_{n+2}(x)=0.$
(102)

Suppose that the following limit exists pointwise everywhere on
$\mathcal{I}_{0}\subseteq\mathcal{I}$, namely

$q(x)=\lim_{n\to\infty}\frac{4q_{n}(x)}{p_{n-1}(x)p_{n}(x)}$ (103)

and define for every $x\in\mathcal{I}_{0}$

$a_{n}(x)=\frac{4q_{n}(x)}{p_{n-1}(x)p_{n}(x)}-q(x).$ (104)

Then, the characteristic roots of (102) are given by

$r_{\pm}(x)=1\pm\sqrt{1-q(x)}\quad\forall x\in\mathcal{I}_{0}.$ (105)

According to Elady  we have the following cases

  1. 1.

If $q(x)<1$ for all $x\in\mathcal{I}_{1}\subseteq\mathcal{I}$ and $p_{n}(x)$,
$q_{n}(x)$ are real-valued functions, then $r_{\pm}(x)$ are real and distinct.
Furthermore, we have the following subcases

    1. (a)

If $a_{n}(x)\to 0$ as $n\to\infty$ pointwise for any
$x\in\mathcal{I}_{2}\subseteq\mathcal{I}$, then the Poincar$\acute{\mbox{e}}$
\- Perron theorem ensures the existence of two linearly independent solutions
$\mathfrak{y}_{\pm,n}(x)$ of (102) such that

$\lim_{n\to\infty}\frac{\mathfrak{y}_{\pm,n+1}(x)}{\mathfrak{y}_{\pm,n}(x)}=r_{\pm}(x)\quad\forall
x\in\mathcal{I}_{0}\cap\mathcal{I}_{1}\cap\mathcal{I}_{2}.$ (106)

Let $\mathcal{I}_{3}\subseteq\mathcal{I}$ be some interval of the real line
where $|r_{-}(x)|<|r_{+}(x)|$ for any $x\in\mathcal{I}_{3}$, then we can
construct some positive functions $f_{\pm}(x)$ such that

$|r_{-}(x)|<f_{-}(x)<f_{+}(x)<|r_{+}(x)|\quad\forall x\in\mathcal{I}_{3}.$
(107)

Moreover, (106) implies that

$\left|\frac{\mathfrak{y}_{-,n+1}(x)}{\mathfrak{y}_{-,n}(x)}\right|\leq
f_{-}(x),\quad\left|\frac{\mathfrak{y}_{+,n+1}(x)}{\mathfrak{y}_{+,n}(x)}\right|\geq
f_{+}(x)\quad\forall x\in\bigcap_{k=0}^{3}\mathcal{I}_{k}$ (108)

from which

$|\mathfrak{y}_{-,n}(x)|\leq
f_{-}^{n}(x)|\mathfrak{y}_{-,0}(x)|,\quad|\mathfrak{y}_{+,n}(x)|\geq
f_{+}^{n}(x)|\mathfrak{y}_{+,0}(x)|$ (109)

and hence,

$\lim_{n\to\infty}\left|\frac{\mathfrak{y}_{-,n}(x)}{\mathfrak{y}_{+,n}(x)}\right|=\left|\frac{\mathfrak{y}_{-,0}(x)}{\mathfrak{y}_{+,0}(x)}\right|\lim_{n\to\infty}\left(\frac{f_{-}(x)}{f_{+}(x)}\right)^{n}=0.$
(110)

This shows that $\mathfrak{y}_{-,n}(x)$ is a minimal solution to (102) and by
Theorem IV.3 the continued fraction (68) converges pointwise to $\alpha$ on
the interval $\bigcap_{k=0}^{3}\mathcal{I}_{k}$.

    2. (b)

Suppose there exists some interval $\mathcal{I}_{4}\subseteq\mathcal{I}$ such
that for any $x\in\mathcal{I}_{4}$

$\sum_{n=n_{0}}^{\infty}|a_{n}(x)|<\infty.$ (111)

Then, Corollary 8.27 in Elady  allows to construct two linearly independent
solutions of (102) having asymptotic behaviour

$\mathfrak{y}_{\pm,n}(x)=r_{\pm}^{n}(x)\left[1+o(1)\right]$ (112)

and by means of (101) the recurrence relation (87) has solutions

$\mathfrak{x}_{\pm,n}(x)=\left(-\frac{1}{2}\right)^{n-1}\prod_{j=n_{0}}^{n-2}p_{j}(x)r_{\pm}^{n}(x)\left[1+o(1)\right]\quad\forall
x\in\bigcap_{k=0}^{4}\mathcal{I}_{k}.$ (113)

Taking into account that the inequality $|r_{-}(x)|<|r_{+}(x)|$ holds on
$\mathcal{I}_{3}$, then

$\lim_{n\to\infty}\left|\frac{\mathfrak{x}_{-,n}(x)}{\mathfrak{x}_{+,n}(x)}\right|=\lim_{n\to\infty}\left|\frac{r_{-}(x)}{r_{+}(x)}\right|^{n}\left[1+o(1)\right]\quad\forall
x\in\bigcap_{k=0}^{4}\mathcal{I}_{k}$ (114)

and we conclude that $\mathfrak{x}_{-,n}(x)$ is a minimal solution.

    3. (c)

If instead

$\sum_{n=n_{0}}^{\infty}\left|a_{n}(x)\right|^{2}<\infty$ (115)

on an appropriate interval $\widehat{I}$, El  showed that (87) has two
linearly independent solutions

$\mathfrak{x}_{\pm,n}(x)=\left(-\frac{1}{2}\right)^{n}\prod_{j=n_{0}}^{n-2}p_{j}(x)\prod_{j=n_{0}}^{n-1}\left[1\mp\sqrt{1-q(x)}\pm\frac{a_{j}(x)}{2\sqrt{1-q(x)}}\right]\left[1+o(1)\right].$
(116)

Then, $\mathfrak{x}_{-,n}(x)$ is a minimal solution and Theorem IV.3 can be
used to conclude that the continued fraction (68) converges pointwise to
$\alpha$.

  2. 2.

This case is relevant when we need to consider those values of $x$ where
$q(x)=1$ and hence, $r_{\pm}(x)=1$. As usual, let $x\in\mathcal{I}$ and
$q(x)=1$ for all $x\in\widehat{\mathcal{I}_{1}}\subseteq\mathcal{I}$. Further
suppose that

$\sum_{n=n_{0}}^{\infty}n\left|a_{n}(x)\right|<\infty\quad\forall
x\in\widehat{\mathcal{I}}_{2}\subseteq\mathcal{I}.$ (117)

Then, Coffman’s Theorem (see Theorem 8.29 in Elady ) implies there exists two
asymptotic solutions $\mathfrak{y}_{-,n}(x)\approx 1$ and
$\mathfrak{y}_{+,n}(x)\approx n$, and hence, we can construct again a minimal
solution.

  3. 3.

In the case $q(x)>1$ on an appropriate subset of $\mathcal{I}$, then
$r_{\pm}(x)$ are complex conjugate. If in addition we require that

$\sum_{n=n_{0}}^{\infty}\left|a_{n+1}(x)-a_{n}(x)\right|<\infty$ (118)

on some subset of $\mathcal{I}$, then, we can construct two linearly
independent solutions as in El1 , namely

$\mathfrak{y}_{\pm,n}(x)=\left[1+o(1)\right]\prod_{m=n_{0}}^{n-1}\beta_{\pm,m}(x),\quad\beta_{\pm,m}(x)=1\pm\sqrt{1-q(x)+a_{n}(x)}$
(119)

provided that

$\Re{\sqrt{1-q(x)+a_{n}(x)}}\geq 0$ (120)

for a fixed branch of the square root, for instance $0\leq\arg{\sqrt{z}}<\pi$.
Also in this case we can extract a minimal solution.

  4. 4.

Suppose that the coefficients in (87) have the asymptotic expansions

$p_{n}(x)\approx a(x)n^{\sigma}(x),\quad q_{n}(x)\approx b(x)n^{\tau(x)}$
(121)

with $a(x)b(x)\neq 0$ on some interval and $\sigma(x),\tau(x)$ real-valued
functions. Then, we have the following criteria to check the existence of a
minimal solution Kr ; Per

    1. (a)

If $\sigma(x)>\tau(x)/2$ on a subset of $\mathcal{I}$, we can find a
fundamental set of solutions $\mathfrak{x}_{\pm,n}(x)$ such that the following
limits hold pointwise

$\lim_{n\to\infty}\frac{\mathfrak{x}_{-,n+1}(x)}{\mathfrak{x}_{-,n}(x)}=-a(x)n^{\sigma(x)},\quad\lim_{n\to\infty}\frac{\mathfrak{x}_{+,n+1}(x)}{\mathfrak{x}_{+,n}(x)}=-\frac{b(x)}{a(x)}n^{\sigma(x)-\tau(x)}.$
(122)

Furthermore, $\mathfrak{x}_{+,n}(x)$ is a minimal solution.

    2. (b)

If at some point $\sigma(x)=\tau(x)/2$, let $r_{\pm}(x)$ denote the roots of
the equation $\lambda^{2}+\sigma(x)\lambda+\tau(x)=0$ with
$|r_{+}(x)|\geq|r_{-}(x)|$ on some subset of $\mathcal{I}$. Then, for
$|r_{+}(x)|\neq|r_{-}(x)|$ the recurrence relation (87) has a fundamental set
of solutions $\mathfrak{x}_{\pm,n}(x)$ such that the following limits hold
pointwise

$\lim_{n\to\infty}\frac{\mathfrak{x}_{-,n+1}(x)}{\mathfrak{x}_{-,n}(x)}=r_{-}(x)n^{\sigma(x)},\quad\lim_{n\to\infty}\frac{\mathfrak{x}_{+,n+1}(x)}{\mathfrak{x}_{+,n}(x)}=r_{+}(x)n^{\tau(x)}.$
(123)

Furthermore, $\mathfrak{x}_{+,n}(x)$ is a minimal solution.

  5. 5.

Suppose that the coefficients in (87) have asymptotic expansions

$p_{n}(x)\approx\sum_{j=0}^{\infty}\frac{a_{j}(x)}{n^{j}},\quad
q_{n}(x)\approx\sum_{j=0}^{\infty}\frac{b_{j}(x)}{n^{j}}$ (124)

with $b_{0}(x)\neq$ on some interval. The characteristic equation associated
to (87) is $r^{2}+a_{0}(x)r+b_{0}(x)=0$ with roots

$r_{\pm}(x)=-\frac{a_{0}(x)}{2}\pm\sqrt{\frac{a_{0}^{2}(x)}{4}-b_{0}(x)}.$
(125)

At this point, the Birkhoff-Adams theorem BA1 ; BA2 ; BA3 ; BA4 ; LI ; Wong
can be used in order to construct asymptotic expansions of the fundamental set
of solutions to (87) allowing to determine whether or not a minimal solution
exists. We have the following cases

    1. (a)

If $a_{0}^{2}(x)\neq 4b_{0}(x)$ on some subset of $\mathcal{I}$, there are two
linearly independent solutions $\mathfrak{x}_{\pm,n}(x)$ given by

$\mathfrak{x}_{+,n}(x)=r_{\pm}^{n}(x)n^{\alpha_{\pm}(x)}\sum_{k=0}^{\infty}\frac{c_{\pm,k}(x)}{n^{k}},\quad\alpha_{\pm}(x)=\frac{a_{1}(x)r_{\pm(x)}+b_{1}(x)}{a_{0}(x)r_{\pm}(x)+2b_{0}(x)}$
(126)

for those values of $x$ such that the denominator in the expression for
$\alpha_{\pm}(x)$ does not vanish. Furthermore, the coefficients $c_{\pm,k}$
satisfy the recurrence relation

$\sum_{j=0}^{s-1}\left[r_{\pm}^{2}(x)2^{s-j}\left(\begin{array}[]{c}\alpha_{\pm}(x)-j\\\
s-j\end{array}\right)+r_{\pm}(x)\sum_{k=j}^{s}\left(\begin{array}[]{c}\alpha_{\pm}(x)-j\\\
k-j\end{array}\right)a_{s-k}(x)+b_{s-j}(x)\right]c_{\pm,j}(x)=0,\quad
c_{\pm,0}(x)=1.$ (127)

    2. (b)

At those points where $r_{-}(x)=r_{+}(x)=r(x)$ with $r(x)=-a_{0}(x)/2$ not a
root of $a_{1}(x)r(x)+b_{1}(x)=0$, (87) has two linearly independent solutions
of the form

$\mathfrak{x}_{+,n}(x)=r^{n}(x)e^{\gamma_{\pm}(x)\sqrt{n}}n^{\widetilde{\alpha}_{\pm}(x)}\sum_{j=0}^{\infty}\frac{c_{\pm,j}(x)}{n^{j/2}}$
(128)

with

$\displaystyle\widetilde{\alpha}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{4}+\frac{b_{1}(x)}{2b_{0}(x)},\quad\gamma_{\pm}(x)=\pm
2\sqrt{\frac{a_{0}(x)a_{1}(x)-2b_{1}(x)}{2b_{0}(x)}},\quad c_{\pm,0}=1,$ (129)
$\displaystyle c_{\pm,1}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{24b_{0}^{2}(x)\gamma_{\pm}(x)}\left(a_{0}^{2}(x)a_{1}^{2}(x)-24a_{0}(x)a_{1}(x)b_{0}(x)+8a_{0}(x)a_{1}(x)b_{1}(x)\right.$
(131)
$\displaystyle\left.-24a_{0}(x)a_{2}(x)b_{0}(x)-9b_{0}^{2}(x)-32b_{1}^{2}(x)+24b_{0}(x)b_{1}(x)+48b_{0}(x)b_{2}(x)\right)$

    3. (c)

At those points where $r_{-}(x)=r_{+}(x)=r(x)$ and
$2b_{1}(x)=a_{0}(x)a_{1}(x)$, we consider instead the equation

$\widetilde{\alpha}(\widetilde{\alpha}-1)r^{2}(x)+\left[a_{1}(x)\widetilde{\alpha}+a_{2}(x)\right]r(x)+b_{2}(x)=0.$
(132)

By $\widetilde{\alpha}_{\pm}$ with
$\Re{\widetilde{\alpha}_{+}}\geq\Re{\widetilde{\alpha}_{-}}$ we denote the
roots of (132). Then, we have the following three scenarios

      1. i.

For those values of $x$ such that
$\widetilde{\alpha}_{+}-\widetilde{\alpha}_{-}\neq 0,1,2,\cdots$ the equation
(87) has two linearly independent solutions of the form

$\mathfrak{x}_{\pm,n}(x)=r^{n}(x)n^{\widetilde{\alpha}_{\pm}(x)}\sum_{j=0}^{\infty}\frac{c_{\pm,j}(x)}{n^{j}}.$
(133)

      2. ii.

For those values of $x$ such that
$\widetilde{\alpha}_{+}-\widetilde{\alpha}_{-}=1,2,\cdots$, then the equation
(87) has one solution $\mathfrak{x}_{-,n}(x)$ given by (133) and another
solution

$\mathfrak{x}_{+,n}(x)=z_{n}(x)+c\mathfrak{x}_{-,n}(x)\ln{n},$ (134)

where $c$ is a constant that may be zero and

$z_{n}(x)=r^{n}(x)n^{\widetilde{\alpha}_{+}(x)}+\sum_{s=0}^{\infty}\frac{d_{s}}{n^{s}}.$
(135)

      3. iii.

For those values of $x$ such that
$\widetilde{\alpha}_{+}=\widetilde{\alpha}_{-}$, then the equation (87) has
one solution $\mathfrak{x}_{-,n}(x)$ given by (133) and a second solution
represented by (135) with

$z_{n}(x)=r^{n}(x)n^{\widetilde{\alpha}_{-}(x)-k_{0}+2}+\sum_{s=0}^{\infty}\frac{d_{s}}{n^{s}}\sum_{s=0}^{\infty}\frac{d_{s}}{n^{s}},$
(136)

where $k_{0}$ is a positive integer such that $k_{0}\geq 3$.

We observe that the above conditions ensuring the existence of a minimal
solution may, in general, have the effect to shrink the interval over which
$x$ must be chosen when we use the Asymptotic Iterative Method . The next
result narrows the interval over which $x$ must be taken so that the continued
fraction (72) and hence, (68) converge pointwise.

###### Theorem IV.4

Let $\mathfrak{K}:=\\{x\in I~{}|~{}p_{n}(x)\geq 1\\}$. Then,the AIM condition
(45) is satisfied for all $x\in\mathfrak{K}$.

  * Proof.

Since (72) and (68) are equivalent, there is no loss in generality in
considering the sequence of approximants of (72). By (67) we have for the
$n$-th and $(n-1)$-th approximants

$C_{n}(x)-C_{n-1}(x)=\frac{(-1)^{n}}{B_{n}(x)B_{n-1}(x)}$ (137)

with $B_{n}(x)$ satisfying the recurrence relation in (48). From (80) we know
that

$B_{n}(x)B_{n-1}(x)>\mu^{2}\sum_{k=0}^{n}p_{k}(x),\quad\mu=\min_{x\in
I}{\\{1,p_{0}(x)\\}}$ (138)

since $p_{n}(x)\geq 1$ for all $n$ ensures that $p_{n}(x)>0$ for all $n$. For
$x\in\mathfrak{K}$ we have

$\sum_{k=0}^{n}p_{k}(x)\geq n+1.$ (139)

Moreover, since all $p_{k}(x)$ are positive, we also have

$\left|\sum_{k=0}^{n}p_{k}(x)\right|\geq n+1$ (140)

and therefore,

$\left|B_{n}(x)B_{n-1}(x)\right|>(n+1)\mu^{2}$ (141)

from which it follows that

$\left|C_{n}(x)-C_{n-1}(x)\right|<\frac{1}{(n+1)\mu^{2}}.$ (142)

Clearly, the above expression vanishes as $n\to\infty$ and the proof is
completed. $\Box$

###### Acknowledgements.

We thank the anonymous referees for the critical reading and useful comments
which helped improve the quality of the paper.

## References

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figure table

# Cryogenic optical beam steering for superconducting device calibration

K. Stifter  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA  Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, CA 94305, USA  H. Magoon  Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  SLAC National Accelerator Laboratory, Menlo Park, CA
94025, USA  Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, CA 94305, USA  Department of Physics, Stanford
University, Stanford, CA 94305, USA  Department of Physics and Astronomy,
Tufts University, Medford, MA 02155, USA  A.J. Anderson  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Kavli Institute for
Cosmological Physics, University of Chicago, IL 60637, USA  Department of
Astronomy and Astrophysics, University of Chicago, IL 60637, USA  D.J. Temples
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  N.A. Kurinsky
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA  Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305,
USA  C. Stoughton  Fermi National Accelerator Laboratory, Batavia, IL 60510,
USA  I. Hernandez  Department of Physics, Illinois Institute of Technology,
Chicago, IL 60616, USA  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  A. Nuñez  SLAC National Accelerator Laboratory, Menlo Park, CA
94025, USA  Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, CA 94305, USA  Department of Physics, Stanford
University, Stanford, CA 94305, USA  K. Anyang  Department of Physics,
Illinois Institute of Technology, Chicago, IL 60616, USA  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  R. Linehan  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  M.R. Young  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Kavli Institute for
Cosmological Physics, University of Chicago, IL 60637, USA  Department of
Astronomy and Astrophysics, University of Chicago, IL 60637, USA  P. Barry
School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3YB UK  D.
Baxter  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Department of Physics and Astronomy, Northwestern University, Evanston, IL
60208, USA  D. Bowring  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  G. Cancelo  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  A. Chou  Fermi National Accelerator Laboratory, Batavia, IL 60510,
USA  K.R. Dibert  Kavli Institute for Cosmological Physics, University of
Chicago, IL 60637, USA  Department of Astronomy and Astrophysics, University
of Chicago, IL 60637, USA  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  E. Figueroa-Feliciano  Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA  Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  L. Hsu  Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  R. Khatiwada  Department of Physics,
Illinois Institute of Technology, Chicago, IL 60616, USA  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  S.D. Mork  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  L. Stefanazzi  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  N. Tabassum  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA  Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305,
USA  Department of Physics, Stanford University, Stanford, CA 94305, USA  S.
Uemura  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  B.A.
Young  Department of Physics, Santa Clara University, Santa Clara, CA 95053,
USA

###### Abstract

We have developed a calibration system based on a micro-electro-mechanical
system (MEMS) mirror that is capable of delivering an optical beam over a
wavelength range of 180 – 2000 nm (0.62 – 6.89 eV) in a sub-Kelvin
environment. This portable, integrated system can steer the beam over a
$\sim$3 cm $\times$ 3 cm area on the surface of any sensor with a precision of
$\sim$100 $\mu$m, enabling characterization of device response as a function
of position. This fills a critical need in the landscape of calibration tools
for sub-Kelvin devices, including those used for dark matter detection and
quantum computing. These communities have a shared goal of understanding the
impact of ionizing radiation on device performance, which can be pursued with
our system. This paper describes the design of the first-generation
calibration system and the results from successfully testing its performance
at room temperature and 20 mK.

###### keywords:

optics, light, MEMS, cryogenics, superconducting sensors, dark matter

##  1 Introduction

Astrophysical evidence indicates that approximately 85% of all matter in the
universe is “dark” [1, 2], yet scientists have yet to uncover the fundamental
nature of this dark matter beyond its gravitational interactions. So far, a
host of experiments aiming to detect dark matter scattering have set strong
bounds on particle-like dark matter models with masses above 1 GeV/c2 in the
absence of a positive dark matter signal [3, 4, 5, 6]. This is prompting new
searches to look over a broader range in dark matter masses [7, 8, 9]. Of
particular interest are searches for particle-like dark matter with masses
well below 1 GeV/c2. Development of sensors with the capability to detect
scatters from this “sub-GeV dark matter” requires thresholds at the eV scale
and below [7]. Phonon- and other quasiparticle-sensitive sub-Kelvin
superconducting devices such as transition-edge sensors (TESs) [10], microwave
kinetic inductance detectors (MKIDs) [11, 12, 13], superconducting nanowire
single-photon detectors (SNSPDs) [14], and superconducting qubits [15, 16] are
advantageous in this regime. In-situ characterization of device response to
both backgrounds and to a putative dark matter signal is essential to
validating the discovery potential of these detectors.

Independently, quantum computing has made great strides in the last decade
through the improvement of coherence times of superconducting qubits, but a
significant challenge remains in the reduction of error-inducing environmental
noise. It has been shown that MeV-scale ionizing radiation produces correlated
qubit errors [17, 18] and shortened coherence times [19, 20]. Though attempts
to mitigate the sensitivity to these effects show great promise [21, 22], a
significantly improved understanding of the underlying mechanisms that cause
superconducting qubit errors is needed [23].

Ionizing radiation is a primary background source for both the dark matter
search and quantum computing communities, which share the need for a compact,
robust, and sensitive calibration setup capable of producing $\sim$eV-scale
energy deposits at a variety of specific locations across the surface of a
device with a precision approaching the scale of features in the qubit
architecture. Furthermore, superconducting qubits, TESs, KIDs, and SNSPDs
typically operate at temperatures well below 1 K, necessitating a calibration
scheme that works at these temperatures or is able to penetrate a cryostat.
Historically, dark matter detectors have been calibrated through various
schemes including radioactive sources, direct exposure to an LED, photons
coupled through a fixed fiber, and filtered blackbodies [24, 11, 25], but each
technique has key drawbacks. Specifically, they either have energies too high
to be useful for some modern applications, create energy deposits with rather
limited position control or flexibility, are difficult to reproduce, and/or
require coupling to a cryostat in ways that often introduce additional
background events. These limitations motivated our development of an in-situ,
steerable, collimated beam of low-energy photons that could be mounted at the
mixing chamber stage (MXC) of a dilution refrigerator (DR).

Optical beam steering in a cryogenic environment is challenging due to both
the dissipation of heat generated when the steering system is operated, as
well as properties of the motion control mechanism that may cause it to stop
functioning at low temperatures. Existing optical steering methods include
micro-electro-mechanical-system (MEMS) mirrors [26, 27, 28, 29], various
piezoelectric systems [26, 30, 31], galvo-scanners coupled to confocal
microscopes [32], and cryogenic actuators [33]. These systems all have trade-
offs between system size, cryogenic functionality, heat production from power
dissipation, scanning speed, range and precision, steering reproducibility,
and introduction of parasitic backgrounds from light leakage.

In this paper, we introduce a full calibration system based on the MEMS mirror
steering technology for use with any photon-sensitive device from room
temperature down to 20 mK and below. The specifications for this design are
shown in Table 1, where we also list relative advantages and disadvantages of
other optical beam steering systems compared to ours. In particular, our
system achieves a steering range that allows for full coverage of cm2\- and
in2-scale devices in a way that limits parasitic backgrounds, while allowing
for sub-Kelvin functionality and a small beam spot size. Section 2 of the
paper details the design of our full calibration system, and Section 3
describes the results from warm and cold operation of the system. This is
followed by a discussion of follow-up work in Section 4 and conclusions in
Section 5.

Table 1: A selection of specific implementations of several optical steering methods. The specifications of the system presented in this paper are given in the top line, followed by some relative advantages and disadvantages of other systems for calibration of sub-Kelvin devices. Steering system |  Steering range | Wavelength | Spot size | Min. Temp. | Power
---|---|---|---|---|---
MEMS mirror (this work) |  $\sim$3 cm $\times$ 3 cm | 180–2000 nm |  $\sim$100 µm | <20 mK |  $\sim$1 µW
| Advantages | Disadvantages
MEMS mirror (prior work) [26, 27, 28, 29] | 60 µm spot size |  >600 mK operation, high light leakage
Piezoelectric systems [26] |  |  High heat production, hysteresis, reduced deflection at low temp.
Attocube nanopositioner [30] |  42 µm spot size, target imaging, focal length control |  0.005 cm2 steering range, 1 K operation
Attocube nanopositioner [31] |  3 µm resolution, target imaging, focal length control, free-space delivery |  0.3 cm2 steering range, high light leakage, high heat production
Scanning confocal microscope [32] |  <1 µm spot size, target imaging, focal length control, free-space delivery |  0.5 cm2 steering range, high light leakage, high heat production
Cryogenic actuator [33] |  47 µm spot size, target imaging | 0.003 cm2 steering range

##  2 Design

To meet the calibration needs of modern sub-Kelvin sensors, we designed and
tested a calibration system capable of steering a collimated beam of light
across the surface of any low-temperature sensor. The compact system is shown
in Figure 1. The deflection angle of the beam is dynamically controlled by a
MEMS mirror located on the base temperature stage of a DR and enclosed in a 11
$\times$ 10 $\times$ 7 cm housing machined from high-conductivity copper.
Communication with the mirror control mechanism is performed using a simple
computer interface at room temperature. Although previous work was done using
these mirrors at $\sim$1 K [26, 27, 28, 29], we needed a system compatible
with testing devices operating at $\sim$10 mK, which required significant
redesign. In our new system configuration, shown on the left in Figure 1, an
assembly of focusing and beam steering components are compactly housed inside
a copper enclosure designed for light tightness, ease of handling, and minimal
footprint within the DR. The system was designed to be used with any fiber-
coupled light source and a wide variety of test devices.

Figure 1:  Left: Internal view of MEMS-based calibration system for sub-Kelvin
operation. The high-conductivity copper enclosure, shown without its lid, has
dimensions of 11 $\times$ 10 $\times$ 7 cm. It includes translation plates for
fine adjustment of optical components and light-tight ridges for reduction of
backgrounds. Light, shown with a red arrow, emerges from a fiber-coupled
refractive focuser (top right) and is reflected off of a steerable MEMS mirror
(bottom right) from Mirrorcle Technologies, Inc. The MEMS mirror is used to
accurately steer the incident beam onto a flat stationary mirror, which then
directs the beam out of the enclosure via an IR filter window onto the surface
of the device under test.

The MEMS mirrors used (shown on the bottom right in Figure 1) are commercial
components purchased from Mirrorcle Technologies, Inc. The deflection angle of
these mirrors can be controlled about two axes, allowing for steering across
two dimensions, referred to as “X” and “Y” in this paper. The circular face of
the mirror is aluminized, allowing for operation over optical wavelengths and
out into the THz regime [34]. The MEMS mirrors are capacitive devices
controlled by four voltages (two for each axis of rotation) in the range of 0
- 180 V, with the angle of deflection set by the voltage difference across
each axis. This capacitive control provides the advantage that under ideal
conditions, no power is dissipated while the mirrors are stationary, and heat
is dissipated only when the mirror is moving. In practice, some dissipation
occurred even with the mirror stationary, as discussed in Section 3. To modify
the mirrors for cryogenic functionality, we worked with Mirrorcle Technologies
to deposit aluminum over the doped silicon control lines. Without this
modification, the control lines would disconnect at cryogenic temperatures due
to carrier freeze out, and the lines would become high impedance, preventing
the electronics from actuating the mirror. This also ensures that the control
lines are superconducting below 1 K and will minimize parasitic heating on the
mirror substrate during charge or discharge of the MEMS.

In our experiments, the control voltages are generated by a commercial driver
box purchased from Mirrorcle Technologies, capable of producing 180 V of
positive bias on each of the four DC lines. The DC control voltages are routed
to the MEMS via cryogenic ribbon cables. With this control system, an (X, Y)
pair of voltage differences reproducibly translates to a precise location on a
detector surface. In this paper, we describe our steering range in terms of
this voltage coordinate system. We define voltage coordinates such that the
mirror’s rest position is assigned [0,0], and voltage differences within the
normalized range [-1,1] can be applied to both the X and Y direction of
motion.

The setup was designed to be utilized with a variety of light sources,
including both pulsed and continuous-wave beams over a wide range of
wavelengths. Optical fibers are used to route the beam from a room-temperature
light source to the cryogenic steering unit. A simple drawing of the optical
path is shown in Fig. 2. The beam is first collimated from a light source at
room temperature into an optical fiber. This fiber passes through a KF40-to-
SMA optical vacuum feedthrough into the cryostat. The fibers are then routed
through custom KF40 copper plates that thermalize the fiber at each cryostat
stage. From here, the fibers are connected to a refractive focuser, and routed
internally in the steering system. We use 100 µm-core diameter multi-mode
fibers that were purchased commercially from Accu-Glass Products, Inc. and
transmit in the UV and visible wavelengths.

Figure 2: The optical path used to route photons from a room-temperature light
source, into the DR cryostat, and to the MEMS enclosure mounted on the mixing
chamber stage (MXC). The optical input line is broken at each DR stage to
ensure thermalization of the fibers. Filtering after the warm light source and
before the device housing reduces IR leakage and attenuates the total photon
flux.

The steering unit housing is designed as a copper box with a user-configurable
optical breadboard-style base-plate. Mounted to the breadboard is the
refractive focuser, the MEMS mirror, and a stationary mirror. Light enters the
enclosure via the optical fiber and is directed through the refractive
focuser. This focuser is an OZ Optics DTS0060 commercial product. It is
coupled to a 10 µm-core optical fiber. Light emerges from the focuser in the
form of an exposed beam incident on the face of the MEMS mirror. The beam is
subsequently reflected off of a secondary, stationary mirror and directed
through a filtered output window and onto the surface of the device under
test. The secondary mirror functions to extend the pathlength of the deflected
beam, thus increasing steering area without requiring a larger physical
enclosure for the system. The entire copper enclosure is mounted to the base
temperature ($\sim 10$ mK) stage of a dilution refrigerator. The steering unit
and test device mount is designed to be entirely light-tight. Copper ridges at
all the joints prevent stray photons from entering unit and interfering with
the operations of the device under test. All of the electrical lines needed
for MEMS control enter the light-tight box via a single MDM25 feedthrough.

To ensure adequate initial alignment of the laser system, the individual
structural mounts for both the MEMS mirror and the stationary mirror were
designed to allow for three degrees of translational freedom. While setting up
the calibration unit, one can use these stages to optimize beam alignment and
focus on the surface of the device under test. The refractive focuser is
mounted to a fixed position in the enclosure to allow for secure coupling to
external fibers. An external contact plate is used to apply clamping force to
the component, maintaining its position and thermal contact with the copper
housing. A front plate with a small cutout is set in front of the focuser to
further control optical alignment.

The laser beam exits the steering unit via a filter window as shown in Figure
3. This window contains two slots for standard 2x2” square infrared (IR)
filters, which are used to filter potential blackbody radiation of the optical
fiber at higher temperature stages. Thermal contact between the filters and
the steering unit is maintained via the installation of a copper “output
plate”. The output plate is mounted to the MEMS enclosure using beryllium-
copper spring washers. In this configuration, pressure is applied uniformly to
the square IR filters, enabling appropriate thermal contact between the filter
and the housing. To ensure sufficient thermalization of the device under test,
copper brackets are used to connect the output plate directly to the mixing
chamber at 10 mK. The thickness of the output plate can be designed such that
the surface of the device under test is positioned exactly at the focal length
of the beam. In our current configuration, this distance is 15cm from the
focuser, but we have also configured the system to work well for 14 cm and 20
cm focal lengths by simply swapping optical components.

Figure 3: Photon path through the MEMS enclosure to the surface of a device
under test. Light enters the MEMS enclosure via fiber-coupled refractive
focuser and emerges as a beam in free space. The beam is reflected off a MEMS
mirror which can sweep the beam across a 2D plane, and is then reflected off a
stationary mirror to redirect it out of the enclosure. The beam passes through
an IR filter and an experiment-specific “output plate” that is thermally
anchored to the MXC of the DR before impinging on the surface of the device
under test, which is located at the focal length of the focuser. A optional
screen-plate can be used to restrict device illumination area.

##  3 Testing Results

In testing the design described above, the primary goal was to prove the
scanning functionality of the adapted MEMS mirror technology at temperatures
in the milliKelvin regime. Additional goals were to demonstrate that the
system is capable of achieving the design specifications of calibration beam
spot size, scanning range, and repeatability.

Unless otherwise specified, the light source used in the tests described below
was a ThorLabs SLS201L/M fiber-coupled, broadband tungsten-halogen lamp that
emits photons over a range of 300 – 2800 nm. ThorLabs NE10A and NENIR60B
neutral density filters were installed at room temperature to reduce the lamp
intensity by a factor of 10 in the optical band and 500,000 in the near IR. In
the DR, the OZ Optics refractive focuser upstream from the MEMS mirror was
designed for a central wavelength of 650 nm and had a focal length of 15 cm.
Lastly, a ThorLabs KG5 colored bandpass filter was inserted into the MEMS
scanning unit housing output filter slot to narrow the wavelength range sent
on to the device under test to 330–665 nm.

###  3.1 Warm Characterization

Before being operated in a dilution refrigerator, the MEMS calibration system
was first tested at room temperature, where we could most easily measure the
final beam spot size and intensity profile, as well as optimize the full
package geometry to achieve the scanning range needed for device calibrations
Bench-top optical measurements were made using a Kralux 2.3 megapixel CMOS
camera placed at the focal point of the focuser. For these room temperature
measurements, a 0.2 m-long custom multi-mode optical patch cable with 100 µm-
core (UM22-100) was used to couple the SMA output of the light source to the
FC/PC input of the focuser.

Using this optical path, the beam spot diameter was measured to be $\sim$170
µm. The spot size was determined by fitting a Gaussian to the one-dimensional
distributions of pixel intensity along the semi-major and semi-minor axes of
the beam spot in a camera image and taking the $\pm$2$\sigma$ width [35], as
shown in Figure 4. It was possible to reduce the beam spot diameter to
$\sim$100 µm using additional low-pass filters that functioned to reduce
chromatic aberration. However, this additional filtering led to low
intensities that were not desirable for early tests. In our initial bench-top
tests, the MEMS mirror was directly connected to its driver via an
$\sim$8”-long 10-pin plastic ribbon cable provided by Mirrorcle.

Figure 4: Distributions of pixel intensity along the semi-major (left) and
semi-minor (right) axes of the beam spot. Data taken from camera images of a
spot from a filtered broadband source used in conjunction with a focuser for
650 nm light. A Gaussian fit is performed and the beam diameter is taken as
the $\pm$2$\sigma$ point. [35]

The maximum scanning area achieved was found to be $\sim$3 cm $\times$ 3 cm at
the focal length of the unit, but the results varied dramatically with
different cabling capacitance. Using the cabling for which the largest
scanning areas were achieved, the location of the beam spot varied linearly
with applied MEMS voltage, and was reproducible upon repeated applications of
the same voltages.

###  3.2 Cold Characterization

The initial cold characterization tests of the basic setup were performed with
the steering unit mounted to a device designed and fabricated by the SPT-3G+
collaboration as a prototype for the SPT-3G+ focal plane.[36] It was designed
for measurement of the cosmic microwave background radiation in the
$\sim$200-240 GHz band [37] and consists of an array of ten microwave kinetic
inductance detectors (MKIDs). The MKIDs on this array are superconducting
resonant circuits, consisting of an absorber, which acts as the inductive
element of the circuit, coupled to an interdigitated capacitor. In this
device, both the absorber and the capacitor are fabricated with aluminum.
Optical power on the silicon substrate around each MKID generates phonons,
which break Cooper pairs in the inductive element. This results in a change in
the MKID’s resonant frequency and quality factor that is dependent on the
optical power incident on the substrate.[38]

The MKID chip was mounted in a copper housing as shown in Figure 5, and
affixed to the steering unit via the “output plate” shown in Figure 3, such
that the beam was incident on the face of the MKID array. The scanning unit
was installed and coupled to the MKID as shown in Figure 3. Data was taken in
two geometric configurations. In the first configuration, a lid was attached
to the MKID housing with a single large opening that exposed the entire
10-MKID array to incident photons (“open-plate configuration”). A second run
was performed later using a machined copper plate containing a known hole
pattern (“screen-plate configuration”).

Figure 5:  Top row: MKID detector array package with the “open-plate”
configuration (left) and the “screen-plate” configuration (right). For
simplicity, only one of ten MKIDs in the array was used for these studies. Its
location is marked with a red star in the left image. Bottom row: Relative
response of a single MKID detector channel as a function of the MEMS mirror
steering normalized control voltage in each configuration. For each data
point, the MEMS mirror is steered to a location and the light source is turned
on. The change in MKID transmission magnitude ($\mathbf{\Delta S_{21}}$) in
response to the photon stimulation is plotted on the z-axis. MKID readout
parameters and physical hardware mounting were adjusted between the two
datasets, so no direct comparison regarding absolute magnitude of device
response can be made between the two plots. On the left, a device response is
present across much of the mapped region, except for an edge, potentially
indicating a device or housing feature. On the right, substantial device
response is only present in a localized region, which we correlate with the
beam passing through a hole in the screen-plate. The distortion of the
circular hole as measured in MEMS voltage space is attributed to non-linear
mapping of voltage space to physical space, and is the subject of future
improvements to the system.

Once the detector and MEMS scanning package was fully assembled, the combined
unit was mounted directly to the mixing chamber plate of a Bluefors LD-400
with a nominal base temperature of $<$10mK and 600uW of cooling power at
100mK. For the duration of these tests, the base temperature of the dilution
unit was maintained between 15 and 35 mK. The MKID was read out using the QICK
instrumentation control package [39, 40]. The QICK system consisted of a
Xilinx ZCU111 evaluation board with a XCZU28DR RFSoC FPGA. The ZCU111 ran
custom firmware based on an overlapped polyphase filterbank [41], which
channelizes data into 2 MHz bandwidth per tone and was capable of streaming
1024 channels per RF line. The interface for controlling the firmware and
streaming data was provided through a Jupyter notebook implemented through the
PYNQ environment [42].

Prior to taking data with the MEMS scanning system, a frequency scan was
performed for all the MKIDs on the array to ensure that they were operational
and to evaluate their basic quality. The MKID with the highest Q (measured to
be $\sim$ $1.2\times 10^{5}$) was used for two initial characterization
studies performed with the “open-plate” and “screen-plate” configurations
respectively. The high-Q pixel on the MKID chip is oriented vertically at the
physical location marked with a star in Figure 5.

The first cryogenic run (“open-plate configuration”) successfully demonstrated
movement of the MEMS mirror at 25 mK. The scanning test consisted of stepping
the MEMS mirror position through the two-dimensional voltage space of its
controller. For each position in voltage space, RF transmission measurements
were made at the MKID resonance frequency both before and after the
continuous-wave light source was switched on. In the presence of incident
photons, we expect the MKID’s resonance to shift downwards in frequency and
decrease in quality factor [11]. Both of these effects will lead to an
increase in transmission magnitude as measured at a fixed readout frequency
located on the original resonance feature. An example of this photon-induced
transmission change is shown in Figure 6. From these measurements, a
differential in the MKID transmission magnitude resulting from the light
source was then constructed. Following each measurement, a wait time of
approximately 20 seconds was observed to allow the chip to cool and the
resonance to return to its original location in frequency space.

Figure 6:  Time-domain MKID response in the “open-plate” configuration as the
light source is turned on. Three traces show that response varies dramatically
when the beam is directed to different locations across the MKID array. Yellow
([0.9, 0.65] in MEMS voltage space) shows the highest response, green
(0.3,0.3) shows a modest response, and blue (0.25,0.15) shows negligible
response.

To map out the response across the plane of the sensor, a thorough scan
through a section of the MEMS voltage space was performed following the data-
taking procedure described above. The mirror was stepped through linear
increments of the voltage in both X and Y, thus creating a grid of MKID
response “pixels”. This two-dimensional response map from the open-plate
configuration run is shown in the left-hand side of Figure 5. The data shows
clear regions of strong response to the photon beam (light colored pixels).
These are delineated from regions of weak response (dark colored pixels).

The first dataset acquired while running with the open-plate configuration is
difficult to fully interpret. We saw varying but repeatable detector response
as a function of mirror position that was suggestive of beam steering, but it
was not immediately obvious which location along the MKID chip was being
targeted by the beam. We hypothesized that this was due to uncontrolled
reflections creating a source of background noise seen by the MKID array. This
hypothesis was supported by warm testing results where we intentionally used
the MEMS mirror to steer the beam off-chip and saw large streaks of light
across the device housing when the beam was reflected off areas of the output
plate of the copper enclosure.

To mitigate these reflections and aid in subsequent data analysis, we
introduced a machined copper “screen-plate” to our detector mounting set-up,
as shown in the upper right of Figure 5. This plate is mounted in front of the
MKID chip as shown in Fig. 3, such that large regions of the MKID chip are
masked. The same MKID device was tested again with much cleaner results, as
shown in the lower right of Figure 5. In particular, the single instrumented
MKID on the detector chip clearly responded when exposed to steered photons,
as evidenced by region of high response when the beam was steered over a hole
in the screen-plate. While these holes were circular, we observed an elongated
density profile in the data, implying a nonlinear mapping from MEMS voltage
coordinates to physical space.

We were able to show that this image distortion was caused by our cabling. In
particular, when using higher capacitance cables, MEMS mirror movement is
reduced near the edges of the scanning region relative to the central region.
The result is that voltage changes made near the center of the scan region
correspond to a larger step in physical position of the beam than the same
voltage change applied near the edge of the scanning range. This model
accounts well for the signal pattern elongation shown in the lower right of
Figure 5. Further work to mitigate these unwanted distortions is ongoing and
will be discussed in Section 4.

The consistency of the MEMS beam-positioning capability at $\sim$20 mK was
studied by scanning the mirror along a diagonal path in voltage space and
monitoring the resulting MKID signal. This scan was repeated multiple times
over the course of an hour. Typical results are shown in Figure 7, where a
similar change in MKID signal is seen to occur at the same location for two
independent mirror scans. This clearly demonstrates active and repeatable
cryogenic beam steering from the MEMS mirror.

To test the power dissipation of the mirror, a heater on the mixing chamber of
the dilution fridge was assigned a temperature setpoint above the base
temperature of the fridge. Once equilibrium was established, the MEMS mirror
was biased and set to trace across its maximum range at its highest allowable
speed. The power of the fridge heater was monitored, and was found to remain
within its level of noise within the duration of the test. In a different MEMS
unit tested in a separate dilution fridge facility, an analysis of thermometry
data demonstrated that the MEMS unit dissipated an average of
0.99$\,\pm\,$0.28 µW of power during a resting power-on state, with minimal
additional dissipation during periodic mirror rotations occurring at a rate of
1 Hz.

Figure 7: Two time traces, acquired $\sim$60 minutes apart, showing the MKID
response as the MEMS mirror was scanned diagonally in voltage space
coordinates from [0.4, 0.4] to [0.1, 0.1]. The observed change in MKID
response occurred when the mirror was positioned at [0.3, 0.3] in voltage
space, corresponding to the steered photon beam passing over a hole in the
screen-plate. The similarity of the two data sets (dark blue and light blue)
suggest a high degree of stability and repeatability in the full MEMS-based
calibration set-up. These data were taken with the same hardware configuration
and readout parameters as the masked “screen-plate” data shown in Figure 5.

##  4 Discussion and Follow-up Work

The initial cold characterization of the prototype MEMS steering unit clearly
demonstrated the successful operation of the MEMS mirror at sub-Kelvin
temperatures. Still, drawbacks to the design and setup were discovered during
the first round of testing. For this reason, several follow-up measurements
were performed and many design upgrades are planned for future iterations of
the calibration system.

###  4.1 Focusing unit

While a beam spot diameter of $\sim$200 µm was achieved in these tests, the
specifications for the focuser give an idealized value of $\sim$80 µm. This
difference is likely due to the mismatch between the relatively wide spectrum
of the filtered broadband light source used for these tests and the narrow
band of wavelengths for which the focuser is designed. To explore whether an
improved beam spot diameter is achievable in our system, a follow-up
measurement was performed using a 625 nm fiber-coupled LED and a OZ Optics
focuser designed for 635 nm light with a 15 cm focal length. In this warm
test, a minimum beam spot diameter of 80 µm was achieved.

Still, this focusing method only works for the single wavelength matched to
the focuser design, and the focuser design is restricted to the wavelengths
between 180 – 2000 nm, which is the current limiting factor for the energy
range of the overall calibration system. If subsequent calibrations want to
use a multiple wavelengths over a wider range, the dilution refrigerator would
have to be warmed up, the calibration system uninstalled, the focuser
replaced, and the calibration system reinstalled. For this reason, the
commercial refractive focusing unit will be replaced in future iterations of
the calibration system with a custom reflective focusing system that provides
comparable or even improved beam spot size diameters and works over a wide
range of wavelengths without needing to be replaced.

###  4.2 Electrical cabling

During characterization following installation into the dilution refrigerator
it was noted that the beam spot position no longer moved linearly with the
applied voltage and the scanning area was significantly reduced as compared to
initial warm bench tests. Additionally, on three separate occurrences during
installation and validation testing, unstable behavior was observed for
specific combinations of bias voltage. In this coordinate of voltage space,
the mirror was unable to hold a steady position, and the beam spot was
observed to oscillate rapidly across a linear path with length on the order of
500 µm. Note that the location of this instability was excluded from the data-
taking process that was used to create Figure 5.

These behaviors were observable only when using the electrical cabling
required for operation in the dilution refrigerator. In followup
investigations, it was discovered that this was due to the intermediate air-
side cable used to carry the DC voltages to the MEMS mirror, which was a set
of twelve, individually shielded twisted pairs terminating in a Fischer
connector. In its current configuration, the commercial DC bias controller is
capable of driving up to 50 pF per channel. Since the capacitance of the MEMS
mirror itself is approximately 20 pF, this leaves a maximum capacitance in the
cabling of about 30 pF before there are adverse effects. We observed that
exceeding this limit led to the observed reduced steering range and non-linear
position response to applied voltage near the edges of the range.

For future experiments, we have developed a simplified electrical path that
removes the need for intermediate air-side cabling and reduces the number of
connections inside the dilution refrigerator itself. The MEMS mirror still
connects to the 10-pin/MDM adapter board embedded in one wall of the MEMS
scanning unit enclosure via a short, plastic 10-pin ribbon cable, but the
adapter board is connected all the way to a DB25 vacuum feedthrough at the
300K stage via a 2 m-long Constantan woven loom with twelve twisted pairs that
is thermally fixed at each stage of the dilution refrigerator. The air side of
the DB25 feedthrough now connects directly to the DB25/10-pin adapter board,
which is connected to the MEMS driver via short, plastic 10-pin ribbon cable.
Warm tests utilizing this setup showed the full expected scanning range of
$\sim$3 cm$\times$ 3 cm and improved linearity of beam spot location with
applied voltage. Final cold tests of the system are underway.

###  4.3 Other improvements

Another challenge to address is the possible reflections of the beam off of
bare copper surfaces in the MEMS steering unit enclosure. To limit this
effect, as well as any other source of stray photons in the MEMS steering unit
housing that may diminish cryogenic device performance, we plan to coat all
inner surfaces with a cryogenic- and vacuum-compatible black paint [43] or an
epoxy mixed with carbon lumped black [44].

Finally, the overall footprint of the this prototype calibration system in a
dilution refrigerator is still relatively large. This leads to inefficient
usage of space, and also can make it challenging to couple the calibration
system to magnetic shielding layers that are required for various sensor
types. For this reason, a second-generation design has been produced. The new
design has a smaller and simplified layout, which allows the entire system to
fit within A4K shielding cans, and also reduces the cost.

##  5 Conclusion

There is a distinct need in both the particle astrophysics and quantum
information science communities to characterize sensors in cryogenic
environments. Current techniques typically focus on energies above the range
of interest for these devices, are limited by a lack of position control,
offer limited coverage area, struggle with position resolution due to large
spot size or hysteresis, or cannot operate at cryogenic temperatures due to
heat dissipation or freeze-out of control mechanisms. The MEMS mirror-based
calibration system presented here overcomes many of these limitations.

Tests of our prototype MEMS scanning unit successfully demonstrated a
$\sim$100 µm beam spot that can be steered over an area of $\sim$3 cm$\times$
3 cm down to 20 mK with $<$1 µW of power dissipation, making this the first
sub-Kelvin operation of the MEMS-mirror technology. The optical system used in
these tests can deliver of photons over an energy range of 0.6 – 5 eV. Despite
some issues with high-capacitance electrical cabling, the positioning of the
beam was found to be reproducible, achieving the goal of repeated photon
injections on the surface of a device through cryogenic beam steering. Given
several small drawbacks of the original design, a next-generation version with
an improved electrical path, optical path, and mechanical design is in
development.

We expect the new iteration of our system to be useful soon for a range of
exciting applications, including the characterization of background events in
dark matter detectors and investigation of underlying physical mechanisms
driving superconducting qubit decoherence.

##  6 Acknowledgements

The authors acknowledge Robert Moffatt and Blas Cabrera as the architects of
the first MEMS-mirror-based beam steering effort, on which this system is
based [26]. We thank the QSC Postdoctoral Research Awards for providing the
seed money that kick-started this project. We thank Mirrorcle Technologies,
Inc. for working with us to produce a version of their MEMS mirror that
functions at cryogenic temperatures, and the DMQIS group at SLAC for their
frequent help with lab space and other logistics. HM would like to thank
Joseph Angelo and Kavin Ammigan for mentorship and CAD advice as part of the
Fermilab Engineering Co-op Program, as well as Tyler Funk for discussions
surrounding cryogenic design. KS would like to thank Alden Fan for his help
with beam spot size analysis and Aviv Simchony for his help with breakout
board construction.

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics. This work was supported by the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Quantum Science Center and the U.S. Department of Energy,
Office of Science, High Energy Physics Program Office. This work was supported
in part by the U.S. Department of Energy, Office of Science, Office of
Workforce Development for Teachers and Scientists (WDTS) under the Science
Undergraduate Laboratory Internships Program (SULI). This work was also made
possible in part through a KIPAC Innovation Grant and support from the KIPAC
Post-baccalaureate Fellowship Program.

This project was originally conceived and developed by NK, and was guided to
completion by KS. HM, KS, and NK designed the physical setup of the scanning
unit, including the housing body, electrical path, and optical path. IH
performed early warm characterization of the MEMS mirrors. HM and KS performed
warm validations of the entire system prior to installation and were also
responsible for installation. AA provided the MKID device, which was designed
and fabricated by KD and PB. AA and MY provided space in their dilution
refrigerator and managed cooldowns. HM, AN, KA, and KS performed the tests to
determine MEMS power dissipation, after which KA and RL analyzed the resulting
data. HM, AA, DT, and KS set up the MKID readout electronics using a QICK
RFSoC package developed by CS, SU, GC, and LS. SU additionally provided
invaluable debugging during early data-taking. HM and KS collected the MKID
data presented in the paper, and AA was instrumental in helping them
understand it. HM was responsible for the final analysis of the MKID data and
for making all plots. KS, HM, LH, BY, NT, SM, and DBa were the primary authors
of this report. DBa, DBo, RK, LH, NK, and DT provided valuable feedback on the
project at weekly meetings. RL provided many useful conversations to help us
understand various signals in the data. DBa, DBo, AC, EFF, LH, RK, NK, and BY
provided research group leadership for the duration of the project.

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